How Permeability Depends on Stress and Pore Pressure in Coalbeds: A New Model
Authors: Palmer Ian , Mansoori John 
Published: December 2007

Abstract
In naturally fractured formations, such as coal, permeability is sensitive to changes in stress or pore pressure (i.e., effective stress). This paper presents a new theoretical model for calculating pore volume compressibility and permeability in coals as a function of effective stress and matrix shrinkage, using a single equation. The equation is appropriate for uniaxial strain conditions, as expected in a reservoir. The model predicts how permeability changes as pressure is decreased (i.e., drawdown). Pore volume compressibility is derived in this theory from fundamental reservoir parameters. It is not constant, as often assumed. Pore volume compressibility is high in coals because porosity is so small. A rebound in permeability can occur at lower drawdown pressures for the highest modulus and matrix shrinkage values. We have also history matched rates from a "boomer" well in the fairway of the San Juan basin using various stress-dependent permeability functions. The best fit stress-permeability function is then compared with the new theory.

Introduction
In naturally fractured formations, such as coal, permeability is sensitive to changes in stress or pore pressure (i.e., changes in effective stress). During drawdown of a reservoir by primary production, effective stress increases and permeability decreases due to cleat compression. However, in coalbeds, drawdown leads to desorption of methane, and this is accompanied by matrix shrinkage which opens the cleats and leads to permeability increase. The two effects of cleat compression and matrix shrinkage act in opposite directions on permeability.
The purpose of this report is to present a new theoretical formulation for stress-dependent permeability, which includes both stress effects and matrix shrinkage in a single equation. The equation is appropriate for uniaxial strain conditions, as expected in a reservoir. The new formulation also predicts pore volume compressibility, which is not constant, as commonly assumed. This work is important in interpreting gas production behavior during drawdown. It may also have implications for enhanced recovery by gas injection.
Seidle et al. measured pore volume compressibility from stress-dependent permeability experiments on cores in the lab. Here we derive stress-dependent permeability from an equation that has the advantage that it applies to uniaxial strain, which is the usual condition in the reservoir, plus it combines cleat compression due to pore pressure falloff, and matrix shrinkage due to gas desorption together in one equation. The matrix shrinkage term is a function of pore pressure, and is incorporated in this way. Finally, we have attempted to history match rates from a "boomer" well in the fairway of the San Juan basin by incorporating various stress-dependent permeability functions.

Theory
The derivation starts from the following equation of linear elasticity for strain changes in porous rock:
(1)
The symbols are given at the end of the text. The incremental pore volume strain d p is a result of a simple volumetric balance. In this equation, changes in porosity are assumed to be small (i.e., linear elasticity). The change in pore volume strain d p leads to a change in porosity as follows:
(2)
where the compressibility of fluid in the pores is assumed very high (i.e., gas saturation is nonzero). M (constrained axial modulus) and K (bulk modulus) are related to Young's modulus, E, and Poisson's ratio, V, via isotropic elasticity theory.
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