Method of finite-element discretization in heterogeneous and highly conductive grid cells

A method for simulating a conductivity measured by a resistivity tool disposed in a borehole penetrating a formation, the method including: constructing a main grid extending from the borehole into the formation; constructing an auxiliary grid that is local to the main grid and fitted to a boundary between a first region having a first conductivity and a second region having a second conductivity; discretizing a problem operator on the auxiliary grid applying Maxwell's Equations to a volume represented by the auxiliary local grid, the Maxwell's Equations including a term representing the conductivity of a material in the volume; calculating one-dimensional functions for points on the auxiliary grid; constructing basic functions from the calculated functions; discretizing by the basic functions for points on the main grid; solving the resulting set of equations to obtain the simulated measured conductivity for the resistivity tool; and providing the simulated measured conductivity as output.