Simple But Accurate Equations for Wellbore Pressure Drawdown Calculation
Authors: Ouyang Liang-Biao , Thomas L. Kent , Evans Clayton E. , Aziz Khalid 
Published: December 2007

Abstract
A novel procedure is proposed for a quick and accurate evaluation of bottomhole wellbore pressure and productivity index. The procedure has led to simple correlations for both horizontal and vertical wells which can be applied to calculate the bottomhole wellbore pressure and productivity index for the entire producing period. The correlations proposed here provide an excellent approximation of the semi-analytical solution at a much reduced computational time. Successful validation of the equations for different well locations and different reservoir geometries has also been achieved.
A simple relationship is also presented to estimate the time to reach pseudo-steady state. In addition, change with time of the horizontal well shape factor is shown. The stabilized shape factor for a horizontal well is decomposed into different components which can be easily evaluated by the equations provided. Values for the stabilized well shape factor for various reservoir dimension ratios of practical interest have also been computed and tabulated.

Introduction
Bottomhole wellbore pressure and productivity index are two important quantities for both horizontal and vertical wells. Approaches reported in the literature to determine these two quantities for horizontal wells can be divided into the following two basic types:
Integral transformation (IT) (such as Laplace Fourier transformation) and separation of variables (SOV) methods. Typical examples for this type of approach include Goode & Thambynayagam, Kuchuk et al., Carvalho & Rosa, Aguilera & Ng, Liu & Yang, Ouyang.–Instantaneous point source/sink superposition and the Newman's product method.
This kind of approach has been used by a number of researchers, including Gringarten & Ramey, Clonts & Ramey, Daviau et al., Babu & Odeh, Rosa & Carvalho, Ozkan et al., Ouyang, and Economides et al.
Although both approaches have been successfully applied to solve a lot of practical reservoir engineering problems, they take nontrivial (and usually long) computing time to get results. As a consequence they are impractical for routine reservoir engineering applications, such as well performance modeling. A novel procedure is proposed here to develop simple correlations for bottomhole wellbore pressure for both horizontal and vertical wells.

Problem Formulation
Consider a horizontal or a vertical well in a homogeneous but anisotropic parallelepiped reservoir. The reservoir has dimensions of xe, ye, ze in x, y, and z directions, respectively. Its outer boundaries can be either constant pressure or impermeable (Fig. 1). Assume that the formation properties are independent of pressure and reservoir fluid is slightly compressible and of constant compressibility. With these assumptions the partial different equation describing isothermal fluid flow in the reservoir can be expressed as:
(1)
Coordinate transformation, such as Besson, can be easily applied to transform an anisotropic reservoir to an isotropic one. Hence, only an isotropic reservoir will be considered in the present paper unless otherwise specified. The results presented in the paper can also be used for anisotropic reservoirs by means of the afore-mentioned coordinate transformation. For an isotropic porous media, Eq. 1 becomes
(2)
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