Summary
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.
Introduction
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.
© 2007. Society of Petroleum Engineers
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History
- Original manuscript received:
27 July 2005
- Revised manuscript received:
4 December 2006
- Manuscript approved:
27 December 2006
- Version of record:
20 June 2007
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