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.

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