Differences in the scale-dependence of dispersivity estimated from temporal and spatial moments in...

Advances in Water Resources 28 (2005) 745–759

www.elsevier.com/locate/advwatres

Differences in the scale-dependence of dispersivity estimatedfrom temporal and spatial moments in chemically and

physically heterogeneous porous media

Daniel Fernandez-Garcia a,*, Tissa H. Illangasekare b, Harihar Rajaram c

a Universidad Politecnica de Valencia, Departamento de Ingenierıa Hidraulica y Medio Ambiente, Camino de vera s/n, 46071 Valencia, Spainb Colorado School of Mines, Environmental Science & Engineering, 1500 Illinois St., Golden, CO 80401, USA

c University of Colorado, Civil, Environmental, & Architectural Engineering, Engineering Center Office Tower 441, 428 UCB, Boulder, CO 80309, USA

Received 19 February 2004; received in revised form 10 November 2004; accepted 28 December 2004

Available online 5 March 2005

Abstract

Tracer tests designed to estimate field-scale dispersivities are commonly based upon the interpretation of breakthrough curves.

Implicitly, no distinction is made between these dispersivity values and those inferred by analyzing the evolution of tracer plumes.

Although this assumption is reasonable in ideal homogeneous media, its applicability to complex geologic formations is unclear.

Recent laboratory tracer tests in a heterogeneous test aquifer have suggested that some differences may exist. This work provides

computational investigations aimed to study the meaning and differences of these two types of dispersivity estimates in three-dimen-

sional chemically and physically heterogeneous porous media. Specifically, the scale-dependence of longitudinal dispersivities for

conservative and linearly sorbing tracers estimated from temporal moments of breakthrough curves are compared with those

obtained from spatial moments of tracer plumes in uniform flow systems. The scale-dependence of dispersivity from spatial and

temporal moments was found to be identical for small r2lnK and r2

R. For larger values of r2lnK and r2

R ðr2lnK > 0:5; r2

R > 0:5Þ, how-

ever, the dispersivities estimated from temporal moments approach a constant value at smaller distances than estimates obtained

from spatial moments. Yet, both dispersivities asymptotically approach the same constant value at large travel distances. From

a practical standpoint, it is also shown that accurate field dispersivity coefficients can be obtained from uniform flow tracer test

by simply using few fully-penetrating observation wells, bypassing the need for more expensive tracer techniques based upon the

spatial description of concentrations.

� 2005 Elsevier Ltd. All rights reserved.

Keywords: Dispersivity; Heterogeneity; Tracer tests; Moment analysis; Particle tracking; Stochastic subsurface hydrology

1. Introduction

Gathering detailed spatial information on a tracer

plume at several times requires large arrays of multilevel

samplers [19,31], requiring prohibitive cost and effort.

0309-1708/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.advwatres.2004.12.011

* Corresponding author.

E-mail addresses: [email protected] (D. Fernandez-Garcia),

[email protected] (T.H. Illangasekare), [email protected] (H. Raja-

ram).

As a result, dispersivities are often estimated in the

field by simply recording breakthrough curves (BTCs)at a few observation wells [47,48,59]. These observation

wells are typically deep-penetrating wells with screens

larger than the vertical heterogeneity scale so that they

provide average flux concentrations along the sampling

depth. Traditionally, it is implicitly assumed that disper-

sivities estimated from BTCs obtained at fixed locations

and values interpreted from spatial moments are equiv-

alent. For instance, the review of data on field-scale dis-persion in aquifers of Gelhar et al. [22] illustrated the

mailto:[email protected]



746 D. Fernandez-Garcia et al. / Advances in Water Resources 28 (2005) 745–759

scale-dependence of longitudinal dispersivity but did not

differentiate between these two dispersivities. In hetero-

geneous porous media where the dispersivity is a scale-

dependent parameter and solute particles may be

favoring different preferential paths, the relationship be-

tween these two dispersivities is unclear. For instance, inrecent uniform flow tracer tests conducted in a well-de-

fined heterogeneous test aquifer [17,18] longitudinal dis-

persivities estimated from BTCs approached a constant

value in less than two horizontal correlation scales kH, in

contrast with the Borden [33] and Cape Cod aquifers

[31] where longitudinal dispersivities estimated from

the evolution of tracer plumes approached constant val-

ues in about 18kH and 15.4kH, respectively. The objec-tive of this paper is to study the differences in the

systematic scale-dependence of these two conceptually

different dispersivity estimates in physically and chemi-

cally heterogeneous porous media in an ensemble aver-

age sense. Only longitudinal dispersivity is considered

because, in general, field tracer test based on BTC meth-

ods cannot infer transverse dispersivity. Monte Carlo

numerical simulations of uniform flow tracer tests weredesigned to simultaneously calculate dispersivities from

temporal moments of the ensemble average BTC ob-

tained at control planes and dispersivities from spatial

moments of the ensemble average tracer plume obtained

at different times across 20 realizations. Transport of

conservative and linearly sorbing tracers were simulated.

The erratic variations of dispersivities estimated from

temporal and spatial moments that should be expectedin single realizations of the random fields are also illus-

trated by plotting the +/� two standard deviation

bounds around the ensemble average (these are 95%

c.i.s. only if the underlying variable has a Gaussian dis-

tribution). The simulations evaluated behavior across

different degrees of heterogeneity ðr2lnK ; r2

RÞ and anisot-

ropy ratios in the correlation scale kH/kV.

Our results show that for both conservative and line-arly sorbing tracers, dispersivities from temporal mo-

ments approach the asymptotic dispersivity value more

rapidly than those dispersivities estimated from spatial

moments. The reason for the difference may be ex-

plained as follows: When dispersivities are estimated

from BTCs observed at control planes, all particles are

required to travel the same longitudinal distance from

the tracer source so that they efficiently sample the sameportion of the heterogeneous porous media, but when

dispersivities are estimated from spatial moments, parti-

cles forming the tail of the tracer plume sample a smaller

portion of the porous media than those comprising the

front of the plume. At small travel distances and for

highly heterogeneous aquifers, this effect is more pro-

nounced and dispersivities from temporal moments

approach a constant value more rapidly than their anal-ogous dispersivities from spatial moments. In all cases,

at large distances from the source, this phenomenon is

irrelevant and both dispersivities approach the same

asymptotic constant value.

The organization of the paper is the following: Sec-

tion 2 describes the transport problem and numerical

features, emphasizing the methodology used to numeri-

cally calculate the temporal moments of ensemble aver-age BTCs in particle tracking transport codes. In

Section 3.1, we demonstrate the influence of degree of

heterogeneity on the differences in behavior between dis-

persivities estimated from temporal moments and esti-

mates obtained from spatial moments. Section 3.2

evaluates the non-Gaussian shape of BTCs based on

the evolution of the coefficients of skewness and kurtosis

with travel distance. In Section 3.3, we present an illus-trative simulation of a field-scale uniform flow tracer

test to demonstrate the approach for estimating disper-

sivities from temporal moments. Finally, Section 4 sum-

marizes the main results and conclusions from this

paper.

2. Design of computational investigations

2.1. Transport problem and heterogeneity

Under steady-state flow conditions, at the representa-

tive elemental volume (REV) scale, linearly sorbing sol-

ute transport through heterogeneous porous media is

governed by the following differential equation [3,20],

/Rð~xÞ oCot

¼ �X3

i¼1

qið~xÞoCoxi

þX3

i¼1

X3

j¼1

o

oxi/Dijð~xÞ

oCoxj

� �

ð1Þ

where C is the dissolved concentration of solute in the

groundwater, / is the porosity, qi is the ith component

of Darcy�s velocity, qið~xÞ ¼ �Kð~xÞoh=oxi, Dij is the dis-

persion tensor, R is the retardation factor, h is the

hydraulic head, and K is the hydraulic conductivity, as-sumed isotropic at the REV scale. Solute transport was

simulated according to (1). Conservative tracers were

only transported by advection and dispersion. Reactive

tracers were further subjected to sorption processes rep-

resented by a reversible linear equilibrium isotherm,

which considers that sorbed solute mass is proportional

to the concentration of dissolve solute by a factor

known as the distribution coefficient Kd [20]. This isfrequently the case for many non-polar organic hydro-

phobic substances dissolved in groundwater [51]. The

retardation factor is expressed as Rð~xÞ ¼ 1þKdð~xÞqb=/,

where qb is the bulk density of the soil.

The natural logs of hydraulic conductivity lnK and

distribution coefficient lnKd were modeled as two spa-

tially correlated field variables. The lnKd variable was

assigned to be perfectly correlated with the lnK variableassuming a linear negative model. Field observations

D. Fernandez-Garcia et al. / Advances in Water Resources 28 (2005) 745–759 747

[5,21,34,43,44] as well as laboratory column and batch

experiments [7,14,15] suggest that sorption properties

in an aquifer are spatially variable. For instance, distri-

bution coefficients for strontium on 1279 subsamples of

cores from the Borden aquifer gave Kd values that ran-

ged from 4.4 to 29.8 ml/g with a mean of 9.9 ml/g andstandard deviation of 2.89 ml/g [44]. In addition, most

of the very low K values were visually associated with

zones of high Kd values for strontium in the Borden

aquifer [44], indicating a negative correlation between

the two variables. Based on these field observations, sev-

eral authors [6,42,56] have used the perfect linear nega-

tive model.

2.2. Computational features

The computational domain is discretized into a regu-

lar mesh formed by 250 · 250 · 200 parallelepiped

cells. The resolution is of five grid cells within a corre-

lation scale in all directions. The lnK and lnKd fields

were generated using the turning bands algorithm

[54] and followed multi-normal distributions withexponential covariance. Each realization is obtained

from 400 randomly placed lines. The maximum nor-

malized frequency was 100 (kmaxk = 100), with a

normalized frequency increment of 0.05 (Dkk = 0.05).

Retardation factors were estimated with the following

expression,

Rð~xÞ ¼ 1 þ qb

/Kdg exp bðlnKð~xÞ � hlnKiÞ½ ð2Þ

where Kdg is the geometric mean of Kd, and b reflects the

relationship between lnKd and lnK. In this work, Kdg,

qb, and / were similar to the Borden aquifer 0.526 ml/

g, 1.81 g/cm3, and 0.35, respectively. The parameter b

was set equal to �0.5. When sorption processes are com-

pletely linked to grain surface areas, Garabedian et al.[21] showed that a power law relationship between con-

ductivity and mean grain radius yields b = �0.5. The

spatial correlation structure of the hydraulic conductiv-

ity field used in our simulations resembles those from the

Borden aquifer [19,33,52]. The geometric mean of the K

field was set equal to 116.7 m/day, which is a reasonable

value for a sandy aquifer.

The hydraulic conductivity field is incorporated intoa seven-point finite difference groundwater flow model,

MODFLOW2000 [24]. Upstream and downstream

boundaries are specified as constant heads, such that

the hydraulic gradient in the mean flow direction is

0.004. No-flow conditions are prescribed at the trans-

verse, top and bottom boundaries. The model calculates

the flow rates at the grid interfaces, which yields the

velocity field. The porosity is assumed to be spatiallyhomogeneous with a value of 0.35. This velocity field

is then used in a random walk particle tracking trans-

port code [28,30,32,55,56,58] that simulates the solute

migration by partitioning the solute mass into a large

number of representative particles; moving particles

within the velocity field simulates advection, whereas a

superposed Brownian motion is responsible for disper-

sion. This method is well suited for large number of sim-

ulations for the simultaneous computation of temporaland spatial moments, and it is also free of numerical

dispersion.

When moving particles, the velocity of a particle at a

given location has two components [28,56]: the flow

velocity, as used in any particle tracking code, plus the

divergence of the dispersion tensor. These two compo-

nents are found from interface grid velocities by using

an interpolation scheme. A linear interpolation of inter-face velocities is used to compute groundwater velocities

carrying a particle within a cell. This methodology leads

to local as well as global divergence-free velocity fields

within the solution domain [40], and it has been proved

to yield accurate solutions to the transport problem in

heterogeneous random fields [8]. The dispersion tensor

field is obtained by first extrapolating interface velocities

to surrounding nodes. This gives all three componentsof the vector pore-velocity to each grid node, which is

then used to estimate the dispersive component of the

random walk using trilinear interpolation. As pointed

out by Labolle et al. [27], this hybrid scheme yields a

continuous dispersion tensor field that satisfies mass bal-

ance at grid interfaces of adjacent cells with contrasting

hydraulic conductivities. The particle displacement was

calculated with [56]

Xp;iðt þ DtÞ ¼ Xp;iðtÞ þ Aið~XpðtÞÞDt

þX3

j¼1

Bijð~XpðtÞÞZj

ffiffiffiffiffiDt

pi ¼ 1; 2; 3 ð3aÞ

where Xp,i(t) is the i-component of the particle location

at time t, Dt is the time increment (time discretization

was calculated based on the grid Courant number,

v Æ Dt/Dx, set equal to 1/80), and Zj is a vector which

contains three normally distributed random numbers

with zero mean and unit variance. The drift term ofthe random walk Ai is given by [50]

Ai ¼vi ~XpðtÞ� �

þP3

j¼1

oDij

oxj~XpðtÞ� �

Rp~XpðtÞ� � i ¼ 1; 2; 3 ð3bÞ

where vi is the i-component of the velocity vector, Dij

is the dispersion tensor with eigenvalues associated

with principal axes parallel and perpendicular to the

direction of flow defined as DI = aLv + Dm, DII = aTv +Dm, DIII = aTv + Dm (aL and aT are respectively the

longitudinal and transverse pore-scale dispersivity, v

is the magnitude of the velocity, and Dm is the molecu-

lar dispersion that was set to zero), and Rp is the retar-

dation factor. The displacement matrix Bij has the form

[32]


fBijg ¼

v1v

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2DI=Rp

p �v1v3ffiffiffiffiffiffiffiffiffiffiffi2DII=Rp

p

vffiffiffiffiffiffiffiffiffiv21þv2

2

p �v2ffiffiffiffiffiffiffiffiffiffiffiffi2DIII=Rp

pffiffiffiffiffiffiffiffiffiv21þv2

2

p

v2v


p �v2v3ffiffiffiffiffiffiffiffiffiffiffi2DII=Rp

p

vffiffiffiffiffiffiffiffiffiv21þv2

2

p v1ffiffiffiffiffiffiffiffiffiffiffiffi2DIII=Rp

p ffiffiffiffiffiffiffiffiffiv21þv2

2

p

v3v


p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

v21þv2

2

v2 DII=Rp

q0

26666664

37777775ð3cÞ

Initially, a large number of particles (10,000) are ran-

domly distributed in a plane transverse to the mean flow

direction. This plane is located three correlation scalesaway from the upgradient boundary to avoid boundary

effects [37,45,46]. The shape of the particle source is a

rectangle centered within this plane. The source size is

40 correlation scales in the transverse direction to the

mean flow and 30 correlation scales in the vertical direc-

tion. This leaves a gap of 5-correlation scales between

boundaries and source. Dispersivities from spatial mo-

ments reported in this work are based on particle distri-butions at times when none of the particles has exited

the domain.

2.3. Evaluation of dispersivities from spatial moments

Spatial moments of aqueous concentrations distrib-

uted in space in a single realization are calculated fol-

lowing the approach developed by Tompson and

Gelhar [55] for conservative tracers and Tompson [56]

for reactive tracers. Spatial moments are calculated from

snapshots of particles at given times as follows:

mðtÞ ¼Z

/Cvð~x; tÞd~x XNPt

k¼1

mðkÞp

Rp X ðkÞp ðtÞ

� � ð4aÞ

XG;iðtÞ ¼1

mðtÞ

Zxi/Cvð~x; tÞd~x

1

mðtÞXNPt

k¼1

mðkÞp X ðkÞ

p;i ðtÞ

Rp X ðkÞp ðtÞ

� �ð4bÞ

SijðtÞ ¼1

mðtÞ

Zðxi � XG;iðtÞÞðxj � XG;jðtÞÞ/Cvð~x; tÞd~x

1

mðtÞXNPt

k¼1

mðkÞp X ðkÞ

p;i ðtÞXðkÞp;j ðtÞ

Rp X ðkÞp ðtÞ

� � � XG;iðtÞXG;jðtÞ ð4cÞ

where m(t) is the total liquid phase solute mass, mðkÞp is

the mass assigned to the kth-particle (set constant for

all particles), / is the porosity, Cv is the concentrationof the solute plume as extracted in a certain volume of

pore water, typically known as resident concentrations

of solute [29,39,50], XG,i is the center of mass, Sij is the

liquid phase second spatial moments associated with

the distribution of particles at a given time, Rp is the

retardation factor at the particle position, NPt is the

number of particles in the system at time t, and X ðkÞp;i is

the i-component of the kth particle location. For conser-

vative transport simulations, retardation factors were set

equal to 1. Following Kitanidis [26], Dagan [14] and

Rajaram and Gelhar [41], the second spatial moment

tensor, Mij, of the ensemble average concentration in

space across all realizations was computed as

MijðtÞ ¼ RijðtÞ þ RijðtÞ ð5aÞ

RijðtÞ ¼ hSijðtÞi ð5bÞ

RijðtÞ ¼ hfXG;iðtÞ � hXG;iðtÞigfXG;jðtÞ � hXG;jðtÞigi ð5cÞwhere Rij(t) is the ensemble average of the second mo-

ment tensor Sij(t), and Rij(t) is the covariance matrix

of the center of mass in a single realization with respect

to the ensemble mean center of mass location. Through-

out this paper, variables in angle brackets refer to theensemble average across all realizations. This approach

avoided small source size effects on dispersivity and lim-

ited the analysis to situations where the initial spatial ex-

tent of the plume is sufficiently large to sample the

porous media. In numerical simulations, it was seen that

the contribution of Rij(t) and RT(x1) to the second spa-

tial and temporal moment, (5) and (10), respectively,

were negligible, meaning that the source dimensionswere sufficiently large to capture most of the random

fluctuations expressed by the velocity spectra. Dispersiv-

ities from spatial moments of ensemble average tracer

plumes were calculated as [19]

A11ðhnGðtÞiÞ ¼M11ðhnGðtÞiÞ

2hnGðtÞið6Þ

where nG(t) is the travel distance of the center of mass ofthe tracer plume in the mean flow direction at time t.

This is applicable to conservative as well as reactive trac-

ers. Dispersivities as given by (6) are viewed as equiva-

lent values in homogeneous porous media that when

used with the classic reactive advection–dispersion equa-

tion, leads to the same first two spatial moments of the

tracer plume as observed in the simulations.

2.4. Evaluation of dispersivities from temporal moments

Temporal moments associated with observed BTCs

at several predefined control planes were computed

using a Lagrangian framework. Monitoring the first-

passage time of particles passing through control planes

allows for the estimation of BTC temporal moments

without having to evaluate the actual shape of theBTC. The nth-absolute temporal moment can be calcu-

lated as the expected value of the arrival time of a par-

ticle at the control plane to the nth power [16,49],

l0nðx1Þ ¼

1

mtot

Z 1

0

tnQCfðx1; tÞdt

1

mtot

XNPa

k¼1

mðkÞp tðkÞp ðx1Þ� �n

; mtot ¼XNPa

k¼1

mðkÞp ð7Þ


where Q is the total water flux passing through the

control plane, x1 is the mean flow direction coordinate,

Cf is the flux concentration of solute passing through

a given surface [29,39,50], tðkÞp is the first arrival passage

time of the kth particle, mtot is the total mass injected,

and NPa is the total number of particles arrived atthe x1-control plane. Note in (7) that QCf(x1, t)/mtot

is the rate of solute mass passing through the x1-control

plane at time t normalized by the total mass injected,

which is the probability density function for the mass

flux [16,49]. The particle retardation factor is not in

(7) as in (4) because we are interested in the total mass

flux BTC. The advantage of this methodology is that

avoids constructing the entire BTCs and subsequentintegration of concentrations over time in (7). Evalua-

tion of the entire BTC from the distribution of particle

arrival times at control planes requires smoothing tech-

niques which could lead to errors in temporal moment

estimates.

First-passage arrival times are computed by finding

the time at which the particle first crosses the control

surface. This was done based on the particle locationinformation right before Xp(t

n) and after Xp(tn+1) pass-

ing through the control surface. In general, this sup-

poses to solve the system of equations

~Xpðtn þ DsÞ ¼ ~XpðtnÞ þ k~Xpðtnþ1Þ � ~XpðtnÞk=ðtnþ1 � tnÞDs; ~Xpðtn þ DsÞ 2 CCS ð8Þ

Here Ds is the unknown incremental time from

the position of the particle previous to intersect the con-

trol surface, tn is the particle travel time previous inter-

section, and tn+1 is the first particle travel time after

passing through the control surface. The second equa-

tion states that the particle is located at the control sur-

face, and its explicit expression depends on the shape ofthe control surface (i.e., observation wells or control

planes).

The nth-absolute temporal moment of the ensem-

ble average BTCs, M 0T ;nðx1Þ, is simply the average

of the nth-absolute temporal moments across all

realizations,

M 0T ;nðx1Þ ¼

1

mtot

Z 1

0

tnhQCfðx1; tÞidt ¼ hl0nðx1Þi

1

N r

XN r

m¼1

l0ðmÞn ðx1Þ ð9aÞ

where l0nðx1Þ is the nth-absolute BTC temporal moment

obtained at the x1-control plane in a single realization of

the aquifer, Nr is the number of realizations, and the

superscript m indicates the realization number. The

nth central temporal moment of the ensemble average

BTCs is then calculated using the relationship betweencentral and absolute temporal moments by Kendall

and Stuart [25],

MT ;nðx1Þ ¼1

mtot

Z 1

0

t �M 0T ;1ðx1Þ

� �nhQCfðx1; tÞidt

¼Xn

r¼0

n

r

� �M 0

T ;n�rðx1Þð�M 0T ;1ðx1ÞÞr ð9bÞ

Using Eqs. (7) and (9), it is interesting to explicitlywrite the second central temporal moments of the

ensemble average BTC as

MT ;2ðx1Þ ¼ RT ðx1Þ þ RT ðx1Þ ð10aÞ

RT ðx1Þ ¼ hl2ðx1Þi ð10bÞ

RT ðx1Þ ¼ l01ðx1Þ �M 0

T ;1ðx1Þh i2

� �ð10cÞ

Analogous to Eq. (5) for spatial moments, it is seen

that the second temporal moment of the ensemble aver-

age BTC is expressed as the ensemble average of the sec-

ond central temporal moment (first term) plus a second

term accounting for the squared deviation of the first

absolute temporal moments in a single realization fromthe ensemble average arrival time. Dispersivities from

temporal moments of ensemble average BTCs were cal-

culated as [2,23,57]

A11ðnP Þ ¼nP

2

MT ;2ðnP ÞðM 0

T ;1ðnP ÞÞ2ð11Þ

where nP is the distance between the tracer source and

the x1-control plane. This is applicable to conservative

as well as reactive tracers. Dispersivities as given by

(11) are viewed as equivalent homogeneous porous

media values that when used with the classic reactive

advection–dispersion equation leads to the same first

two temporal moments of the BTCs as observed in the

simulations.

2.5. Validation of numerical models

Model validation is a crucial step towards under-

standing the numerical errors inherent in the results.

Overall, it will be shown that our numerical simulations

are highly accurate. The process of generating realiza-

tions of random fields is evaluated based on the firsttwo sample statistical moments (mean and covariance

function). In agreement with Ababou et al. [1] and Chin

and Wang [8], parameters selected for the turning bands

algorithm were verified to be adequate. Fig. 1 shows the

average semivariogram and 95% confident intervals

from 20 realizations. The covariance function of the

velocity fields obtained from MODFLOW that will be

used with the transport code are compared to theapproximate analytical expression of Zhang and Neu-

man [60] for statistically isotropic media with r2lnK ¼

0:1. Fig. 2 exhibits a good match between the covariance

function of the numerical velocity field and the analyti-

cal expression. The transport code was validated by

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2 3 6 9 10

Distance [correlation scales]

Sem

ivar

iogr

am

Vertical DirectionHorizontal DirectionExponential Model

1 4 5 7 8

Fig. 1. Validation of the generation of random lnK fields for highly

stratified porous media ðr2lnK ¼ 1:0; kH=kV ¼ 10Þ.

0.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

2.5E-04

3.0E-04

3.5E-04

4.0E-04

4.5E-04

5.0E-04

0 2 8 10 12 14

Dimensionless Distance [x1 / λh]

u 11 -

Flow

Dire

ctio

n

Zhang & Neuman Analytical Solution [1992]

Numerical Velocity Field

σ2

λH/λV = 1

lnK = 0.1

4 6

Fig. 2. Comparison of the covariance function of the simulated

Eulerian velocity field in the mean flow direction to the Zhang and

Neuman [60] analytical solution for statistically isotropic porous media

ðr2lnK ¼ 0:1; kH=kV ¼ 1Þ.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 5 10 15 20 25 30

<ξG>/λH, ξP/λH

A11

/σ2

ln K

λH

Dispersivity from Spatial Moments

Dispersivity from Temporal Moments

Dagan's Isotropic Model [1984]

σ2

λH/λV = 1

ln K = 0.1

σ2

R = 0

Fig. 3. Validation of the transport numerical code with Dagan�s [11]

model for conservative tracers (aL = aT = 0) and statistically isotropic

porous media ðr2lnK ¼ 0:1; kH=kV ¼ 1Þ. It also illustrates that the scale-

dependence of dispersivity estimated from temporal moments of BTCs

and spatial moments of tracer plumes is the same for small r2lnK .


considering both isotropic and anisotropic random lnK

fields: simulated spatial moments of solute transport in

heterogeneous isotropic porous media are compared

with Dagan�s analytical isotropic model [11] in Fig. 3,

whereas simulations for anisotropic lnK fields were ver-

ified by reproducing Naff et al. [38] numerical results

[18], although not shown here. The latter also suggested

that the grid resolution selected (five grid cells per corre-lation scale) was sufficient to accurately predict solute

transport in anisotropic heterogeneous media.

3. Computational results

3.1. Influence of the degree of heterogeneity on

scale-dependence of dispersivity

The scale-dependence of dispersivity as a function of

r2lnK and r2

R is illustrated in Figs. 3 and 4 for conservative

tracers and Fig. 5 for linearly sorbing tracers. Numeri-

cally simulated dispersivities are also compared withfirst-order stochastic theories on solute transport. In

the conservative case, analytical expressions of disper-

sivities from spatial moments of tracer plumes and tem-

poral moments of BTCs were obtained using Dagan�s[11,12] and Naff�s [35] analytical models in Eqs. (6)

and (11), respectively. Analytical expressions of disper-

sivity for sorptive tracers were estimated multiplying

conservative dispersivity values by the index of en-hanced dispersivity (1 � qbbKdg//RA)2 [13,21], where

RA is the arithmetic mean of the retardation factor ran-

dom field. This is strictly valid for imperfectly stratified

aquifers or at large travel distances. Reactive and con-

servative tracers show that simulated dispersivities from

temporal and spatial moments exhibit the same scale-

dependence for small values of r2lnK and r2

lnKd(Figs. 3

and 5a). In agreement with these simulations, in theappendix, it is demonstrated that consistent first-order

stochastic theories also predict that the scale-depen-

dence of dispersivities from spatial and temporal mo-

ments is the same for small r2lnK and r2

R. However,

dispersivities estimated from spatial versus temporal

moments exhibit different scale-dependent behavior at

larger degrees of heterogeneity. Although the asymp-

totic constant values are the same for both types of dis-persivities, there are differences in the distance-scale at

which the dispersivities achieve a constant value. This

is illustrated by increasing the degree of heterogeneity

in a stratified heterogeneous porous media with kH/kV

equal to 10 in Fig. 4 for conservative tracers and Fig.

5 for linearly sorbing tracers. It is evident from Figs. 4

and 5 that for larger degrees of heterogeneity, the

distance-scale at which dispersivities from temporal mo-ments of BTCs approach an asymptotic value is smaller

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Simulated Spatial Moments

Simulated Temporal Moments

Naff's Imperfectly Stratified Model [1992]

Dagan's Anisotropic Model [1988]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Naff's Imperfectly Stratified Model [1992]Simulated Spatial MomentsDagan's Anisotropic Model [1988]Simulated Temporal Moments

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 5 10 15 20 25 30

<ξG>/λH, ξP/λH

0 5 10 15 20 25 30

<ξG>/λH, ξP/λH

0 10 15 20 25

<ξG>/λH, ξP/λH

A11

/σ2

lnK

λH

A11

/σ2

lnK

λH

A11

/σ2

lnK

λH



Dagan's Anistropic Model [1988]

Naff's Imperfectly Stratified Model [1992]

σ2 = 0.25In K σ2 = 0R

σ2 = 0.5In K σ2 = 0R

5

(a)

(b)

(c)

In Kσ2 = 1.0 σ2

= 0R

Fig. 4. Effect of the degree of heterogeneity on the scale-dependence of

dispersivity estimated from temporal moments of BTCs and spatial

moments of tracer plumes for conservative tracers (aL = 10 cm,

aL/aT = 10) and highly stratified porous media (kH = 278 cm, kH/

kV = 10).

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0 5 10 15 20 25

A11

/σ2

lnK λ

HA

11/σ

2 ln

K λ

H

Modified Dagan's Anisotropic Model [1988]

Dispersivity from Spatial Moments

Dispersivity from Temporal Moments

Modified Naff's Imperfectly Stratified Model [1992]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0 10 15 20 25 30

Modified Dagan's Anisotropic Model [1988]Dispersivity from Spatial MomentsDispersivity from Temporal MomentsModified Naff's Imperfectly Stratified Model [1992]

ln Kσ2 = 0.1 Rσ2 = 0.19

ln Kσ2 = 0.5 Rσ2 = 1.12

5

(a)

(b) <ξG>/λH, ξP/λH

<ξG>/λH, ξP/λH

Fig. 5. Effect of the degree of heterogeneity on the scale-dependence of

dispersivity estimated from temporal moments of BTCs and spatial

moments of tracer plumes for sorptive tracers (aL = 2.78 cm, aL/

aT = 10) and highly stratified porous media (kH = 278 cm, kH/kV = 10).


than the corresponding distance-scale for dispersivities

estimated from spatial moments. This discrepancy be-

tween dispersivities estimated from temporal and spatial

moments increases with degree of heterogeneity. For in-

stance, in Fig. 4, when r2lnK ¼ 1:0, dispersivities from

temporal moments approach a constant value in less

than five correlation scales, whereas dispersivities from

spatial moments need about 10 correlation scales. Fig.6 illustrates the same behavior of dispersivity estimates

for statistically isotropic porous media and conservative

tracers moved by advection only. Note that the simula-

tion results in Fig. 6 are in agreement with Chin and

Wang�s [8] computational studies that validated first-

order stochastic analytical solutions for travel distances

less than seven correlation scales.

To explain the behavior shown in Figs. 4–6, Fig. 7ashows the particle cloud of a conservative tracer in plan

view for a given aquifer realization with r2lnK ¼ 1:0. Also

shown is the location of the x1-control plane requisite to

obtain a BTC with the same dispersivity value as that

obtained from the second spatial moment corresponding

to a specific plume centroid location. Fig. 7b display the

scale-dependence of dispersivity estimated from tempo-

ral and spatial moments for the same aquifer realizationas for Fig. 7a.

It is noted from Fig. 4 that for large r2lnK and kH/kV,

Dagan�s analytical solution overestimates equivalent

simulated dispersivities. This is attributed to two factors:

(1) a mass transfer effect among layers due to local

transverse dispersion and (2) the effect of the degree of

heterogeneity. The effect of local transverse dispersion

on effective dispersivity has been thoroughly studied

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 5 10 15 20 25 30<ξG>/ λH, ξP/ λH

A11

/σ2

ln K

λ H


Dagan's isotropic Model [1984]


αL= 0 cm

αT= 0 cm

ln Kσ2 = 1.0

Rσ2

= 1.0 λH/λ

V = 1.0

Fig. 6. Scale-dependence of dispersivities estimated from temporal

moments of BTCs and spatial moments of tracer plumes for conser-

vative tracers (aL = aT = 0) and statistically isotropic porous media

(r2lnK ¼ 1:0, r2

R ¼ 0, kH = 278 cm, kH/kV = 1).

0

10

20

30

40

50

0 5 10 15 20 25x1/λH

ξG/λH, ξP/λH

x 2/λH

A11

/σ2 In

kλ H

ParticlesControl PlaneCenter of MassTracer Source

ξG

P

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 5 10 15 20



ξG

ξP

(a)

(b)

ξ

Fig. 7. (a) Plan view of the particle cloud, showing the location of the

control plane required to obtain the same dispersivity value as that

based on the second spatial moment of the tracer plume with the center

of mass at the location shown. This figure is based on a single aquifer

realization (r2lnK ¼ 1:0, r2

R ¼ 0, kH = 278 cm, kH/kV = 10) and pertains

to conservative tracers (aL = 2.78 cm, aL/aT = 10). (b) The scale-

dependence of dispersivity from temporal moments of BTCs and

spatial moments of tracer plumes associated with this aquifer

realization.


by Naff [36], who showed that when local transverse dis-

persivity is larger than the vertical correlation scale for a

given anisotropy ratio (kH/kV Æ aT/kV > 0.01), mass

transfer among layers causes a decrease in the asymp-

totic dispersivity as given by Dagan�s solution [12],

which does not account for the influence of local disper-sion on macrodispersion. However, it can be that

the overestimation of dispersivity is also partially due

to the small-perturbation approach. Stochastic closed-

form analytical expressions that are based on pertur-

bation methods are strictly valid for small r2lnK and

r2R. When large values of r2

lnK and r2R are applied

under anisotropic conditions, analytical expressions

seem to overestimate dispersivity. On the contrary,Naff�s [36] analytical expressions for arrival times and

temporal moments of BTCs in imperfectly stratified

aquifers seem to underestimate equivalent simulated

dispersivities. Dispersivities estimated with Naff�s [36]

stochastic analytical expressions are smaller than those

estimated with Dagan�s [12] model because the former

uses an inconsistent expansion. Although higher-order

terms in r2lnK were neglected in deriving the covariance

function of the velocity field, higher-order terms were re-

tained in his expressions for the temporal moments. It is

easily shown that dispersivities from temporal moments

obtained retaining only terms to the first-order in Naff�s[36] analytical solutions are equal to dispersivities esti-

mated from spatial moments using Dagan�s [12] model

in imperfectly stratified aquifers.

3.2. Non-Gaussian features in breakthrough curves

By their definition, the fact that dispersivities approach

a constant value at large times only implies that the

growth of spreading measured by the first two statistical

moments approaches an asymptote. It does not provide

information about the actual distribution of concentra-

tions in space or in time. If at large times, these asymp-totic dispersivity values could be directly used to predict

concentrations in conjunction with the advection–disper-

sion equation, the distribution of concentrations should

be close to Gaussian. This is a crucial issue in practice be-

cause these dispersivity estimates will be ultimately used

as input parameters in transport models that commonly

solve the advection–dispersion equation. In this section,

we evaluate the non-Gaussian features of the BTCs ob-tained at the control planes based on the coefficient of

skewnessCsk and the coefficient of kurtosisCk. The skew-

ness characterizes the degree of asymmetry of a distribu-

tion around its mean, where positive values indicate

asymmetry extending towards more positive values (tail-

ing). Kurtosis expresses the relative peakedness or flat-

ness of a distribution relative to a normal (Gaussian)

distribution, where positive values indicate a relativelypeaked distribution. As defined here, a Gaussian distribu-

tion leads to zero skewness and kurtosis,


Cskðx1Þ ¼ MT ;3ðx1Þ=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMT ;2ðx1Þ

q� �3

ð12aÞ

Ckðx1Þ ¼ MT ;4ðx1Þ=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMT ;2ðx1Þ

q� �4

� 3 ð12bÞ

where MT,2(x1), MT,3(x1), and MT,4(x1) are the second,

third and fourth central temporal moment of the ensem-

ble average BTC at the x1-control plane, respectively.

Fig. 8a and b show the coefficient of skewness and kur-

tosis estimated from simulated conservative tracer

ensemble average BTCs as a function of travel distancefor different degrees of heterogeneity (r2

lnK ¼ 0:25; 0:5;1:0) in a highly stratified porous media (kH/kV = 10). It

is seen that concentration distributions close to the

source are significantly non-Gaussian. BTCs exhibit

longer late tails and higher maximum concentrations

that would be predicted using the dispersivities esti-

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 10 20 30 40 50

ξP/λH

0 10 20 30 40 50

ξP/λH

Coe

ffici

ent o

f Ske

wne

ss Var [ln K] = 0.25

Var [ln K] = 0.5

Var [ln K] = 1.0

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

Coe

ffici

ent o

f Kur

tosi

s

Var [ln K] = 0.25

Var [ln K] = 0.5

var [ln K] = 1.0

(a)

(b)

Fig. 8. Non-Gaussian features in ensemble average breakthrough

curves at control planes for conservative tracers (aL = 10 cm, aL/

aT = 10) and highly stratified porous media (kH = 278 cm, kH/kV = 10):

(a) Coefficient of skewness of ensemble average BTCs as a function of

travel distance. (b) Coefficient of kurtosis of ensemble average BTCs as

a function of travel distance.

mated from (11) in the classic advection–dispersion

equation. The skewness and kurtosis of the BTCs de-

crease with time tending to a Gaussian-like behavior.

At small travel distances, the decrease rate is fast, but

non-ideal effects seem to persist over much longer travel

distances relative to the distance-scale at which disper-sivity approaches a constant value. Although dispersiv-

ity shown in Fig. 4c approaches a constant value in

less than 10 correlation scales for heterogeneous aquifers

with r2lnK ¼ 1:0, the coefficient of skewness and kurtosis

are still significantly non-zero at about 45 correlation

scales (Fig. 8) for the same aquifer properties. These

deviations in skewness and kurtosis from the values cor-

responding to a Gaussian distribution increase withheterogeneity.

Anisotropy, kH/kV, and transverse local dispersivity

relative to the vertical correlation scale, aT/kV, are two

important parameters controlling mass transfer between

particle paths [35,38]. Mass transfer between particle

paths is expected to smooth the concentrations out, en-

hance mixing [27], and ultimately diminishes non-

Gaussian effects on the BTCs. Using a small numberof correlation lengths and a small source size, the effect

of transverse dispersivity on the spatial distribution of

concentrations was shown by Naff et al. [38]. Here, we

examine the influence of local dispersion and degree of

anisotropy on the non-Gaussian features of BTCs.

Fig. 9a and b show the coefficient of skewness and kur-

tosis of the ensemble average BTCs as a function of tra-

vel distance for different values of kH/kV and aT/kV at afix r2

lnK equal to 1.0. Comparing the isotropic and highly

stratified cases, it is evident that stratified anisotropy

leads to a more rapid decay in the skewness and kurtosis

coefficients at early times. However, beyond about 45

correlation scales, non-Gaussianity appears to be persis-

tent, regardless of the degree of anisotropy. It is interest-

ing to note that the influence of local dispersion on the

evolution of the skewness and kurtosis coefficients isnot very significant.

3.3. Simulation of a field-scale uniform flow tracer test

From a practical point of view, there are some ques-

tions that still need to be resolved such as how to esti-

mate temporal moments reliably in the field based on

BTCs observed at control planes. The objective of thissection is to show that few deep-penetrating observation

wells, aligned transverse to the mean flow direction, are

sufficient to estimate the first two statistical temporal

moments of mass flux passing through a control plane,

leading to accurate field-scale dispersivity estimates.

This is exemplified in a hypothetical highly stratified

aquifer (kH = 278 cm, kH/kV = 10, similar to the Borden

test site) with r2lnK ¼ 1:0 (much more heterogeneous

than the Borden site). Only conservative tracers are con-

sidered and we use values of aL = 0.01kH and aL/aT = 10.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 10 20 30 40 50

ξP/λH

ξP/λH

Coe

ffici

ent o

f Ske

wne

ss No Local Dispersivity and Isotropic

No Local Dispersivity and Highly Stratified

Local Dispersivity and Highly Stratified

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

0 10 20 30 40 50

Coe

ffici

ent o

f Kur

tosi

s No Local Dispersivity and Isotropic

No Local Dispersivity and Highly Stratified

Local Dispersivity and Highly Stratified

ln Kσ2 = 1.0

Rσ2 = 0

ln Kσ2 = 1.0

Rσ2 = 0

(a)

(b)

Fig. 9. Non-Gaussian features in ensemble average breakthrough

curves at control planes for conservative tracers, as influenced by the

anisotropy ratio, kH/kV, and transverse local dispersivity relative to

vertical correlation scale, aT/kV (closed diamond symbols: aL = aT = 0,

kH = 278 cm, kH/k V = 1; open square symbols: aL = aT = 0, kH =

278 cm, kH/kV = 10; closed triangle symbols: aL = 10 cm, aL/aT = 10,

kH/kV = 10): (a) Coefficient of skewness of ensemble average BTCs as a

function of travel distance; (b) coefficient of kurtosis of ensemble

average BTCs as a function of travel distance.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 10 20 30 40

ξP [m]

Averaged BTC from 5 observation wellsBTCs obtained at control planesIndividual observation wells

50

55

60

65

70

75

80

85

90

0 5 10 15 20 25 30 35 40 45 50

x1 [m]

x 2 [m

]D

ispe

rsiv

ity [m

]

Observation Wells

Tracer Source

MEAN FLOW DIRECTION

(a)

(b)

Fig. 10. Simulation of a field-scale uniform flow tracer test in a

hypothetical heterogeneous aquifer (r2lnK ¼ 1:0, kH = 278 cm, kH/kV =

10, aL = 0.01kH, aL/aT = 10): (a) Plan view of the spatial distribution of

deep-penetrating observation wells and the horizontal location of the

tracer source (9kH · 30kV). (b) Comparison of dispersivities (estimated

from temporal moments of BTCs) obtained from individual observa-

tion wells (crosses symbols) with those for the averaged BTC obtained

from five observation wells with the same x1-distance (closed circle

symbols) and those obtained from control planes BTCs of a large

source simulation with dimensions 40kH · 30kV (line). Dispersivities

estimated from one observation well can be misleading but a few (�5)

observation wells are sufficient.


The test scale is a square of 40 m. A total of 20 observa-

tion wells are placed following a regular grid of 4 · 5 as

depicted in Fig. 10a. There are four batteries of five wells

separated by two correlation scales from each other.Within a battery, wells are two horizontal correlation

scales apart from one another in the transverse direc-

tion. The observation wells are each 40 vertical correla-

tion scales deep with a radius of 0.04kH. The source

(25 m wide and 8.3 m deep) is located upgradient, five

correlation scales away from the nearest well battery.

The number of particles selected was very large

(400,000) so that enough particles passed through allobservation wells. A natural gradient or uniform flow

experiment like this may be conducted in the field, by

injecting the tracer source using for instance a series of

about 10 injection wells, and evaluation of BTCs at a

few deep-observation wells some five correlation lengths

down-gradient. This test is advantageous over other test

cases with detailed spatial sampling in that requires

smaller durations and less equipment and sampling. It

is seen from Fig. 10b that although the variation in dis-

persivity estimates from one observation well to another

is large, the average BTC associated with a battery of

wells yields reasonable dispersivity estimates that areclose to the dispersivity values obtained from control

planes BTCs of a large source simulation with dimen-

sions 40kH · 30kV. That is, even if the mass recovery

from a battery of monitoring wells is not 100%, the esti-

mated temporal moments lead to reasonable dispersivity

estimates. This result is encouraging for the application

of tracer tests in the field, where the objective is to esti-


mate field-scale dispersivities. It should be noted that

although a water flux weighted averaging procedure of

concentrations from many observation wells on the

same control plane is more appropriate for estimating

the control plane mass flux BTC, the water fluxes pass-

ing through the observation wells are normally un-known in the field and hence more practical averaging

procedures should be sought. In Fig. 10, averaged BTCs

from five observation wells were estimated as the arith-

metic average of normalized concentrations, i.e.

Cf(x1, t)/A (where A is the area under the BTC, which

is the mass entering the well divided by the flux rate).

This averaging procedure involves flux rates and seems

more appropriate than a simple arithmetic average.For instance, in a perfectly stratified aquifer with purely

advective transport, this averaging procedure is exact.

The good results in Fig. 10 are mainly attributed to

the fact that fully-penetrating observation wells (espe-

cially in highly stratified aquifers) can sample water

fluxes over many correlation scales in the vertical dimen-

sion, which results in a small variability of fluxes be-

tween wells.

4. Summary and conclusions

The scale-dependence of dispersivities estimated from

two different methodologies: from temporal moments of

breakthrough curves versus spatial moments of tracer

plumes, has been investigated through numerical simu-lations focused on transport of conservative and linearly

sorbing tracer tests in heterogeneous porous media with

spatially variable lnK and lnKd. Our results show that

although the scale-dependence of dispersivity estimated

from spatial and temporal moments is the same for very

small r2lnK and r2

R, dispersivities estimated from spatial

moments approach a constant value at larger distance-

scales than dispersivities estimated from temporalmoments in more heterogeneous aquifers (r2

lnK > 0:5,r2

lnKd> 0:5). In all cases, at large travel distances,

asymptotic dispersivities are the same for both method-

ologies. This discrepancy suggests that dispersive pro-

cesses occurring in the aquifer due to heterogeneity are

captured differently in breakthrough curves than in

snapshots of the tracer plume: When dispersivities are

estimated from BTCs observed at control planes, allparticles are required to travel the same longitudinal dis-

tance from the tracer source so that they efficiently sam-

ple the same portion of the heterogeneous porous media.

However when dispersivities are estimated from spatial

moments at a fixed center of mass location, particles

forming the tail of the tracer plume sample a smaller

portion of the porous media than those comprising the

front of the plume. At small travel distances and forhighly heterogeneous aquifers, this effect is more pro-

nounced and dispersivities from temporal moments of

BTCs approach a constant value more rapidly than their

analogous dispersivities from spatial moments of tracer

plumes. In all cases, at large distances, this phenomenon

is irrelevant and both dispersivities approach the same

asymptotic constant value. The results of this computa-

tional study qualitatively explain recent tracer tests con-ducted in a well-defined heterogeneous test aquifer

[17,18], in which dispersivities estimated from temporal

moments of BTCs were observed to approach a constant

value at surprisingly smaller travel distances than ex-

pected from previous large-scale natural gradient tracer

tests that estimated dispersivity from spatial moments of

tracer plumes [19,31].

It is worth noting that this discrepancy between aqui-fer parameters estimated from temporal moments of

BTCs and spatial moments of tracer plumes has also

been discussed in the literature for other field-scale

transport parameters such as the retardation factor

and effective velocity. For instance, Rajaram [42] dem-

onstrated that when a negative correlation between

lnK and lnKd exists, effective retardation factors esti-

mated as the ratio of the mean arrival time of a reactivesolute to that of a conservative solute decreases with tra-

vel distance (approaching the arithmetic mean of the

retardation factor field asymptotically), while effective

retardation factors estimated as the ratio of the velocity

of the center of mass of the conservative tracer plume to

the reactive tracer plume exhibit an increasing function

with time. Moreover, effective velocities estimated as

the ratio of the travel distance to the mean arrival timewere found to be smaller than the velocity of the center

of mass of the tracer plume for small travel distances

[42]. These results has been recently experimentally val-

idated in a heterogeneous test aquifer [17,18], and com-

putationally verified using stochastic simulations [17].

In addition, a hypothetical field situation was simu-

lated to assess how accurately we can estimate the tem-

poral moments of BTCs obtained at control planes bymeans of strategically placed individual observation

wells. From a practical standpoint, it is shown that accu-

rate field dispersivities are obtained by simply using few

(about five) fully-penetrating observations wells. Non-

Gaussian features associated with ensemble average

BTCs were investigated based on higher-order temporal

moments (coefficients of skewness and kurtosis). Ob-

served BTCs at control planes down-gradient suggest asustained non-Gaussian behavior even for travel dis-

tances larger than 40 horizontal correlation scales.

In summary, perhaps the most significant findings of

this study for their implications to field application of

uniform flow tracer tests are: (1) uniform flow tracer tests

that estimate dispersivities from BTCs at control planes

constitute a more efficient technique compared with tra-

cer tests schemes that estimate dispersivities from spatialmoments of tracer plumes because smaller test scales and

duration times are required, yet both methodologies


consistently provide the same asymptotic dispersivity for

large test scales; (2) the first two temporal moments of

BTCs obtained from total mass fluxes passing through

control planes are accurately estimated with a few

deep-penetrating observation wells situated perpendicu-

lar to the mean flow direction. However, it is noted thatour use of a large plane particle source is a conceptual

convenience, which allows for unambiguous interpreta-

tion of dispersivity estimates that directly relate to quan-

tities predicted by stochastic theory, to evaluate the

theoretical reasons for the difference in scale-dependence

between dispersivities estimated from spatial versus tem-

poral moments. Although the use of smaller pulses of

particles whose principal spreading direction (and maybe even instantaneous direction of mean advection) differ

from the x1 direction is more representative of real situ-

ations, it is worth noting that these ‘‘real situations’’ will

not in general lead to reliable estimates of ‘‘macroscopic’’

field-scale behavior, in that the information gained about

the transport properties of the aquifer may be misleading

in the context of predicting the behavior of large contam-

inant plumes [7,17,18]. In forced-gradient flow configu-rations however, smaller pulses of particles may be

readily used, as shown by Fernandez-Garcia et al. [18],

by simply estimating dispersivity using the averaged

BTC of concentrations from about four multiple but dif-

ferent tracer injections in convergent-flow tracer tests or

from four simultaneous observation wells in divergent-

flow tracer test.

Acknowledgements

The financial support for the research reported in this

paper was received from the Army Research Office

(Awards: DAAD19-99-1-0195 and DAAD19-99-1-

0165). The authors gratefully acknowledge Dr. Russell

S. Harmon, Senior Program Manager for TerrestrialScience Branch of the Engineering and Environmental

Sciences Division of US Army Research Laboratory

for his advice and support. Drs. Stacy Howington and

John Peters of the Coastal and Hydraulic Laboratory

of the US Army Engineers Research and Development

Center helped to identify this research problem. Their

input to this research is gratefully acknowledged. Com-

ments and suggestions from Dr. J. Jaime Gomez-Hernandez, Universidad Politecnica de Valencia, were

greatly appreciated and provided very helpful insight

into the interpretation of the simulation results.

Appendix A

This appendix is intended to demonstrate that first-order stochastic theories also predict that the scale-

dependence of dispersivities from both temporal and

spatial moments is the same for small r2lnK and r2

R in

conservative as well as linearly sorbing solute transport.

Thus, corroborating simulation results for small r2lnK

and r2R. Consider solute particles, undergoing instanta-

neous linear adsorption, moving in a random porous

media with mean flow direction aligned with the x1 coor-dinate axis. Local dispersion is neglected. Assuming that

the x1 component of the velocity field v1 is always posi-

tive in the mean flow direction, the relationship between

the time taken by a particle to cross a control plane per-

pendicular to the mean flow direction, denoted tp(x1),

and the x1 distance from the origin to the control plane

is unique and tp(x1) can be calculated as follows [42,49],

tpðx1Þ ¼Z x1

0

Rðn;Xp;2ðnÞ;Xp;3ðnÞÞv1ðn;Xp;2ðnÞ;Xp;3ðnÞÞ

dn ðA:1Þ

where Xp,2(n) and Xp,3(n) are the random displacements

of a solute particle fluctuating around the mean pathline

passing through the origin at distance n, v1 is the compo-

nent of the velocity field in the mean flow direction at

the particle position, and R is the retardation factor at

the particle position. A macrodispersivity coefficient

that describes the growth of rate of the second central

temporal moments can be derived from Aris [2] work;the dispersivity coefficient that satisfies the advection–

dispersion equation within a region comprised between

two control planes is proportional to the ratio of the

increments of the temporal moments. In the limit (when

the separation distance between the two control planes

approach to zero), this leads to a macrodispersivity coef-

ficient that can be defined as

Aeff11;T ðx1Þ ¼

1

2

hv1ihRi

dMT ;2=dx1

dM 0T ;1=dx1

ðA:2Þ

where M 0T ;1ðx1Þ and MT,2(x1) are respectively the first

temporal moment and the second central temporal mo-

ment of the ensemble average BTC obtained at the x1-

control plane. Angel brackets refer to ensemble average

statistics. Temporal moments are related to the particle

arrival time with the following expressions [49]

M 0T ;1ðx1Þ ¼ htpðx1Þi ðA:3Þ

MT ;2ðx1Þ ¼ hðtpðx1Þ � htpðx1ÞiÞ2i ðA:4ÞAssuming that vi and R are second-order stationary

random fields, expanding Eq. (A.1) in a Taylor series,

and retaining only terms to the first-order, the travel

time of a particle is approximated as [15,42]

tpðx1Þ hRihv1i

x1 þZ x1

0

hRihv1i

� v01ðn; 0; 0Þhv1i

þ R0ðn; 0; 0ÞhRi

� �dn

ðA:5Þwhere v01 and R 0 are the fluctuations of the velocity field

in the mean flow direction and the retardation factor

field around the mean values, respectively. Taking the


derivate of (A.4) and using (A.5), the second central

temporal moment is approximated to the first-order as

dMT ;2

dx1

¼ 2hRi2

hv1i2Z x1

0

Cv1v1ðn; 0; 0Þhv1i2

þ CRRðn; 0; 0ÞhRi2

"

�CRv1ðn; 0; 0ÞhRihv1i

� Cv1Rðn; 0; 0Þhv1ihRi

�dn ðA:6Þ

where Cv1v1 and CRR are the Eulerian covariance func-

tion of v1 and R, CRv1 is the cross-covariance function

between R and v1, Cv1R is the cross-covariance function

between v1 and R. On the other hand, the motion of a

solute particle moving in a heterogeneous porous media,

undergoing linear instantaneous adsorption, is accu-

rately described by

Xp;iðtÞ ¼Z t

0

viðXp;1ðsÞ;Xp;2ðsÞ;Xp;3ðsÞÞRðXp;1ðsÞ;Xp;2ðsÞ;Xp;3ðsÞÞ

ds ðA:7Þ

where Xp,i(s) is the i-component of the particle position

vector at time s. Assuming that vi and R are second-

order stationary random fields, a first-order approxima-

tion of the particle displacement was found by Bellin et

al. [4] as

Xp;1ðtÞ hv1ihRi t þ

hRihv1i

�Z hv1it=hRi

0

hv1ihRi

v01ðn; 0; 0Þhv1i

� R0ðn; 0; 0ÞhRi

� �dn

ðA:8Þ

Applying Taylor�s [53] analysis of diffusion of contin-

uous movements, the macrodispersion coefficient in themean flow direction is written as,

Deff11;SðtÞ ¼

1

2

dM11ðtÞdt

ðA:9Þ

where M11(t) is the ensemble average second central spa-

tial moment defined as

M11ðtÞ ¼ hðXp;1ðtÞ � hXp;1ðtÞiÞ2i ðA:10ÞSubstituting (A.8) into (A.10) and defining macrodis-

persivity as macrodispersion divided by the ratio of the

mean conservative velocity to the mean retardation fac-

tor we write

Aeff11;SðtÞ ¼

hRihv1i

Deff11;SðtÞ

Z hv1it=hRi

0

Cv1v1ðn; 0; 0Þhv1i2

þ CRRðn; 0; 0ÞhRi2

"

�CRv1ðn; 0; 0ÞhRihv1i

� Cv1Rðn; 0; 0Þhv1ihRi

#dn ðA:11Þ

where v1 is the Eulerian velocity component in the direc-

tion of the mean flow, Cv1v1 and CRR are the Eulerian

covariance function of v1 and R, CRv1 is the cross-covari-

ance function between R and v1, and Cv1R is the cross-

covariance function between v1 and R. Using (A.5) in

(A.3), substituting (A.6) and (A.3) into (A.2), and know-

ing (A.11), we obtained

Aeff11;T ðthv1i=hRiÞ ¼

1

2

hv1ihRi

dMT ;2=dx1

dM 0T ;1=dx1

Aeff11;SðtÞ ðA:12Þ

Eq. (A.12) shows that the scale-dependence of macro-

dispersivity estimated from temporal and spatial mo-

ments are the same for small r2lnK and r2

R. This resultsuggests that a simple relationship between temporal

and spatial moments exits for small r2lnK and r2

R. From

Eqs. (A.5) and (A.8), the relationship between temporal

and spatial moments to the first-order is written as

M11ðtÞ hv1i2

hRi2MT ;2

hv1ihRi t

� �ðA:13Þ

This equation states that the ensemble average second

spatial moment of a tracer plume at time t, M11(t), can

be approximated to the first-order by multiplying theensemble average second temporal moment of the con-

centration breakthrough curve obtained at a control

plane transverse to the mean flow direction situated at

the center of mass of the tracer plume, MT,2(hXp,1(t)i),by the square of the mean velocity divided by the mean

retardation factor. Eq. (A.13) is in agreement with ana-

lytical solutions by Shapiro and Cvetkovic [49] and

Cvetkovic et al. [9] for statistically isotropic porous med-ia. Other specific solutions for two-dimensional aquifers

and for stratified aquifers can be found elsewhere [10]

for nonlinear sorption reactions.

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