Extended Gassmann Equation with Dynamic Volumetric Strain: Modeling Wave Dispersion and Attenuation of Heterogenous Porous Rocks

Sedimentary rocks are often heterogeneous porous media inherently containing complex distributions of heterogeneities (e.g., fluid patches, cracks). Understanding and modeling their frequency-dependent elastic and adsorption behaviors is of great interest for subsurface rock characterization from multi-scale 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 derive the theoretical formulations of DVS for patchy saturated rock at mesoscopic scale and cracked porous rock at microscopic grain scales, essentially embodying the wave-induced fluid pressure relaxation process. By incorporating the DVS into the classical Gassmann equation, a simple but practical “dynamic equivalent” modeling approach, extended Gassmann equation, is developed to characterize the dispersion and attenuation of complex heterogeneous rocks at non-zero 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. The proposed 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 fluids patches distribution, the modeling results can better interpret the experimental data sets of dispersion and attenuation for heterogeneous porous rocks.