The Productivity And Optimum Pattern Shape For Horizontal Wells Arranged In Staggered Rectangular Arrays
Authors: Suprunowicz R. , Butler R.M. 
Published: March 2010

Abstract
In a previous paper(1), the authors analyzed the productivity of horizontal wells arranged in a regular rectangular pattern. It was shown that there was an optimum for the pattern shape (i.e. length to width ratio); methods were described for calculating this shape and the productivity of the well.
In this paper, the analysis has been extended to similar patterns in which the rows of wells are staggered. The optimum shape and performance have been calculated in a similar manner to that in the previous paper.
It is found that the staggered pattern has superior characteristics. It has a slightly higher productivity and, more importantly, it is much less sensitive to the shape of the pattern. Essentially, the same performance is found over a wide range of pattern shapes for any particular drainage area. Equations are given for predicting the well productivity in this preferred arrangement as a function of well length and pattern area for pseudo-steady state flow.

Introduction
This paper is a continuation of a previously presented analysis(1) regarding the regular pattern of horizontal wells extended to the pattern in which the rows of wells are staggered. Figure 1 shows the distribution of wells in the reservoir with part la showing a regular pattern, and 1b a staggered pattern. In both cases, each well of length L is placed in the centre of a rectangle having dimensions b × a. In terms of geometry, both patterns are equivalent if only the respective dimensions are equal.
In the previous paper(1), the authors presented a method which made it possible to find the optimum pattern shape and predict the productivity of the well in a two-dimensional reservoir with a modification permitting the productivity evaluation in a three-dimensional reservoir. A similar method will be used to extend the consideration to the staggered pattern.

Description of the Problem
The pattern shown in Figure 1b contains a repeated "pattern cell" that has been shaded. It can be seen that all boundaries of the shaded rectangle are the symmetry axes, thus being no-flaw boundaries. This means the liquid cannot flow across the sides of the rectangle, but can flow along them. Inside the rectangle, there is another no-flow boundary which is shown arbitrarily by the thick, dashed line, and whose exact location is not yet known. It must separate the drainage areas of both wells draining the rectangle.
The shaded area in Figure 1b is, obviously twice as large as that in Figure la.
The problem is treated as two-dimensional, pseudo-steady state flow and thus described by the equation
Equation 1 (available in full paper)
with boundary conditions: zero pressure gradient normal to the sides of the shaded rectangle in Figure 1b.
With constant compressibility of the fluid and constant values of permeability and viscosity, equation(1) can be reduced to Poisson's equation [ef Ref. 1], and further replaced by a system of difference equations. The procedure is described in detail in the literature(1–3). Ultimately, the shaded rectangle is treated as a grid pattern containing mm grid points.