An effective stress law for anisotropic elastic deformation

Abstract

An effective stress law is derived analytically to describe the effect of pore fluid pressure on the linearly elastic response of saturated porous rocks which exhibit anisotropy. For general anisotropy the difference between the effective stress and the applied stress is not hydrostatic. The effective stress law involves two constants for transversely isotropic response and three constants for orthotropic response; these constants can be expressed in terms of the moduli of the porous material and of the solid material. These expressions simplify considerably when the anisotropy is structural rather than intrinsic, i.e., in the case of an isotropic solid material with an anisotropic pore structure. In this case the effective stress law involves the solid or grain bulk modulus and two or three moduli of the porous material, for transverse isotropy and orthotropy, respectively. The law reduces, in the case of isotropic response, to that suggested by Geertsma (1957) and by Skempton (1961) and derived analytically by Nur and Byerlee (1971).

Pore Volume and Porosity Changes under Uniaxial Strain Conditions

“…(The effective stress tensor is sometimes denoted by σ = σ − α P p I.) Constitutive equations and effective stress coefficients for anisotropic poroelastic rocks have been studied by Carroll (1979), Thompson andWillis (1991), andCheng (1997). However, the case of anisotropy will not be pursued further in the present paper.…”

Experimental Verification of the Petroelastic Model in the Laboratory – Fluid Substitution and Pressure Effects

“…This alternative stress is nothing but the so-called "Biot effective stress" (e.g., Carroll, 1979;Bourbié et al, 1987;Thompson and Willis, 1987) and the dimensionless coefficient α is referred to as the Biot effective stress coefficient (Bourbié et al, 1987). The elastic coefficients K (dr) and μ and the dimensionless coefficient α only depend on the porous medium.…”

Hierarchical poroelasticity: movement of interstitial fluid between porosity levels in bones

“…In a porous medium the pores are assumed to be connected; there are no unconnected pores that prevent the flow of fluid through them. The six-dimensional vector P (threedimensional symmetric second rank tensor A P ), introduced by equation (3.2), is called the Biot effective stress coefficient vector (six-dimensional) or tensor (three-dimensional) at the porosity level P. The Biot effective stress coefficient vector P is related to the difference between effective drained elastic constantŝ S d,P and the elastic compliance tensor at the P − 1 porosity level,Ŝ d,P−1 , by the formula (Nur & Byerlee 1971;Carroll 1979;Cowin & Doty 2007;Cowin & Mehrabadi 2007) , 1, 0, 0, 0] T is the six-dimensional vector representation of the three-dimensional unit tensor 1. The Biot effective stress coefficient vector P is so named because it is employed in the definition of the effective stressT eff ,…”