Analysis and Verification of Dual Porosity and CBM Shape Factors
Authors: Mora C.A. , Wattenbarger R.A. 
Published: March 2010

Abstract
A naturally fractured reservoir is characterized as a system of matrix blocks with each matrix block surrounded by fractures. The fluid drains from the matrix block into the fracture system which is interconnected and leads to the well. Warren and Root(1) introduced a mathematical model for this dual porosity matrix-fracture behaviour.
Their model has been widely used for many types of reservoirs, including tight gas and coalbed methane reservoirs. A key part of their model is a geometrical parameter (shape factor) which controls the drainage rate from the matrix to the fractures. Although Warren and Root gave formulas for calculating shape factors, many other authors have presented alternate formulas, leading to considerable confusion.
In addition to the size and shape of a matrix element, two cases are considered by the authors: constant drainage rate from a matrix block and constant pressure in the adjacent fractures.
The current work confirmed the correct formulas for shape factors by using numerical simulation for the various cases. It was found that some of the most popular formulas do not seem to be correct. A summary of the correct shape factor formulas is presented.

Introduction
Naturally fractured reservoirs can be characterized as a system of fractures in very low conductivity rock. The mathematical formulation of this 'dual porosity' or 'double porosity' system of matrix blocks and fractures was presented by Barenblatt et al.(2) The first system is a fracture system with low storage capacity and high fluid transmissibility and the second system is the matrix system with high storage capacity and low fluid transmissibility. The matrix rock stores almost all of the fluid, but has such low conductivity, that fluid just drains from the matrix 'block' into adjacent fractures, as is shown in Figure 1. The fractures have relatively high conductivity, but very little storage.
The drainage from the matrix to the fractures for dual porosity reservoirs was idealized by Warren and Root(1) according to Equation (1).
Equation 1 (available in full paper)
Equation (1) is in the form of pseudosteady-state flow which means that early transient effects have been ignored. Pseudosteady-state also means that the drainage rate is constant. The units of Equation (1) are volume rate of fluid drainage per volume of reservoir. The units of the shape factor, s, are 1/L2.
For dual porosity reservoirs, when pressure test analyses are available, the product s - km can be determined using Equation (2), but cannot be separated.
Equation 2 (available in full paper)
The inter porosity flow coefficient, ?, determines the interrelation between matrix blocks and the fracture system. When km is available from core or log analysis, then shape factor, s, can be estimated. For cases where pressure test analyses are not available, formulas can be used to estimate shape factor. However, there are conflicting equations and values for s in the literature.
Many authors have interpreted Equation (1) to be the equivalent long-term reservoir drainage equation with pf held constant and drainage rate changing with time.