A Mathematical Model of Horizontal Wells Pressure Drawdown and Buildup
Authors: Lu J. , Lin T. , Rogers R. , Lu T. 
Published: March 2010

Abstract
Taking a horizontal well as a uniform line source in a three dimensional space, this paper presents a mathematical model of horizontal well pressure drawdown and buildup in a single porosity reservoir and a double porosity reservoir by developing the necessary mathematical analysis. We first derive the point convergence pressure formulae, then according to the Superposition Principle of Potential, we apply quadrature theorem to derive pressure drawdown and buildup formulae of the uniform line source. Compared with the formulae published in the literature, our formulae, which do not include the sum of infinite series, are more reasonable and are easily used in well testing analysis.

Introduction
For both vertical and horizontal wells, steady-state and unsteady-state pressure-transient tests are useful tools for evaluating in situ reservoir and wellbore parameters that describe the production characteristics of a well. The use of transient well testing for determining reservoir parameters and productivity of horizontal wells has become common because of the upsurge in horizontal drilling. During the last decade, analytic solutions have been presented for the pressure behaviour of horizontal wells.
Determination of transient pressure behaviour for horizontal wells has aroused considerable interest over the past ten years. An extensive literature survey on horizontal wells can be found. Interpretation of well tests from horizontal wells is much more difficult than interpretation of those from vertical wells because of a considerable wellbore storage effect, the three dimensional nature of the flow geometry and lack of radial symmetry, and strong correlations between certain parameters. Analytical solutions for the pressure behaviour of uniform flux, as well as infinite-conductivity horizontal wells have been discussed in the literature(1–5).
In general, the techniques explaining the pressure-transient response in horizontal wells can be grouped into two categories:solutions to the pressure-transient response of a horizontal drainhole based on the use of source and Green's functions; and
Most work dealing with the horizontal well pressure transient problem uses the instantaneous Green's function technique developed by A.C. Gringarten and H.J. Ramey to solve the 3D isotropic diffusivity equation(7, 9, 10). P.A. Goode and R.K. Thambynayagam used finite Fourier transforms to solve the anisotropic problem for the line-source case(1); they presented a solution for an infinite-conductivity horizontal well located in a semi-infinite, homogeneous and anisotropic reservoir of uniform thickness and width. E. Ozkan compared the performances of horizontal wells and fullypenetrating vertical fractures(11–13). F. Daviau also analysed the pressure behaviour of horizontal wells, considering both infiniteconductivity nd uniform-flux inner boundary conditions(14). F. J. Kuchuk extended the previous works(1, 11, 14) on pressure transient behaviour of horizontal wells to include the effects of a gas cap nd/or aquifer(15).
A convenient model to represent the pressure behaviour in a horizontal drainhole is one that assumes no pressure drop in its interior during fluid flow. This means that the pressure is uniform long the wellbore face, and the well is said to have infinite conductivity.