An Infinite Dimensional Technique To Estimate Oil-Water Saturation Functions.

Abstract

We introduce an algorithm to solve an inverse problem for a non-linear system of partial
differential equations, which can be used to estimate oil water displacement functions. The direct model
is non-linear because the sought for parameter is a function of the solution of the system of equations.
Traditionally, the estimation of functions requires the election of a fitting parametric model and thus the
optimum curve depends on that election. We develop an algorithm that does not require a parametric
model and thus provides a more objective fit. 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. We present the partial differential equations that are used to compute the Fréchet
derivative. The resulting method has shown convergence in numerical tests, and because of its general
theoretical formulation has the potential to be extended to solve more complex problems. The main
contribution of this work is the formulation and application of the algorithm described above to estimate
non-linear parameters in functional spaces. This algorithm obtains the sought-after parameters without
the imposition of a priori parametric models. Though the use of such models is currently the common
practice among field engineers, different models yield different results and there is no objective criterion
to choose among them.