Mathematical model of spontaneous potential well-logging and Its numerical solutions / Математическая модель самопроизвольного потенциального каротажа скважин и ее численные решения

The spontaneous potential well-logging is one of the most common and useful welllogging techniques in petroleum exploitation. Since the earth layer can be regarded as a piecewise uniform medium, and due to electrochemical and other factors, there exists a jump of potential (spontaneous potential difference) on each interface of different layers. The corresponding mathematical model should be the boundary value problem of quasi-harmonic partial differential equations with inhomogeneous interface conditions. In axi-symmetric situation, at the crossing point of multiple interfaces, the compatible condition is usually violated so that it is not possible to get a solution to the boundary value problem in the sense of piecewise H1 space, and we are obliged to consider the solution in piecewise W1,p space (1 <p< 2). This is an important feature of spontaneous potential well-logging, which naturally makes the necessity of introducing Sobolev spaces with fractional power, and seriously increases the difficulty of proving the well-posedness and proposing numerical solution schemes.  In this book, in the axi-symmetric situation we demonstrate the well-posedness of the corresponding mathematical model, and develop three efficient schemes of numerical solution to meet the need of practical computation, the soul of which is to eliminate the non-compatibility (singularity) first, and then to put it into the common solution process of the Finite Element Method. Besides, according to the characteristic of the problem, we set up certain results on the limiting behavior of solutions, which notably simplify the computation in many cases. In the meantime, we offer a series of numerical examples according to the practical petroleum well-logging to reveal some important properties of spontaneous potential well-logging and validate some of the theoretic conclusions. <...>