Abstract
Cation exchange in groundwater is one of the dominant surface reactions. Mass transfer of cation exchanging pollutants in groundwater is highly nonlinear due to the complex nonlinearities of exchange isotherms. This makes difficult to derive analytical solutions for transport equations. Available analytical solutions are valid only for binary cation exchange transport in 1-D and often disregard dispersion. Here we present a semi-analytical solution for linearized multication exchange reactive transport in steady 1-, 2- or 3-D groundwater flow. Nonlinear cation exchange mass–action–law equations are first linearized by means of a first-order Taylor expansion of log-concentrations around some selected reference concentrations and then substituted into transport equations. The resulting set of coupled partial differential equations (PDEs) are decoupled by means of a matrix similarity transformation which is applied also to boundary and initial concentrations. Uncoupled PDE’s are solved by standard analytical solutions. Concentrations of the original problem are obtained by back-transforming the solution of uncoupled PDEs. The semi-analytical solution compares well with nonlinear numerical solutions computed with a reactive transport code (CORE2D) for several 1-D test cases involving two and three cations having moderate retardation factors. Deviations of the semi-analytical solution from numerical solutions increase with increasing cation exchange capacity (CEC), but do not depend on Peclet number. The semi-analytical solution captures the fronts of ternary systems in an approximate manner and tends to oversmooth sharp fronts for large retardation factors. The semi-analytical solution performs better with reference concentrations equal to the arithmetic average of boundary and initial concentrations than it does with reference concentrations derived from the arithmetic average of log-concentrations of boundary and initial waters.
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References
Appelo, C.A.J.: Multicomponent ion exchange and chromatography in natural systems. In: Lichtner, P.C., Steefel, C.I., Oelkers, E.H. (eds.) Reactive Transport in Porous Media. Reviews in Mineralogy, vol. 34, pp. 193–227. MSA (1996)
Appelo C.A.J., Postma D. (1993) Geochemistry, Groundwater and Pollution. Balkema, Rotterdam
Bolt G.H. (1979) Movement of solutes in soil: principles of adsorption/exchange chromatography. In: Bolt G.H. (ed) Soil Chemistry B, Physico-chemical Methods. Elsevier, Amsterdam, pp. 285–348
Bond W.J., Phillips I.R. (1990) Approximate solutions for cation transport during unsteady, unsaturated soil water flow. Water Resour. Res. 26:2195–2205
Charbeneau R.J. (1988) Multicomponent exchange and subsurface solute transport: characteristics, coherence and the Reimann problem. Water Resour. Res. 24, 57–64
Clement T.P. (2001) Generalized solution to multispecies transport equations coupled with a first-order reaction network. Water Resour. Res. 37, 157–163
Dai, Z., Samper, J.: Inverse problem of multicomponent reactive chemical transport in porous media: formulation and applications: Water Resour. Res. 40, w07407, doi: 10.1029/2004WR003248 (2004)
Dai Z., Samper J. (2006) Inverse modeling of water flow and multicomponent reactive transport in coastal aquifer systems. J. Hydrol. 327(3–4):447–461
Dai Z., Samper J., Ritzi R. (2006) Identifying geochemical processes by inverse modeling of multicomponent reactive transport in Aquia aquifer. Geosphere 4(4):210–219
Dou W., Jin Y.C. (1996) Analytical solution of the solute transport equation for the binary homovalent ion exchange in groundwater. J. Hydrol. 180, 139–153
Helfferich F., Klein G. (1970) Multicomponent Chromatography. Marcel Dekker, New York
Hunt B.R., Lipsman R.L., Rosenberg J.M. (2001) A Guide to MATLAB for Beginners and Experienced Users. Cambridge University Press, Cambridge, 327 pp
Huyakorn P.S., Ungs M.J., Mulkey L.A., Sudicky E.A. (1987) A three-dimensional analytical method for predicting leachate migration. Ground Water 25(5):588–598
Jin Y.C., Ye S.L. (1999) Analytical solution for monovalent-divalent ion exchange transport in groundwater. Can. Geotech. J. 36:1197–1201
Johnson L.W., Ries L.W. (1981) Introduction to linear algebra. Addison-Wesley-Longman, Reading MA, pp. 155–160
Lichtner P.C., Yabusaki S., Pruess K., Steefel C.I. (2004) Role of competitive cation exchange on chromatographic displacement of cesium in the vadose zone beneath the Hanford S/SX tank farm. J. Vadose Zone 3, 203–219
Molinero J., Samper J. (2006) Modeling of reactive solute transport in fracture zones of granitic bedrocks. J. Cont. Hydrol. 82, 293–318
Molinero J., Samper J., Zhang G., Yang C. (2004) Biogeochemical reactive transport model of the redox zone experiment of the Äspö hard rock laboratory in Sweden. Nuclear Technol. 148, 151–165
Press W.H., Teukolsky S.A., Vetterling W.T., Flannery B.P. (1992) Numerical Recipes in Fortran. Cambridge Univ. Press, NY
Rhee H.K., Aris R., Amundson R. (1989) First-Order Partial Differential Equations: Volume II – Theory and Application of Hyperbolic Systems of Quasi-linear Equations, 1st edn. Prentice Hall, New Jersey
Samper J., Yang C. (2006) Stochastic analysis of transport and multicomponent competitive monovalent cation exchange in aquifers. Geosphere 2, 102–112
Samper J., Yang C., Montenegro L. (2003) CORE2D version 4, A code for non-isothermal water flow and reactive solute transport, Users manual. Universidad de La Coruña, Spain
Smoller J. (1983) Shock Waves and Reaction-Diffusion Equations. Springer-Verlag, New York
Steefel C.I., Carroll S.A., Zhao P., Roberts S. (2003) Cesium migration in Hanford sediment: a multisite cation exchange model based on laboratory transport experiments. J. Contam. Hydrol. 67, 219–246
Sun Y., Clement T.P. (1999) A decomposition method for solving coupled multi-species reactive transport problems. Transp. Porous Med. 37, 327–346
Sun Y., Petersen J.N., Clement T.P., Skeen R.S. (1999) Development of analytical solutions for multi-species transport with serial and parallel reactions. Water Resour. Res. 35(1):185–190
Thoolen P.M.C., Hemker P.W. (1994) Approximate methods for n-component solute transport and ion-exchange. J. Comput. Appl. Math. 53(2):275–290
Valocchi A.J. (1984) Describing the transport of ion-exchanging contaminants using an effective Kd approach. Water Resour. Res. 20, 499–503
van Genuchten, M.T., Alves, W.J.: Analytical solutions of the one-dimensional solute transport equation. Technical Bulletin Number 1661, United States Department of Agriculture (1982)
van Veldhuizen M., Hendriks J.A., Appelo C.A.J. (1998) Numerical computation in heterovalent chromatogrphy. Appl. Numer. Math. 28, 69–89
Wilson J.L., Miller P.J. (1978) Two-dimensional plume in uniform groundwater flow. J. Hydraul. Div. Am. Soc. Civ. Eng. 104, 503–514
Xu T., Samper J., Ayora C., Manzano M., Custodio E. (1999) Modeling of non-isothermal multi-reactive transport in field scale porous media flow system. J. Hydrol. 214, 144–164
Zachara J.M., Smith S.C., Liu C., McKinley J.P., Serne R.J., Gassman P.L. (2002) Sorption of Cs+ to micaceous subsurface sediments from the Hanford site. USA. Geochim. Cosmochim. Acta 66, 193–211
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Samper-Calvete, J., Yang, C. A semi-analytical solution for linearized multicomponent cation exchange reactive transport in groundwater. Transp Porous Med 69, 67–88 (2007). https://doi.org/10.1007/s11242-006-9065-4
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DOI: https://doi.org/10.1007/s11242-006-9065-4