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:1. Introduction
1.1 Biosphere in a Niche Protected by Natural Barriers
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 X1 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 cp 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,
Cj= Kd cj |
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
Cj= Cj(cj) |
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
Cj= Cj(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).
(eq. 1)
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
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 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", Cj) is a function of the concentration of all species:
C1(c1, c2, ...., cNo)
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.
Cj(c) = XT kj cj or
Kd(cj <<1/ki, i = 1, 2) = XT kj.
The retardation of the component in the column is Rj > 1 (as given in eq. 4), i.e. the solute j travels with a speed smaller than the speed of pure water.
Cj(c) = XT or
Kd(cj >>1/kj) = 0.
The retardation of component j is Rj = 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 N0 centered waves, i.e. waves emerging from x = 0, each wave having its own, fixed velocity ξj. 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 = XL as a function of time t.
The abszissa expresses time t as multiples of TL, 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 TL and the one of the Mg concentration rise at time t = 6 TL is not visible.
3.2.2 Simplifying Assumptions: Physics of Flow
These two assumptions mean: Water content (which is equal to the porosity Φ of the porous medium, assuming that the medium is saturated with water) and flow velocity q are the same always and everywhere, i.e. independent from x and t.
3.2.3 Simplifying Assumptions: Chemistry of Flow
3.3 Multicomponent Chromatography and Predictability
Non-linear systems are characterized by thresholds, across which the system behavior cannot be predicted based on information of the behavior on one side of the threshold. Example: behavior of a car with a payload: As soon as the load exceeds the design limits, it is unsafe.
3.4 Chemical Model: Adsorption Isotherms
Such solution methods start from specific simplified laws that describe the chemical interactions immobilizing the chemical components ("adsorption isotherm", the name implying thermodynamic equilibrium):
3.5 The Riemann Problem, Centered Waves
3.5.1 Definition: Riemann Problem
Concentration profiles forming after a single abrupt change of the chemical composition of the water influx ("feed") into a porous medium are called "solutions of the Riemann Problem" or "centered waves".
Initial and boundary condition of the Riemann problem
3.5.2 Assumption: Interacting "centered waves"
It is assumed that every concentration profile in a stream tube can be represented as an interaction of centered waves.
3.5.3 Theorem: Centered waves
As many centered waves emerge from the abrupt concentration change (eq. 11) as there are chemical components in the system. Every centered wave
4. The Jacobian Matrix, Eigenvectors
and Wave Patterns
4.1 Theorem: Jacobian of the System
The matrix of the derivatives of the adsorbed concentrations with respect to the soluble concentrations (eq. 6), the Jacobian of the system, contains all information about the possible centered waves.
4.2 Definition: Street Map of the System
Concentration profiles c(ξj) of the chemical components across rarefaction waves are plotted in concentration space (instead of in space-time). The grid composed of such lines is typical of the chemical interactions in the system ("street map of the system", example).
4.3 Theorem: Eigenvectors of the Jacobian
The Jacobian has as many eigenvectors as the system has chemical components (or -equivalently- centered waves).
5 Examples of Wave Patterns
Fig. 1 shows the construction of two waves in concentration space (upper right diagram) from their representation in space-time (upper left and lower right diagrams).
5.1 Two-Component System with Little Competition
5.2 Two-Component System with Marked Competition
Here is an example of two components (OH- and OAc-) competing strongly with each other for adsorption sites. Adsorption is via ion exchange, thus the adsorption isotherm is a 2-component isotherm of the same type as in the system of Figs. 1 and 4, i.e. as given in eq. 10. Therefore, the waves are straight lines as they are in Fig. 1, but because of the competition both concentrations vary together. Thus, the waves are no longer parallel to the coordinate axes.
5.3 Three-Component System with Competition
When Cl ions are added to the solution of OH and OAc ions and concentration space becomes 3-dimensional, the resulting waves clearly show the presence of concentration thresholds separating regions of concentration space with similar wave structures. The wave structure changes across those thresholds (see Fig. 4 and Fig. 5 of that publication).
5.4 Formation of Secondary Repositories
When chemical interactions are taken into account the way it has been described here (i.e. not simply by applying the linear 1-component Kd-model), it is obvious that during migration the concentration of chemical components will not necessarily decrease continuously.
The solute runs into a "trap" for some chemical component A, i.e. the solute migrates into an area in which component A precipitates (or adsorbes to a larger degree than outside that region).
Consider the following situation:
In some area the chemical component A under consideration has been immobilized either by precipitation or adsorption. Let us call this area "deposit 1 of A". Subsequently, a solute enters deposit 1 and redissolves component A. The solute thus becomes richer in A. In order to be able to dissolve deposit 1 of A, the solute has to carry high concentrations of chemical components that
In both cases, a front of elevated concentration of A migrates through deposit 1. Since the concentration of A in solution is in equilibrium with the concentration of A on adsorbing sites, the front is a traveling secondary deposit of A ("deposit 2"). The concentration peak of A might subsequently be trapped further downstream along the stream tube. In any case, the concentration in deposit 2 may be higher than that in deposit 1.
Fig. 2: Grid of centered waves in aqueous acetate system in equilibrium with an ion exchange resin ("street map of the system" is Fig. 2 in "Verification of Concentration Waves)". A specific Riemann step (-, +) is chosen, and the two waves emerging from that step are emphasized. The state m between the waves is elevated in OAc due to remobilization of previously adsorbed acetate. Abscissa: concentration of OAc in solution (mol/L), ordinate: concentration of OH in solution (mol/L).
From this Riemann step two centered waves emerge: the fast wave is called 2-wave, the slow wave is called 1-wave. In between the waves the concentration is given by the coordinates of the point m.
OAc is accumulated between the waves.
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.
6 Conclusions: Open Questions
Grid topology of multicomponent systems for
Issues:
7 References
Amir, L. and Kern, M. A global method for coupling transport with chemistry in heterogeneous porous media, Comput Geosci 14:465Ð481, 2010 (in cache).
Aris, R. and N.R.
Amundson, Mathematical Methods in Chemical Engineering, Prentice Hall,
Englewood Cliffs, New Jersey, NJ 07632, 1973.
Consider a column filled with an adsorbing porous medium and water ("pore water"). Both the water and the adsorbing medium contain chemical components (like Na or Mg) in addition to some background components Bg. We have a total of N0 chemical components in the column.
8 APPENDIX
8.1 1-component system
The adsorbed concentration C is given
as the following function of the soluble concentration c:
(eq. 2)
C(c) = Kd(c) c ("adsorption isotherm")
8.2 System with N0 Components
C2(c1, c2, ...., cNo)
....
CNo(c1, c2, ...., cNo)
8.2.1 Non-Interacting Components
In the case of independent components, i.e. when Cj = Cj(cj) (similar to the example presented in Fig. 4), the Jacobian is obviously diagonal:
8.2.2 Competing Components
The components compete for adsorption sites as quantified by the multicomponent isotherms below. Specific examples:
8.2.2.1 Example of Competitive Adsorption
In the experiment shown in Fig. 4 adsorption
(eq. 12)
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 XT sites. Thus they compete to a rather limited extent with each other.
8.2.2.2 Transport Properties of Competitive Adsorption
The form of the wave depends on the adsorption properties of the porous medium as specified by the adsorption isotherm eq. 12.
8.3.2 Inherent Assumptions when Solving System with Multicomponent Isotherm
8.3.2.1 Mathematical Model
c(x, t) = c(ξj)
8.3.2.2 Chemical Model
The adsorption processes can be simplified in various ways (see also Gruber und Klein):
(eq. 10)
8.4 Examples of Centered Waves
Centered waves are shown in Fig. 4.
The left side of Fig. 4. shows the concentration jump as a function of time at the column
entrance. The right side shows the concentration waves that originate from the concentration jump at the column entrance.
8.4.1 Rarefaction Wave
Thus, when a wave has a higher concentrations at its front than at its tail, the front concentrations travel faster than the low concentrations at its end, and the wave spreads out as it travels through the column. It it therefore called "rarefaction wave" (see upper part of Fig. 2).
8.4.2 Shock
The reverse situation is interesting:
When the solute with high concentration is located behind the solute with low concentration, it "bumps" into the solute with low concentration. The concentration step does not spread out as it travels through the column. This type of solution to the Riemann problem is called "shock", and it is visualized in the lower part of Fig. 2. The wavefront spreads only due to hydrodynamic dispersion or molecular diffusion as described for a 1-component system with dispersion in sections 12 A 1 and 12 A 2 in J. Gruber, Advective Transport of Interacting Solutes: The Chromatographic Model. A shock that is broadened by dispersion is called "traveling wave" (Liu, 1987).
Location
(URL) of this page
Physics Home
Version: 7 July 2015
Joachim Gruber