Generalized Biot's theory and Mandel's problem of multiple‐porosity and multiple‐permeability poroelasticity

Abstract This paper finds in Biot's theory of poroelasticity a complete and consistent extension to the general case of multiple‐porosity and multiple‐permeability, fluid‐saturated, and linearly elastic media. The constitutive stress‐strain relations for a medium identified with this extension are presented, and the coefficient matrix of mechanical properties associated with these relations is derived from the corresponding intrinsic properties of its single‐porosity constituents. The closed form analytical solution to Mandel's problem is upgraded to the case being considered in this study. This problem addresses the transient consolidation of a porous elastic slab of rectangular geometry, when confined from the top and bottom. A numerical example solution for shale with laboratory setup of Mandel's problem is provided. Results are compared for the cases of single‐, double‐, and triple‐porosity solutions.