The Effects of Non-Darcy Flow on the Behavior of Hydraulically Fractured Gas Wells (includes associated paper 6417 )
- S.A. Holditch (Texas A and M U.) | R.A. Morse (Texas A and M U.)
- Document ID
- Society of Petroleum Engineers
- Journal of Petroleum Technology
- Publication Date
- October 1976
- Document Type
- Journal Paper
- 1,169 - 1,179
- 1976. Society of Petroleum Engineers
- 5.5.8 History Matching, 2.5.2 Fracturing Materials (Fluids, Proppant), 5.6.4 Drillstem/Well Testing, 3 Production and Well Operations, 5.5 Reservoir Simulation, 3.2.3 Hydraulic Fracturing Design, Implementation and Optimisation, 2.4.3 Sand/Solids Control, 5.3.1 Flow in Porous Media, 4.6 Natural Gas
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A reservoir simulator modified to include non-Darcy flow and fracture closure was used to demonstrate the effects of non-Darcy gas flow in a hydraulic fracture on well performance. Results illustrate the effects on the gas-well productivity index and on the analysis of pressure buildup tests.
Laminar flow of fluid through porous media can be described using Darcy's law: (1)
This equation indicates that if the resistance (mu/k) remains constant, the pressure gradient (delta p/delta L) is proportional to the velocity of the fluid (v). However, when the velocity is increased such that the flow is not laminar, the pressure drop will increase more than the proportional increase in velocity.
Fancher et al. recognized this behavior and published a paper in 1933 that gave an analogy between the flow of fluids through porous media and the flow of fluids through pipe. Several authors, including Brownell and Katz and Tek, have since published methods for predicting the laminar and turbulent regions of flow in porous media based on correlations similar to the Reynolds number for flow in pipe.
The generalized equation for flow through porous media may be represented by the following equation suggested by Forchheimer.
If the constant (a) or velocity (i,) approaches zero. then the second term can be ignored and Eq. 2 is equal to Darcy's law (Eq. 1).
Cornell and Katz reformulated Eq. 2 as follows:
In Eq. 3, the constant (a) was replaced by the product of the fluid density (rho) and the beta factor, which is a characteristic of the porous medium. Several authors have published empirical correlations of the beta factor with the porosity and permeability of the porous media.
Geertsma pointed out that the analogy between laminar and turbulent flow of fluids in porous media to the flow of fluids in pipes could be misleading. Geertsma stated that turbulence does not actually occur in the small pore systems of reservoir rock, and the cause of the increased pressure gradients at high fluid velocities is inertial resistance. Consequently, Geertsma defined the parameter beta as the coefficient of inertial resistance. Geertsma's paper, therefore, has created a bit of controversy concerning the terminology of the parameter beta in the Forchheimer equation.
In reality, the excess pressure gradients at high fluid velocities can be caused by either turbulence or inertial resistance, or by a combination of the two, depending on the particular pore configuration of the reservoir rock being considered. In this paper, the parameter beta is referred to as the beta factor. Regardless of its name, the beta factor is a value used to calculate the correct pressure gradients under nondarcy flow conditions.
Cooke investigated nondarcy flow in packed, hydraulically induced fractures. He noted that beta factors for fractures packed with multiple layers of sand had not been reported in the petroleum literature and suggested the following equation for calculating beta factors:
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