Thermodynamically admissible derivation of Biot's poroelastic equations and Gassmann's equations from conservation laws

Abstract. Gassmann's equations, formulated several decades ago, remain a cornerstone in geophysics due to their perceived exactness. However, a concise and rigorous derivation rooted in thermodynamic principles and conservation laws has been missing from the literature. Additionally, recent studies have pointed out potential logical inconsistencies in the original formulation. This paper introduces a derivation of Gassmann’s equations, anchored in fundamental conservation laws and constitutive relations, ensuring their thermodynamic consistency. Alongside this, we extend the discussion to include Biot's poroelastic equations, which are widely used to describe the coupled behavior of fluid-saturated porous media under mechanical deformation. By demonstrating that Gassmann's equations are a specific case within the broader framework of Biot’s theory, we further validate their relevance and applicability in geophysical contexts. Given the numerous independent rederivations and numerical verifications of these equations for diverse pore geometries, we affirm their robustness, provided the underlying assumptions are respected. To facilitate reproducibility and further exploration, symbolic Maple routines are provided for the derivations presented in this study.