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.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 41204082
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11126058
Funding source: Research Fund for the Doctoral Program of Higher Education of China
Award Identifier / Grant number: 20120162120036
Funding source: Special Foundation of China Postdoctoral Science
Award Identifier / Grant number: 2013T60781
Funding source: Central South University
Award Identifier / Grant number: Mathematics and Interdisciplinary Sciences Project
The authors would like to thank Professor Yongji Tan for his helpful suggestions and comments.
© 2014 by Walter de Gruyter Berlin/Boston
