# zbMATH — the first resource for mathematics

The MAST FV/FE scheme for the simulation of two-dimensional thermohaline processes in variable-density saturated porous media. (English) Zbl 1330.76061
Summary: A novel methodology for the simulation of 2D thermohaline double diffusive processes, driven by heterogeneous temperature and concentration fields in variable-density saturated porous media, is presented. The stream function is used to describe the flow field and it is defined in terms of mass flux. The partial differential equations governing system is given by the mass conservation equation of the fluid phase written in terms of the mass-based stream function, as well as by the advection-diffusion transport equations of the contaminant concentration and of the heat. The unknown variables are the stream function, the contaminant concentration and the temperature. The governing equations system is solved using a fractional time step procedure, splitting the convective components from the diffusive ones. In the case of existing scalar potential of the flow field, the convective components are solved using a finite volume marching in space and time (MAST) procedure; this solves a sequence of small systems of ordinary differential equations, one for each computational cell, according to the decreasing value of the scalar potential. In the case of variable-density groundwater transport problem, where a scalar potential of the flow field does not exist, a second MAST procedure has to be applied to solve again the ODEs according to the increasing value of a new function, called approximated potential. The diffusive components are solved using a standard Galerkin finite element method. The numerical scheme is validated using literature tests.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 76M12 Finite volume methods applied to problems in fluid mechanics 76S05 Flows in porous media; filtration; seepage 80A20 Heat and mass transfer, heat flow (MSC2010)
##### Software:
FEFLOW; SEAWAT; ROCKFLOW
Full Text:
##### References:
 [1] Ackerer, P.; Younes, A.; Mosé, R., Modeling variable-density flow and solute transport in porous medium: 1. numerical model and verification, Trans. porous media, 35, 3, 345-373, (1999) [2] Aricò, C.; Tucciarelli, T., MAST solution of advection problems in irrotational flow fields, Adv. water resour., 30, 665-685, (2007) [3] Aricò, C.; Tucciarelli, T., A marching in space and time (MAST) solver of the shallow water equations. part I: the 1D case, Adv. water resour., 30, 1236-1252, (2007) [4] Aricò, C.; Nasello, C.; Tucciarelli, T., A marching in space and time (MAST) solver of the shallow water equations. part II: the 2D case, Adv. water resour., 30, 1253-1271, (2007) [5] Bascià, A.; Tucciarelli, T., An explicit unconditionally stable numerical solution of the advection problem in irrotational flow fields, Water resour. res., 40, 6, W06501, (2004) [6] Bear, J., Hydraulics of groundwater, (1979), McGraw-Hill New York [7] Casulli, V., Eulerian – lagrangian methods for hyperbolic and convection dominated parabolic problems, (), 239-269 [8] Dawson, C.N., Godunov-mixed methods for immiscible displacement, Int. J. numer. meth. fluid, 11, 7, 835-847, (1990) · Zbl 0704.76059 [9] Dawson, C.N., Godunov-mixed methods for advective flow problems in one space dimension, SIAM J. numer. anal., 28, 5, 1282-1309, (1991) · Zbl 0741.65068 [10] Dawson, C.N., Godunov-mixed methods for advection – diffusion equations problems in one space dimension, SIAM J. numer. anal., 30, 5, 1315-1332, (1993) · Zbl 0791.65062 [11] Dawson, C.N., High resolution upwind-mixed finite-elements methods for advection – diffusion equations with variable time-stepping, Numer. meth. PDE, 11, 525-538, (1995) · Zbl 0837.65107 [12] De Josseling de Jong, G., Generating functions in the theory of flow through porous media, () [13] Diersch, H.-J.G., Primitive variables finite-element solutions of free convection flows in porous media, Zschr. angew. math. mech. (ZAMM), 61, 325-337, (1981) · Zbl 0475.76086 [14] Diersch, H.-J.G., Finite-element modeling of recirculating density-driven saltwater intrusion process in groundwater, Adv. water resour., 11, 25-43, (1988) [15] H.-J.G. Diersch, Interactive, graphic-based finite-element simulation systems - FEFLOW - for modelling groundwater flow and contaminant transport process, edited by WASY, Berlin, 1994. [16] Diersch, H.-J.G.; Kolditz, O., Coupled groundwater flow and transport: 2. thermohaline and 3D convection systems, Adv. water resour., 21, 401-425, (1998) [17] H.-J.G. Diersch, FEFLOW finite-element subsurface flow and transport simulation system – user’s manual/reference manual/white papers. Release 5.0, edited by WASY Ltd., Berlin, 2002. [18] H.-J.G. Diersch, FEFLOW finite-element subsurface flow and transport simulation system, white papers II, edited by WASY Ltd., Berlin, 2002. [19] Diersch, H.-J.G.; Kolditz, O., Variable-density flow and transport in porous media: approaches and challenges, Adv. water resour., 25, 899-944, (2002) [20] Elder, J.W., Steady free convection in a porous medium heated from below, J. fluid mech., 27, 29-48, (1967) [21] Elder, J.W., Transient convection in a porous medium, J. fluid mech., 27, 609-623, (1967) [22] Evans, D.G.; Raffensperger, J.P., On the stream function for density-variable groundwater flow, Water resour. res., 28, 8, 2141-2145, (1992) [23] Frind, E.O., Simulation of long-term transient density-dependent transport in groundwater, Adv. water resour., 5, 73-88, (1982) [24] Frolkovic, P.; De Schepper, H., Numerical modelling of convection dominated transport with density-driven flow in porous media, Adv. water resour., 24, 1, 63-72, (2001) [25] Galeati, G.; Gambolati, G., On boundary conditions and point sources in the finite-element integration of the transport equation, Water resour. res., 25, 5, 847-856, (1989) [26] Galeati, G.; Gambolati, G.; Neumann, S., Coupled and partially coupled eulerian – lagrangian model of freshwater – seawater mixing, Water resour. res., 28, 1, 149-165, (1992) [27] Guo, W.; Bennett, G.D., SEAWAT version 1.1: A computer program for simulations of groundwater flow of variable-density, fort myers, (1998), Missimer International Inc. Florida [28] W. Guo, C.D. Langevin, User’s guide to SEAWAT: a computer program for the simulation of three-dimensional variable-density groundwater flow. USGS Techniques of Water Resources Investigations Book 6, USGS, 2002 (Chapter A7). [29] H.R. Henry, Effects of dispersion on salt enrichment in coastal aquifers, in: US Geological Survey Water Supply Paper 1613-C, Sea Water in Coastal Aquifers, 1964, pp. C70-C84. [30] H.R. Henry, J.B. Hilleke, Exploration of multiphase fluid flow in a saline aquifer system affected by geothermal heating, Bureau of Engineering Research, Report No. 150-118, University of Alabama, US Geological Survey Contract No. 14-08-0001-12681, National Technical Information Service Publication No. PB234233, 105, 1972. [31] Herbert, W.; Jackson, C.P.; Lever DA, D.A., Coupled groundwater flow and solute transport with fluid density strongly dependent on concentration, Water resour. res., 24, 1781-1795, (1988) [32] J.D. Hughes, W.E. Sandford, SUTRA-MS, A version of SUTRA modified to simulate heat and multiple-solute transport, Reference Manual, US Department of the interior, US Geological Survey, 2004. [33] P.S. Huyakorn, C. Taylor, Finite-element models for coupled groundwater flow and convective dispersion, in: W.G. Gray, et al. (Ed.), Proceedings of the 1st International Conference Finite-elements in Water Resource, Princeton University, Pentech Press, London, vol. 1, 1976, pp. 131-151. · Zbl 0367.76080 [34] Kolditz, O.; Ratke, R.; Diersch, H.J.G.; Zielke, W., Coupled groundwater flow and transport: 1. verification of variable-density flow and transport models, Adv. water resour., 21, 27-46, (1998) [35] O. Kolditz, A. Habbar, R. Kaiser, T. Rother, C. Thorenz, ROCKFLOW - theory and user’s manual. Release 3.6, Institute of Fluid Mechanics, University of Hannover, Germany, 2001. [36] K.P. Kröhn, Simulation von Transportvorgängen im klüftigen Gestein mit der Methode der Finiten-Elemente. Bericht Nr. 29/1991, Institut für Strömungsmechanik, Universität Hannover, Dissertation, 1991. [37] Mazzia, A.; Bergamaschi, L.; Putti, M., A time-splitting technique for the advection – dispersion equation in groundwater, J. comput. phys., 157, 1, 181-198, (2000) · Zbl 0960.76048 [38] Mazzia, A.; Bergamaschi, L.; Putti, M., On the reliability of numerical solutions of brine transport in groundwater: analysis of infiltration from a salt-lake, Transp. porous media, 43, 65-86, (2001) [39] Mazzia, A.; Putti, M., Higher order Godunov-mixed methods on tetrahedral meshes for density driven flow simulations in porous media, J. comput. phys., 208, 154-174, (2005) · Zbl 1115.76360 [40] Nag Library Manual, 2005. . [41] Nasello, C.; Tucciarelli, T., A dual multi-level urban drainage model, ASCE J. hydr. eng., 131, 9, 743-747, (2005) [42] Neumann, S.P., A eulerian – lagrangian numerical scheme for the dispersion convection equation using conjugate space time grids, J. comput. phys., 41, 270-294, (1981) · Zbl 0484.76095 [43] Noto, V.; Tucciarelli, T., The DORA algorithm for network flow models with improved stability and convergence properties, ASCE J. hydr. eng., 127, 5, 380-391, (2001) [44] Oldenburg, M.; Pruess, K., Dispersive transport dynamics in a strongly coupled groundwater-brine flow system, Water resour. res., 31, 289-302, (1995) [45] Oltean, C.; Bués, M.A., Coupled groundwater flow and transport in porous media. A conservative or non-conservative form, Transp. porous media, 44, 2, 219-246, (2001) [46] Pinder, G.F.; Cooper, H.H., A numerical technique for calculating the transient position of the saltwater front, Water resour. res., 6, 875-882, (1970) [47] Segol, G., Classic groundwater simulations – proving and improving numerical models, (1994), PTR Prentice-Hall Englewood Cliffs [48] Segol, G.; Pinder, G.F.; Gray, W.A., A Galerkin-finite-element technique for calculating the transient position of the saltwater front, Water resour. res., 11, 2, 343-347, (1975) [49] Senger, R.K.; Fogg, G.E., Stream functions and equivalent freshwater heads for modelling regional flow of variable-density groundwater 1. review of theory and verification, Water resour. res., 26, 9, 2089-2096, (1990) [50] Thorne, D.T.; Langevin, C.D.; Sukopc, M.C., Addition of simultaneous heat and solute transport and variable fluid viscosity to SEAWATER, Comput. geosci., 32, 1758-1768, (2006) [51] Tucciarelli, T.; Termini, D., Finite-element modelling of floodplain flows, ASCE J. hydr. eng., 126, 6, 416-424, (2000) [52] I. Voss, A finite-element simulation model for saturated – unsaturated fluid-density-dependent groundwater flow with energy transport or chemically-reactive single-species solute transport, in: US Geol. Surv., Water Resour. Invest., Rep. 84-4369, 1984, p. 409. [53] Voss, I.; Souza, W.R., 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) [54] Younes, A.; Ackerer, P.; Mosè, R., Modelling variable-density flow and solute transport in porous medium: 2. re-evaluation of the salt dome flow problem, Transp. porous media, 35, 3, 375-394, (1999)
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.