Type Curves to Determine the Relative Importance 

of Advection and Dispersion for Solute 

and Vapor Transport 

by John A. Garges3 and Arthur L. Baehrb 

Abstract 

The relative importance of advection and dispersion for both solute and vapor transport can be determined from type curves 

for concentration, flux, or cumulative flux. The dimensionless form of the type curves provides a means to directly evaluate the importance of mass transport by advection relative to that of mass transport by diffusion and dispersion. Type curves based on an analytical solution to the advection-dispersion equation are plotted in terms of dimensionless time and Peclet number. Flux and 

cumulative flux type curves provide additional rationale for transport regime determination in addition to the traditional concentration 

type curves. The extension of type curves to include vapor transport with phase partitioning in the unsaturated zone is a new development. Type curves for negative Peclet numbers also are presented. A negative Peclet number characterizes a problem in which 

the direction of flow is toward the contamination source, and thereby diffusion and advection can act in opposite directions. Examples 

are the diffusion of solutes away from the downgradient edge of a pump-and-treat capture zone, the upward diffusion of vapors 

through the unsaturated zone with recharge, and the diffusion of solutes through a low hydraulic conductivity cutoff wall with an 

inward advective gradient. 

Introduction 

In general, transport in porous media is simulated by simultaneously considering advection, diffusion, and mechanical dispersion. At low-flow velocities, transport may be diffusion dominated, whereas at high velocities, such as those induced by pump 

and treat or vacuum extraction remediation, transport may be 

advection dominated in that local scale mechanical dispersion 

could be neglected. For either case, a transport model that neglects 

insignificant terms would be easier to implement than one that 

simultaneously considers all transport mechanisms, particularly if 

the model is three-dimensional and/or includes simultaneous consideration of multiple and reactive chemical species or parameter 

optimization routines. In the case of advection dominated transport, 

there is no need to obtain effective diffusion and mechanical dispersion coefficients. Transport models based on advection only are 

computationally less demanding because transport along individual flowpaths or characteristic curves can be isolated from the rest 

of the domain. For example, for problems of remedial design 

involving recovery wells or vacuum extraction wells, where relatively high flow rates can be anticipated throughout the simulated 

region, the application of an advective transport model is likely 

appropriate. The unforced movement of gases in the unsaturated 

zone or solutes in low-permeability clay are likely, however, to be 

•conestoga-Rovers and Associates, 559 W. Uwchlan Ave., Ste. 120, 

Exton, PA 19341. 

hU.S. Geological Survey, 810 Bear Tavern Rd., West Trenton, NJ 

08628. 

Received October 1997, accepted March 1998. 

Vol. 36, No. 6--GROUND WATER-November-December 1998 

diffusion dominated. In the case of diffusion dominated transport, 

head and permeability distribution are not required for transport simulation. A formal method for classifying the regime of a particular 

transport problem is presented. The method allows for improved 

model selection, thereby eliminating the use of a needlessly complex model for simple problems. Conversely, the method will alert 

the user when a problem requires inclusion of advection, diffusion, 

and mechanical dispersion. 

Sauty (1980) presented solute concentration breakthrough 

curves in terms of Peclet number and dimensionless time based on 

the solution to the one-dimensional advection-dispersion equation 

presented by Ogata and Banks (1961). The dimensionless form of 

the type curves provides a means to directly evaluate the importance 

of mass transport by advection relative to that of mass transport by 

dispersion (including diffusion). The higher the Peclet number, 

the more dominant advective transport is relative to dispersive 

transport. A more general form of the advection-dispersion equation that includes phase partitioning and the associated generalization 

of the Peclet number and dimensionless time is presented here so 

that type curves can be interpreted to assess the transport regime for 

sorbing solutes and water phase partitioning of vapors. Flux and 

cumulative flux (total mass passing through a point over a specified time) breakthrough curves are derived to provide additional criteria to concentration breakthrough curves for evaluating the relative importance of advection and dispersion. These flux-based 

type curves to date have not been documented in the literature. 

A negative Peclet number characterizes a problem in which the 

direction of flow is toward the contamination source and therefore 

diffusion and advection can act in opposite directions. Type curves 

for negative Peclet numbers are presented for the first time. 

Examples of problems characterized by a negative Peclet number 

959 
include the diffusion of solutes away from the downgradient edge 

of a pump-and-treat capture zone, the upward diffusion of vapors 

through the unsaturated zone with recharge, and the diffusion of 

solutes through a low hydraulic conductivity cutoff wall with an 

inward advective gradient. 

Advection-Dispersion Transport Model 

The mathematical statement of the model is identical for both 

solute transport in an aquifer and vapor transport in the unsaturated 

zone, and is given as follows: 

(PDE) 

ac a2

c ac s---=D~-qat a x2 ax (1) 

(BC) C = C0 atx = 0 

lim C = 0 x~oo 

(I C) C=Oatt=O 

where C is the solute or vapor concentration (M/0); C0 is the 

solute or vapor concentration at the boundary x = 0; D is the dispersion coefficient (including mechanical dispersion and diffusion) in the aqueous or gaseous phase (L2/T); and q is the specific 

discharge of the aqueous or gaseous phase (LIT). The definition of 

s, the storage coefficient, depends on the definition of C (solute or 

vapor), and in either case assumes equilibrium partitioning among 

phases in the porous medium. In general, the storage coefficient 

causes the solute or vapor transport to be retarded because of phase 

partitioning. If C is solute concentration, then the pore space of the 

porous medium is assumed to be saturated and 

s = ($ + KctPb) 

where <(lis porosity, Kct is the adsorption coefficient (0/M), and pb 

is bulk density (M/0). If Cis vapor concentration then the porous 

medium has aqueous and gaseous phases and 

S = (8. + Hwa 8w + HsaPb) 

where ew and ea are the volumetric contents for the aqueous and 

gaseous phases, respectively; ~a is the dimensionless Henry's 

law coefficient; and Hsa is the distribution coefficient between the 

gaseous and adsorbed phases. Note that for the case of unsaturated 

zone transport and the selection of C as vapor concentration, the 

model allows for advection in the water phase and diffusion in the 

air phase simply by interpreting q to be the product of Hwa and the 

specific discharge for the water phase. A semi-infinite domain was 

specified for the general analysis to eliminate sensitivity to an 

arbitrarily located downstream boundary. The only mathematical difference between the model presented here and that of Ogata and 

Banks ( 1961) is the inclusion of the general storage coefficient, s, 

that includes gaseous and adsorbed phases. 

For the case of advection only, D = 0 in Equation 1 and for q 

> 0 the solution to Equation 1 subject to the initial condition and 

boundary condition at x = 0 is as follows: 

=Oiftr<1 

= 1 if tr > 1 

h t qt . d" . 1 . w ere r = - IS 1mens10n ess time. sx 

960 

(2a) 

Equation 2a implies the following solution for flux: 

(2b) 

where J is mass flux (MIL2T). 

For the case of diffusion only, q = 0 in Equation 1 and the solution to the standard diffusion equation subject to the initial and 

boundary conditions specified for the semi-infinite domain is as follows (Carslaw and Jaeger 1959): 

Co c ( = erfc V4m ~) (3a) 

Pick's law (J = - D ~~) and Equation 3a imply the following 

equation for mass flux for the case of diffusion only: 

J ( ~) Co {ill = exp - Y4lli 

\f-;rt (3b) 

Equations 2a, 2b, 3a, and 3c provide solutions for extreme cases 

where either dispersion or advection can be neglected. 

The general solution to Equation 1 that includes both advective and dispersive transport mechanisms is the same as that given 

by Ogata and Banks (1961), and is as follows: 

C _ 1 [ (X - q*t) (q*x) (X + q*t)J -C - -2 erfc 2

, ~ + exp ~ erfc , ~ o vD*t D* 2vD*t 

q D 

where q* = -and D* = -- s s 

Sauty (1980) presented the following dimensionless form of 

Equation 4: 

I 

+ exp(Pe) erfc [ (:~y (1 + tr) j} 

(4) 

(5) 

where the Peclet number, a dimensionless number that scales the 

ratio of advection to dispersion, is defined as follows: 

qx P=-

e D (6) 

In this definition of P e• the spatial coordinate x is the characteristic distance. The definition of dimensionless time tr for Equation 5 

is modified slightly from that defined by Sauty (1980) to include the 

generalized storage coefficient s as follows: 
qt tr =-

sx (7) 

Dimensionless time, tr, is related to the number of pore volume 

flushes over the distance x, adjusted by the storage coefficient. 

For the general advection-dispersion model, dimensionless 

mass flux is defined as follows: 

(8) 

Equations 5 and 8 imply the following expression for dimensionless mass flux: 

I 

_}'_ = _1_ erfc [( pc )

2

(1 - tr)J 

qC0 2 4tr 

+ 2~ (PctJ-f { exp [- a~~) (I - tr)2J 

+ exp (Pc) exp [ - (:;J (1 + tr) 2j} (9) 

The derivation of Equation 9 from Garges ( 1996) is outlined in 

the appendix. Cumulative mass flux over time tr is obtained by 

numerically integrating Equation 9. 

Type Curves for Positive Peclet Numbers 

For the model coordinate system, Pe > 0 if and only if q > 0 

(flow in direction away from source) and advection and diffusion 

act in the same direction. A computer program was written to calculate transient concentration, flux, and cumulative flux type curves 

from Equations 5, 9, and the integration of Equation 9, respectively. 

Figures I through 3 show the concentration, flux, and cumulative 

flux dimensionless type curves for solute or vapor transport. Direct 

comparison of the type curves to the endmember solutions given by 

Equations 2a, 2b, 3a, and 3b would be possible if specific values of 

model parameters comprising the dimensionless numbers were 

known. In general, however, even though the type curves do not 

define distinct transport regimes, diffusion can generally be said to 

be the dominant transport mechanism for Peclet numbers between 

1.0 

0.8 

0.6 

0 

0 

(3 

0.4 

0.2 

0.0 

0.0 0.5 1.0 1.5 2.0 

tr = qt/sx 

Figure 1. Dimensionless concentration type curves for positive Peclet 

numbers. 

0 and 1. Peclet numbers from 1 to 10 characterize a regime where 

both dispersion and advection are important, for Peclet numbers 

greater than 10, transport is dominated by advection. Flux and 

cumulative flux type curves provide additional rationale for transport regime determination. 

The steady-state solution for the case of P e > 0 is as follows: 

J 

qCo 

(lOa) 

(lOb) 

The steady-state solution is not useful in determining the relative 

importance between advection and dispersion; however, it does provide a check for the asymptotic behavior of the type curves. 

Type Curves for Negative Peclet Numbers 

For the model coordinate system, Pe < 0 if and only if q < 0 

(flow in direction toward the source) and advection and diffusion 

2.0 

1.8 

1.6 

1.4 

1.2 

0 

0 1.0 :'; 

0.8 

0.6 

0.4 

0.2 

0.0 

0.0 0.5 1.0 1.5 2.0 

tr = qt/sx 

Figure 2. Dimensionless flux type curves for positive Peclet numbers. 

2.0 

t .8 

1.6 

-0 1.4 

0 

0" 

~ 1.2 >< 

u.. " 1.0 ~ 

~ 0.8 

E " 

0 " 0.6 

0.4 

0.2 

0.0 

0.0 0.5 1.0 1.5 2.0 

tr = qt/sx 

Figure 3. Dimensionless cumulative flux type curves for positive 

Peclet numbers. 

961 
1.0 

0.8 

0.6 

0 

0 

0 

0.4 

0.2 

0.0 

·1 . 2 ·3 ·4 . 5 

tr = qt/sx 

Figure 4. Dimensionless concentration type curves for negative Peclet 

numbers. 

0.0 

·1 . 2 . 3 ·4 ·5 

tr = qt/sx 

Figure 5. Dimensionless flux type curves for negative Peclet numbers. 

-0 

0 -1.0 CT 

-

>< 

=> 

LL 

.2: "' 

~ => 

E 

=> -2.0 0 

tr = qt/sx 

Figure 6. Dimensionless cumulative flux type curves for negative 

Peclet numbers. 

962 

act in opposite directions. Although not previously interpreted for 

the case where Pe < 0, Equation 5 provides the transient concentration type curves and Equation 9 provides transient flux and 

cumulative flux type curves. Figures 4 through 6 show the concentration, flux, and cumulative flux type curves for solute or 

vapor transport. The values for tr on the horizontal axes of Figures 

4 through 6 are negative because q < 0, t > 0, and x > 0. Direct comparison of the type curves to the diffusive endmember solution given 

by Equations 3a and 3b would be possible if specific values of model 

parameters defining the dimensionless numbers were known. For 

Pe <0, the advective endmember solution is zero concentration 

and flux. In general, Figure 4 shows that for Pe < -3 the transport 

is advection dominated, and because flow and diffusion act in 

opposite directions, essentially no solutes or vapors can diffuse into 

the domain under this regime. The regime characterized by -3 < Pe 

s 0.1 is one of transition, as evidenced by the gradual increase in 

steady-state concentrations as Pe increases. Advection is negligible 

in the diffusion-dominated regime characterized by -0.1 < Pe < 0.0. 

The steady-state solution for the case of P e < 0 is as follows: 

c 

- = exp (Pe) (lla) 

Co 

J -=0 

qCo 

(lib) 

The steady-state solution is not useful in determining the relative 

importance between advection and dispersion, but it does provide 

a check for the asymptotic behavior of the type curves. Equation 11 b 

implies no net flux at steady state, therefore Equation lla defines 

the concentration profile for which advective and diffusive fluxes 

are equal and in opposite directions. 

Example - Unsaturated Zone Transport of 

Volatile Organic Compounds (VOCs) 

For this example, the source of VOCs is assumed to be near the 

water table, as would be the case downgradient from a leaking 

underground storage tank. Advection due to recharge acts downward 

toward the source and in the opposite direction of diffusion in 

unsaturated zone gas; therefore, the Peclet number for any VOC will 

be negative. The effective diffusion coefficient for the unsaturated 

zone gas phase is defined as follows: 

D = d S;r (12) 

where d is the diffusion coefficient for the VOC in the bulk gas phase 

(L2 IT), Sa is the air-filled porosity (dimensionless), and -r is the tortuosity (dimensionless). Because advection is occurring in the 

water phase, concentration is the selected dependent variable, the 

value for q used to calculate the Peclet number is defined as follows: 

(13) 

where qw is the recharge rate and Hwa is the Henry's law coefficient 

for the VOC. 

Table 1 provides parameter values for a hypothetical hydrologic 

setting used to calculate Peclet numbers for methyl tert-butyl ether 

(MTBE), benzene, and tetrachloroethene (PCE) which are Pe = -2.9, 

-0.22, and -0.06, respectively, for the reference hydrologic setting 

with an assumed observation location x = 100 em above the water 
Table 1 

Parameter Values for Reference Hydrologic Setting 

X 100 em 

qw 50 em/year 

8a 0.25 

8w 0.15 

'T 0.25 

d 0.08 cm2 I sec at 15°C for all compounds 

Hwa (from Robbins et al. 1993) 

MTBE- 91.7 

Benzene- 6.9 

PCE- 2.0 

table. Figures 4 and 5 indicate that these Pe imply transport regimes 

characterized as advection dominated for MTBE to diffusion dominated for PCE transport, with benzene transport somewhat intermediate but tending toward diffusion dominated. The implication 

is that recharge cannot be neglected in computing the rate of 

volatilization, a natural attenuation pathway at a spill site, for the 

gasoline additive MTBE. Neglecting recharge for benzene (and the 

other BTEX compounds because they have Hwa values similar to 

benzene) would be a meaningful approximation. For chlorinated 

VOCs illustrated by PCE, volatilization off of the water table is 

clearly diffusion dominated. 

Example - Cutoff Wall Design 

The use of cutoff walls made of low hydraulic conductivity 

material to contain contaminated ground water is becoming a common part of remedial solutions at hazardous waste facilities. These 

vertical barriers are used primarily to impede the advective flow of 

contaminated ground water. It has been documented, however, 

that diffusion can cause the transport of solutes outward through cutoff walls even with an inward hydraulic gradient. Devlin and 

Parker (1996) show that hydraulic conductivity is a critical optimization parameter for design of the cutoff wall to achieve the 

desired degree of containment. They concluded that the optimal 

hydraulic conductivity to prevent or minimize the migration of 

solutes through such barriers is not necessarily achieved by the lowest hydraulic conductivity. 

In the case where advection and dispersion are in opposite directions, such as those encountered when advection is inward and dispersion is outward through a cutoff wall, negative Peclet numbers 

result. The dimensionless type curves were used to evaluate transport through cutoff walls. Negative Peclet numbers were calculated 

for a hypothetical cutoff wall 1 m thick, an inward hydraulic gradient of -0.05, a diffusion coefficient of 1.0 X I0-6 cm2/s, and a 

range of hydraulic conductivity values of 1.0 X I 0-6 cm/s and 1.0 

X 10-8 cm/s. On the basis of these input parameters, the negative 

Peclet numbers were calculated according to Equation 6 to be -5 

and -0.05, respectively. For a Peclet number of -5, diffusion is not 

significant and solutes escaping from the containment are not an 

issue. For a Peclet number of -0.05, diffusion is significant and 

solutes may escape containment. These results are consistent with 

those of Devlin and Parker (1996), which show that outward solute 

migration actually is greater at a lower hydraulic conductivity if an 

identical inward hydraulic gradient is assumed. Thus, an optimal 

range of hydraulic conductivity for the cutoff walls exists that will 

permit outward dispersion to be balanced by inward advection 

and dispersion. 

Results of laboratory and theoretical studies indicate that the 

addition of a sorptive phase, such as fly ash, to soil-bentonite cutoff walls may significantly retard solute breakthrough (Mott and 

Weber 1989). For both positive and negative Peclet numbers, the 

type curves can be used to evaluate the sensitivity of the resultant 

concentration and flux to different retardation values. The storage 

term, which accounts for adsorption, is in the denominator of the 

dimensionless time term (x-axis). Thus, the addition of a sorptive 

phase will increase the storage term, s, causing a decrease in the 

dimensionless time term, tr. For any given Peclet number, this 

results in a shift to the left along the dimensionless type curve. For 

example, if the storage term, s, is increased from 1 to 2 by the addition of a sorptive phase, dimensionless time is reduced by a factor 

of 2, or halved. For a Peclet number equal to 10, the result is a 

decrease in normalized concentration from 0.58 to 0.08 (Figure 1). 

Thus, the addition of a sorbent phase to soil-bentonite mixtures can 

significantly improve the performance of such barriers. 

Summary 

This paper describes the use of dimensionless type curves to 

evaluate the relative importance of advection, diffusion, and 

mechanical dispersion for transport of vapors in the unsaturated zone 

and solutes in the saturated zone. Type curves are presented for concentration, flux, and cumulative flux and are defined in terms of the 

Peclet number and dimensionless time. Negative Peclet numbers 

characterize problems in which advection and diffusion act in 

opposite directions. Examples of problems characterized by a negative Peclet number are provided. The type curves are based on an 

analytical solution to the one-dimensional advection-dispersion 

transport equation. The type curves presented herein provide a 

formal method for classifying the regime of a particular transport 

problem, allowing for improved model selection. Such a method is 

beneficial in eliminating the use of a needlessly complex model for 

simple problems. Conversely, the type curves provide a formal 

method to alert the user when a problem requires inclusion of both 

advection and dispersion processes. 

Other factors need to be considered in the model selection 

process. For example, the type curves may show that an advectiononly model is adequate for a particualr flow regime. If the transport 

of nuclear waste is being simulated, however, even a minor amount 

of disprsion may be problematic. The model user must decide 

when advection of dispersion can be neglected. 

References 

Cars1aw, H.S., and J.C. Jaeger. 1959. Conduction of Heat and Solids, 2nd 

ed. New York: Oxford University Press. 

Devlin, J.F., and B.L. Parker. 1996. Optimum hydraulic conductivity to 

limit contaminant flux through cutoff walls. Ground Water 34, no. 4: 

719-726. 

Garges, J.A. 1996. The relative importance between advection and dispersion for solutes in the saturated zone and vapors in the unsaturated 

zone. M.Sc. thesis, Department of Civil Engineering, Drexel 

University, Philadelphia, Pennsylvania. 

Mott, H.V., and W.J. Weber. 1989. Solute migration control in soil-bentonite containment barriers. In Superfund 89, Proceedings of the 

lOth National Conference, sponsored by the Hazardous Materials 

Control Research Institute, Washington, D.C., November 27-29, 

526-533. 

Ogata, A., and R.B. Banks. 1961. A solution of the differential equation 

of longitudinal dispersion in porous media. U.S. Geological Survey 

Professional Paper 411-A. 

963 
Robbins, G.A., S. Wang, and J.D. Stuart. 1993. Using the static headspace 

method to detennine Henry's Law constant. Analytical Chemistry 65, 

no. 21: 3113-3118. 

Sauty, J.P. 1980. An analysis of hydrodispersive transfer in aquifers. 

Water Resources Research 16, no. I: 145-158. 

Appendix 

Derivation of Equation for Flux Type Curves 

The general expression for flux is given by Equation 8: 

(8) 

where C is defined by Equation S: 

C 1{ [(pe)t J c~ = 2 erfc 4t~ (1 - t,) 

l 

+ exp(Pe) erfc [ (::,r (1 + t,) J} (S) 

The term in Equation 8 with ( ~~) is obtained by differentiating 

EquationS. The first step of this differentiation is as follows: 

2 dC 

C0 dx 

I 

+ ddx { exp (P0 ) erfc [ (::,Y (1 + t,) J} 

The definition of erfc (w) is 

2 00 

erfc(w) = Vn L e-~' d-r 

(AI) 

(A2) 

Here w is a function of x and Leibniz's Rule of differentiation 

implies the following: 

d ( ) 2 , dw d~ erfc w = - ~ e- w d~ 

I 

For w = (pe)2 (1 - t,) the following applies: 4t, 

~ dw = ( ____\C P )f - d (1 - t) + (1 - t) ~-d ( ~ P )f dx 4t, dx r r dx 4t, 

(A3) 

which, given the definitions oft, and Pe provided by Equations 6 

and 7, implies 

~ dw = ( ____\C P )t ( ~ qt ) 1 ( +- --"- P )t (1 - t) dx 4t, SX2 X 4t, r 

964 

which upon simplification gives 

I I 

dw = (pc)2 [~ t, + ]_ (l _ t,)J = ]_ (pe)2 dx 4t, X X X 4t, 

I 

Therefore, w = (pe)2 (1 - t,) implies the following: 4t, 

I 

:x { erfc [ (::,Y (1 - t,) J} 

I 

= - __2__ exp [- (pe) (1 - t,)zJ! (pe) 2 

Vn 4t, X 4t, 

Similarly: 

I 

1~ { erfc [ (::,Y (1 + t,) J} 

- - 2 [ exp - -(pe) ( 1 + t, )2J -1 (pe)t - yJt 4t, X 4t, 

The other differention indicated in Equation AI is 

Equations Al, A4, AS, and A6 imply 

I 

- D dC 1 [ ( p ) J 1 ( p )2 qC0 dx = Vn exp - 4:, (l - tY P

0 

4:, 

(A4) 

(AS) 

(A6) 

I 

+ exp (Pe) ( ~) exp [ - (:~) (1 + t,? J ~e (:~) 2 

I 

- ~ exp (Pe) erfc [ (::,Y (1 + t,) J 

Which implies 

~C~ ~~ = ~ (Pet,) q exp [- (::J (1 - tYj 

+ exp (Pe) exp [- (::J (1 + t,)2J} 

I 

- ~ exp (Pc) erfc[ (::,Y (1 + t,) J (A7) 
Substituting Equation A 7 into Equation 8 gives the following 

expression for dimensionless flux: 

1 1 

J- = ! {erfc f(P~) 2 

(I - t,) + exp (P0 ) erfc f(Pe) 2 

(I + t,)]} ~ 2 l~ l~ 

+ exp (Pe) exp[- ( =~) ( 1 + t,Y ]} - i exp (Pe) erfc[ ( !~)(! + t,)] 

(A8) 

Further simplification of Equation A8 gives Equation 9 the equation for dimensionless flux used to generate the type curves: 

l 

_!_ = _1_ erfc I(Pe)

2 

(1 - t,.)J qC0 2 l 4t, 

I - 1

{ f (pe) 2l + l~ (Pet,) 2 expl - 4t, (1 - t,) J 

+ exp (Pe) exp [ - (:~J (I + t,) 2

J} (9) 

965