Rock anelasticity due to patchy saturation and fabric heterogeneity: A double double-porosity model of wave propagation

Heterogeneity of rock's fabric can induce heterogeneous distribution of immiscible fluids in natural reservoirs, since the lithological variations (mainly permeability) may affect fluid migration in geological time scales, resulting in patchy saturation of fluids. Therefore, fabric and saturation inhomogeneities both affect wave propagation. To model the wave effects (attenuation and velocity dispersion), we introduce a double double-porosity model, where pores saturated with two different fluids overlap with pores having dissimilar compressibilities. The governing equations are derived by using Hamilton's principle based on the potential energy, kinetic energy, and dissipation functions, and the stiffness coefficients are determined by gedanken experiments, yielding one fast P wave and four slow Biot waves. Three examples are given, namely, muddy siltstones, clean dolomites, and tight sandstones, where fabric heterogeneities at three different spatial scales are analyzed in comparison with experimental data. In muddy siltstones, where intrapore clay and intergranular pores constitute a submicroscopic double-porosity structure, wave anelasticity mainly occurs in the frequency range (10(4)-10(7)Hz), while in pure dolomites with microscopic heterogeneity of grain contacts and tight sandstones with mesoscopic heterogeneity of less consolidated sands, it occurs at 10(3)-10(7)Hz and 10(1)-10(3)Hz (seismic band), respectively. The predicted maximum quality factor of the fast compressional wave for the sandstone is the lowest (approximately 8), and that of the dolomite is the highest. The results of the diffusive slow waves are affected by the strong friction effects between solids and fluids. The model describes wave propagation in patchy-saturated rocks with fabric heterogeneity at different scales, and the relevant theoretical predictions agree well with the experimental data in fully and partially saturated rocks. Plain Language Summary Wave-induced local fluid flow is widely accepted as the main cause of compressional wave anelasticity (velocity dispersion and attenuation) in fluid-saturated rocks. The two intrinsic causes of wave-induced local fluid flow, rock fabric and fluid patchy saturation, were discussed and analyzed frequently in the literature. However, their combination in one single theory results in much complex geometries, and no poroelasticity theory has been presented so far. In natural tight rocks, the two types of heterogeneities (rock fabric and fluid distribution) generally coexist, since low permeability and capillary forces hinder fluid migration and induce patchy saturation. The exact theoretical equations for wave propagation in this type of medium will provide an important basis for further rock physics studies and field applications of seismic exploration. New theoretical equations are derived by proposing a double double-porosity model and incorporating the two types of heterogeneities into the same set of equations under the framework of Biot poroelasticity. We compare the theory with three sets of experimental data, which contain heterogeneities at the different scales. The model successfully describes wave propagation in patchy-saturated rocks with fabric heterogeneity at different scales, allowing for the most comprehensive description of compressional wave propagation and dissipation characteristics in highly complex in situ reservoir rocks.