ABSTRACT
Stress-induced flow of interstitial fluid in porous solids and the subsequent solid deformations have been used to explain a variety of phenomena observed in geophysics and in engineering practice. The one-dimensional consolidation theory of Terzaghi has been used successfully for predicting subsidence in clay under surface loading (Terzaghi, 1943). However, when the surface loading is of finite size, the clay layer is not thin and not confined, or the clay is of nonuniform thickness, three-dimensional deformation becomes inevitable. In these situations, Terzaghi’s 1-D theory is inapplicable. Biot (1941) extended the formulation for poroelastic solids to three-dimensional and retained the coupling between solid deformation and pore-space pressure. The solutions of Biot theory have been studied by Derski (1964, 1965), McNamee and Gibson (1960a,b), and Schiffman and Fungaroli (1965). Although there have been formulations of mixture theory to model porous geomaterials and they were formulated following more rigorous procedure from thermodynamics (e.g., de Boer, 2000), in practice they do not offer any advantage over Biot’s theory (Detourney and Cheng, 1993). Mathematically, Biot’s theory of poroelasticity is precisely analogous to linear coupled thermoelasticty (e.g., Carslaw and Jaeger, 1959), which is a very well-developed area with an abundance of available solutions in the literature. Unfortunately, in thermoelasticity the adiabatic (no heat loss in solids) and isothermal (no temperature change in solids) Poisson’s ratios of solids under thermal effects are typically indistinguishable, so that the coupling terms between deformation fields and heat conduction are commonly and justifiably dropped. Most of the available solutions are obtained for uncoupled thermelasticity; that is, heat diffusion first can be solved independent of the deformation field of the solids. However, in poroelasticity the undrained and drained Poisson’s ratios differ considerably. Unless the pore fluid is highly compressible, the diffusion process of fluid cannot be uncoupled from the deformation of geomaterials. Therefore, most of the solutions from thermoelasticity cannot be converted to poroelasticity. On the contrary, all solutions for poroelasticity can be employed to consider thermal effect in solids. To solve the coupled equations of Biot’s poroelasticity, displacement functions have been proposed by McNamee and Gibson (1960a,b) for axisymmetric problems, and by Schiffman and Fungaroli (1965) for antisymmetric problems. Such a displacement function approach will be discussed in this chapter, in conjunction with the LaplaceHankel transform method given by Chau (1996) and others.
