Fast modeling of microspherically focused log in a horizontally layered formation

A microspherically focused log is a focused microresistivity device used for evaluation of the electrical property of the subsurface rock formation for oil and gas exploration. The electrodes are mounted on a flexible rubber pad that is applied against the borehole wall to be in close proximity to the formation. Unlike other nonpad-focused electrode tools that are centered in the borehole such as Dual-Laterolog, the modeling of the microspherically focused log is much more complex and requires three-dimensional (3-D) code in general because the electric field is no longer axially symmetric. However, in a horizontally layered medium, the axial symmetry of the earth formation makes the problem two-and-a-half-dimensional (2.5-D), which is much simpler. In this paper, the authors apply the semi-analytic method to tackle this 2.5-D problem and establish a fast-modeling algorithm. First, they expand the Green's function as a Fourier series in order to transform the 2.5-D problem into a sequence of the axially symmetric (2-D) problems. The semi-analytic method is used to obtain the expression of the Green's function. Applying the focusing conditions of the tool and the solution of the Green's function, we then establish the boundary integral equation with respect to the current density, currents, and potentials on the electrode surfaces, and study its numerical solutions. Then, they compare results computed from the semi-analytical method with those from the finite element method (FEM) to verify its accuracy. Finally, they study the response characteristics of the tool in several different environments by the semi-analytical method.