The Seismoelectric Method

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