Stress Dependence Models

Yilmaz and Nur Stress Model

Yilmaz and Nur presented a simple relationship, which uses the concept of a “permeability modulus” or gamma γ which is comparable to the definition of compressibility. As is known, the dependence of permeability on pore pressure is commonly observed in tight gas projects, and is shown to vary exponentially. The dependence of permeability on pore pressure makes flow equations strongly nonlinear. Work by Rodriguez and Barbara demonstrated that there is a greater degree of permeability reduction with low permeability, when discussing tight gas.

and

where:

k_i = initial permeability

k = current permeability at new pore pressure, PR

γ = permeability modulus

Note the similarity to changes in rock porosity due to reduction in pore pressure.

γ is assumed constant for more practical purposes. The choice of a exponential relationship has some experimental evidence, in addition to being a mathematical convenience.

The image below, adapated from Barbara (2004) shows that permeabilty during as an exponential function correlates well with field data.

One can visualize pressure net confining pressure as P_i - P

If pressure dependent of permeability is ignored, erroneous values of permeability, skin, and OGIP will be calculated from history-matching, or erroneous production forecasts predicted.

For Coal Seam Gas (CSG) , as the reservoir pressure decreases, the overburden compresses the cleats thereby reducing the permeability - effectively stress changes the shape of coal cleats, and therefore ultimately cleat permeability. A schematic of this phenomenon is shown in the following image.

More specifically, for a given base, the stresses acting on coals increase with depth leading to the expectation of permeability decrease with depth (see examples of CSG Permeability vs Depth ). For a given coal seam, stresses increase over the course of depletion, leading to the expectation of permeability decrease as gas is produced.

CSG Stress Model

According to Seidle [2011], using a matchstick model, permeability can be shown to be exponentially related to stress:

Where:

k_i = initial permeability

k = current permeability at new hydrostatic stress

o_h = hydrostatic stress

o_hi = initial hydrostatic stress

c_f = Cleat Compressibility

This work has been expanded on by many authors and researchers which is not shown here.

References:

• Rodriguez, Cesar Alexander, Stress-Dependent Permeability on Tight Gas Reservoirs , 2004, Texas A&M University,

• Barbara, Mariela Franquet , Effect of Pressure-Dependent Permeability on Tight Gas Wells , 2004 
  Texas A&M Univesrsity,

• Vairogs, J., Hearn, C. L., Dareing, D. W., and Rhoades, V. W.: “Effect of Rock Stress on Gas Production from Low-Permeability Reservoirs,” JPT (September 1971) 1161-67; Trans., AIME, 251.

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

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