A Simple Productivity Equation for Horizontal Wells Based on Drainage Area Concept
Authors: Elgaghah Shedid A. , Osisanya Samuel O. , Tiab Djebbar 
Published: December 2007

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Abstract
Many flow equations have been developed for horizontal wells but they are complicated in derivation and time consuming in application. A simple productivity equation that incorporates the mechanics of fluid flow in porous media is developed. The equation is based on the drainage area concept. The new equation is validated by using it to study the effect of reservoir height and horizontal well length on the ratio of horizontal well productivity to vertical well productivity when both have the same drainage area. This paper presents the derivation of the equation and its application with several simulated results. Comparisons between the simulated results and results using the more complex equations are provided. Results obtained using the new equation are in agreement with those obtained using Joshi and Borisov equations. The results confirmed the theory that thin reservoirs are good candidates for horizontal wells while thick reservoirs are not. It also showed that an increase in the lateral gives higher ratio of horizontal well productivity to vertical well productivity for the same drainage area. The most advantage of this new equation is that it is very easy to use. The new equation can be used to optimize horizontal well length with respect to horizontal well productivity to vertical well productivity. It is a very useful tool for making decision about application of horizontal well in thin and moderately thick formations.

Introduction
Many equations have been developed to estimate flow rate in horizontal wells. Researchers used a symmetrical geometric shape to describe the horizontal drainage area to simplify the solution. The resultant equations were based on steady-state solutions, but they were complex. The steady-state solution assumes that pressure at any point in the reservoir does not change with time, dP/dt = 0. Table 1 shows the steady-state flow rate equations for horizontal wells developed by various researchers. Because of their complexities, simplifying assumptions are made in application which may lead to some errors in calculations. These in effect may affect decision or judgment concerning the performance of a horizontal well. The equations in Table 1 can be classified into two groups according to the geometry of the horizontal well. The first group by Borisov and Joshi assumed an elliptical shape, and the second group Giger and Giger et al used a rectangle with two semi-circles on both its sides for the same purpose, Fig.1. The reason for using symmetrical geometries is to simplify the solution, but these geometries do not represent the horizontal drainage areas accurately.
The first equation introduced by Borisov was used to calculate steady-state oil flow rate for a horizontal well, but his paper did not show the derivation of that equation. Giger derived his equation and augment the term of replacement ratio to show how many vertical wells can be replaced by horizontal well with the assumption of equal drawdown for horizontal and vertical wells. Joshi introduced an equation with its derivation in his augmentation of well productivity with slant and horizontal wells. The most common feature of all of these equations is that all of them were derived for steady-state and single-phase flow.

Development of The Simplified Productivity Equation For Horizontal Wells
In order to more realistically represent the drainage area of the horizontal well, this study assumes that the flow to the horizontal well from the toe-end side is not the same that from the heel-end side. Therefore, the drainage area is divided into three parts:a rectangle of length (L) and width 2r, for the horizontal section of the well at the center,a semi-circle of radius r, at the toe-end anda small rectangle of length rL/C, and width 2r, at the heel-side, Fig. 2.
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