Streamline Tracing on General Triangular or Quadrilateral Grids
- Sebastien Francois Matringe (Stanford University) | Ruben Juanes (Massachusetts Inst. of Tech.) | Hamdi A. Tchelepi (Stanford University)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- June 2007
- Document Type
- Journal Paper
- 217 - 233
- 2007. Society of Petroleum Engineers
- 5.5 Reservoir Simulation, 5.3.2 Multiphase Flow, 1.2.3 Rock properties, 4.3.4 Scale, 5.1 Reservoir Characterisation, 5.5.7 Streamline Simulation, 4.3.1 Hydrates, 5.5.8 History Matching
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- 473 since 2007
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Streamline methods have received renewed interest over the past decade as an attractive alternative to traditional finite-difference (FD) simulation. They have been applied successfully to a wide range of problems including production optimization, history matching, and upscaling. Streamline methods are also being extended to provide an efficient and accurate tool for compositional reservoir simulation. One of the key components in a streamline method is the streamline tracing algorithm. Traditionally, streamlines have been traced on regular Cartesian grids using Pollock's method. Several extensions to distorted or unstructured rectangular, triangular, and polygonal grids have been proposed. All of these formulations are, however, low-order schemes.
Here, we propose a unified formulation for high-order streamline tracing on unstructured quadrilateral and triangular grids, based on the use of the stream function. Starting from the theory of mixed finite-element methods (FEMs), we identify several classes of velocity spaces that induce a stream function and are therefore suitable for streamline tracing. In doing so, we provide a theoretical justification for the low-order methods currently in use, and we show how to extend them to achieve high-order accuracy. Consequently, our streamline tracing algorithm is semi-analytical: within each gridblock, the streamline is traced exactly. We give a detailed description of the implementation of the algorithm, and we provide a comparison of low- and high-order tracing methods by means of representative numerical simulations on 2D heterogeneous media.
Streamline simulation is now accepted as a practical tool for reservoir simulation. It represents a fast alternative to the classical FD or finite-volume (FV) methods. However, streamline simulation is still a young technology and does not offer the same capabilities as more traditional methods. Here, we investigate the extension of the streamline method to simulate problems on unstructured or highly distorted grids with full tensor permeability fields.
In streamline simulation, the flow problem (pressure equation) and the transport problem (saturation equations) are solved sequentially in an operator-splitting fashion. The transport problem is solved along the streamlines using a 1D formulation of the transport equation expressed in terms of the time-of-flight variable (Bradvedt et al. 1993; Batycky et al. 1997; King and Datta-Gupta 1998). A background simulation grid is used to solve the flow problem and trace the streamlines. Therefore, extension of the streamline method to general triangular or quadrilateral grids hinges on the ability to: (1) properly discretize the pressure equation, and (2) accurately trace the streamlines on these advanced grids.
These two problems are linked. The key link between discretization and streamline tracing resides in the velocity field description. To each discretization corresponds a particular form of velocity field, and the streamline tracing algorithm has to be adapted to each type of velocity field.
|File Size||1 MB||Number of Pages||7|
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