Poro-acoustoelasticity with compliant pores for fluid-saturated rocks

Abstract

Stress-induced influences on elastic wave velocities include elastic and inelastic behaviors. In general, deformation of rocks is primarily linear elastic for small-magnitude stresses; such behavior can be predicted by the conventional poro-acoustoelasticity theory. On the contrary, large-magnitude stresses induce linear elastic deformation in stiff pores and rock grains and nonlinear elastic deformation in compliant pores. Conventional poro-acoustoelasticity combines the kinetic and strain energy functions via the Lagrange equation. This theory reveals the strain energy transformation of the stiff pores and rock grains for velocity variation. The dual-porosity model uses a semiempirical equation to express the influence of the nonlinear elastic deformation of compliant pores on velocity variations; however, this model does not include the strain energy transformation of compliant pores. We incorporate the dual-porosity model into the conventional poro-acoustoelasticity theory to account for linear and nonlinear elastic deformations through the strain energy transformation of rock grains, stiff pores, and compliant pores. We determine that the work of the loading stress is transformed into two parts: the strain energy for the linear elastic deformation of rock grains and stiff pores and the nonlinear elastic deformation of compliant pores. On applying this theory to ultrasonic measurements under different differential pressures for a saturated sandstone sample, we see that the resulting solution of stress-associated velocity variations is more precise than that obtained using the conventional poro-acoustoelasticity theory, especially in the low-effective-pressure regime.