Correlation of the Non-Darcy Flow Coefficient
Authors: Liu X. , Civan F. , Evans R.D. 
Published: March 2010

Abstract
A general correlation for the non-Darcy flow coefficient with respect to permeability, porosity, and tortuosity using a wide variety of dam from consolidated and unconsolidated porous formations is presented. This correlation implicitly includes the effect of the overburden stress through permeability, porosity. and tortuosity and the effect of the existence of other fluid phases through the effective permeability.

Introduction
Because of its simplicity, the Forchheimer equation has been popular to represent the non-Darcy flow behaviour in petroleum reservoir. The Forchheimer equation for horizontal and one dimensional single phase fluid flow through a homogenuous porous media given by Forchheimert (1).
Equation (1) (Available in full paper)
and for multiphase flow by modifying the equation given by Schulenberg and Muller (2).
In Equation (1) and (2), a is the distance, p, u, µ and ρ are the pressure, volumetric flux, viscosity and density of the fluid, k, and β are the the permeability and non-Darcy flow coefficient of the porous media, l, denotes the l db phase: k1 andβ 1 are the effective permeability and the effective non-Darcy flow coefficient..
Development of correlations for the β coefficient has been of continuing interest (Janiek and Katz (3), Geertsma (4), Norman and Archer (5), Evans et al. (6), Evans and Evans (7) and Kutasov (8)). By-and-large, the previous studies dealing with single-phase flow have overlooked the importance of tortuosity, t, and correlated the β coefficient with respect to permeability k and porosity Ø alone. Some of these studies (6,7) included the overburden stress, although its effect is implicitly included through k and Ø and the tortuosity, t. A few studies (4,5) involving multiphase flow have incorporated the effect of the existence of other fluids in terms of the saturation correction on the fractional pore volume occupied by the phase of interest, whereas this effect is already accounted for by means of the effective permeability k1.
FIGURE 1: Correlating non-Darcy flow coefficient with permeability, porosity and tortuosity for a small number of rocks. (Available in full paper)

Formulation
The data such as porosity, permeability and tortuosity can provide a direct measure of the apparent characteristics of the pore structure and can be readily determined by means of well logging and/or well testing at in situ reservoir conditions. The tortuous paths in porous media have been recognized as a major cause of non-Darcy flow (1–9). Tortuosity provides useful information about the representative length of tortuous flowpaths in pore structures. Therefore, it is reasonable to take into account tortuosity, t, in the correlation of the non-Darcy flow coefficient, β, in addition to the porosity, Ø, and the permeability, k.
To demonstrate the importance of including the tortousity, t, the data by Cornell and Katz (10) who measured porosity, permeability, tortousity and non-Darcy flow coefficient for a number of rocks including sandstones, carbonates and dolomites have been used. Their laboratory data we plotted in Figure 1 to correlate the non-Darcy flow coefficient first with permeability, porosity and tortuosity.