SPE Reservoir Evaluation & Engineering
Volume 11, Number 4, August 2008, pp. 735-749

Summary

The shape factor concept, originally introduced by Barenblatt in 1960, provides an elegant and powerful upscaling method for fractured reservoir simulation. The shape factor determines the fluid and heat transfer between matrix and fractures when there is a difference in pressure or temperature between matrix blocks and the surrounding fractures. An appropriate specification of the shape factor is therefore critical for accurate modeling.

Since its introduction, many different values for the shape factor have been proposed in the literature, among which the well-known Warren-Root and Kazemi shape factors. The aim of this paper is to show that the selection of the appropriate shape factor should not only depend on the "shape" and dimensions of matrix blocks, but should also take into consideration the character of the dominant underlying physical recovery mechanisms.

We will show that by taking into account the dominant physical recovery mechanism, the apparent discrepancies in the shape factor values reported in the literature can be overcome. We derive a general expression for the shape factor that not only captures existing shape factor expressions, but also allows extensions to recovery mechanisms requiring a dual permeability approach.

The paper is organized as follows. First, we briefly review the shape factors presented in the literature. We then derive the general expression for the (single-phase) matrix-fracture shape factor. Subsequently, we analytically derive a new shape factor that captures the transient in pressure/temperature diffusion processes. To compare and contrast the impact of the various shape factors, we consider three cases of increasing complexity. First, we consider pressure/temperature diffusion in a single 1D matrix block following a step change in the boundary conditions. Next, we consider isothermal gas/oil gravity drainage from a homogeneous stack. We compare fine-grid single-porosity simulations (in which the matrix is finely gridded and in which the fractures are explicitly represented) with coarse-grid dual-permeability simulations (in which the matrix-fracture interaction is modeled by shape factors). In the third step, we consider gas-oil gravity drainage of the same stack model, but now under steam injection. In this case, steam is injected at the top, and oil recovered from the base of the fracture system. Again, we compare fine-grid single-porosity simulations with coarse-grid dual-permeability simulations. We show that in this case, the constant (asymptotic) shape factor provides a good approximation to the heating of the stack. We will show, however, that with a constant (time-independent) shape factor, the initial fast heating of the matrix blocks cannot be captured. We show that the new transient shape factor, however, enables coarse-grid dual-permeability modeling of thermal recovery processes such that they reproduce fine-grid results.

Introduction

The modeling of matrix-fracture interaction using shape factors has been an active area of research for over 40 years now, and has attracted considerable attention both in the context of single- and multi-phase matrix-fracture modeling (Barenblatt et al. 1960; Warren and Root 1963; Kazemi et al. 1976; Thomas et al. 1983; Coats 1989; Ueda et al. 1989; Zimmerman et al. 1993a; Chang 1993; Lim and Aziz 1995; Gilman and Kazemi 1983; Beckner et al. 1987, 1988; Rossen and Shen 1989; Bech et al. 1991; Bourbiaux et al. 1999).

In their 1960 landmark paper, Barenblatt et al. introduced the shape factor concept to model the (single-phase) fluid transfer between matrix and fractures (1960). The central idea of Barenblatt et al. was not to study the behavior of individual matrix blocks and their surrounding fractures, but instead to introduce two abstract interacting media: one medium, the "matrix," in which the physical matrix blocks are lumped, and one medium, the "fractures," in which the fractures are lumped. Whenever a pressure difference exists between the matrix and the fractures, a fluid flow between the media will occur. The shape factor is then defined by the following relation, which ties the (single-phase) matrix-fracture fluid flow to the instantaneous pressure difference between matrix and fractures:

q = σ( k_m / μ ) V ( p*_m - p_f ) , ....[ EQ. 1 ]

where V denotes the volume of the matrix block. In 1963, Warren and Root used Barenblatt’s shape factor concept in the context of well-testing using dual porosity models. They postulated shape factors for 1-, 2-, and 3D matrix blocks, as given in Table 1. In 1976, Kazemi et al. proposed different shape factors, which were derived using a finite-difference discretization. Kazemi et al. also postulated the generalization of the shape factor concept from single- to multiphase flow by introducing the phase relative permeability into Eq. 1. Thomas et al. (1983) found that they could accurately reproduce fine-grid single-porosity simulation results of water/oil countercurrent imbibition (in cubical blocks) if in their single-cell dual-porosity model they used a shape factor 25 / L² . In their dual-porosity simulation, however, they also used pseudorelative permeability curves and a pseudocapillary pressure, so it is not obvious whether the good fit was mainly caused by the shape factor they used, or by the pseudosaturation functions.

Coats reported that the shape factor proposed by Kazemi is too low by a factor of 2, and derived new 1-, 2-, and 3D shape factors (1989); see Table 1. Ueda et al. (1989) also argued that the Kazemi shape factor should be multiplied by a factor 2 to 3, based on their work in which they compared dual porosity (two-phase) simulations with 1- and 2D fine-grid simulations. In 1993, Zimmerman et al. published a semi-analytical method for modeling of matrix-fracture flow in a dual-porosity model where the matrix blocks are modeled as spherical blocks (1993a). In their paper, they also show that the shape factor for spherical matrix blocks is given by π² / R² where R is the radius of the matrix block. In the same year, Chang derived an explicit formula for the single-phase shape factor for rectangular matrix blocks based on the full transient solution of the diffusion equation introducing new 1-, 2-, and 3D results to the shape-factor literature (1993). The same result was independently obtained in 1995 by Lim and Aziz. Both Chang and Lim and Aziz stressed that the shape factor, which had previously been regarded as a constant, is actually a function of time.

In view of the wide spectrum of results and the apparent lack of consensus regarding which shape factor to use in simulations, a more detailed analysis into the reasons for the different shape factors cited in Table 1 seems desirable. We want to underline that in this paper we focus our attention to single-phase shape factors, thus avoiding the additional complications that arise in the discussion of two-phase matrix-fracture interaction because of relative permeability and capillary pressure. This allows us to more clearly illustrate the different approaches that the previously mentioned authors used.

© 2008. Society of Petroleum Engineers

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History

• Original manuscript received: 5 July 2006
• Meeting paper published: 24 September 2006
• Revised manuscript received: 12 October 2007
• Manuscript approved: 23 December 2007
• Version of record: 20 August 2008