Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol22.Issue3.1998:
Advances in Water Resources Vol. 22, No. 3, pp. 247–256, 1998
q 1998 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
PII: S 0 3 0 9 - 1 7 0 8 ( 9 8 ) 0 0 0 1 1 - 6
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Modeling transient organic vapor transport in
porous media with the dusty gas model
Brent E. Sleep
Department of Civil Engineering, University of Toronto, Toronto, Canada M5S 1A4
(Received 2 June 1997; revised 6 April 1998; accepted 8 April 1998)
The dusty gas model constitutive relationships were incorporated into a numerical
model for three-phase, multicomponent flow and transport in porous media. The dusty
gas model properly accounts for interactions between all gas-phase species in
multicomponent gas mixtures. The model also included Knudsen diffusion, which
becomes important in very fine grained soils. The dusty gas model results were
compared to predictions based on Fick’s law. For the cases studied, Fick’s law
overpredicted flux rates of organic compounds, the effect becoming more pronounced
as the permeability of the soil and the Knudsen coefficient were reduced. Increasing
moisture content also appeared to increase the difference between predictions based
on the dusty gas model and those based on Fick’s law. Hypothetical field-scale
simulations were performed to show the impact of multicomponent effects and
Knudsen diffusion in a sandy soil and in a clay soil. Results showed that remediation
times were significantly underpredicted if Fick’s law was used for gas-phase diffusion.
q 1997 Elsevier Science Limited. All rights reserved
Keywords: dusty gas law, Knudsen diffusion, vapor transport, porous media.
the molecular diffusive fluxes relative to the mean molar
velocity.
Baehr and Bruell6 developed a steady-state one-dimensional model for multicomponent transport, based on the
Stefan–Maxwell equations. They analysed several column
experiments and concluded that Fick’s law did not adequately represent the physics of gas-phase diffusion. They
did find that Fick’s law could be used as an empirical equation for single-component diffusion if the tortuosity was
used as a calibration parameter. Greater tortuosities were
required for compounds with greater saturated vapor concentrations to match experimental results. Abriola et al.2
developed a one-dimensional one-phase model for vapor
transport in a binary component system based on the
dusty gas model. They found that fluxes predicted with a
model based on Fick’s law were always less than those
predicted by the dusty gas model for a sandy soil. For
finer soils, such as a clay soil, Knudsen diffusion became
important and the Fickian model substantially overpredicted
the gas fluxes compared to the dusty gas model. Abu-ElSha’r and Abriola4 reported experiments on single and
binary gases designed to measure Knudsen diffusion coefficients. Massmann and Farrier16 examined flux rates using a
1 INTRODUCTION
In recent years a number of numerical models have been
developed for predicting the movement of organic vapors in
soils. These models have ranged from single-phase singlecomponent models8,11,17 that include dispersion and advection to three-phase (water–organic–gas) multicomponent
compositional models that include advection, dispersion,
capillary forces, and interphase partitioning of organic species between any of the phases present1,10,20. All of these
models are based on the assumption that Fick’s law is an
adequate representation of gas-phase diffusion. Thorstenson
and Pollock21 presented an extensive review of the chemical
engineering literature related to gas-phase diffusion, much
of which was developed for gas-phase diffusion in porous
catalyst pellets. They applied the dusty gas model of Mason
and Malinauskas15 and the simplified version of this model,
the Stefan–Maxwell equations to one-dimensional steadystate multicomponent gas-phase transport. Thorstenson
and Pollock21 showed that for stagnant gases Fick’s law
did not accurately predict the fluxes in multicomponent
systems. For non-stagnant gases, Thorstenson and Pollock21
concluded that Fick’s law provided an accurate estimate of
247
248
B. E. Sleep
steady-state form of the dusty gas model for three-component systems. They concluded that the Fickian and dusty gas
models gave similar steady-state fluxes for permeabilities
greater than 10 ¹10 cm 2. At lower permeabilities the Fickian
model overpredicted the flux rates. Arnost and Schneider5
compared data for transient gas diffusion in porous catalysts
to the predictions of a one-dimensional, transient, dusty gas
model, and found good agreement between experimental
data and model predictions.
The vapor transport models developed to date that incorporate the dusty gas model or the Stefan–Maxwell equations are all one-dimensional, single-phase models, and with
the exception of the binary models of Abriola et al.2 and
Arnost and Schneider5 are steady state. Thorstenson and
Pollock21 pointed out the need to compare Fick’s law predictions with dusty gas model predictions under transient
conditions. At most sites there are significant changes in
source concentrations of organic components as the more
volatile and soluble components are preferentially removed.
The comparisons between Fick’s law and the Stefan–
Maxwell equations made by Thorstenson and Pollock21
and Baehr and Bruell6 neglected gas-phase advection and
the impacts of gas pressure gradients. Massmann and
Farrier16 showed that the Stefan–Maxwell equations gave
erroneous results relative to the dusty gas model for low
permeability soils, and for systems with pressure gradients.
Pressure gradients frequently develop due to variations in
atmospheric pressures as well as a result of remediation
processes such as soil vacuum extraction.
There is a need to incorporate the dusty gas model into a
comprehensive simulator to determine the situations where
Fick’s law is inadequate for gas-phase diffusion. Of particular importance are the predicted rates of spread of organic
vapors, and the rates of diffusion of vapors to the atmosphere at the ground surface, and the predicted rates of disappearance of volatile organic compounds immobilized in
the vadose zone, particularly in fine grained soils.
In this article the formulation of the dusty gas model is
reviewed, and the incorporation of the model into the compositional simulator previously presented by Sleep and
Sykes20 is described. Example simulations of multiphase
multicomponent systems are presented to highlight the
implications of using either the dusty gas model or Fick’s
law for gas-phase diffusion.
2 THE DUSTY GAS LAW MODEL FOR GASEOUS
DIFFUSION IN POROUS MEDIA
the linear adsorption coefficient, r b is the bulk soil density,
r b is the phase molar density, x i b is the species mole fraction, q b is the Darcy velocity vector, J i b is the mechanical
dispersion flux, N i b is the molecular diffusion flux, r i b
represents interphase transfer of species i to, or from,
phase b, and G i b represents sinks and sources of species i
to phase b due to injection or pumping, or biological or
chemical transformations. An overall molar balance for
species i is derived by summing eqn (1) over the water,
gas and organic phases. In this case, the interphase transfer
terms, r i b, for the different phases sum to zero20. As
described in Sleep and Sykes20, the model is based on
assumptions of equilibrium interphase mass transfer and
ideal phase behavior described by Henry’s and Raoult’s
laws.
The Darcy velocity vector for phase b is given by
qb ¼ ¹
þ [=·fSb (Jib þ Nib )] ¹ Gib ¹ rib ¼ 0
Jib ¼ ¹ rb Dmech
ib ·=xib
where f is the porosity, S b is the phase saturation, Kib,
(3)
22
as
The mechanical dispersion tensor, Dmech
ib , is defined
n
n
bk bl
(4)
Dmech
ibkl ¼ aT lnb ldkl þ (aL ¹ aT )
lnb l
where a L and a T are the longitudinal and transverse dispersivities, respectively, n b k and n b l are the components of
the linear porewater velocity, and d kl is the Kronecker delta.
Dispersivities are assumed to be independent of phase
saturation.
The diffusive molar flux for the water phase is given by
Fick’s law:
Niw ¼ ¹ rw tw Diw ·=xiw
(5)
where t w is a tortuosity factor and D i w is the aqueous-phase
molecular diffusion coefficient for species i.
The diffusive molar fluxes for the gas phase, in the presence of a gravitational field, are given by the dusty gas
model22:
nc
X
j ¼ 1, jÞi
¼
ð1Þ
(2)
where k is the intrinsic permeability tensor, k r b is the relative permeability of phase b, m b is the viscosity, p b is the
pressure of phase b, g is the gravitational acceleration, g b is
the mass density, and z is the elevation.
It should be noted that the assumption that Darcy’s law
gives a molar-averaged velocity is consistent with developments of the theory of flow and diffusion of gases in porous
media in the chemical engineering literature 7,15.
The mechanical dispersive molar flux, J i b, for the water
phase is given by
The macroscopically averaged equation for the movement
of a component i in a phase b20 is
]
[(fSb þ Kib, d rb )rb xib ] þ =·(rb xib qb )
]t
kkrb
(=pb þ gb g=z)
mb
xig Njg ¹ xjg Nig
Nig
¹
tg Dijg
tg DK
ig
1
(p =x þ xig =pg þ xig rg g=z)
RT g ig
where D ijg is the binary gas-phase diffusion coefficient for
d
is
Organic vapor transport in porous media
components i and j, DK
ig is the Knudsen coefficient for
component i, R is the gas constant, and T is the temperature
(K). Knudsen diffusion becomes important when the pore
diameters are small enough, or pressures are low enough,
that the gas molecules collide primarily with the pore walls
rather than with other molecules. From eqn (6) it can be
seen that Knudsen diffusion becomes important when DK
ig is
much smaller than D ijg. In this case the first term in eqn (6)
becomes negligible and the diffusive flux is controlled by
the second term. The Stefan–Maxwell equations are a special case of eqn (6) produced by neglecting the pressure
gradient term, =p g, and the Knudsen diffusion term,
21
Nig =DK
discuss a number of
ig . Thorstenson and Pollock
simplifications of eqn (6) such as the pure molecular diffusion regimes, the pure Knudsen regime, and the simplified
equations for binary systems.
In the present model gas-phase diffusion coefficients are
assumed to be a function of pressure and temperature
according to the Fuller, Schettler and Giddings method19:
MA þ MB 1=2
10 ¹ 3 T 1:75
MA MB
(7)
DAB ¼ h X
i2
X
1=3
P ( n)A þ ( n)B1=3
where T is the temperature (K), P is the pressure (atm), M A
and M B are the molecular weights of A and B, and the on
terms are the diffusion volumes determined from the structure of the molecules19.
The Knudsen coefficient is related to the gas molecular
weight and the pore size of the soil. Geankopolis9 recommended the relationship
s
T
3
K
(8)
Dig ¼ 9:7 3 10 r
Mi
where r is the pore radius (cm), M i is the molecular weight
2 ¹1
(g/gmol), and DK
ig in cm s .
Thorstenson and Pollock21 discussed the relationship
between the Knudsen diffusion coefficient and the
Klinkenberg parameter. They derived the relationship
DK
ig ¼
kbi
mig
(9)
where b i is the Klinkenberg parameter that is a characteristic of the soil. For air, Thorstenson and Pollock21 presented the relationship of Heid et al.12:
bair ¼ 0:11k ¹ 0:39
(10)
2
with b air in Pa and k in m . The b values for other components can be determined from
s
mi Mair
(11)
bi ¼ bair
mair Mi
It should be noted that in eqn (6) it has been assumed that
the same tortuosity factor can be used for molecular diffusion and Knudsen diffusion. This may not be appropriate
since the two processes are different phenomena. In
249
particular, the effect of increasing water saturation on tortuosity factors for Knudsen diffusion is unknown. For the
present study the Millington and Quirk18 relationship has
been used to calculate tortuosities.
The discretized form of eqn (6) may be written as
np
X
Vk
(rb xib Sb f)n þ 1 ¹ (rb xib Sb f)n
Dt
b¼1
np
X
m
X
¼
(rb xib qb )l, k þ (fSb Jib )l, k þ (fSb Nib )l,
l ¼ 1, lÞk b ¼ 1
k
where m is the maximum number of neighboring blocks for
block k, and the subscript l, k on the flux terms represents
the fluxes between blocks l and k, evaluated at the interface
between l and k. The calculation of advective, dispersive
and Fickian diffusive fluxes in the simulator is outlined in
Sleep and Sykes20.
To apply the dusty gas model, eqn (6) is used to calculate
the diffusive gas-phase fluxes between neighboring blocks,
(N a b) l , k. For a system of n c components eqn (6) may be
written as a matrix equation16:
AF ¼ B
(13)
where A is a matrix with the terms
nc
X
Aii ¼ ¹
j ¼ 1, jÞi
nc
X
Aij ¼ ¹
j ¼ 1, jÞi
xjg
1
¹
tg Dijg DK
ig
(14)
xig
tg Dijg
(15)
The x ig terms in eqns (14) and (15) represent the gas-phase
mole fractions of species i at the interface between blocks.
Any standard spatial weighting technique such as midpoint
weighting or upstream weighting may be used to calculate
the interface value of x jg (see Sleep and Sykes20 for a discussion of various weighting techniques). In the present
model formulation, midpoint weighting is used. The F
matrix is the vector of molar fluxes between neighboring
blocks:
Fi ¼ (Nig )l,
k
(16)
The B matrix is derived from the right-hand side of eqn (6):
1
(p =x þ xig =pg þ xig rg g=z)
(17)
RT g ig
The gradient terms in eqn (17) are calculated using central
differences. For example, for neighboring blocks k and l the
pressure gradient in the x-direction (x l . x k) is calculated
from
Bi ¼
=pg ¼
2(plg ¹ pkg )
Dxl þ Dxk
(18)
The mole fractions and pressures in eqn (14) may be
evaluated either explicitly or implicitly. If they are evaluated implicitly in a simulator that uses Newton–Raphson
iteration (as is done in this paper), then it is necessary to
250
B. E. Sleep
determine the derivatives of each of the fluxes with respect
to mole fractions and pressures. Once eqn (14) is solved for
the fluxes, using the mole fractions and pressures from the
previous Newton iteration, the derivatives may be calculated from the formula
A
]F ]B ]A
¼
¹ F
]y ]y ]y
(19)
where y is a mole fraction, a pressure, or other state variable that mole fractions, densities, diffusion coefficients or
tortuosities may depend upon. If an LU decomposition is
used to solve eqn (14), then ]F
]y may be found by calculating the right-hand side of eqn (19) and then solving using
forward and backward substitution. Eqn (19) must be
solved for each primary variable to generate all the required
derivatives.
3 MODEL APPLICATIONS
3.1 One-dimensional simulations
A series of simulations of volatilization and subsequent
movement of pentane and hexane in a one-dimensional column of dry soil was performed in order to examine the
differences between predictions of the dusty gas model
and predictions based on Fick’s law. The properties of pentane and hexane are summarized in Table 1. Soil permeabilities of 10 ¹11, 10 ¹13 and 10 ¹16 m 2 were used in the
simulations. The temperature for simulations was 238C
and air was considered as a single species with the properties of nitrogen gas. The corresponding Knudsen coefficients for air, calculated from eqn (10), are 1.05 3 10 ¹2,
6.35 3 10 ¹4 and 9.39 3 10 ¹6 m 2 s ¹1, respectively. In each
simulation a horizontal domain of length 30 m (150 finite
difference blocks of 0.2 m) was used. A liquid mixture of
50 mol% pentane and 50 mol% hexane was injected into the
end of the column for a period of 3 min at a rate of 1.0 3
10 ¹4 m 3 (m 2 s) ¹1. The organic liquid saturation reached a
maximum of 0.21, just slightly greater than the residual
saturation, so the NAPL phase did not move more than
0.4 m (two model cells) in the simulation. Although the
model includes adsorption, adsorption of organic vapors to
the soil was not included in the simulations.
The results of the one-dimensional simulations in dry soil
are shown in Figs 1 and 2. The organic was injected at the
end of the column, corresponding to a horizontal distance of
30 m on Figs 1 and 2. The changes in gas-phase concentrations at this end of the column, where the organic is present
at residual saturation, are reflective of changes in the composition of the organic phase. It may be noted that, as time
proceeds, the concentrations of pentane at the end of the
column decrease while the concentrations of hexane
increase. This is due to the preferential volatilization of
pentane from the residual organic phase, leading to enrichment of the organic phase in hexane.
The differences between the predictions using Fick’s law
for diffusion and predictions using the dusty gas model are
small for a permeability of 10 ¹11 m 2. The difference at 710
days is slightly greater for hexane than it is for pentane since
the binary diffusion coefficients used for each organic compound are the organic–air coefficients. The concentrations
of pentane in air are high due to the higher vapor pressure of
pentane. The binary diffusion coefficient for a hexane–pentane mixture, calculated from eqn (7), is 3.5 3 10 ¹6 m 2 s ¹1,
which is lower than the value of 7.6 3 10 ¹6 m 2 s ¹1 for a
hexane–air mixture. In other words, the high pentane concentrations hinder the diffusion of hexane, a phenomenon
not accounted for when Fick’s law is applied as though each
of the gases were diffusing independently through the air.
With a soil permeability of 10 ¹11 m 2, the ratios of diffusive fluxes to advective fluxes varied with time and location
in the column, but the diffusive fluxes always exceeded the
advective fluxes, except at an early time near the point of
injection of the liquid organic. Typically, at the midpoint of
the column the diffusive fluxes exceeded the advective
fluxes by a factor of four or more. To determine whether
the small difference between the dusty gas model and
Fickian model predictions was due to advective effects,
additional simulations (not shown) were conducted in
which air was allowed to move by advection, but advection
of hexane and pentane was set to zero. The results of these
simulations were very similar to those plotted in Fig. 1a
and 2a, in which advection of hexane and pentane was
included. This indicates that the similarities between predictions of the dusty gas and Fickian models for the soil with a
permeability of 10 ¹11 m 2 are not due to the effects of advection dominating diffusion. Thus, the Fickian model may be
Table 1. Chemical properties
Species
Water
Air
Pentane
Hexane
Benzene
Toluene
Molecular weight
Liquid density (kg m ¹3)
Liquid viscosity (kg (m s) ¹1) 3 10 ¹3)
Aqueous solubility (mg l ¹1)
Vapor viscosity (kg (m s) ¹1 3 10 ¹6)
Diffusion volume
Vapor pressure (Pa)
Air-liquid interfacial tension (dyne cm ¹1)
18.0
1000.0
1.0
29.0
42.0
72.15
626.2
0.240
40
6.0
106.5
58 250
18.43
86.18
660.3
0.326
10
6.5
127.0
16 020
18.43
78.11
876.5
0.652
1780
7.4
91.0
10 130
28.85
92.14
866.9
0.590
515
6.7
111.5
2930
28.5
19
Data from Reid et al. , Lide
13
12.7
72.75
and Mackay et al.14.
18.50
20.1
Organic vapor transport in porous media
Fig. 1. Pentane diffusion in a one-dimensional system. Pentane
concentration profiles for permeabilities of (a) 10 ¹11 m 2, (b)
10 ¹13 m 2, (c) 10 ¹16 m 2.
251
Fig. 2. Hexane diffusion in a one-dimensional system. Hexane
concentration profiles for or permeabilities of (a) 10 ¹11 m 2, (b)
10 ¹13 m 2, (c) 10 ¹16 m 2.
252
B. E. Sleep
Fig. 3. Pentane diffusion in soil with permeability of 10 ¹11 m 2,
water saturation of 0.9.
Fig. 4. System configuration for two-dimensional examples.
adequate for simulating transport of organic vapors such as
pentane and hexane in dry soils with intrinsic permeabilities
of 10 ¹11 m 2 or greater.
For a permeability of 10 ¹13 m 2, the differences between
Fick’s law and dusty gas model predictions are significant,
as shown in Fig. 1b and 2b. In particular, the concentration
of pentane at the end of the column containing the organic
liquid is lower for the Fick’s law case, due to the greater
rates of pentane diffusion predicted by Fick’s law. Thus, at a
permeability of 10 ¹13 m 2, the effects of collisions with pore
walls have an impact on diffusion rates. The Knudsen diffusion coefficients for pentane and hexane for a permeability
of 10 ¹13 m 2 predicted by eqns (9)–(11) are 1.31 3 10 ¹4 and
1.29 3 10 ¹4 m 2 s ¹1 respectively, whereas the respective
air–organic diffusion coefficients are 8.21 3 10 ¹5 and
7.43 3 10 ¹5 m 2 s ¹1. Thus, since the molecular diffusion
coefficients and the Knudsen diffusion coefficients are of
the same order of magnitude, this permeability is in the
transition regime between molecular diffusion and Knudsen
diffusion.
At a permeability of 10 ¹16 m 2 there is a great difference
between the predictions of the dusty gas model and Fick’s
law, as shown in Fig. 1c and 2c. The concentration of pentane left in the organic liquid source is 50% lower at 710
days for Fick’s law than for the dusty gas model prediction.
Hexane concentrations are much greater for the Fick’s law
case caused by the greater preferential stripping of the more
volatile pentane caused by the higher Fick’s law flux rates.
The concentrations are also plotted for a case in which the
Knudsen diffusion coefficients were set to very large values
to make the Knudsen effect insignificant. In this case, the
predicted flux rates are only slightly lower than those predicted by Fick’s law, because of multicomponent effects.
Thus, the major impact of the reduced soil permeability on
vapor diffusion is the result of Knudsen diffusion and not
multicomponent effects. It may be possible, in cases where
pressure gradient effects are not important, to model vapor
diffusion with a Fickian model that incorporates Knudsen
diffusion. This model could be developed from eqn (6).
An additional simulation of pentane and hexane volatilization and transport was conducted in the 10 ¹11 m 2 permeability soil at a water saturation of 0.9. In the absence of
information about the effect of water saturation on Knudsen
coefficients, it was assumed that the Knudsen coefficients
were related to the soil’s intrinsic permeability through eqn
(9), and that the effect of increased water saturation was
accounted for by the reduced tortuosity calculated from
Fig. 5. Benzene distribution (mol%) in gas phase for passive conditions after 250 days, predicted by dusty gas model.
Organic vapor transport in porous media
253
Fig. 6. Benzene distribution (mol%) in gas phase for passive conditions after 250 days, predicted by Fick’s law.
the Millington–Quirk expression18. The same tortuosity
factor was used for both molecular and Knudsen diffusion,
so that the effective Knudsen coefficient was smaller than
the effective molecular diffusion coefficient by the same
factor as in the dry 10 ¹11 m 2 permeability soil. Thus, with
these assumptions, on the basis of the results of the dry
10 ¹11 m 2 permeability soil, Knudsen diffusion would not
were significant relative to molecular diffusion in this
high water saturation simulation.
The results of the simulations for the dusty gas and Fickian models at 140 and 7100 days are shown in Fig. 3. The
diffusion rates predicted by both Fick’s law and the dusty
gas model are much lower than for the dry soil case caused
by the small tortuosity factor associated with a water saturation of 0.9. In this case of high water saturation, where
Knudsen diffusion is assumed to be insignificant, the
dusty gas model predicts greater flux rates than the Fickian
model, as indicated by the lower pentane concentrations at
7100 days at the end of the column where the liquid organic
mixture was injected. The injection of the liquid mixture
and its subsequent volatilization gives rise to a pressure
gradient in the column. As seen from eqn (6), diffusion
fluxes are proportional to pressure gradients as well as concentration gradients in the dusty gas model, whereas they
are assumed to be proportional only to concentration gradients in Fick’s law (see Thorstenson and Pollock21 for a
variety of different simplified forms of the dusty gas
model that illustrate clearly the dependencies on pressure
gradients and the differences between Fick’s law and the
dusty gas model).
If the tortuosities for Knudsen diffusion (and thus the
effective Knudsen diffusion coefficients) decreased more
quickly than tortuosities for molecular diffusion as water
saturation increased, then at high water saturations the
Knudsen effect might become more important. In this case
the dusty gas model fluxes could be less than those predicted
by Fick’s law. However, as water saturation increases the
smaller pores are filled first and the average pore radius of
gas-filled pores, calculated on a volume basis, increases, so
that the Knudsen coefficient might be expected to increase
with increasing water saturation unless there are also substantial increases in the thickness of the water film adsorbed
to the walls of the unsaturated pores. This is clearly an area
that needs some careful experimental study.
3.2 Two-dimensional soil venting
In low permeability soils pressure diffusion and Knudsen
diffusion may be significant even under forced advection
conditions such as soil vacuum extraction (SVE). Simulations of the fate of a mixture of benzene and toluene in a
Table 2. Soil properties
Soil type Permeabi- Porosity
lity(m 2)
1
2
10 ¹13
10 ¹15
0.4
0.4
Brooks
Coreyl
Brooks S wr S or
Coreyp d
(m water)
3.0
2.0
0.1
0.4
0.1 0.2
0.2 0.2
Fig. 7. Mass removal of benzene and toluene predicted by the
dusty gas model and by Fick’s law for passive conditions.
254
B. E. Sleep
Fig. 8. Benzene distribution (mol%) in gas phase for soil venting conditions after 1000 days, predicted by dusty gas model.
layered soil system under passive conditions and under
vacuum extraction were performed to examine the differences in predictions of the Fick-based model and the dusty
gas model. The system configuration is shown in Fig. 4, and
soil properties are given in Table 2. The top and bottom
layers of soil have properties typical of a fine sand, while
the middle layer has properties typical of a silty clay. 172.8 l
of an organic mixture of 0.5 mole fraction benzene and
0.5 mole fraction toluene were injected into the two-dimensional system (1 m thick in the third dimension) at a point
2 m below the ground surface over a period of 10 days. The
organic phase subsequently was allowed to redistribute, dissolve and volatilize. Due to the small volume of organic
mixture released, the organic phase did not imbibe into
the silty clay layer. At 300 days in the SVE case, gas was
injected at the air injection point and extracted from the air
extraction point (see Fig. 4) by prescribing constant node
gas-phase pressures of 4.0 and ¹ 4.0 m water, respectively.
The predicted vapor-phase benzene concentration profiles after 250 days for the dusty gas model and for the
Fickian model are shown in Figs 5 and 6, respectively.
The extent of movement of benzene into the low permeability layer is very small for the dusty gas model compared to
the Fickian model. In Fig. 7 the predicted rates of dissipation of benzene and toluene concentrations in the system
due to diffusion to the atmosphere under passive conditions
are plotted for the two models. The dusty gas model predicts
that more than 5000 days are required for the amount of
benzene remaining to reach less than 1 mole, while the
Fickian model predicts that this level will be reached after
about 3000 days. The predicted rate of dissipation of toluene
is also greater for the Fickian model than for the dusty gas
model.
The vapor-phase benzene concentration profiles with soil
vacuum extraction after 1000 days (700 days of soil
vacuum extraction) for the dusty gas model and for the
Fickian model are shown in Figs 8 and 9 respectively.
There is still considerable benzene left in the low
permeability layer in the dusty gas model simulation,
compared to the Fickian simulation. In Fig. 10 the
predicted rates of dissipation of benzene and toluene concentrations in the system with soil vacuum extraction are
Fig. 9. Benzene distribution (mol%) in gas phase for soil venting conditions after 1000 days, predicted by Fick’s law.
Organic vapor transport in porous media
255
tion. However, the effect of moisture content on Knudsen
coefficients requires more study.
The results of two-dimensional simulations of passive
movement and soil vacuum extraction of benzene and
toluene demonstrated that removal rates predicted by the
Fickian model are much greater than those predicted by
the dusty gas model. Removal rates were limited by the
slow rates of diffusion of the organic vapors through the
low permeability soil layer.
REFERENCES
Fig. 10. Mass removal of benzene and toluene predicted by the
dusty gas model and by Fick’s law for soil venting conditions.
plotted for the two models. The Fickian model predicts
much more rapid removal of benzene and toluene than
does the dusty gas model. In the dusty gas model there is
an initial phase of rapid decline in the amount of benzene
and toluene in the system as these compounds are removed
from the high permeability layer. There is an inflection point
in the toluene removal rates at about 950 days as the organic
in the more permeable zone is depleted. Beyond this point
the removal of organic from the low permeability layer is
much slower due to the Knudsen limitation of diffusion. In
the Fickian model the rate of diffusion from the low permeability layer to the high permeability layer is much higher
and therefore the removal of organic from the system by the
extraction well is much faster.
4 SUMMARY
The dusty gas model for gas-phase diffusion was incorporated into a multiphase, multicomponent compositional
simulator. Simulations in one-dimensional systems showed
that predictions of the dusty gas model were very similar to
those of the Fickian model at permeabilities of 10 ¹11 m 2. As
permeabilities were reduced the differences between the
models increased. At a permeability of 10 ¹16 m 2 the differences in the two models were very large, with much lower
fluxes predicted by the dusty gas model. The major reason
for this was the reduced diffusion rates caused by the Knudsen effect in the finer grained soils. The possibility of incorporating Knudsen diffusion into a Fickian model for
modeling vapor transport in fine grained soils should be
further investigated.
At high moisture contents the dusty gas model predicted
higher flux rates than did the Fickian model, caused by the
dependence of diffusion rates on pressure gradients in the
dusty gas model. The pressure gradients were produced by
the injection of liquid organic and its subsequent volatiliza-
1. Abriola, L.M. and Pinder, G.F. A multiphase approach to the
modeling of porous media contamination by organic compounds: numerical simulation. Water Resour. Res., 1985,
21(1), 19–26.
2. Abriola, L.M., Fen, C.-S. and Reeves, H.W., Numerical
simulation of unsteady organic vapor transport in porous
media using the dusty gas model. In Proceedings of the
IAH Conference on Subsurface Contamination by Immiscible
Fluids, Calgary, Alberta, 18–20 April 1990, ed. K.U. Weyer,
A.A. Balkema, publ., Rotterdam, 1992, pp. 195–202.
4. Abu-El-Sha’r, W. and Abriola, L.M. Experimental assessment of gas transport mechanisms in natural porous media:
parameter evaluation. Water Resour. Res., 1997, 33(4), 505–
516.
5. Arnost, D. and Schneider, P. Dynamic transport of multicomponent mixtures of gases in porous solids. Chem. Eng.
J., 1995, 57, 91–99.
6. Baehr, A.L. and Bruell, C.J. Application of the Stefan–
Maxwell equations to determine limitations of Fick’s law
when modeling vapor transport in sand columns. Water
Resour. Res., 1990, 26(6), 1155–1163.
7. Cunningham, R.E. and Williams, R.J., Diffusion in Gases
and Porous Media. Plenum, New York, 1980.
8. Falta, R.W., Javandel, I., Pruess, K. and Witherspoon, P.A.,
Density-driven flow of gas in the unsaturated zone due to
evaporation of volatile organic compounds. Water Resour.
Res., 1989, 25(10).
9. Geankopolis, C.J., Mass transport phenomena. Holt,
Rinehart and Winston, New York, 1972.
10. Kaluarachchi, J.J. and Parker, J.C. Modeling multicomponent
organic chemical transport in three-fluid-phase porous media.
J. Cont. Hydrol., 1990, 5, 349–374.
11. Gierke, J.S., Hutzler, N.J. and Crittenden, J.C. Modeling the
movement of organic chemicals in columns of unsaturated
soil. Water Resour. Res., 1990, 26(7), 1529–1547.
12. Heid, J.G., MacMahon, J.J., Nielsen, R.F. and Yuster, S.T.,
Study of the permeability of rocks to homogeneous fluids. In
Drilling and Production Practice. American Petroleum
Institute, New York, 1950, pp. 230–244.
13. Lide, D.R., CRC Handbook of Chemistry and Physics, 74th
edn. 1993.
14. Mackay, D., Shiu, W.Y. and Ma, K.C., Illustrated Handbook
of Physical–Chemical Properties and Environmental Fate
for Organic Chemicals, Vol. III, Volatile Organic Chemicals.
Lewis Publishers, 1993, pp. 522–526.
15. Mason, E.A. and Malinauskas, A.P., Gas transport in porous
media: the dusty gas model. Chem. Eng. Monogr., 17.
Elsevier, New York, 1983.
16. Massmann, J. and Farrier, D.F. Effects of atmospheric
pressures on gas transport in the vadose zone. Water
Resour. Res., 1992, 28(3), 777–792.
17. Mendoza, C.A. and Frind, E.O. Advective–dispersive
256
B. E. Sleep
transport of dense organic vapors in the unsaturated zone, 1:
Model development. Water Resour. Res., 1990, 26(3), 379–
387.
18. Millington, R.J. and Quirk, J.M. Permeability of porous
solids. Trans. Faraday Soc., 1961, 57, 1200–1207.
19. Reid, R.C., Prausnitz, J.M. and Sherwood, T.K., The Properties of Gases and Liquids, 3rd edn. McGraw-Hill, New York,
1977, 688 pp.
20. Sleep, B.E. and Sykes, J.F. Compositional simulation of
groundwater contamination by organic compounds, 1:
Model development and verification. Water Resour. Res.,
1993, 29(6), 1697–1708.
21. Thorstenson, D.C. and Pollock, D.W. Gas transport in unsaturated porous media: Multicomponent systems and the adequacy of Fick’s law. Water Resour. Res., 1989, 25(3), 477–
507.
22. Bear, J., Dynamics of Fluids in Porous Media, Elsevier, New
York, 1972.
q 1998 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
PII: S 0 3 0 9 - 1 7 0 8 ( 9 8 ) 0 0 0 1 1 - 6
0309-1708/98/$ - see front matter
Modeling transient organic vapor transport in
porous media with the dusty gas model
Brent E. Sleep
Department of Civil Engineering, University of Toronto, Toronto, Canada M5S 1A4
(Received 2 June 1997; revised 6 April 1998; accepted 8 April 1998)
The dusty gas model constitutive relationships were incorporated into a numerical
model for three-phase, multicomponent flow and transport in porous media. The dusty
gas model properly accounts for interactions between all gas-phase species in
multicomponent gas mixtures. The model also included Knudsen diffusion, which
becomes important in very fine grained soils. The dusty gas model results were
compared to predictions based on Fick’s law. For the cases studied, Fick’s law
overpredicted flux rates of organic compounds, the effect becoming more pronounced
as the permeability of the soil and the Knudsen coefficient were reduced. Increasing
moisture content also appeared to increase the difference between predictions based
on the dusty gas model and those based on Fick’s law. Hypothetical field-scale
simulations were performed to show the impact of multicomponent effects and
Knudsen diffusion in a sandy soil and in a clay soil. Results showed that remediation
times were significantly underpredicted if Fick’s law was used for gas-phase diffusion.
q 1997 Elsevier Science Limited. All rights reserved
Keywords: dusty gas law, Knudsen diffusion, vapor transport, porous media.
the molecular diffusive fluxes relative to the mean molar
velocity.
Baehr and Bruell6 developed a steady-state one-dimensional model for multicomponent transport, based on the
Stefan–Maxwell equations. They analysed several column
experiments and concluded that Fick’s law did not adequately represent the physics of gas-phase diffusion. They
did find that Fick’s law could be used as an empirical equation for single-component diffusion if the tortuosity was
used as a calibration parameter. Greater tortuosities were
required for compounds with greater saturated vapor concentrations to match experimental results. Abriola et al.2
developed a one-dimensional one-phase model for vapor
transport in a binary component system based on the
dusty gas model. They found that fluxes predicted with a
model based on Fick’s law were always less than those
predicted by the dusty gas model for a sandy soil. For
finer soils, such as a clay soil, Knudsen diffusion became
important and the Fickian model substantially overpredicted
the gas fluxes compared to the dusty gas model. Abu-ElSha’r and Abriola4 reported experiments on single and
binary gases designed to measure Knudsen diffusion coefficients. Massmann and Farrier16 examined flux rates using a
1 INTRODUCTION
In recent years a number of numerical models have been
developed for predicting the movement of organic vapors in
soils. These models have ranged from single-phase singlecomponent models8,11,17 that include dispersion and advection to three-phase (water–organic–gas) multicomponent
compositional models that include advection, dispersion,
capillary forces, and interphase partitioning of organic species between any of the phases present1,10,20. All of these
models are based on the assumption that Fick’s law is an
adequate representation of gas-phase diffusion. Thorstenson
and Pollock21 presented an extensive review of the chemical
engineering literature related to gas-phase diffusion, much
of which was developed for gas-phase diffusion in porous
catalyst pellets. They applied the dusty gas model of Mason
and Malinauskas15 and the simplified version of this model,
the Stefan–Maxwell equations to one-dimensional steadystate multicomponent gas-phase transport. Thorstenson
and Pollock21 showed that for stagnant gases Fick’s law
did not accurately predict the fluxes in multicomponent
systems. For non-stagnant gases, Thorstenson and Pollock21
concluded that Fick’s law provided an accurate estimate of
247
248
B. E. Sleep
steady-state form of the dusty gas model for three-component systems. They concluded that the Fickian and dusty gas
models gave similar steady-state fluxes for permeabilities
greater than 10 ¹10 cm 2. At lower permeabilities the Fickian
model overpredicted the flux rates. Arnost and Schneider5
compared data for transient gas diffusion in porous catalysts
to the predictions of a one-dimensional, transient, dusty gas
model, and found good agreement between experimental
data and model predictions.
The vapor transport models developed to date that incorporate the dusty gas model or the Stefan–Maxwell equations are all one-dimensional, single-phase models, and with
the exception of the binary models of Abriola et al.2 and
Arnost and Schneider5 are steady state. Thorstenson and
Pollock21 pointed out the need to compare Fick’s law predictions with dusty gas model predictions under transient
conditions. At most sites there are significant changes in
source concentrations of organic components as the more
volatile and soluble components are preferentially removed.
The comparisons between Fick’s law and the Stefan–
Maxwell equations made by Thorstenson and Pollock21
and Baehr and Bruell6 neglected gas-phase advection and
the impacts of gas pressure gradients. Massmann and
Farrier16 showed that the Stefan–Maxwell equations gave
erroneous results relative to the dusty gas model for low
permeability soils, and for systems with pressure gradients.
Pressure gradients frequently develop due to variations in
atmospheric pressures as well as a result of remediation
processes such as soil vacuum extraction.
There is a need to incorporate the dusty gas model into a
comprehensive simulator to determine the situations where
Fick’s law is inadequate for gas-phase diffusion. Of particular importance are the predicted rates of spread of organic
vapors, and the rates of diffusion of vapors to the atmosphere at the ground surface, and the predicted rates of disappearance of volatile organic compounds immobilized in
the vadose zone, particularly in fine grained soils.
In this article the formulation of the dusty gas model is
reviewed, and the incorporation of the model into the compositional simulator previously presented by Sleep and
Sykes20 is described. Example simulations of multiphase
multicomponent systems are presented to highlight the
implications of using either the dusty gas model or Fick’s
law for gas-phase diffusion.
2 THE DUSTY GAS LAW MODEL FOR GASEOUS
DIFFUSION IN POROUS MEDIA
the linear adsorption coefficient, r b is the bulk soil density,
r b is the phase molar density, x i b is the species mole fraction, q b is the Darcy velocity vector, J i b is the mechanical
dispersion flux, N i b is the molecular diffusion flux, r i b
represents interphase transfer of species i to, or from,
phase b, and G i b represents sinks and sources of species i
to phase b due to injection or pumping, or biological or
chemical transformations. An overall molar balance for
species i is derived by summing eqn (1) over the water,
gas and organic phases. In this case, the interphase transfer
terms, r i b, for the different phases sum to zero20. As
described in Sleep and Sykes20, the model is based on
assumptions of equilibrium interphase mass transfer and
ideal phase behavior described by Henry’s and Raoult’s
laws.
The Darcy velocity vector for phase b is given by
qb ¼ ¹
þ [=·fSb (Jib þ Nib )] ¹ Gib ¹ rib ¼ 0
Jib ¼ ¹ rb Dmech
ib ·=xib
where f is the porosity, S b is the phase saturation, Kib,
(3)
22
as
The mechanical dispersion tensor, Dmech
ib , is defined
n
n
bk bl
(4)
Dmech
ibkl ¼ aT lnb ldkl þ (aL ¹ aT )
lnb l
where a L and a T are the longitudinal and transverse dispersivities, respectively, n b k and n b l are the components of
the linear porewater velocity, and d kl is the Kronecker delta.
Dispersivities are assumed to be independent of phase
saturation.
The diffusive molar flux for the water phase is given by
Fick’s law:
Niw ¼ ¹ rw tw Diw ·=xiw
(5)
where t w is a tortuosity factor and D i w is the aqueous-phase
molecular diffusion coefficient for species i.
The diffusive molar fluxes for the gas phase, in the presence of a gravitational field, are given by the dusty gas
model22:
nc
X
j ¼ 1, jÞi
¼
ð1Þ
(2)
where k is the intrinsic permeability tensor, k r b is the relative permeability of phase b, m b is the viscosity, p b is the
pressure of phase b, g is the gravitational acceleration, g b is
the mass density, and z is the elevation.
It should be noted that the assumption that Darcy’s law
gives a molar-averaged velocity is consistent with developments of the theory of flow and diffusion of gases in porous
media in the chemical engineering literature 7,15.
The mechanical dispersive molar flux, J i b, for the water
phase is given by
The macroscopically averaged equation for the movement
of a component i in a phase b20 is
]
[(fSb þ Kib, d rb )rb xib ] þ =·(rb xib qb )
]t
kkrb
(=pb þ gb g=z)
mb
xig Njg ¹ xjg Nig
Nig
¹
tg Dijg
tg DK
ig
1
(p =x þ xig =pg þ xig rg g=z)
RT g ig
where D ijg is the binary gas-phase diffusion coefficient for
d
is
Organic vapor transport in porous media
components i and j, DK
ig is the Knudsen coefficient for
component i, R is the gas constant, and T is the temperature
(K). Knudsen diffusion becomes important when the pore
diameters are small enough, or pressures are low enough,
that the gas molecules collide primarily with the pore walls
rather than with other molecules. From eqn (6) it can be
seen that Knudsen diffusion becomes important when DK
ig is
much smaller than D ijg. In this case the first term in eqn (6)
becomes negligible and the diffusive flux is controlled by
the second term. The Stefan–Maxwell equations are a special case of eqn (6) produced by neglecting the pressure
gradient term, =p g, and the Knudsen diffusion term,
21
Nig =DK
discuss a number of
ig . Thorstenson and Pollock
simplifications of eqn (6) such as the pure molecular diffusion regimes, the pure Knudsen regime, and the simplified
equations for binary systems.
In the present model gas-phase diffusion coefficients are
assumed to be a function of pressure and temperature
according to the Fuller, Schettler and Giddings method19:
MA þ MB 1=2
10 ¹ 3 T 1:75
MA MB
(7)
DAB ¼ h X
i2
X
1=3
P ( n)A þ ( n)B1=3
where T is the temperature (K), P is the pressure (atm), M A
and M B are the molecular weights of A and B, and the on
terms are the diffusion volumes determined from the structure of the molecules19.
The Knudsen coefficient is related to the gas molecular
weight and the pore size of the soil. Geankopolis9 recommended the relationship
s
T
3
K
(8)
Dig ¼ 9:7 3 10 r
Mi
where r is the pore radius (cm), M i is the molecular weight
2 ¹1
(g/gmol), and DK
ig in cm s .
Thorstenson and Pollock21 discussed the relationship
between the Knudsen diffusion coefficient and the
Klinkenberg parameter. They derived the relationship
DK
ig ¼
kbi
mig
(9)
where b i is the Klinkenberg parameter that is a characteristic of the soil. For air, Thorstenson and Pollock21 presented the relationship of Heid et al.12:
bair ¼ 0:11k ¹ 0:39
(10)
2
with b air in Pa and k in m . The b values for other components can be determined from
s
mi Mair
(11)
bi ¼ bair
mair Mi
It should be noted that in eqn (6) it has been assumed that
the same tortuosity factor can be used for molecular diffusion and Knudsen diffusion. This may not be appropriate
since the two processes are different phenomena. In
249
particular, the effect of increasing water saturation on tortuosity factors for Knudsen diffusion is unknown. For the
present study the Millington and Quirk18 relationship has
been used to calculate tortuosities.
The discretized form of eqn (6) may be written as
np
X
Vk
(rb xib Sb f)n þ 1 ¹ (rb xib Sb f)n
Dt
b¼1
np
X
m
X
¼
(rb xib qb )l, k þ (fSb Jib )l, k þ (fSb Nib )l,
l ¼ 1, lÞk b ¼ 1
k
where m is the maximum number of neighboring blocks for
block k, and the subscript l, k on the flux terms represents
the fluxes between blocks l and k, evaluated at the interface
between l and k. The calculation of advective, dispersive
and Fickian diffusive fluxes in the simulator is outlined in
Sleep and Sykes20.
To apply the dusty gas model, eqn (6) is used to calculate
the diffusive gas-phase fluxes between neighboring blocks,
(N a b) l , k. For a system of n c components eqn (6) may be
written as a matrix equation16:
AF ¼ B
(13)
where A is a matrix with the terms
nc
X
Aii ¼ ¹
j ¼ 1, jÞi
nc
X
Aij ¼ ¹
j ¼ 1, jÞi
xjg
1
¹
tg Dijg DK
ig
(14)
xig
tg Dijg
(15)
The x ig terms in eqns (14) and (15) represent the gas-phase
mole fractions of species i at the interface between blocks.
Any standard spatial weighting technique such as midpoint
weighting or upstream weighting may be used to calculate
the interface value of x jg (see Sleep and Sykes20 for a discussion of various weighting techniques). In the present
model formulation, midpoint weighting is used. The F
matrix is the vector of molar fluxes between neighboring
blocks:
Fi ¼ (Nig )l,
k
(16)
The B matrix is derived from the right-hand side of eqn (6):
1
(p =x þ xig =pg þ xig rg g=z)
(17)
RT g ig
The gradient terms in eqn (17) are calculated using central
differences. For example, for neighboring blocks k and l the
pressure gradient in the x-direction (x l . x k) is calculated
from
Bi ¼
=pg ¼
2(plg ¹ pkg )
Dxl þ Dxk
(18)
The mole fractions and pressures in eqn (14) may be
evaluated either explicitly or implicitly. If they are evaluated implicitly in a simulator that uses Newton–Raphson
iteration (as is done in this paper), then it is necessary to
250
B. E. Sleep
determine the derivatives of each of the fluxes with respect
to mole fractions and pressures. Once eqn (14) is solved for
the fluxes, using the mole fractions and pressures from the
previous Newton iteration, the derivatives may be calculated from the formula
A
]F ]B ]A
¼
¹ F
]y ]y ]y
(19)
where y is a mole fraction, a pressure, or other state variable that mole fractions, densities, diffusion coefficients or
tortuosities may depend upon. If an LU decomposition is
used to solve eqn (14), then ]F
]y may be found by calculating the right-hand side of eqn (19) and then solving using
forward and backward substitution. Eqn (19) must be
solved for each primary variable to generate all the required
derivatives.
3 MODEL APPLICATIONS
3.1 One-dimensional simulations
A series of simulations of volatilization and subsequent
movement of pentane and hexane in a one-dimensional column of dry soil was performed in order to examine the
differences between predictions of the dusty gas model
and predictions based on Fick’s law. The properties of pentane and hexane are summarized in Table 1. Soil permeabilities of 10 ¹11, 10 ¹13 and 10 ¹16 m 2 were used in the
simulations. The temperature for simulations was 238C
and air was considered as a single species with the properties of nitrogen gas. The corresponding Knudsen coefficients for air, calculated from eqn (10), are 1.05 3 10 ¹2,
6.35 3 10 ¹4 and 9.39 3 10 ¹6 m 2 s ¹1, respectively. In each
simulation a horizontal domain of length 30 m (150 finite
difference blocks of 0.2 m) was used. A liquid mixture of
50 mol% pentane and 50 mol% hexane was injected into the
end of the column for a period of 3 min at a rate of 1.0 3
10 ¹4 m 3 (m 2 s) ¹1. The organic liquid saturation reached a
maximum of 0.21, just slightly greater than the residual
saturation, so the NAPL phase did not move more than
0.4 m (two model cells) in the simulation. Although the
model includes adsorption, adsorption of organic vapors to
the soil was not included in the simulations.
The results of the one-dimensional simulations in dry soil
are shown in Figs 1 and 2. The organic was injected at the
end of the column, corresponding to a horizontal distance of
30 m on Figs 1 and 2. The changes in gas-phase concentrations at this end of the column, where the organic is present
at residual saturation, are reflective of changes in the composition of the organic phase. It may be noted that, as time
proceeds, the concentrations of pentane at the end of the
column decrease while the concentrations of hexane
increase. This is due to the preferential volatilization of
pentane from the residual organic phase, leading to enrichment of the organic phase in hexane.
The differences between the predictions using Fick’s law
for diffusion and predictions using the dusty gas model are
small for a permeability of 10 ¹11 m 2. The difference at 710
days is slightly greater for hexane than it is for pentane since
the binary diffusion coefficients used for each organic compound are the organic–air coefficients. The concentrations
of pentane in air are high due to the higher vapor pressure of
pentane. The binary diffusion coefficient for a hexane–pentane mixture, calculated from eqn (7), is 3.5 3 10 ¹6 m 2 s ¹1,
which is lower than the value of 7.6 3 10 ¹6 m 2 s ¹1 for a
hexane–air mixture. In other words, the high pentane concentrations hinder the diffusion of hexane, a phenomenon
not accounted for when Fick’s law is applied as though each
of the gases were diffusing independently through the air.
With a soil permeability of 10 ¹11 m 2, the ratios of diffusive fluxes to advective fluxes varied with time and location
in the column, but the diffusive fluxes always exceeded the
advective fluxes, except at an early time near the point of
injection of the liquid organic. Typically, at the midpoint of
the column the diffusive fluxes exceeded the advective
fluxes by a factor of four or more. To determine whether
the small difference between the dusty gas model and
Fickian model predictions was due to advective effects,
additional simulations (not shown) were conducted in
which air was allowed to move by advection, but advection
of hexane and pentane was set to zero. The results of these
simulations were very similar to those plotted in Fig. 1a
and 2a, in which advection of hexane and pentane was
included. This indicates that the similarities between predictions of the dusty gas and Fickian models for the soil with a
permeability of 10 ¹11 m 2 are not due to the effects of advection dominating diffusion. Thus, the Fickian model may be
Table 1. Chemical properties
Species
Water
Air
Pentane
Hexane
Benzene
Toluene
Molecular weight
Liquid density (kg m ¹3)
Liquid viscosity (kg (m s) ¹1) 3 10 ¹3)
Aqueous solubility (mg l ¹1)
Vapor viscosity (kg (m s) ¹1 3 10 ¹6)
Diffusion volume
Vapor pressure (Pa)
Air-liquid interfacial tension (dyne cm ¹1)
18.0
1000.0
1.0
29.0
42.0
72.15
626.2
0.240
40
6.0
106.5
58 250
18.43
86.18
660.3
0.326
10
6.5
127.0
16 020
18.43
78.11
876.5
0.652
1780
7.4
91.0
10 130
28.85
92.14
866.9
0.590
515
6.7
111.5
2930
28.5
19
Data from Reid et al. , Lide
13
12.7
72.75
and Mackay et al.14.
18.50
20.1
Organic vapor transport in porous media
Fig. 1. Pentane diffusion in a one-dimensional system. Pentane
concentration profiles for permeabilities of (a) 10 ¹11 m 2, (b)
10 ¹13 m 2, (c) 10 ¹16 m 2.
251
Fig. 2. Hexane diffusion in a one-dimensional system. Hexane
concentration profiles for or permeabilities of (a) 10 ¹11 m 2, (b)
10 ¹13 m 2, (c) 10 ¹16 m 2.
252
B. E. Sleep
Fig. 3. Pentane diffusion in soil with permeability of 10 ¹11 m 2,
water saturation of 0.9.
Fig. 4. System configuration for two-dimensional examples.
adequate for simulating transport of organic vapors such as
pentane and hexane in dry soils with intrinsic permeabilities
of 10 ¹11 m 2 or greater.
For a permeability of 10 ¹13 m 2, the differences between
Fick’s law and dusty gas model predictions are significant,
as shown in Fig. 1b and 2b. In particular, the concentration
of pentane at the end of the column containing the organic
liquid is lower for the Fick’s law case, due to the greater
rates of pentane diffusion predicted by Fick’s law. Thus, at a
permeability of 10 ¹13 m 2, the effects of collisions with pore
walls have an impact on diffusion rates. The Knudsen diffusion coefficients for pentane and hexane for a permeability
of 10 ¹13 m 2 predicted by eqns (9)–(11) are 1.31 3 10 ¹4 and
1.29 3 10 ¹4 m 2 s ¹1 respectively, whereas the respective
air–organic diffusion coefficients are 8.21 3 10 ¹5 and
7.43 3 10 ¹5 m 2 s ¹1. Thus, since the molecular diffusion
coefficients and the Knudsen diffusion coefficients are of
the same order of magnitude, this permeability is in the
transition regime between molecular diffusion and Knudsen
diffusion.
At a permeability of 10 ¹16 m 2 there is a great difference
between the predictions of the dusty gas model and Fick’s
law, as shown in Fig. 1c and 2c. The concentration of pentane left in the organic liquid source is 50% lower at 710
days for Fick’s law than for the dusty gas model prediction.
Hexane concentrations are much greater for the Fick’s law
case caused by the greater preferential stripping of the more
volatile pentane caused by the higher Fick’s law flux rates.
The concentrations are also plotted for a case in which the
Knudsen diffusion coefficients were set to very large values
to make the Knudsen effect insignificant. In this case, the
predicted flux rates are only slightly lower than those predicted by Fick’s law, because of multicomponent effects.
Thus, the major impact of the reduced soil permeability on
vapor diffusion is the result of Knudsen diffusion and not
multicomponent effects. It may be possible, in cases where
pressure gradient effects are not important, to model vapor
diffusion with a Fickian model that incorporates Knudsen
diffusion. This model could be developed from eqn (6).
An additional simulation of pentane and hexane volatilization and transport was conducted in the 10 ¹11 m 2 permeability soil at a water saturation of 0.9. In the absence of
information about the effect of water saturation on Knudsen
coefficients, it was assumed that the Knudsen coefficients
were related to the soil’s intrinsic permeability through eqn
(9), and that the effect of increased water saturation was
accounted for by the reduced tortuosity calculated from
Fig. 5. Benzene distribution (mol%) in gas phase for passive conditions after 250 days, predicted by dusty gas model.
Organic vapor transport in porous media
253
Fig. 6. Benzene distribution (mol%) in gas phase for passive conditions after 250 days, predicted by Fick’s law.
the Millington–Quirk expression18. The same tortuosity
factor was used for both molecular and Knudsen diffusion,
so that the effective Knudsen coefficient was smaller than
the effective molecular diffusion coefficient by the same
factor as in the dry 10 ¹11 m 2 permeability soil. Thus, with
these assumptions, on the basis of the results of the dry
10 ¹11 m 2 permeability soil, Knudsen diffusion would not
were significant relative to molecular diffusion in this
high water saturation simulation.
The results of the simulations for the dusty gas and Fickian models at 140 and 7100 days are shown in Fig. 3. The
diffusion rates predicted by both Fick’s law and the dusty
gas model are much lower than for the dry soil case caused
by the small tortuosity factor associated with a water saturation of 0.9. In this case of high water saturation, where
Knudsen diffusion is assumed to be insignificant, the
dusty gas model predicts greater flux rates than the Fickian
model, as indicated by the lower pentane concentrations at
7100 days at the end of the column where the liquid organic
mixture was injected. The injection of the liquid mixture
and its subsequent volatilization gives rise to a pressure
gradient in the column. As seen from eqn (6), diffusion
fluxes are proportional to pressure gradients as well as concentration gradients in the dusty gas model, whereas they
are assumed to be proportional only to concentration gradients in Fick’s law (see Thorstenson and Pollock21 for a
variety of different simplified forms of the dusty gas
model that illustrate clearly the dependencies on pressure
gradients and the differences between Fick’s law and the
dusty gas model).
If the tortuosities for Knudsen diffusion (and thus the
effective Knudsen diffusion coefficients) decreased more
quickly than tortuosities for molecular diffusion as water
saturation increased, then at high water saturations the
Knudsen effect might become more important. In this case
the dusty gas model fluxes could be less than those predicted
by Fick’s law. However, as water saturation increases the
smaller pores are filled first and the average pore radius of
gas-filled pores, calculated on a volume basis, increases, so
that the Knudsen coefficient might be expected to increase
with increasing water saturation unless there are also substantial increases in the thickness of the water film adsorbed
to the walls of the unsaturated pores. This is clearly an area
that needs some careful experimental study.
3.2 Two-dimensional soil venting
In low permeability soils pressure diffusion and Knudsen
diffusion may be significant even under forced advection
conditions such as soil vacuum extraction (SVE). Simulations of the fate of a mixture of benzene and toluene in a
Table 2. Soil properties
Soil type Permeabi- Porosity
lity(m 2)
1
2
10 ¹13
10 ¹15
0.4
0.4
Brooks
Coreyl
Brooks S wr S or
Coreyp d
(m water)
3.0
2.0
0.1
0.4
0.1 0.2
0.2 0.2
Fig. 7. Mass removal of benzene and toluene predicted by the
dusty gas model and by Fick’s law for passive conditions.
254
B. E. Sleep
Fig. 8. Benzene distribution (mol%) in gas phase for soil venting conditions after 1000 days, predicted by dusty gas model.
layered soil system under passive conditions and under
vacuum extraction were performed to examine the differences in predictions of the Fick-based model and the dusty
gas model. The system configuration is shown in Fig. 4, and
soil properties are given in Table 2. The top and bottom
layers of soil have properties typical of a fine sand, while
the middle layer has properties typical of a silty clay. 172.8 l
of an organic mixture of 0.5 mole fraction benzene and
0.5 mole fraction toluene were injected into the two-dimensional system (1 m thick in the third dimension) at a point
2 m below the ground surface over a period of 10 days. The
organic phase subsequently was allowed to redistribute, dissolve and volatilize. Due to the small volume of organic
mixture released, the organic phase did not imbibe into
the silty clay layer. At 300 days in the SVE case, gas was
injected at the air injection point and extracted from the air
extraction point (see Fig. 4) by prescribing constant node
gas-phase pressures of 4.0 and ¹ 4.0 m water, respectively.
The predicted vapor-phase benzene concentration profiles after 250 days for the dusty gas model and for the
Fickian model are shown in Figs 5 and 6, respectively.
The extent of movement of benzene into the low permeability layer is very small for the dusty gas model compared to
the Fickian model. In Fig. 7 the predicted rates of dissipation of benzene and toluene concentrations in the system
due to diffusion to the atmosphere under passive conditions
are plotted for the two models. The dusty gas model predicts
that more than 5000 days are required for the amount of
benzene remaining to reach less than 1 mole, while the
Fickian model predicts that this level will be reached after
about 3000 days. The predicted rate of dissipation of toluene
is also greater for the Fickian model than for the dusty gas
model.
The vapor-phase benzene concentration profiles with soil
vacuum extraction after 1000 days (700 days of soil
vacuum extraction) for the dusty gas model and for the
Fickian model are shown in Figs 8 and 9 respectively.
There is still considerable benzene left in the low
permeability layer in the dusty gas model simulation,
compared to the Fickian simulation. In Fig. 10 the
predicted rates of dissipation of benzene and toluene concentrations in the system with soil vacuum extraction are
Fig. 9. Benzene distribution (mol%) in gas phase for soil venting conditions after 1000 days, predicted by Fick’s law.
Organic vapor transport in porous media
255
tion. However, the effect of moisture content on Knudsen
coefficients requires more study.
The results of two-dimensional simulations of passive
movement and soil vacuum extraction of benzene and
toluene demonstrated that removal rates predicted by the
Fickian model are much greater than those predicted by
the dusty gas model. Removal rates were limited by the
slow rates of diffusion of the organic vapors through the
low permeability soil layer.
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Fig. 10. Mass removal of benzene and toluene predicted by the
dusty gas model and by Fick’s law for soil venting conditions.
plotted for the two models. The Fickian model predicts
much more rapid removal of benzene and toluene than
does the dusty gas model. In the dusty gas model there is
an initial phase of rapid decline in the amount of benzene
and toluene in the system as these compounds are removed
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4 SUMMARY
The dusty gas model for gas-phase diffusion was incorporated into a multiphase, multicomponent compositional
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