Effects of Stress, Pore Pressure, and Pore Fluids on Bulk Strain, Velocity, and Permeability in Rocks

The static strain response to solids to combinations of confining stress and pore pressure is investigated both theoretically and experimentally. The theoretical analysis is a synopsis of linear elastic relations for porous media taken mainly from Biot (1941), Biot and Willis (1956), and Geertsma (1957). Experimental strain measurements on a suite of rocks as a function of hydrostatic confining stress and pore pressure are presented. Equilibrium strain at any combination of confining stress and pore pressure is predicted on the basis of 1) the zero pore pressure or drained, jacketed, stress-strain relation, and 2) the unjacketed strain relation. Although an "effective stress law" for bulk strain (Nur and Byerlee, 1971; Garg and Nur, 1972; Carroll,1979) can be formulated by associating terms in the basic elasticity relations, the prediction of strain proceeds more directly from the aforementioned two sets of measurements. Unjacketed strain measurements with a confining pressure fluid are emphasized as a means of directly measuring the intrinsic strains of aggregate minerals in rocks and for calculating the intrinsic bulk moduli. A technique is outlined for experimentally obtaining pore volume or porosity as a function of confining pressure from unjacketed and jacketed strain data. An argument is made, based on linear elasticity analysis for strain response, that difference between external stress and internal pore pressure, often called the effective stress or effective pressure if external stresses are uniform, predicts many physical properties exclusive of bulk strain because of 1) the large intrinsic moduli of minerals, and 2) the definition of stress as a force per unit area is maintained during deformation because of the small strains normally encountered in consolidated rocks and sediments. Jacketed and unjacketed stress-strain data, jacketed and unjacketed bulk moduli, and porosity calculations, all as a function of confining pressure, are presented for a suite of igneous, metamorphic, and sedimentary rocks.

The permeability of rocks to combinations of confining pressure and pore pressure is investigated. Simple elasticity considerations, which exclude irreversible effects such as hysteresis, indicate that differential pressure, the difference between confining and pore pressure, should determine the permeability in homogeneous porous materials in which the solid phase has a bulk modulus of typical rock-forming minerals. Experimental measurements of permeability for a sample of Chelmsford granite at two different pore pressures support this conclusion since the differential pressure determines permeability to within experimental error. Also investigated is the anomalous high pore pressure dependence of permeability in Berea sandstone as previously observed by Zoback and Byerlee (1975) with a light machine oil and by Walls and Nur (1979) with distilled water. The effect is observed with a 50,000 ppm NaCl solution, but only on the first cycle of variation in pore and confining pressures. Further cycles result in reduction and perhaps disappearance of the effect. Permeability measurements with distilled water as the pore fluid indicate substantial reduction due to mechanical blockage of fluid pathways as measured and interpreted by Khilar and Folger (1983). Interpretation of permeability measurements and pore pressure effects are complicated by the pressure of natural hydrocarbons in Berea sandstone and by its water sensitivity.

The dependence of P- and S-wave velocities on the combined influence of pore and confining pressure is investigated. Simple considerations of rock elasticity similar to those in permeability analysis indicate that differential pressure, the difference between confining pressure and pore pressure, should determine velocity in the absence of effects due to pore fluid properties and hysteresis. Measured P- and S-wave velocities in granites and sandstones, and a limestone were made as a function of pore and confining pressures systematically varied as to avoid hysteresis. Pore fluids used include nitrogen gas, benzene, and water. Experimental velocities indicate that the variation pore fluid bulk modulus and density at different pore pressures often cause small (several percent or less) but systematic deviations of velocities from being determined by the differential pressure. In measurements where variation in pore fluid properties are not important, as in fluid-saturated S-wave velocities or fluid saturated P-wave velocities in one sandstone measured with water, or where the variation in pore fluid properties can easily be accounted for, as in the nitrogen- saturated S-wave velocities in the porous sedimentary rocks, there is evidence that the shear and bulk moduli are being determined by differential pressure. In general, however, the determination of velocities by differential pressure can only be considered an approximation, although probably a good one in water-saturated rocks.

Measurements of ultrasonic (1Mhz) P- and S-wave velocities for a suite of rocks including sandstones, limestone, granitic rocks, and a metamorphic dolomite are presented. Measurements were made on vacuum-dry (20mm Hg) and benzene-saturated (100 bars pore pressure) samples for all of the rocks, and also water-saturated (100 bars pore pressure) for most of the samples. Measurements were made at ambient laboratory temperatures. The effects of the water at reducing shear moduli for a number of the rocks at higher confining pressures are noted. For this reason benzene-saturated measurements made. Dry versus saturated measurements indicate that both shear and bulk moduli increase upon saturation in all rocks except for the limestone. The increase is greatest for the bulk moduli and the greatest at lower confining pressures for both bulk and shear moduli. In all the low porosity rocks (granites and dolomites) and in the lowest porosity sandstone (9.5%) the saturated S-wave velocities are higher than dry. For the higher porosity sandstones the saturated S-wave velocities are higher than dry at low pressures and cross over at the higher pressures as the effect of density supersedes the effect of saturation on the shear modulus. Velocity data are compared with various models. The Gassmann equation for saturated effective bulk modulus does not predict saturated P-wave velocities for either low porosity granitic rocks or higher porosity sedimentary rocks. Predicted velocities are consistently low even if the shear modulus is used. Using the fluid density to predict saturated S-wave velocities cannot account for the higher shear moduli. The effects of pore fluid inertia on velocities as treated by Biot (1956a,b,) appear to be negligible for the sandstones and perhaps nonexistent for the Bedford limestone as saturated S-velocities are predicted exactly from dry with the saturated bulk density. The increase of saturated over dry shear moduli is consistent with the crack models of Budiansky and O’Connell (1974) and Kuster and Toksöz (1974). Although the velocity data can be interpreted on the basis of these crack models it is not the purpose here to judge whether or not they are appropriate for all rocks, only to note the consistency between observations and these models as constructed. The Budiansky and O’Connell model with isolated cracks fits dry and saturated velocities for two of the low porosity rocks (Westerly granite and Webatuck dolomite) with low crack densities. Dry and saturated P- and S-wave velocities for Westerly granite are consistent with the model of Kuster and Toksöz for a spectrum of ellipsoidal pore shapes, as implemented by Cheng (1978), particularly with regards to higher saturated shear moduli.