Matrix Shrinkage Models

Introduction

The term “matrix shrinkage” effect occurs when the volume of the coal matrix decreases during desorption; this in turn causes fractures to dilate (it may counter reduction in permeability due to increased effective stress). It is believed that this phenomenon is related to surface energy of coal during desorption—release of gas increases coal surface energy, which causes coal surface to contract during desorption. This leads to an increase of permeability.

It is different than loss of permeability due to pressure depletion (i.e. stress sensitivity permeability).

Palmer and Mansoori

Palmer and Mansoori described combined stress and matrix shrinkage influences on coal permeability using a ratio of porosity at a given pressure to the porosity at initial pressure. In the early years of CBM exploration, it was a common model.

Two competing effects, cleats compression due to depletion and matrix shrinkage, result in the permeability rebound on the lower pressure.

The Palmer and Mansoori equation is given below, starting with normalized porosity:

Where:

Φ = porosity at current pressure, dim

Φ_i = porosity at initial pressure

K = Bulk Modulus , psia

M = constrained axial modulus, psia (and is a function of Poisson's Ratio and E)

P = current reservoir pressure, psia

P_i = initial reservoir pressure, psia

c_m = psia^-1 and is sometimes shown as c_f (or formation compressibility)

c_o = volumetric strain coefficient, psia^-1 and is sometimes shown as:

b = 1/P_L is also shown often as B or β = 1/P_L (see other forms below)

The constrained axial modulus is given by:

and v = Poisson's Ratio (dimensionless)

The ratio of K/M is given by:

And cm is give by:

Where:

β_g = grain compressibility, and is often assumed to be zero. Seidle (XXX) also shows c_m = 1/M

f= fraction 0 → 1. An empirical factor

Other forms and nomenclature of the Palmer and Mansoori equation include:

Where:

β = 1/ PL and is different from the β_g term found in c_m above

Assuming β = b gives, and using “o” instead of “i”, we have

Where:

Φ_o = porosity at initial pressure

P_o = initial reservoir pressure, psia

See Also:

References:

• Clarkson, C. R.  Unconventional Reservoir Rate-Transient Analysis: Volume I and II  2021,  Gulf Professional Publishing.

• Gierhart, R.R., Clarkson, C.R., and Seidle, J.P. 2007. Spatial Variation of San Juan Basin Fruitland Coalbed Methane Pressure Dependent Permeability: Magnitude and Functional Form. Paper IPTC 11333 presented at the International Petroleum Technology Conference held in Dubai, U.A.E., 4-6 December. Full-length conference paper.

• Salmachi, A., Clarkson, C., Zhu, S., Barkla, J.A., 2018. Relative permeability curve shapes in coalbed methane reservoirs. In: Paper SPE 192029 presented at the SPE Asia Pacific Oil and Gas Conference and Exhibition, Brisbane, Australia, 23–25 October.

• John Seidle, Fundamentals of Coalbed Methane Reservoir Engineering, 2011 PennWell Corporation

• Lin, Wenjuan, Gas sorption and the consequent volumetric and permeability change of coal 
  2010, University of Stanford.

• R. R. Tonnsen, A Study of the Relationships Between Permeability, Stress, & Pore Volume Compressibility in Deep Coalbed Methane Environments, M. Sc., Colorado School of Mines, 2006.