Unified fluid flow model for pressure transient analysis in naturally fractured media

Abstract

Naturally fractured reservoirs present special challenges for flow modeling with regards to their internal geometrical structure. The shape and distribution of matrix porous blocks and the geometry of fractures play key roles in the formulation of transient interporosity flow models. Although these models have been formulated for several typical geometries of the fracture networks, they appeared to be very dissimilar for different shapes of matrix blocks, and their analysis presents many technical challenges. The aim of this paper is to derive and analyze a unified approach to transient interporosity flow models for slightly compressible fluids that can be used for any matrix geometry and fracture network. A unified fractional differential transient interporosity flow model is derived using asymptotic analysis for singularly perturbed problems with small parameters arising from the assumption of a much smaller permeability of the matrix blocks compared to that of the fractures. This methodology allowed us to unify existing transient interporosity flow models formulated for different shapes of matrix blocks including bounded matrix blocks, unbounded matrix cylinders with any orthogonal crossection, and matrix slabs. The model is formulated using a fractional order diffusion equation for fluid pressure that involves Caputo derivative of order 1/2 with respect to time. Analysis of the unified fractional derivative model revealed that the surface area-to-volume ratio is the key parameter in the description of the flow through naturally fractured media. Expressions of this parameter are presented for matrix blocks of the same geometrical shape as well as combinations of different shapes with constant and random sizes. Numerical comparisons between the predictions of the unified model and those obtained from existing transient interporosity ones for matrix blocks in the form of slabs, spheres and cylinders are presented for linear, radial and spherical flow types for both fixed boundary pressure and rate.