Stability analysis-based reformulation of wave equations for poro-elastic media saturated with two fluids

Abstract

SUMMARY The stability of partial differential equations determines the properties of their solutions. This study focuses on the stability analysis of the equations describing wave propagation in fluids-saturated porous media. We briefly introduce the stability analysis method for the wave propagation equations and discuss the adverse effects on the solutions. In this way, the first part of this paper is mainly devoted to the analysis of the Tuncay and Corapcioglu's (TC) model, which describes the dynamic behaviour of porous media saturated with two immiscible fluids. It is pointed out that the TC model allows spatially bounded but time-exponentially exploding solutions and may yield unstable numerical results. Based on the deduced unstable factors, we construct a stable equivalent fluid (SEF) model. We rigorously analyse the stability of the SEF model using the energy method. For predicting the influence of saturation on wave velocity, the robustness of this model is preserved due to its consistency with the original TC model. Furthermore, the numerical simulations of the wavefields show that the results of the TC model exponentially increase with time after the initial effective wave signal, which does not occur in the SEF model curves. This indicates the necessity of considering the stability from a mathematical point of view during the construction of physical model. It could be useful to merge the mathematical stability theory with the geophysical wave propagation modelling theory.