Motivated by an urgent need in petroleum exploitation, both the mathematical modeling and the mathematical and numerical method of resolution for several kinds of electrical well-loggings such as resistivity well-logging (by direct current), spontaneous potential (SP) well-logging and complex resistivity well-logging (by alternating current) etc. are investigated in this talk. This study leads to some new concepts and problems in applied partial differential equations, for instance, boundary value problems with equivalued surface, boundary homogenization, necessity of using Sobolev spaces with fractional power and Maxwell-Ohm system in a piecewise homogeneous medium etc.