Abstract
Accurate streamline tracing and travel time computation are essential ingredients of streamline methods for groundwater transport and petroleum reservoir simulation. In this paper we present a unified formulation for the development of high-order accurate streamline tracing algorithms on unstructured triangular and quadrilateral grids. The main result of this paper is the identification of velocity spaces that are suitable for streamline tracing. The essential requirement is that the divergence-free part of the velocity must induce a stream function. We recognize several classes of velocity spaces satisfying this requirement from the theory of mixed finite element methods and, for each class, we obtain the precise functional form of the stream function. Not surprisingly, the most widely used tracing algorithm (Pollock’s method) emanates in fact from the lowest-order admissible velocity approximation. Therefore, we provide a sound theoretical justification for the low-order algorithms currently in use, and we show how to achieve higher-order accuracy both in the streamline tracing and the travel time computation.
Similar content being viewed by others
References
Ackerer, P., Mosé, R., Siegel, P., Chavent, G.: Application of the mixed hybrid finite element approximation in a groundwater flow model: Luxury or necessity? Reply to the Comment by C. Cordes and W. Kinzelbach. Water Resour. Res. 32(6), 1911–1913 (1996)
Aziz, K., Settari, A.: Petroleum Reservoir Simulation. Elsevier, Amsterdam (1979)
Babuška, I.: The finite element method with Lagrangian multipliers. Numer. Math. 20, 179–192 (1973)
Batycky, R.P., Blunt, M.J., Thiele, M.R.: A 3D field-scale streamline-based reservoir simulator. SPE Reserv. Eng. 11(4), 246–254 (1997)
Bear, J.: Dynamics of Fluids in Porous Media. Environmental Science Series. Elsevier, Amsterdam (1972). Reprinted with corrections, Dover, New York (1988)
Bratvedt, F., Bratvedt, K., Buchholz, C.F., Gimse, T., Holden, H., Holden, L., Risebro, N.H.: Frontline and Frontsim, two full scale, two-phase, black oil reservoir simulators based on front tracking. Surv. Math. Ind. 3, 185–215 (1993)
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, vol. 15. Springer, New York (1994)
Brezzi, F.: On the existence, uniqueness and approximation of saddle point problems arising from Lagrange multipliers. RAIRO Anal. Numér. 8, 129–151 (1974)
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991)
Brezzi, F., Douglas, J. Jr., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47, 217–235 (1985)
Brezzi, F., Douglas, J. Jr., Duran, R., Fortin, M.: Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51, 237–250 (1987)
Cai, Z., Douglas, J. Jr.: Park, M. Development and analysis of higher order finite volume methods over rectangles for elliptic equations. Adv. Comput. Math. 19, 3–33 (2003)
Chavent, G., Jaffré, J.: Mathematical Models and Finite Elements for Reservoir Simulation. Studies in Mathematics and Its Applications, vol. 17. Elsevier, Amsterdam (1986)
Chen, Z., Ewing, R.E.: Comparison of various formulations of three-phase flow in porous media. J. Comput. Phys. 132, 362–373 (1997)
Cordes, C., Kinzelbach, W.: Continuous groundwater velocity fields and path lines in linear, bilinear, and trilinear finite elements. Water Resour. Res. 28(11), 2903–2911 (1992)
Cordes, C., Kinzelbach, W.: Comment on “Application of the mixed hybrid finite element approximation in a groundwater flow model: Luxury or necessity?” by R. Mosé, P. Siegel, P. Ackerer, and G. Chavent. Water Resour. Res. 32(6), 1905–1909 (1996)
Durlofsky, L.J.: Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities. Water Resour. Res. 30(4), 965–973 (1994)
Feng, D., Wang, X.M., Cai, W.L., Shi, J.: A mass conservative flow field visualization method. Comput. Graph. 21(6), 749–756 (1997)
Guadagnini, A., Sánchez-Vila, X., Riva, M., De Simoni, M.: Mean travel time of conservative solutes in randomly heterogeneous unbounded domains under mean uniform flow. Water Resour. Res. 39(3), 1050 (2003). doi:10.1029/2002WR001811
Hægland, H., Dahle, H.K., Eigestad, G.T., Lie, K.A., Aavatsmark, I.: Improved streamlines and time-of-flight for streamline simulation on irregular grids. Adv. Water Resour. 30(4), 1027–1045 (2007). doi:10.1016/j.advwatres.2006.09.002
Jimenez, E., Sabir, K., Datta-Gupta, A., King, M.J.: Spatial error and convergence in streamline simulation. SPE Reserv. Evalu. Eng. 10(3), 221–232 (2007)
Kaasschieter, E.F.: Mixed finite elements for accurate particle tracking in saturated groundwater flow. Adv. Water Resour. 18(5), 277–294 (1995)
King, M.J., Datta-Gupta, A.: Streamline simulation: A current perspective. In Situ 22(1), 91–140 (1998)
Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Prentice-Hall, Englewood Cliffs (1983). Reprinted with corrections, Dover, New York (1994)
Matringe, S.F., Gerritsen, M.G.: On accurate tracing of streamlines. In: SPE Annual Technical Conference and Exhibition, Houston, TX (SPE 89920) (2004)
Matringe, S.F., Juanes, R., Tchelepi, H.A.: Robust streamline tracing for the simulation of porous media flow on general triangular and quadrilateral grids. J. Comput. Phys. 219, 992–1012 (2006). doi:10.1016/j.jcp.2006.07.004
Matringe, S.F., Juanes, R., Tchelepi, H.A.: Tracing streamlines on unstructured grids from finite volume discretizations. In: SPE Annual Technical Conference and Exhibition, San Antonio, TX (SPE 103295) (2006). To appear in Soc. Pet. Eng. J.
Matringe, S.F., Juanes, R., Tchelepi, H.A.: Mixed finite element and related control volume discretizations for reservoir simulation on three-dimensional unstructured grids. In: SPE Reservoir Simulation Symposium, Houston, TX (SPE 106117) (2007)
Matringe, S.F., Juanes, R., Tchelepi, H.A.: A new mixed finite element on hexahedra that reduces to a cell-centered finite difference method. Numer. Math. (2008, submitted)
Matringe, S.F., Juanes, R., Tchelepi, H.A.: Streamline tracing on general triangular or quadrilateral grids. Soc. Pet. Eng. J. 12(2), 217–223 (2007). doi:10.2118/96411–PA
Mosé, R., Siegel, P., Ackerer, P., Chavent, G.: Application of the mixed hybrid finite element approximation in a groundwater flow model: Luxury or necessity? Water Resour. Res. 30(11), 3001–3012 (1994)
Nedelec, J.C.: Mixed finite elements in ℝ3. Numer. Math. 35, 315–341 (1980)
Pollock, D.W.: Semianalytical computation of path lines for finite difference models. Ground Water. 26, 743–750 (1988)
Prevost, M., Edwards, M.G., Blunt, M.J.: Streamline tracing on curvilinear structured and unstructured grids. Soc. Pet. Eng. J. 7(2), 139–148 (2002)
Raviart, P.A., Thomas, J.M.: A mixed finite element method for second order elliptic problems. In: Galligani, I., Magenes, E. (eds.) Mathematical Aspects of the Finite Element Method. Lecture Notes in Mathematics, vol. 606, pp. 292–315. Springer, New York (1977)
Russell, T.F., Wheeler, M.F.: Finite element and finite difference methods for continuous flows in porous media. In: Ewing, R.E. (ed.) The Mathematics of Reservoir Simulation, pp. 35–106. SIAM, Philadelphia (1983)
Weiser, A., Wheeler, M.F.: On convergence of block-centered finite differences for elliptic problems. SIAM J. Numer. Anal. 25(2), 351–375 (1988)
Wheeler, M.F., Yotov, I.: A multipoint flux mixed finite element method. SIAM J. Numer. Anal. 44(5), 2082–2106 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Juanes, R., Matringe, S.F. Unified Formulation for High-Order Streamline Tracing on Two-Dimensional Unstructured Grids. J Sci Comput 38, 50–73 (2009). https://doi.org/10.1007/s10915-008-9228-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-008-9228-2