Chapter 7 - Biot Theory for Porous Media

Abstract

Dynamic porous media behaviour is described by means of Biot theory of poroelasticity. However, many developments in the area of porous media existed before Biot introduced the theory in the mid-50s. These include, for instance, Terzaghi law, Gassmann equation and the static approach leading to the concept of effective stress, much used in soil mechanics. The simple asperity-deformation model is useful to explain the physics of porous and cracked media under confining and pore-fluid pressures. Moreover, I consider a model for pressure build-up due to kerogen-oil conversion. The dynamical problem is analyzed in detail using Biot approach: that is, the definition of the energy potentials and kinetic energy and the use of Hamilton principle to obtain the equation of motion. The coefficients of the strain energy are obtained by the so-called jacketed and unjacketed experiments. The theory includes anisotropy and dissipation due to viscodynamic and viscoelastic effects. A short discussion involving the complementary energy theorem and volume-average methods serves to define the equation of motion for inhomogeneous media. The interface boundary conditions and the Green function problem are treated in detail, since they provide the basis for the solution of wave propagation in inhomogeneous media. The mesoscopic loss mechanism is described by means of white theory for plane-layered media developed in the mid-70s. The theory is applied to layered and fractured media in order to obtain the five stiffness components of the equivalent transversely isotropic medium. Then, I analyze the physics of diffusion fields resulting from Biot equations. An energy-balance analysis for time-harmonic fields identifies the strain- and kinetic-energy densities and the dissipated-energy densities due to viscoelastic and viscodynamic effects. The analysis allows the calculation of these energies in terms of the Umov-Poynting vector and kinematic variables, and the generalization of the fundamental relations obtained in the single-phase case (Chapter 4). Measurable quantities, like the attenuation factor and the energy velocity, are expressed in terms of microstructural properties such as tortuosity and permeability. Finally, I derive Gassmann equation for an anisotropic frame and a solid pore infill.