Transport with spatially variable kinetic
sorption: recursion formulation
A. K. Mishra
1 a ,*, A. Gutjahr
b& H. Rajaram
c aDepartment of Earth and Environmental Science, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA b
Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA c
Department of Civil, Environmental and Architectural Engineering, University of Colorado, Boulder, CO, USA
(Received 19 May 1997; revised 2 April 1998; accepted 2 June 1998)
A recursion formulation for the transport of linearly sorbing solutes undergoing non-equilibrium sorption is developed. Constant or spatially varying sorption kinetics can be modeled using the recursion approach. The sorption and desorption rates are modeled as two independent random processes with a prescribed mean and covariance structure with spatial variability in the rate parameters included as well. The recursion solution, in terms of the probability density function for solute travel times, is derived by specifying transition probabilities for moving between the aqueous and sorbed phases. A few simple examples are used to illustrate the approach. The computer implementa-tion leads to a very rapid algorithm that is easily extended to cover cases beyond the basic model presented here.q1999 Elsevier Science Ltd. All rights reserved.
Keywords: recursion formulation, sorption kinetics, spatial variability, solute trans-port, heterogeneity of sorption and desorption rates, Markov process.
1 NOMENCLATURE
C(y) Covariance function at a lagy
F Mean of lnkf process
Kd Distribution coefficient
kf Sorption rate coefficient, [day¹1]
kr Desorption rate coefficient, [day¹1]
L Length of the column [m]
Pjn,k Probability that a solute particle takesnspace steps to movektime steps starting from statej;j¼1 aqueous state;j¼2 sorbed state
r11 Probability of remaining in the aqueous phase
r22 Probability of remaining in the sorbed phase
r12 Transition probability to move from the aqueous to the sorbed phase
r21 Transition probability to move from the sorbed to the aqueous phase
v fluid velocity [m/day]
Dt Time step [day]
Dx Space step [m]
mkf Mean ofkf[day¹1]
j2 Variance of lnkfor lnkr
l Correlation length [m]
2 INTRODUCTION
Several contaminants found in the subsurface exhibit a tendency to adsorb onto the aquifer solids. The adsorption of organic and metallic species is vividly observed in recent
large-scale field tracer tests such as the Borden21and Cape
Cod tracer tests.10,25 The influence of adsorption on
con-taminant transport is often modeled by using a distribution
coefficient (Kd), which strictly applies only in case of linear
equilibrium adsorption. However, significant evidence has accumulated over the last decade suggesting that adsorption isotherms for several common contaminants are
non-linear,2,29 and furthermore, that the rates of adsorption
are relatively slow.11,16,18,19The transport of sorptive
con-taminants in subsurface environments is further complicated by heterogeneity in physical and chemical properties
of natural earth materials. For instance, Robin et al.22
estimated a variance of 0.52 for lnKd, and significant spatial
correlation in lnKdvariations at the Borden site.
Several recent theoretical and computational studies have focused on reactive transport in heterogeneous media.
Valocchi28 presented an analytical solution for reversible
linear kinetic sorption in a stratified aquifer. Cvetkovic and
Shapiro,6Selroos and Cvetkovic,23and Dagan and Cvetkovic9
Printed in Great Britain. All rights reserved 0309-1708/99/$ - see front matter PII: S 0 3 0 9 - 1 7 0 8 ( 9 8 ) 0 0 0 2 3 - 2
549
*Corresponding author. E-mail: akmishra@lbl.gov
1
consider the same problem in three-dimensionally hetero-geneous media. In all of the above studies, the effect of spatial variability in the reaction rates was not considered.
Garabedian et al.,10 Dagan,8 Chrysikopoulos et al.,5 and
Bellinet al.3examined the influence of spatial variability
in Kd on sorptive solute transport, demonstrating that a
negative correlation between lnK and lnKd results in
increased macrodispersion. This feature is also evident in
field data from the Cape Cod site10 and the numerical
simulations of Burret al.4While these studies incorporate
heterogeneity in sorption characteristics of soils, sorption
kinetics were not considered. A recent study by Huet al.14
that did incorporate heterogeneity in lnK and in sorption
kinetics revealed a complex interaction between kinetic sorption and the velocity variations.
Particle-tracking algorithms (e.g.26) are computationally
efficient and offer an attractive option for large-scale simulation of the processes described above. Recently,
Tompson27 extended the particle-tracking approach to
model equilibrium sorption, by mapping the particle distributions onto a grid for defining local concentrations.
Andricevic and Foufoula-Georgiou1 and Quinodoz and
Valocchi20 proposed particle-tracking algorithms that can
include kinetic sorption. In these approaches, a particle is allowed to transition back and forth between a ‘sorbed’ and an ‘aqueous’ state, while the probability density of the times spent in each phase is related to the sorption kinetics. However, spatial variability in sorption kinetics was not considered in these previous works. In this paper we present a recursion-based approach that does include spatial variability and that is very efficient for calculating break-through curves by combining analytical and numerical models.
The paper is organized as follows: in the next section, the Markov process model of sorption–desorption is described, followed by the development of the recursion relations. In Section 3, some comparisons to analytical solutions for transport with constant kinetic parameters are presented and the influence of spatially variable sorption kinetics on concentration breakthrough illustrated. The paper concludes with a discussion of the results and avenues for further research.
3 MARKOV PROCESS MODEL OF SORPTION– DESORPTION AND A RECURSION
FORMULATION
3.1 Markov process model
Andricevic and Foufoula-Georgiou1modeled the sorption
process as a birth–death Markov process with constant rate parameters. In the birth–death model, the number of transi-tions between the aqueous and sorbed phases during a fixed time period will be a Poisson process. We examine this case below for spatially varying coefficients.
We start by discretizing time into steps of length Dt
and space into steps of length Dx. For simplicity, the
velocityvis 1 in the arguments below, while in the general
case, Dx¼vDt. We index the variables starting from the
influent end and let kf(n) and kr(n) be the sorption and
desorption rate parameters at location nDx. We model the
one-dimensional column as a collection of small discrete segments and consider the fate of the sorbing mass advected through the column allowing for sorption–desorption processes. The rate at which the solute is sorbed onto the solids is kf(n), and the rate of desorption is kr(n); for a linear sorption isotherm, the distribution coefficient is
Kd¼kf=k r.
The probability that a particle at location nDxis sorbed
during time-stepDtis
Pr(sorption)¼kf(n)Dtþo(Dt) (1)
where o(Dt) indicates a negligible component as Dt
approaches 0. Namely
LimDt→0
o(Dt)
Dt ¼0 (2)
If solute is sorbed, the time spent in the sorbed phase is assumed to have an exponential distribution with mean 1=kr(n). This again implies
Pr(desorption)¼kr(n)Dtþo(Dt) (3)
3.2 Development of a recursion formulation
For linear kinetic models where the sorption–desorption process only depends on the forward and backward rate coefficients, recursive equations are developed below for the probability distribution of the number of time steps taken to reach the effluent end. The probability distribution of the travel time can then be related to the concentration breakthrough curve.
We denote the state of a solute particle as either mobile
ðj¼1Þor sorbedðj¼2Þand letP(nj,)kbe the probability that it
takesktime-steps for a solute particle to movenspace steps,
starting from statej. The probability that the solute remains
in the aqueous phase during a time-step is denoted byr11(n),
while r12(n)¼1¹r11(n) is the probability that the solute
moves from the aqueous to the sorbed phase. Similarly, the probability that solute initially in the sorbed phase
remains in the sorbed phase after a time stepDtis denoted
by r22(n), while r21(n)¼1¹r22(n)is the transition proba-bility for moving from the sorbed to the aqueous phase.
The transition probabilities are
r12(n)¼1¹r11(n)¼kf(n)Dtþo(Dt)>kf(n)Dt (4)
r21(n)¼1¹r22(n)¼kr(n)Dtþo(Dt)>kr(n)Dt (5)
We can derive the probability P(nj,)k that a solute particle
travels n space steps in k time steps and is in state j as
follows. For j¼ 1 (namely starting in state 1, the mobile
state) the following cases hold:
(a) Solute stays mobile during the next time step;
the particle needs to move n ¹1 space steps in the
nextk¹1 time steps;
(b) Solute becomes sorbed during the next time
step; this occurs with probability 1¹r11(n) in which
case the particle needs to move n space steps in the
nextk¹1 time steps.
We assume further that the ‘decision’ about sorption/ desorption occurs at the beginning of the time step and
then arrive at the following recursion relation, fork.n,
P(n1,)k¼r11(n)P(n1¹)1,k¹1þ(1¹r11(n))P(n2,)k¹1 (6)
By a similar argument,
P(n2,)k¼r22(n)P(n2,)k¹1þ(1¹r22(n))Pn(1¹)1,k¹1 (7)
Appropriate boundary conditions for these equations can be derived directly as follows:
P(n1,)n¼r11(1)r11(2)…r11(n)¼Pni¼1r11(i) (8)
P(n2,)n¼P n¹1
i¼1r11(i)(1¹r22(n)) (9)
P(11,)k¼(1¹r11(1))r22(1)k¹2(1¹r22(1)), k$2 (10)
P(12,)k¼r22(1)k¹1(1¹r22(1)), k$2 (11)
P(nj,)k¼0, k,n,j¼1,2 (12)
For example, eqn (9) refers to the case wherentime steps
are needed to travelnspace steps where the solute particle
starts from the sorbed state. This can only occur if
(a) the particle enters the mobile state in step 1, and
(b) never leaves the mobile state in the next (n ¹1)
steps.
The other boundary conditions are derived in a similar manner. While these equations are for transitions occurring
at the beginning of a time step, similar equations can be derived for transitions at the end of a period.
For example, suppose the distance of the effluent point
from the inlet is 1 m, the velocity is 0.1 m day¹1
andDtis 0.1 day, so that the distance moved in one time unit is 0.01 m. The total number of steps required to cover a distance of 1 m is 100 space steps and the earliest time at which a particle can reach the effluent end is 10 days or
100 time steps if it was never sorbed. We calculate P(n1,)k
for n ¼100 and values ofk$ 100 up to a suitably large
number (kmax) where the concentration reaches a negligible
value. We first obtainP(11,)kandP1(2,)kfork¼2 tokmaxand then use the recursion equation to findP(n1,)kforn¼2 to 100.P(n1,)k
for n ¼100 and for k¼100 to kmax(kmax ¼1000) in the
examples below gives the breakthrough curve in normalized concentration units.
A novel feature of the recursion method is that it effectively simulates the transport of an infinite number of particles through the flowpath. By way of contrast, the resolution and smoothness of the breakthrough curve from the particle-tracking approach depends on the number of particles used. This is especially true in the case of non-equilibrium sorption, where the breakthrough curve exhibits a long tail. Accurate resolution of the tail requires a very large number of particles, and even then oscillations are observed. On the other hand, the recursion formulation is capable of resolving the tail very accurately. This feature is illustrated in Fig. 1, where the breakthrough curve obtained using the recursion approach is compared to curves with particle-tracking simulations that used 1000,
2000 and 5000 particles. The values of kf and kr in
this case are constant at 0.5 and 0.1 day¹1
, while the other parameters are the same as discussed above. The break-through curve obtained using the recursion approach is very smooth, in contrast to the noisy behavior of the breakthrough curves obtained with particle-tracking.
Fig. 1.Comparison with particle tracking algorithm for a flow system with constant sorption and desorption rates in a one-dimensional flow
Furthermore, the recursion method is computationally more efficient than direct particle-tracking. This is illustrated in Table 1 where a comparison of the number of operations required to obtain the breakthrough curves above is given. The number of operations required to obtain the breakthrough curve by recursion depends on the time step chosen and the maximum time to which we want
to compute the breakthrough curve. The choice ofDtalso
dictates the resolution of the breakthrough curve. The values ofP(n1,)k, which denote the concentration, are obtained at a
given spatial location at time intervalsDtunits apart. For the
recursion results, a maximum time of 250 days and a time step of 0.5 day were used while a time step of 0.1 day was used for particle tracking. There is an order of magnitude difference in computational effort between the methods. The resolution and computational efficiency of the recursion approach are especially important in the context of transport with spatially variable sorption kinetics, when multiple-realization simulations are necessary to quantify the ensemble averages.
For constant kf and kr the recursion formulation is a
discrete analog of the results of Giddings and Eyring.12
The results of the two models were compared by Mishra17
and agreed closely with an appropriate choice of the time step. ForDt varying between 0:1=kf and 0:5=kf, the break-through curves for the two models are almost identical.
These results were used as guidelines for choosing Dt in
the spatially variable cases.
4 INFLUENCE OF SPATIALLY VARIABLE
KINETICS ON THE TRANSPORT OF A LINEARLY SORBING SOLUTE
Breakthrough curves in a single realization of spatially
variable kf andkr can be obtained directly by using eqns
(6)–(12) and the ensemble average breakthrough curve as an average over a number of realizations. In the examples presented below a Fast Fourier Transform random field
generator13 was used to generate independent realizations
kf and kr. The recursion method was used to obtain the
breakthrough curves for 30 realizations. The parameters
used were the length of the column, L ¼1 m, velocity of
the fluid, v ¼0.1 m day¹1
, Dt ¼0.1 day. The covariance
function for lnkf and lnkr was an exponential function of
the form
C(y)¼j2e¹
lyl
l (13)
The correlation lengthlused for the simulations was 0.2 m
and the mean values of kf andkr were 1.0 and 0.2 day¹1
, identical to the values of the constant rates used in Fig. 1.
The lnkfand lnkr processes were generated so that the
mean values of kf and kr were as desired by using the
following relationships between the statistics of lognormal and normal distributions:
eFþj2=2¼mkf (14)
whereFandj2were respectively the mean and variance of
lnkf. Similar conversions were made forkr.
In Fig. 2, the breakthrough curve for the case of constant rates is compared to the ensemble mean breakthrough curve Table 1. Comparison of flops for obtaining the breakthrough
curve between recursion formulation and particle tracking.
(1 flop¼1 computational operation)
Description Number of flops
Recursion formulation 28 406
Particle tracking (1000 particles) 774 624
Particle tracking (2000 particles) 1 541 241
Particle tracking (5000 particles) 3 899 537
Fig. 2.Mean breakthrough curve obtained with spatially variablekfandkrand comparison to results with constantkf¼1.0 day¹1and
kr¼ 0.2 day¹1
. The mean breakthrough curve is based on 30 realizations. The variance of lnkfand lnkrare 0.5. The61jbounds indicating the
variability between realizations are also shown. The column length is 1 m, and flow velocity is 0.1 m day¹1
for the spatially variable rates when the variances of lnkf
and lnkr are both 0.5. The ensemble mean curve for the
spatially variable rates shows a small shift in the position of the peak concentration when compared to the constant rate case signifying a slight reduction in the effective retardation factor. The peak concentration in the ensemble mean breakthrough curve is about 0.01 compared to about 0.02 for the constant rates case. Similarly, the breakthrough curve ends at approximately 150 days for the constant rate case while for the spatially variable rates it ends at 250 days. One of the most important consequences of spatially variable rates shown in Fig. 2 is that the breakthrough curve has longer tails than in the constant rate case. The mean curve also shows extensive tailing caused by the spatial variability of the rates, illustrating the importance of understanding the heterogeneity in the rates.
Fig. 2 includes 61j limits on the mean breakthrough
curve. At later times these bounds narrow. To emphasize the variability that can occur, breakthrough curves from five different realizations are presented in Fig. 3. Note that individual realizations vary significantly from the mean ensemble curve as well as from each other. The peak con-centrations (presented as normalized units relative to a input concentration of 1) range widely from 0.044 to 0.005 and the time for complete breakthrough ranges from 90 days to over 250 days.
5 CONCLUSION AND DISCUSSION
The recursion method presented is relatively easy to
implement and effectively mimics particle-tracking
involving an infinite number of particles. For this reason, it is very efficient and holds promise in connection
with large-scale simulations of transport involving
kinetic sorption. Spatial variability of sorption kinetics is incorporated easily within the recursion formulation.
Application of the recursion formulation along
numerically computed streamtubes or pathlines enables an extension of this formulation to multidimensional transport simulation. Unlike some previous streamtubes analyses
(e.g.24), the computational advantages and analytical
simplicity of the recursion formulation are not lost when the reaction rates and velocity are variable along the
stream-tubes.17One limitation of the recursion formulation is that it
was derived ignoring the influence of local dispersion. This assumption is not likely to be very limiting in the context of transport in heterogeneous media if the small-scale variations in the velocity field are resolved adequately.
The brief results included here show that in the non-equilibrium case, the individual rate coefficients can have an important effect on the breakthrough curve. The spatial variability of the rates can have a significant effect on the breakthrough curve and on field-scale behavior.
Mishra17 compared the results of the model with that of
Chrysikopoulos et al.5who used a perturbation model for
Kdand showed that including the spatial variability for both
rates led to a shift in the mean breakthrough curve. In
addition, Mishra17 did extensive sensitivity studies to see
which parameters are controlling factors for the behavior of the breakthrough curve. The results of those studies indicate that the desorption rate and its variance are important factors that can lead to the longer tails seen in the breakthough curves.
It is also interesting to see if the constant rate case would lead to differential equations similar to those used in more
conventional cases.9Standard partial differential equations
for solute in the aqueous phase[C(x,t)] and in the sorbed
phase [S(x,t)] are given, for example, by Dagan and
Cvetkovic:9
]C
]t þv ]C
]x¼krS¹kfC (15)
]S
]t¼kfC¹krS (16)
In these equations velocity,v, is constant as are the rateskf andkr.
We sketch an argument showing that P(n1,)k corresponds
to C(x,t) and KdP(n2,)k corresponds to S(x,t). First let x ¼ nDx,t¼kDtand
P(j)(x,t);P(nj,)k: (17)
Then eqns (6) and (7) become
P(1)(x,t)¼(1¹kfDt)P(1)(x¹Dx,t¹Dt) not the concentration in the sorbed state.
While we only present cases for constant velocity in this paper, two points should be noted. First, one can extend this method readily to include variable velocity. Second,
Cvetkovic and Dagan7for the case of constant rates give
analytical formulae for the moments of a plume in a spatially variable velocity field that are one-dimensional integrals involving the non-reactive moments and a function
g(t,x). That function corresponds to the continuous version
ofP(n1,)k. Analogous to their results we can incorporate spatial
variability forkfandkrby using the values ofP(n1,)kin their
moment equations.
6 EXTENSIONS AND APPLICATIONS
Several extensions have either been developed or are being
pursued. In particular, Mishra17treats the following cases:
1. Non-exponential residence times using a method of
stages similar to approaches used in queuing theory;15
2. Two-dimensional problems using streamlines to study both longitudinal and transverse dispersion; and 3. Biodegradation wherein decay can occur in the sorbed
phase.
Other possibilities include incorporation of
cross-correlation betweenkfandkr; studies of three-dimensional
models, and development of analytical results for dispersion using a recursion approach.
ACKNOWLEDGEMENTS
This work was supported in part by grants from the US Department of Energy through the New Mexico Waste
Management Education and Research Consortium
(WERC). AKM would also like to acknowledge the assistance of C.V. Chrysikopoulos in providing his code for comparison with our method.
REFERENCES
1. Andricevic, R. and Foufoula-Georgiou, E. Modeling kinetic non-equilibrium using the first two moments of the residence time distribution. Stochastic Hydrology and Hydraulics, 1991,5,155–171.
2. Ball, W.P. and Roberts, P.V. Long-term sorption of halogenated organic chemicals by aquifer materials: 2. Intra-particle diffusion. Environ. Sci. Technol., 1991, 25, 1223– 1237.
3. Bellin, A., Rinaldo, A., Bosma, W.J.P., van der Zee, S.E.A.T.M. and Rubin, Y. Linear equilibrium adsorbing solute transport in physically and chemically heterogeneous porous formation: 1. Analytical solutions. Water Resour. Res., 1993,29(12), 4019–4030.
4. Burr, D.T., Sudicky, E.A. and Naff, R.L. Nonreactive and reactive solute transport in three-dimensional heterogeneous porous media: mean displacement, plume spreading and uncertainty.Water Resour. Res., 1994,30,791–816. 5. Chrysikopoulos, V.D., Kitanidis, P.K. and Roberts, P.V.
Analysis of one-dimensional solute transport through media with spatially variable retardation factor.Water Resour. Res., 1990,26(3), 437–446.
6. Cvetkovic, V. and Shapiro, A.M. Mass arrival of sorptive solute in heterogeneous porous media. Water Resour. Res., 1990,26(5), 2057–2067.
7. Cvetkovic, V. and Dagan, G. Transport of kinetically sorbing solute by steady random velocity in heterogeneous porous formations.J. Fluid Mechanics, 1994,265,189–215. 8. Dagan, G.,Flow and Transport Through Porous Formations.
9. Dagan, G. and Cvetkovic, V. Spatial moments of a kinetically sorbing plume in a heterogeneous aquifer.Water Resour. Res., 1993,29(12), 4053–4061.
10. Garabedian, S. P., Gelhar, L. W. and Celia, M. A., Large scale dispersive transport in aquifers: field experiments and reactive transport theory, Report. No. 315. Ralph M. Parsons Laboratory, Department of Civil Engineering, M.I.T., Cambridge, 1988.
11. Goltz, M.N. and Roberts, P.V. Interpreting organic solute transport data from a field experiment using physical none-quilibrium models.J. Contam. Hydrol., 1986,1(1/2), 77–93. 12. Giddings, J.C. and Eyring, H. A molecular dynamic theory of chromatography.Journal of Physical Chemistry, 1955, 59,
416–421.
13. Gutjahr, A., Bullard, B. and Hatch, S., Joint conditional simulations and flow modeling. In Geostatistics for the Next Century, ed. R. Dimitrakopoulos. Kluwer, 1994, pp. 185–196.
14. Hu, B.X., Deng, F. and Cushman, J.H. Nonlocal reactive transport with physical and chemical heterogeneity: linear nonequilibrium sorption with random Kd. Water Resour. Res., 1995,31,2239–2252.
15. Kleinrock, L.,Queuing Systems, Vol. 1. Wiley, New York, 1975.
16. Miller, C.T. and Weber, W.J. Sorption of hydrophobic organic pollutants in saturated soil system. J. Contam. Hydrol., 1986,1,243–261.
17. Mishra, A. K., Reactive transport in chemically and physically heterogeneous porous media: effect of non-equilibrium linear sorption. Ph.D. dissertation, New Mexico Institute of Mining and Technology, Socorro, NM, 1997. 18. Nkedi-Kizza, P., Biggar, J.M., van Genuchten, M.Th.,
Wierenga, P.J., Selim, H.M., Davidson, J.J. and Nielson, D.R. Modeling tritium and chloride36 transport through an aggregated oxisol. Water Resour. Res., 1983,19, 691– 700.
19. Ptacek, C.J. and Gilham, R.W. Laboratory and field measure-ments of non-equilibrium transport in the Borden aquifer, Ontario, Canada.J. Contam. Hydrol., 1982,10,119–158. 20. Quinodoz, H.M. and Valocchi, A.J. Stochastic analysis of the
transport of kinetically sorbing solutes in aquifers with randomly heterogeneous hydraulic conductivity. Water Resour. Res., 1993,29(9), 3227–3240.
21. Roberts, P.V., Goltz, M.N. and Mackay, D.M. A natural gradient experiment on solute transport in a sand aquifer: 3. Retardation estimates and mass balances for organic solutes.Water Resour. Res., 1986,22(13), 2047–2058. 22. Robin, M.J.L., Gutjahr, A.L., Sudicky, E.A. and Wilson, J.L.
Cross-correlated random field generator with direct Fourier Transform method.Water Resour. Res., 1993,29(7), 2385– 2397.
23. Selroos, J.O. and Cvetkovic, V. Modeling solute advection coupled with sorption kinetics in heterogeneous formations.
Water Resour. Res., 1992,28,1271–1278.
24. Simmons, C.S., Ginn, T.R. and Wood, B.D. Stochastic-convective transport with nonlinear reaction: mathematical framework.Water Resour. Res., 1995,31,2675–2689. 25. Stollenwerk, K.G. Modeling adsorption of molybdate.Water
Resour. Res., 1995,31(2), 347–357.
26. Tompson, A.F.B. and Gelhar, L.W. Numerical simulation of solute transport in three-dimensional, randomly heterogeneous porous media.Water Resour. Res., 1990,26(10), 2541–2562. 27. Tompson, A.F.B. Numerical simulation of chemical migration in physically and chemically heterogeneous porous media.Water Resour. Res., 1993,29(11), 3709–3726. 28. Valocchi, A.J. Spatial moment analysis of the transport of kinetically adsorbing solutes through stratified aquifers.
Water Resour. Res., 1989,25(2), 273–279.
29. Weber, W.J. Jr., McGinley, P.M. and Katz, L.E. Sorption phenomena in subsurface systems: concepts, models and effects on contaminant fate and transport.Water Res, 1991,
