Squirt flow in partially saturated cracks: a simple analytical model

Abstract

Obtaining the seismic response of rocks containing cracks whose scales are much smaller than the prevailing wavelengths is a classic and important problem in rock physics. Seminal analytical models yield the seismic signatures of cracked rocks saturated with a single fluid phase. However, in a wide variety of practically relevant scenarios, cracks may be partially saturated with multiple immiscible fluids of contrasting compressibilities, such as gas and water. When a passing seismic wave deforms the medium, fluid pressure gradients arise within such partially saturated cracks, which, in turn, tend to relax through a process commonly known as squirt flow. The corresponding viscous dissipation may greatly affect the seismic amplitudes and velocities, as well as the anisotropic behaviour of the medium. To date, extensions of classical analytical models to include squirt flow occurring within isolated partially saturated cracks remain limited either in the saturation or in the frequency range. In this work, we present a simple analytical model to compute the seismic response of rocks containing partially saturated aligned cracks accounting for squirt flow effects. First, we solve the linearized Navier-Stokes equations within a partially saturated penny-shaped crack subjected to an oscillatory strain. Then, we obtain a closed analytical expression for a complex-valued frequency-dependent effective fluid bulk modulus which accounts for the stiffness variations of each crack due to squirt flow. Using classic effective medium models, together with such an effective saturating fluid, we retrieve the effective compliance matrix of the probed partially saturated cracked rock. The proposed analytical solution is validated by comparison with corresponding 3-D numerical simulations and existing analytical models.