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Detournay and Cheng (1993) use yet another deﬁnition, putting 1/Kφ = 1/K − 1/(Ks vs ), where vs is the volume fraction of the solid phase. A Review of the Linear Theory of Anisotropic Poroelastic Solids 1 ∂V εkk 1 , =− =− Ks V0 ∂p Δσij =−Δpδij Δp 1 ∂Vφ 1 v − v0 1 , =− =− Ks Vφ0 ∂p Δσij =−Δpδij v0 Δp 1 ∂Vs 1 vs − vs0 1 . =− =− Ks Vs0 ∂p Δσij =−Δpδij 1 − v0 Δp 29 (86) (87) (88) The fact that these moduli are deﬁned as properties of a representative macroscopic rock sample implies in particular that Ks is to be viewed as an average bulk modulus of a skeleton body which in general may be quite heterogeneous and may also contain ﬂuid inclusions.
G. Berryman (1995). Realizability of negative pore compressibility in poroelastic composites. J. Appl. Mech. 62, 1053–1062. G. (1995). Mixture theories for rock properties. In: Rock Physics and Phase Relations. J. Ahrens, Am. Geophys. , pp. 205–228. A. (1941). General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164. A. (1955). Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26, 182–185. A. (1956a). Thermoelasticity and irreversible thermodynamics.
Moreover, dividing (106) by (105) and using (80), we obtain Ku Sσ = (107) K Sε as a further useful relationship and counterpart of the well-known thermodynamic relation cp /cv = κT /κS between the ratio of the speciﬁc heats at constant pressure and volume and the ratio of the isothermal and adiabatic compressibilities. Corresponding results for the diﬀerences between the Poisson ratios ν and νu and Young’s moduli E and Eu may now be obtained by expressing K (Ku ) in terms of G and ν (νu ) in (106)15 .