A General Single-Phase Wellbore/Reservoir Coupling Model for Multilateral Wells
- Khalid Aziz (Stanford U.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- August 2001
- Document Type
- Journal Paper
- 327 - 335
- 2001. Society of Petroleum Engineers
- 2.2.2 Perforating, 5.5 Reservoir Simulation, 3.3.1 Production Logging, 4.1.2 Separation and Treating, 5.6.4 Drillstem/Well Testing, 4.3.4 Scale, 5.2.1 Phase Behavior and PVT Measurements, 5.2 Reservoir Fluid Dynamics, 1.10.1 Drill string components and drilling tools (tubulars, jars, subs, stabilisers, reamers, etc), 5.3.2 Multiphase Flow
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A general single-phase flow model that rigorously couples flow in the reservoir with flow in the wellbore is presented. The model can be applied to a parallelepiped drainage domain consisting of any number of wells containing any number of laterals of arbitrary configurations. The fluid is assumed to be slightly compressible, and gravity head is included. Reservoir inflow for the entire time period (from early transient to late pseudosteady state) is handled in the coupling model by integrating the instantaneous point source/sink solution over both time and space. Pressure drop along any lateral in the well system is the sum of three components: wall friction, gravity, and acceleration owing to wall inflow or outflow. Effects of wall mass transfer on wall friction and kinetic energy change are taken into account.
A computer package has been developed and used to perform a series of sensitivity studies that lead to several important and interesting observations. The coupling model may be applied to determine well productivity, well index for multiphase flow simulation, wellbore pressure profile, and wellbore inflow or outflow distribution for any time in the well's production/injection life.
The use of horizontal-well technology has become a standard practice in the oil and gas industry. Because of pressure drop along horizontal wells and other practical and economic concerns, multilateral horizontal wells have been introduced since the early 1990's. It is recognized that, in some situations, pressure drop along a horizontal well may be a crucial parameter affecting well performance and hence should be considered through a coupling model.1-7 Most available coupling models handle wellbore flow in the same way as flow in a regular pipe. However, according to our investigations (Ouyang et al.8,9), there are significant differences between fluid flow in a wellbore where mass transfer occurs through the pipe wall and fluid flow in a regular pipe. Primary factors contributing to these differences are the boundary-layer effect, the kinetic-energy effect, and the flow-pattern effect. Proper treatment of reservoir inflow and wellbore flow for multilateral horizontal wells and for wells with arbitrary configurations creates special problems for reservoir simulators.
In the present paper, a simple coupling model is proposed. It can be applied to different parallelepiped reservoir-drainage volumes consisting of any number of wells with any number of laterals of arbitrary configurations. A computer software package based on this simple coupling model has been developed. Reservoir inflow for the entire time period (from early transient to late pseudosteady state) is handled in the coupling model by integrating the instantaneous point source/sink solution over both time and space. Because pressure drop along a lateral can be high owing to gravity, wall friction, and acceleration, wellbore pressure at each location in a lateral is determined by considering pipe geometry and local flow conditions. Pressure drop along any lateral in the well system is evaluated by applying the wellbore flow model proposed by Ouyang et al.,8 which accounts for pressure drops caused by wall friction, gravity, and acceleration owing to wall inflow or outflow. Consequently, effects of wall mass transfer on wall friction and kinetic-energy change are taken into account.
Consider one or more horizontal, slanted, or vertical wells in a parallelepiped drainage volume under the following constraints.
The reservoir is homogeneous but anisotropic, with the coordinates aligned with the principal permeability directions.
The reservoir has dimensions of xe, ye, and ze in the x, y, and z directions, respectively.
The outer boundaries of the reservoir are either at constant potential, or they are impermeable.
Formation properties are independent of pressure.
Reservoir fluid is single-phase and slightly compressible with a constant compressibility.
For a system of this kind, the governing equation for fluid flow in the reservoir can be expressed as
where ?=the fluid potential.10 For a coordinate system where z is defined as positive upward, the fluid potential is related to reservoir pressure p by
The initial condition and boundary conditions are
where xi=x, y, or z.
As a reasonable approximation, each well or lateral is represented by a line source or sink. With this assumption, the following approach, as described in detail in Ouyang et al.,11 can be applied to deal with reservoir inflow during the entire time period (from early transient to steady-state or pseudosteady-state flow) for deviated, multilateral, or undulating horizontal wells, or for wells with curved or irregular trajectories.
Start with a 1D instantaneous plane source solution (Carslaw and Jaeger12).
Account for the boundary effects by using the method of images (MOI) and the principle of superposition.
Develop a 3D instantaneous point source/sink solution within the parallelepiped reservoir by means of the Newman's product method (Gringarten and Ramey13).
Integrate over time to get the 3D continuous point source/sink solution within the reservoir.
Integrate over well trajectory to get potential drawdown.
As mentioned by Ouyang et al.,11 this instantaneous point source/sink solution approach is very flexible and may be applied to reservoir problems with complex well configurations. Successful applications of this approach have been reported by Economides et al.14 and Maizeret.15
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