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Table of contents :
Abstract ... III
Résumé ... V
Chapter 1 ... 1
1.1. Motivation ... 1
1.2. Geophysical Methods in Hydrology ... 3
1.3. Physical Background to Induced Polarization ... 6
1.4. Electrochemical Background to Induced Polarization ... 8
1.5. Induced Polarisation and Relaxation Time ... 10
1.5.1. Time domain and frequency domain measures ... 10
1.5.2. Relaxation time ... 11
1.5.3. Relaxation length scale ... 17
1.5.4. Petrophysical parameters and relaxation time ... 19
1.6. Review of the Development of the Induced Polarization Method until today ... 21
1.7. Review of State of the Art IP-models ... 24
1.8. Objectives and Outline of the Thesis ... 27
Chapter 2 ... 29
Impact of changes in grain size and pore space on the hydraulic conductivity and spectral induced polarization response of sand
Abstract ... 30
2.1. Introduction ... 31
2.2. Electrochemical polarization and electrical relaxation ... 32
2.3. Experimental procedure and data analysis ... 35
2.4. Results ... 41
2.4.1. Hydraulic characterization ... 41
2.4.2. SIP measurements ... 44
2.5. Conclusions ... 48
Acknowledgments ... 49
Chapter 3 ... 51
Relating the permeability of quartz sands to their grain size and spectral induced polarization characteristics
Abstract ... 52
3.1. Introduction ... 53
3.2. Estimation of Permeability of Granular Media ... 55
3.2.1. Grain-Size-Based Permeability Estimation ... 55
3.2.2. Determination of Permeability from SIP Parameters ... 57
3.3. Experimental Background ... 59
3.4. Results ... 69
3.5. Discussion ... 75
3.6. Conclusions ... 78
Acknowledgments ... 79
Chapter 4 ... 81
Summary and Conclusions
4.1. Summary ... 81
4.2. Conclusions ... 82
4.3. Outlook ... 84
References ... 87
Appendix A ... 95
Joint interpretation of hydrological and geophysical data: Electrical resistivity tomography results from a process hydrological research site
in the Black Forest Mountains, Germany
Appendix B ... 125
Is it the grain size or the characteristic pore size that controls the induced polarization relaxation time of clean sands and sandstones?
Acknowledgments ... 143
Abstract ............................................................................................................ III Résumé................................................................................................................ V Chapter 1 ............................................................................................................. 1 Introduction 1.1. Motivation ..................................................................................................................... 1 1.2. Geophysical Methods in Hydrology ............................................................................. 3 1.3. Physical Background to Induced Polarization .............................................................. 6 1.4. Electrochemical Background to Induced Polarization .................................................. 8 1.5. Induced Polarisation and Relaxation Time ................................................................. 10 1.5.1. Time domain and frequency domain measures ..................................................... 10 1.5.2. Relaxation time ..................................................................................................... 11 1.5.3. Relaxation length scale .......................................................................................... 17 1.5.4. Petrophysical parameters and relaxation time ....................................................... 19 1.6. Review of the Development of the Induced Polarization Method until today ............ 21 1.7. Review of State of the Art IP-models ......................................................................... 24 1.8. Objectives and Outline of the Thesis .......................................................................... 27
Chapter 2 .......................................................................................................... 29 Impact of changes in grain size and pore space on the hydraulic conductivity and spectral induced polarization response of sand Abstract ....................................................................................................................... 30 2.1. Introduction ................................................................................................................. 31 2.2. Electrochemical polarization and electrical relaxation ............................................... 32 2.3. Experimental procedure and data analysis .................................................................. 35 2.4. Results ......................................................................................................................... 41 2.4.1. Hydraulic characterization .................................................................................... 41 2.4.2. SIP measurements ................................................................................................. 44 2.5. Conclusions ................................................................................................................. 48 Acknowledgments ....................................................................................................... 49
Chapter 3 .......................................................................................................... 51 Relating the permeability of quartz sands to their grain size and spectral induced polarization characteristics Abstract ....................................................................................................................... 52 3.1. Introduction ................................................................................................................. 53 3.2. Estimation of Permeability of Granular Media ........................................................... 55 3.2.1. Grain-Size-Based Permeability Estimation ........................................................... 55 3.2.2. Determination of Permeability from SIP Parameters ............................................ 57 3.3. Experimental Background ........................................................................................... 59 3.4. Results ......................................................................................................................... 69 3.5. Discussion ................................................................................................................... 75 3.6. Conclusions ................................................................................................................. 78 Acknowledgments ....................................................................................................... 79
Chapter 4 .......................................................................................................... 81 Summary and Conclusions 4.1. 4.2. 4.3.
Summary ..................................................................................................................... 81 Conclusions ................................................................................................................. 82 Outlook ........................................................................................................................ 84
References ......................................................................................................... 87 Appendix A ....................................................................................................... 95 Joint interpretation of hydrological and geophysical data: Electrical resistivity tomography results from a process hydrological research site in the Black Forest Mountains, Germany Appendix B ..................................................................................................... 125 Is it the grain size or the characteristic pore size that controls the induced polarization relaxation time of clean sands and sandstones? Acknowledgments ........................................................................................... 143
The spatial resolution visualized with hydrological models and the conceptualized images of subsurface hydrological processes often exceed resolution of the data collected with classical instrumentation at the field scale. In recent years it was possible to increasingly diminish the inherent gap to information from point like field data through the application of hydrogeophysical methods at field-scale. With regards to all common geophysical exploration techniques, electric and electromagnetic methods have arguably to greatest sensitivity to hydrologically relevant parameters. Of particular interest in this context are induced polarisation (IP) measurements, which essentially constrain the capacity of a probed subsurface region to store an electrical charge. In the absence of metallic conductors the IPresponse is largely driven by current conduction along the grain surfaces. This offers the perspective to link such measurements to the characteristics of the solid-fluid-interface and thus, at least in unconsolidated sediments, should allow for first-order estimates of the permeability structure. While the IP-effect is well explored through laboratory experiments and in part verified through field data for clay-rich environments, the applicability of IP-based characterizations to clay-poor aquifers is not clear. For example, polarization mechanisms like membrane polarization are not applicable in the rather wide pore-systems of clay free sands, and the direct transposition of Schwarz‟ theory relating polarization of spheres to the relaxation mechanism of polarized cells to complex natural sediments yields ambiguous results.
In order to improve our understanding of the structural origins of IP-signals in such environments as well as their correlation with pertinent hydrological parameters, various laboratory measurements have been conducted. We consider saturated quartz samples with a grain size spectrum varying from fine sand to fine gravel, that is grain diameters between 0,09 and 5,6 mm, as well as corresponding pertinent mixtures which can be regarded as proxies for widespread alluvial deposits. The pore space characteristics are altered by changing (i) the grain size spectra, (ii) the degree of compaction, and (iii) the level of sorting. We then examined how these changes affect the SIP response, the hydraulic conductivity, and the specific surface area of the considered samples, while keeping any electrochemical variability during the measurements as small as possible. The results do not follow simple assumptions on relationships to single parameters such as grain size. It was found that the complexity of natural occurring media is not yet sufficiently represented when modelling IP. At the same time simple correlation to permeability was found to be strong and consistent. Hence, adaptations with the aim of better representing the geo-structure of natural porous media were applied to the simplified model space used in Schwarz‟ IP-effect-theory. The resulting semiempiric relationship was found to more accurately predict the IP-effect and its relation to the parameters grain size and permeability. If combined with recent findings about the effect of pore fluid electrochemistry together with advanced complex resistivity tomography, these results will allow us to picture diverse aspects of the subsurface with relative certainty. Within the framework of single measurement campaigns, hydrologists can than collect data with information about the geo-structure and geo-chemistry of the subsurface. However, additional research efforts will be necessary to further improve the understanding of the physical origins of IP-effect and minimize the potential for false interpretations.
Dans l‟étude des processus et caractéristiques hydrologiques des subsurfaces, la résolution spatiale donnée par les modèles hydrologiques dépasse souvent la résolution des données du terrain récoltées avec des méthodes classiques d‟hydrologie. Récemment il est possible de réduire de plus en plus cet divergence spatiale entre modèles numériques et données du terrain par l‟utilisation de méthodes géophysiques, notamment celles géoélectriques. Parmi les méthodes électriques, la polarisation provoquée (PP) permet de représenter la capacité des roches poreuses et des sols à stocker une charge électrique. En l‟absence des métaux dans le sous-sol, cet effet est largement influencé par des caractéristiques de surface des matériaux. En conséquence les mesures PP offrent une information des interfaces entre solides et fluides dans les matériaux poreux que nous pouvons lier à la perméabilité également dirigée par ces mêmes paramètres. L‟effet de la polarisation provoquée à été étudié dans différentes études de laboratoire, ainsi que sur le terrain. A cause d‟une faible capacité de polarisation des matériaux sableux, comparé aux argiles, leur caractérisation par l‟effet-PP reste difficile a interpréter d‟une manière cohérente pour les environnements hétérogènes. Pour améliorer les connaissances sur l‟importance de la structure du sous-sol sableux envers l‟effet PP et des paramètres hydrologiques, nous avons fait des mesures de laboratoire variées. En détail, nous avons considéré des échantillons sableux de quartz avec des distributions de taille de grain entre sables fins et graviers fins, en diamètre cela fait entre 0,09 et 5,6 mm. Les caractéristiques de l‟espace poreux sont changées en modifiant (i) la distribution de taille des grains, (ii) le degré de compaction, et (iii) le niveau d‟hétérogénéité dans la distribution de taille de grains. En suite nous étudions comment ces changements
influencent l‟effet-PP, la perméabilité et la surface spécifique des échantillons. Les paramètres électrochimiques sont gardés à un minimum pendant les mesures. Les résultats ne montrent pas de relation simple entre les paramètres pétro-physiques comme par exemples la taille des grains. La complexité des media naturels n‟est pas encore suffisamment représenté par les modèles des processus PP. Néanmoins, la simple corrélation entre effet PP et perméabilité est fort et consistant. En conséquence la théorie de Schwarz sur l‟effet-PP a été adapté de manière semi-empirique pour mieux pouvoir estimer la relation entre les résultats de l‟effet-PP et les paramètres taille de graines et perméabilité. Nos résultats concernant l‟influence de la texture des matériaux et celles de l‟effet de l‟électrochimie des fluides dans les pores, permettront de visualiser des divers aspects du sous-sol. Avec des telles mesures géo-électriques, les hydrologues peuvent collectionner des données contenant des informations sur la structure et la chimie des fluides des sous-sols. Néanmoins, plus de recherches sur les origines physiques de l‟effet-PP sont nécessaires afin de minimiser le risque potentiel d‟une mauvaise interprétation des données.
Chapter 1 Introduction
Being influenced by the work done at UNESCO's International Hydrological Program, I learned about some of the multitude of concerns involving a single resource. I learned that the proper understanding of the involved political, socioeconomic and scientific problems is the vital parameter in offering the right solutions to the involved groups of people. Besides life threatening risks from flooding and draught, water related issues are manifold and are of concern to every nation in the world. In Norway, Austria, and Switzerland hydro-power supplies most of the electricity consumed by these countries population and in Brazil and Turkey energy from water is key to regional development programmes. In the Gulf States, energy from fossil fuel is used to produce fresh water from salt water, whereas France needs sufficient surface water in rivers to cool nuclear power plants during summer. While in Germany over 80 % of the resources annual recharge does not even enter the chain of agricultural, industrial, public and private use (Statistisches Bundesamt, 2009), countries in southern Europe perceive availability of water as major limiting factor for economic growth and well being of their inhabitants. In regions where water is scarce, sectors like tourism, industrial production lines, and agriculture compete against each other for the availability of the resource. In many cases competition affects basic needs of local eco-systems as well as the human population. With such developments water scarcity is not necessarily a prerequisite but is better understood as a factor of multiplication. The management of all parts of the above incomplete list, benefits from improved short term and long term predictions. However, in a time where climate change is eminent we need to question long term meteorological and hydrologic trends. Classic hydrological 1
rainfall-runoff models based on the long term observation of European catchments for example need to be critically questioned with regards to estimates of future developments, as the water cycle undergoes ongoing change until a new atmospherically stable balance may one day establish. Natural drainage systems, their visual part represented by surficial river networks, were formed over long time periods. They will need time to adapt to e.g. changing precipitation patterns due to the local effects of climate change. Even since before climatic change, one of the biggest problems related to water is the variability in temporal and spatial distribution of this vital resource. While due to climate change the meteorological input to the water cycle will remain difficult to predict for future years at the local scale, at global scale we know that increasing world population makes sustainable use of resources without alternative. The exact estimation of what is sustainable use is the question scientific research needs to be able to answer. In order for society to adapt in the best possible way, we need to provide means for predictions which are directly based on the physical understanding of the processes in the hydrologic system. Without spatial context to structural information of the subsurface, physical understanding of subsurface processes and the application of it remain incomplete. Today, the ability to get spatial precise pictures of subsurface properties by installing sets of electrodes on the earth‟s surface, temporarily, only for the time of the measurement, and than studying the quality of electric through-flow is simply amazing. While the broad use of geoelectrical methods is strongly related to developments in inversion techniques and simplicity of use of the measuring set-up, the basis for its reasonable interpretation is, as Andreas Kemna wrote in his thesis‟ conclusions in 2000, a sufficient knowledge of the relationship between the observed physical parameters and underlying structural characteristics. Though our work started seven years after this statement was formulated, the understanding of this relationship was still insufficient with respect to the interpretation of an 2
IP-signal of some naturally occurring sediments, e.g. alluvial deposits without clay. Given new encouraging developments in the geoelectrical method of induced polarization allowing for higher precision with weakly polarisable materials, the decision was taken to perform various tests under controlled laboratory conditions with varieties of quartz sand. The results of these measurements were than taken to provide further insight on the structural characteristics controlling the IP-effect.
Geophysical Methods in Hydrology
The method applied in this study, namely induced polarization, is part of the geoelectrical methods in geophysics. Today a wide range of geophysical methods are available which can provide spatial information useful with aquifer characterization (Table 1.1). However, it is electrical methods which are most sensitive to parameters relevant to the hydrological conductivity of a probed subsurface material. In the absence of electric conductors in the matrix of a saturated porous material, electric conductivity is limited to the fluid in the pore system (Archie, 1942) and to surface conductivity at the interface between the electrolyte and non conductive matrix. The basic link between electrical and hydraulic conductivity becomes apparent when comparing electrical conductivity as expressed in Ohm‟s law by J E , where J is current density, the electric conductivity, and E the electric field, to hydraulic conductivity K, expressed in Darcy‟s law with q Kh , where q is the specific discharge, and h the hydraulic head. Obviously, though many similarities between flow of water and electricity in a porous system exist, not every change in structural characteristics at matrix level has similar effects on both, the electrical and the hydraulic properties of the material. The key to good application of geoelectrical methods is hence a good understanding of those differences and relationships.
In the past, several studies gave rise to the perspective to effectively estimate hydrological relevant subsurface parameters, notably permeability, from the use of the induced polarization method (e.g. Kemna, 2000; Kemna et al., 2004, 2005; Binley et al., 2005; Tong et al., 2006; Hördt et al., 2007; Slater, 2007; Koch et al., 2011; Revil and Florsch, 2010; Weller et al., 2010; Zisser et al., 2010). All of these studies focus on the relaxation time parameter, which can be inferred from time or frequency domain induced polarization data. While the exact relationship between relaxation time and different subsurface parameters remains difficult to identify, it is clear that this direct relationship to subsurface structural parameters is a principal advantage of the induced polarization method and needs better understanding with regards to textural aspects in porous media.
Table 1.1. Listing of some of the commonly used geophysical methods with regard to their application in hydrological surveys (after Hubbard and Rubin, 2005). Method
Electrical resistivity, gamma radiation, magnetic and gravitational field, thermal radiation, electromagnetic reflectivity
Mapping of bedrock, fresh-salt water interfaces, faults, and assessment of regional water quality
Mapping of top of bedrock, water table, and faults
P-wave reflectivity and velocity
Mapping of stratigraphy and water table, top of bedrock, delineation of faults/fracture zones
Groundpenetrating radar (GPR)
Dielectric constant values and dialectric contrasts
Mapping aquifer zonation, water table, top of bedrock, fresh-salt water interfaces, estimation/monitoring of water content and quality
Mapping aquifer zonation, water table, top of bedrock, fresh-salt water interfaces and plume boundaries, estimation of hydraulic anisotropy, estimation/monitoring of water content and quality
Induced polarization (IP)
Complex electrical resistivity, chargeability, relaxation time
Gravitational field, density variations
Nuclear magnetic resonance (NMR)
Total amount of hydrogen atoms, relaxation time
Back-scattered neutron counts
Time domain reflectrometry
Dielectric constant values and dialectric contrasts
Mapping of stratigraphy and water table, estimating and monitoring of water content
Mapping aquifer zonation, estimation of grain size distribution and other structural parameters Mapping of fractures, ground water infiltration zones, monitoring/estimation of water/contaminant flow velocity and water table
Mapping of stratigraphy, top of bedrock, monitoring of aquifer recharge, water table Mapping of aquifer zonation, estimation/monitoring of water content, hydraulic conductivity
Physical Background to Induced Polarization
The IP effect is formed by an electric field applied to a mix of electrical conductive and nonconductive media. In porous media, the EDL formed around the non-conductive parts of the pore-system describes the area where dielectric polarization processes take place. Electric charges in the EDL are forced by the electric field and re-oriented according to field-lines. Maxwell‟s equations express the way electric currents and charges interact with electric and magnetic fields. Electric fields are related through current flux in conductive media and magnetic fields through movement of charge or a static magnet. Ampère‟s law altered by Maxwell (1892)
D , t
denotes the intensity of the magnetic field H, where D is the electric displacement field related to the electric Field E over the electric permittivity ,
D E .
The temporal change in the electric displacement field is also known as displacement current density JD. Jf is free current density defined by Ohm‟s law of electric conduction
J f E ,
with electric conductivity , and E the electric field, which after Faraday‟s law of induction is quantified by the rate of change in the magnetic field
H , t
with 0 being the permeability of free space. Total current density J T is defined as the sum of displacement and free current density and can be expressed as J T eff i eff E ,
in an electric field dependent on frequency, with the complex number i2=-1, and =2f the angular frequency. Effective conductivity eff and effective permittivity eff express complex 6
conductivity given by *=eff+ieff. We can now express total current density as a complex quantity
J T *E ,
with complex conductivity decomposed in real and imaginary part by =‟+i”,
or expressed as amplitude | and phase by
* * exp i .
Complex conductivity can be measured with SIP measurement devices and is used to describe induced polarization in the frequency domain. Analysis and interpretation of the conductivity related parameter values is hence related to the distribution of the electrical field in the EDL and generally in porous media. We note that the electric field in the Stern layer is parallel to the mineral surface. In the diffuse layer we assume already variations in the field angel, which in a micro scale relate to the variations in surface roughness of the solid and in a more macroscopic context are related to changes in the width of pores (Figure 1.2).
Figure 1.1. Electric field in porous media (after Clenell, 1997).
Electrochemical Background to Induced Polarization
The electrical double layer (EDL, Figure 1.1) is the defined process area of induced polarization in granular porous media (e.g., Titov et al., 2010; Vaudelet et al., 2011). It describes the interface formed between solid and liquid phase. It relates to pore-characteristics as it does to grain characteristics. Negatively charged silica surfaces in a conductive liquid characterized by the presence of anions and cations are the basis for the formation of an electrical double layer. It comprises (i) the Stern layer, where bound cations partly balance the negative charge of the mineral surface, and (ii) the diffuse layer where the remaining charge discrepancies between solid and free electrolyte are balanced. The separation of the two layers is characterised by limited exchange between the two layers. The sorption/desorption processes are responsible for the formation of the more tightly fixed counterions in the Stern layer. Vaudelet et al. (2011) argue that the kinetics of these processes which enable exchange between Stern and diffuse layer are slow, and they conclude that such an effect can be neglected at the frequencies typically considered for induced polarization.
Figure 1.2. Double layer conceptual model (Revil and Florsch, 2010).
Any polarization process of the EDL is related to the ionic mobility i of the dominant species i in the electrolyte which is related to the diffusion coefficient Di of the species by
i kT qi
where k is the Boltzman constant, T is temperature in Kelvin, and qi = zie is the charge with elementary charge e and zi the unsigned valance of species i. Vaudelet et al. (2011) further state that the flux of species in the Stern layer driven by an electric field can be described by the Nernst-Planck local equation, an extension of Fick‟s law, which relates the diffusive flux to the concentration in relation to the concentration gradient., given by
i i qi
where s denotes the surface gradient along the mineral surface, i the concentration, and i is the electrochemical potential of species i. The above equations control the general aspects of the influence of electro-chemical characteristics in the studied media on the relationship between spatial and temporal IP-measures. While Revil and co-workers (e.g. Vaudelet et al., 2011) relate relevant polarization mechanisms principally to the Stern layer, the classic Dukhin and Shilov (1969, 1974) polarization theory or standard model for colloidal suspensions is based on the assumption of a thin double layer. Under steady state conditions, the ion concentrations C and valences of ions z, electrical potential , fluid velocity V, and pressure P can be determined using the Nernst-Planck equation as well as the following set of equations: Condition of incompressible fluid given by
Continuity equation given by
Poisson equation given by 9
2 ( z C z C )
Navier-Stokes equation given by
e 2V ( z C z C )e ,
where e is the viscosity of the electrolyte. Though various extensions for the standard model exist, including the generalized standard model (Duhkin & Shilov, 2001; Shilov et al., 2001), all are limited to the ideal situation of particles in suspension, and hence none exists which describe the process in natural granular media with particles in contact with one another.
Induced Polarisation and Relaxation Time 1.5.1. Time domain and frequency domain measures
Figure 1.3. IP-signals in a) the time domain, and b) in the frequency domain.
The induced polarization signal can be measured either in time domain with DC-resistivity (Figure 1.3a) or in frequency domain using alternating current (Figure 1.3b). Both approaches have their advantages and disadvantages, mainly with measurement complexity (AC-current) and resolution of results (DC-current). For conceptual interpretation of polarizing the subsurface induced through current-flux however, it is helpful to recognize that data gathered either way yields the same information.
1.5.2. Relaxation time The parameter relaxation time is used to describe the temporal aspect of ion movement under the IP-effect. Various studies show high correlation between these temporal aspects and permeability in a variety of natural porous media (Figure 1.4).
Figure 1.4. Relaxation time - permeability relationships examples in the literature (after, Zisser et al., 2010).
In order to derive values of relaxation time from IP response of natural materials, a variety of approaches has been tested. While in time domain IP such a step is in principle rather self explanatory and is defined as the time it takes for the measured potential to reach initial values after current is switched of, in frequency domain IP we convert frequency measures f to a relaxation time equivalent with
In time as well as in frequency domain the temporal resolution of the data is relevant for its further interpretation. In natural media, signals of different origins overlap. In Figure 1.5 we see the different processes related to frequency they occur in.
Figure 1.5. Polarization mechanisms above 1 kHz: a) electronic, atomic and ionic polarization: quantum resonances, b) molecular polarization: polar molecules in fluids (e.g. water dipole), and c) space charge polarization: material interfaces and in fluids (after Clenell, 2010).
While the resolution of the measurement defines the theoretical possibility to differentiate between processes of different origin, processes in the low frequency area between 1 mHz and 1 kHz may not yet be fully identified. Figures 1.6, and 1.7 are examples of modern IPmeasurements, from time domain (compare to 1.3a) and frequency domain measurements (compare to Fig. 1.3b). These studies mainly focus on effects of polarization in a time range slower than polarization associated with Maxwell-Wagner type mechanisms. In frequency domain we can roughly translate this to polarization taking place at frequencies below 100
Hz. Figure 1.6. High resolution analysis of the induced polarization decay curve a), yielding b) the relaxation time spectra from a time domain measurement (Tong and Tao, 2007).
Figure 1.7. High resolution frequency domain measurement.
As an example for polarization in the range below the Maxwell-Wagner polarization, Leroy et al. (2008) found first experimental evidence for polarized surface roughness of glass beads with detailed laboratory studies (Figure 1.8). In this example the potential process areas, grain size as well as surface roughness are clearly separate, and can be individually addressed in data-interpretation. For natural materials however, this often is not the case, due to superimposition of effects. In current practice hence, a single characteristic or dominant relaxation time 0 is identified which can than be compared to values of other characteristic parameters describing porous media (e.g. Fig. 1.4).
Figure 1.8. Glass bead surface roughness identified as process area for electrochemical (EC) polarization (after Leroy et al., 2008).
Pelton et al. (1978) demonstrate the use of the Cole-Cole model (Cole-Cole, 1948) with IP-data based on the phenomenological description of a single polarizing process (Kemna, 2000).
Table 1.2. Listing of some of the existing models with references, equivalent circuit analogs, and associated complex resistivities (after Dias, 2000).
In Table 1.1, a variety of phenomenological models are listed in chronological order, together with their electric equivalent circuit. Though a variety of models exist, the most widely used model till today is still the Cole-Cole, or, Multi-Cole-Cole (MCC) model in cases where a single model may not describe the data sufficiently well. Somewhat comparable to a multiCole-Cole approach is the more recent approach of Debye decomposition (DD, Weller et al., 2008). Here, the data is modelled via superposition of multiple Debye models (Tab. 1.1). 15
MCC and DD have in common that it is possible to model complex data sets with supposedly overlapping of sources for polarization processes. In case of the less complex Debye model a multitude of relaxation times can hence be computationally estimated and a relaxation time distribution drawn. The letter enables estimation of mean, median, and modal values for relaxation time, and should provide values for which are closer to a direct transfer frequency to time constant compared to values for the time constant derived using the Cole-Cole phenomenological equivalent model. For example,
1 2 peak
with peak denoting the model value of the time constant distribution derived from DD. We need to note here, that the time constant derived via the application of these phenomenological approaches is not necessarily equal to the relaxation time we‟d like to estimate. The description of the exact relaxation processes however, is ongoing research. As an example, Revil and Florsch (2010) assume in their work that the distribution of relaxation times from a sample of granular media is inversely related to its grain size distribution. The step of assigning a geometrical parameter to this temporal parameter is the definition of a process area for the relaxation process. Only if this step is a true description of the measured environment, the correct model representation, notably the number of different relevant polarization processes, is possible. As long as such is still unclear, the inverse approach remains to find a good model fit and possibly draw conclusions on the assumptions it is based on.
1.5.3. Relaxation length scale Relaxation time , the temporal component measured with the IP-method, can be brought in relation to a geometrical size, the relaxation length scale, r2. This follows the general understanding of normal diffusion processes D, where
r 2 D .
In order to use IP-measurements for example for estimation of permeability we need to understand how the relaxation length scale is related to geometrical parameters in a porous media. In IP literature two principal process areas are discussed (Figure 1.9, Figure 1.10). For the simplified case of (i) particles in suspension, Schwarz (1962) formulates the theory for polarization of the surface of the spheres
d2 , 8 Di
where d is the spheres diameter, and Di is the diffusion coefficient of species i. In this set-up the altering impact of the solid, non conducting parts to the electric field of the studied particle is arguably neglected, and the EDL is assumed infinitely thin. For the special case of (ii) porous media with tight pore throats, membrane polarization (Marshall and Madden, 1959) is often identified as the process responsible for local charge accumulation. Membrane polarization describes processes in pores as thin that parts of the EDL can act as an ion selective barrier. Whether, the relaxation time is related to the geometric length, or the diameter of pore throats, or e.g. the length between two pore throats, is an ongoing discussion.
Figure 1.9. Polarization mechanisms below Maxwell-Wagner (interfacial) polarization mechanism (after Zisser, 2010). Membrane polarization is associated with ion selective current transport in tight pore space. Electrochemical polarization mechanisms include the polarization of the Stern layer through movement of cations, and polarization of the diffuse layer through a deformation of a cloud of ions.
Figure 1.10. Detailed scheme of electrochemical processes around a single silica grain in a NaCl background electrolyte (Revil, 2010).
While today (i) is discussed with regards to Stern-layer polarization (e.g. Leroy et al., 2008; Revil and Florsch, 2010; Vaudelet et al., 2011), (ii) is addressed with regards to local ion concentration gradients in the EDL, including the diffuse layer (Titov et al., 2010). Both approaches are used (e.g. Titov et al., 2002; Titov et al. 2004; Ulrich and Slater, 2004; Kruschwitz et al., 2010; Revil and Florsch, 2010) in trying to explain induced polarization in 18
natural environments much more complex than the above described structural conditions. For such complex cases, sound physical process understanding is still missing. For (i), this manifests in an incomplete understanding of the influence of neighbouring spheres on the polarization process on the surface of individual spheres in complex granular media. For (ii), with regards to polarization in complex systems which are described by a multitude of different pore sizes and/or an unspecified sequence of similar sized pores, a situation where pore throats act as ion-selective membranes is complicated in many ways. The necessary presence of ion-selective pore throats alone makes it in-applicable to a range of sandstones and to all granular media in the range of sandy grain sizes, simply because pore throats are too wide to act ion-selective; notably in the absence of clayish particles.
1.5.4. Petrophysical parameters and relaxation time Relaxation time is arguably the most promising IP-parameter for estimates of permeability. With regards to general classification of petrophysical parameters we can state the following: they describe
interface characteristics (surface).
There is a heavy interlink between petrophysical parameters (Figure 1.11). With the above classification we are able to visualize some of these, as for example grain size is clearly a granular characteristic which also governs the size of the surface in a granular matrix. The sorting of the grain size distribution likewise, is an important granular characteristic which, unlike the size of grains, is also decisive for the porosity in a granular media. Results from studies testing the correlation between IP-parameters and petrophysical ones, therefore often present a rather homogeneous picture of generally good correlation
without an important preference for one parameter pair against another (e.g. Binley et al., 2005). With regards to IP relaxation time, the process area is located at the solid-fluid interface, in other words the surface of granular media. At the same time, we know that the absolute value of the surface area doesn‟t play a role for the time granular media takes for relaxation. Instead sections of the interface which can be put in relation to grain or pore characteristics, govern the quality of the parameter.
Figure 1.11. Map of parameter interdependencies. Interesting are the few interlinks related to the size of grains.
Review of the Development of the Induced Polarization Method until today
The following history of the induced polarization method is based on the work with the same title from Collet, 1990. Electrical polarization effects in soils and rocks were first recognized by Schlumberger in 1911. He conducted laboratory studies of the electrical properties of ore samples and rocks using alternating current. Schlumberger shifted to apply direct current as an answer to problems with line interferences when using AC in the field. In 1913, when prospecting for highly conductive ore bodies, Schlumberger noticed that after the current was switched off, small but measureable potential differences continued for some time. He concluded that a polarization of the subsurface had occurred. He further reasoned that metallic bodies could be distinguished from other subsurface material by this type of effect. In 1920, he first published these findings under the name of induced polarization. A new prospecting method was hence developed based on interruption of injected current and measurement of a subsequent polarization effect. Inadequacy of technical equipment was still a big issue during that time and as a consequence progress was limited. During the 1930s and 40s the IP method was tested widely in oil exploration. With regards to environmental studies, Vaquier et al., 1957 made the attempt to apply IP with groundwater exploration in clay rich material. Than, in 1959, Wait laid the mathematical foundation for the time and frequency domain for IP prospection, while Seigel with his work established the term chargeability in the IP-domain. At the same time, Marshall and Madden (1959) conducted a range of laboratory and basic theoretical studies on polarization of nonmetallic origin, defining in their work terms like electrode and membrane polarization. In line with the work of Schwan et al. (1962), Schwarz (1962) and Kormiltsev (1963) develop relationships between grain-/pore characteristics and relaxation time. These studies remain amongst the most important for over 50 years with regards to polarization mechanisms of non-conductive solids in a background electrolyte, though the various improvements made in
colloid science based on the standard model by Duhkin and Shilov (1969, 1974) need also to be mentioned in this context. In 1978 than, after in the 1960s IP investigations began to look at complex resistivity, Pelton et al. found that the parameters chargeability and time constant showed a wide variation between types of mineralization in an applied Cole-Cole impedance model. Their findings are based on small scale in-situ measurements in ore outcrops in order to discriminate sulphide ore from graphite and porphyry copper mineralization by its ore-mineral grade and texture. Since the 1980s, SIP was increasingly used for petroleum industry and environmental applications. Klein and Sill (1982) studied the effect of grain size, electrolyte conductivity and clay content on the resistivity spectrum of artificial samples composed of clay, glass beads and water, and found that values of time constant follow in general the size of glass beads. With regards to IP applications in petroleum, research was done in shaly reservoirs and specific models developed (Vinegar and Waxman, 1984; de Lima and Sharma, 1992). Towards the end of the eighty‟s and during the ninety‟s, new laboratory SIP instruments allowed wide band frequency measurements for environmental studies. For example, change in phase spectra due to contamination with organic material was found within clay by Olhoeft (1984, 1985). Vanhala et al. (1992), Börner et al. (1993), and Vanhala (1997) measure complex resistivity with regards to mapping of oil contamination or general identification of contaminants with IP. Börner et al. (1996) study the SIP relationship with hydraulic conductivity in the laboratory and in the field. Though, an evolutionary development of the method is evident throughout the years, from the year 2000 on, there was a new substantially increased interested in the application of SIP related to hydrogeophysics (Binley, personal communication 2009). Based on the developments made with the imaging of DC-resistivity, Kemna (2000) with his work on tomographic inversion of complex resistivity put computational developments made with 22
electrical resistivity tomography (ERT) together with a future perspective for SIP in fieldapplications, at that time principally for applications in hydrogeophysics. In the following years the details of the relation of non-metallic polarization in the subsurface and petrographic parameters is subject to various studies. Titov (2002, 2004) propose models of the conceptual understanding of polarization mechanism to explain their data from studies of sieved sands with varying degrees of saturation. However, Ulrich and Slater at the same time (2004) with similar studies state that a final conclusion about the exact localisation of the polarization processes remains enigmatic. IP-results from samples of sandstone (Binley et al., 2005) compared to a wide set of petrophysical parameters further monitored the close link of the IPmethod with different hydrologically relevant parameters. Though a universal understanding of the IP-effect in complex natural environments was still missing, a variety of studies monitored the close link between the relaxation mechanism in polarized porous media and their capability to transport water (e.g. Slater and Lesmes, 2002; Kemna et al., 2004; Binley et al., 2005; Tong et al. 2006; Hördt et al., 2007; Slater, 2007), which also motivated the start of the studies for this thesis in 2007. In parallel to the developments in hydrogeophysics, biogeophysical application of the IP-method receives increased interest, e.g. with regards to the metabolite of bacteria in inaccessible regions of the subsurface (e.g. Ntarlagiannis et al., 2005; Abdel Aal et al., 2006). Together with improved imaging techniques, IP response is related to the localization of contaminated cites, and monitoring of microbiological processes in aquifers (e.g. Williams et al., 2009). Recently, a number of published studies represent the increased research effort put into analyzing the structural aspects of the IP-effect. Kruschwitz et al. (2010) focus in general on the textural controls, while Titov et al. (2010), Weller et al. (2010), and Zisser et al. (2010a), discuss permeability estimation in sandstones. Revil and Florsch (2010) provide permeability estimates based on Stern-layer polarization mechanisms in granular media, and parameter relationships with electric tortuosity and formation factor are discussed in Binley et 23
al. (2010) and Weller et al. (2010). IP-related electrochemical aspects which are linked to the mobility of charges, are studied in the form of effects from changing temperature (Binley et al., 2010; Zisser et al., 2010b), sorption behaviour of species in saturated sands (Vaudelet et al., 2011), and changes in fluid conductivity or pH (Revil and Skold, subm. 2011).
Review of State of the Art IP-models
At present the IP phenomenon is often described through the use of electric equivalent circuits. Most common is the use of Cole-Cole or Generalized-Cole-Cole type models (Pelton, 1977, 1978). This phenomenological approach to model IP-results is first published by Wait (1959), and can also be logically related to the work of Marshall and Madden (1959). The latter describe the polarization process as a consequence of alternating zones with variations in ion-mobility (their Figure 1), notably a zone with ion-selective characteristics followed by a zone which is not. This approach is generally linked to understanding polarization processes as membrane-like effect and is further developed by Titov et al. (Short-Narrow-Pore model, 2002). Volkmann and Klitzsch (2010) state that the existing models based on membrane polarization do not take into account the electrochemical processes taking place in the EDL. This translates in a certain difficulty to assign a size range in which the membrane effect may occur and the models application is valid. Today, the spatial understanding of electrochemical processes is arguably best explained in connection to grain size based polarization processes (Revil and Cosenza (2010), their Figure 1). Electrochemical processes become first accounted for in the theory of Schwarz (1962). Schwarz used some of the results published in Schwan et al. (1962), notably a quadratic relationship between characteristic frequency and the inverse of a polarized particles radius, to experimentally justify his theory. The observed form of proportional dependence is similar to those in general diffusion processes and polarization was attributed to processes of the electrochemical diffusion type. Processes which take place mainly in the EDL, describe 24
electrochemical polarization mechanisms as e.g. discussed by Leroy et al. (2008). This includes both, membrane, as well as grain size based relaxation mechanisms. As a consequence from knowing the speed of polarization mechanisms, a time or frequency measured with IP is attributed to a length, the so called “relaxation length scale”. It is an open question whether this length is best attributed to the size of grains or to pore-characteristics like the length or radii of pores and pore-throats. Revil and Florsch (2010), who attribute relaxation time to the size of grains and the distribution of phase shift frequency spectra with the inverse distribution of grain size, limit relevant relaxation processes to the counterion movement in the Stern layer of the EDL. Volkmann and Klitzsch (2010) propose the model of a simple pore system with EDL. In their approach they address the problem that models describing membrane polarization (e.g. Titov et al., 2002) previously didn‟t employ diffusion processes and thus were not working with parameters relevant to the electrochemical nature of low frequency polarization, including temperature (Binley et al., 2010; Zisser et al., 2010) and mobility of species (Vaudelet et al., 2011). Principally following the ideas presented with the membrane approach (Marshall and Madden, 1959; Titov et al., 2002), Volkmann and Klitzsch use the Debye length to assign dimensions to the EDL. This measure is employed in order to account for the basic assumption that small enough pores show ion selective characteristics. The fact that they do not make any distinction between Stern and diffuse layer parts of the EDL is similar to what is proposed in the standard model. The standard model (Duhkin & Shilov, 1969, 1974) is based on the thin double layer assumption similar to what was proposed by Schwarz (1962) and the induced variations of the charges in the electrolyte, surrounding of spherical particles in suspension. In the model, predicted surface conductivity corresponds solely to the diffuse layer of the EDL. However, various extensions have been made to the original version of the model, e.g. for non-spherical particles (Eremova and Shilov, 1975, 1995; Grosse and Shilov, 1997; Grosse et al., 1999) and the effect of increased concentration of suspended particles 25
(Delgado et al., 1998), an effect already observed by Schwan et al. (1962). The generalized standard model (Duhkin and Shilov, 2001; Shilov et al., 2001) further includes a term for conduction of the stagnant or Stern layer. Recently, Grosse (2011) proposed a more detailed approach to incorporate stagnant layer properties for ions not limited to a single sign. Though Grosse (2011) takes into account an effect of the Stern or stagnant layer without using further simplification of the original Dukhin and Shilov double layer polarization theory, the model is limited to particles in suspension. No assumption is made with regards to actual contact between grains and how this might influence polarization. Geometric factors like limitations of ion movement at the interface today are not understood well enough. It shows that the general debate is still open concerning e.g. the composition of the Stern layer with regards to co- and counter-ions, and in particular the interaction between Stern and diffuse layer. However, the latter is set to zero in all current descriptions of the IP-effect. For spheres in suspension sufficient approximation is possible and leads to good results in colloid science. However, in complex pore space where e.g. the diffuse layer is interconnected over the whole pore space as expressed by Revil and Florsch, while mobility on the Stern layer is more restricted to smaller structures. Further, the exact mobility of ions in the EDL is still subject to discussion (Revil and Florsch, 2010). In conclusion, neither Revil and co-workers nor the various extensions of the standard model can explain effects from compaction, or contact between neighboring grains in granular porous media in general. Further, the underlying assumptions of grain size based polarization processes cannot be used to explain IP-results in connected pore space as e.g. modeled in Volkmann and Klitsch (2010), or polarization associated to surface roughness (Leroy et al., 2008). Likewise, theories relating polarization length scale to pore throat characteristics, on a conceptual visualization-type basis, remain strongly associated with the membrane effect which lacks applicability in larger pore-throat type media such as unconsolidated alluvial sediments and is not at all applicable in the case of particles in suspension. 26
Objectives and Outline of the Thesis
Based on the early results of the IP-method in mineral exploration and with mineralogical clay, research soon became focused on better understanding the origin of the observed polarization effect. From a hydrological point of view, it seemed important to understand the structural mechanisms at the basis of the polarization in the subsurface, while for interpretation at field scale the geochemical view point can neither be neglected. During the last years technical progress in the measurement of the IP-effect, allowed us now to study weakly polarisable alluvial sediments, which form important aquifers in many parts of the world. Trying to remove electrochemical characteristics from the equation within a controlled laboratory set-up, we aim in analysing the IP-effects structural origin in saturated proxies of alluvial sediments in order to allow for estimates of permeability from relaxation time. In the following we will describe the basic idea and first results from our laboratory studies in chapter 2. Here, we discuss the impact of grain size distribution sorting as well as the influence of different degrees of compaction of the samples on the IP-results. Of the two state of the art concepts explaining polarization at low frequencies in granular media on a physical basis, none seems adequate to predict the recorded frequency-phase spectra. Assumptions necessary to employ membrane polarization are not fulfilled in a model space with grains the size of sand, and the perceived IP‟s sensitivity to compaction of samples contradicts a simple grain size based polarization mechanism. In chapter 3, we model electrochemical polarization for samples of sand with different grain size and degrees of compaction. We discuss the assumed direct relationship between grain size distribution and the distribution of relaxation times from SIP measurements together with the underlying simplification of electrochemical processes of the applied model. One result of the studies described in chapter 3 is that the relaxation time corrected by the diffusion coefficient, is proportional to grain size however, with other influencing factors which are not represented in the physics of the model. The use of the physics from 27
polarization process at a singles spheres surface cannot be transposed unaltered for polarization of a pack of grains. In chapter 4, we summarize the studies results and discussions. Concluding remarks are followed by an outlook on future work and application of the results in hydrogeophysics.
Impact of Changes in Grain Size and Pore Space on the Hydraulic Conductivity and Spectral Induced Polarization Response of Sand Kristof Koch, Andreas Kemna, James Irving, and Klaus Holliger
Published in Hydrology and Earth System Sciences, 15, 1785-1794, 2011
Abstract Understanding the influence of pore space characteristics on the hydraulic conductivity and spectral induced polarization (SIP) response is critical for establishing relationships between the electrical and hydrological properties of surficial unconsolidated sedimentary deposits, which host the bulk of the world‟s readily accessible groundwater resources. Here, we present the results of laboratory SIP measurements on industrial-grade, saturated quartz samples with granulometric characteristics ranging from fine sand to fine gravel. We altered the pore space characteristics by changing (i) the grain size spectra, (ii) the degree of compaction, and (iii) the level of sorting. We then examined how these changes affect the SIP response, the hydraulic conductivity, and the specific surface area of the considered samples. In general, the results indicate a clear connection between the SIP response and the granulometric as well as pore space characteristics. In particular, we observe a systematic correlation between the hydraulic conductivity and the relaxation time of the Cole-Cole model describing the observed SIP effect for the entire range of considered grain sizes. The results do, however, also indicate that the detailed nature of these relations depends strongly on variations in the pore space characteristics, such as, for example, the degree of compaction. The results of this study underline the complexity of the origin of the SIP signal as well as the difficulty to relate it to a single structural factor of a studied sample, and hence raise some fundamental questions with regard to the practical use of SIP measurements as site- and/or sampleindependent predictors of the hydraulic conductivity.
Knowledge of the distribution of the hydraulic conductivity within an aquifer is a key prerequisite for reliable predictions of groundwater flow and contaminant transport. This information is in turn critical for the effective protection, remediation, and sustainable management of the increasingly scarce and fragile groundwater resources in densely populated and/or highly industrialized regions throughout the world. To this end, geophysical constraints with regard to aquifer structure and the distribution of hydraulic parameters are considered to be particularly valuable. The primary reasons for this are that geophysical methods are less expensive than other direct investigation methods and non-invasive in nature, and that they have the potential to bridge an inherent gap which exists in terms of spatial resolution and coverage between traditional hydrogeological methods such as core analyses and tracer or pumping tests (e.g., Rubin and Hubbard, 2005; Koch et al., 2009). Although standard geophysical techniques cannot in general provide any direct information regarding the hydraulic conductivity in the subsurface, there are a number of more specialized approaches that exhibit a more-or-less direct sensitivity to this important parameter (e.g., Slater, 2007; Holliger, 2008). In particular, induced polarization (IP) measurements and spectral induced polarization (SIP) represent promising methods. A clear link between hydrological properties and IP/SIP parameters has been empirically documented by various studies (e.g., Börner et al., 1996; Slater and Lesmes, 2002; Kemna et al., 2004; Binley et al., 2005; Hördt et al., 2007; Slater, 2007). However, recent studies of Binley et al. (2010), Kruschwitz et al. (2010), and Titov et al. (2010) also point out that the detailed nature and origin of such linkages remain enigmatic. In particular, it is not clear how basic changes in the pore space and/or grain size characteristics affect the SIP response and its relation to the hydraulic conductivity. In this paper, we address the question of whether characteristics related to grain or to pore size affect most the SIP relaxation time. We do so through SIP measurements on 31
saturated industrial-grade granular quartz samples with effective grain sizes ranging from fine sand to fine gravel. The pore space and grain size characteristics of the original samples are modified through compaction and sieving, while the chemistry of the saturating pore fluid is kept constant in order to examine the impact of structural changes on the SIP response. We express the impact of these variations with regard to the hydraulic conductivity and subsequently discuss its relation with the recorded SIP response, in particular the determined relaxation time. In the following, we first describe the basic principles of electrochemical polarization and relaxation, followed by our experimental setup, the granulometric properties of the samples used in this study, and the basic data analysis methodology.
Electrochemical polarization and electrical relaxation
The IP-type polarization of non-metallic minerals is generally referred to as interface or membrane polarization (Marshall and Madden, 1959; Vinegar and Waxman, 1984) and takes place in the lower frequency range up to the Maxwell-Wagner effect (Maxwell, 1892; Wagner, 1914; Chen and Or, 2006). In the absence of metallic conductors, such as ore minerals or graphite, SIP phenomena are commonly associated with polarization effects related to a polarized electrical double layer (EDL). The EDL schematically describes the organization of ionic charges at the interface between solid and fluid and was first introduced by Helmholtz in the middle of the 19th century. The inner layer is given by the typically negatively charged mineral surface attracting positively charged ions contained in the pore fluid to form the supposedly firmly attached Stern layer (Stern, 1924). Beyond the Stern layer, positively charged ions continue to be attracted by the negatively charged mineral surface, but at the same time are repelled by each other and the Stern layer. The resulting dynamic equilibrium is referred to as the diffuse layer and represents the transition zone between the Stern layer and the neutral part of the pore water. The individual elements of the diffuse layer 32
are hence in equilibrium with the neutral part of the pore water as is the basis for the concentration profiles of anions and cations from the Boltzmann distributions derived via the equality of the electrochemical potentials between any distance in the diffuse layer and infinity (e.g., Revil & Glover, 1997). The EDL provides the conceptual background for the electrochemical processes considered to be responsible for the observed SIP response. Today‟s conceptual understanding of the origin of the SIP response is strongly based on the work of Schwarz (1962) and his interpretation of the polarization effect as a result of the redistribution of counter-ions surrounding spherical particles in suspension. He describes the system as counterions on the surface of a highly-charged colloidal particle which are strongly bound by electrostatic attraction. In order to escape from the surface into the free solution, they have to overcome a high potential barrier. Along the surface, however, they can be moved much more easily. Thus they will be moved tangentially by an external field, polarizing the ion atmosphere and inducing an electric dipole moment of the particle. Schwarz‟ model, therefore, comprises the entire double layer, while the source process is allocated within the tangential movement of a single layer of counterions, which in turn effects the ion atmosphere around the particle. The only geometric factor involved in this model is the size of the sphere. Translating this geometrically simple analytical model to texturally complex porous media is not evident, as it simply is too strong a simplification. The commonly applied models are based on polarization of the diffuse layer in the pore system (e.g., Dukhin and Shilov, 1974), where Titov et al. (2004), amongst others, attempted to provide a visualization of the two basic concepts with regard to the origin of the SIP effect in porous media by linking the relaxation length scale to either granular or capillary models. A number of studies have tried to gain further insight into these matters by attributing the polarization to the EDL surrounding the individual grains (e.g., Lesmes & Morgan, 2001) and to excesses and deficiencies in ion concentrations along pore throats (e.g., Titov et al., 2002). It is only based on the recent work of Leroy et al. (2008), Leroy & Revil (2009), Jougnot et al. 33
(2010), and in particular Revil & Florsch (2010) that a more and more refined picture of polarization processes taking place in the Stern layer has started to emerge. Indeed, this work provides strong evidence to suggest that the Stern layer is potentially the region where the most important processes associated with the observed SIP response take place. In virtually all of these studies, the relaxation time or relaxation frequency has been theoretically related to a certain length scale representing either the grain radius (Schwarz, 1962) or a pore length scale (Kormiltsev, 1963). With regards to the observed electrical relaxation, a number of workers found consistently good correlations between hydraulic conductivity and the Cole-Cole time constant for a range of geological materials (e.g., Binley et al., 2005; Kemna et al., 2005; Tong et al., 2006; Zisser et al., 2010). Pelton et al. (1978) were arguably first to illustrate the adequacy of Cole-Cole-type models (Cole and Cole, 1941) for phenomenological description of the monitored SIP responses (e.g., Vanhala, 1997; Dias, 2000). Recently, Revil and Florsch (2010) supplied a corresponding theoretical justification, which is based on the polarization of the Stern layer and supported by the results of several recent studies (Leroy et al., 2007; Jougnot et al., 2010; Schmutz et al., 2010).
Experimental procedure and data analysis
Figure 2.1. Grain size distribution curves for the different samples considered in this study. Left: Six sieved fractions originating from sands F36 and WQ1. Right: Grain size distributions of unsieved industrial granular quartz samples. See also Table 2.1.
Table 2.1. Table of the original, unsieved samples and their effective and medium grain sizes d10 and d50, respectively, and their level of sorting as defined by d60/d10. F36
We have performed laboratory-based measurements on water-saturated industrial-grade granular quartz samples over a very broad range of average grain diameters from fine sand to fine gravel (Table 2.1, Figure 2.1). For these measurements, the pore space characteristics of the samples were modified by varying (i) the grain size, (ii) the degree of compaction, and (iii) the degree of sorting. We then examined how these changes affected the hydraulic conductivity, specific surface area, and the SIP response of the considered samples. Compaction was achieved through handheld multidirectional vibration. The shaking was continued until the sample volume stabilized at 90% of its original volume. Sieving the sands 35
F36 and WQ1 provided extremely well sorted samples of grain sizes: 0.09 - 0.125 mm, 0.125 - 0.18 mm, 0.18 - 0.25 mm, 0.25 - 0.5 mm, 0.5 - 0.71 mm, and 0.71 - 1.0 mm. Measurements of the hydraulic conductivity were made using the constant head method. The porosity was determined by weighing the saturated samples and thus determining its density given as total=(1-) SiO2 + H2O, where total is the inferred density of weighted sample and SiO2 and H2O are the a priori known densities of quartz and water, respectively. Specific surface area analysis was undertaken through the use of laser diffractometry. Specific surface measurements based on laser diffraction methods aim at estimating the grain diameters through the assumption of a specific geometrical form factor of the grain, which in our case is spherical. The signal of the diffracted light from a measured grain is thus fitted to that of an equivalent sphere. Hence, specific surface area measurements of this type provide us with the so-called geometric surface area of a grain, which is equal to the surface of the equivalent sphere. The corresponding specific surface area, given by the surface of this equivalent sphere divided by its volume, does therefore neither account for the small-scale grain surface roughness nor for the porosity. For a granulometrically heterogeneous sample, this parameter thus represents a combined measure of sorting and grain size of the overall distribution. The SIP measurements were conducted over a frequency range from 1 mHz to 45 kHz and electrical conductivities of ~60 µS/cm and ~300 μS/cm were considered for the saturating pore fluids. In this context, it is important to note that in general the signal-to-noise ratio of the observed phase spectra improved significantly with decreasing electrical conductivity of the pore fluid. This is due to the fact that the SIP phase spectrum essentially corresponds to the ratio of the complex and real parts of the electrical conductivity and thus reflects the increasing relative importance of the complex surface conductivity processes taking place in the EDL at low conductivities of the saturating pore fluid.
Figure 2.2. Schematic illustration of the high-sensitivity impedance spectrometer used for the SIP measurements presented in this study. The sample holder corresponds to a plexiglas cylinder with an inner diameter of 6 cm and a length of 30 cm. The current electrodes, C1 and C2, are made of circular porous bronze plates with a diameter of 6 cm and are located at the top and bottom of the sample holder. The potential electrodes, P1 and P2, correspond to rings made of silver wire located at a distance of 10 cm from the top and bottom of the measurement cylinder. These silver wires are embedded in grooves to keep them outside the actual sample and the electric field associated with the current electrodes and thus to minimize electrical polarization effects. Adapted from Zimmermann et al. (2008).
The SIP measurements were carried out using the highly sensitive impedance spectrometer for weakly polarizeable media developed by Zimmermann et al. (2008). The device is based on the four-point measurement method, and allows phase measurements in the required frequency range with an accuracy better than 0.1 mrad. Our experimental setup was designed to minimize polarization of the electrode materials. The cylinder holding the sample has a length of 30 cm and a diameter of 6 cm. The current electrodes, consisting of porous bronze plates with an effective pore diameter of 15 μm, form the upper and lower boundaries of the sample volume. The potential electrodes correspond to rings made of silver wire placed into grooves along the inner wall of the 30-cm-long measurement cylinder (Figure 2.2). These grooves are located at a distance of 10 cm from either end of cylinder. This results in an equidistant spacing of 10 cm between individual electrodes and a geometric factor given as k=r2/l with r and l denoting the radius of the measurement cylinder and the distance between 37
the potential electrodes, respectively. Fixing the potential electrodes into grooves aims at avoiding polarization effects by keeping the silver wire outside of the electric field associated with the current electrodes. The corresponding electrical connection to the electrodes is naturally provided through the conducting pore fluid, which fully saturates the sample filling the cylinder (Zimmermann et al., 2008). The sample‟s response to the applied current is recorded and provides information about the real and imaginary parts of the electrical resistivity. To analyze the measured complex resistivity data, we first fit them using the so-called Cole-Cole model (Cole and Cole, 1941) given by
0 1 m 1
where ω denotes angular frequency, 0
1 , c 1 i
the low-frequency asymptote, or direct-current
value, of the electrical resistivity ρ(ω), m the chargeability, c the Cole-Cole exponent, the time constant or relaxation time, and i 1 . The chargeability m describes the magnitude of the polarization effect, the Cole-Cole exponent c determines the width of the peak of the phase curve described by equation 1, and the time constant is related to the location of this peak in the frequency band (e.g., Lesmes and Friedman, 2005; Cosenza et al., 2009, p. 573). The exponent c typically takes values in the range from ~0.2 to ~0.7. Here, smaller values of c are associated with broader resonance peaks. In a number of previous laboratory-based SIP studies, relaxation time has been found to exhibit a more-or-less clear correlation with the hydraulic conductivity (e.g., Pape and Vogelsang, 1996; Kemna et al., 2005; Binley et al., 2005). To deal with the strong non-linearity and non-uniqueness of the inverse problem of estimating Cole-Cole model parameters that fit the measured frequency-domain SIP data, we 38
use a Markov chain Monte Carlo (McMC) inversion approach in this study. With standard Monte Carlo parameter estimation methods, random sets of parameter values are generated and then either accepted or rejected based on how well they allow us to fit the measured data. McMC importance sampling intensifies the formerly completely random sampling in more “probable” areas of the model space, which allows for increased efficiency of the procedure. Sets of feasible model parameters are generated with a frequency according to their probability of occurrence. The inversion procedure is based on the work of Mosegaard and Tarantola (1995). For a detailed description of the application to SIP data, and a comparison to a deterministic approach of fitting the Cole-Cole model, please see Chen et al. (2008). We considered uniform prior distributions for all parameters except for the low-frequency asymptote of the resistivity ρ0, which we set to the resistivity value observed at the lowest measurement frequency of 1 mHz. Experimenting with the width of these prior distributions, we found that it had essentially no influence on the final result and only a relatively minor influence on the speed of convergence, which points to the inherent robustness of the inversion procedure. Figure 2.3 shows a representative log-log plot of the observed phase as a function of the frequency from one of our SIP measurements. The corresponding saturated quartz sand sample was measured three times up and down the considered frequency range and clearly displays the resonance or relaxation effect described through the Cole-Cole model given by equation 1. For the considered data set, the resonance peak is located at ~1 Hz. At ~50 Hz the graph shows some erratic noise, which is most likely related to the local power supply. For comparison, we also show the corresponding measurements for the saturating pore fluid only, which in this case was water with an electrical conductivity of ~60 S/cm.
Figure 2.3. Example of phase spectra obtained from SIP measurements on a saturated sample of sand F36 (Table 2.1) as well as on the corresponding saturating pore water alone. The latter had an electrical conductivity of ~60 S/m. Note the negative scale for the phase.
To get a feeling for the data errors associated with the SIP measurements, which must be taken into account in the McMC stochastic inversion procedure, we consider the error characteristics provided for the impedance-meter (Zimmerman et al. (2008), along with the results of repeated measurements that we have made on a number of different samples. Figure 2.3 shows measurements that were repeated six times on one of the SIP samples (sand F36). Clearly the results are consistent, and the random measurement errors can be seen to be very small. This is of course dependent upon the length of time that has lapsed between measurements, as the SIP readings will be affected by temperature, carbon dioxide levels, and any reactions occurring inside the cylinder. Systematic errors are to be expected for large lag times between measurements. Overall, we found that data quality is a significant issue for SIP measurements on weakly polarizable samples. In particular, we have noticed that our SIP measurements were adversely affected in the lower frequency range between ~0.001 and 0.01 Hz. Although similar observations have been made by other researchers (Jougnot, personal communication, 2009; Okay, personal communication, 2009), the causes of these noise 40
problems remain enigmatic. As a consequence, SIP measurements on the more coarse-grained samples, whose relaxation peaks are expected to be located at the lower end of the considered frequency range, must probably be regarded as representing the limits of current measurement capabilities. In this context, it is, however, important to note that due to the use of multiple repeated measurements over the entire frequency range and variable electrical conductivities of the saturating pore fluid, meaningful quantitative interpretations were, unless explicitly mentioned otherwise, possible for most of our samples.
Results 2.4.1. Hydraulic characterization
Figure 2.4. Plot of the saturated hydraulic conductivity K versus porosity for all of the samples considered in this study. Black squares represent compacted and unsieved samples, while full green circles represent uncompacted and unsieved samples. Sieved, well sorted sands are represented as nonsolid black squares for compacted, and non-solid green circles for uncompacted samples.
Figure 2.4 shows a semi-log plot of the saturated hydraulic conductivity versus porosity for the samples considered in this study. The samples are distinguished in terms of being 41
compacted, non-compacted, sieved, and non-sieved. As expected, we observe that the compaction of a sample generally results in a reduced hydraulic conductivity and porosity due to smaller pore diameters. Changing of the grain size distributions through the use of sieved fractions, on the other hand, shows a tendency towards greater porosity values for comparable values of the hydraulic conductivities. At similar porosity, well-sorted samples tend to drain less efficiently than their more heterogeneous counterparts, and in other words need more pore space to conduct the same amount of water. It is assumed that pore size distribution is more homogeneous in well-sorted samples, and discrepancies between smallest and biggest pore sizes are smaller than with materials of broad grain size distributions. The above finding is consistent with the fact that a squared increase of the pore surface area, which governs frictional properties, is opposed by a cubic gain in pore volume, which governs the transport volume. Meaning that one pore with the same volume as two smaller pores combined imposes less friction on the water flux and hence transports more water in the same time. Although this statement seems to be largely self-evident and a principal factor in the formation of preferential, channelized flow patterns, this effect is rarely considered in current hydrological research (e.g., Hillel, 2004). Figure 2.5 shows a semi-log plot of the saturated hydraulic conductivity versus the specific surface area. The specific surface area was evaluated with a Beckman Coulter LS™ 13320 Laser Diffraction Particle Size Analyzer. The method is applicable for non-compacted samples with grain diameters smaller than 2 mm and hence the total amount of measureable samples is somewhat limited. Nevertheless, the corresponding results demonstrate that well sorted samples show a much stronger and clearer correlation between hydraulic conductivity and specific surface area compared to their more poorly sorted, more heterogeneous counterparts. Overall, we see that for comparable surface areas, well sorted samples are characterized by higher hydraulic conductivity than most of the more poorly sorted samples. Keeping in mind the above mentioned basic relationship of surface to volume, this 42
discrepancy between sieved and non-sieved sands should be entirely related to the heterogeneity of the samples‟ pore structure. As a consequence, the discrepancy between the maximum value of the specific surface area for a certain value of the hydraulic conductivity and the actually observed value for a given sample should provide us with a parameter related to the actual width of a samples pore size distribution. This parameter is related to the dynamic storage and hydraulic transport behavior of a material and might be of use in hydrological modeling. In the considered case, the maximum values are given by the results for the well sorted samples, but could also be inferred through theoretical considerations.
Figure 2.5. Plot of the saturated hydraulic conductivity K versus specific surface area (SSA) for noncompacted samples having grain sizes smaller than 2 mm. Green solid circles denote unsieved samples, while non-solid green circles represent sieved sand fractions. The power law fit refers only to the data from sieved sand fractions.
2.4.2. SIP measurements
Figure 2.6. Plot of the saturated hydraulic conductivity K versus Cole-Cole relaxation time for sieved and unsieved samples using a conductivity of ~300 µS/cm for the saturating pore water. In both cases the samples were compacted. The power law fit refers only to the data from unsieved sands, which is displayed as black full squares.
Figure 2.6 shows a log-log plot of the saturated hydraulic conductivity versus the estimated relaxation time obtained by inverting the measured SIP data based on the Cole-Cole model given by equation 1. Only compacted samples are considered and the least squares fitting was applied separately to the results for sieved and non-sieved samples. The latter exhibit a power law relation between the hydraulic conductivity, K, and the relaxation time, tau. The correlation between K and is much stronger for the non-sieved samples than for the sieved samples, as a possible consequence of the relatively poor quality of the SIP measurements for the latter. This finding, which has been corroborated through repeated measurements, is enigmatic when interpreting the SIP-response based on the simple sphere model from Schwarz (1962). When considering the grain diameter to be one of the most decisive parameters we would expect to see a trend towards sharper, more clear, relaxation peaks, 44
along with increasing granulometric homogeneity of the samples, which indirectly should result in a more clear K- correlation for better sorted sand.
Figure 2.7. Plot of the saturated hydraulic conductivity K versus Cole-Cole relaxation time for compacted and uncompacted, unsieved, samples measured at ~300 µS/cm electrical conductivities of the saturating pore water.
Figure 2.7 shows a log-log plot of the saturated hydraulic conductivity versus the estimated Cole-Cole relaxation time, where compacted or non-compacted samples have now been separated. With regard to time constant , the overall effect of compaction seems to be a shift towards smaller time constants, associated with smaller length scales of the underlying polarization process (Schwarz, 1962; Kormiltsev, 1963). The primary effect of compaction is the reduction of the porosity but likewise it results in increasing a sample‟s specific surface area per unit pore volume: for example, decreasing the pore space between uniform spheres by 10% through compaction results in an increase of the specific surface area per unit pore 45
volume of almost 17%. This demonstrates that together with the well known sensitivity of the SIP response to changes in grain size (e.g., Schwartz, 1962; Leroy et al., 2008; Revil and Florsch, 2010), the specific surface area per unit pore volume also seems to be a good indicator of the size of the polarization cells sensed by the inferred relaxation processes (e.g., Kormiltsev, 1963; Börner and Schön, 1991).
Figure 2.8. Plot of the saturated hydraulic conductivity K versus the Cole-Cole relaxation time for uncompacted samples measured at an electrical conductivity of the the saturating pore water of ~60 µS/cm. K either was calculated from the grain-size distribution measurements using the empirical, grain-size-based formulas of Beyer (1964) and Hazen (1892) or experimentally determined through constant-head-type measurements.
Finally, Figure 2.8 compares the observed K-values and the ones inferred based on common granulometric models (Hazen, 1892; Beyer, 1964) with the -values of the corresponding samples. Interestingly, we find a systematically stronger correlation for the measured K-values compared to the K-values inferred from empirical relations. In this context, it is interesting and important to note that these relationships are based on well sorted 46
sand samples comparable to the ones considered here. Hence, the fact that the time constant data fit the actual measurements of hydraulic conductivity more consistently than granulometrically-based estimates suggests that the phenomenology of the SIP response cannot be reduced to a single textural factor, such as, for example, the grain size. As a consequence, this is once again an indication that consideration of multiple structural parameters will be necessary when trying to identify parameters governing the source processes of induced polarization. Please note that the values of all data points shown in Figure 2.8, including those of the apparent outlier in the grain-size-based estimates with the smallest time constant, have been confirmed by repeated measurements. In this context, Binley et al. (2005) pointed out that the Cole-Cole model time constant appears to be better correlated with K than with other measures of the interfacial surface, such as, for example, the surface area per unit pore volume. In their work, values of the time constant are compared to median grain size d50 (r2=0.62; as compared to r2=0.64 in our study (graph not shown)), pore throat diameter (r2=0.61), surface area per unit volume (r2=0.75), and hydraulic conductivity (r2=0.78; compared to r2=0.77 for uncompacted samples and r2=0.94 for compacted samples in our study (Figs. 2.6, 2.7)). Although these findings by Binley et al. (2005) are for consolidated sandstone samples, the results are in qualitative agreement with our findings. However, results shown in Figures 2.4 and 2.5 indicate that changes in the sorting have a direct impact on these relationships. This is a further indication of the actual complexity of petrophysical parameter relationships and the processes which dominate hydraulic conductivity as well as induced polarization measures. Following Reppert and Morgan (2001), Scott (2006), and Revil and Florsch (2010), the impact of the pore width on the recorded SIP response can be considered of secondary importance. This is primarily due to the comparatively large sizes of the pore throats of both non-compacted and compacted samples considered in this study, which in turn points to the generally subordinate contribution of pore throat and membrane effects to the observed SIP 47
response for unconsolidated sandy sediments in the frequency range below the MaxwellWagner effect. At the same time, the observed impact on a samples‟ SIP-signature related to the transformation of pore space through compaction clearly indicates that individual structural aspects like grain size or surface roughness (e.g., Leroy et al., 2008) are not the only parameters determining SIP response. Compaction changes the specific surface area per unit pore volume, which in turn is expected to find its distinct expression in corresponding changes of the formation factor. As a consequence, the recently proposed theoretical model from Revil and Florsch (2010), in which the formation factor plays an essential role, seems to be indirectly supported by our findings. In detail, however, the link between the formation factor and the SIP response of granular sandy media is as of yet largely unexplored and, in our view, represents an important topic for future research in this domain.
The goal of this study was to improve our understanding of the polarization processes through experimental control of different pore space characteristics in laboratory measurements. Results show a clear relation of the SIP response with the hydraulic conductivity, demonstrate the variability of this relation in response to changes in pore size and grain size characteristics, and hence demonstrate potential of SIP-based methods for remote-sensingtype first-order hydraulic characterizations of the shallow subsurface. In agreement with previous findings, the results of our measurements indicate a power-law-type correlation between the inferred Cole-Cole relaxation time and hydraulic conductivity for the considered broad range of saturated sand samples. Changes in compaction and sorting of the samples resulted in a certain shift in the SIP response but did not fundamentally alter this overall picture. With regard to an improved understanding of the underlying physical properties, the strong interdependencies between grain size distribution and pore characteristics complicate the inference of the origin of the corresponding SIP effect. Yet, increasing the degree of 48
compaction while leaving the grain size distribution unchanged showed a systematic effect on the SIP response towards smaller relaxation times, which in turn are associated with lower degrees of heterogeneity in the probed material. The phenomenological approach of relating values of the relaxation time to a change in the size of heterogeneities in the sampled material showed for both changes in grain size and pore size of our samples, while the change of pore size through compaction also effects the ratio pore interface versus porosity. Due to this interlink of specific surface area with compaction and pore width, multiple explanations are possible and hence fundamental question regarding the interdependencies of the polarization spectra and pore width remains as of yet unresolved.
Acknowledgments This research has been funded by a grant from the Swiss National Science Foundation. We would like to thank the four reviewers André Revil, Lee Slater, and Maosong Tong, as well as an anonymous reviewer, for their constructive comments which helped to improve the manuscript. We greatly profited from collaboration with and logistical support from the Forschungszentrum Juelich, Germany. We would also like to thank Quarzwerke Frechen for providing the samples used for this study. Andreas Kemna acknowledges support by the SFB/TR32 "Patterns in soil-vegetation-atmosphere systems: Monitoring, modeling, and data assimilation" funded by the German Science Foundation (DFG).
Relating the permeability of quartz sands to their grain size and spectral induced polarization characteristics Kristof Koch, André Revil, and Klaus Holliger
Accepted in Geophysical Journal International, April 2012
Abstract Recently, Revil & Florsch (2010) proposed a novel mechanistic model based on the polarization of the Stern layer relating the permeability of granular media to their spectral induced polarization (SIP) characteristics based on the formation of polarized cells around individual grains. In order to explore the practical validity of this model, we compare it to pertinent laboratory measurements on samples of quartz sands with a wide range of granulometric characteristics. In particular, we measure the hydraulic and SIP characteristics of all samples both in their loose, non-compacted and compacted states, which might allow for the detection of polarization processes that are independent of the grain size. We first verify the underlying grain size/permeability relationship upon which the model of Revil & Florsch (2010) is based and then proceed to compare the observed and predicted permeability values for our samples by substituting the grain size characteristics by corresponding SIP parameters, notably the so-called Cole-Cole time constant. In doing so, we also asses the quantitative impact of an observed shift in the Cole-Cole time constant related to textural variations in the samples and observe that changes related to the compaction of the samples are not relevant for the corresponding permeability predictions. We find that the proposed model does indeed provide an adequate prediction of the overall trend of the observed permeability values, but underestimates their actual values by approximately one order-ofmagnitude. This discrepancy in turn points to the potential importance of phenomena, which are currently not accounted for in the model and which tend to reduce the characteristic size of the prevailing polarization cells compared to the considered model, such as, for example, membrane polarization, contacts of double-layers of neighbouring grains, and incorrect estimation of the size of the polarized cells due to the irregularity of natural sand grains.
One of the basic difficulties in groundwater resources management is the inherently elusive nature of the spatial distribution of the pertinent hydraulic properties in general and of the permeability in particular (e.g., Gelhar, 1993). An adequate hydrological characterization of the subsurface, notably with regard to its permeability structure, is, however, a prerequisite for effectively addressing problems regarding groundwater flow and contaminant transport (e.g., Domenico & Schwartz, 1998). Surface- and borehole-based geophysical measurements allow for imaging the geological structure of the shallow subsurface as well as the distribution of some pertinent rock physical properties (e.g., Rubin & Hubbard, 2005; Linde et al., 2006; Robinson et al., 2008; Hubbard & Linde, 2011). Geophysical approaches also have the potential of bridging the inherent gap in terms of spatial resolution and coverage that tends to exist for classical hydrological measurements (e.g., Koch et al., 2009). While a number of standard geophysical methods exhibit a direct sensitivity to the water content, such as groundpenetrating radar (e.g., Annan, 2005), or to ground water flow, such as self-potential (e.g., Malama et al., 2009a,b), only a few methods, such as nuclear magnetic resonance (e.g., Stingaciu et al., 2010), seem to exhibit a more or less direct sensitivity to permeability (e.g., Hubbard & Linde, 2011). Amongst the latter, spectral induced polarization (SIP) measurements seem to offer particularly significant potential with regard to a wide range of practical hydrological applications (e.g., Slater, 2007). SIP is a low-frequency geoelectrical method based on the observation of effects related to the temporary storage of electrical charges in the probed subsurface region in response to the injection of an alternating current. This reversible storage of electrical charges produces a phase lag between the current injected using two current electrodes and the voltage difference measured between two potential electrodes (e.g., Binley & Kemna, 2005). A number of researchers have tried to connect the key parameters describing the observed SIP phase spectra to various textural characteristics of porous media, which in turn tend to be 53
more or less strongly related to the permeability. Indeed, several studies have documented reasonably strong relationships between permeability and parameters derived from induced polarization (IP) measurements in porous media (e.g., Börner et al., 1996; Slater & Lesmes, 2002; Kemna et al., 2004; Binley et al., 2005; Hördt et al., 2007; Slater, 2007; Revil & Florsch, 2010; Weller et al. 2010). Slater (2007) provides a comprehensive review of the corresponding methodological foundations and of the pertinent literature. Arguably, the most important textural characteristics in the given context are (i) the pore size or the raw moments of pore size distribution (e.g., Kormiltsev, 1963; Titov, 2002, 2004; Revil et al., 2011) and (ii) the grain size or the raw moments of the grain size distributions (e.g., Schwarz, 1962; Lesmes & Morgan, 2001; Revil & Florsch, 2010). At low frequencies, IP- and SIP-type phenomena are related to the existence of polarization length scales associated with the accumulation or depletion of charge carriers, notably ions in the case of saturated porous media, under the effect of an imposed electrical field. If, and only if, these polarization length scales can be associated with the geometrical parameters controlling permeability, then IP- or SIP-type measurements over a broad range of frequencies, typically 1 mHz to 100 Hz, can be used to assess permeability. Schwarz (1962) developed the theoretical foundations for linking the diameter of suspended spheres with the relaxation time. This theoretical framework was adapted by Leroy et al. (2008) to account for polarization processes in the Stern layer formed by counterions, which maintain their hydration shells and are weakly sorbed to the mineral surface. Revil & Florsch (2010) then proposed a mechanistic theory to predict the permeability of granular media from some raw moments of the grain size distribution and the intrinsic formation factor corrected for surface conductivity. This model is based on the assumption that the diffuse layers around grains are inter-connected in granular porous media such as sand. Hence, only the polarization of the Stern layer at the surface of the grains and the related polarization length scales can be directly associated with the size of individual grains. Charge movements related to 54
polarization processes in the Stern layer are predominantly parallel to the grain surfaces, as slow diffusion processes hinder perpendicular movements, which implies that the dominant sizes of the corresponding polarization cells, and hence the associated SIP relaxation times, can be directly related to the prevailing grain size. Enhanced knowledge regarding such a relation between the size of polarization cells and size of grains could then pave the way to a better understanding of the potential and limitations of SIP measurements, especially, when connected to the grain-size-based permeability estimation proposed by Revil & Florsch (2010). The data presented in this study correspond to samples of quartz sands, which vary significantly in terms of their grain size and textural characteristics and thus allow for exploring the effects of different levels of sorting and compaction on the phase amplitude in SIP (Koch et al., 2011). A particular emphasis of this study is to assess the impact of changes in sorting on the SIP characteristics, where sieved versions of the same original samples allows for a more direct comparison. In the following, (i) we first briefly outline the essential aspects of the mechanistic model proposed by Revil & Florsch (2010), whose practical validity we seek to test in this study, (ii) present the experimental procedures and the resulting data, and (iii) finally compare the laboratory data with the corresponding theoretical predictions.
3.2. Estimation of Permeability of Granular Media 3.2.1. Grain-Size-Based Permeability Estimation The motivation to connect the permeability to the textural parameters of a porous material is not new, the Kozeny-Carman model (Kozeny, 1927; Carman, 1937, 1956; Freeze & Cherry, 1979) being a well known and still pertinent example. If the porous material is granular with all the grains being perfect spheres of uniform diameter d, the Kozeny-Carman equation can
be written as (e.g., Freeze & Cherry, 1979; Revil & Cathles, 1999; Chapuis & Aubertin, 2003; McCabe et al., 2005a,b)
d 2 3 , 180(1 ) 2
where k is permeability and porosity. The Kozeny-Carman model tends to overestimate the permeability, especially at low porosities, as only a fraction of the total pore space actually contributes to the medium‟s permeability (e.g., Revil & Cathles, 1999). To alleviate this problem, a number of authors proposed to work with parameters that are dynamically weighted by the norm of the local electrical field in the absence of surface conduction at the pore water/mineral interface (Johnson et al., 1986; Avellaneda & Torquato, 1991)
2 , 8F
where F is the formation factor and Λ is the characteristic pore size of the porous material. Archie‟s (1942) law then connects the formation factor to the porosity as
F m ,
where the so-called cementation exponent m can be regarded as a grain shape parameter for granular media (Sen et al., 1981). According to Revil & Cathles (1999), the characteristic pore size can be quantified as by d (2m( F 1))1 , which in turn allows for transforming Eq. 2 to 56
d2 2 32m2 F F 1 .
This relation was then used by Revil & Florsch (2010) as the basis for their mechanistic model relating the permeability to the key SIP parameters, notably the relaxation time. Please note that Eq. 4 again refers to the idealized case of perfectly spherical grains with a uniform diameter d.
3.2.2. Determination of Permeability from SIP Parameters The resistance and the phase associated with SIP measurements can be expressed as a complex impedance, which, once corrected by a geometrical factor accounting for the experimental setup, may be written in terms of the complex conductivity
* ( )
1 ' ( ) i " ( ) * ( ) expi ( ) , ( )
where ω denotes the angular frequency, the complex resistivity, i 1 the imaginary number, φ the phase difference between the injected current and the observed voltage, and ' and '' the real and imaginary parts of the complex conductivity , respectively. Revil & Florsch (2010) then use a Cole-Cole-type model for relating the distribution of the SIP relaxation times to the observed complex resistivity
* 0 1 M 1
, c 1 i CC
where M is the dimensionless chargeability characterizing the magnitude of the polarization effect, c is the dimensionless Cole-Cole exponent and determines the width of the peak on the corresponding phase curve with small values of c being associated with wider peaks and vice versa, CC is the Cole-Cole time constant and is related to the location of this peak in the frequency band, and 0 the low-frequency asymptote, or direct-current value, of the electrical resistivity (e.g., Cole & Cole, 1941; Pelton et al., 1978; Dias, 2000; Lesmes and Friedman, 2005; Cosenza et al., 2009). Following the original model of Scharz (1962), for the IP relaxation time, Revil & Florsch (2010) assume that the Cole-Cole time constant CC can be related to the square of the grain diameter d as
d2 , 8Di
where Di represents the diffusion coefficient of species i in the Stern layer, which is temperature-dependent and varies ~2% per degree Kelvin. For the quartz sand samples considered in this study, we shall assume that the idealized uniform grain diameter d in Eq. 7 can be approximated by median grain diameter d50. The pore fluid used is a sodium chloride electrolyte. Following Revil & Florsch (2010), we assume DNa+ = 1.3210-9 m2 s-1 at 25 °C (Tong et al., 2006) and an equivalent ion mobility Na+ = 5.1910-8 m2s-1V-1 at 25 °C (Revil et al., 1998) on the surface of the grains. Assuming that the uniform diameter d in Eq. 4 can be approximated by the median grain diameter d50, this then results in the following relation between the permeability and SIP relaxation time
CC Di 4m F ( F 1) 2 2
The thus resulting proportionality between the permeability k and the relaxation time CC was indeed already proposed by Pape & Vogelsang (1996) in relation with their work on borehole logs from the German continental deep drilling program (KTB).
3.3. Experimental Background The granulometric and hydraulic characteristics of the quartz sand samples considered in this study are given in Table 3.1. All measurements were carried out both in a loose, noncompacted state (Table 2a) as well as in a compacted state (Table 2b). The total porosity is estimated based on the observed bulk density (1 ) S f , where s = 2650 kg/m3 and
f = 1000 kg/m3 denote the mass densities of the quartz grains and of the pore water, respectively. For all samples considered in this study, we assume the total porosity to be equal with the effective porosity. Compaction of the samples was achieved by continued shaking and refilling of the sample holder. Vertical sorting effects, with bigger grains moving towards the surface and smaller grains accumulating at the bottom, were minimized, if not largely eliminated, by continuously changing the orientation of the vessel during the compaction process. The average difference in porosity between the non-compacted and compacted states is of the order of 7% (Tables 3.2a and 3.2b). The compaction of the samples leaves their grain size distributions and some of their surface characteristics, such as, for example, the surface roughness of the grains, unchanged, but increases their surface area relative to the pore volume. The reduction in porosity involves a corresponding increase of the electrical formation factor F (Eq. 3) of the compacted samples so that the influence of this parameter on 59
the SIP response can be tested independently. As illustrated in Figure 3.1, F is estimated by plotting the bulk conductivity of the sample as a function of the pore water conductivity. The cementation exponent m can then be inferred from the porosity and the formation factor F as m = -ln F/ln . The corresponding values for F and m are given in Tables 3.2a and 3.2b.
Figure 3.1. Bulk conductivity as a function of fluid conductivity for the compacted version of sample F36 illustrating the use of the inverse of the linear fit‟s slope given by = (1/F) f for estimating the formation factor F. In this example, the samples linear fit yields a value of the formation factor F of 4.1.
Table 3.1. Grain size characteristics from laser diffraction measurements. u = d60/d10 is the sorting of the grain size distribution. Samples SP1 to SP6 correspond to sieved fractions of sand F36 (SP1, SP2, SP3) and sand WQ1 (SP4, SP5, SP6). Sample d10 [mm]
F36 F32 WQ1 SP1 SP2 SP3 SP4 SP5 SP6
0.10 0.18 0.47 0.13 0.17 0.23 0.38 0.52 0.68
0.18 0.27 0.66 0.18 0.24 0.32 0.50 0.68 0.87
0.20 0.29 0.71 0.19 0.25 0.34 0.53 0.72 0.91
1.91 1.62 1.51 1.43 1.44 1.47 1.40 1.38 1.34
Mean [mm] Mode [mm]
0.18 0.27 0.66 0.18 0.23 0.32 0.50 0.68 0.86
0.19 0.27 0.68 0.19 0.25 0.32 0.52 0.68 0.91
In addition to the original quartz sand samples, which differ in terms of their granulometric characteristics, we also consider additional samples obtained by sieving of the samples F36 and WQ1 (Table 3.1). The sieving resulted in six well-sorted sub-fractions, denoted as samples SP1 to SP6, whose SIP spectra are expected to be associated with well defined phase peaks (Leroy et al., 2008; Revil & Florsch, 2010). Please note that taking sieved fractions of a particular sample allows for exploring the effects of sorting on the SIP characteristics, which is a particular emphasis of this study. Due to constraints in the experimental setup, we had to use different sample holders for the SIP and permeability measurements. This change of sample holders has the potential drawback that there can be slight differences in the underlying porosity values, although great care was taken to achieve the same final compaction state. For clarity, we denote the porosities prevailing for the permeability and SIP measurements as k and SIP, respectively (Tables 3.2a and 3.2b). Permeability was inferred from constant head measurements in a dedicated sample holder with a diameter 5.1 cm and a length of 5.0 cm.
Table 3.2. Electrical and hydraulic parameters for a) uncompacted and b) compacted samples. Permeability k was measured with the constant head method at porosity k. Porosity SIP denotes the porosity measured in the sample holder used for SIP measurements using the impedance meter developed by Zimmerman et al. (2008) (Figure 2). s.d. denotes the standard deviation of the various parameters as inferred from the posterior distribution resulting from the Bayesian MCMC inversion of the complex resistivity data. The electrical measurements were taken at a temperature of 21.6 °C with a standard deviation of 0.9 °C.
a) Sample F36 F32 WQ1 SP1 SP2 SP3 SP4 SP5 SP6
CC [s] s.d. CC [s]
1.76E-11 5.31E-11 1.29E-10 2.08E-11 3.30E-11 6.75E-11 1.71E-10 2.80E-10 3.94E-10
0.47 0.45 0.47 0.48 0.49 0.49 0.49 0.48 0.49
3.77 3.55 3.25 3.14 3.40 3.26 3.12 3.10 3.34
0.44 0.44 0.47 0.46 0.44 0.49 0.49 0.48 0.49
1.59 1.54 1.56 1.48 1.49 1.65 1.58 1.54 1.70
0.0119 0.0022 0.0232 0.0048 0.0066 0.0051 0.0063 0.0121 0.0113
0.0303 0.0005 0.0011 0.0010 0.0007 0.0007 0.0013 0.0022 0.0024
0.439 0.508 2.127 0.304 0.297 5.133 0.840 4.684 12.422
1.11E-11 2.40E-11 7.50E-11 1.17E-11 1.98E-11 3.81E-11 1.05E-10 1.96E-10 2.56E-10
0.39 0.39 0.41 0.40 0.40 0.42 0.42 0.42 0.41
4.12 3.75 3.97 3.23 3.55 3.64 3.52 3.36 3.63
0.38 0.39 0.42 0.41 0.39 0.42 0.44 0.43 0.43
1.48 1.40 1.59 1.30 1.35 1.49 1.54 1.44 1.53
0.0232 0.0038 0.0082 0.0068 0.0050 0.0073 0.0087 0.0122 0.0080
0.0303 0.0005 0.0011 0.0010 0.0007 0.0007 0.0013 0.0022 0.0024
0.511 0.067 0.285 0.002 0.057 0.585 0.187 1.628 5.351
0.47 0.58 0.32 0.57 0.32 0.47 0.37 0.33 0.22
0.03 0.01 0.02 0.02 0.03 0.02 0.02 0.06 0.05
b) Sample F36 F32 WQ1 SP1 SP2 SP3 SP4 SP5 SP6
CC [s] s.d. CC [s] 0.231 0.142 1.856 0.142 0.399 5.485 0.802 3.654 3.42
0.511 0.067 0.285 0.002 0.057 0.585 0.187 1.628 5.351
0.32 0.35 0.44 0.50 0.40 0.45 0.28 0.33 0.28
0.16 0.02 0.03 0.02 0.02 0.02 0.03 0.05 0.05
The grain size distributions of our samples were determined using a Beckman Coulter LSTM 13320 Laser Diffraction Particle Size Analyzer, which uses an indirect method of estimating grain size via light diffraction. Based on the instrument‟s specifications, which conform to the ISO 13320 standards, the corresponding error margins are estimated to be 3% for d50 and 5% for d10/d90, respectively, with dx referring to the grain diameter for which x% of the grains have a diameter smaller than this value (ISO 13320, 1999).
Figure 3.2. Sketch of the high-sensitivity impedance meter used for the SIP measurements presented in this study (adapted from Zimmerman et al. (2008)). The measurements are done with a Wennertype array: C1 and C2 correspond to the current electrodes while P1 and P2 correspond to the potential electrodes.
The SIP measurements were performed using the high-accuracy impedance spectrometer developed by Zimmermann et al. (2008) (Figure 3.2). This instrument provides an accuracy with regard to the phase of 0.1 mrad over the frequency range of interest, which extends from approximately 1 mHz to 100 Hz. Tests under more or less identical conditions did indeed demonstrate that it was possible to closely replicate the experimental results in this frequency range at different laboratories with local specimens of this impedance spectrometer (Figure 3.3). The SIP response of a probed sample provides information on the bulk electrical conductivity and the phase delay, and thus on the real and imaginary parts of the complex electrical conductivity, over the considered frequency range (Eq. 5). A representative example of an observed SIP spectrum is shown in Figure 3.4. Both the phase and the imaginary part of the electrical conductivity exhibit a peak at lower frequencies and a steady increase of their values for frequencies in excess of 100 Hz.
Figure 3.3. Results for repeated measurements of the phase as a function of frequency made with local specimens of the same model of the impedance spectrometer developed by Zimmerman et al. (2008). Black squares show the results from sample #F36 measured at Forschungszentrum Jülich (FZJ) with pore water conductivity at 0.0045 S/m. Grey circles correspond to measurements made with pore water conductivity at 0.006 S/m at the Institute of Geophysics of the University of Lausanne (UniL-IG).
Figure 3.4. Example of the real and imaginary parts of the complex conductivity and corresponding phase spectra obtained from the SIP measurement of sample SP5 saturated with an NaCl electrolyte with an electrical conductivity of 0.0035 S/m. The low-frequency response of the phase and the imagninary conductivity corresponds to the polarization of the electrical double layer while the higherfrequency responses, that is, above ~100 Hz, are related to the effects of Maxwell-Wagner polarization (e.g., Lesmes & Morgan, 2001) and coupling effects associated with instrumentation.
The low-frequency peak is indicative of the SIP relaxation phenomena we seek to model with Eq. 6. Conversely, the gradual increase of the phase and the imaginary part of the electrical conductivity at higher frequencies is related to Maxwell-Wagner polarization as well as instrument-related coupling effects (e.g., Lesmes & Morgan, 2001). Repeated experiments showed that measurements in the frequency range between 1 mHz and 10 mHz were not of the same consistency as those for higher frequencies. Similar observations have been made by other researchers (Jougnot, personal communication, 2009; Okay, personal communication, 2009). While the causes for these problems remain enigmatic, they are likely to be related to the inherently very long measurement times associated with the low end of the SIP frequency spectrum. We recorded minor interferences with the local electricity grid in the vicinity of 50 Hz (Figure 3.5).
Figure 3.5. Zoom-in of an SIP measurement showing the typical interference effects with local electricity grid around 50 Hz.
In order to compare our measurements with the corresponding predictions of the model by Revil & Florsch (2010), which focuses on the low-frequency relaxation mechanism, we 65
consider SIP spectra in the range between 10 m Hz and 10 Hz and determine the Cole-Cole time constant of the so-called low-frequency peak. The only exception is sample WQ1, for which it was necessary to use frequencies down to 1 mHz, because the spectrum at higher frequencies was entirely flat. We fit the observed SIP data with the Cole-Cole model given by Eq. 6 using a Bayesian Markov-chain-Monte-Carlo (MCMC) inversion approach (e.g., Mosegaard & Tarantola, 1995; Chen et al., 2008). This approach is particularly suitable for low-dimensional, strongly non-linear problems of the type considered here and allows for a statistical assessment of the parameter estimation process based on the corresponding posterior distributions. In the absence of a priori information on the general characteristics, of the Cole-Cole parameter distributions, the MCMC inversions were executed using uniform prior distributions for c and 0 and log-uniform prior distributions for CC and M. The logarithmic scale of the uniform prior distributions for CC and M was chosen in order to account for the fact that likely values of these parameters may extend over several orders-ofmagnitude. Table 3.3 gives the lower and upper ranges of the corresponding prior distributions. With the exception of the prior distribution for 0, which was well constrained by the corresponding measurements of the bulk resistivity at the low end of the considered frequency spectrum, the considered prior distributions are quite broad and thus allow for assessing the information content of the SIP data with regard the Cole-Cole parameters.
Table 3.4. Lower and upper limits of MCMC uniform prior distributions.
lower limit c 0 [m] M CC [s]
0 400 0.0001 0.00001
upper limit 0.7 1100 10 100
We estimate the data uncertainties for the MCMC inversion procedure based on the output errors from the impedance meter, which in turn are based on repeated, three-fold measurements at each frequency. The results for the mean values and the standard deviations of CC, M, and, c are given in Tables 3.2a and 3.2b. Another critical aspect of the MCMC procedure is to determine the length of the so-called burn-in period, the number of iterations required for the Markov chain to converge and subsequently generate samples from the Bayesian posterior distribution. Only after these first iterations are removed from the dataset, the data can be used for analysis of the posterior statistics of the Cole-Cole parameters. In our case, the end of the burn-in phase, which was determined by comparing the results of three independent Markov chains with random starting points, was uniformly reached after less than 5000 iterations. After burn-in, each chain was allowed to run for 500,000 iterations, which is deemed to be largely sufficient to adequately sample the posterior distribution for this low-dimensional, four-parameter inverse problem. Representative examples of observed and fitted SIP spectra are shown in Figure 3.6. Depending on the number of frequencies considered, the computations took between 1 and 5 days on a standard desktop computer with a 2.6 GHz dual core processor.
Figure 3.6. Examples of Cole-Cole models fitted to parts of the observed SIP data using a Bayesian MCMC inversion approach. Black dots denote the observed data, solid red lines the mean of the group of accepted models, and dashed black lines the corresponding standard deviations.
3.4. Results As a basis for analyzing the SIP-related aspects of the model of Revil & Florsch (2010), we first need to validate the underlying relation between permeability and grain size (Eq. 4). To this end, Figure 3.7 shows a comparison between observed permeability data (Tables 3.2a and 3.2b) and the corresponding predictions based on Eq. 4 (Revil & Cathles, 1999), again assuming that the idealized uniform grain diameter d can be approximated by the median grain diameter d50 of the considered samples. For completeness, we also show of the corresponding predictions based on the Kozeny-Carman equation (Eq. 1). The corresponding error estimates are based on the full differentials of the governing equations. The uncertainties of the input parameters were inferred for each sample individually and are based on the estimated errors of the various measurements involved: 3% error in the measurements of grain size with the LSTM 13320 Laser Diffraction Particle Size Analyzer, 0.0005 kg for the scale used for weighting the samples, and 0.0001 m accuracy of the caliper gauge for the measurement of diameter and height of the sample holder, 0.5 % for both fluid electric resistivity measured with the electrical conductivity meter and bulk electric resistivity measured using the SIP-device. These uncertainties in the input parameters then translate into error estimates of 1.4% for porosity, 1.1 % for the formation factor, and 2.2 % for the cementation exponent. Please note that the formation factor is recomputed using Eq. 3 at the porosities at which the permeability measurements are performed for the individual samples. As expected, the Kozeny-Carman model (Eq. 1) predicts too high values for the permeability, because it overestimates the size of the pore throats and the fraction of the pore space that controls the flow. Conversely, the model of Revil & Cathles (1999), which represents the foundation for the SIP model by Revil & Florsch (2010) based on the substitution of grain size information with SIP parameters, provides remarkably good predictions of the permeability. 69
Figure 3.7. Predicted versus measured permeabilities comparing the Kozeny-Carman equation (Eq. 1; black circles) and the Revil & Cathles (1999) model (Eq. 4; open squares). As expected, the KozenyCarman equation overpredicts the permeability, because of its inherent overestimation of the fraction of the connected porosity. The formation factor F was computed from the cementation exponent m and the porosity k at which the permeability is measured.
In agreement with previous studies (e.g., Schwarz, 1962; Binley et al., 2005; Leroy et al., 2008), we generally observe a systematic dependence of the frequency of the relaxation peaks on the grain size for our samples with smaller grains being associated with higher relaxation frequencies and vice versa (Figure 3.8). However, an interesting, and as of yet enigmatic, difference is observed in the amplitude of phase shift between samples with the original grain size distribution (Figure 3.8, black data points) and the sieved parts of the same samples (Figure 3.8, colored data points). This effect cannot be explained using currently available 70
models and is most likely to have an influence on permeability estimations via the imaginary part of conductivity. This effect was independently observed for sample F32 by Odilia Esser at the Juelich Research Center and by the lead-author for sample WQ1 . In both cases, the electrical conductivity of the pore water was kept rather constant at 3-3.5 μS/cm for sample F32 and its sieved parts and at ~50 μS/cm for sample WQ1 and its sieved parts.
Figure 3.8. Granulometry (top) and SIP responses (bottom) of a) sample F32 and two sieved fractions thereof at a pore water conductivity of 3 to 3.5 μS/cm and b) sample WQ1 and three sieved fractions thereof at pore water conductivity of ~50 μS/cm. In both cases, we observe a significantly smaller amplitude of the phase for the well sorted, sieved fractions (coloured data points) of the samples than for the original grain size distribution (black circles) of the samples.
Tighter packing results in a small but clearly discernible shift of the peaks of the phase (Figure 3.9a) and the imaginary part of conductivity (Figure 3.9b) peaks towards higher frequencies, which in turn are associated with shorter relaxation times.
Figure 3.9. Effect of compaction on a) the phase and b) the imaginary part of the electrical conductivity ” for samples WQ1 and F36.
As this source of variation in relaxation time is not related to grain size, it will to some degree alter the permeability estimates based on Eq. 8, which accounts for the impact of compaction on permeability via the formation factor (Eq. 4), but does not take into account the impact of compaction on SIP relaxation time (Eq. 7). While this observation and its discrepancy with the mechanistic model of Revil & Florsch (2010) is interesting per se, the corresponding shifts in relaxation time are too small to have any significant implication with regard to corresponding permeability prediction. A comparison between the measured permeability values and the corresponding predictions based on Eq. 8 is shown in Figure 3.10. As for the grain-size-based model considered above, the formation factor was recomputed at the porosity at which the permeability measurements are performed (Tables 3.2a and 3.2b). The corresponding error estimates are again based on the full differential of the governing equations. The largest part of uncertainty originates from the standard deviation of the estimates relaxation times (Tables 3.2a and 3.2b). The errors for temperature and the electrical resistivity of the fluid of 0.1 °K and of 0.5%, respectively, are based on the specifications of the electrical conductivity meter and its built-in thermometer meter and the error for bulk electric resistivity of 0.5% correspond to the precision of the SIP device. The porosity error of 1.4 % is based on the full differential equations of the governing equations for our porosity estimates with a 0.0005 kg accuracy for the scale used for weighing the samples and 0.0001 m accuracy of the caliper gauge for the measurement of diameter and height of the sample holder. The latter uncertainties also contribute to the errors shown in Figure 3.10, but their contribution remains relatively small in comparison with the standard deviation of the relaxation. The results shown in Figure 3.10, illustrate that Eq. 8 correctly reproduces the trend of the measured permeabilities, but systematically under-predicts their actual values by approximately one order-of-magnitude, which is indeed within the common uncertainty range 73
associated with predictions and measurements of permeability. However, there are also indications, such as the dependence of the relaxation times on compaction, that some of the rather strong assumptions inherent to the model of Revil & Florsch (2010), such as its focus on the polarization of the Stern layer, could be at the source of this observation.
Figure 3.10. Predicted (Eq. 8) versus measured permeabilities using the mean Cole-Cole time constant based on the spectral induced polarization data, the formation factor, and the cementation exponent. The results for compacted and loose, non-compacted samples are displayed as open circles and solid black squares, respectively.
3.5. Discussion A seemingly obvious candidate for explaining the systematic under-prediction of the observed permeability would be a corresponding error in the diffusion coefficient (Eq. 8). Indeed, previous workers have put forward the idea of using the diffusion coefficient as a fitting parameter for closely related problems (e.g., Schwarz, 1962; Kruschwitz et al., 2010). In our case, this approach would, however, lead to values of ion mobility that exceed those used in related works by one order of magnitude (e.g., Leroy et al., 2008). This discrepancy is quite unrealistic and thus indicates that fundamental inadequacies in the underlying mechanistic model, rather than inaccuracies in the estimation of the diffusion coefficient, are likely to be at the source of the mismatch between the observed and the modelled permeabilities. For the following, it is therefore important to remember that the model by Revil & Florsch (2010) is essentially based on the assumption that the polarization process is dominated by the Stern layer. In the given context, this dominance of the Stern layer implies that the diffuse layer is assumed to be interconnected over the entire pore space and hence polarization processes taking place in the diffuse layer are not considered in the model. Notably membrane polarization, which is widely regarded as an important polarization mechanism (e.g., Marshall & Madden, 1959; Vinegar & Waxman 1974; de Lima & Sharma, 1992; Lesmes & Morgan, 2001; Titov et al., 2002) and is not accounted for by this model. Not accounting for the polarization processes in the diffuse layer may indeed result in a systematic underestimation of the observed permeability by the corresponding SIP-based model (Eq. 8). The fact that the underlying grain-size-based model (Eq. 4) predicts the observed permeability of our samples quite adequately supports this view. To explain the systematic under-prediction of the observed permeability by this model, we therefore need to look for realistic, but unaccounted, polarization mechanisms, which cause the observed SIP relaxation times to be shorter than would be expected from the Stern
layer polarization of individual spherical grains. Quite importantly, these polarization mechanisms should also be compatible with our observation that compaction of the samples results in a small but systematic reduction of their Cole-Cole time constants (Tables 3.2a and 3.2b). As described above, an important polarization mechanism in porous media, which works in this direction and which is not accounted for in the considered model, is membrane polarization, which is commonly associated with the pore length (e.g., Volkmann & Klitzsch, 2010). The pore length in unconsolidated granular media is generally smaller than the average diameter of the surrounding grains and hence is likely to be associated with systematically shorter relaxation times than those assumed by the model of Revil & Florsch (2010). Moreover, compaction tends to further reduce the pore length, which in turn is compatible with our observation that the compacted samples exhibit systematically shorter relaxation times than their non-compacted, loose counterparts (Tables 3.2a and 3.2b). In the context of this study, it is also important to remember that the individual grains of our quartz sand samples differ quite significantly in terms of their shape and their surface roughness from the underlying assumption on ideal spheres (Figure 3.11). These shape characteristics might allow for the development of multiple polarized cells on the surfaces of the individual grains. In well controlled laboratory experiments involving spherical glass beads, Leroy et al. (2008) did indeed observe that edging of the beads resulted in the development of additional smaller polarized cells in response to the corresponding increase of the surface roughness. While this process can be explained based on the Stern layer polarization processes, it is not accounted for in the current model of Revil & Florsch (2010). Moreover, the size of the polarized cells may also be reduced with respect to the considered Stern-layer-based model as a result of contacts between the electrical double layers surrounding the individual grains.
Figure 3.11. Electron microscope images of a quartz sand grain from sample F32 at various levels of magnification.
This phenomenon and its effects on the governing polarization mechanisms are not well understood and have only been studied with regard to spheres in suspension (e.g., Duhkin & Shilov, 1969, 1974, 2001; Shilov et al., 2001; Grosse, 2011). It has, however, been shown that the distance between particles in suspension has an effect on polarization processes and that the relaxation times do indeed decrease with increasing particle concentration (e.g., Schwan et al., 1962; Borkovskaja & Shilov, 1992; Shilov & Borkovskaya, 1994; Delgado et al., 1998). If the correspondingly reduced size of the polarized cells is linked to the distance between contacts of multiple individual grains, we can assume it to be similar to the pore length in unconsolidated granular media and thus to be similar to the size of the polarization cells associated with membrane polarization. In this context, it is interesting to note that the recently proposed hydraulic model of Bernabé et al. (2010, 2011) does indeed use the pore
length l together with the hydraulic radius rH as the key parameters for permeability estimation. Finally, it also important to remember that the decrease in porosity as a result of compaction is associated with a corresponding increase of the tortuosity T F 1m (Clennell, 1997). Together with recent findings of Binley et al. (2010) for sandstone samples, which indicate a linear relationship between the square root of the tortuosity and the relaxation length scale, this may point to the possibility and/or the necessity of correcting Revil & Florsch‟s (2010) mechanistic model for the tortuosity. That said, the corresponding variations of the formation factor remain relatively small for our samples and hence the formation factor would need to be multiplied by four in order to adequately represent compaction related variations in relaxation time.
3.6. Conclusions We explored the practical validity of the mechanistic model based on the polarization of the Stern layer proposed by Revil & Florsch (2010) relating SIP relaxation time to the mean grain size and the permeability. To this end, we compared the theoretical predictions with corresponding laboratory measurements for quartz sands with relatively broad and varying granulometric characteristics. The grain-size-based model, upon which the SIP model considered in this study is based, predicts the permeability very adequately. Correspondingly, we attribute the observed discrepancies arising in the permeability prediction from the SIP relaxation time to inadequacies in the part of the model concerning polarization processes. The trend of the permeability predictions based on the observed Cole-Cole time constants inferred from SIP data agrees well with the corresponding measured data. The model does, however, systematically underestimate the observed permeability values by approximately one order-of-magnitude. While the magnitude of this systematic mismatch is within the 78
common uncertainty range associated with predictions and measurements of this inherently elusive hydraulic parameter, there are indications, such as the dependence of the relaxation times on compaction, that some of the rather strong assumptions inherent to this model, such as its limitation to the polarization of the Stern layer, could be at the source of this observation. From a practical point of view, the effects of compaction seem to be negligible for the prediction of permeability from SIP data. However, the very observation of this effect points to the existence and the potential importance of polarization processes that are not accounted for in the considered model, such as, for example, membrane polarization and/or the presence of multiple and hence smaller polarized cells in response to surface roughness and inter-grain contacts.
Acknowledgments. This research has been supported by a grant from the Swiss National Science Foundation. AR thanks the National Science Foundation for the support of the project „Spectral Induced Polarization to invert permeability‟ (NSF award NSF EAR-0711053), whose results fertilized this study. We thank Harry Vereecken and the Forschungszentrum Juelich for facilitating collaborations. Special thanks go to Sander Huisman, Egon Zimmermann, Odilia Esser, Franz-Hubert Haegel, Katrin Breede, Marie Scholer, and James Irving for their help with various aspects of this project and to Quarzwerke Frechen for providing the samples used in this study. We also gratefully acknowledge the thorough and constructive reviews by Jörg Renner (editor), Andrew Binley, Lee Slater, and an anonymous referee as well as fruitful discussions with Andreas Kemna and Damien Jougnot.
Chapter 4 Summary and Conclusions
The initial planning was to establish a solid understanding for the complete range of sandy grain sizes of the relationship between hydraulic conductivity and the SIP-parameter relaxation time. The good correlation between these two parameters was confirmed for the measured samples. However, it became soon evident that state of the art knowledge about electrochemical polarization processes, namely membrane polarization and grain size polarization, in the studied range of material does not sufficiently describe our findings. In particular with regards to the ongoing controversy about whether grain size or pore size significantly affects polarization, the previous chapters deliver the following understandings: Chapter 2, describes the empirical determination of the previously unknown impact of structural changes through changes in packing of the material on the process of polarization in the water saturated pore space. One consequence of the studies-results is that solely grain size characteristics do not fully describe the recorded data. It was shown that with constant grain size a change of the pore structure has a measurable impact on the polarization processes, and hence apparently affects the characteristics of the process area. Together with the theoretical understanding that membrane like polarisation processes are fairly unlikely to affect IP-processes in pores of sandy, unconsolidated porous media, the physically well based Revil and Florsch model was considered in chapter 3. It includes the most comprehensive combination of the state of the art induced polarization electrochemical understanding, and allowed for the refined analysis of the most basic findings from the conducted laboratory studies. The purely grain size related understanding of the polarization processes is directly based on the work of Schwarz (1962), Schwan et al. (1962), Vinegar and 81
Waxman (1984), Leroy et al. (2008), as well as work on permeability (Revil and Cathles, 1999) and electrochemistry (Revil, 1999). It became apparent, that though the physics seem well described for a simplified model-space, the complexity of natural material is not completely covered by the model. Other physical factors influencing the IP-effect need to be better identified to explain the discrepancy between predicted and true results. Application of the Revil and Florsch model revealed that the grain size based processes taken into account sufficiently represent the qualitative approximation of observed trends with the IP-effect in granular media. However, quantitative analysis revealed unresolved open questions in today‟s conceptual understanding of the IP-processes, such as the IP effect from compaction, and in addition the new observation of unexplained shifts in the amplitude of polarization of samples, possibly related to the samples sorting of grains.
Today two major tendencies exist to conceptually explain the effect of polarization in porous media. At the one hand, based on Schwarz‟s work in the early sixties and on colloidal science in general, there is the idea of a sphere polarized by an electric field. On the other hand there is the influence of pore characteristics, as for example pore-throat size or pore length. Both parameter classes i) achieve good results in connection to hydraulic conductivity estimation, ii) correlate relatively well with the parameters describing the IP-effect. However, iii) both failed until now to fully convince, or give a full explanation of the IP-effect signal recorded in various different laboratory studies. The latter, iii), might be based on the incomplete understanding of any other, non-structural, effects on IP, such as the known influence electrochemical parameters which was not included in the IP-models of Marshall and Madden or Titov for example. Further, ii), the strong inter-link, between pore- and grain characteristics makes discrimination in laboratory studies almost impossible. However, such interlink between parameters turns to be a benefit when it comes to the usability of empiric results, as 82
with regards to the availability of parameters in field-studies. Consequently, the quality of our models should not vary dramatically whether one identifies grain size or pore characteristics as principle process driver. After all, both characteristics provide description of one and the same interface. Unlike such geometric factors, electrochemical factors are connected to the properties of the solid-fluid interface, the EDL, but not to hydraulic conductivity. The form of the EDL is governed by, i) the pore surface characteristics, ii) the electrochemistry of the pore fluid, and iii) the dynamic distribution of electromagnetic fields. These factors effectively influence the shape-factor of the EDL. This global interface consists of a globally distributed, interconnected, diffuse layer, as well as smaller local structures as the multiple Stern layers around e.g. individual grains. In-between those main structures the capacity of charge displacement is dynamic. Where charge movement is not free but also not completely fixed, it is inert, and polarization can occur. The exact origin of a reduced mobility of charges remains unknown till today, and only simplified model spaces such as particles in suspension allow the physically correct interpretation of IP-processes. This leads us to some of the potential benefits and difficulties with the application of the IP-method:
Due to the relevance of subsurface electrochemistry, and geo-structure, the sound interpretation of geoelectrics yields tremendous potential to picture diverse aspects of the subsurface within the framework of single measurement campaigns. This becomes particularly relevant with biological prospection, where temporal changes in structure and fluid chemistry occur alongside.
At field sites where the fluid in the pore system is not static, the knowledge of local self potential should be considered for physically based IP-interpretation.
If we now consider the principal geometrical origin of the IP-effect to be identified in the wall length between e.g. pore-throats than we judge the superposition of signals
from different wall length as one of the strongest concerns for correct interpretation of IP-data from an unknown subsurface.
In order to better understand superposition of polarization effects in SIP computational modelling and laboratory studies need to focus on the IP-signals from mixtures of materials. For example a comparison of IP-signals from two materials mixed together and measured compared to measuring both materials in row in a single measurement, would help understand the nature of superposition of IP-signals as well as it would help improve understanding on the geo-structural origin of the phenomenon. Re-visiting results from existing studies on material mixtures as well as variations in saturation will provide first results on this matter. Likewise, IP-laboratory studies with materials like syporex, an artificial material used by Revil and Cathles for studies on permeability with continuous pore walls, as well as studying particles of mica which exceed the size of clay, will possibly allow widening the knowledge on the IP-effects structural origin. One of the problems when studying a process in the laboratory with the aim of later use in field-applications is the uncertainty to not cover the ensemble of influential parameters. This is mainly due to the constraints in laboratory conditions. The difference between processes in nature and the model-space established in laboratory or computational analysis of processes is in part referred to as model error and is highly relevant for later usability of results. For example, studying sands instead of glass beads in laboratory measurements obviously helped to not mistake process observation on a rather homogenous structure with those relevant for the polarization of a natural sand grain environment. At the same time however, the studies presented in this work did not take into account potential movement in the pore fluid. Future studies will be necessary to evaluate how fluid movements can impact IP results. 84
Apart from such characteristics which are relevant for IP field application, one of the remaining theoretically important questions might be whether the IP-effect is mainly governed by differences in ion mobility between EDL and free electrolyte, or by barriers which hinder ion-movement along the EDL and hence create local charge accumulations inside of the EDL. The difficulty herewith is to discern one effect from the other, as it is not possible to systematically exclude one from the other. Further, local barriers are perceived as a general reduction of ion mobility as long as the resolution of observation techniques is not detailed enough to identify their exact origin. Since the physics of charge-movement and the behaviour of the electric field are known, these questions can in theory be answered within a detailed computational model space. The computational problem lies within the dynamic changes in the electromagnetic field and the reciprocal effect with charge movement and local accumulations of charge in time-dynamic modelling. The scale at which we discuss IP-processes in the laboratory exceeds measurement resolution. At the scale applied in field ERT/IP surveys, data interpretation like using the relationship permeability to relaxation time as was part of our study will continue to depend on the individual case, namely local conditions and measurement set-up. With
time-laps geoelectrical measurements, it is important to note that changes to previous measurements are relatively well visible inside the respective resolution of the applied electrical measurements and demand less supplementary information about the subsurface for interpretation. Likewise, available borehole logs and other geological data is relatively constant with time in the subsurface environment, except for particular cases of e.g. subsurface erosion (e.g. Schmalkalden-Incident, Nov. 2010, Germany), which is also an area of research potentially benefiting from the application of geophysical subsurface prospection of the geo-structure. In most case-studies knowledge of basic subsurface structures is readily available, and many suitable applications are related to the extension of information from a point like source into a spatial context. 85
On a hydrological note, future work needs to raise awareness that any solid or fluid parameter is only indirectly related to the interface specific parameters permeability and IPeffect. Together with the understanding of IP as an interface parameter, a principal question for application will be whether the user is sufficiently aware of problems and benefits in the source characterisation of IP-data. While in the laboratory this is easy in comparison, the field user prior to interpretation and along with aspects of technical equipment, needs to be aware and account for estimates of two of the three subsurface conditions of i) fluid chemistry, ii) interface structure, and iii) sources of dynamic electromagnetic fields, e.g. through fluid movement. However, hydrogeophysical studies can benefit from the fact that chemical changes in pore water can often be related to changes in the geological structure. Only strong differences as from clay barriers might provoke a more extreme shift of pore water conductivity in that case than mostly dependent on diffusion processes. The presence of clay is well detected with the available IP measurement devices. Future research should focus on the improved understanding of the potential benefits comprised within the two-tier measurement of the in-phase conductivity which is fluid volume related, and the out-of-phase conductivity which is sensitive to interface properties. Making use of a single measurement device and electrode setup yields information on fluid movement, fluid chemistry, and aquifer structure. In this context, in particular joint-inversion data treatment techniques need to be mentioned. A hydro- as well as biogeophysical approach with combined 3D self potential, resistivity, and induced polarization measurement potentially makes worthwhile strong effort put into the development of 3D joint-inversion data treatment, and tools for simplified data acquisition. Together with non-destructive spatial resolution of subsurface permeability patterns based on the structural interpretation of the IPeffect, the potential for geochemical and biological applications cannot be evaluated high enough.
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Joint interpretation of hydrological and geophysical data: Electrical resistivity tomography results from a process hydrological research site in the Black Forest Mountains, Germany
Kristof Koch, Jochen Wenninger, Stefan Uhlenbrook, Mike Bonell
Published in Hydrological Processes, 23(10), 1501-1513, 2009
ABSTRACT The use of electrical resistivity tomography (ERT; non-intrusive geophysical technique) was assessed to identify the hydrogeological conditions at a surface water/groundwater test site in the southern Black Forest, Germany. A total of 111 ERT transects were measured, which adopted electrode spacings varying from 0.5 to 5 m as well as using either Wenner or dipoledipole electrode arrays. The resulting two-dimensional electrical resistivity distributions are related to the structure and water content of the subsurface. The images were interpreted with respect to previous classical hillslope hydrological investigations within the same research basin using both tracer methods and groundwater level observations. A raster-grid survey provided a quasi 3D resistivity pattern of the floodplain. Strong structural heterogeneity of the subsurface could be demonstrated, and (non)connectivities between surface and subsurface bodies were mapped. Through the spatial identification of likely flow pathways and source areas of runoff, the deep groundwater within the steeper valley slope seems to be much more connected to runoff generation processes within the valley floodplain than commonly credited in such environmental circumstances. Further, there appears to be no direct link between subsurface water-bodies adjacent to the stream-channel. Deep groundwater sources are also able to contribute towards stream flow from exfiltration at the edge of the floodplain as well as through the saturated areas overlying the floodplain itself. Such exfiltrated water then moves towards the stream as channelised surface flow. These findings support previous tracer investigations which showed that groundwater largely dominates the storm hydrograph of the stream, but the source areas of this component were unclear without geophysical measurements. The work highlighted the importance of using information from previous, complementary hydrochemical and hydrometric research campaigns to better interpret the ERT measurements. On the other hand, the ERT can provide a better spatial understanding of existing hydrochemical and hydrometric data.
INTRODUCTION An assessment of the spatial context of subsurface water contributions to the stream hydrograph is now more easily facilitated due to considerable progress in the application of geophysical techniques (Rubin and Hubbard, 2005). Today, geoelectrical methods are successfully applied operationally in geological engineering as well as in multiple domains of scientific research. Images of the electrical resistivity of the subsurface help to locate shallow archeological remnants (i.e. Neighbour et al., 2001; Barton and Fenwick, 2005) as well as they help with the monitoring of subsurface solute transport (e.g. Kemna et al., 2002). Furthermore, the estimation of hydrological parameters from geophysical data as for example reviewed in Slater (2007) and Hubbard and Rubin (2000) is an integral part of past and present hydrogeophysical research (e.g. Kelly and Fröhlich 1985; Mazac et al. 1985; Huntley 1986; Fröhlich et al. 1996; de Lima and Niwas (2000); Binley et al., 2005; Lesmes and Friedman 2005; Singha et al. 2007). As formulated by Robinson et al. (2008) in their review on a vision for hydrogeophysics, there is an interest for both hydrologists and geophysicist to make use of the synergistic effects from combined thinking and methods to advance process based watershed research. Unlike in the applications of geochemistry in hydrology though, the current approaches in hydrogeophysics rarely consider a “joint interpretation” of geophysical and hydrological methods. Rather, the common approach taken is to combine the separately discussed results at the end of the interpretation process. The latter of course is well suited to answer open hydrological questions when sufficient ground truth for the detailed interpretation of geophysical and hydrological data is available. However, any supplementary information on the subsurface is most needed when obtaining such ground truth is a problem be it due to a lack of resources or difficult environmental conditions such as those often encountered in the headwaters of drainage basins.
Overall, there seem to be at least three major fields for geophysical applications in catchment hydrology: i) Full coverage of a catchment via for example airborne geophysics as a direct (input) data source for GIS-based hydrological models (cf. Robinson et al., 2008). ii) Relatively precise spatial deduction of hydrological parameters in the subsurface from 2D or 3D geophysical measurements when combined with subsurface material which is hydrologically and geophysically well understood. iii) Relating data from such features as a well, spring, stream channel or saturated area within the context of the hydrogeology setting based on relative changes in the geophysical data. As shown in Figure A.1, from a process hydrological perspective such geophysical applications provide information on: I) process areas (monitoring spatial structure and properties; e.g. Uhlenbrook et al., 2008; Wenninger et al., 2008) II) process dynamics (visualising propagation of changes in fluid property, e.g. tracer propagation or saturation; Kemna et al., 2002; Singha and Gorelick, 2005; Deiana et al., 2008; Looms et al., 2008). In this study the above mentioned point iii) and point I) describe our particular interest with regards to the conditions in the research area and available instrumentation.
methodological aim of this study is hence to provide a subsurface spatial context to classic hydrological methods and their sparsely distributed 1D measurements; and further to Figure A.1. Hydrogeophysics in a process hydrological context.
apply cross interpretation of relative changes in electrical properties of the subsurface with hydrological facts in order to identify particular process areas in a highly heterogeneous environment. Thus we chose to cover the flood plain quasi 3D, make detailed samples of particular areas of interest like saturated areas or springs and cover the hillslope via a crosssection as it corresponds well to the typical Freeze (1972) concave conceptual model. The application of 2D electrical resistivity tomography (ERT) as the surrogate measure along hillslope transects can provide new insights into spatial patterns of the geological structure of subsurface (e.g. minerals, texture, depth of layers of soil and drift cover) and the occurrence of groundwater recharge (e.g. Berthold et al. 2004; Wenninger et al. 2008). Such geoelectrical patterns, however, provide only a descriptive perspective of subsurface structures and water occurrence and they require supporting „ground-truthing‟. Consequently, such investigations need to be linked with additional information of both surficial (e.g. glacial drift deposits) and deeper solid rock structures. However, studies of the vertical spatial variability of electrical resistivity of the subsurface are allowing us to „see‟ the heterogeneity that exists below the surface, and re-evaluate our traditional, conceptual understanding of storm runoff generation processes for different geologic and geomorphic situations (Freeze, 1972; Beven, 1986; Bonell 1998).
Figure A.2. Detailed map of the investigation area within the Bugga catchment (40 km2), including location of the ERT profiles.
The use of ERT was done on the basis of many years of process hydrological research in our experimental basin (Brugga basin; 40 km2; Southern Black Forest, Germany; Figure A.2) with advances in the work with environmental tracers and process based hydrological modelling (Uhlenbrook et al. 2002, 2004, 2005, 2008, Wenninger et al. 2004). One important finding arising from the previous research was the detection of large parts of pre-event water dominating storm flow and the resulting question arose of how this links with our conception of hillslope hydrological processes. The challenge in answering this question however, is dealing with the many unknowns of the subsurface, which can in part be addressed through the use of electrical resistivity tomography (ERT) within the Brugga experimental catchment. By connecting our interpretation of DC-resistivity with classical hydrological methods, we improved our hillslope concept and set the findings from previous tracer and hydrometric studies in an improved spatial context. Thus the objectives of this paper are to demonstrate the following: a. The classic conceptual models for runoff generation in concave valleys do not conform to the situation provided in this study. 100
b. A deep groundwater body, that normally is undetected on the steep valley sides by classic, hillslope hydrology experimentation (due to the difficult field logistics of inserting boreholes through a boulder-strewn field at the surface and subsurface hard rock), is (locally) much more coupled to the surficial hydrology of the lower concave slope (floodplain) than is usually reported in the literature. c. The process hydrology of the valley floor of the main river is highly complex and involves a diversity of runoff sources and runoff components.
THE BRUGGA EXPERIMENTAL BASIN AND STUDIED HILLSLOPE SITE A detailed description of the Brugga experimental basin has been outlined in several sources (e.g. Uhlenbrook et al., 2002). The basin is widely covered by mixed deciduous coniferous forests (75%); 22% is pasture land and urban land use is less than 3%. Glacial morphology is evident in landforms such as moraines, cirques and the U-shaped valleys. Annual precipitation is 1750 mm, and the mean annual stream discharge is about 1220 mm. The area is underlain by metamorphic and intrusive rocks. This bedrock material is mostly covered by brown soils developed in the glacial and periglacial drift covers. The bedrock, however, is exposed on steep slopes. So far, it was concluded that the most important source area for runoff generation is the drift cover (GLA, 1981; Uhlenbrook et al., 2002), as little is known about the hydraulic conductivity of the bedrock. The hydraulic conductivity is generally described as low in this region (between 10-10 and 10-5 m s-1; Stober, 1995) based on the determination of saturated hydraulic conductivity Ks, in the laboratory (cores) and in the field. The drift cover was mainly formed by solifluction processes in the Pleistocene under widely occurring periglacial conditions at that time. The upper St. Wilhelmer valley, a U-shaped glacial valley, was chosen for detailed experimental investigations (Figure A.2). The study site at about 800 metres above mean sea level, and is characterized by a relatively flat valley floor. The uphill part of the investigation 101
area (topographic catchment area: 0.21 km2) is dominated by a steep slope (up to ~45°) with periglacial drift and bedrock outcrops with a boulder field below. In previous investigations Wenninger et al. (2004) provided baseline information on the isotopes and hydrochemistry (deuterium, dissolved silica, and major anions and cations), changes in shallow groundwater levels within ten wells (depth 1.4 to 1.7 m; located within the floodplain) and the discharge of the mentioned tributary creek. The latter is initiated through a little wetland near the main stream.
Figure A.3. Conceptual model of the flow dynamics at the investigation site “Hintere Matte” (after Wenninger et al. 2004).
It was shown that the well water levels are highly responsive to storm events (Wenninger et al., 2004; Figure A.3). Large proportions of groundwater are mobilized within the storm hydrographs of the little creek (i.e. about 80% during a double peak event, estimated using tracers). However, a lack of knowledge concerning the heterogeneous nature of the drift and solid rock geology along the well transect made it difficult to attribute the observed hydrometric-hydrochemical responses to the result of piston flow. Hydraulic approximations assuming realistic permeability values (based on soil texture) and the measured geometry and slope led to the conclusion that a pure Darcy-type flow cannot
explain the rapid and significant increase of groundwater discharge observed at the wetland area, if the groundwater flow field is not changed significantly during the event (Wenninger et al., 2004). Therefore, through the subsequent use of ERT, a better understanding of the subsurface structure (i.e. geology, soil and drift cover, and existence of water bodies) may lead towards a more reasoned assessment of the various mechanisms for subsurface water transfer.
APPLICATION OF ELECTRICAL RESISITIVITY TOMOGRAPHY (ERT) General The electrical resistivity of the subsurface is mainly related to geological and hydrological parameters, i.e. rock/soil type, porosity and fluid properties (degree of water saturation and solute content of the pore water). Due to multiple geological and hydrological subsurface properties controlling the electrical properties of the subsurface, there is a varying level of ambiguity with regards to the interpretation of the sampled electrical data. Electrical resistivity surveys have been made for decades in hydrogeology and geotechnical investigations; and more recently they have been used in environmental hydrology (e.g. Berthold et al., 2004; Binley and Kemna, 2005; Wenninger et al., 2008). The determination of subsurface resistivity is based on Ohm‟s law, which describes the relationship between the current density, the electrical field (voltage) and the resistivity (e.g. Binley and Kemna, 2005). In the absence of significant amounts of mineralogical clay, ore minerals or graphite the resistivity of the pore-filling fluid determines the effective measured resistivity. Electrical resistivity decreases with increasing temperature, increasing saturation, increasing salinity and/or increasing pore volume. The influence of saturated pore space for example is used in Archie‟s equations (Archie, 1942) to relate saturation to porosity and resistivity. However, there is no universal correlation ready for application in all environments and especially hydraulic conductivity and electrical resistivity do not share an identical response to changes 103
in pore space characteristics over the whole range of possible variations. Further, the effect of parameters such as fluid and bulk/matrix chemistry on the electrical resistivity might be difficult to incorporate. However, the influence of the latter can be measured, understood and integrated in the cross-interpretation of electrical resistivity data with hydrological information. As an example, a local alteration of the pore fluid‟s chemistry must also be related to changes in flow pathways and/or travel time of the fluid. The configuration used in this study was the so-called electrical resistivity tomography (ERT) method that results in an image of the subsurface direct current (DC) - resistivity. For an introduction on electrical resistivity in geophysics see e.g. Herman (2001), Barker (2007) and the “Basic principles”, “Resistivity prospecting/Electrical tomography” chapters from the online course of geophysics (UNIL & IFP, 2007).
Figure A.4. Understanding of the hydrologic system.
To improve our understanding of the hydrologic system, we have the geological, geographical, meteorological background information, plus hydrochemistry and hydrometric data (cf. Wenninger et al., 2004). With the geophysics we have additional information to gain further insights into the system (Figure A.4). Before being able to express our understanding of the system via mathematical descriptions or perform research on data-integration, it is necessary to identify the links and interdependencies between the different experimental 104
methods as well as the identified elements and dynamic processes of the studied system. The gap in spatial information evident with the point-like sampling of water from streams or wells is to some extent addressed when using geophysical methods. At the same time, geophysical data represents the output response of, for example, an electric input which is transformed through the influence of the subsurface volume, and hence is only indirectly related to hydrological relevant information. Thus, the combination of information-sources with such a difference in character is a good basis for an analytical process where new hypotheses are developed which cover the dynamic as well as the spatial aspects of the organisation of a particular system. As an example, the active use of basic information on process hydrological understanding, and limitation of the geophysical data interpretation to what is hydrologically relevant information, the multiple interpretations of a geoelectrical image can, in some case, be narrowed down to a convenient number of possible hydrologically relevant interpretations. Important are also the plausibility checks of the different methods (incl. checking the methods against each other). Almost any implementation of hydrological research has an important subsurface spatial component, which unfortunately often has to be based on poor spatial sampling due to the difficulties to access the subsurface process areas directly. Thus, the most basic benefit from the use of geophysical methods in hydrology is their subsurface spatial context information, added to the detailed hydrochemical, hydrometrical or other data sources (often point or integral/lumped data sets). It is difficult to correctly interpret and translate the geophysical response towards an image of hydrologically relevant information (Figure A.5). Part of these difficulties are related to the problem of fully understanding the physical interactions of the input (in case of ERT: electric current) with the subsurface materials and further to correctly link the output data (in case of ERT: voltage electrical resistivity) to hydrological relevant properties of the subsurface. Other problems are bound to the treatment, in particular the inversion, of the 105
raw data necessary to obtain two dimensional (2D) tomographic images. For example, the best mathematical solution of a defined problem does not always reflect closest the true natural conditions and therefore computational artefacts may be part of the calculated images.
Figure A.5. Difficulties in the use of ERT.
Both, inversion uncertainties and physically based uncertainties determine the resolution of the measurement. There are some options which allow adjustment to different prevailing conditions and study objectives. On the one hand the geometrical set up of the measurement, the applied electrode array and the inter-electrode spacing, account for the actual penetration into the subsurface by the applied electrical current and thus the physical information being stored in the raw data. On the other hand, the inversion of the collected raw data files includes further options, providing e.g. the possibility to accentuate certain features of the raw data. The possible adaptations of the method for each measurement, within the physically based margins of the method in general, set the actual resolution of the ERT. The estimate of uncertainty in the final image of electric resistivity in order for correct interpretation is highly related to its resolution.
Compared to other methods in hydrology, where resolution is an important factor (e.g. spatio-temporal rainfall distribution from metrological network data), the resolution issue is more complex using ERT. For rainfall data one might use the point information from multiple pluviographs and integrate them towards a 2D rainfall-distribution image. Within ERT the final images can be compared to the 2D rainfall-distribution. The ERT image is the inversionmodelled 2D image from point-like resistivity data. Yet, this point-like resistivity data represents 3D information from electrical current flow in the subsurface.
Specific details of the resistivity survey experimental strategy In this study, resistivity surveys were carried out using the Syscal Junior Switch System with 24 electrodes and two multi-core cables. The electrodes were set along transects and a rollalong procedure was applied to investigate transects of several hundred meters (Figure A.2). The spacing between two electrodes was varied between 0.5 and 5 m, which resulted in a different spatial resolution and penetration depth of the mapping procedure. Four electrode Wenner and dipole-dipole arrays were used for data collection (Binley and Kemna, 2005). A total of 111 measurements were undertaken using different electrode spacings (0.55 m) depending on slope position, and the assumed depths of the investigated subsurface elements and the groundwater system (max. penetration of ERT signals: 1.2-18 m). This included a single 260 m roll-along measurement with 5 m spacing, which incorporated the steep valley slope as well as the valley bottom. Intensive measurements were made at the floodplain, between the steep slope and the main stream, using a raster survey (electrode spacing of 4 m). The raster-grid featured 17 cross-sections separated by a distance of 15 m. All raster survey measurements were executed during dry conditions, preceded by at least three consecutive rainless days to ensure that the soil moisture status was comparable. To better understand the value of each ERT taken at the research site, the measurements were further cross-checked against each other. Therefore, transects were 107
measured multiple times varying between the two geoelectrical arrays „Wenner‟ and „dipoledipole‟, applying electrode-spacings from 0.5 to 5 meters (i.e. to cross-check resolution according to depths) and under changed hydroclimatic conditions (temperature, moisture). The dipole-dipole array was found to provide more detailed 2-D images, as it was possible to detect anomalies which were averaged out in a Wenner array tomography (compare with Binley and Kemna, 2005 p.138-139). An advantage of a Wenner array however, is that it gives a better signal-to-noise-ratio (Dahlin and Zhou, 2004).
RESULTS Figure A.6 shows the long ERT survey for the main slope transect using a Wenner array and 5 m spacing of electrodes (54 in total). Also included are the land cover and other drainage features that help to interpret the measurements. The root-mean-square (RMS) errors for the inverted models, provided in the figure captions are a measure of the difference between the observed resistivity data and those data predicted using the inverted models. Thus, the RMS can be high if the observed data contain more noise or if the true 3-D subsurface isn‟t homologue to the assumed 2-D formulated model. It can also be larger when the true subsurface structure contains sharp contrasts in resistivity, as the resistivity inversion algorithm seeks to find a model that satisfies a trade-off between fitting the observed data, but also conforming to a constraint on its smoothness.
Figure A.6. ERT slope profile (Wenner array, 5 m spacing, 54 electrodes, RMS error 4.5%) with surface facts. (Vertical scale is two times exaggerated).
Several features emerge at this coarse scale: i.
Patterns of high electrical resistivity values (>6000 Ωm) occur upslope associated with boulder-strewn forest cover of the valley sides;
within the floodplain (shown as meadow) are surficial pockets of higher resistivity;
beneath the surficial zones of high resistivity in the floodplain exists a very strong gradient towards a zone of much lower resistivity (>1/CC) electrical conductivities, respectively. The low and high frequency components entering into Eq. (9) are given by [Revil and Florsch, 2010]: 1 2 w d , F 1 2 w d S , F 1 w ( F 1) S , F
(10) (11) (12)
where d and S denote the specific surface conductivities of the diffuse and Stern layers, respectively, w and w denote the electrical conductivity and dielectric constant of the pore water, and S denotes the dielectric constant of the solid phase. From Eqs. (10) and (11), the chargeability is approximately given by M (2/) ( S / w ). Note that S depends on salinity as modeled by Revil and Florsch . The angular frequency of the phase peak is given by 1 2c
1 1 M
If the chargeability M is very small, which is indeed the case for the clean saturated sands investigated in the following (Table 1), we have peak 1/ CC . Note that there are alternative approaches in fitting polarization spectra to avoid the use of a specific parametric function. For example, Nordsiek and Weller  and Tong et al.  used Debye decomposition approaches in the frequency- and time-domains, respectively. In the induced polarization model developed by Revil and Florsch , the Cole Cole time constant can be approximately related to the grain diameter d by
d2 . 8Di
The coefficient Di represents the diffusion coefficient of the counterions, generally cations. This coefficient is temperature-dependent with an increase of ~2% per degree Kelvin. In Koch et al. , the pore fluid was a NaCl electrolyte with D+ = 1.3210-9 m2 s-1 at 25 °C and the mobility of the cations given by += 5.1910-8 m2s-1V-1. From the point of view of dimensional analysis, Eq. (14) corresponds to a simple scaling law between the diffusion coefficient Di in units of m2 s-1, a relaxation time CC in units s, and the squared pore diameter d2 in units of m2. Another acceptable scaling law from the point of view of dimensional analysis would be 131
2 , 2 Di
(the factor of 2 in Eq. 15 is obtained by replacing the grain diameter by its relationship to the grain radius and stating that the polarization length scale is not the grain radius but the pore throat size ). Eq. (15) implies that the dynamic pore radius corresponds to the fundamental scaling parameter for the relaxation time. In other words, is assumed to be the characteristic polarization length scale of the porous material. At this point, Eq. (15) is still in need of a mechanistic model to justify its existence. That said polarization of the electrical double layer is able to polarize the adjacent pore water in the standard model for colloidal suspensions (Grosse ). Eq. (15) is also consistent with the observations made by Scott and Barker  and Krushwitz et al., . They observed that for granular material with large pore sizes (> 10 μm), a positive correlation exists between the time constant and the pore throat size. This observation is also consistent with the dataset of Binley et al.  as discussed below. Eqs. (14) and (15) both predict that the time constant is nearly independent on the salinity as shown by Revil and Skold  and Weller et al.  (a small salinity dependence in Eqs. 14 and 15 can arise because the diffusion coefficient may be slightly dependent on salinity). In the following, we consider that Eq. (15) is our starting hypothesis and we are going to build additional relationships based on this concept. Replacing in Eq. (15) with its expression in Eq. (7), we obtain
d2 . 4m2 ( F 1) 2 Di
Eq. (16) predicts a very specific dependence of the Cole-Cole relaxation time CC with the formation factor F and therefore with the connected porosity and, as shown later, with the water saturation. This also means that the Cole-Cole relaxation time CC increases with compaction. This is consistent with the measurements reported in Table 1 and the results of 132
previous work [Klitzsch, 2004, Halisch and Weller, 2006]. We can now follow previous attempts and use the time constant to determine the permeability (e.g., Titov et al. , Revil and Florsch ). Substituting Eq. (15) into Eq. (6) yields the following expressions for the permeability
Di CC . 4F
The limitations of Eq. (17) are the following: (1) we are dealing only with clean sands and clean sandstones (for clayey sands see Revil ), (2) the pore size distribution needs to be unimodal and reasonably symmetric for the Cole-Cole model to apply, and (3) the pore size is >10-15 m or so. In the following, we shall test the newly developed relationships based on the characteristic pore size with regard to two set of laboratory measurements and compare the corresponding predictions with those of the grain-size-based model of Revil and Florsch .
UNSATURATED CASE By analogy to Archie‟s first and second laws [Archie, 1942] and in agreement with the Vinegar and Waxman  dependence of surface conductivity with saturation, we can perform the following change of variables (1/ F ) (1/ F ) swn and swn (with sw denoting the water saturation and n the saturation exponent) to extend our model to the general case of partial saturation. This yields (e.g., Linde et al. ) 1 n 2 sw w d , F 1 2 swn w d S , F
1 swn w (1 swn ) a ( F 1) S , (20) F denotes the dielectric constant of air. The chargeability should be given by
M (sw ) M (sw 1)sw n and therefore should increase with decreasing saturation. The same
change of variables yields the following scaling relationship between the Cole-Cole time constant and the saturation: CC (sw ) CC ( sw 1) sw2n , where CC ( sw 1) represents the value of the Cole-Cole time constant at full water saturation. The exponent 2n is justified because the Cole-Cole time constant is roughly inversely proportional to square of the formation factor when F >> 1 and, more directly, by the used change of variables swn .
TESTS ON LABORATORY MEASUREMENTS FREQUENCY-DOMAIN
CLEAN SATURATED SANDS The frequency-domain induced polarization measurements considered here were acquired in a frequency range from 1 mHz to 45 kHz [Koch et al., 2011] using a highly sensitive impedance spectrometer with an accuracy of 0.1 mrad for the phase for frequencies between 1 mHz and 100 Hz. The samples were saturated with a NaCl electrolyte with an electrical conductivity w ranging 0.004 and 0.006 S/m. The properties of the considered samples are listed in Table 1. To fit the Cole-Cole model to the observed data, we used a Bayesian Markov-chain Monte Carlo (McMC) sampler [Koch et al., 2011]. All the samples from Koch et al.  were considered with the exception of sample SP3, which contains plate-like particles of mica with a length of approximately one milimeter. A comparison between the prediction of the model of Revil and Florsch  and the measured Cole-Cole relaxation times is shown in Figure 1a, whereas the corresponding comparison for Eq. (17) is shown in Figure 1c. There is a good agreement between the experimental data of Table 1 and the prediction of Eq. (17) and but not with regard to the model of Revil and Florsch . We then proceed compare the measured permeability (Table 1) with predicted permeability using the model of Revil and Florsch  (Figure 134
1b) and Eq. (17) (Figure 1d). Again, the model based on the characteristic pore size (Eq. 17) developed in this study provides a more favorable prediction than the grain-size-based model of Revil and Florsch .
Figure B.1. Predicted versus measured parameters from the grain size and the pore size for the clean saturated sand samples list in Table 1 [Koch et al. 2011]. Test of the grain size model for the ColeCole time constant (a) (powerlaw fit: R2 = 0.70) and for the permeability (b) (powerlaw fit: R2 = 0.61). Test of the pore size model for the Cole-Cole time constant (c) (powerlaw fit: R2 = 0.69) and for the permeability (d) (powerlaw fit: R2 = 0.88).
Table B.1. Grain size characteristics of loose and compacted quartz sands as inferred from laser diffraction measurements [Koch et al., 2011]. The grain size d50 denotes the median of the particle size distribution and is the length scale used for the grain diameter d. The electrical measurements were taken at temperature of 21.6°C. Samples denoted as "*-C" represent compacted sands. The cementation exponent m is determined as m = -ln F/ln and the length scale from Eq. (7) using the values of d50, m, and F. The uncertainty of F is of the order of 3%, the uncertainty of is of the order of 2%, and the uncertainty of m is of the order of 5%. "sd" stands for standard deviation. The ColeCole parameters have all been obtained by fitting Eq. (9) to the data.
Sample F36 F32 WQ1 SP1 SP2 SP3 SP4 SP5 SP6 F36-C F32-C WQ1-C SP1-C SP2-C SP3-C SP4-C SP5-C SP6-C
0.0119 0.0022 0.0232 0.0048 0.0066 0.0051 0.0063 0.0121 0.0113
0.158 0.035 0.060 0.051 0.046 0.103 0.048 0.077 0.096
0.44 0.51 2.13 0.30 0.30 5.13 0.84 4.68 12.4
0.065 0.030 0.054 0.065 0.094 0.042 0.044 0.163 0.173
20.4±1.6 34.4±2.8 94.0±7.8 28.4±2.3 33.6±2.7 42.9±3.4 74.6±6.0 105±8.4 109±8.7
1.98E-11 3.55 3.81E-11 3.64
1.05E-10 3.52 1.96E-10 3.36 2.56E-10 3.63
0.44 0.43 0.43
1.54 1.44 1.53
0.0087 0.0122 0.0080
0.059 0.072 0.115
0.80 3.65 3.42
0.091 0.160 0.409
64.4±5.2 100±8.0 108±8.6
0.18 0.27 0.66 0.18 0.24 0.32 0.50 0.68 0.87 0.18 0.27 0.66 0.18 0.24 0.32 0.50 0.68 0.87
1.76E-11 5.31E-11 1.29E-10 2.08E-11 3.30E-11 6.75E-11 1.71E-10 2.80E-10 3.94E-10
3.77 3.55 3.25 3.14 3.40 3.26 3.12 3.10 3.34
0.44 0.44 0.47 0.46 0.44 0.49 0.49 0.48 0.49
1.59 1.54 1.56 1.48 1.49 1.65 1.58 1.54 1.70
1.11E-11 4.12 2.40E-11 3.75
7.50E-11 3.97 1.17E-11 3.23
. At brine conductivity (NaCl) 0.004 - 0.006 S/m.
. Determined from Eq. (7). Relative error determined from d50 m F .
PARTIALLY SATURATED SANDSTONES The second test of the newly proposed relationships concerns the dependence of the ColeCole relaxation time on the saturation. If the proposed scaling relationship between the ColeCole relaxation time and the formation factor is correct, we can easily predict the dependence of the Cole-Cole relaxation time on the saturation. We selected the dataset of Binley et al.  because it corresponds to a clean sandstone. The Cole-Cole time constant for this
dataset is pretty well predicted by our model (Figure 2a) except for very small pore sizes (