In this paper, the algorithms for the numerical determination of electrophysical properties of drilling mud invasion zones in the formation during well drilling are presented. This problem is considered as one of the stages in solving the multiphysics problem of the electromagnetic monitoring of the drilling fluid invasion. The purpose of the study is to develop a technique to obtain the effective values of electrical conductivity based on a set of numerically determined saturations for fluid components. The electrical characteristics of fluid components in separate, formation layers' porosity and the rock matrix properties are known. These effective electrical conductivities are then used in numerical modeling of the electromagnetic signal responses in the borehole to identify the colmatation processes in the near-well bottom zone. The finite element method and its non-conformal modifications are used in all steps of numerical simulation both for fluid dynamics and electromagnetic problems, including the Discontinuous Galerkin method and the heterogeneous multiscale finite element method.