Extended Gassmann equation with dynamic volumetric strain: Modeling wave dispersion and attenuation of heterogeneous porous rocks

Abstract

Sedimentary rocks are often heterogeneous porous media inherently containing complex distributions of heterogeneities (e.g., fluid patches and cracks). Understanding and modeling their frequency-dependent elastic and adsorption behaviors is of great interest for subsurface rock characterization from multiscale geophysical measurements. The physical parameter of dynamic volumetric strain (DVS) associated with wave-induced fluid flow is proposed to understand the common physics and connections behind known poroelastic models for modeling dispersion behaviors of heterogeneous rocks. We have derived the theoretical formulations of DVS for patchy saturated rock at the mesoscopic scale and cracked porous rock at microscopic grain scales, essentially embodying the wave-induced fluid-pressure relaxation process. By incorporating DVS into the classic Gassmann equation, a simple but practical “dynamic equivalent” modeling approach, the extended Gassmann equation, is developed to characterize the dispersion and attenuation of complex heterogeneous rocks at nonzero frequencies. Using the extended Gassmann equation, the effect of microscopic or mesoscopic heterogeneities with complex distributions on the wave dispersion and attenuation signatures can be captured. Our theoretical framework provides a simple and straightforward analytical methodology to calculate wave dispersion and attenuation in porous rocks with multiple sets of heterogeneities exhibiting complex characteristics. We also demonstrate that, with the appropriate consideration of multiple crack sets and complex fluid patches distribution, the modeling results can better interpret the experimental data sets of dispersion and attenuation for heterogeneous porous rocks.