Modeling and Simulation of Horizontal Wellbore-Reservoir Flow Equations
Authors: Dickstein F. , Lara A.Q. , Neri C. , Peres A.M. 
Published: December 2007

Abstract
A system of conservation equations is considered for the solution of the coupled horizontal well-reservoir flow for a monophasic, slightly compressible fluid. A totally implicit numerical scheme, locally refined around the well, is proposed. We show that this scheme is able to solve the different (well and reservoir) time scales in a robust and efficient way. Numerical experiments give qualitative and quantitative insight into the nature of this coupled flow.

Introduction
In Well Testing, as well as in Reservoir Simulation, it is usual to model the flow within vertical wells in a rather simplified manner. In particular, the pressure loss due to friction forces is not taken into account and it is assumed that pressure distribution is hydrostatic along the well. It does not seem reasonable to consider the same assumptions for horizontal wells, which are much longer than vertical wells. The objective of this work is to investigate the nature of this coupled horizontal well- reservoir flow. To do so, we have considered a single phase, slightly compressible fluid being injected (produced) through a horizontal well. This is modeled by a system of fundamental conservation equations for the well and for the reservoir. Usual mass conservation, plus Darcy's law, describes the reservoir flow. Wellbore equations, considering friction, inertial and acceleration effects, are one dimensional and express 1-D mass and momemtuni conservation. The coupling is taken into account by pressure and mass flow continuity at the wellbore-reservoir interface. Analogous set of equations has been previously considered in Ref. 2.
The resulting system is discretized through a totally implicit scheme on a cartesian grid, locally refined around the well. A Finite Volume scheme is formulated for the reservoir equations. Well equations are approximated by a first order upwind-like method for systems of Conservation Laws. The resulting nonlinear algebraic system is solved using Newton's method. Our numerical experiments give insight into the nature of this coupled flow. In particular, they indicate that the pressure loss along the well may be important in some cases.

The Model
We consider the reservoir as the parallelepiped, containing a horizontal well aligned to the × direction. The system of equations modeling the problem is obtained frim the usual conservation laws for the reservoir and the well, under the hypothesis that the flow within the well is axially symmetric. Momentum loss in the wellbore due to wall friction, inertial and acceleration effects are taken into account. Wellbore-reservoir coupling is performed through pressure and the mass flux continuity at the wellbore surface wall. Let P(x, y, z, t) be the reservoir pressure, p(x, t) and m(x, t) be the well pressure and momentum, respectively. We then write,
reservoir mass conservation
(1)
Darcy's law
(2)
where is a known space function, , , are known functions of the pressure P.
well mass conservation
(3)
where the mass flow density is equal to at the point of the well surface open to the flow, and is equal to 0 otherwise, n being the external normal to the surface.