Rocks response to stress is governed by its microstructure and constituent minerals. This is manifested in pressure dependence of elastic waves velocity, i.e. velocity behavior as a function of effective stress. In the lower pressure regime, the increase in velocity with increasing pressure is very steep and nonlinear. This is due to the closure of microcracks of low aspect ratios, which dramatically affects the elastic properties of rock and thereby the velocities. In the higher pressure regime, the increase in velocity (with increasing effective pressure) is moderate and somewhat linear in nature, as fewer numbers of cracks with high aspect ratios get closed. Several efforts have been made in the past to model the pressure dependence of velocity. However, there is a little attempt to relate this pressure dependence with petrophysical parameters like rock composition and microstructure.
In this study, we have tried to correlate the pressure dependence of velocity with rock microstructure and composition. A simple functional form was developed and fitted to the compressional and shear wave velocities for 145 different rock samples (both saturated and unsaturated) from six different formations. where V is velocity in km/sec and P is effective pressure in MPa. The function allowed extrapolation/interpolation (at higher/intermediate pressure) of velocities to within ±4%.
Velocities were inverted using differential effective medium model, Cheng-Toksoz (1979), to generate pore crack aspect ratio concentration spectrum of rocks. Thin section and SEM images were taken for direct evidence of rock microstructure. The exponential coefficient ‘D’ was found to be related with rock microstructure. Samples with low values of exponential coefficient were low porosity rocks that had mainly thin cracks and a small fraction of spherical pores. These samples have relatively higher concentration of clays. On the other hand, rocks with high values of exponential coefficient are high porosity rocks that have higher fraction of high aspect ratio pores and comparatively little concentrations of thin cracks. Clay content of these samples is relatively low. The other coefficients of the fitting function showed a statistical correlation with porosity, but not with total clay content. This is probably because we have used total clay content and have not differentiated among laminar, structural or dispersed clays.
In addition, a multivariate nonlinear regression analysis was done to analyze the effect of clay content, effective stress, and porosity on elastic wave velocities in sandstones. Laboratory measurements on 76 different water-saturated sandstones from three different locations (Oklahoma, California, and Ecuador), with porosities ranging from 1 to 20 percent and clay content varying from 0 to 34 percent, were statistically analyzed. We have found that velocity as a function of porosity, clay content, and effective pressure can be modeled as where P is effective pressure in MPa. This equation is in agreement with observed linear dependence of velocity on clay and porosity and non linear dependence of velocity on pressure. Han et al. (1986) data was also fitted using this functional form and reduction in data variance is 96 percent for V and 90 percent for V.
Three different formations used for this study showed different sensitivities to porosity and clay content. These differences were analyzed and related to depositional environment and presence of clay within the rock.
