Characterization of Swelling Modulus and Effective Stress Coefficient Accommodating Sorption-Induced Swelling in Coal

Guijie Sang, Derek Elsworth, Shimin Liu, Satya Harpalani

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24 Scopus citations


The effective stress law transforms external stress (σ) and pore pressure (p), into a single equivalent variable (σeffective), expressed as σeffective = σ - αp, where α is the effective stress coefficient. For porous media, every property such as drained deformability, permeability, storage capacity, and acoustic velocity has its own particular effective stress coefficient. We extend the effective stress law for deformation in sorbing porous media (coal and organic-rich shales), accommodating sorption-induced swelling, by introducing the concept of an effective modulus of swelling/shrinkage. This attributes the volumetric strain (εv) of the sorbing medium to changes in the effective stress as εv = (σ - αsp)/K, with the effective stress coefficient αs = 1 - K/Ks + K/Zp, in terms of the bulk modulus (K) of the sorbing porous medium, the bulk modulus (Ks) of the solid grains and the swelling modulus (Zp). Thus, the static problem of deformation in sorbing porous media can be simplified into an elastic problem in nonporous and nonsorbing media with merely one variable: effective stress (σ - αsp). Unconstrained experiments on coal define the swelling modulus (Zp) and its effective stress coefficients for CH4 and CO2. At low gas pressures (<7 MPa), the swelling modulus (Zp) is an order of magnitude lower than the bulk modulus of solid grains (Ks) as ∼4 < Ks/Zp < 30, depending upon the particular gas (CH4/CO2) and gas pressure. This is consistent with the dominant influence of sorption-induced swelling at low gas pressures and its important effect on stress-permeability evolution during depletion of coalbed methane. Where the influence of swelling is included, the effective stress coefficient may exceed the normal bound (0 < α < 1) of unity for CO2CO2) and for CH4CH4). For the stronger affinity of CO2 to coal, αCO2 is ∼2-3 times larger than αCH4, varying with gas pressure. As anticipated, for relatively "stiff" sorbing media, αCO2 and αCH4 are much larger than 1 and decline more rapidly with an increase in gas pressure, compared to relatively "soft" sorbing media, where αCO2 ≈ αCH4 ≈ 1, where the decline is less rapid with gas pressure. The effective stress coefficient for non/lightly sorbing helium remains constant at αHe less than ∼1 for both "soft" and "stiff" sorbing media. This effective stress law also applies to uniaxial conditions-and, appropriately, is shown to be independent of mechanical boundary conditions. Experiments on coal samples under uniaxial strain conditions validate this. The results indicated that the effective stress, accommodating sorption effects, may be transformed among different mechanical boundary conditions as a unified effective stress coefficient. Under uniaxial strain, the effective stress can be expressed as a function of overburden stress and pore pressure as σeffective = σv - αsp, which attributes to the changes in volumetric strain as εv = σeffective/M, where M is the constrained axial modulus.

Original languageEnglish (US)
Pages (from-to)8843-8851
Number of pages9
JournalEnergy and Fuels
Issue number9
StatePublished - Sep 21 2017

All Science Journal Classification (ASJC) codes

  • Chemical Engineering(all)
  • Fuel Technology
  • Energy Engineering and Power Technology


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