A Supplemental Discussion of Productivity Formulae of Horizontal Wells

Abstract This paper presents a supplemental discussion of productivity formulae for horizontal wells using a uniform line sink model. Formulae are given for the productivity of horizontal wells in steady state in an infinite lateral extent drainage reservoir with impermeable top and bottom boundaries. This paper also focuses on the effects of different boundary conditions on the productivity of a horizontal well. Significant differences in the productivity of a horizontal well in a reservoir with different drive mechanisms are demonstrated. Introduction In Reference (1), Jing Lu presented new productivity formulae for horizontal wells, which are derived from the three-dimensional solution of the Laplace Equation, as opposed to two-dimensional or pseudo-three-dimensional solutions used in the previous Equations in References (2), (3), (4), and (5). Lu's formulae are given for the productivity of horizontal wells in ellipsoidal drainage reservoirs and infinite lateral extent drainage reservoirs with gas cap or bottom water. In this paper, formulae are given for the productivity of horizontal wells in steady state in an infinite lateral extent drainage reservoir with impermeable top and bottom boundaries. The significant effects of different boundary conditions on the productivity of a horizontal well are also demonstrated. Horizontal Well Model Figure 1 is the model of a horizontal well in an infinite lateral extent drainage reservoir. The following assumptions are made:The reservoir is horizontal, homogeneous, anisotropic, and has constant Kx, Ky, Kz permeabilities, thickness H, and porosity Φ. The reservoir is with infinite lateral extension, i.e., the boundaries of the reservoir in the horizontal directions are so far away that the pressure disturbance does not travel far enough to reach the boundaries during the well production.The reservoir pressure is initially constant. The pressure remains constant and equal to the initial value at an infinite distance from the well. The reservoir is bounded by top and bottom impermeable formations.The production occurs through a well of radius Rw, represented in the model by a uniform line sink located at a distance zw from the lower boundary. The length of the well is L.A single-phase fluid, of small and constant compressibility Cf, constant viscosity µ, and formation volume factor B, flows from the reservoir to the well at a constant rate Q. Fluids properties are independent of pressure. Equations for Horizontal Wells As Figure 1 shows, the horizontal well is a uniform line sink in 3D space, and the coordinates of the two ends are (-L/2, 0, zw) and (L/2, 0, zw). We suppose the point (xO, 0, zw) is on the well line, and its point convergence intensity (point flow rate) is q. In order to obtain the point convergence pressure of (xO, 0, zw), we have to obtain the basic solution of the partial differential equation below: Equations (available in full paper)

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