A parameter identification problem for spontaneous potential logging in heterogeneous formation

Abstract

In this paper, we want to estimate the true resistivity of permeable bed using the spontaneous potential on the measuring electrode in heterogeneous formation, where the resistivity of the objective layer varies continuously as a function of position. The problem is a generalization of Peng's work for the traditional three-parameter invasion model, where the resistivity of formation is piecewise constant. The mathematical model of the problem is described by a variational problem of elliptic type, where the formation resistivity appears as a nonlinear coefficient. We study an extended problem and then apply its results to prove the existence of solutions to the parameter identification problem arising from SP logging. Finally, a sufficient condition is given to ensure the uniqueness of solutions.