INTRODUCTION

Rapid progress in development of rigorous bounding methods for material coefficients in heterogeneous media (Milton, 2002; Torquato, 2002) has been made over the last fifty years. Effective medium theory, although very useful in many practical circumstances, nevertheless has not made such rapid progress. So a question that naturally arises is whether it might be possible to construct new effective medium formulas directly from the known bounds? Skeptics will immediately ask: Why do I need to do this at all if bounds are available? But the answer to this question is most apparent in poromechanics, where the bounds are frequently too far apart to be of much use in engineering and, especially, in field applications. Hill (1952) was actually the first to try constructing estimates from bounds. First he showed that the Voigt (1928) and Reuss (1929) averages/estimates in elasticity were in fact upper and lower bounds, respectively, on stiffness. Then he proceeded to suggest that estimates of reasonable accuracy were given by the arithmetic or geometric means obtained by averaging these two bounds together. Thus, the Voigt-Reuss-Hill estimates were born. Better bounds than the Voigt and Reuss bounds are now known and no doubt some attempts to update Hill's approach have been made. However, to make a direct connection to traditional approaches of effective medium theory, I apply a more technical procedure here in order to obtain estimates of up-scaled constants using the known analytical structure of the bounds, especially for Hashin-Shtrikman (1962) bounds. When this mathematical structure is not known -- as might be the case if the bounds are expressed algorithmically rather than as analytical formulas -- then I will see that it proves very worthwhile to expend the additional effort required to determine this structure. Whenever it is possible to carry the analysis further than has been done in the published literature, a self-consistent effective medium formula is fairly straightforward to obtain from the resulting expressions. The self-consistent predictions then lie within the bounds, as might be desired and expected. In the next section, one particular class of double-porosity models (Berryman, 2002; Berryman and Pride, 2002) is considered. [Other classes of models with different microstructures many also be of interest and some of these have also been discussed in previous work (Berryman, and Wang, 1995; Berryman and Pride, 2002), but other microstructures generally have less analytical structure that can be exploited, so unfortunately much less detailed information can be obtained about these models from analysis alone.] Results from double-porosity geomechanics analysis are presented. These results are general (for the model under consideration), and do not depend explicitly on generally unknown details of the spatial arrangement or microstructure of the porous constituents. Microstructure enters these formulas only through the overall drained bulk modulus K_d^*. Then, in the following section, a preferred model microstructure -- that of a locally layered medium -- is imposed. This microstructure has the advantage that it forms hexagonal (or transversely isotropic) ``crystals'' locally. Then, if I assume these crystals, or grains, are jumbled together randomly so as to form an overall isotropic medium, I have the ``random polycrystal of porous laminates'' reservoir model. Hashin-Shtrikman bounds are known for such polycrystals composed of grains having hexagonal symmetry. So bounds are easily found. From the form of the bounds, I also obtain estimates of both overall bulk modulus and shear modulus (Berryman, 2005), thus completing the semi-analytical poromechanics model. The final two sections show examples, and summarize my results. Although the language I use here tends to emphasize the analogy to polycrystals of laminates, the reader should keep in mind that the equations of elasticity -- and for present purposes (I do not treat permeability here) also the equations of poroelasticity -- are scale invariant. So the mathematics is the same whether the layering I are considering takes place at the scale of microns, meters, or kilometers. However, there is an obvious but implicit limitation that the scale considered cannot be so small that the continuum hypothesis fails to be valid.