Summary
The accuracy of streamline reservoir simulations depends strongly on the
quality of the velocity field and the accuracy of the streamline tracing
method. For problems described on complex grids (e.g.,corner-point geometry or
fully unstructured grids) with full-tensor permeabilities, advanced
discretization methods, such as the family of multipoint flux approximation
(MPFA) schemes, are necessary to obtain an accurate representation of the
fluxes across control volume faces. These fluxes are then interpolated to
define the velocity field within each control volume, which is then used to
trace the streamlines. Existing methods for the interpolation of the velocity
field and integration of the streamlines do not preserve the accuracy of the
fluxes computed by MPFA discretizations.
Here we propose a method for the reconstruction of the velocity field with
high-order accuracy from the fluxes provided by MPFA discretization schemes.
This reconstruction relies on a correspondence between the MPFA fluxes and the
degrees of freedom of a mixed finite-element method (MFEM) based on the
first-order Brezzi-Douglas-Marini space. This link between the finite-volume
and finite-element methods allows the use of flux reconstruction and streamline
tracing techniques developed previously by the authors for mixed finite
elements. After a detailed description of our streamline tracing method, we
study its accuracy and efficiency using challenging test cases.
Introduction
The next-generation reservoir simulators will be unstructured. Several
research groups throughout the industry are now developing a new breed of
reservoir simulators to replace the current industry standards. One of the main
advances offered by these next generation simulators is their ability to
support unstructured or, at least, strongly distorted grids populated with
full-tensor permeabilities.
The constant evolution of reservoir modeling techniques provides an
increasingly realistic description of the geological features of petroleum
reservoirs. To discretize the complex geometries of geocellular models,
unstructured grids seem to be a natural choice. Their inherent flexibility
permits the precise description of faults,flow barriers, trapping structures,
etc. Obtaining a similar accuracy with more traditional structured grids, if at
all possible, would require an overwhelming number of gridblocks.
However, the added flexibility of unstructured grids comes with a cost. To
accurately resolve the full-tensor permeabilities or the grid distortion, a
two-point flux approximation (TPFA) approach, such as that of classical finite
difference (FD) methods is not sufficient. The size of the discretization
stencil needs to be increased to include more pressure points in the
computation of the fluxes through control volume edges. To this end, multipoint
flux approximation (MPFA) methods have been developed and applied quite
successfully (Aavatsmark et al. 1996; Verma and Aziz 1997; Edwards and Rogers
1998; Aavatsmark et al. 1998b; Aavatsmark et al. 1998c; Aavatsmark et al.
1998a; Edwards 2002; Lee et al. 2002a; Lee et al. 2002b).
In this paper, we interpret finite volume discretizations as MFEM for which
streamline tracing methods have already been developed (Matringe et al. 2006;
Matringe et al. 2007b; Juanes and Matringe In Press). This approach provides a
natural way of reconstructing velocity fields from TPFA or MPFA fluxes. For
finite difference or TPFA discretizations, the proposed interpretation provides
mathematical justification for Pollock’s method (Pollock 1988) and some of its
extensions to distorted grids (Cordes and Kinzelbach 1992; Prévost et al. 2002;
Hægland et al. 2007; Jimenez et al. 2007). For MPFA, our approach provides a
high-order streamline tracing algorithm that takes full advantage of the flux
information from the MPFA discretization.
© 2008. Society of Petroleum Engineers
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History
- Original manuscript received:
10 July 2006
- Meeting paper published:
24 September 2006
- Revised manuscript received:
17 January 2008
- Manuscript approved:
8 February 2008
- Version of record:
15 December 2008
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