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.
Similar content being viewed by others
References
Post, V., Abarca, E.: Preface: saltwater and freshwater interactions in coastal aquifers. Hydrogeol. J. 18, 1–4 (2010)
Bear, J., Cheng, A.D.: Theory and applications of transport in porous media: modeling groundwater flow and contaminant transport. Springer, New York (2010)
Diersch, H.J., Kolditz, O.: Variable-density flow and transport in porous media: Approaches and challenges. Adv. Water Resour. 25, 899–944 (2002)
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)
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)
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)
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)
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)
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)
Ackerer, P., Younes, A.: Efficient approximations for the simulation of density driven flow in porous media. Adv. Wat. Resour. 31, 15–27 (2008)
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)
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)
Larson, M., Niklasson, A.: A conservative flux for the continuous Galerkin method based on discontinuous enrichment. CALCOLO 41, 65–76 (2004)
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)
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)
Diersch, H.G.: FEFLOW finite element subsurface flow and transport simulation system. Reference manual. Germany: WASY GmbH, Berlin (2005)
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)
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)
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)
Hassanizadeh, S.: Modeling species transport by concentrated brine in aggregated porous media. Transport Porous Med. 3, 299–318 (1988)
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)
Bear, J.: Dynamics of Fluids in Porous Media. Elsevier, New York (1972)
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)
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)
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)
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)
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
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)
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)
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)
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)
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)
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)
NEA: The international hydrocoin project, level 1 code verification. Tech. rep., Swedish Nuclear Power Inspectorate and OECD/Nuclear Energy Agency, Paris (1988)
Goswami, R., Clement, T.: Laboratory-scale investigation of saltwater intrusion dynamics. Water Resour. Res. 43, 1–11 (2007)
Oswald, S., Kinzelbach, W.: Three-dimensional physical benchmark experiments to test variable-density flow models. J. Hydrol. 290, 22–44 (2004)
Henry, H.: Effects of dispersion on salt encroachment in coastal aquifers, sea water in coastal aquifers. Geol. Surv. Water-supply Pap. 1613-C (1964)
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)
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)
Abarca, E., Carrera, J., Sánchez-Vila, X., Dentz, M.: Anisotropic dispersive Henry problem. Adv. Water Resour. 30, 913–926 (2007)
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)
Post, V., Kooi, H., Simmons, C.: Using hydraulic head measurements in variable-density ground water flow analyses. Ground Water 45, 664–671 (2007)
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)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Povich, T.J., Dawson, C.N., Farthing, M.W. et al. Finite element methods for variable density flow and solute transport. Comput Geosci 17, 529–549 (2013). https://doi.org/10.1007/s10596-012-9330-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-012-9330-2