Abstract
Two basic ingredients present in modeling the transport of reactive multi-components: the transport is described by a set of advection-dispersion-reactive partial differential equations (PDEs) based on the principle of mass balance; the chemical reactions, under the assumptions of local equilibrium, are described by a set of highly nonlinear algebraic equations (AEs) base on the principles of mole balance and mass action. For a typical application, the complete set of nonlinear PDEs and AEs consist of more than one hundred simultaneous equations. Thus, it is impractical to solve this set of equations simultaneously. General practice is to divide this set of equations into two subsets: one is the primary governing equations (PGEs) consisting of mainly the transport equations and the other one is the secondary governing equations consisting of mainly the geochemical reaction equations. The PGEs are solved for the chosen primary dependent variables (PDVs) and the SGEs are used to compute for the secondary dependent variables (SDVs). The major difficulties in simulating the reactive transport is the numerical solution of PGEs. From the computational point of view, the solution of the set of highly nonlinear PDEs are solved either with the direct substitution approach (DSA) or with the sequential iteration approach (SIA). For DSA, geochemical equilibrium reaction equations are substituted into the hydrologic transport equations to results in a set of nonlinear partial differential equations. These nonlinear PDEs are solved simultaneously for the chosen PDVs. For SIA, the procedure consists of iterating between sequentially solving the hydrologic transport equations for the chosen PDVs and solving geochemical equilibrium equations for the SDVs. The most important consideration in choosing PDVs is: it should decouple the simultaneity of nonlinear PDEs as much as possible and it should yield nonlinear source/sink terms in any PDE as small as possible. This paper presents a discussion on the diagonalization of the simultaneous nonlinear PDEs.
Research supported by the Office of Health and Environmental Research, Department of Energy, Grant No. DE-FG02-91-ER61197
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
LANGMUIR, I. 1918. The adsorption of gases on plane surface of glass, mica, and platinum, J. Amer. Chem Soci. 40: 1361–1402.
FREUNDLICH, H. 1926. Colloid and Capillary Chemistry, Methuen, London.
TRUESDELL, A.H. and B.F. JONES 1974. WATEQ, A Computer Program for Calculating Chemical Equilibria of Natural Waters. J. Res. U.S. Geol. Surv. 2 (2): 233–248.
WESTALL, J.C., J.L. ZACHARY, and F.M.M. Morel 1976. MINEQL: A Computer Program for the Calculation of Chemical Equilibrium Composition of Aqueous System. Technical Note 18, Department of Civil Engineering, MIT, Cambridge, MA, 91 pp.
WOLERY, T.J. 1979. Calculation of Chemical Equilibrium Between Aqueous Solution and Minerals: The EQ3/6 Software Package. UCRL-52658, Lawrence Livermore Laboratory, Livermore, CA.
SPOSITO, G. and S.V. Mattigod 1980. GEOCHEM: A Computer Program for the Calculation of Chemical Equilibria in Soil Solutions and Other Natural Water Systems. Department of Soil and Environmental Sciences, University of California, Riverside, CA, 92 pp.
PARKHURST, D.L., D.C. Thorstenson and L.N. Plummer 1980. PHREEQE — A Computer Program for Geochemical Calculations. U.S. Geological Survey Water Resources Investigations 80–96, USGS, Reston, VA.
FELMAY, A.R., D.C. GIRVIN, and E.A. JENNE 1984. MINTEQ: A Computer Program for Calculating Aqueous Geochemical Equilibria, EPA-600/3-84-032, National Technical Information Services, Springfield, VA.
YEH, G.T. 1992. EQMOD: A Chemical Equilibrium Model of Complexation, Adsorption, Ion Exchange, Precipitation/Dissolution, Redox, and Acid-Base Reactions. Lectural Note, Part I. Short Course on Modeling Multispecies-Multicomponent Transport through Saturated-Unsaturated Media, Penn State University, University Park, PA, November, 1992.
VALOCCHI, A.J., R.L. STREET, and P.V. ROBERTS 1981. Transport of ion-exchanging solutes in groundwater: Chromatographic theory and field simulation. Water Resour. Res. 17(5):1517–1527.
JENNINGS, A.A., D.J. KIRKNER, AND I.L. THEIS 1982. Multicomponentequilibrium chemistry in groundwater quality models. Water Resour. Res. 18(4):1089–1096.
MILLER, C.W. 1983. CHEMTRN USER’s MANUAL, LBL-16152, Lawrence Berkeley Laboratory, University of California, Berkeley, CA.
CEDERBERG, G.A. 1985a. TRANQL: A Ground-water Mass-Transport and Equilibrium Chemistry Model for Multicomponent Systems, Ph.D. Thesis, Department of Civil Engineering, Stanford University, Stanford, CA.
CEDERBERG, G.A. 1985b. A groundwater mass transport and equilibrium chemistry model for multicomponent systems. Water Resour. Res. 21(8):1095–1104.
YEH, G.T. and V.S. TRIPATHI. 1991. A model for simulating transport of reactive multispecies components: Model development and demonstration. Water Resour. Res. 27(12): 3075–3094.
YEH, G.T. and V.S. TRIPATHI. 1989. A critical evaluation of recent developments of hydrogeochemical transport models of reactive multi-chemical components. Water Resour. Res. 25(l):93–108.
RUBIN, J. and R.V. JAMES. 1973. Dispersion-affected transport of reacting solutes in saturated porous media: Galerkin method applied to equilibrium-controlled exchange in unidirectional steady water flow. Water Resour. Res. 9(5):1332–1356.
THEIS, T.L., D.J. KIRKNER, and A.A. JENNINGS. 1982. Multi-solute Subsurface Transport Modeling for Energy Solid Wastes. Tech. Progress Rept., C00- 10253 - 3, Dept. of Civil Engrg., Univ. of Notre Dame, Notre Dame, IN.
YEH, G.T. and V.S. TRIPATHI. 1990. HYDROGEOCHEM: A Coupled Model of HYDROlogic Transport and GEOCHEMical Equilibria in Reactive Multicomponent Systems. ORNL-6371. Oak Ridge National Laboratory, Oak Ridge, TN., 312 pp.
YEH, G.T. 1987. FEMWATER: A Finite Element Model of WATER flow through Saturated-Unsaturated Porous Media — First Revision, ORNL-5567/R1, Oak Ridge National Laboratory, Oak Ridge, TN.
RUBIN, J. 1983. Transport of reacting solutes in porous media: Relation between mathematical nature of problem formulation and chemical nature of reactions. Water Resour. Res. 19(5):1231–1252.
HUYAKORN, P.S., J.W. MERCER, and D.S. WARD. 1985. Finite element matrix and mass balance computational schemes for transport in variably saturated porous media. Water Resour. Res. 21(3):346–358.
BJERRUM, N. 1926. Ionic association I. Influence of ionic association on the activity of ions at moderate degrees of association. Kgl. Danske Videnskab. Selskb. Math-fys. Medd. 7: 1–48.
FUOSS, R.M. 1935. Properties of electrolyte solutions. Chem. Rev. 17: 27–42.
REILLY, P. J., R.H. WOOD, and R.A. ROBINSON. 1971. Prediction of osmotic and activity coefficients in mixed-electrolyte solutions. J. Phys. Chem. 75: 1305–1315.
NORDSTROM, D.K., L.N. PLUMMER, T.M.L. WIGLEY, T.J. WOLERY, J.W. BALL, E.A. JENNE, R.L. BASSETT, D.A. CRERAR, T.M. FLORENCE, B. FRITZ, M. HOFFMAN, G.R. HOLDREN, JR., G.M. LAFON, S.V. MATTIGOD, R.E. MC-DUFF, F. Morel, M.M. REDDY, G. SPOSITO, and J. THRAILKILL. 1979. A Comparison of Computerized Chemical Models for Equilibrium Calculations in Aqueous Systems, Proc. Chemical Modeling in Aqueous Systems, 176th Meetings of the Am. Chemical Soci., Miami Beach, Sept. 11–13, 1978, ACS Symp. Series 93, Am. Chemical Soci. Washington, D.C.
MOREL, F. and J. MORGAN. 1972. A numerical method for computing equilibria in aqueous chemical systems. Environ. Sci. Technol. 6(1):58–67.
PAULING, L. 1959. General Chemistry, 2nd Ed., W.H. Freeman and Company, San Francisco, 710 pp.
WALSH, M.P., S.L. BRYANT, AND L.W. LAKE. 1984. Precipitation and dissolution of solids attending How through porous media. AIChE J. 30(2):317–328.
STUMM, W. and J.J. MORGAN. 1981. Aquatic Chemistry An Introduction Emphasizing Chemical Equilibria in Natural Waters. John Wiley & Sons, New York. 780 pp.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Springer-Verlag New York, Inc.
About this paper
Cite this paper
Yeh, GT., Cheng, HP. (1996). On Diagonalization of Coupled Hydrologic Transport and Geochemical Reaction Equations. In: Wheeler, M.F. (eds) Environmental Studies. The IMA Volumes in Mathematics and its Applications, vol 79. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8492-2_19
Download citation
DOI: https://doi.org/10.1007/978-1-4613-8492-2_19
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4613-8494-6
Online ISBN: 978-1-4613-8492-2
eBook Packages: Springer Book Archive