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.
We thank the anonymous reviewer for their insightful comments and suggestions. AC thanks to DIUBB 172409 GI/C, DIUBB 183309 4/R, and FAPEI at U. del Bío-Bío, Chile. RL thanks to PY-F1-01MF16 at U. de Magallanes, Chile. PM thanks to Spanish MINECO projects MTM2014-54388-P and MTM2017-83942-P and Conicyt PAI-MEC folio 80150006. MS thanks to Fondecyt 1140676 and BASAL project CMM, U. de Chile and CI2MA, U. de Concepción, and by Conicyt project Anillo ACT1118 (ANANUM).
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