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.

[1]  L. Eldén.  Algorithms for the regularization of ill-conditioned least squares problems , 1977 .

[2]  P. Hansen.  Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion , 1987 .

[3]  T. Ishido,et al.  Experimental and theoretical basis of electrokinetic phenomena in rock‐water systems and its applications to geophysics , 1981 .

[4]  Michael A. Mooney,et al.  Saline pulse test monitoring with the self‐potential method to nonintrusively determine the velocity of the pore water in leaking areas of earth dams and embankments , 2012 .

[5]  Burke J. Minsley,et al.  Hydrogeophysical Investigations at Hidden Dam, Raymond, California , 2011 .

[6]  Salvatore Straface,et al.  Self‐potential signals associated with pumping tests experiments , 2004 .

[7]  A. Revil,et al.  Self-potential signals associated with preferential groundwater flow pathways in sinkholes , 2006 .

[8]  André Revil,et al.  Tomography of the Darcy velocity from self‐potential measurements , 2007 .

[9]  F. B. Hildebrand,et al.  Methods of Applied Mathematics , 1966 .

[10]  D. Oldenburg,et al.  3-D inversion of gravity data , 1998 .

[11]  M. Riva,et al.  Joint Inversion of Steady-State Hydrologic and Self-Potential Data for 3D Hydraulic Conductivity Distribution at the Boise Hydrogeophysical Research Site , 2011 .

[12]  Domenico Patella,et al.  Self‐potential global tomography including topographic effects , 1997 .

[13]  Burke J. Minsley,et al.  Three-dimensional source inversion of self-potential data , 2007 .

[14]  T. Ishido,et al.  Self-potential generation by subsurface water flow through electrokinetic coupling , 1989 .

[15]  André Revil,et al.  Streaming potential in porous media: 2. Theory and application to geothermal systems , 1999 .

[16]  Cass T. Miller,et al.  Reconstruction of the Water Table from Self‐Potential Data: A Bayesian Approach , 2009, Ground water.

[17]  W. Menke.  Geophysical data analysis : discrete inverse theory , 1984 .

[18]  D. Oldenburg,et al.  3-D inversion of gravity data , 1998 .

[19]  Bernd Kulessa,et al.  Understanding biogeobatteries: Where geophysics meets microbiology , 2010 .

[20]  Behavior near zero of the distribution of GCV smoothing parameter estimates , 1995 .

[21]  D. Oldenburg,et al.  Three-dimensional modelling of streaming potential , 2007 .

[22]  A. Revil,et al.  Streaming electrical potential anomaly along faults in geothermal areas , 1998 .

[23]  G. Smith,et al.  Numerical Solution of Partial Differential Equations: Finite Difference Methods , 1978 .

[24]  Burke J. Minsley,et al.  Three‐dimensional self‐potential inversion for subsurface DNAPL contaminant detection at the Savannah River Site, South Carolina , 2007 .

[25]  Changsong Deng,et al.  Statistics and Probability Letters , 2011 .

[26]  Nelson J. Trujillo-Barreto,et al.  Bayesian model averaging in EEG/MEG imaging , 2004, NeuroImage.

[27]  A. N. Tikhonov,et al.  Solutions of ill-posed problems , 1977 .

[28]  Kurt M. Strack,et al.  Society of Exploration Geophysicists , 2007 .

[29]  Niklas Linde,et al.  Estimation of the water table throughout a catchment using self-potential and piezometric data in a Bayesian framework , 2007 .

[30]  Nicolas Florsch,et al.  Least squares inversion of self‐potential (SP) data and application to the shallow flow of ground water in sinkholes , 2006 .

[31]  S. Sunoj,et al.  Statistics and Probability Letters , 2012 .

[32]  A. Ogilvy,et al.  DEFORMATIONS OF NATURAL ELECTRIC FIELDS NEAR DRAINAGE STRUCTURES , 1973 .

[33]  A. Revil,et al.  Stochastic inversion of permeability and dispersivities from time lapse self‐potential measurements: A controlled sandbox study , 2010 .

[34]  R. Lytton,et al.  Finite-Element Modeling of Streaming Potentials Due to Seepage: Study of a Dam , 1991 .

[35]  B. Hubbard,et al.  Cross‐coupled flow modeling of coincident streaming and electrochemical potentials and application to subglacial self‐potential data , 2003 .

[36]  El-Sayed M. Abdelrahman,et al.  Shape and depth solutions from numerical horizontal self-potential gradients , 1997 .

[37]  W. R. Sill,et al.  Self-potential modeling from primary flows , 1983 .

[38]  Laurent Spinelli,et al.  Analyse spatiale de l'activité électrique cérébrale : nouveaux développements. (Spatial analysis of cerebral electrical activity : new developments) , 1999 .

[39]  N. Linde,et al.  Chemico-electromechanical coupling in microporous media. , 2006, Journal of colloid and interface science.

[40]  Roberto D. Pascual-Marqui,et al.  Functional imaging with low resolution brain electromagnetic tomography (LORETA) : review, new comparisons, and new validation , 2002 .

[41]  Philippe Labazuy,et al.  The volcano‐electric effect , 2003 .

[42]  M. Voltz,et al.  Monitoring of an infiltration experiment using the self‐potential method , 2006 .