# Type Curves to Determine the Relative Importance ofAdvection and Dispersionfor Solute and Vapor Transport

by John A. Garges3 

### and Arthur L. Baehrb



### Abstract The relative importance ofadvection and dispersion for both solute and vapor transport can be determined from type curves forconcentration, flux, orcumulativeflux. The dimensionlessform ofthe type curves provides ameans to directly evaluate the importance ofmass transport by advection relative to that ofmass 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 provideadditional rationalefor transportregime determination in

### addition to the traditional concentration type curves. The extension oftype curves to include vapor transport with phase partitioningin the unsaturated zone is anew development. Type curves for negative Peclet numbers also are presented.Anegative Pecletnumber characterizes aproblem in which the direction offlow is 

### toward the contamination source, and thereby diffusion and advection canactin opposite directions. Examples are the diffusion ofsolutes away from the downgradient edge ofa pump-and-treat capture zone, the upward diffusion ofvapors through the unsaturated zone with recharge, and the diffusion ofsolutes through alowhydraulic conductivity cutoffwall with an inward advective gradient.


### Introduction In general, transport in porous mediais simulatedby 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 couldbeneglected. Foreithercase, 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 ofmultiple and reactive chemical species orparameter optimizationroutines. In the case ofadvection 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 orcharacteristic 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 orsolutes in low-permeability clay are likely, however, to be

•conestoga-Rovers andAssociates, 559 W. Uwchlan Ave., Ste. 120, Exton, PA 

### 19341. hU.S. Geological Survey, 810 BearTavern Rd., West Trenton, NJ 08628. 

### Received October 1997, accepted March 1998.
Vol. 36, No. 6--GROUNDWATER-November-December 1998 diffusion dominated. In the case ofdiffusion dominated transport, head and permeability distribution are not required for transport simulation. Aformal method for classifying the regime ofa particular transport problem is presented. The method allows for improved model selection, thereby eliminating the use ofa needlessly complex model for simple problems. Conversely, the method will alert the user when aproblemrequires inclusion ofadvection, diffusion, and mechanical dispersion. Sauty (1980) presented solute concentration breakthrough curves in terms ofPeclet number and dimensionless time based on the solution to the one-dimensional advection-dispersion equation presented by Ogata and Banks (1961). Thedimensionless form of the type curves provides a means to directly evaluate the importance ofmass transport by advection relative to that ofmass transport by dispersion (including diffusion). The higher the Peclet number, the more dominant advective transport is relative to dispersive transport. A more general form ofthe advection-dispersion equation thatincludes phase partitioning and the associated generalization ofthe Peclet number and dimensionless time is presented here so that type curves can be interpreted to assess the transportregime for sorbing solutes and water phase partitioning ofvapors. Flux and cumulative flux (total mass passing through a point over a specified time) breakthrough curves are derived to provide additional criteria to concentrationbreakthrough curves for evaluating the relative importance of advection and dispersion. These flux-based type curves to date have not been documented in the literature. Anegative Pecletnumber characterizes aproblem in which the direction offlow 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 ofproblems characterized by a negative Peclet number
959

### include the diffusion ofsolutes away from the downgradient edge ofa pump-and-treat capture zone, the upward diffusion ofvapors 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 TransportModel The mathematical statementofthe model is identicalfor both solute transportin an aquifer and vaportransport in the unsaturated zone, and is given as follows: (PDE) 

# ac a 2

# c ac

**s---=D~-q**



# at a x2 

# ax (1) 

### (BC) C 

## = C0 

### atx = 0 lim C = 

### 0

**x~oo** 

### (IC) 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 orgaseous phase (L2/T); and q is the specific discharge ofthe aqueous orgaseous phase (LIT). The definition of s, the storage coefficient, depends on the definition ofC (solute or vapor), and in either case assumes equilibriumpartitioning among phases in the porous medium. In general, the storage coefficient causes the solute orvapor transport to be retarded because ofphase partitioning. If

### C is solute concentration, then the pore space ofthe porous medium is assumed to be saturated and
s = 

### ($ 

### + KctPb) where <(lis porosity, Kct is the adsorption coefficient (0/M), and pb is bulkdensity (M/0). If

### Cis vaporconcentrationthen 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 coefficientbetweenthe gaseous and adsorbed phases. Note thatfor the case ofunsaturated zone transport and the selection ofC as vapor concentration, the model allows for advection in the waterphase and diffusion in the airphase simply by interpreting q to be the product ofHwa and the specific discharge for the waterphase.

### A semi-infinite domain was specified for the general analysis to eliminate sensitivity to an arbitrarily located downstreamboundary. The only mathematical difference between the model presented here and that of Ogata and Banks (1961) is the inclusion ofthe general storage coefficient, s, that includes gaseous and adsorbed phases. For the case ofadvection 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

= 

### 1iftr 

### > 1 h t 

### qt . d" . 1 . w ere r = - IS 

### 1mens10n ess time. sx
960 (2a) Equation 2aimplies the following solution for flux:
(2b)
where J is mass flux (MIL2

### T). Forthe case ofdiffusion only, q

## = 0 in Equation 1and 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): 

# c (



# ~) Co = 

### erfc 

# V4m (3a)

Pick's law (J = - 

### D



# ~~) and Equation 3a imply the following equation for mass flux for the case ofdiffusion 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 numberthat scales the ratio ofadvection to dispersion, is defined as follows: qx P=-

e 

### D (6)


### In this definition ofPe• 

### the spatial coordinate x is the characteristic distance. The definition ofdimensionless time tr for Equation 5 is modified slightly from that definedby Sauty (1980) to include the generalized storage coefficient s as follows:

### qt tr =- 

### sx 

### (7)
Dimensionless time, tr, is related to the number ofpore 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)2

### J


## + exp (Pc) exp [

- 

# (:;J (1 

## + tr) 2

# j} (9)
The derivation ofEquation 9 from Garges (1996) is outlined in the appendix. Cumulative mass flux over time tr is obtained by numerically integrating Equation 9. 

## Type Curves forPositive Peclet Numbers For the model coordinate system, Pe 

### > 0 if and only ifq 

### > 0 (flow in direction away from source) and advection and diffusion act in the same direction. Acomputer program was written to calculate transient concentration, flux, and cumulative flux type curves from Equations 5, 9, and the integration ofEquation 9, respectively. Figures I through 3 show the concentration, flux, and cumulative flux dimensionless type curves for solute orvapor transport. Direct comparison ofthe type curves to the endmember solutions given by Equations 2a, 2b, 3a, and 3b would be possible ifspecific values of model parameters comprising the dimensionless numbers were known. In general, however, even though the type curves do not define distincttransport regimes, diffusion can generally be said to bethe dominant transport mechanismfor 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 curvesfor positive Peclet numbers. 

### 0 and 

### 1. Pecletnumbers from 1to 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 ofPe

### >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 ofthe type curves. 

## Type Curvesfor 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. 

### 9611.0

0.8

0.6
0 0 0 0.4

0.2

0.0 ·1 . 2 ·3 ·4 . 5 tr = qt/sx Figure4. Dimensionless concentrationtype curvesfor negative Peclet numbers.

0.0

·1 . 2 . 3 ·4 ·5 tr = qt/sx Figure5. Dimensionless flux type curvesfor negative Peclet numbers.

-0 -1.0 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 notpreviously interpretedfor 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 vaportransport. The values for tr on the horizontal axes ofFigures 

### 4 through 

### 6 are negative because q 

### <0, t 

### >0, and x 

### >0. Directcomparison ofthe type curves to the diffusive endmember solution given by Equations 3aand 3bwould be possibleifspecific values ofmodel 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 orvapors can diffuse into the domain underthis regime. Theregime characterizedby -3 

### <Pe s 0.1 is one oftransition, as evidenced by the gradual increase in steady-state concentrations as Peincreases. Advection is negligible in the diffusion-dominated regime characterizedby

### -0.1 

### <Pe 

### <0.0. The steady-state solution for the case ofP 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 itdoes provide a checkfor the asymptotic behaviorofthe typecurves. Equation 11b implies no netflux 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) Forthis example, the source ofVOCs is assumed to benearthe 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 ofdiffusion in unsaturated zone gas; therefore, thePeclet numberfor any VOC will be negative. Theeffective diffusion coefficientfor the unsaturated zone gas phase is defined as follows: 

### D 

## = d S;r (12) whered is the diffusion coefficientfor 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 waterphase, 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 1provides parametervalues for a hypotheticalhydrologic setting used to calculatePecletnumbers for methyl tert-butyl ether (MTBE), benzene, andtetrachloroethene (PCE) which are Pe = -2.9, 

### -0.22, and

### -0.06, respectively, for thereference hydrologic setting with an assumed observation location x

## =100em above the water

### Table 1 ParameterValues 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 MTBEto 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 - CutoffWall Design The use of cutoff walls made oflow hydraulic conductivity material to contain contaminated ground water is becoming a commonpart ofremedial 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 ofsolutes 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 casewhere advection and dispersion are in opposite directions, such as those encountered when advection is inward anddispersion is outward through a cutoffwall, negative Pecletnumbers result. The dimensionless type curves were used to evaluate transportthrough cutoffwalls. Negative Peclet numbers were calculated for a hypothetical cutoffwall 1 m thick, an inward hydraulic gradient of-0.05, a diffusion coefficient of 1.0 X I0-6 cm2/s, and a range ofhydraulic conductivity values of 1.0 X I0-6 cm/s and 1.0 X 10-8 cm/s. Onthe basis ofthese 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 ofDevlin and Parker (1996), which show that outward solute migration actually is greater at a lower hydraulic conductivity ifan identical inward hydraulic gradient is assumed. Thus, an optimal range ofhydraulic conductivity for the cutoffwalls exists that will permit outward dispersion to be balanced by inward advection and dispersion. Results oflaboratory and theoretical studies indicate that the addition ofa 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 ofthe resultant concentration and flux to different retardation values. The storage term, which accounts for adsorption, is in the denominator ofthe dimensionless time term (x-axis). Thus, the addition ofa 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, ifthe storage term, s, is increased from 1to 2 by the addition ofa sorptivephase, 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 ofa sorbent phase to soil-bentonite mixtures can significantly improve the performance ofsuch barriers. 

### Summary This paper describes the use ofdimensionless type curves to evaluate the relative importance of advection, diffusion, and mechanical dispersion for transport ofvapors 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 ofthe Peclet number and dimensionless time. Negative Peclet numbers characterize problems in which advection and diffusion act in opposite directions. Examples ofproblems 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 ofa particular transport problem, allowing for improved model selection. Such a method is beneficial in eliminating the use ofa needlessly complex model for simple problems. Conversely, the type curves provide a formal method to alert the user when a problemrequires inclusion ofboth advection and dispersion processes. Other factors need to be considered in the model selection process. Forexample, the type curves may show that an advectiononly model is adequate for a particualrflow regime. Ifthe transport ofnuclear waste is being simulated, however, even a minor amount of disprsion may be problematic. The model user must decide when advection ofdispersion can be neglected. 

### References Cars1aw, H.S., and J.C. Jaeger. 1959. Conduction of Heat andSolids, 2nd ed. New York: Oxford University Press. Devlin, J.F., and B.L. Parker. 1996. Optimum hydraulic conductivity to limit contaminantflux through cutoffwalls. 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 ofthe 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 ofthe differential equation oflongitudinal dispersion in porous media. U.S. Geological Survey Professional Paper411-A. 963Robbins, 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. WaterResources Research 16, no. I: 145-158. 

## Appendix Derivation ofEquationfor FluxType 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 ofthis differentiation is as follows: 

### 2 dC 

### C0 dx
I 

### + ddx { exp (P 0 ) erfc [

# (::,Y (1 

### + t,)J} The definition oferfc (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 ( P

# )f d d ( P

# )f**~** = ____\C

- 

### (1

- t) 

### + (1

- 

### t)**~- ~**  dx 4t, dx r r dx 4t, (A3)

which, given the definitions oft, and 

### Pe provided by Equations 6 and 7, implies

dw ( P

# )t( qt ) 1 ( P

# )t**~** = ____\C**~** 

## +- --"- (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 [ 

# (pe) ( )2J 1

# (pe)t

- - exp - - 

### 1 

### + 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 P0 

# 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,)2

# J}

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 (P 0 ) 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

qC 0 2 l 4t, I - 1 { f (pe) 2l + **l~** (Pet,) 2 expl

- 4 t, (1
- t,) J + exp (Pe) exp [
- **(:~J** (I + t,) 2 J} (9) 965