An inverse problem in petroleum exploitation

The spontaneous potential well logging is an important technique in petroleum exploitation. The goal of this method is to identify the resistivity of the objective layer when the potential on the electrode is measured. The mathematical model of this problem is described by a variational problem for an elliptic equation with various boundary conditions, in which the resistivity of the objective layer appears as a coefficient. The identification of this parameter is a nonlinear inverse problem. We first solve an extended inverse problem and then apply its result to prove the existence of solutions to the model of the SP well logging. Furthermore, we give a general sufficient condition on the given data to ensure the uniqueness of solutions.