An Evaluation of Kinetic and Equilibrium Equations for the Prediction of Pesticide
Movement Through Porous Media1

M. TH. VAN GENUCHTEN, J. M. DAVIDSON, AND P. J. WiERENGA2
ABSTRACT
Miscible displacement techniques were used to study the
movement of picloram (4-amino-3,5,6-trichloropicolinic acid)through a water-saturated Norge loam soil. The equilibrium
adsorption and desorption isotherms for picloram and Norgeloam soil were not single-valued relations. Picloram mobility
was reduced significantly when the average pore-water velocity
was decreased from 145 to 14.2 cm/day. Observed and predicted effluent concentration distributions were compared.
Predictions were made with a S/360 CSMP simulation model,using two kinetic rate equations and an equilibrium Freundlich
equation. The two kinetic models and the equilibrium model
each satisfactorily described the observed effluent concentration
distributions at low pore-water velocities provided the nonsingle-valued character of the adsorption-desorption process
was included in the calculations. At high pore-water velocities,
the kinetic adsorption models were found inadequate to predictthe picloram movement. An empirical model was then developed, based on the assumption that equilibrium existed during
displacement and that only a fraction of the soil participatedin the adsorption process. This fraction was found to be a function of the average pore-water velocity. With the empiricalmodel, a reasonable fit between data and calculated effluent
curves was obtained for all pore-water velocities.
Additional Index Words: herbicide movement, adsorption,
molecular diffusion, miscible displacement, computer simulation, S/360 CSMP.

several mathematical models have been proR ECENTLY,
posed to describe the movement of chemicals throughadsorbing porous media (7, 12, 13, 14). One of the earliest
solutions for equilibrium adsorption between the solution
and adsorbed phases was presented by Lapidus and Amundson (10). Hashimoto et al. (7) obtained a similar solution
which included the longitudinal mixing process as well as
equilibrium and linear adsorption during one-dimensional
flow. Their solution, including the "retardation factor," waslater used by Kay and Elrick (9). Lindstrom et al. (13)and Davidson and Chang (2) used a similar solution to describe the movement of several organic chemicals throughporous media. These solutions assumed instantaneous adsorption and a linear and single-valued adsorption-desorption relation. These assumptions may not be valid for all
conditions that occur in a soil-water system.
Kay and Elrick (9) and Davidson and Chang (2) showed
considerable deviation between the mathematical model
based on the assumption of equilibrium adsorption (7) andexperimental data, especially at the high flow velocities.
The two studies suggested that for high pore-water velocities, the use of a kinetic rate equation to describe the adsorption process may be necessary. Lapidus and Amundson(10), Oddson et al. (14) and Lindstrom et al. (12) each presented solutions that included a rate equation in a convective type transport equation to describe the displacement ofa chemical in a sorbing porous medium. Oddson et al. (14)and Lapidus and Amundson (10) used a first-order rateequation, but assumed that the diffusion and dispersion
processes were negligible in the transport equation in orderto obtain an analytical solution. Lindstrom et al. (12) used anadsorption rate equation as well as a "sticking probability"
factor.
The presence of a non-single-valued (hysteretic) adsorption-desorption relation has received little attention in earlier studies. Swanson and Dutt (16), studied the movementof 2-chloro-4-ethylamino-6-isopropylamino-i'-triazme (atrazine) through different soils and encountered such a
non-single-valued adsorption-desorption relationship. They
found that both the adsorption and desorption isotherms
were of the Freundlich form, but that the coefficients foradsorption were not equal to those for desorption. Thesesame authors were able to predict the movement of atra-

30

SOIL SCI. SOC. AMER. PROC., VOL. 38, 1974

zine reasonably well with a computer model that assumed
instantaneous adsorption and desorption.
The purpose of this study was to use some of the available adsorption models in a transport equation to describe
experimental laboratory data. A S/360 CSMP (Continuous
System Modeling Program) computer program (8) was developed for three different adsorption models, based onequations presented by Lindstrom et al. (12), Oddson et al.
(14), and an equilibrium Freundlich adsorption relationship, respectively. Computer results were then comparedwith experimentally determined picloram effluent concentration distributions from a water-saturated Norge loam soil.

The values of klt kz, and b may not be the same for adsorption and desorption if these processes are not single-valued, andhence may change with the sign of the adsorption rate, e.g.:
b=
"des

pdS

Adsorption Model II
The second form of Eq. [2] considered is:

CN — S

St

ac

dC

[i]

[5]

where frads and frdes represent the values of b used to describe
the adsorption and desorption of the chemical, respectively.

ADSORPTION EQUATIONS
The movement of adsorbed chemicals through a porous medium under steady state soil-water conditions was assumed toobey the following differential equation:

when 35 / dt > 0
when SS / dt < 0][6]Equation [6] is a first-order kinetic rate equation, with first-orderforward, k\, and backward, k'2, kinetic rate coefficients (day-1).
Lapidus and Amundson (10) and Oddson et al. (14) used thisadsorption relation (for N = 1.0) to obtain an exact solution to
Eq. [1]. Their solution assumed the dispersion coefficient D0
to be equal to zero. For equilibrium adsorption (dS/dt = 0),
Eq. [6] reduces towhere C is the concentration of the chemical in solutioncm3), S is the amount of chemical adsorbed per gram of soilG^g/g). D0 is the dispersion coefficient (cmVday), v0 is theaverage pore-water velocity (cm/day), p is the bulk density(g/cm3), 0 is the saturated water content (cm3/cm3), x is thelinear distance in the direction of flow (cm), and t is time (days).
Exact solutions to Eq. [1] for specific initial and boundary
conditions are only available when assumptions are made concerning the mechanisms of adsorption or diffusion and dispersion (7, 10, 13, and 14). Although it would be desirable, as indicated by Lapidus and Amundson (10), to use a general relation for adsorption-desorption of the form:

The Freundlich equation has been shown to describe for several organic pesticides (3, 9) the relationship, at equilibrium, between the amount in solution (C) and the amount adsorbed(S). The Freundlich equation is as follows:

dS/dt = f(C, S)

S - KCN.

S = (k'^61 k'2p) CN = KCN
where A! is a distribution coefficient.
Adsorption Model III[2]

only three special cases of [2] will be considered in this manuscript.

dS

—
dt

—
at

*i

c

Q

i

—- exp [—2bS] — — S [[3]

where k^ and k2 are the forward and backward kinetic rate
coefficients (day"1), respectively, b is similar to the surface
stress coefficient (g/pg) described by Fava and Eyring (6),and 5, C, 9, and p are as defined previously for Eq. [1]. Notethat in Eq. [3], the backward rate coefficient, kz, and the partition coefficient, k^/kz, are independent of concentration. For
equilibrium adsorption (dS/dt — 0), Eq. [3] reduces to thefollowing:
S = (ek^C/ pk2) exp (—2bS).

[4]

The parameters b and ek^/pk^ may be obtained from the slope
and intercept of the curve hi (S/C) vs. S for various equilibrium concentrations.

[8]

Differentiating Eq. [8] with respect to time gives

Adsorption Model I
A rate expression for the adsorption process which considers
both the forward and backward kinetic rate coefficients as wellas the sticking probability of the chemical on the surface of theadsorbent has been derived by Lindstrom et al. (12). This adsorption equation allows the energies of activation for adsorption and desorption to change with surface coverage. This special case of Eq. [2] is

[7]

dC

—.
dt

[9]

Equation [9] substituted into Eq. [1] allows this equation to be
expressed in terms of one dependent variable, C. The solution
to Eq. [1] and [9] with N = 1.0 for single-valued adsorption
and desorption has been used by Davidson and Chang (2) and
Kay and Elrick (9).
SOLUTION PROCEDURE
A procedure similar to that used to describe water movement[De Wit and van Keulen (5); van Keulen and van Beek (17);Bhuiyan et al. (1) ] and heat transfer in soils [Wierenga and
De Wit (18) ] was used to find a solution for Eq. [1]. The differential equation was solved by dividing the soil column into
N layers of thickness Ax. The mass flow across each boundary
was calculated and the mass balance equation for each layer
employed. A schematic representation of the procedure is given
in Fig. 1, together with some of the variables associated witheach layer. The flow of herbicide (HFLW, jig/day) in and outof each layer per unit time was calculated with the following
equation:
HFLW,
*
Ax

* AREA[11]

VAN GENUCHTEN ET AL.: EQUATIONS FOR PREDICTING PESTICIDE MOVEMENT
where C{ is the concentration in each layer or compartment attime t, i is the index for depth (i — 1,2, ... n) and AREA is
the cross-sectional area of the column most conveniently set atunity, as was done in this study. The net flux of herbicide into
each compartment is given by Eq. [12].

ADSORPTION i

31

FLOWING PART

1 HFLWj

Cj-|

NHFLWj = HFLW4 — HFLWi+ j —ADSR, * VOLL * RHO[12]

where ADSR; is the adsorption rate, calculated with the available variables (C, S, . . .) at time /, VOLL is the volume of eachlayer (Ax * AREA), and RHO is the bulk density. ADSR isexpressed in /tg/g of soil per day, and is calculated as follows:

ADSRj

DL,
HFLWH

[13]
where dS/dt is the adsorption rate, calculated from Eq. [3], [6]or [9]. The net flow of herbicide per unit time into each layer(NHFLWj) gives the change in the amount of herbicide in solution in each layer, AMSOL{. Using the rectangular integration method, we may now calculate the amount of herbicide attime t + Ann layer i with the following equation:AMSOU +At = AMSOL^ + NHFLWj * Af.

[14]

The amount of herbicide adsorbed in every layer (AMADS{)
was calculated in the same way
AMADS'. +At = AMADSJ +ADSR, * VOLL * RHO * A?.

[15]

The adsorbed, 5j G"g/g soil), and solution, C{ Gig/cm3 soil
solution), concentrations in each layer were calculated with theequationso__1

AMADS,
VOLL•RHOAMSQLjC, = -

VOLW[16]

where VOLW equals the volume of water in each layer,
VOLW = VOLL * e.
MATERIALS AND METHODS
The procedure described by Davidson and Santelmann (4)was used to study the movement of 4-amino-3,5,6-trichloropicolinic acid (picloram) through Norge loam. The carboxyl groupon the picloram was 14C-labeled and had a specific activity of
4.1 /tCi/mg. The displacement rate of the picloram through thesoil was varied with a constant volume pump. The pore-watervelocities used to displace the herbicide ranged between 145
and 14.2 cm/day. The soil-water flux was divided by the watercontent 0, determined after each displacement, to give the average pore-water velocity v0. Five-milliliter effluent water samples were collected sequentially with a fraction collector andanalyzed for picloram and chloride. The picloram concentration in each 5-ml effluent sample was determined from 14C activity using a liquid scintillation counter. The relative concentration (C/C0) was calculated from the picloram concentrationin the effluent (C) and the initial concentration (C0) of the
picloram at the inflow location. The study was conducted in atemperature-controlled room at 25 ± 0.5C.
The soil, initially calcium saturated, was air-dried and thecomplete sample forced through a 2-mm sieve. The soil was
packed into a glass cylinder to an average bulk density of 1.54
g/cm3. The cross-sectional area and length of the soil columns
were 45 cm2 and 30 cm, respectively. The soil was saturated
with 0.0\N CaSO4 and the desired soil-water velocity was estab-Fig. 1—Schematic representation of the CSMP-model for
pesticide movement in soil.
lished. The number of pore volumes (V/V0) displaced throughthe soil was calculated by dividing the volume of effluent (V)by the volumetric water capacity (V0) of the soil. The pH, cation exchange capacity, and organic matter content of the Norgeloam soil were 6.6, 9.2 meq/100 g, and 1.7%, respectively. The
soil had 46% sand, 38% silt, and 16% clay.
The picloram solution was prepared using 0.0 IN CaCl2 and10.6 juCi/liter of 14C-labeled picloram. Two hundred ml (approximately 0.41 pore volume) of picloram solution wereadded to the initially water-saturated soil at the desired soilwater flow rate and displaced through the soil with 0.01N CaSO4
using the same flow rate. The picloram application was equivalent to 1.11 kg/ha (1.0 Ib/acre). The time, T (pulse period),
required to introduce the 200-ml of picloram solution at eachflow rate is given in Table 2.
The equilibrium adsorption isotherm for picloram was determined using duplicate 1:1 mixtures of 5 g of soil in 5 ml of various picloram concentrations in 0.0 IN CaCl2. Each sample wasshaken for 5 hours and then centrifuged at 1,250 times the forceof gravity for 15 min. Earlier tests had shown that the 5 hoursof shaking exceeded that required for equilibrium adsorption.
One-half ml samples of the clear supernatant were analyzed for
14
C activity and the count ratios converted to herbicide concentration. The difference between the original picloram concentration and that in the supernatant was assumed to be the result ofadsorption.
Equilibrium desorption isotherms for picloram were obtainedusing the duplicate 1:1 soil-to-herbicide solution mixtures prepared for the adsorption study. A known volume of solution was
extracted from each mixture following the initial adsorption
and replaced with herbicide free 0.01N CaCl2. The concentration of picloram in the extract was measured and the micrograms of picloram removed from the system by the extraction
calculated. After each extraction, the samples were shaken for
5 hours, centrifuged, and another extraction and analysis of thesupernatant made. The 5-hour shaking period was experimentally determined to be sufficient time to obtain equilibrium between the solution and adsorbed phases. This procedure was followed for each successive desorption step.
RESULTS AND DISCUSSION
The equilibrium adsorption and desorption data for picloram and Norge loam soil are given in Fig. 2. It is obvious from the data that these two processes, adsorption anddesorption, cannot be described by one equation and hence
are not single-valued. Also, note that the adsorbed concentration does not appear to approach zero as the solution

32

SOIL SCI. SOC. AMER. PROC., VOL. 38, 1974
1.0

0.40-

EQUILIBRIUM

0.8

0.6
« ADSORPTION EXPERIMENTAL° DESORPTION EXPERIMENTAL—— S=KCN° EXPERIMENTAL——— CALCULATED, REVERSIBLE——— CALCULATED, NON REVERSIBLE

0.4

0.2
1.0

1.5

CONCENTRATION Ug/cm3)

Fig. 2—Equilibrium adsorption and desorption data for picloram in Norge loam.
concentration, C, goes to zero during desorption. The solidlines in Fig. 2 were calculated using the Freundlich equation (Eq. 8), and the dashed lines were calculated using
the adsorption equation (Eq. [4]) proposed by Lindstrom
et al. (12). The parameters for desorption are a function of
the maximum adsorbed concentration prior to the initiationof desorption. These parameters are defined below andwere substituted into Eq. [4] and [8] to calculate the desorption curves.
Writing Eq. [4] for adsorption and desorption and equating the two relations gives
l—2baAs Smaxmax] Cmax[17]
where ads and des refer to adsorption and desorption andSmax and Cmax represent the adsorbed and solution concentrations prior to the initiation of desorption. Solving Eq.
[17] for Z>des gives:
r-

n

[18]

\- Cj/ SK

The values of (&2)ads and (k2)des in Eq. [4] used for calculating the adsorption rate were selected on the basis ofdata fit. The other parameters for Eq. [4] are given in
Table 1.
Table 1—Parameters used in Eq. [4] and [8] to describe theadsorption and desorption isotherm data (Fig. 2). Smax isthe adsorbed concentration (/ug/g) when desorption
is initiated

0.094

cm'/g
0. 188

2.34

0.967

S/KS

Adsorption
Desorption

0. 1800. 268

0. 94

3.73

0.967

0. 204

0.402

Desorption
Smax =0. 0767 ns/S
™' °

10.7

0.967

0. 101

0.315

2.0

3.0

2.5

3.5

Fig. 3—Observed and computed relative effluent curves for
picloram movement through Norge loam. Calculations weremade assuming equilibrium during adsorption and desorption.
The desorption parameters for the Freundlich relationship (Eq. [8]) were determined in a similar manner tothose in Eq. [17] and [18].
= *ads X (CmaxFads = Kdes X (CmaxFdes

[19]

where
[20]

X

The ratio N&As/N6es was found to be a function of the
maximum adsorbed concentration[21]

NBdt/NiM = 2.105 +0.062 S

The Nais/Ndes values for the three picloram desorption
curves ranged from 2.3 to 3.1 (Table 1). Swanson and Dutt(16) found that an average Nads/Ndes value of 2.3 satisfactorily described atrazine adsorption and desorption in several soils.
Equations [4] and [8] fit the adsorption data reasonably
well. However, the two equations differ in their ability todescribe the desorption data at low picloram solution concentrations. The Freundlich relation appears to fit the lower
desorption data somewhat better. The physical parameters
associated with each picloram displacement are presented
in Table 2. The dispersion coefficients (D0) for chloride
and picloram were determined using a procedure similar tothe one proposed by Rose and Passioura (15).
Table 2—Soil physical data for various picloram displacement
experiments through Norge loam soil
Dispersion
coefficient,

Average
Pore- water

Soilwater

Bulk

no.

YQ

g

P

Chloride

Plcloram

T

1

14.2
31.2
66.5
132
145

0. 3626
0.36980. 3649
0. 3634
0. 3508

1.53
1.54
1.53

1. 5

1.55

11.6
8. 8

2. 8
4.5
6.0
9.5
10. 8

0.896
0.372
0.181
0.091
0.088

—
0.424

Desorption* ads and des refers to adsorption and desorption, respectively.

1.5

2

3
4

5

1.63

Pulse

VAN GENUCHTEN ET AL.: EQUATIONS FOR PREDICTING PESTICIDE MOVEMENT
1.01

33

1.0 r

< k 2'ads = 3 ° DAY~'

0.8

OAY

0.6
fc

v0= 132 (cm/day)—— CALCULATED= 000 EXPERIMENTAL

V0=I4.2 (cm/day)-— CALCULATED•«• EXPERIMENTAL

0.4

0.2

0.5

1.0

1.5

2,0

V

Fig. 4—Observed and computed relative effluent curves for
picloram movement through Norge loam at two pore-watervelocities. Calculations were based on the kinetic rate equation (Eq. [3]).

Figure 3 presents observed and computed relative effluent concentration distributions for experiment 1 (Table 2)
assuming equilibrium conditions during the movement (v0= 14.2 cm/day). The figure illustrates the importance ofusing both adsorption and desorption characteristics. The
solid line was calculated using the adsorption parameters
for both adsorption and desorption. The dashed line wascalculated using the same equilibrium Freundlich relation(Eq. [8]), but with equations [20] and [21] in Eq. [8] fordesorption. The data and dashed line are in agreement, bothin shape and position. The maximum effluent concentration occurs earlier and is reduced in magnitude comparedto the single-valued adsorption-desorption case (solid line).
Also, the area under the dashed line between 0 and 2.5
pore volumes is less than that under the solid line. The difference in area under the two curves represents the quantityof picloram remaining in the soil after 2.5 pore volumes asa result of the nonsingular relationship between adsorption
and desorption. The assumption that the adsorption and desorption processes were at'equilibrium when the averagepore-water velocity was 14.2 cm/day appears justified
based upon the agreement between the calculated dashedline and data.
In Fig. 4 the effluent data of experiment 1 are comparedwith effluent data of experiment 4. Increasing the averagepore-water velocity from 14.2 to 132 cm/day significantly
changed the position and shape of the picloram concentration distribution. The shift to the left of the effluent curve
relative to VIV0 was also observed by Kay and Elrick (9)for 1, 2, 3, 4, 5, 6-hexachlorocyclohexane (lindane). Sucha shift is not expected under equilibrium conditions.
Changes in pore-water velocity may affect the slope of thefront portion of the effluent curve but should not displace
the curve if equilibrium conditions exist. Apparently, theassumption of equilibrium adsorption-desorption is invalid
at a pore water velocity of 132 cm/day.
Figure 4 also presents calculated effluent curves for pi-

Fig. 5—Computed relative effluent curves for picloram movement through Norge loam using a kinetic rate coefficientfo, (Eq. [3]), which varied from zero to infinity.

cloram. The calculations were based on the kinetic adsorption-desorption model (Eq. [3]). The rate coefficients described in Eq. [3] were estimated from the shape of theeffluent curve of experiment 1. The values obtained were
30 and 10 day'1 for (&2)ads and tides' respectively. Because of the difficulty in measuring such values, theyshould be considered estimates.
The relative picloram concentration of the effluent inexperiment 1 (v0 = 14.2 cm/day) is adequately describedusing the kinetic adsorption-desorgtion model (Fig. 4).
However, the calculated line does not fit the data as well as
when the equilibrium Freundlich adsorption-desorption relation was assumed (Fig. 3). The calculated effluent curvefor the fast displacement of experiment 4 (va — 132 cm/day) deviates substantially from the data, and has a shape
typical to that generally found for nonequilibrium systems[Davidson and McDougal (3) ]. In calculating the lattercurve, the same rate coefficients were used (30 and 10
day"1, respectively) as those used for the slow displacement since rate coefficients are considered to be independent of the pore-water velocity. When rate coefficients wereselected to fit the fast displacement in Fig. 4 (3 and 1 day"1for (&2)ads and (fr2)des, respectively), the prediction of theeffluent curve for the slow displacement failed. It thus appears impossible to adequately predict picloram effluent
curves in this soil with Eq. [3] using rate coefficients thatare independent of pore-water velocity.
When Eq. [6] rather than Eq. [3] was used with values of
30 and 15 (day"1) for (&2')ads and ^aOdes- respectively, approximately the same results were obtained. This indicatesthat Eq. [3] may have little advantage over Eq. [6] in describing picloram displacement. The adsorption model described by Eq. [6] is the same as that used by Lapidus andAmundson (10) and Oddson et al. (14), except for the dispersion term and the nonsingular adsorption-desorption
characteristics which the latter did not include.
Figure 5 shows the relative effluent concentration distributions obtained for various kinetic rate coefficients at an

34

SOIL SCI. SOC. AMER. PROC., VOL. 38, 1974

1.0

" EXP I Vo = l4.2 cm/DAY• • EXP 2 Vo = 31.2 cm/DAY«« EXP4\A>=I32 cm/OAY'<> EXP 5\A>= 145 cm/DAY

0.8

1.0
.8

NO ADSORPTION

.6

c_

0.6

CO

fo
0.4

0.5

0.2

°-°oNo

L25

1.5

1.75

2.0

2.25

V

Fig. 6—Observed and computed relative effluent curves for
picloram movement through soil at four pore-water velocities. Calculations were made by assuming equilibrium in the
FREQ part of the soil, and using the nonsingular, adsorption-desorption equilibrium model.

average pore-water velocity of 14.2 cm/day. The distribution coefficient, kj/ktf (Eq. [3]), is 0.188 cm3/g for both
adsorption and desorption (assumed here to be singlevalued); the other parameters necessary for the calculations
are the same as those associated with experiment 1 (Table2). The line calculated with k2 — oo represents equilibrium
adsorption. Note that as the rate coefficients decrease to 2.0
day"1, the peak concentration is reduced and the shape ofthe curve is flattened. Further decreases in k2 resulted in
the distributions moving to the left and in an increase in the
maximum concentration. When kz = 0, the distribution is
the same as that for a non-adsorbed material. Similar results to those shown in Fig. 5 would be obtained if the rate
coefficient were held constant and the average pore-watervelocity were varied from very large (v0 —> oo) to very smallvalues (v 0 —» 0). Lindstrom et al. (12) calculated the chemical distribution inside a soil column along the vertical axis.
Their results, when plotted as effluent concentration versus
pore volume, look very similar to those in Fig. 5, including
the low peak concentrations at intermediate kinetic rate coefficients. However, when comparing observed and computed ^effluent data in Fig. 4 and the shapes and positionsof the observed effluent curves in Fig. 6 (e.g., the absenceof a lower peak concentration at intermediate pore-watervelocities and a gradual increase in peak concentration withincreasing velocity), it becomes evident that the kinetic
models alone are unable to predict the effluent curves for
all pore-water velocities.
The data in Fig. 4 and 6 show that an increase in the
pore-water velocity causes the chemical to appear at lower
VI'V0 values in the effluent. This would suggest that thetime for adsorption at higher velocities is too short, and thata kinetic rate model needs to be employed. It has beenshown above that this was not the case. A more likely reason is that at high pore-water velocities less herbicide willreach the adsorbing sites inside the aggregates. The residence time of the herbicide in the soil is too short to allow

1.0

1.5
V

2.0

Fig. 7—Computed relative effluent curves for movement of anadsorbing chemical through soil. The ratio of B over A isdefined as the fraction near equilibrium (FREQ).

diffusion to all adsorbing sites. Therefore, only a fraction of
the adsorbing sites of the soil is in direct contact with thedisplacing solution. The movement of picloram to thesmaller pores (e.g., interparticle and intraparticle) is primarily by diffusion from the larger pores. This agrees with
Leenheer and Ahlrichs (11), who concluded that the rate
limiting step in the adsorption of O,O-Diethyl-o-p-nitrophenylphosphorothioate (parathion) on soil was the diffusionof the insecticide molecule into the interior of the adsorbing material. At low pore-water velocities, the time available for diffusion is greater than at high velocities, and as aresult, a greater portion of the adsorbing sites can bereached by the pesticide.
In order to model the latter system, the soil may, as a first
approach, be divided in two fractions, i.e. the fraction of
the soil or the adsorbing sites that is readily adsorbing picloram and that is in equilibrium with the displacing solution, and the fraction of the soil where adsorption is diffusion controlled.
The first fraction, which is near equilibrium (FREQ),
may be determined from the position of the picloram effluent concentration distribution. This is illustrated in Fig. 7,
where the left hand curve was obtained by assuming no adsorption (FREQ = 0), and the right hand curve was obtained for equilibrium adsorption (FREQ =1.0). The
FREQ term is defined as the ratio of the retardation distances B and A, and is a function of the average pore-watervelocity. With increasing pore-water velocities, the residence time of the herbicide in the soil decreases resulting
in less soil in direct contact with the pesticide. Therefore
the values of FREQ decrease with increasing pore-watervelocity. Using the breakthrough data from experiments
1-5 (Table 2), the relation between FREQ and flow rate
was established. FREQ values varied between 0.92 (v0 =14.2 cm/day) and 0.23 (v0 = 145 cm/day). Using thesecalculated FREQ values and the data in Tables 1 and 2,
four picloram effluent concentration distributions were calculated with the computer model based on Eq. [1] and [9](Fig. 6). The computer model was identical to the one used
for calculating the solid line in Fig. 3, except that for Fig. 6
the value of the bulk density, which is the mass of soil perunit volume, was multiplied by the appropriate FREQVAN GENUCHTEN ET AL.: EQUATIONS FOR PREDICTING PESTICIDE MOVEMENT

value. It was assumed that the fraction of the adsorbing sites
in equilibrium with the picloram was proportional to themass of the soil in equilibrium with the picloram. No effort
was made to calculate the amount of picloram diffusing
into, and out of the smaller pores and aggregates. This may
be the reason why the prediction of the effluent curves during desorption was not good.
The calculated lines and experimental data in Fig. 6
agree reasonably well. The predicted curves certainly aremuch closer to the data than for the fast displacement in
Fig. 4. The calculated peak concentration is too high insome instances, but can be considered reasonable. Also, theleft hand (breakthrough) portion of each curve agrees, ingeneral, surprisingly well with the data, indicating conditions near equilibrium during adsorption, even at the highflow rates. The largest deviation between the calculatedline and data occurs during desorption. This deviation also
may be caused by desorption not being an equilibrium
process as was assumed in the model.
The success of the FREQ approach to describe picloram
displacements raises some doubts about the usefulness of
the kinetic rate models in describing picloram movement
through soil. The accurate description of the adsorption sideof the effluent curves suggests that conditions near equilibrium exist, even at high flow velocities (132 cm/day). It
may also indicate that the rate coefficients used in thisstudy for adsorption were underestimated. The difficulty ofmeasuring these coefficients is a serious problem in studying the kinetics of adsorption-desorption.
Because of the complexity of the soil-water system, theprocedure of defining a function that relates the surface
area readily available for adsorption to pore-water velocity
may be a reasonable approach to the problem at the present
time. In the model described here, the soil was divided into
two fractions. A better approach may be to divide the soil
into three fractions, e.g. a fraction at equilibrium, a fraction
with a slow adsorption rate, and a fraction where adsorption is diffusion controlled. With the approach used in thispaper it is not difficult to model such a system. The maindifficulty is in obtaining the necessary soil parameters.
CONCLUSIONS
1) The adsorption and desorption of picloram by Norgeloam soil is not a single-valued function. Excluding thisnonsingularity of the adsorption and desorption process incalculating effluent concentration distributions resulted in aserious disagreement between the calculated and observed
distributions.
2) Kinetic adsorption-desorption models were evaluatedand found inadequate when predicting picloram mobility
for high average pore-water velocities. These models, however, agreed with the equilibrium adsorption model whenused to describe the picloram effluent concentration distribution at low pore-water velocities.
3) An empirical model was developed that could describe the movement of picloram at high flow rates. The
procedure defines a fraction of the adsorbing sites (alonglarge pores) that is in equilibrium with the picloram during
displacement. This fraction is a function of the average

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pore-water velocity. The procedure provided a reasonable
fit between data and calculated effluent concentration distributions for four pore-water velocities. The adsorption in
the remaining part of the adsorbing sites is slow and controlled by physical processes (diffusion, intra-aggregatetransport).
4) The simulation program (CSMP) is constructed insuch a way that other processes influencing the movementof picloram can be included (unsaturated flow, decay).