Unified Fractional Derivative Transient Inter-porosity Flow Model for Naturally Fractured Media

Abstract

A unified approach to modeling flows of slightly compressible fluids through naturally fractured media using transient interporosity models is presented. The unified fractional differential transient interporosity flow model is derived using the methodology of asymptotic analysis for problems with small parameters arising based on the assumption about much lower permeability of matrix blocks compared to fractures. The model is established for any shape of matrix blocks and unifies existing transient interporosity flow models formulated for matrix spheres, cylinders and slabs. It is formulated in the form of a fractional order partial differential equation with the Caputo derivative of order 1/2 with respect to time. Explicit solutions for the unified model are derived for different axisymmetrical domains using Hankel-Weber transforms. The surface area-to-volume ratio is shown to be the fundamental parameter in the flow description. The rescaling technique that involves this parameter allows observing similarities of flow for any matrix geometries.