Abderrahim Jardani

The self-potential method : theory and applications in environmental geosciences

Foreword Preface 1. Fundamentals of the self-potential method 2. Development of a fundamental theory 3. Laboratory investigations 4. Forward and inverse modeling 5. Applications to geohazards 6. Application to water resources 7. Application to hydrothermal systems 8. Seismoelectric coupling Appendix A: a simple model of the Stern layer Appendix B: the u-p formulation of poroelasticity References Index.

Stochastic joint inversion of 2D seismic and seismoelectric signals in linear poroelastic materials: A numerical investigation

The interpretation of seismoelectrical signals is a difficult task because coseismic and seismoelectric converted signals are recorded simultaneously and the seismoelectric conversions are typically several orders of magnitude smaller than the coseismic electrical signals. The seismic and seismoelectric signals are modeled using a finite-element code with perfectly matched layer boundary conditions assuming a linear poroelastic body. We present a stochastic joint inversion of the seismic and seismoelectrical data based on the adaptive Metropolis algorithm, to obtain the posterior probability density functions of the material properties of each geologic unit. This includes the permeability, porosity, electrical conductivity, bulk modulus of the dry porous frame, bulk modulus of the fluid, bulk modulus of the solid phase, and shear modulus of the formations. A test of this approach is performed with a synthetic model comprising two horizontal layers and a reservoir partially saturated with oil, which is embedded in the second layer. The result of the joint inversion shows that we can invert the permeability of the reservoir and its mechanical properties.

Hydraulic conductivity field characterization from the joint inversion of hydraulic heads and self-potential data

Pumping tests can be used to estimate the hydraulic conductivity field from the inversion of hydraulic head data taken intrusively in a set of piezometers. Nevertheless, the inverse problem is strongly underdetermined. We propose to add more information by adding self-potential data taken at the ground surface during pumping tests. These self-potential data correspond to perturbations of the electrical field caused directly by the flow of the groundwater. The coupling is electrokinetic in nature that is due to the drag of the excess of electrical charges existing in the pore water. These self-potential signals can be easily measured in field conditions with a set of the nonpolarizing electrodes installed at the ground surface. We used the adjoint-state method for the estimation of the hydraulic conductivity field from measurements of both hydraulic heads and self potential during pumping tests. In addition, we use a recently developed petrophysical formulation of the streaming potential problem using an effective charge density of the pore water derived directly from the hydraulic conductivity. The geostatistical inverse framework is applied to five synthetic case studies with different number of wells and electrodes and thickness of the confining unit. To evaluate the benefits of incorporating the self-potential data in the inverse problem, we compare the cases in which the data are combined or not. Incorporating the self-potential information improves the estimate of hydraulic conductivity field in the case where the number of piezometers is limited. However, the uncertainty of the characterization of the hydraulic conductivity from the inversion of the self-potential data is dependent on the quality of the distribution of the electrical conductivity used to solve the Poisson equation. Consequently, the approach discussed in this paper requires a precise estimate of the electrical conductivity distribution of the subsurface and requires therefore new strategies to be developed for the joint inversion of the hydraulic and electrical conductivity distributions.

SP2DINV: A 2D forward and inverse code for streaming potential problems

The self-potential method corresponds to the passive measurement of the electrical field in response to the occurrence of natural sources of current in the ground. One of these sources corresponds to the streaming current associated with the flow of the ground water. We can therefore apply the self-potential method to recover non-intrusively some information regarding the ground water flow. We first solve the forward problem starting with the solution of the ground water flow problem, then computing the source current density, and finally solving a Poisson equation for the electrical potential. We use the finite-element method to solve the relevant partial differential equations. In order to reduce the number of (petrophysical) model parameters required to solve the forward problem, we introduced an effective charge density tensor of the pore water, which can be determined directly from the permeability tensor for neutral pore waters. The second aspect of our work concerns the inversion of the self-potential data using Tikhonov regularization with smoothness and weighting depth constraints. This approach accounts for the distribution of the electrical resistivity, which can be independently and approximately determined from electrical resistivity tomography. A numerical code, SP2DINV, has been implemented in Matlab to perform both the forward and inverse modeling. Three synthetic case studies are discussed.

Joint inversion of hydraulic head and self‐potential data associated with harmonic pumping tests

Harmonic pumping tests consist in stimulating an aquifer by the means of hydraulic stimulations at some discrete frequencies. The inverse problem consisting in retrieving the hydraulic properties is inherently ill-posed and is usually underdetermined when considering the number of well head data available in field conditions. To better constrain this inverse problem, we add self-potential data recorded at the ground surface to the head data. The self-potential method is a passive geophysical method. Its signals are generated by the groundwater flow through an electrokinetic coupling. We showed, using a 3D saturated unconfined synthetic aquifer, that the self-potential method significantly improves the results of the harmonic hydraulic tomography. The hydroelectric forward problem is obtained by solving first the Richards equation, describing the groundwater flow, and then using the result in an electrical Poisson equation describing the self-potential problem. The joint inversion problem is solved using a reduction model based on the principal component geostatistical approach. In this method, the large prior covariance matrix is truncated and replaced by its low-rank approximation, allowing thus for notable computational time and storage savings. Three test cases are studied, to assess the validity of our approach. In the first test, we show that when the number of harmonic stimulations is low, combining the harmonic hydraulic and self-potential data does not improve the inversion results. In the second test, where enough harmonic stimulations are performed, a significant improvement of the hydraulic parameters is observed. In the last synthetic test, we show that the electrical conductivity field required to invert the self-potential data can be determined with enough accuracy using an electrical resistivity tomography survey using the same electrodes configuration as used for the self-potential investigation. This article is protected by copyright. All rights reserved.

HT2DINV: A 2D forward and inverse code for steady-state and transient hydraulic tomography problems

Abstract Hydraulic tomography is a technique used to characterize the spatial heterogeneities of storativity and transmissivity fields. The responses of an aquifer to a source of hydraulic stimulations are used to recover the features of the estimated fields using inverse techniques. We developed a 2D free source Matlab package for performing hydraulic tomography analysis in steady state and transient regimes. The package uses the finite elements method to solve the ground water flow equation for simple or complex geometries accounting for the anisotropy of the material properties. The inverse problem is based on implementing the geostatistical quasi-linear approach of Kitanidis combined with the adjoint-state method to compute the required sensitivity matrices. For undetermined inverse problems, the adjoint-state method provides a faster and more accurate approach for the evaluation of sensitivity matrices compared with the finite differences method. Our methodology is organized in a way that permits the end-user to activate parallel computing in order to reduce the computational burden. Three case studies are investigated demonstrating the robustness and efficiency of our approach for inverting hydraulic parameters.

Self-potential: A Non-intrusive Ground Water Flow Sensor

ABSTRACT The flow of the ground water in an aquifer or during pumping test generates an electrical current (called the streaming current), which is of advective nature. The resulting electrical field (streaming potential field, one of the components of the self-potential field) can be remotely measured at the ground surface or in boreholes. We first discuss the underlying physics of this electrokinetic effect and the role of the electrical double layer coating the surface of the grains. We show how the drag of the excess of electrical charge of the pore water by the flow is equivalent to a source current density. Then, we discuss the metrological aspects, the type of voltmeter and electrodes required to carry out good measurements in field conditions. Two applications are discussed in steady-state conditions. The first is dedicated to the flow of water in shallow aquifers. In this case, the streaming current and the conduction current are nearly balanced and, inside the aquifer, the electrical equipotenti...

A numerical investigation of cross‐hole seismoelectric conversion

We consider the case of two wells in parallel to each other. A seismic source is located in the first well and electrodes are located in a second well. For times comprised between the time of the seismic source and the time of the first arrival of the seismic waves in the second borehole, all the recorded electrical disturbances are associated with seimoelectric conversion of the seismic energy when the seismic P wave reaches heterogeneities located between the two wells. These heterogeneities need to correspond to change in electrical conductivity and electrokinetic properties between the materials. We use a finite element method to solve both for seismic wave equation and the associated seismoelectric disturbances in a piecewise homogeneous porous material. We consider the case corresponding to a vertical contact between two homogeneous materials. The seismoelectric data recorded in the second well can be inverted using a deterministic algorithm to retrieve the position of the vertical heterogeneity. This is a part of an ongoing study.

Stochastic joint inversion of 2D seismic and seismoelectric signals for reservoir characterization

We use the finite element method to solve both for seismic wave equation and the associated seismoelectric disturbances in a piecewise homogeneous porous material containing an oil reservoir. The seismic and seismoelectric data recorded at the top surface of the medium are jointly inverted using an adaptative Metropolis algorithm to retrieve the posterior probability distribution of the material properties. Comparison with the true material properties shows the ability of our approach to retrieve the electrical and mechanical material properties of each formation including the permeablity of the oil reservoir.