Generalized Darcy's Law Including Source Effect

Abstract

Conventional Darcy's law for flow description in porous media was derived from a simple experiment carried out by Darcy in 1856 on a linear, homogeneous-and sourceless porous, medium. In this paper starting from basic Navier-Stokes momentum balance principle, a generalized Darcy's law has been derived that includes the effect of a source/sink term. Navier-Stokes momentum balance of similar nature was attempted by Casulli et al. ⁽¹⁾, also in their studies on porous media flow even though many authors have objected to such analysis. Brinkman ⁽²⁾ and Hubbert ⁽³⁾ have also hinted at such .. application of Navier-Stokes equation to derive the traditional Darcy's law. In deriving the generalized Darcy's law, described in this paper, a similar to Casulli et al. ⁽¹⁾, linear approximation of, the stress term has been used.

A numerical investigation of the error resulting from not considering the source effect reveals that errors in pressure prediction may range to as high as 500 psi; This is a significant error and cannot be overlooked.

Introduction

Darcy's law that is widely used in petroleum engineering, hydrology and soil science mathematics has its origin in Darcy's ⁽⁴⁾ simple experiment on a linear, homogeneous, sourceless porous medium. To this date effect of a source term on Darcy velocity term has not been explored barring one work ⁽⁵⁾, In numerical reservoir simulation both source and heterogeneity are involved inthe input data, but not in the form of altered momentum expression such as a modified Darcy' s law. In this paper effect of source on Darcy flow velocity has been derived.

It is opined that Navier-Stokes equation widely referred to in continuum fluid mechanics cannot exactly describe porous media flow, primarily because of the very structure of porous medium which offers a guided motion to permeating fluid. However, such analysis has been attempted by Casulli et al. ⁽¹⁾, Brinkman ⁽²⁾ and Hubbert ⁽³⁾. There are possibly more references on such applications that have not come across this author's notice. In this paper a full Navier-Stokes type momentum balance has been considered as a starting equation and from this a generalized Darcy's law with source term has been derived. Before illustrating the derivation, it appears necessary to take note of Marle's ⁽⁶⁾ work which describes macroscopic averaging of microscopic (pore level) equations for representing porous media flow. From this standpoint it may appear that the current derivation in this paper or the starting Navier-Stokes equation is not valid. To this end, this author argues that the main motivation of averaging, which is avoiding of a meaningless concept of porosity at a point, is not important actually. In application phase such as in numerical simulation we consider a finite elemental volume with non-zero Δx, Δy, Δz of the elemental block and therefore a finite pore volume is considered and so meaningless point porosity situation is not encountered at all.

Derivation of Generalized Darcy's Law

Consider the elemental volume of size Δx, Δy, Δz. Then the accumulation term will be Δx Δy Δz( δ/ δt)( ρ φ> u _x). The equation of motion in x direction will be :