Biot constitutive relation and porosity perturbation equation

Abstract

In a porous medium, the porosity perturbation, i.e., the change in porosity, is an integral part of a deformation process. Yet, there is no explicit statement about that in the Biot theory. By linking its constitutive relation to the continuity equations, the tacit assumption about the porosity perturbation in this theory is inferred. The linear dependence of the porosity perturbation on the pressure difference of the two phases is embedded in its constitutive relation. The solid and fluid pressures affect the change in porosity in equal magnitude but in opposite sense. By assuming that the fluid pressure may affect the porosity perturbation to an extent different than that of the solid pressure, the Biot constitutive relation is generalized. This introduces a nondimensional parameter. It could be named the porosity effective pressure coefficient, because the measure of the extent the fluid pressure affects the change in porosity relative to the solid pressure. In the regime in which the fluid pressure affects to a lesser extent, this parameter spans from unity, the state in which fluid resists the change in porosity in equal but opposite manner to solid, to zero, the state in which fluid ultimately ceases to affect the porosity change at all. As this parameter diminishes from unity, the undrained bulk modulus drops from being the Gassmann modulus. Ultimately, it becomes the series combination of the dry frame bulk modulus with the bulk modulus of fluid weighted by the Biot coefficient when the parameter is vanishing. The other regime is the one in which the fluid pressure affects to an extent greater than the solid pressure. Here, the parameter may span from unity to the ratio of bulk modulus of constituent solid mineral to fluid, which is its upper limit. At the upper limit, the undrained bulk modulus is the Voigt average: the upper bound of the modulus of a composite medium.

On wave propagation characteristics in fluid saturated porous materials by a nonlocal Biot theory

“…However, it no longer keeps standing because of the introduction of nonlocal parameter into the coefficient of the polynomial Eq. (19). If we examine the coefficient MH 1 þαMH 2 , it is observed that when critical circular frequency ω c has no effect on fast wave.…”

Drained pore modulus determination using digital rock technology

“…Here η is the porosity at the current state of deformation and η0 is the porosity at the reference (undeformed) state. While the change of porosity does not appear explicitly in Biot's theory, Sahay (2013) has shown that there is always an underpinning porosity equation. These three kinematic variables can be changed by a combination of the fluid pressure (p f ) and the confining pressure (p c ).…”

“…The bulk moduli of the solid and fluid phase are Ks and Kf, respectively. The porosity perturbation equation entails the parameter n as a macroscopic measure for micro-inhomogeneity and differs from the Biot porosity equation in that the porosity change is not simply determined by the difference pressure (Sahay, 2013). The constitutive equations (2) and (3) also differ from the Biot constitutive equations in that the Biot coefficient (α) and the effective pressure coefficient for the bulk volume (α*) are taken as distinct quantities (Müller and Sahay, 2016b) as well as the fluid storage coefficient involves the effective pressure coefficient for the bulk volume.…”

“…The constitutive equations (2) and (3) also differ from the Biot constitutive equations in that the Biot coefficient (α) and the effective pressure coefficient for the bulk volume (α*) are taken as distinct quantities (Müller and Sahay, 2016b) as well as the fluid storage coefficient involves the effective pressure coefficient for the bulk volume. These constitutive equations are based on the volume averaging approach of poroelasticity developed in Sahay et al (2001); see also Sahay (2013) for further details. …”

“…Since the introduction of the poroelasticity theory by Biot (1941), the definition of an effective deformation moduli for the constitutive relations has been sought (Biot and Willis, 1957;Makhnenko and Labuz, 2016;and Cheng, 2016 for an overview). We use volume averaging of the pore-scale equations to define this parameter (Sahay et al, 2001;Sahay, 2013;Müller and Sahay, 2013;Müller and Sahay, 2016a), which describes the effects of pore-scale heterogeneities.…”