Revisiting Gassmann-type relationships within Biot poroelastic theory

Abstract. Gassmann's equations have long served as a cornerstone of geophysics and rock physics, widely regarded as exact within their domain of applicability. However, recent studies have questioned their validity, arguing that Gassmann's derivation involves a logical error and that an additional solid modulus is needed even for monomineralic materials. In this work, we present a general derivation of the extended Biot poroelasticity equations, grounded in conservation laws and classical irreversible thermodynamics (CIT). We show that the formulations of Gassmann (1951), Brown and Korringa (1975), Detournay and Cheng (1993), and Rice and Cleary (1976) emerge as special cases of this unified framework. While previous studies have analyzed the thermodynamic admissibility of standard Biot and Gassmann models, we extend this analysis to the more general theory by explicitly incorporating the off-diagonal terms arising from the second partial derivatives (Hessian) of internal energy. A key finding is that Gassmann's self-similarity condition – that porosity remains unchanged under equal changes in fluid and total pressure – is a sufficient but unnecessary condition for the derivation of Gassmann-type relationship between undrained and drained bulk moduli. It holds if and only if the matrix of the second partial derivatives of internal energy is diagonal. When the off-diagonal terms in this matrix are retained, a generalized form of Gassmann's equations is required, which we derive. To promote transparency and support further research, we provide symbolic Maple routines with thermodynamic consistency checks, ensuring full reproducibility and accessibility.