Productivity Formulae for Horizontal Wells in a Circular Cylinder Drainage Volume
Authors: Lu J. 
Published: March 2010

Abstract
By the similar methods in References (1), (2), and (3), this paper presents productivity formulae for horizontal wells with circular cylinder drainage volume, using a uniform line sink model.
If top and bottom reservoir boundaries are impermeable, the circular cylinder drainage volume radius has significant effects on the horizontal well flow rate. If the reservoir has a gas cap or bottom water, the effect of the circular cylinder radius on flow rate can be ignored, and, under the same top and bottom boundary conditions, the productivity formulae for a circular cylinder drainage reservoir will reduce to the corresponding formulae for an infinite lateral extent reservoir, respectively.
Therefore, for a horizontal well, different productivity formulae should be used under different reservoir boundary conditions.

Introduction
In References (1) and (2), J. Lu presented productivity formulae for horizontal wells in an ellipsoidal drainage reservoir and an infinite lateral extent reservoir.
This paper presents productivity formulae for horizontal wells in steady-state with circular cylinder drainage volume. The significant effects of drainage volume radius and boundary conditions on the flow rate of a horizontal well are also demonstrated.
The new formulae in this paper, 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 literature in References (4) - (7), are more accurate in describing the three-dimensional percolation flow through porous media into horizontal wells.

Horizontal Well Model
Figure 1 is a schematic of a horizontal well. A horizontal well of length L drains a circular cylinder with height H and radius Re. The following assumptions are made:The reservoir is horizontal, homogeneous, anisotropic, and has constant Kx, Ky, Kz permeabilities, thickness H, and porosity Φ. During production, the horizontal well has a circular cylinder drainage volume with height H and radius Re. The middle point of the horizontal well is the centre of the drainage circle.The reservoir pressure is initially constant. The reservoir has edge water, the pressure remains constant and equal to the initial value at lateral surface of the drainage circular cylinder, and the reservoir is bounded by top and bottom impermeable or constant pressure boundaries.The production occurs through a horizontal 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 Qw. Fluid properties are independent of pressure.There is no water encroachment and no water/gas coning. Edge water, gas cap, and bottom water are taken as constant pressure boundaries, and multiphase flow effects are ignored.

Horizontal Well Productivity
The equations for the horizontal well and the equations for top and bottom boundary conditions are the same as those in References (1) and (2).