Micromechanics

Abstract

The bulk continuum model presented in the preceding chapter relates the externally observable quantities of a deforming porous specimen, such as the total stress, pore pressure, frame deformation, and fluid expelled from the frame, to each other, in order to construct constitutive relations that can be used to predict material behaviors. There are times, however, it is desirable, or even necessary, to learn what is happening inside a porous medium in terms of the solid and fluid phase, and the porous structure, such as change in porosity. For example, when we observe an external volume change of a porous frame, we may want to know how much of it is derived from the solid deformation, and how much is due to the pore space being taken out? When we measure a volume of fluid being expelled from a porous frame, how much of it is due to the reduction of the internal pore space, and how much is due to the expansion of fluid itself? For material constants, such as the undrained bulk modulus, how much of its apparent compliance is attributed to the compressibility of the solid constituent (which is typically small), and how much is to the pore space (which can be much larger)? Or, given an undrained bulk modulus, in what proportion does it draw its strength from the porous frame and the fluid?Gassmann in 1951 (Veirteljahrsschrift der Naturforschenden Gesellschaft in Zürich 96:1–23, 1951) presented a model intended to partially answer these questions. In the model, Gassmann partitioned the total volume of the frame into a part occupied by the solid, and a part by the pores. In an effort to construct constitutive equations that relate the volumetric deformations to the applied stresses, he identified three micromechanical material constants, a solid, a fluid, and a pore compressibility. The Gassmann model, however, assumed that at the grain level (microscopic scale), the solid phase is homogeneous and isotropic, though at the macroscopic scale, the material can be heterogeneous and anisotropic. This model has been called the ideal porous medium model. Many porous material, particularly geomaterials, however, are not homogeneous and/or isotropic at the grain level. For example, rocks at the microscopic level are made of grains of different minerals, such as quartz, calcite, mica, and even clay minerals; hence are heterogeneous at that level. This suggests that Gassmann model is a special model.For a general model, the microhomogeneity and microisotropy assumptions were removed by Biot and Willis in 1957 (J Appl Mech ASME 24:594–601, 1957). The resultant micromechanics model contains four independent material constants associated with volumetric deformation, one more than the ideal porous medium model. This micromechanical analysis has been widely accepted, and reformulated by many others (Brown, Korringa, Geophysics 40(4):608–616, 1975; Carrell, Mechanical response of fluid-saturated porous materials. In: Rimrott FPJ, Tabarrok B (eds) Theoretical and applied mechanics, 15th international congress on theoretical and applied mechanics, Toronto, pp 251–262, 1980; Detournay, Cheng, Fundamentals of poroelasticity. In: Fairhurst C (ed) Comprehensive rock engineering: principles, practice and projects, vol. II, Analysis and design method. Pergamon Press, Oxford/New York, pp 113–171, 1993; Nur, Byerlee, J Geophys Res 76(26):6414–6419, 1971; Rice, Cleary, Rev Geophys 14(2):227–241, 1976; Wang, Theory of linear poroelasticity: with applications to geomechanics and hydrogeology. Princeton University Press, Princeton, 287pp, 2000), in ways that are consistent with the original model. In this chapter, the Biot-Willis micromechanics model is presented.

We propose to show that a linear theory of consolidation can be established by combining the theory of elasticity with Darcy’s law of flow of a fluid through a porous material. Suppose the porous skeleton considered is a continuous and homogeneous medium that satisfies the usual assumptions of the theory of elasticity. Let’s ignore for the moment the liquid fills. Tension acting on a cubic unit-element side of the porous medium are similar to those that are defined in usual elasticity. The element is assumed to be large compared to the pore size. —Maurice Biot (1935)