῏
J. Jpn. Soc. Soil Phys.
῏῍ῌῐῑ῎
No. 3., p. -ῌ3 ῍,**-῎Review of Recent Insights on Physical Behavior of Gases in
Porous Media : Theoretical and experimental Investigations
on Density Driven Gas Flow
Dennis E. ROLSTON* and Andrew S. ALTEVOGT*** Land, Air and Water Resources, University of California, Davis** Civil & Environmental Engineering, Princeton University
Abstract
Under natural conditions, forces such as gravity, temperature, and pressure gradients in the soil
coupled with di#usive ﬂuxes will control the fate of gaseous contaminants. The assumptions
underlying the basic equations of di#usive and advective gas transport processes in soil are
discussed.

To test the traditional equations for gas transport, laboratory experiments wereconducted to explore the transport of a dense gas (freon-++-) through columns of air-dry Oso-Flacosand with a large concentration of freon-++- maintained at the inlet to the columns. Gas densities(concentrations) were monitored at the inlet and the outlet and within the columns duringtransport.

Signiﬁcant di#erences in ﬂuxes and density proﬁles were observed for the three

primary ﬂow directions (horizontal, vertically upward, vertically downward) at high source
densities. Numerical models based on the standard Darcy-Fickian transport equation did not ﬁt
the measured ﬂuxes. Slip ﬂow was found to be signiﬁcant relative to Darcy advective ﬂow, but
did not account for the discrepancy between model simulations and data.

Further theory

development was necessary in order to ascertain why the standard equations did not adequately
describe the di#usive and advective transport processes for dense gases. New equations governing the transport of gaseous chemicals in porous media were derived by applying the method ofvolume averaging to the point equations for mass and momentum ﬂux. The form of the newtransport parameters provide possible explanations for discrepancies between experimental andnumerical modeling results for systems where neither di#usive nor advective driving forces
dominate.
Key words : dense gas, gaseous contaminant, slip ﬂow, di#usive and advective gas transport,
volume averaging techniqueIntroduction
An understanding of the transport of gasphase chemicals in porous media is importantin determining the fate of many soil and vadosezone gases and contaminants. Numerous physical and chemical properties and the way theseproperties interact and vary through time andspace will control gas-phase contaminant transport. Gas-phase ﬂuxes in the vadose zone will
generally be controlled by some combinationof driving forces due to gravity and gradientsin pressure, temperature, and concentration.
Physical and chemical parameters such as permeability, porosity, di#usion, and sorption coe$cients will exert control over the magnitudeof the ﬂuxes of interest.
Modeling of gas transport in situations ofenvironmental concern has generally been accomplished by utilizing equations that havebeen successfully employed to describe liquidtransport processes. Liquids and gases will, for

ῐ

4



example, behave di#erently in their interactions with the solid-phase in porous media.
This becomes quite clear in examining the assumptions which underlie Darcy’s Law, thestandard liquid phase advective velocity expression. Darcy’s Law is based on the assumption that velocity at the pore walls is equal tozero. In the gas phase, however, if transport
occurs in the region where neither advection
nor di#usion is dominant, this assumption will
be invalid. The form of the advective velocity
expression may need to be altered to accountfor the non-zero velocity at the pore walls (slipvelocity). The slip velocity accounts for only
one of a number of factors that must be examined in order to arrive at a thorough understanding of the physics of gas transport inporous media.
The objectives of this study were to examine
the validity of the traditional gas-phase transport equations for porous media and to explore
any modiﬁcations that must be made to theequations based on an in-depth examination of
the assumptions which underlie the traditional
equations. Transport experiments were conducted and compared with the output of anumerical model based on traditional and newtransport equations.
Physics of Gas Transport
Di#usion velocities of gas mixtures in soil
are related to each other in a complex mannerdependent upon the mole fraction of each gas,
the molar ﬂuxes of each gas, and the binary
di#usion coe$cient of each gas pair. Curtissand Hirschfelder (+3.3) gave the general equations for steady transport of a multicomponent
mixture of gases, based on gas kinetic theory.
If gravity e#ects are ignored or di#usion
occurs only horizontally (and di#usion occurs
for vertical distances no greater than +** m),
the well-known Stefan-Maxwell (SM) equations provide the theoretical framework for
di#usion of gases in soils. The SM equations(Amali and Rolston, +33-) are :3.  ,**-v

ῌ

yi Jjyj Ji
P dyj

RT dx
Dij

+

j+, jiwhere Jj is the molar ﬂux density [moles/(soilarea x time)] for species j, yj is the vapor mole
fraction for species j, Dij is the e#ective binary
di#usion coe$cient in soil of species i in j (orvice versa), P is the total gas mixture pressure(Pa), T is the mixture temperature (K), R is theuniversal gas constant (2.-+. J mol+ K+), andx is the spatial coordinate.
Fick’s law for di#usion is a restrictive case of
the SM equations. Jaynes and Rogowski (+32-)
concluded that Fick’s law would result in incorrect estimations of mass distributions andmass ﬂuxes for systems containing more than
two vapor components when the concentrations of the di#using species are not negligible
compared to the bulk phase gas. Amali and
Rolston (+33-) showed that if the total mole
fraction of volatile organic chemical (VOC)
vapors in air exceeded about *.*/, Fick’s law
ﬂuxes deviated from the true di#usive ﬂux foreach species by up to /ῌ. This indicated thatthe total mole fraction of the di#using species
also needs to be considered in addition to therequirement of Le#elaar (+321) that the binary
di#usion coe$cients of the di#using species
must be su$ciently di#erent for multicomponent e#ects to be observed.
Thus, Fick’s law is generally applicable for
only a few special cases. One of these cases isfor the di#usion of a trace gas in a binary
mixture, meaning that the mole fraction of the
tracer gas is small. The second special case isfor di#usion of two gases in a closed system(total pressure remains constant). In this case,neither gas needs to be in trace amounts.
Therefore, if Fick’s law is to be used as thebasis for any method of measuring the soil gas
di#usivity, the method should satisfy one ormore of these conditions.
Assuming that the special case conditions
are met, Fick’s law is given by
Mg(Cg
fgDp
,At(x
where Mg is the amount of gas di#using (g gas),῔῕ : Review of Recent Insights on Physical Behavior of Gases in Porous Media : Theoretical and experimental Investigations on Density Driven Gas Flow

A is the cross-sectional area of the soil (m, soil),t is time (s), fg is the gas ﬂux density (g gas mῑ,soil sῑ+), Cg is concentration in the gaseous
phase (g gas mῑ- soil air), x is distance (m soil),and Dp is the soil-gas di#usion coe$cient (msoil air mῑ+ soil sῑ+). The unsteady di#usion ofa gas, which is nonreactive (physically, biologically, and chemically), is described by the
combination of Fick’s law (Eq. [,]) and the continuity equationfation (m/s,), and a is a factor in the slip velocity
expression. Adsorption of the gas can be described by a Langmuir isotherm. The Darcy
velocity term takes into account an advective
component due to di#erences in gas density.
The slip velocity term arises from the nonequimolar ﬂux for gases that are not in trace
amounts and have very di#erent di#usion co-

῎-῏

where fa is the soil-air content (m- air mῑ- soil).
For cases where external forcing occurs
causing pressure gradients or driving forces
due to density di#erences, advective processes
need to be considered in addition to the di#usive processes given by Eq. [,] and [-]. The
traditional equations for di#usive and advective gas-phase mass and momentum conservation in a binary system can be represented, in
one dimension as (Altevogt et al., ,**- a)fuῐrbulkphase pressure (Pa), g is gravitational accelera-

e$cients.

(,Cg
(Cg
ΐDp ,
(t
(x

Mass :

5aQ
ῌ+ῐarF῍,

(rF
(
ΐ
(t
(z

ῌ῏῏῏ῌ
῍῍῏῏῏ῌ῍῎Materials and Methods
Experiments were conducted in an apparatus consisting of a stainless steel cylinder
packed with Oso-Flaco sand with large chambers on either end. The volumes of the inlet
and outlet chambers were -+ and +0.- L, respectively, and were constructed of --mm thickstainless steel. The soil cylinder was *..* mlong with an I. D. of 11 mm. Freon-++- (+,+,,
Trichloro- +,,,, triﬂouroethane) was utilized as
the tracer gas due to its relatively high molecular weight and vapor pressure. Table + presents the physical and chemical properties ofRctardation Coc$cicnt

rDP

(wF
(z

ῑ

ῌ῏ῌ
῍῍῏ῌ
῍῎

(
῎rF vz῏(z ῌ῍῎῎.῏Table +

Advcctive Flux

Physical and chemical characteristics ofthe experimental system

Fickian Di#usivc Flux

Momentum : vzΐῑ

Kz
m

dwF
dz
dP
ῐrg ῐ
῎/῏
dz
wFῐa

ῌ῏῏῍῏῏῎Darcy Vclocity

Dp

ῌῌ
῍῍ῌ
῍῎
Slip Vclocity

where rbulk is the bulk density of the soil (kgsoil/m- soil), a and Q are measured adsorption/
desorption rate constants for the Langmuir isotherm (m- gas/kg freon and kg freon/kg soil,

Packed Column Properties
Bulk Density
Permeability

Oso-Flaco Sand Properties
Mineral Density
Mean Pore Diameter

Freon ++-

is mass fraction of the tracer gas (kg freon/kg
gas), vz is total advective velocity (m- gas mῑ,“Air”freon/m gas), t is time (s), z is distance (m soil),-r is total gas phase density (kg gas/m- gas), wF

soil sῑ+), m is mixture viscosity (Pa s), Kz ispermeability (m- gas/m soil), P is total gas,40+ g soil/m- soil particles/4-ῒ+*ῑ/ m gas

Chemical Properties

Molecular Weight
Vapor Pressue
Viscosity
Binary Soil Gas
Di#usion Coefficient

respectively), rF is density of the tracer gas (kg+4/1 g soil/m- soil
,41.ῒ+*ῑ++ m- gas/m soil

Molecular Weight
Viscosity+214-3 g/mol*4.. atm (.4./ῒ+*. Pa)+4*-ῒ+*ῑ/ Pa s
+4-ῒ+*ῑ0 m- gas mῑ+ soil sῑ+,242+ g/mol+420ῒ+*ῑ/ Pa s



6

the experimental system. All parameters were
measured independently.
Experiments were run in a ,/ constanttemperature room. Experiments were initiatedby injecting a measured amount of liquid freon
into the vented inlet chamber and allowing it
to vaporize. A plunger connecting the chamber to the soil cylinder was kept closed until
the freon had fully vaporized and its concentration had stabilized. The plunger was thenpulled and freon was allowed to enter the soil.
During transport experiments, gas samples
were taken with /* mL gas sampling syringes(Hamilton) through mininert (VICI Inc.) ports
located along the soil cylinder and in the inlet
and outlet chambers and analyzed on a gas
chromatograph. Experiments were conductedin three separate conﬁgurations : verticallyupward (inlet at the bottom of the system),vertically downward (inlet at the top), andhorizontally. For more information on the experiments, see Altevogt et al. (,**- a, ,**- c).
Results and Discussion
Comparison of Numerical Modeling Output
with Experimental Data
Figure + presents model output plotted withmeasured data for a vertically downward transport experiment conducted at an approximate
inlet freon-++- density of +.20 mg/mL. Figure +a contains the results for ﬂux at the soil

Fig. +

3.

,**-

column outlet. Fig. +b compares the measured
freon density proﬁles at / and ,/ cm from theinlet chamber with model output. In bothﬁgures, it is clear that the traditional transport
equations (Eqs. [.] and [/]) are far from adequate for describing experimental data. It
should be noted that many previous studies
might have missed this result due to the use of
ﬁtted dispersion coe$cients. It should be
noted that these equations do not include amechanical dispersion term. The highest velocity encountered in all of our experiments
was 0.,+*. m/s. Utilizing this value alongwith the mean pore diameter (estimated frommean particle size) and di#usion coe$cientfrom Table + yields a Peclet number of *.*,/.
This is the maximum value of the Peclet
number that was encountered, and it lies well
below the criteria value of + (Aris +3/0, Bear
+322, Auriault and Adler +33/). This means thatmechanical dispersion will be negligible for allof the ﬂow regimes of interest in this study.
The slip velocity was accounted for only byutilizing the expression in Eqs. [.] and [/].
Volume Averaged Transport Equations
Based on the poor ﬁt of the traditional equations to the experimental data, additional
theoretical analysis was conducted. The fully
coupled equations derived in the volume
averaging work of Altevogt et al. (,**- b) canbe represented as

Comparison of data and numerical models for vertically downward data. Flux to the outlet
chamber (a) and density proﬁles (b). q*.0/ for new model output.

῔῕ : Review of Recent Insights on Physical Behavior of Gases in Porous Media : Theoretical and experimental Investigations on Density Driven Gas Flow

Mass :

fuῐrbulkaQ
ῌ+ῐarF῍,

(rF
(
ῒ
(t
(z

῏ῒῒῒ῍
῎ῐῒῒῒ῍῎ῑ
Rctardation Coc$cicnt

rDP

(wF
(z

ῑ

῏ῒ῍
῎ῐῒ῍
῎ῑ
Fickian Di#usivc Flux((῎rF vz῏ῐ(z ῏ῐῑ (zTraditional
Advcctive Flux

Couplcd
ClosureVariablc
Estimatc

῎0῏

ῌῌ῍ῌ῎

qrF vz῏ῒῐῒῑ
Newly Dcrivcd
Slip Coupling
Tcrm

Momentum : vzῒῑ

Kz
m

dwF
dz
dP
ῐrg ῐ
ῐ
dz
wFῐa

῏ῒῒῐῒῒῑ
Darcy Vclocity

Dp

῏῍
῎ῐ῍
῎ῑ
Slip Vclocity

ῌ rbulkaQ ῎ῌ wFῑ+ ῎ (rF
῍ῌ+ῐarF῍,῏῍rFῐra῏ (t῎1῏῏ῒῒῒῒῒ῍
῎ῐῒῒῒῒῒ῍
῎ῑ
Sorptive Vclocity

The term q appears in Eq. [0] as a result of theorder of magnitude estimate (Altevogt et al.,,**- b) that the slip coupling term will be a
linear function of the species density within
the range *.+ΐqΐ+*. Di#erences between Eqs.
[.] and [/] and Eqs. [0] and [1] lie in the reduction in magnitude of the advective ﬂux and theinclusion of a sorptive contribution to the velocity, both of which arise directly from microscale coupling of the mass and momentum
equations. Equations were solved using a onedimensional ﬁnite di#erence numerical technique.
The newly derived mass and momentum
equations ﬁt the data remarkably well with aﬁtted q value of *.0/. This value of qῒ*.0/ wasequally applicable to cases of upward and horizontal transport (not shown). The model
output from the new transport equations withqῒ*.0/ provides a very good match to the data.
This corresponds well with our estimate that q
should be of the order *.+ΐqΐ+*. In order toobtain an exact expression for the slip coupling
term in Eq. [0], it will be necessary to fully

7

solve the coupled closure equations presented
in Altevogt et al. (,**- b).
It must be noted that due to the high molecular mass of freon-++- relative to that of “air”,
the value of a will be minimized and the e#ects
of slip magniﬁed. For species with smaller
di#erences in molecular mass, the value of theslip velocity will be greatly reduced. Secondly,
freon-++- adsorbs weakly to the Oso-Flaco sandused in these studies as compared with otherorganic chemical/porous media combinations.
Thus, the values of sorptive velocity will undoubtedly be higher in many situations.
Summary
Comparison of simulations using the traditional transport equations with data fromcolumn experiments using a dense gas (freon++-) clearly show that these equations are notadequate for describing transport of volatile
chemicals in the region where neither di#usion
nor advective processes dominate. Sensitive
measurement of pressure within the soilcolumn (Altevogt et al., ,**- a) showed that theinteraction between the non-equimolar di#usive ﬂux and advection due to gravity produces signiﬁcant pressure gradients within thesoil that di#er depending on ﬂow direction.
Newly derived transport equations, using
volume averaging techniques, gave additionalterms arising directly from microscale coupling of the mass and momentum equations.
Simulations using the newly derived equations
compared remarkably well with data. Further
research is needed on the coupling term toevaluate the physical signiﬁcance and degreeof transferability to other systems.
References
Altevogt, A.S., D.E. Rolston, and R.T. Venterea.
(,**- a) : Density and pressure e#ects on thetransport of gas-phase chemicals in unsaturated porous media. Water Resour. Res., -3 (-).
Available from : doi : +*.+*,3/,**,WR**+--2.
Altevogt, A.S., D.E. Rolston, and S. Whitaker. (,**b) : New equations for binary gas transport inporous media, Part + : equation development.

*5Cº » 3. ¼ j,**-m

8Adv. Water Resour. ,0 : 03/ῌ1+/.
Altevogt, A.S., D.E. Rolston, and S. Whitaker. (,**c) : New equations for binary gas transport inporous media, Part , : experimental validation.
Adv. Water Resour. ,0 : 1+1ῌ1,-.
Amali, S., and D.E. Rolston. (+33-) : Theoretical investigation of multicomponent volatile organic vapor di#usion : Steady-state ﬂuxes. J. Environ. Qual. ,, : 2,/ῌ2-+.
Aris, R. (+3/0) : On the dispersion of a solute in a
ﬂuid ﬂowing through a tube. Proceedings of
the Royal Society London, Ser. A, ,-/ : 01ῌ11.
Auriault, J.L., and P.M. Adler. (+33/) : Taylor dispersion in porous media : Analysis by multiple

scale expansions. Advances in Water Resources +2 : ,+1ῌ,,0.
Bear, J. (+322) : Dynamics of Fluids in PorousMedia. Dover Publications Inc., New York.
Curtiss, C.F., and J.O. Hirschfelder. (+3.3) : Transport properties of multicomponent gas mixtures. J. Chem. Phys. +1 : //*ῌ///.
Jaynes, D.B., and A.S. Rogowski. (+32-) : Applicability of Fick’s law to gas di#usion. Soil Sci. Soc.
Am. J. .1 : .,/ῌ.-*.
Le#elar, P.A. (+321) : Dynamic simulation of multinary di#usion problems related to soil. Soil
Sci. +.- : 13ῌ3+.

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