An exact effective stress law for elastic deformation of rock with fluids

Abstract

The exact expressions for the effective stress (•r•) and, in particular, pressure (P> that cause elastic strain of material with pore fluids are, assuming only that Hook's law is valid, (•r•) -o'• --aPO,• and (P) --Pc --aP• where a --I --(K/K.), Pc and P,are confining and pore pressures, and K and K. are the bulk moduli of the rock and grain, respectively. The equation for (P> was first suggested by Geertsma (1957) and by Skempton (1960) on empirical grounds. The expression does not depend directly on porosity, but when t•ores vanish the effective pressure (P> equals the confining pressure Pc, because then K --K.. Thus the strain of a porous solid with pore pressure can be completely determined from the elastic modulus of the solid without pore pressure, if the effective stress law in the equation for (•> is used.The exact expression for the effective stress describes quite accurately the measured strains in sandstone and granite samples at confining and pore pressures to 2.5 kb. The results are not applicable to inelastic processes, such as fracture, or elastic processes other than strain.The classical work by Terzaghi [1923] revealed that the application of pore pressure to porous elastic earth materials causes an apparent increase of volume similar in magnitude, but reversed in sign, to the volume change caused by a confining pressure. Similar results were obtained later by numerous investigators. However, there is a great deal of disagreement on the theoretical accuracy and validity of Terzaghi's relation. Part of the difficulty arises from occasionally vague definitions of strains and stresses in a porous aggregate. Following Biot [1941] we will define the strain in an aggregate as a local average over a small, but finite, area, such as measured with a strain gauge on rock samples in the laboratory. The area over which the strain is taken is large enough to average over a sufficient number of grains and pores. Using this definition, we do not consider the local deviations of strain from the local average. Similarly we also define the stress a acting on a porous solid in the way it is done in the laboratory, namely, by the ratio of force F to a small, but finite, area A, to which this force is applied; thus (r = F/A. Note that porosity V does not enter the expression for the stress when it is so defined. The stresses, like the strains, are therefore also local averages taken over a small, but finite, area. ELASTICITY OF PoRous MATERIALSWe now make the important assumption, following Terzaghi [1923], Blot [1941], and others, that the strains can be expressed as linear combinations of the stresses within the

The Mechanical Coupling of Fluid-Filled Granular Material Under Shear

“…Nur and Byerlee [1971] assert their law by a series of experiments showing that there is no correlation between the applied stress and the resulting measured volumetric strain, while they find a linear relation between the effective stress and the measured volumetric strain, with a stress-strain curve similar to dry samples [Nur and Byerlee, 1971, their figure 2]. In our test we reproduce numerically the experimental series of Nur and Byerlee [1971]. We perform a series of numerical simulations where in each simulation a system of variable size grains is packed under confining stress.…”

“…The first test verifies that the model reproduces correctly the law of effective stress. Nur and Byerlee [1971] develop an effective stress law for volumetric strain of fluid-filled porous material: σ ′ ij = σ ij − αδ ij P , where α, the effective stress coefficient, is a function of the compressibility of the solid grains and of the matrix. For the case of incompressible grains α = 1.…”

“…These dependencies were formulated using an effective stress coefficient, 0 < α ≤ 1, that multiplies P , where α was found to be different for different physical quantities [Wang, 2000, Pride, 2005. Still, it was shown that generally α is very close to one and when the material composing the solid matrix is incompressible relative to the pore fluid, α = 1, and Terzaghi's formulation is valid [Nur and Byerlee, 1971, Robin, 1973, Wang, 2000, Pride, 2005.…”

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

“…The constant n (Π) is called the effective stress coefficient relative to the physical property Π . For the bulk volumetric strain n (Π) is equal to Biot's poroelastic coefficient (e.g., Nur and Byerlee, 1971;Robin, 1973), which is the case in Equation (4). In contrast, for the porosity and for the drained bulk modulus n (Π) is equal to 1 in idealized model (e.g., Zimmermann, 1991;Gurevich, 2004).…”

Two-Dimensional Fractional Order Generalized Thermoelastic Porous Material

“…A short list of papers pertinent to the present study includes Biot(1941Biot( , 1956, Gassmann (1951), Biot and Willis (1957), Biot (1962), Deresiewicz and Skalak (1963), Mandl (1964), Nur and Byerlee (1971), Brown and Korringa (1975), Rice and Cleary (1976), Burridge and Keller (1981), Zimmerman et al (1986Zimmerman et al ( ,1994, Berryman and Milton (1991), Thompson and Willis (1991)], Pride et al (1992), Berryman and Wang (1995), Tuncay and Corapcioglu (1995), Alexander and Cheng (1991), Charlez, P. A., and Heugas, O. (1992), Abousleiman et al (1998), Ghassemi and Diek (2002), Tod (2003).…”