×

zbMATH — the first resource for mathematics

Finite element methods for variable density flow and solute transport. (English) Zbl 1392.76031
Summary: 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.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Post, V; Abarca, E, Preface: saltwater and freshwater interactions in coastal aquifers, Hydrogeol. J., 18, 1-4, (2010)
[2] Bear, J., Cheng, A.D.: Theory and applications of transport in porous media: modeling groundwater flow and contaminant transport. Springer, New York (2010) · Zbl 1195.76002
[3] Diersch, HJ; Kolditz, O, Variable-density flow and transport in porous media: approaches and challenges, Adv. Water Resour., 25, 899-944, (2002)
[4] Brooks, AN; Hughes, TJR, 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) · Zbl 0497.76041
[5] Hughes, TJR; 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) · Zbl 0622.76074
[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, 1851-1866, (1987)
[7] Knabner, P; Frolkovič, P; Aldama, AA (ed.); etal., Consistent velocity approximation for finite volume or element discretizations of density driven flow in porous media, No. 1, 340-352, (1996), Southhampton
[8] Farthing, MW; 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)
[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) · Zbl 1194.76123
[10] Ackerer, P; Younes, A, Efficient approximations for the simulation of density driven flow in porous media, Adv. Wat. Resour., 31, 15-27, (2008)
[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) · Zbl 1058.76034
[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) · Zbl 1017.65525
[13] Larson, M; Niklasson, A, A conservative flux for the continuous Galerkin method based on discontinuous enrichment, CALCOLO, 41, 65-76, (2004) · Zbl 1168.65415
[14] Franca, LP; Hauke, G; Masud, A, Revisiting stabilized finite element methods for the advective-diffusive equation, Comput. Methods Appl. Mech. Engrg., 195, 1560-1572, (2006) · Zbl 1122.76054
[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)
[16] Diersch, H.G.: FEFLOW finite element subsurface flow and transport simulation system. Reference manual. Germany: WASY GmbH, Berlin (2005)
[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)
[18] Frolkovič, P; Keer, R (ed.); etal., Consistent velocity approximation for density driven flow and transport, 603-611, (1998), Maastricht
[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) · Zbl 1157.76388
[20] Hassanizadeh, S, Modeling species transport by concentrated brine in aggregated porous media, Transport Porous Med., 3, 299-318, (1988)
[21] Herbert, A; Jackson, C; Lever, D, Coupled groundwater flow and solute transport with fluid density strongly dependent upon concentration, Water Resour. Res., 24, 1781-1795, (1988)
[22] Bear, J.: Dynamics of Fluids in Porous Media. Elsevier, New York (1972) · Zbl 1191.76001
[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)
[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) · Zbl 0497.76041
[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)
[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)
[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] 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)
[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) · Zbl 0882.65154
[30] Demmel, JW; Eisenstat, SC; Gilbert, JR; Li, XS; Liu, JWH, A supernodal approach to sparse partial pivoting, SIAM J. Matrix Anal. Appl., 20, 720-755, (1999) · Zbl 0931.65022
[31] Li, XS; Demmel, JW, Superlu_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems, ACM Trans. Math. Softw., 29, 110-140, (2003) · Zbl 1068.90591
[32] Kolditz, O; Ratke, R; Diersch, HJ; Zielke, W, Coupled groundwater flow and transport: 1. verification of variable density flow and transport models, Adv. Water Resour., 21, 27-46, (1987)
[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)
[34] NEA: The international hydrocoin project, level 1 code verification. Tech. rep., Swedish Nuclear Power Inspectorate and OECD/Nuclear Energy Agency, Paris (1988)
[35] Goswami, R; Clement, T, Laboratory-scale investigation of saltwater intrusion dynamics, Water Resour. Res., 43, 1-11, (2007)
[36] Oswald, S; Kinzelbach, W, Three-dimensional physical benchmark experiments to test variable-density flow models, J. Hydrol., 290, 22-44, (2004)
[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)
[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)
[39] Simpson, MJ; 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)
[40] Abarca, E; Carrera, J; Sánchez-Vila, X; Dentz, M, Anisotropic dispersive henry problem, Adv. Water Resour., 30, 913-926, (2007)
[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)
[42] Post, V; Kooi, H; Simmons, C, Using hydraulic head measurements in variable-density ground water flow analyses, Ground Water, 45, 664-671, (2007)
[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.