Unsaturated Soils: Behavior, Mechanics and Conditions 9781536159868, 1536159867

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ENVIRONMENTAL REMEDIATION TECHNOLOGIES, REGULATIONS AND SAFETY

UNSATURATED SOILS BEHAVIOR, MECHANICS AND CONDITIONS

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ENVIRONMENTAL REMEDIATION TECHNOLOGIES, REGULATIONS AND SAFETY

UNSATURATED SOILS BEHAVIOR, MECHANICS AND CONDITIONS

MARTIN HERTZ EDITOR

Copyright © 2019 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. We have partnered with Copyright Clearance Center to make it easy for you to obtain permissions to reuse content from this publication. Simply navigate to this publication’s page on Nova’s website and locate the “Get Permission” button below the title description. This button is linked directly to the title’s permission page on copyright.com. Alternatively, you can visit copyright.com and search by title, ISBN, or ISSN. For further questions about using the service on copyright.com, please contact: Copyright Clearance Center Phone: +1-(978) 750-8400 Fax: +1-(978) 750-4470 E-mail: [email protected].

NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the Publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book.

Library of Congress Cataloging-in-Publication Data ISBN: HERRN

Published by Nova Science Publishers, Inc. † New York

CONTENTS Preface Chapter 1

Chapter 2

Chapter 3

Chapter 4

vii Unsaturated Soils: Aspects That Control Moisture Content Teresa M. Reyna, Santiago M. Reyna and María Lábaque Some Aspects of Numerical Modelling of Hydraulic Hysteresis of Unsaturated Soils Javad Ghorbani and David W. Airey Temperature Distribution and Water Migration in Unsaturated Soil Bingxi Li, Yiran Hu, Fei Xu, Yaning Zhang, Shuang Liang, Wenyu Song and Zhongbin Fu Chemical Potential in Soils and Complex Media Jean-Claude Bénet

1

41

79

133

Index

193

Related Nova Publications

199

PREFACE This compilation opens with an exploration of the vadose, or unsaturated zone, which is of utmost importance as the nexus between surface water and groundwater. It is the link between what happens on the surface and what happens below, inside the aquifers. As such, understanding this underground natural environment is essential for the sustainable development of society. Due to the complexities involved in considering the hysteretic response of the Soil Water Characteristic Curve and its dependency on volume changes, these two features are often ignored in numerical studies of unsaturated soils. To facilitate their use in numerical modeling, a model for the Soil Water Characteristic Curve equation based on the bounding surface concept is proposed. The authors go on to focus on the temperature distribution and water migration in unsaturated soil, which is of significant importance because these factors are related to the thermal-physical properties of soil. The changes in these properties may cause disastrous engineering problems such as the cracking of pavement, damage of structure foundation and fracture of pipelines. Lastly, Unsaturated Soils: Behavior, Mechanics and Conditions addresses the measurement and expression of the mass chemical potential in the presence of superficial layers induced by a complex structure. By

viii

Martin Hertz

definition, the mass chemical potential of a constituent in a soil represents the variation of the internal energy of the medium when a unit mass of the constituent is transferred to a reference state with the entropy, volume and mass of the other constituents remaining constant. Chapter 1 - Humankind increasingly puts more pressure on natural resources. The preservation of the underground natural environment is becoming a necessity to guarantee the progress of humanity. The evaluation of soil conditions in the field requires considerable time and effort. This effort, time and equipment needed will depend on the range of uses required for the data. The area and depth to be studied must be carefully defined. The area to study will depend on the variability existing in the place. In certain occasions the characteristics of the soil - water interactions vary more with depth than with the area considered; some cases are just the opposite. The study of the vadose, or unsaturated zone, is of utmost importance because this zone is the nexus between surface water and groundwater. It is the link between what happens on the surface and what happens below, inside the aquifers. Improving the knowledge of this underground natural environment is essential to guarantee the sustainable development of society. Chapter 2 - The Soil Water Characteristic Curve (SWCC) plays an essential role in the response of unsaturated soils as it quantifies the ability of soils to maintain or lose their available moisture. It is generally understood that the SWCCs exhibit hysteresis when subjected to cycles of wetting and drying, and more generally the relationship between suction, volume changes, and the saturation degree for a given soil is not unique and can be path-dependent. In particular, the SWCC may show strong sensitivity to volume changes in cohesive soils. Due to the complexities involved in considering the hysteretic response of the SWCC and its dependency on volume changes, these two features are often ignored in numerical studies of unsaturated soils. To facilitate their use in numerical modeling, a model for the SWCC equation based on the bounding surface concept is proposed. The resulting non-linear model is presented in an incremental form relating the rate of saturation degree to the rate of suction and volume changes. It is shown that the proposed model has the ability to

Preface

ix

replicate the shape and the evolution of the scanning curves observed in the experiments compared to currently available alternatives. The model has benefits for numerical application as it improves the convergence properties of the overall numerical analysis and includes a robust integration scheme with automatic error control for updating the saturation degree during the analyses. Results are presented to show the ability and the robustness of the proposed scheme in modeling the unsaturated soil response under various loading conditions including static and dynamic analyses. Chapter 3 - Temperature distribution and water migration in unsaturated soil are of significant importance because they are related to the thermal-physical properties of soil and the changes in these properties may cause disastrous engineering problems including cracking of pavement, damage of structure foundation, fracture of pipelines, etc. This chapter focuses on the temperature distribution and water migration in unsaturated soil. Temperature distribution and water migration in unsaturated soil during the freezing and thawing processes are experimentally investigated and the results are presented and discussed. Based on the soil structure, multiphase flow, temperature distribution and water potential, a mesoscopic lattice Boltzmann model (LBM) is developed to simulate the temperature distribution and water migration in unsaturated soil. Based on the soil structure, multiphase flow, temperature distribution, water potential, melting film and interactive force, a modified LBM model is also developed to simulate the temperature distribution and water migration in unsaturated soil. Main conclusions and future outlook are also briefly stated and summarized. The contents included in this chapter not only detail the experimental procedures and the results of temperature distribution and water migration in unsaturated soil, but also introduce the models for estimating the temperature distribution and water migration in unsaturated soil. Chapter 4 - By definition, the mass chemical potential (CP) of a constituent in a soil represents the variation of the internal energy of the medium when a unit mass of the constituent is transferred to a reference state with the entropy, volume and mass of the other constituents

x

Martin Hertz

remaining constant. It quantifies the action of the rest of the environment on the constituent considered. For water, it results from mechanical or electrical actions generated by the superficial layers which are responsible for this retention. CP is a fundamental quantity in the same way as temperature or pressure This chapter addresses the measurement and expression of the CP in the presence of superficial layers induced by a complex structure. Measuring CP in soils, gels and various media of biological origin, is based on one of its fundamental properties, namely that, at equilibrium, the chemical potential of a component i has the same value on either side of a membrane permeable to i, temperature being uniform. The proposed analysis revisits conventional methods such as tensiometry or the use of filter paper and exploitation of the desorption isotherm. A mechanical method has been developed to measure the CP of volatile organic compounds in soils. The combined effects of the interfaces and a solute on the CP of constituents of an aqueous solution in the capillary domain are analyzed. In the case of an ideal solution of large molecules (e.g., polyethylene glycol) in a porous medium with a pore size small enough to hinder diffusion, a solute gradient is established. The Gibbs Duhem relationship induces a chemical potential gradient of water. The phenomenon of osmotic dehydration appears and is accompanied by a loss of water and stresses. The effect of gas phase pressure in the capillary medium refers to tensiometric measurement methods, with an important nuance: capillary potential and suction being empirical formulations of the CP, these concepts can then benefit from the body of mechanics, chemistry, thermodynamics, thus allowing a very broad approach to soil function and the expression of state functions, i.e., chemical potential, entropy and internal energy. The authors then examine the case of a saturated medium (solid phase + water) subjected to deformations that may result from external actions or internal water transfer. In the context of elastic behavior of the whole medium, increments of CP and stresses can be expressed as a function of thermomechanical coefficients. Soil is a very special container generating charged surface layers, open on the outside, capable of generating temporospatial CP variations and fluctuations of the constituents of the aqueous solution. Passage of a solution from the free

Preface

xi

state to that of a solution in soil opens perspectives on applications such as plant nutrition, water transfer in hygroscopic environments, pollution/soil remediation, geochemistry, analysis of water transfers in arid zones (desertification) and prebiotic chemistry.

In: Unsaturated Soils Editor: Martin Hertz

ISBN: 978-1-53615-985-1 © 2019 Nova Science Publishers, Inc.

Chapter 1

UNSATURATED SOILS: ASPECTS THAT CONTROL MOISTURE CONTENT Teresa M. Reyna*, PhD, Santiago M. Reyna, PhD and María Lábaque, MS Department of Production, Management and Enviroment, Department of Hydraulics, National University of Cordoba, Cordoba, Argentina

ABSTRACT Humankind increasingly puts more pressure on natural resources. The preservation of the underground natural environment is becoming a necessity to guarantee the progress of humanity. The evaluation of soil conditions in the field requires considerable time and effort. This effort, time and equipment needed will depend on the range of uses required for the data. The area and depth to be studied must be carefully defined. The area to study will depend on the variability existing in the place. In certain occasions the characteristics of the soil *

Corresponding Author’s E-mail: [email protected].

2

Teresa M. Reyna, Santiago M. Reyna and Maria Labaque water interactions vary more with depth than with the area considered; some cases are just the opposite. The study of the vadose, or unsaturated zone, is of utmost importance because this zone is the nexus between surface water and groundwater. It is the link between what happens on the surface and what happens below, inside the aquifers. Improving the knowledge of this underground natural environment is essential to guarantee the sustainable development of society.

Keywords: unsaturated soils, moisture content, unsaturated media

INTRODUCTION Along with air and water, soil is the basis of life on Earth. It has an extremely important role in the production of food, and in the cycles of water and nutrients as well. The formation of soils depends on a long and complex process. It is sluggishly renewable since it continues to be generated through natural processes, but very slowly, and it is not renewable when its destruction is faster than its formation. Humankind is exerting more pressure on natural resources, so the preservation of the natural environment is becoming more and more a necessity to guarantee the sustainable development of a community. In this context, the protection of soil resources has become a priority. When we talk about unsaturated soils, we refer to the portion of soil that is above the groundwater table, which interacts with surface hydrological processes and is very susceptible to be affected by the anthropic activities carried out on the surface. The most outstanding feature of unsaturated soils corresponds to the amount of water they have, which generates a different behavior of them not only from the mechanical point of view but also from the point of view of water, air and water flows, and pollutants transport. This characteristic means that their study can be carried out with different approaches for different problems. Agronomists, geotechnicians, geologists, hydrogeologists, hydrologists, engineers, sanitarists and environmentalists, among others, study unsaturated soils

Unsaturated Soils

3

where the selection of representative variables and hypotheses are adjusted to the interaction phenomenon to be studied. From the agronomical point of view, the amount of water in soil, as well as the infiltration capacity, is a basic input for the calculation of crop water needs and to determine the amount of irrigation water. Currently, the basis for improvements in irrigation efficiency is based on improvements in the calculation of soil moisture conditions and the application of models that allow an efficient application given the costs of irrigation, especially when they are the source as groundwater. From a hydrological point of view, incorporating an infiltration model to determine the amount of water retained in the soil, allowing continuous simulations, permits to obtain hydrographs of surface floods and their flood areas. From a geotechnical point of view, Dudley (1970) commented that Terzaghi had already called attention to the tendency of unsaturated soils to experience volume changes when they were flooded. Partially saturated soils are the most frequent in many arid and semi-arid regions. Much of the behavior experienced by a partially saturated soil is related to volumetric deformation. On the other hand, there is a great diversity of partially saturated soils with particular characteristics or behaviors under different moisture contents, such as very plastic expansive clays (with swelling and shrinkage in wet and dry cycles), alluvial deposits (collapsible soils when they present an open structure), colluvial and aeolian soils, compacted soils, etc. Many of these problems have been tried and solved separately as special ones (Alonso et al., 1987). From the environmental point of view, soil pollution affects the transport of water in the unsaturated zone and contaminates the lower layers, and also affects the surface waters that are linked to groundwater. On the other hand, the effects of the transport of substances in soils affect their quality and therefore their agronomic use. Solid waste landfills and the use of land as the final destination of effluents produce soil contamination. The transport of these pollutants has to be studied because the remediation processes can be expensive and complex. The problem and management of its protection and care by the authorities requires, therefore, a complete vision of its environment and the

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Teresa M. Reyna, Santiago M. Reyna and Maria Labaque

interactions that can generate contamination and deterioration. Pollution and deterioration situations will directly impact on agricultural activities, on surface and underground water resources, including civil infrastructure and urbanization works. This chapter presents a brief description of the water content characteristics of unsaturated soils, the complicated issue of the measurement of humidity, and introduces the issue of impacts produced in the context of climate change.

UNSATURATED SOILS To understand the processes that occur in the soil and how they interact with surface processes, it is necessary to know the characteristics of the soil, the ways to measure the amount of water they have, and the hydraulic functions of the soil that will allow studying different areas. The area from the water table to the floor surface is called the “aeration” zone. In this area the pores contain air and water. It is subdivided into zones (Bear, 1972), capillary, vadose and water - soil. The “capillary” zone, where the water rises by capillarity from the water table, the “vadose” or intermediate zone, where the water is motionless, governed by hygroscopic and capillary forces, and the “ground water” zone, which is the area adjacent to the soil surface and extends below it through the area of plant roots. This area is affected by the conditions of the soil surface, seasonal and diurnal fluctuations, irrigation, precipitation, humidity and air temperature, etc. In cases where the phreatic level is at shallow depth, the vadose zone disappears, since the water ascends by capillarity to the surface. Below the aeration zone, under the phreatic surface to the impermeable mantle, there is the zone of “saturation”; in this area the pores are completely filled with water. On the other hand, three phases coexist in the soil: solid (50%), liquid (20-30%) and gaseous (20-30%) and depending on the type of soil is the percentage of the different phases. These components are interrelated, so

Unsaturated Soils

5

the organization of the solid components together determines the amount of porous space allocated to air and water. As stated, the amount of water and air in the ground is subjected to great fluctuations, especially due to the influence of climatic conditions and movements of the groundwater. In unsaturated soils we can observe three phases: 





Solid Phase: The main constituents of the solid phase are mineral or inorganic species and organic matter. The inorganic fraction corresponds to a mixture of several primary components (quartz, feldspars, etc.) and secondary (silicate clays), varying its size from clay to rock fragments. They determine, in relation to other components, the physical and physical chemical properties of soils. In addition, they constitute a source of nutrients. The organic fraction is composed of organic substances in varying degrees of decomposition, including soil organisms, living and dead. The humid colloidal fraction affects the physical and chemical properties of soils, such as porosity, water retention and cation exchange capacity among others. It is one of the sources of nutrients such as nitrogen, phosphorus and sulfate. Liquid Phase: This phase is a solution containing dissolved salts in small amounts in the form of ions, variable in quantity, and has the ability to dissolve different solutes, in a dynamic relationship between water and soil, producing chemical reactions. In some cases, it is also possible to observe the presence of polluting substances such as organic compounds and pesticides to which the variation in temperature could also be added. Gas Phase: Corresponds to the product of the exchange between numerous living organisms of the soil and the “atmosphere” of the soil and, like the external atmosphereit is a mixture of gases such as O2, N2, CO2 and minor gases. The soil has higher CO2 content and a lower N2 and O2 content than the external air, due to the biological activity of the soil. The relative humidity of this atmosphere can reach 100% since part of the air is dissolved in the

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Teresa M. Reyna, Santiago M. Reyna and Maria Labaque

soil solution. In addition, it provides the oxygen necessary for the life of most soil organisms and plants (Honorato, 2000). The knowledge of the existing interactions between the three phases of the system constitutes the basic point for understanding the unsaturated soil behavior. Two of the most important parameters for studying unsaturated soils are their porosity (typical of the nature of the soil) and the representative elementary volume (related to the appropriate scale to define the average values of the properties).

Porosity As it was presented in previous paragraphs, the soil is constituted by three phases: Solid, Liquid and Gaseous (Air). The pores can be classified into macropores and micropores according to their size. The porosity of a material represents a percentage that relates the volume occupied by the pores in a unit volume of soil; thus, if the porosity is 50%, it means that half of the rock is made up of pores and the other half is solid particles. Table 1. Porosity values for different soil types (Davis, 1969) Material Clays Fine sands Medium sands Thick sands Gravel Sand and gravel mixed Glacial deposits Solid rocks and dense rocks Unequal fractured and weakened rocks Recent permeable basalts Vesicular lava Volcanic rock Sand conglomerate Carbonate rock with original and secondary porosity

Porosity percentage 50-60 40-50 35-40 25-35 20-30 10-30 25-45 1 h  1

(1)

where θr and θs, residual and saturated water content, , empirical parameter, , pore distribution index, and h, indicates suction. Van Genuchten (1980) presented a smooth function for moisture as a function of suction with more attractive properties. This function depends, in addition to θr and θs, on three empirical parameters α, m and n that affect the shape of the humidity curve. This equation has as limit the expression of Brooks and Corey for λ equal to m times n. Van Genuchten (1980) presented an equation for calculating the effective degree of saturation, which has advantages for its implementation in flow calculation models in unsaturated porous media,

Se=

1 [1+ ( h )n ] m

(1)

where , n and m are empirical constants. Van Genuchten equation with m = 1 was used by Ahuja and Swartzenruber (1972), Endelman and others (1974) and Varallyay and Mironenko (1979). To improve the description of water retention in the soil near saturation, several continuously differentiable (smooth) equations have been proposed. These include functions presented by King (1965), Visser (1968), Laliberte (1969), Su and Brooks (1975), and Clapp and Hornberger (1978). These functions were able to more accurately reproduce the observed water retention data in the soil, but were mathematically too complicated to be incorporated into models to determine hydraulic conductivity, or they had other characteristics that made them less attractive for the study of water infiltration in soils (such as the lack of a simple inverse relationship). If constraints are imposed on parameters m and n (e.g., m = 1-1/n, or m = 1-2/n), an expression of the relatively simple hydraulic conductivity function can be reached. Conversely, considering variables α and n leads to mathematical expressions for K and D (hydraulic

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Teresa M. Reyna, Santiago M. Reyna and Maria Labaque

diffusivity, D = K dh/dq) that are very complicated to study the flow of water in soils. Van Genuchten et al. (1991) concluded that the model presented in 1980 fits very well the moisture data observed for most soils if m and n variables are considered. The only exceptions are certain structures or aggregates of soils characterized by a mainly bimodal pore size distribution. Even so, these authors do not recommend the use of this function for all observed moisture data sets, as in the case of field measurements, where the available data correspond only to a narrow range of humidity. Unless further measurements are made at relatively low water contents (laboratory measurements), such data can lead to the definition of an unsuitable retention curve in the dry range (Van Genuchten et al., 1991). In the models presented by Brooks and Corey and Van Genuchten we find the values of θr and θs defined in a traditional way, that is: Parameter θr is the content of water in the soil that will not participate in the flow of the liquid phase because its flow paths are blocked or because the adsorption produced by the solid phase is very strong (Luckner et al., 1989). Formally, θr, can be defined as the water content at which dθ/dh and K tend to zero when h tends to large values. The residual water content is an extrapolated parameter, and therefore, does not necessarily represent the lowest water content that a soil can have. This is especially true in arid regions where the transport of the gas phase can dry soils at water contents much lower than θr. The saturated water content, θs, denotes the maximum volumetric water content of a soil. The saturated water content is not equal to the porosity of the soil; θs in the field is generally between 5 to 10% less than the porosity due to trapped or dissolved air. In the water retention functions in the soil of these models, parameters, θr and θs, are constants obtained essentially from statistical adjustments made on measured values. The Vogel and Cislerova model (1988) modifies Van Genuchten’s (1980) equations to add flexibility in the description of hydraulic properties near saturation. The proposed model replaces θs in the Van Genuchten model by a fictitious parameter extrapolated θm, slightly larger than θs, for a minimum capillary height, h, other than zero. Vogel and Cislerová (1988)

Unsaturated Soils

15

(presented by Simunek and others, 1996), modified Van Genuchten’s equations (1980) adding flexibility in the description of hydraulic properties near saturation. The water retention function of the soil,  (h) is given by the equation:

 (h)=

r+

 m - r m (1 +| h |n ) s

h < hs

(3) h  hs

where hs is the air input value and m, n are the same defined in van Genuchten’s expression, m is a fictitious parameter a little larger than s. Parr, Zou and Mc. Enroe (1998) use Van Genuchten’s model to represent the hydraulic functions of soil in a study of the effects of infiltration on the transport of pollutants in agriculture. The model of Celia et al. (1987) proposes an exponential function of conductivity that is a reasonable model for data obtained from the laboratory. Fredlund et al. (1994) developed equations to describe characteristic water retention curves similar to those of Van Genuchten (1980) and combine them with the hydraulic conductivity model of Childs and Collis (1950). They showed that the proposed equation to describe retention curves is effective in predicting the permeability coefficient in most soils. In cases where there is no retention curve data for high suction values, this equation can be used to estimate the behavior of the retention curve in these ranges. Mualem (1976a) presents a model in which hydraulic conductivity is expressed as a function of saturated hydraulic conductivity, and a parameter l estimated by Mualem as 0.5 being the average value for many soils. In this model hydraulic conductivity decreases when n decreases, parameters n and m are the same parameters that were defined in the Van Genuchten model, and when n is equal to 1 relative hydraulic conductivity is identical to zero. When n is less than 1, the conductivity function cannot be predicted; this characteristic is an important limitation of the case of variables m, n. For this reason, van Genuchten, Leij and Yates recommend

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Teresa M. Reyna, Santiago M. Reyna and Maria Labaque

the use of variables m, n only for the case of having well-defined moisture data, and the use of the restriction m = 1 - 1/n for all other cases. The Mualem model (1976a) expresses hydraulic conductivity as:

K( S e ) = K s S le [

f( S e ) 2 ] f(1)

(4)

where

f( S e ) =  0S e

1 dx h(x)

(5)

Ks is the hydraulic conductivity in saturation state and l is a parameter of pore connectivity. Equation (4) can be solved using the complete functions Beta ((p,q)) and incomplete Beta (I(p,q)). The equations for conductivity and hydraulic diffusivity assume that the value of hydraulic conductivity in a saturated state is well defined and can be easily measured. This is true for granular soils, but for soils in the natural state, this is not true. The inspection of the conductivity and diffusivity curves shows that a small change in the moisture content produces changes of several orders in K and D, which indicates that small errors in the measurement of moisture content near saturation can produce large errors in the estimation of the saturated hydraulic conductivity of the soil. Stankovich and Lockington (1995) propose the use of a method to convert Van Genuchten’s model (1980). First used by Klenhard and others (1989), this model presents results suitable for soils with a relatively small pore distribution and less suitable for use in clay soils. In the Burdine hydraulic conductivity model, parameter l has the value of 2. One of the most important differences between the model proposed by Burdine and the model of Mualem is that in the Burdine model the value of n> 2 is maintained, while Mualem is only valid for all n> 1. Since

Unsaturated Soils

17

many soils have values less than 2, the Burdine model is less applicable than the Mualem expression. The model of Brooks and Corey (1964, 1966) poses the following expression for the function of hydraulic conductivity as a function of suction:

K ( h) =

K s(

hb 2+( 5/2) ) h

h > hb

(6)

h  hb

Ks

where hb: suction, : pore distribution index. Table 3. Parameters for water content and hydraulic conductivity models Moisture models

Parameters for water content models

Parameters for models of hydraulic conductivity

Brooks and Corey (1964,1966) Van Genuchten (1980)

r, s, , 

Ks, hb, 

s, , n, m, r s, a, p, q r, m, s, , n, m

Ks, , n, m, l

Fredlund et al. (1994) Vogel and Cislervá (1998)

Ks, Kk, r, k, m, (h) Ks, l, 0, r, s, (h)

Mualem (1976a)

Ks, l, (h), s

Burdine (1953) Celia et al. (1987)

Ks, ha, 0, (h)

K0/A, 

K0, 

Boadu (2000) proposes the use of new regression models that take into account the distribution of grain size in compacted soils for the determination of saturated hydraulic conductivity. Table 3 shows the necessary parameters for each model presented in this chapter. The model by Celia et al., similar to the one proposed by Chen et al. (2001), proposes a linear relationship between hydraulic conductivity of unsaturated soils and moisture content, and a decreasing exponential relation for the function of unsaturated hydraulic conductivity in function

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Teresa M. Reyna, Santiago M. Reyna and Maria Labaque

of the suction and for the function of moisture depending on the suction. In those soils where the relationship between hydraulic conductivity and humidity is linear, the function proposed by Celia and others is adequate to represent them (Reyna, 2000). For example, in Wyckoff and Botset sand experiments (1936), and Irmay’s theoretical analysis (1955), shown by Bear (1972), it was observed that for saturation values above 40%, the linear adjustment is adequate, and therefore the functions of Celia and others can be applied. This model or similar ones, such as that presented by Chen et al. (2001), are used for theoretical studies because they allow Richards equation to be linearized. Hydraulic functions depend on some parameters they need for their calibration of the hydraulic properties, determined by means of tests. There are complications for measuring the hydraulic properties of soils due to two important factors: non-linearity of conductivity and non-linearity of moisture as a function of suction. A solution to the problem of hydraulic properties in all suction values results in using mathematical models that allow representing the hydraulic properties of unsaturated soils. The data obtained from the measurements made in the field or in the laboratory have the problem that they cannot cover the entire range of moisture that the soil suffers. Mathematical models have the advantage of representing in an approximate way the soil at the points where measurements were made and, allow obtaining values in all the states of the soil where no data were obtained through measurements. Within all the mathematical models that can be used, semi-empirical models have the great advantage that they represent the functions of hydraulic conductivity and moisture in function of suction in an appropriate form and they need less measurements for their definition. The choice of the best model to represent the hydraulic properties of unsaturated soils will depend on the subsequent use to be made with these functions. It is also important to remember that soils undergo the hysteresis phenomenon during wetting and drying processes and, on the other hand, the non-uniformity of the pore distribution causes that there is always some air and water left in the wetting and drainage processes. The lack of definition of the curves for hysteresis cycles can generate some problems

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19

in infiltration studies, especially when it is desired to try to couple the equations of surface runoff and infiltration to perform hydrological modeling.

PROCESSES UNDER WHICH THE SOLUTE IS SUBJECTED DURING ITS TRANSPORT ON THE GROUND The process of transporting a solute in the soil where volatilization, which brings the solute into the atmosphere, runoff to water surfaces, and leaching into groundwater (Cheng, 1990), will appear, are affected by diffusion, convection and dispersion processes. In addition, there are other processes that, added to the above, affect some solutes in their interaction with the geological environment such as retention (adsorption or sorption), and transformation. The processes of chemical transformation can be catalyzed by the constituents of the soil or photochemically induced.

MEASUREMENT OF SOIL MOISTURE CONTENT The evaluation of soil moisture content at different suction conditions in the field requires considerable time and effort, as well as equipment. The effort, time and equipment needed will depend on the range of interest required for the data, be it between -15 to -20 x 102 kPa or only between 0 to -50 kPa. (Linares, 2012). The area of the soil and the depth for which the determination of the relationship is necessary must be carefully defined. The selection of the size of the area where the studies will be carried out will depend on the variability existing in the place whose characterization is the final objective of the study. In certain occasions the characteristics of the soil - water vary more with depth than with area. The moisture in the soil depends mainly on texture or particle size distribution. On the other hand, the content of organic matter and the

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Teresa M. Reyna, Santiago M. Reyna and Maria Labaque

composition of the solution phase can play a determining role in soil moisture function or retention function. Organic matter has a direct effect on the retention function due to its hydrophilic nature, and an indirect effect due to the modification of soil structure that can be affected by the presence of organic matter (Reyna, 2000). Currently, there are different equipment and techniques for measurements among which neutron probe, tensiometers, sensors, reflectometry and others can be mentioned.

Neutron Probe A neutron probe is a sophisticated and accurate piece of equipment that measures the moisture content in soil. It requires calibration and operation by a licensed operator. This technique is based on the theory that fast neutrons are thermalized when they collide with a body of similar mass, such as hydrogen nuclei. The energy of the neutrons is transmitted to the protons which causes the neutron “bounce or shock” to be much lower. The application of this technique consists of three steps: 1) emission of fast neutrons from a radioactive source, 2) attenuation of the velocity of the neutrons after successive collisions with the atoms at the point of emission and 3) accounting for neutrons with attenuated velocity by a detector near the source. By means of electrical impulses, the neutrons captured by the detector are translated into a digital reading. To convert the reading of the neutron probe to volumetric moisture, a calibration model is necessary, where the volumetric moisture of the soil is the main factor. There are some factors that influence measurements such as hydrogen from organic matter, chlorine, iron and boron present in the soil, capable of attenuating neutrons and absorbing thermally neutron nuclei (Díaz Trujillo, 2007).

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Tensiometers Tensiometers are widely used to measure the available water content of the soil when the matrix potential is high (Reyna, T. 2000). These devices are simple, they are not very expensive, and they are very practical in agricultural systems. They consist of a porous ceramic capsule permeable to water and solutes, connected to a pressure gauge by means of a transparent plastic tube that is filled with water, in such a way that the column of water in its interior forms a continuous with the water of the solution of the soil in the surrounding space through the porous capsule. The values obtained reflect the soil tension; are negative, and the operating range is from 0 to -80 kPa; below this value the water column breaks, penetrating the air and invalidating the following measurements. The tensiometers are insensitive to the osmotic potential of water in the soil and therefore do not provide an adequate measurement of the water potential in soils with significant salinity. Tensiometers are often used in combination with the neutron sprayer, resistance blocks or psychrometers to cover the full range of soil moisture. They require relatively frequent maintenance, which consists of adding water plus a solution for the control of algae (Diaz Trujillo, 2007).

Granular Matrix Sensors They have been developed more recently (they were patented in 1985 and manufactured commercially since 1989). They measure the electrical resistance between two electrodes inserted in a small cylinder composed of a porous material. Each device is covered by a membrane consisting of a stainless steel coupling, externally covered by a rubber that makes the sensor more durable than the plaster block. However, the recorder is calibrated to give the value in water tension, by means of an equation that takes into account the temperature of the soil estimated or measured near the sensor. The size of the pores in the matrix is greater than that of the

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pores in the gypsum blocks, allowing greater sensitivity in the range of more water content in the soil (Diaz Trujillo, 2007).

Dielectric Sensors The TDR and FDR probes measure the dielectric constant of the medium, which is an intrinsic property of the medium. The FDR system calculates the moisture of a soil by responding to changes in the dielectric constant of the medium, using a frequency domain reflectometry technique known as capacitance, while the TDR uses time domain reflectometry (Robinson, et al. 2008).

Time Domain Reflectometry (TDR) The TDR system consists of an oscilloscope connected to two or three metal rods that are inserted parallel to the ground. If a difference of potential is applied to one end of the rods, the energy is transmitted along the ends to the end, where they are reflected to the oscilloscope. In it, the evolution of the potential over time is measured. Some equipment consists of two main parts: the electronic unit, and the waveguides. The electronic unit contains the oscilloscope and the central processor, which controls all measurement, display, and storage functions. The waveguides can be installed horizontally or vertically and remain permanently on the ground to make periodic measurements in the same location or be used in a portable way. The TDR uses a series of conversion tables to convert the dielectric constant to a percentage of moisture in the soil. There are different conversion tables to be used with the different types of waveguides. It is not necessary to have a different table for different soil types since the dielectric constant depends more on the amount of water than on the other soil components. The apparatus calculates the average value of humidity over the total length of the waveguides. The apparatus allows manual

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measurements or continuous measurements by connecting the fixed sensors to a data-logger.

Reflectometry in the Frequency Domain (FDR) The FDR method is also known as a capacitance probe. The electrodes and the adjacent soil form a capacitor whose capacity is a function of the dielectric constant of the soil. This is related empirically to the volumetric content of water. A capacitance sensor requires a probe calibration for each floor and horizon to obtain an optimal measurement of volumetric moisture. The volume of soil measured is not dependent on the type of soil or water content and approaches a cylinder 10 cm high with a diameter of about 25 cm, assuming there are no spaces with air (Velez et al. 2007). All capacitance sensors installed in soils, even with similar characteristics, must be calibrated with the aim of improving their accuracy given the influence on the measurement of other factors independent of the moisture content such as pH variability or electrical conductivity inside the porous matrix. Capacitance sensors are the most economical and easy to install. In addition, they allow a continuous recording of moisture values in the soil, enabling direct information and in real time. They are very useful for the planning of alert monitoring systems. They can be used as substitutes for neutron probes.

CLIMATE CHANGE EFFECTS ON SATURATED AND UNSATURATED SOILS Global warming, environmental degradation, the loss of biodiversity, the pollution of inland waters and the growth of conflict potential due to competition for the use of natural resources are serious problems affecting

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land. Climatic demands, such as drought, can lead to increased pumping of groundwater or investment in new storage areas, which have long-term effects. The increase in uncertainty, regarding climate outcomes and, therefore, the supply of fresh water can change the directions of investment and use in regional resources. Increasing the magnitude and frequency of floods can result in changes in erosion and sedimentation processes and in the mobility of biological and chemical pollutants in both surface water, soil, and groundwater. Population density and the use of resources per capita have increased dramatically during the last century, and watersheds, aquifers and associated ecosystems have undergone important modifications that affect the vitality, quality, and availability of the resource (UNESCO, 2011). The existence of reliable data sources and access to them are fundamental for the evaluation, prediction and adaptation of global change. A generalized decrease in the entry of water into the subsoil and the storage of groundwater will also lead to a reduction in the natural discharges of these, affecting, therefore, the ecosystems that depend on groundwater. It is expected that, in some regions of the planet, a greater intensification of the risks associated with extreme climatic phenomena such as floods and, especially, droughts will happen. In addition to affecting the amount of water that reaches the surface of the land in the form of rain or snow, the increase of greenhouse gases in the atmosphere will also influence the concentration of CO2 that is dissolved in the rainwater. Associated with this phenomenon, the type of vegetation cover will be modified and the functioning of the ecosystems associated with groundwater (wetlands, rivers, springs, etc.), could also be modified by the decrease in water content in the soils. The increase in temperature also causes the increase of wild fires and therefore affects the soils. Meteorological conditions can cause ignition of fires as well as rapid expansion of the fires as a consequence of strong winds.

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The heating of the surface of the earth and the atmosphere is mainly a consequence of solar radiation from the Sun, however, on a smaller scale, heat can be caused by a large fire. In the environment of forest fires, direct sunlight and high temperatures can preheat fuels, bringing them closer to their point of ignition. The physical characteristics of the soil are affected by the heating of the soil during forest fires. These characteristics include: texture, clay content, structure, bulk density, and porosity (quantity and size). Physical properties such as wettability and structure are affected at relatively low temperatures, while the content of quartz sand that contributes to the texture is less affected and only occurs at extreme temperatures. The components of soil texture (sand, silt and clay) have high temperature thresholds and are not affected by fire unless they are subject to high temperatures on the soil surface. Fires and associated soil warming can destroy the structure of the soil, affecting both the total porosity and the pore size distribution in the surface horizons of a soil. Climate change represents a serious threat to world food security, in large part due to its effects on soils. Changes in temperature and rainfall patterns can have a great impact on organic matter and the processes that take place in our soils, as well as on the plants and crops that grow in them. In order to respond to the challenges related to global food security and climate change, agriculture and land management practices must undergo fundamental transformations. Improving agricultural practices and soil management that increase soil organic carbon-such as agroecology, organic farming, conservation agriculture, and agroforestry-bring multiple benefits. They produce fertile soils that are rich in organic matter (carbon), maintain soil surfaces with vegetation, require less chemical inputs, and promote crop rotation and biodiversity. These soils are also less susceptible to erosion and desertification, and will maintain vital ecosystem services such as hydrological and nutrient cycles, which are essential to maintain and increase food production. (FAO, 2015).

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CONCLUSION The concept of how we should use nature is evolving rapidly. The resources provided by the planet are important for the development of life, some of them are renewable and others non-renewable according to their availability in time, their rate of generation (or regeneration), and their rate of use or consumption. In this chapter, concepts were developed that clarify and allow establishing bases of these water-soil relationships and allow them to be modeled. It is necessary to continue carrying out new studies, in particular in the subject of obtaining data in order to characterize unsaturated soils in greater depth, especially since they must be applied by different scientists with different approaches. Regarding the study of soils, this is fundamental since food production activities are developed in them. Therefore, soils care depends on knowledge, and to take care of something needs this knowledge basis first. Soils must also be protected from both point and non-point contamination. Unsaturated soils play an essential role in the care of soil and water resources, so their nature and vulnerabilities must be studied and modeled in order to anticipate the impacts that these resources could suffer.

REFERENCES Alonso, E. E., Gens, A., Hight, D. W. 1987. Special problem soils. General report. In proceedings of the 9th European Conference on Soil Mechanics and Foundation Engineering. 3: 1087-1146. Bear, J. 1972. Dynamics of Porous Media. Dover Publications, Inc. N.Y., U.S.A. Bishop, A. W. 1959. The Principle of Effective Stress. Teknik Ukeblad. Norwegian. Brooks, R. H. y Corey, A. T. 1966. Properties of Porous Media Affecting Fluid Flow. Journal of Irrigation and Drainage Division. 92: 61-90.

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Brooks, R. H., Corey, A. T. 1964. Hydraulic Properties of Porous Media. Hydrology Paper. No 3, Civil Engineering Department, Colorado State University, Fort Collins. USA. Celia, M. A., Ahuja, R., Lajpat; Pinder, G. F. 1987. Orthogonal Collocation and Alternating-Direction Procedures for Unsaturated Flow Problems. Journal of Advances in Water Resources, 10: 178-187. Chen, J.-M., Tan, Y.-Chi, Chen, Chu-Hui. 2001. Multidimensional Infiltration with Arbitrary surface fluxes. Journal of Irrigation and Drainage Engineering. 127: 370-377. Davis, S. N. 1969. Porosity and permeability of natural materials. Flow Through Porous Media, edited by R. J. M. De Wiest. 53-89. Academic Press, New York. Díaz Trujillo, M. V. 2007 Studies of the soil-surfactant-pesticide system in the adsorption and desorption processes of atrazine, MBT and chlorpyrifos. MD diss., University of Chile. Dudley, J. H. 1970. Review of collapsing soils. Journal of the Soil Mechanics and Foundations Division, 96: 925 – 947. FAO, 2015 Soils help combat and adapt to climate change. http://www. fao.org/3/a-i4737s.pdf. Fredlund, D. G., Rahardjo, H. 1993. Soil Mechanics for Unsaturated Soils. Wiley-Interscience Publication. John Wiley & Sons, INC. Fredlund, D.; Xing. A y Huang, S. 1994. Predicting the Permeability Function for Unsaturated Soils Using the Soil-Water Characteristic Curve. Canadian Geotechnical Journal. 32: 533-546. Honorato R. 2000. Manual of Edaphology. Catholic University of Chile. Irmay, S. 1955. Flow of Liquid Through Cracked Media. Bulletin of the Research Council of Israel, Nº 1, 5ª, pp. 84. U.S.A. Jennings, J. E. B. and Burland, J. B. 1962. Limitations to the use of effective stress in partly saturated soils. Geotechnique 12-2: 125 - 144. Klute, A. editor. 1986. Physical and Mineralogical Methods. Methods of Soil Analysis. SSSA Book Series 5.

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Linares, J. 2012. Application of a One-dimensional Model for Flow in Saturated and Unsaturated Soils and Transport of Herbicides. Application in the Central Zone of the Province of Cordoba. MD diss. National University of Cordoba. Lu Ning y Likos W. J. 2004. Unsaturated Soils Mechanics. John Wiley & Sons Inc. Reyna, T. 2000. Hydraulic functions in unsaturated soils. Application to the Pampean loess. PhD diss., National University of Cordoba. Robinson, D., Jones, S., Wraith, J., Firedmena, S. 2008. A review of advances in dielectric and previous thermoelectrical conductivity measurements in soils using time domain reflectometry. Vadose Zone Journal. 2-4: 444–475. Doi 10.2113/2.4.444. Stankovich, J. M., Lockington, D. A. 1995. Brooks-Corey and van Genuchten soil water-retention models. Journal of Irrigation and drainage Engineering. 121 (1) :1-7. UNESCO. 2011. The Impact of global change on water resources: the response of UNESCO’s International Hydrological Programme. https://unesdoc.unesco.org/ark:/48223/pf0000192216. Van Genuchten, M. Th., 1980. A Closed Form Equation for Predicting the Hydraulic Conductivity of Unsaturated. Soil Science Society of America Journal. 2: 444-475. Doi: 10.2136/sssaj1980.03615 995004400050002x. Vélez, J., Intrigliolo, D., Castel, J. 2007. Programming irrigation in citrus fruits based on sensors measuring the water status of the soil and the plant. Water Engineering Journal, 14 (2). 127-135. Doi: https:// doi.org/10.4995/ia.2007.2907. Vogel, T. y Cislerova, M. 1988. On the Reliability of Unsaturated Hydraulic Conductivity Calculated from the Moisture Retention Curve. Transport in Porous Media. 3 (1), 1-15. Wyckoff R. D. y Botset H. G. 1936. The Flow of Gas-Liquid Mixtures Through Unconsolidated Sands. Journal of General and Applied Physics. 7: 325-345. Doi.org/10.1063/1.1745402.

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BIOGRAPHICAL SKETCHES Teresa Maria Reyna Affiliation: Department Production, Managament and Enviromental and Department of Hydraulics, National University of Cordoba, Cordoba, Argentina Education: PhD in Engineering Sciences. National University of Cordoba. Master’s in Engineering Sciences – Specialization in Water Resources. National University of Cordoba. Civil Engineer. National University of Cordoba Research and Professional Experience: 

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2017 to the present: Member of the Board of Directors of the High Institute of Environmental Studies of the National University of Cordoba. Professor in Charge of Technology, Environment and Society. Department of Production, Management and Environment. School of Exact, Physical and Natural Sciences. National University of Cordoba. 2018-Present. 2009 to the present: Director of Postgraduate Specialization in Hydraulics. Director of the Department of Hydraulics, Faculty of Exact, Physical and Natural Sciences. May 2016-December 2016. Sub-director of the Department of Hydraulics of the School of Civil Engineering, Faculty of Exact, Physical and Natural Sciences. August 2014 - May 2016. Adjunct Professor through competitive examination of the Chair of “Hydrology and Hydraulic Processes”. Department of Hydraulics. School of Exact, Physical and Natural Sciences. National University of Cordoba. 2005-Present.

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Adjunct Professor of the Chair of “Hydrology and Hydraulic Processes”. Department of Hydraulics. School of Exact, Physical and Natural Sciences. National University of Cordoba. 1998 2005. Adjunct Professor of the Chair of “Hydraulics Works”. Department of Hydraulics. School of Exact, Physical and Natural Sciences. National University of Cordoba. 1998 - Present. Professor of the Master’s Program in Engineering Sciences (Water Resources; Transportation; and Management). School of Exact, Physical and Natural Sciences. National University of Cordoba. 2001 – Present.

Professional Appointments: 

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Managing Partner of CEAS SA, consulting company dedicated to studies and projects on hydraulics engineering, environmental engineering and sanitation. Partner of GCIS, Consortium of Consulting Engineering Companies, Cordoba, Argentina. Advisor of the Sub-Secretariat of Water Resources, National Government, Argentina. March of 2006 to March 2010. Member of the Editorial Board of Pinnacle Journal Publication. Pinnacle Agricultural Research and Management. Online Edition. ISSN: 2360-9451. Associate Editor of the Journal Ambiente & Água - An Interdisciplinary Journal of Applied Science/Institute of Environmental Research in Bacias Hidrográficas. Taubaté Taubaté: IPABHi. ISSN Quarterly Edition 1980-993X, 2014/2015.

Honors: She has been awarded numerous schollarships by prestigious official institutions as: National Board of Scientific and Technological Research (CONICET), and the Secretary of Science and Technology (SECYT) of the National University of Córdoba. Besides, she has received awards as first candidate in the selection of the Graduate Program of

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Masters’ in Structural Engineering, Honor Diploma of the Engineering Association of Córdoba and Special Mention (“Silver Medal”) for second best GPA of her class of Civil Engineering, School of Exact, Physical and Natural Sciences. Publications from the Last 3 Years: Editor of book Garcia-Chevesich, Pablo A., Neary, Daniel G., Scott, David F. Benyon, Richard G., Reyna, Teresa. Editors. Forest Management and the Impact on Water Resource: A Review of 13 Nations. IHP VIII/Technical document Nº 37. Latin America and the Caribbean. (2017). ISBN 978-92-3-100216-8. http://unesdoc.unesco.org/ images/0024/002479/247902E.pdf Author of Book Chapters Reyna, Teresa, Labaque, María; Reyna, Santiago; Funes, Fernanda. 2018. Flow in unsaturated soils and transport of herbicides in agricultural areas. In “Soil Contamination”, ISBN 978-953-51-6385-5. Reyna, Teresa, Reyna, Santiago, Labaque Maria. 2017. Chapter 1. Forest Management and Water in Argentina. Forest Management and the Impact on Water Resource: A Review of 13 Nations. IHP VIII/Technical document Nº 37. Latin America and the Caribbean. ISBN 978-92-3-100216-8. http://unesdoc.unesco.org/images/0024/ 002479/247902E.pdf. Publications in Magazines with Reference Reyna, Santiago; Teresa Reyna; Fabián Fulginiti; María Lábaque. 2018. “Aplicación de Métodos Optimización para Hidrogeneración. Estudio del Sistema de los Ríos Las Cañas - Gastona -Medina”. EPIO. Investigacion Operativa - Año XXVI - Nº 43 – 24-36. [Application of

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Optimization Methods for Hydrogeneration. Study of the Las Cañas River System - Gastona -Medina. EPIO. Operative investigation] Reyna, Teresa, Santiago Reyna, María Lábaque, César Riha, Carlos Góngora. 2017. Desafíos ambientales para uso de la hidrogeneración. Avances en Ciencias e Ingeniería (ISSN: 0718-8706). Vol. 8, N° 3. Chile. [Environmental challenges for the use of hydrogeneration. Advances in Science and Engineering] Reyna, Teresa; Belén Irazusta, Maria Lábaque, Santiago Reyna, and Cesar Riha. 2019. “Hydraulic Microturbines, Design, Adaptations for Teaching of Microgeneration”. Modern Environmental Science and Engineering. ISSN: 2333-2581. Reyna, Teresa; Santiago Reyna, María Lábaque, César Riha, Belén Irazusta, and Agustín Fragueiro. 2018. “Design of Microturbines Kaplan and Turgo for Microgeneration Systems: Challenges and Adaptations”. Modern Environmental Science and Engineering Volume 4, Number 8. ISSN 2333-2581. Reyna, Teresa; Santiago Reyna, María Lábaque, César Riha, Florencia Grosso. 2016. Applications of Small Scale Renewable Energy. Journal of Business and Economics ISSN 2155-7950, USA. Vol. 7, No. 2, pp. 250-258. DOI: 10.15341/jbe(2155-7950)/02.07.2015/008. Santiago Ochoa,. Teresa Reyna, Carlos M. García, Horacio Herrero, José Manuel Díaz, and Ana Heredia 2017. Análisis de la implementación de un modelo hidrodinámico tridimensional al flujo de un cauce natural. Revista Ingeniería del Agua. Universitat Politècnica de València. Fundación para el Fomento de la Ingeniería del Agua. e-ISSN: 18864996 - ISSN: 1134-2196. Vol. 21 N° 2. [Analysis of the implementation of a three-dimensional hydrodynamic model to the flow of a natural channel. Water Engineering Magazine.] She has numerous refereed publications and has actively participated in national and international congresses and conferences.

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Santiago Maria Reyna Affiliation: Department Production, Managament and Enviromental and Department of Hydraulics, National University of Cordoba, Córdoba, Argentina Education: PhD, MSCE, MSCE, Purdue University, USA. Civil Engineer. National University of Córdoba Research and Professional Experience: 

 

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Managing Partner of CEAS SA, consulting company dedicated to studies and projects on hydraulics engineering, environmental engineering and sanitation. Partner of GCIS, Consortium of Consulting Engineering Companies, Cordoba, Argentina. Advisor of the Minister of Water, Environment and Public Services, Province of Córdoba, Argentina. March of 2014 to present. Advisor of the Sub-Secretary of Water Resources, National Government, Argentina, (liaison with the ORSEP, INA, ENOHSA; cooperation in the assessment and follow-up of Projects of Hydraulic Works; in charge of Hydroelectric Projects; liaison with the Secretariat of the Environment). March of 2006 to March 2014. He has represented the Republic of Argentina at the Plenary Meeting in Nairobi of the United Nations Environment Program (UNEP) and at the G5, in Geneva, on topics of the Environment and Water Resources. He was, until Dec. 2013, Alternate Member of the Adaptation Fund Board (Kyoto Protocol) representing GRULAC (Latin America and the Caribbean Group).

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Teresa M. Reyna, Santiago M. Reyna and Maria Labaque 

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2008 to the present: Plenary Professor of the Chairs of Environmental Engineering and Hydraulics Works. National University of Córdoba. 1997 to 2008: Full Professor through competitive examination of the Chair of Hydraulics Works. Department of Hydraulics. School of Exact, Physical and Natural Sciences. National University of Cordoba. 2000 to the present: Full Professor of the Chair of Environmental Engineering. Department of Hydraulics. School of Exact, Physical and Natural Sciences. National University of Córdoba. Resolution 24 - HCD-00. 2001 to 2017: Head of the Graduate School of the School of Exact, Physical and Natural Sciences, National University of Córdoba. 2006 to the present: Director of two Masters’ Programs in Engineering Sciences: Environmental Engineering and Water Resources. 2013 to the present: Coordinator of the Career of Environmental Engineering, National University of Córdoba. 2006 to 2008: Member of the Board of Directors of the High Institute of Environmental Studies of the National University of Córdoba.

Professional Appointments:  

 

Consultant engineer for national and international organizations. President and Partner of CEAS SA, consulting company dedicated to studies and projects on hydraulics engineering, environmental engineering and sanitation, Córdoba, Argentina. CEO of GCIS, Consortium of Consulting Engineering Companies, Córdoba, Argentina. Managing Partner of “Ambientes y Sistemas SRL” (until December 2013), consulting company dedicated to the area of

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environmental and sustainable development studies and the application of their management, Córdoba, Argentina. Honors:   

 

Full Academic of the Pan American Academy of Engineering (2010). Full Academic of the “Academia del Plata”, Argentina (2013). Diplomate, Water Resources Engineer (DWRE), of the American Academy of Water Resources Engineers (AAWRE), American Society of Civil Engineers (2007). “Medalla de oro” (gold medal) for obtaining his degree in Civil Engineering with highest honors (1984). University Award for Professors, National University of Córdoba (1995).

Publications from the Last 3 Years: Author of book chapters Reyna, Teresa, Labaque, María; Reyna, Santiago; Funes, Fernanda. 2018. Flow in unsaturated soils and transport of herbicides in agricultural areas. In Soil Contamination, ISBN 978-953-51-6385-5. Reyna, Teresa, Reyna, Santiago, Labaque Maria. 2017. Chapter 1. Forest Management and Water in Argentina. Forest Management and the Impact on Water Resource: A Review of 13 Nations. IHP VIII/Technical document Nº 37. Latin America and the Caribbean. ISBN 978-92-3-100216-8. http://unesdoc.unesco.org/images/0024/ 002479/247902E.pdf. Publications in Magazines with Reference Reyna, Santiago; Teresa Reyna; Fabián Fulginiti; María Lábaque. 2018. “Aplicación de Métodos Optimización para Hidrogeneración. Estudio

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Teresa M. Reyna, Santiago M. Reyna and Maria Labaque

del Sistema de los Ríos Las Cañas - Gastona -Medina”. EPIO. Investigacion Operativa - Año XXVI - Nº 43 – 24-36. [Application of Optimization Methods for Hydrogeneration. Study of the Las Cañas River System - Gastona -Medina. EPIO. Operative investigation] Reyna, Teresa, Santiago Reyna, María Lábaque, César Riha, Carlos Góngora. 2017. Desafíos ambientales para uso de la hidrogeneración. Avances en Ciencias e Ingeniería (ISSN: 0718-8706). Vol. 8, N° 3. Chile. [Environmental challenges for the use of hydrogeneration. Advances in Science and Engineering] Reyna, Teresa; Belén Irazusta, Maria Lábaque, Santiago Reyna, and Cesar Riha. 2019. “Hydraulic Microturbines, Design, Adaptations for Teaching of Microgeneration”. Modern Environmental Science and Engineering. ISSN: 2333-2581. Reyna, Teresa; Santiago Reyna, María Lábaque, César Riha, Belén Irazusta, and Agustín Fragueiro. 2018. “Design of Microturbines Kaplan and Turgo for Microgeneration Systems: Challenges and Adaptations”. Modern Environmental Science and Engineering Volume 4, Number 8. ISSN 2333-2581. Reyna, Teresa; Santiago Reyna, María Lábaque, César Riha, Florencia Grosso. 2016. Applications of Small Scale Renewable Energy. Journal of Business and Economics. ISSN 2155-7950, USA. Vol. 7, No. 2, pp. 250-258. DOI: 10.15341/jbe(2155-7950)/02.07.2015/008. He has numerous refereed publications and has actively participated in national and international congresses and conferences.

Maria Labaque Affiliation: Department Production, Managament and Enviromental and Department of Hydraulics, National University of Cordoba, Córdoba, Argentina

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Education: Master’s in Engineering Sciences – Specialization in Water Resources. National University of Córdoba. Civil Engineer. Catolic University of Córdoba Research and Professional Experience: 

 





Professor of Technology, Environment and Society. Department of Production, Management and Environment. School of Exact, Physical and Natural Sciences. National University of Cordoba. 2018-Present. 2015 to the present: Technical Secretary of Postgraduate Specialization in Hydraulics. Adjunct Professor of the Chair of “Hydraulics Works”. Department of Hydraulics. School of Exact, Physical and Natural Sciences. National University of Cordoba. Adjunct Professor to the Chairs of: Fluid Mechanics”. Department of Hydraulics. School of Exact, Physical and Natural Sciences. National University of Cordoba. Professor of the Master’s Program in Engineering Sciences (Water Resources; Transportation; and Management). School of Exact, Physical and Natural Sciences. National University of Cordoba. 2015 – Present.

Professional Appointments: 

Managing Partner of CEAS SA, consulting company dedicated to studies and projects on hydraulics engineering, environmental engineering and sanitation.

Honors: 

She has been awarded scholarship by the Department of Science and Technology (SECYT).

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Teresa M. Reyna, Santiago M. Reyna and Maria Labaque  

She represented the Catholic University of Córdoba before the Argentine Committee of Dams during 2003. She represented the National University of Córdoba before the Committee of the Salí-Dulce Basin in the Workshop of Contamination. El Mollar, Province of Tucumán. June 1998.

Publications from the Last 3 Years: Author of book chapters Reyna, Teresa, Labaque, María; Reyna, Santiago; Funes, Fernanda. 2018. Flow in unsaturated soils and transport of herbicides in agricultural areas. In Soil Contamination, ISBN 978-953-51-6385-5. Reyna, Teresa, Reyna, Santiago, Labaque Maria. 2017. Chapter 1. Forest Management and Water in Argentina. Forest Management and the Impact on Water Resource: A Review of 13 Nations. IHP VIII/Technical document Nº 37. Latin America and the Caribbean. ISBN 978-92-3-100216-8. http://unesdoc.unesco.org/images/0024/ 002479/247902E.pdf. Publications in Magazines with Reference Reyna, Santiago; Teresa Reyna; Fabián Fulginiti; María Lábaque. 2018. “Aplicación de Métodos Optimización para Hidrogeneración. Estudio del Sistema de los Ríos Las Cañas - Gastona -Medina”. EPIO. Investigacion Operativa - Año XXVI - Nº 43 – 24-36. [Application of Optimization Methods for Hydrogeneration. Study of the Las Cañas River System - Gastona -Medina. EPIO. Operative investigation] Reyna, Teresa, Santiago Reyna, María Lábaque, César Riha, Carlos Góngora. 2017. Desafíos ambientales para uso de la hidrogeneración. Avances en Ciencias e Ingeniería (ISSN: 0718-8706). Vol. 8, N° 3. Chile. [Environmental challenges for the use of hydrogeneration. Advances in Science and Engineering]

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Reyna, Teresa; Belén Irazusta, Maria Lábaque, Santiago Reyna, and Cesar Riha. 2019. “Hydraulic Microturbines, Design, Adaptations for Teaching of Microgeneration”. Modern Environmental Science and Engineering. ISSN: 2333-2581. Reyna, Teresa; Santiago Reyna, María Lábaque, César Riha, Belén Irazusta, and Agustín Fragueiro. 2018. “Design of Microturbines Kaplan and Turgo for Microgeneration Systems: Challenges and Adaptations”. Modern Environmental Science and Engineering Volume 4, Number 8. ISSN 2333-2581. Reyna, Teresa; Santiago Reyna, María Lábaque, César Riha, Florencia Grosso. 2016. Applications of Small Scale Renewable Energy. Journal of Business and Economics. ISSN 2155-7950, USA. Vol. 7, No. 2, pp. 250-258. DOI: 10.15341/jbe(2155-7950)/02.07.2015/008. She has numerous refereed publications and has actively participated in national and international congresses and conferences.

In: Unsaturated Soils Editor: Martin Hertz

ISBN: 978-1-53615-985-1 © 2019 Nova Science Publishers, Inc.

Chapter 2

SOME ASPECTS OF NUMERICAL MODELLING OF HYDRAULIC HYSTERESIS OF UNSATURATED SOILS Javad Ghorbani and David W. Airey School of Civil Engineering, The University of Sydney, Camperdown, NSW, Australia

ABSTRACT The Soil Water Characteristic Curve (SWCC) plays an essential role in the response of unsaturated soils as it quantifies the ability of soils to maintain or lose their available moisture. It is generally understood that the SWCCs exhibit hysteresis when subjected to cycles of wetting and drying, and more generally the relationship between suction, volume changes, and the saturation degree for a given soil is not unique and can be path-dependent. In particular, the SWCC may show strong sensitivity to volume changes in cohesive soils.



Corresponding Author’s E-mail: [email protected].

42

Javad Ghorbani and David W. Airey Due to the complexities involved in considering the hysteretic response of the SWCC and its dependency on volume changes, these two features are often ignored in numerical studies of unsaturated soils. To facilitate their use in numerical modeling, a model for the SWCC equation based on the bounding surface concept is proposed. The resulting non-linear model is presented in an incremental form relating the rate of saturation degree to the rate of suction and volume changes. It is shown that the proposed model has the ability to replicate the shape and the evolution of the scanning curves observed in the experiments compared to currently available alternatives. The model has benefits for numerical application as it improves the convergence properties of the overall numerical analysis and includes a robust integration scheme with automatic error control for updating the saturation degree during the analyses. Results are presented to show the ability and the robustness of the proposed scheme in modeling the unsaturated soil response under various loading conditions including static and dynamic analyses.

Keywords: unsaturated soils, hysteresis, finite element, multiphase flow, porous media, plasticity

INTRODUCTION Over the last twenty years, there has been a growing attempt to explain and predict the response of unsaturated soils, which often exist in the upper zone over much of the earth’s surface. In unsaturated soils, solid, gas and liquid phases are simultaneously present. Numerical equations governing solid and fluid interactions in a porous medium were first developed by extending Biot’s theory to unsaturated soils (Lewis and Schrefler, 1982; Li et al., 1989; Zienkiewicz et al., 1990 and Li and Zienkiewicz, 1992). The authors assumed that neither phase transfer nor any chemical reactions were possible during fluid flow, which was referred to as “immiscible.” In numerical analyses based on this approach, simplifying assumptions have been made to ignore the effect of volume changes on variations of the degree of saturation and to neglect the hysteretic behaviour of the SWCCs during the analysis (Schrefler and Scotta, 2001; Khalili et al., 2008; Khoei and Mohammadnejad, 2011; Ghorbani et al., 2016a and Ghorbani, 2016).

Some Aspects of Numerical Modelling of Hydraulic Hysteresis …

43

While the assumptions lead to simpler numerical implementation, they can be in disagreement with experimental evidence, (Sheng and Zhou, 2011) which shows that SWCCs can depend strongly on volume change (most importantly for fine-grained soils) and exhibit hysteretic responses during wetting and drying. In this chapter, a dynamic fully coupled model which does not involve the aforementioned simplifications will be presented and details of important aspects of the modelling of hydraulic hysteresis in unsaturated soils will be given.

MATHEMATICAL FRAMEWORK FOR UNSATURATED SOIL DYNAMICS The effective stress suggested by Bishop (1960) has been adopted (Equation (1)) which provides the capability of smooth transitions between saturated and unsaturated states. 𝛔′ = 𝛔 + 𝜒𝑝𝑤 𝐦 + (1 − 𝜒)𝑝𝑔 𝐦

(1)

In the above equation, 𝛔′ is the effective stress, 𝑝𝑔 denotes the air pressure and 𝑝𝑤 is the pore water pressure. In addition, 𝜒 is the effective stress or Bishop’s parameter, which ranges from 0 to 1 for dry to saturated conditions, respectively. To maintain consistency with the authors’s earlier publication (Ghorbani et al., 2014); Ghorbani et al., 2016a; Ghorbani et al., 2016b; Ghorbani et al., 2016c and Ghorbani et al., 2017) the water and air pressures are assumed positive in compression; whereas stress components are assumed negative in compression. Moreover, 𝜒 is considered as saturation degree, 𝑆𝑤 (e.g., Schrefler, 1984). In addition, suction, 𝑝𝑐 , is defined as 𝑝𝑐 = 𝑝𝑔 − 𝑝𝑤

(2)

44

Javad Ghorbani and David W. Airey

Solid (s), water (w) and gas (g) phases in unsaturated soils are assumed to be continuously spread throughout space. The degree of saturation, 𝑆𝛽 (𝛽 = 𝑤, 𝑔), the density of each phase 𝜌𝛼 (𝛼 = 𝑠, 𝑤, 𝑔), and the volume fraction of each phase in the soil 𝑛𝛼 are obtained from: 𝑆𝛽 =

𝛺𝛽 , 𝛺−𝛺𝑠

𝜌𝛼 =

𝑀𝛼 , 𝛺𝛼

and 𝑛𝛼 =

𝛺𝛼 𝛺

(3)

where 𝛺𝛼 and 𝛺 are the volume occupied by each phase and the total volume of the unsaturated soil, respectively; and 𝑆𝑤 + 𝑆𝑔 = 1. 𝑀𝛼 is the mass of each phase inside the soil. Porosity of the unsaturated soil, 𝑛 is defined by 𝑛 = 𝑛w + 𝑛g = 1 − 𝑛s

(4)

The average density of the soil, 𝜌, is defined as 𝜌 = (1 − 𝑛)𝜌𝑠 + 𝑛𝑆𝑤 𝜌𝑤 + 𝑛𝑆𝑔 𝜌𝑔

(5)

Conservation of Mass and Momentum Balance Based on the principle of conservation of mass, inside an arbitrary volume 𝛺, the flows of the soild, water, and gas phases through a surface, d𝛤 can be described as 𝜌𝛼 𝑛𝛼 . 𝑽𝜶 . 𝐧∗ . d𝛤, where 𝐧∗ is the unit vector normal to the surface, 𝜌𝛼 𝑛𝛼 is the partial mass density of each material, and 𝑽𝜶 is the velocity of each phase. The principle of mass conservation is defined by 𝑑 ∫ 𝑑𝑡 𝛺

𝜌𝛼 𝑛𝛼 𝑑𝛺 = − ∮𝛤 𝜌𝛼 𝑛𝛼 . 𝐕 𝜶 . n∗ . 𝑑𝛤

(6)

By applying the Gauss theorem, the conservation of mass can be written in the form of a differential equation, as follows:

Some Aspects of Numerical Modelling of Hydraulic Hysteresis … 𝜕(𝜌𝛼 𝑛𝛼 ) + div(𝜌𝛼 𝑛𝛼 𝐕 𝜶 ) 𝜕𝑡

=0

45 (7)

By considering 𝐕 𝒔 = 𝐮̇ for the solid phase, ignoring the spatial variations of the solid phase (𝛁𝜌𝑠 = 0, with 𝛁 as the gradient operator) and neglecting the compressibility of solid particles, the following will be obtained: 𝜕𝑛

−𝜌𝑠 𝜕𝑡 − 𝜌𝑠 𝐮̇ . 𝛁𝑛 + (1 − 𝑛)𝜌𝑠 div(𝐮̇ ) = 0

(8)

Dividing Equation (8) by 𝜌𝑠 and considering the concept of material time derivative, 𝐷𝑛 𝐷𝑡

𝐷(∗) 𝐷𝑡

=

𝜕(∗) 𝜕𝑡

+ 𝐮̇ . 𝛁 ∗ results in the following equation:

= (1 − 𝑛)div(𝐮̇ )

(9)

Also, the conservation of mass for the fluid phases can be written as Ghorbani et al., (2018c): 1 𝐰̇ 𝑤 div(𝐰̇ 𝑤 ) + . 𝛁𝜌𝑤 𝑆𝑤 S𝑤 𝜌𝑤

𝑛𝐷(𝜌𝑤 ) 𝑛𝐷(𝑆𝑤 ) + 𝜌𝑤 𝐷𝑡 𝑆𝑤 𝐷𝑡

+ div(𝐮̇ ) +

𝑛𝐷(𝜌𝑔 )

+ div(𝐮̇ ) + 𝑆 div(𝐰̇𝑔 ) + 𝑆

=0

(10)

and

𝜌𝑔 𝐷𝑡

+

𝑛𝐷(𝑆𝑔 ) 𝑆𝑔 𝐷𝑡

1

𝑔

𝐰̇ 𝑔 . 𝛁𝜌𝑔 𝑔 𝜌𝑔

=0

(11)

In an isothermal environment, the rate of change in the fluid density is: 1 𝐷𝜌𝛽 𝜌𝛽 𝐷𝑡

1 𝐷𝑝𝛽 , (𝛽 𝛽 𝐷𝑡

=𝐾

= 𝑤, 𝑔)

(12)

where 𝐾𝛽 is the bulk modulus of the fluid phases. Substituting Equations (12) into (10) and (11) leads to the following:

46

Javad Ghorbani and David W. Airey 𝑛 𝐷𝑝𝑤 𝐾𝑤 𝐷𝑡

+

𝑛𝐷𝑆w 𝑆𝑤 𝐷𝑡

𝑛 𝐷𝑝𝑔 𝐾𝑔 𝐷𝑡

+𝑆

𝐰̇ 𝑤 . 𝛁𝜌𝑤 𝑤 𝜌𝑤

1

+ div(𝐮̇ ) + 𝑆 div(𝐰̇ 𝑤 ) + S 𝑤

=0

(13)

and 𝑛𝐷𝑆𝑔 𝑔 𝐷𝑡

1

𝐰̇ 𝑔 . 𝛁𝜌𝑔 𝑔 𝜌𝑔

+ div(𝐮̇ ) + 𝑆 div(𝐰̇𝑔 ) + 𝑆 𝑔

=0

(14)

To consider the dependency of saturation degree on suction and porosity (or void ratio), we can describe the rate of saturation change by 𝐷𝑆𝑤 𝐷𝑡

=

𝜕𝑆𝑤 𝐷𝑝𝑐 𝜕𝑝𝑐 𝐷𝑡

+

𝜕𝑆𝑤 (1 − 𝜕𝑛

𝑛)div(𝐮̇ )

(15)

By adding Equation (15) to (13) and (14) and multiplication of (13) and (14) by 𝑆𝑤 and 𝑆𝑔 and defining 𝐶1 =

𝑛𝑆𝑤 𝐾𝑤

(16)

𝜕𝑆

𝐶2 = 𝑛 𝜕𝑝𝑤

(17)

𝑐

𝐶3 =

𝑛𝑆𝑔

(18)

𝐾𝑔

𝐶4 =

𝑛 𝑆 𝐾𝑔 𝑔

𝐶5 =

𝜕𝑆𝑤 (1 𝜕𝑛

𝐶6 = 𝑆𝑔 −

𝜕𝑆w 𝜕𝑝𝑐

(19)

− 𝑛) + 𝑆𝑤

(20)

−𝑛

𝜕𝑆𝑤 (1 − 𝜕𝑛

𝑛)

the following governing equations will be obtained:

(21)

Some Aspects of Numerical Modelling of Hydraulic Hysteresis … 𝐰̇ 𝑤

𝐶1 𝑝̇𝑤 + 𝐶2 𝑝̇𝑐 + 𝐶5 div(𝐮̇ ) + div(𝐰̇ 𝑤 ) + 𝜌 . 𝛁𝜌𝑤 = 0

47 (22)

𝑤

and 𝐶3 𝑝̇𝑤 + 𝐶4 𝑝̇𝑐 + 𝐶6 div(𝐮̇ ) + div(𝐰̇𝑔 ) +

𝐰̇ 𝑔 . 𝛁𝜌𝑔 𝜌𝑔

=0

(23)

where the super-imposed dot represents the time derivative of a variable. For the non-solid phases, the linear momentum balance equation is following Li and Zienkiewicz (1992) as 𝐰̇𝛽 = 𝐤𝛽 [−𝛁𝑝𝛽 + 𝜌𝛽 (𝐛 − 𝐮̈ ) −

𝐷 𝐰̇ 𝛽 ( )] , (𝛽 𝐷𝑡 𝑛𝑆𝛽

= 𝑤, 𝑔)

(24)

where 𝐤 𝑤 and 𝐤𝑔 are the permeability matrices of unsaturated soils, defined by 𝐤𝛽 = 𝐤 𝑖𝑛𝑡 .

𝑘𝑟𝛽

(25)

𝜂𝛽

with 𝑘𝑟𝑤 and 𝑘𝑟𝑔 being the relative permeability of the water phase and the air phase, respectively; 𝐤 𝑖𝑛𝑡 is the intrinsic or absolute permeability matrix of the soil, 𝜂𝛽 denotes the viscosity of non-solid phases. The linear momentum balance equation for the unsaturated soil can be obtained by 𝐷

𝐰̇ 𝑔

𝐷

𝐰̇ 𝑤

𝐋𝑇 𝛔 + 𝜌𝐛 − 𝜌𝐮̈ − 𝑛 [(1 − 𝑆𝑤 )𝜌𝑔 𝐷𝑡 (𝑛(1−𝑆 )) + S𝑤 𝜌𝑤 𝐷𝑡 (𝑛𝑆 )] = 0 𝑤

𝑤

(26) where

48

Javad Ghorbani and David W. Airey 𝜕 𝜕𝑥

0

0

𝐋𝑇 = 0

𝜕 𝜕𝑦

0

[0

0

𝜕 𝜕𝑧

𝜕 𝜕𝑦 𝜕 𝜕𝑥

0

0 𝜕 𝜕𝑧 𝜕 𝜕𝑦

𝜕 𝜕𝑧

0

(27)

𝜕 𝜕𝑥 ]

According to Lewis and Schrefler (1999) and Zienkiewicz et al., (1999), the relative acceleration of the fluids can be assumed to be negligible, so that: 𝐷 (𝐰̇𝛽 ) 𝐷𝑡

= 0, (𝛽 = 𝑤, 𝑔)

(28)

The imposed Dirichlet boundary conditions for the primary variables on the boundaries are: ̅ on 𝛤𝑢 𝐮=𝐮

(29)

𝑝𝑤 = 𝑝̅𝑤 on 𝛤𝑝𝑤

(30)

𝑝𝑤 = 𝑝̅𝑐 on 𝛤𝑝𝑐

(31)

whereas the Neumann boundary conditions on the prescribed tractions and fluxes are: 𝐈𝜎𝑇 𝛔 = 𝐭̅ on 𝛤𝑡

(32)

̅̇ 𝑤 on 𝛤𝑞 𝐤 𝑤 [−𝛁𝑝𝑤 + 𝜌𝑤 (𝐛 − 𝐮̈ )]. 𝐧∗ = 𝑤 𝑤

(33)

̅̇ 𝑔 on 𝛤𝑞 𝐤𝑔 [−𝛁𝑝𝑔 + 𝜌𝑔 (𝐛 − 𝐮̈ )]. 𝐧∗ = 𝑤 𝑔

(34)

̅̇ 𝛽 (𝛽 = 𝑤, 𝑔) are the prescribed values of the outflow rate of nonwhere 𝑤 solid phase on the permeable boundaries 𝛤𝑞𝛽 (𝛽 = 𝑤, 𝑔). Moreover, we define,

Some Aspects of Numerical Modelling of Hydraulic Hysteresis …

𝐈𝜎𝑇

𝑛𝑥 =[0 0

0 𝑛𝑦 0

0 0 𝑛𝑧

𝑛𝑦 𝑛𝑥 0

0 𝑛𝑧 𝑛𝑦

𝑛𝑧 0] 𝑛𝑥

49

(35)

and 𝑛𝑥 𝐧∗ = [ 𝑛 𝑦 ] 𝑛𝑧

(36)

with 𝑛𝛼 (𝛼 = 𝑥, 𝑦, 𝑧) as unit outward normal vector to the boundaries in x, y and z directions. Note that 𝛤𝑢 ∪ 𝛤𝑡 = 𝛤 , 𝛤𝑞𝑤 ∪ 𝛤𝑝𝑤 = 𝛤 and 𝛤𝑞𝑔 ∪ 𝛤𝑝𝑐 = 𝛤 . The weak form of the governing equation for the unsaturated soils can then be written as: ∫𝛺 𝛿𝑝𝑤 (𝐶1 𝑝̇𝑤 + 𝐶2 𝑝̇𝑐 + 𝐶5 div(𝐮̇ ) + div(𝐤 𝑤 [−𝛁𝑝𝑤 + 𝜌𝑤 (𝐛 − 1

𝐮̈ )]) + 𝜌 𝐤 𝑤 [−𝛁𝑝𝑤 + 𝜌𝑤 (𝐛 − 𝐮̈ )]. 𝛁𝜌𝑤 ) 𝑑𝛺 + ∫𝛺 (𝛿𝑝𝑐 + 𝑤

𝛿𝑝𝑤 ) (𝐶3 𝑝̇𝑤 + 𝐶4 𝑝̇𝑐 + 𝐶6 div(𝐮̇ ) + div(𝐤𝑔 [−𝛁𝑝𝑔 + 𝜌𝑔 (𝐛 − 𝐮̈ )]) + 1 𝐤 [−𝛁𝑝𝑔 𝜌𝑔 𝑔

+ 𝜌𝑔 (𝐛 − 𝐮̈ )]. 𝛁𝜌𝑔 ) 𝑑𝛺 + ∫𝛤 𝛿𝑝𝑤 [𝐤 𝑤 [−𝛁𝑝𝑤 + 𝑞𝑤

∗

̅ 𝑤]

𝜌𝑤 (𝐛 − 𝐮̈ )]. 𝐧 − 𝑤̇

𝑑Г − ∫𝛤 (𝛿𝑝𝑐 + 𝛿𝑝𝑤 )[𝐤𝑔 [−𝛁𝑝𝑔 + 𝜌𝑔 (𝐛 − q𝑔

̅̇ 𝑔 ] 𝑑Г = 0 𝐮̈ )]. 𝐧 − 𝑤 ∗

(37)

∫𝛺 𝛿𝑢(𝐋𝑇 𝛔 + 𝜌𝐛 − 𝜌𝐮̈ )𝑑𝛺 + ∫𝛤 𝛿𝑢(𝐭̅ − 𝐈𝜎𝑇 𝛔)𝑑Г = 0 𝑡

(38)

Spatial and Time Discretisations by Finite Element Approach The finite element discretisation interpolates the unknown displacements, pore water pressures and suctions, respectively shown by 𝐮, 𝑝𝑤 and 𝑝𝑐 , per:

50

Javad Ghorbani and David W. Airey 𝐮 = 𝐍𝐮 𝐔, 𝑝𝑤 = 𝐍𝐩𝐰 𝐏𝐰 and 𝑝𝑐 = 𝐍𝐩𝐜 𝐏𝐜

(39)

where 𝐍𝐮 , 𝐍𝐩𝐰 and 𝐍𝐩𝐜 are the shape functions for the displacement, pore water pressure and suction, respectively. Based on the above interpolation functions, and considering pore water pressure as positive in compression, the following system of fully-coupled algebraic equations can be derived in matrix form as: 𝐌𝐮 𝐔̈ + 𝐂𝐔̇ + ∫𝛺 𝐁𝑇 𝛔′ d𝛺 − 𝐐𝐰 𝐏𝐰 − 𝐐c 𝐏𝐜 = 𝐅𝐮

(40)

𝐌𝐰 𝐔̈ + 𝐐𝑇𝐰 𝐔̇ + 𝐂𝐰𝐰 𝐏̇𝐰 + 𝐂𝐰𝐜 𝐏̇𝐜 + 𝐇𝐰𝐰 𝐏𝐰 + 𝐇𝐰𝐜 𝐏𝐜 = 𝐅𝐰

(41)

𝑇 𝐌𝐜 𝐔̈ + 𝐑𝐓𝐜 𝐔̇ + 𝐂𝐜𝐰 𝐏̇𝐰 + 𝐂𝐜𝐜 𝐏̇𝐜 + 𝐇𝐰𝐜 𝐏𝐰 + 𝐇𝐜𝐜 𝐏𝐜 = 𝐅𝐜

(42)

where 𝐁 is the strain-displacement matrix and 𝐂 is the damping matrix. The definitions of the various matrices and vectors are given below: 𝐌𝐮 = ∫𝛺 𝐍𝐮𝑇 𝜌𝐍𝐮 𝑑𝛺 𝑇

(43)

𝐌𝐰 = ∫𝛺 (𝛁𝐍𝐩𝐰 ) ( 𝐤𝑔 𝜌𝑔 + 𝐤 𝑤 ρ𝑤 )𝐍𝐮 𝑑𝛺

(44)

𝐐𝐜 = ∫𝛺 𝐁𝑇 (1 − 𝑆𝑤 )𝐦𝐍𝐩𝐜 𝑑𝛺

(45)

𝐐𝐰 = ∫𝛺 𝐁𝑇 𝐦𝐍𝐩𝐰 𝑑𝛺

(46)

𝐌𝐂 = ∫𝛺 (𝛁𝐍𝐩𝐜 )𝑇 𝐤𝑔 𝜌𝑔 𝐍𝐮 𝑑𝛺

(47)

𝑇

𝐇𝐰𝐰 = ∫𝛺 (𝛁𝐍𝐩𝐰 ) ( 𝐤𝑔 + 𝐤 𝑤 )(𝛁𝐍𝐩𝐰 )d𝛺 𝑇

𝐇𝐜𝐜 = ∫𝛺 (𝛁𝐍𝐩𝐜 ) ( 𝐤𝑔 )(𝛁𝐍𝐩𝐜 )𝑑𝛺

(48) (49)

Some Aspects of Numerical Modelling of Hydraulic Hysteresis … 𝑇

51

𝐇𝐰𝐜 = ∫𝛺 (𝛁𝐍𝐩𝐰 ) ( 𝐤𝑔 )(𝛁𝐍𝐩𝐜 )𝑑𝛺

(50)

𝑇 𝐂𝐰𝐰 = ∫𝛺 𝐍𝐩𝐰 𝐶1∗ 𝐍𝐩𝐰 𝑑𝛺

(51)

𝑇 ∗ 𝐂𝐜𝐜 = ∫𝛺 𝐍𝐩𝐜 𝐶2 𝐍𝐩𝐜 𝑑𝛺

(52)

𝑇 𝐂𝐰𝐜 = ∫𝛺 𝐍𝐩𝐰 𝐶3∗ 𝐍𝐩𝐜 𝑑𝛺

(53)

𝑇 ∗ 𝐂𝐜𝐰 = ∫𝛺 𝐍𝐩𝐜 𝐶3 𝐍𝐩𝐰 𝑑𝛺

(54)

𝐑 𝐜 = ∫𝛺 𝐁𝑇 𝐶6 𝐦𝐍𝐩𝐜 𝑑𝛺

(55)

It is assumed that 𝐶1∗ = 𝐶1 + 𝐶3 =

𝑛𝑆𝑤 𝐾𝑤

+

𝑛

𝑛𝑆𝑔

(56)

𝐾𝑔

𝜕𝑆

𝐶2∗ = 𝐶4 = 𝑆𝑔 𝐾 − 𝑛 𝜕𝑝𝑤 𝑔

(57)

𝑐

𝑛

𝐶3∗ = 𝐶4 + 𝐶2 = 𝑆𝑔 𝐾

(58)

𝑔

The load and the flow vectors are defined by: ̅ 𝛤 𝐅𝐮 = ∫𝛺 𝐍𝐮T 𝜌𝐛𝑑𝛺 + ∫𝛤 𝐍𝐮𝑇 𝐭𝑑

(59)

𝑡

𝑇 𝑇 ̅ 𝐅𝐰 = ∫𝛺 (𝛁𝐍𝐩𝐰 ) ( 𝐤𝑔 𝜌𝑔 + 𝐤 𝑤 𝜌𝑤 )𝐛 𝑑𝛺 − ∫𝛤 𝐍𝐩𝐰 𝑤̇𝑤 𝑑𝛤 − 𝑞𝑤

∫𝛤

𝑞𝑤 ∩ 𝛤𝑞𝑔

𝐰̇𝑔𝑇 𝜌𝑔

𝑇 ̅ 𝐍𝐩𝐰 𝑤̇𝑔 𝑑𝛤 − ∫𝛤

𝛁𝜌𝑔 ) 𝑑𝛺

𝑃𝑐 ∩ 𝛤𝑞𝑤

𝐰̇ 𝑇

𝑇 𝑇 𝐍𝐩𝐰 𝐰̇𝑔𝑇 𝐧∗ 𝑑𝛤 − ∫𝛺 𝐍𝐩𝐰 ( 𝜌 𝑤 𝛁𝜌𝑤 + 𝑤

(60)

52

Javad Ghorbani and David W. Airey 𝐰̇ 𝑇

𝑇

𝑇 ̅ 𝑇 𝑔 𝐅𝐜 = ∫𝛺 (𝛁𝐍𝐩𝐜 ) 𝐤𝑔 𝜌𝑔 𝐛 𝑑𝛺 − ∫𝛤 𝐍𝐩𝐜 𝑤̇𝑔 𝑑𝛤 − ∫𝛺 𝐍𝐩𝐜 𝛁𝜌𝑔 𝑑𝛺 𝜌 𝑞𝑔

𝑔

(61) where 𝐛 represents the vector of body force. Also by using Voigt notation in this paper, we define matrix 𝐦 in three-dimensional space as follows 𝐦𝑇 = [1

0 0]

1 1 0

(62)

The generalised-α method (originally introduced by Chung and Hulbert (1993) for structural dynamics and later modified for the case of partially saturated soils by Ghorbani et al., (2014) and Ghorbani et al., (2016)) is chosen. The integration parameters in this method satisfy the following relations: 𝛼𝑚 =

2𝜌∞ −1 , 𝜌∞ +1

𝜌∞ ∞ +1

𝛼𝑓 = 𝜌

1

and 𝛽𝑡 = 4 (1 − 𝛼𝑚 + 𝛼𝑓 )2

and the following condition yields second order accuracy 1

𝛾𝑡 = 2 − 𝛼𝑚 + 𝛼𝑓

(63)

where 𝜌∞ is the desired value of the spectral radius at infinity, and 𝛽𝑡 and 𝛾𝑡 are Newmark’s parameters. The unconditional stability of the method is guaranteed if 1

1

𝜃𝑡 ≥ 0.5, 𝛼𝑚 ≤ 𝛼𝑓 ≤ 0.5 and 𝛽𝑡 ≥ 4 + 2 (𝛼𝑓 − 𝛼𝑚 ) where 𝜃𝑡 is an integration parameter. Coupling the generalized–α method with a Newton-Raphson iteration gives the following equations for Jacobian matrix, 𝐉

Some Aspects of Numerical Modelling of Hydraulic Hysteresis … 1 𝛾 (1 − 𝛼𝑚 )𝐌𝐮 + 𝑡 (1 − 𝛼𝑓 )𝐂 + (1 − 𝛼𝑓 )𝐊 𝛽𝑡 Δ𝑡 2 𝛽𝑡 Δ𝑡 1 𝛾 (1 − 𝛼𝑚 )𝐌𝐰 + 𝑡 (1 − 𝛼𝑓 )𝐐𝑇𝐖 𝐉= 𝛽𝑡 Δ𝑡 2 𝛽𝑡 Δ𝑡 1 𝛾 (1 − 𝛼𝑚 )𝐌𝐜 + 𝑡 (1 − 𝛼𝑓 )𝐑𝑇𝐜 [ 𝛽𝑡 Δ𝑡 2 𝛽𝑡 Δ𝑡

53

−𝐐𝐰 (1 − 𝛼𝑓 )

−𝐐𝐜 (1 − 𝛼𝑓 )

1 (1 − 𝛼𝑓 )𝐂𝐰𝐰 + (1 − 𝛼𝑓 )𝐇𝐰𝐰 𝜃𝑡 Δ𝑡 1 𝑇 (1 − 𝛼𝑓 )𝐂𝐜𝐰 + (1 − 𝛼𝑓 )𝐇𝐰𝐜 𝜃𝑡 Δ𝑡

1 (1 − 𝛼𝑓 )𝐂𝐰𝐜 + (1 − 𝛼𝑓 )𝐇𝐰𝐜 𝜃𝑡 Δ𝑡 1 (1 − 𝛼𝑓 )𝐂𝐜𝐜 + (1 − 𝛼𝑓 )𝐇𝐜𝐜 𝜃𝑡 Δ𝑡 ]

MODELING HYSTERETIC BEHAVIUOR OF THE SWCC It is well known that the Soil Water Characteristic curves (SWCCs) show hysteresis during wetting and drying. For instance, experimental data of hysteretic behaviour of the SWCC have been presented by researchers such as Fredlund and Rahardjo (1993). This gives a non-unique and pathdependent relationship between the saturation degree 𝑆𝑤 , porosity 𝑛 and suction 𝑝𝑐 , for a soil. There are various approaches for modelling the SWCC that will be explained in coming sections.

Approach 1: Assumption of Elasticity for Scanning Curves In this method, the hysteretic SWCC equation is described by the main wetting and drying relations playing a role similar to the yield surface in elasto-plastic mechanical constitutive models. The hydraulic state cannot exceed the two yield surfaces in the space defined by the logarithm of suction and the saturation degree. A schematic representation of the intermediate paths between the yield surfaces, known as “scanning curves,” are shown in Figure 1. In this figure, the wetting and drying lines have been generated using the empirical equation proposed by Brooks and Corey (1964) which can be approximated as follows

𝑆𝑒 =

𝑏−𝛼 𝑃𝛼 𝜈 ( ) { 𝑝𝑐∗

(𝛼 = 𝑤, 𝑑) 𝑝𝑐 > 𝑃𝑑 1

𝑝𝑐 ≤ 𝑃

𝑑

(64)

54

Javad Ghorbani and David W. Airey

where 𝑃𝛼 (with 𝛼 = 𝑤 and 𝑑 for the wetting and drying lines, respectively) is known as the air-entry value and 𝜈 𝑏−𝛼 is a fitting parameter. 𝑆𝑒 is the effective degree of saturation defined by 𝑆 −𝑆

𝑆𝑒 = 𝑆 𝑤 −𝑆𝑟𝑎 𝑟𝑤

(65)

𝑟𝑎

where 𝑆𝑟𝑎 is the residual degree of saturation at extremely dry conditions, and 𝑆𝑟𝑤 is the residual degree of saturation (normally =1.0) when fully saturated (Ghorbani et al., (2018a)). In addition, Equation (64) incorporates a modified suction, 𝑝𝑐∗ defined by ′

𝑝𝑐∗ = 𝑒 𝛺 𝑝𝑐

(66)

where 𝛺′ is an additional fitting parameter quantifying the role of volume changes in altering the saturation degree. This definition of 𝑝𝑐∗ was also used by Ghorbani et al., (2018c) for modification of the SWCC equation by Van Genuchten (1980). For the scanning curve shown in Figure 1, it has been assumed that a relationship can be established between any two arbitrary initial and final 𝑝∗ 𝑖 + 𝛥𝑝∗ 𝑝∗ 𝑖 points along this line, located at | 𝑐𝑖 and | 𝑐 𝑓 𝑐 , respectively. The 𝑆𝑒 𝑆𝑒 transition between these two points can be approximated by the following relationship 𝑓

𝑖

𝑝𝑐∗ +𝛥𝑝𝑐∗

𝑆𝑒 = 𝑆𝑒𝑖 + 𝑐 ℎ ln (

𝑝𝑐𝑖

)

(67)

where 𝑐 ℎ is the slope of the scanning curve. The main advantage of using this method is the simplicity of implementation. However, the approach has the following shortcomings:

Some Aspects of Numerical Modelling of Hydraulic Hysteresis …

55

1) An inability to model arbitrary paths of scanning curves as noted by Li (2005). 2) The presence of critical points at the intersections of the scanning curves and the main wetting and drying line which leads to a numerical problem as outlined below. 1.2

Saturation Degree

1 Scanning curves

0.8 0.6

Drying line

0.4 Wetting line

0.2 0 1

10

100

1000

Ln (p*c) Figure 1. A typical SWCC considering the effect of hysteresis.

The latter is identified by Ghorbani et al., (2018c). To elaborate this further, it should be realized that in the preceding sections, a fully implicit time integration scheme was presented for solving the equations of unsaturated soil dynamics, by combining the generalised-α method and Newton-Raphson iteration. Because the derivatives of saturation degree with respect to suction and porosity,

𝜕𝑆𝑤 𝜕𝑝𝑐

and

𝜕𝑆𝑤 , 𝜕𝑛

are present in the fully

coupled equations, it can be expected that the Newton-Raphson iteration, may fail to converge in the neighbourhood of the critical points. This can occur if the initial guess for the solution (required at the initiation of the Newton-Raphson algorithm) is far from the correct solution.

56

Javad Ghorbani and David W. Airey

Approach 2: Arbitrary Evolution of Scanning Curves This approach was originally proposed by Li (2005) where the main wetting and drying lines were considered as two bounding lines determining the area (in the space defined by the saturation degree, and the modified suction) within which the material state can be located. A simpler version of this approach was presented by Zhou et al., (2012) where the evolution of the scanning curves was defined based on the distance from the bounding lines. Unlike the elastic approach, this method yields an incremental relationship relating the changes of the saturation degree to the changes of volume and suction. The rate of change of the effective saturation degree, 𝑆𝑒 (or the saturation degree 𝑆𝑤 ) can be described by 𝑑𝑆𝑒 = 𝑀∗ . 𝑑𝑝𝑐∗

(68)

where 𝑑𝑝𝑐∗ =

𝜕𝑝𝑐∗ 𝑑𝑝𝑐 𝜕𝑝𝑐

+

𝜕𝑝𝑐∗ 𝑑𝑛 𝜕𝑛

(69)

To define the equation of 𝑀∗, we can suppose that, for a certain value of 𝑝𝑐∗ , the slope of the SWCC in 𝑆𝑒 − 𝑝𝑐∗ space should approach the slope 𝜕𝑆 𝑤

of the main wetting, (𝜕𝑝∗𝑒𝑤), and the slope of the main drying, ( 𝑐

𝜕𝑆𝑒𝑑

𝜕𝑝𝑐∗

𝑑

),

curves once we get close to these two lines. Then, an interpolation function can be employed by which the slope of the SWCC at any arbitrary points between these two lines is obtained as follows 𝑝∗

𝑏𝛼 𝜕𝑆 𝛼 𝑒 , (𝛼 𝜕𝑝𝑐∗

𝑀∗ = (𝑝∗𝑐𝛼 ) 𝑐

= 𝑤, 𝑑)

(70)

with 𝑝𝑐∗ 𝛼 = 𝑆𝑒𝛼 −1 , (𝛼 = 𝑤, 𝑑)

(71)

Some Aspects of Numerical Modelling of Hydraulic Hysteresis …

57

and 𝑏𝑤 and 𝑏𝑑 are model parameters with negative and positive values in the wetting and the drying processes, respectively. 𝜕𝑆 𝛼

Note that once 𝑝𝑐∗ reaches 𝑝𝑐∗ 𝛼 , we get 𝑀∗ = 𝜕𝑝∗𝑒𝛼. 𝑐

Finding a closed-from solutions for Equation (69) and (70) is generally not possible, and therefore, a numerical integration scheme may be required to solve them. One such scheme is given by Ghorbani et al., (2018c) who suggested to use an adaptive explicit integration method with automatic error control. The scheme breaks the suction and void ratio increments, in the time step 𝑡, into several subincrements in pseudo time steps. The method evaluates the error in calculating the saturation degree using the Modified Euler method and uses the obtained errors to set the size of the subincrements during the integration process. Eventually, the effective saturation degree will be updated in each time step of the analysis as follows Δ𝑝𝑐∗

𝑆𝑒𝑡+Δ𝑡 = 𝑆𝑒𝑡 + ∫0

𝑀∗ . 𝑑𝑝𝑐∗

(72)

In the initial step of the algorithm, we identify the current position in 𝑆𝑒 − 𝑝𝑐∗ space. In this space, we assume that the main wetting and drying lines are two bounding lines. Hence, at any time of the analysis, the state can only be located within (on a scanning curve) or on the main wetting and drying lines. If the state is situated outside the area enclosed by the wetting and drying lines it should be discarded and the analysis should be ended. Then, there is a need to check if the previous wetting/drying process is reversed in the current time step as follows Δ𝑝𝑐∗ 𝑡−∆𝑡 . Δ𝑝𝑐∗ 𝑡 < 0

(73)

where Δ𝑝𝑐∗ 𝑡−∆𝑡 and Δ𝑝𝑐∗ 𝑡 are the change of the scaled suction, 𝑝𝑐∗ , in the previous and the current time steps, respectively. The algorithm is described hereinafter.

58

Javad Ghorbani and David W. Airey 1. Initial input: 𝑛𝑡 , Δ𝑛𝑡 , Δ𝑛𝑡−Δ𝑡 , 𝑆𝑒𝑡 , 𝑝𝑐𝑡 , Δ𝑝𝑐𝑡 , Δ𝑝𝑐𝑡−Δ𝑡 , 𝑝𝑐𝑟 , 𝑛𝑟 and 𝑆𝑊𝑇𝑜𝑙. 2. Find the location of the material in 𝑆𝑒 − 𝑝𝑐∗ space; stop the analysis if it is outside the region encompassed by the main wetting and drying curves. Else: 3. Check if a reverse process happens 4. Set 𝑡̃ = 0 and Δ𝑡̃ = 1 5. While 𝑡̃ < 1 perform the following: Compute Δ𝑆𝑒𝑖 , for 𝑖 = 1,2, using: Δ𝑆𝑒𝑖 = Δ𝑡̃. 𝑀𝑝𝑐 𝑖 . Δ𝑝𝑐𝑡 + Δ𝑡̃. 𝑀𝑛 𝑖 Δ𝑛𝑡

(74)

with 𝜕𝑝𝑐∗ 𝑖 ) 𝜕𝑝𝑐

𝑀𝑝𝑐 𝑖 (𝑆𝑒𝑖 , 𝑝𝑐𝑖 , 𝑛𝑖 , 𝑝𝑐𝑟 , 𝑛𝑟 ) = 𝑀∗ 𝑖 ( and

𝜕𝑝∗ 𝑖

𝑀𝑛 𝑖 (𝑆𝑒𝑖 , 𝑝𝑐𝑖 , 𝑛𝑖 , 𝑝𝑐𝑟 , 𝑛𝑟 ) = 𝑀∗ 𝑖 ( 𝜕𝑛𝑐 ) Also,

𝑆𝑒1 = 𝑆𝑒𝑡̃ , 𝑝𝑐1 = 𝑝𝑐𝑡 + 𝑡̃. Δ𝑝𝑐𝑡 , 𝑛1 = 𝑛𝑡 + 𝑡̃. Δ𝑛𝑡 𝑆𝑒2 = 𝑆𝑒𝑡̃ + Δ𝑆𝑒1 , 𝑝𝑐2 = 𝑝𝑐𝑡 + (𝑡̃ + Δ𝑡̃). Δ𝑝𝑐𝑡 , 𝑛2 = 𝑛𝑡 + (𝑡̃ + Δ𝑡̃). Δ𝑛𝑡 and lastly

Some Aspects of Numerical Modelling of Hydraulic Hysteresis … 1

𝑆𝑒𝑡̃+Δ𝑡̃ = 𝑆𝑒𝑡̃ + 2 (Δ𝑆𝑒1 + Δ𝑆𝑒2 )

59 (75)

𝑝𝑐𝑡̃+Δ𝑡̃ = 𝑝𝑐2 𝑛𝑡̃+Δ𝑡̃ = 𝑛2 6. Calculate the relative error for the current substep:

𝐸 = 𝑚𝑎𝑥 {

|Δ𝑆𝑒2 −Δ𝑆𝑒1 | ̃

̃

|𝑆𝑒𝑡+Δ𝑡 |

, 𝐸𝑃𝑆}

(76)

where EPS is a machine constant denoting the smallest relative error that can be calculated. 7. If 𝐸 > 𝑆𝑊𝑇𝑜𝑙 the substep has failed. First calculate: 𝑆𝑊𝑇𝑜𝑙 , 0.1} 𝐸

𝒜 = 𝑚𝑎𝑥 {0.9√

(77)

then set: ∆𝑡̃ ← 𝑚𝑎𝑥{𝒜∆𝑡̃, 𝛥𝑡̃𝑚𝑖𝑛 }, 8. and go to step 5. 9. The substep is accepted, update porosity, suction and the effective saturation degree. 10. Extrapolate to get the size of next subincremenet as follows: 𝑆𝑊𝑇𝑜𝑙 , 1.1} 𝐸

𝒜 = 𝑚𝑖𝑛 {0.9√

then set

60

Javad Ghorbani and David W. Airey ∆𝑡̃ ← 𝒜∆𝑡̃, 𝑡̃ ← 𝑡̃ + ∆𝑡̃. 11. Do not let the next step size drop below the minimum step size and ensure that the pseudo-time does not proceed beyond 𝑡̃= 1 by setting: ∆𝑡̃ ← 𝑚𝑎𝑥{∆𝑡̃, Δ𝑡̃𝑚𝑖𝑛 } and then ∆𝑡̃ ← 𝑚𝑎𝑥{∆𝑡̃, 1 − 𝑡̃}. 12. Exit with the updated effective saturation degree once 𝑡̃ = 1.

A desirable integration tolerance is considered in the integration algorithm by 𝑆𝑊𝑇𝑜𝑙 which can be used as a tool to control the integration error in the analysis. In the following algorithm 𝑡̃ is the pseudo-time (used for substepping) and should not be confused with the actual time 𝑡 in the analysis. A recommendation for the minimum substep size (Δ𝑡̃𝑚𝑖𝑛 ), based on the authors’ experience, is of the order of 10−4 and is considered in this paper. Throughout this chapter 𝑆𝑊𝑇𝑜𝑙 is of the order of 10−5 unless stated otherwise. To demonstrate the performance of this approach, the following equation is chosen as the equation of the main wetting and drying curves:

𝑆𝑒 = (1 +

𝛼 𝛼 −𝑚 𝑝𝑐∗ 𝑛 (𝑃𝑑 ) )

(𝛼 = 𝑤, 𝑑)

(78)

where 𝑛𝛼 and 𝑚𝛼 (𝛼 = 𝑤, 𝑑) are two fitting parameters and 𝑃𝛼 (𝛼 = 𝑤, 𝑑) are the air-entry values of the main wetting and drying curves. Also, the parameters given in Table 1 are selected to simulate a multi-stage wetting/drying process as follows.

Some Aspects of Numerical Modelling of Hydraulic Hysteresis …

61

Table 1. The SWCC parameters 𝒏𝒅 20

𝒎𝒅 0.1

𝑷𝒅 (kPa) 100

𝒃𝒅 5.0

𝒏𝒘 20

𝒎𝒘 0.1

𝑷𝒘 (kPa) 50.0

𝒃𝒘 -5.0

Figure 2. Simulation of a four-stage process of wetting and drying with the approach proposed by Zhou et al., (2012).

The starting values of 𝑆𝑒 and 𝑝𝑐∗ are 0.4 and 150.00 kPa, respectively. The following four-stage process has been modelled:    

Wetting with Δ𝑝𝑐∗ = −72.0 kPa Drying with Δ𝑝𝑐∗ = 37.0 kPa Wetting with Δ𝑝𝑐∗ = −51.0 kPa Drying with Δ𝑝𝑐∗ = 60.0 kPa

62

Javad Ghorbani and David W. Airey

The result of the simulation is presented in Figure 2. It is noticed that the chosen procedure can allow arbitrary paths for scanning curves in simulating a multi-step wetting and drying process. Nevertheless, the approach fails to avoid the generation of critical points at the wetting/drying reversals. The points are circled in the figure.

Approach 3: Smooth Evolution of Scanning Curves This approach was introduced by Ghorbani et al., (2018c) with the motivation to eliminate the critical points in Approach 2. Ghorbani et al., (2018c) showed that this can be achieved by storing the values of 𝑝𝑐∗ and 𝑆𝑒 at the initiations of reversals, referred to by 𝑝𝑐∗ 𝑟 and 𝑆𝑒𝑟 , respectively. Moreover, the slope of the SWCC at the beginning of a reverse process, is needed and is given by 𝑀∗ 𝑟 . This allows the definition of 𝑀∗ in Equation (70) to be altered such that at the start of a reverse process 𝑀∗ 𝑟 is obtained 𝑝𝑐∗ 𝑏𝛼 𝜕𝑆𝑒𝛼 𝛼) 𝑝𝑐∗ 𝜕𝑝𝑐∗

and at the end of the process (

is acquired. Here, an interpolation

function is introduced to gradually damp 𝑀∗ 𝑟 as 𝑝𝑐∗ changes from 𝑝𝑐∗ 𝑟 to 𝑝𝑐∗ 𝛼 as follows 𝑝∗ −𝑝∗

𝛼

𝑏𝑠𝑐

𝑀∗ 𝑠𝑐 = 𝑀∗ 𝑟 (𝑝∗𝑐𝑟 −𝑝𝑐∗ 𝛼) 𝑐

𝑐

(79)

Note that Equation (79) ensures that once 𝑝𝑐∗ → 𝑝𝑐∗ 𝑟 , we get 𝑀∗ 𝑠𝑐 → 𝑀∗ 𝑟 ; and once 𝑝𝑐∗ → 𝑝𝑐∗ 𝛼 , we get 𝑀∗ 𝑠𝑐 → 0.0. Another parameter (𝑏𝑠𝑐 ) is added, which is a positive quantity, to have extra control on the shape of scanning curves near the start of a reverse process. By having this definition, the updated form of Equation (70) becomes 𝑝∗

𝑏𝛼 𝜕𝑆 𝛼 𝑒 𝜕𝑝𝑐∗

𝑀∗ = (𝑝∗𝑐𝛼 ) 𝑐

+ 𝑀∗ 𝑠𝑐

(80)

Some Aspects of Numerical Modelling of Hydraulic Hysteresis …

63

The result of the simulation of the four-stage process (demonstrated in Figure 2, previously) with the modified approach is presented in Figure 3. The material parameters and the initial conditions are the same and 𝑏𝑠𝑐 = 20.0. It is clear that the approach has succeeded in making smooth variations of the SWCC slopes upon the initiation of reverse processes (circled in Figure 2) which was the main motivation of the presented modifications.

Figure 3. Simulation of the four-stage process by the modified approach.

From Equation (79), it can be seen that as 𝑏 𝑠𝑐 → ∞, we get 𝑀∗ 𝑠𝑐 → 0. In this case, the model by Zhou et al., (2012) will be recovered. This is shown in Figure 4 where three analyses were performed with 𝑏 𝑠𝑐 being set to 10, 20 and 50. Other parameters are set to be the same as those provided in Table 1. The initial saturation degree and suction are set to 0.53 and 74 kPa, respectively. It is demonstrated that all the studied cases follow the same initial wetting path (∆𝑝𝑐∗ = −10 kPa) since 𝑏 𝑠𝑐 does not change the shape of the scanning curves in this situation. It is also shown that as 𝑏 𝑠𝑐 increases, a sharper change of the slope of SWCC at the transition point is obtained.

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Javad Ghorbani and David W. Airey

Figure 4. The effect of 𝑏 𝑠𝑐 on the shape of scanning curve.

Figure 5. Simulation of the test reported by Kwa et al., (2018).

Table 2. The SWCC parameters for the test performed by Kwa et al., (2018) 𝒏𝒅

𝒎𝒅

2.1

0.0105

𝑷𝒅 (kPa) 2.0

𝒃𝒅

𝒏𝒘

𝒎𝒘

𝑷𝒘 (kPa)

𝒃𝒘

𝒃𝒔𝒄

𝜴′

𝑺𝒓𝒂

𝑺𝒓𝒘

5

2.3

0.08

0.1

-5

20

0

0

1

Some Aspects of Numerical Modelling of Hydraulic Hysteresis …

65

Applied pressure

smooth / impermeable

smooth / impermeable

Drainage boundary

H=10 m

rough / impermeable displacement nodes

displacement, pore pressure and suction nodes

Figure 6. Soil column and boundary conditions.

The performance of the model in reproducing the evolution of the scanning curve during a wetting/drying cycles has been evaluated against the data provided by Kwa et al., (2018) of a soil containing basalt aggregates with particle sizes from 9.5mm to 2 𝜇m and 18% of feldspar fines. As mentioned by the authors, the dependency of the SWCC equation

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Javad Ghorbani and David W. Airey

on void ratio is small; therefore, it has been ignored in the presented simulations. The results of simulations together with the data are presented in Figure 5 corresponding to a wetting/drying process. The main wetting and drying curves are approximated by taking the equation proposed by Van Genuchten (1980). The parameters of the model are shown in Table 3. In obtaining the curves, it is assumed that the initial saturation degree and suction are 0.336 and 189.1 kPa, respectively as done in the test. It is seen that good agreement with the experimental data can be achieved with the selected approach, in particular in the way the scanning curves evolve during cycles of wetting and drying.

APPLICATION TO BOUNDARY-VALUE PROBLEMS Two example problems are selected to demonstrate the performance of the proposed model and the accompanying integration scheme in solving boundary value problems. The first studies the response of an elastic one-dimensional soil column to a loading/unloading process and the subsequent wetting/drying cycle. An unsaturated soil column with a height of 10 m is considered. The boundary conditions and the geometry of the column are shown in Figure 6. The initial saturation degree, 𝑆𝑤 is set to 0.3 everywhere. The material properties and SWCC parameters are given in Table 3 and Table 4, respectively. A preesure ramp is applied on the top boundary so that the pressure increases from 0 to 10 MPa over 1000 seconds and then reduced to zero over further 1000 seconds as shown in Figure 7. Two analyses have been carried out with this pressure, time history. In the first, the SWCC formulation is assumed to not have any hysteresis, whereas in the second analysis the hysteresis is considered. In both analyses, the initial material state is considered to be on the main drying curve (𝑆𝑤 = 0.3).

Some Aspects of Numerical Modelling of Hydraulic Hysteresis …

67

Table 3. Material parameters Parameter type Porosity Density of the solid skeleton Density of the water Density of the air Elastic modulus of the mixture Poisson’s ratio Bulk modulus of the water Bulk modulus of the air

Symbol 𝑛 𝜌𝑠 𝜌𝑤 𝜌𝑎 𝐸 𝜈 𝐾𝑤 𝐾𝑎

Intrinsic permeability Viscosity of the water Viscosity of the air

𝑘 𝜂𝑤 𝜂𝑎

Value 0.375 2650 997 1.01 40.0 0.02 2.25 × 109 1.10 × 105 2.5 × 10−12 1.0 × 10−3 1.8 × 10−5

Unit kg m−3 kg m−3 kg m−3 MPa Pa Pa Pa m2 Ns m−2 Ns m−2

There are two mechanisms associated with the response of the column. During the loading, the column shows compression. As the column settles, the saturation degree increases. Thus, the suction decreases and consequently water pressure increases. On the other hand, throughout the unloading, the column expands and saturation degree decreases. In this condition, the suction increases, and the water pressure decreases. The variations of the saturation degree against the suction at the bottom of the column in both analyses are plotted in Figure 8 along with the main drying and wetting lines. It is demonstrated that in the absence of hysteresis) the wetting and the drying paths will be the same and the 𝑆𝑤 − 𝑝𝑐 state moves up and down the drying line. However, in the presence of hysteresis (second analysis), as compression begins the material gradually moves away from its initial position on the main drying curve. In such a condition, the material is located on a scanning curve and tends to reach the bounding line shown by the main wetting curves. In the second step of the analysis where unloading begins, the material moves back toward the drying curves. The hysteresis loop is clearly seen in the graph. Also, both transitions from the main wetting and drying curves to the scanning curves have occurred smoothly.

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Javad Ghorbani and David W. Airey Table 4. The SWCC parameter

𝒏𝒅

𝒎𝒅

25.0

0.4

𝑷𝒅 (kPa) 500.0

𝒃𝒅

𝒏𝒘

𝒎𝒘

𝑷𝒘 (kPa)

𝒃𝒘

𝒃𝒔𝒄

𝜴′

𝑺𝒓𝒂

𝑺𝒓𝒘

5.0

4.0

0.7

250.0

-5

20

0

0

1

Figure 7. The pressure applied on the top of the unsaturated column.

Figure 8. Comparison of the saturation degree variation in the bottom of the unsaturated column in the analyses with and without hysteretic SWCC.

Some Aspects of Numerical Modelling of Hydraulic Hysteresis …

69

Table 5. Comparison of the relative error, 𝑬𝑺𝒆 , and computational time for the simulations with different integration tolerances 𝑬𝑺𝒆 3.55e-3 2.91e-3 8.45e-4 4.10e-4 5.69e-5 -

𝑺𝑾𝑻𝒐𝒍 10e-1 10e-2 10e-3 10e-4 10e-5 10e-12

CPU time 0.94 0.95 0.95 0.95 0.95 1

To evaluate the integration process, a tight tolerance of 10−12 for 𝑆𝑊𝑇𝑜𝑙 is selected as a benchmark by which the relationships of 𝑆𝑊𝑇𝑜𝑙 with the accuracy of the intergration scheme and the computational times are investigated. The relative error, 𝐸𝑆𝑒 , in the analysis is given by 𝑟𝑒𝑓

𝐸𝑆𝑒 =

∑𝑙𝑖=1|𝑆𝑒

−𝑆𝑒 |

𝑖

𝑟𝑒𝑓 ∑𝑙𝑖=1 𝑆𝑒 𝑖

(81)

𝑟𝑒𝑓

where 𝑆𝑒 is the effective saturation degree calculated by taking −12 𝑆𝑊𝑇𝑜𝑙 = 10 ; and the subscript 𝑖 denotes the ith integration point in the mesh and 𝑙 is the total number of integration points. Five tolerances of 10−1, 10−2, 10−3, 10−4 and 10−5 were chosen for 𝑆𝑊𝑇𝑜𝑙. The relative error (𝐸𝑆𝑒 ) and CPU time for these analyses are compared in Table 5. The results indicate that tighter integration tolerance can decrease the relative error 𝐸𝑆𝑒 without significant changes of the speed of the analysis as the corresponding values of CPU time suggest in the table.

Coupling the hysteretic SWCC model with elasto-plasticity For the final example, the capability of the proposed approach in modelling the elasto-plastic response of unsaturated soils during an impact load is demonstrated.

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Javad Ghorbani and David W. Airey Table 6. The SWCC parameter for the impact problem

𝒏𝒅

𝒎𝒅

20.0

0.14

𝑷𝒅 (kPa) 200.0

𝒃𝒅

𝒏𝒘

𝒎𝒘

𝑷𝒘 (kPa)

𝒃𝒘

𝒃𝒔𝒄

𝜴′

5

20

0.14

100

-5

20

0.05

𝑺𝒓𝒂

𝑺𝒓𝒘

0

1.0

Figure 9. The schematic finite element model for the problem of a rigid footing.

The geometry of the plane strain model is presented in Figure 8. In this example, a rigid strip footing is placed on top of the soil. The top boundary is permeable; and the side boundaries are restrained so that no lateral displacements are possible. The overall width of the footing is 2 m. The Updated Lagrangian framework (Ghorbani et al., 2016b and Ghorbani et

Some Aspects of Numerical Modelling of Hydraulic Hysteresis …

71

al., 2018b) is selected to consider the occurrence of large deformations in this example. The SWCC parameters for this analysis are listed in Table 6. The selected constitutive model is the same as that used by Ghorbani et al., (2018c) (given in Appendix A) and its parameters are shown in Table 7. In this example, the saturation degree of 0.5 and suction value 180 kPa were prescribed initially on the domain. A triangular dynamic pressure with the maximum of 90 kPa is applied on the footing (Figure 9.a). For the given load, a settlement of 33.4 cm is predicted by the model (Figure 9.b) during the impact. During the loading, the saturation degree below the impact area rises because of the compression of the soil. The process is inverted once the unloading step is begun. Table 7. Constitutive model and material parameters Parameter type Slope of critical state line The slope of loading line in fully saturated state Poisson’s ratio of the mixture The slope of unloading/reloading line in fully saturated state Specific volume at 𝑝′ = 1 Over Consolidation Ratio (OCR) Initial yield surface size Unsaturated material parameter Unsaturated material parameter Density of the solid skeleton Density of the water Density of the air Bulk modulus of the water Bulk modulus of the air Intrinsic permeability Viscosity of the water Viscosity of the air

Symbol 𝑀𝑚𝑎𝑥 𝜆0

Value 0.9833 0.2

Unit -

𝜇 𝜅

0.3 0.04

-

𝛤∗ 𝛼 𝑏 𝑟 𝜌𝑠 𝜌𝑤 𝜌𝑎 𝐾𝑤 𝐾𝑎 𝑘 𝜂𝑤 𝜂𝑎

2.5 1.0 10 5.4 0.75 2700 997 1.1

kPa 1/(kPa) kg m−3 kg m−3 kg m−3 Pa Pa

2.25 × 109 1.01 × 105 2.5 × 10−12 1.0 × 10−3 1.8 × 10−5

m2 Ns m−2 Ns m−2

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Javad Ghorbani and David W. Airey

(a)

(b)

(c)

(d)

Figure 10. The dynamic applied load and the model response during the impact. a) Applied pressure. b) Settlement of the ground during the impact. c) Saturation degree changes at 25 cm below the centre of the footing. d) Suction changes at 25 cm below the centre of the footing.

This is illustrated in Figure 9.c at a point located at 25 cm under the centre of the footing. Saturation degree and suction have an inverse relationship. This is illustrated in Figure 9.d where, at the same point, suction initially decreases and then increases. The variations of the scaled suction with respect to the saturation degree is given in Figure 10. The figure clearly illustrates the hysteretic

Some Aspects of Numerical Modelling of Hydraulic Hysteresis …

73

behavior predicted by the model during the impact with a smooth transition at the wetting/drying reversals.

Wetting

Initial point

Figure 11. Hysteretic response of SWCC at 25 cm below the centre of the footing.

CONCLUSION This chapter initially has provided the finite element formulations for modelling dynamic response of unsaturated soils. Then, three approaches for modelling the hysteretic response of the SWCCs have been presented and the numerical problems associated with each method along with their capabilities have been demonstrated. It has been shown that the approach presented by Ghorbani et al., (2018c) can provide robust solutions in solving complex boundary-value problems in unsaturated soil mechanics involving large deformations, elasto-plasticity and dynamic loadings.

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Javad Ghorbani and David W. Airey

APPENDIX A. ELASTO-PLASTIC CONSTITUTIVE MODEL An extension of the Modified Cam Clay (MCC) model first proposed by Roscoe (1968) for unsaturated soils is selected. The yield surface, 𝑓, and the plastic potential surface, 𝑔, are described by 1

(1+𝛽 ′ )𝑝′ 𝛼∗

𝑓 = 𝑔 = 𝛽2 (

2

(1+𝛽 ′ )𝑞

2

− 1) + ( 𝑀(𝜃)𝛼∗ ) − 1

(82)

where 𝛼 ∗ determines the size of the yield surface for unsaturated conditions, 𝑝′ represents the effective mean stress and 𝑞 is the deviatoric stress. 𝛽 ′ and 𝛽 as two parameters adjusting the shape of the yield surface. 𝑀 is the slope of the critical state line in 𝑝′ − 𝑞 stress space and can change with the Lode angle, 𝜃 according to the rule suggested by Sheng et al., (2000) where

𝑀 = 𝑀𝑚𝑎𝑥 (

2𝛼′ 4

1/4

4 4

1+𝛼 ′ −(1−𝛼′ ) sin 3𝜃

)

(83)

where 𝑀𝑚𝑎𝑥 is the slope of the CSL under triaxial compression when Lode angle equals to 30°; and 𝛼 ′ can have the following relationship with friction angle, 𝜙 3−sin 𝜙

𝛼 ′ = 3+sin 𝜙

(84)

The elastic s bulk modulus 𝐾 is defined as follows 𝐾=

−(1+𝑒)𝑝′ 𝜅

(85)

where 𝑒 is the void ratio, 𝜅 is a model parameter. The evolution of the yield surface is expressed by an equation suggested by Zhou et al., (2012) where the hardening parameter for unsaturated soil, 𝛼 ∗ is expressed by

Some Aspects of Numerical Modelling of Hydraulic Hysteresis … 𝛼∗ 𝑝𝑟

𝛼 (𝜆0 −𝜅)/(𝜆−𝜅)

= (𝑝 )

75 (86)

𝑟

where 𝛼 is the yield surface size at the fully saturated condition, 𝑝𝑟 is a reference mean effective stress, 𝜅 represents the slope of the compressionrecompression line, and 𝜆 and 𝜆0 are the slopes of the normal compression lines for the unsaturated and saturated conditions, respectively. The variation of 𝜆 with respect to the effective saturation degree, 𝑆𝑒 , is described by: 𝜆 = 𝜆0 (1 − (1 − 𝑆e )𝑏1 (1 − 𝑟))

(87)

APPENDIX B. ADDITIONAL DETAILS ON THE EQUATIONS OF SWCC The equation of the SWCC is

𝑆𝑒𝛼

= (1 +

𝛼 𝛼 −𝑚 𝑝𝑐∗ 𝑛 (𝑃𝛼 ) )

(𝛼 = 𝑤, 𝑑)

(88)

which yields

𝑆𝑒𝛼 −1

=

𝑝𝑐∗ 𝛼 (𝑆𝑒 )

𝛼

−1 𝑚𝛼

= 𝑃 ((𝑆𝑒 )

1 𝑛𝛼

− 1)

(𝛼 = 𝑤, 𝑑)

(89)

In addition, 𝛼 ∗ 𝑛

𝜕𝑆𝑒𝛼 𝜕𝑝𝑐∗

=

𝑝𝑐 −𝑚𝛼 𝑛𝛼 𝑆𝑒𝛼 ( 𝛼 )

𝑃 ∗ 𝑛𝛼 𝑝 𝑐) 𝑝𝑐∗ (1+( 𝛼 ) 𝑃

(𝛼 = 𝑤, 𝑑)

(90)

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REFERENCES Bishop, Alan W. 1960. The principles of effective stress, Norges Geotekniske Institutt. Brooks, Royal H, and Corey, Arthur T. 1964. “Hydraulic properties of porous media and their relation to drainage design.” Transactions of the ASAE 7(1): 26-0028. Chung, Jintai, and Hulbert, Gregory M. 1993. “A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method.” Journal of applied mechanics 60(2): 371-375. Fredlund, Delwyn G, and Rahardjo, Harianto. 1993. Soil mechanics for unsaturated soils, John Wiley & Sons. Ghorbani, Javad. 2016. “Numerical simulation of dynamic compaction within the framework of unsaturated porous media.” Civil Engineering, University of Newcatsle. Doctor of Philosophy. Ghorbani, Javad, Nazem, Majidreza, and Carter, John P. 2014. “Application of the generalised-α method in dynamic analysis of partially saturated media.” Computer Methods and Recent Advances in Geomechanics: 129. Ghorbani, Javad, Nazem, Majidreza, and Carter, John P. 2016a. “Numerical modelling of multiphase flow in unsaturated deforming porous media.” Computers and Geotechnics 71: 195-206. Ghorbani, Javad, Airey, David W, and El-Zein, Abbas. 2018c. “Numerical framework for considering the dependency of SWCCs on volume changes and their hysteretic responses in modelling elasto-plastic response of unsaturated soils.” Computer Methods in Applied Mechanics and Engineering 336: 80-110. Ghorbani, Javad, El-Zein, Abbas, and Airey, David W. 2018a. “Thermoelasto-plastic analysis of geosynthetic clay liners exposed to thermal dehydration.” Environmental Geotechnics: 1-15. Ghorbani, Javad, Nazem, Majidreza, and Carter, John P. 2016b. “Dynamic Analysis of Unsaturated Soils Subjected to Large Deformations.” Applied Mechanics & Materials 846.

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Ghorbani, Javad, Nazem, Majidreza, and Carter, John P. 2016c. “Numerical Study of Dynamic Soil Compaction at Different Degrees of Saturation.” The Twenty-fifth International Ocean and Polar Engineering Conference, International Society of Offshore and Polar Engineers. Ghorbani, Javad, Nazem, Majidreza, Carter, John P, and Airey, David W. 2017. “A numerical study of the effect of moisture content on induced ground vibration during dynamic compaction.” 3rd International Conference on Performance Based Design in Earthquake Geotechnical Engineering, Vancouver. Ghorbani, Javad, Nazem, Majidreza, Carter, John P., and Sloan, Scott W. 2018b. “A stress integration scheme for elasto-plastic response of unsaturated soils subjected to large deformations.” Computers and Geotechnics 94: 231-246. Khalili, Nasser, Habte, Michael A., and Zargarbashi, Saman. 2008. “A fully coupled flow deformation model for cyclic analysis of unsaturated soils including hydraulic and mechanical hystereses.” Computers and Geotechnics 35(6): 872-889. Khoei, Amir R., and Mohammadnejad, Toktam. 2011. “Numerical modeling of multiphase fluid flow in deforming porous media: a comparison between two-and three-phase models for seismic analysis of earth and rockfill dams.” Computers and Geotechnics 38(2): 142166. Kwa, Katherine A., Ghorbani, Javad, and Airey, David W. 2018. “The effect of fines on the hydraulic properties of well graded materials.” 7th International conference on unsaturated soils, Hong Kong. Lewis, Roland W., and Schrefler, Bernhard A. 1982. “A finite element simulation of the subsidence of gas reservoirs undergoing a water drive.” Finite element in fluids 4: 179-199. Lewis, Roland W., and Schrefler, Bernhard A. 1999. “The finite element method in the static and dynamic deformation and consolidation of porous media.” Meccanica 34(3): 231-232. Li, Xikui, Ding, D, Chan, Andrew H. C., and Zienkiewicz, Olgierd C. 1989. “A coupled finite element method for the soil-pore fluid

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interaction problems with immiscible two-phase fluid flow.” Proc. 5th Int. Symp. on Numerical Methods in Eng., Lausanne. Li, Xikui, and Zienkiewicz, Olgierd C. 1992. “Multiphase flow in deforming porous media and finite element solutions.” Computers & structures 45(2): 211-227. Li, Xikui S. 2005. “Modelling of hysteresis response for arbitrary wetting/drying paths.” Computers and Geotechnics 32(2): 133-137. Roscoe, Kenneth H. 1968. “On the generalised stress-strain behaviour of’wet’clay.” Engineering Plasticity: 535-609. Schrefler, Bernhard A. 1984. University College of Swansea. PhD. Schrefler, Bernhard A., and Scotta, Roberto. 2001. “A fully coupled dynamic model for two-phase fluid flow in deformable porous media.” Computer Methods in Applied Mechanics and Engineering 190(24– 25): 3223-3246. Sheng, Daichao, Sloan, Scott W., and Yu, Hai-Sui. 2000. “Aspects of finite element implementation of critical state models.” Computational mechanics 26(2): 185-196. Sheng, Daichao, and Zhou, An-Nan. 2011. “Coupling hydraulic with mechanical models for unsaturated soils.” Canadian Geotechnical Journal 48(5): 826-840. Van Genuchten, Martinus Th. 1980. “A closed-form equation for predicting the hydraulic conductivity of unsaturated soils.” Soil science society of America journal 44(5): 892-898. Zhou, An-Nan, Sheng, Daichao, Sloan, Scott W., and Gens, Antonio. 2012. “Interpretation of unsaturated soil behaviour in the stress – Saturation space, I: Volume change and water retention behaviour.” Computers and Geotechnics 43: 178-187. Zienkiewicz, Olgierd C., Xie, Y. M., Schrefler, Bernhard A., Ledesma, A., Bi, x, ani, x, and N. 1990. “Static and Dynamic Behaviour of Soils: A Rational Approach to Quantitative Solutions. II. Semi-Saturated Problems.” Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 429(1877): 311-321.

In: Unsaturated Soils Editor: Martin Hertz

ISBN: 978-1-53615-985-1 © 2019 Nova Science Publishers, Inc.

Chapter 3

TEMPERATURE DISTRIBUTION AND WATER MIGRATION IN UNSATURATED SOIL Bingxi Li, Yiran Hu, Fei Xu, Yaning Zhang*, Shuang Liang, Wenyu Song and Zhongbin Fu School of Energy Science and Engineering, Harbin Institute of Technology (HIT), Harbin, Heilongjiang, China

ABSTRACT Temperature distribution and water migration in unsaturated soil are of significant importance because they are related to the thermal-physical properties of soil and the changes in these properties may cause disastrous engineering problems including cracking of pavement, damage of structure foundation, fracture of pipelines, etc. This chapter focuses on the temperature distribution and water migration in unsaturated soil. Temperature distribution and water migration in unsaturated soil during the freezing and thawing processes are experimentally investigated and the results are presented and discussed. Based on the soil structure, multiphase flow, temperature distribution and water potential, a *

Corresponding Author’s E-mail: [email protected].

80

Bingxi Li, Yiran Hu, Fei Xu et al. mesoscopic lattice Boltzmann model (LBM) is developed to simulate the temperature distribution and water migration in unsaturated soil. Based on the soil structure, multiphase flow, temperature distribution, water potential, melting film and interactive force, a modified LBM model is also developed to simulate the temperature distribution and water migration in unsaturated soil. Main conclusions and future outlook are also briefly stated and summarized. The contents included in this chapter not only detail the experimental procedures and the results of temperature distribution and water migration in unsaturated soil, but also introduce the models for estimating the temperature distribution and water migration in unsaturated soil.

Keywords: unsaturated soil, temperature distribution, water migration, freezing, thawing, lattice Boltzmann model

1. INTRODUCTION Frozen grounds (permafrost and seasonally frozen grounds) occupy approximately 54 million km2 of the exposed land areas of the northern hemisphere [1], and these grounds (especially the seasonally frozen grounds) freeze and thaw in response to the temperature change and water variation, causing many serious engineering problems including pavement cracking, building foundation damage, pipelines fracture, etc. [2, 3]. During the soil freezing and thawing processes, heat transfer and water migration coexist and they occur in a coupled manner. Due to the temperature difference between different soil zones, especially between the ground surface and the interior, a temperature gradient forms, driving the heat to flow from the higher temperature zone towards the lower one. Meanwhile, pore water migrates from the unfrozen zone to the freezing fringe and builds the frozen front, influencing the heat conduction process due to the effect of convection and latent heat of phase change [4]. Thus, systematic experiments and simulations of temperature distribution and water migration during the freezing and thawing processes are of significant importance.

Temperature Distribution and Water Migration…

81

2. TEMPERATURE DISTRIBUTION AND WATER MIGRATION IN UNSATURATED SOIL For unsaturated soil, it initially contains a certain amount of water which then will redistribute along the temperature gradient during the freezing and thawing processes. When the soil starts to freeze, temperature gradient forms from the surface to the deeper soil which leads the water to freeze from the surface, and then the soil water potential gradient causes the migration of water from the unfrozen region to the frozen region. During the soil thawing process, water is obstructed by the middle frozen layer while the ice layer starts to melt from both upper and lower sides. To study the basic properties of temperature distribution and water migration in unsaturated soil, experiments are considered in this section. The temperature distribution and water migration in unsaturated soil can be studied based on experimental device which is mainly composed of soil storage system, water supply system, temperature control system, parameter measurement system and so on. The soil storage system is used to put the soil sample, the water supply system is used to supply water to the bottom of the soil sample as groundwater, the temperature control system is used to control the freezing and thawing temperature of the soil sample, and the parameter measurement system is used to measure the temperature and the other parameters of the soil sample along the depth during freezing and thawing processes.

2.1. Experiment Devices Used The experimental devices are usually divided into two types based on the scales: the small-scale experimental device and the feet-scale experimental device. In small-scale experiment device, the depth and cross section diameter of soil sample is smaller than 1 feet due to experimental operability, but to some extent it's different from the actual situation due to

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Bingxi Li, Yiran Hu, Fei Xu et al.

the scale. In feet-scale experimental device, the depth and cross section diameter of soil sample is larger than 1 feet. Figure 1 shows a small-scale experiment device established by Bai et al. [5]. The small-scale experimental apparatus is composed of temperature control system, water supply system and data acquisition system. The temperature control system includes a top plate, a bottom plate, a test box and a visualization software. The water supply system is a Mariotte flask connected with the bottom plate via a plastic tube. The data acquisition system is a DT80 data logger, the apparatus is instrumented along the vertical direction to measure the soil temperatures at different depths and one displacement sensor at the top of the sample to measure the amount of frost heave in vertical direction. Figure 2 shows a feet-scale soil freezing and thawing system (SFTS) constructed by Xu et al. [6] in Harbin Institute of Technology (HIT), China. The SFTS consists of two subsystems: (a) the soil test system and (b) the temperature control system. The soil test system comprises several parts: a sample loading barrel (inner diameter of 1 m, height of 2.5 m), eight temperature-humidity sensors (placed along the axis of the container), and two temperature walls at the top and the bottom separately. The temperature control system is a hydraulic refrigeration cycle system which supplies cold and warm fluid flows to the top and bottom walls to control the temperatures, respectively.

Figure 1. Small-scale freezing and thawing test chamber [5].

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Figure 2. Feet-scale unsaturated soil freezing–thawing experimental device [6].

2.2. Temperature Distribution during Freezing Process During the freezing process, the temperatures of the soil at different depths generally decrease with varying degrees. The top soil is the first one to freeze, so the temperature there decreases initially and it is reduced with a rapid rate. Along the depth, the deeper the soil is, the lower the temperature decreasing rate shows. After freezing or declining to a certain extent, the temperature of soil at different depths may stay stable and change slightly.

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Figure 3. Temperature distribution at different times during freezing process [6].

Figure 3 shows the temperature distribution at different times during freezing process, layer 1-8 distribute from the shallow to the deep. Initially, the temperature distribution along the depth was uniform and the temperature walls at the top and bottom were set at -20 C and 10 C to start the freezing process respectively. The whole freezing process lasted for 20 days. It is observed that the temperature at the shallowest layer 1 decreased sharply in the first 7 days and then kept almost unchanged in the rest of the freezing process. The temperature at the deepest layer 8 decreased more slowly and after about 14 days it reached 10 C. During the freezing process, all the temperatures at layers of different depths decreased in varying degrees, from the shallow to the deep, the temperature decline rate slows down gradually. On the other hand, the temperature distribution also shows differences along the depth at different days. On the 7th day, the difference in temperature in the shallower depth is much larger than that in the deeper depth, and the temperature at layer 2 is close to the bulk water freezing point while the temperatures at layers 3-8 are higher than that. Then on the 14th and 20th days, the soil temperatures along layers 1-8 go up almost linearly in the range of -25 C to 10 C, the temperatures at layers 1-3 are lower than the freezing point which means the freezing may reach and stop at around layer 3.

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Figure 4. Temperature variations at different depths [5].

Figure 4 shows the temperature variations at different depths during the freezing process [5]. It is observed that the temperature changes with time at different depths, and the changes can be divided into a cooling stage and a steady stage. The cooling stage is from 0 h to about 10 h and the steady stage is from 10 h to the end of the measurement (96 h). In the cooling stage, the temperatures in the top decrease first from 12 C and then keep steady at -3 C. The temperatures at deeper depths also reduce with the temperatures decreasing in different degrees, and the closer to the top the soil is, the larger the cooling rate shows. In the steady stage, the temperatures at different depths show slight increases and they then keep steady while the frozen front also keeps steady in this stage.

2.3. Water Migration during Freezing Process During the freezing process, the initial water contents at different soil depths may show differences due to the scale of the experiment. Figure 5 shows the initial water contents at different soil depths in a feet-scale experiment [6], it is almost linear from the shallow to the deep. Researchers [7-9] explained that the pore water evaporation and gravity

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force lead to the decrease of water content in upper soil, meanwhile the capillary force among upper soil particles sucks the pore water from the lower soil. The dynamic equilibrium between the decrease of water content in upper soil and migration from the lower results in the linear water content gradient as shown in Figure 5. Figure 6 shows the total volumetric water content change during freezing process in a small-scale experiment [10]. At 0 h, the initial water contents at different soil depths are generally uniform, which is different from the initial water content distribution in the feet-scale experiments. This is mainly because in small-scale device, the evaporation, gravity force and capillary force are not so obvious to show differences in depths at the beginning of freezing.

Figure 5. Initial water contents at different soil depths [6].

When the freezing process starts, water in the top soil freezes, and water in the deeper soils flows toward freezing fronts where it changes from liquid to solid. This will result in a sharp increase in the water content in the freezing area, then water changes into ice and is fixed in the top area, due to the evaporation, sometimes water content in the surface may show a little decrease. As the freezing front moves to the deeper soil, the precipice of the water content caused by the freezing fringe will also move to the deeper place and stop at the end of the freezing process.

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Figure 6. The total volumetric water content 0, 12, 24, and 50 h after freezing [10].

Figure 7. Water contents at different times during freezing process [6].

Figure 6 shows the measured values of the total volumetric water contents at 0, 12, 24, and 50 h after freezing started [10]. At the beginning of the freezing, the water content is almost uniform along the depth as mentioned above, then as the freezing starts, the water content of soil in 0 cm-5 cm shows significant increase, and a precipice of the water content

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forms around the depth of -5 cm that indicates where the freezing front is. When the freezing moves deeper, the freezing front moves from -5 cm to 19 cm as the freezing time changes from 12 h to 50 h. When freezing stops, water content in the frozen region is significantly higher than the unfrozen region. Figure 7 shows the water contents along the depth at different times during the freezing process [6]. When freezing began, the water contents in the shallow showed a rise at first and then fell back (layer 1 and 2), and the water contents in the deep soils changed a little. On 7th day, water content decreased initially and then increased again. On 14th day, the water content increased initially, then decreased slightly and increased again. The general water content distribution on the 20 day is similar to that on 14 day. From 0 day to 20 day, the peak of the water content moved from layer 1 to layer 3 which shows the frozen front moving from surface to the deeper soil. The water content peak is mainly caused by cryo-suction in the frozen front. As a result, there is no water content decrease in frozen front while there is a slight increase in contrary, meanwhile the water contents in the deeper soils (layer 6-8) show a decrease with the water supply from the bottom.

2.4. Temperature Distribution during Thawing Process Generally, the temperature in the unsaturated frozen soil distributes almost linearly in the beginning of thawing process. When the environment temperature at the top rises from the bulk water freezing point below and the freezing state of the top soil can no longer be maintained, thawing starts at the top and the temperature of the top soil rises with a rate. Then heat travels to the deep with ice in the upper soil melting, resulting in the temperature rise of the deeper soil. Due to the heat attenuation, the temperature rise rate is usually lower than the upper one. Thus, during the thawing process, the temperature curve appears a concave peak along the depth and the concave peak gradually diminishes with the temperature curve usually becoming a stable straight line at the end of the thawing

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process. Generally, the temperature of the deeper soil keeps almost unchanged owing to the groundwater with basic constant temperature.

Figure 8. Temperature distribution at different times during thawing process [6].

Figure 9. Soil temperatures at Valletta site during the freeze-thaw period of autumn 1998 [11].

Figure 8 shows the temperature distributions along soil depth at different times of the experiments [6]. At the beginning of the thawing process, the top layer was around -10 C while the deepest layer was about

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10 C, then the temperatures of top and bottom walls were fixed at about 15 C to start the thawing. It can be easily seen that during the thawing process, a downward temperature curve peak is shown along the depth. Figure 9 shows the soil temperatures at Valletta site during the freeze-thaw period of autumn 1998 given by Matsuoka et al. [11] and Figure 10 shows the rock temperature during pre-macrocracking given by Murton et al. [12]. During the thawing period of the temperature curve, the temperature of shallow soil rises notably faster than the deeper. The difference is that at the end of the thawing period, the temperatures of the surface and upper soils are much higher than the deeper, which may be because the experiments carried out by Matsuoka et al. and Murton et al. were in the outside and the measured depths were less than 50 cm.

Figure 10. Rock temperature during pre-macrocracking. Freezing periods are numbered ‘1F’ and ‘2F’. Thawing periods are numbered ‘1T’ and ‘2F’ [12].

2.5. Water Migration during Thawing Process At the end of the freezing process, the frozen region has gone into the soil at a certain depth, and water in the upper frozen soil has turned into ice. In the freezing front area, water is sucked upward due to the capillary force which results in water content at a certain level in the freezing front.

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When the thawing process begins, ice in the top soil starts to melt and then the melting will go downward to the deep region of the frozen soil. Ice in the frozen area changes into water after melting, and then the melting water will be affected by the evaporation and gravity. Water content in the top soil may rise initially due to the melting and then fall back because of the evaporation and gravity. Water contents in the middle and lower regions of the frozen area are hard to predict because water in this area is not only converted from ice, but also supplied by the melting water from the upper area. Meanwhile, water in this area will migrate downward under the effect of the gravity. Water in the unfrozen deeper soil normally shows insignificant change, there may be little rise of the water content in this area due to the melting water from above. Migration of water in the frozen soil is coupled with the temperature in the upper frozen soil especially around the bulk water freezing point, when temperature rises from below to above the freezing point, usually the water content may fluctuate meanwhile.

Figure 11. Water content variations at different times during thawing process [6].

Figure 11 shows the water content variations at different times along the depth during thawing process and Figure 12 shows the volumetric water content variation with temperature during the soil thawing process for soils at layer 1 and 2 [6]. In Figure 11, the top layer 1 of the frozen region shows a rise after the thawing and then falls back, the water

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contents of layer 2-4 all fall to the linear moisture distribution of all depth, water contents of layer 5-8 all change with a little rise. In Figure 12, water contents in layer 1 and 2 show obvious fluctuations around the freezing point 0 C.

Figure 12. Volumetric water content variation with temperature during the soil thawing process for soils at layer 1 and layer 2 [6].

Figure 13. Unfrozen water content dynamics during freezing and thawing [13].

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Figure 13 shows the unfrozen water content dynamics during the freezing and thawing [13]. During soil thawing, both the unfrozen water contents at depths of 7 cm and 11 cm increase rapidly and after the thawing water content at the depth of 7 cm is much higher than that at the depth of 11 cm because of the moisture migration from warm area to cold area.

3. LBM MODEL FOR TEMPERATURE DISTRIBUTION AND WATER MIGRATION IN UNSATURATED SOIL Water migration and heat transfer in soil are coupled processes. During the freezing and thawing processes, the temperature, ice content and liquid water content show highly nonlinear relationships [14]. Fundamentally, soil is particulate porous material, and researches show that the heat and mass transport in porous material is strongly affected by its pore structure [15, 16]. Lattice Boltzmann methods (LBM) which is originated from the lattice gas automata (LGA) method, is a simplified fictitious molecular dynamics model in which space, time, and particle velocities are all discrete [17]. In LBM, the macroscopic dynamics of fluids are the statistical average results of the microscopic thermal behavior of fluids molecules and the macroscopic dynamics behaviors are insensitive to the movement details of each specific molecule. Thus, the LBM can be considered as a mesoscopic numerical methodology between microscopic molecular dynamics and macroscopic fluid dynamics. Therefore, the LBM shows capability of simulating transport phenomena which involves complex boundaries and interfacial dynamics. Song et al. [18] presented a mesoscopic numerical model based on LBM for simulating the heat and mass transfer phenomena with phase transformation in frozen soil during freezing process. Then, Xu et al. [4, 19] developed a modified LBM model to simulate the temperature distribution and water migration during unsaturated soil freezing.

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3.1. Reconstruction of Soil Structure Typically, unsaturated soil is consisted of solids (mineral and organic matter) and voids (or pores) of which a part is occupied by water and the other part by gas [20]. To simulate the coupled complex heat and mass transport process in the soil, the first thing is to reconstruct the soil structure. Because soil consists of particles of various sizes and these particles are in completely random distribution, stochastic approaches are considered reliable [21, 22]. The physical properties of soils usually considered are structure, bulk density, porosity, temperature, resistivity and so on [23]. To rebuild the soil structure, porosity is usually the first essential element to be considered. Based on the given certain porosity and some other properties, the pore structure is created randomly and the rest parameters are determined. To reflect the stochastic phase distribution characteristics of most porous media, Wang et al. [24] proposed a random internal morphology and structure generation-growth method, which was termed the quartet structure generation set (QSGS) based on the stochastic cluster growth theory for generating more realistic microstructures of porous media. The flowchart of QSGS process is shown in Figure 14. Based on the QSGS method, Li et al. [25] rebuilt a two-dimensional soil structure occupied by single-phase fluid and Figure 15 gives the random two-dimensional pore structure with different porosities  in which the black part is the solid skeleton and the white part is the pore. Song et al. [18] adopted the QSGS method to reconstruct soil structure based on a given porosity. Figure 16 gives the soil structure generated by stochastic generationgrowth method and connectivity of pores under different porosities  and core distribution probabilities d . In the figure, black represents solid elements and white represents fluid elements in the first row and black represents impermeable region and white represents permeable region in the second row. Porosity  determines the final void spaces in the soil and core distribution probability d determines the quantity of particles.

Temperature Distribution and Water Migration…

Figure 14. Flowchart of the QSGS process [24].

(a)   0.43

(b)   0.52

Figure 15. Schematics of generated porous media using QSGS method [25].

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(a)

(b)

(c)

Figure 16. Soil structure generated by stochastic generation-growth method and connectivity of pores. (a)   0.55 , d  0.5 ; (b)   0.40 , d  0.3 ; (c)   0.40 ,

d  0.35 [18].

3.2. Multiphase LBM Model Flows in unsaturated soil are usually consisted of multiphases including water, air and so on. Shan el al. [26-29] proposed and modified a pseudopotential LBM model to describe the multiphase flow in porous media. Due to its simplicity and versatility, the pseudopotential model is considered as the most widely used multiphase LBM model [30]. Based on the simple and popular Bhatnagar-Gross-Krook (BGK) collision operator [31], the single relaxation time LB equation can be expressed as follows

f ,i (x, t  t )  f ,i (x, t ) 

1

  ,v

( f ,i (x, t )  feq,i (x, t ))

(1)

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f ,i (x  ei t , t  t )  f ,i (x, t  t )

(2)

where f ,i (x, t ) is the density distribution function of the ith direction at the lattice site x and time t (see Figure 17); f eq is the equilibrium distribution function;

c  x / t

is the lattice speed, in which

x  t  1 represents the lattice spacing and time step respectively;  v is the dimensionless relaxation time. Eq. (2) represents the streaming step in which fluid particles are streamed from one lattice to a neighboring lattice according to the discrete velocity ei in the lattice. For a D2Q9 lattice,  0, 0    ei   cos   1  / 2  ,sin   1  / 2    2 cos   5   / 2   / 4  ,sin   5   / 2   / 4 



i0





i  1, 2,3, 4



i  5, 6, 7,8

(3) eq The equilibrium distribution function f is given by

2 2 3 9 3  feq,i  i  1  2  ei  ueq   4  ei  ueq   2  ueq   2c 2c  c 

(4)

where the weight factor  i in Eq. (4) is given by

4 / 9 i  0  i  1/ 9 i  1, 2,3, 4 1/ 36 i  5, 6, 7,8 

(5)

The viscosity in lattice unit is given by

   cs2   ,  0.5  t

(6)

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where cs is the lattice sound speed. The macroscopic density and momentum at each lattice node are given by

   f ,i

(7)

 u   f ,iei

(8)

i

i

The mass fraction

 

 of phase  is calculated by



  

(9)

In the soil, different bulk forces exist such as fluid–fluid interaction force, fluid–solid interaction force, gravitational force and others. Velocity shift scheme provided by Shan et al. [26] is used to describe the force term, the parameter u eq in Eq. (4) in the scheme is given by

ueq  u 

  , F 

(10)

where u  is the common velocity of the fluid mixture which is

 u   , u       ,



(11)

In Eq. (10), F is the sum of all interactive forces between fluid particles. In this model, F  Fcoh  Fads  Fb  Ftmp , Fcoh is the cohesive

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force between particles from the same phase; Fads is the adhesive force between fluid particles and solid boundary; Fb is the sum of bulk forces such as gravitational force and buoyancy force; Ftmp is the bulk force caused by temperature potential change. To simplify the question, vapor in soil is neglected and then the fluid system only consists of water and air. Therefore, only repulsion is needed to consider for different components, and the interactive force applied on the particles of one component is given by

Fcoh  x      x, t  Gcoh    x  ei ei

(12)

i

where

 represents the other component of the fluid mixture; Gcoh is an

adjuster to tune the surface tension between two components. The cohesive force between fluid particles and solid boundary is given by

Fads  x      x, t  Gads i si  x  ei ei

(13)

i

where si works as a tag, and the value of si is

0 node x+ei is fluid lattice node si  x  ei    1 node x+ei is solid boundary

(14)

Gads adjusts the interaction force between the particles of component

 and solid boundary, and it is positive for wetting fluid while negative for non-wetting fluid. The effect mass is



     0  



  0 1  exp   

(15)

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Fb   g

(16)

Figure 17. The layout of velocity directions of a D2Q9 lattice.

3.3. LBM Model for Temperature Distribution In LBM model, another layer of thermal lattice is used to describe the heat conduction process. A D2Q9 thermal lattice is adopted and the LB equation for thermal diffusion can be expressed as

gi  x  ei t , t  t   gi  x, t  

1



 g  x, t   g  x, t  i

eq i

(17)

The discrete velocity ei in the lattice is expressed in Eq. (3). The equilibrium distribution function g ieq is given by

g ieq  iT

(18)

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The weight factor  i is given in Eq. (5). The thermal diffusivity  is given by

  cs2    0.5 t

(19)

The macroscopic temperature can be calculated from

T   gi

(20)

i

In order to solve the heat conduction problem with phase change, Jiaung et al. [32] added a source term to Eq. (17) and proposed a new LB equation, which is given by

gi  x  ei t , t  t   gi  x, t  

1



 g  x, t   g  x, t   i

eq i

L  i  fl ,i  t  t   f l ,i  t   C

(21)

3.4. LBM Model for Water Potential In soil, water migrates under the influence of different forces such as gravity, pressure and capillary force. Water potential  is used to quantify the tendency of water movement and the total water potential  Tot is given by

 Tot   P   Z   S   A   tmp

(22)

where  P ,  Z ,  S ,  A and  tmp are pressure, gravitational, solute (osmotic), air pressure and temperature potentials, respectively.  S and

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 A can be neglected for low salinity soil under standard atmospheric pressure, then Eq. (22) is simplified to

 Tot   P   Z   tmp

(23)

In the LBM model,  P is represented by Fcoh and Fads (caused by capillary action),  Z is represented by Fb and  tmp is represented by

Ftmp . The movement of water in soil can be quantified by the gradient of water potential  caused by temperature change and it can be derived from Clapeyron’s equation when the soil temperature is above freezing point which can be written as

 

Ll T T Vl

(24)

where Ll and Vl represent the latent heat of condensation and specific volume change during vapor condensation, respectively. When the soil temperature reaches the freezing point, the gradient of water potential caused by temperature is given by

 

Ls T T Vs

(25)

where Ls and Vs represent the latent heat of fusion and specific volume change during freezing, respectively. Ls and Vs in Eq. (25) are the property of water and T  273K is the freezing point of water in Kelvin. Therefore, Eq. (25) becomes

  13.48 106 T

(26)

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The temperature water potential has the same value with the suction pressure caused by freezing process which is given by

Pr 

Ls  u2  r r T Vs 2

(27)

In lattice unit the suction pressure is given by

P

u 2 2

(28)

Then the suction pressure in lattice unit becomes

P

u 2 P  r ur2 r

(29)

It is known that

ur csr  u cs

(30)

where csr is the real speed of sound,  r is the real density of water. However, Eq. (26) needs to be revised for real soil. The correction factor depends on the type of soil [33]. Song et al. [18] proposed a relation to implement  in lattices which is given by

Ftmp  G tmpx where G

tmp

is determined by the type of soil.

(31)

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3.5. Validation of the LBM Model In 1990, Mizoguchi [34] performed a classic laboratory experiment, the experiment results have been adopted to validate numerical models in various studies [10, 35]. Though the previous models are not based on lattice Boltzmann method, it does not hinder the experiment results being appropriate to validate the LBM model. In Mizoguchi’s experiment, he packed four identical cylinders with Kanagawa sandy loam, and each cylinder was 20 cm long and had an internal diameter of 8 cm. The samples were prepared with the same initial conditions involving a uniform temperature of 6.7 C and a close to uniform volumetric water content of 0.33. Three samples were prepared for the freezing test while the fourth was prepared to measure the initial condition. The sides and bottoms of the cylinders were thermally insulated. The temperature of the top was controlled at -6 C by circulating fluid. Thus the samples were frozen from the top down. Thermal couples were installed in the cylinders at different depths to obtain the temperature distributions. The samples were taken from the freezing apparatus after different hours. When being removed from the cylinders, the samples were divided into 1 cm thick slices and dried in an oven to obtain total water content (liquid water plus ice) distributions. Figure 18 shows the comparisons of temperature distribution in experimental results and simulation results after 12, 24 and 45 h freezing. It can be observed that the temperature of simulation in frozen soil is lower than those of experiment at 12 h, and the simulation results show good agreements with the experimental results at 24 and 45 h. The reason for the difference between simulation data and experimental results is that the heat flux caused by water flux is neglected due to the considerably low water flow speed in this model while the water flow flux is relatively high during the first hours of freezing. Figures 19-21 show the comparisons of experimental results and simulation results after 12, 24 and 50 h freezing, respectively. All the simulation results show good agreements with the experimental results.

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Figure 18. Comparison between experimental and numerical results of temperature distribution after 12, 24 and 45 h freezing [18].

Figure 19. Comparison between experimental and numerical results of water content distribution after 12 h freezing [18].

The freezing front is clearly visible in Figures 19-21 where the water content decreases rapidly along the depth, and with the freezing time increases the freezing front moves to a deeper position. Thus in the model the rapid decrease of water content in frozen fringe and slow recovery in

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deeper position are well predicted. However, the size of the frozen fringe in simulation is smaller than that in the experiment which is because that the water in simulation freezes instantly in frozen fringe while it is not the same in reality.

Figure 20. Comparison between experimental and numerical results of water content distribution after 24 h freezing [18].

Figure 21. Comparison between experimental and numerical results of water content distribution after 50 h freezing [18].

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4. MODIFIED LBM MODEL FOR TEMPERATURE DISTRIBUTION AND WATER MIGRATION IN UNSATURATED SOIL 4.1. Physical Model The physical model of the modified LBM model is schematically given in Figure 22 [4]. When the soil freezing starts, temperature gradient forms in the soil due to the difference of temperature along the depth, and heat transfers to low temperature region from the higher one while water moves to frozen region from the unfrozen one. According to the freezing situation, the soil region is divided into frozen zone, freezing fringe and unfrozen zone. In frozen zone, Wettlaufer et al. [36, 37] pointed out that pre-melted films exist between the ice lens and soil particles which is caused by certain amount of unfrozen water fixed due to the intermolecular forces. Style et al. [38] pointed out that these pre-melted films provide channels for water from unfrozen region moving to freezing front to supply the growth of ice lenses. In freezing fringe, ice cores form due to the temperature there which is lower than freezing point Tm , on the other hand, the temperature there is also higher than the ice entering point Tie which stops the ice invasion from frozen zone. Due to the Gibbs-Thomson effect [39, 40], a certain amount of unfrozen water exists between the soil particles and ice grains. In the unfrozen zone, water exits in the voids among soil particles where the adhesion force and water-air interface tension keep balance with gravity. In section 3.1, it’s mentioned that soil consists of particles of various sizes which are in completely random distribution and Wang et al. [24] proposed a QSGS method to reflect the stochastic phase distribution characteristics of most porous media. Because the QSGS method is simple, it is also used in the modified LBM model to generate the stochastic pore structure.

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In the modified LBM model, attempts are made to find functional correlations between the particle size distribution and some macroscopic properties of the porous medium. Koponen et al. [41] pointed that the porosity  and the specific surface area S are two important macroscopic properties of the porous medium, which give the ratios of the total void volume and the total interstitial surface area to the bulk volume, respectively. Xu et al. [4] related the porosity  and the specific surface area S to the average volume fraction of obstacles f v . The value of porosity is equal to the probability that a given point in volume V is not overlapped by any of the obstacles, the porosity  is given by

  1  f v 

K

Figure 22. Diagram of physical model for unsaturated soil freezing [4].

(32)

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where K is the number of obstacles, f v  V0 / V is the average volume fraction of obstacles, V0 is the average volume of obstacles. The specific surface area is S  A / V while A is the value of the total surface area of all obstacles, A is given by

A  KA0 1  f v 

K 1

(33)

where A0 is the average surface area of obstacles. Then the porosity

 and the specific surface area S are related to the

average volume fraction of obstacles f v . Moreover, the average volume fraction of obstacles f vl and the average number of obstacles in a single lattice N are related to f v , where f vl  Nf v .

4.2. Governing Equations In 1992, a series of DdQm models (d-dimensional space, m-discrete velocities) are proposed by Qian et al. [42], and they constitute the basic models of LB method. Due to its common for solving fluid flow problems, the D2Q9 lattice model is also used in modified LBM model and the layout is shown in Figure 23. The streaming velocity for the D2Q9 model is given by

i 1  0, 0   ei   1, 0  ,  0, 1 i  2  5  1, 1 i  69  The associated lattice weights wi are

(34)

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Figure 23. D2Q9 lattice model for modified LBM model.

4 9 i  1  1 wi   i  25 9 1  36 i  6  9 

(35)

The sound speed ( cs ) is equal to 1 / 3 . The mass conservations for each of two fluids and the solid fraction are given by

 g t

     g ug   0

s  m t

(36a)

(36b)

Temperature Distribution and Water Migration…

l     l ul    m t

111 (36c)

where  m is the melting rate, u g and ul are the local fluid velocities of liquid and gas, respectively. The momentum equations for the fluid mixture are treated as a single one which is given by

 u    u   u   p      v  u  u    g  t 



where  

 

(37)

is the total density of the mixture. The total momentum

of the fluid mixture is

u   fi vi  1 2  F 

i

(38)



The initial conditions are given by

u g  ul  0   0  s  g  g0   l  l 0

(39)

Figure 24 shows the fluid flow boundary conditions. Bounce back boundaries are applied on the top, the down and the ice-fluid interfaces to maintain the mass conservation. Periodic boundaries are applied on the sides to maintain the continuity of flow.

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Figure 24. Fluid flow boundary conditions [4].

The enthalpy conservation of the three phases is

   s CsTs   ks Ts t

(40a)

  l ClTl   l ClTl  kl Tl t

(40b)

   g Cg Tg  t

  g Cg Tg  k g Tg

(40c)

where ki and Ti are respectively the thermal conductivity and temperature of phase i, Ci is the specific heat,  s ,  l and  g are the local density of solid, liquid and gas, respectively. The initial condition is

T  Tcold The boundary conditions are given by

(41)

Temperature Distribution and Water Migration…

113

Tup  Tcold at t  0

(42)

Tdown  Thot at t  0

(43)

The side boundary conditions are periodic as shown in Figure 25. The usual Lattice Boltzmann equation is applicable for the thermal diffusion during soil freezing process which is given by

Figure 25. Temperature boundary conditions [4].

gi  X  ei t , t  t   gi  X , t  

t

 g  X , t   gi h  i

eq

 X , t 

(44)

where g i is the temperature distribution function in the i-th velocity direction, and  h is the relaxation time related to the thermal diffusion by

  cs2  h  0.5 . The equilibrium distribution function for g ieq can be eq

obtained by: gi

 X , t   wi g .

4.3. Multiphase Fluid Due to its simplicity and versatility, the Shan-Chen pseudopotential model is also used in the modified LBM model to describe the multiphase

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fluid. In Shan-Chen model, the usual Lattice Boltzmann equation is used as the distribution function of all components during soil freezing which is given by

fi  X  ei t , t  t   fi  X , t  

t

 fi  X , t   f i ,eq  X , t   (45)

where f i  is the density distribution function of the  -th component in the i-th velocity direction, and   is the relaxation time related to the 2 kinematic viscosity for the component  by v  cs    0.5 . The

equilibrium distribution function for fi

fi

 , eq

 ,eq

 x, t  can be obtained by

2 2  eq ei  ueq  ueq   e  u i   2  X , t   i   1  2   cs 2cs4 2cs   

(46)

where the density and momentum of the  fluid component are given by

  i fi and  u  i Vi fi

(47)

u eq in equation (46) is determined by ueq  u 

  F 

(48)

where u  is the common velocity of the fluid mixture, and it is given by

Temperature Distribution and Water Migration…

115

 u

    u  

(49)

And F  Fcoh  Fads  Fb is the sum of the interactive force between fluid particles, Fcoh stands for cohesion force between solid boundary and fluid particles, Fads stands for adhesion forces and Fb stands for external body forces such as gravity and buoyancy forces. The interactive force which acts on the particles of species  located at position x can be obtained by

Fcoh  X , t      X , t  G coh i   X  ei t  ei

(50)

i

 are the first and second components of the fluid mixture coh respectively, G is the adjustment parameter of surface tension between where  and

the two modeled fluids. The cohesion force between particles of fluid  and the solid boundary can be obtained by

Fads  X , t      X , t  Gads i si  X  ei t  ei

(51)

i

where si  1 if the lattice node i belongs to the solid boundary and si  0 if i belongs to a fluid lattice node. The body force to mimic the effect of a buoyancy force can be obtained by

Fb  X , t    g

(52)

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where g is the gravity and  is the difference in density between the two components.

4.4. Melting Problem Bulk water freezes in the soil porous medium, this Neumann-Stefan problem including half space conduction melting with thermal diffusivity has been solved analytically by Neumann [43]. Jiaung et al. [32, 44] proposed an enthalpy-based lattice Boltzmann method and it has been successfully used for solid-liquid phase change problem in Huo’s work [45]. In the modified LBM model, Jiaung’s enthalpy-based method is used for melt-solid moving boundary in porous media that has not been considered in section 3. The melting term is treated as a source for crystallization or sink term for melting. At the time-step n and iteration kn, the macroscopic temperature is obtained by

T n,k   gin,k

(53)

i

where Tn,kn  Tkn  t  n  . The local enthalpy is obtained by

Enn,kn  cT n,kn  L f fl n,kn1

(54)

The enthalpy is used to linearly interpolate the melt fraction

fl n ,kn

0  n ,kn  En  Ens   Enl  Ens 1 

En n ,kn  Ens  cTm 0 Ens  En n ,kn  Ens  L f En n ,kn  Ens  L f

(55)

Temperature Distribution and Water Migration…

117

where Ens and Enl are the enthalpies of the solid and liquid at the melting temperature Tm . Saruya et al. [46] pointed out that the particle size distribution shows important relationship with the Gibbs-Thomson effect in the freezing fringe. Saruya et al. [46] also pointed out that T f which describes the difference between the warmest temperature and the normal bulk melting temperature

Tm 0  273.15 K

can

be

obtained

by

T f  Tm 0  Tm  2 slTm 0 /   L f R p  , where R p is the characteristic radius of a pore throat,  L f  3.110 J/m is the latent heat of fusion 8

3

per unit volume and  sl  29  103 J/m 2 is the ice-liquid surface energy. So, the actual water melting temperature in porous media is

Tm  Tm 0 

2 slTm 0  L f Rp

(56)

R p is given by R p   p R where R is the particle radius and  p is a correlation coefficient given by  p  0.7 . The average characteristic radius of a single lattice is given by 1

 3 f 3 Rp   p  v   4 

(57)

In the frozen zone, Engemann et al. [47] pointed out that pre-melted film exists between ice and soil particle surface below bulk melting point due to the intermolecular forces between components (particles, liquid, and ice), and the thickness follows a logarithmic growth law when approaching the melting point

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L T   a  0  ln

Tm  T0 Tm  T

(58)

where T0  Tm  17 K . The constant a  0  0.84 nm corresponds to the decay length of the non-ordering (average) density. Figure 26 shows the schematic of pre-melted film, the average thickness of the pre-melted film in a single lattice is given by

Figure 26. Schematic of pre-melted film [4]. 1

 3 3 L   1  fl  fl f v   4 

(59)

The relationship between the average obstacle volume fraction f v and pre-melted film melting temperature is given by

Temperature Distribution and Water Migration…

T T

Tmi  Tm 

e

m 0 1    3 1 f  f f  3 l l v    4    a 0    

       

119

(60)

In the modified LBM model, the bulk melting temperature Tm in equation (55) is substituted by the variable melting temperature Tmi to make the formation of pre-melted possible. The collision is given by

gin,k 1  X  ei   gi  X  

L 1 gi  X   gieq  X    ti f  f l n,k  X   f l n1  X    h c (61)

where  h is the relaxation time related to the thermal diffusion of the 2 component  by k  cs  h  0.5 .

The macroscopic temperature is given by

T   Ti

(62)

i

4.5. Interactive Force During soil freezing, water moves towards the frozen front through the freezing fringe from the unfrozen zone, overcoming the resistance along the path at the same time. In the modified LBM model, two more interactive forces are considered based on the formula

F  Fcoh  Fads  Fb given in section 4.3. One is Ffre which describes the suction in frozen front by treating as a kind of interactive force between ice and water particles, and it is similar to the adhesion force between fluid

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particles and solid boundaries in the Shan-Chen model. The other is a body force Fblo which describes the resistance along the path in unfrozen region, and it is related to the distance from freezing fringe according to Darcy’s law. Thus, the sum of the interactive forces for mixture fluid flow during soil freezing is given by

F  Fcoh  Fads  Ffre  Fblo  Fb

(63)

The adhesion force Fads in soil is related to the soil particle distribution which is obtained by

Fads  X , t      X , t  Gads  i fi  X  ei t  ei

(64)

i

where f is a flag variable equal to the soil particle volume fraction of the lattice node i. The suction force Ffre is caused by the pore water pressure difference among frozen zone, freezing fringe and unfrozen zone. Water moves from unfrozen zone towards the freezing fringe due to the pore water pressure decrease between ice grain surfaces and soil particle surfaces and then from the freezing fringe towards the frozen front due to water pressure deep decrease of pre-melted film in frozen front. This suction behavior only forms in the freezing fringe and pre-melted film near the ice particles and it is reasonable to be treated as an adhesion force between ice particles and fluid particles. The suction force Ffre can be obtained by

Ffre  X , t      X , t  G fre ili  X  ei t  ei

(65)

i

where l is a flag variable that equals to fl for the particles of fluid  with phase change happening near the phase change boundary, and it is equal to

Temperature Distribution and Water Migration…

121

0 for the particles of fluid  without phase change happening. G is the parameter that describes the magnitude of suction force which is related to the pore water pressure drop in freezing fringe and pre-melted film. The generalized Clapeyron equation can be described by fre

Pcl  P   L f

Tm  T Tm

(66)

where Pcl is the pore water pressure in the freezing fringe and pre-melted film at temperature T, P is the soil particle pressure in warm region, Tm is the bulk water freezing point, L f is the latent heat of ice melting,

 is the

ice density. When ground water supplies at pressure PR , the pore water pressure difference between ground water and pore water near ice particles is

P  PR  P   L f

Tm  T Tm

(67)

fre

The parameter G is related to the maximum pressure difference that locates in the frozen front where the temperature is equal to the ice entering temperature Tie while Tie is given by

Tie  Tm 

Tm sl  L f Rp

(68)

Consequently, the suction force can be obtained by

PR  P  Ffre 

Ac

 sl Rp

(69)

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where Ac is the cross-sectional area of flow. fre

Thus, G can be considered as a parameter relating to the surface tension at ice-water interface and the pore structure of soil particles. According to the kinetics of ice growth in porous media described by Style et al. [33], two kinds of resistance exist along the path way for water moving from unfrozen zone to the freezing fringe then to the frozen front: (a) the hydraulic resistance due to the flow in the porous medium and (b) the resistance due to the flow in the thin films around the particles. The hydraulic resistance is given by

Fh 

h kp

(70)

where k p is the permeability from Carman-Kozeny equation

R 2 1    kp  100 2

3

(71)

The resistance in the thin films is given by

F  f

3k p R 2 2 L3

(72)

where  p is the viscosity of water in thin films. Then the whole resistance along the path is given by

Fblo  Fh  Ff

(73)

Figure 27 shows the stress of fluid particles in unfrozen zone. In Figure 27 (a), the adhesion force and surface tension at the water-air interface keep balance with gravity as Ft   ghA while there is no pressure

Temperature Distribution and Water Migration…

123

gradient between pore water and ground water before soil freezing. In Figure 27 (b), pressure gradient forms in the pore water by which the tension force is gradually replaced to keep the balance with gravity during soil freezing and water flow begins after the pressure gradient overcomes the gravity. Thus the body force Fb is thought as a block force by this varying force which is given by

   ,initisl  3 gh 0 Fb     gh    ,initisl  3 gh

(a)

(74)

(b)

Figure 27. The stress of fluid particles in unfrozen zone [4].

4.6. Validation of the Modified LBM Model Mizoguchi’s experiment is also used to validate the modified LBM model, and the details of the experiment are described in section 3.5. The comparisons of experimental and numerical results of water contents after 12, 24, 50 h freezing are shown in Figure 28-30 [19]. From Figures 28-30, it is observed that all the simulation results are in good agreements with the experimental results.

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In Figure 28 (a), after soil freezing for 12 hours, along the depth of soil sample from 0 to 20 cm, the water content decreases slightly from 0.425 to 0.39, then decreases sharply from 0.39 to 0.29 and increases from 0.29 to initial water content 0.33. The soil freezing front locates nearly at the depth of 6.5 cm where the water content shows sharp decrease in Figure 28 (a). In Figure 29 (a), after the soil freezing for 24 hours, water content varies in the range from 0.42 to 0.39 in the shallow depths, then decreases sharply from 0.4 to 0.27 and increases from 0.27 to 0.32. The water content sharp decrease at around the depth of 9 cm in Figure 29 (a) indicates the freezing front locates here. In Figure 30 (a), after the soil freezing for 50 hours, the water content decreases slightly from 0.41 to 0.37, then decreases sharply from 0.37 to 0.25 and increases from 0.25 to the initial water content of 0.30. The freezing front locates at around the depth of 14 cm in Figure 30 (a). In these three figures, water contents along the depth of experiments and simulations are generally close, the freezing front moving trend and the freezing front location of experiments and simulations during the freezing process are similar. Besides, in Figures 28 (b), 29 (b) and 30 (b), the correlation coefficients between the simulated values and experimental data after 12, 24 and 50 hours freezing are 0.92, 0.83 and 0.92 respectively. The high correlation coefficients also indicate the reliability and availability of the modified LBM model.

(a) Volumetric water content distribution

(b) Correlation coefficient

Figure 28. Experimental and numerical results of water contents after 12 h freezing [19].

Temperature Distribution and Water Migration…

(a) Volumetric water content distribution

125

(b) Correlation coefficient

Figure 29. Experimental and numerical results of water contents after 24 h freezing [19].

(a) Volumetric water content distribution

(b) Correlation coefficient

Figure 30. Experimental and numerical results of water contents after 50 h freezing [19].

CONCLUSION AND FUTURE OUTLOOK Temperature distribution and water transfer in the unsaturated soil are mainly related to the soil movement phenomenon which may be harmful to the road and building ground safety. Experiment researches on temperature distribution and water migration in unsaturated soil during the freezing and thawing processes, LBM model establishment and validation, modified LBM model establishment and validation are presented and discussed. Experiment devices in small-scale and feet-scale are briefly introduced, the constituent parts of the two scales are generally similar

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while the most obvious difference is the size of the soil sample size. The temperature decreases faster in the lower depths as compared with the interior during the freezing process, and it increases initially and then the temperature curve appears a concave peak along the depth with the peak gradually diminishing during the thawing process. The water content is generally uniform along the depth in small-scale while it is almost linear in feet-scale at the beginning of the freezing process. Water content in the top soil may rise initially while in the unfrozen deeper soil it normally shows insignificant change during the thawing process. A D2Q9 mesoscopic lattice Boltzmann model (LBM) is established, and it includes: (a) soil structure reconstruction, (b) multiphase flow calculation, (c) temperature distribution calculation and (d) water potential calculation. (a)~(c) are based on the quartet structure generation set (QSGS) method, Shan-Chen pseudopotential model, double distribution function model respectively, while (d) considers cohesive force between particles from the same phase, adhesive force between fluid particles and solid boundary, external bulk forces and buoyancy force and temperature potential change bulk force. A modified D2Q9 LBM model is developed, and it includes: (a) soil structure reconstruction, (b) multiphase flow calculation, (c) temperature distribution calculation, (d) water potential calculation, (e) melting film consideration and (f) interactive force reconstruction. As compared with the LBM model, the modified model adopts Jiaung’s enthalpy-based method for melt-solid moving boundary in porous media, moreover it considers the suction in frozen front and the resistance along the path in unfrozen region for the interactive force reconstruction. Currently, experiments and simulation works have yielded some good results. However, the temperature distribution and water transfer in unsaturated soil are complex phenomena, more work is needed to put forward in the future: (a) more elaborate experiments to explore the water migration in the freezing fringe, (b) further experiment researches on the interaction between temperature distribution and water transfer, (c) new models to predict temperature distribution and water transfer in unsaturated soil, etc.

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ACKNOWLEDGMENTS This study is supported by Natural Science Foundation of China (Grant No. 51776049) and Special Foundation for Major Program of Civil Aviation Administration of China (Grant No. MB20140066).

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[16] Vadasz, P. (2008). Emerging Topics in Heat and Mass Transfer in Porous Media: From Bioengineering and Microelectronics to Nanotechnology. Springer Science Business Media. [17] Frisch, U., Hasslacher, B., and Pomeau, Y. (1986). Lattice-gas automata for the Navier-Stokes equation. Physical Review Letters, 56(14): 1505-1508. [18] Song, W., Zhang, Y., Li, B. X., and Fan, X. (2016). A lattice Boltzmann model for heat and mass transfer phenomena with phase transformations in unsaturated soil during freezing process. International Journal of Heat and Mass Transfer, 94: 29-38. [19] Zhang, Y., Xu, F., Li, B. X., Kim, Y. S., Zhao, W., Xie, G., and Fu, Z. (2018). Three phase heat and mass transfer model for unsaturated soil freezing process: part 2 - model validation. Open Physics, 16(1): 84-92. [20] Wikipedia contributors. Soil. In Wikipedia, The Free Encyclopedia. Retrieved 03:15, March 15, 2019, from https://en.wikipedia.org/w/ index.php?title=Soil&oldid=887118956. [21] Yaron, B., Calvet, R., and Prost, R. (1996). The soil as a porous medium, in: Soil Pollution. Springer, Berlin, Heidelberg, 3-24. [22] Schlüter, S., and Vogel, H. J. (2011). On the reconstruction of structural and functional properties in random heterogeneous media. Advances in Water Resources, 34(2): 314-325. [23] Gardner, C. M. K., Laryea, K. B., and Unger, P. W. (1999). Soil physical constraints to plant growth and crop production. Rome: Food and Agriculture Organization of the United Nations. [24] Wang, M., Wang, J., Pan, N., and Chen, S. (2007). Mesoscopic predictions of the effective thermal conductivity for microscale random porous media. Physical Review E, 75(3): 036702. [25] Li, S. F., and Xue, F. (2015). Numerical simulation of seepage and heat transfer in saturated soils based on lattice Boltzmann method. Journal of Thermal Science and Technology, 14(6): 445-455. [26] Shan, X., and Chen, H. (1993). Lattice Boltzmann model for simulating flows with multiple phases and components. Physical review. E, 47(3): 1815-1819.

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[27] Shan, X., and Chen, H. (1994). Simulation of nonideal gases and liquid-gas phase transitions by the lattice Boltzmann equation. Physical review. E, 49(4): 2941-2948. [28] Shan, X., and Doolen, G. (1995). Multicomponent lattice-Boltzmann model with interparticle interaction. Journal of Statistical Physics, 81(1): 379-393. [29] Shan, X., and Doolen, G. (1996). Diffusion in a multicomponent lattice Boltzmann equation model. Physical review. E, 54(4): 36143620. [30] Chen, L., Kang, Q., Mu, Y., He, Y. L., and Tao, W. Q. (2014). A critical review of the pseudopotential multiphase lattice Boltzmann model: methods and applications. International Journal of Heat and Mass Transfer, 76: 210-236. [31] Bhatnagar, P. L., Gross, E. P., and Krook, M. K. (1954). A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Physical Review, 94(3): 511525. [32] Jiaung, W. S., Ho, J. R., Kuo, C. P. (2001). Lattice Boltzmann method for the heat conduction problem with phase change. Numerical Heat Transfer Fundamentals, 39(2): 167-187. [33] Style, R. W., and Peppin, S. S. L. (2012). The kinetics of ice-lens growth in porous media. Journal of Fluid Mechanics, 692: 482-498. [34] Mizoguchi, M. (1990). Water Heat and Salt Transport in Freezing Soil (PhD thesis). University of Tokyo, Tokyo. [35] Tan, J. X., Chen, W., Tian, H. M., and Cao, J. (2011). Water flow and heat transport including ice/water phase change in porous media: numerical simulation and application. Cold Regions Science and Technology, 68(1-2): 74-84. [36] Wettlaufer, J. S., and Worster, M. G. (1995). Dynamics of Premelted Films: Frost Heave in a Capillary. Physical Review E, 51(5): 46794689. [37] Wettlaufer, J. S., Worster, M. G., Wilen, L. A., and Dash, J. G. (1996). A theory of premelting dynamics for all power law forces. Physical Review Letters, 76(19): 3602-3605.

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[38] Style, R. W., Peppin, S. S. L., Cocks, A., and Wettlaufer, J. S. (2011). Ice-lens formation and geometrical supercooling in soils and other colloidal materials. Physical Review E, 84(4): 1-13. [39] Cahn, J. W., Dash, J. G., Fu, H. (1992). Theory of Ice Premelting in Mono-sized Powders. Journal of Crystal Growth, 123(1-2): 101-108. [40] Ishizaki, T., Maruyama, M., Furukawa, Y., and Dash, J. G. (1996). Premelting of ice in porous silica glass. Journal of Crystal Growth, 163(4): 455-460. [41] Koponen, A. I., Kataja, M., and Timonen, J. (1997). Permeability and effective porosity of porous media. Physical Review E, 56(3): 3319-3325. [42] Qian, Y. H., Dhumieres, D., and Lallemand, P. (1992). Lattice BGK models for navier-stokes equation. Europhysics Letters (EPL), 17(6): 479-484. [43] Weber, H. (1901). Partial differential equations of mathematical physics II. Vieweg, Braunschweig. [44] Chatterjee, D., and Chakraborty, S. (2005). An enthalpy-based lattice Boltzmann model for diffusion dominated solid-liquid phase transformation. Physics Letters A, 341(1-4): 320-330. [45] Huo, Y., and Rao, Z. (2015). Lattice Boltzmann simulation for solidliquid phase change phenomenon of phase change material under constant heat flux. International Journal of Heat and Mass Transfer, 86: 197-206. [46] Saruya, T., Kurita, K., and Rempel, A. W. (2014). Indirect measurement of interfacial melting from macroscopic ice observations. Physical Review E, 89(6): 060401. [47] Engemann, S., Reichert, H., Dosch, H., Bilgram, J., Honkimäki, V., and Snigirev, A. (2004). Interfacial melting of ice in contact with SiO2. Physical Review Letters, 92(20): 205701.

In: Unsaturated Soils Editor: Martin Hertz

ISBN: 978-1-53615-985-1 © 2019 Nova Science Publishers, Inc.

Chapter 4

CHEMICAL POTENTIAL IN SOILS AND COMPLEX MEDIA Jean-Claude Bénet*, PhD Laboratoire de Mécanique et Génie Civil, Université de Montpellier, Montpellier, France

ABSTRACT By definition, the mass chemical potential (CP) of a constituent in a soil represents the variation of the internal energy of the medium when a unit mass of the constituent is transferred to a reference state with the entropy, volume and mass of the other constituents remaining constant. It quantifies the action of the rest of the environment on the constituent considered. For water, it results from mechanical or electrical actions generated by the superficial layers which are responsible for this retention. CP is a fundamental quantity in the same way as temperature or pressure

*

Corresponding Author’s E-mail: [email protected].

134

Jean-Claude Bénet This chapter addresses the measurement and expression of the CP in the presence of superficial layers induced by a complex structure. Measuring CP in soils, gels and various media of biological origin, is based on one of its fundamental properties, namely that, at equilibrium, the chemical potential of a component i has the same value on either side of a membrane permeable to i, temperature being uniform. The proposed analysis revisits conventional methods such as tensiometry or the use of filter paper and exploitation of the desorption isotherm. A mechanical method has been developed to measure the CP of volatile organic compounds in soils. The combined effects of the interfaces and a solute on the CP of constituents of an aqueous solution in the capillary domain are analyzed. In the case of an ideal solution of large molecules (e.g., polyethylene glycol) in a porous medium with a pore size small enough to hinder diffusion, a solute gradient is established. The Gibbs Duhem relationship induces a chemical potential gradient of water. The phenomenon of osmotic dehydration appears and is accompanied by a loss of water and stresses. The effect of gas phase pressure in the capillary medium refers to tensiometric measurement methods, with an important nuance: capillary potential and suction being empirical formulations of the CP, these concepts can then benefit from the body of mechanics, chemistry, thermodynamics, thus allowing a very broad approach to soil function and the expression of state functions, i.e., chemical potential, entropy and internal energy. We then examine the case of a saturated medium (solid phase + water) subjected to deformations that may result from external actions or internal water transfer. In the context of elastic behavior of the whole medium, increments of CP and stresses can be expressed as a function of thermomechanical coefficients Soil is a very special container generating charged surface layers, open on the outside, capable of generating temporospatial CP variations and fluctuations of the constituents of the aqueous solution. Passage of a solution from the free state to that of a solution in soil opens perspectives on applications such as plant nutrition, water transfer in hygroscopic environments, pollution/soil remediation, geochemistry, analysis of water transfers in arid zones (desertification) and prebiotic chemistry.

INTRODUCTION So-called complex media consist of three phases (solid, liquid and gas), finely divided and nested, and separated by large extension interfaces

Chemical Potential in Soils and Complex Media

135

that influence the thermal, mechanical and thermodynamic behavior of the medium as a whole. The solid phase confers a shape specific to the building; the superficial phases induce self-stressing forces that contribute to maintaining the shape of the object. The pores of the solid phase host solutions and gas mixtures in communication with the outside. A complex medium is an open reactor that can be the seat of chemical reactions. These media are highly represented in our environment and constitute everything that cannot be described as a continuous environment, including soils, rocks, porous materials, gels and biological media. Complicated calculation is not needed to find that it is easier to extract a kilogram of water from a bottle than the same mass of water contained in a cubic meter of clay with little moisture. In the first case, the binding forces of the water with the container are low and tilting the bottle is enough to remove the water. In clay, the binding forces involved between atoms, ions and molecules are all the stronger as the surface layers sequester much of the water. It is then necessary to supply energy to the system in order to obtain a kilogram of water of quality equivalent to that contained in the bottle. The surface layers determine the availability of water. This can be extended to any chemical species in the medium. In order to live and grow, plants must extract water from the soil. Water in the soil is translated by a value of its chemical potential. If this very low, roots cannot extract moisture and we must water the plants. To subsist and develop, micro-organisms must be able to extract water from food, the preservation of which exploits an ancient technique, namely drying. These situations can be approached in a rigorous way using the idea of chemical potential [1]. After presenting the theoretical bases that lead to the concept of chemical potential, currently available chemical potential measurement techniques are presented. Water, in three states, is the most abundant substance on the surface of the globe; it is the preferred solvent for many organic and inorganic substances. The main measurement techniques for water in soils are discussed and are extended to volatile organic compounds (VOCs) in soils. Various solid phases are also considered, including gels, food products and plant tissues

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Jean-Claude Bénet

After this experimental approach, the calculation of the chemical potential will be extended to more complex case    

the combined effect of a solute and superficial liquid-gas layers, the case of the impeded diffusion of large molecules in solution, osmotic potential, the effect of gas phase pressure, the effect of a stress tensor.

THEORETICAL BASES Gibbs and Gibbs-Duhem’s Relationship We consider a complex system (SC) contained in a volume V subjected to exchanges of heat, work, and material with the outside (Figure 1). There are two scales to describe how the SC changes. The so-called micro scale corresponds to the characteristic dimensions of the singularities of the medium: size of the grains, thickness of the liquid pockets and size of the gas bubbles. The macro scale contains a large number of singularities. To define the macro scale, average extensive quantities—mass, internal energy, entropy, etc.—are calculated on a sphere centered at a point M (Figure 1). The representative elementary volume (VER) is the quantity from which the mean values vary little. Continuous displacement of the point M generates continuous fields of mass or volume quantities assigned to the point M. The size of the VER—given by measurement of the radius of the sphere—defines the macro scale. Subsequently it is assumed that the medium is macroscopically homogeneous meaning that the size of the VER is independent of the position of the point M on the geometric space occupied by the SC. In addition, the fields of variables and macroscopic functions are uniform over this space.

Chemical Potential in Soils and Complex Media

137

Complex medium MASS

WORK

Pressure Stress



Chemical reactions

P



Chemical potential



i

M VER

HEAT Temperature T

Figure 1. Open complex system.

If U, V, S, mi are the internal energy, volume, entropy and mass of the constituent i in the SC, the principles of thermodynamics make it possible to write the Gibbs relation [2, p.23]: m

dU  TdS  PdV   i dmi

(1)

i 1

The increment of U is an exact total differential of the variables V, S, mi which results in:

T (

U )V , mi S

(2)

P(

U ) S , mi V

(3)

i  (

U ) S ,V ,m j , j i mi

(4)

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Jean-Claude Bénet

T and P are temperature and pressure evaluated with constant composition. It is possible to introduce strain and stress tensors to express work as will be done later. The intensive variable, evaluated at constant S and V since only constituent i is exchanged with the outside, is the chemical potential of component i in the complex system. Relationship (4) defines the chemical potential. It represents the action of the whole of the medium on the constituent i. The variables U, V, S, mi being extensive and the system homogeneous, by modifying its size in the parameter  : U (  S ,  V ,  m j )   ( S , V , mi )

(5)

Euler’s theorem implies [3, p.59]: m

U  TS  PV    i mi

(6)

i 1

This relationship combined with (1) gives the relation of Gibbs Duhem [3,p.60]: m

SdT  VdP    i dmi

(7)

i 1

Knowing the state function of a pure phase, this relationship will yield an expression of the chemical potential in simple cases, at constant temperature.

Chemical Potential in Soils and Complex Media

139

Thermodynamic Equilibrium Relative to the Transport of a Constituent An isolated system consisting of two subsystems 1 and 2 separated by a rigid wall, diathermy, permeable to a single chemical species i; mi1 and mi2 are the numbers of moles of species i in systems 1 and 2. A reversible transformation during which dmi1 the mass of i which passes from 1 to 2, is considered: dmi 2   dmi1

(8)

dU1  dU i 2  0

(9)

Any infinitesimal transformation from a state of equilibrium being reversible: dS = 0

(10)

By expressing dS from (1): dS  (

S1 S2 S S  )dU1  ( 1  2 )dmi1  0 U1 U 2 mi1 mi 2

(11)

What is written with (2) and (3): dS  (

1 1    )dU1  ( i1  i 2 )dmi1  0 T1 T2 T1 T2

(12)

Since this relationship is true for all infinitesimal variations of dU and dmi1 [2. p.34]: T1  T2 and 1   2

(13)

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Jean-Claude Bénet

If both systems are at the same temperature, the equilibrium between two phases separated by a membrane that is semi permeable to a given species results in equal chemical potentials of said species on either side of the membrane.

Chemical Equilibrium in a Phase A reactive system is considered in a reactor whose walls are rigid and adiabatic. The reaction involving chemical species is written in symbolic form: r

 i Ri  0

(14)

1

Where Ri are the chemical species involved in the reaction and  i are the stoichiometric coefficients counted negatively for the species that are consumed and positively in the opposite case. Given the constraints imposed (dU = 0 and dV = 0), T being the temperature at equilibrium, the entropy variation (1) is of the form: r

i m

1

T

dS  

dmi

(15)

The mass variation dmi is proportional to the stoichiometric coefficient of i in the reaction and the coefficient of proportionality is the degree of progress of the reaction: dmi   i d

The entropy variation dS takes the form:

(16)

Chemical Potential in Soils and Complex Media dS  

d T

r





141 (17)

im i

1

At equilibrium [2, p. 167]: r



 0

(18)

im i

1

This relationship is at the origin of the introduction of chemical potential by Gibbs (1985). It is only applicable if the reactor is rigid and adiabatic.

Expressions of Chemical Potential Simple Cases We limit ourselves to the case of a perfect gas whose equation of state is of the form: PVm  RT

(19)

Vm is the molar volume and R is the perfect gas constant (R = 8,314 J

mole-1 K-1). According to the Gibbs-Duhem relationship (7) along an isothermal transformation, the change in molar chemical potential is written: dm  Vm dP  RT

dP P

(20)

By integrating this relationship between a reference pressure P  and the pressure P, the temperature being assumed constant:

m (T , P)  m  (T )  RT ln(

P ) P

(21)

142

Jean-Claude Bénet Table 1. Expression of molar chemical potential for pure phases or ideal mixtures

System Perfect gas [2, p. 106]

State function

Molar chemical potential

PVm  RT

 m (T , P)   m  (T )  RT ln(

Van der Waals gas [4., p.155]

PVm  RT  B' ( T )P  C' ( T )P 2

R: perfect gas constant

B' and C ' functions of Van der Waals coefficients

 m   m  ( T )  RT (ln

P ) P (22)

P ( P ) ) P

(23)

 (P) expressed in terms of B' and C'

Perfect gas mixture [2, p. 207] Ideal solution [2,p.220]

PVm  RT ; Pi   Pi

 im  im (T )  RT ln

Pi P

(24)

Pi : partial pressure of i G   mi ( i

Gi m  RT ln i ) Ni  mi i

 im  im (T )  RT ln xi

(25)

xi : molar fraction of i

G : free enthalpy of the mixture Pure, slightly compressible liquid [2,p.122]

Vm (T , P)  Vm (T , P  )(1  T ( P  P  )

 T : Isothermal compressibility

 im   im  (T )  P /  (T )

(26)

 (T ) : density of the liquid

m (T ) is the reference chemical potential at temperature T and pressure reference. Table 1 summarizes the main cases for which the chemical potential can be calculated using Gibbs-Duhem relation..

Laplace and Kelvin’s Relations The Laplace model applies to the liquid gas interface. On the mechanical level, it is modelled by a curved surface of small thickness which separates the two fluids at different pressures. The mechanical equilibrium of the interface is translated by Laplace’s relation [2, p. 44]:

Chemical Potential in Soils and Complex Media

pG  pL   LG (

pG

and

pL

143

1 1  ) R1 R2

(27)

are the pressures in the gaseous and liquid phases,

 LG is

the surface tension of the liquid-gas interface (0.072 N/ m 2 for water at 20°C),

R1 and R2 the main radii of curvature at a point on the surface. This

relation applied to a spherical gas bubble of radius R gives:

pG  pL  2 LG

1 R

(28)

It is found experimentally that curved liquid-gas surface phenomena directed towards the gas phase have the effect of lowering the equilibrium vapor pressure of the liquid. To analyze the effect of curvature of the interface, three surface geometries are considered: a flat surface, the bubble and the droplet (Figure 2). P' , v' , ' Pvs( T ) P " , v" ,  "

P " , v" ,  " r

Figure 2. Vapor-liquid equilibrium in bubbles and droplets.

In the first case, the vapor pressure is equal to the saturation vapor pressure at the temperature T: We note P, v,   the pressure, mass volume, chemical potential on the side of the concavity and P,v,   the same quantities on the outside of the bubble or droplet. The radius of the bubble or droplet is r . In an infinitesimal variation, the mechanical equilibrium obeys the Laplace relationship:

144

Jean-Claude Bénet

dP"  dP'  d (

2 LG ) r

(29)

At a fixed temperature, the equilibrium with respect to the transfer of matter obeys (13):

d m"  d m'

(30)

At constant temperature, the Gibbs-Duhem relationship applied to the liquid and gaseous phases gives:

d m'  v ' dP'

(31)

d m"  v" dP"

(32)

So: v' dP'  v" dP"

(33)

Substituting into (29) ' " ( dP' ( v " v )  d ( 2 LG )

v

dP" (

r

v'  v" 2 )  d ( LG ) v' r

(34)

(35)

For the droplet, v  is the molar volume of the liquid phase which is negligible compared to the molar volume of its vapour. By adopting the law of perfect gases:

Chemical Potential in Soils and Complex Media

v' 

RT P'

145 (36)

Relationship (34) gives:

dP' (

RT 2 )  d ( LG ) ' " Pv r

(37)

Assuming constant surface tension and integrating between an infinite radius: 1/r = 0 for which P'= Pvs (T)) and any radius r corresponding to an equilibrium pressure P drop , we obtain the Kelvin formula:

ln(

2 v m P drop) )  LG Pvs(T ) RrT

(38)

Soil geometry is of the bubble type and, in the same way:

ln(

P bulle 2 v m )   LG Pvs( T ) RrT

(39)

It can be seen that the curvature of the interface increases the equilibrium pressure of the vapor in the case of a droplet and decreases it in the case of a bubble.

MEASURING CHEMICAL POTENTIAL IN COMPLEX MEDIA Table 2 gives some characteristics of a loamy clay soil (SLA is the reference medium for this study).

146

Jean-Claude Bénet Table 2. Some characteristics of silty clay soil (SLA)

Real Density of the solid phase 2650 kg/m3

Apparent density of solid phase 1500 kg/m3

Porosity

43%

Granulometry D = diameter of grains 0.001 mm < D