Computational Geosciences

, Volume 16, Issue 4, pp 1125–1134 | Cite as

Pathline tracing on fully unstructured control-volume grids

  • S. L. Painter
  • C. W. Gable
  • S. Kelkar
Original Paper


The trend toward unstructured grids in subsurface flow modeling has prompted interest in the issue of streamline or pathline tracing on unstructured grids. Streamline tracing on unstructured grids is problematic because a continuous velocity field is required for the calculation, while numerical solutions to the groundwater flow equations provide velocity in discretized form only. A method for calculating flow streamlines or pathlines from a finite-volume flow solution is presented. The method uses an unconstrained least squares method on interior cells and a constrained least squares method on boundary cells to approximate cell-centered velocities, which can then be continuously interpolated to any point in the domain of interest. Two-dimensional tests demonstrate that the method correctly reproduces uniform and corner-to-corner flow on fully unstructured grids. In three dimensions using regular hexahedral grids, the method agrees well with established semianalytical methods. Tests also demonstrate that the method produces physically realistic results on fully unstructured three-dimensional grids.


Streamline Pathline Particle tracking Control volume Unstructured grid Finite volume Groundwater flow 


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  1. 1.
    Amemiya, T.: Advanced Econometrics. Harvard University Press, Cambridge (1985)Google Scholar
  2. 2.
    Crane, M.J., Blunt, M.J.: Streamline-based simulation of solute transport. Water Resour. Res. 35(10), 3061–3078 (1999). doi: 10.1029/1999WR900145 CrossRefGoogle Scholar
  3. 3.
    Cordes C., Kinzelbach, W.: Continous groundwater velocity fields and path lines in linear, bilinear, and trilinear finite elements. Water Resour. Res. 28(11), 2903–2911 (1992)CrossRefGoogle Scholar
  4. 4.
    Cvetkovic V., Dagan, G.: Transport of kinetically sorbing solute by steady random velocity in heterogeneious porous formations. J. Fluid Mech. 265, 189–215 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Dagan G., Cvetkovic V.: Reactive transport and immiscible flow in geological media. I. General theory. Proc. R. Soc. Lond. A 452, 285–301 (1996)zbMATHCrossRefGoogle Scholar
  6. 6.
    Finkel, M, Liedl, R., Teutsch, G.: Modelling surfactant-enhanced remediation of polycyclic aromatic hydrocarbons. Environ. Model. Softw. 14(2–3), 203–211 (1999)Google Scholar
  7. 7.
    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)CrossRefGoogle Scholar
  8. 8.
    Hammond, G.E., Lichtner, P.C., Lu, C., Mills, R.T.: PFLOTRAN: reactive flow and transport code for use on laptops to leadership-class supercomputers. In: Zhang, F., Yeh, G.T., Parker, J.C. (eds.) Ebook: Groundwater Reactive Transport Models. Bentham Science Publishers. ISBN 978-1-60805-029-1 (2010)Google Scholar
  9. 9.
    Jimenez, E., Sabir, K., Datta-Gupta, A., King, M.J.: Spatial error and convergence in streamline simulation. SPE 92873. In: SPE Reservoir Simulation Symposium, Woodlands, Texas, 31 Jan–2 Feb (2005)Google Scholar
  10. 10.
    King, M.J., Datta-Gupta, A.: Streamline simulation: a current perspective. In Situ 22(1), 91–140 (1998)Google Scholar
  11. 11.
    Los Alamos Grid Toolbox: LaGriT, Los Alamos National Laboratory. (2011)
  12. 12.
    Lu, N.: A semianalytical method of path line computation for transient finite-difference groundwater flow models. Water Resour. Res. 30(4), 2449–2459 (1994)CrossRefGoogle Scholar
  13. 13.
    Malmström, M., Destouni, G., Martinet, P.: Modeling expected solute concentration in randomly heterogeneous flow systems with multi-component reactions. Environ. Sci. Technol. 38, 2673–2679 (2004)CrossRefGoogle Scholar
  14. 14.
    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. Comp. Physics 219(2), 992–1012 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Matringe, S.F., Juanes, R., Tchelepi, H.A.: Tracing streamlines on unstructured grids from finite volume discretizations. Soc. Pet. Eng. J. 6(3), 423–431 (2008)Google Scholar
  16. 16.
    Morel-Seytoux, H.J.: Unit mobility ratio displacement calculations for pattern floods in homogeneous medium. Soc. Pet. Eng. J. 13(4), 217–227 (1966)Google Scholar
  17. 17.
    Naff, R.L., Russell, T.F., Wilson, J.D.: Shape functions for velocity interpolation in general hexahedral cells. Comput. Geosci. 6(3–4), 285–314 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Narasimhan, T.N., Witherspoon, P.A.: An integrated finite difference method for analyzing fluid flow in porous media. Water Resour. Res. 12(1), 57–64 (1976)CrossRefGoogle Scholar
  19. 19.
    Park, C.H., Beyer, C., Bauer, S., Kolditz, O.: Using global node-based velocity in random walk particle tracking in variably saturated porous media: application to contaminant leaching from road constructions. Environ. Geol. 55, 1755–1766 (2008). doi: 10.1007/s00254-007-1126-7 Google Scholar
  20. 20.
    Pollock, D.W.: Semi-analytical computation of path lines for finite-difference models. Ground Water 26(6), 743–750 (1988)CrossRefGoogle Scholar
  21. 21.
    Prevost, M., Edwards, M.G., Blunt, M.J.: Streamline tracing on curvilinear structured and unstructured grids. SPE J. 7(2), 139–148 (2002)Google Scholar
  22. 22.
    Pruess, K., Oldenburg, C., Moridis, G.: TOUGH2 User’s Guide, Version 2.0. LBNL-43134. Lawrence Berkeley National Laboratory, Berkeley California (1999)CrossRefGoogle Scholar
  23. 23.
    Yeh, G.T.: On the computation of Darcian velocity and mass balance in the finite-element modeling of groundwater-flow. Water Resour. Res. 17(2), 1529–1534 (1981)CrossRefGoogle Scholar
  24. 24.
    Zyvoloski, G.A.: FEHM: A Control Volume Finite Element Code for Simulating Subsurface Multi-phase Multi-fluid Heat and Mass Transfer. Los Alamos Unclassified Report LA-UR-07-3359, Los Alamos National Laboratory, Los Alamos, NM (2007)Google Scholar

Copyright information

© Springer Science+Business Media B.V. (outside the USA) 2012

Authors and Affiliations

  1. 1.Computational Earth Sciences GroupLos Alamos National LaboratoryLos AlamosUSA

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