The estimation of oil water displacement functions

we introduce an algorithm to solve an inverse problem for a non-linear hyperbolic partial differential equation. it can be used to estimate the oil-fractional flow function from the buckley-leverett equation. the direct model is non-linear: the sought for parameter is a function of the solution of the equation. traditionally, the estimation of functions requires the election of a fitting parametric model. the algorithm that we develop does not require a predetermined parameter model. therefore, the estimation problem is carried out over a set of parameters which are functions. the parameter is inferred from measurements of saturation at different spatial points as a function of time. the estimation procedure is carried out linearizing the solution of the direct model with respect to the parameter and then computing the least-squares solution in functional spaces. the sensitivity equations are derived. we test the algorithm with several numerical experiments.