Compressional and shear‐wave velocities were measured in the laboratory from 1 bar to 4 kbar confining pressure for wet, undrained samples of Cretaceous shales from depths of 3200 and 5000 ft in the Williston basin, North Dakota. These shales behave as transversely isotropic elastic media, the plane of circular symmetry coinciding with the bedding plane. For compressional waves, the velocity is higher for propagation in the bedding plane than at right angles to it, and the anisotropy is greater for the 5000-ft shale. For shear waves, the SH‐wave perpendicular to bedding and the SV‐wave parallel to bedding propagate with the same speed, which is about 25 percent lower than that for the SH‐wave parallel to bedding. In general, compressional and shear velocities are higher for the indurated 5000-ft shale than for the friable 3200-ft shale. All velocities increase with in‐increasing confining pressure to 4 kbar. The 3200-ft shale exhibits velocity hysteresis as a function of pressure, whereas this effect is almost nonexistent for the 5000-ft shale. Many features of the dependence of velocity on pressure can be explained by consideration of effective pressure and the degree of water saturation. For both shales, laboratory compressional wave velocities are on average 10 percent higher than log‐derived velocities. The discrepancy cannot be explained completely, but likely contributing factors are sampling bias, velocity dispersion, and formation damage in situ.
Phenomenological equations (with coefficients to be determined by specified experiments) for the poroelastic behavior of a dual porosity medium are formulated, and the coefficients in these linear equations are identified. The generalization from the single‐porosity case increases the number of independent coefficients for volume deformation from three to six for an isotropic applied stress. The physical interpretations are based upon considerations of different temporal and spatial scales. For very short times, both matrix and fractures behave in an undrained fashion. For very long times, the double‐porosity medium behaves like an equivalent single‐porosity medium. At the macroscopic spatial level, the pertinent parameters (such as the total compressibility) may be determined by appropriate field tests. At an intermediate or mesoscopic scale, pertinent parameters of the rock matrix can be determined directly through laboratory measurements on core, and the compressibility can be measured for a single fracture. All six coefficients are determined from the three poroelastic matrix coefficients and the fracture compressibility from the single assumption that the solid grain modulus of the composite is approximately the same as that of the matrix for a small fracture porosity. Under this assumption, the total compressibility and three‐dimensional storage coefficient of the composite are the volume averages of the matrix and fracture contributions.
Measurements have been completed for eight different poroelastic moduli of water‐saturated Berea sandstone and Indiana limestone as a function of confining pressure and pore pressure. The poroelastic moduli for Indiana limestone are generally consistent to ±10%, which was verified by a formal inversion procedure for independent moduli from the eight measurements. For Indiana limestone, best fit values were drained bulk modulus, 21.2 GPa; the undrained bulk modulus, 31.7 GPa; drained Poisson's ratio, 0.26; undrained Poisson's ratio, 0.33; and pore pressure buildup coefficient, 0.47 at 20–35 MPa effective stress. The poroelastic moduli for Berea sandstone are generally consistent to ±20%. The greater inconsistency is most likely caused by the nonlinear variation of the moduli at different strains. For Berea sandstone, best fit values were drained bulk modulus, 6.6 GPa; undrained bulk modulus, 15.8 GPa; drained Poisson's ratio, 0.17; undrained Poisson's ratio, 0.34; and pore pressure buildup coefficient, 0.75 at 10 MPa effective stress.
The pore pressure response of saturated porous rock subjected to undrained compression at low effective stresses are investigated theoretically and experimentally. This behavior is quantified by the undrained pore pressure buildup coefficient, [Formula: see text] where [Formula: see text] is fluid pressure, [Formula: see text] is confining pressure, and [Formula: see text] is the mass of fluid per unit bulk volume. The measured values for B for three sandstones and a dolomite arc near 1.0 at zero effective stress and decrease with increasing effective stress. In one sandstone, B is 0.62 at 13 MPa effective stress. These results agree with the theories of Gassmann (1951) and Bishop (1966), which assume a locally homogeneous solid framework. The decrease of B with increasing effective stress is probably related to crack closure and to high‐compressibility materials within the rock framework. The more general theories of Biot (1955) and Brown and Korringa (1975) introduce an additional parameter, the unjacketed pore compressibility, which can be determined from induced pore pressure results. Values of B close to 1 imply that under appropriate conditions within the crust, zones of low effective pressure characterized by low seismic wave velocity and high wave attenuation could exist. Also, in confined aquifer‐reservoir systems at very low effective stress states, the calculated specific storage coefficient is an order of magnitude larger than if less overpressured conditions prevailed.
Summary Differential effective medium (DEM) theory is applied to the problem of estimating the physical properties of elastic media with penny‐shaped cracks, filled either with gas or liquid. These cracks are assumed to be randomly oriented. It is known that such a model captures many of the essential physical features of fluid‐saturated or partially saturated rocks. By making an assumption that the changes in certain factors depending only on Poisson's ratio do not strongly affect the results, it is possible to decouple the equations for bulk (K) and shear (G) modulus, and then integrate them analytically. The validity of this assumption is then tested by integrating the full DEM equations numerically. The analytical and numerical curves for both K and G are in very good agreement over the whole porosity range. Justification of the Poisson ratio approximation is also provided directly by the theory, which shows that as porosity tends to unity, Poisson's ratio tends towards small positive values for dry, cracked porous media and tends to one‐half for liquid‐saturated samples. A rigorous stable fixed‐point is obtained for Poisson's ratio, νc, of dry porous media, where the location of this fixed‐point depends only on the shape of the voids being added. Fixed‐points occur at for spheres and νc≃πα/18 for cracks, where α is the aspect ratio of penny‐shaped cracks. These theoretical results for the elastic constants are then compared and contrasted with results predicted by Gassmann's equations and with results of Mavko and Jizba, for both granite‐like and sandstone‐like examples. Gassmann's equations do not predict the observed liquid dependence of the shear modulus G at all. Mavko and Jizba predict the observed dependence of the shear modulus on the liquid bulk modulus for a small crack porosity and a very small aspect ratio, but fail to predict the observed behaviour at higher porosities. In contrast, the analytical approximations derived here give very satisfactory agreement in all cases for both K and G. For practical applications of this work, it appears that the ratio of compliance differences is approximately independent of the crack porosity for a given rock, but the constant is usually greater than for granites, while general statements concerning sandstones are more difficult to make.
Gassmann’s original equation provides a means to relate bulk elastic parameters of a porous material with the compressibility of the pore fluid. The original analysis assumed microhomogeneity and isotropy, which assumed that pore compressibility was equal to grain compressibility. Although subsequent theoretical arguments have shown that Gassmann’s original assumption is violated for most rocks and that pore compressibility need not equal grain compressibility, few experimental studies have compared the two compressibilities; the assumption that pore compressibility equals grain compressibility is still commonly made. We measured hydrostatic poroelastic constants of Berea sandstone and Indiana limestone under drained, undrained, and unjacketed conditions over a range of confining and pore pressures to test the assumption that pore compressibility equals grain compressibility. These two rocks were chosen because they havesimilar values of porosity but different elastic behaviors: Berea sandstone is nonlinearly elastic, especially at low effective stresses, but Indiana limestone is linearly elastic at nearly all stresses. At low effective stresses below [Formula: see text], the pore compressibility for Berea sandstone does not equal grain compressibility but approaches fluid compressibility. Even at higher effective stresses, pore compressibility for Berea sandstone does not equal bulk grain compressibility but approaches a value approximately two to three times the bulk grain compressibility. In contrast, pore compressibility for Indiana limestone does seem to be equal to grain compressibility except perhaps at low effective stresses below [Formula: see text]. The difference between pore compressibilities of these two rocks is likely from the presence of more compliant clay minerals mixed with quartz grains with more microcracks in the Berea sandstone as compared to the well-cemented Indiana limestone.
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