Pressure-Transient Behavior of Horizontal Wells With and Without Gas Cap or Aquifer

Abstract

Summary. New analytic solutions are presented in real time and as Laplacetransforms for horizontal wells in reservoirs bounded at the top and bottom byhorizontal planes. Two types of boundary conditions are considered at theseplanes. and the Laplace-transform pressure solutions are used to includewellbore-storage and skin effects. Solutions are based on the uniform-flux, line-source solution, but differ from most existing solutions owing to the use of pressure averaging to approximate the infinite-conductivity wellborecondition and use of the correct equivalent wellbore radius for an anisotropicreservoir. New flow periods (regimes) are identified, and simple equations andexistence criteria are presented for the various flow periods that can occurduring a transient test. Introduction Determination of transient pressure behavior for horizontal wells hasaroused considerable interest over the past 10 years. An extensive literaturesurvey on horizontal wells can be found in Ref. 2. Most work dealing with thehorizontal-well problem uses the instantaneous Green's function techniquedeveloped by Gringarten and Ramey to solve the 3D isotropic diffusivityequation. Goode and Thambynayagam used finite Fourier transforms to solve theanisotropic problem for the line-source case. Because the infinite-conductivityinner-boundary condition (uniform pressure over the sand face) poses a verydifficult boundary-value problem. a uniform-flux condition on the innerboundary is commonly used. The infinite-conductivity solution is thenapproximated with either an equivalent-pressure-point or pressure-averagingtechnique. We prefer the pressure-averaging method because it requires no apriori information. is exact in the limit of a small wellbore radius, and ismore accurate at intermediate times than the equivalent-pressure-point method. These reasons are discussed further in Appendix A. Another feature of thesolutions presented in this paper is the use of the correct equivalent wellboreradius for an anisotropic formation, which guarantees that elliptical-floweffects near the well are treated correctly at late times. At early times it ispreferable to use the elliptical-cylinder solution. Solutions presented in thispaper. however, are sufficient for most practical problems. Using thesetechniques. we extend the work of Goode and Thambynayagam and Clonts and Rameyto obtain new analytic solutions for horizontal wells with and without theeffects of gas cap or aquifer. The wellbore-storage effect is accounted for, and new formulas are presented for the determination of reservoir parametersfrom the characteristics of different flow regimes. Solutions With and Without Gas Cap or Aquifer First we discuss the basic solutions for horizontal wells for theconstant-rate case without wellbore-storage and skin effects. These solutionswill then be combined with constant wellbore storage and/or measured downholeflow rate. The horizontal well shown in Fig. 1 is considered to be completed inan infinite anisotropic medium bounded above and below by horizontal planes. The boundaries of the reservoir in the horizontal directions are considered tobe so far away that they are not seen during the test. The permeabilities inthe principal directions are denoted by kx, ky, and kz. We develop thesolutions for the general case where the three permeabilities are all differentin Appendices A and B, but in the text we consider a transversely isotropicmedium and write k, kv - kH and kz -kv. The flow of a slightly compressiblefluid of constant compressibility and viscosity is assumed throughout themedium. Gravity effects are neglected. Two types of top and bottom boundaryconditions are considered. In the first case. both the top and the bottomboundaries have no-flow conditions. In the second case, one of the boundariesis at constant pressure, while the other is a no-flow boundary as before; thiscase can represent either a gas cap at the top boundary or an active aquifer(in which the water mobility is high compared with the mobility of thereservoir fluid) at the bottom. For convenience, we refer to the first model asthe no-flow-boundary model and to the second as the constant-pressure-boundarymodel. The notation of this paper assumes that in the latter model, theconstant-pressure boundary is at the top (the gas-cap case). but the formulasmay be readily adapted for the case of an aquifer at the bottom. During thelast few years, several solutions for horizontal wells have been presented. Most of these solutions are for the no-flow-boundary model, and apart from thework of Goode and Thambynayagam. none present solutions in the Laplace-transform domain. A solution for the constant-pressure condition atboth the top and bottom boundaries was presented by Daviau et al. This solutionis different from the constant-pressure-boundary solution presented here, inwhich one of the boundaries (top or bottom) is no-flow. This flexibility isimportant because if we have a constant-pressure boundary such as a gas cap, the well is usually drilled close to the other (no-flow) boundary. The solutionmethod is discussed in Appendix A, and the actual solutions are developed in Appendix B. Our solutions differ somewhat from other solutions given in theliterature because we approximate the infinite-conductivity condition byaveraging the pressure along the well length instead of using an equivalentpressure point. A discussion of the pressure-averaging technique is given in Appendix A, together with a derivation of the correct equivalent wellboreradius to be used for an anisotropic formation. We define dimensionless timeand pressure (in field units) by ................(1) ................(2) and other dimensionless parameters ............................(3a) ............................(3b) ............................(3c) In the time domain, the dimensionless pressure response, pD, forconstant-rate drawdown is most conveniently given as a time integral over theinstantaneous Green's function (see Appendix B): ...................(4) .......(5) .........................(6) SPEFE P. 86^