The Seismoelectric Method - Theory and Application

Contents 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