is contained in an e ective adsorption isotherm, depending on the carrier con- ... The sorption processes, a ecting the transport of HOC, will be treated in the ..... 4] Ruthven Douglas M. : Principles of adsorption and adsorption processes,.
Qualitative Properties of a Model for Carrier Facilitated Groundwater Contaminant Transport P. Knabner E. Schneid Institut for Applied Mathematics, University Erlangen, Germany
1 Introduction Contaminants with very low water solubilities (e.g. polycyclic aromatic hydrocarbons) play an important role in risk assessment of dangerous wastes and development of soil remediation. The mobility of such hydrophobic substances can be strongly a ected by the existence of carriers (e.g. dissolved organic carbon), which can adsorb the contaminant and thereby enhance or reduce its velocity. The numerical simulation of the spreading of these contaminants, requires the solution of reactive transport equations for all involved components, coupled by the contaminant's sorption to the carrier. Our development is based on a model [2], in which all the carrier's in uence on the contaminant transport is contained in an e ective adsorption isotherm, depending on the carrier concentration and thereby also on space and time. First we shortly summarize the modelling of reactive transport of a single component (carrier, contaminant, carrier bound contaminant) in a porous medium, then in section 3 we combine the two equations for the contaminant components. The properties of the contaminant's e ective isotherm and its in uence on the transport equation are discussed in section 4. In the following the contaminant is abbreviated as HOC (hydrophobic organic chemical), the carrier as DOC (dissolved organic carbon) and the carrier bound contaminant as HOC-DOC. Of course, the model is not restricted to the transport of only these particles but may also be applied to others.
2 Reactive Transport The modelling of transport of a single component (HOC, DOC or HOC-DOC) is based on the following assumptions: The porosity, the di usion-dispersiontensor and the advection are equal for all components and the water ow is not in uenced by the transport of the components. These assumptions should be sensible, if one deals with very small concentrations of particles, with diameters 1
that are very much smaller than the mean poresize. Further we assume, that transport of HOC has no in uence on transport of DOC, i.e. the amount of HOC adsorbed on DOC has not any e ect, neither on transport nor on sorption properties of the DOC. This assumption may be justi ed, if carrier particles are very much larger than contaminant particles. These assumptions enable us to separate the solution of carrier transport and to insert the free and adsorbed carrier's concentration distribution over space and time as parameters into the contaminant transport equation. The equation of reactive transport of a single component in a porous medium, made up of a balance law
~ ~j = P @t (C ) ? r
and the ux consisting of advection, di usion and dispersion,
~j = ?Dr~ C + C~v
is of the form:
@t (C ) ? r~ (Dr~ C ? C~v ) = P
The term P represents the sum of possible sources, like equilibrium sorption (? @t (C )), non-equilibrium sorption (?' @t S ) or the transition rate of HOC from its free to the carrier bound state (F ). As the two sorption processes of DOC are not a ected by the amount of sorbed HOC, the reactive transport equation for DOC is of the form:
~ (Dr~ CD ? CD~v ) = ?'D@t SD ? D @t D (CD ) @t (CD ) ? r
(1)
The equations for the two contaminant components (HOC and HOC-DOC) are:
@t (CHf ) ? r~ (Dr~ CHf ? CHf ~v) = ?F + PHf ~ (Dr~ CHb ? CHb~v) = F + PHb @t (CHb ) ? r
(2)
(3) The sorption processes, a ecting the transport of HOC, will be treated in the following section.
3 Carrier Facilitated Transport Addition of the equations (2) and (3) gives an equation for the whole contaminant, CH = CHf + CHb (4) because of their linearity in the nonreactive parts, thereby eliminating the transition rate F .
~ (Dr~ CH ? CH~v ) = PHf + PHb @t (CH ) ? r 2
(5)
HOC can be sorbed by the carrier, in addition to the equilibrium and nonequilibrium sorption sites of the soil. This process is, in accordance with experimental results, modelled only as equilibrium sorption by an isotherm CHb = CD (CHf ) (6) and exhibits the governing connection between the transport of HOC and HOCDOC. Sorption to DOC can always take place, independent of its state (free, bound to equilibrium or nonequilibrium sorption sites). In (5) we have one source term for equilibrium sorption of HOC to the soil, a second term for nonequilibrium sorption of the whole contaminant and in addition a third source term for the sorption of HOC to DOC, sorbed at equilibrium sorption sites. This one is derived from (6) by substituting the immobile carrier concentration D D (CD ) for the mobile carrier concentration CD and taking the time derivative, to get the transition rate from immobile to mobile phase. PHf + PHb = ? Hf @t Hf (CHf ) ? 'H @t SH ? D @t ( D (CD )(CHf )) (7) The source terms cannot be expressed as functions, depending only on CH , because they do not have the same simple form as the transport terms, and the concentrations CHf and CHb have to be expressed as functions of CH and CD . This can be achieved under minimal assumptions [2] on the isotherm with (4) and (6). CHf = G(CH ; CD ) CHb = CH ? G(CH ; CD ) Together with (7) equation (5) becomes:
@t (CH ) ? r~ (Dr~ CH ? CH~v) = (8) ? Hf @t Hf (G(CH ; CD )) ? D @t ( D (CD )(G(CH ; CD ))) ? 'H @t SH
The equilibrium parts of (8), called the e ective isotherm, di er from that one discussed in [3] with respect to its dependence on CD and CH . While the second source term of (8) is a product of two isotherms, one of DOC and one of HOC (introduced in [2] as scenario B), in [3] an isotherm (introduced in [2] as scenario A) was used, that depends directly on the concentration of HOC-DOC because of the choice of di erent sorption sites for DOC and HOC-DOC.
4 E ective Isotherm Now we will neglect all nonequilibrium processes and investigate the in uence of the carrier on the contaminant transport, which is represented by the e ective isotherm. (CH ; CD (x; t)) = (9) Hf (G (C ; C (x; t))) + D (C (x; t)) (G (C ; C (x; t))) H D H D Hf D D
3
A vanishing carrier concentration in (9) reproduces the isotherm of ordinary transport of HOC. The comparison of the e ective sorption (CH ; CD ) with the elementary sorption Hf Hf (CHf ) at equal concentrations CH = CHf shows whether DOC leads to co-sorption ( (CH ; CD ) > Hf Hf (CHf )) or to cotransport ( (CH ; CD ) < Hf Hf (CHf )). The derivative of the e ective isotherm with respect to CD shows the change of the in uence of DOC on the transport of HOC. @ = ? Hf ? D ( ? ( ? C ) ) D D D @CD 1 + CD Hf D = ?1 + (10) C A(CH ; CD ) 0
0
0
0
0
D
0
We speak of mobilization, if the e ect of an increasing carrier concentration is a reduction of the e ective isotherm (A > 0), i.e. the sorption of HOC to mobile DOC is increasing, while the sorption to immobile DOC and to the soil is decreasing. In the opposite case, where the increasing carrier concentration causes an increase of the e ective isotherm (A < 0), we speak of immobilization. For suciently small values of CD the quotient CH ; 0)) = Hf Hf (G((0) D D shows whether DOC causes co-transport ( > 1 ) A > 0) or co-sorption ( < 1 ) A < 0) of HOC. Sorption isotherms for the class of components, we focus on, have been investigated in numerous publications [1]. The qualitative behaviour of Hf , D and in the range from vanishing to medium concentrations can be characterized in the following way: At rst they are linear either because of the validity of Henry's Law [4] in the range of small concentrations or because of a partitioning equilibrium [1]. In a range from small to medium concentrations these isotherms exhibit a concave curvature [5], which is most often approximated by an isotherm of Freundlich type. Based on this characterization of the elementary isotherms, we can make further statements about the in uence of the carrier on the e ective isotherm. In the range, where the isotherms Hf , D and are concave, the isoterm Hf is a monotone increasing function, while the term D ? ( D ? CD D ) is a monotone decreasing function of CD . In the case of > 1 the in uence of DOC (co-transport) does not change qualitatively by an increase of CD , because the carrier causes always mobilization (A > 0). In the case of < 1 an increasing concentration of DOC can cause a transition from co-sorption to co-transport, because the immobilizing e ect of DOC can change to a mobilizing e ect. One can see from (9) that for < 1 the transition from co-sorption to co-transport exists, if the sorption isotherm D has a saturation. To illustrate these properties with an example, we make some simpli cations. The formation of HOC-DOC is modelled with a linear isotherm = 1 and the concentration of HOC is set constant CH = 1. 0
0
0
0
0
0
4
0
A(1,Cd)
0.4
0.2
0.001
0
0.1
0.5
1
1.5
2
0 Cd 0.5 -0.2 1 -0.4
-0.6 2 -0.8
Figure 1: Co-sorption and co-transport for the contaminant This implies CHf = 1=(1 + CD ). The sorption to the soil is represented by Langmuir isotherms. = a akC + kC
The two parameters of this isotherm are its slope k at C = 0, characterizing an approximatively linear range for small concentrations, and the saturation a, which can be used to describe the range of concave curvature. Hf = 1 CHf D = 1 kCD Hf 2 1 + CHf D 2 1 + kCD In gure (1) A(1; CD ) is plotted as a function of CD for di erent parameters (k 2 f0:001; 0:1; 0:5; 1; 2g). The e ective isotherms resulting from the three bigger k-values show the transition from immobilization to mobilization with increasing carrier concentration. The rst derivative of the e ective isotherm with respect to CH @ = 1 Hf + D @CH 1 + CD Hf D shows: If in the range of small CH the isotherms and Hf are linear, is also linear and consists of the weighted sum of the two sorption isotherms, reduced 0
0
5
0
2
B(Ch,1)
1 0.99 0
0.2
0.4
0.6
0.8
Chf
1
0 0.8 0.6 -1 0.4 -2
-3
0.1
Figure 2: Curvature of the e ective isotherm by the factor 1=(1 + CD ). The second derivative @2 = Hf ( + C ( ? )) + D 1 Hf Hf @CH2 (1 + CD )3 Hf D D = (1 + C1 )3 B (CH ; CD ) 0
00
0
00
00
0
00
0
D
0
shows, that the e ective isotherm has not necessarily the same curvature as the elementary isotherms and Hf . We will illustrate this by an example. Therefore we use Langmuir isotherms for the two sorption processes of HOC = 0:05:5+CCHf ; Hf = 2 2+CCHf Hf Hf and the parameter q governs the relative density of sorption sites Hf = q ; D = 1 ? q :
The concentration of mobile and immobile DOC is set constant with CD = 1 and D(1) = 1. In gure (2) B (CH ; 1) is plotted as a function of CHf for di erent parameters (q 2 f0:1; 0:4; 0:6; 0:8; 0:99g). The change of sign of B with increasing concentration of HOC corresponds to a transition from convex to concave curvature of the e ective isotherm, although and Hf are concave. 6
5 Summary The model introduced in [2] and discussed in [3] has been modi ed in two ways: The sorption of the carrier to the soil is not in uenced by the contaminant and thereby the contaminant can be adsorbed either to the mobile or to the immobile carrier. The qualitative behaviour of the e ective isotherm has been discussed with the use of Langmuir isotherms for the sorption of carrier and contaminant to the soil and for the sorption of the contaminant to the carrier. The transition from co-sorption to co-transport depending on the carrier concentration and the change in curvature of the e ective isotherm are present in the shown model, too.
Notation Variables: space and time coordinates porosity C concentration in the uid phase ~j
ux density P ,F source and sink terms D di usion-dispersion-tensor ~v advection velocity mass density of sorbent S mass density of sorbate at nonequilibrium sorption sites , isotherms of equilibrium sorption sites a, k parameters for the Langmuir isotherm x,t
Indices: Hf HOC Hb HOC-DOC H HOC and HOC-DOC D DOC equilibrium sorption ' nonequilibrium sorption
References [1] Brusseau Mark L., P.S.C. Rao : Sorption nonideality during organic contaminant transport in porous media, Critical Reviews in Environmental Control, 19 1 33-99 (1989) [2] Knabner Peter, K. U. Totsche, I. Koegel-Knabner : The modelling of reactive solute transport with sorption to mobile and immobile sorbents. Part I: Experimental evidence and model development, to appear in: Water Resources Research, (1995) [3] Knabner Peter, K. U. Totsche : The modelling of reactive solute transport with sorption to mobile and immobile sorbents. Part II: Model discussion and numerical simulation, to appear in: Water Resources Research, (1995) 7
[4] Ruthven Douglas M. : Principles of adsorption and adsorption processes, John Wiley & Sons, New York 1984 [5] Schuth Christoph : Sorptionskinetik und Transportverhalten von polyzyklischen aromatischen Kohlenwassersto en (PAK) im Grundwasser Laborversuche Tubinger Geowissenschaftliche Arbeiten - Reihe C, 19 (1994)
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