Models of Contaminant Migration:
The Role of Chromatographic Models

Joachim Gruber

Soil Physics, Institute for Soil Sciences, Hohenheim University
D - 70599 Stuttgart, Germany
(affiliation at time of this seminar, June 1996)

Abstract

Some wastes have to be kept isolated from the biosphere for times so long that during their migration aside from the well known dilution processes other processes might re-accumulate them, thus creating secondary waste deposits. Of the models we have been using to predict future waste concentrations major ones are unable to deal with such accumulation processes. The more comprehensive models applied for some two decades provide us numerical solutions of the equations describing migration. Because of the non-linear chemical interactions their results can only be inter- or extrapolated, when we know where to expect transitions. The chromatographic model reveals more of the structure of the system of equations. Some of its basic features are described, among them its method to visualize transitions between regions within which waste migration can be inter- or extrapolated.

A. Introduction

Biosphere under the Shield of Concentration Thresholds

Tasks of Transport Models

Because of the nonlinearities in the mass action laws used to formulate the chemical interactions

B. Limits to our Understanding of Contaminant Transport - the Role of Geochemistry

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		 Cederberg 1985
Gruber 1987
Ortoleva et al. 1987
Runkel et al. 1997
... adsorption: non-linear multicomponent-adsorption isotherms

Cj= Cj(c)
analytical, multicomponent chromatography (D = 0), approximation
-----
shocks (implosion waves, calculated in Los Alamos, 1943),
chromatographic accumulation after remobilisation (applied in chemical engineering) 		 Aris & Amundson 1973 
Gruber 19941994a1996
Helfferich & Klein 1970 
Rhee et al. 1970, 1989
Schweich 1986
natural geochemical barriers empirically observed maximum concentrations in potable groundwater
----
time asymptotic concentrations applicable to estimate period of isolation of radioactive waste 		 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

C. Visualizing Migration of Chemical Components in Porous Media

Summary

What might multicomponent chromatography (or -equivalently- the mathematics of non-linear hyperbolic differential equations) contribute to our understanding of chemical transport (migration)? We expect this theory to help us understand how secondary deposits may form far from the orignal waste repository, possibly within the biosphere. We imagine this to be comparable perhaps to clouds forming and decomposing again in the drifting packages of air masses in the sky.

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 (flow path), the x-coordinate extending along the tube axis. It is visualized as solute flow and fixation to the solid surfaces presented by the porous medium (adsorption, surface precipitation).

  1. The migration of a chemical component j in a stream tube is mathematically described by the transport equation.
  2. A series of assumptions will reduce the complexity of the transport equation, simplifying
    1. the physical processes and
    2. the chemistry of solute fixation,
    thus placing the migration problem in two well established fields:
    1. the chromatography (= migration along a stream tube) of multiple chemical components ("multicomponent chromatography"), or -equivalently-
    2. the mathematics of non-linear hyperbolic differential equations.
  3. The nonlinearity is due to the non linear concentration dependence of the chemical interactions.
  4. Chromatogaphic transport starts with a perturbation of a stream tube.
    1. One basic perturbation is the abrupt change of the chemical composition of the solute entering the stream tube ("the Riemann Problem").
      1. The abrupt composition change ("concentration step") at the stream tube entrance develops into a concentration wave that travels along the stream tube. Perhaps in analogy to the circular wave system that we generate by throwing a stone into a pond, the wave is called "centered wave", its "center" being the location of the perturbation in space and time.
  5. The mathematics of hyperbolic systems
    1. tells us that there are basically two types of centered waves, depending on the type of abrupt change:
      1. the "rarefaction wave" which becomes flatter while traveling, and
      2. the shock (sometimes called "traveling wave") which is a propagation of the concentration step that tends to conserve the steepness of the step.
    2. helps us visualize the complete wave pattern that belongs to a given chemical interaction: The waves are plotted in concentration space instead of in space-time. That way, a "street map" of all possible waves is generated that is typical of the chemical interaction. One could compare it with the street map of a city being typical of the civilization that built it.
    3. lends us thus a systematic method to understand the influence of chemical model parameter uncertainty on solute migration. This seems essential in view of the large extent of our lack of knowledge,
      1. both on the microscopic, process related scale and on the field scale, as well as
      2. on the parameter variation in the large time frame within which some contaminants have to isolated.

I. Basic Assumptions

(nomenclature
for details see J. Gruber, "Waves in a Two-Component System: The Oxide Surface as a Variable Charge Adsorbent")

I. 1 Transport Equation

conservation of masss

(eq. 1)

I. 2 Simplifying Assumptions: Physics of Flow

These 2 assumptions mean: Water content (which is equal to the porosity in the water saturated porous medium considered) and flow velocity q are the same always and everywhere, i.e. independent from x and t.

I. 3 Simplifying Assumptions: Chemistry of Flow

II. 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,<

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

III. Chemical Model: Adsorption Isotherms

Such solution methods start from specific simplified laws describing the chemical interactions immobilizing the chemical components ("adsorption isotherm", the name implying thermodynamic equilibrium):
  1. 1-component system,
  2. system with non-interacting components,
  3. system with several components competing for adsorption sites.

IV. The Riemann Problem, Centered Waves

IV. 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

(eq. 11)
Riemann Step

IV. 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.

IV. 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

V. The Jacobian Matrix, Eigenvectors and Wave Patterns

V. 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.

V. 2 Definition: Street map of the system

Concentration profiles  c(xj,) 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").

V. 3 Theorem: Eigenvectors of the Jacobian

The Jacobian has as many eigenvectors as the system has chemical components (or -equivalently- centered waves).

VI. Examples of  Wave Patterns

Fig. 2 shows the two waves in concentration space that were previously shown in space-time in Fig. 1.

VI. 1 Two-Component System with Little Competition

Fig. 2: Construction of waves in concentration space. The waves in space-time are the same as in Fig. 1. 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.

VI. 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 2, i.e. as given in eq. 10. Therefore, the waves are straight lines as they are in Fig. 2, but because of the competition both concentrations vary together. Thus, the waves are no longer parallel to the coordinate axes.

VI. 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).

VI. 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.

Here are two examples:

  1. Precipitation/adsorption:

    2. The solute runs into a "trap" for some chemical component, i.e. the solute migrates into an area in which that component precipitates (or adsorbes to a larger degree than outside that region).
  2. Remobilization:

    3. 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
    • form soluble complexes with A, thus "dissolving" the precipitate of A or
    • compete with A for adsorption sites, thus "desorbing" A.
    In both cases, a front of elevated concentration of A migrates through deposit 1.

    The concentration peak of A might subsequently be trapped further downstream along the stream tube, forming a secondary deposit, deposit 2 of A, the concentration of which is may be higher than that of deposit 1 of A.


Fig. 3: Grid of centered waves in aqueous acetate system in equilibrium with an ion exchange resin as shown in Fig. 2. 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).

Fig. 3 shows such a remobilization process in the aqueous acetate solution in equilibrium with an ion exchange resin discussed above. A specific Riemann Problem concentration step has been superimposed on the wave grid of Fig. 2 of the above mentionaed paper:

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.
  1. OAc concentration at m = 0.095 mol/L
  2. OH concentration = 0.015 mol/L
OAc is accumulated between the waves.


Fig. 4: Centered waves (heavy lines in Fig. 3) emerging from an initial concentration step at the column entrance. System is the same as in Fig. 3, 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: centered1- 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 remobilizes.

Fig. 4 is the space-time representation of the two centered waves into which the initial concentration step develops.

D. Conclusions: Open Questions

Grid topology of multicomponent systems for
  1. ion exchange, of e.g. alkali- and earth alkali ions, on surfaces with constant charge (clay minerals),
  2. complex formation of metal cations on surfaces with variable charge (metal oxides, organic surfaces)
Issues:

E. References

Aris, R. and N.R. Amundson, Mathematical Methods in Chemical Engineering, Prentice Hall, Englewood Cliffs, New Jersey, NJ 07632, 1973.

Bear, J., Hydraulics of Groundwater, McGraw Hill Book Company, New York, 1979.

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.

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

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.

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.

APPENDIX

App. 1  1-component system

The adsorbed concentration C is given as the following function of the solubl:e concentration c:
(eq. 2)
                C(c) = Kd(c) c ("adsorption isotherm")

If the distribution coefficient Kd is independent from the solute concentration, then

(eq. 3)
eq. 3

and this simplifies the expression for the retardation of the contaminant (retardation with respect to the water, the solvent of the components):

(eq. 4)
retardation

App. 2 System with N0 Non-Interacting Components

If the components do not compete for adsorption sites, we have a system of N0 non-interacting equations:

(eq. 5)
                C1(c1) = Kd(c1) c1
                C2(c2) = Kd(c2) c2
                ....
                CNo(cNo) = Kd(cNo) cNo

The derivatives of adsorbed concentrations with respect to the soluble concentrations can be expressed in vector form (the equivalent of eq. 3)


(eq. 6)
eq. 6

This Kd matrix is called Jacobian of the system. In this case of indepedent components the Jacobian is diagonal:

(eq. 7)
eq. 7

The retardation is expressed by the matrix R , which is diagonal ,too, and composed of the retardaions of each independent component (the equivalent of eq. 4)

(eq. 8)
eq. 8

App. 3 Multicomponent System with Competition

The components compete for adsorption sites as quantified by the multicomponent isotherms, also called "surface speciation". Often the surface speciation is calculated numerically in speciation programs like MINEQL. Here are the multicomponent isotherms (analogous to eq. 5)

(eq. 9)

                C1(c1, c2,  ...., cNo)
                C2(c1, c2,  ...., cNo)
                 ....
                CNo(c1, c2,  ...., cNo)
 

The Jacobian of the interacting system, the matrix of the derivatives of the vector of adsorbed concentrations, C', has been defined by eq. 6.

Solution:

  1. The eigenvalues of the Jacobian are Kd1, Kd2, ..., KdNo.
  2. With these, eq. 8 gives the retardations of the No centered waves.

App. 3.1 Example of competitive adsorption

In the experiment shown in Fig. 1 adsorption
  • .. follows a multicomponent Langmuir isotherm (eq. 10) with the the components competing for adsorption sites being Na, Mg and possibly some poorly adsorbing background components

    • (eq. 12)
    eq. 12

    where j = Na, Mg. Bg is the constant background ion concentration, consisting of any number of ions.
    If 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 compete to a rather limited extent for adsorption sites. This case has been chosen in the experiment in Fig. 1. for

  • ... follows the laws of ion exchange, which in the case of Fig. 1 would be mathematically equivalent to a system of 2 little competing components in front of a background.<

App. 3.1.1 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).
  1. At low concentrations (i.e. for kj cj << 1 with j = 1, 2) the adsorbed concentration Cj increases linearly with the soluble concentration cj, i.e. 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.
  2. At high concentrations (i.e. for kj cj near 1), the adsorbed concentration is nearly independent from the soluble concentration, i.e roughly equal to the concentration XT of adsorption sites: Cj(c) = XT or Kd(cj >>1/kj) = 0. The retardation of the component in the column 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).

    3. At intermediate concentrations the retardation of component j decreases with increasing concentration of component j.

App. 3.2 Inherent Assumptions in Solution for System with a Multicomponent Isotherm

App. 3.2.1 Assumption in Mathematical Model
c(x, t) = c(x)

This means: a single abrupt change of the concentraton of the incoming water at x = 0 develops in spacce-time in a deterministic way. In other words: Each component j of the solution concentration vector c (a centered wave, i.e. a wave emerging from x = 0) has its own, fixed velocity xj. In multicomponent chromatography this is called "coherence".

App. 3.2.2 Possible Simplifying Assumptions in Chemical Model
The adsorption processes can be simplified in various ways:
  1. multicomponent isotherm (Langmuir-isotherm): competition for a limited number XT of adsorption sites

    2. (eq. 10)

    eq. 10

    • ion exchange on surfaces with constant charge
    • complex formation on surfaces with variable charge: "surface complexation models" describe adsorption reaction with mass action laws involving surface sites and solute radicals:
      • "Triple Layer Model"
      • "Generalized 2-Layer Model" - site types with differing affinities (Dzombak and Morel)
  2. "Ideal Adsorbed Solution Theory": adsorption reaction are described with an analogue of Raoult's law instead of with mass acion law.
  3. kj(c) = const, kn(c) cn << Max(1, kj(c) cj), i.e. limitation to regions in concentration space in which the multicomponent isotherm eq. 10 is -to a good approximation- composed of single component isotherms eq. 5

App. 4 Examples of Centered Waves

Centered waves are shown in Fig. 1.
  • Consider a column of porous medium, the inlet of which lies at x = 0, the outlet at x = XL.
  • Let Na+ mean the concentration of Na+ ions and Mg++   the concentration of Mg++ ions.
  • The porous medium is initially filled with an aqueous solution containing 25 mmol Na+ ions per liter and no Mg++ ions, i.e. c+ = {Na+, Mg++} = {25, 0} mmol/L ("pre-equilibrant").
  • Suddenly, let's say at t = 0, the composition of the aqueous solution entering the column, the feed, switches from c+ to c- = {Na+, Mg++} = {0, 25} mmol/L.
The left side of Fig. 1. 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.
 

App. 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. 1).
 

App. 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. 1. 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).

Figure 1

Fig. 1: Concentration waves result from a sudden concentration jump at the entrance of a column filled with a porous medium, i.e. the column entrance is the origin of 2 waves (the waves are "centered" about x = 0). y-axis: concentration (units mmol per liter) measured at column inlet x = 0 (left side of figure) and column exit x = XL (right side of figure) 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.

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