Solution of the fundamental problem in resistivity logging with a hybrid method

Abstract

The fundamental resistivity logging problem of a resistivity tool in the presence of both vertical and horizontal boundaries is solved with a hybrid method. The hybrid method combines the mode concept in wave-guide theory together with the finiteelement method. In the mathematical formulation, the horizontal boundaries are used to separate the geometry of the problem into different regions. In each region, the waveguide modes are obtained through the solution of an equivalent variational problem. The solutions are calculated by a one-dimensional finite-element method. The vertical boundaries are taken into account in these calculations. The orthonormality of modes in each region allows a series representation of the potential in the regions. Boundary conditions at horizontal bed boundaries then couple the modes between different regions and enable the solutions for the potential to be expressed in terms of reflection and transmission matrices of modes. The source excitation determines the amplitudes of the modes. The results of the hybrid method are in excellent agreement with those of the integral transform solution. Numerical results of the apparent resistivity are illustrated as a function of formation properties. The effects of an invaded zone are also examined by considering radial inhomogeneous profiles in the formation. The results of the hybrid method are numerically efficient because it reduces the two-dimensional finite-element problem into a one-dimensional one. It also provides a physical interpretation of the solution in terms of modes.