Stationary And Non-stationary Flow of Fluid In a Limited Reservoir Towards Horizontal And Slanting Wells
Authors: Ibragimov A.I. , Baganova M.N. , Nekrasov A.A. 
Published: March 2010

Abstract
In this paper a 3D analytical simulator for modelling horizontal well (HW) and slanting well (SW) performance is presented. Hydrodynamic characteristics of the development process are treated in terms of Newton's potentials. The developed simulator provides the ability to model both steady and transient states. The computational experiments allowed to analyse the impact of the geometry of the horizontal well inside the reservoir on the performance index and to infer new flow rate formulae for steady and transient drainage.
The proposed method is based on iterative algorithms and domain decomposition principles reducing the drainage problem in domains with complicated structure to problems in standard (simple) domains, such as ball, cylinder, etc. The developed numerical methods and flow rate formulae have been validated and proved to be highly precise.

Introduction
The drainage of fluid in the neighborhood of one or several horizontal wells is still less fully studied than in the case of vertical wells. This default may be accounted for by the essentially 3D nature of hydrocarbon influx toward the well in a reservoir limited by its top, bottom, and the no-flow boundary. The main problem consists in determining the dependency of the well's productivity on the reservoir geometry and on the interference between several wells.
A number of analytical formulae and charts(1–8) linking the production rate and the differential pressure in relation to geometric and hydrodynamic parameters of the horizontal well inside a reservoir are available. Analytical formulae are mostly obtained by reduction of the 3D model to 2D flow. A precise solution of the essentially 3D drainage problem is very complicated and can be constructed only for special cases(9, 10).
Another efficient approach consists in the development of computational models to simulate drainage processes in the 3D reservoirs(11–13). In this paper an explicit method(14, 16) is presented to solve the problem of one-phase incompressible and elastic fluid inflow toward a single horizontal well or a group of horizontal wells in a 3D reservoir limited by top, bottom and the no-flow boundary.
The wellbore is modelled by a sequence of linear discrete segments distributed along the axis of the well. The pressure function is presented as a superposition of Source Functions (SF) defined on these segments with unknown density distributions and satisfying the following conditions:In the stationary case the Laplace equation and in the nonstationary case the heat equation is satisfied in each inner point of the reservoir;At each point on the top and bottom of the reservoir the noflow condition is satisfied and on the external reservoir boundary the pressure attains the given values.
The solution is sought in terms of discrete density distributions, subject to one of the following conditions: either the pressure at the wellbore or the flow rates at discrete points of the well are given.
In view of the maximum principle(15) it is easy to check that for the approximate solution the deviation from the exact solution in each inner point of the reservoir doesn't exceed the deviation from the given value at the boundary of the well.