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

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 problemis 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.