Geometrical models for poroelastic behaviour

Abstract

S U M M A R YPoroelasticity implicitly incorporates pore structure information through the use of empirically determined macroscopic parameters; hence, quantitative analysis of pore geometry effects on poroelastic behaviour cannot be performed. Analogues for poro-. elastic parameters with explicit dependence on pore structure are derived here by using an inclusion-based model where inclusions represent individual pores. The inclusionbased formulation used in this paper permits uniform pore fluid pressure throughout the pore space, a requirement of poroelasticity. When specific inclusion-based approximations resulting from different descriptions of inclusion interactions were considered, it was found that the analogue quantities obtained from the dilute volumetric average, Kuster-Toksoz and equivalent inclusion-average stress approximations replicated the relationships between poroelastic parameters predicted by poroelasticity. This equivalence implies that pore geometry information can be consistently incorporated into poroelastic parameters with an inclusion-based model when one of these approximations is used.This connection is used to examine pore geometry effects on poroelastic behaviour by considering a simple model with identically shaped pores. The results of this modelling study show that variations in pore shape significantly affect poroelastic parameters and that the nature of these effects is controlled by the specified applied stress-strain and fluid pressure conditions. The following observations were made about specific poroelastic quantities. The Skempton ratio is relatively insensitive to porosities below 0.1. Its minimum value for a given solid matrix, pore fluid and total porosity level is obtained when all pores are spherical; this value can be significantly greater than the theoretical lower limit of zero. Specific storage terms increase by several orders of magnitude as pores become more crack-like. The difference between the traditional definition of specific storage and that proposed by Green & Wang ( 1990) for 'normal' aquifer conditions grows as the pore aspect ratio decreases.

Constitutive mechanical relations of solid‐liquid composites in terms of grain‐boundary contiguity

“…These models successfully predict that the macroscopic properties depend not only on the volume fraction of the liquid phase but also strongly depend on the shape of the inclusions, such as spheres [e.g., Mackenzie, 1950; Eshelby, 1957], oblate spheroids with variable aspect ratio [Wu, 1966;Walsh, 1965Walsh, , 1969Kuster and ToksSz, 1974; ToksSz e! al., 1976; $chraeling, 1985;Endres, 1997], and tube geometry with variable pore shape [Mavko, 1980]. These shapes, however, have been chosen for the sake of mathematical simplicity to calculate the mechanical effects of the inclusions.…”

The interacting inclusion model of wave-induced fluid flow

“…structural clay in a reservoir of sandstone). The formulation is inspired by the works of Sheng (1990), Hudson et al (1996), Endres (1997), Xu (1998) and many others (see Jakobsen et al 2003b). The population of inclusions is divided into families of inclusions having the same shape/orientation, t‐matrices t ( n ) (discussed below), and volume concentration v ( n ) , labelled by n = 1, 2, … , N .…”

“…In any case, one has to transform from the stiffness to the compliance domain if the goal is to demonstrate that an inclusion‐based model of rocks as viscoelastic composites is consistent with the relations of Brown & Korringa (1975) (or their isotropic version discovered by Gassmann (1951)) for low‐frequency dependence of the elastic properties of a porous rock on the compressibility of the pore fluid (e.g. Thomsen 1985; Endres 1997).…”

“…Until quite recently, it appeared virtually impossible to design an inclusion‐based model of real media which is not only consistent with the Brown–Korringa relations but also allows for non‐dilute inclusion concentrations. Consistency with Brown and Korringa is required in order to mimic the behaviour of real media characterized by a connected pore space (see Thomsen 1985; Endres 1997). The problem is to achieve this consistency with Brown and Korringa without having to ignore the essential storage porosity (e.g.…”

The acoustic signature of fluid flow in complex porous media

“…Dvorkin and Nur (1993) have developed a unified model with the local and global flow mechanisms, which is [not consistent with the theory of Gassmann (1951), as it should be (see Thomsen, 1985), according to Chapman et al (2002)] valid for porous media with a single solid constituent only and have another serious drawback in common with Biot's original model: microstructural information is simply incorporated through the use of empirically determined macroscopic parameters. In principle, it is possible to remove this drawback (see Burridge and Keller, 1981) and/or extend Biot's theory to complex porous media (see Berryman and Milton, 1991;Berryman, 1992Berryman, , 1998Berryman and Wang, 1998), but the works of many scientists (e.g., O'Connell and Budiansky, 1977;Budiansky and O'Connell, 1980;O'Connell, 1983;Hudson et al, 1996;Ravalec and Gueguen, 1996;Endres and Knight, 1997;Endres, 1997;Xu, 1998;Pointer et al, 2000;Tod, 2001Tod, , 2002 suggest that a good theory for rocks need not necessarily be based on Biot's approach at all.…”