The elastic moduli of a solid permeated with an isotropic distribution of flat cracks have been calculated from the energy of a single crack by use of a self-consistent approximation. The results are applicable for a dense network of cracks and give physically reasonable results up to the point that the shear modulus vanishes. Results for both circular and elliptical cracks are essentially the same if the crack density is characterized by 2N〈A2/P〉/π, where N is the number of cracks per unit volume, A is the area of crack, and P is the perimeter of cracks; for circular cracks of radius a this becomes N〈a3〉. This crack density parameter can be related quantitatively to crack traces observed in thin section. Results for completely dry or saturated cracks, for mixtures of dry and saturated cracks, and for cracks saturated with a compressible fluid are presented. For all cases, both seismic wave velocities decrease with increasing crack density. The velocity ratio VP/VS decreases for dry cracks and increases for saturated cracks. For the analysis of data a plot of VP/VS versus VS uniquely specifies the crack density. Comparison of the theory with wave velocities measured in laboratory rock samples demonstrates its validity for large crack densities. Interpretation of velocity changes before the San Fernando earthquake indicates that the region contained a substantial density of cracks at all times, that the anomalous decrease in VP/VS was due to the vaporization of pore fluid in nearly all of the previously saturated cracks without the introduction of new dry cracks, and that during the period of the recovery of the velocities to previous values the number of cracks in the region away from the epicentral zone decreased as they were resaturated, whereas the crack density increased following resaturation in the epicentral zone. Such use of the theoretical results may be useful in further investigations of preseismic phenomena.

Seismic velocities in dry and saturated cracked solids

The determination of the elastic properties of composite materials (multiphase aggregates, polycrystals, and porous or cracked solids) from the elastic properties of the components may be approached in several ways. The problem may be treated statistically, via scattering theory, through variational principles, or by the assumption of specific geometries for the material under consideration. Each of these methods is reviewed in turn. The widely used Voigt-Reuss-Hill average can be a poor approximation for both two-phase composites and polycrystals, and its replacement by the two Hashin-Shtrikman bounds is recommended. For pore-containing or crack-containing media, specific geometry models must be considered if useful results are to be obtained. If aggregate theory is used to estimate the moduli of individual components of a composite whose bulk properties are known, the shear moduli of the component phases must be matched (within a factor of 2 or 3) for the method to be useful. Results for nonlinear composites (which allow calculation of the pressure variation of aggregate moduli) have been obtained for only a few special cases.

The Elastic Properties of Composite Materials

Velocities of seismic compressional and shear waves in porous rocks under different saturation conditions are calculated theoretically and compared with laboratory data. For theoretical formulations, the rocks are represented by a solid matrix and pores of spherical and oblate spheroidal shapes. The effect of confining pressure on velocities is calculated by taking into account pore closing and saturant compressibilities.The theoretical calculations show that with all other parameters fixed, thin pores (small aspect ratios) have much greater effects on elastic moduli and velocities than rounded pores at the same concentration. The properties of the saturating fluid (gas, oil, or water) have greater effects on the compressional velocities than on shear velocities. The velocities of compressional waves are higher when the rock is saturated with water than when it is dry or gas-saturated. For shear waves the behavior is generally opposite, with shear velocities higher in the dry or gas-saturated case than in the water-saturated case.Compressional and shear velocities measured as a function of pressure in laboratory samples of granite, limestone, and sandstone, under dry and water-saturated states, are fitted with theoretical curves and pore shape spectra which fit the data are calculated. A spectrum of pore shapes ranging from spheres to very fine cracks (aspect ratios 1 to 10 (super -5) ) is required to fit the data. Theoretical velocities calculated using these models fit the measured velocities in water-saturated and frozen rocks, as well as the compressional velocities in partially saturated rocks.With the rock models derived on the basis of laboratory data, theoretical seismic velocities are calculated for various pressures and temperatures for reservoir rocks fully or partially saturated with gas, oil, or brine. Compressional velocities are highest for brine saturation and lowest for gas saturation. The difference decreases with increasing pressure. The presence of a small amount (5 percent) of gas in brine as an immiscible mixture reduces the compressional velocities significantly, even below those of fully gas-saturated values at some pressures.The reflection coefficients for compressional waves at a gas-brine interface in a model of a sandstone are high at pressures corresponding to shallow and moderate (less than about 8000 ft) depths. At greater confining pressures, reflection coefficients become small, except when the pore fluid pressure (gas pressure) is very high. Thus, large reflections or 'bright spots' from great depths may indicate overpressured formations. The reflection coefficients from mixed gas-brine interfaces are lower than those of pure gas interfaces. A combination of interval velocities and reflection amplitudes may help identify the mixed gas-brine reservoirs. Poisson's ratios for gas-saturated rocks are lower than those for brine-saturated. This difference persists to great depths.

Velocities of seismic waves in porous rocks

In this paper we report laboratory acoustic velocity and electrical resistivity measurements on Berea Sandstone and Austin Chalk samples saturated with a stoichiometric mixture of tetrahydrofuran (THF) and water. THF and water is an excellent experimental analogue to natural gas hydrates because THF solutions form hydrates similar to natural gas hydrates readily at atmospheric pressures. Hydrate formation in both the chalk and sandstone samples increased the acoustic P wave velocities by more than 80% when hydrates formed in the pore spaces; however, the velocities soon plateaued and further lowering the temperature did not appreciably increase the velocity. In contrast, the electrical resistivity increased nearly 2 orders of magnitude upon hydrate formation and continued to increase slowly as the temperature was decreased. In all cases resistivities were nearly frequency independent to 30 kHz, and the loss tangents were high, always greater than 5. The dielectric loss showed a linear decrease with frequency suggesting that ionic conduction through a brine phase dominates at all frequencies, even when the pores are nearly filled with hydrates. We find that resistivities were strongly a function of the dissolved salt content of the pore water. Pore water salinity also influenced the sonic velocity, but this effect is much smaller and only important near the hydrate formation temperature.

Acoustic and resistivity measurements on rock samples containing tetrahydrofuran hydrates: Laboratory analogues to natural gas hydrate deposits

. P-wave refraction seismics is a key method in permafrost research but its applicability to low-porosity rocks, which constitute alpine rock walls, has been denied in prior studies. These studies explain p-wave velocity changes in freezing rocks exclusively due to changing velocities of pore infill, i.e. water, air and ice. In existing models, no significant velocity increase is expected for low-porosity bedrock. We postulate, that mixing laws apply for high-porosity rocks, but freezing in confined space in low-porosity bedrock also alters physical rock matrix properties. In the laboratory, we measured p-wave velocities of 22 decimetre-large low-porosity ( 100 micro-fissures) from 25 °C to −15 °C in 0.3 °C increments close to the freezing point. When freezing, p-wave velocity increases by 11–166% perpendicular to cleavage/bedding and equivalent to a matrix velocity increase from 11–200% coincident to an anisotropy decrease in most samples. The expansion of rigid bedrock upon freezing is restricted and ice pressure will increase matrix velocity and decrease anisotropy while changing velocities of the pore infill are insignificant. Here, we present a modified Timur's two-phase-equation implementing changes in matrix velocity dependent on lithology and demonstrate the general applicability of refraction seismics to differentiate frozen and unfrozen low-porosity bedrock.

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P-wave velocity changes in freezing hard low-porosity rocks: a laboratory-based time-average model

Compressional and shear wave velocities of water-saturated rocks were measured as a function of both pressure and temperature near the melting point of ice to confining pressure of 2 kb. The pore pressure was kept at about 1 bar before the water froze. The presence of a liquid phase (rather than ice) in microcracks of about 0.3% porosity affected the compressional wave velocity by about 5% and the shear wave velocity by about 10%. The calculated effective bulk modulus of the rocks changes rapidly over a narrow range of temperature near the melting point of ice, but the effective shear modulus changes gradually over a wider range of temperature. This phenomenon, termed elastic anomaly, is attributed to the existence of liquid on the boundary between rock and ice due to local stresses and anomalous melting of ice under pressure.

Shozaburo Takeuchi

Papers

Elasticity of water‐saturated rocks as a function of temperature and pressure