Two Types of Transient Phenomena and Full Relaxation Macroscale Model for Single Phase Flow through Double Porosity Media

Darcy's flow of a weakly compressible fluid through double porosity media is studied in the framework of the homogenization theory. In previous papers, various classes of single-phase flow have been detected with various determination of the effective permeability tensor for each class. In this paper, the full model including transient phenomena is developed, where the macroscale momentum balance equation represents a modification of Darcy's law with a nonequilibrium term. The effective permeability tensor appears to be nonstationary and is changing during the system evolution in time. Three relaxation times characterize the transient transformations of each component of the macroscale flow velocity.This effect is superposed with the second relaxation phenomenon caused by the exchange flow between dense blocks and the highly conductive matrix. The relaxation times for the effective permeability and for the exchange flow are shown to have different orders.All relaxation parameters are explicitly determined through solutions of cell problems.