DIVISION S-l—SOIL PHYSICS
Flux-Averaged Concentrations for Transport in Soils Having
Nonuniform Initial Solute Distributions
Nobuo Toride, Feike J. Leij, and Martinus Th. van Genuchten*
ABSTRACTThe need to distinguish between volume-averaged or resident concentrations (cr) and flux-averaged or flowing concentrations (c,) is now
widely accepted. Flux-averaged concentrations associated with theconvection-dispersion equation (CDE) have been mostly used for solute transport problems involving uniform initial distributions. We
present flux-averaged concentrations for nonuniform initial distributions using analytical solution methods for a semi-infinite soil system
and numerical methods for a finite system. Mathematically, c, isequivalent to cr associated with a first-type inlet condition (rather thana third-type condition) only for semi-infinite soil profiles having uniform initial conditions. We show that, for a stepwise initial distribution, c, can be both negative or much greater than the initialconcentration of cr, especially during the early stages of solute displacement. This physically odd situation results from the fact that c,represents a solute flux rather than a directly measurable volumetricconcentration. Flux-averaged concentrations at the exit of a finite soilcolumn with a uniform initial distribution are nearly identical to c,
for a semi-infinite system when the column Peclet number is greaterthan ~5. However, if the initial distribution involves a high gradientin cr near the exit, c, values for finite and semi-infinite systems at theexit can be very different, similarly as those for cr, because of theadoption of different outlet conditions.

ute transport where area-averaged solute fluxes areusually of primary concern (Dagan et al., 1992; Destouni and Cvetkovic, 1991; Jury and Roth, 1990).
Flux-averaged concentrations have been used mostly
for solute applications to a soil with a uniform initial
distribution (Kreft and Zuber, 1978). For a semi-infinite system, cf is then mathematically equivalent toCT if a first- or concentration-type rather than a thirdtype inlet condition is used. Van Genuchten and Parker (1984) showed that ct for semi-infinite and finite
systems are nearly identical at the exit for column
Peclet numbers greater than ~5.
Although nonuniform initial solute distributions arequite common, relatively little attention has been paid
to cf for such conditions. Our objective was to further
investigate ct for soils having a nonuniform initial solute distribution. Typical examples of cf for a stepwise
initial distribution in semi-infinite and finite systems
will be presented.

MATHEMATICAL PROBLEM
One-dimensional solute transport in a homogeneous
soil during steady-state flow is generally described withthe CDE:
rT~'HE CONVECTION-DISPERSION EQUATION has long been
JL used to predict solute transport in soils. Concentration distributions obtained with the CDE depend
greatly on the formulation of the boundary and initialconditions. Considerable attention has therefore been
paid to the proper formulation of boundary conditions
and the description of solute injection and detection
modes (Danckwerts, 1953; Wehner and Wilhelm,
1956). The need to distinguish between cr and cf has
now been widely recognized (Kreft and Zuber, 1978;
Parker and van Genuchten, 1984; Jury and Roth, 1990).
The resident concentration is defined as the massof solute per unit volume of soil solution; this definition is the conventional interpretation of a concentration. Solute transport problems involving cr areusually solved for third- or flux-type inlet conditions
since such conditions ensure that mass balance requirements are satisfied (van Genuchten and Parker,1984). On the other hand, the flux-averaged concentration is defined as the ratio of the solute mass andwater flux densities (Kreft and Zuber, 1978). If soluteconcentrations are obtained from effluent curves, which
are encountered in column displacement experiments,
the use of cf is generally preferable. The use of cf isalso convenient in stochastic models of field-scale sol-dtdx2dx[1]

where R is the retardation factor, cr is the volume-averaged or resident concentration (M L~3), D is the dispersion coefficient (L2 T~'), v is the pore-water velocity(L T -I ), x is distance (L), and t is time (T).
The initial and boundary conditions for finite and semiinfinite systems arec,(x,0) = /(*)

[2]dcr(0,t)
- Ddx

[3]
[4a]
[4b]

where f(x) is the initial concentration, g(t) is the input
concentration, € is outlet of the soil system under consideration, and Eq. [4a] and [4b] give the outlet condition for finite and semi-infinite systems, respectively.
Although no "correct" outlet condition can probably beformulated for finite systems (Parlange et al., 1992), azero concentration gradient (Eq. [4a]) is often used basedon the assumption that cr is macroscopically continuous

U.S. Salinity Lab., USDA-ARS, 4500 Glenwood Drive, Riverside, CA 92501. Received 30 Nov. 1992. 'Corresponding author.
Published in Soil Sci. Soc. Am. J. 57:1406-1409 (1993).

1406

1407

TORIDE ET AL.: FLUX-AVERAGED CONCENTRATIONS FOR TRANSPORT

at the outlet and that no dispersion occurs outside thesoil (Danckwerts, 1953; Wehner and Wilhelm, 1956).
Solutions for semi-infinite systems may also be applied
to finite systems if the assumption is made that soluteconcentrations upstream of the boundary are not affectedby the boundary itself.
The flux-averaged concentration, c(, is defined as (Kreftand Zuber, 1978)
D dc.
2132dx2cs\x,t) =[11]where the superscript SI is used to indicate a semi-infinite system using Eq. [4b] as the outlet condition, andwhere i/f for cr is given by[5]R(x -- vt

1

where / (M L~ T- ) and q (L L~ T- ) are the soluteand water flux densities, respectively. Application of theabove transformation to Eq. [1] through [4] yields thefollowing transport problem in terms of cf (Parker andvan Genuchten, 1984):
dt

If the input is solute-free, i.e.,g(t) = 0, the solution for
the resident concentration, cf , can be expressed as (vanGenuchten and Alves, 1982)

dxV2t\VXTrDR ^\D[R(x + Xj) + vt]
4DRt[6]

DRsubject to[12]

x exp
[7]

= g(t)[8][9a]£,.,,-0[9b]Equation [7] shows that the flux-mode initial profile
will differ from the resident-mode initial profile (Eq. [2])unless a uniform initial condition (df/dx = 0) is present.
The solution for cr using a first-type instead of a thirdtype inlet condition (Eq. [3]) is often used to obtain cf
for a semi-infinite soil (Kreft and Zuber, 1978; Parkerand van Genuchten, 1984). Equations [7] through [9b]indicate that this is only permissible for a semi-infinite
soil with a uniform initial concentration. A more generalapproach for obtaining an expression for cf is to use Eq.
[5] to transform the solution for cr, solved for a thirdtype inlet condition. Below we explore the differences
between cf and c, obtained in this manner for nonuniform
initial concentrations assuming both semi-infinite and finite soil systems.
FLUX-AVERAGED VS. RESIDENT
CONCENTRATIONS

Semi-infinite System
When the solute is initially distributed between xl andx2 with a unit concentration, the initial condition may bewritten as0f(x) = 1 10 < x < X!
x, < x < x2lO

X2 < X < oc

[10]

Substituting Eq. [11] into Eq. [5] leads to an expression
for the flux-averaged concentration, cf , which is also
given by Eq. [11] except that i/r is now (Toride et al.,1993)\jj(x,t;x^ = 1 — -2 erfc

R(x - Xj) - vt
rfim

L'R(x + Xi) + vtl
V4DRiJI DR

4DRt[R(x + xi) + vt]24DRt

4TJV2t[13]

It is convenient to introduce the following dimensionless variables:D'[14a,b]

where € is now an arbitrary length (the column length
in the case of a finite system), P is the Peclet number,and T is dimensionless time.
Figure 1 shows cf1 and cf distributions vs. relative
distance, x/€, at T = 0.05 for two values of P whensolute-free water is applied to a soil having a stepwise
resident initial distribution as indicated by the dashedline. For P = 2 (Fig. la), dispersive transport becomesdominant with respect to convective transport and significant spreading can be observed in cf1 in both theupstream and downstream directions. Notice that, forsmall T, Cf' is negative for x ~ 0.5€, and greater thanunity (the initial resident concentration) for x/€ ~ 1. This
result is not surprising, at least in a mathematical sense,

1408

SOIL SCI. SOC. AM. J., VOL. 57, NOVEMBER-DECEMBER 1993

1.2
P=2

1.0

T=00.8o 0.60.40.20.0
0.4
0.6
0.8
1.0x/SL
Fig. 2. Flux and resident concentrations vs. dimensionlessdistance, xl€, for semi-infinite (cf, cf) and finite (cf, cf)
systems at reduced time, T = 0.2, for Peclet number P =2 and a retardation factor R = 1 assuming solute-free input
to a soil having a stepwise initial distribution as shown by
the dashed line.
0.0

0.2

initial conditions, the aforementioned anomaly in ct has,
therefore, largely been ignored.

-0.5

0.5

1.0

2.5
1.5
2.0
x/A
Fig. 1. Flux (csf) and resident (csr') concentrations vs.
dimensionless distance, xl€, for a semi-infinite system atreduced time, T = 0.05, for two values of the Peclet numberOP) and a retardation factor R = 1 assuming solute-free
input to a soil having a stepwise initial distribution as shownby the dashed line

if we recall the definition of cf as given by Eq. [5], i.e.,
cf is directly proportional to the solute flux / for steadywater flow. Because/ and the water flux, q, are vectors,
cf becomes negative when the directions of these twofluxes are opposite; this is the case when large positive
gradients in c, are present. Hence, the negative cf nearthe surface is the result of an upward solute flux in spiteof a downward water flux. For similar reasons, cf becomes larger than cr when the gradient of cr is negative.
For relatively large negative gradients such as those inFig. 1 around xl€ = 1, cf can initially be significantly
greater than the initial resident concentration cr(x,0).1 Notice from Fig. Ib that the differences between cf andc;51 become relatively small when the Peclet number increases.
We emphasize here that Fig. la illustrates a somewhat
extreme example. First, the application of the CDE isprobably questionable during the early stages of thetransport process. This is because the concept of hydrodynamic dispersion may be valid only after a sufficient(travel) time has elapsed (Taylor, 1953; Jury et al., 1991,p. 218-267). Second, the only plausible physical mechanism for backward solute movement is ionic or molecular diffusion and not hydrodynamic dispersion. Finally,
initial concentration distributions of the type shown inFig. 1 may be very difficult to implement experimentally
during solute displacement studies.
As mentioned above, for solute application to a semiinfinite soil with a uniform initial distribution, c{ is identical to c, obtained for a first-type inlet condition. Except
for the difficult problem of actually establishing such an
inlet condition, no physically unrealistic values for cr will
occur. Because most of the literature deals with uniform

Finite System
Van Genuchten and Parker (1984) compared cf for asemi-infinite system with the analytical solution for a
finite system, as derived by Brenner (1962), for a step
input to a soil profile having a homogeneous initial condition. They concluded that c{ for semi-infinite and finite
systems are almost identical at the exit, x = €, for P >5. To our knowledge, no analytical solution for a finite
system involving a nonuniform initial distribution has
been derived. We therefore numerically calculated cr fora finite system with the program HYDRUS of Kool andvan Genuchten (1991), and subsequently evaluated cf
numerically by substituting the results for c, into Eq. [5].
Figure 2 shows concentration profiles for four concentration modes when solute-free water is applied to asoil with a stepwise initial distribution c,(x,Q) as givenby the dashed line. The superscript F is used for concentrations of a finite system. Since the zero-gradientcondition Eq. [4a] does not allow dispersive transport
across the exit, cf is greater than cf1 near the outlet. The
zero-gradient condition also implies that cf is always
equal to cf at A: = € (Eq. [5]). Finally, notice that thegradient of cf at x = € is not generally zero in accordance with Eq. [9a].
Figure 3 shows breakthrough curves for three concentration modes at x = € when P is 1 or 5, using the same
initial condition as in Fig. 2. Notice again that cf andcf are identical at x = €. The results also show that,when P = 1, cf' is much greater than cf for relatively
small T, and lower than cf for larger T. The differences
between cf and cf decrease for larger P but are stillpresent when P — 5. The differences among the three
curves depend also on the initial distribution. This isshown in Fig. 4, which presents breakthrough curves forP = 5 and x = € when the initial concentration is givenby

0 < x < 0.9€[15]
0.9 € < x
Because of the extreme gradient at x = 0.9f close tothe exit boundary, the differences between cf and cf

TORIDE ET AL.: FLUX-AVERAGED CONCENTRATIONS FOR TRANSPORT

1409

finite and finite soil systems. Because the initial andoutlet conditions are influenced by the transformationfrom c, to cf, cf is only equivalent to c,, as derived witha first-type inlet condition, when the soil is semi-infinite
and has a uniform initial condition. For stepwise resident
initial distributions, ct can be negative or greater than
the initial values for c, in the early stages of the displacement process because cf represents a solute flux rather
than a measurable concentration.
As long as the initial concentration is uniform, cf values for semi-infinite and finite systems are almost identical at the exit. If the initial distribution involves a steep
concentration gradient near the exit, however, large differences between these two concentrations can still exist
for relatively large column Peclet numbers as a result ofdifferent outlet conditions. Although further investigations are still needed to find proper descriptions of thedifferent solute detection modes and the corresponding
outlet conditions, cf for semi-infinite systems cannot be
automatically applied to effluent curves for finite soil
systems when a nonuniform initial solute distribution ispresent.

o.o
1.5
1.0
2.0
T=vt/A
Fig. 3. Flux and resident concentrations for semi-infinite(cf, cf1) and finite (cf, cf) systems at or = € as a function ofdimensionless time, T = vt/€, for two values of the Peclet
number (P) and a retardation factor R = I assuming solutefree input to a soil having the same stepwise initial distribution
as given in Fig. 2.
0.0

0.0

1.0
1.5
2.0
T=vt/A
Fig. 4. Flux and resident concentrations for semi-infinite(cf1, cf1) and finite (cf = cf) systems at x = € as a functionof dimensionless time, T = vt/€, for Peclet number, P = 5
and a retardation factor S = 1 assuming solute-free input
to a soil having a stepwise initial concentration distribution(Eq. [15]) near the exit boundary.
0.0

increase, compared with those shown in Fig. 3d, especially at relatively small times. These differences remainquite significant for P values exceeding 5.
CONCLUSIONSThis study shows that nonuniform initial conditions
can significantly affect cf distributions in both semi-in-