Mass Transfer Studies in Sorbing Porous Media: II. Experimental Evaluation with Tritium
M. TH. VAN GENUCHTEN 2 AND P. J. WlERENGA3
and immobile (Cim) concentration distributions in a semi-infinite
column, resulting from a pulse input of solute.

ABSTRACT
A comparison is made between observed tritium effluent concentration distributions and those calculated with a previously published analytical solution for the movement of chemicals through unsaturated,aggregated sorbing media. In the analytical model the liquid phase of
the soil is divided into mobile and immobile regions, with transfer between the two regions diffusion controlled. Effluent data obtained
from several displacements of tritium through 30-cm long columns of
Glendale clay loam were used to determine the different parameters in
the analytical solution by curve fitting. The data indicate some adsorption or isotopic exchange of tritium during its flow through the soil columns. The amount of immobile water increases with decreasing flow
velocity and increasing aggregate size, and varies between 6 and 45%of the total volume of water in the columns. The analytical solution
provides an excellent description of the experimental effluent data,and shows that tailing can be explained satisfactorily by diffusional exchange of tritium between mobile and immobile regions of the soil.

[1]

[2]
[

dCml

f V C

0

lim v m C m - D -r— = \ m 0
z—»o+L
"£ J
t^

[3]

t-

[4]

im[C m (z,t)] =[5]

Additional Index Words: intra-aggregate diffusion, immobile water,
miscible displacement

The variables in these and the following equations are defined in
the Appendix.
Assuming a linear adsorption isotherm (N= 1), the following analytical solutions were derived for the relative concentrations in
the mobile and immobile liquids:

N PART i OF THIS STUDY (van Genuchten and Wierenga,1976), a mathematical model was developed to describe
the movement of chemicals through unsaturated, aggregated sorbing porous media. The medium was thereby
analyzed by dividing the liquid phase into mobile and immobile regions. Transfer between the two liquid regions,presumably a diffusion process, was assumed to be proportional to the concentration difference between the mobile
and immobile liquid phases. Sorption processes in both thestagnant and dynamic regions of the medium were further
assumed to be instantaneous. The resulting differential
equations were solved analytically assuming a linear equilibrium adsorption isotherm.
The experiments to be discussed in this study were carried out to evaluate the model of van Genuchten and
Wierenga (1976), and to obtain estimates of the different
parameters in the adsorption model. The effects of flow velocity, bulk density, and aggregate size on the amount ofimmobile water (and hence, on the amount of tailing andasymmetry) were also studied. In a subsequent paper (vanGenuchten et al., 1977) the theoretical model will be used
to describe the movement of 2,4,5-T (2,4,5-trichlorophenoxy acetic acid) through the same soil columns.

It-n _c2x,Cim(x,''> ~{c2(x,T)-c,(x,T-Tl)

L/J

T^T,

~
G(x, r)H,(T, r)dr
K Jo

cl(x,T) = G(x,T)[8]

C2 (x, T) = a [ T G(x, r)H2 (T, r)dr
J o
- T)}

G(x, T) = I erfc-

[9]

(1 + Px + PT//3R) exp (fx) erfc {(P/4/3/?7)"2 (QRx + T)}+ (PT/TT/3R)112

exp{-P(J3Rx-T)2/4/3RT}[10]

Hi(T,r] = e x p ( - M - v )MATHEMATICAL DESCRIPTION

I 1 (£K«/v)" 2 /(l - |8)}

[11]

}

[12]

,r) = exp(-«- v)

We refer to the original paper of van Genuchten and Wierenga(1976) for a more extensive discussion of the physical system. The
following general system of equations describes the mobile (Cm)'Journal article no. 565, Agric. Exp. Stn., New Mexico State Univ., Las
Cruces, NM 88003. The work upon which this report was based was supported in part by funds obtained from the U.S. Dep. of the Interior, Officeof Water Resources, as authorized under the Water Resources Act of 1964.
Received
7 July 1975. Approved 9 Nov. 1976.
2
Formerly Graduate Student, Dep. of Agronomy, New Mexico State
Univ.; presently Research Associate, Dep. of Civil Engineering, Princeton
Univ., Princeton, NJ 08540.
'Associate Professor, Dep. of Agronomy, New Mexico State Univ., Las
Graces, NM 88003.

= d(T-T)/(\-= 2(«v)"2 [13]MATERIALS AND METHODS
Table 1 summarizes some physical and chemical properties of
the Glendale clay loam soil used in this study. This soil is a subsample of the calcareous Glendale series, a fine silty mixed ther-

272

273

VAN GENUCHTEN & WIERENGAI MASS TRANSFER STUDIES IN SORBING POROUS MEDIA! II
Table 1—Physical and chemical properties of the
Glendale clay loam soil.
AnalysisAmountSand, %
Silt, %
Clay, %
CEC, meq/100 g
Exchangeable ions (meq/100 g)
Ca
Mg
Na
K
NH,
EC X 103 (saturation extract)
pH (saturation extract)
pH (saturated paste)
CaC03 equivalent, %Organic matter, %Gypsum, %

38.8
29.8
31.4
31.1
22.0
3.6
0.8
1.5
1.2
3.3
7.5
7.7
7.0
0.8
0.0

mic typic torrifluvent. The soil was air dried and passed througheither a 6.3 or 2.0 mm screen.
A solution of 0.0IN CaCl 2 in tritiated water was used for the
displacement studies. Ten millicuries of tritium were added toeach liter of water resulting in a final tritium activity of approximately 23,000 cpm/ml solution. Activities of tritium were determined on a Packard Tricarb Model 3310 liquid scintillation
counter, using Aquasol (New England Nuclear, Boston, Mass.) as
the counting medium.
Soil columns, 30 cm long, were prepared by carefully packing
air-dried Glendale clay loam soil into plexiglass cylinders of 5.4cm inside diameter (ID). A fine fritted glass porous plate was attached to the outflow end of each cylinder. The column was positioned vertically on top of, and connected to, a vacuum chamber
enclosing an automatic fraction collector (Instrumentation Specialties Co., Lincoln, Nebraska). Through the use of the vacuum
chamber and fritted glass porous plate, a constant vacuum could bemaintained at the lower end of the soil column, using a vacuum
regulator (Moore Products Co., Springhouse, Pennsylvania). The
leaching solution was applied to the top of the soil column using amodified syringe pump described by Wierenga et al. (1973). A
constant inflow could be maintained with the pump over long
periods of time. The columns were first leached extensively with
0.01/V CaCl2. After steady state flow conditions were established,
the 0.01/V CaCl2 solution was replaced with the tritiated CaCl2
solution. The number of pore volumes (T) displaced through thesoil was calculated by dividing the volume of effluent by the volumetric water capacity of the soil column. The relative effluent concentration (cm) was calculated from the tritium activity in theeffluent and the initial activity of the tritium at the inflow location.
The average water content of the soil during a displacement wasdetermined by weighing the column before leaching and several
times during the displacement experiment. Water contents of thecolumns were regulated within a narrow range by adjusting thepressure in the vacuum chamber. However, to assure uniform
water content distributions with depth, only suctions between 20
and 75 cm H2O were applied. Gravimetrically measured watercontents at the end of the leaching studies showed < 0.02 cm3/cm3
difference between the upper and lower parts of the columns, theaverage water contents being identical to those determined byweighing the columns. As was shown by Cassel et al. (1975), variations in water contents of this magnitude have very little effect on
effluent concentration distributions. A total of 15 runs were madewith five columns, using soil of two different aggregate sizes, andemploying soil water fluxes (q), ranging from 1.3 to 34 cm/day.
The experimental data for the different displacement studies are
given in Table 2 where the experiments are numbered by column
and flux.
RESULTS AND DISCUSSION
The data from the various displacement experiments wereused to determine for each run the values of the different un-

Table 2—Soil-physical data for tritium displacements
through Glendale clay loam.
Bulkdensity, p

Water
content, 8Flux, <?

r,t,

Largestaggregate
size

g/cm 3

cm3/cm3

cm/day

-

days

mm

1-1
1-2
1-5

1.360
1.360
1.360

0.460
0.460
0.460

5.09
5.09
2.55

2.334 6.331
0.512
1.389
1.990 10.76

2.0
2.0
2.0

2-1
2-2
2-5

1.361
1.361
1.361

0.464
0.468
0.467

4.55
4.57
2.29

2.361 7.215
0.452 1.389
1.762 10.77

2.0
2.0
2.0

3-2
3-4
3-6

1.222
1.222
1.222

0.454
0.445
0.434

17.0
4.20
1.32

3.102
1.909
1.788

2.475
6.076
9.653

6.3
6.3
6.3

4-2

1.309

0.433

4.04

1.871

6.021

6.3

5-1
5-2
5-3
5-4
5-5

1.126
1.126
1.126
1.126
1.126

0.421
0.401
0.395
0.399
0.393

33.9
16.6
8.22
5.54
2.29

3.145
1.172
2.080 1.508
2.763 3.987
6.822
3.160
2.419 12.45

6.3
6.3
6.3
6.3
6.3

Exp.
no.

Pulse period

known parameters (P, R, a, and /3) in the linear adsorption
model (Eq. [6]-[13]). The values of these parameters wereobtained by means of fitting Eq. [8] to the experimental
effluent data. Because four dimensionless parameters had tobe evaluated for each effluent curve, a nonlinear curve-fitting technique was used for this determination rather than atrial and error procedure. The program used in this study isthe nonlinear least-squares (GAUSHAUS) curve-fitting program, originally written by Meeter (1964). This program4 isbased on the maximum neighborhood method of Marquardt(1963), which combines the Gauss (Taylor series) method
and the method of steepest descent. A detailed treatment of
this technique is given by Daniel and Wood (1973).
Figure 1 shows the results obtained when the curve-fitting program is applied to the experimental data of experiment 5-3 (Table 2). The open circles in the figure represent
the observed data points, symbolically given by the data set{T,cm(l,T)}. Recall that the effluent concentration distribution c m (1,7) is given by Eq. [6], with x = 1 (z=L). Hence
0 =s T < 7,[14]

Forr>3.1,c 1 (l,T)inEq. [14] is very close to one (Fig. 1),
and we may approximate the effluent concentration by

[15]
or
J.I.

[16]

Equation [16] states that when the linear model is used witha large enough pulse T^/the break-through part of theeffluent concentration curve, i.e. where dcjdt > 0, can be
represented also by the data set {T,\-cm (l.TM-T1,)}, obtained
from the elution side of the curve, i.e., where dcmldt < 0.
Both data sets {T, cm(l,7)} and {T,l - cJlJ+Tj)} areplotted in Fig. 1, and define approximately the same curve.
Some discrepancies occur at the lower concentrations but••Available upon request.

274

SOIL SCI. SOC. AM. J . , VOL. 41,

1977

Exp. 5-3
1.0

g- 8
(-

-6

CALCULATED
P = 35•*EXPERIMENTAL{T,Cm(l,T)},#,=0.610

EXPERIMENTAL{T,l-C m (l,T+T,)}a = 0.27

2.0

1.0

3.0

l

l

4.0

5.0

I/day

6.0

PORE VOLUME-T

Fig. 1—Example of curve fitting with the non-linear least squares GAUSHAUS curve fitting program, using the effluent date of experiment 5-3.

are primarily caused by the assumption of Eq. [16] that Cj(1,7) = 1 for7>3.1. By using Eq. [16] an accurate description of the break-through and elution sides of the curve
is obtained. Also the tailing at the higher concentrations
can be extended towards higher pore volumes, as shown in
Fig. 1. The latter procedure is useful since the experimental
errors during scintillation counting of the tritium are usuallyquite large at the higher concentrations.
The following procedure was used to obtain values of thefour dimensionless variables /3, P, R, and a.
1) Approximately 15 points were chosen from the twodata sets given above, and fitted to Eq. [8]. In this way initial estimates were obtained for the variable /3, the retardation factor R, the mass transfer coefficient a and the Peclet
number P. Estimates of these parameters were obtained for
all tritium experiments.
2) The initial estimates of the retardation factor R for
the different experiments were used to estimate the adsorption constant K for tritium by using the definition of R andthe values of the water contents (6) and bulk densities (p)listed in Table 2. R values found ranged from 1.007 to
1.043, while the estimates of A" ranged from 0.003 to 0.017,
the average being 0.009. The impact of these low values ofK on the displacements is small; the retardation is only
about 0.025 pore volumes. Attempts were made to alsomeasure the adsorption constant independently by equiliTable 3—Summary of various tritium (3H2O) displacements throughGlendale clay loam.
Exp.

a

P
-

R

P

cm /day

-

-

a
-

0
-

1-5

5.09
5.09
2.55

0.926
0.926
0.928

1.027
1.027
1.027

95
95
95

1.47
1.47
0.56

0.940
0.940
0.942

3.7
3.7
1.9

0.25
0.25
0.048

2-1
2-2
2-5

4.55
4.57
2.29

0.850
0.887
0.830

1.026
1.026
1.026

45
45
45

1.50
1.31
1.05

0.862
0.900
0.841

7.6
7.2
3.9

0.23
0.20.
0.080

3-2
3-4
3-6

17.0
4.20
1.32

0.841
0.872
0.717

1.024
1.025
1.025

56
56
56

0.39
0.72
1.96

0.852
0.884
0.725

24.0
5.7
2.2

0.22
0.10
0.086

no.

1-1
1-2

Flux, gD

cm 2 /day I/day

4-2

4.04

0.788

1.027

57

0.93

0.798

6.2

0.12

5-1
5-2
5-3
5-4
5-5

33.9
16.6
8.22
5.54
2.29

0.723
0.704
0.605
0.531
0.694

1.024
1.025
1.026
1.025
1.026

35
35
35
35
35

0.33
0.55
1.00
1.54
1.46

0.731
0.711
0.610
0.534
0.702

94.0
50.0
29.0
22.0
7.1

0.37
0.31
0.27
0.28
0.11

brating increasing concentrations of tritium with soil over a
10-day period. No significant adsorption could be detected.
The batch equilibration technique is apparently not sensitive
enough to measure small values of K. In this study, thevalue of 0.009 for K will be used for all experiments, resulting in the retardation factors as given in Table 3. Other studies have also revealed some retardation of tritium (Nielsen
and Biggar, 1962; Heemstra et al., 1961; Mansell et al.,
1973; Wierenga et al., 1975). The retardation may becaused by some isotopic exchange of tritium with crystallattice hydroxyls of the soil clay fraction and/or by the replacement of some exchangeable cations by 3H+.
3) An average Peclet number P was assumed for eachcolumn, and was obtained by taking the arithmetic averageof the P-values of the different experiments of the same column. For example, for the five experiments involving column 5, the initial estimates were 48, 27, 29, 40, and 32, respectively, resulting in an average Peclet number ofapproximately 35. The use of a constant P for each column
is consistent with the assumption that the dispersion coefficient D varies approximately linearly with the flow velocity
in the mobile liquid, provided longitudinal diffusion has aneglible influence on the dispersion coefficient. Earlier
studies have shown that this assumption is correct as long as
the water content does not vary appreciably among the different experiments (Raimondi et al., 1959; Harleman et al.,
1963; Villermaux and van Swaay, 1969; Bear, 1972).
Simpson (1969) concluded that the dispersion coefficient(D) changed approximately linearly with flow velocity andthat deviations from this assumption were probably notmuch larger than the experimental error in the work of asingle investigator.
4) In order to obtain final values for the two remaining
parameters /3 and a, a second 2-parameter curve fitting was
carried out, using the previously determined values of P and
R as known variables in the fitting program. We believe thatby carrying out a second curve fitting, more exact estimatesof /3 and a are obtained than by using the initial estimates.
Changes in the value of P, for example, are directly reflected in the value of a, while experimental or fitting errors
in R are carried over into /?.
When the four parameters P, R, /3, and a, obtained withthe two curve fittings, are substituted in Eq. [8], the solidline in Fig. 1 results. In Fig. 2, a comparison is shown be-

VAN GENUCHTEN & WIERENGA.' MASS TRANSFER STUDIES IN SORBING POROUS MEDIA: II

275

\

Exp.

5-3

3H

1.0

o
i

V

1 -

8

•a.
'

i

RELATIVE C(

.

3 N

S

(D

^^"
4

1.0

\ \

————— EQUATION (6)————— LINOSTROM ETALII967)

1

,

2S>

q°8.22 cm/day
8=0.395

v\

>.

1

I
3.0

\

^=0.610

V

;\

4.0

o=0.27

1 /doy

P=35

5.0

6.

PORE VOLUME-T

Fig. 2—Observed and calculated tritium effluent curves for experiment 5-3.

tween the entire effluent concentration distribution of experiment 5-3, and the calculated curve using Eq. [6] for thepulse. The figure shows an excellent agreement between
calculated and observed results. The dashed line in Fig. 2
represents the solution of the convective-dispersive equation obtained by Lindstrom et al. (1967) for the same initialand boundary conditions as adopted here (including the observed tritium retardation). This solution requires all thewater in the column to be mobile (<j>= 1 ,/= 1). No satisfactory description of the effluent data could be obtained, evenby allowing the value of P to deviate from 35. Somewhat
better results were obtained with the solutions of Brenner(1962) and Lapidus and Amundson (1952). However, forthese latter solutions Peclet numbers in the order of 3 or 4
had to be used, leading to dispersion coefficients of about
175 cnvYday.
We emphasize here that the linear adsorption model (Eq.
[6] - [13]) depends on four dimensionless variables (/3, R,
P, a), while the original model in dimensionalized form
contains five parameters (<$>,/, K, D, a). This difference inthe number of parameters is due to the fact that the dimensionless variable /3 contains both the parameter <f> (fraction
mobile liquid) and the parameter/(fraction of the adsorption sites in the dynamic region of the soil). Both <j> and/have similar effects on the displacements; a decrease in either parameter results in an increased mobility of the chemical during the displacements as was shown earlier (vanGenuchten and Wierenga, 1976). Although an estimate of/3
suffices to complete the mathematical description, the physical picture of the soil is clarified when also an estimate of
the fraction mobile water is available. The following procedure was used to obtain an estimate of <f> from the curve fitted value of f3. From the definition of /3 (Appendix) we have(t)Rm = <t, +f(R-[17]= pR-f(R-\).

[18]

or
When we use the fitted values of /3 and R from experiment
5-3 (Table 3), Eq. [18] reduces to:0 = 0.620 - 0.0256/.

[19]

Because the retardation factors for tritium are very close tounity, it is clear from Eq. [19] that fairly imprecise estimates off will still yield accurate values of <j>. A value of

0.4 for/was used in Eq. [18], obtained from data on the
movement of 2,4,5-T through the same soil (van Genuchten
et al., 1977). This value of/is obviously only an estimate
since the adsorption mechanism for 2,4,5-T and tritium are
not necessarily the same.
The effluent concentration distributions for column experiments 5-1, 5-2, and 5-4 are shown in Fig. 3, 4, and 5,
respectively, and are compared with the analytical solution(Eq. [6]). The agreement between data and the curves
calculated with Eq. [6] are good in each case. Similar
agreement was found for all other tritium displacements(figures not presented). The solution of Lindstrom et al.
(1967) again fails to accurately describe the experimental
data of Fig. 3, 4, and 5. No significant improvements could
be obtained by adjusting the value of P in the solution ofLindstrom et al. However, with the solutions of Lapidus
and Amundson (1952) and Brenner (1962), choosing values
for P of approximately 2-5 (resulting in dispersion coefficients of 150-500 cm2/day), differences between calculated
and experimental curves could be made quite small.
The following observations can be made when examining
Fig. 2-5 and the data presented in Table 3.
1) The amount of immobile water generally increases
with decreasing flow velocity. This effect is clearly shownby the displacements through column 5, except for experiment 5-5 (Table 3). Between experiments 5-4 and 5-5, thecolumn was allowed to dry for several days, which may
have had an effect on the amount of immobile liquid. For
column 5 the amount of immobile water increases fromabout 25% of the total amount of water in the column at aflux of 33.9 cm/day, to approximately 45% at a flux of 5.5
cm/day. Similar effects of flow velocity on the amount ofimmobile water have also been observed by Nielsen andBiggar(1962); Coats and Smith (1964); Villermaux and van
Swaay (1969); and by Bennet and Goodridge (1970).
2) The mass transfer coefficient a decreases with flow
velocity, indicating a slower mixing between the two liquid
regions with decreasing velocity. Bennet and Goodridge(1970) and Coats and Smith (1964) noted similar effects.
The decrease in a with decreasing flow velocity may becaused by an increase in the amount of immobile water, andhence by an increase in the diffusion length for the chemicalfrom the mobile liquid to the center of the immobile liquid.
In addition, convective mixing may have occurred inside
the immobile liquid at higher flow velocities. Such a mixing

276

SOIL SCI. SOC. AM. J . , VOL. 41,

RELATIVE CONCEINTRATION-Cm••i
at CD c

Exp.

5-1

1977

3H

/sFr°"°'~~~

\ %x>

f" j

\

\

q=33.9 cm /day

o'° /
J
I

o o o o o o EXPERIMENTAL

\ \
^ \

/

/

————— EOUATION (6)

\

/

/

————— LINDSTROM ET ALII967)

8 = 0.421^^

a=0.37
P=35

I/day

D fo

\
N? \

2.0

1.0

3.0
PORE

4.0

5.0

6.0

VOLUME-T

Fig. 3—Observed and calculated tritium effluent curves for experiment 5-1.
Exp. 5-2

3H

6i.o

*^~

u

0 .8
IE

I •«
1 2
^
ui
<E

n

'
'

\\

1

q=!6.6 cm/day

//
/ /

S'6

.-o—o ^\

O^^

•/.-

8 = 0.401
o=0.3l

//

\

\

\
I/day

\

o . o o o o EXPERIMENTAL————— EOUATION (6)

V\

P'35

JJ

— -—— LINDSTROM ET AL ( 1967)

\i

/ /
!

,4

1.0

2.0

,

"\"°^°---o__

,

3.0

4.0

6.0

5.0

PORE VOLUME-T

Fig. 4—Observed and calculated tritium effluent curves for experiment 5-2.
Exp. 5-4•^ ^-'-ra~~~ ° °>. \

1.0
u

\\

0 .6
&

: /i
Iu
j> i s i i

I,

\

\

q=5.54 cm/day

\ \
\o \

o o o o o o EXPERIMENTAL

O
O

4————— EOUATION (6)UJ

2— — — LINDSTROM ET AL«967)

8=0.399

\^

B,028

\=s<>

)/day

P=35

-J

UJ

"=

n

1.0

1

1

i

l

l

3.0

2.0

1

4.0

1

\—

I

5.0

°7°—— 0— o.

6.0

PORE VOLUME-T

Fig. 5—Observed and calculated tritium effluent curves for experiment 5-4.

process would effectively increase the value of a, because
the present model assumes that solute exchange between
mobile and immobile liquid regions occurs only by diffusion and not by convection.
3) A decrease in the size of the aggregates decreases
the amount of dead water in the column. An analysis of theeffluent data of column 1 showed that only 6% of the waterin the column was immobile (Table 3). This column was
uniformly packed with aggregates of < 2 mm. Column 2
showed a slightly higher immobile water content than column 1, although both were packed with the same aggregatesized material and at the same bulk density. It is important
to note here that the effluent data of columns 1 and 2,

packed with aggregates < 2 mm, could be described
equally well with the two-parameter model of Lindstrom etal. (1967) as with the four-parameter model discussed inthis paper. The solution of Lindstrom et al. gave inaccurate
descriptions of the data only when large percentages of immobile water were present, such as is the case with column
5, and to a lesser degree with columns 3 and 4.
4) A reduction in bulk density appears to increase theamount of immobile water, as shown by the data in Table 2
and 3. Soil column 5 had the lowest bulk density, butshowed the most dead water and hence the most tailing. An
increase in bulk density is likely to cause a compaction of
the soil, resulting in a narrower pore-size distribution. The

VAN GENUCHTEN & WIERENGA: MASS TRANSFER STUDIES IN SORBING POROUS MEDIA! II

277

20
DISTANCE

COLUMN -Z(cm)

Fig. 6—Concentration profiles inside a 30-cm long soil column at different times during leaching with a short pulse (Y, = 1 day) of tritiated water.
Values of the parameters used to construct the curves are given in Fig. 5. Numbers on the curves indicate times (days) after leaching was initiated.
narrower the pore-size distribution, the narrower is thepore-water velocity distribution, resulting in relatively lessimmobile water.
5) The dispersion coefficient D appears to increase
somewhat with increasing aggregate size and decreasing
bulk density (Tables 2 and 3). Further experiments, however, are necessary to clearly separate the effects of these
two soil-physical properties on the magnitude of the dispersion coefficient.
An important feature of the model is that the solute concentrations in both the mobile and immobile regions of thesoil can be calculated. Figure 6 shows concentration distributions inside a 30-cm long soil column for both themobile and immobile liquid at different times during leaching with a 1-day pulse of tritium. Values of the different parameters used to construct the curves were obtained fromexperiment 5-4 (see Tables 2 and 3). The effluent curve of
this experiment was described accurately with the linear adsorption model (Eq. [6]), as shown by Fig. 5. Hence someconfidence in the accuracy of the calculations seems justified. The concentrations in the mobile and immobile liquids were calculated with Eq. [6] and [7], respectively. Figure 6 shows considerable differences between theconcentrations in both liquid regions. Due to the slow exchange by diffusion between mobile into the immobile liquids, the immobile concentration lags behind the mobile
concentration. The average concentration cav at any pointin the column is given byments about whether this is true however, are difficult toverify. In fact, the ratio of mobile vs. immobile water extracted and thus the concentration of the extract will dependon the applied vacuum, the duration of the applied vacuum
and the way in which the porous cups is installed. Solute
distributions such as those shown in figure 6 may be helpful
in interpreting soil solution extracts, as they define upperand lower limits of the solute concentrations in the soil.
From the discussions above it appears that the linear adsorption model may provide a physically realistic description of the tailing phenomena in unsaturated soils. The
model may prove to be useful in describing the removal byleaching of various chemicals from field soils. This leachingis often accomplished under unsaturated conditions, whenlarge percentages of immobile water may be present.

[20]

APPENDIX - NOTATION
C0= influent concentration (/ag/cm3)
Ci= concentration breakthrough side effluent curve
Cm.Cim = concentration ofmobile and immobile liquid, respectively, (/ag/cm3)
cm,C|m = relative concentrations (CJC
0; C im /C 0 , respectively).
D= dispersion coefficient (cm2/day)/= fraction adsorption sites in dynamic regionla,ll= modified Bessel functions
K, N= constants in Freundlich isotherm: S = K CN
L
= length of column (cm)P
= peclet number of column: P = vmL/Dq= volumetric flux (cm/day)R= average retardation factor: R = 1 +pK/8
Rm= retardation factor of dynamic region: Rm = 1 +fpK/Om
Rim= retardation factor of stagnant region:

Since the fraction mobile liquid ((/>) was 0.534 for experiment 5-4, it follows that the average concentration isroughly the average of cm and c im . It may be argued that thesoil solution withdrawn through porous cups or similar
devices represents mainly mobile liquid, since it is this liquid which most readily moves within the soil. Definite state-

Sm,Sim = adsorption in dynamic and stagnant region respectively (/ug/g)t= time (days)?!
= pulse period (days)T= pore volume: T = vjlL — vmt(f>/LTl— dimensionless pulse period: 7, = v^/L
v0
= average pore-water velocity (cm/day): v0 = qlO

Cav = 4>Cm + (1 - </>)Cj,/?,„,= i +0 -f)pKielm

vm
x
z
a
TU
13
6
Om
0im
<{>
p

= average pore-water velocity in dynamic region(cm/day): vm = qldm
— dirnensionless distance: x = zIL= distance (cm)= mass transfer coefficient (I/day)= dimensionless mass transfer coefficient: TS = aLlq
= 4>RJR= water content: 6 = 0m+0lm(cm3/cm3)= mobile water content (cm3/cm3)= immobile water content (cm3/cm3)= fraction mobile water: <f> = 0J0= bulk density (g/cm3)