Productivity of a Horizontal Well (includes associated papers 20306, 20307, 20394, 20403, 20799, 21307, 21610, 21611, 21623, 21624, 25295, 25408, 26262, 26281, 31025, and 31035)

Abstract

Summary An equation for calculating the pseudosteady-state flow-i.e., theproductivity-of a horizontal well is presented that is easy to use andproductivity-of a horizontal well is presented that is easy to use andidentical in form to the well-known productivity equation for a vertical well. The equation was extracted from a very complex general solution. Application of the equation requires the determination of two parameters:a geometricfactor that accounts for the effect of permeability anisotropy, well location, and the relative dimensions of the drainage volume andthe skin caused byrestricted entry, which accounts for the effect of well length. The paper givessimple equations for calculating these two parameters. The equation is used tostudy the effects on well productivity of well length, location, and degree ofpenetration; vertical productivity of well length, location, and degree ofpenetration; vertical and horizontal permeabilities; and the horizontaldimensions of the drainage volume. It is concluded that, though each of thesevariables has a significant effect, the well length and degree of penetrationhave the strongest. Introduction A worldwide interest exists today in drilling horizontal wells to increaseproductivity. Because of its large flow area, a horizontal well may be severaltimes more productive than a vertical one draining the same volume. However, itis more expensive to drill a horizontal well than a vertical one. Therefore, todetermine the economic feasibility of drilling a horizontal well, the engineerneeds a reliable method to estimate its expected productivity. There are alsoother reasons to drill a horizontal well that are not discussed here-e.g. tointersect and to drain a formation's natural fracture system moreeffectively. The solution to the partial-differential equation that describes the flowbehavior of a horizontal well and that preserves the physics is very complex. Because of this, simplifying assumptions are frequently introduced, such asinfinite extension of the drainage volume in one or two dimensions andconstant-pressure boundaries at finite distances. In spite of these simplifyingassumptions, some of the solutions have quite complicated structures, makingthem unsuitable for routine application. Recently several authors publishedformulas based on various assumptions and approximations for calculating theproductivity of horizontal wells. In this paper, we present a solution to thepartial-differential equation for a finite present a solution to thepartial-differential equation for a finite reservoir. We reduce the complexsolution to an easy-to-use equation for calculating productivity. This equationis identical in form to the well-known equation for a vertical well. In ourtreatment, we require only that the drainage volume be approximately boxshaped. Otherwise, the drainage volume can be of any dimension; the permeabilities inthe x, y, and z directions can be equal or different; and the well can be ofany length and located anywhere in the drainage volume. Thus, our equation isvery general. Because our simple equation was derived from complicated expressions, it isnot exact. In most, if not all, cases of interest where the well penetration isgreater than or equal to 50%, however, the error in the productivitycalculation is less than 3 %. This error may increase to about 10% as thepenetration decreases. We believe this is quite adequate for engineeringcalculation. In this work, we discuss the effects of some critical parameters on theproductivity of horizontal wells and illustrate these effects graphically. Theoretical Analysis Physical Model. The physical model, Fig. 1. consists of a well PhysicalModel. The physical model, Fig. 1. consists of a well of radius rw and lengthL. The well is drilled in a box-shaped drainage volume, parallel to theydirection. The dimensions of the drainage volume are: thickness=h, length (xdirection) = a, and width (y direction)= b. The well has a length L greaterthan or equal to b, and extends between Y: and y2. Its × and z locations areindicated by x0 and z0, respectively. It produces at a constant rate q (uniformflux). A discussion of the uniform-flux boundary condition is given in AppendixA. The permeabilities in the x, y, and z directions are, respectively, kx, ky, and kz. The porosity, phi, is constant, and the fluid is slightly compressible. All the boundaries of the drainage volume are sealed. The pressure in thedrainage volume before the well is produced at to is uniform and is equal topi. At time t=0+, we start to withdraw a quantity of fluid at the rate q fromthe well. We want to find the pressure drop delta p=pi-p as a function of timeand space for t greater than 0. Related discussions and replies: 20306, 20307, 20394, 20403, 20799, 21307,21610, 21611, 21623, 21624, 25295, 25408, 26262, 26281, 31025, and 31035