Analysis of the simulation of single phase flow through a naturally fractured reservoir

A general form of the double porosity model for single phase flow through a naturally fractured reservoir is derived by explicitly considering fluid flow in individual matrix' blocks. The Warren and Root model is shown to be a crude approximation to this model. The general model consists of a parabolic equation coupled to a series of parabolic equations. It is shown that the coupling term can be viewed as a positive-semidefinite perturbation of the time derivative, and hence it is verified that the model is well posed. A finite element method is presented to approximate the solution, and optimal order $L^2 $-error estimates are derived.