James G. Berryman

It is demonstrated that effective medium theories for poroelastic composites such as rocks can be formulated easily by analogy to well-established methods used for elastic composites. An identity analogous to Eshelby’s classic result has been derived previously for use in composites containing arbitrary ellipsoidal-shaped inclusions. This result is the starting point for new methods of estimation, including generalizations of the coherent potential approximation, differential effective medium theory, and two explicit schemes. Results are presented for estimating drained shear and bulk modulus, the Biot–Willis parameter, and Skempton’s coefficient. Three of the methods considered appear to be quite reliable estimators, while one of the explicit schemes is found to have some undesirable characteristics. Furthermore, the results obtained show that the actual microstructure should be taken carefully into account when trying to decide which of these methods to apply in a given situation.

A general analysis of poroelasticity for hexagonal, tetragonal, and cubic symmetry shows that four eigenvectors are pure shear modes with no coupling to the pore-fluid mechanics. The remaining two eigenvectors are linear combinations of pure compression and uniaxial shear, both of which are coupled to the fluid mechanics. The analysis proceeds by first reducing the problem to a 2 × 2 system. The poroelastic system including both anisotropy in the solid elastic frame (i.e., with ‘hard anisotropy’), and also anisotropy of the poroelastic coefficients (the ‘soft anisotropy’) is then studied in some detail. In the presence of anisotropy and spatial heterogeneity, mechanics of the pore fluid induces shear dependence on fluid bulk modulus in the overall poroelastic system. This effect is always present (though sometimes small in magnitude) in the systems studied, and can be comparatively large (up to a maximum increase of about 20%) in some porous media - including porous glass and Schuler-Cotton Valley sandstone presented here. General conclusions about poroelastic shear behavior are also related to some recently derived product formulas that determine overall shear response of these systems. Another method is also introduced based on rigorous Hashin-Shtrikman-style bounds for nonporous random polycrystals, followed by related self-consistent estimates of mineral constants for polycrystals. Then, another self-consistent estimation method is formulated for the porous case, and used to estimate drained and undrained effective poroelastic constants. These estimates are compared and contrasted with the results of the first method and a unified picture of the overall behavior is found in three computed examples for polycrystals of grains having tetragonal symmetry.

Four methods of upscaling coupled equations at the microscale to equations valid at the mesoscale and/or macroscale for fluid-saturated and partially saturated porous media will be discussed, compared, and contrasted. The four methods are: (1) effective medium theory, (2) mixture theory, (3) two-scale and multiscale homogenization, and (4) volume averaging. All these methods have advantages for some applications and disadvantages for others. For example, effective medium theory, mixture theory, and homogenization methods can all give formulas for coefficients in the up-scaled equations, whereas volume averaging methods give the form of the up-scaled equations but generally must be supplemented with physical arguments and/or data in order to determine the coefficients. Homogenization theory requires a great deal of mathematical insight from the user in order to choose appropriate scalings for use in the resulting power-law expansions, while volume averaging requires more physical insight to motivate the st...

To provide quantitative measures of the importance of fluid effects on shear waves in heterogeneous reservoirs, a model material called a “random polycrystal of porous laminates” is introduced. This model poroelastic material has constituent grains that are layered (or laminated), and each layer is an isotropic, microhomogeneous porous medium. All grains are composed of exactly the same porous constituents and have the same relative volume fractions. The order of lamination is not important because the up‐scaling method used to determine the transversely isotropic (hexagonal) properties of the grains is Backus averaging, which, for quasi‐static or long‐wavelength behavior, depends only on the volume fractions and layer properties. Grains are then jumbled together totally at random, filling all space, and producing an overall isotropic poroelastic medium. The poroelastic behavior of this medium is then analyzed using the Peselnick‐Meister‐Watt bounds (of Hashin‐Shtrikman type). We study the dependence of the shear modulus on pore fluid properties and determine the range of behavior to be expected. In particular, we compare and contrast these results to those anticipated from Gassmann's fluid substitution formulas and to the predictions of Mavko and Jizba for very low porosity rocks with flat cracks. This approach also permits the study of arbitrary numbers of constituents, but for simplicity, the numerical examples are restricted here to just two constituents. This restriction also permits the use of some special exact results available for computing the overall effective stress coefficient in any two‐component porous medium. The bounds making use of polycrystalline microstructure are very tight. Results for the shear modulus demonstrate that the ratio of compliance differences R (i.e., shear compliance changes over bulk compliance changes when going from drained to undrained behavior, or vice versa) is usually nonzero and can take a wide range of values, both above and below the value R = 4/15 valid for low porosity, very low aspect ratio flat cracks. Results show that the overall shear modulus in this model can depend relatively strongly on mechanical properties of the pore fluids, sometimes (but rarely) more strongly than the dependence of the overall bulk modulus on the fluids.

Laboratory experiments on wave propagation through saturated and partially saturated porous media have often been conducted on porous cylinders that were initially fully saturated and then allowed to dry while continuing to acquire data on the wave behavior. Since it is known that drying typically progresses from outside to inside, a sensible physical model of this process is concentric cylinders having different saturation levels—the simplest example being a fully dry outer cylindrical shell together with a fully wet inner cylinder. We use this model to formulate the equations for wave dispersion in porous cylinders for patchy saturation (i.e., drainage) conditions. In addition to multiple modes of propagation obtained numerically from these dispersion relations, we find two distinct analytical expressions for torsional wave modes. We solve the resulting torsional wave dispersion relation for two examples: Massillon sandstone and Sierra White granite. One essential fact that comes to light during the analysis is that the effective shear moduli of the gas- and liquid-saturated regions must differ, otherwise it is impossible to account for the laboratory torsional wave data. Furthermore, the drainage analysis appears to give improved qualitative and quantitative agreement with the data for both of the materials considered.

A new application of time-reversal processing of wave scattering data permits characterization of scatterers by analyzing the number and nature of the singular functions (or eigenfunctions) associated with individual scatterers when they have multiple contributions from monopole, dipole, and/or quadrupole scattering terms. We discuss acoustic, elastic, and electromagnetic scattering problems for low frequencies. Specific examples for electromagnetic scattering from one of a number of small conducting spheres show that each sphere can have up to six distinct time-reversal eigenfunctions associated with it.