[1] We derive the equations of motion of a double-porosity medium based on Biot's theory of poroelasticity and on a generalization of Rayleigh's theory of fluid collapse to the porous case. Spherical inclusions are imbedded in an unbounded host medium having different porosity, permeability, and compressibility. Wave propagation induces local fluid flow between the inclusions and the host medium because of their dissimilar compressibilities. Following Biot's approach, Lagrange's equations are obtained on the basis of the strain and kinetic energies. In particular, the kinetic energy and the dissipation function associated with the local fluid flow motion are described by a generalization of Rayleigh's theory of liquid collapse of a spherical cavity. We obtain explicit expressions of the six stiffnesses and five density coefficients involved in the equations of motion by performing "gedanken" experiments. A plane wave analysis yields four wave modes, namely, the fast P and S waves and two slow P waves. As an example, we consider a sandstone and compute the phase velocity and quality factor as a function of frequency, which illustrate the effects of the mesoscopic loss mechanism due to wave-induced fluid flow.Citation: Ba, J., J. M. Carcione, and J. X. Nie (2011), Biot-Rayleigh theory of wave propagation in double-porosity media,
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 (104–107 Hz), while in pure dolomites with microscopic heterogeneity of grain contacts and tight sandstones with mesoscopic heterogeneity of less consolidated sands, it occurs at 103–107 Hz and 101–103 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.
The standard Biot‐Gassmann theory of poroelasticity fails to explain strong compressional wave velocity dispersion experimentally observed in 12 tight siltstone with clay‐filled pores. In order to analyze and understand the results, we developed a new double‐porosity model of clay squirt flow where wave‐induced local fluid flow occurs between the micropores in clay aggregates and intergranular macropores. The model is validated based on the combined study of ultrasonic experiments on specimens at different saturation conditions and theoretical predictions. The presence of a sub‐pore‐scale structure of clay micropores contained in intergranular macropores, where the fluid does not have enough time to achieve mechanical equilibrium at ultrasonic frequencies and thus stiffens the rock matrix, provides a suitable explanation of the experimental data. Moreover, the model provides a new bound for estimating the compressional wave velocity of tight rocks saturated with two immiscible liquids. The theoretical predictions indicate that the velocity variation between gas‐ and liquid‐saturated specimens is predominantly induced by the clay squirt stiffening effect on the rock matrix and not by fluid substitution. The effect contributes more than 90% to the variation in the porosity range of 0–5%. Thus, clay squirt flow dominates the relationships between compressional wave velocity and pore fluid in tight rocks.
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