Variational Energy Formulation

In Chap. 2, the constitutive equations for poroelasticity were constructed using the phenomenological approach. In such approach, we attempt to model a new phenomenon, such as the deformation of a saturated porous body, by drawing an analogy with a familiar phenomenon, such as the deformation of an elastic body. In the analogy, we define stresses and strains for a porous body following the elasticity concept, even though their interpretation may not be clear. (For example, see the illustration in Fig. 2.1 for the lack of a clear definition for a continuous stress field in a porous body.) We then bring in the additional force component, namely the pore pressure, and its conjugate deformation, the fluid strain (or actually the relative fluid to solid strain), to build a linear relation that is similar to that of the elasticity theory. This type of ad hoc construction of a working theory is typically motivated first by the observation, and then supported by physical insight, without formally resorting to the laws of physics. To gain additional physical insight, in Chap. 3 we utilized the micromechanics approach to explicitly model the material phases in a porous medium, not only the solid and the fluid, but also the pore space as an additional “phase”, together with their interactions. These micromechanical constitutive laws were assembled and matched up with the bulk continuum theory to provide physical insight. However, these constitutive laws were largely constructed using the “effective stress” concept; hence their theoretical basis is still phenomenological. The material constants associated with the theory, such as K, K′ s , and K″ s , are still empirical constants, as their physical mechanisms are based on composite responses, and are not fully isolated to tie to the equation of the state of the phases. There are many attempts to build porous medium constitutive models that go beyond the phenomenological model. Among the more widely pursued approaches are the theory of mixtures (Atkin and Craine, Q J Mech Appl Math 29:209–244, 1976; Bowen, Int J Eng Sci 18(9):1129–1148, 1980; Bowen, Int J Eng Sci 20(6):697–735, 1982; Crochet and Naghdi, Int J Eng Sci 4:383–401, 1966; de Boer R, Theory of porous media, highlights in the historical development and current state. Springer, Brelin/New York, 2000, 618pp; Morland, J Geophys Res 77(5):890–900, 1972), and the homogenization theory (Auriault and Sanchez-Palencia, J de Mec 16(4):575–603, 1977; Chateau and Dormieux, Int J Numer Anal Methods Geomech 26(8):831–844, 2002; Mei and Auriault, Proc R Soc Lond Ser A Math Phys Eng Sci 426(1871):391–423, 1989; Terada et al., Comput Methods Appl Mech Eng 153(3–4):223–257, 1998). The theory of mixtures uses the mathematical assumption that the solid and fluid phases occupies the same space, whose presence and influence are weighted by a volume fraction, in order to fulfill the requirement of a continuous mathematical function. This approach is largely mathematical, and the “material coefficients” generally do not possess physical meaning until a comparison is made with a theory that is in use, such as a phenomenological theory. The homogenization approach, on the other hand, explicitly recognizes solid and fluid phases occupying different space at the microscopic level, with an assumed periodic pore geometry. The full partial differential equations, such as Navier-Stokes equation for the fluid and elasticity equation for the solid, are prescribed, together with boundary conditions. Effort is then made to simplify these equations based on the perturbation of small parameters, in order to extract mathematical terms, and physical phenomena, of the first order, second order, etc. The homogenization approach typically involves heavy mathematics. Different judgement on the mathematical treatment sometimes leads to different, and even inconsistent results. Its ultimate justification still requires the validation by physical experiments. In this chapter we take a different approach from those above. We shall build the poroelasticity theory based on the thermodynamics principles of work and energy, first based on the reversible processes for the elastic constitutive laws, and then in the later chapters on the irreversible processes for the fluid, heat flux, and chemical diffusion laws. To cope with the highly heterogeneous nature of porous materials, the volume averaging method developed in the flow through porous medium theory (Bear J, Bachmat Y, Introduction to modeling phenomena of transport in porous media. Kluwer, Dordrecht/Boston, 1990, 553pp), and the theory of heterogeneous (composite) materials (Christensen RM, Mechanics of composite materials. Wiley Interscience, New York, 1979, 384pp; Nemat-Nasser S, Hori M, Micromechanics: overall properties of heterogeneous materials, 2nd edn. North-Holland, 1999, 810pp), are used to define the continuous and smooth functions needed in the construction of partial differential equations. The porous materials are considered as consisting of a heterogeneous solid phase and a homogeneous fluid phase (Lopatnikov and Cheng, Mech Mater 34(11):685–704, 2002). The macroscopic stresses and strains are defined as microscopically volume or surface averaged quantities depending on whether internal energy or external work is concerned. Constitutive equations are constructed using the variational energy principle stemming from the classical physics law of minimum potential energy (Landau LD, Lifshitz EM, Theory of elasticity, 3rd edn. Course of theoretical physics, vol 7. Butterworth-Heinemann, Oxford/Burlington, 1986, 195pp). The resultant model consists of a set of intrinsic material constants (Cheng and Abousleiman, Int J Numer Anal Methods Geomech 32(7):803–831, 2008), which are directly associated with the equations of state of the solid and fluid phase, and the fundamental deformation modes of the pore structure.