COMMENTS AND LETTERS TO THE EDITOR
may be hard to implement. Rather for vLD >4 any outlet
condition, which will normally be given by Eq. [4], corrects
the profile upstream for a short distance only and the correction can usually be obtained readily using boundary layer
theory (Parlange and Starr, 1975,1978). We have also shownthat when Eq. [1] and [4] hold, and for vL/D >4, an appropriate breakthrough curve is often obtained by considering a semi-infinite column and taking Cr = Co, at theinlet (Parlange and Starr, 1975, 1978). Our point of departure with van Genuchten and Parker (1984) is that we believe that the simpler solutions can be used safely only for
Peclet numbers larger than four, which is the usual case formost practical situations.
Finally physical uncertainties may affect not only theboundary condition at the outlet, Eq. [4], but also at theinlet, Eq. [1]. The latter presumes that the flow is perfectly
one-dimensional, which is not necessarily the case. For instance Starr and Parlange (1976) considered a case when astagnant phase develops at the inlet and Eq. [1] does nothold there. In that case and more generally when the solute
moves in two different phases, e.g., one rather mobile andone rather stagnant, not only the boundary conditions may
be affected but also the transport equation. That is, it isdangerous to assume a priori that a single transport equation
can be used to describe the flow if two phases are present.
Rather, an equation for each phase must then be considered.
Received 14 Mar. 1985.
J.-Y. PARLANGE
Griffith University,
Brisbane, Queensland, 4111
D. A. BARRY
Australia
J.L. STARR
VSDA-ARS-BARC
Beltsville, MD 20705

Comments on "Boundary Conditions for Displacement

Experiments Through Short Laboratory Soil Columns"
In a recent paper van Genuchten and Parker (1984) discussed various boundary conditions which have been usedin miscible displacement studies. Although we agree withmost of their points, we believe that they overemphasized
the shortcomings of Brenner's solution (Brenner, 1962) which
was obtained following Danchwerts' (1953) prescription forboundary conditions. At the inlet boundary the condition is-D(dCr/dx) + vCr = vCo, x = 0, t > 0.

[1]

Except for limiting cases, Eq. [1] will normally yield Cr=£ Co, leading to an apparent discontinuity in concentration
at the inlet, i.e., between the soil column and the inlet reservoir. The authors use this as an argument to justify a possible discontinuity at the outlet. We shall show that it isphysically possible to have a discontinuity in concentration
at the inlet but not at the outlet of the column. Consider thesteady state flow

vdCr/dx = DtfCr/dx*,[2]which applies within the column. Eq. [2] has the solution

Cr = A + B exp [-(L - x)v/D][3]

where L is some distance and A and B are constants. The
exponential term is vanishingly small for x < L and increasingly large for x > L, as long as the corresponding
Peclet numbers are large. An increasingly large term is not
physical, thus B = 0 for x > L. A rapid variation in Cr is
possible for x < L but not for x > L. Parlange and Starr(1975) utilized such a boundary layer for x < L to accommodate the boundary condition (Danckwerts, 1953),

dCr/dx = 0 at x = L[4]where L then stands for the column length. In general, aboundary layer is possible upstream from a given position
but not downstream. When a reservoir is in contact with thecolumn inlet (x = 0) a boundary layer will form for x < 0,i.e., the concentration can vary rapidly for x < 0 in thereservoir from Cr (x = 0) to Co. In that case D may standfor the molecular diffusion, Do, with Do «. D in the column,
as long as the flow in the reservoir is laminar, i.e., if it is
not well mixed. Thus when Do is considered negligible, thethickness of the boundary layer is infinitely thin and an apparent, or macroscopic; discontinuity between Co and Cr issustainable at x = 0.
In contrast, at the column outlet (x > 0) no boundary
layer can develop so there must be continuity in concentration at x = L. For nonsteady state conditions a growing
boundary layer must be considered but this does not affectits existence, which is all that concerns us here (cf. Wehnerand Wilhelm, 1956). If there is continuity at the outlet thenEq. [4] holds although van Genuchten and Parker (1984)
observe that there is often some uncertainty of the exact
physical processes at the outlet. The argument here is notabout this real difficulty but in pointing out that the analysisof the boundary layers near the inlet and outlet show noinconsistency between the derivations of Eq. [1] and [4].
van Genuchten and Parker (1984) conclude that the solution of the transport equation using Eq. [1] and assuming
a semi-infinite column is fundamental to predict the profile
for all Peclet numbers because they "assume that solute distributions inside the finite column are unaffected by thepresence of an outflow boundary". We believe that the experimental conditions necessary for the assumption to hold

Reply to "Comments on Boundary Conditions for Displacement Experiments Through Short Laboratory SoilColumns"
Parlange et al. in essence comment on three issues: 1. ouroveremphasis in their view of the shortcomings in Brenner'ssolution, 2. the need to restrict column displacement experiments to Peclet numbers greater than about four, and 3.
the effects of physical disturbances due to imperfect inlet
and outlet boundaries. We wish to reply to their comments.
For convenience we use the same abbreviations for the 4
analytical solutions as in our paper (van Genuchten andParker, 1984): BR for the solution of Brenner (1962), CAfor the solution of Cleary and Adrian (1973), LA for thesolution of Lapidus and Amundson (1952), and LB for thesolution of Lindstrom et al. (1967).
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SOIL SCI. SOC. AM. J., VOL. 49, 1985

1. We indeed raised questions regarding the applicabilityof Brenner's solution BR to transport through a finite system. Brenner's solution imposes a third-type boundary condition at x = 0 and a zero-gradient condition at the columnexit, x = L. The last condition is based on the intuitive
assumption (e.g., see Wehner and Wilhelm, 1956) that concentrations should be continuous across the exit boundary.
While this requirement certainly must be met at the microscopic level, continuity of concentration does not necessarily
follow at the macroscopic level when discontinuities in medium properties are imposed. The analysis by Parlange etal. (like the more general discussion by Wehner and Wilhelm, 1956) has not convinced us otherwise. In our opinion,
continuity cannot be deduced from the observation that nophysical boundary layer will develop in the exit reservoir x> L. Even if a zero gradient is present outside the column,
it tells nothing about the mathematical boundary layer associated with a discontinuity in medium properties. Macroscopically and from an analytical perspective, it is convenient to consider an infinitesimally thick boundary region
within which the dispersion coefficient and porosity change
from porous medium to bulk solution values. Imposing discontinuities in these parameters must yield a macroscopic
concentration discontinuity. The dilemma of flux vs. concentration discontinuity at porous media boundaries can beavoided by not imposing the infinitesimal boundary region
assumption. As the window of an averaging volume glides
through the boundary region, hypothetical discontinuities inmedium properties will be smoothed and macroscopic concentrations as well as fluxes become continuous. Inasmuch
as boundaries are mathematical approximations of reality,
it is also important to consider the physical implications ofinvoked simplifications. For example, combining diffusion
and hydrodynamic dispersion in a single coefficient can leadto difficulties since diffusive transport may occur in both
upstream and downstream directions at boundaries that arecontinuous with a fluid phase, while hydrodynamic dispersion will be constrained to the downstream direction. Finally, we note that Brenner's solution specifies volume-averaged concentrations C,, As we discussed in our paper andelsewhere (Parker and van Genuchten, 1984), alternative solutions can be obtained when the transport equation is expressed in terms of the flux-averaged concentration Cfi whichis related to Cr by
Cf=Cr- (D/v) (dCr/dx).
[1]
Using a first-type inlet boundary condition and again a zerogradient at x = L, solution CA results. As shown in our Fig.
2, this solution differs from the flux-concentration that canderived indirectly from BR upon application of Eq. [1] above.
This inconsistency may reflect inherent weaknesses in theeither CA and/or BR, although we tend to believe that CAis at fault because of its failure to preserve mass (see ourFig. 4). Nevertheless, the discussion above shows some perils in Brenner's solutions that are not easily solved. As amore plausible alternative to Brenner's approach, we suggest
to regard concentrations inside the column to be unaffectedby the exit boundary; this allows us to use solutions for asemi-infinite system, notably LA for flux-averaged and LB
for volume-averaged concentrations. Contrary to CA andBR, these two solutions are properly related through transformation [1] above. Thus, LA correctly preserves mass when
integrated vs. time at any point in the profile (or effluent),
while LB always preserves mass when integrated versus
depth.
2. From the discussion above and our Fig. 4, it follows
that both LA and BR correctly predict the retardation factor
R in the system, and hence that both solutions can be usedsafely to derive adsorption or exchange coefficients from column displacement experiments (as opposed to batch equilibration). This conclusion is valid independently of the valueof P. However, we do acknowledge that application of BRand LA will lead to different dispersion coefficients (or Pecletnumbers) when short laboratory columns are used (notably
when P is less than about 5; see our Fig. 3). Fortunately,
this last issue is mostly academic since dispersion coefficients measured from laboratory columns are generally nonrepresentative of the field scale anyway. The problem of massconservation nevertheless remains important. Column displacement experiments are often used to study the transportof heavy metals, radionuclides, or other strongly adsorbed
solutes. Because of large retardation factors, any practical
study must be limited to relatively small soil columns. Our
study shows that LA and BR can be used safely to measureR, irrespective of the P-value. Contrary to the suggestion byParlange et al., this conclusion holds also for P < 4 provided
the exact series solution of BR is used. Mass conservation
may become a problem when approximate solutions (Brenner, 1962; Parlange and Starr, 1975) are used at relatively
small P-values. These approximations were derived because
available series solutions were found to diverge for large Pvalues. Use of the approximate solutions hence should be
restricted to the larger P-values. Because the series solutions
converge rapidly for P-values less than about 20, and because computationally efficient algorithms are available (vanGenuchten and Alves, 1982), we also see no need to extrapolate the approximate solutions to P-values for which they
are known to be inaccurate.
3. Finally, we agree with Parlange et al. that physical disturbances at or near the inlet and exit boundaries may affectthe value of the measured dispersion coefficient. However,
one should still be able to correctly measure the retardation
factor provided (i) the inlet or outlet screens/porous plates
are chemically inert and do not hold water, and (ii) retardation factors are measured by fitting theoretical curves tothe entire effluent curve and not just to a limited portion ofthe curve (notably the initial breakthrough part or to thatpart of the curve where the relative concentration is approximately 0.5). In instances where severe tailing of breakthrough curves is observed, the use of two-region ("mobileimmobile" type) models may be required.
Received 18 Mar. 1985.
U.S. Salinity Laboratory
M. TH. VAN GENUCHTEN
4500 Glenwood Drive
Riverside, CA 92501
J. C. PARKER
Dep. of Agronomy
Virginia Polytechnic Inst.
and State Univ.
Blacksburg, VA 24061