Numerical and Experimental Modeling of Non-Darcy Flow in Porous Media

Abstract

Abstract A numerical simulation of the newly derived Forchheimer's diffusivity equation has been achieved, careful selection of the non-Darcy coefficient (ß) and a fruitful discussion on the use of this coefficient has been presented. Although the proposed model has been designed for single phase flow in porous media, with minor modification it can be applied to multi-phase flow cases. Results show that the proposed numerical model is valid for single-phase flow and is functioning in all ranges of flow in porous medium. The point at which the flow breaks from the Darcian trend has been determined and a new dimensionless term, "Be", has been suggested to define the point of deviating from Darcian flow. "Be" has been found to be 0.0526 at 5% deviation from the Darcian behavior. An experimental model has been designed to examine certain critical parameters experimentally for the purpose of comparing results with the numerical model predictions. A horizontal fracture has been induced to a homogeneous synthetic sample in the direction of flow with changing aperture to study the effect of fractures on flow behavior. Experimental results have been compared to the numerical model predictions; a satisfactory agreement within the domain of testing has been concluded which in turn encourages the implementation of this model on a field scale. Unlike other approaches, the non-Darcy coefficient "ß" has been determined experimentally which reflects the authors' belief that when possible, "ß" should be evaluated based on the same set of data used to determine permeability in the laboratory. Introduction The behavior of fluid flow in porous media depends on the flowing fluid and the porous medium properties as well as the flow characteristics - basically, velocity or flow rate - of the fluid. For many years Darcy's law has been considered the fundamental equation that governs liquid flow in porous media. Flow behavior of gas reservoirs where high-velocity flow is expected around the wellbore has been treated differently, using the Forchheimer's1 equation. An inertia term consists of density, velocity and the non-Darcy coefficient - a factor representing mainly the porous medium characteristics - are added to the existing viscous term in Darcy's equation to form the Forchheimer's equation. The non-Darcy coefficient "ß" appearing in Forchheimer's equation has the unit of inverse length, and so many researchers attributed its dependency on porous media characteristics2. Investigators3 in gas flow technology have used frequently the term "turbulent" and "non-Darcy" to describe visco-inertial flow at high velocities near the wellbore of a gas well. Although, turbulence might be conceivable in gas reservoirs, it can only develop in liquid reservoirs under very special conditions. Liquids under similar conditions of turbulence found in gas reservoirs most likely will flow in a laminar nonlinear mode, this cannot be considered turbulence; rather, it is better recognized as "non-Darcy" flow4. Numerical modeling is favored for its general solutions and its flexibility in representing all ranges of reservoir features. Experimental modeling is preferred in specific situations that have to be demonstrated physically. In this paper, a numerical simulation of Forchheimer's-based diffusivity equation is presented and an experimental modeling technique has been utilized to verify the proposed numerical model. Non-Darcy Coefficient Different names have been used for the term ß appearing in Forchheimer's equation. It has been called the turbulence factor, the coefficient of inertial resistance, the velocity coefficient, the non-Darcy flow coefficient, the Forchheimer flow coefficient, the inertial coefficient, the non-Darcy coefficient and the Beta factor. In this paper we shall use the term non-Darcy coefficient when referring to ß as it appears in Forchheimer's equation. Numerous attempts have been made in the past to derive the Forchheimer equation from theoretical point of view and hereby evaluate the coefficient ß. These attempts showed that ß can be related in various ways to rock properties rather than fluid properties such as porosity, permeability, tortuosity, specific surface area, grains and pores size distribution, surface roughness etc.