Modeling ion transport mechanisms in unsaturated porous media

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Modeling ion transport mechanisms in unsaturated porous media

Marchand, J.; Samson, E.; Beaudoin, J. J.

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Modeling ion transport mechanisms in

unsaturated porous media

Marchand, J. ; Samson, E. ; Beaudoin, J. J.

A version of this paper has been submitted to / Cet article a été soumis au

Journal of Colloid and Interface Science, 2001, pp. 1-18

www.nrc.ca/irc/ircpubs

porous media J. Mar hand 12 E. Samson 12 J.J. Beaudoin 3 1

CRIB Département de génie ivil

Université Laval, Ste-Foy (Q ), Canada, G1K 7P4

2

SIMCO Te hnologies in .

1400, boul. du Par Te hnologique, Québe (Q ),Canada, G1P4R7

3

Materials Laboratory - Institutefor Resear h in Constru tion,

National Resear h Coun il,Ottawa(On), Canada, K1A 0R6

January 17, 2001

INTRODUCTION

The transport of ions in olloids, granular and onsolidated porous media is important

to a wide variety of environmental and engineering problems. Typi al examples are the

transport of ontaminants in marine sediments, the ontainment of hazardous waste in

soils, ioni ex hange in biologi al membranes and pro esses related to the durability of

building materials [1-4℄. Re ently, there has been a great interest in understanding the

fundamental me hanisms of ioni transport in rea tive porous systems. Comprehensive

analyses [5,6℄ and the development of new modeling te hniques [7,8℄ have largely

on-tributed tothe improvementof theknowledgeof the variousphenomena that ontrolthe

anisms of multi-ioni spe ies transport in rea tive and unsaturated porous media. The

mathemati aldes ription of the mass onservation equations in isothermal onditions at

the lo al (nanos opi ) s ale are dis ussed rst. Sin e, in many pra ti al ases, porous

solids are often (partially) saturated with on entrated ele trolytes, emphasis is pla ed

on the behavior of non-idealioni solutions. Te hniques to average the equations over a

Representative ElementaryVolume (REV)of the materialare then presented. The main

hypotheses underlying the appli ation of the homogenization approa h are emphasized.

Problems related to the treatment of hemi al and physi al intera tion phenomena are

also reviewed. Finally, the appli ation of the formation fa tor on ept and the relative

importan eof surfa e (grain-boundary) diusion phenomenaare dis ussed.

IONIC TRANSPORTAT THEPORE SCALE

The transportof ioni spe ies ina porous solid o urs inthe liquid-saturatedfra tion of

the pore spa e in whi h ions are free to move. A s hemati representation of the pore

spa e of a material is given in gure 1. For this spe i example, the surfa e of the

material in onta t with the pore solution is assumed to bear a negative harge. This is

oftenthe asefor al iumsili atehydratesand layminerals[9,10℄. Thissurfa e hargeis

usuallydue toa ombinationoftheintrinsi harge ofthemineralnetworkandtosurfa e

rea tions at the solid/liquidinterfa e.

As shown in gure 1, ounterions are dire tly adsorbed onthe surfa e of the solid. This

layer of ions, alled the Stern layer, ompensates only partially the surfa e harge of the

solid. The remainingportion of the surfa e harge is neutralizedby anex ess of positive

harges distributed in a diuse layer, forming the so- alled double layer. The thi kness

of the double layer is generally assumed to be small ompared to the hara teristi s

dimensions of the pore spa e [6℄. Ions lo ated in the free water are onsidered to be

lo ated beyond that plane are onsidered to be mobile (i.e. relatively free to driftin the

poresolution)[11℄. Inmanypra ti al ases,thetransportofionsinaporousmaterialwill

o ur predominantlybeyond the shearplane(i.e. inthe ele tri aldiuselayerand inthe

freewater). Eviden eofioni transportatthesolid parti le/solutioninterfa e(i.e. inthe

Sternlayer) has alsobeen reported[5,6℄. Therelativeinuen e of thelatter phenomenon

onthedes riptionof thetransport me hanismsatthe ma ros opi s alewillbedis ussed

in moredetail in the lastse tion of this paper.

In many pra ti al ases, the pore spa e of the solid is not fully saturated by the liquid

phase. Parts of the pore spa e an be o upied by a gas, su h as water vapor(see gure

2). Lo algradients inthe liquid ontent of the materialmay trigger the transportof the

liquid phase by apillary su tion. This movement of the liquid may have a signi ant

inuen e on the distributionof ions throughoutthe material.

Numerous authors have attempted to develop ioni transport models in porous solids

on the basis of more orless detailed des riptions of the mi rostru ture of the materials.

Given the intrinsi omplexity of most porous systems, the ee tive appli ation of these

data onstitutesaformidabletask[12℄. As emphasizedby Revil[6℄, thesemodels usually

oversimplifythemi rostru tureby onsideringporestobeformedby apillariesorparallel

plates. Consequently, many of these models rely on tting parameters (determined on

the basis of simplied assumptions whi h may not physi ally represent the system well)

tofully des ribethe intri atenature of the mi roporoussolid.

Otherauthorshaveele tedtoaveragethevariablesanequationsofinterestovera(REV).

The main advantage of this approa h ( alled the homogenization te hnique) is that it

does not require any detailed knowledge of the material inner stru ture. Furthermore,

the new averaged variablesappearing inthe equationsrepresent the quantities measured

s ale of the transport me hanisms of the multi-ioni spe ies system. As previously

men-tioned, various phenomena an ontribute to the transport of ions in solution. For

in-stan e, ions willmove under the inuen e of an ele tro hemi alpotential gradient. Ions

an also be transported by adispla ement of the uid itself. The ux j

i of a given ioni spe ies i is given by [13℄: j i = B i grad i + i v (1) where i isthe on entration, i

istheele tro hemi alpotential,B isaphenomenologi al

onstant and v isthe uid velo ity. The ele tro hemi al potentialisdened as:

 i = o i +R Tln( i i )+z i F (2) where o i

is the onstant standard potential,R isthe ideal gas onstant,T is the

temper-ature,

i

is the hemi al a tivity oe ient,

i

isthe on entration of the ioni spe ies i,

z i

is the valen e number, F is the Faraday onstant, and isthe ele tri alpotential.

Substituting equation(2) inequation (1)gives:

j i = BR Tgrad i BR T i grad(ln i ) Bz i F i grad + i v (3)

The rstterm onthe right-handsideof equation(3) an beasso iatedtoFi k'sdiusion

law, whi h relates a ux of parti les to a on entration gradient through the diusion

oe ient D. Equation (3) an be modiedby onsidering that:

B = D p i R T (4) with D p i

being the diusion oe ient of the ioni spe ies i. The subs ript p identies a

variable dened at the s ale of the pore. Repla ingequation(4) inequation (3)yields:

j i = D p i grad i D p i i grad(ln i ) D p i z i F R T i grad + i v (5)

The rst term on the right-hand side of equation (5) orresponds to the lassi al Fi k

ions. This ouplingis involved when, for example,two spe ies are diusing in a solution

with one of the spe ies having a higher diusion oe ient. In order to maintain the

ele troneutrality, a diusion potential arises to slow the faster ions and a elerate the

slower ones.

As indi ated, the rst terms on the right-handside of equation (5) are often referred to

as the extended Nernst-Plan k equation. The adve tion term added at the end of the

equation takes into onsideration the inuen e of uid displa ement on the transport of

ions.

The oe ient D

p i

appearing in equation (5) represents the diusion oe ient of the

ioni spe ies in free water and in a ideal (i.e. very diluted) solution. For a given ioni

spe ies, D p i is a onstant. Values of D p i

for the most ommon ioni spe ies an be found

in textbooks.

Forvery dilutedele trolyti solutions, the a tivity and ele tri al ouplingtermsin

equa-tion (5) are very small and an be negle ted. However, as the ioni for e of the solution

in reases, these terms are be oming more signi ant. They an be viewed as orre tion

fa torstothe onstantdiusion oe ientD

p i

. This anbemoreeasilyviewedbyapplying

the followingtransformation toequation (5):

j i = D p i  1+ dln i dln i + z i F R T i d d i  grad i + i v (6)

However, this latter formofthe uxequation isratherdi ulttosolvenumeri ally. It is

only useful toshow the physi almeaning of the diusion oe ient.

The mass onservation equation atthe pore s ale levelis given by [12℄:

 i t +div(j i )+r i =0 (7) where r i

is a sour e/sink term a ounting for the hemi al rea tions undergone by the

theheterogeneousone,willae tthetransportofioni spe iesinporousmaterials. These

rea tions, that o ur between the aqueous and the solid phase [14℄, an be divided into

two distin t ategories: dissolution/pre ipitation, and ioni ex hange. At the pore s ale,

they an be handled by boundary onditions at the solid/liquidinterfa e.

Substituting equation(5) inequation (7)one nds:

 i t div  D p i grad i +D p i i grad(ln i )+ D p i z i F R T i grad i v  +r i =0 (8)

This equation must bewritten forea h ioni spe ies present in the system.

Theele trostati potential( )appearinginthepreviousequation an be al ulatedusing

Poisson's equation [3℄: r 2 + F  N X i=1 z i i =0 (9)

where N is the total number of ioni spe ies and  is the diele tri permittivity of the

aqueous phase.

The hemi al a tivity oe ients an be al ulated with various models su h as

Debye-Hü kel or Davies, depending on the level of on entration involved. A review of su h

models an be found inreferen e [15℄. All these models relate

i

to the on entration of

a given ioni spe ies and the ioni for eof the ele trolyti solution.

TRANSPORTMODEL ATTHE MATERIAL SCALE

All the equations presented in the previous se tion were derived at the pore s ale. In

orderto omeup withama ros opi des riptionof thevarioustransport pro esses,these

equations have to be averaged over a Representative Elementary Volume (REV) of the

material. In the following paragraphs, the basi on epts behind the homogenization

te hnique are des ribed. A omprehensive des ription of the te hnique an be found in

larger than the s ales of the mi ros opi stru ture of the porous solid, and mu h smaller

than the s ale of the ma ros opi phenomena. A ordingly, its size depends on the

in-trinsi properties of the material (e.g. porosity, solid matrix ontent, air-void ontent,

...).

A REV,inwhi hafra tion ofthe porespa e iso upied by agaseousphase,is shown in

gure 2. The total volume of the REV is given by V

o

. The part of the volume o upied

by the liquidphase isdesignated by V

L o

. The volumetri fra tion of liquidis dened as:

 L = V L o V o (10)

When the materialisfully saturated,the volumetri fra tionof liquid orresponds tothe

total porosity . The gaseous phase o upies a volume V

G o

. It is a mixture of air and

water vapor. It is assumed that both air and water vapor ll the entire gaseous phase

volume. The volumetri fra tion of gas is dened as:

 G = V G o V o (11)

Inthe remainderofthetext, the subs riptsLand Gwilldesignatethe liquidand gaseous

phases respe tively.

Let a

denote the amountperunit volume of some extensive quantity A in the phase ,

either solid, liquid or gas. The al ulation of on entration ormass density an serve as

examples for a

. Two types of average an be dened. The volumetri phase average is

given by: a = 1 V o Z V L o a dV (12)

The volumetri intrinsi phase average is dened as:

a = 1 V L o Z V L o a dV (13)

The two values are relatedby the following expression:

a = a (14)

derivative,adivergen e,agradientandaprodu t[12℄,equations(8)and(9)areintegrated

overthe REVtoyieldthe ma ros opi equationsdened overthe materials ale. Toease

the reading, upper ase symbols will be used to designate the volumetri intrinsi phase

average of a given variablein the aqueous phase. It gives:

( L C i ) t div   L D i gradC i + D i z i F R T C i grad + L D i C i gradln i  L C i V  + L R i + 1 V o Z S LS o j i
n LS dS =0 (15)

As an be seen, the averaging pro edure introdu es new parameters in the transport

equations. The parameter D

i

is the diusion oe ient at the ma ros opi level. It is

relatedto D p i by the expression: D i = L D p i (16) where L

is the tortuosityof the aqueous phase. The latter isa purely geometri alfa tor

a ountingforthe omplexityoftheporousnetwork. Forun hargedporousmaterials(for

whi h surfa e diusion phenomena an be negle ted), the diusion paths of the various

ioni spe iespresentinthe systemallhavethe sametortuosityintheinter onne ted pore

spa e [12℄. In this ase, the tortuosity an be related to the so- alled formation fa tor

(F)by the following expression:

 L = 1 F (17)

Foranon ondu ting poroussolidsaturated withanioni solution,theformationfa tor is

the ratioofthe poresolution ondu tivity

p

tothe bulkmaterial ondu tivity

b [17,18℄: F =  p  b (18)

Forgranular porousmaterials,theele tri alformationfa torisrelatedtothe porosity by

a power relationship alled Ar hie's law[6,18℄:

F =

m

threshold, the formation fa tor an be al ulated using the followingexpression [6℄: F =(  m p ) (20) where  p

stands for the per olated porosity. In pra ti e, the value of the ementation

exponent m usually varies from 1.5to2 [6,18℄.

It should be emphasized that the previous denition of the tortuosity is only valid for

saturatedsystems. Tortuosityvaluesarelikelyto hangewitharedu tionofthe degreeof

saturationofthematerial(parti ularlywhenthedegreeofsaturationisbelow60per ent)

[6℄. It is also subje ted to the lo al hange in porosity if dissolution or pre ipitation

rea tions o ur. A more omplete dis ussion on the tortuosity fa tor an be found in

referen es [12,16℄.

Thehomogenizationte hniquealsogeneratestheparameter

i

inequation(15). Thisnew

parameter orresponds to the hemi al a tivity oe ient al ulated with the averaged

on entrations C

i .

Finally,the average of the divergen e ofthe uxleads tothe surfa e integralin equation

(15). The ve tor n

LS

is an outward (to the L-phase) unit ve tor on the solid/aqueous

phaseinterfa e(designatedasS

LS

). Theterm(j

i

n

LS

)givestheamountofions rossing

the solid/aqueousphase interfa e,asa resultof dissolution/pre ipitationorionex hange

rea tions. However, thisintegralmightprovedi ulttoevaluate. Itispossibletoexpress

it dierently by performing the averaging operation on the ions in the solid phase [12℄.

The onservation equation atthe mi ros opi s ale is:

 is t +div(j is )=0 (21)

where the subs ript s designates the solid phase. Contrary to equation (7), it is

as-sumed that no hemi al rea tions o ur within the solid phase, and that all

ver-( s C is ) t +div( s j is s )+ 1 V o Z S SL o j is
n SL dS =0 (22) where s

isthe volumetri fra tionof solidphaseand n

SL

isanoutward(tothe S-phase)

unit ve tor on the solid/aqueous phase interfa e (designated as S

SL

). The integral in

equation (22) has the same value as the one in equation (15) but with an opposite sign

sin e the ions oming out of the aqueous phase are being bound by the solid phase.

Furthermore, the ux j

is

is zero sin e there is no ioni movement in this phase. This

allows one to write:

( s C is ) t = 1 V o Z S LS o j i
n LS dS (23)

Substituting equation(23) inequation (15) yields the followingexpression for the

trans-port ofioni spe ies:

( s C is ) t + ( L C i ) t div   L D i gradC i + D i z i F R T C i grad + L D i C i gradln i  L C i V  + L R i =0 (24)

Having averaged the transport equations, the same pro edure is applied for Poisson's

equation. This gives:

div( L  L grad )+ F  N X i=1 z i C i =0 (25)

The system of non-linear equations made of expressions (24) and (25), as well as its

pore s ale ounterpart (equations 8 and 9) an be solved numeri ally when no hemi al

rea tions nor adve tion is involved. More information on the subje t an be found in

[19,20℄. The hemi al rea tion and the adve tion terms require spe ial treatments that

are dis ussed inthe next paragraphs.

Two terms now appear in equation (24) to model the hemi al rea tions. The term R

i

a ounts for the homogeneous rea tions, i.e. the ones involvingonly the aqueous phase.

Thereisalsotheterm(

s C

is

)=ttomodeltheex hangeofionsbetweentheaqueousand

mustbeusedtosolvethetransportequation. Dis ussionsonthesealgorithmsarefoundin

referen es[25,26℄. Othershaveusedde oupledalgorithms[27℄wherethe hemi alrea tion

termsareeliminatedfromthetransportequationunderthehypothesisthatduringasmall

timestep,theyhaveanegligibleee tonthetransportofions. The on entrationproles

are orre ted afterward with a separate hemi alequilibrium ode.

Finally,one has totreat the adve tion term 

L C

i

V. For ases where the apillary for es

are dominant, su h as in drying or apillary absorption situations, the transport of the

uid phase an be modeled with Ri hard'sequation [21℄:

 L t div(D  grad L )=0 (26) where D 

is the total moisture diusivity, a ounting for water under both liquid and

gaseous form. The velo ity of the uid phase is given by:

 L V = D L grad L (27) where D L

isthe liquidwater diusivity. This model, whenever appli able, is noteworthy

be ause the variable

L

alreadyappears in the transport equation(24).

FUNDAMENTAL ASSUMPTIONS AT THE BASIS OF THE HOMOGENIZATION

PROCEDURE

It should be emphasized that the validity of equations (24) and (25) rests on a series

of lassi al assumptions. For instan e, the derivation of the equation is based on the

hypothesis that the porous material is in me hani al equilibrium and kept in isothermal

onditions. The porous solid is onsidered as an open system that an ex hange ions,

liquidsand gases with its environment.

The validity of equations (24) and (25) alsorests on a series of simplifying assumptions.

It is hypothesized that the ele tri al harges at the surfa e of the solid do not have

as a result of the double layer ee t [22℄. The derivation of equation (24) is thus based

on the hypothesis that these perturbations do not extend far from the interfa e and

have onlynegligibleee ts onthe transport me hanisms at a ma ros opi s ale. Similar

assumptions are made for the diusion potential . The ee t of the surfa e harges on

itsdistribution a ross a pore is negle ted.

In a very omprehensive investigation of ioni transport me hanisms in porous media,

Revil[5,6℄ ould learlyestablishedthat the hargednatureofthe surfa eof mostporous

materials an haveasigni antinuen e onthetransportofions atthenanos opi s ale.

For instan e, his results indi ate that surfa e transport phenomena in the Stern Layer

tendtohaveasigni antinuen eontheee tivetortuosityofthesystem. Furthermore,

this ee t appears to be dierent for ations than for anions. The author has proposed

a very elegant model that takes into a ount the inuen e of surfa e (grain-boundary)

phenomenaonma ros opi transportme hanisms. Unfortunately,themodelisonlyvalid

for granular porous media saturated with a binary symmetri 1:1 ele trolyte (su h as

NaCl).

Revil'sstudy indi atesthatgrain-boundaryee ts anbenegle tedforsystemssaturated

with a on entrated ele trolyti solutions[5,6℄. This is very fortunate sin emany porous

systems are inpra ti e saturated with relatively on entrated solutions. This on lusion

isingoodagreement withthe on lusionof otherstudies [1,2℄. This observation wasalso

onrmed by a series of numeri al simulations performed at the pore s ale for various

multi-ioni systems [23℄.

Another assumption on erns the adve tion term

i

v. Its averaging gives a dispersion

term that, as dis ussed in referen e [24℄, is similar to a Fi kian diusion term. This

dispersion ee t will be important for the ases where the uid ow in the material is

signi ant. To model its ee t, the diusion oe ient in the term a ounting for the

lassi al diusion ee t (D

i gradC i ) is repla ed by: D  i = D i +D disp . For ases where

dispersion term vanishes.

CONCLUSION

The me hanismsof ioni diusion in unsaturatedporous media have been the subje t of

alotofattentionoverthepastde ade. Resear hdoneoverthisperiodhas learlyallowed

to better des ribing the various transport pro esses at the pore s ale. Signi ant eort

has alsobeen madetowards abetter des riptionof ma ros opi transportme hanismsof

multi-ioni systems.

More resear h is needed to reliable des ribe the inuen e of grain-boundary transport

phenomena onthe transport me hanismsinmulti-ioni systems. Resear his alsoneeded

tounderstand the inuen e of temperature variationson the behaviorsof these systems.

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during arti ial re harge, Lubbo k, Texas. Groundwater 1979, Vol. 17, No. 3, pp.

Toward the center of the pore

Surface of the material

Stern

layer

Diffuse

layer

Free

water

Solid phase

Aqueous phase

Shear plane

Figure 1: Distributionof ions in a typi al pore. The ions adsobed in the Stern layer are

onsideredtobepartofthesolidphase. Theions inthefreewateraswellasinthediuse

Solid

Solid

Solid

Solid

Liquid

Gas

Vo

Vo

Vo

L

G