The Seismoelectric Method
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Foreword by Bernd Kulessa xi
Foreword by Niels Grobbe xii
Preface xiv
1 Introduction to the basic concepts 1
1.1 The electrical double layer 1
1.1.1 The case of silica 2
1.1.1.1 A simplified approach 2
1.1.1.2 The general case 8
1.1.2 The case of clays 10
1.1.3 Implications 14
1.2 The streaming current density 15
1.3 The complex conductivity 17
1.3.1 Effective conductivity 18
1.3.2 Saturated clayey media 19
1.4 Principles of the seismoelectric method 22
1.4.1 Main ideas 22
1.4.2 Simple modeling with the acoustic approximation 25
1.4.2.1 The acoustic approximation in a fluid 25
1.4.2.2 Extension to porous media 26
1.4.3 Numerical example of the coseismic and seismoelectric conversions 27
1.5 Elements of poroelasticity 28
1.5.1 The effective stress law 28
1.5.2 Hooke's law in poroelastic media 31
1.5.3 Drained versus undrained regimes 31
1.5.4 Wave modes in the pure undrained regime 33
1.6 Short history 34
1.7 Conclusions 36
2 Seismoelectric theory in saturated porous media 42
2.1 Poroelastic medium filled with a viscoelastic fluid 42
2.1.1 Properties of the two phases 42
2.1.2 Properties of the porous material 45
2.1.3 The mechanical equations 49
2.1.3.1 Strain-stress relationships 49
2.1.3.2 The field equations 51
2.1.3.3 Note regarding the material properties 52
2.1.3.4 Force balance equations 53
2.1.4 The Maxwell equations 53
2.1.5 Analysis of the wave modes 54
2.1.6 Synthetic case studies 56
2.1.7 Conclusions 59
2.2 Poroelastic medium filled with a Newtonian fluid 59
2.2.1 Classical Biot theory 59
2.2.2 The u-p formulation 60
2.2.3 Description of the electrokinetic coupling 61
2.3 Experimental approach and data 62
2.3.1 Measuring key properties 62
2.3.1.1 Measuring the cation exchange capacity and the specific surface area 62
2.3.1.2 Measuring the complex conductivity 63
2.3.1.3 Measuring the streaming potential coupling coefficient 63
2.3.2 Streaming potential dependence on salinity 63
2.3.3 Streaming potential dependence on pH 66
2.3.4 Influence of the inertial effect 66
2.4 Conclusions 69
3 Seismoelectric theory in partially saturated conditions 73
3.1 Extension to the unsaturated case 73
3.1.1 Generalized constitutive equations 73
3.1.2 Description of the hydromechanical model 77
3.1.3 Maxwell equations in unsaturated conditions 81
3.2 Extension to two-phase flow 81
3.2.1 Generalization of the Biot theory in two-phase flow conditions 81
3.2.2 The u-p formulation for two-phase flow problems 83
3.2.3 Seismoelectric conversion in two-phase flow 85
3.2.4 The effect of water content on the coseismic waves 86
3.2.5 Seismoelectric conversion 90
3.3 Extension of the acoustic approximation 91
3.4 Complex conductivity in partially saturated conditions 92
3.5 Comparison with experimental data 93
3.5.1 The effect of saturation 93
3.5.2 Additional scaling relationships 93
3.5.3 Relative coupling coefficient with the Brooks and Corey model 95
3.5.4 Relative coupling coefficient with the Van Genuchten model 96
3.6 Conclusions 97
4 Forward and inverse modeling 101
4.1 Finite-element implementation 101
4.1.1 Finite-element modeling 101
4.1.2 Perfectly matched layer boundary conditions 102
4.1.3 Boundary conditions at an interface 104
4.1.4 Description of the seismic source 104
4.1.5 Lateral resolution of cross-hole seismoelectric data 104
4.1.6 Benchmark test of the code 105
4.2 Synthetic case study 105
4.2.1 Simulation of waterflooding of a NAPL-contaminated aquifer 105
4.2.2 Simulation of the seismoelectric problem 107
4.2.3 Results 110
4.3 Stochastic inverse modeling 112
4.3.1 Markov chain Monte Carlo solver 112
4.3.2 Application 115
4.3.3 Result of the joint inversion 118
4.4 Deterministic inverse modeling 118
4.4.1 A statement of the problem 118
4.4.2 5D electric forward modeling 121
4.4.3 The initial inverse solution 125
4.4.4 Getting compact volumetric current source distributions 126
4.4.5 Benchmark tests 126
4.4.6 Numerical case studies 127
4.4.7 Discussion 133
4.5 Conclusions 133
5 Electrical disturbances associated with seismic sources 136
5.1 Theory 136
5.1.1 Position of the problem 136
5.1.2 Forward modeling 137
5.1.3 Modeling noise-free and noisy synthetic data 141
5.1.4 Results 141
5.2 Joint inversion of seismic and seismoelectric data 145
5.2.1 Problem statement 145
5.2.2 Algorithm 146
5.2.3 Results with noise-free data 147
5.2.4 Results with noisy data 148
5.2.5 Hybrid joint inversion 150
5.2.6 Discussion 154
5.3 Hydraulic fracturing laboratory experiment 155
5.3.1 Background 155
5.3.2 Material and method 156
5.3.3 Observations 159
5.3.4 Electrical potential evidence of seal failure 164
5.3.5 Source localization algorithms 165
5.3.5.1 Electrical and hydromechanical coupling 166
5.3.5.2 Inversion phase 1: gradient-based deterministic approach 167
5.3.5.3 Inversion phase 2: GA approach 169
5.3.6 Results of the inversion 170
5.3.6.1 Results of the gradient-based inversion 170
5.3.6.2 Results of the GA 175
5.3.6.3 Noise and position uncertainty analysis 181
5.3.7 Discussion 183
5.4 Haines jump laboratory experiment 185
5.4.1 Position of the problem 185
5.4.2 Material and methods 186
5.4.3 Discussion 187
5.5 Small-scale experiment in the field 190
5.5.1 Material and methods 191
5.5.2 Results 191
5.5.3 Localization of the causative source of the self-potential anomaly 192
5.6 Conclusions 194
6 The seismoelectric beamforming approach 199
6.1 Seismoelectric beamforming in the poroacoustic approximation 199
6.1.1 Motivation 199
6.1.2 Beamforming technique 200
6.1.3 Results and interpretation 202
6.2 Application to an enhanced oil recovery problem 203
6.3 High-definition resistivity imaging 208
6.3.1 Step 1: the seismoelectric focusing approach 208
6.3.2 Step 2: application of image-guided inversion to ERT 212
6.3.2.1 Edge detection 212
6.3.2.2 Introduction of structural information into the objective function 214
6.3.2.3 Results 215
6.3.3 Discussion 216
6.4 Spectral seismoelectric beamforming (SSB) 217
6.5 Conclusions 219
7 Application to the vadose zone 220
7.1 Data acquisition 220
7.2 Case study: Sherwood sandstone 223
7.2.1 Experimental results 223
7.2.2 Results 224
7.2.3 Interpretation 225
7.2.3.1 Seismoelectric signal preprocessing 225
7.2.3.2 Seismoelectric-water content relationship 226
7.2.4 Empirical modeling 227
7.2.5 Discussion 228
7.3 Numerical modeling 229
7.3.1 Theory 229
7.3.2 Description of the numerical experiment 231
7.3.3 Model application and results 231
7.4 Conclusions 235
8 Conclusions and perspectives 237
Glossary: the seismoelectric method 240
Index 243
CHAPTER 1
Introduction to the basic concepts
The goal of the first chapter is to introduce some of the key concepts required to understand the seismoelectric theory that will be developed for the saturated case in Chapter 2 and for the partially saturated and two-phase flow cases in Chapter 3. These key concepts include the electrical double layer theory and the reasons why an electrical (streaming) current density is produced when the pore water flows relative to the skeleton formed by the solid grains. In the context of the seismoelectric theory, the propagation of seismic waves will be responsible for the relative flow of pore water, and the resulting source current density will be responsible for electromagnetic (EM) disturbances. We will provide a short history of the seismoelectric method as well as its basic concepts. We will also give an introduction to wave propagation theory. At the end of this chapter, we will also provide some simulations using a simplified version of the seismoelectric theory that is based on the acoustic approximation. These models will illustrate, in a simple way, the key concepts behind the seismoelectric method, especially the difference between coseismic signals and seismoelectric conversions. Finally, we will present a preliminary model of seismoelectric phenomena pertaining to the Biot-Frenkel theory of linear poroelasticity.
1.1 The electrical double layer
As discussed later in Section 1.4, the existence of seismoelectric effects is closely related to the existence of the electrical double layer at the interface between the pore water and the skeleton (made of the elastic minerals). In the presence of several immiscible fluids in the pore space, seismoelectric effects can be also associated with the existence of an electrical double layer at the interface between the pore water and these other fluids such as air or oil. Therefore, we believe that it is important to start this book with an extensive description of what the electrical double layer is for silica and clay minerals that are in contact with an electrolyte composed of water molecules and ions. We will focus on silica and clays but the electrical double layer theory has been also developed for carbonates (Cicerone et al., 1992; Strand et al., 2006; Hiorth et al., 2010) and other types of aluminosilicates such as zeolites (van Bekkum et al., 2001).
The electrical double layer is a generic name given to electrochemical disturbances existing at the surface of minerals in contact with water containing dissolved ions. The electrical double layer comprises (1) the Stern layer of sorbed ions on the mineral surface (Stern, 1924) and (2) the diffuse layer of ions bound to the surface through the coulombic force associated with the deficiency or excess of electrical charges on the mineral surface and the Stern layer (Gouy, 1910; Chapman, 1913). The sorbed ions of the Stern layer possess a specific affinity for the mineral surface in addition to the coulombic interaction (specific is usually used to include all types of interactions that are not purely coulombic). In the case of the diffuse layer, the ions are interacting with the mineral surface only through the coulomb interaction.
The readers that are interested to understand the seismoelectric effect but that are not interested by the interfacial electrochemistry can skip Sections 1.1.1 and 1.1.2 and can go directly to Section 1.1.3 of this chapter.
1.1.1 The case of silica
1.1.1.1 A simplified approach
Figure 1.1 sketches the surface of a silica grain coated by an electrical double layer. When a mineral like silica is in contact with water, its surface becomes charged due to chemical reactions between the available surface bonding and the pore water as shown in Figure 1.2. For instance, the silanol groups, shown by the symbol >SiOH, of the surface of silica (where > refers to the mineral crystalline framework), behave as weak acid-base (amphoteric sites). This means that they can lose a proton when in contact with water to generate negative surface sites (>SiO-). They can also gain protons to become positive sites (>SiOH2+). Putting water in contact with a fresh silica surface leads to a slight acidification of the pore water, as shown in Figure 1.2, which explains why silica is considered to be an acidic rock. At the opposite end, a mineral like carbonate will generate a basic pH (>7.0) in the pore water.
Figure 1.1 Sketch of the electrical double layer at the pore water-mineral interface coating a spherical grain (modified from Revil & Florsch, 2010). The local conductivity s(?) depends on the local distance ? from the charged surface of the mineral. The pore water is characterized by a volumetric charge density corresponding to the (total) charge of the diffuse layer per unit pore volume (in coulombs (C) m-3). The Stern layer is responsible for the excess surface conductivity SS (in siemens, S) with respect to the conductivity of the pore water sf, while the diffuse layer is responsible for the excess surface conductivity Sd. These surface conductivities are sometimes called specific surface conductance because of their dimension, but they are true surface conductivities. The Stern layer is comprised between the o-plane (mineral surface) and the d-plane, which is the inner plane of the electrical diffuse layer (OHP stands for outer Helmholtz plane). The diffuse layer extends from the d-plane into the pores. The element M+ stands for the metal cations (e.g., sodium, Na+), while A- stands for the anions (e.g., chloride, Cl-). In the present case (negatively charged mineral surface), M+ denotes the counterions, while A- denotes the coions. The fraction of charge contained in the Stern layer with respect to the total charge of the double layer is called the partition coefficient f.
Figure 1.2 Formation of the electrical double layer in the case of silica. In the present case, a neutral silica surface is brought in contact with a neutral pore water solution composed of cations M+ and anions A-. The silanol surface groups at the surface of silica release a certain number of protons in the pore water, making the solution slightly acidic. Some of the cations from the pore water are adsorbed in the Stern layer. The surface charge density and the Stern layer charge density are compensated in the diffuse layer. In a sandstone, the bulk pore water is neutral (no net charge density), and only the diffuse layer is not neutral and more precisely characterized usually by an excess of (positive) charges.
It follows that the mineral surface charge of silica appears to be pH dependent. It is typically negative at near-neutral pH values (pH 5-8) and possibly positive or neutral for very acidic conditions (pH <3). The simplest complexation reactions at the surface of silica can be summarized as (e.g., Wang & Revil, 2010, and references therein)
(1.1) (1.2)where are the two equilibrium constants associated with the surface sorption and desorption of protons. This 2-pK model considers that two charged surface species, namely, >SiO- and >SiOH2+, are responsible for the surface charge density of silica. That said, the reaction in Equation (1.1) is often neglected in a number of studies because the occurrence of the positive sites, >SiOH2+, can only happen at low pH values (typically below pH <3 as mentioned briefly previously).
We also assume that the pore water contains a completely dissociated monovalent salt (e.g., NaCl providing the same amount of cations Na+ and anions Cl-). In the following, a "counterion" is an ion that is characterized by a charge opposite to the charge of the mineral surface, while a "coion" has a charge of the same sign as the mineral surface. The typical case for silica is to have a negative surface charge, and therefore, the counterions are the Na+ cations and the coions are the Cl- anions. Note however that the sorption of cations is characterized by a high valence and a strong affinity for the silica surface (for instance, Al3+) and can reverse the charge of the mineral surface (surface and Stern later together) and therefore can reverse the sign of the charge of the diffuse layer. The sorption is described by the following reaction:
(1.3)where KM corresponds to the equilibrium constant for this reaction. Sorption is distinct from precipitation, which involves the formation of covalent bonds with the mineral surface. This sorption can be strong (formation of an inner-sphere complexes with no mobility along the mineral surface) or weak. In the "weak case," the formation of the Stern layer is a kind of condensation effect demonstrated by molecular dynamics. A weak sorption example is the case of a hydrated sodium. In this example, the sorbed counterion Na+ keeps its hydration sphere, and it forms a so-called outer-sphere complex with the mineral surface (e.g., Tadros & Lyklema, 1969). Such counterions are expected to keep some mobility along the mineral surface, responsible (as briefly explained in Section 1.3) for a low-frequency polarization of the mineral grains in an alternating electrical field. The layer of ions formed by the sorption of these...
