Bounds on geomechanical constants for a model of heterogeneous reservoirs

James G. Berryman

berryman@sep.stanford.edu

ABSTRACT

A well-known result due to Hill provides an exact expression for the bulk modulus of any multicomponent elastic composite whenever the constituents are isotropic and the shear modulus is uniform throughout. Although no precise analog of Hill's result is available for the opposite case of uniform bulk modulus and varying shear modulus, it is shown here that some similar statements can be made for shear behavior of random polycrystals composed of laminates of isotropic materials. This model is intended to incorporate characteristics that mimic geomechanical properties of heterogeneous earth reservoirs, including local layering due to sedimentary processes. In particular, the Hashin-Shtrikman-type bounds of Peselnick, Meister, and Watt for random polycrystals composed of hexagonal (transversely isotropic) grains are applied to our model of polycrystals of laminates. An exact product formula relating the Reuss estimate of bulk modulus and an effective shear modulus (of laminated grains composing the system) to products of the eigenvalues for quasi-compressional and quasi-uniaxial shear eigenvectors also plays an important role in the analysis of the overall shear behavior of the random polycrystal. When the bulk modulus is uniform in such a system, the equations are shown to reduce to a simple form that depends prominently on the uniaxial shear eigenvalue - as expected from physical arguments concerning the importance of uniaxial shear in these systems. Applications of the analytical results presented here include benchmarking of numerical procedures used for studying elastic behavior of complex composites, and estimating coefficients needed in up-scaled equations for elasticity and/or poroelasticity of heterogeneous reservoirs.