Generalized Darcy's Law With Source And Heterogeneity Effects
Authors: Das A.K. 
Published: March 2010

Abstract
Steady state Darcy's law has long been used (since 1856) for pressure prediction from continuity equation in porous media fluid flow. Effects of a source term, heterogeneity such as spatial porosity variation and unsteady nature of flow on Darcy-equivalent velocity have been investigated in this paper. In absence of Darcy type experimental observation in presence of these three factors, an analytical Navier-Stokes equation type investigation has been presented. These model equations are based on a new "pore average velocity" concept rather than conventional superficial velocity concept. Large pressure prediction errors are indicated for individual factors (source or heterogeneity) ranging to as high as 1,000 psi (7,000 kpa) for typical time steps (10 days) used in petroleum reservoir simulation. For implicit formulation unbounded pressure errors may result. There appears to be a severe need for time step limitation which makes modelling difficult and impractical.

Introduction
Fluid motion through porous substances has been a subject of keen interest since the time of Darcy(1). Over the years, many experimental and theoretical works have been performed leading to interesting resolutions on the subject. The applications of these resolutions today are in four main areas viz. petroleum engineering, soil science, hydrology, and chemical engineering. The major areas of application in petroleum engineering are reservoir simulation and well testing.
In this paper, a fresh new approach to the study of porous media fluid dynamics has been presented keeping in mind, especially, the petroleum engineering applications. The need for this method arose from a motivation to generalize the original Darcy experiment to include the effects of porosity variation(2), source effect(3) and unsteady nature(4) of flow.
To date, all the works on porous media flow can be divided into two major groups : those falling in line with Darcy's experimental conclusion that flow velocity (called superficial velocity) in pores is proportional to pressure gradient(1) and those reconciling with the more general Navier-Stokes type analysis commonly found in the subject of fluid mechanics(5). While the former has enjoyed more widespread acceptance, the latter is fraught with some major criticisms. The criticism to Navier-Stokes type analysis relates to the special structure of porous media that offers a guided motion to permeating fluid which breaks down the true "continuum" attribute of the flowing medium.
A series of recent studies by this author revealed that considerations of the presence of a source term, heterogeneity (such as porosity variation spatially) and unsteady fluid motion in the traditional Darcy sand element lead to considerable error in calculations of pressure in common applications such as reservoir simulation and well testing. No matter whether Navier-Stokes approach is objectionable or not, the very fact that presence of these three effects will alter the Darcy derived ū ∝? ∇ Prelation can not be denied, at least until new experiments are performed to investigate this aspect. In the absence of immediate experimental means of investigation, present analytical investigation has been taken up and the present paper is the outcome of these analytical investigations.