A logical error in Gassmann poroelasticity

Abstract

AbstractThe well‐known ‘Biot–Gassmann’ formula for the undrained compressibility of porous rock is in error. The error was introduced by Gassmann (implicitly revising earlier work by Biot) and results in an expression which is simpler than Biot's corresponding expression, containing only one material parameter (KS) in place of Biot's two (H and R). The error occurred when Gassmann applied a hydraulically open theorem to a hydraulically closed system, thus violating the assumptions of the undrained compressibility. Similar errors have marred the many subsequent derivations of Gassmann's result. Biot's (prior) correct formulation was subsequently rederived exactly (in different notation) by Brown and Korringa, and separately by Rice and Cleary. The inconsistency between Gassmann and these three analyses is caused by Gassmann's logical error, rather than by the issue of solid micro‐homogeneity. The additional parameter may be experimentally determined via the ‘Skempton B‐coefficient’ (the ratio of fluid pressure to external pressure in an undrained quasi‐static compression test). This correction resolves a long‐established inconsistency between Gassmann's result and effective‐medium theory. A new generation of experimentation is required to determine the systematics of the magnitude of the correction to the Gassmann result.