Modeling and Homogenization of Filtration Processes in Periodic Media with Sources

Periodic media formed by a large number of cells, which can be inhomogeneous and defines a multiscale inhomogeneous porous media, will be discussed. Filtration problems with sources are considered in the porous media, where sources are modeled by delta-functions. Depending on coefficient magnitudes compared to the microscale parameter, three filtering modes are identified and described for the filtration problems. The modes describe, respectively, weak transfer and filtration, homogenized transfer and weak filtration, homogenized filtration and transfer. This classifies and simulates probably possible modes, for example, of modern oil production in problem wells, at least for the model inhomogeneous porous media under consideration. In all these cases, asymptotic approximations of solutions to such problems are obtained. Estimates of the accuracy of such asymptotic approximations are also presented. The estimates are necessary for computer simulation for the filtration problems with suitable accuracy. In turn, this makes it possible, instead of a direct (for example, numerical) solution to a rather complex problem for small microscales, to solve a simpler homogenized problem and receive a description of processes with guaranteed accuracy.