### Abstract

The effective stress law transforms external stress (σ) and pore pressure (p), into a single equivalent variable (σ_{effective}), expressed as σ_{effective} = 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 (ε_{v}) of the sorbing medium to changes in the effective stress as ε_{v} = (σ - α_{s}p)/K, with the effective stress coefficient α_{s} = 1 - K/K_{s} + K/Z_{p}, in terms of the bulk modulus (K) of the sorbing porous medium, the bulk modulus (K_{s}) of the solid grains and the swelling modulus (Z_{p}). 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 - α_{s}p). Unconstrained experiments on coal define the swelling modulus (Z_{p}) and its effective stress coefficients for CH_{4} and CO_{2}. At low gas pressures (<7 MPa), the swelling modulus (Z_{p}) is an order of magnitude lower than the bulk modulus of solid grains (K_{s}) as ∼4 < K_{s}/Z_{p} < 30, depending upon the particular gas (CH_{4}/CO_{2}) 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 CO_{2} (α_{CO2}) and for CH_{4} (α_{CH4}). For the stronger affinity of CO_{2} to coal, α_{CO2} is ∼2-3 times larger than α_{CH4}, varying with gas pressure. As anticipated, for relatively "stiff" sorbing media, α_{CO2} and α_{CH4} are much larger than 1 and decline more rapidly with an increase in gas pressure, compared to relatively "soft" sorbing media, where α_{CO2} ≈ α_{CH4} ≈ 1, where the decline is less rapid with gas pressure. The effective stress coefficient for non/lightly sorbing helium remains constant at α_{He} 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 σ_{effective} = σ_{v} - α_{s}p, which attributes to the changes in volumetric strain as ε_{v} = σ_{effective}/M, where M is the constrained axial modulus.

Original language | English (US) |
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Pages (from-to) | 8843-8851 |

Number of pages | 9 |

Journal | Energy and Fuels |

Volume | 31 |

Issue number | 9 |

DOIs | |

State | Published - Sep 21 2017 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Energy and Fuels*,

*31*(9), 8843-8851. https://doi.org/10.1021/acs.energyfuels.7b00462