Advertisement

Computational Geosciences

, Volume 17, Issue 3, pp 529–549 | Cite as

Finite element methods for variable density flow and solute transport

  • T. J. PovichEmail author
  • C. N. Dawson
  • M. W. Farthing
  • C. E. Kees
Original Paper

Abstract

Saltwater intrusion into coastal freshwater aquifers is an ongoing problem that will continue to impact coastal freshwater resources as coastal populations increase. To effectively model saltwater intrusion, the impacts of increased salt content on fluid density must be accounted for to properly model saltwater/freshwater transition zones and sharp interfaces. We present a model for variable density fluid flow and solute transport where a conforming finite element method discretization with a locally conservative velocity post-processing method is used for the flow model and the transport equation is discretized using a variational multiscale stabilized conforming finite element method. This formulation provides a consistent velocity and performs well even in advection-dominated problems that can occur in saltwater intrusion modeling. The physical model is presented as well as the formulation of the numerical model and solution methods. The model is tested against several 2-D and 3-D numerical and experimental benchmark problems, and the results are presented to verify the code.

Keywords

Saltwater intrusion Variable density Stabilized FEM Velocity post-processing 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Post, V., Abarca, E.: Preface: saltwater and freshwater interactions in coastal aquifers. Hydrogeol. J. 18, 1–4 (2010)CrossRefGoogle Scholar
  2. 2.
    Bear, J., Cheng, A.D.: Theory and applications of transport in porous media: modeling groundwater flow and contaminant transport. Springer, New York (2010)CrossRefGoogle Scholar
  3. 3.
    Diersch, H.J., Kolditz, O.: Variable-density flow and transport in porous media: Approaches and challenges. Adv. Water Resour. 25, 899–944 (2002)CrossRefGoogle Scholar
  4. 4.
    Brooks, A.N., Hughes, T.J.R.: Streamline upwind/petrov-galerkin formulations for convection dominated flows with particular emphasis on the incompressible Naviar-Stokes equations. Comput. Methods Appl. Mech. Engrg. 32, 199–259 (1982)CrossRefGoogle Scholar
  5. 5.
    Hughes, T.J.R., Mallet, M., Mizukami, A.: A new finite element formulation for computational fluid dynamics: II. beyond supg. Comput. Methods Appl. Mech. Eng. 54, 341–355 (1986)CrossRefGoogle Scholar
  6. 6.
    Voss, C., Souza, W.: Variable density flow and solute transport simulation of regional aquifers containing a narrow freshwater-saltwater transition zone. Water Resour. Res. 23(10), 1851–1866 (1987)CrossRefGoogle Scholar
  7. 7.
    Knabner, P., Frolkovič, P.: Consistent velocity approximation for finite volume or element discretizations of density driven flow in porous media. In: Aldama, A.A., et al. (ed.) Computational Methods in Water Resources XI, vol. 1: Computational methods in subsurface flow and transport problems., pp. 340–352. Southhampton: Computational Mechanics Publication (1996)Google Scholar
  8. 8.
    Farthing, M.W., Kees, C., Miller, C.: Mixed finite element methods and higher order temporal approximations for variably saturated groundwater flow. Adv. Water Resour. 27, 373–394 (2004)Google Scholar
  9. 9.
    Kees, C., Farthing, M., Dawson, C.: Locally conservative, stabilized finite element methods for variably saturated flow. Comput. Methods Appl. Mech. Eng. 197, 4610–4625 (2008)CrossRefGoogle Scholar
  10. 10.
    Ackerer, P., Younes, A.: Efficient approximations for the simulation of density driven flow in porous media. Adv. Wat. Resour. 31, 15–27 (2008)CrossRefGoogle Scholar
  11. 11.
    Mazzia, A., Putti, M.: Mixed-finite element and finite volume discretizations for heavy brine simulations in groundwater. J. Comput. Appl. Math. 147, 191–213 (2002)CrossRefGoogle Scholar
  12. 12.
    Hughes, T., Feijóo, G., Mazzei, L., Quincy, J.: The variational multiscale method—a pardigm for computational mechanics. Comput. Methods Appl. Mech. 166, 3–24 (1998)CrossRefGoogle Scholar
  13. 13.
    Larson, M., Niklasson, A.: A conservative flux for the continuous Galerkin method based on discontinuous enrichment. CALCOLO 41, 65–76 (2004)CrossRefGoogle Scholar
  14. 14.
    Franca, L.P., Hauke, G., Masud, A.: Revisiting stabilized finite element methods for the advective–diffusive equation. Comput. Methods Appl. Mech. Engrg. 195, 1560–1572 (2006)CrossRefGoogle Scholar
  15. 15.
    Lin, H.C., Richards, D.R., Yeh, G.T., Cheng, J.R.C., Cheng, H.P., Jones., N.L.: Femwater: A three-dimensional finite element computer model for simulating density-dependent flow and transport in variably saturated media. Report chl-97-12, U.S. Army Research & Development Center (1997)Google Scholar
  16. 16.
    Diersch, H.G.: FEFLOW finite element subsurface flow and transport simulation system. Reference manual. Germany: WASY GmbH, Berlin (2005)Google Scholar
  17. 17.
    Voss, C.: A finite-element simulation model for saturated–unsaturated fluid-density-dependent ground-water flow with energy transport or chemically-reactive single-species solute transport. US Geol. Surv. Water Resour. Invest. (Rep 84-4369) (1984)Google Scholar
  18. 18.
    Frolkovič, P.: Consistent velocity approximation for density driven flow and transport. In: Van Keer R. et al. (ed.) Advanced Computational Methods in Engineering, Part 2., pp. 603–611. Maastricht: Shaker Publishing (1998)Google Scholar
  19. 19.
    Dentz, M., Tartakovsky, D., Abarca, E., Guadagnini, A., Sanchez-Vila, X., Carrera, J.: Variable-density flow in porous media. J. Fluid. Mech. 561, 209–235 (2006)CrossRefGoogle Scholar
  20. 20.
    Hassanizadeh, S.: Modeling species transport by concentrated brine in aggregated porous media. Transport Porous Med. 3, 299–318 (1988)Google Scholar
  21. 21.
    Herbert, A., Jackson, C., Lever, D.: Coupled groundwater flow and solute transport with fluid density strongly dependent upon concentration. Water Resour. Res. 24(10), 1781–1795 (1988)CrossRefGoogle Scholar
  22. 22.
    Bear, J.: Dynamics of Fluids in Porous Media. Elsevier, New York (1972)Google Scholar
  23. 23.
    Bear, J., Cheng, A.D., Sorek, S., Ouazar, D., Herrera, I. (eds.): Theory and Applications of Transport in Porous Media: Seawater Intrusion in Coastal Aquifers - Concepts, Methods and Practices, chap. 5. Kluwer Academic, Dordrecht (1999)Google Scholar
  24. 24.
    Lever, D., Jackson, C.: On the equations for the flow of concentrated salt solution through a porous medium. Tech. Rep. DOE/RW/85.100, U.K. DOE Report (1985)Google Scholar
  25. 25.
    Farthing, M., Kees, C.E.: Evaluating finite element methods for the level set equation. Technical Report TR-09-11, USACE Engineer Research and Development Center (2009)Google Scholar
  26. 26.
    Kees, C.E., Farthing, M.W.: Parallel computational methods and simulation for coastal and hydraulic applications using the Proteus toolkit. In: Supercomputing11: Proceedings of the PyHPC11 Workshop (2011)Google Scholar
  27. 27.
    Balay, S., Brown, J., Buschelman, K., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Smith, B.F., Zhang, H.: PETSc Web page. http://www.mcs.anl.gov/petsc (2011). Accessed 12 Dec 2011
  28. 28.
    Balay, S., Brown, J., Buschelman, K., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Smith, B.F., Zhang, H.: PETSc users manual. Tech. Rep. ANL-95/11—Revision 3.2, Argonne National Laboratory (2011)Google Scholar
  29. 29.
    Balay, S., Gropp, W.D., McInnes, L.C., Smith, B.F.: Efficient management of parallelism in object oriented numerical software libraries. In: Arge, E., Bruaset, A.M., Langtangen, H.P. (eds.) Modern Software Tools in Scientific Computing, pp. 163–202. Birkhäuser, Basel (1997)Google Scholar
  30. 30.
    Demmel, J.W., Eisenstat, S.C., Gilbert, J.R., Li, X.S., Liu, J.W.H.: A supernodal approach to sparse partial pivoting. SIAM J. Matrix Anal. Appl. 20(3), 720–755 (1999)CrossRefGoogle Scholar
  31. 31.
    Li, X.S., Demmel, J.W.: SuperLU_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems. ACM Trans. Math. Softw. 29(2), 110–140 (2003)CrossRefGoogle Scholar
  32. 32.
    Kolditz, O., Ratke, R., Diersch, H.J., Zielke, W.: Coupled groundwater flow and transport: 1. Verification of variable density flow and transport models. Adv. Water Resour. 21(1), 27–46 (1987)CrossRefGoogle Scholar
  33. 33.
    Voss, C., Simmons, C., Robinson, N.: Three-dimensional benchmark for variable-density flow and transport simulation: matching semi-analytical stability modes for steady unstable convection in an inclined porous box. Hydrogeol. J. 18, 5–23 (2010)CrossRefGoogle Scholar
  34. 34.
    NEA: The international hydrocoin project, level 1 code verification. Tech. rep., Swedish Nuclear Power Inspectorate and OECD/Nuclear Energy Agency, Paris (1988)Google Scholar
  35. 35.
    Goswami, R., Clement, T.: Laboratory-scale investigation of saltwater intrusion dynamics. Water Resour. Res. 43, 1–11 (2007)CrossRefGoogle Scholar
  36. 36.
    Oswald, S., Kinzelbach, W.: Three-dimensional physical benchmark experiments to test variable-density flow models. J. Hydrol. 290, 22–44 (2004)CrossRefGoogle Scholar
  37. 37.
    Henry, H.: Effects of dispersion on salt encroachment in coastal aquifers, sea water in coastal aquifers. Geol. Surv. Water-supply Pap. 1613-C (1964)Google Scholar
  38. 38.
    Simpson, M., Clement, T.: Improving the worthiness of the Henry problem as a benchmark for density-dependent groundwater flow models. Water Resour. Res. 40, 1–11 (2004)CrossRefGoogle Scholar
  39. 39.
    Simpson, M.J., Clement, T.: Theoretical analysis of the worthiness of Henry and Elder problems as benchmarks of density-dependent groundwater flow models. Adv. Water Resour. 26, 17–31 (2003)CrossRefGoogle Scholar
  40. 40.
    Abarca, E., Carrera, J., Sánchez-Vila, X., Dentz, M.: Anisotropic dispersive Henry problem. Adv. Water Resour. 30, 913–926 (2007)CrossRefGoogle Scholar
  41. 41.
    Guo, W., Langevin, C.D.: User’s guide to SEAWAT: a computer program for simulation of three-dimensional variable-density groundwater flow. US Geol. Surv. Water-Resour. Invest. (Book 6, Chapter A7) (2002)Google Scholar
  42. 42.
    Post, V., Kooi, H., Simmons, C.: Using hydraulic head measurements in variable-density ground water flow analyses. Ground Water 45, 664–671 (2007)CrossRefGoogle Scholar
  43. 43.
    Johannsen, K., Kinzelback, W., Oswald, S., Wittum, G.: The saltpool benchmark problem—numerical simulation of saltwater upconing in a porous medium. Adv. Water Resour. 25, 335–348 (2002)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht (outside the USA) 2012

Authors and Affiliations

  • T. J. Povich
    • 1
    Email author
  • C. N. Dawson
    • 2
  • M. W. Farthing
    • 3
  • C. E. Kees
    • 3
  1. 1.Department of Mathematical SciencesUnited States Military AcademyWest PointUSA
  2. 2.Computational Hydraulics GroupThe University of Texas at AustinAustinUSA
  3. 3.Coastal and Hydraulics LaboratoryU.S. Army Engineer Research and Development CenterVicksburgUSA

Personalised recommendations