The Use of an r-z Model To Study the Effect of Completion Technique on Gas Well Deliverability
Authors: Crichlow Henry B. , Root Paul J. 
Published: December 2007

Abstract
A digital computer model of a radial gas reservoir was constructed to investigate the effect of completion techniques on gas well deliverability. The model was a standard r-z model divided linearly in the z-direction and logarithmically in the r-direction. Individual reservoir properties were assigned to each element of the model grid. These include porosity, radial and vertical permeability, and water saturation. A finite-difference approach was used to set up the flow equations, and both alternating direction implicit procedure (ADIP) and line successive overrelaxation (LSOR) were used to set up the system of simultaneous equations. The Thomas algorithm was used to solve the tridiagonal systems. From this research the following conclusions were drawn:(1)The real gas potential is effective in linearizing the gas flow equation. For nonturbulent flow the coefficient of performance in the backpressure equation, Q = C [ (Pe) - (Pw)]n can be evaluated independently oil the fluid properties of the gas.(2)Partially producing properties of the gas.(2)Partially producing intervals constitute a skin, the magnitude of which depends on the location of the perforations and the anisotropic nature of the medium.(3)In a damaged or stimulated well, within limits, the significant factor in deliverability reduction is the kind rather than the extent of the damage.(4)From the numerical standpoint ADIP is a more efficient method in "well-behaved" problemsthat is, in homogeneous systemswhereas LSOR is better suited to partially open and nonhomogeneous systems.

Introduction
Calculation of the flow rate and prediction of the deliverability of gas wells are factors of great economic importance to the natural gas industry. Consequently, the accurate analysis of gas flow in producing gas wells has been a subject of considerable interest, and many papers dealing with it may be found in the literature. One of the earliest methods for calculating gas flow, that of Jenkins and Aronofsky, involved the succession of steady states. Janicek and Katz, using a similar assumption that the rate of pressure change with time is independent of the radius at any given time, derived a set of relatively straightforward predictive equations. Other calculational methods are based on solutions to the partial differential equation describing gas flow in a porous medium. Until recently the analysis was based on linearizations that required evaluation of the gas properties at some average pressure. As a result, these solutions can be applied only when the flow gradients are small. Today gas reservoirs are being discovered at much greater depths and at relatively higher pressures. In many cases the formation permeability pressures. In many cases the formation permeability to gas is quite low. Thus, solutions to be linearized equation can lead to serious errors in predicting deliverability (and, hence, reserves) predicting deliverability (and, hence, reserves) because of the large drawdowns occurring in these systems. The simplifying assumptions implied by the linearized equations are not necessary when the real gas potential proposed by Al-Hussainy et al. is used. This function greatly facilitates the incorporation of the pressure-dependent variables, viscosity, and gas deviation factor into a mathematical model of gas flow. Its use reduces the unsteady-state flow equation directly to a form analogous to that of the diffusivity equation without the tacit assumptions that the pressure gradients within the flow system are small. Furthermore, the coefficients of the spatial derivatives no longer contain the pressure-dependent fluid properties. Because of these advantages the (p) function was used in this investigation of gas well deliverability.
SPEJ
P. 259