Most models applied to deal with a more complex geochemistry provide us numerical solutions of the equations describing radionuclide migration. Because of the non-linear geochemical interactions their results can only be inter- or extrapolated when we know the system structure. The chromatographic model reveals us this structure.
In this paper some of the basic features of the chromatographic model are described, among them a method to visualize regions within which waste migration behaves normally, meaning that migration behavior can be inter- or extrapolated.
An example might be the following specific scenario, where inventory X is a repository of hazardous waste:
Application:
Inventory X is radioactive -
Conservative estimate of the necessary isolation time of radioactive waste based on the "specific activity concept"
If X is radioactive, composed of stable and radioactive isotopes of the element x, the isotopic compositions of x in X and X_{1} might differ only insignificantly (processes 3 and 4 minimize mixture of radwaste elements with natural elements). Thus the radioactive waste element x needs to be isolated from the sequence of processes 1 - 6 as long as the bioavailable concentration of x (wherever it may actually be) is radiotoxic, meaning: its permissible concentration c_{p} is smaller than the naturally observed one ("reference concentration level").
geochemistry
is being considered via ... |
method of model solution ----- transport property inherent in model |
early literature |
... adsorption,
ignoring competition: (linear) 1-component system with distribution
coefficient, Kd,
C_{j}= Kd c_{j} |
analytical ----- dilution increasing with travel time, similar to diffusion, analogous to atmospheric transport calculated in Los Alamos, 1944 |
text
book: Bear 1979,
Gruber 1978 |
... adsorption, ignoring competition: non-linear 1-component-adsorptions-isotherm
C_{j}= C_{j}(c_{j}) |
analytical or numerical ----- travelling wave having constant concentration over time |
v. Duijn & Knabner 1991, Liu 1987, Tondeur 1987, van der Zee 1990 |
... complex formation and precipitation calculated with numerical chemical speciation model | coupled
geochemical-geohydraulic computer program ----- conditional results of a non-linear model |
Amir and Kern 2010 Cederberg 1985 Gruber 1987 Ortoleva et al. 1987 Runkel et al. 1997 |
... adsorption: non-linear multicomponent-adsorption isotherms
C_{j}= C_{j}(c) |
analytical, multicomponent chromatography (dispersion/diffusion coeff. D = 0), approximation ----- shocks (similar mathematics as used in shaping implosion triggers for nuclear weapons, Los Alamos, 1943), chromatographic accumulation after remobilization (applied in chemical engineering) |
Aris & Amundson 1973 Gruber 1994, 1994a, 1996 Helfferich & Klein 1970 Rhee et.al.1970, 1989 Schweich 1986 |
natural geochemical barriers |
empirically observed maximum concentrations in potable groundwater ---- environmental concentrations of stable isotopes used to estimate period of isolation of corresponding radioactive isotopes |
Gruber 1983, 1988 |
precipitation |
consideration of the generation of spatial and temporal structures ---- self-organisation with spatial concentration oscillations |
Lichtner 1985 Ortoleva et al. 1986 |
Sections 3 and 4 will give a short overview of multicomponent chromatography as used in describing contaminant transport. Details have been presented in my following papers:
In this introduction into the field of multicomponent chromatography, we will approximate the porous medium with an ensemble of mutually independent stream tubes. Migration of chemical components will be considered then in one such stream tube, the x-coordinate extending along the tube axis. It is visualized as solute flow and immobilization on the solid surfaces presented by the porous medium (adsorption, surface precipitation).
Because the adsorbed concentration C is a non-linear function of the soluble concentrations c, the migration behavior is non-linear. Usually, a coupled, numerical geochemical-geohydraulic model is used to compute migration phenomena. Because of the nonlinearities, interpretation (i.e. inter- and extrapolation) of the results of numerical solutions of the transport equation is problematic, as long as it is unknown at what concentrations or parameter values the system switches from one type of behavior to another.
More conceptual insight provide the solution methods employed in the well established fields of
Initial and boundary condition of the Riemann problem
Fig. 1: Construction of waves in concentration space (central diagram) from their representation in space-time. The waves are the same as in Fig. 4. Abszissa in concentration space: Na-concentration, ordinate in concentration space: Mg-concentration. The Na-wave (rarefaction wave) arrives before the Mg-wave (shock).
Because Na and Mg ions compete very little with each other for adsorption sites, the waves are roughly parallel to the coordinate axes in concentration space.
Here are two examples:
Fig. 2 shows such a remobilization process in the aqueous acetate solution in equilibrium with an ion exchange resin. A specific Riemann Problem concentration step has been superimposed on the wave grid (street map of the system):
Fig. 3: Centered waves (heavy lines in Fig. 2) emerging from an initial concentration step at the column entrance. System is the same as in Fig. 2, only the representation has been switched to concentration vs. space-time. Abscissa: distance from column entrance (measured in e.g. cm), ordinate: concentration of OAc and of OH in solution (measured in e.g. mol/L). Upper part of Figure: initial concentration step(at t = 0), lower part of Figure: centered 1- and 2-wave with intermediate state m elevated in OAc. The state between the waves is elevated in OAc because previously adsorbed acetate has been remobilized.
Fig. 3 is the space-time representation of the two centered waves in Fig. 2 into which the initial concentration step develops.
Bear, J., Hydraulics of Groundwater, McGraw Hill Book Company, New York, 1979.
Merkel, B.J. and B. Planer-Friedrich, Groundwater geochemistry: a practical guide to modeling of natural and contaminated aquatic systems, D.K. Nordstrom (ed.), Springer, 2008.
Cederberg, G.A.; Street, R.L.; Leckie, J.O. A Groundwater Mass Transport and Equilibrium Chemistry Model for Multicomponent Systems. Water Resour. Res. , 21, 1095-1104, 1985.
van Duijn, C.J. and P. Knabner, Solute transport in porous media with equilibrium and non-equilibrium multiple-site adsorption: travelling waves, J. reine angew. Math., 415, 1-49, 1991.
Dzombak, D.A. and F.M.M. Morel, Surface Complexation Modeling: Hydrous Ferric Oxide, John Wiley, 1990
Gruber, J. and A.A. Moghissi: Methodology for hazard assessment of environmental tritium, in: Internat. Conf. Behaviour of Tritium in the Environemnt, San Francisco, CA, U.S.A., October 17 - 21, 1978.
Gruber, J., High-level radioactive waste from fusion reactors, Environ. Sci. Tech. 17, 425 - 431, 1983. 1997 version
Gruber, J., Contaminant Accumulation During Transport Through Porous Media, Los Alamos National Laboratory, 1987
Gruber, J., Destabilization of waste plumes, in: Waste Management '87, Tucson, AZ, U.S.A., 1. - 5. March, 1987.
Gruber, J., Natural geochemical isolation of neutron-activated waste: scenarios and equilibrium models, Nuclear and Chemical Waste Management, 8, 13 - 32, 1988.
Gruber, J., Contaminant Accumulation During Transport Through Porous Media, Water Resourc. Res., 26, 99 - 107, 1990.
Gruber, J., Waves in a Two-Component System: The Oxide Surface as a Variable Charge Adsorbent, Ind. Eng. Chem. Res., 34, 8, 1994. Abstract
Gruber, J., Advective Transport of Interacting Solutes: The Chromatographic Model, Springer, Heidelberg, 1994a. Abstract
Gruber, J., Transport in wandernden Fronten, in: Umweltverhalten von Sedimenten, Abschlußbericht, BMFT-Verbundprojekt 02WT90143, 1994b. Abstract
Gruber, J., Advective transport of interacting solutes: the chromatographic model, Chapter 11 in: U. Förstner and W. Calmano (eds.), "Sediments and Toxic Substances", Environmental Sciences Series, Springer, Heidelberg, 1996.
Gruber, J., Concentration waves: chromatographic theory and experimental verification
Gruber, J. und R. Klein, Remobilisierende Wellen in Strompfaden bei chemischem Gleichgewicht: Anreicherung im Rahmen der Vielkomponenten-Chromatographie, 1997.
Helfferich, F. and G. Klein, Multicomponent Chromatography - Theory of Interference, Marcel Dekker, New York, 1970.
Lichtner, P.C., Continuum model for simultaneous chemical reactions and mass transport in hydrothermal systems, Geochim. Cosmochim. Acta, 49, 779-800, 1985.
Liu, T.-P., Hyperbolic conservation laws with relaxation, Commun. Math. Phys. 108, 153-175 , 1987.
Ortoleva, P. et al., Redox front propagation and banding modalitites, Physica, 19D, 334-354, 1986.
Ortoleva, P., E. Merino, G. Moore, and J. Chadam, Geochemical self-organization I: Reaction-transport feedbacks and modeling approach, Am. J. Sci., 287, 979-1007, 1987.
Ortoleva, P. "NONLINEAR PHENOMENA AT GEOLOGICAL REACTION FRONTS WITH ENERGY APPLICATIONS", Report Prepared by Peter Orto;eva, Geo-Chemical Research Assoc., 1105 Brooks Dr., Bloomington, IN 47401, DOE/ER/13802--2, DE93 008375, Final Report, DOE Grant #DE-FG02-87ERI3802-A000, for the Basic Energy Sciences Program, US Department of Energy, 1993 (in cache).
Ortoleva, P., principal investigator, Self-Organized Mega-Structures in Sedimentary Basins", Department of Energy/Basic Energy Sciences, Contract No. DE-FG02-91ER14175, Final Report, November 2000 - October 2003 (in cache).
Rhee, H.-K., R. Aris and N.R. Amundson, On the theory of multicomponent chromatography, Philos. Trans. Roy. Soc. London, A267, 419 - 455, 1970.
Rhee, H.-K., R. Aris and N.R. Amundson, First-Order Partial Differential Equations: Volume II, Theory and Application of Hyperbolic Systems of Quasilinear Equations, Prentice Hall, Englewood Cliffs, New Jersey, NJ 07632, 1989.
Runkel, R.L., Bencala, K.E., Broshears, R.E., Chapra, S.C., Reactive Solute Transport in Streams: I. Development of an Equilibrium-based Model, U.S. Geological Survey, University of Colorado, June 18, 1997
Schweich, D., J. Villermaux, M. Sardin, An introduction to the non-linear theory of adsorptive reactors, AIChE Journal, 26, 3, 477-486, 1980Tondeur, D., Unifying concepts in non-linear unsteady processes, Part I: Solitary travelling waves, Chem. Eng. Process., 21, 167-178, 1987.
Tondeur, D., Unifying concepts in non-linear unsteady processes, Part I: Solitary travelling waves, Chem. Eng. Process., 21, 167-178, 1987.
van der Zee, S.E.A.T.M., Analytical traveling wave solution for transport with nonlinear nonequilibrium adsorption, Water Resour. Res., 26, 2563-2577, 1990.
Consider further that everywhere in the column each chemical component has its own concentration, and that this "initial concentration spectrum" is the same throughout the column and does not change with time.
Consider now water with a new concentration spectrum ("feed") being continuously fed into the column entrance (x = 0) while the same amount of water with the initial concentration spectrum leaves the column exit. This abrupt change of the chemical composition of the water propagates through the column: One notices that the change splits into a number of changes separated by areas of constant concentration. Changes and constant states migrate through the column. The migrating changes will be called "waves" and the constant concentrations between the waves will be called "constant states".
This process is usually calculated by numerically solving the equations describing the transport of the chemical components: For every initial and feed concentration spectrum the computer comes up with a set of waves and constant states. Because the chemical interactions with the porous medium depend non-linearly on the component concentrations, one principally cannot inter- or extrapolate between computed results.
Another way of calculating waves and constant states is solving the transport equations analytically. This way we get one complete solution for all initial and feed concentrations. The solution is a graphical map of all waves and constant states that the chemical system can generate. In "Multicomponent Chromatography" Helfferich (Helfferich & Klein, 1970) calls it the "street map of the system". In mathematics (Hyperbolic Systems of Conservation Laws, Lax 1957, 1973) it is called a "hodograph".
This paper will explain waves using the latter method, i.e. using analytical solutions of the transport process.
If the distribution coefficient Kd is independent from the solute concentration, then
and this simplifies the expression for the retardation of the contaminant (retardation with respect to the water, the solvent of the components):
The concentration of any species j on the adsorption sites ("adsorbed concentrations", C_{j}) is a function of the concentration of all species:
C_{1}(c_{1}, c2, ...., c_{No})
C_{2}(c_{1}, c2, ...., c_{No})
....
C_{No}(c_{1}, c2, ...., c_{No})
The derivatives of adsorbed concentrations with respect to the soluble concentrations can be expressed in matrix form (the equivalent of eq. 3)
The Kd matrix is called Jacobian of the system.
The retardation is expressed by the matrix R, which is diagonal, too, and composed of the retardations of each independent component (the equivalent of eq. 4)
Solution:
where component j = 1 is Na, and j = 2 is Mg. Bg is the constant background ion concentration, consisting of any number of ions.
If -as is the case in the experiment of Fig. 4- the background components bind poorly to the adsorption sites and if there are plenty of adsorption sites occupied with background ions, Na and Mg ions have an ample choice of adsorbing sites, i.e. they can choose from any of X_{T} sites. Thus they compete to a rather limited extent with each other.
C_{j}(c) = X_{T} k_{j} c_{j} or
Kd(c_{j} <<1/k_{i}, i = 1, 2) = X_{T} k_{j}.
The retardation of the component in the column is R_{j} > 1 (as given in eq. 4), i.e. the solute j travels with a speed smaller than the speed of pure water.
C_{j}(c) = X_{T} or
Kd(c_{j} >>1/k_{j}) = 0.
The retardation of component j is R_{j} = 1 (see eq. 4), i.e. at those concentrations the solute j travels with nearly maximum speed (i.e. with nearly the speed of pure water).
This means: a single abrupt change of the concentraton of the incoming water at x = 0 develops in space-time into N_{0} centered waves, i.e. waves emerging from x = 0, each wave having its own, fixed velocity ξ_{j}. In multicomponent chromatography this is called "coherence".
Fig. 4: Concentration waves result from a sudden concentration jump at the entrance of a column filled with a porous medium. In other words: the column entrance is the origin of 2 waves, the waves being "centered" about x = 0. There is very little competition between Na and Mg for adsorption sites, thus in concentration space the waves are almost parallel to the coordinate axes. y-axis: concentration (units mmol per liter) measured at column inlet x = 0 and exit x = X_{L} as a function of time t. The abszissa expresses time t as multiples of T_{L}, the time necessary for pure (ion free) water to travel from the column entrance to the column exit.
The concentration of adsorption sites on the porous medium covered with (more loosely bound) background ions is larger than the concentration of adsorbed Na and Mg ions, thus there are plenty of adsorption sites available for Na and Mg. Therefore the influence of the Na concentration drop at time t = 3 T_{L} and the one of the Mg concentration rise at time t = 6 T_{L }is not visible.