Applicability of Gassmann’s equations to cracked media

Abstract

Gassmann’s equations are a cornerstone of rock physics, widely used for fluid substitution modeling and reservoir characterization. However, recent studies have raised concerns about their applicability to cracked media. To address this issue, we carried out a 3D numerical investigation together with an analytical analysis. Using a 3D finite-element approach, we solved the coupled system incorporating constitutive stress-strain relations for the solid elastic phase and a quasistatic, linearized compressible formulation of the Navier-Stokes equations for the fluid phase. Our numerical results, together with comparisons based on effective medium theory (EMT) and its integration with Gassmann's equations, confirm the validity of Gassmann’s equations when applied to cracked media. Moreover, we show that Gassmann’s equations are fully consistent with EMT when the latter is formulated using the complete Eshelby tensor, whereas simplified EMT formulations, such as those employing penny-shaped approximations, lead to inconsistencies. These findings reinforce the theoretical foundation of Gassmann’s equations and clarify their role within the broader EMT framework.