Enhanced oil recovery : resonance macro-and micro-mechanics of petroleum reservoirs 9781119293835, 1119293839, 9781119293859, 1119293855, 978-1-119-29382-8

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Enhanced Oil Recovery

Scrivener Publishing 100 Cummings Center, Suite 541J Beverly, MA 01915-6106 Publishers at Scrivener Martin Scrivener ([email protected]) Phillip Carmical ([email protected])

Enhanced Oil Recovery Resonance Macro- and Micro-Mechanics of Petroleum Reservoirs

O. R. Ganiev, R. F. Ganiev and L. E. Ukrainsky

Copyright © 2017 by Scrivener Publishing LLC. All rights reserved. Co-published by John Wiley & Sons, Inc. Hoboken, New Jersey, and Scrivener Publishing LLC, Beverly, Massachusetts. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. For more information about Scrivener products please visit www.scrivenerpublishing.com. Cover design by Kris Hackerott Library of Congress Cataloging-in-Publication Data: ISBN 978-1-119-29382-8

Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

Abstract This monograph discusses the scientific fundamentals of resonance macro- and micro-mechanics of petroleum reservoirs and its petroleum industry applications. It contains an overview of the research and engineering results of resonance macro- and micro-mechanics of petroleum reservoirs, which provides the scientific and applied foundations for the creation of groundbreaking wave technologies for production stimulation and enhanced oil recovery. The monograph is intended for a wide audience: students, teachers, scientists and practitioners who are interested in the fundamentals, the development and application of leading-edge technologies in the petroleum industry and other industrial sectors.

Contents Preface

xiii

Introduction: A Brief Historical Background and Description of the Problem

xvii

1

Scientific Foundation for Enhanced Oil Recovery and Production Stimulation 1.1 The Practical Results of Near-Wellbore Formation Cleaning by Wave Stimulation 1.2 The Scientific Fundamentals of the First-Generation Wave Technology for Stimulation of Production Processes 1.2.1 Large-Scale Laboratory Experiments at Shell Test Facilities 1.2.2 Resonances in Near-Wellbore Formation. Resonances in Perforations 1.2.3 Excitation of Oscillations in Micro-Pores by One- Dimensional Longitudinal Macro-Waves in a Medium. Resonances. Transformation of Micro-Oscillations in Pores to Macro-Flows of Fluid. The Capillary Effect 1.2.4 Cleaning of Horizontal Wells 1.2.5 Preliminary Results 1.3 Stimulation of Entire Reservoirs by First-Generation Wave Methods for Enhanced Oil Recovery. Resonance Macro- and Micro-Mechanics of Petroleum Reservoirs: A Scientific Foundation for Enhanced Oil Recovery

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1 1

7 8 12

15 18 20

21

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Contents

2 Remove Micro-Particles by Harmonic External Actions 2.1 An Analysis of the Forces Acting on Pore-Contaminating Particles under a Harmonic External Action 2.2 Conditions for the Detachment of a Solid Particle from the Wall of a Pore under Harmonic External Action 2.3 The Criterion of Successful Harmonic Wave Stimulation. Criterion Determination Procedure 2.4 Summary

27

3 Remove Micro-Particles by Impact Waves 3.1 Determining Flow Parameters behind an Impact Wave 3.2 Assessing the Forces That Act on a Particle as the Front of an Impact Wave Is Passing 3.3 Conditions for the Detachment of a Solid Particle from the Wall of a Pore under the Action of a Passing Impact Wave 3.4 The Criterion for Successful Wave Stimulation by Impact Waves. Criterion Determination Procedure 3.5 Summary

45 46

4

63

The Wave Mechanisms of Motion of Capillary-Trapped Oil 4.1 The Conditions for the Detachment of a Droplet from the Wall of a Pore 4.2 The Case of Harmonic Action on a Capillary-Trapped Droplet 4.3 The Case of Impact Wave Action on a Capillary-Trapped Droplet 4.4 Summary

5 Action of Wave Forces on Fluid Droplets and Solid Particles in Pore Channels 5.1 The Mechanism of Trapping of Large Oil Droplets in a Waterflooded Reservoir. Propulsion of Droplets by One-Dimensional Nonlinear Wave Forces 5.2 The Average Flow of Fluid Caused by Oscillations in a Saturated Porous Medium with a Stationary Matrix and Inhomogeneous Porosity 5.2.1 The Statement of the Problem 5.2.2 Calculation Results

27 30 39 43

51

53 58 61

64 66 70 72

73

73

76 76 79

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5.3

6

Fluid Flows Caused by Oscillations in Cone-Shaped Pores 5.3.1 The Statement of the Problem 5.3.2 Calculation Results

The Mobilization of Droplets and Blobs of Capillary-Trapped Oil from Microcavities 6.1 The Mathematical Statement of the Problem 6.2 The Natural Frequency of Gravity-Capillary Waves on Oil-Water and Oil-Surfactant Interfaces in Pores 6.3 Interface Instability Range 6.4 Oil-Water Interface Instability 6.5 Oil-Surfactant Interface Instability

84 84 88

91 91 95 97 98 102

7 Statements and Substantiations of Waveguide Mechanics of Porous Media 105 7.1 Resonance Mechanisms Possible in Fluid-Saturated Porous Media 105 7.2 Resonance of Two-Dimensional Axially Symmetric Waves in Horizontal Layers of Reservoir. Efficient and Directed Excitation of Wave Energy in Target Sub-Layers 108 7.3 Resonance of Two-dimensional Plane Waves in Reservoir Compartmentalizing Strike-Slip Faults and Fractured Zones 114 7.3.1 The Mathematical Model of a Fluid-Saturated Porous Medium 115 7.3.2 The Statement of the Problem and Solution Procedure 118 7.3.3 Damping Decrements of Waves in a Natural Vertical Waveguide 121 7.3.4 Statement of a Resonance Waveguide Problem and Its Substantiation for Porous Media. Introduction 127 7.3.5 Resonances. Waveguide Processes in Porous Media with Heterogeneities. The Distribution of Forces Acting on Pore-Contaminating Solid Particles and Capillary-Trapped Oil Droplets in a Waveguide 132

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7.4

Linked Waveguides in Compartmentalized Reservoirs. The Transfer of Oscillations into Reservoir Inner Zones under Multidimensional Resonance Conditions 7.4.1 The Statement of the Problem of Forced OneDimensional Oscillations in Linked Sections of a Multi-Phase Medium under Resonance Conditions 7.4.2 The Results of Mathematical Simulation 7.5 Experimental Determination of Resonant Frequencies of a Reservoir. Practical Recommendations for Selecting Controlled Means and Oscillation/Wave Generators

8

The Resonant and Waveguide Characteristics of a Well 8.1 Selecting Wave Parameters for Stimulation of Horizontal Wells 8.1.1 Scientific Fundamentals 8.1.2 Practical Recommendations on Stimulation of Horizontal Wells 8.2 Near-Wellbore Stimulation. The Induction of Resonance 8.2.1 Resonances in the Wellbore Section between the Oscillation Generator and the Bottom. Using Waves to Transfer Wave Energy 8.2.2 Practical Recommendations for Stimulation of the Near-Wellbore Formation Zone

9 Experimental Study of Wave Action on a Fluid-Filled Porous Medium 9.1 Experimental Study of the Potential to Clean up the Near-Wellbore Formation Zone from Contamination using Wave Stimulation 9.1.1 Test Equipment and Methodology 9.1.2 The Results of Cleanup from Clay Mud 9.1.3 The Results of Cleanup from Clay-Polymer Mud 9.1.4 Summary 9.2 The Experimental Study of the Effect of Shock Waves on the Displacement of Hydrocarbons by Water in a Porous Medium. Connected Wells 9.2.1 The Test Equipment

141

142 144

145 151 153 153 158 159

159 162

165

165 166 169 171 173

173 174

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9.2.2

A Theoretical Analysis of the Propagation of Waves Generated by a Shock-Wave Valve in the Test Facilities and Evaluation of the Forces Caused by the Wave Action 9.2.3 The Methodology of Tests 9.2.4 Results of Flow Acceleration Tests 9.2.5 The Effect of Wave Stimulation on Connected Wells 9.2.6 Summary

177 180 181 185 186

Conclusion

189

References

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Index

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Preface This monograph discusses one of the most important present-day research and engineering problems that affect the growth of the country’s economy: cost-effective oil production stimulation and enhanced oil recovery. The authors and other scientists at the Scientific Center for Nonlinear Wave Mechanics and Technologies of the Russian Academy of Sciences (NC NVMT RAN) have developed the scientific and applied foundations for what is known as resonance macro- and micro-mechanics of petroleum reservoirs, a novel and challenging path to efficient oil recovery enhancement, which in some cases may incorporate and amplify the effects of other well-known conventional enhanced oil recovery methods (chemical, thermal, hydraulic fracturing, horizontal wells, etc.). The truth of this statement is backed by the results of both theoretical research at the leading edge of the theory of nonlinear oscillations and experimental studies conducted in laboratories and in the field. This is described in detail in the introduction and in the following chapters of the monograph. Resonance macro- and micro-mechanics of petroleum reservoirs is a new area of fundamental and applied research in nonlinear wave mechanics, ahead of the international state-of-the-art and led by Russia. The proposed field of resonance mechanics of petroleum reservoirs is based on the recently discovered multi-dimensional largescale resonance phenomena in heterogeneous oil reservoirs that have significant effect on the motion of various micro-inclusions, such as solid particles and droplets of fluids (water, oil, etc.) in the micro-pores of an oil formation (both near and far from the wellbore). In turn, the motion of microinclusions can drastically change the macro-mechanics of the porous medium. Therefore, there are dynamically linked resonance macro- and micro-processes in petroleum reservoirs (porous media), which can be controlled for the purposes of both oil production stimulation and enhanced oil recovery. xiii

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What is totally new and important about it is the theoretical and practical discovery of the phenomenon of significant amplification of multidimensional waves in porous media as the waves propagate and are transmitted over long distances (which is the second aspect of the discovery), which permits stimulation of large areas of reservoirs with various heterogeneities, contaminated by solid particles both near and far from the wellbores or containing capillary-trapped oil, for enhanced oil recovery purposes. These factors are the most common causes of declining production and recovery of oil in many cases, and it is difficult to select an economical stimulation method for their removal. Th e undertaken research and field trials have shown that the reservoir damage control methods proposed in this monograph (based on the science of nonlinear resonance mechanics of porous media) may be more efficient and economical than others. To be able to create such multi-dimensional resonance conditions in the field, various controlled appliances and devices, broadband oscillation and wave generators with instrumentation and mathematical control systems have been developed. The corresponding software packages for typical field applications are constantly improved and updated depending on the geological settings of specific oil fields. This scientific base is used to design and build petroleum industry-oriented controlled appliances and devices that form a field of the so-called wave machine engineering sector. Wave machine engineering has also been founded by the NC NVMT RAN team and is rapidly evolving to the benefit of various industries [1, 2, 3, 4]. This book mostly discusses the fundamentals of resonance macro- and micro-mechanics of petroleum reservoirs, substantiation of its scientific and applied aspects, and prospects of its use in petroleum industry applications. However, it should be noted that the initial research and development base for the statement and solution of the problem of resonance macro- and micro-mechanics of petroleum reservoirs was formed by earlier results of the so-termed wave technology. The wave technology, created by the NC NVMT RAN team for a wide range of applications in various industries including oil and gas, was quickly acclaimed by experts. As far back as in 1990 it was approved by a panel of experts of the USSR Ministry of Petroleum Industry for use at Soviet oil fields for production stimulation purposes. At the initial stage of wave technology development by the academic research team as a new field of mechanics (the theory of nonlinear oscillations and waves and their technological applications), it was actively and specifically supported by the leaders of the USSR and by eminent

Preface

xv

progressive statesmen. It was their idea to set up a research council (in 1984–1985) within the USSR Ministry of Petroleum Industry on the subject matter of using wave and vibrational processes in the petroleum industry (the council was chaired by R. F. Ganiev, then a professor and now a Member of the Russian Academy of Sciences) whose aim was to coordinate the research efforts of teams in various sectors of the industry. A trial site was established at Nizhnevartovsknefnegas Oil Production Association and about 100 wells were provided for tests by various subsidiaries of the association. The tests involved various aspects of the wave technology, including gas-lift applications (for a lower gas consumption), drilling applications (for cleaner reservoir penetration through the use of bridging capabilities), etc. It was first-generation wave technology, developed and extensively tested by the industry in 1985–1990. More than 3,000 wells located in different regions of the USSR, mainly in Western Siberia, were stimulated with very good results. As mentioned above, the wave technology was acclaimed by oil industry experts (including refining and petrochemical) and was also actively copied by various empirical inventors (not always petroleum engineers), especially after 1991. Unfortunately, when some of these people started using the science-driven wave technology without understanding its scientific fundamentals and principles, it resulted in incorrect use, negative results in some cases, and even partial damage to reputation of the technology (as explained in more detail in the Introduction and the first chapter of the book). R. Kh. Muslimov, a prominent scientist and a practicing expert in geology, a professor and a member of the Academy of Sciences of Tatarstan, reasonably and impartially wrote in his monograph that experts in oscillation mechanics, reservoir engineers and geologists should all joint their efforts to implement this high-end and promising technology [5]. Meanwhile, the NC NVMT RAN team continued to invest effort in the wave technology on a broad scale. A number of wave resonance effects in near-wellbore formation zones were found, wave capillary effects of multiple acceleration (by 100s to 1,000s times or more) of fluids (water and oil) in micro-pores were identified, along with other wave phenomena in porous media. First-generation wave technology was then quite fully developed and verified by practice, progressing further on a new scientific basis. Cooperation (contracts) with Western oil companies, such as British Petroleum, Shell, Smith International (a drilling company) also played an important role in the development and perfection of the wave technology. Field trials of the enhanced oil recovery technique were conducted in

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Alaska, the North Sea, the Sultanate of Oman (drilling improvement), and laboratory tests were carried out at the highly valuable and unique Shell test facilities in the company’s research center in Holland. Theoretical determinations of nonlinear dynamic characteristics of oscillation and wave generators were verified at these facilities in conditions as close as possible to real wells. Efficient removal of various types of contaminants from near-wellbore formation zones and multiple acceleration of flow processes in porous media were confirmed. It should be noted that these results were obtained on full-size models of near-wellbore formation, rather than on quite small core samples (as it is usually done). The main results (scientific and practical) were published in various periodicals, backed by dozens of patents (including overseas patents) and inventors’ certificates of NC NVMT RAN scientists, and summarized in the authors’ monographs [6, 7, 8, 9, 10, 11]. That is why the Introduction and the first chapter contain only a brief overview of the principal scientific fundamentals and some results (scientific and practical) of wave technology application, as needed to substantiate the formulation of the main problem discussed here, resonance macro- and micro-mechanics of petroleum reservoirs, as a research and practical basis for oil production stimulation and enhanced oil recovery. Several problems of resonance macro- and micro-mechanics of petroleum reservoirs and the wave technology have been solved in collaboration with our colleague I. G. Ustenko, an NC NVMT RAN staff member and senior research scientist (references to these studies are provided in the corresponding chapters of the book). NC NVMT RAN researchers Yu. S. Kuznetsov, D.Sc. (Eng.), N. A. Shamov D.Sc. (Eng.), S. A. Kostrov, Ph.D. (Eng.), G. A. Kalashnikov, Ph.D. (Eng.), Yu. B. Malykh, Ph.D. (Eng.), and others, as well as many reservoir engineers and production geologists from Western Siberia took active participation in field trials of drilling improvement and production stimulations techniques at the initial stage of wave technology use. A. I. Petrov, D.Sc. (Geo.), a prominent expert in geology and mineralogy, was our consultant in this area throughout our work. In preparing this monograph, the authors drew upon petroleum industry knowledge provided in the generalizing monographs of renown petroleum geologists R. Kh. Muslimov, D.Sc. (Geo.) [5] and R. S. Khisamov, D.Sc. (Geo.) [12]. The authors are very grateful to all those who are mentioned above. The authors would like to thank R. I. Nigmatullin, a Member of the Academy of Sciences, for his review of this book and helpful advice. The authors

Introduction: A Brief Historical Background and Description of the Problem The state of the Russian economy depends, to a large extent, on the efficient and stable functioning of the petroleum industry, one of few sectors able to meet the demands of both the internal market and exports. In the present and coming decades, the task of increasing hydrocarbon recovery efficiency, ultimate oil (for oil fields) and component (for gas condensate and gas condensate/oil fields) recovery factors is and will be one of the key challenges in achieving the country’s energy security. For this reason, an efficient use of various improved recovery techniques along with a science-based search for novel enhanced oil recovery methods is critical for the oil industry to grow in the current conditions. Reservoir flow characteristics (porosity and permeability), pore space contamination, reservoir fluid composition, its viscosity and capillary properties at the fluid-rock interface are among the key factors controlling oil recovery processes. The most common cause for declining flow rates of oil and gas production wells is contamination of the pore space in near-wellbore formation zones. Pore space contamination may occur from various causes such as invasion of drilling mud clay particles into the formation while drilling; mobilization of rock fines with the extracted reservoir fluids while producing; deposition of resins, asphaltenes and paraffines in the pore space; chemical processes in the rock, etc. Traditionally, a number of techniques have been used to remediate nearwellbore formation damage: injection of special solutions, thermochemical and electro-chemical stimulation. In heavily contaminated lowpermeability rocks, performance of these methods depends on the chemistry of contaminants and on the correct selection of treatment fluids. xvii

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Introduction

In 1984–85, a near-wellbore cleanup method using oscillations and waves was proposed by the A. A. Blagonravov Institute of Machines Science of the Academy of Sciences of the USSR (reorganized later into the Scientific Center for Nonlinear Wave Mechanics and Technologies of the Russian Academy of Sciences (NC NVMT RAN)). The method consists in placing a purpose-designed oscillation generator that excites pressure waves in the near-wellbore formation region into a well close to perforations. Passing through contaminated pores in near-wellbore formation, the waves act upon the contaminating particles stuck to pore walls and, provided they are of a sufficiently high power, can detach the particles from pore walls thereby ensuring cleanup. The success of wave stimulations obviously depends on the magnitude by which the force acting on a particle stuck to a pore wall is greater than the force of adhesion of the particle to the pore wall. These stimulations are called first-generation wave technologies. They have become quite common. Such stimulation jobs have been performed in Western Siberia, Tataria, Bashkiria and other regions of Russia, as well as in Oman, USA (in Alaska), Norway (on a North Sea platform), and in China. More than 3,000 wells have been treated. As a result of the stimulations, flow rates of production wells increased by 70–80% (in some cases, 2 to 5-fold); injectivity of injection wells increased by 80–90% (documents issued by the Ministry of Petroleum Industry are available to confirm the performance of the wave technology). In the 1990s, first-generation wave stimulations were accepted by the Ministry of Petroleum Industry of the USSR as a technology recommended for application throughout the Soviet Union. A detailed review of the results obtained during the application of first-generation wave technology is provided in Chapter 1 below. Thanks to its widespread use in the country, the first-generation wave technology was soon acclaimed enthusiastically by many inventors specializing in well workovers. They tried to modify the original technology that was built around the use of vortex and cavitation generators of various designs, running within certain operating envelopes dictated by the properties of the nearwellbore zone to be treated; the operating envelopes were defined through complex research aimed at determining the rates of fluid flow through the generator, input pressures, and geometric arrangements. A distinctive feature of these generators is a wide range of frequencies and the high amplitude of the excited pressure oscillations. For example, tests conducted at Shell test facilities in the Netherlands showed that pressure amplitudes of some spectral components in the 2–5 kHz band were greater than 15 atmospheres. Attempts were made, most of which failed, to replace the proposed generator with other types such as rotary-pulse or ultrasonic generators, because radiation from rotary-pulse sources is mono-harmonic, while

Introduction xix the amplitude of ultrasonic generators is not high enough and, moreover, ultrasonic waves attenuate very quickly in near-wellbore formation. Even vortex cavitation generators, if running outside of pre-determined operating envelopes or not in a precise geometric arrangement, did not always bring positive results due to insufficient pressure wave amplitudes within a reservoir’s formation damage zone. To summarize: the first-generation wave technology invented at NC NVMT RAN proved to be efficient in multiple field tests, however, a number of superficially similar near-wellbore formation wave stimulation techniques appeared under the name of “wave technology”. In most cases, these techniques fail to bring positive results because they either use inadequate generators or their generators run outside of the operating envelopes that ensure success. Either way, they fail to take into account the scientific basis of the technology. Meanwhile, the originators of the first-generation wave technology continued to improve it. To date, they have made significant progress, capitalizing on recent advances in the science of resonance effects. The next step in the development of this technology was the idea of using near-wellbore resonance phenomena to amplify wave amplitudes, thereby augmenting wave cleanup processes in the near-wellbore formation zone. In its simplest form, the idea was implemented for resonances at perforations [13]. As far as we know, the simple idea of using resonant and waveguide properties of near-wellbore formation zones has not been previously contemplated by anyone. Although, as mentioned in [14], the cleaning efficiency and cleaning rate are improved significantly with increased wavefield amplitude. And it is exactly resonance that allows achieving the highest amplitude with minimum energy, while waveguiding properties point to the wave excitation frequencies at which their amplitudes decay with distance slower than at others. Apparently, the fact that no one has tried to look at the problem at this angle can be explained by the prevailing opinion that the structure of a reservoir as a whole or even only of the near-wellbore zone is so complex that it is practically impossible to determine, with a sufficient accuracy, its resonant frequencies and to build a wave excitation source (generator) that can generate exactly one of these frequencies. However, as studies conducted at NC NVMT RAN have shown, the use of vortex cavitation generators with a wide multi-harmonic (practically continuous) radiation spectrum permits coverage of entire frequency bands, including nearwellbore formation resonant frequencies. As far as approximate determination of resonant frequency values is concerned, it was shown in [13] that they can be found quite accurately if every perforation hole that

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is filled with a fluid interacting with a porous medium at the interfaces is considered to be a resonator. It was later discovered that resonant frequencies, as well as the so-called critical waveguiding frequencies that ensure the lowest attenuation, can be approximated not only for the near-wellbore formation zone (covering the entire thickness of the formation) but for entire formations having a certain structure. As shown below, the new generation wave technology is based upon this discovery. The next step in the evolution of first-generation wave technology became possible thanks to the development of drilling techniques. To be more specific, the enabling method behind this step was the creation in near-wellbore formation of networks of extended perforation tunnels providing reliable connectivity between the reservoir and the wells. In particular, it is proposed to use a perforation drilling system to create small-diameter perforation tunnels/waveguides, extending deep into the formation from the wellbore, followed by wave stimulation with a cavitation wave generator whose frequency band is within the passband of the created network of perforations/waveguides. Testbed trials have confirmed the feasibility of drilling deep perforation tunnels [15]. It should be noted that, unlike the near-wellbore resonant wave stimulation where the geometry of the perforations and hence the resonant frequencies of wave stimulation are fixed, this method opens up totally new possibilities. For example, it becomes possible to create a system of perforations with desired resonant frequencies, selecting the frequencies from a range close to the most powerful emissions in the spectrum of the available generator. On the other hand, first-generation wave technology was tested in combination with chemical cleaning methods, i.e. injection of various chemical agents that react with near-wellbore contaminants and transform them into easily removable solutions. Combined application of wave technology and chemical methods has produced some techniques that perform much better than each initial method alone [9]. A combination of near-wellbore wave stimulation with jet pump operation laid the groundwork for yet another method of near-wellbore formation remediation called “Overbalanced/underbalanced wave cleaning of near-wellbore formation”. The method permits significant improvement of the cleaning of near-wellbore formation zones around the main borehole and side tracks, as well as special completion screens. Unique equipment for these operations has been designed and successfully tested in the field [16]. It has been prepared for extensive commercial use.

Introduction xxi Along with the aforementioned studies that have significantly advanced near-wellbore formation wave stimulations, NC NVMT RAN scientists have made a major, groundbreaking step forward and have actually come up with new generation wave technologies that are an alternative to the best enhanced oil recovery methods (including hydraulic fracturing and other leading-edge techniques). This step in wave technology advancement became possible thanks to the discovery of multi-frequency resonances and critical waveguiding frequencies associated with a formation’s natural waveguiding properties that are controlled by its structural heterogeneities: horizontal and vertical stratification or compartmentalization. Moreover, multi-dimensional spatial resonance forms are capable of multi-fold amplification and critical waveguiding forms propagate in formations to considerable distances. The discovery of multi-frequency spatial resonance waveforms and critical waveguiding forms of motion in formations has allowed us to broaden significantly the wave technology’s potential for improving production rates and enhancing oil recovery. NC NVMT RAN possesses unique software for computing resonances and critical waveguiding frequencies for formations with known structural heterogeneities. Optimal designs of wave stimulation devices and oscillation generators have been developed. One of such approaches was tested on fields operated by Tomskneft in Russia, as well as on fields in Texas and California and proved to be many times less expensive than hydraulic fracturing, with an on-par performance but without the risk of reservoir flooding. In order to use natural resonant and waveguiding properties of formations, controlled by their structural heterogeneities including horizontal and vertical stratification or compartmentalization caused by vertical naturally-fractured zones and faults, it is proposed to conduct wave stimulations in a frequency band corresponding to the resonant frequencies of formations with structural heterogeneities. Thanks to the use of waveguiding properties of formations and the discovery of the resonant wave amplification phenomenon in spatial structures, it has become possible to stimulate much larger areas, to transmit resonant wave energy accurately to a predetermined zone containing capillary-trapped oil, and to mobilize capillary-trapped oil into the fluid flow stream towards production wells. In field conditions, this translates to a higher oil recovery and a lower water cut of fluid produced from a particular reservoir. The unique equipment (and corresponding software) that has been developed to create resonant multi-frequency waves in rock formations, accompanied by significant changes in velocities and pressures of the

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reservoir fluid and the rock matrix, permits efficient stimulation of large areas within an oilfield. In addition to the mobilization of capillarytrapped oil, near-wellbore formation zones around wells located far from the wave source within a treatment site can be stimulated too. The results obtained show that this technology is a cost effective and environmentally friendly alternative to hydraulic fracturing and other leading edge enhanced oil recovery methods. These results have provided a basis for the creation of resonance macroand micro-mechanics of petroleum reservoirs. This is a groundbreaking new method of enhanced oil recovery that can be combined with other well-known techniques such as chemical, thermal, horizontal well and other stimulations to significantly improve their performance.

Enhanced Oil Recovery: Resonance Macro- and Micro-Mechanics of Petroleum Reservoirs. O. R. Ganiev, R. F. Ganiev and L. E. Ukrainsky. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

1 Scientific Foundation for Enhanced Oil Recovery and Production Stimulation First-Generation Wave Technologies for Improving Near-Wellbore Fluid-Flow Capacity of Oil-Bearing Formations and Enhanced Oil Recovery. Resonances in Near-Wellbore Formation. Large-Scale Laboratory Effects. The Statement and Substantiation of the Problem of Resonance Macro- and Micro-Mechanics of Petroleum Reservoirs: A Scientific Foundation for Enhanced Oil Recovery and Production Stimulation

1.1 The Practical Results of Near-Wellbore Formation Cleaning by Wave Stimulation The wave technology that was introduced in the 1980s by NC NVMT RAN has been used for many years to remediate near-wellbore damage in productive reservoirs. A wide range of field tests of wave principle-based devices has been conducted, mostly on oil fields operated by Nizhnevartovskneftegas, Yuganskneftegas, Langepasneftegas, Kogalymneft and Tomskneft, as well as on fields located in the Republics of Bashkortostan and Tatarstan, in the Perm Territory [9], and others. These tests have been conducted on both injection and production wells. In a number of cases, the tests were performed on several dozens of injection wells while monitoring response in offset production wells. Increased injectivities of injection wells as well as increased flow rates and lower water cuts of the responding production wells indirectly indicate that oil recovery can be enhanced by wave stimulation of remote and stagnant reservoir zones. 1

2

Enhanced Oil Recovery

Most of the tests were conducted on oil fields operated by Nizhnevartovskneftegas, where a total of about 400 wells (injection and production) were subjected to wave stimulation by the end of 1987. An analysis of the results of the 1986 field tests of wave principle-based equipment has shown that the equipment is quite efficient in terms of stimulation of near-wellbore formation zones of production and injection wells. Average incremental injectivity and flow rate after wave stimulation jobs was 255 m3/day of water and 23.4 t/day of oil per well, respectively (from a report dated 1 January 1987 on wave stimulation performance at Nizhnevartovskneftegas oil production facilities in April/December 1986). It has been noted that the best performance is provided by combined well stimulation that includes hydrochloric acid treatment and the use of a downhole hydro-impact generator tool developed at NC NVMT RAN, with acid injection through the oscillation generator. An analysis of the results of production well stimulations has shown that oil flow rates increased by an average of 15-20 t/day after combined well stimulation jobs (i.e. flow rates increased by 1.5–2.5 times). Downhole hydro-impact generator treatments account for 50–60% of the increase, i.e. 8–11 t/day of oil. This conclusion is confirmed by data on the wells that were stimulated using only downhole hydro-impact generators without any other stimulation methods. To hone the wave stimulation procedure, a dedicated program of injection and production well treatments was designed and implemented. The purpose of the program was to identify response to downhole hydro-impact tool stimulations only, without effects from other treatments such as acidizing, acid pickling, cleaning circulation, etc. The program involved a package of logs run into the stimulated wells. A bulk of the research scope was completed by the mid-June, 1987. An analysis of the results of production well stimulations shows that the flow rates of all the tested wells increased. For example, one well (No. 12104) demonstrated an increase in the oil flow rate from 90 to 147 t/day; another well (No. 297) showed an increase in total fluid flow rate from 86 to 150 m3/day with an unchanged water cut; while yet another well (No. 12397) showed an increase in total fluid flow rate from 38 to 48 m3/day with the water cut dropping from 5% to 1–2%. These results along with results of several other tests allowed us to confirm the earlier conclusion that wave stimulations are an efficient method of near-wellbore formation treatment. The near-wellbore cleanup field tests were conducted on the basis of the authors’ theoretical studies. The results are described in [8, 9, 10, 11].

Scientific Foundation for Enhanced Oil Recovery 3

Fig. 1.1 shows a typical production history of a production well. A production well operated by Priobye oil company is taken as an example.

Fig. 1.1. Priobye Oil Company Production Well Wave Stimulation Results

As we see, the oil flow rate was declining and by the time of the stimulation job it had dropped more than 2 times below its initial value. It is an indication of near-wellbore formation damage. After the treatment which consisted in placing a source of pressure oscillations (with frequencies matching the estimated resonant frequencies for a particular well in order to drive contaminating particles from the near-wellbore zone into the wellbore) close to perforations in front of the productive reservoir, flow rates were practically restored to their initial levels and remained stable during the observation period (10 months). These results are typical for the proposed technology. Many such tests were conducted in different regions of the Russian Federation and abroad. Based on results of the stimulation jobs conducted in Russia, this technology was officially accepted (a certificate of acceptance by the USSR Ministry of Petroleum Industry dated 1990 is available). Let us cite some of the data available on two stimulation jobs performed on West Siberian production wells. 1. The production method was an electric submersible pump. Duration of well stimulation treatment: 14 hours. After the stimulation job, oil flow rate increased from 8 t/day to 25.6 t/day, while water cut dropped from 89% down to 66%. Incremental oil production during a 3-month period amounted to 2,219.0 tons.

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2. The production method was gas lift. Duration of the well stimulation treatment: 13 + 3 = 16 hours. After stimulation, the payzone net thickness increased from 4.4 m to 5.4 m due to addition of a reservoir layer. Oil flow rate increased from 11.1 t/day to 12 t/day. Water cut decreased from 40% to 10%. Incremental oil production during a 3month period amounted to 100 tons. Gas consumption for gas lift operation was the same before and after the stimulation job: 8,000 nm3/day. Provided below are well log curves for a number of injection wells that underwent wave treatment in different regions of Russia. The most typical West Siberian and Bashkirian were chosen as examples. In all of the cases, injectivity profiles before and after the stimulation are shown. As can be seen, the stimulations helped improving conformance in some cases, opened non-contributing sub-layers and cleaned near-wellbore reservoir zones. Fig. 1.2 shows injectivity profiles of two injection wells operated by Chernogorneft. After the wave stimulation treatment, their injection capacity Q increased from 101 m3/day to 493 m3/day and from 350 m3/day to 580 m3/day, respectively. The vertical columns of figures are the distances from the wellheads in meters. The intervals marked by arrows are perforated intervals. P is the wellhead injection pressure. As can be seen, wave stimulations in both wells opened new sub-layers and increased sweep efficiencies K sweep by 1.65 and 2.18 times, respectively. Fig. 1.3 and Fig. 1.4 show injectivity profiles of two wells operated by Kogalymneft. After the wave stimulation treatment, their injection capacity increased from 175 m3/day to 418 m3/day and from 350 m3/day to 740 m3/day, respectively. We can see improved conformance after the wave stimulation treatment in the first case, and activation of new sub-layers (injection along the entire perforated interval) in the second case. The perforated interval on Fig. 1.4 is marked by horizontal dash lines. Fig. 1.5 shows injectivity profiles before and after wave stimulation of a well operated by Samotlorneft. After the wave stimulation treatment, its injection capacity increased from 239 m3/day to 668 m3/day. Fig.1.6 shows injectivity profiles of a well operated by Arlanneft, a subsidiary of Bashneft Oil Company, before and after wave stimulation, at different wellhead pressures. As we can see, the injection capacity increased in all cases. The active section of the perforated interval became much longer, and injectivity of the most permeable sub-layer increased noticeably.

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Fig. 1.2. Chernogorneft Oil Company Injection Well Stimulation

Fig. 1.3. Kogalymneft Oil Company Injection Well Stimulation

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Fig. 1.4. Kogalymneft Oil Company Injection Well Stimulation

Fig. 1.5. Samotlorneft Oil Company Injection Well Stimulation

Fig. 1.6. Arlanneft Oil Company Injection Well Stimulation

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Fig. 1.7. Arlanneft Oil Company Injection Well Stimulation

Fig.1.7 shows injectivity profiles for another well operated by Arlanneft. Its injection capacity increased from 38 m3/day to 137 m3/day. As the selected data indicate, stimulation of near-wellbore formation zones provides a significant positive effect in various geological conditions. Provided below is a broader selection of data on typical near-wellbore stimulations of injection wells. 1. Duration of well stimulation: 5.5 hours. As a result of the stimulation job, diffusivity increased from 1495 to 4150 cm2/sec, permeability increased from 0.021 to 0.058 D, transmissibility increased from 40 to 109.9 D*cm/cP, and injectivity increased from 755 m3/day (at reservoir pressure maintenance pressure P = 97 atm) to 894 m3/day (P = 98 atm). 2. Duration of well stimulation: 6 hours. As a result of the stimulation job, injectivity increased from 303 m3/day (at P =120 atm) to 600 m3/day. It should be noted that injection well stimulations can bring about higher production rates just as good as production well stimulations. For instance, an analysis of displacement parameters for the two aforementioned injection wells has shown that four production wells responded to the injectivity increase. Incremental oil production from these wells after the stimulation treatment of the two aforementioned injection wells reached 600 tons over a period of two months. The results of some stimulation jobs performed on West Siberian wells are summarized in Table 1.1.

1.2 The Scientific Fundamentals of the FirstGeneration Wave Technology for Stimulation of Production Processes Provided below is a brief overview of the results that form the basis of the first-generation wave technology for near-wellbore stimulation. In greater detail they are described in [8, 9, 10, 11].

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Luginets Field 1 2 3 4 5

64 875 1,219 1,102 885

9.4 4.8 6 12 8

195 189 220 216 205

Tevlin Field 6 7,778 10 215 7 7,660 9.2 190

Kogalym Field 8 5,519 5 270 9 2,063 7.3 275 Pokomasov Field 10 263

7 262

Well Status after 9 Months in Operation

Before After Flow Water ProducStimu- Stimu- Rate Cut tion lation lation (t/day) (%) Method

Incremental Oil Production (t)

Flow Rate (t/day) Water Cut (%)

Production Method

Reservoir Pressure

Net Oil Thickness (m)

Well No.

Item No.

Table 1.1

It is a gas / oil field; Jurassic sediments (the target reservoir is J1/З); tight sandstones (average permeability is 20 mD, porosity is 16%). The reservoir has interlayers of shales and siltstones. inactive 0 0 13 6.4 0 SRP 1,585 nat. flow 20 0.8 3 4.2 20 nat. flow 1,511 nat. flow 15 1.7 5 5.2 15 nat. flow 719 nat. flow 15 1.7 4 10.6 15 SRP 1,975 nat. flow 0 1 4.2 4.2 0 SRP 965 Total: 6,755 SRP SRP

0 0

12 5

40 35

50 30

0 0

ESP 4,800 SRP 3,800 Total: 8,600 Medium- to fine-grained sandstones and coarse-grained siltstones. Arkose, shaly, heterogeneous. Shaly sandstone cement, occasionally interlayers with shaly/carbonate cement. Low permeability: 25 mD on average, porosity: 20%. injector 100 80 490 230 100 45,000 injector 100 250 540 400 100 41,500 Total: 86,500 Polymictic sandstones; porosity: 20.7%; permeability: 50–150 mD inactive 58 7.6 18.75 1.9 45 SRP 171 Total: 171

First, we would like to present the results of experiments conducted at Shell test facilities in Rijswijk, the Netherlands.

1.2.1 Large-Scale Laboratory Experiments at Shell Test Facilities Here below is a description of an experimental program in which a hydro-impact generator was used. A test drilling apparatus capable of simulating the processes that take place near an oil well (with real-life

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operating parameters) was used in the experiments. A schematic of the apparatus is shown in Fig. 1.8 and its general view is provided in Fig. 1.9.

Fig. 1.8

Fig. 1.9

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Enhanced Oil Recovery

The test procedure was as follows. Upon completion of drilling, the rate of flow of fluid through the zone of rock adjacent to the drilled hole and the differential pressure across the zone were measured. After that, the drill bit was replaced with a hydro-impact tool and wave stimulation of the walls of the drilled hole was conducted. Then, the rate of fluid flow through the same rock zone and the differential pressure across it were re-measured. A plot of measured flow rate versus differential pressure is provided in Fig. 1.10. Judging by the slope of the curves, initial permeability of invaded specimens is estimated to be around 300 mD, while after the wave treatment it is about 1000 mD (with a 20% accuracy). Initial permeability of clean specimens was 800–900 mD. Therefore, the wave stimulation brought about a roughly 4-fold increase in permeability.

Fig. 1.10

Tests were also conducted on mud-invaded hexagonal sandstone blocks measuring 39 cm by 80 cm. A hole was drilled through the center of the blocks. A conventional bit was used to drill the hole under conditions that were as close as possible to field drilling conditions. In particular, the pressure of drilling mud near the bit was 45 bar. The diameter of the drilled hole was about 22 cm. In the process of drilling, the drilling mud invaded the near-wellbore zone and clay particles contained in the mud, apart from precipitating on the surface of the rock specimen, penetrated inside the block to a certain depth. Upon completion of drilling, the rock specimen was removed from the high-pressure vessel and was cut up along a plane intersecting its axis. After that, near-wellbore zone

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wave cleanup tests were conducted on the drilled rock specimen. For this purpose, the test drilling apparatus was rearranged. The drill bit was removed and the hydro-impact generator was installed in its place at the end of the supply pipe. The supply pipe was connected to a tank filled with water. Water was pumped through the generator as described above. The generator was positioned at a distance of 12 cm from the bottom of the hole in the block. The flow rate was increased to 300 liters per minute.

Fig. 1.11

After the wave treatment the rock specimen was cut up. A photograph of the cut is provided in Fig. 1.11 (right). For comparison, a sample that was cut up after drilling without wave treatment is shown on the left side. This sample was drilled under exactly the same conditions as the one that underwent wave stimulation. A layer (3–4 cm thick) of a lighter color than the rest of the invaded zone was found around the borehole. As we can see, the contamination has disappeared. Therefore, a conclusion can be drawn that the waves cleaned the near-wellbore zone from the invading drilling mud clay particles. The results of these experiments have provided the foundation for the development of first-generation wave technology for petroleum industry applications.

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1.2.2 Resonances in Near-Wellbore Formation. Resonances in Perforations As mentioned above, near-wellbore formation wave stimulation intensifies when resonance modes of wave motion are achieved. In the general case, resonance occurs when frequencies of external wave action coincide with the natural frequencies of objects exposed to the action. In actual field conditions, various sources of resonances can be found both at the bottom of a production well and in the near-wellbore formation zone. We can investigate the onset of resonance in an individual perforation hole at the bottom of a well as one of such sources [13]. Let the perforation hole be a fluid-filled horizontal channel in the shape of a straight circular cylinder with a cross-section radius R0 and a length l. One end of the channel connects to the wellbore, while the other end and the enveloping surface, together with the impermeable plane z = 0, are the boundaries of the fluid-saturated porous medium that fills the semi-space z > 0. The channel occupies a volume measuring r < R0, 0 < z < l (Fig. 1.12).

Fig. 1.12. Wave Resonance in a Perforation Hole at the Bottom of a Well. The Model Setup

Steady-state oscillations in such a system caused by the action of harmonic pressure oscillations at the end of a channel connected to the wellbore are studied in [13]. The computed results are shown in Figs. 1.13–1.14. Fluid parameters correspond to crude oil parameters while porous medium parameters correspond to in-situ oil saturated sandstone.

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Fig. 1.13 shows channel wall oscillation amplitude versus distance to the wellbore. The curves differ in frequency ω. As can be seen from the plot, the shape of wall boundary oscillation and its amplitude depend essentially on bottomhole pressure oscillation frequency. The resonant nature of channel wall oscillations is demonstrated by the curves on Fig.1.14. These curves are channel wall oscillation amplitudes at midpoint z = l/2 (curve 1) and at channel end z = l (curve 2) versus pressure oscillation frequency for an l = 0.3 m.

Fig. 1.13

Fig. 1.14

Let us call the frequency that corresponds to the main peak of the curves in Fig. 1.14 the main resonant frequency ωres. At this frequency, the maximum oscillation of the channel wall is almost an order of magnitude greater than at other frequencies. In order to achieve resonant oscillation of the channel wall, bottomhole pressure oscillation frequency needs to be chosen very precisely. This requirement follows from the behavior of the curve near the main resonant frequency. Thus, the presence of a finite-length channel near the wellbore significantly increases the amplitude of oscillations. The main resonant frequency depends on the parameters of the porous medium and the geometry of the channel. However, numerical simulation has shown that changing the parameters of the porous medium and the radius of the channel within a wide range of values has little effect on the main resonant frequency. In the course of simulation, the channel radius R0 was changed in the range from 2 mm to 10 mm. Significant oscillations of the channel wall are observed at other frequencies as well, but their amplitude is much lower than at the main resonant frequency. Permeability has the largest spread in values. Permeabilities between layers can differ by an order of magnitude or more. Fig.1.15 shows the

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Enhanced Oil Recovery

main resonant frequency versus channel length. Curve 1 corresponds to a permeability k = 1 10–10 m2, and curve 2 corresponds to a permeability k = 5 10–12 m2. Curve 1 permeabilities are typical for Bashkirian oilfield formations, whole curve 2 permeabilities are commonly observed in West Siberian formations. As we can see, there is almost no difference between the values of resonant frequencies for perforation channels of the same length.

Fig. 1.15

Fig. 1.16

The main resonant frequency mostly depends on the length of the channel. As can be seen from Fig. 1.15, the smaller the channel length, the higher the frequency. Fig. 1.16 shows the amplitude of porous medium matrix oscillations versus the distance to the channel axis r at the main resonant frequencies for various channel lengths. The mutual arrangement of the curves indicates that for oscillations at the main resonant frequency the affected zone grows with the length of the channel. Therefore, a frequency of bottomhole pressure oscillations corresponding to the highest amplitude of oscillations in the near-wellbore zone can be chosen. There is a weak dependence between this pressure oscillation frequency and the parameters of the porous medium but a strong dependence on the length of the channel. We note that the foregoing problem involves only a single perforation hole. However, several or even all perforation holes of a well may resonate simultaneously. This happens when a generator’s radiation frequencies are resonant for all of the perforation holes. Besides, in this case the entire near-wellbore formation zone may be involved in resonance. In practical terms, this is the most preferable case for near-wellbore wave stimulation.

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1.2.3 Excitation of Oscillations in Micro-Pores by OneDimensional Longitudinal Macro-Waves in a Medium. Resonances. Transformation of Micro-Oscillations in Pores to Macro-Flows of Fluid. The Capillary Effect This sub-section describes the results achieved by the authors in cooperation with S. A. Korneyev and published in [10, 11, 17]. We investigated the possibility of transforming the wave motions of a porous fluid-saturated medium into a unidirectional monotonous motion of fluid such as the flow of fluids from a productive reservoir to a wellbore. These types of motions can be realized in porous fluidsaturated media because in heterogeneous wave fields an impulse transferred from the matrix to the fluid during a period of oscillation can be non-zero. This is caused by various nonlinearities as well as by heterogeneities of wave fields inside the pores. When macro-waves propagate in a porous fluid-saturated medium, they excite oscillations inside each pore. If certain conditions are met, the oscillatory motions in the micropores transform into strong unidirectional monotonous macro-motions of the fluid. We studied the simplest one-dimensional wave field (a compressional wave) in a porous fluid- saturated medium, located on a segment 0 < x < L and defined as follows. The left end of the segment is described by the fictitious tension f 0 and the pressure in the fluid as the sum of the constant pressure component p0 and disturbance in the form of harmonic oscillations; the right end of the segment is described by the absence of matrix shifts and a pressure that is equal to its undisturbed value p0. The conditions at the left end correspond to a wellbore with an oscillation generator, while on the right end they correspond, for instance, to a boundary between reservoir zones with highly different matrix porosities and densities, or to another wellbore. The solution of the linearized equations is a standing compressional wave. In addition to nonlinearities such as convection terms commonly observed in hydrodynamic systems, unidirectional seepage flows are also controlled by nonlinearities specific to porous fluid-saturated media and associated with porosity oscillations in wave fields. The thing is that porosity oscillations are a superposition of terms whose frequencies and phases coincide with the oscillations of densities and pressures in the fluid. Therefore, situations are possible when superposition of porosity oscillations with fluid pressure and density oscillations in a wave field

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Enhanced Oil Recovery

leads to the initiation of unidirectional motion of fluid in the pores of a fluid-saturated porous medium. Mathematically, it is described by nonlinear terms in equations of motion that contain products of cyclic in time porosity disturbances by cyclic disturbances of pressure gradients in the fluid and fictitious tension in the matrix, or by cyclic disturbances of fluid flows and matrix in interfacial interaction forces, in particular in equations for frictional forces and forces of added masses. From the applied point of view, the most interesting thing is to find out whether waves can be used to achieve directional motions, in particular a fluid flow that is nonzero in time on the average. The relationship between fluid flow and frequency shows that there are frequency bands within which considerable fluid flows may occur. It is caused by resonances of the initial compressional standing macro-wave. Fig. 1.17 shows fluid flow velocities versus the excitation frequency ω for certain values of parameters of the porous medium. As we can see (Fig. 1.17a), fluid flow velocities are quite high in a number of narrow frequency bands. In what follows, the frequencies that correspond to maximum fluid flow velocities are referred to as resonant frequencies.

Fig. 1.17

For comparison purposes, Fig. 1.17b shows pressure gradient versus Darcy steady-state fluid flow velocity for the medium. As we can see, to achieve a fluid flow velocity equal to the velocity achieved by external pressure oscillation stimulation with an amplitude of the first resonant frequency of just 1 bar, the steady-state case requires a pressure gradient of about 20 bar/m, while water injection through dedicated wells usually provides pressure gradients of 0.3–0.4 bar/m [18]. Therefore, wave stimulation is capable of initiating flows at such velocities that in the steady-state case require practically unreachable enormous pressure gradients.

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We note that resonant frequencies of the one-dimensional (in this case) wave strongly depend on the length of the interval L. As analyses have shown, the smaller L the higher the resonant frequency. For L = 1 m, the resonant frequency р is approximately 1.0 kHz. In summary, we can say that nonlinear wave mechanisms of motion have been found that can create additional fluid flows in porous fluidsaturated media. Velocities of such flows are similar to the velocities of steady-state flows that can be created in the same medium by significant steady-state pressure gradients. The magnitude of these gradients is such that it cannot be achieved by any other currently known method. As far as it is applicable to oil production, it means that wave fields created in a medium can be used to clean up near-wellbore formation zones, mobilize capillary-trapped oil, drive hydrodynamically unconnected hydrocarbon blobs towards wellbores, etc. A similar phenomenon of the transformation of micro-oscillations of pores into significant fluid macro-flows was found by the authors as far back as in 1989 [19]. It was demonstrated that traveling waves that propagate in the walls of a micro-capillary (Fig. 1.18) – a model of a pore in a porous medium – can cause non-oscillatory unidirectional motions of the fluid filling the capillary along its axis. The parameters of such flows are shown in Table 1.1. For instance, a traveling shear translational wave with an amplitude of just 10–2 μm propagating in the wall of a capillary with an undisturbed radius of 10 μm can cause steady-state flow inside the capillary with a cross-sectional average velocity of 4.5 cm/sec. To create the same flow by applying steady-state pressures at the ends of the capillary, a pressure gradient of 36 bar/m would be required [10, 11]. Such a gradient cannot be achieved in subsurface formations using conventional methods [18].

Fig. 1.18

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Enhanced Oil Recovery

Table 1.2 Capillary Radius (m) 10–2 10–3 10–5

Wave Frequency (Hz) 550 5·104 2·107

Average Velocity (m/s) 0.39 0.344 0.045

Equivalent Steady-State Pressure Gradient (mPa/m) 0.312·10–4 0.275·10–2 3.6

1.2.4 Cleaning of Horizontal Wells It was proposed to use first-generation wave technology to clean sand control screens used in horizontal wells. Typical, commonly used screens are shown in Fig. 1.19. The one on the left is a pre-pack screen in which sand and gravel are used as the filtering components (here below this type of screens is referred to as S1), while on the right is a strata-pack screen that uses pre-perforated multilayer metal cylinders (hereinafter referred to as S2). A detailed description of the relevant experiments conducted by the authors in cooperation with NC NVMT RAN scientists I. G. Ustenko and V. N. Ivanov is provided in [10, 11]. In this paper, we offer only a brief overview of the results.

Fig. 1.19

Sand control screens are commonly used in horizontal well completion strings to prevent drilling mud solids and reservoir fluid solids from plugging the wellbore. To ensure reliable protection, it is important that the screens remain clean during and after the well completion process. It is obvious that the simplest method of cleaning is washing. In this case, the washing fluid is circulated through the screen’s polluted zone. Note that the polluted zone has to be isolated (for instance, by packers) for the duration of the washing job. The thing is that if the polluted zone is not isolated, the washing fluid would circulate only through clean sections of the screen, leaving the polluted section untouched.

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The wave method of cleaning screens in horizontal wells does not require isolation of polluted zones as in this case cleaning is provided by waves and cavitation created by wave generators. A schematic of a screen-cleaning wave generator assembly in a horizontal well is shown in Fig. 1.20, where 1 is the reservoir, 2 is the screen, 3 is the tubing containing the washing fluid, 4 is the wave generator, 5 is the cavitation zone, 6 are the fluid flow lines downstream of the generator, 7 is the lowpressure zone, 8 are the pressure waves, 9 are the cavitation bubbles, 10 is the flow of the reservoir fluid through the screen, 11 is the flow of the washing fluid through the screen.

Fig. 1.20

The washing fluid is pumped through the generator. If the flow rate and the pressure are high enough for the generator to operate, then waves and cavitation appear in the flow downstream of the generator. Cavitation bubbles collapse at a distance from the generator and create highamplitude pressure pulses that do the cleaning. Figs. 1.21–1.22 show the results of screen washing tests. As can be seen, the wave cleaning method outperformed conventional washing with respect to both screen types. In the case of screen S1 (Fig. 1.21), it took less time to clean the screen with a better quality of cleaning. Screen S2 (Fig. 1.22) was cleaned even faster. Moreover, the conventional washing method was unable to clean this screen at all. It was determined that the screen’s permeability began to drop, and the test was stopped. We would like to note that although the upstream pressure in the wave cleaning case was higher than in the conventional washing case, the total energy consumption was lower due to a shorter duration of the cleaning job. The tests show that wave stimulation offers better performance and cost efficiency compared to conventional washing. This method allowed us to clean screen S1 two times faster. The final screen permeability

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Enhanced Oil Recovery

achieved by the wave cleaning method is much closer to the initial permeability than the final permeability achieved by washing. It took even less time to clean screen S2 by the wave method than screen S1, while the washing method was unable to clean screen S2 at all.

Fig. 1.21

Fig. 1.22

In summary, the wave screen cleaning technology has the following advantages of over conventional washing: 1) better quality of cleaning, 2) shorter procedure duration, 3) smaller power consumption, 4) no need to isolate the polluted zone with packers, 4) no invasion of the washing fluid into the productive reservoir. Therefore, the wave cleaning method has proven its high performance.

1.2.5 Preliminary Results However, despite numerous successes, there were some failures in the application of wave technologies for near-wellbore zone cleaning – mostly, as mentioned above, because the method was used incorrectly, without understanding its scientific basis, without inducing resonance in the near-wellbore zone, etc. Failures of first-generation wave stimulations had various causes. In some cases, they were caused by reservoir depletion, while in other cases they were caused by in-situ chemical processes that led to significant permeability impairment. In yet other cases, the wave forces acting on the contaminating particles in near-wellbore formation were not high enough to overcome the force of adhesion, etc. Even in cases where the matrix was contaminated by plugging solids, wave stimulation failure analysis was complicated by an inaccurate understanding of the mechanisms and criteria of near-wellbore formation cleaning by the wave method, etc. There was no accurate understanding of how the wave cleaning mechanism worked. What wave parameters are critical for the cleaning process? What governs the wave parameters critical for the

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cleaning action? This study is an attempt to find answers to these questions that are linked to – as it will be shown below – the development of the so-called micro- and macro-mechanics of petroleum reservoirs. Accordingly, the first-generation wave methods designed mainly for near-wellbore formation stimulation will also advance significantly.

1.3 Stimulation of Entire Reservoirs by FirstGeneration Wave Methods for Enhanced Oil Recovery. Resonance Macro- and MicroMechanics of Petroleum Reservoirs: A Scientific Foundation for Enhanced Oil Recovery In a number of cases where near-wellbore wave stimulation jobs were conducted using a generator based on vortex flow effects, increases in flow rates were observed not only on the well in which the generator was installed but on offset wells too. Therefore, apart from wave stimulation of near-wellbore formation zones, this method can be used to stimulate entire reservoirs or large sections measuring several square kilometers. In this case the generator is installed in one of the wells within the selected field section, and the offset wells are influenced by the stimulation (Fig. 1.23).

Fig. 1.23

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Enhanced Oil Recovery

As an example, Fig. 1.24 shows the results of a stimulation job conducted on a section of the Luginets oil field located in the Tomsk region. As we can see, production flow rates were restored not only on the stimulated well but on several offset wells too: the offset wells and the stimulated well form a pattern.

Fig. 1.24

We would also like to provide here some results of field tests conducted at the Elk Hills oil field in California [20]. The responding wells were located within 800-meter radius circles whose centers were the wells in which the generators were installed. A map of the field is shown in Fig. 1.25.

Fig. 1.25

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The 73 responding wells were located within two circles as shown on the map. Stimulation results are shown in Fig. 1.26, which is the combined response of the 73 responding wells located inside the circles. The simulation program started in December 2003. By January 2005 combined flow rates of the 73 wells (oil production) increased by 42% compared to the initial level at the beginning of the stimulation campaign. Compared to trend-based January 2005 projected production, combined flow rates increased by 60%. According to the oil cut trend (a parameter inverse to water cut), the oil cut was expected to drop. However, a 28% increase in the oil cut compared to the initial level at the beginning of the stimulation program, or a 47% increase compared to the trend-projected value was observed.

Fig. 1.26

Fig. 1.27 is a comparison of responses to two enhanced oil recovery methods: hydraulic fracturing and impact wave stimulation of a large section of an oil field. These jobs were conducted at the Lost Hill oil field in Texas. It is a diatomite formation that has 0.1–2,000 mD permeability, 45–55% porosity, 700–1200 m productive depth, 30% water cut and 26o API oil gravity. In October 1999, a 7.2-point Richter scale earthquake took place. We can see on the diagram that the 7 wells that were fractured in November 1999 responded to the earthquake. In August 2000, a section of the field was stimulated using an impact wave generator.

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Enhanced Oil Recovery

Fig. 1.27

The response to hydraulic fracturing is obvious in terms of increases in oil production and oil cut. However, the response to wave stimulation is comparable to hydraulic fracturing. Note that the 7-well hydraulic fracturing cost was US$2 million, while the hydro-impact wave stimulation cost only US$ 100,000. These data prove that in some cases stimulation of entire reservoirs can bring very good results. However, just like in the case of nearwellbore formation treatment, the stimulations were not always successful. Failures of first-generation wave stimulations of entire reservoirs had various causes, including the application of the method without a clear understanding of its scientific basis. In some cases, there was not enough residual oil in the stimulated section; in other cases, the wave forces acting on capillary-trapped oil droplets and reservoir-contaminating solids were not high enough to overcome the capillary forces and the force of adhesion, etc., just like it was mentioned above for the nearwellbore formation wave treatment case. Wave stimulation failure analysis was complicated by an inaccurate understanding of the causes behind oil recovery impairment in productive reservoirs, the lack of stimulation success or failure analyses, and the lack of a rigorous theory to explain the observations. Moreover, it was not clear until recently what wave parameters are critical for the stimulation of entire reservoirs, etc. Summarizing, the principal mechanisms of oil production and oil recovery impairment were not investigated thoroughly enough, the criteria (conditions) for enhancing the recovery of oil from heterogeneous reservoirs were not identified, etc. The resonance macro- and micro-mechanics of petroleum reservoirs, which is the subject of this study, provides answers to the questions posed above. It serves as the scientific basis for the development of con-

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trolled machines and devices, including oscillation and wave generators with proper measuring and control systems, for stimulation of nearwellbore zones and entire reservoirs to increase production and to enhance oil recovery. Wave stimulation equipment is based on mathematical software products that have been created thanks to the development of the resonance macro- and micro-mechanics of a petroleum reservoir – a growing field of fundamental science.

Enhanced Oil Recovery: Resonance Macro- and Micro-Mechanics of Petroleum Reservoirs. O. R. Ganiev, R. F. Ganiev and L. E. Ukrainsky. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

2 Remove Micro-Particles by Harmonic External Actions The Micro-Mechanics of a Porous Medium and the Criterion for Successful Wave Stimulation to Remove Contaminating Micro-Particles Stuck to Pore Walls by Harmonic External Actions

This chapter describes theoretical investigation of the phenomena that take place in the wave fields in fluid-saturated porous media. We consider the effect of these phenomena on fluid flow processes in productive reservoirs and on pore space cleaning, that is on the motion of microinclusions in the pores under the action of forces of the wave nature.

2.1 An Analysis of the Forces Acting on Pore-Contaminating Particles under a Harmonic External Action In this sub-section, we consider the case where the flow capacity of nearwellbore formation suffers from the precipitation in the near-wellbore zone of small rock particles that were either carried in from the reservoir with the produced fluids or invaded the near-wellbore zone during drilling. We assume that the contaminating solids are retained in the pore space by the action of adhesion forces that make the particles cling to rock walls. We consider the wave action to be monoharmonic. The main idea behind using wave stimulation to clean the pore space from any contaminants of this type is to use wave action to transmit oscillation motions to the solid particles that contaminate rock matrix pores, reservoir fluid and pore walls. It will result in the detachment of contaminating particles and oil droplets from pore walls and their mobilization with the seepage fluid flow towards production wells.

27

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Enhanced Oil Recovery

Let Oxyz be a system of Cartesian coordinates with the origin at an arbitrary point O, fixed in the absolute space. Let the unit vectors of this system – ex, ey, ez – be also fixed, and let the unit vector ez be vertical. We consider an arbitrary pore with a solid particle retained on the surface of the pore by the force of adhesion. At the point of contact between the particle and the pore, let us introduce the normal to the surface of the pore np directed inside the pore, and the vectors τp1 and τp2 that form an orthonormal basis on the plane tangent to the surface of the pore at the point of contact with the retained particle.

Fig. 2.1 We assume that the point on the boundary of the pore to which the contaminating particle is attached moves together with the matrix of the porous fluid-saturated medium with a velocity of w w x e x w y e y w z ez . We also assume that the particle is acted upon by the following forces: 1) the sum of reservoir pressure gradients – grad P and the wave forces that occur in a wave field (such as those described in Section 1.2.3). We consider this sum to be independent of time; 2) the force of gravity and the force of buoyancy directed along ez; 3) the non-steady-state fluid pressure gradient due to wave action – dw , where p is the density of the grad p; 4) the force of inertia p dt particle; 5) a Stokes type force proportional to the difference between velocities of the fluid and the matrix: u w , where ηf is the viscosity of the fluid, α is a coefficient that depends on the shape and dimensions of the pore and the contaminating particle (in the simplest case 4.5rp 2 , where rp is the mean radius of particles); we note that the

Remove Micro-Particles by Harmonic External Actions

29

latter force does not have a non-zero component along the normal np due to the medium persistence assumption; 6) an added mass type force proportional to the difference between accelerations of the fluid and the dw du particle: , where β is the added mass coefficient, f is f dt dt the density of the fluid; 7) the force of adhesion Fad that retains the particle on the pore wall. We assume that the force of adhesion Fad balances off any force f that is applied to the particle and tries to detach the particle from the wall, i.e. whose projection on the normal to the pore surface n p directed inside the pore is positive if the modulus of the force f does not exceed some limiting value Flim that depends on the force of cohesion between the particle and the pore surface and the normal pressure N that presses the particle to the pore surface. We assume that Flim MS kNS , where M is a coefficient of the force of cohesion that defines the maximum specific force of cohesion between the particle and the pore per a unit of the contact area that is independent of the normal pressure; S is the contact area between the particle and the pore wall; k is a coefficient of proportionality between the normal pressure that presses the particle to the pore wall and the increase in Flim . If f Flim , then the particle is detached from the pore wall and can be carried away with the fluid. Let the condition f Flim be called the detachment condition. Its realization is a prerequisite for removing the contaminating particle from the pore. In the general case, the detachment force f can have a component f which lies on a plane tangent to the pore wall, and a component fn which is perpendicular to this plane and acts along the direction of the normal n p . It is evident that when fn > 0, the component fn contributes to the detachment of the particle from the pore wall, while when fn < 0 the component presses the particle to the pore wall. We assume that in the coordinate system {ex, ey, ez} the normal n p is defined by its projections n px , n py , n pz on the unit vectors e x , e y , e z , respectively, and the two

unit orthogonal vectors τp1 and τp2 that lie on a plane tangent to the pore wall are defined by their projections p1x , p1 y , p1z and p 2 x , p 2 y , p 2 z on the unit vectors e x , e y , e z , respectively. The component f is defined by its projections fτ1 and fτ2 on τp1 and τp2, respectively. Then, when

Enhanced Oil Recovery

30

the particle is acted upon by an arbitrary force f , the force of adhesion Fad can be found as follows: f *2

f , if

Fad

0, if

f

fn , if fn 0 , Flim 0, if fn 0

where f *

f 21

2 *

f

2 1

f 22 f

2 2

Flim

MS k f n , if f n MS, if fn

(2.1)

,

Flim

0

.

0

2.2 Conditions for the Detachment of a Solid Particle from the Wall of a Pore under Harmonic External Action Let us write out the condition for the detachment of a particle from the surface of a pore. The normal component of the forces acting on the particle under the aforesaid assumptions can be written as follows:

fn

P np

V

p

dw x dt

f

p np

g ez n p

f

dux npx dt

dw y

du y

dt

dt

f

dw y

dw x npx dt

p

dt

npy

npy

dwz npz dt

dw z dt

f

duz npz dt

,

where V is the volume of the particle, g is the free fall acceleration and the projection of the gravity force acting on the particle on the unit vector ez . Now we can write the detachment condition as follows: If fn > 0, then P np

p

f

p np

g ez n p

f

dw y

du y

dt

dt

npy

P p

f

p

f

g ez

dw z dt

pi

p pi

ux w x

f

uy w y

pix

dt

dwz npz dt

npy

f

dw x dt

piy

dw x dt

dw x dt

dux dt

dw y pix

f

2

2

dw y pix

dt

uz w z

piy

dw z dt

piz

M2 , R2

piz

i 1,2 f

dux npx dt

duz npz dt

p

pi

dw y

dw x npx dt

dt

du y dt

piy

f

dw z dt

duz dt

piz

(2.2) where R is the characteristic linear dimension of the particle.

Remove Micro-Particles by Harmonic External Actions

31

If fn < 0, then P p

f

dw x dt

p

g ez

pi

pi

p pi

ux w x

f

uy w y

pix

piy

2

dw y pix

piy

dt

uz w z

dw z dt

piz

piz

i 1,2 f

M R

k

dw x dt

dux dt

P np

p

f

dw x dt

pix

f

f

g ez n p

dux npx dt

dw y

du y

dt

dt

p np

p

f

piy

dw x npx dt

dw y

du y

dt

dt

dw z dt

f

dw y dt npy

duz dt

piz 2

npy

f

dwz npz dt dw z dt

duz npz dt

.

(2.3) Note that the detachment conditions define not only the condition for the detachment of a single particle from the wall of a pore. They also define the condition for the destruction of agglomerates of particles in restrictions in pore channels. Specifically, an agglomerate is destructed if it is acted upon by a force that is sign-alternating in time. Under the action of such a force whose magnitude is sufficient for the detachment condition to be met, the particles periodically detach and shift towards the restriction or away from it depending on the sign of the force. The particles may become mobile, turn over and pass through the pore restriction at a certain time point. This phenomenon somewhat resembles the process of sifting of loose media through a vibrating sieve: grains slip through sieve openings under the action of not a high but an alternating force. We expand the expressions that appear in square brackets in the detachment conditions (2.2) or (2.3) and may depend on wavefield characteristics into Taylor series about the wavefield amplitude ε, and in the obtained expansions we limit ourselves to the terms that contain ε with an exponent not higher than 1. Additionally, we neglect any terms containing the harmonic terms with frequencies that do not coincide with the frequency of the wave field (it needs to be reminded that here we consider only monoharmonic excitation). The obtained approximate expressions are the sums of two types of summands: time-independent summands and monoharmonic functions of time. Physically, the steadystate summands in the considered approximation do not depend on the wave field. They depend only on the steady-state external forces: the reservoir pressure and the flow in the reservoir caused by it. These sum-

32

Enhanced Oil Recovery

mands may appear in the expressions for the steady-state pressure gradient grad P and fluid velocity u. Additionally, we assume that fluid velocity is the sum of the same two types of summands: a time-independent summand that defines the flow process, which in what follows is marked by a straight line over the symbol of every fluid velocity component, and a harmonic summand proportional to the harmonic function of time with a frequency equal to the excitation frequency, which is marked by a tilde over above the symbol of every fluid velocity component. So, here we neglect the harmonics that do not coincide with the main harmonic of the wave field, i.e., it is assumed that the action of nonlinear parts of the wave fields on the motion of particles can be neglected. That is, in this statement of the problem we neglect the wave forces whose examples are described above in Section 1.2.3. For this approximation, all distributions of the amplitudes of oscillation velocities of the matrix, the fluid and the pressure along spatial coordinates are proportional to the oscillation excitation amplitude ε. In the expressions for matrix and fluid acceleration we also neglect the harmonics of the frequency that does not coincide with the frequency of the monoharmonic wave field. With these assumptions, the expression for the normal component of a force applied to a particle on the part of the fluid and the pore wall can be written as follows: fn

P np

where С3

D3 sin

f

t

p

3

wx t

ux npx t

V С3 f

D3 sin

t

3

,

(2.4)

gez n p ,

p np

p

wx npx t wy

f

t

uy t

wy t npy

wz npz t

npy

f

wz t

uz npz . t

The detachment condition (2.2) can be formulated as follows. If С3 D3 sin t 3 0, then the particle detaches from the pore wall if the following condition is met: 3 2 M2 (2.5) Сi Di sin t i , R2 i 1 P where Сi ux pix u y piy uz piz , i 1,2; p f g ez pi pi

Remove Micro-Particles by Harmonic External Actions

Di sin

wx t

f

uy w y

piy

p

i

ux w x

f

pi

wx t

p

t

pix

wy pix

pix

f

t

uz w z

wy

ux t

wz t

piy

piy

t

piz

piz

uy

t

33

wz t

f

uz t

piz

,

i 1,2. The detachment condition (2.3) is formulated as follows. If С3 D3 sin t 3 0, then the particle detaches from the pore wall if the following condition is met: 2

Сi

Di sin

2

M k С3 R

2

t

i

i 1

D3 sin

t

. (2.6)

3

We transform (2.5) to write the detachment condition for the case of a positive normal component of the force acting on the particle as follows: С3

D3 sin

a

2

2b

t

0,

3

(2.7)

c 0,

(2.8)

where 3

3

Di2 sin2

a

t

; b

i

i 1

M2 . R2

3

Сi Di sin

t

i

Сi2

; c

i 1

i 1

The detachment condition (2.6) for the case of a negative normal component acting on a particle is written as follows:

С3

A

D3 sin 2

2B

t

(2.9)

0,

3

G 0,

(2.10)

2

Di2 sin2

where A

t

i

k 2 D32 sin2

t

3

;

i 1 2

B

Сi Di sin

t

Mk D3 sin R

i

i 1 2

Сi2

G i 1

M R

t

3

kD3C3 sin

t

3

;

2

kC3

.

The sign of the coefficients A and a is of primary importance for an analysis of the conditions (2.7)–(2.10).

34

Enhanced Oil Recovery

As far as the coefficient a is concerned, it is evidently non-negative. If the wave field is non-one-dimensional, then theoretically the sign of the m 3 , m 1,2,3... coefficient a can be zero at time points t Tm only if the differences between the phases 1 2 , 1 3 are multiple of π. But if the wave field is one-dimensional, then the projections of the forces acting on the particle are co-phased harmonic functions, i.e., Tm a 0. Consequently, the sign of 1 2 3 . In this case, when t the coefficient a is almost always positive, except for the case of a onedimensional wave field or satisfaction of the above stated conditions for the phases of the projections of the forces acting on a particle in nonone-dimensional wave fields at the discrete time points t Tm . Here below we assume that the condition a > 0 is met, and the abovementioned exceptions are considered separately below. Depending on the value of k, which is a coefficient of proportionality between the normal pressure that presses the particle to the pore wall and the increase in the ultimate detachment force that appears in the definition of the force of adhesion, the coefficient A can have any sign at various points of time. We note that the variables c and G describe the impact on particle detachment from pore walls of time-independent actions, namely the reservoir pressure and the force of gravity. If they are positive, then the time-independent actions exceed the adhesive forces and the particle detaches without any wave impact. In practice, during the many millennia of the reservoir’s existence all the particles for which this condition holds have detached from pore walls under the action of timeindependent forces (reservoir pressure, weight and buoyancy force). Therefore, if we neglect the particles that have been detached from the walls of a pore and pressed to the wall of another pore (where the detachment condition by the time-independent forces is met) by wave action, it is natural to limit ourselves to the case where c and G are negative. The exceptions that have been just mentioned above will be considered separately. With negative values of c and G particles do not detach from pore walls without wave impacts. The conditions (2.7)–(2.10) are written for a specific pore that occupies a specific position in the space and is acted upon by a specific wave action for which all the coefficients Ci and Di (i = 1, 2, 3) are defined for each specific point of time. A simple analysis produces the following results for the first case that is considered here, a > 0, c < 0 and G < 0.

Remove Micro-Particles by Harmonic External Actions

35

We introduce the notations:

b2 ac

B2

AG M R M R

M2 R2

3

Di2 sin2

t

i

i 1

C2 D1 sin( t

1

) C1D2 sin( t

2

)

C3 D1 sin( t

1

) C1D3 sin( t

3

)

C2 D3 sin( t

3

) C3 D2 sin( t

D1C2 sin( t

1

) D2C1 sin( t

2

)

2 2 2

F,

2

)

2 2

kC3 D1 sin( t

1

) kC1D3 sin( t

3

) 2

kC3 D2 sin( t min1

2

) kC2 D3 sin( t

C3 D3 sin t

3

)

H,

. 3

. We note that when t Tm , min1 Let us remark that in the case in question when a > 0, c < 0, the following condition holds F > 0. We limit ourselves to positive values of the amplitude ε. First, we consider the time points when the following condition holds:

D3 sin

t

3

0.

(2.11)

Then, the condition (2.7) holds in the following cases: ; 1) for С3 > 0 with any positive amplitude values: 0 2) for С3 < 0 with ; min1 The condition (2.9) holds if the condition (2.11) is satisfied, only if С3 < 0 and 0 min1 . For the time points that satisfy a condition inverse to (2.11), specifically D3 sin t 3 0, (2.12) we obtain that (2.7) holds only if С3 > 0 and 0 min1 . The condition (2.9) holds if the condition (2.12) is satisfied in the following cases: ; 1) for С3 < 0 with any positive amplitude values: 0 . 2) for С3 > 0 with min1

36

Enhanced Oil Recovery

The detachment conditions (2.7) and (2.8) are met in the following cases: 1) for time intervals when (2.11) is met, С3 > 0, min ; 2) for time intervals when (2.11) is met, С3 < 0, max( min1 , min ) ; 3) for time intervals when (2.12) is met, C3 > 0, min1 min , , min1 min where

b

F

. a The detachment conditions (2.9) and (2.10) are met in the following cases: 4) for time intervals when (2.12) is met and С3 ≤ 0, A > 0, min 2 ; 5) for time intervals when (2.12) is met and С3 < 0, A < 0, H > 0, B > 0, min2 min3 ; 6) for time intervals when (2.12) is met and С3 > 0, A > 0, max( min1 , min 2 ) ; 7) for time intervals when (2.12) is met and СС3 > 0, A0, B>0, max( min1 , min 2 ) min 3 , max( min1 , min2 ) min3 ; 8) for time intervals when (2.11) is met, С3 < 0, A > 0, min 2 min1 , min2 min1 ; 9) for time intervals when (2.11) is met and С3 < 0, A < 0, H > 0, B > 0, min 2

min

min1

B

,

min 2

H

min

min1

,

min3

,

where

B min2

H A

,

. A In other cases, when the coefficients satisfy neither of the latter conditions 1) to 9), the detachment conditions are not met. The determined conditions have a simple physical sense. Not only can they be obtained formally, mathematically, but also by considering the projections of the wave forces and static forces acting on the particle on the normal to the surface of the pores, as well as by comparing the absolute values of the forces that contribute to the detachment of the particle from the wall of the pore on the one hand and to the pressing of the particle to the wall of the pore on the other hand. An interesting case here is when the detachment takes place, and the projection of the resultant of the wave forces acting on the particle on the normal to the surface of the pore is negative, that is, the wave force presses the particle to the pore wall. However, they also have tangential components that act in a plane tangent to the surface of the pore. These components contribute to the min 3

Remove Micro-Particles by Harmonic External Actions

37

detachment of the particle from the pore wall. The higher the amplitude of the action, the bigger both the normal and tangential components. Since the absolute values of the projections may depend nonmonotoniously on the amplitude of the actions, a situation then becomes possible where tangential components prevail for some amplitude values, while normal components prevail for others. The dependence between the detachment force and the amplitude is also non-monotonic. When the amplitude increases above zero, the detachment force first increases and then, after reaching the maximum at which the detachment condition is met, it begins to decrease. It is easy to show that for a > 0, c < 0 and G < 0 (this is the case considered here), this can take place only when the following conditions are met: A 0, B 0, H 0. Apart from the case considered above, a non-monotonic dependence of the absolute values of the projections of the force on the local coordinate axes {np, τp1, τp2} may also be seen for a > 0, c > 0 and G > 0. As mentioned above, this is a case where the detachment conditions are satisfied by the action of the steady-state forces only, without any wave action. It will be shown below that in this case wave action may cause the detachment conditions cease to hold at certain values of the amplitude, i.e. wave action can contribute to retaining the particles on the surface of some pores. For example, for time intervals satisfying the condition (2.11) with С3 > 0, c > 0, b < 0 and F > 0, the detachment condition is met when the amplitude of wave action is zero. As the amplitude ε grows from zero to

b

F

, the absolute value of the resultant force a that detaches the particle from the pore wall decreases, and at min 4 it M . As the wave force equals the limit value of the force of adhesion R amplitude increases further within the range min 4 min , the absolute value of the force of adhesion exceeds the absolute value of the resultant of the external forces acting on the particles. The detachment b condition is not satisfied. If the amplitude exceeds , the absolute a value of the resultant of the external forces acting on the particle starts to grow, and when it reaches min it becomes equal to the limit value of M . Further increase in the wave force amplitude the force of adhesion R causes satisfaction of the condition of detachment of particles from the pore wall. So, in this case there are two possible ways to achieve particle min 4

38

Enhanced Oil Recovery

detachment from the pore wall. Wave action can be stopped. Then the detachment condition will be satisfied by the action of static forces. Conversely, the amplitude can be increased to meet the condition min . The latter approach is associated with an increase in energy consumption and is not always feasible. Furthermore, there may be cases where an increase in the amplitude of wave action does not increase the detachment force, but rather decreases it. For example, when А < 0, G > 0 the detachment condition is satisfied as long as the amplitude is in the range 0 min 3 . However, for any amplitudes min 3 the detachment condition ceases to hold. In this case, it happens because the absolute value of the normal component of the wave force pressing the particle to the pore wall increases with the amplitude of wave action, and the corresponding increase in the limit value of the force of adhesion is so high that at any amplitude min 3 it exceeds the absolute value of the tangential component of the wave force. Consequently, a radical way to ensure that the detachment condition is met in the cases for which the conditions c > 0 or G > 0 are satisfied is combined stimulation when wave action alternates with the absence of any external artificial actions. We note that when A < 0, as the amplitude of wave action increases, the left-hand side of the detachment condition (2.10) decreases, starting from a certain value of the amplitude, and tends to –∞, which guarantees that the detachment condition is not satisfied. So, in certain modes of wave action and for certain orientations of the pores, increasing the amplitude of the action not only fails to contribute to the detachment of a particle from the pore wall, but, on the contrary, presses the particle to the wall more strongly. This case is realized when 2

k 2 D32 sin2

t

Di2 sin2

3

t

i

.

(2.13)

i 1

An increase in the parameter k that appears in the expression for the force of adhesion contributes to the realization of the relation (2.13). Its specific value for a known type of rock that forms the porous medium and for known contaminants can be determined experimentally. If the condition (2.13) is not satisfied, it follows that in order to achieve the detachment conditions the amplitude of the wave force needs to be as high as possible. If the condition (2.13) is satisfied, the amplitude needs to be within certain ranges. Therefore, to ensure efficient wave stimulation we need to have the opportunity to change the excitation amplitude over a fairly wide range. In particular, if the condition (2.13) does not

Remove Micro-Particles by Harmonic External Actions

39

hold for a medium, for example, due to small values of the parameter k or such an orientation of the pores in the wave field when the tangential components of the wave forces are considerably higher than the normal ones, there needs to be an opportunity to significantly increase the amplitude. Resonant modes of motion are the most efficient and energy effective way to achieve it. These modes are discussed in the following sections. Let us note that – as it is mentioned in the Introduction – petroleum scientists have long believed that resonance modes in productive reservoirs are unfeasible. For example, an article by a leading Shell specialist S. W. Wong [14] says that “optimal resonance excitation in reservoirs is unlikely or even impossible”. In this monograph we provide a definite answer to this question: resonance wave modes do exist and can be excided in many reservoirs with quite common geological settings.

2.3 The Criterion of Successful Harmonic Wave Stimulation. Criterion Determination Procedure Even if the conditions for the detachment of a particle from a pore wall that have been defined in the previous section are satisfied, it cannot guarantee that the pore space will be actually cleaned from particles stuck to the pore walls. The thing is that these conditions have been obtained only for an individual pore that is oriented in a certain manner relative to the fixed coordinates. In an actual medium the pores are oriented differently in every point of space. There are no reliable methods to determine the actual orientation of a pore in every point of space. Apparently, it can be assumed that any spatial orientation of a pore is equally probable. Therefore, the fulfillment of the conditions for the detachment of a particle from a pore wall is also probabilistic. Specifically, pores of all possible orientations need to be considered to determine the probability of detachment of a particle from a pore wall in a particular point of space. The space of elementary events in this case consists of all possible pore orientations. The probability of satisfaction of the conditions for the detachment of a particle from a pore wall equals to the ratio of the number of pore orientations for which the detachment condition is met to the total number of all possible orientations. If pore orientations can be somehow parameterized, then in the space of the parameters the ratio of a measure of the domain occupied by the values of the parameters that

40

Enhanced Oil Recovery

define the orientations of a pore for which the detachment condition is met to a measure of the domain occupied by all permissible values of the parameters is the sought probability. The variables Ci and Di (i = 1, 2, 3) and all those defined by them – с, b, F, B, G, H – depend on the orientation of the pore in space. An arbitrary orientation of a pore and, hence, of the coordinate system p1 , p2 , n p rigidly attached to it, is defined through the unit vectors of the fixed local coordinate system ex , e y , ez with the help of three independent parameters that can be, for example, the Euler angles , arbitrary position of a pore can be defined as follows: p1

cos cos

ex

sin sin cos

cos sin

and . An

sin cos cos

ey

sin sin ez , sin cos

p2

cos sin cos

cos sin ez , n p sin sin e x sin cos e y

ex

sin sin

cos cos cos

cos e z .

ey

(2.14) The variables Ci and Di (i = 1, 2, 3) that appear in (2.5) are defined for an arbitrarily oriented pore through the Euler angles that define its orientation (2.14) as follows: P x

С1 P y

f

P z

uy p

g

ux

f

f

cos cos

cos sin

p

P x

P y

C3

uy

f

P z C2

ux

f

f

P sin sin x

g

sin cos cos

uz sin sin ,

sin cos

sin sin

f

sin sin cos

f

cos sin cos

cos cos cos uz cos sin ,

P sin cos y

P z

p

f

g cos ,

Remove Micro-Particles by Harmonic External Actions D1 sin

t

p x

p

t wz t

t

p x

ux w x

f

f

uy w y

f

f

uz w z

f

f

ux w x

f

f

uy w y

f

wx t

f

uz w z

f

wy p

p z

p

t wz t

t

wx t

ux w x

f

wy p

p

ux t

wy

uy

t

t

wz t

uz t

cos sin ,

wx t

ux t

sin cos

cos sin cos

wy

uy

t

t

sin sin

cos cos cos

wz t

uz t

sin cos

cos sin cos

sin sin

cos cos cos

cos sin ,

3

p

p y

wx t

2

p

p y

p z

f

wy

D2 sin

p x

wx t

p

p z

D3 sin

1

p

p y

41

f

t wz t

f

uy w y uz w z

wx t

f

sin sin

wy

uy

t

t

f

f

ux t

wz t

uz t

(2.15) sin cos cos .

We note that gradients of matrix pressure, velocity and acceleration, as well as gradients of fluid velocity and acceleration, are functions of spatial coordinates and time. Their values that appear in the right parts of (2.15) are found for a specific point in space in which the pore in question is located. Inserting (2.15) in the detachment conditions (2.5) and (2.6) for each set of the Euler angles that define the orientation of a pore in space, we can determine whether any of the nine conditions necessary for the detachment conditions formulated in the previous sub-section are met, and find the values of the wave excitation amplitude for which the condition is satisfied. We assume that any orientation of a pore defined by three values of the Euler angles are equally probable. Any arbitrary pore position is defined by values of the Euler angles from the following inter0,2 , 0,2 , 0, . Therefore, the volume of a dovals: main of valid values of the Euler angles in the 3D space {φ, ψ, θ} is 4π3.

42

Enhanced Oil Recovery

As the actions discussed here are periodic in time, it is sufficient to consider one period, i.e., the domain of valid values of the dimensionless parameter t equals 2 . The domain of values of the Euler angles and of the dimensionless parameter of time t in the 4D space {φ, ψ, θ, ωt} for which the detachment condition is met at a given value of the amplitude ε is located within a restricted domain that contains all valid values of the Euler angles and any values of the dimensionless parameter of time with a period 2 that define any arbitrary pore orientation at any time in the oscillation period. This external domain measures 8π4. The volume of the domain for which the detachment condition is satisfied is a certain portion of 8π4. At points of time for which the detachment condition is met, pores whose orientation is defined by the Euler angles from the domain of values will be freed from particles stuck to their walls in the process of wave stimulation. Let the volume of this domain be V(ε) < 8π4. In accordance to the aforementioned assumptions, we believe that the probability of satisfaction of the detachment conditions in a particular pore, or the probability of successful stimulation, is the ratio V(ε)/8π3. For practical use of the proposed method, the following steps can be taken. First, geophysical logging methods shall be used to identify the region of a reservoir where the flow properties have been impaired due to the clogging of the reservoir pores with solid particles. It is most likely that such regions will be located in near-wellbore formation zones around wells whose flow rates have dropped significantly. For definiteness, let us call such a reservoir region “G”. Next, the probability of successful stimulation shall be determined for each point of the region G. To this end, theoretical assessment calculations shall be performed to determine the steady-state hydraulic characteristics of the forces acting on a particle located in a randomly oriented pore at a selected point. The orientation of the pore in space is defined by the values of the Euler angles. All intervals of valid Euler angle values shall be divided into n parts. The variation interval of the dimensionless parameter of time shall also be divided into n parts. On the intersection of the division planes we obtain (n + 1)4 points. For each of the obtained points, we determine the coefficients Ci (i = 1, 2, 3) and the amplitudes of wave action Di (i = 1, 2, 3). A procedure for such calculations for certain types of reservoirs and their wave stimulations is detailed below. All calculations for determining Ci (i = 1, 2, 3) and Di (i = 1, 2, 3) are performed on the basis of the linearized equations. All these amplitudes are proportional to the amplitude of pressure oscillations ε created by the generator. Then, using the obtained expressions for Ci (i = 1, 2, 3) и Di

Remove Micro-Particles by Harmonic External Actions

43

(i = 1, 2, 3), values of с, G, b, B, F, and H can be determined for a given orientation of the pore. The obtained values allow us to verify satisfaction of the conditions for particle detachment from the pore walls for a given orientation of the pore. If the detachment condition is satisfied for n1 orientations that the pore had at fixed points of time from the (n+1)4 tested ones, then the probability of successful stimulation of a particular point in the reservoir region G equals S = n1/(n+1)3. Note that the value of n1 depends on the method of excitation of harmonic oscillations of the medium and on the amplitude ε. This way, the probability of successful stimulation S can be calculated for each point in the reservoir region G and for each method of stimulation with a high level of credibility. The S-value describes the probability of satisfaction of the condition for the detachment of particles from the walls of variously oriented pores. Thus, it is a characteristic of successful wave cleaning of productive reservoirs from solid particles under the action of continuous harmonic waves.

2.4 Summary 1. The conditions for the detachment of solid particles that cling to pore walls by the force of adhesion have been defined for an arbitrary monoharmonic wave field in a monoharmonic approximation linear with respect to the amplitude of the wave field. 2. Assuming equal probability of any orientation of the pores in a porous medium, a probabilistic criterion of successful satisfaction of the condition for the detachment of a contaminating solid particle from reservoir pore walls under any monoharmonic wave stimulation has been formulated. 3. A method for calculating the probability of satisfaction of the condition for the detachment of a contaminating solid particle from the walls of reservoir pores has been proposed for random monoharmonic stimulation of reservoirs of a random geometry. 4. The obtained results provide a mathematical basis for the creation of a software product for assessing the probability of success of monoharmonic wave stimulation of a selected reservoir region by a particular method of wave excitation and for selecting wavefield parameters (such as wave amplitude and arrangement of generators) to ensure the highest probability of successful satisfaction of the condition for the detachment of solid particles from reservoir pore walls.

Enhanced Oil Recovery: Resonance Macro- and Micro-Mechanics of Petroleum Reservoirs. O. R. Ganiev, R. F. Ganiev and L. E. Ukrainsky. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

3 Remove Micro-Particles by Impact Waves The Wave Mechanisms of Motion of Inclusions in Micro-Pores of a Porous Medium and the Criterion for Successful Wave Stimulation to Remove Contaminating Micro-Particles Stuck to Pore Walls by Impact Waves

In addition to harmonic wave action, impact waves are considered here. Impact waves or impact momenta, excited by some means or other, pass through each pore of the porous matrix and act on particles stuck to pore walls. If the wave forces (created by the passage of the front of an impact wave through a pore) acting on contaminating particles are sufficient to overcome the force of adhesion of the pore-contaminating solid particles to the pore walls, then these particles detach from the pore walls, the flow capability of the matrix is improved, and new volumes of oil are mobilized with fluid flows. The latter contributes to enhanced oil recovery. To assess the forces that act on pore-contaminating particles in an impact wave, we need to know the parameters of the flow that are realized behind the impact wave front. The front of an impact wave is a zone where the parameters of the medium change significantly at distances so small that they can be considered zero as far as a continuous medium is concerned. In this zone, the continuous heterogeneous medium representations and the differential equations resulting from them make no sense. It is therefore necessary to introduce a surface of discontinuity of the flow parameters, where the continuous motion equations hold on both sides of the surface. 45

46

Enhanced Oil Recovery

3.1 Determining Flow Parameters behind an Impact Wave We consider a plane impact wave propagating in a fluid-saturated porous media (see Fig. 2.1). The normal to the wave propagation front in a fixed Cartesian coordinate system {ex, ey, ez} is written as follows: wn = wnxex + wnyey + wnzez.

(3.1)

The wave front is a surface of discontinuity propagating at a velocity Dwn in a resting (velocities before the passage of the front are zero) elastic porous medium saturated with a viscous compressible fluid. From the integral equations, we can obtain basic conditions on the discontinuity surface to relate the parameters of the medium before the wave front (as indicated by the subscript “0”) to the ones behind it [21] (if we use v1 and v2 to denote the fluid velocity and the matrix velocity, respectively, then the medium before the wave front is at rest and in equilibrium, v10 = v20= 0): – from the mass-conservation equations we obtain the following relationships: D) m1 , 1 (w n v 1 10 D D) m2 , 2 (w n v 2 20 D – from the general medium momentum-conservation equation, it follows that: wn wn w n wn m1 v 1 m2 v 2 w n . 0 Here the subscripts 1 and 2 denote the fluid and the matrix, respectively; ρi is the reduced density, vi are the velocities of the fluid and the matrix, p1 is the pressure in the fluid, 1 is the volumetric fraction of pores,

1 1 , 2 2wnwn , wnwn are components of the stress tensor in the porous matrix and the full stress tensor in the medium, respectively. Here and in what follows it is assumed that when a wave front passes, the only thing that changes are the projections of the velocities along the normal to the wave front wn, while other velocity components remain unchanged. From the first three equations, we can obtain an expression for the velocities of the phases after the passage of the front and for the velocity of the impact wave: 2

1

v1 1

2

D, v 2 2

D, (

0

),

Remove Micro-Particles by Impact Waves wn wn

wn wn 0

D2

47

,

1

2

10

20 1

2

where vi (i = 1, 2) are the projections of the velocities along the normal wn. To determine the flow parameters behind the front, we need to add the equations of state of the medium and the closure relations [21]: kl

kl f kl f

kl 2

kl f 1

1

10

2

1

p1

p2

2

mm 2

f

p1

kl

p10 ,

pf

p1

kl

, kl

p

1 1

2

f

2

,

pf

mm 20

1

2 mm 2

p1

,

kl 2

f

2

20

1 2

p1

kl

1

p2

mm f

3

, p2

p20 ,

,

,

20

where

i

,

pi

are the true density and the compressibility factor of the

i- phase, p2 is the average pressure in the porous matrix, klf is the tensor of fictitious stresses that is physically interpreted as part of the tensor of mean stresses in the matrix caused by a fluid-independent mechanism of force transfer – through contacts between matrix grains; kl mm are the strain tensor of the second phase and its spherical part 2 , 2 th

(

mm 2

xx 2

yy 2

zz 2

).

It follows from the one-dimensionality of motion that mm 2

mm 20

wn wn 2

wn wn 20

.

p10 and assuming Considering that at the initial point of time 0wnwn that in the planes parallel to the wave front, in which the orthogonal w2 w2 0 , we Cartesian coordinates {τw1; τw2} are introduced, 2 w 1 w 1 2 mm 2

wnwn 2

wnwn 20

f

p10 . It follows from the condition 2 f f that the normal stresses in the matrix and in the fluid behind the impact p1 that: wave are equal 2wnwn 2 f f wn wn 2 . p1 p10

obtain

,

f

20

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Enhanced Oil Recovery

After appropriate transformations we can obtain a closed system of two algebraic equations that allow us to use the values of the parameters before the wave to obtain the values of the flow parameters behind the wave as a function of the impact wave velocity D (hugoniots):

p1

D2

p10

;

1

2

10

20 1

p2 where

1

p1 (1

1, K

2

1

v1

2

) K

f

2

D, v 2

D,

i

i

2

f

p10

f

2

1 2

wnwn f

, i 1,2 ,

,

20

f

2

i0

2

p1

2

,

,

f

1

1

wn wn 20

K

20

2 3

f

(3.2)

2

10

1

p1

p1

p10 ,

10

20

1

p2

p2

p20 ,

20

1 p1

2

f

2

1

f

10

10

20

(

20

2

f

, , f

wnwn 20

)

,

20 wn wn

2

f

1 p1

2

f

2

1

f

2

(

f

2

f

)

wn wn 20

,

20 wn wn 0

p10 .

From the first equation of the system we can obtain an expression for the minimum possible velocity of propagation of a compressional impact wave in a medium depending on the parameters of the medium and the initial state (the explicit expression is not provided because of its awkwardness): Dmin lim D. p1 p10 ( p1 p10 )

Below are the results of calculations for a medium with the following parameters: the fluid (water) – ρ100 = 1000 kg/m3, β1 = 4.44 10 10 m∙sec2/kg, p10=100 bar; the matrix – 200 = 2650 kg/m3, β2 = 2.94 10 11 m∙sec2/kg, 1.25 1010 Pa, f 1.25 1010 Pa, f ( f 2 / 3 f ) 2 . f

Remove Micro-Particles by Impact Waves

Fig. 3.1

Fig. 3.2

Fig. 3.3

49

50

Enhanced Oil Recovery

Fig. 3.4

Fig. 3.5

Figs. 3.1–3.4 show the hugoniots: dependencies of the dimensionless pressure and fluid velocity, the matrix velocity and the difference between phase velocities behind the impact wave from the impact wave velocity for different values of the initial volumetric fraction of pores in the porous medium 10 0.1, 0.2, 0.3, all calculated according to (3.2). It can be seen that the higher the porosity of the rock, the lower the minimum velocity of the wave. This is caused by an increase in the compressibility of the medium. The dependence between the minimum wave velocity and the porosity is shown in Fig. 3.5. Impact waves in this medium can propagate with velocities whose values lie above this curve (in the shaded area).

Remove Micro-Particles by Impact Waves

51

3.2 Assessing the Forces That Act on a Particle as the Front of an Impact Wave Is Passing As the front of an impact wave is passing, a particle stuck to the wall of a pore is acted upon by wave forces of the same nature as that in a continuous wave field (see Chapter 2). However, here we need to take into account some of the forces that are neglected in Chapter 2. First of all, these are the forces associated with the dynamic head and the difference between the matrix and fluid velocities. So, in order to assess the action of an impact wave on a particle in a pore we take into account the following forces here: 1) static forces: – the reservoir pressure gradient –grad P; – the force of gravity and the force of buoyancy that act in the direction of the unit vector ez; 2) forces associated with impact wave front passage: – the fluid pressure gradient in the impact wave –grad p1; – the dynamic head

1 wn 2

1

v1 v 2 wn

2

that acts in the direction of

wave propagation; – the force of inertia

v2 w n (ρ is the density of the material of the t

particle); – a Stokes type force proportional to the difference between velocities of the fluid and the matrix; as the wave front passes through the pore to which the particle is stuck, this force increases from zero at the time of arrival of the fore edge of the impact wave front t0 to v1 v 2 w n , where η is the viscosity of the fluid, α is a coefficient that depends on the shape and dimensions of the pore and the contaminating particle at the time of arrival of the back edge of the impact wave front t1 (note that this force in this case results from the different velocities of the pore and the fluid during the passage of the impact wave, and the initially zero difference v1 v 2 can reach significant values, as shown in Fig. 3.4). So, assuming that the increase in matrix and fluid velocities during impact front passage is linear in time, we obtain an expression for the Stokes force that acts on the particle as the front of the impact wave passes t t0 through the pore in question: Fs v1 v 2 wn ; t1 t 0

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Enhanced Oil Recovery

– an added mass type force proportional to the difference between acv2 v1 celerations of the fluid and the particle: w n , where β is f t t the added mass coefficient, f is the density of the fluid (just like in the case harmonic actions, the convective terms are neglected). The force of adhesion acting on a particle in a pore is still defined by the relation (2.1). Note that all the forces associated with impact wave front passage act in the direction of the normal to the front wn. To be able to calculate these forces, we need to know the actual thickness of the impact wave (though small, the front of a wave in fact does have a non-zero thickness). This thickness can be determined, for example, by experiments on core samples. It is desirable to ensure that experimental parameters (primarily, the maximum pressure and the pressure rise time) are as close as possible to the values that are expected to be achieved in field conditions using some source of pressure rise or other. Let the thickness of the wave front be l, then, knowing the velocity of the wave and the parameters of the flow behind the wave (see Section 3.1) and considering the state of rest ahead of the wave, forces acting on the particle during wave front passage can be estimated at points of time t within the interval {t0 < t < t1 = t0 + l/D}: -grad p1

1 wn 2

1

1

v2 t

Fs

v1 v 2

p1

p10 l

w n,

2 1

2

v1 v 2 w n ,

wn

2l

v2 wn t v1 wn t

v2 Dw n , l v 2 v1 Dw n , 1 l

v1 v 2

t t0 D

wn. (3.3) l Note that because of the relationships (3.2), all the expressions that appear on the right-hand side of (3.3) can be expressed in terms of a single parameter of the impact wave, namely, the rise of pressure in the fluid that occurs as the front of the impact wave is passing: p1 – p10. After the impact wave passes through the pore in question, at points of time t > t1 the forces associated with the reservoir pressure gradient, the

Remove Micro-Particles by Impact Waves

53

force of gravity, the force of buoyancy, and the Stokes force remain nonzero. All other forces are zeroed. Processes associated with the arrival of reflected waves are not considered here.

3.3 Conditions for the Detachment of a Solid Particle from the Wall of a Pore under the Action of a Passing Impact Wave Considering the definition of the force of adhesion used in this study (2.1) and the forces that act on a particle in a pore during the passage of an impact wave as defined in the preceding paragraph (3.3), let us write down the condition for the detachment of a particle from the wall of a pore arbitrarily oriented in space. Similar to the consideration of harmonic actions on particles in pores in Chapter 2, we consider an arbitrarily oriented pore whose orientation is defined by the normal np to the surface of the pore at the point of location of the particle, directed inside the pore, and by two orthogonal unit vectors located on a tangent plane to the surface of the pore at the particle location point τp1 and τp2. For points of time t satisfying the condition for the passage of an impact wave through the pore {t0 < t < t1 = t0 + l/D}, the normal component of the forces acting on the particle during the passage of the impact wave can be written as follows:

fn V

P e gr np

1

ge zn p Gw nn p ,

where V is the volume of the particle, e gr is a unit vector whose direction coincides with the vector of reservoir pressure gradient, G

2 p1

p10

1

v1 v 2

2

2D

1

v2

2l

v

1 1

v1 v 2 t t0

.

(3.4) At the points of time t > t1 after the passage of the impact wave front in the problem formulation provided here, the particles are acted upon by all the steady-state forces and by a Stokes force defined by the constant difference between matrix and fluid velocities after the passage of the front. For these points of time the force action on the particle is described not by the function defined by (3.4), but by the following function: G

v1 v 2 .

(3.5)

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Enhanced Oil Recovery

In many cases, a simplified expression for G can be used. Here we mean the cases where one or several terms in the expression for G is (are) significantly bigger than the others. For example, practical experience shows that the forces caused by the effect of added masses and the Stokes force (the latter holds as long as a high-viscosity fluid is not considered) are much smaller that the forces caused by the pressure gradient, the force of inertia and the force of dynamic head that arise as an impact wave passes through a pore. The expression for G (3.4) can be simplified as follows: G

2 p1

p10

v1 v 2

1

2

2D v 2

2l

.

(3.6)

l G versus the wave p10 intensity ( p1 p10 ) / p10 for different porosity values (the numbers are on the figure) for the cases when the expression (3.6) for G holds. The density of the particle is assumed to be equal to the density of the material of matrix grains.

Fig. 3.6 shows the dimensionless variable F

Fig. 3.6

The detachment condition for the time interval of the passage of the impact wave front through the pore can be written as follows: If fn > 0, then

P e gr n p P e gr i 1,2

1

pi

ge zn p Gw nn p 1

g ez

pi

2

Gw n

2 pi

M2 , R2

(3.7)

Remove Micro-Particles by Impact Waves

55

where R, like in Chapter 2, is the characteristic linear dimension of the particle, G is defined by one of the expressions (3.4) or (3.6) depending on the ratio between the values of the summands of which G consists of. If fn < 0, then

P e gr

pi

g ez

1

Gw n

pi

2 pi

i 1,2

(3.8)

2

M R

P e gr n p

k

1

ge zn p Gw nn p

.

If the simplified equation for G (3.6) cannot be used, other data shown in Fig. 3.6 can be used to estimate G. Then, the conditions (3.7) and (3.8) can be written as follows: l p10

l p10

P e gr n p 1

l

g ez n p

1

ge zn p F w nn p 2

P e gr n p

Fw nnp

p10 g ez

1

l

pi

P e gr

2 pi

p10

i 1,2

0,

Fw n

pi

(3.9)

l2 M 2 R 2 p102

and l p10

l p10

P e gr n p

l

1

pi

ge zn p F w nn p

P e gr

k l

P e gr n p

1

p10

0,

2 pi

p10

i 1,2

lM Rp10

g ez

1

Fw n g e zn p

pi

(3.10)

2

Fw nn p

.

An analysis of the relationships (3.9) and (3.10) shows that decreasing М l and increasing the dimensional variable F shown the ratios and R p10 on Fig.3.6 helps satisfying the condition for the detachment of particles from pore walls. Therefore, in order to clean the pore space with the help

p1 p10 should be increased p10 and the thickness of the front should be decreased. To verify if the condiof impact waves the wave intensity A

56

Enhanced Oil Recovery

tions (3.9) and (3.10) can be met using a certain source of impact waves, preliminary tests should be conducted on core samples characteristics of the site to be stimulated. The tests should be performed to experimentally determine the maximum possible intensity of the excited wave A and the minimum possible length of the front l, followed by the characteristic contamination radius R and the coefficient of adhesion M. Having found the value of F is found versus the wave intensity A with the aid of Fig. 3.6, and using available information about reservoir pressure gradients in the stimulated formation and densities of the pore-contaminating particles and the fluid, the cleaning criterion (3.9) and (3.10) can be verified. If the criterion is met, it can be concluded that the proposed impact wave generator is suitable for the cleaning of the reservoir section in question with a high probability of success. Success probability can be assessed theoretically. An assessment method is outlined in the next section. The detachment conditions (3.7) in the general case can be written as follows: P e gr n p

1

ge zn p Gw nn p

0,

2

M , R2

G 2 2Gb1 b2

where b1

P

e gr n p

w nn p

e gr

(3.11)

wn

pi

pi

i 1,2

1

g

ez n p

w nn p

ez

pi

wn

,

pi

i 1,2

b2 2 P

1

P g

2

2 1

g2

e gr n p e zn p

e gr

pi

ez

pi

.

i 1,2

The second of the conditions (3.11) allows us to determine the range of values of G, defined by the intensity of the impact wave, that satisfy the detachment conditions for each pore orientation that satisfies the first of the conditions (3.11). From physical considerations it is clear that in the absence of an impact wave a particle retained on the pore wall cannot detach from it only under the action of the reservoir pressure gradient, the force of gravity and the force of buoyancy. The thing is that these forces have been constantly acting over many centuries or even millennia or more, since the productive reservoir came into existence, and if their action were suffi-

Remove Micro-Particles by Impact Waves

57

cient for the detachment, the particle would have detached and would have not been able to cling to the pore wall. Therefore, when G = 0 the M2 second of the conditions (3.11) is not satisfied, i.e., b2 . With this in R2 mind, and assuming that the total force action on the particle during the passage of an impact wave through it is towards the direction of propagation of the impact wave, i.e., that G is positive, it is easy to show that the second detachment condition (3.11) is satisfied if M2 . (3.12) R2 Note that if the steady-state components of the forces acting on a particle stuck to the wall of a pore, i.e. the force of gravity, the force of buoyancy and the reservoir pressure gradient, are neglected, then the conditions (3.11) are simplified and resolve themselves to: M2 Gw n n p 0; G 2 . (3.13) R2 The detachment conditions (3.8) in the general case are written as follows: P n gr n p Gw nn p 0, 1 ge zn p b12 b2

G b1

B0G 2 2 B1G B2

where B0

1

M2 , R2

(3.14)

2

1 k 2 w n np ;

B1

P

wn

e gr

pi

pi

k e gr n p w nn p

i 1,2

1

ez

g

pi

wn

pi

k ez n p w nn p

i 1,2

k

M w nn p . R

Just like for the detachment condition (3.6), only the case where the steady-state forces acting in the reservoir are not sufficient to detach the particle from the pore wall is of any interest to consider. In this case, for the detachment condition (3.7) to be satisfied, according to the second condition (3.14) the force action on the particle in the pore during the passage of an impact wave through the pore must satisfy the following condition:

B1 G

B12 B0 B2 B0

M2 R2

.

(3.15)

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Enhanced Oil Recovery

If the steady-state forces acting on the particle in question that is stuck in the pore are neglected, the conditions (3.14) take the form: 2 M M2 Gw n n p 0; G 2 1 1 k 2 w nn p 2k Gw nn p . (3.16) R R2

3.4 The Criterion for Successful Wave Stimulation by Impact Waves. Criterion Determination Procedure Similarly to the case of harmonic wave impacts, whether the particle detachment condition is satisfied when an impact wave passes through a pore depends on the orientation of the pore in space. Just like in Chapter 2, we introduce the Euler angles. The orientation of the pore is defined by the relationships (2.27). We write the directions of the reservoir pressure gradient e gr in the fixed Cartesian coordinate system ex , e y , ez as

e gr

grx ex

gry e y

grz ez .

(3.17)

The problem is formulated as follows. Determine the probability of satisfaction of the condition for the detachment of a particle from the wall of a pore for a given impact wave generating apparatus with a known

p1 p10 and a known wave front p10 thickness l, for a given reservoir pressure gradient, parameters of the rock, the fluid and the contaminating particles of known dimensions, and for known values of the coefficients of adhesion M and k. We assume here that we may restrict ourselves to the approximation (3.6) in order to estimate the forces acting on a particle in a pore during the passage of an impact wave. This restriction is not essential, as the forces that were neglected in (3.6) can easily be added if we set values of fluid viscosity and the added mass coefficient that is defined by the effective shape of the particles. The methodology for calculating the probability of satisfaction of the detachment condition is as follows. Use the given values of wave intensity and medium porosity to find the dimensionless parameter F in accordance to the data shown in Fig. 3.6. Then, using known values of l and p10, find the value of G. intensity of the generated wave А =

Remove Micro-Particles by Impact Waves

59

Choose pore orientation by choosing certain values of the Euler angles. Insert the values of the reservoir pressure gradient at the point of location of the pore in question, the difference between the densities of the particle and the fluid, and (2.14) for the chosen values of the Euler angles into the first of the conditions (3.11). If the condition is satisfied, insert (2.14), (3.1) and (3.17) into the expression for the coefficients b1 and b2 that appear in the second condition (3.11), and use the value of G found above to check whether the condition (3.12) is met. If it is met, then, consequently, the condition for particle detachment from the pore wall is satisfied. If not, then the detachment condition is not satisfied. The particle does not detach from the pore wall for the given pore orientation, for the given impact wave generator, for the given medium and other conditions. If the first condition (3.11) is not satisfied, i.e., the projection of the force acting on the particle in the pore along the normal to the pore at the point of location of the particle is not positive. It presses the particle to the pore. Therefore, the first condition (3.14) is satisfied automatically, and satisfaction of the condition (3.15) must be checked to verify the detachment criterion. Insert (2.14), (3.1) and (3.17) into the expression for the coefficients B0, B1 and B2 that appear in the second condition (3.14), and use the value of G found above to check whether the condition (3.15) is met. If it is met, then it is indicative of satisfaction of the detachment condition. If it is not met, then, consequently, the particle does not detach from the wall for the pore orientation in question. Now, assuming that the pores are randomly oriented in space so that any position of the normal to the surface of the pore at the point of contact with the particle is equiprobable, we can determine the probability of successful stimulation. Assuming that all pore orientations are defined by a single unique set of values of the Euler angles, a rectangular grid can be applied to the domain of definition of the Euler angles as it was done for the case of harmonic actions in Chapter 2, and each point can be verified for satisfaction of the detachment conditions. Then, the probability of successful stimulation S can be calculated as a ratio of the volume of the domain occupied by the points for which the detachment condition is satisfied to the volume of the entire domain of valid values of the angles. For each value of the parameter G, the obtained value of S will be different. Analyzing the dependence S(G), we can evaluate the opportunities offered by impact wave stimulation for pore space cleaning. To illustrate, we consider the case of neglecting the action of steadystate forces in comparison to the forces caused by the action of the impact wave described by the conditions for the detachment of particles

60

Enhanced Oil Recovery

M , RG cos w n np , where the symbol β denotes the angle between the unit vectors w n and w n . Any possible mutual orientation of the vectors w n and w n can be determined if the angle β takes on values from the interval {0; π}. The conditions (3.13) and (3.16) take the following form:

from the pore walls (3.13) and (3.16). We write down

0

2

; cos2

2

2

;

2

2k 1 k

1,

2

cos

1

1 k2

0.

A simple analysis of the latter relationships shows that the detachment condition is met only at a sufficiently small value of the parameter γ, specifically the relation 2 1 should hold, and moreover, even when the latter condition is satisfied, particles will detach only from those pores whose inward normal to the surface makes with the direction of impact wave propagation an angle β that satisfies the following conditions: 0

arccos

k

1 k2 1 k2

2

.

So, if we assume that all the pore orientations defined by the values of the angle β are equiprobable, then the probability of successful stimulation S for a given value of the parameter 2 1 is as follows: S=

1

arccos

k

1 k2 1 k2

2

.

(3.18)

It is not difficult to show that as the pressure at the impact wave front rises tending to infinity, the parameter decreases tending to zero, while the argument of the arccos function that appears in (3.18) tends to “–1”, and therefore, the probability of success S tends to unity. On the other hand, if γ tends to 1 while remaining less than 1, then the length of the range of angles β for which the detachment condition is

. Therefore, the probability of successful stimulation 2 tends to ½. This minimum success probability is achieved when the function G tends to M/R while remaining greater than this value.

satisfied tends to

Remove Micro-Particles by Impact Waves

61

As soon as G becomes less than this value, the detachment condition is not satisfied for any of the orientations of the pore, and the probability of success drops abruptly to zero. Therefore, for the probability of successful stimulation to be non-zero, an impact wave generator should be provided so that the parameter G remains greater than M/R. In this case, the probability of successful stimulation exceeds ½. As G increases further, the probability of success grows tending to 1 with a G tending to infinity.

3.5 Summary 1. The conditions for the detachment of a solid particle, retained by the force of adhesion, from the pore wall by an impact wave passing through the pore have been defined. 2. A probabilistic criterion of successful satisfaction of the detachment conditions for a given direction of propagation of the front of an impact wave and its intensity has been established, assuming equiprobable orientation of pores in the porous space. 3. A method for calculating the criterion of successful satisfaction of the condition for the detachment of contaminating solid particles from the walls of reservoir pores has been proposed for stimulation of any reservoir region by an arbitrary impact wave. 4. The obtained results provide a mathematical basis for the creation of a software product for assessing the probability of successful satisfaction of the condition for the detachment of contaminating solid particles from reservoir pore walls in a given reservoir region and for selecting impact wave parameters (such as wave intensity and arrangement of generators) to ensure the highest probability of success.

Enhanced Oil Recovery: Resonance Macro- and Micro-Mechanics of Petroleum Reservoirs. O. R. Ganiev, R. F. Ganiev and L. E. Ukrainsky. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

4 The Wave Mechanisms of Motion of Capillary-Trapped Oil The Wave Mechanisms of Motion of Capillary-Trapped Oil. Wave Action on Fluid Droplets Retained on Pore Walls by Capillary Forces

This chapter focuses on the behavior of small oil droplets stuck to the walls of pores as shown in Fig. 2.1. The remaining pore volume of the porous medium is considered to be occupied by another fluid such as water in the case of flooded reservoirs. In contrast to the problems discussed above where solid particles are retained on pore walls by the force of adhesion, in this case droplets cling to pore walls due to capillary forces. This formulation can be used to describe the motion of a capillary-trapped oil droplet in a waterflooded reservoir. We assume that a droplet in a pore is acted upon by the same forces that are discussed in Chapter 2 in the statement of the problem of the forces acting on a solid particle with one exception: the force of adhesion in this case is replaced by the force of surface tension according to the law of Laplace [22], which creates a differential pressure between the opposite sides of fluid (oil/water) interfaces thereby retaining droplets on pore walls. According to the well-known Laplace’s equation the additional 2 where downforce pressure p can be expressed as follows: p p Rc/o p is the pressure in the water surrounding the droplet, α is the coefficient of surface tension at the water-oil interface, Rc/o is the radius of curvature of the spherical surface of the oil-water interface. Surface tension forces act on the surface of the droplet. Assuming that the droplet is a spherical segment with a radius of the spherical surface Rc/o, we define a main vector F of the downforce pressure p acting on the spherical surface of the segment. The vector F is normal to the wall of the pore and directed from the fluid to the wall of the pore. The magnitude of this force equals 63

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Enhanced Oil Recovery

F Rc/o 1 cos 2 , where Θ is the contact angle between the surface of the pore and the tangent to the surface of the droplet at points of interface between the droplet and the pore. Deformation of the droplet in the course of its motion until detachment from the surface of the pore is neglected. We assume that the contact angle remains constant until the time of detachment and equals its static value defined by the properties of the materials of the pore and the droplet. Under these assumptions, the projection of the resultant vector of the surface tension forces pressing the droplet to the pore along the normal to the surface of the pore, per unit volume of the droplet, is expressed as follows:

Fn p V where Fcap

Fcap ,

(4.1)

3 sin2

, rd is the radius of a sphere 2 rd 2 3 2 1 cos 2 cos whose volume is equal to the volume of the droplet. Fig. 4.1 shows Fcap versus Θ for the case of an oil droplet in water, which corresponds to α = 0.03 N/m, with the droplet radius rd = 0.1 mm.

Fig. 4.1

4.1 The Conditions for the Detachment of a Droplet from the Wall of a Pore Let us write out the condition for the detachment of a droplet from the surface of a pore.

The Wave Mechanisms of Motion of Capillary-Trapped Oil 65

The normal component of the forces acting on the particle under the above-stated assumptions (2.4) can be written as follows:

fn

V

P np f

p

wx t

f

g ez n p

p np

ux npx t

f

p

wy

wx npx t

wy

uy

t

t

t

npy

wz npz t

npy

f

wz t

uz npz t

,

where V is the volume of the particle, g is the free fall acceleration and the projection of the gravity force acting on the particle on the unit vector ez . Now we can write the detachment condition as follows: P np f

p

wx t

f

g ez n p

ux npx t

p np

p

wy f

t

wx npx t uy t

npy

wy t

wz npz t

npy

f

wz t

uz npz t

Fcap .

(4.2) This way, we can use (4.1) and the curve shown in Fig. 4.1 to determine the value of the amplitude of the wave field that guarantees the detachment of the particle from the wall for any droplet/rock contact angle, which can be determined by core sample analysis. We note that in contrast to the case of a solid particle stuck to the wall of a pore, discussed in Chapters 2 and 3, the tangential forces acting on the droplet have no effect on satisfaction of the detachment condition. They only affect the motion of the droplet along the pore wall. These motions do not affect directly the detachment of the droplet from the pore wall. However, motions of droplets along smooth parts of the pore surface with a velocity that depends on the applied tangential forces and viscous friction forces can affect droplet dimensions. If the detachment condition is not satisfied, the droplets keep moving until a droplet is caught by any irregularity in the pore wall. Then the droplets merge, and their characteristic dimension rd increases. This is conducive to the reduction of the absolute value of the parameter Fcap (4.1) and, accordingly, facilitates satisfaction of the detachment condition for the oil that is capillary-trapped in the pores. An n-fold increase in the characteristic dimension of droplets causes an n2-fold reduction of the resultant of the static and wave forces that is necessary to detach the droplet from the wall. Therefore, for any value of the resultant force that has a nonzero projection on the normal to the wall of the pore and for every contact angle there exists a droplet radius rd such that droplets with a radius

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Enhanced Oil Recovery

bigger than rd detach from the pore wall. In addition, the lower the surface tension coefficient the lower the detachment force, i.e., the use of surfactants with low surface tension coefficients instead of water for reservoir flooding purposes facilitates the detachment of oil droplets from the surface of pores by wave action.

4.2 The Case of Harmonic Action on a Capillary-Trapped Droplet Just like in the case of solid particles stuck to pore walls, the left-hand side of the inequality (4.2) under the assumptions made in Section 2.2 can be expressed in the form of two summands in the case of action of a continuous, monoharmonic in time wave field with a frequency ω. The first one defines the action of the time-independent static forces associated with the action of the reservoir pressure gradient, the force of gravity and the force of buoyancy, and is a time-independent term. The second one describes approximately, in an approximation linear with respect to the amplitude of action, the action of the wave field, which is a harmonic term. We have: P (4.3) С, p f g ez n p np p np f

p

wx npx t

wy

uy

t

t

npy

wy t

npy

f

wz npz t wz t

uz npz t

f

wx t D sin

ux npx t t

(4.4)

.

The detachment condition (4.2) is now formulated as follows:

С

D sin

t

Fcap .

(4.5)

Assuming that the amplitude ε is positive, from (4.5) we obtain the following conclusions. D , then the detachment condition (4.5) is met at any If Fcap C point of time with any wave action amplitude or even without wave action. This condition holds for any pores with such an orientation in space that the static forces are enough to satisfy the condition for the detachment of droplets from pore walls. For pores having such a spatial orientation that the following condition is met Fcap C D , the detachment condition (4.5) is not satisfied

The Wave Mechanisms of Motion of Capillary-Trapped Oil 67

at any point in time. In such pores the static and capillary forces press the droplet to the wall of the pore, while wave actions are insufficient to detach the droplet from the wall. D , the detachment condition for pores with If D Fcap C such an orientation in space is satisfied in a time interval within the period of oscillation. It is not difficult to notice that if the capillary forces exceed the static forces, i.e. Fcap C 0 , then the higher the amplitude the greater the length of this interval and it tends to the oscillation halfperiod

in which sin

t

0. That is, increasing amplitude is

conducive to satisfaction of the detachment condition. If the static forces exceed the capillary forces, i.e. Fcap C 0 , and the condition for the detachment of a droplet from the pore wall is satisfied even in the absence of wave action at any given time, as the amplitude increases there appear such time intervals in which the detachment condition is not satisfied, then these intervals become bigger tending to the oscillation half-period, whereas the time intervals in which the droplet detachment condition is satisfied become smaller and tend to the oscillation half-period

on which sin

t

0.

Thus, increasing the amplitude of wave action facilitates the detachment of capillary-trapped droplets from the pore walls for such pores to whose walls the droplets are pressed by the static forces. By contrast, increasing the amplitude of wave action helps retain the droplets that detach from the pore walls due to the action of the static forces. This fact allows us to draw a practical conclusion about how a stimulation job should be set up to maximize the mobilization of capillary-trapped oil droplets with the flow of reservoir fluid. Prior to stimulation, all the droplets on pore walls for which the condition С Fcap is satisfied will be detached from the walls by the action of static forces and will be mobilized with the fluid motion. At the same time, any droplets on pore walls for which the opposite condition С Fcap is satisfied will remain pressed to the pore walls. After the start of wave stimulation, during certain time intervals the wave forces whose projections along the normal to the pore wall are positive exceed the resultant vector of the static forces pressing the droplet to the wall and detach the droplet from the wall. At the same time, any droplets that come into contact with the pore walls at the points where the static forces are conducive to their detachment can be pressed back to the pore walls

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Enhanced Oil Recovery

by the wave forces. If wave stimulation stops now, the detachment condition for the droplets that are pressed to the pore walls by the wave forces will be satisfied by the action of static forces, and the droplets will detach from the pore walls. Therefore, to maximize performance of wave stimulation, wave action needs to be alternated with time intervals during which there is no stimulation. All of the above concerns the detachment of a particle from the wall of a specific pore. The parameter C is defined by Eq. (4.3), and the parameter D is defined by Eq. (4.4). Both of these parameters depend on the spatial coordinates of the point of contact between the droplet and the pore wall and on the spatial orientation of the pore defined by the directional cosines npx , npy , npz . There are only two independent cosines among them. Using a parameterization of the directional cosines of the orientation of the normal to the pore surface at the droplet location point in terms of the Euler angles (2.14), we find:

npx

sin sin , npy = sin cos , npz =cos .

(4.6)

Inserting (4.6) in (4.3) and (4.4), we obtain: P sin sin x

C

D sin p y

p x wy

t

p

p z

P sin cos y

f

p

t f

wz t

p

P cos z f

uy f

t f

wx t sin cos

p

f

f

g cos , (4.7)

ux sin sin t (4.8)

uz cos . t

Eq. (4.8) indicates that any of the expressions in square brackets on the right-hand side of Eq. (4.8) can be linearly expressed in terms of the other w u and are linearly detwo expressions if the three vectors p, t t pendent. There can always be found Euler angle values such that the right-hand side of (4.8) is reduced to zero. In other words, in this case there can always be found a direction of the normal to the pore wall such that the wave force does not act on a droplet pressed to the pore wall in the direction of the normal. That is, the wave force cannot detach the

The Wave Mechanisms of Motion of Capillary-Trapped Oil 69

droplet from the pore wall with the specified direction of the normal at any wave action amplitude. Therefore, the probability of successful wave stimulation by such a wave field cannot be 100%. If two of the expressions in square brackets on the right-hand side of Eq. (4.8) are linearly expressed in terms of the third one, then there is a whole surface of directions of the normals to the pores for which the wave action on the droplet neither detaches nor presses it to the wall. The probability of successful wave stimulation by such a field becomes even lower. Hence follows a practical conclusion that in the case of wave stimulations designed to detach droplets of capillary-trapped oil from pore walls it is w desirable to excite such wave fields for which the vectors p, t u are linearly independent in reservoir sections to be stimulated. and t The analytical condition that guarantees it can be written as follows:

det

p x p y p z

wx t wy

ux t uy

t wz t

t uz t

0.

(4.9)

In practice, the probability of successful wave stimulation for the detachment of oil droplets from pore walls should be determined in the same way as it is described at the end of Chapter 2 for the case of cleaning the pore space from solid particles. First, geophysical logging methods should be used to identify the reservoir region where the presence of capillary-trapped residual oil is most likely. For definitiveness, let us denote it by N. Then geological data is used to determine, at least qualitatively, the reservoir pressure gradient P in the selected region as a function of spatial coordinates of the region N. Now we can use (4.7) to calculate the values of C for the points of the region N. The value of Fcap is determined by core analysis. Then the value of (Fcap – С) is calculated, also as a function of spatial coordinates of region N. Then, select a type of the wave field to be excited for wave stimulation of the region N, based on generator placement options and generator availability. After that, prepare theoretical estimates of the distribution of wave amplitudes in the chosen wave field in the region N. A procedure for such calculations for certain types of reservoirs is detailed below in the following chapters. As a result of the calculation, we

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Enhanced Oil Recovery

w u and as functions of t t spatial coordinates in the region N. The obtained distributions and Eq. (4.8) allow us to determine D as a function of spatial coordinates in the region N. The following calculations are performed for each point of the region N. In the 3D space {θ, ψ, ωt}, the coordinate definition domains are: 0,2 , 0, , t [0,2 ). All intervals of valid Euler angle values shall be divided into n parts. The variation interval of the dimensionless parameter of time shall also be divided into n parts. On the intersection of the division planes we obtain (n + 1)3 points. The calculated values of (Fcap – С) and D for each of the obtained points and the chosen value of the amplitude ε allow us to verify satisfaction of the droplet detachment conditions for a given orientation of the pore and for a given point of time. If the detachment condition is satisfied for n1 orientations that the pore has at fixed points of time from the (n+1)3 tested ones, then the probability of successful stimulation of a particular point in the reservoir region N equals S = n1/(n+1)3. Note that the value of n1 depends on the method of excitation of harmonic oscillations of the medium and on the amplitude ε. This way, the probability of successful stimulation S can be calculated for each point in the reservoir region N and for each method of stimulation. As a result, we obtain the dependence of the probability of successful stimulation on the amplitude S = S(ε) for the chosen stimulation method. Based on this relationship stimulation amplitude can be selected. obtain distributions of the vectors

p,

4.3 The Case of Impact Wave Action on a Capillary-Trapped Droplet According to what is said in Chapter 3, for points of time t satisfying the condition for the passage of an impact wave through the pore {t0 < t < t1 = t0 + l/D}, the normal component of the forces acting on the particle during the passage of the impact wave can be written as follows:

fn V

P e gr np

1

ge zn p Gw nn p ,

where V is the volume of the particle, e gr is a unit vector whose direction coincides with the vector of the reservoir pressure gradient, G is defined by one of Eq. (3.4) or Eq. (3.6) depending on the conditions specified in Chapter 3, and wn is the vector of the normal to the impact wave front.

The Wave Mechanisms of Motion of Capillary-Trapped Oil 71

The condition for the detachment of droplets from pore walls then becomes: P e gr n p Gw nn p Fcap , (4.10) 1 g e zn p where Fcap is as defined above in (4.1). The left-hand side of (4.10) is the sum of projections of the forces acting on the droplet along the normal to the pore wall at the point of location of the droplet. The first summand is controlled by the action of the reservoir pressure gradient, the second summand is controlled by the forces of gravity and buoyancy, and, finally, the third one describes the action of the impact wave. Note that if the normals to the impact wave front and to the pore wall are orthogonal (this is true if there is a vector, collinear to wn, that lies in a plane tangent to the pore wall and passing through the droplet location point), then the impact wave has not effect on whether the droplet is detached from or pressed to the pore wall. Its action comes down to a tangential displacement of droplets along the wall. Droplets can be detached from the walls of such pores either by the reservoir pressure gradient or by the forces of gravity and buoyancy. Therefore, for an impact wave to be able to detach droplets from the pore walls, its front (as defined by the normal to the front wn) shall be directed so that neither the reservoir pressure gradient (with the normal egr) nor the vertical ez are collinear to wn. The conditions necessary for an impact wave to contribute to the detachment of droplets from the pore walls are as follows: G cos 0, (4.11) where ϑ is the angle between the vectors wn and np. In the general case, the condition (4.10) can be written as follows:

P grx sin sin G w nx sin sin

gry sin cos w ny sin cos

+ grz cos + w nz cos

1

g cos

Fcap ,

where θ and ψ are the Euler angles. So, the condition for the detachment of a droplet from the wall of a pore can be written as follows: w ny sin cos + wnz cos 0, then the droplet deIf w nx sin sin taches provided that

G

Fcap

P grx sin sin w nx sin sin

gry sin cos

+ grz cos

w ny sin cos

+ w nz cos

1

g cos

,

(4.12)

Enhanced Oil Recovery

72

w ny sin cos If w nx sin sin taches provided that G

Fcap

P grx sin sin w nx sin sin

+ wnz cos

gry sin cos

0, then the droplet de+ grz cos

w ny sin cos

+ w nz cos

1

g cos

.

(4.13) l G versus the wave p10 intensity ( p1 p10 ) / p10 for different porosity values (the numbers are on the figure) for the cases when the expression (3.6) for G holds. The density of the particle is assumed to be equal to the density of the material of matrix grains. Using these data, we can determine the domains of values of the Euler angles for which the detachment condition (4.12) or (4.13) is satisfied, and then, the probability of successful stimulation depending on the intensity of the wave ( p1 p10 ) / p10 and its direction wn. Knowing wave intensity, we can determine the optimum direction of the wave at for given directions of the reservoir pressure gradient and the force of gravity.

Fig. 3.6 shows the dimensionless variable F

4.4 Summary 1. The conditions for the detachment of a droplet pressed to the wall of a pore by capillary forces under the action of arbitrary forces have been established. 2. The conditions for the detachment of a droplet from the wall of a pore under the action of monoharmonic forces have been established. An action algorithm for selecting parameters of stimulation by monoharmonic waves that ensure the highest probability of satisfaction of the conditions for the detachment of a droplet pressed to the wall of a pore by capillary forces has been formulated. 3. An algorithm for selecting impact wave direction and intensity that ensure the highest probability of detachment of capillary-trapped droplets has been established.

Enhanced Oil Recovery: Resonance Macro- and Micro-Mechanics of Petroleum Reservoirs. O. R. Ganiev, R. F. Ganiev and L. E. Ukrainsky. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

5 Action of Wave Forces on Fluid Droplets and Solid Particles in Pore Channels

5.1 The Mechanism of Trapping of Large Oil Droplets in a Waterflooded Reservoir. Propulsion of Droplets by One-Dimensional Nonlinear Wave Forces Along with the cases considered above in the previous chapters where droplets of capillary-trapped oil or pore-contaminating solids are acted upon by forces that periodically change in time, the so-called permanent “wave forces” may exist in wave fields. These forces are generated by the transformation of oscillations into steady-state forces that do not depend on time due to nonlinearities [8, 10, 11]. Examples of such forces are provided above in Section 1.2.3. The magnitude of these wave forces depends on the parameters of the wave field. These wave forces control the motions of elements suspended in the wave fields, such as droplets of foreign fluids, bubbles and solids. Several examples of such forces with an assessment of their actual values are provided here below. Such forces, in contrast to the sign-alternating oscillation forces discussed above, can move solid particles stuck in pore channels or capillary-trapped droplets whose dimensions are commensurate with the flow cross-sections of the pore channels. These cases have a high practical significance, for example, for the recovery of residual oil that remains in waterflooded reservoirs in the form of isolated droplets. Oil droplets whose dimensions are substantially smaller than the diameter of a pore channel typical for a particular reservoir can flow freely through the porous medium unless they are pressed to the wall by capillary forces. This case was considered in the previous chapter. The conditions for detachment of such droplets from the walls by harmonic actions and by impact actions have been 73

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Enhanced Oil Recovery

established. However, if the diameter of the droplet is greater than the diameter of the pore channel, the situation is different. A droplet that gets into a restriction in a channel of a smaller diameter stops moving. In Fig. 5.1 this situation corresponds to the motion of the oil droplet from left to right. To propel a droplet, the pressure gradient needs to be

2 1 1 P , i.e., Fmin , L r1 r2 np where np is the unit vector directed along the axis of the pore in the direction of reduction of the radius of the pore channel. According to [18], Fmin = 115 bar/m for r1 = 10 μm, r2 = 40 μm, L = 400 μm, with the value of the surface tension coefficient at the oil-water interface σ = 0.03 N/m. greater than the following variable: Fmin

Fig. 5.1

Let us remind that flow is induced by a one-dimensional compressional resonant wave in a porous saturated medium (see section 1.2.3.). To realize such flow in steady-state conditions, we need a gradient P = 19 bar/m, where nw is a unit vector along the normal to the nw wave front in the direction of the induced flow velocity; with an amplitude of pressure oscillations of 1 bar at one the ends (see Fig. 1.17 b), it is sufficient to propel through said restriction a droplet of a length L = 4000 μm. If the amplitude of pressure oscillations at one of the ends is 2.45 bar, the induced gradient will be sufficient to propel a droplet of a length L = 400 μm. Such a droplet can also be propelled using the capillary effect (see section 1.2.3) thanks to which a traveling shear translational wave with an amplitude of just 10–1 μm propagating in the wall of a capillary with an undisturbed radius of 10 μm can cause steady-state flow inside the capillary with a cross-sectional average velocity of 450 cm/sec. To create the same flow by applying steady-state pressures to

Action of Wave Forces on Fluid Droplets 75

P = 3,600 bar/m [10, nw 11] is required. This gradient is sufficient for the propulsion of not only a 400-μm long droplet, but much smaller (by an order of magnitude or more) droplets as wells. It is unrealistic to create such a gradient in a reservoir using the conventional methods [18]. We note, however, that to actually ensure that droplets are propelled through restrictions, the waves excited in the reservoir should be oriented so that their front is perpendicular to the axis of the pore. Considering the chaotic orientation of pores in the reservoir, it would be very difficult to achieve this with the help of the above-considered examples (a one-dimensional longitudinal wave and a shear traveling wave propagating in the cylindrical wall of the pore). Indeed, if we denote by φ the angle between the normal to the wave front nw directed towards the induced flow and the axis of the pore np directed towards the smaller cross-section of the pore, then the projection of the force that acts on the droplet along the pore axis is P np grad P nw n p grad P cos . The propulsion conditions nw Fmin . Assuming that any pore can be written as follows: grad P cos orientation is equiprobable, we can calculate the probability W of satisfaction of the propulsion condition. It is not difficult to establish that the probability W of satisfaction of the propulsion condition takes the followF 1 ing value: W arccos min . Hence it follows that when grad P grad P Fmin , W = 0; and when Fmin grad P , 0 < W < 1/2. That is,

the ends of the capillary, a pressure gradient

as grad P tends to infinity, the probability of droplet propulsion by onedimensional waves tends to 1/2. This is because the latter inequality is impossible for any values of the angle φ from the range of values 3 that corresponds to obtuse angles. The thing is that the 2 2 direction of the fluid flow induced by the wave in the saturated porous medium makes an obtuse angle with the direction of motion of the oil droplet in a tapering pore (Fig. 5.1), which prevents propulsion of oil droplets through the pores. Therefore, regardless of the value of grad P the propulsion condition is not satisfied for this interval of angle values. Since the length of the segment π is half the length of the domain of definition of the angle φ, the probability of satisfaction of the propulsion condition cannot exceed 1/2.

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The wave forces considered in this section act in a direction that is not related to the medium and are fully controlled by the wave field. In the following sections, we consider such wave action cases where the direction of action of the wave forces is controlled by the properties of the porous medium.

5.2 The Average Flow of Fluid Caused by Oscillations in a Saturated Porous Medium with a Stationary Matrix and Inhomogeneous Porosity The results provided below were published in [23].

5.2.1 The Statement of the Problem We consider the one-dimensional (along the x axis in Fig. 5.2) steadystate motion of an elastic porous medium saturated with a viscous incompressible fluid. We assume that the porous matrix of the medium is spatially heterogeneous, incompressible and stationary. Periodic external actions are applied to the fluid.

Fig. 5.2

The physical properties of the porous matrix – true density, bulk density and permeability 2 , i 0 , k0 – are assumed to be spatially heterogeneous. Then: v2 0, 2 const, 1 const, 1

10

(x ), 1

2

20

1

m 1

2

( x ), k

k0 ( x ),

dq 2 1 d 1 q 2 dt 1 dx

v x K

1 1

0 ( 1 1

k

q

v

1 1

q(t ) ),

p1 . x

(5.1)

Here, v i , i , i are the velocity, volume concentration and true density of the i-th phase (1 – fluid and 2 – matrix), p1 is the pressure in the fluid, k is the matrix permeability, 1 is the dynamic viscosity of the fluid, K , m are the friction and added mass coefficients.

Action of Wave Forces on Fluid Droplets 77

At x = L let a constant pressure p1 (L, t ) be maintained, and at x = 0 let a periodic pressure variation p1 (0) p1 (L) P0 sin(2 t ) be given. Then, integrating Eq. (5.1) with respect to x from 0 to L, we obtain a nonlinear ordinary differential equation for q:

dQ A2Q A3Q 2 A4 sin(2 ), (5.2) d where dimensionless variables are introduced in accordance with the q 1L x k following relationships ,K ,Q ; the following t, y 0 L k0 1 k0 P0 expressions are true for the coefficients Ai (i =1, 2, 3, 4): A1

1

A1

1

m

1 0 1 0 0 2 1 0

kP L

0 1 0 0 2 1

kP L

A2

dy ,

1

1

m 3 1

2

1

0 1

k0

0

1 dy , 1K

d 1 dy dy

(1) 1 (0) 1 2 1 (1) 1 (0)

m

1

1

A4

1

K

1

0

A3

2

0 1

k0

0 1

(1) 1 (0) 1 (1) 1 (0)

m

,

.

We use the following characteristic values of the parameters [21, 24]:

1 105 Hz, P0

105 Pa,

(0.2 3) 10 12 m2 , 10 0.01 0.3, kg kg 0 103 3 , 1 10 3 , 0.5, K 1 2. 1 m m×s m We base the length scale L on the fluid incompressibility condition (the compressibility of the matrix is lower than that of the fluid). For this, the wavelength in the medium must be greater than L. For the speed of sound in a saturated porous medium with a stationary incompressible matrix we have the following expression [25]: 2 K 1 C1 C , , k 10 2 i Re 1 k0

2

m

where C1 is the speed of sound in the pure fluid. The wavelength is found C 1 . Fig. 5.3 shows a plot of wavelength versus frequency as (k0 10 12 m2 , 2 0,7 , C1 1500 ms 1 , m 0.5, K 1).

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Enhanced Oil Recovery

Fig. 5.3. Wavelength

versus frequency

The wavelength for the frequency of 100 kHz (L = 2.8 mm) can be taken as the length scale. It can be shown that in the chosen parameter range A3 Ai (0 Ai , i 1,2, 4) . Hence, choosing A3 as a small parameter, we can solve Eq. (5.2) by expanding in the small parameter. Representing the solution Q0 A3Q1 and collecting terms of (5.2) in the form Q with like powers of A3 , for the first two terms of the expansion we obtain the following system of equations: dQ A1 0 A2Q0 A4 sin(2 ), d (5.3) dQ1 2 A1 A2Q1 Q0 . d Once this system is solved, the sum of the first two terms of the expansion gives the following expression for the steady-state solution of Eq. (5.2): A4

Q

2

4 A3

A12

A22

sin 2

A42 2 A2 4

2

A12

A22

A sin 4

2

,

where φ is a function of the amplitudes A1 and A2, A and θ are arbitrary constants that can be determined from the initial conditions. After time averaging we obtain:

Q

A3 A42 2 A2 4

2

A12

A22

I3 2 I 23

k02 P0 0 1

2 1

0 1

L2 K 3

,

(5.4)

Action of Wave Forces on Fluid Droplets 79 1

I2 0

1

1 dy , I 3 y K y

1

(L) 1 (0) 1 2 1 (L) 1 (0)

m

1

(L) 1 (0) 1 (L) 1 (0)

m

or, in a dimensional form:

q t

3

k0

I P 33 2I2 2 0

1

K

0 1 3

L

,

(5.5)

Here we have taken into account the fact that 4 2 A12 A22 over almost the entire parameter range considered (except for frequencies close to 100 kHz).

5.2.2 Calculation Results As can be seen from expression (6), under a periodic action the spatial inhomogeneity of the medium porosity 1 leads to the formation of a fluid flow that is unidirectional on average. The fluid flows in the direction of higher porosity (determined by the sign of the difference 1 (L ) 1 (0) in the expression for I 3 ), i.e., in the direction of increase in the fluid flow area. This means that wave action on, for example, an oil reservoir can cause fluid flow from low-porosity, low-permeability reservoir regions to high-porosity zones, in particular, from compartments into fractures in fractured / porous reservoirs, which makes it possible to enhance the production of hard-to-recover oil. If the porosity values on the boundaries are the same, an average flow does not develop (given the boundary conditions for the pressure considered). As it follows from (5.5), the realized flow is the more intense the greater the value of I 3 , the smaller I 2 and the distance L at which the given porosity difference is realized, and the greater the amplitude of the action P0 (a quadratic dependence). Let the medium permeability be constant K ( y ) 1 , and the porosity distribution be defined by the following step function: 1

( y)

1

(0), 0

y h;

1

( y)

1

(1), h

y 1.

Fig. 5.4 shows, for the following values of the parameters k0

10

12

m2 ,

1

(0) 0.3 , L

2.8 mm, P0

105 Pa,

m

0.5, K

1

the dependence of the absolute value of the daily flow rate q (m3/day) per unit surface area on 1 (L) for different values of h = 0.5, 0.8, 0.9

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(curves 1, 2 and 3, respectively). It can be seen that the curves are nonmonotonic, and the flow rate grows as h increases, i.e. with the length of the interval section occupied by a medium whose porosity equals the initial porosity 1 (0) 0.3, and, accordingly, as the length of the interval section occupied by a medium with lower porosity 1 (L) decreases.

Fig. 5.4

Fig. 5.5 shows the same dependences for h = 0.9 and for various action amplitude P0 1, 1.2, 1.5 105 Pa values (curves 1, 2 and 3, respectively). The higher curves correspond to higher amplitudes.

Fig. 5.5

Let us evaluate which constant pressure gradients are required to create the same flows. To do this, we consider the steady-state flow described by the equation: A2Q A3Q 2 A4 P ,

Action of Wave Forces on Fluid Droplets 81

Fig. 5.6

Fig. 5.7

where p P0 P is the dimensional differential pressure. We take the obtained averaged flows to calculate the necessary pressure gradients p / L . Figs. 5.6 and 5.7 show the pressure gradients that correspond to the flows shown in Figs. 5.4 and 5.5. In order to compare the flow rate due to oscillations versus the flow rate at a constant differential pressure, we define the following boundary p P0 sin(2 t ) , where the condition at the left end: p1 (0) p1 (L) sign of p determines the sign of the constant pressure gradient. Typically, differential reservoir pressures range from one to thirty atmospheres on a distance of several hundred meters. We take a differential pressure of 2 106 Pa over a distance of 200 m as the scale for the reservoir pressure. Then the characteristic value of p for the parameters considered equals p 28 Pa. Eq. (5.2) takes the form: dQ A1 A2Q A3Q 2 A4 P sin(2 ) , p P0 P . d

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Repeating the solution procedure described above, we obtain the following expression: A4 P A2

Q

A3 A42 A2

P2 A22

1 2

2 4

A12

2

or, in a dimensional form (taking into account that 4 q t

p k0 L

1 K I2

1

p

P02 I 3 2 I 23

2

A12 3

k0 1

(5.6)

A22

A22 ): 0 1 3

K

L

.

(5.7)

As can be seen from these equations, the first term in Eq. (5.6) and Eq. (5.7) corresponds to Darcy’s law, while the second term is caused by the spatial heterogeneity of porosity, the inertial effects, and the periodic action. The porosity and pressure gradients may “operate” either in the same direction or in opposite directions (it depends on the signs of p and I 3 ). Let us divide expression (5.7) into two parts: q1 q2

P02 I 3 2 I 23

k0 1

p k0 L

1

K

1 K I2

3

0 1 3

L

,

I p 33 I2 2

k0 1

K

3

0 1 3

L

,

where the first expression corresponds to the absence of a constant differential pressure (it is a purely vibrational component) and the second one corresponds to its presence. For the case when the porosity gradient and the constant pressure gradient operate in the same direction, Fig. 5.8 shows the plots of the ratio q1 q2 versus 1 (L) for various values of h = 0.5, 0.8, 0.9 (curves 1, 2, and 3, respectively) (the parameter values are the same as in Fig. 5.4). The wave component can contribute a substantial proportion of the total flow, especially on the interface with low-porosity layers or compartments. As the amplitude of the action increases, this contribution may considerably exceed the other component (see Fig. 5.7, curves 1, 2, 3 – P0 1, 1.2, 1.5·105 Pa , respectively, all other parameter values are the same as in Fig. 5.4). Summarizing the interim results for the problem considered in this section, we can make the following conclusions.

Action of Wave Forces on Fluid Droplets 83

Fig. 5.8. P0 105 Pa

Fig. 5.9. h = 0.9

It has been shown analytically that under wave action on an incompressible viscous fluid saturating a stationary porous medium, spatial heterogeneity of porosity 1 leads to the development of an average fluid flow in the porous medium under a time-periodic action (with a flow rate that depends quadratically on the amplitude of the action). The fluid flows in the direction of higher porosity (determined by the sign of the difference between porosities at the ends of the interval), i.e., in the direction of increase in the fluid flow area. In particular, it means that wave action on an oil reservoir can cause fluid flow at interfaces from low-porosity, low-permeability reservoir regions to high-porosity zones, for example, from compartments into fractures in fractured/porous reservoirs, which makes it possible to enhance oil production. If the porosity values on the boundaries are the same, an average flow does not develop.

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Dependencies have been established between the absolute value of the daily flow rate per unit surface area and the value of porosity at the righthand end of the interval. These curves are nonmonotonic. As can be seen from the obtained expressions, if there is a constant differential pressure (for instance, a difference between the reservoir pressure and the bottomhole pressure), the first term in Eq. (5.6) and Eq. (5.7) corresponds to Darcy’s law, while the second term is caused by the spatial heterogeneity of porosity and contains components corresponding to the inertial effects and the periodic action. The gradients of porosity and constant pressure may “operate” either in the same direction or in opposite directions. It has been shown that the wave component can contribute a substantial proportion of the total flow, especially on the interface with lowporosity layers or compartments. As the amplitude of the action increases, this contribution may considerably exceed the component associated with the constant pressure gradient.

5.3 Fluid Flows Caused by Oscillations in Cone-Shaped Pores Section 5.2 discusses fluid flows caused by fluid oscillations in a medium with a stationary matrix whose porosity is heterogeneous in space.

5.3.1 The Statement of the Problem As mentioned above (see Section 1.2.3 and the authors’ other studies [26, 8, 10, 11]), periodic and wave actions on various systems can create unidirectional (on average) monotonic motion. With reference to a situation commonly seen in oil reservoirs, one more mechanism of conversion of oscillations and waves into a unidirectional motion is considered here. The mechanism of initiation of a unidirectional (on average) flow operates at the micro level: in a variable cross-section pore whose walls make small periodic oscillations. In the one-dimensional formulation, we consider the axisymmetric flow of a viscous incompressible fluid in a cylindrical, variable cross-section channel that is caused by a periodic variation of the cross-section of the pore. We write down the equations describing the flow: s

s w t

x

0,

const,

Action of Wave Forces on Fluid Droplets 85

w

w

t

w

1

x 0

p 8 w, x s

(5.8)

L.

x

Here is the density of the fluid, w(x, t ) is the velocity, p( x , t ) is the pressure, s( x , t ) is the cross-sectional area of the channel (defined by is kinematic viscosity. When deriving a time-periodic function), Eq. (5.8) it was assumed that the force of friction at each cross-section of the channel is described by Poiseuille’s law for viscous flow in a cylindrical tube (the second term on the right-hand side of the equation of motion).

Fig. 5.10

After the introduction of a new variable of flow q sw and some simple rearrangements, the equation can be rewritten as follows:

s t 2 (q / s)

q

q x

0,

p 8 q. t x x s We introduce the dimensionless variables as follows: t, x

LX , q

s

L s0 Q , p

p0 P , s

s0 S ,

where L, s0= s0(0,0), p0 and ω are the length of the channel, the area of the initial cross-section of the channel at the initial point of time, the characteristic pressure, and the angular frequency, respectively. Then, the equations can be written in the following dimensionless form:

S (Q 2 / S)

Q

Eu

Q X

0, Eu S

X p0 , Re L2 2 0 X 1.

P

1 Q , Re S

X s0 8

,

(5.9)

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Enhanced Oil Recovery

Integrating the first equation with respect to X from 0 to X, we obtain: X

Q( X , ) Q0 ( )

S( y , ) dy , Q0 ( ) Q(0, ).

0

Inserting this expression into the second equation of the system (5.9) and integrating with respect to X from 0 to 1, we obtain an ordinary nonlinear differential equation for Q0(τ): A( )

dQ0 ( ) Q0 ( ) B1 ( ) B2 ( ) C( )Q02 ( ) d Eu P (1, ) P (0, ) D1 ( ) D2 ( ) D3 ( ),

(5.10)

where the 2π-periodic coefficients 1

1

dy , B1 ( ) S( y , ) 0

A( )

1 dy , 2 Re 0 S ( y , )

1

B2 ( ) 2

C( )

x

1 1 S( y , ) dy dx , S( x , ) x S( x , ) 0 0 1

1 1 1 , D1 ( ) 2 S(1, ) 1

D2 ( )

1 1 S( x , ) x S( x , ) 0 1

D3 ( )

x

1 0 S( x , ) 0 x

S( y , )

2

S( y , ) 2

dydx ,

2

dy

dx ,

0

x

S( y , ) 1 1 dydx 2 Re 0 S ( x , ) 0

are defined by the given time-periodic function S(X,τ). We assume that S( X , )

(1) 1,

f ( X ) (1 f (0) 1,

sin( )), 0

X 1,

f (1) SL , (S(0,0) 1, S(1,0) SL ),

i.e. the cross-section of the channel is defined by the product of the function f(X) that describes the steady-state shape of the channel and the function ϕ(τ) that describes periodic oscillations with a relative amplitude ε. The results for different types of functions are provided below. We are interested in the average flow that is caused in the channel by pulsations of the channel walls in the absence of a differential pressure across the channel, i.e., the first term on the right-hand side of Eq. (5.10) equals zero. We assume that the amplitude of oscillations is small, 1. Then, using the small parameter method, all variables can be expanded

Action of Wave Forces on Fluid Droplets 87

(the unknown flow and all the coefficients in Eq. (2)) in series in powers of the amplitude i

Y( )

Y (i ) ( )

i 0

and a system of equations can be obtained to find the terms of the flow expansion in series. We limit ourselves to the first three terms of the expansion. It is easy to notice that Q0(0) ( ) 0 is the solution of the first equation of this system. For the next two terms of the expansion we obtain the following two equations: dQ0(1) ( ) B1(0)Q0(1) ( ) D1(1) ( ) D2(1) ( ) D3(1) ( ), d dQ (2) ( ) A(0) 0 B1(0)Q0(2) ( ) G( ), d dQ (1) ( ) G( ) D1(2) ( ) D2(2) ( ) D3(2) ( ) A(1) ( ) 0 d A(0)

(5.11)

2

( B1(1) ( ) B2(1) ( ))Q0(1) ( ) C (0) Q0(1) ( ) ,

A(0) , B1(0) , C (0)

consts.

The period-average flow in the channel is of interest: 2

2

1 1 Q0 ( )d Q0(0) ( ) Q0(1) ( ) 2Q0(2) ( ) d . 2 0 2 0 The first term of the integrand sum equals zero, the second term is a 2π-periodic function and its integral over the period is also zero. Therefore, the average flow is determined by the third term of the expansion. Consequently, to determine the average flow is necessary to solve the first equation of the system (5.11) and insert its solution in the second equation. There is no need to solve the second equation. It is sufficient to apply the averaging operation to it (integration over the period) and we obtain: 2 2 2 2 1 1 2 Q0 ( ) Q0 ( )d Q0(2) ( )d G( )d . 2 0 2 0 B1(0) 2 0 Q0 ( )

Using Eq. (5.10), we can calculate the effective pressure gradient required to create a steady-state flow equal in size to the obtained average flow. Removing the dependence on time from Eq. (5.10), we obtain:

P

P(1) P (0)

1 Q0 ( ) B1(0) Eu

2

Q0 ( ) C (0) .

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Enhanced Oil Recovery

Or, in a dimensional form p0 1 p Q0 ( ) B1(0) L L Eu

2

Q0 ( ) C (0) .

5.3.2 Calculation Results We consider four options (in a dimensionless form) to see how the shape of the channel affects the solution: 2

S1 ( X , )

f1 ( X ) 2 ( )

1

1

S2 ( X , )

f2 ( X ) ( )

1

1 SL X 1

S3 ( X , )

f1 ( X ) ( )

1

1

S4 ( X , )

f3 ( X ) 2 ( ) e

SL X

2

1

sin( ) ,

sin( ) ,

2

X ln SL

SL X 1

1

sin( ) , 2

sin( ) .

In the first case the radius of the channel is linearly dependent on X and oscillates with the amplitude ε; in the second case the area of the channel is linearly dependent on X and oscillates with the amplitude ε; in the third case the radius of the channel is linearly dependent on X while the area oscillates with the amplitude ε; and in the fourth case the radius of the channel is exponentially dependent on X and oscillates with the amplitude ε. Fig. 5.11 shows plots of channel radius versus X for differrent fi(X) (for the case of a 4-fold increase in the radius at the right-hand end of the channel).

Fig. 5.11

The analytical expressions obtained for the average flow and the effective pressure gradient are not provided here due to their clumsiness. Fig. 5.12 shows dimensionless average flow versus Re for various shapes of

Action of Wave Forces on Fluid Droplets 89

the channel and values of the dimensionless cross-sectional area at the right end of the channel SL (the tapering ratio). With small Re numbers, the flow is negative, i.e., it is directed to the restriction in the channel; as Re grows, the direction of the flow changes. As the tapering ratio SL grows, the flow increases, and the point of sign change tends to zero. As SL tends to unity, the flow tends to zero, and the point of sign change tends to 2 . It is obvious that the flow is dependent on channel geometry.

Fig. 5.12

Fig. 5.13 shows an example of calculation of the effective pressure gradient as a function of frequency for various values of the characteristic area s0. The shape of the channel is defined as S2(X,τ). The parameters are: ρ = 103 kg/m3, μ = 10–3 Pa·sec, L = 4·10–5 m, SL =16, ε = 0.05. As we can see, oscillations in tapering pores may cause strong flows of fluid through the pores. The rates of such flows, according to the calculation results provided above, reach the same values as steady-state flows that occur in a medium with similar resting pores under the action of enormous pressure gradients of 500 bar/m or more that cannot be achieved in practice by conventional methods. We note, however, that in

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Enhanced Oil Recovery

pores with a radius of 1–10 μm such flows occur under the action of ultrasonic oscillations at frequencies greater than 50 kHz. It appears to be very difficult to excite such oscillations at long distances from the wells. However, the calculations have shown that strong flows can also be caused by oscillations at significantly lower frequencies. For example, oscillations excited by a vortex cavitation generator have a continuous spectrum in the frequency range of 1–15 kHz.

Fig. 5.13

Fig. 5.14 shows one more example of calculation of the effective pressure gradient as a function of frequency for various values of the characteristic area s0. The shape of the channel is defined as S2(X,τ). The parameters are: ρ = 103 kg/m3, μ = 10–3 Pa·sec, L = 4·10–4 m, SL =16, ε = 0.15.

Fig. 5.14

Therefore, stimulation of a porous medium at a frequency of 10– 12 kHz can cause unidirectional (on average) flows that in the steadystate case would require a pressure gradient of about 100 bar/m.

Enhanced Oil Recovery: Resonance Macro- and Micro-Mechanics of Petroleum Reservoirs. O. R. Ganiev, R. F. Ganiev and L. E. Ukrainsky. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

6 The Mobilization of Droplets and Blobs of Capillary-Trapped Oil from Microcavities

Wave stimulation may equally prove to be a useful method when it comes to mobilizing oil droplets trapped in irregularities in the walls of a water/surfactant-filled pore. However, waves act on a droplet trapped in a pore through a completely different mechanism. In this case, wave stimulation is used to detach wetting oil droplets from the walls of pores and cause them to be entrained in fluid flow towards production wells. Provided certain frequency and amplitude criteria are met, continuous waves may be conducive to mobilizing oil droplets trapped in the pores of a stimulated reservoir. Excitation of gravity-capillary waves on an oilwater interface inside rock matrix pores may become one of possible ways of bringing about such effects. As shown below, the frequency of such waves depends on the size of oil-filled pores and cavities, and on the interfacial tension between the oil and fluid in the pore. In reality, it may vary from fractions of a kilohertz to tens of megahertz. One of possible mechanisms to excite such waves is a parametric excitation caused by an external wave field. There are other ways to excite oscillations in pores that are based on various types of external wave action. However, we review the above as a typical example.

6.1 The Mathematical Statement of the Problem We consider two-dimensional motion (the XZ-plane, see Fig. 6.1) of oil and water in a pore of a fluid-saturated porous medium. The XZ Cartesian system is rigidly attached to the pore in the motion plane. We assume that the porous matrix of which the pore walls consist is nondeformable and flatly oscillating in the XZ-plane as a solid body. A force of inertia proportional to the acceleration of the pores acts on the fluid in the pores.

91

92

Enhanced Oil Recovery

Inertia forces projected on corresponding axes, referred to as a volume unit of the fluid with a density ρ, are expressed as follows: Fx

2

dx ei

t

dx e

i t

;

Fz

2

dz e i

t

dz e

i t

,

(6.1)

where Ω is the oscillation frequency of the pore; dx and dz are complex oscillation amplitudes of the pore in directions X and Z, respectively; and the upper line represents the complex conjugation. We assume that there is a cavity in the pore wall of a depth hu and a width L. The cavity is filled with oil which, in unperturbed motion, remains still relative to the pore. The space outside of the cavity is filled with water (Fig. 6.1). In unperturbed motion, both fluids (water and oil) are still relative to pore walls, i.e.:

ux0

uz0

0 for 0 z hu ; ux0

uz0

0 for

hd

z 0.

(6.2)

Fig. 6.1

We consider the problem of small deviations from unperturbed motion using a linear approximation and ignoring viscous forces. We assume that all fluids, both inside and outside of the pore, are incompressible. We also assume that the motion does not cause the fluid to separate from pore walls, and that the weight of the water in the pore under the oil-filled cavity (i.e. within 0 < X < L) remains unchanged during the motion. This condition obviously keeps the oil in the cavity. Below, we consider only such small perturbations that don’t cause the displacement of the oil by water from its cavity when the oil-water interface deforms. Besides, we concern ourselves only with stability of the interface under infinitely small perturbations. We consider a loss of the

The Mobilization of Droplets and Blobs of Capillary-Trapped Oil 93

oil-water interface stability as a prerequisite for the excitation of oscillations whose propagation is able to destroy the interface and cause the oil to flow out from the cavity in the pore wall. However, we will leave oscillations alone in this chapter. We assume that the flow is potential on the flow segment 0 < x < L. For the perturbation potential of the oil velocity x, z , t in the pore cavity 0 z hu related to oil velocity perturbation components by relations ux

, uz , and for the potential of the water x z x , z , t in the pore outside of the cavity hd z 0 relat-

velocity

ed to water velocity perturbation components by relations ux

x

,

uz

, we derive the following boundary value problem for the Laz place equation: 0 in the range: 0 x L; 0 z hu ; (6.3) 0 in the range: 0 x L; hd z 0 with the following kinematic boundary conditions at a rigid boundary and at an interface with the water volume between planes x = 0 and x = L: 0x

x xx

L

0, L

, 0 z

xx

; 0

hu ; hd

z z

0;

0z z

hu

, 0 x

L;

(6.4) 0z

hd

, 0 x

L.

The oil-water interface is a discontinuity surface of tangential displacements and oil and water velocities. In this plane, normal components of the displacement and oil and water velocities remain continuous since no voids form between those. With z = 0, the oil-water interface should satisfy kinematic and dynamic boundary conditions. In unperturbed motion, the oil-water interface corresponds to a portion of the plane z = 0. As the motion progresses, the interface deforms but remains cylindrical with its element perpendicular to the XZ-plane. We assume that the displacement vector of interface points in XZ coordinates is approximately parallel to Z-axis and, consequently, as dictated by the assumed approximation, has the only nonzero component ξ = ξ(x, t), that is Z-component. Let the oil particle at the oil-water interface, in unperturbed motion, remain still. Meanwhile, in perturbed motion, interface points move. The relations between interface point velocities and these of oil and water give kinematic boundary conditions. By applying

Enhanced Oil Recovery

94

Taylor expansion to these values by perturbation, carrying them to the unperturbed interface surface z=0 and leaving only linear terms in relation to perturbations, we derive: (6.5) . t zz 0 zz 0 In a linear approximation relative to perturbations and with account taken of the assumptions made, a dynamic condition at the interface that shows a relation between pressures at both sides of the oil-water interface may be written as follows (the gravity vector faces strictly downward along the Z-axis negative direction, g is the gravity acceleration): 2 2

dz e i

t

w

dz e

i t

0. x2 (6.6) Where α is the oil-water interface surface tension , ρн is the density of oil and ρв is the density of water. Note that, with the assumptions made, the problem does not depend on oscillations of the pore in X-direction. This is because it is assumed for this purpose that only displacements in Z-direction relative to the pore exist at the oil-water interface and all displacements in X-direction relative to the pore are ignored. Based on the oil and water flow potential equations and boundary conditions at a solid surface and oil-water interface, we find potentials and displacements at the boundary in the following equations: o

g

w

( an cos kn x bn sin kn x ); n 1

an

t

chkn z hu

n 1

an

o

z 0

knshknhu

t

z 0

cos kn x ,

(6.7)

chkn z hd

cos kn x , knshknhd 2 n , n is an integer; an are wave amplitudes to be deterwhere kn L mined and are functions of time; ch is a hyperbolic cosine; sh is a hyperbolic sine, and the superposed dot represents time differentiation. Note that Eqs. (6.7) satisfy all kinematic boundary conditions of Eq. (6.5) at the oil-water interface, and satisfy Eqs. (6.3) and boundary conditions (6.4). We substitute Eqs. (6.7) into Eq. (6.6). By setting cos knx, expressions equal to zero for each n, we find: n 1

an

2 gkn n

a

2

dz e i

t

dz e

i t

an ,

(6.8)

The Mobilization of Droplets and Blobs of Capillary-Trapped Oil 95

where 2 gkn

g

kn

; w

w

cth kn hd

o o

cth knhu

; .

kn3 w

cth knhd

o

cth kn hu

6.2 The Natural Frequency of Gravity-Capillary Waves on Oil-Water and Oil-Surfactant Interfaces in Pores 2 We note that the factor gkn appearing in Eq. (6.8) corresponds to a natural frequency of the gravity-capillary waves on an oil-water interface. The equation that links it to other parameters of the problem resembles the Thomson equation (W. Thomson, 1871) [22] for a frequency of a plane wave traveling over an infinite fluid layer overlying a solid 2 surface. This equation reads: gkt k w k 3 . As distinct from o the expression for gkn , the above k represents a random wave number whereas in the equation for gkn , the quantity kn is determined from the length of the cavity in the pore wall L: kn = 2πn/L, where n is an integer. Physically, this distinction lies in the fact that due to the rigid walls in the cavity, with x = 0 and x = L, waves on the oil-water interface are standing rather than traveling. Another distinction is the multiplier 1

that describes the dependence of natural cth kn hd o cth kn hu frequency from ratios of pore wall cavity depth hu and pore flow channel width hd to cavity length L, as well as from the densities of the fluids inside and outside of the cavity. The functions gkn L are shown on Fig. 6.2. The solid curves 1–3 on Fig. 6.2 are plotted for the surface tension α = 0.03 N/m typical of the oilwater interface. The dotted curves 4–6 are plotted for α = 10–5 N/m typical of the oil-surfactant interface [18]. The curve groups 1–3 and 4–6 are distinct from each other in pore configurations. For the curves 1 and 4, we assume relations hu /L and hd /L equal to 0.1; for the curves 2 and 5, equal to 10–3 and, finally, for the curves 3 and 6, equal to 10–6. That is, the higher the number of the curve, the longer the cavities in which oil becomes film-like. w

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96

Fig. 6.2

As we can see, with α = 0.03 N/m for hu /L = hd /L = 0.1 (Curve 1), for cavities with the length L below 10 μm, the natural frequency of gravitycapillary waves exceeds 100 kHz, and with the cavity length L = 1 μm, it reaches 10 MHz. Notice that, in this case, an increase in the cavity depth and flow channel width has almost no effect on the natural frequency. This means that for all relations hu /L and hd /L greater than 0.1, Curve 1 remains in place. Instead, as the oil inclusion filling the cavity in the pore wall changes geometrically to a thin film, that is when the relation of the cavity depth to length, and that of the flow channel width to cavity length decrease, the natural frequency of gravity-capillary waves goes down. This is evident from the layout of curves 1, 2 and 3. Notice that with hu/L = hd /L = 10–6 for the cavity L = 1 μm, the natural frequency of gravity-capillary waves does not exceed 25 kHz. Mathematically, this 2 is because the expression for gkn (see Fig. 6.6) has a denominator with hyperbolic cotangents of relations cth kn hd o cth kn hu hu/L and hd /L, and as they tend to zero, the hyperbolic cotangent tends to 2 infinity, and gkn tends to zero. The natural frequency of gravity-capillary waves also decreases if we replace water with the surfactant solution. This is clearly obvious from the layout of dotted and solid curves on Fig. 6.2. Further investigations will need to clarify the issue of stability of the zero solution for Eq. (6.8) which is a linear equation with periodic coefficients. Specifically, we will have to determine whether such quantities of the amplitudes dz and excitation frequencies Ω exist that can make w

The Mobilization of Droplets and Blobs of Capillary-Trapped Oil 97

Eq. (6.8) unstable. This will be a clear sign of the oil-water or oilsurfactant interfacial instability.

6.3 Interface Instability Range At a first approximation, an analysis of Eq. (6.8) using the averaging method failed to find their instability in a non-resonance case, i.e. with such quantities Ω and ωgkn, that form no vanishing linear combination with integer coefficients. We seek to find modes of motion close to oscillations with natural frequencies of gravity-capillary waves ωgkn.. Let us consider the case where the pore oscillation frequency Ω and the n-th form interface natural frequency ωgkn satisfy the following relation: 2 2 gkn

, (6.9) 4 where is the small frequency mismatch, i.e. the natural frequency of the oil-water or oil-surfactant interface is close to a half of the pore oscillation frequency. We substitute Eq. (6.9) into Eq. (6.8). Assuming that in the future we consider modes of motion close to oscillations with the frequency equal to a half of the pore oscillation frequency, we introduce a small parameter into the derived equation and write it as follows: 2

an 2

where

dz e i

t

4

dz e

an

(6.10)

,

i t

an .

We normalize the system (6.10) to the standard averaging form [27] by the following change of variables:

an

Ce

i t 2

Ce

i t 2

; an

i

2

Ce

i t 2

Ce

i t 2

,

(6.11)

where С = С(t) is the new complex variable. Eq. (6.10) for the new unknown component С will read:

C

i

e

i t 2

; C

i

The averaged equations will be as follows: i i 2 С C dz C ; С

e

i t 2

2

.

dz C

(6.12)

C .

(6.13)

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The characteristic equation of the system (6.13) is as follows: 2 2

2 2

4

2

dz dz

.

(6.14)

The condition for roots of the characteristic equation (6.14) to have some with positive real parts is as follows: 2

4

dz d z

2

.

(6.15)

If the condition (6.15) is satisfied for any n, this means, according to the second N. N. Bogolyubov theorem [27], that the initial equation (6.8) is unstable, and, consequently, that the n-th form oil-water interface is equally unstable. Taking Eq. (6.9) into account, the instability condition (6.15) may read as follows: 4 2 2 4 1 16 2 2 8 gkn 16 gkn 0, (6.16) where 2 dz dz . Hence, the instability condition (6.15) or (6.16) boils down to the following conditions: 2 gkn 2 gkn 1 , , then (6.17) – if 4 1 4 1 4 – if

1 4

, then

2

gkn

.

(6.18)

1 4

6.4 Oil-Water Interface Instability Figure 6.3 shows instability ranges for the first three forms (n = 1, 2, 3) plotted from (6.17) and (6.18) for the case where the pore flow channel contains water, and the cavity contains oil. In this case, the surface tension α is equal to 0.03 N/m. The cavity length L = 0.1 mm and relations of the cavity depth to L, and the flow channel width to L are equal to 1. As is seen from the plot, the instability range for the first form with real frequency amplitudes of pore oscillations (tens of micrometers) covers a bandwidth around 20.467 kHz. Figure 6.4 shows an enlarged view of the portion of the first form instability range for small amplitudes below 5 μm. As we can see, excitation of vertical oscillations of the pore with frequencies of 20,467 ± 500 Hz generates instability on the oil-water interface even if the minimum amplitude is as low as 1 μm. The only concern

The Mobilization of Droplets and Blobs of Capillary-Trapped Oil 99

is how to excite oscillations at so high a frequency in reservoir zones far away from the generator under real damping conditions existing in the rock. Besides, at small amplitudes frequencies are fairly dependent on the cavity length L. So at small amplitudes these should be close to twice of the natural frequency of gravity-capillary waves on a fluid-fluid interface 2 gkn shown for some cavity lengths L in Table 6.1.

Fig. 6.3

Fig. 6.4 Table 6.1. hu /L = hd /L = 1 L (mm) 0.1 0.15 0.05

2

gk1

(Hz)

20,467 11,140 57,888

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As we can see, changes in the cavity length by only 0.05 mm bring changes in the resonant frequency by over 9 kHz. Figure 6.3 shows that for the instability range to cover a wide scope of pore dimensions it is necessary to significantly increase the amplitude of oscillations of pores. For example, to excite unstable oscillations on a fluid-fluid interface in pores where cavity lengths lie within the range 0.05 mm < L < 0.15 mm, amplitudes of pore oscillations should exceed 40–50 μm on a 20 kHz ultrasound bandwidth. Alternatively, it requires a polyharmonic generator with a continuous spectrum over the range 10–60 kHz. If the pore dimensions range proves to be even wider in the reservoir in reality, then amplitudes required to excite unstable oscillations on a fluid-fluid interface should be significantly larger. With such amplitudes, generation of instability over an ultrasound bandwidth is really problematic. The reason being that this requires high-power ultrasound generators which would be able to transmit waves with frequencies of 20 kHz and higher over a significant distance in a fluid-saturated porous medium and excite vertical oscillations of the matrix with amplitudes 40–50 μm at a significant distance from the source. Minimum frequencies required to excite unstable waves on an interface (see Fig. 6.3) are 50 kHz for the second form and over 100 kHz for the third. The analysis of results shows that in reality excitation of instability and thereby mobilization of capillarytrapped oil is unattainable for the entire reservoir. However, if the length of cavities in pores is several orders greater than the depth of cavities and the width of pore flow channels, then this significantly decreases natural frequencies of gravity-capillary waves on an oilwater interface (see Figure 6.2), and the excitation frequency (around 2 times as large as the natural frequency of gravity-capillary waves) can be significantly reduced too. Relations between cavity depths and lengths in pores as well as relations between flow channel width and lengths of cavities in pore walls can be investigated in detail by core analysis. If residual oil is capillary-trapped in the rock as a film, excitation of oscillations at the oil surface is possible using vortex generators. Indeed, Figure 6.2 shows that for hu /L = hd /L = 10–3, which corresponds to the Curve 2, the natural frequency on an oil-water interface in pores with cavities with L = 100 μm is equal to 0.7–0.9 kHz, meaning that excitation of oscillations on the interface can be reached at excitation frequencies of 1.4–1.8 kHz. Figure 6.5 shows instability ranges for the first four forms plotted from (6.17) and (6.18) for the case where the pore flow channel contains water, and the cavity contains oil. In this case, the surface tension α

The Mobilization of Droplets and Blobs of Capillary-Trapped Oil 101

is equal to 0.03 N/m. The cavity length L = 0.1 mm and relations of the cavity depth to L, and the flow channel width to L are equal to 0.001 (hu /L = hd /L = 10–3). Figure 6.6 shows an enlarged view of the portion of the first form instability range for the pore oscillation amplitudes below 100 μm. As we can see, for oil capillary-trapped in pores as a film, instability ranges occur at significantly lower frequencies than in the case above (see Figures 6.3 and 6.4). However, these ranges proved to be significantly tighter. For amplitudes below 20 μm, the frequency bandwidth that accounts for excitation of first form unstable waves is 1622 ± 2 Hz. For amplitudes below 100 μm, the frequency bandwidth is 1622 ± 15 Hz. Frequencies that excite first form unstable waves permit the use of vortex generators in this case. Table 6.2 illustrates dependence of excitation frequencies of first form unstable waves at small excitation amplitudes on the length of the cavity in the pore wall L for the case in question.

Fig. 6.5

Fig. 6.6

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Table 6.2. hu /L = hd /L = 10–3 2

L (mm) 0.1 0.15 0.05

gk1

(Hz)

1622 883 4588

Therefore, to excite unstable waves on an oil-water interface in pores where cavity lengths lie within the range 0.05 mm < L < 0.15 mm, amplitudes of pore oscillations should exceed 4 mm, as it follows from data used to plot instability ranges on Fig. 6.5, which appears hardly achievable in practice. The alternative is more feasible and consists in using a broadband generator with a continuous spectrum over the range of 0.8–5 kHz. Note that such a characteristic is available in vortex generators. They can be recommended for use to mobilize capillary-trapped oil if core analyses verify that pore dimensions in the stimulated zone fall within the range of 0.05-0.15 mm.

6.5 Oil-Surfactant Interface Instability Surfactant solutions are often used to displace oil instead of water. They have a low surface tension at the oil-surfactant interface α = 10–5 N/m [18]. Of course, such change in the surface tension against the above case where we investigated instability on an oil-water interface with the surface tension of α = 0.03 N/m leads to a significant change in the natural frequency of gravity-capillary waves on a fluid-fluid interface. Figure 6.7 shows instability ranges on an oil-surfactant interface for the first four forms (n = 1, 2, 3) plotted from (6.17) and (6.18) for the case where the pore flow channel contains surfactant solution, and the cavity contains oil. In this case, the surface tension α is equal to 10–5 N/m. The cavity length L = 0.1 mm and relations of the cavity depth to L, and the flow channel width to L are equal to 1. As is seen from the plot, the instability range for the first form that does exist at real frequency amplitudes of pore oscillations (tens of micrometers) covers, according to the amplitude, a bandwidth around 0.383 kHz. As the amplitude tends to zero, the range collapses to the point Ω = 0.383 kHz; for ε = 10 μm, instability occurs for frequencies 0.35 kHz < Ω < 0.45 kHz, which means that the instability range significantly expands and makes up 100 Hz; and, finally,

The Mobilization of Droplets and Blobs of Capillary-Trapped Oil 103

for amplitudes in excess of 35 μm, instability occurs at all frequencies in excess of 200 Hz. Therefore, this permits the use of a fairly wide variety of known generators ranging from vortex generators to rotary vibrators for excitation of unstable waves on a fluid-fluid interface in pores and mobilization of capillary-trapped oil.

Fig. 6.7

Fig. 6.8

Figure 6.8 shows instability ranges on an oil-surfactant interface for the first form (n = 1) plotted from (6.17) and (6.18) for the case where the pore flow channel contains surfactant solution, and the cavity contains oil. In this case, the surface tension α is equal to 10–5 N/m. Ranges were plotted for three geometrically different pores with various cavity lengths L = 0.1 mm; L = 0.5 mm and L = 1 mm; and relations of the cavity depth to L, and the flow channel width to L were equal to 1 in all studied cases. As is seen from Fig. 6.8, all three instability ranges overlap. This common

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range is colored yellow. Consequently, for any point within this range, waves are excited on an oil-surfactant interface in any pore where cavities in walls vary in size over the range 0.1 mm < L < 1 mm. Hence, it is possible in the case under consideration to excite waves on a subject fluid-fluid interface in a wide variety of pores, and, provided that amplitudes are sufficient, cause capillary-trapped oil to flow out from these pores, thus, entraining it in the fluid flow. To make this happen, we need to excite oscillations in the area in question with a frequency in excess of 0.2 kHz and with the matrix oscillation amplitude of 0.35 mm. It appears quite feasible in practice for large reservoir areas. Likewise, we can consider such pores that are, for example, close to symmetric (with circular or elliptical cross-sections) at the oscillatory excitation acting perpendicular to the element of the cylinder or in a more complex manner. In this case, the system in question remains under the influence of dual internal resonance; it is possible to demonstrate that excitation conditions for resonance oscillations at the fluid surface are much wider, and that an external broadband vibrator is able to generate strong resonance oscillations at the fluid surface and cause fluid droplets to detach [28]. The above-mentioned cases are fairly typical and cover most of micropores of the porous medium. For our purpose, we can also apply wave effects of multi-fold acceleration of fluid motion in capillaries and other mechanisms of motion described in Chapters 1 and 5 above which are conducive to breaking away capillary-trapped oil from various micropores of the porous medium. These various wave motions create the fundamentals for the micromechanics of petroleum reservoirs and contribute to increased oil production and enhanced oil recovery by near-wellbore stimulations and by exciting resonance in reservoirs.

Enhanced Oil Recovery: Resonance Macro- and Micro-Mechanics of Petroleum Reservoirs. O. R. Ganiev, R. F. Ganiev and L. E. Ukrainsky. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

7 Statements and Substantiations of Waveguide Mechanics of Porous Media Resonances. Statements and Substantiations of Waveguide Mechanics of Porous Media. Results. Algorithms and Software Products. Principles of Controlled Means and Oscillation Generators for Reservoir Stimulation

The above-derived criteria of near-wellbore cleanup and detachment of capillary-trapped oil show that for the cleanup to be efficient under harmonic excitation of a continuous wave field, we need to increase the amplitudes of matrix oscillation and fluid pressure gradient. For energy saving considerations, it is proposed to excite a wave field in reservoirs using a resonance phenomenon. This raises the question of whether it is possible at all times. For example, for a spatially infinite reservoir in the absence of any reflections at a linear approximation, traveling onedimensional waves would not cause any resonance. This made some authors [14] mistakenly think that no resonance is possible in reservoirs at all. This is key to the statement of this problem.

7.1 Resonance Mechanisms Possible in Fluid-Saturated Porous Media In infinite space, a nonlinear parametric interaction of two onedimensional waves excited by two monoharmonic generators of different frequencies is possible in a homogeneous fluid-saturated porous medium. This was established by authors and first published in [29]. As an example, Figure 7.1 shows dependence of a high-frequency wave amplitude on coordinate under various excitation conditions. Parameters of the porous medium were selected identical to in-situ parameters of an oil reservoir in Western Siberia. The porous medium matrix is a coarsegrained consolidated sandstone with permeability k = 1 D and porosity т0 = 0.25. The amplitude of a high-frequency oscillation generator was

105

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Enhanced Oil Recovery

assumed to be equal to 1 atm, and that of a low-frequency was assumed 10 atm. The parameter p1 on the plot is a pressure amplitude for propagation of a wave from a monoharmonic generator and a maximum pressure amplitude of a high-frequency wave for the two-frequency excitation. Frequencies of the high-frequency and low-frequency waves are equal to f2 = 25 kHz and f1 = 1 kHz, respectively.

Fig. 7.1

The dotted curve on Fig.7.1 corresponds to the propagation of a highfrequency monoharmonic wave if nonlinearity is ignored. Curves 1–3 describe amplitudes of a high-frequency wave from a monoharmonic generator with account taken of the medium nonlinearity for the generator amplitudes equal to 1, 5 and 8 atm, respectively. In all cases, we observe a regular attenuation of the wave amplitude as the distance to the generator grows. As is seen from the behavior of curves 1–3, for the selected parameters of porous medium, if a wave propagates from a monoharmonic generator, the effect of nonlinearity is insignificant when generator amplitudes are below 1 atm. It is also seen from the behavior of curves that whenever nonlinearity is ignored at large amplitudes, it may cause a significant error in calculations. Curve 4 on Fig. 7.1 demonstrates distribution of an amplitude-modulated high-frequency wave when it nonlinearly interacts with the low-frequency wave. As we can see, the result considerably differs from the distribution of monoharmonic wave amplitudes. The distribution peaks at the point outside of the location of the generator, at some (in places fairly significant) distance from it. The shape of this curve shows that at sufficiently large distances from the oscillation generator the amplitude of a highfrequency component is significantly greater than the amplitude of a high-frequency wave under monoharmonic conditions. The maximum of wave amplitude and the maximum coordinate depend on the frequencies and amplitudes of the oscillation generator. As the amplitude of the

Statements and Substantiations of Waveguide Mechanics

107

low-frequency oscillation generator goes down, the peak shifts to the origin of coordinates. The described phenomenon is a result of nonlinearities in fluid-saturated porous media motion equations. It takes place even in spatially unlimited domains and for homogeneous media. This is a promising new way ahead for nonlinear wave mechanics of fluidsaturated porous media [8, 10, 11, 28]. However, if any heterogeneity occurs in a reservoir at any distance from the oscillation generator, it may cause resonances associated with reflection of waves from heterogeneities. Studies [8, 10, 11, 17] consider this situation for one-dimensional waves in spatially finite areas of a reservoir. Results of these studies are described in Section 1.2.3 (see Fig. 1.17). Another example of a possible resonance in the near-wellbore zone of reservoir is observed from pressure oscillations in fluid-filled perforated holes of a finite length. A relevant problem was solved by authors in their studies [8, 9, 10, 11, 13]. Some of the above results are set out in Chapter 1 (see Section 1.2.2). Even in the case of infinite-length reservoirs without any horizontal heterogeneities, presence of horizontal boundaries may cause resonances even at a linear approximation. A similar situation was already mentioned in Chapter 1 (see Fig. 1.23). For reservoirs describable using a mathematical model as a layer of a horizontally infinite homogeneous elastic medium confined between absolutely rigid horizontal planes, the authors [30] were able to validate the statement and demonstrate that a resonance phenomenon does actually occur. There are some excitation frequencies (hereinafter referred to as resonant frequencies) at which energy flow and oscillation amplitudes are maximal, and wave attenuation is minimal compared to other excitation frequencies. The main prerequisites for such resonances are wave structure of several dimensions and the presence of several wave types propagating in the medium at various phase velocities. For example, for solid bodies, these can include compressional and shear waves (body and rotational waves, respectively). For fluid-saturated elastic porous media, these include fast and slow compressional waves and a shear wave [31, 32, 33]. Spatially multidimensional waves such as two- and three-dimensional waves that correspond to a superposition of waves propagating at various velocities may experience resonances in infinite domains. The physical nature of such resonances is associated with the fact that at some excitation frequencies some waves may overlap with any of the free waves that may occur in infinite domains. These may be free waves similar to Rayleigh or any other free waves. Below we consider some statements that constitute a problem of resonance waveguide mechanics of fluid-saturated porous media.

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Enhanced Oil Recovery

Finding values of resonant frequencies for multidimensional waves of different nature that propagate in the medium at various velocities is instrumental in generating effective wave fields to be used for a cleanup and the increase of oil recovery in reservoir zones far away from the ‘exciting’ well.

7.2 Resonance of Two-Dimensional Axially Symmetric Waves in Horizontal Layers of Reservoir. Efficient and Directed Excitation of Wave Energy in Target Sub-Layers First of all, in accordance with the study [30], we detail the results of the above-described case concerning resonance in horizontal reservoir layers. We consider a horizontal reservoir which could be divided into horizontal homogenous layers separated from each other by horizontal flat boundaries distinct from each of layers in elastic and damping properties. Each layer is assumed homogenous. We ignore any interaction between the layers. Hence, we choose for consideration one homogeneous layer (Fig. 1.23). We consider a zone in a homogenous reservoir that lies in the deep subsurface between two horizontal flat boundaries distinct from producing reservoir in elastic and damping properties. A well is drilled from the surface that penetrates through to the lower boundary of such reservoir (Fig. 1.23). Downhole, the well has a means capable of exciting harmonic pressure oscillations at the amplitude P. The reservoir thickness is H, and the well diameter is 2R0. To study excitation and propagation of waves in the reservoir, it suffices to consider a model where the reservoir is represented as a homogenous attenuating elastic body. Elasticity and attenuation parameters should be averaged from core flow tests with core sampled from various zones of the reservoir (these parameters will be hereinafter referred to as effective). These parameters also take into account the presence of produced fluids and porosity. With no oscillations, the stress in the reservoir and at its boundaries depends on the overburden pressure, reservoir occurrence depth and rock properties as well as on the properties of surrounding subsurface formations. In our model, we assume that no tangential stress at the boundary of the reservoir exist at all times of wave excitation and propagation.

Statements and Substantiations of Waveguide Mechanics

109

If pressure oscillates at the reservoir/wellbore boundary, then boundary conditions at other reservoir boundaries will depend only on the properties of the reservoir and surrounding rocks, hence, they may significantly vary. For example, if overburden pressure at a reservoir boundary is significantly greater than reservoir excitations caused by pressure oscillations in well while the rigidity of reservoir base and top rocks is only slightly higher than the rigidity of the producing reservoir, then normal stress excitations at horizontal reservoir boundaries may remain equal to zero when pressure oscillates in the well. This may involve normal displacements of reservoir boundaries. Alternatively, if overburden pressure at a reservoir boundary is comparable with reservoir excitations caused by pressure oscillations in the well while the rigidity of the reservoir base and top rocks is significantly greater than the rigidity of the reservoir, then reservoir boundaries should remain still and normal stress excitations at a boundary may be distinct from zero. Suitable boundary conditions are selected by analysis of all the above essential parameters. It is important to note that any one-dimensional wave field can not satisfy all boundary conditions, hence, the solution of the problem is of several dimensions. Study [30] considers a case where the stress at the upper reservoir boundary maintains the value it had without oscillations, meaning that stress excitations are set to zero, and the lower boundary remains still, i.e. rigidly fixed. We prescribe the pressure at the circular-shaped well boundary to be equal to the sum of unperturbed reservoir pressure in a well at reservoir depth and pressure oscillations generated using dedicated means. We assume that the reservoir is infinite in the horizontal direction. As a result of the solution of a boundary problem from the physical statement above, we obtain the following results. Numerical solutions are derived for Poisson ratios v = 0.42 of normalized height H = 2R0 (that equates to the thickness of reservoir equal to the 2,3 G , where G 2 R0 is an effective shear rigidity of the layer material, ρ is its effective density, and ω is the frequency in hertz. For selected material parameters and within the specified frequency range, an isotropic ideally elastic waveguide may have up to two propagating modes.

well diameter) over a frequency range of 0

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Based on conducted calculations, we derive a frequency dependence for an oscillation- time-averaged power flow of a rate of W/m2 per unit of cylindrical surface {r = 1, 0 < z < 2} upstream of the reservoir, where dimensionless variables r and z are introduced by the following relations: r = R/R0, z = Z/R0 (see Fig. 7.2). Figure 7.3 shows this frequency dependence, referred to as the amplitude of pressure oscillations in well P in N/m2, under forced oscillations of the waveguide for various effective attenuations ε from the range 0.001 < ε < 0.1, which can be found from tests for longitudinal oscillations of the reservoir rock core. Along the X-axis on Fig.7.3, we plot a dimensionless frequency of forced oscillations 2

R0 / c , where с

G

.

Fig. 7.2

Fig. 7.3

Statements and Substantiations of Waveguide Mechanics

111

Curve 1 corresponds to the attenuation coefficient ε = 0.1, Curve 2 to ε = 0.05, Curve 3 to ε = 0.01, Curve 4 to ε = 0.005, and Curve 5 to ε = 0.001. As is seen from the plot, if the frequency of oscillations verges towards some almost fixed frequency, the energy absorption of the waveguide under forced oscillations increases. Even with no internal friction, that is at ε = 0, the amount of energy remains limited due to the radiation damping caused by propagation waves existing within the specified frequency range which expand in space because of the cylindrical shape of the domain. The dimensionless frequencies 2πR0ωmax/c = Ωmax, at which energy peaks under forced oscillations (points at which curves on Fig. 7.3 peak), for R0 = 0.1 m, с = 2627 m/s, are equal (for curves 1-5 on Fig. 7.3) to ωmax of 6.73 kHz; 6.74 kHz; 6.75 kHz; 6.76 kHz; and 6.77 kHz, respectively. In what follows, they are called maximum energy absorption frequencies. They depend on reservoir properties. Here and below, all data are given for a fairly thin productive sub-layer with a thickness two times greater than the well radius. The higher the thickness of the producing reservoir, the lower the frequencies at which the amount of absorbed energy reaches its maximum. As we can see, at the frequency of oscillations P of only 1 bar and with the maximum attenuation of ε = 0.1 picked from values under consideration, the energy flow into the reservoir under resonance is 300 kW/m2. In reality, frequencies of pressure oscillations may prove to be much higher than 1 bar. Specifically, if the amplitude of pressure oscillations is 10 bar at the frequency ωmax of around 6735 Hz, then the flow of energy into reservoir is 3 MW/m2 with the attenuation of ε = 0.1. The frequency of maximum energy absorption and maximum amount of absorbed energy are both dependent on the attenuation coefficient ε. The higher the attenuation coefficient, the lower the intensity of a wave field excitation, hence, the resonance Q-value degrades. With the growth of ε > 0, the frequency of maximum energy absorption also goes slightly down. Since the amount of energy upstream of the waveguide is limited both by internal friction ε ≠ 0 and radiation damping, the change in these characteristics is not proportional to the change in the attenuation coefficient. With small values of attenuation coefficient ε = 0.001, the overall waveguide attenuation is largely driven by radiation damping which is caused by expansion of waves in space because of the cylindrical shape of the domain. With the amplitude of pressure oscillations in the well equal to 1 bar, given the above attenuation, the maximum power delivered into reservoir is 2.5 MW/m2. If we use Ωe to denote the dimensionless frequency of maximum energy absorption with ε = 0, then, as is seen from

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Fig. 7.3, at ε = 0.1, the frequency of maximum energy absorption shifts by 11 % from Ωe; with ε = 0.05, it shifts by 3.6 %; with ε = 0.01, by 0.9 %, and with ε = 0.005 and ε = 0.001, it virtually remains the same. With the increase of ε, the obtained shift of resonant frequency becomes consistent with the shift under forced oscillations of the single-degree-of-freedom system with viscous damping [34]. The decrease in the amount of energy is also not proportional to the growth in the attenuation coefficient ε. As the attenuation coefficient ε > 0 decreases by a factor of 10 from 0.1 to 0.01, the energy amount upstream of the waveguide increases by a factor of 4.7, and with the decrease of ε > 0 by a factor of 10 from 0.01 to 0.001, the energy amount only increases by a factor of 1.9. In this range, the attenuation is mainly driven by the radiation damping. To conduct analysis of the dependence of energy upstream of the waveguide on the frequency, we take into account the specifics of the dispersion spectrum in the boundary problem under consideration [35]. In the frequency range below the cutoff for the first traveling wave 2πR0ω/c < Ω1 = π/4, the energy upstream of the waveguide is extremely low. Within this frequency range, waves in the layer and energy behavior are totally dependent on the internal friction, so no resonance traveling waves occur. Consequently, if for reservoir excitation we use frequencies of 2πR0ω/c < 0.78, then we fail to utilize resonance properties of the reservoir, and hence, the flow of energy into the reservoir will not depend on the frequency of external excitation. In such a case, the flow of energy into the reservoir will be dependent only on the amplitude of excitation. For optimization of the process in terms of response characteristics, we need to increase the frequency of excitation to satisfy inequality 2πR0ω/c > Ω1 = π/4. The energy increases with the growth of frequency, and forced oscillations of the waveguide with free surfaces [36] did not cause any resonance phenomena similar to the edge resonance. We note that the system under consideration may appear to experience edge resonance under other boundary conditions at horizontal surfaces that confine the reservoir. The energy maximum is observed at the frequency 2πR0ω/c = Ωmax, that depends on the quantity of ε. A further increase in the frequency results in a dramatic drop in the amount of energy upstream of the waveguide. From the frequency 2πR0ω/c = Ω = 1.86 onwards, the energy upstream of the waveguide begins to grow and reaches another maximum at some greater excitation. However, the value of such second maximum is significantly lower than that of the first maximum. So at further maxima, amounts of energy upstream of the reservoir will be even less.

Statements and Substantiations of Waveguide Mechanics

113

It is known that under forced oscillations of semi-bounded elastic bodies [35] resonance phenomena may depend on the distribution of the load along the loaded boundary. So this brings about a problem of changes in the load distribution along the circular surface {r = 1, 0 < z < 2} which we consider. We assume that in the boundary condition the quantity P depends on the spatial coordinate z rather than denotes the amplitude of oscillation excitations constant over the circular surface. In particular, we consider the following distribution r 1, z P cos z 2 that corresponds to a mode where the amplitude of pressures oscillations is anti-symmetrically distributed along the height of the excited cylinder. Its solution shows that the resonant frequency and field phase response behavior are the same as under original load but that the quantity of the resonance peak considerably decreases (more than one order). This fact is clear evidence that modes where distribution of exciting pressure amplitudes is symmetrical relative to the plane z = 1, and in particular, modes with equilibrium distribution account for a higher resonance peak than modes with anti-symmetrical distribution of exciting pressure amplitudes. It should be taken into account in development of pressure oscillation generators for an energy emitting well. However, it should be borne in mind that the obtained result relates only to the type of boundary conditions that is under consideration here. For the other type of boundary conditions (existing both at horizontal surfaces that confine the layer and at the circular surface where oscillations are excited), resonance may be possible at other frequencies and with other values. Consequently, we may state that resonance plays a significant role in wave stimulation in axially symmetric homogeneously layered reservoirs. To optimize wave stimulation, it is preferable to select a frequency of excitation close to the lowermost value of resonance. Meanwhile, the flow of energy into the reservoir should be maximal. This will not only ensure maximum values for amplitudes of a wave field excited in a reservoir but will expand the area surrounding the exciting well where waves play a significant role. We should note that resonant frequencies depend on the reservoir thickness and also on boundary conditions selected for the flat surfaces that confine the reservoir. To select the specific stimulation pattern, we need to conduct theoretical research similar to the research described herein but with other boundary conditions and other reservoir parameters, or alternatively, determine resonant frequency experimentally using one of the approaches set out below (see Section 7.5).

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7.3 Resonance of Two-dimensional Plane Waves in Reservoir Compartmentalizing Strike-Slip Faults and Fractured Zones Prior to development of an area in a field, a pre-drilling geophysical analysis is normally conducted to establish the structure of heterogeneities in the reservoir to be developed, and to determine whether reservoir is compartmentalized into compartments (typically rectangular shaped) bound by local faults of increased porosity and permeability [37]. A local fault close to a near-wellbore zone helps to increase the fluid flow from the reservoir into the wellbore. As an example, Figure 7.4 gives a view of the heterogeneity in the structure of reservoir deformation field based on 3D seismic survey data [38] and shows a fracture network.

Fig. 7.4

The local fault is a narrow and long layer of a porous medium bound at the sides by walls of tighter compartments of low porosity and permeability, hence, is a natural waveguide. As is known, there is an infinite discrete set of two- and three-dimensional compressional/shear waves that propagate in waveguides without any change in its structure. The solution of the problem concerning propagation of such waves in a fluidsaturated elastic porous medium is a very labor-consuming matter. If we consider a network of linked waveguides, the complexity rises dramatically. The above problem for porous media has not been stated to date, let alone solved.

Statements and Substantiations of Waveguide Mechanics

115

Below, at first we discuss waveguide processes in one isolated fault linked with the generator, and then in Section 7.4 we discuss wave processes in linked waveguides. The selection of a fluid-filled porous medium model which we use here in below discussions was a joint effort that involved A. V. Zvyagin.

7.3.1 The Mathematical Model of a Fluid-Saturated Porous Medium The mathematical model proposed in [39] is used to describe a fluidsaturated medium. This model that follows the concept developed by Ya. I. Frenkel [31] and Biot [32, 33] is based on the following assumptions. With respect to wave propagation, motions and deformations of the matrix material and the liquid component are considered to be small. It allows us to consider matrix and fluid motions as the main unknown fields. It is assumed that the force of impact of the first component on the second one can be regarded as a mass force proportional to the relative acceleration of the components, and vice versa, and the friction of fluid against the matrix is also taken into account. In addition, the viscous friction in the matrix is taken into account, while the internal viscosity of the fluid is neglected. We note that simulation of dissipation processes in poroelastic media is a high-priority problem of geophysics. Here, we do not consider the well-known approaches, described in the literature on the subject. A review of the approaches can be found, for example, in [40]. We use the chosen model to demonstrate how natural and artificial reservoir heterogeneities can be used to enhance wave stimulation of various reservoir zones. The results obtained and presented here below are for the most part qualitative in nature and should serve as the basis for setting up targeted field tests in order to be refined and used in practice. We introduce the following notation: s , f are the true densities of the solid and fluid phases, respectively; f is the porosity of the medium (

s

1

f

is the volumetric content of the solid phase); u

u1 , u2 , u3

is the displacement vector of the solid phase; U U1 ,U 2 ,U 3 is the displacement vector of the fluid phase; 11 , 12 , 13 , 22 , 23 , 33 are stress tensor components in the solid phase (the porous matrix); f kl is the stress in the fluid phase. We assume that the shear strength of the fluid is to be considered only when assessing the forces of phase interaction, and, compared to them,

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all other shear resistances in the fluid phase are neglected. We also assume that the equation of state proposed in [39] is written as follows: f

λ *  div u

kl

kl

R div U

pf

f

Q div U

Q  div u ,

uk xl

1 2G *   2

kl

ul , xk

(7.1)

where

R

(

f

)2

f

,  Q

s

ef

1

*

, 

ef

1

*

s

s

f

f

,

*

1 1 , λ* λ ,  2 t 3 t pf is the pressure in the fluid, βs and βf are the compressibilities of the matrix material and the fluid, respectively, νef is a coefficient that describes the bonds between matrix grains (consolidation of the matrix) [21], G is the shear modulus of the porous matrix, λ is Lame’s first constant of the porous matrix, η and ξ are the shear and bulk viscosities of the porous matrix, respectively. Note that it is implied physically that the porous matrix is governed by the Voigt’s rheological law. The force of phase interaction is described by the following relations [41]: G*

G

R fs

p

R (fsp) R

( ) fs

s

Fm F

R (fsp ) m

R (fs ) ,

1 2

K m a*

f

s

f

U u ,

f

s

2 f

f

(7.2)

U u ,

0 1, m where νf is the kinematic viscosity of the fluid phase; values of the dimensionless coefficients m and Km depend on the structure of the medium and characteristics of the flow such as frequency, and in the quasi-steadystate approximation for frequency tending to zero m 1, K m 4,5 ; for the case where the pores are cylindrical channels with an average radius a* : m 0, K m 8 ; for dispersed solid spherical particles a*2 K(1 – αf ), K is the permeability. In a more general case, values of the coefficients m and Km can be determined experimentally and may depend on the frequency. By introducing the notions 1 11 12 s s 1, 22 12 f f 2, 12 f s f m , (7.3) 2

Statements and Substantiations of Waveguide Mechanics

117

the equations of motion can be transformed, according to [32], as follows: 11

u

12

G 12

22

G

6

div u Q div U

t

u F ,

2 t

u

F

U grad

(7.4)

U grad Q div u R div U

F,

M U u ,

where M K ma* 2 f f f s . In Eqs. (7.4), the variables 11 0, 22 0, 12 0 determined from (7.3) have a density dimension. We use the displacement vector representation in a form similar to the Lame representation:

u grad U

grad

1

grad

1

1

2

rot ψ1 , div ψ1

2

grad

0,

rot ψ 2 , div ψ 2

2

(7.5)

0,

where φ1 and φ2 are the scalar potentials – functions of the spatial coordinates and time, ψ1 and ψ2 are the vector potentials – vector functions of the spatial coordinates and time, β1 and β2 are the constants to be determined. In what follows we limit ourselves to considering steady-state timeharmonic solutions of Eqs. (7.4). Substituting Eq. (7.5) in Eqs. (7.4) and (7.1), and representing the potentials in the form of products of functions of the spatial coordinates (in what follows they are called amplitudes) and the multiplier e i t , we can obtain the following equations for the amplitudes that are denoted here by the same letters as the full variables, and obtain a quadratic equation whose roots are β1 and β2: 2 j

a 2j 1

a 2j

Q

11 2 1

i

j j

12

b2

b2

i

12

G 2

11

22

g

12

; g

cj

1

, j 1,2;

j

M (1

j

1

22

j

; dj

iM

22

dj

j

11

12

f

1

)

2

; cj

j

11

;

2

1

iM ; M

11

22

2 2

11

22

12

12

;

;

; 12

j

(7.6)

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MG 22

2

f

11

M

i

Q

11

2

22

2

M

R i

M 2 ; 2

12

1

i

i

R Q

M

R

Q

12

R 1

11

12 1

Q

12

2

2

M

22

22

22 1

2

M 2

i 1

2G;

M

2

R Q

R

12

Q

0,

(7.7)

22

2 . 3

The obtained wave equations take into account the viscosity of a viscoelastic porous medium saturated with a resiliently compressible fluid, and the viscosity of phase interaction. If both these viscosities tend to zero, then in the limit we obtain equations describing a three-wave process that consists of two compressional waves with phase velocities a1 and a2 and one shear wave with a phase velocity b, where a1, a2 and b are the limit values of the eponymous coefficients in Eqs. (7.6) when all the viscosities tend to zero.

7.3.2 The Statement of the Problem and Solution Procedure We consider non-one-dimensional waves in zones that separate a reservoir into compartments as described in section 7.3.1. When wave processes are considered in faults and fractured zones that divide a reservoir into compartments, we assume that motions of the fluid and the porous medium in the compartments can be neglected in comparison to motions of the fluid and the porous medium in the fault and fractured zones. In addition, we neglect the tangential stresses at the boundaries between a fault zone or a highly fractured zone and a compartment. Note that since the normal stresses at the boundaries between the zones of faults or fractured zones and the compartments are assumed to be non-zero, a wave process excited in a fault or fractured zone can excite waves in the compartments and provide their stimulation as well as stimulation of the fault or fractured zones. To consider wave excitation processes in compartments, we need to consider the problem in the next approximation

Statements and Substantiations of Waveguide Mechanics

119

when motions of the matrix and the fluid at the borders of the faults are not neglected. Fig. 7.5 shows such a fault that crosses a productive reservoir. We assume that the thickness of the fault 2h is considerably smaller than the thickness of the productive reservoir H. We limit ourselves to the case when oscillations are excited along the entire thickness of the reservoir (the length of the perforated interval in the well is the same as the thickness of the reservoir), and we assume that motions of the fluid and the porous medium occur only in planes parallel to the plane Оху in Fig. 7.4, while projections of the motions on the vertical axis Oz can be neglected. We assume that all nonzero tensions and motions depend only on the coordinates x, y and time t. Therefore, in what follows we limit ourselves to considering a plane problem. It needs to be noted that in this case it is done only to shorten the description of the results. The complete threedimensional problem in a similar mathematical formulation to the one provided below can be solved by analogy to the two-dimensional problem described here.

Fig. 7.5

Eqs. (7.7) describe the distributions of amplitudes of the potentials in the three-dimensional case. In the plane case considered here, the amplitudes of the potentials φ1 and φ2 and the flow functions ψ1 and ψ2 are functions of only x and y. In addition, the flow functions ψ1 and ψ2 have only one non-zero component, namely the projection on the axis Oz. Below we designate these projections as the scalars ψ1 and ψ2,, and considering the simple relation between them according to (7.7) we designate the amplitude of potential ψ1 as ψ, then 2 .

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Let us consider steady-state wave action on a fault that is shown as a green semiband dividing the productive reservoir into compartments in Fig. 7.5. It also shows the coordinate system Oxyz in which the statement of the mathematical problem is provided. A close view of the same coordinate system Oxyz and boundary conditions is shown in Fig. 7.6.

Fig. 7.6

The region x 0, y h , which is filled with a homogeneous elastic porous medium saturated with a compressible fluid, is acted upon by a periodic end load. The boundary conditions for the stress amplitudes and the pressure at the end x = 0 are written as follows: xx x 0

s 0

( y ),

xy x 0

0,

f

(0, y , t )

f

0

( y ).

(7.8)

h According to the above, we assume that at the side borders y there are no transverse motions of the elastic porous medium or the fluid, nor any tangent stresses. Therefore, for the amplitudes of these variables we have: 2

y

h, u y

0, U y

0,

xy

0

u2

.

(7.9)

At infinity, as x all the motions and stress disturbances decrease tending to zero. Amplitudes of the steady-state time-periodic solution of the boundarylimit problem (7.8), (7.9), (7.7) are found by expansion into forms. Forms for each of the potentials 1 , 2 , are represented by a product of the functions f1n , f2n , f3n of x, respectively, by a linear combination of the trigonometric functions that ensures satisfaction of the boundary condi-

Statements and Substantiations of Waveguide Mechanics

121

y y and cos n , h h where n = 1, 2, 3 ... is the number of the form. The conditions (7.9) define the eigenvalues of the problem λn that are the following according to 2n 1 Eq. (7.9): 0 0, n n 1,2,3... . The functions f1n x , 2 f 2n x , f 3n x are chosen so as to ensure that the conditions of limited motions and stresses with x are satisfied and the equations derived from (7.7) hold. According to this, the amplitudes of the potentials φj (j = 1, 2) and ψ are presented as follows: h : sin

tions on the side surfaces of a layer with y

sin

1 n 1

n

h

1 ik1n x n

y

e

,

n

sin

2

cos n 1

n

h

y

2 ik2n x n

y

h

n 1

n

,

e

n

n

e ik3 x .

(7.10)

According to (7.7), the wave numbers k nj and frequencies must satisfy the following relationships: 2

k nj

i dj

a 2j i c j

2 n 2

h

2

,

j 1,2; k3n

b2

i g i f

2 n 2

h

,

(7.11)

where the sign of the root is chosen so that the imaginary part of each of the wave numbers k nj is non-negative, which ensures that the conditions , and the coefficients of the problem statement are met as x a j , d j , c j , b, g , f are defined by Eqs. (7.7).

7.3.3 Damping Decrements of Waves in a Natural Vertical Waveguide A damping decrement of a wave equals to an imaginary part of quantities in Eq. (7.11). According to the theory of Y. I. Frenkel [31] and Biot [32], under zero viscosity in each of the waveforms determined by the transverse wave number n , three waves are possible: two compressional propagating at phase velocities a1 and a2 in Eq. (7.6) and one shear propagating at the phase velocity b in Eq. (7.6). If we consider the problem at nonzero viscosities, each of waves becomes damped. Nonzero damping decrements for each of waves in each waveform are dependent not only on the viscosity in the matrix and inter-phase friction but also on the

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possible multidimensional nature of waves, specifically, on the absence or presence of transverse components in the wave vector. Figures 7.7 to 7.11 below show dependences of damping decrements on the frequency for various waves calculated for the following parameters that describe dynamic properties of the medium: αs = 0.9, αf = 0.1, ρf = 1,000 kg/m3, ρs = 3,200 kg/m3, G = 5.54∙108 Pa, λ = 1.18∙1010, βs = 2.7·10–11 Pa–1, f

Km

f

1

C2

4.5,

m

, C = 1,500 m/s, νef = 0.029079, η = ξ = 105 Pa∙s, 1, a*2

K (1

f

),

f

10

of the medium, K = 100 mD, R

6

m2/s, K is the permeability (

f

)2 *

*

1

ef s

s

f

f

, 

Q

f

s

1

ef

, 

*

. Note that this way to account for dissipation

is one of the simplest. More complex models to describe dissipative process in Biot’s media [40] are available in geophysics to date and, for some, the development is underway. However, to derive high-quality regularities of wave effects on reservoirs with natural heterogeneities before field tests for a subsequent update upon their completion, the use of the assumed model may be deemed justified. For one-dimensional compressional waves in Eq. (7.11), the transverse wave number, n 0, becomes zero. Therefore, at 0 0, imaginary 0 parts of expressions (7.11) are equal to damping decrements of onedimensional waves that are independent of the wave transverse coordinate y. Figure 7.7 shows dependencies of damping decrements of the first compressional (Curve 1) and shear (Curve 2) one-dimensional waves ( 0 0). As we can see, at frequencies below 3 kHz, the damping decrement of the first compressional wave is below 1 m–1. Hence, for assumed parameters of the medium, a 3.5 kHz first compressional wave begins to measurably damp (more specifically, shrinks by a factor of e) fairly fast, at a distance of 1 meter. The shear wave damps even more. At a frequency of 1 kHz, the decrement makes up 3 m–1. The amplitude contracts by a factor of e at a distance of 0.33 m from the oscillation generator. As concerns the second compressional one-dimensional wave, its attenuation is considerably higher. Fig. 7.8 shows dependence of the second compressional wave attenuation on frequency. As we can see, for this wave, at frequencies of around 10 Hz the damping decrement nears 100. This means that for a 10 Hz one-dimensional second compressional wave, the amplitude contracts by a factor of e at a distance below 1 cm. For frequencies of around 1 kHz, the damping decrement is over

Statements and Substantiations of Waveguide Mechanics

123

1,000 m–1 which corresponds to the e-fold contraction of wave amplitude at a distance below 1 mm.

Fig. 7.7

Fig. 7.8

As shown above in chapters 2 and 4, the best candidates for wave stimulation are spatially heterogeneous waves. Consequently, two and three-dimensional waves are of the most practical interest for the natural waveguide under consideration that corresponds to a fault or a highly fractured zone in the reservoir. We now consider the waves that constitute the first two-dimensional waveform. It relates to a transverse wave

. Similarly to the above-described zero one2 dimensional waveform, this waveform consists of three waves: first compressional (for parameters under consideration at the limit of all viscosities tending to zero, it tends to a shear wave of a phase velocity

number

n

1

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a1 = 2567 m/s) and second compressional (where the phase velocity cutoff is 329.7 m/s), and a shear wave (where the phase velocity cutoff is 478.7 m/s). However, as distinct from the one-dimensional compressional waveform, in case of a two-dimensional waveform, the direction of a wave vector in each of these three waves does not match the axis of the waveguide. Each of these waves propagates in the waveguide, reflecting from its side walls. As is known from acoustics [42], one-dimensional traveling waves may occur in waveguides only at frequencies in excess of the so called cutoff value that depends on physical properties and geometry of the waveguide. For each wave that constitutes a multidimensional waveform, we find such cutoff frequencies from relations derived from a1 a2 n n Eq. (7.11) when all viscosities tend to zero: c/o1 n; n; c/o2 h h b n For a 2-meter wide natural waveguide (h = 1 m) n . c/o3 h 1 1 1 641.75 Hz; c/o2 82.4 Hz; c/o3 119.7 Hz, for a 20-m wide c/o1 natural waveguide (h = 10 m), cutoff frequencies of each of three waves 1 64.2 Hz; that constitute the waveform will be 10 times less, that is c/o1 1 1 8.2 Hz; c/o3 12 Hz. c/o2 Below we give data on damping decrements of these waves for two natural waveguides: 2 meter wide (h = 1 m) and 20 m wide (h = 10 m). Figures 7.9 and 7.10 show dependencies of damping decrements of the first compressional wave (Curve 2) and shear wave (Curve 1) for h = 1 m and h = 10 m, respectively.

Fig. 7.9

Statements and Substantiations of Waveguide Mechanics

125

Fig. 7.10

As we can see, a hallmark of spatially heterogeneous waves (first compressional and shear) is that damping decrement of spatially heterogeneous waves is large at low frequencies. Besides, there’s a bandwidth for which an increase in the frequency reduces the damping decrement. The lowest damping is reached at some finite frequency values that tend to cutoffs under a zero viscosity rather than at a zero frequency as in case of one-dimensional waves. Differences between cutoffs and data on Fig. 7.9 and 7.10 are due to the effect of viscosity. Bandwidths with the lowest attenuation are green color. In this case, the green stripe to the left corresponds to the minimum attenuation of the shear wave, and the one to the right corresponds to the minimum attenuation of the right-hand compressional wave. On the other hand, for the red color bandwidth where the intersection of damping decrement curves of shear and first compressional waves lies, decrements of both waves are close to each other and each wave is characterized by almost the same attenuation that significantly exceeds the minimum. Consequently, in this bandwidth, both waves, the first compressional and the shear wave, damp much faster than the minimally damped waves in green color bandwidths on Fig. 7.9 and 7.10. For a 2 meter wide fault (Fig. 7.9), the lowest damping decrement of the first compressional wave equal to 0.085 m–1 is at 770 Hz (e-fold attenuation occurs at a distance of 11.8 meters from the oscillation generator).

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As concerns the shear wave, the lowest damping decrement equal to 0.136 m–1 in a 2 meter wide waveguide is at 144.5 Hz (e-fold attenuation occurs at a distance of 7.35 meters from the oscillation generator). If we compare data on Fig. 7.9 and Fig. 7.10, we may conclude that an increase in the width of the fault reduces frequencies for which the damping decrement is minimal. Specifically, for a 20 meter wide fault or fractured zone (see Fig. 7.10), the cutoff frequency for the shear wave (Curve 1 on Fig. 7.10) is 14.5 Hz with the damping decrement equal to 0,001366 m–1 (e-fold attenuation occurs at a distance of 732 meters). For the first compressional wave (Curve 2 on Fig. 7.10), the cutoff frequency for the shear wave is 75 Hz with the damping decrement equal to 0,00086 m–1 (e-fold attenuation occurs at a distance of 1162 meters). As it follows from the obtained results, to cause a spatially heterogeneous wave process in a natural waveguide that would cover the widest area, oscillations need to be excited within bandwidths where the damping decrement is minimal for the given waveguide, that is within green color bandwidths on Fig. 7.9 and 7.10. The methodology of determination of these frequencies boils down to a determination of a minimum modulus of the real part in the right member in Eq. (7.11). At the end of this section, we give data on damping decrements of second heterogeneous compressional waves. Curve 1 on Fig. 7.11 demonstrates dependence of the damping decrement on frequency for the second spatially heterogeneous compressional wave of the first form (λ1 = π/2) for a 2 meter wide waveguide, and Curve 2 shows the same dependence for a 20 meter wide waveguide. As is seen from these curves, both waves show fairly significant attenuation. The viscosity affects the second compressional wave much stronger than the first compressional wave and shear wave. Even at a fairly low frequency of 0.02 Hz, the amplitude contracts by a factor of e at a distance of a quarter of a meter for both waveguides. As the frequency grows, the difference between the wide and the narrow waveguides virtually disappears. Curve 1 and Curve 2 on Fig. 7.11 merge into a single curve. The dependence of damping decrement on frequency differs from that of the first compressional wave and shear wave in that it is monotonically increasing at all frequencies. This is due to the fact that viscositydependent terms enter into the expression for the damping decrement in such a manner that even at ω = 0, the damping decrement is a monotonically increasing frequency function without a decreasing interval for parameters under consideration.

Statements and Substantiations of Waveguide Mechanics

127

Fig. 7.11

7.3.4 Statement of a Resonance Waveguide Problem and Its Substantiation for Porous Media. Introduction The waveguide under consideration is a fluid-saturated viscoelastic porous layer. We assume no fluid or matrix displacements normal to the planes of the layer’s flat lateral surfaces nor tangential stresses exist in the layer. At the end surface, we prescribe a time-periodic load in the form of a normal stress of the matrix and fluid pressure. Tangential stresses at the end surface remain equal to zero. We now consider the same layer with the same boundary conditions at lateral boundaries but with fully zero load at the end surface x = 0, and with the absence of any viscous forces in the matrix and the fluid as well as in inter-phase interactions between the matrix and the fluid. Are any undamped oscillations possible in such layer? As is seen from simple calculations, there is an infinite set of such oscillations, and potentials that describe these must satisfy the following relations: 1

An sin

n

h

y e

i k1n x

nt

Cn cos

where k3n

0 2 n 2

b

2n 1

0,

n 2 n 2

h

2

, n

h

2

y e

Bn sin i k3n x

nt

n 1,2,3... , k nj

n

h

y e

i k2n x

nt

,

, (7.12)

2 n 2 j

a

2 n 2

h

, j 1,2;

, and the signs before the roots should be selected so as

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to keep the potentials bounded at x → ∞; n = 0,1, 2, …; for each n, invariables An, Bn and Cn are expressed linearly through one random invariable; ωn are the oscillation frequencies uniquely determined for each n through physical parameters and geometrical parameters of the layer under consideration. Frequencies ωn in Eq. (7.12) are determined uniquely, meaning that each n and each fixed set of physical and geometrical parameters has a single oscillation frequency. These oscillations are the so called natural oscillations that a layer can make in the absence of external normal stresses and excitations of fluid pressure at the end surface. In case of an elastic layer, such type of natural oscillations is best represented by a Rayleigh wave. With account taken of links between displacements of the matrix and potentials, we can show that under natural oscillations of the layer (7.12), at its end surface, displacements of the matrix at frequencies ωn occur with nonzero amplitudes, specifically, that the oscillai t tion amplitude, that is the coefficient of e n , in the expression for matrix displacements reads as follows:

un

sin cos

n

h n

h

y y

ikn1 An ikn2 Bn n

h

An Bn

n

h

Cn ex

ikn3Cn e y .

This means that oscillations at the frequency ωn are possible at the end surface of the free (unaffected) layer. It is similar to natural oscillations of a mass on spring which in the absence of friction and external forces is able to make undamped oscillations at its natural frequency (frequency of natural oscillations). If a time-periodic force is applied to the spring-hung mass at a frequency equal to the natural frequency of the mass on spring, then the amplitude of oscillations begins to increase with time proportional to the time. The above phenomenon is called a resonance. The same effect is observed in case of a naturally oscillating layer. So, if a time-periodic load is applied to the end surface of the layer at the frequency equal to the frequency of natural oscillations ωn, the amplitude of oscillations will be growing, thus, causing the resonance. In the event that the system is affected by friction forces, amplitudes do not grow infinitely but oscillation amplitudes show a considerable increase. We now consider the same layer but with existing viscous forces acting in the matrix and in inter-phase interactions between the matrix and fluid. At the end surface, the layer experiences a loading in the form of periodic oscillations of normal stresses and pressure at the frequency ω.

Statements and Substantiations of Waveguide Mechanics

129

For unique determination of amplitudes of the wave process 1n , n2 and n expanded into expansions (7.10), we need to satisfy conditions (7.8) of the equality between normal stresses and pressure, and external forcing normal and tangential stresses and pressure at the layer’s end surface at x = 0. For this purpose, we expand the function τ0(y) that defines the amplitude of external normal load into a Fourier series y for sin n . The satisfaction of conditions at the layer’s end for each n h number boils down to a solution of a heterogeneous system of linear equations where the determinant depends on the frequency, physical and geometrical parameters of the problem, and where the n-th Fourier coefficient of expansion of the function τ0(y) is involved in the system’s right member. It should be noted that for each of forms, determinants of this systems prove to be distinct from zero at any frequencies if at least one viscosity (of viscoelastic matrix or inter-phase interaction) is nonzero. At a certain frequency for each n number, the determinant reaches a minimum near ω ≈ ωn. If both viscosities (of viscoelastic porous matrix and inter-phase interaction between the matrix and fluid) equal to zero, then each form has such value of frequency n at which the determinant of the system under consideration becomes zero. This mathematical fact shows that at this frequency, the system unaffected by external loading has a non-zero solution (7.12). Consequently, there should be non-zero values of 1n An , n2 Bn and n Cn that determine amplitudes of a natural wave process unaffected by external loading at the frequency ωn. If at least one of the above-mentioned viscosities is distinct from zero, then coefficients 1n , n2 and n have finite maximum values at the frequency ω, close to n . This means that the system experiences a resonance. In what follows, we will call the frequency ω at which coefficients 1n , n2 and n peak a resonant frequency. Resonance values of amplitudes 1n , n2 and n determine the n-th form resonance values of amplitudes of potentials 1 , 2 and at the waveguide’s end surface at x = 0. Specific calculations performed for the first form (n = 1) where

, show that normal displacements of the matrix and normal 2 stresses in the matrix along the transverse coordinate y are proportional y to the sine or, more specifically, to sin (Fig. 7.12), meaning that we 2h get a distribution anti-symmetric relative to the layer’s axis. This is be1

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cause the two-dimensional form requires an anti-symmetric distribution of normal matrix displacements at the end surface which would ensure propagation of an oblique wave of matrix displacements reflecting from the walls within the layer.

Fig. 7.12. 1 – first form, 2 – second form

Figure 7.13 shows dependencies of the first form amplitudes and 1 on the frequency.

1 1

,

2 1

Fig. 7.13

On Fig. 7.13a the half-width of a fault is h = 10 m; on Fig. 7.13b, it is h = 1 m, and Fig. 7.13c shows amplitudes of the second compressional wave (slow wave) for the two values of the half-width of a fault h. As we can see, the dependence is not monotonic. So, a resonance associated with matching external load and natural frequencies does actually occur. We note that for such resonance to take place in reality, the amplitude of external normal stresses needs to be distributed as shown on Fig. 7.12, or at least the first coefficient of Fourier expansion of the normal load in sines should be distinct from zero. This would cause oblique

Statements and Substantiations of Waveguide Mechanics

131

waves reflecting from side walls in their propagation along the layer that are required for the first waveform. Comparing Fig. 7.13a, 7.13b and Fig. 7.13c, we can see that amplitudes of the three waves that constitute the first form greatly vary in value. Amplitudes of the first compressional and shear waves are more than six orders greater than that of the second compressional wave. As concerns the first compressional and shear waves, they are comparable in order. However, the amplitude of shear wave is somewhat higher than the amplitude of the first compressional wave. Consequently, the first compressional and shear waves account for the main share of contribution into the wave field. The resonant frequency for h = 10 and parameters selected here is 11.25 Hz. It is the same for all three waves that constitute the form under consideration. As concerns the dependence of resonant frequency on h, we may conclude that the resonant frequency behaves almost inversely proportional to h. Specifically, for h = 10 m, the resonant frequency is 11.25 Hz, and for h = 1 m, it is 112.5 Hz. Besides, with decrease of h the amplitude of resonance oscillations significantly reduces. As we go further into the waveguide, amplitudes of potentials as well as amplitudes of pressure, displacements of the matrix and fluid, and stresses in the matrix damp with the damping decrement depending on the frequency, as it was shown in the previous section. Whether the distribution of amplitudes vs. frequencies sticks to the profile conducive to resonance in inner points of the waveguide at x > 0 depends on what kind of attenuation occurs at the resonant frequency and whether it is above or below the attenuation at other non-resonant frequencies. In particular, if the resonant frequency is significantly lower than the cutoff for the value form under consideration, then the damping decrement at the resonant frequency will be considerably greater than the damping decrement at frequencies beyond the cutoff, so the resonance-conducive profile of amplitudes inside the waveguide will smooth out to disappear at some distance from the waveguide end. Some other boundary conditions are also considered alongside with the above-described. This includes, for example, a case where the medium is completely stuck to the side walls, meaning that instead of the condition where σxy = 0 at y h we consider the condition where ux = 0 at y h . This causes significant changes in the results. In particular, this brings about significant changes in resonant frequencies. It is clear evidence that changes in boundary conditions involve changes in the nontrivial solution of the free system and changes in forms of natural oscillations.

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7.3.5 Resonances. Waveguide Processes in Porous Media with Heterogeneities. The Distribution of Forces Acting on Pore-Contaminating Solid Particles and Capillary-Trapped Oil Droplets in a Waveguide Below we consider a fault in a reservoir between two compartments. The fault is unbounded along x (Fig. 7.14). At the fault boundary at x = 0, there is a well with an oscillation generator that generates oscillations of pressure in the fluid and normal stresses in the porous medium. The boundaries between the fault and the surrounding compartments are still. There are no displacements of the fluid and the matrix normal to the boundaries (at the selected approximation). That is, the boundary conditions are exactly the same as in Eqs. (7.8)–(7.9). After finding solutions to a boundary problem for Eqs. (7.7) with boundary conditions (7.8)–(7.9), we find the oscillation amplitudes for all quantities involved in the equations and boundary conditions.

Fig. 7.14

The solution for potentials reads as shown in (7.10). Based on the expression (7.10), we may write a solution for any unknown involved in the problem. As an example, we write a solution for displacements of the matrix: u grad 1 grad 2 rot . un e i t ,

u n 0

Statements and Substantiations of Waveguide Mechanics

133

where sin

un

+ cos

0

0,

2n 1 n

n

h

n

h

1 ik1n x n

ikn1

y y

e

n

h

ikn2

1 ik1n x n

2 ik2n x n

h

2 ik2n x n

e

n

n

e

n

e ik3 x e x

ikn3Cn e y ,

e

n 1,2,... , k nj j 1,2,3 are derived from rela-

2 tionships (7.11). Likewise, we may write a solution for amplitudes of fluid displacements, fluid pressures, and for stresses in a viscoelastic porous layer. This makes it possible to determine the amplitude of a force acting on porecontaminating particles and capillary-trapped oil droplets in a flooded reservoir. We already discussed this force f in Section 2.1. In what follows, we provide an analysis of the distribution of this force amplitude over the fault shown on Fig. 7.14. According to data in Section 2.1, a portion of the force f caused by the wave field can be roughly approximated by the following expresu U . We note that the sion: f grad f f u U f pu symbols used in the latter to denote velocities and accelerations of the fluid and medium matrix are introduced in Chapter 7 and differ from the notation agreed in Chapter 2. Equally, it is taken into account here that σf = – p. At the same time, the coefficients p , , f , f , have the same meanings as in Chapter 2. We note that, in what follows, we assume an average particle radius involved in the expression for a as follows: rp 0,1a* for K = 1 mD and rp 0,01a* for K = 100 mD, where a*2

K (1

f

) as per the assumed interfacial friction model [41]. Bear-

ing in mind that all the variables

f

, u , U, u and U contain the time

i t

factor e , the force f acting on particles in the pores may also be written in the same form. Similarly to the expansion of expression for displacements of the matrix, stresses in the matrix as well as displacements of the fluid and pressure into forms, it is possible to expand the expression for the force acting on particles in the pores into the same forms. With reference to (7.12) and omitting the multiplier that precedes each summand e i t , we obtain: fn

ex sin

n

h

y Fxn e y cos

n

h

y Fyn ,

(7.13)

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where, 2

F = n x

ik nj

Q R

j

j 1,2 2 p

a

i dj

2 j

i cj 2

1

f

2 n

F = h n y

Q R j 1,2 2 p

1

j

a

i

i

f

1

j

n j ik j x n

e

, f

h

e

ikn3 x n

2

2

p

i cj i

2 f

n

i dj

2 j 2

f

2 p

f

f

ikn3e

i

f

1

j

n j ik j x n

e

.

ikn3 x n

As we can see, the force acting on particles in the pores consists of two components: the unit vector ex, component directed from the well along the fault axis into the reservoir and the unit vector ey component directed orthogonally to the fault (Fig. 7.14). The ex components of the force f for all forms n =1, 2,… are distributed anti-symmetrically relative to the axis of the fault under consideration, whilst ey components are distributed symmetrically. The expression (7.13) may be used to calculate the probability of successful stimulation as per the methodology set out in Section 2.3. For a qualitative evaluation of the wave forces acting on particles and capillary-trapped oil droplets in the pores, we now consider moduli of functions Fxn and Fyn . To some extent, these are suitable for approximate qualitative evaluation of the possibility to use the obtained wave field to remove solid particles from the pores and to mobilize capillary-trapped oil. Figure 7.15 provides a logarithmic scale view of moduli of functions Fx1 (to the left) and Fy1 (to the right) calculated per (7.14) versus the frequency for various distances x from the oscillation generator that are indicated in each figure for a 10-meter half-width (h) fault with a permeability of 1 mD (dotted curves) and 100 mD. A particle to which forces with amplitudes Fx1 and Fy1 are applied is assumed to be as large as 3∙10–8 m (the superscript 1 is omitted in symbols Fx1 and Fy1 on Fig. 15). As we can see, with x = 0 (Fig. 7.15a and Fig. 7.15b) the amplitude components of oscillations of the force applied to particles in pores have a clearly pronounced resonance peak for both the longitudinal and transverse components at the fault face where the oscillation generator is located.

Statements and Substantiations of Waveguide Mechanics

(а)

(b)

(c)

(d)

(e)

(f) Fig. 7.15

135

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(g)

(h)

(i)

(j)

(k)

(l) Fig. 7.15 (continued)

Statements and Substantiations of Waveguide Mechanics

(m)

(n)

(o)

(p)

(q)

(r) Fig. 7.15 (continued)

137

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(s)

(t) Fig. 7.15 (end)

The longitudinal component of the force at x = 0 increases with frequency. Consequently, in proximity to the generator high-frequency oscillation components provide a strong impact on particles in pores. This proves the possibility of using high-frequency oscillation generators for wave stimulation of pores near the generator. We note that the amplitude of longitudinal oscillations near the generator is significantly greater (by almost 2–3 orders of magnitude) than the amplitude of transverse oscillations. This may have negative impact on the cleanup quality of such pores whose axes are transverse to the fault. As the distance from the generator located at the fault face grows, the amplitude of the force decreases due to attenuation whilst amplitudes of longitudinal and transverse oscillations equalize. Specifically, at a distance of 1 meter from the generator (Fig. 7.15c and Fig.7.15d), a resonance peak of the amplitude of longitudinal oscillations of the force acting on particles in pores of the medium with a permeability of 100 mD experiences a decrease of almost three orders, and in pores of the medium with a permeability of 1 mD it decreases by almost four orders of magnitude. The resonance peak of the transverse oscillation amplitude decreases by less than one order of magnitude. As concerns the high-frequency bandwidth, due to the intensive attenuation at frequencies above 10 kHz the oscillation amplitudes of the force longitudinal component peak near 2 kHz (Fig. 7.15c). Consequently, for stimulation of the fault near-wellbore zone within 1 meter from the well, we may recommend using frequencies from two bandwidths: nearresonant frequencies (11 Hz) and the high-frequency bandwidth between 1 and 3 kHz. We note, however, that only the high-frequency peak is of practical interest as the low-frequency resonance bandwidth is very

Statements and Substantiations of Waveguide Mechanics

139

narrow and needs special fine tuning which may prove to be impossible sometimes. A peak begins to form within the low-frequency bandwidth which is associated with the lowest attenuation band of the shear wave shown on Fig. 7.10 as a left-hand green strip. However, within 10–30 Hz and 100– 500 Hz bandwidths associated with the lowest attenuation bands of compressional and shear waves, respectively, oscillation amplitudes of both longitudinal and transverse components of the force acting on particles in pores remain fairly high, so these bandwidths may be recommended as candidates for wave stimulations in the case under consideration. Amplitudes dip between the above bandwidths which is associated with the high attenuation band for both waves (shear and first compressional) shown on Fig. 7.10 as a red color strip. At further distances from the well along the fault, the oscillation amplitudes of components of the force acting on particles in pores damp even more. However, even at a distance of 1 km (Fig. 7.15s and Fig. 7.15t), the oscillation amplitudes of the force components remain fairly high within 10–30 Hz and 70–130 Hz bandwidths and exceed 108 N/m3 for the medium with a permeability of 1000 mD and 106 N/m3 for the medium with a permeability of 1 mD. Calculations for faults of other widths have shown qualitative similarities (a rapidly disappearing low-frequency resonance with increasing distance from the generator, and two peaks occurring within the lowest attenuation bandwidths). ‘Peak’ frequencies grow as fault width decreases. At the same time, the oscillation amplitudes of the force acting on particles and droplets in pores decrease slightly. The surface of the generator is dominated by high-frequency components of the force. As the distance from the generator grows, the peak amplitudes of the force are primarily observed within two bandwidths. As concerns the high-frequency peak, it persists even at a significant distance from the generator. Its behavior follows two trends. It decreases in value and verges towards the lowest attenuation band of the first compressional wave. At the same time, a peak remains which is associated with the lowest attenuation band of the shear wave. Thus, even at a distance of 1 km from the oscillation generator, both peaks are clearly visible. These bandwidths may be recommended as candidates for wave stimulations of the waveguide under consideration at large distances from the oscillation generator. Figure 7.9 shows actual damping decrements for the first compressional and shear waves and Figure 7.11 shows actual damping decre-

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ments for the second compressional wave. In both cases, the waveguide is 2 m wide (h = 1 m) and viscosity effects are taken into account. The second compressional wave damps much more than the two other waves, hence, similarly to the case of h = 10 meters, its contribution into amplitudes of oscillations of the force acting on particles in pores is negligibly small. The lowest attenuation of the two remaining waves is observed between 130 and 150 Hz for the shear wave and between 700 and 800 Hz for the first compressional wave. These bandwidths may be recommended as candidates for wave stimulations of the waveguide under consideration. However, we need to ensure the conditions so that the maximum possible portion of the oscillation energy would fall on the form under consideration. For this purpose, the amplitudes of oscillations of normal stresses at the fault’s end with x = 0 should be distributed in such a manner as to ensure that they are anti-symmetrical relative to the fault axis (similarly to Curve 1 on Fig. 7.12). We note that all data provided on Fig. 7.15 are obtained for the case where the amplitude of normal stresses applied to the waveguide’s end for the form under consideration is only 1 Pa. Existing oscillation generators are capable of generating much higher amplitudes of the external load. Specifically, studies of authors [8, 10, 11] provide data from tests carried out on Shell test facilities where it was established that a vortex generator is capable of generating a pressure amplitude up to 15 bar, that is 15∙105 Pa. This means that amplitudes of the force achievable in reality may exceed data from Fig. 7.15 by over 6 orders of magnitude. To evaluate the oscillation amplitude of components of the force acting on particles in pores, we refer to the study by G. G. Vakhitov [18] that provides data on the additional pressure gradient obtainable in the reservoir by injection of water. According to G. G. Vakhitov, the above value is 0.3–0.4 bar/m. A simple calculation shows that such pressure gradient corresponds to the force acting on the unit of a particle volume equal to (1.5–2)∙107 N/m3. Consequently, even at a distance of 1 km from the ‘exciting’ well, particles in pores of a 20 meter wide fault in the medium with a permeability of 100 mD (see Fig. 7.15g–t) will experience the action of a periodic force whose amplitude is two orders greater than the force driven by injection of water into the reservoir if the excitation frequency from the bandwidths recommended above for wave stimulations and the amplitude of pressure oscillations generated by the generator is only 1 Pa. For a real life vortex generator, the amplitude of this force will exceed the conventionally generated amplitude by

Statements and Substantiations of Waveguide Mechanics

141

9 orders of magnitude. For a medium with a permeability of 1 mD, the amplitude of the force will exceed the amplitude conventionally generated by injection of water into the reservoir by over four orders of magnitude. Using data similar to those shown on Fig. 7.15, we can determine the amplitude which a generator should generate to ensure a desired level of the force at the prescribed distance in the waveguide. Thus, for example, in case of a 2 meter wide waveguide with a permeability of 100 mD, to generate the force acting along the waveguide length of 500 m that is three orders greater than the conventional effect from injection of water, we need to select the generator capable of generating 800 Hz oscillations with an amplitude in excess of only 0.01 bar. For the sake of simplicity, we set out the results that relate to the cases where the thickness of the reservoir is significantly greater than the fault’s width and oscillations are excited over the entire reservoir thickness. In practice, it is preferable to consider a fault that is bounded not only by its width but also by the reservoir thickness. The waves propagating in such a fault of rectangular cross-section will be of three dimensions. Additional resonant frequencies will appear. Besides, we can consider a problem of a distance from the reservoir top or base at which stimulation will be the most efficient. All the above aspects are taken into account in the software currently developed by the authors.

7.4 Linked Waveguides in Compartmentalized Reservoirs. The Transfer of Oscillations into Reservoir Inner Zones under Multidimensional Resonance Conditions Normally, reservoir-compartmentalizing faults are connected with each other. They form networks deeply penetrating into the reservoir (see Fig. 7.4). In order to show that wave field energy can be transferred into various reservoir zones through a network of such linked waveguides (local faults), we consider a subpopulation of possible wave types, i.e. onedimensional compressional waves. Hence, we have a network of onedimensional sections of the medium (lines) in which compressional waves propagate.

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As an example, we consider a simple network that is T-shaped in plan view. Wells are put at each of three ends of the network: one exciting and the two others are production and injection wells. Waves are expected to be excited for a wave stimulation of local faults. For this purpose, we calculate the force needed for particle or droplet detachment from the pore walls at various stimulation frequencies. From this, we state a condition for the excitation amplitude that would ensure a wave stimulation of any given compartmentalizing fault section. We find the frequency at which the above amplitude is minimal. On the basis of these calculations, we select a generator required for wave stimulations.

7.4.1 The Statement of the Problem of Forced One-Dimensional Oscillations in Linked Sections of a Multi-Phase Medium under Resonance Conditions We consider a system of three segments that constitute a T-shape configuration with lengths L1 , L2 and L3 . In each of these segments, motion of the elastic porous medium takes place which is described by the system of equations (7.10). We can consider a system of linked waveguides with a multidimensional motion in each. In what follows, for simplicity, we concern ourselves only with consideration of one-dimensional waves. We assume that all potentials are functions of only the longitudinal coordinate x, and only displacements of the porous medium ux and fluid Ux are distinct from zero. With this, Eqs. (7.10) read as follows: 2 2

a 2j

j

a

x R

1

2 j

11

2

i

j

2

j

dj

x2 j)

M (1

; dj j

12

cj

j

11

12

,

j

j 1,2,

(7.14) 2

; cj

j

11

. j

12

We can write relations with physical quantities for a one-dimensional case as follows: 1

u

2

x

x

1

, U

1

2 f

Q

x

2

2 2

x

x

,

2 1

R

2

1 1

x2

2

2 11

2G

s

x2

2 22

x2

,

2

1

R

2

1 1

x2

2 1

2

R

(7.15)

2

x2

1 1

2

2

.

2

2

,

Statements and Substantiations of Waveguide Mechanics

143

where u and U are the longitudinal displacements of the porous medium and fluid, respectively. We also need to prescribe boundary conditions at section ends and conditions for the connection of sections at the point of their contact. For each of segments shown on Fig. 7.16, we introduce an individual system of coordinates: Ох1, Ох2 and О3х3, respectively. The point O where all sections cross is the origin of coordinates for the first and the second section (on Fig. 7.16 their lengths are denoted L1 and L2, respectively), and the point О3 is the origin for the third section. We need to replace x, φ1, φ2, u, U, σs and σf in equations (7.14) for each of the sections by xi, φ1i, φ2i, ui, Ui, σsi and σfi,, where i is the number of the section. Equally, index i needs to be assigned to all coefficients involved in equations (7.14) and (7.15) as it may be that physical properties of rocks in each section are different.

Fig. 7.16

We put wells at free ends of sections x1 = L1, x2 = L2 and x3 = 0. At its free end with x3 = 0, the third segment houses an oscillation generator. Hence, the problem’s boundary conditions may read as follows: at segments’ ends u2 0, x1= L1, x2= L2: f 1 f2 f p0 , u1 x3= 0:

s3

s

p0

A exp(i t ) ,

f3

f

p0

A exp(i t ) .

At the central point x1= 0, x2 = 0, x3 = L3: s1 s2 s3 ; s1

f1 s3

f1 f3

f1

f1

U12

f1 f3

U1 u1

U 32

f3

s1

2 s1 1

u

s3

s2

f2

U 22

f2

f2

s2

2 s2 2

u

2 s3 3

u ,

U1 U 2 U 3 u2 f 2 f 2 U2

0, f3

f3

U 3 u3 ,

(7.16)

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where A is the amplitude of pressures oscillations in the fluid at the external end of the third segment. We now concern ourselves only with finding a time-periodic steadystate specific solution of a boundary problem (7.14), (7.16) with a period of

2

. We note that in this case we consider motion in a homogeneous

medium at each individual section. Therefore, all coefficients involved in equations and boundary conditions (7.14) and (7.16) are time and spatial coordinate independent constants.

7.4.2 The Results of Mathematical Simulation Chapters 1–5 describe expressions for the forces to be acting on porecontaminating solids and capillary-trapped oil droplets that are needed to break the bond between the particles or droplets and the pore walls in the matrix. These forces are manifested at the microlevel, i.e. inside each of pores. The physical forces to be applied to particles may include powerful (high amplitude) periodic pulsations in the fluid pressure drop in pores and inertia forces driven by oscillations of the pore walls in the matrix onto which particles obstructing pore channels are stuck. The abovedescribed effects may also be conducive to the detachment of capillarytrapped oil droplets from the pore walls. An increase in the amplitude of the force acting on droplets and particles in pores may be possible not only by an increase in the amplitude of external excitation but also by introducing resonance conditions to the medium. Below we set out the results of wave force calculations under resonance conditions within the section which is schematically shown on Fig. 7.16 in accordance with the above-described mathematical model. Section lengths: L1 = 727 m, L2 = 263 m and L3 = 103 m. Figures 7.17–7.19 illustrate our calculations by showing wave force amplitude 3D distribution graphs for three sections at various resonant frequencies. For the case under consideration shown schematically on Fig. 7.16, the mathematical modeling makes it possible to find, firstly, the groups of resonant frequencies and, secondly, the distribution of the amplitude of the force F acting on micro-inclusions inside pores for each of our sections of interest. As concerns resonant frequencies, their values may be verified experimentally during field tests prior to wave stimulations using one of the approaches set out below in Section 7.5. Further, with given physical properties of the rock, pore clogging particles and the fluid, and

Statements and Substantiations of Waveguide Mechanics

145

using geological data on the structure of compartments, we check lengths of sections Li, to ensure that computed frequencies match experimental frequencies. Following this, we calculate detachment forces at each section and determine the distribution of the F amplitude. Then, using the results in chapters 1–5, we determine the requisite quantity of A providing us with the information needed for the selection of a generator that would most likely be capable of cleaning the target reservoir section. Figures 7.17–7.19 below show the results of calculations from the above data. The Y-axis plots a value of parameter F. The X-axis is a dimensionless coordinate along the corresponding section of the medium. The analysis of results shows that, amongst resonant frequencies, you may find such at which the wave field and the detachment forces will be maximal within one of pre-selected sections. For example, at the frequency ω = 0.745 Hz, the maximum level of the force and detachment F parameter are observed in the third and the first sections; at the frequency ω = 1.715 Hz, the maximum level of the force is observed in the second and the third sections; and, finally, at ω = 11.994 Hz, the maximum level is observed in some zones of the second section, while in some zones of the third section, the detachment force level is lower than in the first and the second. Consequently, the selection of excitation frequencies from the resonant frequency group can help excite maximum oscillations in any desired section. Besides, in some cases, the excitation of oscillations may be carried out in a section that does not have a common boundary with the section in which the maximum amplitude of the detachment force can be excited. The proposed method makes it possible to extend wave stimulations to a system of wells by exciting oscillations in one or several wells.

7.5 Experimental Determination of Resonant Frequencies of a Reservoir. Practical Recommendations for Selecting Controlled Means and Oscillation/Wave Generators The structure of a reservoir and its heterogeneities may prove to be very diverse in actual field conditions. For this reason, the problem of the practical determination of the resonant properties of a reservoir is critical. In a specific reservoir, resonance mechanisms may include a variety

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Fig. 7.17. ω = 0.745 Hz

Fig. 7.18. ω = 1.715 Hz

Fig. 7.19. ω = 11.9 94 Hz

Statements and Substantiations of Waveguide Mechanics

147

of patterns, for example, resonances associated with a system of fractures or a separate field compartment. Without some experimental data, the theoretical analysis of all such mechanisms is not possible. In some cases, the theoretical results are of qualitative importance. They are required to substantiate set-up of the full-scale experiments. Besides, we should also take into account possible multimode oscillation conditions and nonlinear interactions between wave processes of various natures that may take place. We should also note that normally it is extremely difficult to obtain accurate data for all parameters involved in our mathematical models under consideration. This makes the theoretical analysis significantly difficult and reduces the reliability of calculations performed using simple models. All these necessitate the development of a fairly simple method to verify the theoretical results of resonant frequencies in a producing reservoir against experimental data. At present, the existing methodologies of a search and identification of resonance conditions for oscillations excited in reservoirs are largely underdeveloped and do not extend as far as commercial applications; many technological problems associated with it are equally unsolved. Since the use of resonance conditions is instrumental in successful wave stimulations, this issue should merit special attention in the future. Below we discuss possible ways to solve this problem. First of all, we need to determine using geological field data what resonance mechanism can be or should be employed in our well or field stimulations. At this phase, we need to review the history of well performance, estimate residual oil in place and roughly locate its positions. Following this, we have to identify any existing geological heterogeneities, faults and fractures in the reservoir, and select the target zone to be stimulated. At the following stage, we need to conduct a theoretical analysis using the corresponding model and determine the possible bandwidth of resonant frequencies. For correct calculations, we also need to take into account the reservoir’s physical properties and geometry such as porosity, permeability, elasticity and viscosity moduli, thickness, properties of hydrocarbons contained in the reservoir and some other parameters. Depending on the obtained resonant frequency bandwidth, we should select an appropriate design of oscillation/wave controlled means and generators. Based on the analysis, various types of oscillation/wave controlled means and generators with the corresponding measuring system may be recommended. These may include hydrodynamic, including widebandwidth vortex generators, shock-wave generators etc.

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These units employ different designs with differently operating wave stimulation processes and differently propagating waves. Nevertheless, common methods of the test signal sensing may be recommended to verify resonant frequencies. One of such methods can be described as follows. An adjustable oscillation frequency wave generator is installed in the ‘exciting’ well. Its bandwidth should match that theoretically derived. Receivers are installed in the studied wells at perforations. To determine possible resonances, we need to obtain amplitude-frequency response characteristics of the well or reservoir. To this end, we scan the generator frequency over the entire bandwidth while recording the signal from receivers. The resonant frequencies will correspond to points where the signal strength peaks in the amplitude-frequency response characteristics. An alternative method may involve the use of a broad band generator and a spectral analyzer. The emission spectrum of the generator should be determined in advance and should fully cover the studied frequency bandwidth. An acoustic signal recordable using a sound level meter is transmitted to the spectral analyzer. It is convenient if the emission strength of the generator is the same for all spectral components (white noise). In the case of a white noise generator, a spectrum of oscillations at the analyzer will correspond to the required well amplitude-frequency response. In case of a more complex spectrum, the amplitude-frequency response can be easily established from data obtained using a spectrum analyzer. The type of such sensors depends on the output frequencies of the generator. For example, the best candidates for the acoustic bandwidth are downhole sound level meters [43, 44]. For frequencies between 100 Hz and 10 kHz, the best candidates are represented by piezo pressure sensors. For shock waves and for oscillations of several kilohertz, piezoelectric accelerometers can be used. Seismic detectors may also be recommended for use with shock waves. We propose another possible method for verification of resonant frequencies. It is sonic logging. A sonic detector is run into the hole to a perforated interval depth where it then records noises from the bottom of the hole. These noises will largely have a random cause and will be characterized by various strengths and spectral compositions. However, if the noise spectrum contains any frequencies that match the natural resonant frequencies, such frequencies may be identified by their amplitude and persistence. After statistical analysis of the experimental data, we will be able to identify the required resonant frequencies. Development of this technology for commercial use should focus on mathematical methods used to process recorded signals.

Statements and Substantiations of Waveguide Mechanics

149

In case of simultaneous stimulation of a group of wells, more sophisticated state-of-the-art systems for control and measurement of wave processes should be developed alongside with associated software products. There are various software products now available from NC NVMT RAN. They are being continuously improved on the basis of the developed theory.

Enhanced Oil Recovery: Resonance Macro- and Micro-Mechanics of Petroleum Reservoirs. O. R. Ganiev, R. F. Ganiev and L. E. Ukrainsky. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

8 The Resonant and Waveguide Characteristics of a Well The Resonant and Waveguide Characteristics of a Well. Using Well Geometry to Select Resonant Frequencies and Wave Amplitudes, Generator Locations and Characteristics for Successful Stimulation of Near-Wellbore Formation Zones and Treatment of Horizontal Wells

In the previous chapter, we showed that oil reservoirs had various mechanisms available to excite waves at the macro level. Traveling through the micropores in the reservoir, the waves can have a significant impact on them which may be used in commercial oil production. In the first six chapters, we discussed the forces acting on solid particles and oil/water droplets in the reservoir micropores. We established the criteria for cleaning solid particles from the pores, detachment of capillary-trapped oil droplets from the pore walls, excitation of perturbations on the oilwater interface for oil droplets that fill cavities in the pore walls. Equally, we established algorithms to calculate the success criteria for wave stimulations in the reservoirs. A software product in which we can implement all these results is currently available. As shown by calculations, waves of maximally possible amplitudes and spatial (three-dimensional) waves with the wavefront changing its position over time at each point in the space give the best stimulation results. For practical applications, it is proposed to use the resonant properties inherent in the nature of reservoirs. This primarily concerns those compartmentalized reservoirs divided into compartments by horizontal and vertical faults. Each of those faults represents a natural waveguide. As shown in two routine problems of Chapter 7 (a horizontal axially symmetrical layer between two flat boundaries and a vertical layer of a rectangular cross-section), a wavefield excited in such natural environment is significantly dependent on fre151

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quency. There are some frequencies that show a multifold increase in oscillation amplitudes as compared to other frequencies. These resonant frequencies depend not only on the physical characteristics of the reservoir but also on its configuration. It does not involve, as is conventionally assumed, a simple reflection of a one-dimensional wave from a heterogeneity. Rather, multidimensional waves (two and three-dimensional) are involved that are capable of ensuring a significant growth in the wave amplitude even in infinite domains of the reservoir at certain frequencies dependent on the reservoir configuration, geometry and physical characteristic, provided the theoretically derived criteria are satisfied. The nature of such resonance is associated with the matching frequency of the exciting force and the natural frequency of a multidimensional wave process that can take place in a reservoir without any external excitation. Depending on the distribution of wave amplitudes in a free wave process, a resonance-generating forced wave process may ensure the deepest penetration into the reservoir from the oscillation generator which cannot be achieved by conventional one-dimensional waves. A number of methods and algorithms are available now for calculation of resonant frequencies in reservoirs of various configurations and the development of some methods and algorithms is ongoing. A universal software product is being developed for the most typical configurations with a corresponding controlled wave equipment and measuring systems. We note that alongside natural waveguides other types such as artificially created waveguides are also available. These artificially created features include wells, both vertical and horizontal. Similar to those described in Chapter 7 for natural waveguides filled with a fluidsaturated porous medium, in the case of a cylindrical oil/water-filled well, there are some frequencies dependent on geometrical and physical well parameters that may be used to excite oscillations so that these oscillations would be capable of ensuring the lowest attenuation and maximum amplitudes of wave processes and could transfer waves to the desired location. Relevant studies are currently ongoing to cover various well types and entire fields from real life. A software product is being developed to enable calculations of wavefields in waveguide networks of both natural and artificial origin. The main focus should be on the resonance forms of oscillations that ensure the maximum wave amplitude in the target zone of the field, often, quite far away from the oscillation generator. The obtained results significantly extend the applicability of such multiple resonances to stimulating oil production and enhancing oil recovery. Various set-ups are possible for development of these depending

The Resonant and Waveguide Characteristics of a Well 153

on practical needs, especially in case of horizontal wells and laterals, for a cleanup to increase oil production and enhance oil recovery from reservoirs.

8.1 Selecting Wave Parameters for Stimulation of Horizontal Wells 8.1.1 Scientific Fundamentals Horizontal wells are horizontal channels in productive reservoirs that may be surrounded by penetrable metal casings or may have no casing at all (Fig. 8.1). Lengths of horizontal wells are much bigger than their diameters. In the course of oil production horizontal wells become contaminated. Wave generators can be used to clean them. Cleaning of wellbore sand screens was discussed in Chapter 1. We consider here a horizontal production well in which a wave generator is to be run on a coiled tubing string. The permeability of wellbore walls is decreasing. We consider here the problem of choosing wave parameters for the best wellbore zone cleaning.

Fig. 8.1

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We assume that a horizontal well is a cylinder that is unbounded from one side. At a first approximation, we can ignore the fluid flow component directed through the walls of the cylinder for the sake of fluid velocities inside it. Therefore, we model the zone in question as a semi-infinite impermeable tube of a radius R with absolutely rigid impermeable walls; the tube is filled with an ideally compressible barotropic fluid. We assume that time-periodic pressure disturbance occurs at the left end of the tube. Let us consider the oscillations of fluid that take place. We shall investigate the small perturbations in the state of rest that correspond to the acoustical approximation [42]. The linearized equation system for the perturbation of variables is reduced to the Helmholtz equation [42] for pressure: 2 p p 1 2 1 1 2p (8.1) 0, p r 2 2 2 2 r r r C t r z2 where r, φ, z are respectively the radial, tangential and axial coordinates of the cylindrical system of coordinates with the origin at the tube end. The boundary conditions corresponding to the above formulation can be written as follows. At the tube wall when r = R: Vr = 0. (8.2) At the tube end when z = 0: p

ei t .

p0 r ,

(8.3)

Let us find a time-periodic bounded solution of Eq. (8.1) with the i

t kz m

. It boundary conditions (8.2), (8.3) in the form: p F r e follows from (8.1) and (8.2) that the solution of the problem has the following form: Pmn

p

n m

r,

e

i

t kz

.

(8.4)

m 0,n 1

r im e , mn is the n-th root of the equation R J m 0, the amplitude Pmn is a coefficient of expansion of the function p0 , appearing in the boundary condition at the well’s end sur-

where

n m

r,

Jm

n m

face (8.3), in a Fourier-Bessel series using the form ber k satisfies the following dispersion ratio:

k

2

n m

C2

R2

2

n m

, the wave num-

2

.

(8.5)

The Resonant and Waveguide Characteristics of a Well 155

A fact well known in acoustics is that spatial traveling waves may be realized in a waveguide filled with acoustic medium not for all excitation frequencies but for some of them satisfying several criteria [42] follows from (8.5). Specifically, the form nm may be realized only for the frequencies that satisfy the following condition: n C . (8.6) m R Following the terminology accepted in acoustics, we refer to the rightmember quantities (8.6) as frequency cutoffs for the form (m, n). Once the excitation frequency exceeds the corresponding cutoff value, the wave number k, according to Eq. (8.5), becomes real. This means that a travelling wave with the form nm can occur in a waveguide. Figure 8.2 shows red rectangles of the frequency cutoffs in kilohertz versus circumferential wave number m for the sound velocity С = 1500 m/s (distilled water) and R = 0.05 m (a typical radius of a horizontal well equal to 5 cm).

Fig. 8.2

For each of m, the lower cutoffs correspond to the lower n of the radial form. Figure 8.2 demonstrates the first two frequency cutoffs (n = 1, 2) for the axially symmetrical case (m = 0). The lowest, zero, value corre1 0

i

*

t

z C

sponds to a one-dimensional compressional wave P e independent of r and φ and realizable at any frequency provided that the coeffi-

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cient P01 0 . This cutoff represents 01 0 , i.e. the first (least) root of the equation J 0 0 and the form 10 1 . The second frequency cutoff value for the axially symmetrical case is 18.39 kHz. This cutoff represents 2 3,85 , i.e. the second root of the equation J 0 0 and the form 0 r . Figure 8.3 shows the distribution of the pressure ampliR tude over a cross-section of the horizontal hole that houses the generator. A traveling wave with the same form may be realized only if the excitation frequency ω exceeds the above-established cutoff value of 18.39 kHz. As is seen from Fig. 8.3, the largest portion of pressure oscillations takes place in the central part of the pipe. The pressure oscillation amplitude on walls is significantly lower than on axis. The observed distribution makes the application of this form for the cleanup not quite suitable. 2 0

J0

2 0

Fig. 8.3

r i e represented by the R quantities m =1, n =1 and 11 1,84 , the cutoff value equals 8.79 kHz. Consequently, for the frequencies in excess of such cutoff, for example, for the value ω* shown on Fig. 8.2, in a horizontal well, a traveling well

For the axially asymmetric form

1 1 1

PJ

1 1

r i e e R

i

*

*

t

C

2

1 1

R

1 1

J1

1 1

2

z

may

propagate

dimensional compressional traveling wave P01e

i

*

t

alongside z C

a

one-

. No other forms of

The Resonant and Waveguide Characteristics of a Well 157

traveling waves are possible in the horizontal well under consideration at this frequency. The remaining terms of the series (8.4), at ω = ω* shown as a horizontal orange straight line on Fig. 8.2, are the in-phase oscillations exponentially damped along z (Fig. 8.1). Figure 8.4 shows the distribution of pressure oscillation amplitudes in r i a wave of an axially asymmetric form 11 J1 11 e over a front-end R cross section.

Fig. 8.4

As we can see, for this form, the amplitude of pressure oscillations peaks at r = R on the pipe wall which ensures the advantage of this form for the cleanup of hole walls from contaminants. An increase in the frequency ω* represented by a shift of the horizontal orange line upwards on Fig. 8.2 may cause some other traveling waves in the horizontal well model under consideration. C Thus, if ω* exceeds the quantity 21 shown on Fig. 8.2 by a red recR tangle that lies along the vertical straight line m=2 and that is marked by n=1, then with such ω* it may be possible to excite a traveling wave that i

*

*

t

2

1 2

2

z

C R r 2i may be written as follows: P J e e . R In some cases, where the excitation frequency ω* takes on values equal to cutoffs, that is when any of the red rectangles on Fig. 8.2 that determine cutoff frequencies come to be on the horizontal orange line, it causes a form of oscillation that is invariant along z in the pipe (horizontal well) (see Fig. 8.1). The same radial and circumferential (for asymmet-

1 2 2

1 2

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rical forms) oscillations defined by the boundary condition at z = 0 with the amplitudes equal to the oscillation amplitudes through the front-end hole cross section and corresponding to the form under consideration take place through any cross section over the hole. It should be noted that all the described results are derived based on the ideal model of the acoustic medium. The results ignore attenuations that always take place in real life fluids. A scattering of waves on fluidcontaminating solid particles is equally ignored. These factors must be taken into account in real calculations.

8.1.2 Practical Recommendations on Stimulation of Horizontal Wells Based on the data set out in chapters 1–2 and 4–5 above, the efficiency of wave stimulations of horizontal wells depends largely on the amplitudes of pressure pulsations, pressure gradients and velocities in the porous medium. Depending on waveforms realized in the well, the amplitudes of pressure waves may peak in various well zones. The axially symmetrical pressure forms show the wave amplitude maximum along the flow axis while axially asymmetrical forms show this on the hole wall. We note that according to the problem statement the hole walls are assumed absolutely rigid and any motions of the hole-surrounding medium are ignored. However, assuming that the distribution of pressure over the hole wall is the same as over an absolutely rigid wall, we can determine the distribution of pressures and velocities in the hole-surrounding medium. Assuming that a growth of the pressure amplitudes on the hole wall causes an increase in the pressure amplitudes of the medium, we conclude that the most efficient is to excite axially asymmetric forms. For a practical application of this model, we need to select the generator such that its spectrum would contain a frequency exceeding the cutoff for the first axially asymmetrical form. Specifically, for the well under consideration filled with water, the excitation frequency should exceed 8.79 kHz. Besides, the pressure distribution upstream of the well should be of axially asymmetrical pattern, meaning that the boundary condition expanded at z = 0 should only contain axially asymmetrical components and no axially symmetrical components should be present. Considering the m-th order Bessel function properties [45], we may note that the greatest value from points where J m 0, for any m, is at the point of the first maximum of J1 function. Therefore, the best conditions for wave stimulation of horizontal wells are when the distribution of pressure over the hole’s front-end

The Resonant and Waveguide Characteristics of a Well 159

r i e . Besides, the expanR sion component P11 is distinct from zero and equal to A, and the remaining expansion components return to zero.

surface satisfies the following ratio: AJ1

1 1

8.2 Near-Wellbore Stimulation. The Induction of Resonance Let us now consider oscillations in the wellbore section located below the generator. These oscillations can be used for wave stimulation of the near-wellbore formation zone.

8.2.1 Resonances in the Wellbore Section between the Oscillation Generator and the Bottom. Using Waves to Transfer Wave Energy We consider oscillations of a compressible ideal barotropic fluid within the confined wellbore space between the oscillation generator and the bottom (Fig. 8.5). At a first approximation, we may ignore the fluid flow component directed through circular-shaped wall perforations for the sake of fluid velocities inside the well. We model the domain under consideration as a rigid tight pipe of a radius R and length L. At the upper end of the pipe, we prescribe time-periodic pressure perturbations to be excited by the generator while the lower end of the pipe is closed and models the bottom of the well. We introduce the cylindrical coordinate system Orφz where the axis matches that of the wellbore (Fig. 8.5) The linearized continuity and flow equations of a compressible barotropic fluid boil down to a reduced wave (Helmholtz) equation (8.1). The boundary conditions in a cylindrical system on a circular-shaped surface of the hole under condition that the fluid flow through perforations is ignored for the sake of oscillations inside the hole may be written as (8.2). The boundary conditions at the generator assembly surface with z = L may be written as (8.3). Finally, the boundary conditions at the well bottom with z = 0 are as follows: Vz = 0.

(8.7)

We seek to find a solution, time-periodic at the frequency ω, of a boundary problem (8.1)–(8.3), (8.7) as a sum of the outgoing and

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t kz m

arriving traveling waves p F1 r e lem’s solution may be written as follows Pmn

p

n m

r,

m 0,n 1

ei

F2 r e t

e e

ikz ikL

i

t kz m

e ikz , e ikL

. This prob-

(8.8)

r im e , mn is the n-th root of the equation R J m 0 , the amplitude Pmn is a coefficient of expansion of the function p0 , appearing in the boundary condition at the well’s end sur-

where

n m

r,

Jm

n m

face (8.3), in a Fourier-Bessel series using the form ber k satisfies the dispersion ratio (8.5).

n m

, the wave num-

Fig. 8.5

Depending on the ratio between the parameters , mn , R, C , summands of the series (8.8) may be fundamentally different. Below we detail a list of possible variants. 1. If the condition (8.6) is satisfied, the corresponding summands are standing time-harmonic waves for which resonant frequencies are available.

The Resonant and Waveguide Characteristics of a Well 161

1.1. We note that the group of such summands at any ω ≠ 0, С and R includes a series term for which m = 0, n = 1. For this member 01 0, 1 1. It represents a one-dimensional standing wave for which, ac0 cording to ratio (8.5), k

k01

C

. Hence, this summand of the sum (8.8), cos

accurate to a multiplier e

i t

1 0

, reads as follows: P

C

cos

z . Obviously, if L

C the following condition is satisfied, the system experiences a resonance due to the summand under consideration:

C 2l 1

, l = 0, 1, 2… (8.9) 2L With С = 1500 m/s and L = 10 m, the resonant frequencies corresponding to our summand are as follows: 37.5 Hz, 112.5 Hz, 187.5 Hz, 262.5 Hz etc. In this case, one-dimensional compressional waves reflect from the well bottom, thus causing standing resonant waves between the well bottom and the generator. 1.2. If the frequency ω is such that the condition (8.6) is satisfied not only for 01 0 but also for some other roots of the equation J m 0, the summands of the sum (8.8) will have some more of those representing standing waves. Assuming that, for instance, the condition (8.6) is satisfied for the root mn (n + m > 1), then the sum (8.8) will include a summand that may written, accurate to a multiplier e i t , as follows: n

2

2 r im cos kmn z m n k . The resonant P J e where , m n 2 R C R2 cos km L frequencies corresponding to our summand of the sum (8.8) are derived from the following relations: n m m

n m

C

2l 1 2L

2

n m

R2

2

, l = 0, 1, 2 …

(8.10)

For the first axially symmetric form distinct from one-dimensional compressional wave represented by 02 3,84 and for С = 1,500 m/s, L = 10 m and R = 0.075 m, the resonant frequencies for l varying from 0

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to 10 near 12.2 kHz. An increase in l causes insignificant growth in the resonant frequencies. The reason being that the l-independent second summand under the root in the expression for resonant frequencies (8.10) is significantly greater than the first summand within the above range of l. For the first axially asymmetric form represented by 11 1,84 and for С = 1,500 m/s, L = 10 m and R = 0.075 m, the resonant frequencies for l ranging from 0 to 10 vary from 5.857 kHz to 5.91 kHz. The analysis of numerical values of resonant frequencies for spatially heterogeneous forms shows that the lowest values are reached in case of r i cos k11z i t e e . Its hallmark is an axially asymmetric form P11 J1 11 R cos k11L that the pressure amplitudes peak on the pipe wall at r = R. 2. Unless the condition (8.6) is satisfied, we have no resonance. 2.1 For such summands of the sum (8.8) for which from Eq. (8.5) we n C , the distribution of the pressure amplitude obtain an equality m R is independent of the longitudinal coordinate z. It is the same in each of planes z = const and overlaps with the distribution over a cross-section of the interval housing the generator. Oscillations of the velocity components Vr and Vφ take place in each of planes z = const. n C 2.2 The summands of the sum (8.8) for which the condition m R n ch k z r m e im e i t , where is satisfied may be written as follows: Pmn J m mn R chkmn L n m

k

n m

2 2

. They are represented by such forms of pressure R2 C2 oscillations for which the amplitude of oscillations damps with growth of the distance to the generator.

8.2.2 Practical Recommendations for Stimulation of the Near-Wellbore Formation Zone The obtained results allow us to devise some practical recommendations for stimulation of the near-wellbore formation zone. Firstly, similarly to horizontal wells, stimulations using axially asymmetric resonant waveforms appear to be the most practical. The oscillation generator should be designed so that the generated wavefield

The Resonant and Waveguide Characteristics of a Well 163

is axially asymmetric in space. The best results will be achieved when the distribution of pressure over a front-end surface z = L satisfies the followr i ing ratio: AJ1 11 e . Besides, the expansion component P11 should R be distinct from zero and equal to A, and the remaining expansion components should return to zero. In practice, this means that the distribution of pressure amplitudes over a cross-section where the generator is installed should be axially asymmetric. On method that permits achieving this is as follows. The circular cross-section at the location where the generator is installed is divided by the diameter

into two semicircles. In each of the semicircles, 2 oscillations should be in antiphase. Besides, the wave front generated by the generator will make an acute angle with the pipe axis so this causes a spatially heterogeneous standing resonant wave. Secondly, resonant frequencies corresponding to an axially asymmetric resonant form should be included into the generator bandwidth. These frequencies should satisfy the equation (8.10), where m = 1 and n = 1:

2l 1

2

1 1

2

. If we consider an example of a reser2L R2 voir of a thickness L = 10 m, with a hole radius of R = 0.075 m, sound velocity of С = 1,500 m/s, then taking into account that 11 1.84, the resonant frequencies for l ranging from 0 to 10 will vary from 5.857 kHz to 5.91 kHz. Thirdly, the generator frequencies should contain such frequencies at which the resonance is achieved in perforation channels as described in Chapter 1 (see Fig. 1.5). We should note that the resonance in a perforation channel depends on a one-dimensional compressional standing wave established in a fluid-filled perforation hole. As shown by calculations set out in Chapter 1, resonant frequencies of a perforation channel depend on the length of a perforation hole Lp, in which a onedimensional compressional wave sets up. Such resonant waves in the space between the cross-section of the generator location and the well bottom are described above (see (8.9)). Eq. (8.9) is equally suitable for calculation of resonant frequencies in perforation holes in which onedimensional resonant standing waves establish themselves, if we replace the reservoir thickness L by the length of a perforation channel Lp. in the equation. Besides, according to the data provided in Chapter 1, the best stimulation results are achieved using the first resonant frequency deC

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rived from the equation (8.9) where L is replaced by Lp and the index l is set to l = 0. To determine the resonant frequency of a resonant waveform that is best suited for stimulations, we use the equation (8.10), where m=1 and 2

2l 1

1 1

2

. In this case, an axially asymmetric 2L R2 spatially heterogeneous waveform manifested as a resonant standing wave will be realized in the space between the generator and the well bottom separated by the distance L equal to the reservoir thickness.

n = 1:

C

P11 J1

1 1

cos k11z r cos cos t . R cos k11L

For the frequencies exciting resonance in a perforation channel to be within the range of frequencies

2n 1

2

1 1

2

generated 2L R2 in this case by the generator, the length of perforations Lp must satisfy the C

L

following ratio: L p

2n 1

2

2L

2

2

1 1 2

Hence, for a reser-

2

1

2 R 2l 1 2l 1 voir thickness L =10 m, a well radius of 0.075 m, and the first axially asymmetric form 11 1,84 , we have: Lp ≈ 0.8 m. In some cases, to satisfy the resonance ratio, we can either reduce the length of a perforation channel or increase the well diameter.

Enhanced Oil Recovery: Resonance Macro- and Micro-Mechanics of Petroleum Reservoirs. O. R. Ganiev, R. F. Ganiev and L. E. Ukrainsky. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

9 Experimental Study of Wave Action on a Fluid-Filled Porous Medium

Below we discuss experimental investigations conducted to explore the effect from continuous polyharmonic waves and shock waves on the cleanup of fluid-filled porous media.

9.1 Experimental Study of the Potential to Clean up the Near-Wellbore Formation Zone from Contamination using Wave Stimulation The results described in this section were obtained by the authors in cooperation with N. V. Gun and I. G. Ustenko. The plugging of the pore space in the reservoir is one of reasons of the declining production from oil/gas wells. Such contamination may specifically result from the ingress of clay mud particles into the reservoir while drilling the well [46]. We investigated possibilities to clean the pore space from plugging clay particles and recover the original permeability of the porous medium using wave stimulations. For our investigations, we used a model that simulates the near-wellbore formation zone and run tests for different types of contaminants. The tests demonstrate that in case of a montmorillonite clay contamination the wave stimulation result in an abrupt, sharp increase in permeability. Besides, even if the time of wave stimulation is not sufficient to ensure an appreciable cleanup in the model, the stimulation does actually significantly reduce the contamination. A positive effect from wave stimulation on the recovery of permeability is also observed where we added polymer to the modeled mud and injected it into the porous medium. There are different ways to approach the problem of cleaning the nearwellbore formation zone from plugging contaminants, including treat165

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ment with dedicated solutions, thermochemical and shock-wave treatments. Of all these methods, wave stimulations [47, 48] proved to be more promising due to its low cost, efficiency and a wide range of possible wave generators. The application of wave stimulation for a cleanup of near-wellbore zones began in the middle of 1980s, and, to date, around 3000 wells in the Russian and foreign fields have been treated. An average increase in the production of wells after using this method varied from 50 to 60 %, and, in some cases, the increase was multifold [10, 11]. However, the task of increasing the efficiency of stimulation of the near-wellbore reservoir zone becomes extremely critical under current conditions of the oil industry development. A further development of the wave technology requires a base of theoretical and experimental knowledge on interaction of waves with the fluid-filled porous medium containing various inclusions. The subject investigation is mainly aimed at verifying and further studying some theoretically established effects that should ensure an efficient cleanup of the pore space from contaminants, increase the flow of fluid in the near-wellbore formation zone and the recovery of oil.

9.1.1 Test Equipment and Methodology These investigations were conducted in cooperation with the team headed by A. E. Sheidlin, a member of the Russian Academy of Sciences, using the test facility called ‘Reservoir’ at OIVT RAN. The facilities are designed for studying flow processes of the formation fluid under actual insitu pressure and temperature conditions and make it possible to conduct experiments with fluids and gases of various fractional compositions. The general view of the facilities is shown on Fig. 9.1, where 1 indicates the flow section; 2 indicates the ‘wave’ section; 3 is the shock wave generator; and 4 is the injection pump. During experiments, we used distilled water as a reservoir fluid. We modeled drilling mud using water-based montmorillonite clay mud and clay mud with polymer added as a binding agent. The flow section (FS) consisted of a 2.2 m long 16 mm stainless steel pipe filled with pre-washed sand with grain sizes varying from 0.09 to 0.125 mm. In conducted tests, the FS permeability in the absence of contaminants varied from 1.5 to 4.5 D. To fill the FS with clay mud and water, we used two 6 l receiving vessels connected via the three-way valve to one of the FS ends. One of receivers was for clay mud and another for water. To maintain constant pressure in receivers, we used a pressure vessel connected with receivers via a pressure controller.

Experimental Study of Wave Action

167

Fig. 9.1

Three heaters were used to heat up the flow section to a required temperature. The temperature control was performed using 8 thermocouples evenly spaced along the FS length. We measured the filtrate efflux rate using a gauge tank installed at the opposite open end of the flow section. The permeability was calculated from Darcy law using the derived filtrate efflux rate and the measured pressure differential. For wave stimulation of the tested section, we used a vortex generator [49, 50, 51, 52, 53, 54] capable of generating oscillations within the bandwidth from 2 to 50 kHz. A nipple was installed in the generator chamber to connect it to the open end of the flow section. The vortex generator was activated using a 1 m3/h pump with a discharge pressure of up to 80 atm. The fluid flow rate and pump discharge pressure were controlled using a variable speed drive which permitted controlling the operation of the generator. To control the distribution of pressure pulsations in the FS, we used piezoelectric pressure indicators evenly spaced along the FS length. Fig. 9.2 shows the pressure pulsation spectra for the generator chamber and for the flow section at a distance of 70 cm from the inlet hole (curves 1 and 2, respectively). As is seen from the plot, the distribution of waves in the FS pipe shows a clearly pronounced resonant behavior. This is indicated by existing narrow-band components within the oscillation

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range (Curve 2). No oscillations with frequencies over 15 kHz are present. It may be assumed that these components were either reflected or absorbed at a fairly short distance from the FS inlet hole. The amplitude of pressure pulsations in the generator chamber reached 1.5–2.5 atm and same in the FS at the specified distance was 0.1–0.2 atm.

Fig. 9.2

The tests were carried out as follows. At first, we injected plugging contaminants into the flow section. For this purpose, we filled the receiver connected to the flow section with prepared, properly agitated, filtered clay mud. We left the opposite FS end open and installed the gauge tank underneath. Then, we gradually went up with pressure in the receiver by a 20–40 atm step at a time. At each step, we recorded the volume of filtrate efflux from the flow section. As soon as the filtrate ceased to flow from the FS outlet, we raised the pressure in the receiver. In most cases, we let the pressure differential reach 100–120 atm, thus, causing the decrease in the FS permeability down to 0.04–0.08 D. Based on the amount and consistency of the filtrate at the FS outlet, we evaluated the type and distribution of the deposited substrate over the FS length. While injecting clay mud, we made sure the substrate deposits along the entire pipe length, thus, ensuring the plugging is stable and enduring and also ensuring a positive effect on the repeatability of test results. At the end of the injection/plugging process, we reversed the pipe. We connected the formerly open pipe end to the distilled water receiver and measured the FS permeability. Following this, we connected the free end

Experimental Study of Wave Action

169

of the pipe to the generator chamber and carried out wave stimulation during the set time period. At the end of this, we measured the permeability. If necessary, we repeated the process of wave stimulation a required number of times.

9.1.2 The Results of Cleanup from Clay Mud To contaminate the flow section, we used a single-component 3–4 % montmorillonite clay mud. Lower concentrations did not yield any contamination as the filtrate flowed through the pipe without reducing its permeability. With higher concentrations, the process was too fast: an interval with low permeability formed near the FS end so the the volume of mud penetrated into the flow section remained small even after increasing the injection pressure up to 200 atm. As a result, the plugged interval did not extend any further and the obtained contamination was unstable and easily removable. We conducted some tests in order to clean up the flow section using a static pressure differential without any wave stimulation. A typical result is illustrated on Fig. 9.3. It shows the change in FS permeability vs. time. The plot demonstrates permeability relative to the original permeability prior to contamination. Curve 1 corresponds to the pressure differential of 40 atm as measured at the FS ends. As is seen from the plot, despite of the existing fluid flow, the permeability does not change meaning that our efforts yielded no cleanup.

Fig. 9.3

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However, once the pressure differential exceeds some cutoff value of P*, the fluid breaks through. The fluid entrains the deposited substrate and carries it from the flow section, thus, recovering the permeability. The described scenario is illustrated by Curve 2. On the tenth minute, we abruptly increased the original pressure differential of 100 atm to 120 atm. Let us now discuss the test results obtained after wave stimulations. Wave stimulation proved to be efficient in removing our contamination. Figure 9.4 shows the obtained results.

Fig. 9.4. 1 – Р = 5 atm, 2 – Р = 10 atm, 3 – Р = 20 atm

Curves from the plot are represented by various external constant pressure differentials that we generated using the receiver. As is seen from the plot, the smaller the applied constant pressure differential, the more efficient the cleanup becomes. The smaller the differential, the faster the fluid breaks through. Such result may seem illogical and, therefore, needs some explanation. Indeed, the higher pressure differential in conducive to maintaining the higher fluid flow rate and, therefore, facilitates the faster carryover of contaminants from the pore space and, hence, earlier breakthrough. If the fluid flow rate, however, remains insufficient for the breakthrough, then, as follows from Fig. 9.3 (Curve 1), the permeability shows almost no increase over time. The fluid flow at low rates is incapable of cleaning the pores from heavy contamination. Moreover, our previous results shown on Fig. 9.3 demonstrate that an increase in the pressure differential will result in the fluid breakthrough only if the value of pressure

Experimental Study of Wave Action

171

differential exceeds some cutoff. As follows from the obtained results, the high pressure differential below the cutoff has negative effect on the recovery of permeability as it neutralizes the effect of wave stimulation. The phenomenon may be explained if we analyse the forces acting on contaminating particles and it was discussed above in Chapter 5. The external constant pressure differential makes the clay particles clog tight spots in pore channels and strongly presses them against the pore walls inclined at a right angle (or close to a right angle) to the acting pressure gradient. With the pressing force present, the adhesive forces escalate accordingly and contribute to retaining the particles in pores. Wave stimulation is able to induce an alternating force acting on particles. So, if at any spot the wave-induced forces exceed particle-retaining forces, the particles release and become mobile. They start to rotate, move and, at some point, can pass through the tight spot in the pore. This results in an emerging ‘cleanup zone’. Since the permeability is recovering in this zone, the constant pressure differential goes down there which contributes to relieving the force that presses the particles against the pore walls and to further cleaning. The increasing permeability also has a positive effect on propagation of waves deeper into the reservoir. As a result, the ‘cleanup zone’ gradually extends. The above process takes the longer, the stronger is the initial steady force that presses the particles against the pore walls. We studied the effect of the wave stimulation on the cutoff value of the pressure differential P*. The test procedure was as follows. With constant external pressure differential, we carried out wave stimulation of the contaminated flow section. In a set time interval, we stopped the process and gradually went up with the external pressure differential until the filtrate breakthrough was achieved. Fig. 9.5 shows the results of such tests. During the wave stimulation, we used P = 10 atm. As is seen from the plot, the wave stimulation is conducive to reducing the pressure differential cutoff value, that is, the longer the wave stimulation, the lower pressure differential is needed to remove the contamination.

9.1.3 The Results of Cleanup from Clay-Polymer Mud To obtain a stronger plugging effect, we used a two-component mud model containing 3% of clay and 0.1% of a binding polymer KMTs-500. The resultant mud weight was 1 g/cm3, and its viscosity – 20 mPa·s. Due to the increased adhesive properties, the mud contaminants were impossible to remove using the treatment process under constant pressure differential conditions. The time of wave stimulation required for

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Enhanced Oil Recovery

an FS cleanup also considerably increased. The obtained results are shown on Fig. 9.6. It demonstrates the dependence of the FS permeability relative to the original permeability on the duration of wave stimulation.

Fig. 9.5

Fig. 9.6. 1 – Р = 5 atm, 2 – Р = 20 atm, 3 – Р = 40 atm

Similarly to one-component clay mud, the cleanup process begins the faster, the lower is the constant pressure differential Р. There is, however, a significant difference. It takes considerably longer for the permeabil-

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173

ity to recover, especially at small pressure differential values. With higher Р, the permeability recovers faster, and albeit it takes longer for the cleanup process to begin, these conditions ultimately prove more efficient.

9.1.4 Summary We conducted the study of the potential to clean up the near-wellbore formation zone from contamination using wave stimulations. For this purpose, we compared the efficiency of wave stimulation with the treatment of the porous medium under constant pressure differential conditions. In case of a clay mud contamination, we observed that the pressure differential at which contaminants are removed from the porous space is lower by an order of magnitude for wave stimulation than for the treatment. We also demonstrated that even if the duration of wave stimulation was insufficient to ensure an appreciable cleanup of the pore space, the wave stimulation did actually considerably reduce the contamination. We showed that an enhancement in adhesive characteristics of the clay mud increases the effect from a unidirectional fluid flow on the cleanup process as it contributes to the carryover of contaminating particles from the pore space. This permits recommending that whenever the wave stimulation are used to recover the porosity and permeability of the near-wellbore formation zone, it is advisable to carry out this process alternately with a drawdown in order to initiate the inflow of filtrate. Besides, it is recommended to carry out wave stimulation when the difference between the formation pressure and bottom hole pressure is small.

9.2 The Experimental Study of the Effect of Shock Waves on the Displacement of Hydrocarbons by Water in a Porous Medium. Connected Wells The test results described below were obtained by the authors in cooperation with I. G. Ustenko. We investigated the effect of shock waves on the potential to recover capillary-trapped hydrocarbons from the reservoir. For our tests, we used the test facilities that simulate the fluid flow process in wells with hydrodynamic connections. We assumed that the wave action was transferred

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from the ‘exciting’ well deep into the reservoir via the systems of fractures and faults of increased permeability. These tests showed that the shock wave stimulation was able to increase the fluid flow, mobilize capillarytrapped hydrocarbons and raise the permeability. At present, unconventional hydrocarbons account for around 60 % of our country’s mineral reserves and this share is steadily growing all the time. This category of reserves primarily includes formations with tight heavy oil reservoirs. It should be noted that all efforts to improve the applied technologies failed to deliver the required increase in the unconventional oil recovery efficiency. The flooding is the enhanced oil recovery method that is most commonly used in our country. An extensive use of the flooding some time ago made it possible to significantly improve the reservoir performance in oil fields. However, over the sixty years of commercial development of oil fields, the main production zones (reservoirs) have reached the mature stage characterized by the high level of reservoir depletion and significant water cut in the well stream. The RF average water cut is estimated at around 86%. This value is expected to reach 89% by 2030 [55]. While the water displaces oil during the oil production process, some hydrocarbons are trapped in the pore space due to capillary forces. Depending on the specific rock wettability, oil can form a thin film on the pore walls held in place by adhesive forces and not entrained in the fluid flow. The hydrocarbons may equally form fine hydrocarbon droplets filling the pore space and obstructing the fluid flow. The above effects result in a considerable decrease in the reservoir permeability and oil recovery. Below we set out the results of tests conducted to study the effects of wave stimulation on the increase of reservoir permeability and recovery of capillary-trapped oil. This study was aimed at detecting the mechanisms involved in a wave/rock/fluid interaction and at identifying the effects conducive to improving the reservoir poroperm properties.

9.2.1 The Test Equipment Similarly to the above described, these investigations were conducted using the test facilities ‘Reservoir’, that were created at OIVT RAN under the guidance of A. E. Sheidlin, the member of the Russian Academy of Sciences. The facilities are designed for studying the flow processes of formation fluids under actual in-situ pressure and temperature conditions. The facilities are able to provide the test parameters, including the

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pressure of up to 40 MPa and temperature of up to 400°С. Such parameters make it possible to simulate a wide range of in-situ formation conditions and conduct experiments with fluids and gases of various fractional compositions. Fig. 9.1 shows a general view of the test facilities, and Fig. 9.7 shows its layout.

Fig. 9.7. 1 – flow section (FS); 2 – wave section (WS); 3 – ball valve; 4 – receiving vessels with test fluids; 5 – three-way changeover valve; 6 – pressure gauges; 7 – gauge tank; 8 – wave generator; 9 – pressure indicators; 10 – nitrogen pressure vessel; 11 – wave generator drive; 12 – variable speed drive

The test section consisted of two cylindrical pipes made of stainless steel and filled with pre-washed quartz sand with grain sizes varying from 0.09 to 0.125 mm. The internal diameter of pipes was 10 mm, and the filler porosity varied from 0.3 to 0.35. A pipe used to simulate the fluid flow, the “flow section”, served as one-dimensional model of connected wells. The length of the flow section (FS) was 3 m. For model fluids, we used paraffinic oil, kerosene, pentane and their mixtures and we used distilled water as a displacement fluid. To fill the flow section with the hydrocarbon mixture and to carry out its flooding, we used receiving vessels (4) connected via the three-way valve (5) with the flow section. To maintain constant pressure in receivers, we used a pressure vessel (10) connected with receivers via a pressure controller. To control the pressure at the FS ends, we used the pressure gauges (6). We measured the filtrate efflux rate using the gauge tank (7) installed at the opposite end of the flow section. The permeability was calculated from Darcy law using the derived filtrate efflux rate and the measured pressure differential.

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The second sand-filled pipe, the wave section (WS), was used to transfer the wave action from the wave generator to the flow section. The length of the WS pipe was 1.5 m. The wave section was connected to the center of the flow section via the ball valve that made it possible to hydrodynamically separate the test sections from each other. The permeability of the test sections as measured to water under conditions of completely flooded pipes was 1.2 D for the wave section and between 0.18 and 0.35 D for the flow section. We used piezoelectric pressure indicators to control the pressure distribution in the test sections and generation of waves by generators. The pressure indicators were evenly spaced along the length of the test sections (Fig. 9.8). Data from the pressure indicators were transmitted to a digital oscillograph for recording and analysis.

Fig. 9.8

To heat up the flow section to a required temperature, we used three ribbon heaters: two end heaters and one main heater. The temperature control was performed using 8 thermocouples evenly spaced along the FS length. For the shock-wave treatment of the test section, we used the generator (8) whose schematic is shown on Fig. 9.9. The generator generates shock waves with amplitudes of up to 60 atm and frequencies varying from unit fractions to integral units of hertz. It operates as follows. From

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the tank (1) water is pumped over using pump (2) to the generator chamber (3). The pump maintains the constant fluid flow rate. The pump rate is controlled by the variable speed drive (11). From the chamber the water can flow back to the tank through two pipelines. One of the pipelines has a solenoid-operated valve (4). With the valve closed, the water flows though one pipeline, thus, causing the increase of pressure in the chamber. The control valve (6) permits setting a desired maximum pressure level that is controlled using the pressure gauge (7). Once the valve opens, the pressure in the chamber drops down, and the lowest pressure level can be set using the second control valve. The valve operation is controlled using the timing relay (5). The relay makes it possible to set the duration of the pressure pulse and the process period within the range from 0.1 to 100 s. The valve operating time of ~ 10 ms is sufficient to ensure a fairly sharp increase in the shock front. A nipple (8) is provided in the chamber to connect it to the test sections. Besides, a pressure indicator (9) may be installed to control and record the process.

Fig. 9.9. The hydraulic schematic of the shock-wave generator: 1 – water tank; 2 – pump; 3 – chamber; 4 – solenoid-operated valve; 5 – control relay; 6 – control valves; 7 – pressure gauge; 8 – nipple to connect the test section; 9 – pressure indicator; 10 – oscillograph; 11 – variable speed drive

9.2.2 A Theoretical Analysis of the Propagation of Waves Generated by a Shock-Wave Valve in the Test Facilities and Evaluation of the Forces Caused by the Wave Action We consider one-dimensional motion of one-component fluid in the porous medium. Bearing in mind a relatively short length of the pipes and that the filling sand was not consolidated, we ignore the effect of the porous matrix elasticity and the action of associated waves. We assume also that the medium porosity and permeability are constant.

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We assume that the change in the pipe pressure depends solely on the fluid flow. Hence, according to the Darcy flow law, we have: k P . x

V

(9.1)

The law of conservation of fluid mass for the fluid flow may be written as: V

P p

x

t

(9.2)

.

where P is the pressure; V is the average filtrate rate (by flow and section); k is the permeability; is the viscosity; p is the fluid bulk modulus. By eliminating the rate in the equations (9.1)–(9.2), we obtain:

P x, t t where

k

2

k p

x2

P (x , t ),

(9.3)

is the transmissibility factor. p

To derive the initial and boundary conditions, we determine the following boundary value problem. We consider the section of a length L. A shock-wave generator is connected to one of the section ends at the point x 0 . The other end is open. Originally, no pressure differential exists at the section and the fluid does not flow through it. From the point of time t 0 , the generator periodically generates rectangular pressure pulses of a duration Timp. Hence, we obtain the initial and boundary conditions for the function P ( x , t ) : P(x,0) 0 , (9.4)

P (0, t )

Pmax

0 t Timp

Pmin Timp

where T0 is the period

t T0

P(L, t ) 0 , where t

0.

(9.5) (9.6)

Solution of the problem (9.3)–(9.6) helps us find the force caused by P the shock wave and determined as F . x We compare this force with the force F0 created under action at the point x 0 of a constant steady pressure which we equate with the period-averaged pressure pulse: P0

Pmax Timp Pmin (T0 Timp ) T0

.

(9.7)

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We perform the calculations for the following parameters: k = 0.15 D, Pmax = 30 atm, T0 = 15 s,

Timp = 2 s,

p

= 10–3 Pa·s, Pmin = 5 atm,

= 4.4·10–10 Pa–1, L = 3 m.

The results are shown on Fig. 9.10–9.11. Fig. 9.10 shows the shockwave pulse waveform at various distances from the source. Fig. 9.10 compares the force F created by the shock wave in the medium with the force F0 caused by the steady pressure differential P0 . The plot demonstrates how the relative difference of the forces changes with time. The positive quantities on the plot correspond to the case where the wave force exceeds the steady force and its direction overlaps with the filtrate flow course. As is seen from the plots, there are some points of time at which the force caused by a propagating shock wave exceeds the force existing under steady flow conditions. We assume that this may be conducive to a displacement of capillary-trapped hydrocarbons. Besides, a case may exist where the wave force is in opposition to the average fluid flow which is illustrated by Curve 1 on Fig. 9.11. For comparison, Fig. 9.12 shows experimentally obtained oscillograms of the rectangular shock-wave pulse propagating in the flow section at various distances from the wave generator. The pulse amplitude is 30 atm and its duration is 2 s. The permeability is the same as in the calculations. As is seen from Fig. 9.10 and Fig. 9.12, the calculated and experimental pulse waveforms look similar. The proposed model also rather credibly describes the attenuation of the shock-wave pulse amplitude.

Fig. 9.10. The shock-wave pulse at various distances from the source: 1 – 0.5 m; 2 – 1.0 m; 3 – 1.5 m; 4 – 2.0 m

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Fig. 9.11. The force caused by the shock wave at various distances from the source: 1 – 0.5 m; 2 – 1.0 m; 3 – 1.5 m; 4 – 2.0 m

Fig. 9.12. The rectangular shock-wave pulse at various distances from the wave generator: 1 – 0.15 m; 2 – 0.95 m; 3 – 1.35 m; 4 – 2.15 m

9.2.3 The Methodology of Tests The tests were conducted in two phases. The first phase included the reference test, specifically, a steady-state flow test at a constant pressure differential to simulate the conventional flooding process. It was followed by a series of tests that involved a shock-wave treatment under varying conditions. The obtained results were compared and conclusions on the process efficiency were made.

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To conduct a steady-state flow test, we connected a distilled water receiver to the flow section pre-filled with hydrocarbons. We applied a pressure P0, normally below 30 atm, to the receiver. We left the opposite FS end open and installed the gauge tank there. By measuring the volume of the fluid at the FS outlet at certain time intervals, we found the permeability. Knowing the amount of water in the filtrate, we calculated its water cut. Depending on the specific experiment, the shock-wave treatment of the flow section was carried out using different approaches. The wave action was applied either directly to the FS, and, therefore, the shockwave generator was connected to one of the FS ends, or via the wave section one of whose ends was connected with the generator, and the other with the FS center. The residual hydrocarbon content was evaluated indirectly by changes in the FS permeability. Forming the film on the pore walls, the capillary-trapped petroleum products are known to narrow down the flow section of the pore channels, thus, reducing the permeability of the porous medium. As hydrocarbons are displaced, the permeability grows tending to the cutoff value that equates to the complete recovery of hydrocarbons from the flow section. Consequently, by monitoring the changes in the permeability, it is possible to evaluate the residual hydrocarbon content. We conducted some preliminary tests to determine the appropriate chemical composition of the model fluid. In so doing, we pursued three goals: secure the capillary trapping effect for the hydrocarbon fraction in the FS porous space, ensure the presence of hydrocarbons has a noticeable effect on the water relative permeability and enable the recovery of residual hydrocarbons to a sufficiently full extent. As a result of the conducted tests, a mixture of paraffinic oil and kerosene (up to 15 vol. %) was proposed as a model hydrocarbon fluid. The appropriate viscosity was found to be within the range of 20 to 30 mPa·s.

9.2.4 Results of Flow Acceleration Tests As earlier mentioned, the tests were conducted in two phases. At first, the reference test was performed without any wave stimulation at a constant pressure differential, and then, the following test involved the use of a shock-wave generator connected to one of the FS ends. Thus, we modeled the case of the two connected wells where one (‘production’) was at the right FS end, and the other (‘injection’) was at the left end. The reservoir treatment is conducted via a injection well.

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The generator generated rectangular shock-wave pulses of prescribed amplitude and duration. The amplitude and duration are selected such that the pressure differential as measured at the FS ends and wave-period averaged matches that of the steady pressure differential of the reference test. Pmax ·Timp Pmin (T0 Timp ) P0 , T0 where Pmax and Pmin are the maximum and minimum pressures in the pulse; P0 is the steady pressure in the receiver; Timp is the pulse duration; and T0 is the wave period. Fig. 9.13 and 9.14 show the pressure pulses in the flow section at various distances from the generator and at different phases of the wave stimulation process. Initially, the height of the pulse Pmax Pmin was 35 atm and its duration was 1 s.

Fig. 9.13. The duration of wave treatment: 1 –9 min, 2 – 67 min, 3 – 105 min, 4 – 190 min

Fig. 9.14. The duration of wave treatment: 1 – 9 min, 2 – 67 min, 3 – 105 min, 4 – 190 min, 5 – 240 min

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As is seen from the plots, as the more viscous oil is displaced by water, the shock-wave dissipation energy decreases, the pressure conductivity factor increases, the wave front becomes sharper and the pulse amplitude grows. This process takes as long as is needed for the waterflood front to reach the indicator location. During this time, the pulse amplitude was close to its maximum. As the front moves further beyond the indicator, the pulse amplitude decreases as measured by the indicator. It is illustrated by Curves 4 and 5 on Fig. 9.14. Such an effect is due to the fact that with increase in the permeability, more fluid penetrates further into the flow section towards the opposite pipe end, thus, reducing the differential pressure at closer locations. Fig. 9.15–9.17 show how the water relative permeability changes in the process of displacement of the hydrocarbon mixture from the FS. The graphs were plotted for various pressure differentials at the FS ends, various viscosities and for different wave action parameters. As is seen from the plots, the shock-wave treatment accelerates the flow of hydrocarbons as compared with the steady flow conditions. We believe this is associated with the action of additional forces caused by the shock wave on the capillary-trapped fluid. Besides, in most cases, the wave treatment yields the higher ultimate permeability which is indicative of the lower content of residual hydrocarbons. The tests showed that the effect of the shock-wave treatment became more notable with increase of the hydrocarbon fluid viscosity. It is obvious if we compare the results on Fig. 9.16 and Fig. 9.17. The effect of the

Fig. 9.15. The change in FS permeability during the flooding. Differential pressure P0 = 30 atm, viscosity ~0.028 Pa·s: 1 – no wave treatment, 2 – Pmax = 50 atm, Pmin = 30 atm, Timp = 0.3 s, T0 = 5.3 s; 3 – Pmax = 60 atm, Pmin = 25 atm, Timp = 0.5 s, T0 = 4.5 s

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shock-wave treatment also grows with decrease in the steady pressure differential, that is with declining contribution of the steady forces into the fluid flow. This may be concluded if we compare Fig. 9.15 and Fig. 9.16. The shock-wave treatment becomes more efficient with increase in the amplitudes of pulses and frequencies which is illustrated by Curves 2 and 3 on Fig. 9.15, and by Curves 2 and 3 on Fig. 9.16.

Fig. 9.16. The change in FS permeability during the flooding. Differential pressure P0 = 20 atm, viscosity ~0.028 Pa·s: 1 – no wave treatment, 2 – Pmax = 50 atm, Pmin = 14 atm, Timp = 1 s, T0 = 6 s; 3 – Pmax = 60 atm, Pmin = 10 atm, Timp = 1.5 s, T0 = 7.5s

Fig. 9.17. The change in FS permeability during the flooding. Differential pressure P0 = 20 atm, viscosity ~0.02 Pa·s: 1 – no wave treatment, 2 – Pmax = 35 atm, Pmin = 5 atm, Timp = 2 s, T0 = 4 s; 3 – Pmax = 50 atm, Pmin = 14 atm, Timp = 0.4 s, T0 = 2.4 s

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The duration of shock pulses also influences the process. If the duration is too small, the efficiency of wave treatment declines. When the permeability of the porous medium is low, the short pressure pulse fails to propagate over a sufficient distance, and the shock wave, therefore, quickly damps. For practical applications of the shock-wave treatment, it should be borne in mind that the pulse duration must be determined with account taken of the rock permeability and required range of treatment.

9.2.5 The Effect of Wave Stimulation on Connected Wells We conducted some tests to simulate ‘areal’ wave treatments. During these tests, we modeled the technologies that attempt to stimulate the field areas drilled by a group of wells. In this case, the generator was installed in one well of the selected field area and then the surrounding wells were stimulated. The layout of test facilities is provided on Fig. 9.7. In the conducted test, we modeled the three connected wells scenario where one (‘production’) well was at the open FS end, the other (‘injection’) well was at the opposite end and the third ‘exciting’ well was at the lower end of the wave section. A special attention in this experiment was given to monitoring such parameter as the period-averaged shock wave pressure. The FS connection to the totally permeable wave section and additional pressure source could lead to a significant alteration in the fluid flows. Therefore, we selected such shock-wave parameters that minimized the intersectional flows between the flow section and the wave section. This takes place if connection/disconnection of the wave section only slightly changes the average pressure at the FS center. In such a case, during the shock-wave propagation the fluid flow would alter twice. The propagating pressure pulse causes the flow from the flow section to the wave section and boosts the fluid flow in the left portion of the filtration section while the decaying pressure causes the backflow from the wave section to the filtration section and boosts the fluid flow in the right portion. It may be assumed that the described scenario will be conducive to the most efficient stimulation of the flow section. Indeed, if the average pressure at the FS center gets significantly low after connection to the wave section, then the pressure differential in the left portion of the FS will increase and in the right portion of the FS will decrease, thus, causing the increase in the fluid flow at the FS left-hand side and decrease at the FS right-hand side, respectively. This will contribute to better cleanup in the FS left side but will decrease the effect in the right. Likewise, an increase in the average pressure results in the decreasing fluid flow in the left portion of the flow section.

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Enhanced Oil Recovery

The test results shown on Fig. 9.18 prove the conclusions made. The pressure differential at the FS – P0 = 20 atm. The mixture viscosity is 20 mPa·s. As the permeability in the FS portions was known, we could calculate the pressure at the center which made up around 12 atm. Taking into account the drop in the WS pressure, we predicted that the optimum average pressure at the shock-wave generator would be close to 15 atm. We conducted the tests at = 10; 15; and 20 atm. Indeed, the best result was achieved at 15 atm. The results at the other pressures were somewhat lower but still higher than in the scenario of the steady-pressure induced flow.

Fig. 9.18

9.2.6 Summary We conducted a series of tests to study the effects of wave and shockwave stimulation on the increase of reservoir permeability, potential to recover capillary-trapped oil from the reservoir and increase in the oil recovery factor. These tests involved studying the effect of physical properties of hydrocarbons on their ability to remain inside the pores and capillaries and resist water displacement. This phenomenon proved to strongly depend on the viscosity of the hydrocarbon fluid. The share of capillary-trapped residues dramatically drops with decrease in the viscosity. Thus, the conducted experiments showed that with the viscosity below 3 mPa∙s the share of hydrocarbon residues was below 3% of the pore volume.

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We studied the effect of shock waves on the displacement of hydrocarbons by water. We managed to demonstrate that the shock wave stimulation is able to increase the fluid flow and mobilize capillary-trapped hydrocarbons causing the increase in the permeability and recovery of oil from the reservoirs. We modeled ‘areal’ wave treatments to stimulate field areas drilled by a group of wells. We equally managed to demonstrate a positive effect of shock waves on the flow acceleration and increase in the porous medium permeability for this scenario. We proved, basically, the possibility of wave and shock-wave stimulation of a system of wells in the field when one or several wells are directly treated.

Enhanced Oil Recovery: Resonance Macro- and Micro-Mechanics of Petroleum Reservoirs. O. R. Ganiev, R. F. Ganiev and L. E. Ukrainsky. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

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Enhanced Oil Recovery: Resonance Macro- and Micro-Mechanics of Petroleum Reservoirs. O. R. Ganiev, R. F. Ganiev and L. E. Ukrainsky. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

References 1

Ganiev, R.F., Problems of machine mechanics and technology. Prospects of the development of the Blagonravov Institute of Engineering Science. Journal of Machinery Manufacture and Reliability, 1, 3–20, 2010. 2 Ganiev, R.F., Problems of machine mechanics and technology. Prospects of the development of the Blagonravov Institute of Engineering Science. Part 2. Journal of Machinery Manufacture and Reliability, 3, 3–17, 2010. 3 Ganiev, R.F., Ganiev, S.R., Kasilov, V.P., Pustovgar, A.P., Wave technology in innovative mechanical engineering, p. 64, Regular and Chaotic Dynamics, Moscow, 2010. 4 Ganiev, R.F., Ganiev, S.R., Kasilov, V.P., Pustovgar, A.P., Wave technology in innovative mechanical engineering. Wave and oscillation phenomena for the creation of high-technology industrial applications, p. 92, Regular and Chaotic Dynamics, 2012. 5 Muslimov, R.K. Oil recovery: The past, the present and the future, ANRT, Kazan, 2012. 6 Ganiev, R.F. et al, Wave equipment and technology. Scientific fundamentals, field trials and their results, potential for application, p. 126, Logos, Moscow, 1993. 7 Ganiev, R.F., Wave machines and technology (Introduction to wave technology), p. 192, Regular and Chaotic Dynamics, Moscow, 2008. 8 Ganiev, R.F., Ukrainskiy, L.E., Nonlinear wave mechanics and technologies, p. 712, Regular and Chaotic Dynamics, Moscow, 2008. 9 Ganiev, R. F., Ukrainskiy, L.E., Andreyev, V.E., Kotenev, Yu.A., Problems and prospects of multiphase systems wave technologies in the oil and gas industry, p. 185, Nedra, St. Petersburg, 2008. 195

196 10

References

Ganiev, R.F., Ukrainskiy, L.E., Nonlinear wave mechanics and technologies. Wave and oscillatory phenomena on the basis of high technologies. Second enlarged edition, p. 780, Regular and Chaotic Dynamics, Moscow, 2011. 11 Ganiev, R.F., Ukrainskiy, L.E., Nonlinear wave mechanics and technologies. Wave and oscillatory phenomena on the basis of high technologies, p. 583, Begell House, Inc., New York, Connecticut, 2012. 12 Khisamov, R.S., Highly efficient technologies for development of oil fields, p. 549, Techinput, Moscow, 2005. 13 Ganiev, R.F., Petrov, S.A., Ukrainskiy, L.E., On the resonant nature of distribution of wave field amplitudes in near-wellbore formation. Vibrotechnika, 62 (1), 1989. 14 Wong, S.-W., van der Bas, F., Groenenboom, J., Zuiderwijk P., Near Wellbore Stimulation by Acoustic Waves. SPE 82198. 15 Shamov, N.A. et al, The equipment and technology for ultra-deep perforation tunnels. Oil and Gas Business, 2, 131–176, 2012. 16 Shamov, N.A. et al, The technology and equipment for improving connectivity between wellbores and reservoirs. Oil and Gas Business, 1(4), 47–57, 2006. 17 Ganiev, R.F., Ukrainskiy, L.E., Ganiev, O.R., Resonant seepage flows in a fluid-saturated porous medium. Transactions of the Academy of Sciences, 412(1), 1–4, 2007. 18 Vakhitov, G.G., Application of surfactants for enhanced oil recovery in the USSR and abroad, p. 16, VNII of the USSR Ministry of Petroleum Industry, Moscow, 1981. 19 Ganiev, R.F., Ukrainskiy L.E., Frolov, K.V., Wave mechanism of fluid acceleration in capillaries and porous media. Transactions of the USSR Academy of Sciences, 306(4), 1989. 20 Kostrov, S., Wooden, W., In-situ seismic stimulation shows promise for revitalizing mature fields. Oil & Gas Journal, April 18, 43–49, 2005. 21 Nigmatulin, R.I., Fundamentals of the mechanics of heterogeneous media, p. 336, Nauka, Moscow, 1978. 22 Landau, L.D., Livshitz, E.M., The hydrodynamic, p. 733, Moscow, Nauka, 1986. 23 Ganiev, O.R., The effect of a periodic action on average flow in an inhomogeneous liquid-saturated porous medium. Proceedings of the Russian Academy of Sciences. Fluid Dynamics, 2, 98–104, 2006.

References 197 24

Volarovich, M.P. (Ed.), Physical Properties of Minerals and Rocks at High Thermodynamic Parameters: Handbook, p. 255, Nedra, Moscow, 1988. 25 Gubaidullin, A.A., Kuchugurina, O.Yu., Specifics of linear wave propogation in double porous media. Transport in Porous Media, 34 (1–3), 29–45, 1999. 26 Ganiev, R.F., Ukrainskiy, L.E., The dynamics of particles under the action of vibration, p.169, Naukova Dumka, Kiev, 1975. 27 Mitropolsky, Yu.A., Averaging method in nonlinear mechanics, p. 440, Naukova Dumka, Kiev, 1971. 28 Ganiev, R. F., Nonlinear resonance and catastrophes. Reliability, safety and noiselessness, p. 592, Regular and Chaotic Dynamics, Moscow, 2013. 29 Ganiev, R.F., Petrov, S.A., Ukrainskiy, L.E., Nonlinear wave effects in a fluid-saturated porous medium. Proceedings of the Russian Academy of Sciences, Fluid Dynamics, 1, 74–79, 1992. 30 Gomilko, A.M., Gorodetskaya, N.S., Grinchenko, V.T., Ukrainskiy, L.E., An axially symmetric mixed problem of stationary dynamical theory of elasticity for a layer with a parallel hole. Applied Mechanics, 34(1), 39–46, 1998. 31 Frenkel, Ya.I., On the theory of seismic and seismoelectric phenomena in a moist soil. Proceedings of the Russian Academy of Sciences, Geography and Geophysics Series, 8(4), 133–149, 1944. 32 Biot, M.A., Theory of propagation of elastic waves in fluid-saturated porous solid. Part I. Low frequency range. J. Acoust. Soc. Amer., 28(2), 169–178, 1956. 33 Biot, M.A., Theory of propagation of elastic waves in fluid-saturated porous solid. Part II. Higher frequency range. J. Acoust. Soc. Amer., 28(2), 179–191, 1956. 34 Timoshenko, S.P., Oscillations in engineering, p. 444, Nauka, Moscow, 1967. 35 Martynenko, O.M., Wave processes in a semi-infinite layer with combined boundary conditions. Ph.D. thesis in physics & mathematics, p. 48, Institute of Hydrodynamics, National Academy of Sciences of Ukraine, Kiev, 1990. 36 Gomilko, A.M., Gorodetskaya, N.S., Meleshko, V.V., Edge resonance in the forced flexural vibrations of a half-strip. Journal of Acoustics, 37(5), 908–913, 1991.

198 37

38

39

40

41

42 43

44

45

46

47

48

49

50

References

Kleschev, K.A., Petrov, A.I., Shein, V.S., The geodynamics and new types of natural reservoirs of oil and gas, p.285, Moscow, Nedra, 1995. Glukhmanchuk, E.D., Vasilevskiy, A.N., Regularities in fault/fracture structures in the evolution of tectonic deformations of West Siberian fields, in: Way Forward in Utilizing Petroleum Potential of the KhantyMansiysk Autonomous Okrug (VIII Workshop Conference), vol. 2, pp. 67–76, Khanty-Mansiysk, 2005. Rakhmatulin, K.A, Saatov, Ya.U., Filippov, I.G., Artikov, G.U., Waves in two-component media, FAN, Tashkent, 1974. Kovtun, A.A., On equations of the Biot model and their modifications. Problems of geophysics, 44, 3–26, St. Petersburg, 2011. Nigmatulin, R.I., The dynamics of multiphase media. Parts 1 & 2, Nauka, Moscow, 1987. Isakovich, M.A., The general acoustics, p. 495, Nauka, Moscow, 1973. Afanasiev, E.F. et al, Field management using the acoustic method, p. 36, VNIIEgazprom, 1987. Yu.G. Astrakhantsev and A.K. Troyanov, Measuring device to measure geo-acoustic noises in a well, RF Patent 2123711, G 01 V 1/4, 1998. Yanke, E., Emde, F., Lesh, F., Special functions, p. 344, Nauka, Moscow, 1968. Bulatov, A.I., The theory and practice of well completion, Nedra, Moscow, 1997. R.F. Ganiev, L.E. Ukrainskiy, G.A. Kalashnikov, S.A. Kostrov, S.A. Petrov, A method of saturated porous medium treatment, Inventors’ certificate 1727431, 1991. Ganiev, O.R., Ganiev, R.F., Ukrainskiy, L.E., An experimental study of fluid flow stimulation in near-wellbore zones using wave effects, in: Problems of Mechanics, D.M. Klimov (Ed.), pp. 215–220, Fizmatlit, Moscow, 2003. W.S. Avdujewski, R.F. Ganiev, G.A. Kalashnikov, S.A. Kostrov, R.S. Mufazalow, Hydrodynamic oscillation generator, RF Patent 2015749, 1994. W.S. Awdujewski, R.F. Ganiev, R.S. Mufasalow, J.P. Sacharow, Drilling apparatus of the cutting and shearing type, US Patent 5220966, 1993.

References 199 51

52

53

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55

R.F. Ganiev, R.S. Mufasalow, J.P. Sacharow, R.C. Wasiliew, R.G. Mu– chutdinow, Installation for clearing the zone near the drill hole, US Patent 5311955, 1994. W.S. Awdujewski, R.F. Ganiev, R.S. Mufasalow, J.P. Sacharow, Drilling apparatus, US Patent 5303784, 1994. R.F. Ganiev, G.S. Abdulmanov, R.S. Mufazalow, G.A. Kalashnikov, S.A. Kostrov, Downhole wave generator, RF Patent 1833664, 1992. R.F. Ganiev, V.P. Vagin, G.A. Kalashnikov, S.A. Kostrov, A.V. Rast– vortsev, Method of stimulation of the near-wellbore zone in a development well and the means for its implementation, RF Patent 2047754, 1995. Maksimov, V.M., The current state of oil production, oil recovery and enhanced oil recovery methods, Drilling and Oil, 2, 2011.

Enhanced Oil Recovery: Resonance Macro- and Micro-Mechanics of Petroleum Reservoirs. O. R. Ganiev, R. F. Ganiev and L. E. Ukrainsky. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

Conclusion This monograph summarizes the science behind the resonant macro and micromechanics of the oil reservoir that may be used as fundamental and applied knowledge base in developing the wave technologies to stimulate oil production and enhance oil recovery from the reservoirs. When developing this applied discipline we relied on both the sufficiently large expertise from the previous oil field tests in Russia and theoretical results of nonlinear wave mechanics created by us, the authors, and our work teams that forms a novel branch of mechanics [8, 10, 11]. While exploring the resonance micro and macromechanics of the oil reservoir we managed to state and then solved a new problem, never dealt with before, concerning the resonant oscillations and waveguide mechanics of the fluid-saturated porous media with various heterogeneities. Besides, we managed to identify several resonant oscillation and waveguide effects that we used as a basis for fundamentally new methods of the oil field development. These results are way ahead of the relevant world science and engineering and lie in the primary focus of the Russian science. They are properly detailed in the previous chapters of the monograph and below we only summarize some of their basic characteristics. So in our efforts, we managed to: – show the possible nonlinear mechanisms (caused by a resonant delivery of the oscillation and wave energy) that generate the constant directional forces in the porous media and give rise to capillary effects able to accelerate the fluid flow 100–1000 times. These mechanisms create additional high-value pressure differentials in the porous media that are impossible to achieve using any other known means (thermal, chemical, etc.) and that enable to carry out an efficient cleanup of the pores in the near-wellbore zone from solid par189

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ticles and liquid droplets to stimulate oil production, and that enable to generate fluid flows by displacement of capillary-trapped oil from various spatially oriented pores, thus, increasing the recovery of oil from the reservoirs. – identify various mechanisms (forces) that retain solid particles and fluid/oil droplets in the pores of the porous medium, and, on the basis of strictly mathematical models, derive the success criteria for the cleanup of the pores in the near-wellbore zone from solid particles and fluid droplets; also determine the wave mechanisms to displace (create a directional motion of) capillary-trapped oil from the porous media (consisting of the pores variously spatially oriented within the pore space and variously shaped, and of variously shaped capillaries). To utilize the above-described effects and satisfy the success criteria (conditions) of wave treatments in both the near-wellbore formation zone and the far-field zones, it is necessary to create spatial wavefields (including resonance wavefields) of a certain strength (dependent on the relevant success criteria of the treatments) in the near-wellbore zone and in far-field zones. The problem thus stated led to the development of a theory of resonant oscillation and waveguide mechanics of the fluidfilled porous media where the oscillations and waves (heterogeneous and spatial) are excited and propagate through vertical and horizontal wells (with their waveguide characteristics determined) into the oil reservoirs (including those with heterogeneities). The developed theory of porous media, including oscillation and wave processes in wells, in turn leads to defining new physical principles for engineering of controlled technical means and a measurement system. We now point out only some, the most fundamental concepts: the study of resonant oscillation and wave processes in the porous medium in the near-wellbore zone with account taken of perforations (associated resonances) and in the far-field zones (in reservoirs with heterogeneities) as a multiphase system shows that the regularities in the oscillation and wave (waveguide) process behavior in some cases do not conform with the traditional common understanding. Below we discuss some of such hallmarks. We discovered waves which by analogy, at a certain approximation, can be called Rayleigh surface waves and which, when resonance enhanced, are able to penetrate deep into the reservoir; in the meantime, their frequencies can be comparatively low (depending on the width of a waveguide, i.e. an unconsolidated reservoir zone with heterogeneities). After we considered multi-dimensional oscillation motions (multiphase media) as a system with hydrodynamic connections, we established

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the potential to treat a system of wells (a field) under conditions of possible resonances. In this case, the treated well may be used directly to transmit the energy of oscillations to remote wells using the resonance effect (i.e. more actively). The developed waveguide mechanics of the porous medium made it possible to identify various wave types (spatial) that propagate fairly deep subsurface (from 10–100 m to 1–2 km) and, furthermore, propagate at higher frequencies, meaning that the high-frequency waves penetrate deeper than low-frequency waves of the same type (depending on the reservoir damping characteristics). Hence, we obtained a seemingly illogical result whereby it takes less time for the low-frequency waves to attenuate than for the high-frequency waves which attenuate much longer and penetrate fairly deep into the farfield zones. Besides, it is the nature of excitation of oscillations and waves that defines the rate at which the wave energy is fed into the reservoir (how efficient is the transfer of energy to the reservoir, that is what portion of energy with minimum attenuation is fed into the reservoir). We also determined the conditions under which the maximum amount of energy is delivered into the reservoir (layers) to enhance oil recovery. The spatial configuration of the waves makes them particularly efficient in acting on solid particles, water droplets and oil droplets capillarytrapped in the randomly oriented pores of the porous medium. For example, the study of dynamics of solid particles and fluid droplets (water and oil) in the porous medium demonstrates that spatial waves are the one that create strong motion mechanisms for these matters (thus, causing cleanup from solid particles, displacement of fluid droplets etc.). As noted above, high-frequency waveguide processes may appear to be the most efficient means in the reservoirs with heterogeneities. We also managed to derive the conditions for the efficient use of shock-wave stimulation of the reservoirs, including those containing viscous oil, to enhance oil recovery. As concerns the near-wellbore formation zone, resonant oscillations in the porous medium prove to work most efficiently for it (interconnected resonances in the well and in some perforation channels or in a system of perforations create the internal multiple resonance; besides, the entire near-wellbore zone may remain under conditions of the multiple resonances). The use of such oscillations and waves may equally prove to be efficient for the production of heavy crudes due to the oscillation-induced decrease in the so called ‘effective’ viscosity (in the near-wellbore zone and for capillary-trapped oil in the reservoirs).

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The long-propagating spatial waves are primarily possible due to a special nature by which oscillations and waves are excited (at the oscillation/wave generator), including certain oscillation forms that create the spatial waveguides (specifically, waveguides in the wells and spatial waveguides in the porous media with heterogeneities). The knowledge of excitation conditions (wave characteristics such as oscillation form, frequency spectrum etc.) of the resonant oscillation and spatial waveguide processes, in turn, helps in determining the physical principles behind the development of relevant wave equipment to be used for generation of the optimal oscillation and wave processes. For example, one of such characteristics consists in the excitation of oscillations in the medium (a well or porous medium with heterogeneities) with axially asymmetric oscillation forms, and at certain bandwidth frequencies and with other specifics of wave processes which can be established from the developed theory. In some typical cases of the wave process excitation, the characteristic form of oscillations (at the generator) is what drives the spatial oscillation and waveguide processes in wells and porous media with relevant resonance and amplitude-frequency waveguide characteristics for the efficient treatment of wells and fields. In other cases, if these principles are not followed, the oscillations and waves with requisite characteristics may sometimes fail to propagate in the target direction to the required location and, hence, yield no result, meaning that while oscillations and waves are excited, they do not propagate in the well and into the reservoir. This essentially renders this technique invalid. The above is one of key challenges to be addressed in development of the controlled wave means and oscillation/wave generators and their subsequent application in the described processes. Therefore, as discussed above in the introduction and in Chapter I, at the very beginning of the wave technology development, its application by practical experimentalists and inventors without due understanding and without reference to the fundamentals failed to yield the desired outcome and, in some cases, compromised the science of wave technology. However, the wave technology developers have recently made some fundamentally new breakthrough and showed that it is able to bring about much more opportunities, in particular, for the enhanced oil recovery from the reservoirs with various heterogeneities. Consequently, to be able to use this applied science in the oil industry practice, we face the need for development of appropriate wave equipment, the so called wave generators and associated means, including

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various-type oscillation/wave generators (with a specific oscillation frequency bandwidth and amplitude-frequency characteristics matching the parameters of specific typical fields) and control and measuring systems both for separate wells and groups thereof (fields), to tune to the resonance conditions, stabilize waves and oscillations and control their dynamics. This primarily involves building the wave equipment engineering tools of a certain class for oil industry, specifically automatic tools, able to meet modern scientific and engineering requirements and based on the existing software solutions. These should be designed for typical wells and oil fields using the developed resonant micro and macromechanics of the oil reservoir. In the development of the subject equipment, the account should be also taken of the derived criteria for the detachment of solid particles and fluid droplets in the near-wellbore zone and capillarytrapped oil in the porous media from the pore walls, bearing in mind that these matters are responsible for the declining production from the well and decreasing oil recovery from the reservoirs. Consequently, the first and foremost task to be solved at the first stage is to modify the earlier means proposed by the wave technology developers and full-scale tested in order to create the controlled wave generators, associated means, oscillation/wave generators of a new type using the resonant macro and micromechanics of the reservoir. To substantiate the practical application of the developed technology, it is necessary to study the geological and geophysical parameters of the near-wellbore formation zones, fields and oil reservoirs (as a fluid-filled porous medium) with heterogeneities. The solution of this task is a joint effort that should involve mechanical experts, field geologists and geophysicists. The work is currently underway to develop software packages, design and engineer appropriate wave equipment with associated control and measuring systems based on the developed science and with reference to the geological and geophysical characteristics. The basic technological concepts of the developed resonant macro and micromechanics of the oil reservoir and the standard wave equipment based on these principles have been experimentally validated in various laboratories by the pilot-scale and full-scale tests. The work is ongoing at present to develop the technology and engineer the controlled means and generators for the oil industry. It is necessary to carry out broad-scale tests of these equipment in the future in dedicated oil field sites with various standard geological and geophysical characteristics and subject to the main scientific and technical concepts.

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It needs also to be noted that the described results of resonant macro and micromechanics of the porous media constitute a new fundamental branch of mechanics and engineering that appears to have been pioneered by the authors and their colleagues. These results may have a wide range of applications in various engineering disciplines (not only in the petroleum industry) which may include, for example, the material science, specifically for impregnation of the porous materials (composites) with nano and microfillers (this is a critical problem in the aerospace industry and rocket engineering and, thus far, has not been fully solved); catalyst technologies and membrane technologies in order to increase performance; manufacturing of foam composites with high-performance mechanics; decontamination of the facilities (especially those with the porous surfaces) used in the nuclear fuel cycle; medical technology for development of the new models of human cardiovascular diagnostic equipment based on the studies of regularities in the pulse wave propagation (as helicon waves) in multiple vessels as well as in macro and microcapillaries and in cells (as a porous medium) etc.

Enhanced Oil Recovery: Resonance Macro- and Micro-Mechanics of Petroleum Reservoirs. O. R. Ganiev, R. F. Ganiev and L. E. Ukrainsky. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

Index Average Flow, 76, 79, 83, 87 Axially Symmetric Waves, 108

Harmonic Wave Stimulation, 39, 43 Impact Wave Stimulation, 23, 58, 59

Capillary Effect, 15, 74, 189 Capillary-Trapped Oil, xiv, xxi, 17, 24, 63, 67, 69, 73, 91, 103, 104, 105, 132, 144, 151, 174, 186, 190 Cleaning, xix, 1, 18, 27, 43, 56, 69, 145, 151, 153, 165, 170 Cleaning of Horizontal Wells, 18 Cone-Shaped Pores, 84 Criterion of Successful, 39, 43, 61 Damping Decrements, 121

Mechanism of Trapping of Large Oil Droplets, 73 Mobilization of Droplets and Blobs, 91

Natural Frequency, 95–97, 100, 102, 128, 152 Near-Wellbore Formation, xv, xvii, 1–3, 7, 12, 17, 20, 42, 151, 159, 162, 165, 173, 190 Near-Wellbore Resonances, 12 Enhanced Oil Recovery, xiii, xvii, Nonlinear Wave Forces, 73 xxi, 21, 23, 45, 104, 174, 192 Oil-Surfactant Interface Instability, 102 First-Generation Wave TechnoloOil-Surfactant Interfaces, 95 gies, 1–25 Fluid Droplets, 63, 73, 104, 190, Oil-Water Interface, 63, 74, 92–95, 98, 102 191, 193 Fluid-Saturated Porous Medium, Oil-Water Interface Instability, 98 12, 16, 91, 100, 105, 115 Plane Waves, 115 Gravity-Capillary Waves, 91, 95, Production Stimulation, xiii, 7 96, 100, 102 Propulsion of Droplets, 73

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Resonances, xix, 12, 15, 105, 127, 148, 152, 159, 190 Wave Сleaning, xx, 19, 43 Wave Stimulation, xviii, xx, 1–4, 10, 16, 19, 21, 24, 27, 39, 42, 45, 58, 67, 91, 113, 123, 138, 142, 151, 158, 165, 173, 185 Wave Stimulation of Entire Reservoirs, 21, 24 Waveguides, xix, 105, 121, 127, 132, 141, 151, 155, 189–192