Modelling quasi-static poroelastic propagation using an asymptotic approach

An asymptotic technique, valid in the presence of smoothly varying flow properties, allows for the construction of a semi-analytic solution to the governing equations of quasi-static poroelasticity. In this formulation flow properties are variable while mechanical properties are constant within a specified formation. However, both mechanical and flow properties are allowed to vary discontinuously across formation boundaries. The asymptotic analysis determines that two longitudinal modes of propagation are possible: the Biot fast and slow waves. The Biot slow wave propagates as a diffusive disturbance, similar in nature to a pressure disturbance. The Biot fast wave has the characteristics of an elastic wave, and decays much more slowly with distance than does the slow wave. The Biot slow wave is found to generate elastic displacements, in the form of fast waves, as it propagates. This generation of fast waves explains the rapid onset of displacement at a remote observation point due to the injection of fluid into a poroelastic layer. A comparison of the asymptotic solution with analytic and numerical (finite-difference) solutions indicates overall agreement in both homogeneous and heterogeneous media.