Gassmann Consistency for Different Inclusion‐Based Effective Medium Theories: Implications for Elastic Interactions and Poroelasticity

Abstract

By evaluating the consistency of the Gassmann theory with various inclusion‐based effective medium theories, we investigate the impact of elastic interactions between ellipsoidal pores on the poroelasticity. To rule out any factors that can violate the Gassmann condition, other than elastic interactions, we first construct idealized models that contain only a single set of isolated, identical, and vertically aligned ellipsoidal pores. The numerical simulation suggests that the periodic distribution of ellipsoidal pores generate uniform pore pressure distribution, whereas random distribution of ellipsoidal pores generates heterogeneous pore pressure distributions. Then we analyze the precise conditions under which the underlying Gassmann relationship is valid for various inclusion‐based models. The results reveal the following: (1) Noninteracting effective medium theories are always consistent with the Gassmann prediction, simply because the elastic interactions are ignored. (2) The elastic interactions between randomly distributed pores cause heterogeneous pore pressure that violates the essential requirement of the Gassmann theory. The differential effective medium and self‐consistent approximation theories corresponding to this model thus are inconsistent with the Gassmann prediction. (3) The elastic interactions between periodically distributed pores cause uniform pore pressure; therefore, the Gassmann condition is fully satisfied. The T‐matrix approach explicitly takes into account such elastic interactions and thus is consistent with the Gassmann theory. It is interesting to notice that on top of other well‐known common types of heterogeneities, like pore structure or fluid heterogeneities, the distribution of pores and its associated elastic interactions can be a separate source of heterogeneity, and this makes Gassmann equations not valid anymore.

Effect of stress interactions on anisotropic P‐SV‐wave dispersion and attenuation for closely spaced cracks in saturated porous media

“…Theoretical and numerical predictions of effective compressibility for rocks with finite concentrations of cracks have been extensively studied in rock physics, referring to Mavko et al (2009) for a comprehensive review. One of the most popular approaches to describe the effective properties of cracked model is the non-interaction approximation (NIA) (Grechka and Kachanov, 2006a;Zhao et al 2020), taking the effective compliance tensor as the sum of the background tensor and the contribution of individual cracks. The NIA works well under the dilute assumption of crack densities where cracks are located at such distances that their interactions can be ignored, thus providing a reference to estimate the effect of stress interactions.…”

Estimation of shear‐wave velocities in unconventional shale reservoirs

“…Zhao et al . (2020) demonstrate that distribution of pore and cracks is essential to satisfy the pore pressure equilibrium condition and the underlying Gassmann equation, rather than the pore connectivity. Grechka (2009) found that aligned disconnected non‐interacting low aspect ratio pores do indeed conform to Gassmann's equations.…”

Experimental study on frequency‐dependent elastic properties of weakly consolidated marine sandstone: effects of partial saturation

“…9a). This further confirms that the partially saturated rock at the measured frequency band (2–500 Hz) is prone to be in a relaxed or partially relaxed status, where the pore pressure is approximately equilibrated (Batzle et al ., 2006; Zhao et al ., 2020). Moreover, ultrasonic velocities at varying saturation levels are systematically higher than the Gassmann–Wood predictions, demonstrating that the partially saturated rock is in the unrelaxed status, where the two immiscible fluids tend to be isolated from each other.…”