Theory and analytical solution to Cryer's problem of N-porosity and N-permeability poroelasticity

Self-consistent extension of linear poroelasticity (Biot 1941) to overlapping systems or scales of porosity and permeability within fluid-saturated, elastic materials is presented. An arbitrary number of N porosity systems with linear relations for fluid flow within and in between the porosities is considered. The theory is applied to Cryer's problem of poroelasticity. The problem considers sudden confinement of a spherical sample of porous material while the pore fluid is allowed to drain from the surface boundary. The closed-form solution to Cryer's problem is developed in Laplace transform domain for the general case where N porosity systems with full inter-porosity fluid exchange properties are present in the spherical sample. Numerical results are retrieved in the time domain for a hierarchical class of multiple-porosity models describing the commonly observed porosity systems of organic-rich shale. The stress and pore fluid pressure solutions pertaining to the cases of up to five coexisting porosity systems are demonstrated.