Poromechanics Constitutive Relations and Analytical Solution for Nonlinear Gas Transport and Sorption in Deformable Rocks

Flow of gas through porous media can be coupled with sorption against the pore walls and deformation of the solid phase. This paper presents a thermodynamic description of the described three-way coupling in porous elastic media with two distinct networks or scales of porosity. The developed framework is next applied to the problem of natural gas sorption and flow toward a producing boundary in fractured rocks. The considered problem involves a set of strong and entangled nonlinearities arising from the coupled processes of sorption and solid strain, resulting changes in rock permeability, as well as pressure dependence of gas compressibility. An analytical solution to this problem is developed using the homotopy-perturbation method. The obtained solution compares well against a numerical solution to the same problem. The solution rigorously accounts for local dependence of the involved parameters on the pore fluid pressure. Results from a case study indicate that compaction of the pore space and the consequential permeability loss would cause substantial reduction in gas production rate. However, the same process triggers desorption of excess gas molecules from the pore walls serving to negate the compromised flow rate due to rock compaction. The arisen entanglement between gas desorption and bulk deformation in rocks is further studied by identifying the pertinent dimensionless variable groups. Gas flow rate for a given pressure disturbance at the producing boundary is shown to exhibit nontrivial variations with changes in the values of two dimensionless groups describing the porous frame compressibility and sorption strength at the pore walls.