STRESS SENSITIVITY OF SANDSTONES

Our observations made on dry-sandstone ultrasonic velocity data relate to the variation in velocity (or modulus) with effective stress, and the ability to predict a velocity for a rock under one effective press ure when it is known only under a different effective pressure. We fin d that the sensitivity of elastic moduli, and velocities, to effective hydrostatic stress increases with decreasing porosity. Specifically, we calculate the difference be tween an elastic modulus, M(P-1, phi), of a sample of porosity phi at effective pressure P-1 and the same mod ulus. M(P-2, phi), at effective pressure P-2. If this difference, Delt a M = M(P-1, phi) - M(P-2, phi), is plotted versus porosity for a suit e of samples, then the scatter of Delta M is close to zero as porosity approaches the critical porosity value, and reaches its maximum as po rosity approaches zero. The dependence of this scatter on porosity is close to linear. Critical porosity here is the porosity above which ro ck can exist only as a suspension-between 36% and 40% for sandstones. This stress-sensitivity pattern of grain-supported sandstones (clay co ntent below 0.35) practically does not depend on clay content. In prac tical terms, the uncertainty of determining elastic moduli al a higher effective stress from the measurements at a lower effective stress is small at high porosity and increases with decreasing porosity. We exp lain this effect by using a combination of two heuristic models-the cr itical porosity model and the modified solid model. The former is base d on the observation that the elastic-modulus-versus-porosity relation can be approximated by a straight line that connects two points in th e modulus-porosity plane: the modulus of the solid phase at zero poros ity and zero at critical porosity, The second one reflects the fact th at at constant effective stress, low-porosity sandstones (even with sm all amounts of clay) exhibit large variability in elastic moduli. We a ttribute this variability to compliant cracks that hardly affect poros ity but strongly affect the stiffness. The above qualitative observati on helps to quantitatively constrain P- and S-wave velocities at varyi ng stresses from a single measurement at a fixed stress. We also show that there are distinctive linear relations between Poisson's ratios ( nu) of sandstone measured at two different stresses. For example, in c onsolidated medium-porosity sandstones nu(40) = 0.018 + 0.913 nu(20), where the subscripts indicate hydrostatic stress in MPa. Linear functi ons can also be used to relate the changes (with hydrostatic stress) i n shear moduli to those in compressional moduli. For example, G(40) - G(20) = 0.084 + 0.344 (M(40) - M(20)), where G = rho V-S(2) is shear m odulus and M = rho V-P(2) is compressional modulus. both in GPa. and t he subscripts indicate stress in MPa.