Multiscale Numerical Simulation for Assessing Effective Electrical Conductivity Estimations of Geological Medium

The paper considers methods for estimating the effective electrical conductivity of a geologic medium with an internal structure represented as a set of layer-by-layer tomographic images. Based on this type of input data, a discrete hierarchical mesh model of the sample is constructed. It is used to describe the internal singularities of the investigated medium with significant accuracy. To obtain the effective electrical conductivity of the geologic medium, a direct problem of electric field potential distribution in the sample volume is solved numerically. Since the computational domain is defined as a hierarchy of mesh partitions, a multiscale modification of the scalar finite element method is applied. The averaging procedure of the properties is based on Ohm's law. The possibility of step-by-step averaging of the electrical properties of a rock sample is also investigated. To obtain the electrical effective characteristic of the entire sample, the effective properties of individual macroelements in the mesh partitioning are determined and further averaged using the algorithms proposed in the paper. The variation of the effective electrical conductivity over certain sub-domains of the sample is given.