Abstract
Previous researchers have used the cutting and rejoining model to represent a porous body when attempting to place Archie's law on a theoretical basis and calculate rock permeabilities (F = phi m, where F = formation factor, phi = porosity and m = a constant). In using the model, a porous solid is assumed to consist of sections with parallel sides containing straight, continuous pores of total porosity phi (pore to total volume or, for a homogeneous body, pore to total area). The joining of pores at an interface between two sections is assumed to be random which results in a common pore area phi. This reduced area phi is then taken as that area which limits electrical conduction or fluid flow in the body. Agreement between experimental results and results calculated using a model is not always good. Because of this, various modifications of the model are presentedfor instance, the random nature of the joining is questioned. Formation factor measurements on specially constructed porous specimens are described in this paper. It is concluded that the common pore area limits flow completely only if sections containing straight pores are very thin, and that, in most natural materials, the common pore area at an interface must be greater than that given by random joining. However, the model is still useful in permeability calculations, provided that the fractional area limiting flow is determined from formation factor measurements.
Introduction
Two important properties of porous media are permeability k, which describes the ease with which a fluid can flow through a porous medium, and formation factor F, defined as the ratio of the resistivity of a saturated specimen to the resistivity of the saturating liquid which describes the electrical properties of the saturated material. k is defined by Darcy's law,
(1)
where mu = liquid viscosity, q = flow rate, L = length of specimen, p = pressure across the specimen and Ae = the effective area over which flow occurs. F is given experimentally by Archie's law,(2)
where phi is porosity defined as the pore to total volume or, for a homogeneous body, the pore to total area; and m lies between 1.3 for loose sand and 2 for cemented sand. Consider a porous body of total cross-sectional area A and length L, saturated with a liquid of resistivity R and with an effective area for conduction of Ae. Then, from the definition of F given previously,
(3)
or, from Eqs. 2 and 3,
(4)
Thus, in any attempt to calculate k or F, some assumptions must be made concerning the pore structure of the body and a value derived for Ae.
To determine Ae, Childs and George and Wyllie and Gardner suggested an elementary statistical representation often referred to as the cutting and rejoining model of a porous body. The body is assumed to be composed of two or more parallel-sided sections of porosity phi, containing straight, continuous pores with a common pore area of phi at an interface between two sections. The value is given by a random joining of the pores in either section.
SPEJ
P. 13ˆ
Characteristic form of dynamics equations of Cosserat medium
