Sébastien Penz

Finite difference resistivity modeling on unstructured grids with large conductivity contrasts

The 3-D geo-electrical forward problem solved with a finite difference approach faces several difficulties. Besides the singularity at the source location, major issues are caused by the definition of the computational domain to match a particular topography, and by high conductivity contrasts. To address these issues, we combine here two methods. First, we implement a specific finite difference method that takes into account specified interfaces in elliptic problems. Here, the contrasts are defined along grid lines. Second, we extend the method to unstructured meshes by integrating it to the generalized finite difference technique. In practice, once the conductivity model is defined, the approach does not need to explicitly specify where the large contrasts are located. Several numerical tests are carried out for various Poisson problems and show a high degree of accuracy.

Variational Bayesian inversion of synthetic 3D controlled-source electromagnetic geophysical data

Inversion of controlled-source electromagnetic (CSEM) data is dealt with for a geophysical application. The goal is to retrieve a map of conductivity of an unknown body embedded in a layered underground from measurements of the scattered electric eld that results from its interaction with a known interrogating wave. This constitutes an inverse scattering problem whose associated forward problem is described by means of electric field domain integral equations. The inverse problem is solved in a Bayesian framework where prior information is introduced via a Gauss-Markov-Potts model. This model describes the body as composed of a finite number of different materials distributed into compact homogeneous regions. The posterior distribution of the unknowns is approached by means of the variational Bayesian approximation as a separable distribution that minimizes the Kullback-Leibler divergence with respect to the posterior law. Thus, we get a parametric model for the distributions of the induced currents, the ...

Practical Inversion of Electric Resistivity in 3D from Frequency-domain Land CSEM Data

EM prospection are method of choice for many applications such as deep water or geothermal prospection because of their sensitivity to electrical resistivity and their potential to investigate at depths of 500m or even more. However, the investigated areas in Europe are usually urbanised and industrialised so high level of cultural noise prevents from the use of MT. Land CSEM is an alternative. But cost and logistical constrains may limits to the use of a small number of transmitter positions, often only one. The inversion of CSEM data in the near field using a single transmitter position suffers from critical sensitivity singularities due to the unsymmetrical illumination. To overcome this problem we proposed an inversion framework adapted to this ill-conditioned inversion problem. The framework relies specifically on a robust Gauss-Newton solver, on model parameter transformations and data reformulation under the form of pseudo-MT tensors. We describe here the modelling and inversion approach implemented in our code POLYEM3D and describe the framework proposed for its practical application. We illustrate its application on synthetic cases and then show the application of the process to a real CSEM dataset acquired for thermal water prospection at a few kilometer from a nuclear power plant in France.

Resistivity Modelling: Handling Source Singularity and Topography within a Single Step

The singularity of the potential occurring at the source location is a key point of electrical resistivity forward modelling because it might lead to large numerical errors. To tackle this problem a classical method consists of splitting the total potential into a primary part containing the singularity and a secondary part. The primary potential is defined analytically for flat topography but requires numerical computation in the presence of topography. In that case, an accurate solution happens to be computationally expensive. For any geometry we propose to keep for the primary potential the analytic solution defined for homogeneous models and flat topography, and to modify accordingly the free surface boundary conditions for the secondary potential. The primary potential still contains the singularity and new free surface conditions ensure that the total potential still satisfies the Poisson equation. The modified singularity removal technique thus remains fully efficient even in the presence of topography, without additional numerical computation. The modified secondary potential in a homogeneous model is not null in the case of topography as it would be in the classical approach. We implement the approach with a Finite Difference method. We present potential distributions computed with this technique to illustrate its versatility.

High Resolution Electrical Resistivity Tomography in Superficial Limestones at Tournemire Site, France

Deep argillaceous formations are considered in many countries as potential host media for high-level long-life radioactive waste due their confining properties. The precise sedimentary, structural and hydrogeological characterization of such potential host sites is a key point in determining their appropriateness for the long-term deep underground disposal of radioactive waste in geological formations. The presence of faults in clay–rock formations should be carefully assessed, since these features could modify the confining properties. This study focuses on testing the potential of the electrical resistivity method to detect fault or fractured zones in the near subsurface layers above an argillaceous formation. We present in this paper results from a high-resolution electrical resistivity survey carried out at the IRSN Tournemire Experimental Platform (TEP). The electrical resistivity profile was located transversely to the fault and fractured zones location, inferred from geological data, that affect the Jurrassic formations at the TEP. Electrical resistivity data were successively acquired with 8m, 4m and 2m-electrode spacing. This multi-resolution acquisition allows to investigate the near subsurface limestones and dolomites to a depth of 100 metres. In particular, two sub vertical conductive corridors reaching the surface through higher resistive layers are correlated with fractured zones.

2D Acoustic Wave Propagation on Unstructured Grids Using the Generalized Finite Difference Method

Finite difference schemes on regular grids are efficient to accurately predict the seismic wave field, at least for smooth models. However, spurious diffractions become visible when the topography or internal interfaces do not align with the Cartesian grid. We propose to use the generalized finite difference scheme to propagate the wave field on unstructured meshes. We study the accuracy of the proposed solution on simple applications for the 2D acoustic constant density case and show that triangular meshes allow to reduce the imprint of diffractions.

Resistivity Modelling with Topography

A major difficulty of electrical resistivity forward modelling is caused by the singularity of the potential occurring at the source location. To avoid large numerical errors, the total potential is split into a primary part containing the singularity and a secondary part. The primary potential is defined analytically for flat topography, but is classically computed numerically in the presence of topography: in that case, an accurate solution requires expensive computations. We propose to select for the primary potential the analytic solution defined for homogeneous models and flat topography, and to modify accordingly the free surface boundary condition for the secondary potential, such that the total potential still satisfies the Poisson equation. The modified singularity removal technique thus remains fully efficient even in the presence of topography, without any additional numerical computation. The modified secondary potential in a homogeneous model is not null in the case of topography as it would be in the classical approach. We implement the approach with the Generalized Finite Difference method. We present a 2.5D inversion example on a simple synthetic data set.