The Effects of Non-Darcy Flow on the Behavior of Hydraulically Fractured Gas Wells (includes associated paper 6417 )

Abstract

A reservoir simulator modified to include non-Darcy flow and fractureclosure was used to demonstrate the effects of non-Darcy gas flow in ahydraulic fracture on well performance. Results illustrate the effects onthe gas-well productivity index and on the analysis of pressure builduptests. Introduction Laminar flow of fluid through porous media can bedescribed using Darcy's law:(1) This equation indicates that if the resistance (mu/k)remains constant, the pressure gradient (delta p/delta L) isproportional to the velocity of the fluid (v). However, when thevelocity is increased such that the flow is not laminar, thepressure drop will increase more than the proportional increase in velocity. Fancher et al. recognized this behavior and publisheda paper in 1933 that gave an analogy between the flow offluids through porous media and the flow of fluidsthrough pipe. Several authors, including Brownell andKatz and Tek, have since published methods forpredicting the laminar and turbulent regions of flow inporous media based on correlations similar to the Reynoldsnumber for flow in pipe. The generalized equation for flow through porousmedia may be represented by the following equationsuggested by Forchheimer. (2) If the constant (a) or velocity (i,) approaches zero. thenthe second term can be ignored and Eq. 2 is equal toDarcy's law (Eq. 1). Cornell and Katz reformulated Eq. 2 as follows: (3) In Eq. 3, the constant (a) was replaced by the product ofthe fluid density (rho) and the beta factor, which is acharacteristic of the porous medium. Several authors havepublished empirical correlations of the beta factor with theporosity and permeability of the porous media. Geertsma pointed out that the analogy betweenlaminar and turbulent flow of fluids in porous media to theflow of fluids in pipes could be misleading. Geertsmastated that turbulence does not actually occur in the smallpore systems of reservoir rock, and the cause of theincreased pressure gradients at high fluid velocities isinertial resistance. Consequently, Geertsma defined theparameter beta as the coefficient of inertial resistance. Geertsma's paper, therefore, has created a bit ofcontroversy concerning the terminology of the parameter betain the Forchheimer equation. In reality, the excess pressure gradients at high fluidvelocities can be caused by either turbulence or inertialresistance, or by a combination of the two, depending onthe particular pore configuration of the reservoir rockbeing considered. In this paper, the parameter beta isreferred to as the beta factor. Regardless of its name, the beta factoris a value used to calculate the correct pressure gradientsunder nondarcy flow conditions. Cooke investigated nondarcy flow in packed, hydraulically induced fractures. He noted that beta factors forfractures packed with multiple layers of sand had notbeen reported in the petroleum literature and suggested the following equation for calculating beta factors: (4) JPT P. 1169^