Dual-Porosity Dual-Permeability Poroelastodynamics Analytical Solutions for Mandel’s Problem

Analytical solutions to the classical Mandel’s problem play an important role in understanding Biot’s theory of poroelasticity and validating geomechanics numerical algorithms. In this paper, existing quasi-static poroelastic solutions to this problem are extended to the dual-porosity dual-permeability poroelastodynamics solution which considers inertial effects for a naturally fractured and fluid-saturated sample subjected to a harmonic excitation. The solution can generate the associated elastodynamics and poroelastodynamics solutions as special cases. A naturally fractured Ohio sandstone is selected to demonstrate the newly derived solution. The elastodynamics, poroelastodynamics, and dual-porosity poroelastodynamics solutions are compared to illustrate the effects of fluid–solid coupling and the natural fractures. The rock sample behaves in drained condition at low frequencies when the oscillation has insignificant impedance effects on fluid movement. Compared to the other two solutions, the dual-porosity solution predicts the largest amplitude of displacement at low frequencies when the response is predominantly controlled by the stiffness. The Mandel–Cryer effect is observed in both rock matrix and fractures and occurs at a lower frequency in rock matrix because it is easier to build up pore pressure in lower-permeability rock matrix. At high frequencies, pore fluids are trapped and the rock sample behaves in an undrained state. At the resonance frequencies, the elastodynamics solution provides the largest amplitude of displacement, followed by the poroelastodynamics and dual-porosity poroelastodynamics solution. This is because of the dissipation caused by the presence of both fluid and fractures.