Green's Functions for Solving Unsteady Flow Problems in Naturally Fractured Reservoirs With Arbitrary Fracture Connectivity: Part I-Theory

A general theory of Green's functions for solving boundary-initial value problems governed by two linear pressure diffusion equations coupled through a fluid transfer term which depends on the pressure histories, is presented. The governing equations, developed previously by Wijesinghe and Culham (1984), describe the flow of fluid in naturally fractured reservoirs with arbitrary fracture connectivity simultaneously taking into account both fracture and pore system permeabilities and unsteady interporosity fluid transfer. In this study, the relationship between the Green's functions (ie. generalized source solutions) and the adjoint Green's functions is determined, and the boundary conditions which must be imposed on the adjoint Green's functions to obtain explicit solutions for the fracture and pore pressures, are presented. The fundamental Green's functions, which characterize the basic properties of the two coupled differential equations, are derived and examples of their application to reservoir fluid flow problems are given. As in the case of the classical diffusion equation for fluid flow in single porosity formations, the Green's functions based on the present theory enable solutions to complex fluid flow problems in naturally fractured reservoirs to be systematically constructed from simpler source solutions.