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 (≤10−3) or high crack density (≥10−1). 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).
