Measuring the Biot Coefficient for a Fluid-Saturated Crystalline Rock

Abstract

ABSTRACT: Biot coefficient α relates the increment of fluid content Δζ to an increment of bulk volume strain Δεv during the application of an increment of mean stress ΔP under drained conditions i.e. no change in cavity fluid pressure Δp = 0. The Biot coefficient α is measured as (i) the ratio of increment of fluid content to bulk volumetric strain α = Δζ/Δεv; (ii) the ratio of increment of mean stress to cavity fluid pressure α = ΔP/Δp under the conditions of zero volumetric strain; and (iii) (Equation), which requires the measurement of two other elastic coefficients, K = drained bulk modulus and (Equation) = unjacketed bulk modulus. Measurement of α using methods (i) and (ii) is challenging for crystalline rock. Hence, method (iii) is often promoted in the literature. The objective of this paper is to use methods (i), (ii), and (iii) to determine α at a Terzaghi effective mean pressure P′ = P − p = 15.0 MPa for Westerly blue granite, a typical crystalline rock. The specimen is saturated using the back pressure saturation approach, where a constant Skempton B-value at a specific Terzaghi effective mean pressure ensures full saturation. The experiments demonstrate that the three methods yield similar results for a at a given P′. 1. INTRODUCTION A theory for deformation of a porous elastic solid containing a viscous compressible fluid was proposed by Biot (1935, 1941, 1956). Considering only volumetric response, with dynamic variables P and p and kinematic variables εv and ζ (Detournay and Cheng 1993), the constitutive relations are: (Equation) (Equation) where εv = bulk volumetric strain, ζ = fluid content, P = mean stress, p = cavity fluid pressure, K = drained bulk modulus, B = Skempton cavity fluid pressure coefficient, and α = Biot coefficient. In a jacketed test on a fluid-saturated specimen where Δp = 0, Biot and Willis (1957) showed the constitutive equations of poroelasticity yield three relationships for the elastic coefficient α: (Equation) (Equation) (Equation) where ΔVf = ΔVp = change in fluid volume contained within rock cavities under drained conditions, i.e. Δp = 0. Measurement of ΔVf requires calibration for system compliance involving careful evaluation of fluid exchange between the fluid-saturated rock and the reservoir. Under drained conditions, Eq. (3) is a representation of the ratio of cavity fluid volume change ΔVp to bulk volume change ΔV while Eq. (4) presents α as a measure of the competition between the change in mean stress ΔP and cavity fluid pressure change Δp to maintain zero bulk volumetric strain. The third interpretation of Biot coefficient α, i.e. Eq. (5), involves the measurement of two other poroelastic coefficients K and (Equation).