Simulation of Filtration Processes for Inhomogeneous Media and Homogenization

The authors analyze dynamic processes of filtration in porous media and consider periodic porous media formed by a large number of “blocks” with low permeability, separated by a connected system of “faults” with high permeability. Taking into account the structure of such media in the modeling determines the dependence of filtration equations and their coefficients on small parameters characterizing the microscale of the porous medium and permeability of the blocks. The initial–boundary-value problems for nonstationary equations of filtration in such porous media are considered and the homogenized problems that determine the approximate asymptotic solutions to such problems are given. The homogenized problems are formulated as initial–boundary-value problems for integro-differential equations with convolutions. The estimates for the accuracy of the asymptotics and relevant convergence theorems are proved. Statements about solvability and regularity have been established for the problems that are optimal and do not depend on parameters.