A dual-porosity model for gas reservoir flow incorporating adsorption behaviour—part I. Theoretical development and asymptotic analyses

In this paper a rigorous dual-porosity model is formulated, which accurately represents the coupling between large-scale fractures and the micropores within dual porosity media. The overall structure of the porous medium is conceptualized as being blocks of diffusion dominated micropores separated by natural fractures (e.g. cleats for coal) through which Darcy’s flow occurs. In the developed model, diffusion in the matrix blocks is fully coupled to the pressure distribution within the fracture system. Specific assumptions on the pressure behaviour at the matrix boundary, such as step-time function employed in some earlier studies, are not invoked. The model involves introducing an analytical solution for diffusion within a matrix block, and the resultant combined flow equation is a nonlinear integro-(partial) differential equation. Analyses to the equation in this text, in addition to the theoretical development of the proposed model, include: (1) discussion on the “fading memory” of the model; (2); one-dimensional perturbation solution subject to a specific condition; and (3) asymptotic analyses of the “long-time” and “short-time” responses of the flow. Two previous models, the Warren-Root and the modified Vermeulen models, are compared with the proposed model. The advantages of the new model are demonstrated, particularly for early time prediction where the approximations of these other models can lead to significant error.