High‐ and low‐frequency elastic moduli for a saturated porous/cracked rock‐Differential self‐consistent and poroelastic theories

Abstract

Although P‐ and S‐wave dispersion is known to be important in porous/cracked rocks, theoretical predictions of such dispersions have never been given. We report such calculations and show that the predicted dispersions are high in the case of low aspect ratio cracks [Formula: see text] or high crack density [Formula: see text]. Our calculations are derived from first‐principle computations of the high‐ and low‐frequency elastic moduli of a rock permeated by an isotropic distribution of pores or cracks, dry or saturated, with idealized geometry (spheres or ellipsoids). Henyey and Pomphrey developed a differential self‐consistent model that is shown to be a good approximation. This model is used here, but as it considers cracks with zero thickness, it can not account for fluid content effects. To remove this difficulty, we combine the differential self‐consistent approach with a purely elastic calculation of moduli in two cases: that of spherical pores and that of oblate spheroidal cracks with a nonzero volume. This leads to what we call the “extended differential, self‐consistent model” (EM). When combining these EM results with the Gassmann equation, it is possible to derive and compare the theoretical predictions for high‐ and low‐frequency effective moduli in the case of a saturated rock. Since most laboratory data are ultrasonic measurements and in situ data are obtained at much lower frequencies, this comparison is useful for interpreting seismic data in terms of rock and fluid properties. The predicted dispersions are high, in agreement with previous experimental results. A second comparison is made with the semi‐empirical model of Marion and Nur, which considers the effects of a mixed porosity (round pores and cracks together).

Transient creep, aseismic damage and slow failure in Carrara marble deformed across the brittle‐ductile transition

“…This gives rise to elastic strain incompatibilities and microcracks are needed in order to accommodate plastic deformation. Both for AeWet and AeDry, the evolution of crack density was calculated from P wave velocities evolution (Figure 2) using two well-established Effective Medium theories: the NonInteracting Cracks [Kachanov, 1994] and the Extended Differential Self-Consistent (DEM) [Le Ravalec and Guéguen, 1996] methods. The DEM has been proven to be the most reliable method to calculate the effective elastic properties of cracked rocks [Orlowsky et al, 2003] as it takes into account the existing stress interactions between cracks.…”

Static and Dynamic Moduli of Malm Carbonate: A Poroelastic Correlation

“…Fjaer (1999) and Holt et al (2012) or by defining compressional and shear wave velocities in terms of effective moduli and randomly distributed inclusions (Le Ravalec and Guéguen, 1996). Messop (2012) ascribed the difference between static and dynamic moduli to the time scale of pressure diffusion processes and the applied boundary conditions in a poroelastic medium.…”

“…The drained and undrained dynamic moduli are characterized by time dependent processes (e.g., the pore pressure diffusion and crack closure) within an REV and by hydraulic and mechanical boundary conditions. Therefore, not only heterogeneities and spatial distribution of the microscopic properties (e.g., crack aspect ratio and pore radius) are of importance but also the diffusivity coefficient of the rock under consideration as well as the applied loading rate play an important role (Detournay and Cheng, 1993;Le Ravalec and Guéguen, 1996). That is, how fast the induced pore pressure due to the applied stress wave equilibrated with pressures at the boundaries.…”

“…Several other dissipation mechanisms can be assumed, such as viscoelastic and solid dissipation (Biot, 1962), and different wave propagation theories can be derived for porous and cracked media as reviewed by Le Ravalec and Guéguen (1996) and Sarout (2012). Each characteristic (cutoff) frequency is related to a dissipation mechanism and categorizes the wave propagation theories into different domains.…”

“…Each characteristic (cutoff) frequency is related to a dissipation mechanism and categorizes the wave propagation theories into different domains. In addition, each characteristic frequency determines the transition from a low to a high frequency range, corresponding to relaxed and unrelaxed dynamic moduli, respectively (Le Ravalec and Guéguen, 1996;Mavko et al, 2003) which should be distinguished from drained and undrained moduli.…”