A Numerical Method for an Inverse Problem Arising in Two-Phase Fluid Flow Transport Through a Homogeneous Porous Medium

In this paper we study the inverse problem arising in the model describing the transport of two-phase flow in porous media. We consider some physical assumptions so that the mathematical model (direct problem) is an initial boundary value problem for a parabolic degenerate equation. In the inverse problem we want to determine the coefficients (flux and diffusion functions) of the equation from a set of experimental data for the recovery response. We formulate the inverse problem as a minimization of a suitable cost function and we derive its numerical gradient by means of the sensitivity equation method. We start with the discrete formulation and, assuming that the direct problem is discretized by a finite volume scheme, we obtain the discrete sensitivity equation. Then, with the numerical solutions of the direct problem and the discrete sensitivity equation we calculate the numerical gradient. The conjugate gradient method allows us to find numerical values of the flux and diffusion parameters. Additionally, in order to demonstrate the effectiveness of our method, we present a numerical example for the parameter identification problem.