Direct and inverse methods for determining gas flow properties of shale

Gas flow in shale is a poorly understood and potentially complex phenomenon. It is currently being investigated using a variety of techniques including the analysis of transient experiments conducted on full core and crushed shale using a range of gases. A range of gas flow mechanisms may operate including continuum flow, slippage, transitional flow and Knudsen diffusion. These processes as well as gas sorption need to be taken into account when interpreting experimental data and extrapolating the results to the subsurface. A finite volume method is developed in this paper to mathematically model gas flow in shale. The finite volume method combines the efficiency/simplicity of finite difference methods with the geometric flexibility of the finite element approach. The model is applicable to non-linear diffusion problems, in which the permeability and fluid density both depend on the scalar variable, pressure. The governing equation incorporates the Knudsen number, allowing different flow mechanisms to be addressed, as well as the gas adsorption isotherm. The method is validated for unsteady-state problems for which analytical or numerical solutions are available. The method is then applied for solving a pressure-pulse decay test and a comparison with an alternative finite-difference numerical solution is made. An inverse numerical formulation is generated, using a minimisation iterative algorithm, to estimate different number of unknown parameters. Both numerically simulated noisy and experimental data are input into the formulation of the inverse problem. Error analysis is undertaken to investigate the accuracy of results. A good agreement between inverted and exact parameter values is obtained. Results for inversions done for practical laboratory pressure-pulse decay tests of samples with very low permeability are also presented.