Upper and lower stiffness bounds for porous anisotropic rocks

Abstract

S U M M A R YWe derive double inequalities providing the bounds for components of the effective stiffness tensor of a two-phase, porous-cracked medium with aligned ellipsoidal inclusions. The bounds are derived on the basis of the Hashin-Shtrikman variational principle, and the conditions for positive semi-definiteness of quadratic forms. Inequalities are presented for isotropic, cubic, hexagonal and orthorhombic overall symmetries. The results obtained for orthorhombic symmetry are valid for the general determination of transport properties (effective permeability, thermal and electrical conductivity). We conclude that inequalities for diagonal components of the effective tensor have the form of bounds, whereas in general these bounds do not exist for off-diagonal components. One important implication of this is that Voigt-Reuss averages do not provide the upper and lower bounds for off-diagonal components of the effective tensor, as is sometimes assumed. We also present numerical results for stiffness bounds obtained by modelling various shales with isotropic, transverse isotropic and orthorhombic overall symmetries.Rocks can be considered as microscopically inhomogeneous composites, whose components are mineral grains, pores and cracks. The problem of determination of macroscopic (effective) physical properties of such a composite is a many-body problem that, in general, can only be solved approximately. Application of the various theoretical methods for solving the problem gives different results. However, based on common physical principles, it is possible to obtain rigorous bounds for effective parameters. Hill (1952) has shown that based on the principle of energy minimum at the equilibrium state, the Voigt and Reuss averages provide the upper and lower bounds, respectively, for the effective bulk and shear moduli of isotropic composites. Narrower bounds can be obtained from the Hashin-Shtrikman variational principle (Hashin & Shtrikman 1963) that was later developed for composites of the 'matrix-inclusions' type (Walpole 1966;Willis 1977). Kröner (1977) derived bounds for heterogeneous media taking into account the spatial correlation functions of the stiffness tensor. He showed that the Hashin-Shtrikman bounds apply if the medium is statistically homogeneous, that is, the properties depend only on the distance between two points, and its properties are described by the two-point correlation functions. Ponte Castaneda & Willis (1995) have extended the Hashin-Shtrikman bounds to the case of arbitrary shape and orientation of inclusions with taking into account their spatial distribution. The shape of inclusions is assumed to be ellipsoidal, and the distribution of inclusions' centres in space is statistically described by an ellipsoid as well, which has a shape that may be different form the inclusions' shape. If the inclusions are of the same shape and aligned, and the shape of ellipsoid describing the inclusions' spatial distribution is similar to the inclusions' shape, the formulas are reduced to the Wi...

Automatic generation and discretization of fully periodic representative volume elements of plain woven composites

“…In case of anisotropic materials, the applicability is guaranteed only for main diagonal elements. 27 The comparison of the given numerical examples with the analytical solutions of Voigt and Reuss bounds shows the applicability and trustworthiness of our approach.…”

Rock Physics Model of Shale: Predictive Aspect

“…Since the moduli and anisotropy parameters calculated using Voigt and Reuss schemes could be considerably different from each other (Dutta, 2018), the relative proportion within the two averaging schemes is also considered as a free parameter in this study. The relative proportion lies between 0 and 1 (0 coincide with the Reuss scheme while 1 gives the Voigt scheme) although the Voigt and Reuss schemes are not strict bounds for anisotropic domains (e.g., Bayuk et al., 2008). Note that the choice of Voigt versus Reuss averaging scheme would depend on configuration of clay platelets, which is very difficult to assess.…”