analytic solutions to the advective contaminant transport equation with non-linear sorption

*Correspondence to: D. W. Smith Department of Civil, Surveying and Environmental Engineering, The University ofNewcastle, Callaghan, NSW 2308, Australia

CCC 0363}9061/99/090853}27$17.50 Received 24 March 1998Copyright ( 1999 John Wiley & Sons, Ltd. Revised 8 June 1998

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS

Int. J. Numer. Anal. Meth. Geomech., 23, 853}879 (1999)

ANALYTIC SOLUTIONS TO THE ADVECTIVECONTAMINANT TRANSPORT EQUATION WITH

NON-LINEAR SORPTION

DAICHAO SHENG AND DAVID W. SMITH*

Department of Civil, Surveying and Environmental Engineering, The University of Newcastle, Callaghan, NSW 2300 Australia

SUMMARY

This paper considers advective transport of a soluble contaminant through saturated soil with non-linearsorption of the contaminant onto a stationary porous media. The non-linear sorption isotherms consideredin the transport analysis are the Langmuir and Freundlich sorption isotherms. A special case of theFreundlich sorption isotherm is the linear sorption isotherm, and it is shown that in this case transportthrough a homogeneous soil results in the initial concentration pro"le simply being translated in thedirection of the groundwater #ow. However, when the sorption isotherm is non-linear the initial concentra-tion pro"le distorts as it is translated with the groundwater #ow, leading to the development of concentra-tion shock fronts and rarefactions. Analytic solutions to the non-linear "rst-order hyperbolic equations aredeveloped for a number of contaminant transport problems of practical signi"cance. It is shown that in thecase of the Langmuir sorption isotherms, shock fronts develop at the leading edge of the concentrationpro"le while for the Freundlich sorption isotherm shock fronts may develop at either the leading or trailingedge of the concentration pro"le. Copyright ( 1999 John Wiley & Sons Ltd.

Key words: advective contaminant transport; non-linear analytic solution; shock front

1. INTRODUCTION

Contaminant transport through a porous media is usually mathematically modelled by means ofthe dispersion}advection equation.1,2 A measure of the relative importance of dispersive trans-port relative to advective transport is provided by the Peclet number (PN), which is for ourpurposes de"ned as,

PN"l¸/D (1)

where l is the average true velocity of the pore #uid within a porous media, D is the coe$cient ofhydrodynamic dispersion, and ¸ is a characteristic length.

When the Peclet number is zero, the dispersion-advection equation reduces to the transientdi!usion equation. An example where this equation "nds application is the analysis of con-taminant migration beneath an engineered land"ll.3 On the other hand, when the Peclet numberis very large, advective transport of contaminant dominates hydrodynamic dispersion transport.

While it is an experimental fact that dispersive processes will always accompany advectivetransport, for some applications a su$ciently close approximation to contaminant transportbehaviour may be found when dispersive transport is neglected. For example, the advectivetransport equation may "nd application when considering contaminant transport througha uniform aquifer. For this reason, solutions to the non-linear transport equation are of interest intheir own right.

For those cases where both dispersion and advection are of importance, it is often theadvection component that complicates a numerical solution technique. This is due to that theadvective transport equation has a fundamentally di!erent solution compared to the di!usive ordispersive transport equation. Dispersion or di!usion always tends to smooth the concentrationpro"le while advection maintains or creates discontinuities in the concentration pro"le.4 Conse-quently, most e!ective numerical techniques employed for the solution of the dispersion}advec-tion equation, such as Petrov}Galerkin methods,5,6 characteristic-Galerkin methods,7,8 upwind"nite di!erence methods9,10 and the Laplace transform methods,11,12 have to pay specialattention to the behaviour of the transport equations when advective transport is dominant. Forcertain numerical solution techniques such as the Eulerian}Lagrangian "nite elementmethod13,14 an explicit analytic solution of the advective transport equation is required, whichgives an additional motivation to "nd analytic solutions to the non-linear advective transportequation.

While books and advanced research monographs in applied mathematics do give analyticsolutions to linear advection problems and certain types of non-linear advection problems,15~17

those solutions cannot be easily extended to non-linear advective problems occurring in relationto contaminant transport through porous media. For example, a well-known non-linear trans-port equation is Burger's equation, but the solution to this equation shares little in common tothe advective transport equation with non-linear sorption isotherms which are of interest togeotechnical and environmental engineers.

In this paper, non-linear sorption isotherms investigated include the Langmuir and Freundlichsorption isotherms. Two practical boundary value problems encountered in geoenvironmentalengineering are examined in detail. The #uid velocity is assumed to be constant in space and time.Only one-dimensional solutions will be presented in this paper.

In order to appreciate the e!ects of non-linear sorption during advective transport, it is ofbene"t to "rst review the behaviour of linear sorption during advective transport. This approachprovides a convenient way of introducing the contaminant transport equation, and also providesreference point with which to compare and contrast later solutions to the advective transportequation with non-linear sorption.

2. ADVECTIVE CONTAMINANT TRANSPORT WITH LINEARSORPTION ISOTHERM

A consideration of mass balance of a contaminant entering and leaving a one-dimensionalRepresentative Elementary Volume (REV) in a saturated non-deforming porous media leads tothe following well known mass conservation equation:1

!

L f

Lx"n

Lc

Lt#r (2)

854 D. SHENG AND D. W. SMITH

Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech., 23, 853}879 (1999)

Figure 1. Linear sorption isotherm

where t denotes the time, x the spatial co-ordinate and increases from left to right in this paper,f the mass #ux of contaminant out of the representative elementary volume, c the concentrationof contaminant in pore #uid, and r the rate of contaminant mass sink per unit volume.

The advective mass #ux is given by

f"nlc (3)

where n is the porosity of the soil, l the average true linear velocity of pore #uid in the soil andl taken as positive in direction of increasing x.

The mass sink rate may represent sorption of the contaminant onto the solid phase (that is,onto the soil skeleton). This may be represented by the equation

r"o$

LS

Lt(4)

where o$is the dry density of the soil, and S the mass of contaminant sorbed per unit mass of dry

soil.Employing equations (3) and (4) in equation (2) and assuming the soil porosity is spatially

invariant leads to the one-dimensional advection transport equation with sorption

!nlLc

Lx"n

Lc

Lt#o

$

LS

Lt(5)

Then employing the chain rule in equation (5) leads to the one-dimensional advective transportequation with reversible equilibrium controlled sorption, viz,

!nlLc

Lx"An#o

$

LS

LcBLc

Lt(6)

In this section we consider the advective transport equation with linear sorption, so in this casethe rate of change of sorbed contaminant with respect to the contaminant concentration is takento be constant (see Figure 1), viz,

LS/Lc"K$

(7)

where K$

is the equilibrium partitioning coe$cient of the contaminant between the #uid andsolid phases, and may be de"ned as the increment in mass of contaminant sorbed onto the soilskeleton per unit increment of concentration per unit dry mass of soil.18

ADVECTIVE CONTAMINANT TRANSPORT EQUATION WITH NON-LINEAR SORPTION 855


Substituting equation (7) into equation (6) leads to the linear advective transport equation:

!

lR

Lc

Lx"

Lc

Lt(8)

where R is the retardation coe$cient of the contaminant and R"(1#o$K

$/n).

Equation (8) must be solved subject to a boundary condition and an initial condition whichmay be expressed as

c (x, 0)"g (x ) (9)

c(0, t)"h(t ) (10)

The boundary condition is only needed at the in#ow boundary, here taken to be at x"0 as theadvective velocity is taken to be positive.

The general solution of equation (8) is

c"f (x!lt/R) (11)

where f is an arbitrary function determined by the initial and boundary condition. For a homo-geneous soil, this solution implies that the concentration pro"le at time zero is unchanged inshape at later times, and is simply translated to the right as time increases. The movingcontaminant pro"le is sometimes described as a &wave' representing the time-dependent spatialvariation of the contaminant concentration pro"le. The concentration wave moves from left toright at speed l/R, thereby explaining why R is known as the retardation coe$cient.

If the contaminant moves into another homogeneous soil with a greater linear partitioningcoe$cient, the retardation is increased, the concentration wave is slowed down and is longitudi-nally compressed. If the contaminant then re-enters a soil with the original sorption properties,the contaminant wave resumes its original shape and transport speed. This behaviour of the waveis a characteristic feature of a linear advective transport equation.

Next a zero-order chemical reaction is introduced, representing possible biodegradation of thecontaminant at a rate that is independent of the contaminant concentration. For simplicity it isassumed that the zero-order rate of reaction is the same for the contaminant in the pore #uid as itis for the contaminant sorbed onto the solid phase. In this case, the sink term in equation (2)becomes

r"(n#o$K

$) A

Lc

Lt#jB (12)

where j is the rate constant for the zero-order reaction.Substituting equations (3) and (12) into equation (2) leads to the following linear advective

transport equation:

!

lR

Lc

Lx"

Lc

Lt#j (13)


c"f (x!lt/R)!jt (14)



where f is an arbitrary function again depending on the initial and boundary condition. Thissolution implies that the initial contaminant pro"le maintains a similar shape, but diminishes insize at a constant rate as it is translated to the right at speed l/R.

Now consider the e!ect of a linear "rst-order irreversible reaction, representing possibleradioactive decay, hydrolysis or biodegradation, acting simultaneously with linear reversiblesorption. For simplicity it is assumed that the "rst-order rate of reaction is the same for thecontaminant in the pore #uid as it is for the contaminant sorbed onto the solid phase. In this case,the sink term in (2) becomes

r"(n#o$K

$) A

Lc

Lt#jcB (15)

where j is the rate constant for the "rst-order reaction.Substituting equations (3) and (15) into equation (2) leads to the following linear advective

transport equation:

!

lR

Lc

Lz"

Lc

Lt#j (16)


c"e~jt f (x!lt/R) (17)

where f is an arbitrary function again depending on the initial and boundary condition. Thissolution implies that the initial contaminant pro"le maintains a similar shape over time, butdiminishes in size at an exponential rate as it is translated to the right at speed l/R.

Having considered the behaviour of the concentration pro"le as it is transported by thegroundwater through a saturated homogeneous soil with linear sorption, attention is now turnedto advective transport with non-linear sorption.

3. ADVECTION WITH A NON-LINEAR SORPTION ISOTHERM

3.1. Langmuir sorption isotherm

A well known non-linear sorption isotherm is the Langmuir sorption isotherm shown inFigure 2. The mathematical equation describing this isotherm can under certain assumptions bederived from the chemical equilibrium equations, while in other cases it is simply employed as anapproximation of experimental data. In either case, the form of the non-competitive sorptionisotherm is taken to be

S"BS

.c

1#Bc(18)

and hence

LS

Lc"

BS.

(1#Bc)2(19)

where S.

is the maximum adsorption capacity, and B is a material constant for a soil-solutesystem.



Figure 2. Langmuir sorption isotherm

Substitution of equation (19) into equation (6) leads to the following advective transportequation:

!nlLc

Lx"An#

o$BS

.(1#Bc)2B

Lc

Lt(20)

Again, the solution of equation (20) depends on the boundary and initial condition of thecontaminant concentration, as described by equations (9) and (10). Taken together, equations (9),(10) and (20) de"ne a non-linear hyperbolic wave propagation problem. The speed of the wavepropagation at concentration c is equal to

dx

dt"

nl

An#o$BS

.(1#Bc)2B

(21)

It is clear that the speed de"ned by equation (21) is a function of the concentration c, due to thenon-linear sorption term. We can also see that contaminants with a Langmuir sorption isothermmove more rapidly as the concentration increases. This implies that if the functions g (x) and h (t)are not equal or constant, regions of higher concentrations will translate to the right at a fasterspeed than regions of lower concentration. In pictorial terms, a region of higher concentrationmay try to &overtake' a region of lower concentration, leading to the formation of a &shock front'.A shock front is simply a discontinuity in the concentration pro"le. Alternatively, a region ofhigher concentration may &pull away' from a region of lower concentration, leading a &rarefac-tion'. A rarefaction is simply a region in the concentration wave where the concentrationdecreases over time.

The main point of this description of wave behaviour is that unlike the advective transportequation with linear sorption, the concentration wave distorts as is translated from left to right(see Figure 3). The distortion in the concentration wave is due to the non-linearity of the sorptionof the contaminant onto the soil skeleton.

Equation (21) de"nes a group of lines in the x}t plane, and along these lines the wavepropagates with a constant concentration. These lines are called characteristic lines or character-istic directions.

In the following section, two practical problems of advective contaminant transport withsorption described by the Langmuir isotherm are employed to illustrate the non-linear transportbehaviour. In the "rst case there is a continuous injection of the contaminant from the left



Figure 3. Distortion of a wave pro"le during non-linear transport

boundary at a higher concentration than the initial concentration. This problem illustrates thebehaviour of a concentration &shock front'. In the second case, the injection of contaminant at theleft boundary is maintained for a "nite length of time, then reduces to the initial concentration atthe upstream boundary. This problem illustrates the formation of a rarefaction along the trailingedge of the concentration wave as it propagates through the soil. It is noted here that in thefollowing section, the same problems are investigated for contaminant sorption according toa Freundlich sorption isotherm.

3.1.1. Case I: continuous injection (shock front)

g(x )"a, 0(x(R

h (t )"b, 0)t(R

In this case, the incoming concentration (Figure 4) is larger than the initial concentration and so itis transported with a higher speed. The transport speed of the incoming concentration b is givenby

lb"

dx

dt"

nl

n#o$

BS.

(1#Bb)2

(22)

which corresponds to the slope of dashed lines in the plan view of the x}t plane (see Figure 5). Thetransport speed of the initial concentration a is given by

la"

dx

dt"

nl

n#o$

BS.

(1#Ba)2

(23)

which corresponds to the slope of the solid lines in Figure 5. Since the upstream speed lbis higher

than the downstream speed la, a shock in the concentration will develop (or in this case be

maintained). To "nd the speed of the shock front, the following approach may be adopted.Substituting equation (18) into equation (5) leads to a statement of mass conservation fora representative elementary volume, viz,

!nlLc

Lx"

LLt

F (c) (24)



Figure 4. Initial concentration and boundary condition for Case I: Continuous injection

where

F (c )"nc#o$S"nc#o

$

BS.c

1#Bc(25)

Equation (24) holds even if c is not a smooth function. Now let,

x"Xs(t) (26)

denote the location of the shock front (that is, the location of the concentration discontinuity).The concentration on the left side of X

s(t) is b and on the right side it is a (see Figure 5).

Integrating equation (24) from x"x1

to x"x2

leads to

!Px2

x1

nlLc

Lxdx"

d

dt Px2

x1

F (c ) dx (27)

where x1

and x2

are respectively on the left and right side of Xs(t) at time t. The left-side term of

equation (27) is simply !nl(c(x2, t )!c (x

1, t)"!nl (a!b). The right-side term of equation

(27) can be evaluated using the Leibnitz rule for di!erentiating integrals which shows

d

dt Px2

x1

F (c) dx"PXs (t)

x1

LF(c)

Ltdx#F(b)

dXs

dt#P

x2

Xs (t)

LF(c)

Ltdx!F (a)

dXs

dt(28)

Both integrals on the right side of equation (28) tend to zero in the limit as x1PX

s(t)~ and

x2PX

s(t )`. Therefore, equation (27) "nally becomes

!nl(a!b)"F (b )dX

sdt

!F (a)dX

sdt

(29)

Substituting F (a) and F(b) according to equation (25) into equation (29) gives the shock frontspeed;

lsba

"

dXs

dt"

nl (b!a)

F (b)!F (a )"

nl

n#o$BS

.(1#Ba)(1#Bb)

(30)

where lsba

denotes the speed of a shock front with a concentration step from b to a. The speed ofthe shock front is intermediate between the speeds l

band l

a. The position of shock front at some

time is represented by the enhanced solid line in the x}t plane shown in Figure 5.



Figure 5. Characteristic lines and shock front for Case I: Continuous injection, Langmuir sorption isotherm

For later purposes it is noted here that if the concentrations b and a at the upstreamand downstream sides of a shock front respectively change with time, (30) may be ex-pressed as

ls"

nl

n#o$BS

.(1#Bc

6)(1#Bc

$)

(31)

where c6

is the concentration at the upstream side of the shock front at time t, and c$

is theconcentration at the downstream side of the shock front at time t.

3.1.2. Case II: Temporary injection (shock front and rarefaction)

g (x )"a, x'0

h(t)"b, 0)t)t*

h(t)"a, t't*

As for Case I, the shock front at the leading edge of the injected contaminant patch willbe maintained. Once the injection stops, the incoming concentration (Figure 6) will besmaller than the injected concentration and so will move at a lower speed. Therefore, an areaof rarefaction in concentration will develop (see Figure 7). The length of the rarefaction areaat time t can be determined from the speed di!erence at the downstream and the up-stream boundary. The upstream speed is l

aat any time t*t

*. The downstream speed is l

buntil

a certain time tb

when the width of the injected concentration patch becomes zero due to thespeed di!erence between l

band l

sba(see Figure 7 and 8). Therefore, the length of the rare-

faction area before tb

is the distance from la(t!t

*) to l

b(t!t

*). After time t

b, the downstream

speed of the rarefaction area will be the speed of the shock front which gradually decreasesfrom l

sbaat t

bto l

aat in"nite time. The length of the rarefaction area will be the distance from

la(t!t

*) to l

sca(t!t

*), where l

scais the speed of the shock front with a concentration step from

c(x, t) to a (see Figure 7). The concentration c(x, t) in the rarefaction area can be determined asfollows.



Figure 6. Initial concentration and boundary condition for Case II: Temporary injection

In the rarefaction area, the contaminant transports with a velocity lcalong the characteristic

lines determined by the concentration c and so the distance travelled is given by

x"lc(t!t

*)"

nl (t!t*)

n#o$BS

.(1#Bc)2

, R6(t))x)R

$(t) (32)

where R6(t) is the &upstream' boundary of the rarefaction area, and R

$(t) is the &downstream'

boundary of the rarefaction area.For the case shown in Figure 7, R

6(t) is given by l

a(t!t

*) and R

$(t) by l

b(t!t

*) for t)t

band

by Xs(t ) for t't

b. Equation (32) can be solved for c in terms of x and t:

c(x, t )"1

BSo$BS

.x

nl(t!t*)!nx

!

1

B, R

6(t))x)R

$(t ) (33)

Equation (33) shows that the concentration in the rarefaction area is not a linear function of x.The time t

bafter which the upstream concentration at the shock front will be less than the

injected concentration b can be deduced from the geometry of characteristic lines shown in Figure7. The initial patch width at time t

*is l

sbat*, and the speed di!erence between the shock front and

the patch is lb!l

sba. Therefore, the time taken before the upstream concentration at the shock

front is reduced from the injected concentration b is equal to

tb"t

*#

lsba

t*

(lb!l

sba)

(34)

where the speeds lband l

sbaare determined according to equations (22) and (30), respectively.

The shock front Xs(t) is a straight line de"ned by X

s"l

sbat for t)t

bin the x}t plane, but

curves for t'tb. The curve of the shock front trajectory in the x}t plane arises because of the

decrease in speed of the shock front as the maximum concentration in the shock front decreases.To determine the location of X

s(t) for t't

b, it is necessary to solve the following integral

equation:

PXs

la (t~t*)CAnc#

o$BS

.c

1#BcB!Ana#o$BS

.a

1#Ba BD dx"nl (b!a)t*

(35)



Figure 7. Characteristic lines, shock front and rarefaction area for Case II: Temporary injection, Langmuir sorptionisotherm

Equation (35) states that the total mass within the rerefaction area minus the initial mass in thisarea must equal the injected mass. The concentration c in equation (35) can be replaced byequation (33). The solution of (35) for X

s(t) is given in Appendix I. With an explicit expression of

Xs(t), it is possible to determine the time when the maximum concentration is reduced to a certain

value, for example, to half the injected concentration (this is, b/2).The solution for the case of temporary injection of a contaminant transport through a soil with

a Langmuir sorption isotherm (18) is shown pictorially in Figure 8. This "gure clearly illustratesthe development of the rerefaction area on the trailing edge of the concentration wave, and thechanging trajectory of the shock front in the x}t plane as the maximum concentration in theshock front decreases with time.

3.2. Freundlich sorption isotherm

The Freundlich sorption isotherm sometimes provides a good approximation to the experi-mental data for contaminant sorption isotherms. The general form of the Freundlich sorptionisotherm is (see Figure 9),

S"K&cm (36)

and so in this case,

LS/Lc"K&mcm~1 (37)

where K&and m is a material constant for a soil}solute system.

Substitution of equation (37) into equation (6) leads to the following governing advectivetransport equation:

!nlLc

Lx"(n#o

$K

&mcm~1 )

Lc

Lt(38)



Figure 8. Schematic view of solution of Case II: Temporary injection, Langmuir sorption isotherm

Figure 9. Freundlich sorption isotherm

The solution of equation (38) again depends on the boundary and initial conditions de"ned byequations (9) and (10). Taken together, equations (9), (10) and (38) de"ne a non-linear hyperbolicwave propagation problem. The speed of the wave propagation at concentration c is equal to

dx

dt"

nln#o

$K

&mcm~1

(39)

For m'1, a higher concentration corresponds to a lower transport speed, while for m(1 thetransport speed decreases with increasing concentration. For the special case m"1, equation (37)reduces to the linear sorption isotherm, and the contaminant will be transported at a constantspeed. Once again, if the boundary conditions g(x ) and h (t) are not equal or constant, shocks andrarefactions will develop in the concentration pro"le during the transport process.

To illustrate the development of shock fronts and rarefactions when sorption of the con-taminant onto the soil skeleton is described by the Freundlich equation, the two practical casesinvestigated previously for the Langmuir sorption isotherm are considered again. In the "rst casecontinuous injection at a higher concentration occurs at the left boundary, while in the secondcase, the injection is maintained for a "nite period of time.

3.2.1. Case I: Continuous injection. Depending on the value of m, either a shock front orrarefaction area in the concentration wave will develop at the leading edge of the concentration



Figure 10. Rarefaction area and shock front in concentration for Freundlich sorption isotherm in Case I: Continuousinjection, (a) m'1, (b) m(1

pro"le. For m'1, a rarefaction develops at the leading edge of the concentration pro"le and soan initial discontinuity in the concentration pro"le will be smoothed with time, as illustratedusing characteristics in Figure 10(a). If m(1, a shock front develops and so an initially smoothconcentration pro"le will develop a discontinuity in concentration at the leading edge, asillustrated using characteristics in Figure 10(b).

The speed of the injected concentration b is given by

lb"

dx

dt"

nln#o

$K

&mbm~1

(40)

which corresponds to the slope of dashed lines in Figure 10. The transport speed of the initialconcentration a is given by

la"

dx

dt"

nln#o

$K

&mam~1

(41)

which corresponds to the slope of the solid lines in Figure 10.The rarefaction area in Figure 10(a) at time t is the distance from l

bt to l

at. A concentration c in

this rarefaction interval transports with a speed lcso that

x"lct"

nltn#o

$K

&mam~1

, lbt)x)l

at (42)

This equation can be solved explicitly for c in terms of x and t as

c (x, t )"Anlt!nx

o$K

&mx B

1@(m~1), l

bt)x)l

at (43)

The location of the shock front in Figure 10(b) can be determined in a similar way to that for theshock front shown in Figure 5 (for the Langmuir sorption isotherm), viz,

lsba

"

dXs

dt"

nl(b!a)

F(b )!F (a)"

nl(b!a)

n (b!a)#o$K

&(bm!am)

(44)



Figure 11. Characteristic lines, shock front and rarefaction area for Case II: Temporary injection, Freundlich sorptionisotherm with m'1

3.2.2. Case II: Temporary injection. As for the example of continuous injection, two di!erentwave pro"les can develop, dependent on the value of m. For m(1, the injected concentrationpatch will move in a similar way to that shown in Figures 7 and 8, except that the downstreamshock front will have a speed de"ned by equation (44) and the concentration in the upstreamrarefaction area will be de"ned by equation (43) with time t starting from the end of injection t

*.

Perhaps unexpectedly, for m'1, the shock front will develop at the upstream side of theinjected concentration patch whereas a rarefaction area will develop at the downstream side ofthe patch (as we have seen in Case I). For time t before t

b(see Figure 11 for de"nition of t

b), the

location of the shock front in the x}t plane is given by

x"Xs(t)"l

sba(t!t

*)"

nl(b!a)(t!t*)

n (b!a)#o$K

&(bm!am)

(45)

After time tb, the speed of the shock front will gradually increase from l

sbaat t

bto l

aas time

becomes in"nitely large. An explicit function for the location of this shock front is given inAppendices II and III. The time t

bcan be computed using equation (34) with l

band l

sbadeter-

mined according to (40) and (44), respectively.The rarefaction area is the distance from l

bt to l

at for t)t

band from X

s(t) to l

at for t't

b. The

concentration in the rarefaction area is determined by equation (43). The characteristic lines aswell as the location of the shock front and the rarefaction area are shown in Figure 11. Theanalytic solution of this example problem is pictorially illustrated in Figure 12.

4. EFFECTS OF IRREVERSIBLE REACTION

4.1. Zero-order reaction

The governing equation for advective transport with a simultaneous zero-order chemicalreaction is given by equation (12), viz,

!nlLc

Lx"An#o

$

LS

LcB ALc

Lt#jB (46)



Figure 12. Schematic view of solution of Case II: Temporary injection: Freundlich sorption isotherm

where j is a constant representing the rate of reaction. Along the characteristic directions de"nedby

dx

dt"

nl

An#o$

LS

LcB(47)

the partial di!erential equation (46) reduces to

dc

dt"

Lc

Lx

dx

dt#

Lc

Lt"

Lc

Lx

nl

An#o$

LS

LcB#

Lc

Lt"!j (48)

Equation (48) indicates that along its characteristic directions de"ned by equation (47), theunknown concentration c is a linear function of time t, viz,

c"!jt#c0

(49)

where c0

is the initial concentration at t"0. From equation (49), it is apparent that a concentra-tion wave transports along its characteristic direction with a linearly decreasing amplitude overtime, since j is usually a positive number. For a non-linear sorption isotherm, the characteristicdirections dx/dt are in general functions of c, and so these directions change as c changes duringthe transport process. This means that (47) de"nes a group of curved characteristics, instead ofstraight lines as was the case when there was no chemical reaction. The explicit expressions ofcharacteristic curves for a Langmuir isotherm can be found by substituting (49) into (47) andintegrating with respect to time. This leads to

x"lt!So$S.

nB

lj

arctanjB Jno

$BS

.t

o$BS

.#n (1#Bc

0) (1#Bc

0!jBt)

#m (50)

where m is an arbitrary x position when t"0.



Figure 13. E!ect of zero order reaction on the concentration transport, uniform concentration

For the Freundlich sorption isotherm a general explicit form describing the characteristiccurves for any value of m cannot be obtained analytically, so an example solution is given form"2. For this case,

x"nl

2o$jK

&

(ln(n#2o$K

&c0)!ln(n#2o

$K

&c0!2o

$jK

&t ))#m (51)

The distances de"ned by equation (50) and (51) are relatively complicated functions of t, whichmakes the solution of equation (46) more complex than discussed in the previous section. In thesimplest case when the initial concentration c

0is uniform for !R(x(R, the solution of

equation (46) is an inclined surface in the three-dimensional space as shown in Figure 13. Ata certain time t

s, the initial concentration c

0will decrease to zero and the contaminant will have

disappeared. This time is given by,

ts"c

0/j . !R(x(R (52)

However, if the initial concentration is not uniform or a di!erent concentration is applied ata boundary, shocks and rarefactions in concentration pro"le can in general develop and thesolutions to equation (46) are much more complex than that shown in Figure 13.

To illustrate the solution of equation (46), the same example problems of continuous andtemporary injection will be employed again, but now the initial concentration will be set to zerofor the sake of simplicity (i.e. c(x, 0)"g (x)"0).

4.1.1. Case I: Continuous injection. For a Langmuir sorption isotherm, the speed of concentra-tion transport increases with increasing concentration. Therefore, at the front of the injectedconcentration patch, the upstream speed is higher than the downstream speed and a shock frontwill develop (or be maintained). The speed of the shock front can be calculated from (31), withc$"0 and c

6"b!jt, viz,

vs"

dXs

dt"

nl

n#o$BS

.(1#Bc

6)(1#Bc

$)

"

nl

n#o$BS

.(1#Bb!Bjt)

(53)



Figure 14. Schematic view of solution of Case I: Continuous injection, Langmuir sorption isotherm and Freundlichsorption isotherm with m"2, Zero-order reaction

Integrating (53) gives the explicit expression of the shock front

Xs"lt#

o$S.l

njln

n#nBb#o$BS

.!nBjt

n#nBb#o$BS

.

(54)

The concentration c decreases linearly from c"b at x"0 to c"b!jt at position Xs. The

solution is illustrated in Figure 14. At time ts"b/j, the concentration at X

sdecreases to zero and

the shock front disappears. This location where the shock front disappears can be found bysubstituting t with b/j in equation (54), viz,

Xs"

lbj#

o$S.l

njln

n#o$BS

.n#nBb#o

$BS

.

(55)

For the contaminant sorption onto the soil skeleton described by the Freundlich sorptionisotherm, with a particular value of m chosen for simplicity (that is m"2), the speed of the shockfront is given by

ls"

dXs

dt"

nl(c6!c

$)

n (c6!c

$)#o

$K

&(cm

6!cm

$)"

nln#o

$K

&(b!jt )2

(56)

and so the shock front location is given by

Xs"S

n

o$K

&

lj

arctanj Jno

$K

&t

n#o$K

&b(b!jt)

(57)

As shown pictorially in Figure 14, the solution of equation (46) for the Freundlich sorptionisotherm with m"2 is similar to that for the Langmuir sorption isotherm. The position x

swhere

the concentration decreases to zero and the shock front disappears (see Figure 14) can again befound by substituting b/j for t in equation (57).



Figure 15. Schematic view of solution case II: Temporary injection, Langmuir sorption isotherm or Freundlich sorptionistherm with m'1, Zero-order reaction

4.1.2. Case II: Temporary injection. For a Langmuir sorption isotherm or a Freundlichsorption with m'1, the solution of equation (46) for a temporary injection of contaminant isschematically shown in Figure 15. The time t

"(shown in Figure 15) is once again the time when

the maximum concentration in the shock front is reduced by the rarefaction. For time t(tbthe

location of the shock front location, Xs(t), is de"ned by equations (54) and (57) for the Langmuir

and Freundlich sorption isotherm respectively. For time t'tb, the location of the shock front,

Xs(t), will be a complicated function of t.Explicit expressions of the concentration in the rarefaction area can be obtained by solving

equations (50) or (51) for c0

in terms of x and t and then substituting c0(x, t) into equation (49).

This leads to

c(x, t)"1

2 ASj2t2#So$S.

Bn

4jt

tan A(lt!x)j

l So$S.

Bn B!jtB!1

B(58)

for the Langmuir sorption isotherm and to

c(x, t)"jt

exp A2jK

&x

nl B!1

!

n

2K&

(59)

for the Freundlich sorption isotherm with m"2.The time t

bcan be found by solving for the crossing point in the x}t plane between the shock

front Xsand the characteristic curve through the point (x"0, t"t

*). Unfortunately the explicit

expressions for tbare too complex to be presented in this paper. Once t

bis known, the maximum

concentration at tbis then given by

c.!9

(tb)"Max(b!jt

b,0) (60)



4.2. First-order reaction

The governing equation for the advective transport with a "rst-order reaction takes the form

!nlLc

Lx"An#o

$

LS

LcB ALc

Lt#jcB (61)

Along the characteristic directions de"ned by

dx

dt"

nl

An#o$

LS

LcB(62)

the partial di!erential equation (46) reduces to

dc

dt"

Lc

Lx

dx

dt#

Lc

Lt"

Lc

Lx

nl

An#o$

LS

LcB#

Lc

Lt"!jc (63)

This indicates that along its characteristic directions, the unknown concentration c is anexponential function of time t, viz,

c"c0

exp(!jt ) (64)

where c0

is the initial concentration at t"0.The characteristics de"ned by equation (61) for a Langmuir or a Freundlich sorption isotherm

are in general complex non-linear functions of t and explicit expressions cannot be obtainedeasily. However, the e!ect of the "rst-order reaction on the advective transport is much the sameas for the zero-order reaction, but with an exponential decay rate. The solution of equation (61),for the case of a temporary contaminant injection, is shown pictorially in Figure 16.

5. NUMERICAL EXAMPLE

A simple problem of practical signi"cance to environmental engineers interested in predicting thedevelopment of a contaminant plume is described here. Fluoride contaminated waste is place inan unlined pit and is migrating vertically through an initially uncontaminated clayey sand to anunderlying aquifer (see Figure 17). The amount of #uoride waste in the pit is taken to besu$ciently large so that the leachate from the unlined pit has essentially a constant concentrationof #uoride, taken to be 300 mg/l. An important problem is to estimate the time for the #uoridecontaminant to move from the unlined pit to the aquifer below.

The properties of the clayey sand between the pit and the aquifer are taken to be,

Dry density o$"1500]103 mg/l

Porosity n"0)44

Permeability k"10~6 m/s

Experimental investigations using batch testing and di!usion cells have established that thesorption isotherm for #uoride onto kaolinite is of a competitive Langmuir type,19 and theamount of sorption is strongly dependent on pH values (see Figure 18). The competitive sorption



Figure 16. Schematic view of solution of Case II: Temporary injection, Langmuir sorption isotherm or Freundlichsorption isotherm with m'1, First-order reaction

Figure 17. Advective transport of #uoride contaminant from an unlined pit to an undelrying aquifer

Figure 18. Fluoride sorption isotherms in kaolinite, Kau (1997)19

isotherm is described by the equation

S"S.B1c

1#B2

101H~7#B1c

(65)



where

S."3)33 mg/g (for 100% kaolinite)

B1"0)0216 l/mg

B2"8)05

It is assumed that the sorption of the #uoride onto the sand is purely due to clay content(S

."0)333 mg/l, for 10 per cent kaolinite content), and that the pH of the leachate does not vary

with depth. The e!ective dispersion coe$cient of the soil varies between 10~6 and 10~9 m2/s,20which gives a Peclet number in the range of 20}20 000, with reference to l in equation (67) anda characteristic length of 5 m. Due to the relatively large Peclet number, the transport process willbe advection dominated, and so a good estimation of the transport time to the aquifer fordi!erent pH's may be made using the non-linear advective transport solutions developed inprevious sections.

Substituting equation (65) into equation (25) and then into equation (30) shows the transportspeed of the #uoride front from the unlined pit is given by

lsba

"

dXs

dt"

nl(b!a )

F (b)!F (a)"

nl

n#o$B1S.(1#B

2101H~7)

(1#B2

101H~7#B1a) (1#B

2101H~7#B

1b)

(66)

The average true downward velocity of the pore #uid within the clayey sand is, according to theDarcy law

l"l$n"

1

nk

*H

*t"

10~6

0)44

10

5"0)45]10~5 m/s (67)

and employing this result in equation (66) leads to the transport speed of the #uoride front

lsab

"

nl

n#o$S.B

11#B

2101H~7#B

1b

"

2]10~6

0)44#1)079]104

7)48#8)08]101H~7

(68)

Therefore, the time for the #uoride contaminant to reach the aquifer is given by

t"5

lsab

"2)5]106 A0)44#1)079]104

7)48#8)08]101H~7B (69)

The time calculated for various pH values using equation (69) is shown in Figure 19, identi"ed bythe legend &non-linear'.

It is of interest to compare the non-linear solution given by equation (69) with a solution fora linear sorption isotherm. One approach to linearize the sorption isotherm would be to use theinitial tangential slope of the S}c curve (K

$0) to approximate the derivative LS/Lc (see Figure 18).

For this approach

K$0"

LS

Lc Kc/0

"

S.B1(1#B

2]101H~7)

(1#B2]101H~7#B

1c)2 K

c/0

"

S.B1

1#B2

101H~7(70)



Figure 19. Time for #uoride arrival at the aquifer versus pH (Non-linear; using non-linear sorption isotherm, Linear:using linear sorption approximated by the initial tengantial slope K

$0)

and the transport speed of the #uoride contaminant through the clayey sand is given by

lc"

nln#o

$K

$0

The time for the #uoride contaminant to reach the aquifer would therefore be (see Figure 19).

t"5

lc

"2)5]106 A0)44#1)079]104

1#8)08]101H~7B (71)

Comparing the two plots in Figure 19 it is clear that using the initial tangent sorption isothermleads to signi"cant larger estimated transport times compared to the non-linear transportpredictions, particularly at low pH values. At high pH values the non-linear and linear solutionsconverge due to sorption of the contaminant onto the soil skeleton reducing to zero.

An alternative linear approximation of equation (65) is to use the secant slope (K$4

) for LS/Lc(see Figure 18). The linear sorption isotherm is then given by

K$4"

S.B1

1#B2101H~7#B

1b

(72)

and the transport speed of the #uoride contaminant through the clayey sand is

lc"

nl

n#o$S.B

11#B

2101H~7#B

1b

(73)

Equation (73) predicts exactly the same shock front speed lsab

as given by equation (68) for thenon-linear sorption isotherm. Therefore, in this example, this simple secant linear approximation



leads to the exact solution. It is cautioned that this coincidence of solutions would not necessarilyfollow for more complicated initial and boundary conditions.

6. CONCLUSION

In this paper, analytic solutions to the advective transport equation with non-sorption has beendeveloped for a several contaminant transport problems of practical signi"cance. It has beenshown that advective transport of a contaminant in a porous media with non-linear sorptionisotherm is a complex process. A number of important behaviours may be observed, namely,

(1) When an initially continuous contaminant concentration pro"le is transported througha porous media with a non-linear sorption isotherm, concentration shocks and rarefactionsmay develop.

(2) The transport speed of a concentration shock front is intermediate between the transportspeed of the concentration at its downstream and upstream sides.

(3) In the case of the Langmuir sorption isotherm, a shock front may develop at the leadingedge of the concentration pro"le and a rarefaction area may develop at the trailing edge.

(4) For the Freundlich sorption isotherm, shock fronts and rarefaction areas may develop ateither the leading or trailing edges of the concentration pro"le, depending on the value ofthe material parameter m.

It has been demonstrated by the solution of a simple problem that the analytic solutionspresented in this paper can be employed to solve contaminant transport problems of practicalinterest. Finally, it is expected that the analytic solutions presented in this paper may also beemployed in the future development of more accurate numerical techniques of the non-lineardispersion-advection transport equation.

NOTATION

a initial concentrationB coe$cient in Langmuir sorption isothermb injected concentrationc concentration in pore #uidD coe$cient of hydrodynamic dispersionf mass #ux of contaminant out of the representative elementary volumeK

$distribution coe$cient in a linear sorption isotherm

K&

material parameter in Freundlich sorption isotherm¸ characteristic lengthm material parameter in Freundlich sorption isothermn soil porosityPN Peclet numberR retardation coe$cientR

$downstream boundary of rarefaction area

R6

upstream boundary of rarefaction arear rate of contaminant mass sink per unit volumeS sorbed concentrationS.

maximum sorption capacity in Langmuir sorption isotherm



t timet"

time at which the width of the injected concentration patch becomes zerot*

end time of injectionl average true velocity of pore #uidla

transport speed of concentration alb

transport speed of concentration blc

transport speed of concentration cls

transport speed of shock front slsba

transport speed of shock front with a concentration step from b to aX

sshock front location

x space coordinateo$

soil dry densityj rate constant of reaction

APPENDIX

A.1. Location of shock front in the case of temporary injection with a Langmuir sorption isotherm

For t)tb, the shock front location X

s(t) is a straight line de"ned by

Xs"l

sbat"

nlt

n#o$BS

.(1#Ba)(1#Bb)

, t)tb

(74)

Fo t'tb, the shock front will curve gradually in the x}t plane from speed l

sbaat t

bto speed l

aat

time in"nity. To "nd the location of Xs(t) for t't

b, mass balance considerations indicate that the

total mass within the rarefaction area minus the initial mass in this area must equal the injectedmass, viz,

PXs

la (t~ti)Anc#

o$BS

.c

1#BcB dx!Ana#o$BS

.a

1#Ba B (Xs!l

a(t!t

*))"nl (b!a)t

*(75)

The concentration c in (75) can now be replaced according to (33). Integrating the "rst term on theleft-hand side of (75) leads to, after some rearrangement

Xs"

1

2(A21#A2

2)(A2

1l(t!t

*)!2A

2A

4(t!t

*)#2A

2A

3)

#

1

2(A21#A2

2)JA

5(t!t

*)2#A

6(t!t

*)#A

7, t't

b(76)

where

A1"2S

no$S.

B

A2"o

$S.!

n

B!na!

o$S.Ba

1#Ba

A3"nl (b!a)t

*



A4"A

1J(l!l

a)l

a!A

2la

A5"A4

1l2!4A2

1A

2A

4l!4A2

1A2

4

A6"4A2

2A

3A

4#8A2

1A

3A

4#2A2

1A

2A

3l

A7"!4A2

1A2

3

With this explicit expression of Xs(t), it is possible to determine the time when the maximum

concentration is reduced to a certain value, for example, to b/2.

A.2. Location of shock front in case of temporary injection with a Freundlich sorption isotherm(m"0.5)

For the Freundlich sorption isotherm, equation (36), when m(1, the injected concentrationpatch will distort in a similar way to that for the Langmuir sorption isotherm, i.e. a shcok frontforms at the downstream side and rarefaction area develops at the upstream side of the injectedconcentration patch. The characteristic lines and the pictorial depiction of the solution are similarto those shown in Figure 7 and 8, respectively. The shock front location for t)t

bis a straight line

given by

Xs"l

sbat"

nl (b!a)t

n (b!a)#o$K

&(bm!am)

(77)

while for t'tb

the shock front location will curve due to the decreasing concentration at theupstream side of the front. Applying the principle of mass balance in the rarefaction area leads tothe following integral equation:

PXs

la (t~ti)

(nc#o$K

&cm) dx!(n a#o

$K

&am)(X

s!l

a(t!t

*))"nl(b!a)t

*(78)

The concentration c in equation (75) is the concentration within the rarefaction area which isgiven by equation (43) with t replaced by (t!t

*), i.e.

c(x, t)"Anl (t!t

*)!nx

o$K

&mx B

1@(m~1), l

b(t!t

*))x)l

a(t!t

*) (79)

Substituting equation (79) into equation (78) leads to an implicit equation for Xs. For a given m,

the integral equation (78) can be solved explicitly for Xs. For example, for m"0)5, the shock front

Xsis de"ned by

Xs"

1

2A8

(!A9(t!t

*)!A

10#JA

11A

10(t!t

*)#A2

10), t't

b(80)

where

A8"

a

n An#o$K

&2JaB

2

A9"!2al An#

o$K

&2JaB



A10"n (b!a)lt

*

A11"4ao

$K

&l

n#o$K

&n

A.3. Location of shock front in the case of temporary injection with a Freundlich sorption isotherm(m"2)

For Freundlich sorption isotherm with m'1, a shock front will develop at the upstream sideof the injected concentration patch whereas a rarefaction area will develop at the downstreamside of the injected concentration patch. For time t before t

bin Figure 11, the location of the shock

front is a line in the x}t plane by

Xs(t )"l

sba(t!t

*)"

nl(b!a)(t!t*)

n(b!a)#o$K

$(bm!am)

(81)

After time tb, the location of the shock front is de"ned by the mass balance equation,

Plat

Xs

(nc#o$K

&cm) dx!(na#o

$K

&am)(X

s!l

at)"nl(b!a)t

*(82)

where c is given by (43). Equation (82) implicitly de"nes Xsin terms of t. For a given m, the

integral in (82) can be solved for Xs. For example, for m"2, the shock front location X

sis

explicitly de"ned by

Xs"

1

2A12

(A13

t#A14#JA

13A

14t#A2

14), t't

b(83)

where

A12"

(n#2o$K

&a )2

4o$K

&

A13"

nl(n#2o$K

&a )

2o$K

&

A14"n (b!a)lt

*

REFERENCES

1. J. Bear, Dynamics of Fluids in Porous Media, Elsevier, New York, 1972.2. M. T. van Genuchten, &Recent progress in modelling water #ow and chemical transport in the unsaturated zone', Proc.

20th General Assembly of the International ;nion of Geodesy and Geophysics, Vienna, Austria, IAHS, Institute ofHydrology, Wallingford, UK, 1991, pp. 169}183.

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5. I. Christie, F. Gri$ths, A. R. Mitchell and O. C. Zienkiewicz, &Finite element methods for second order di!erentialequations with signi"cant "rst derivatives', Int. J. Numer. Meth. Engng., 10, 1389}1396 (1976).

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14. D. Sheng and K. Axelsson, &Uncoupling of coupled #ows in soil, A "nite element method', Int. J. Numer. Anal. Meth.Geomech., 19, 537}553 (1995).

15. J. D. Logan, Applied Mathematics, A Contemporary Approach, Wiley, New York, 1987.16. J. Smoller, Shock =aves and Reaction}Di+usion Equations, Springer, New York, 1983.17. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II, Wiley-Intersience, New York, 1962.18. R. H. Freeze and J. A. Cherry, Groundwater, Prentice-Hall, Englewood Cli!s, NJ, 1979.19. P. M. H. Kau, &The experimental investigation of #uoride migration in clay soils', Ph.D. ¹hesis, University of

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analytic solutions to the advective contaminant transport equation with non-linear sorption

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