An inverse problem arising from the displacement of oil by water in porous media

The purpose of the present work is to analyze an inverse problem whose forward model is a non-linear hyperbolic–elliptic system. We present theoretical results on the inverse problem and prove them. The forward model arises from the displacement of oil by water in porous media. The inverse problem consists in the estimation of oil–water relative permeability functions, which are coefficients of the forward model differential equations. We also formulate an algorithm on functional spaces prior to any discretization. The algorithm is based on the linearization of the parameter-to-output mapping. This technique is new in the case where the parameter depends on the solution of the forward model. It has the advantage of laying aside parametric models. The algorithm is iterative and is of prediction–correction type. We use synthetic data to test the new algorithm.