Tracing Streamlines on Unstructured Grids From Finite Volume Discretizations
- Sebastien F. Matringe (Chevron ETC) | Ruben Juanes (Massachusetts Inst. of Tech.) | Hamdi A. Tchelepi (Stanford University)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- December 2008
- Document Type
- Journal Paper
- 423 - 431
- 2008. Society of Petroleum Engineers
- 4.3.1 Hydrates, 5.3.2 Multiphase Flow, 5.5.1 Simulator Development, 5.5 Reservoir Simulation, 4.1.2 Separation and Treating, 5.3.1 Flow in Porous Media, 5.1.1 Exploration, Development, Structural Geology, 5.5.7 Streamline Simulation, 5.1.5 Geologic Modeling, 4.3.4 Scale, 5.6.5 Tracers
- 2 in the last 30 days
- 491 since 2007
- Show more detail
- View rights & permissions
|SPE Member Price:||USD 5.00|
|SPE Non-Member Price:||USD 35.00|
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.
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.
|File Size||1 MB||Number of Pages||9|
Aavatsmark, I. 2002. Anintroduction to multipoint flux approximations for quadrilateral grids.Computational Geosciences 6 (3-4): 405-432.doi:10.1023/A:1021291114475.
Aavatsmark, I., Barkve, T., and Mannseth, T. 1998a. Control-Volume Discretization Methodsfor 3D Quadrilateral Grids in Inhomogeneous, Anisotropic Reservoirs.SPEJ 3(2): 146-154. SPE-38000-PA. doi:10.2118/38000-PA.
Aavatsmark, I., Barkve, T., Ø. Bøe, and Mannseth, T. 1996. Discretization onnon-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media.Journal of Computational Physics 127 (1): 2-14.doi:10.1006/jcph.1996.0154.
Aavatsmark, I., Barkve, T., Ø. Bøe, and Mannseth, T. 1998b. Discretization onunstructured grids for inhomogeneous, anisotropic media. Part I: Derivation ofthe methods. SIAM Journal on Scientific Computing 19 (5):1700-1716. doi: 10.1137/S1064827595293582.
Aavatsmark, I., Barkve, T., Ø. Bøe, and Mannseth, T. 1998c. Discretization onunstructured grids for inhomogeneous, anisotropic media. Part II: Discussionand numerical results. SIAM Journal on Scientific Computing19 (5): 1717-1736. doi: 10.1137/S1064827595293594.
Aziz, K. and Settari, A. 1979. Petroleum Reservoir Simulation.London: Elsevier.
Babuška, I. 1973. The finiteelement method with Lagrangian multipliers. Numerische Mathematik20 (3): 179-192. doi:10.1007/BF01436561.
Batycky, R.P., Blunt, M.J., and Thiele, M.R. 1997. A 3D Field-Scale Streamline-BasedReservoir Simulator. SPERE 12 (4): 246-254. SPE-36726-PA.doi: 10.2118/36726-PA.
Bratvedt, F., Bratvedt, K., Buchholz, C.F., Gimse, T., Holden, H., Holden,L., and Risebro, N.H. 1993. Frontline and Frontsim: Two full scale, two-phase,black oil reservoir simulators based on front tracking. Surveys onMathematics for Industry 3 (3): 185-215.
Brenner, S.C. and Scott, L.R. 1994. The Mathematical Theory of FiniteElement Methods. New York: Texts in Applied Mathematics,Springer-Verlag.
Brezzi, F. 1974. On the existence, uniqueness and approximation of saddlepoint problems arising from Lagrange multipliers. RAIRO Anal. Numér.8: 129-151.
Brezzi, F. and Fortin, M. 1991. Mixed and Hybrid Finite ElementMethods, Vol. 15. Berlin: Series in Computational Mathematics,Springer-Verlag.
Brezzi, F., Douglas, J. Jr., and Marini, L.D. 1985. Two families of mixed finiteelements for second order elliptic problems. Numerische Mathematik47 (2): 217-235. doi:10.1007/BF01389710.
Cordes, C. and Kinzelbach, W. 1992. Continuous groundwater velocityfields and path lines in linear, bilinear, and trilinear finite elements.Water Resources Research 28 (11): 2903-2911.doi:10.1029/92WR01686.
Datta-Gupta, A. and King, M.J. 1995. A semianalytic approachto tracer flow modeling in heterogeneous permeable media. Advances inWater Resources 18 (1): 9-24. doi:10.1016/0309-1708(94)00021-V.
Dormand, J.R. and Prince, P.J. 1980. A family of embeddedRunge-Kutta formulae. J. Comput. Appl. Math. 6 (1): 19-26.doi:10.1016/0771-050X(80)90013-3.
Edwards, M.G. 2002. Unstructured, control-volumedistributed, full-tensor finite-volume schemes with flow based grids.Computational Geosciences 6 (3-4): 433-452.doi:10.1023/A:1021243231313.
Edwards, M.G. and Rogers, C.F. 1998. Finite-volume discretizationwith imposed flux continuity for the general tensor pressure equation.Computational Geosciences 2 (4): 259-290.doi:10.1023/A:1011510505406.
Hægland, H., Dahle, H.K., Eigestad, G.T., Lie, K.-A., and Aavatsmark, I.2007. Improvedstreamlines and time-of-flight for streamline simulation on irregulargrids. Advances in Water Resources 30 (4): 1027-1045.doi:10.1016/j.advwatres.2006.09.002.
Jimenez, E., Sabir, K., Datta-Gupta, A., and King, M.J. 2007. Spatial error and convergence instreamline simulation. SPEREE 10 (3): 221-232. SPE-92873-PA.doi: 10.2118/92873-PA.
Juanes, R. and Matringe, S.F. In press. Unified formulation forhigh-order streamline tracing on two-dimensional unstructured grids.Journal of Scientific Computing (accepted 2006). doi:10.1007/s10915-008-9228-2.
King, M.J. and Datta-Gupta, A. 1998. Streamline simulation: A currentperspective. In Situ 22 (1): 91-140.
Lee, S.H., Jenny, P., and Tchelepi, H.A. 2002a. A finite-volume method withhexahedral multiblock grids for modeling flow in porous media.Computational Geosciences 6 (3-4): 353-379.doi:10.1023/A:1021287013566.
Lee, S.H., Tchelepi, H.A., Jenny, P., and DeChant, L.J. 2002b. Implementation of a flux-continuousfinite-difference method for stratigraphic, hexahedron grids. SPEJ7 (3): 267-277. SPE-80117-PA. doi: 10.2118/80117-PA.
Marsden, J.E. and Hughes, T.J.R. 1994. Mathematical Foundations ofElasticity. Mineola, New York: Dover Publications, Inc.
Matringe, S.F., Juanes, R., and Tchelepi, H.A. 2006. Robust streamline tracingfor the simulation of porous media flow on general triangular and quadrilateralgrids. J. Comput. Phys. 219 (2): 992-1012.doi:10.1016/j.jcp.2006.07.004.
Matringe, S.F., Juanes, R., and Tchelepi, H.A. 2007a. A new mixed finiteelement on hexahedra that reduces to a cell-centered finite difference method.Numerische Mathematik (submitted October 2007).
Matringe, S.F., Juanes, R., and Tchelepi, H.A. 2007b. Mixed Finite-Element andRelated-Control-Volume Discretizations for Reservoir Simulation onThree-Dimensional Unstructured Grids. Paper SPE 106117 presented at the SPEReservoir Simulation Symposium, Houston, 26-28 February. doi:10.2118/106117-MS.
Matringe, S.F., Juanes, R., and Tchelepi, H.A. 2007c. Streamline Tracing on GeneralTriangular or Quadrilateral Grids. SPEJ 12 (2): 217-233.SPE-96411-PA. doi: 10.2118/96411-PA.
Mosé, R., Siegel, P., Ackerer, P., and Chavent, G. 1994. Application of the mixed hybridfinite element approximation in a groundwater flow model: Luxury ornecessity? Water Resources Research 30 (11): 3001-3012.doi:10.1029/94WR01786.
Pollock, D.W. 1988. Semianalyticalcomputation of path lines for finite-difference models. Ground Water26 (6): 743-750. doi:10.1111/j.1745-6584.1988.tb00425.x.
Prévost M., Edwards, M.G., and Blunt, M.J. 2002. Streamline Tracing on CurvilinearStructured and Unstructured Grids. SPEJ 7 (2): 139-148.SPE-78663-PA. doi: 10.2118/78663-PA.
Raviart, P.A. and Thomas, J.M. 1977. A mixed finite element method forsecond order elliptic problems. In Mathematical Aspects of the FiniteElement Methods: Proceedings of the Conference Held in Rome, December 10-12,1975, ed. I. Galligani and E. Magenes, Vol. 606, 292-315. New York: LectureNotes in Mathematics, Springer-Verlag.
Verma, S. and Aziz, K. 1997. AControl Volume Scheme for Flexible Grids in Reservoir Simulation. Paper SPE37999 presented at the SPE Reservoir Simulation Symposium, Dallas, 8-11 June.doi: 10.2118/37999-MS.