, I 47 1

Centrifugal modelling of nonsorbing, nonequilibrium solutk'tran~~ort in a locally inhomogeneous soil B

L. LI AND D.A. BARRY

Department of Environmental Eilgirleering, Cerltre for Water Research, University of Western Australia, Nedlands, Western Australia 6009, Australia

AND

K.J.L. STONE

Geotnechanics Group, Department of Civil Engitzeering, University of Western Australia, Nedlands, Western Australia 6009, Aclstralia

Received May 27, 1993

Accepted March 7, 1994

This paper presents results of centrifugal modelling of physical nonequilibrium transport of nonsorbing solute in a locally inhomogeneous soil. Mathematical modelling of this class of transport process is restricted by the dif- ficulties in determining the model parameters. The modelling results suggest that physical modelling on a geo- technical centrifuge may offer another approach to tackle this problem under certain conditions.

Key words: tracer transport, centrifuge, physical modelling, heterogeneous soil, two-region model, scaling.

On prCsente les rCsultats d'essais en centrifugeuse destinCs B modCliser le transport physique transitoire d'un solutC non adsorb6 dans un sol localement hCtCrogbne. La modClisation mathCmatique de cette class9 de processus de transport est restreinte par les difficult& B determiner les parametres du modble. Les rCsultats des essais suggkrent que la modClisation physique en centrifugeuse peut offrir une approche diffkrente pour rCsoudre ce problbme sous certaines conditions.

Mots cle's : transport de traceur, centrifugeuse, modClisation physique, sol hCtCrogbne, modble B deux regions, mise B 1'Cchelle.

Can. Geotech. J. 31. 471-477 (1994)

Introduction Solute transport in natural soils is usually coupled with

sorption processes. The assumption of local equilibrium sorp- tion has been employed often to simplify the complexity of mathematical modelling of solute transport in natural soils. However, many experiments (e.g., Brusseau and Rao 1990) have shown that this assumption is not always valid. Indeed, Barrow (1989) and Brusseau (1992) suggest that the occur- rence of pure equilibrium sorption is relatively infrequent in natural soils. Instead, the sorption process is time-dependent, in which case it is called nonequilibrium sorption.

Nonequilibrium sorption can be caused by both physical and chemical factors (Brusseau and Rao 1990), including hetero- geneous hydraulic conductivity, heterogeneous soil chemi- cal properties, and chemical nonequilibrium sorption. The physical factors, such as an aggregated or locally inhomo- geneous soil structure, can lead to nonequilibrium sorption of a nonsorbing tracer solute, as observed in laboratory exper- iments (Murali and Aylmore 1980; Li 1993). This "sorption" is termed physical nonequilibrium sorption. A locally inho- mogeneous soil is one in which the hydraulic conductivity ~ ~ r r i e s at the laboratory scale, i.e., the characteristic length of the variations is of the order of centimetres rather than metres. It has been suggested that the physical factors are the pri- mary causes of the nonequilibrium transport behaviour observed in soils (Brusseau and Rao 1990). In the present paper, we consider the physical nonequilibrium process imposed on a tracer solute by a locally inhomogeneous soil.

The general mathematical model describing this class of transport process is, in dimensional form (Coats and Smith 1964), Printed in Can:~da 1 lrnpnrnd au Cenadn

[Traduit par la ridaction]

acb [lb] 0 , - = a ( c , - c,) at

where 0 , and 0, are, respectively, the water content of mobile and immobile regions; c , and cb are, respectively, the solute concentration in these two regions; D is the longitudinal dispersion coefficient; V is the pore-water velocity in the mobile region; a is the sorption rate; t is time; and x is the distance coordinate. Equations [ l ] are commonly termed the mobile-immobile region model. They have been used frequently to simulate solute transport in natural soils (Barry and Sposito 1989). van Genuchten (1981) showed that the mobile-immobile region model is mathematically identical to the "two-site sorption" model. The different sites in the latter model consist of type 1 sites, where the sorption is instantaneous, and type 2 sites, where the sorption is time dependent. The latter process is termed chemical nonequi- librium sorption to distinguish it from the apparent time- dependent sorption process due to physical phenomena (i.e., physical nonequilibrium sorption). The equivalence of the models implies that the mechanism of nonequilibrium sorp- tion (i.e., whether it is due to chemical or physical phe- nomena) cannot be determined by the models themselves through curve fitting.

In locally inhomogeneous soils, transport of a nonsorb- ing solute will be affected by physical nonequilibrium sorp- tion, resulting from the combined effects of local flow vari- ations and interregion diffusion. The sorption rate a (termed

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47 2 CAN. GEOTECH. J. VOL. 3 1 , 1994

the mass-transfer coefficient) is then a lumped parameter. For natural soils, it cannot be determined independently, but only through curve fitting, and so it needs to be cali- brated for different soils. The same conclusion applies to 0, and 0,. The difficulties in determining the model param- eters limit the application of the model and suggest that physical modelling may be of considerable practical use as a complimentary technique with mathematical modelling. If a physical model is used, then the question of scaling the model results to the prototype naturally arises.

In the following, we investigate the feasibility of cen- trifugal modelling of solute transport with physical non- equilibrium sorption under the condition of saturated and steady flow. The scale factors for the centrifugal model are derived from inspectional analysis, and the possibility of achieving similitude between the model and prototype is discussed. Results from "modelling of models," using an artificial locally inhomogeneous soil and conducted on a geotechnical centrifuge, will be compared with the scaling predictions.

type due to the increas'e of a&eleration in the model, i.e., V, = NV,.

Inspectional analysis (Patricia and Donald 1984) is now applied to the nonequilibrium transport equations [2]. We have, for the model,

where the subscript m denotes model. For the prototype, the corresponding equations are

Scaling of the transport process for the physical model: Inspectional analysis

To simulate the prototype transport process, the physical model must be set up with scale factors that relate the prop- erties of the model to those of the prototype. The scale fac- tors, which are determined by maintaining similarity between the model and prototype, can be derived from inspectional analysis. The principle is that identical solute transport processes in the physical model and prototype must satisfy the same dimensionless form of the governing transport equation, including the boundary and initial conditions (Patricia and Donald 1984; Li et al. 1993).

Consider the physical nonequilibrium transport of non- sorbing solute in a soil column of locally inhomogeneous soil. The governing transport equations [ l ] can be written in dimensionless form as

and

1-p ac, [2b] -- = o(C, - C,)

P a7

where P = VLID is the column PCclet number (L is the soil column length); o = aLIV0, is the Damkohler number; P = 8,/(8, + 0,) is the ratio of mobile region volume to total volume; T = tVIL is the nondimensional time; X = xlL is the dimensionless distance; and C, = c,lco and C, = cblco are, respectively, the nondimensional solute concentration in mobile and immobile regions (c, is the input solute concentration).

The same centrifugal modelling approach (small length scale physical modelling) as used by previous researchers (e.g., Cooke and Mitchell 1991; Hensley and Schofield 1991) is employed here to model the physical nonequilibrium transport process. The length and acceleration scales are thus set as L, = LpIN, where L, is the model soil column length; L, is the prototype soil column length; N is the scale factor; and g,, = Ng,, where g, is the model acceleration and g, is the normal gravitational acceleration. The pore- water velocity in the model is N times that in the proto-

where the subscript p denotes prototype. Given that the grain PCclet number P, (= dVIIS*, where d is the grain par- ticle size, the value of which is taken equal to d,,; and D* is the molecular diffusion coefficient) is suitably small (e.g., less than unity), then dispersion is velocity independent (Bear 1972), and so P, = P,. Equations [3] and [4] are identical if prn = Pp and o, = w,. In other words, these conditions must be satisfied for exact centrifugal modelling of C,. If this is the case, then centrifugal modelling results are similar and directly transferable to the prototype, i.e., in dimensional form:

Cam(tm, xtn) = C,\p(tp, xp) where x, = x,lN and t , = $1~'. The length and time scales are, not surprisingly, the same as those for the equilibrium sorption process (e.g., Hensley 1987).

It is of importance to examine more closely the implica- tion of the equality of equations [3] and [4]. Although it has been suggested already that 0, and 8, are apparent param- eters, it is reasonable to expect that their values are functions of the soil structure alone (Li 1993). Thus, for the same soil, p will be identical in the model and prototype, i.e., p,, = Pp. On the other hand, a is a parameter that measures the interaction of the solute between the regions. So, for a given soil structure, it will vary with flow conditions. The scaling condition of o, = op reduces to the requirement that the sorption rate a in the model increases by a factor of N'. Thus, modelling is dependent on the scaling of a , i.e., if the scale of a(S, = all l /ap) equals N', physical mod- elling is possible. Whether this condition will hold is deperi- dent on how a changes in the centrifugal model of the prototype.

In locally inhomogeneo~~s soils, the apparent sorption r&te (a) is mainly affected by two factors, namely, the local flow variations and interregion diffusion (Li 1993). Under suitable conditions, either factor can dominate. When velocity (or flow rate) is relatively low, the dominant factor is the flow variations, and a relates to velocity according to a = V'ID. On the other hand, interregion diffusion will dominate the nonequilibrium sorption process when the ~ e l o c i t y ~ i s rela- tively high; and a is a constant proportional to D'*lhu, where h is the characteristic length of the local flow variafions. Flow in natural soils is very slow; therefore, the first. type of

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LI ET AL. r )

TABLE L . List of tests and comparison of models 1 and 2 test results u'sing time inomeds , 1

Tests: set 1 Tests: set 2 Tests: set 3 Acceleration

and MT 1 MT2 MT3 MT4 MT5 MT6 moments Model I Model 2 Model I Model 2 Model 1 Model 2

NS 20 40 3 0 M d % (h) 3.33 3.14 1.76 M,lc, (h') 69.17 64.26 10.64 MOlc, (h3) 1448.04 1334.29 65.67 M80 1 .o0 1 .00 1 .oo M82 (h) 19.11 18.89 5.17 M 8 , ( t 2 ) 399.35 391.36 31.73 cr' (h-) 34.16 34.52 5 .OO

NOTE MOr MI , an"dM?, raw moments; M60, M6,. ,

relationship between a and velocity ( a = v2/D) is likely dominant. As mentioned above, the velocity in the centrifugal model increases by a factor of N, i.e., V,,, = NV,. Thus, so long as a = V?D remains valid, then a,, = V;,/D, and a, = v?,/D,. Theref6re, the scale factor for a is S, = (v:,/v~) (D,/D,,,) = S,?/S,. The dispersion coefficient has been taken to be a constant under the condition that P, is less than unity, so S, = 1 and S, = S,i = N' (S, = N in the centrifugal model). Then, the scaling conditions are satisfied in the centrifugal model, and centrifugal modelling of the physical nonequilibrium transport of nonsorbing solute in locally inhomogeneous soils may be feasible.

Modelling of models A method of verifying the feasibility of centrifugal mod-

elling is to perform modelling of models. The idea is based on the fact that if the physical model with certain scale changes can replicate the corresponding prototype, the model should be able to replicate a corresponding model with dif- ferent scales, and vice versa (Arulanandan et al. 1988). For example, model 2 should replicate model 1 if L,,, = L,,,IN and gm2 = Ngml (subscripts ml and m2 denote, respectively, models 1 and 2).

Exper-itnental inaterials The soil used in the experiments was a commercially

available crushed silica flour, Silicon 200. It is mainly fine silt, with about 17% clay (clay particle size is defined as less than 20 pm; Bear 1972). The particle-size distribution determined by using the sedimentation method (Head 1984) is shown in Fig. 1. The hydraulic conductivity of this mate- rial is around 3.5 X 10-%IS, according to the falling-head and constant-head permeameter tests (Head 1984), both of which were conducted on a consolidated sample of the mate- rial. The porosity was estimated to be about 0.43. The arti- ficial porous material used to simulate the local inhomo- geneity in soils was polyethylene porous cylinders (PPC, diameter 13.5 mm, length 10 mm). The hydraulic conductivity of PPC is 4.2 X 10-5 m/s from the falling-head test, which is 12 times higher than that of the soil (Silicon 200). Its porosity is about 0.5. Sodium chloride was used as the tracer.

Dry PPC (the volume percentage of PPC in the tests is 14%) was thoroughly mixed with dry silt to simulate a locally inhomogeneous soil. The mixture was then packed into the column and saturated fully with very slow flow from the bottom. The soil column was washed using deion- ized water to achieve zero initial solute concentration and consolidated on the centrifuge prior to testing.

and MS2, Dirac source moments; cr2, variance

Particlc size (pm)

FIG. 1. Silt particle-size distribution.

Modelling of models Modelling of models was conducted with two different

length physical models to verify the feasibility of centrifu- gal modelling of the prototype transport process. Both mod- els were constructed in a plastic-lined aluminium column of 140 mm diameter. For model 1, the soil column is 24 cm long, which is twice that used for model 2 (Table 1). Model 1 was run under three different centrifugal accelerations (g, = log, g, = 30g, g, = jog), and model 2 was run under three corresponding accelerations (g, = 20g, g, = 6Og, g3 = 1005). The configuration of each model is shown in Fig. 2. The surface and bottom heads are maintained by two preset over- flows. The concentration of the effluent at the bottom was measured continuously with a conductivity probe.

After setup, the model was placed in a rectangular strong- box and moved onto the centrifuge platform. The model is then accelerated to lOOg for 60 min to allow consolidation of the soil (very little consolidation occurred). The solute or deionized water was fed to the model surface through a hydraulic slip ring. Once sufficient consolidation time had elapsed, a pulse of solute was introduced to the surface of the soil column, followed by deionized water. The pulse time (To) of solute input for both physical models was set accord- ing to the time scale To ,,,, = N'T, ,,,, (N = 2). The signal sensed by the conductivity probe reflected the changes of the effluent solute concentration and was logged continu- ously. These data allow the solute's breakthrough curve

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CAN GEOTECH J VOL 31, 1994 \

I i

k

I D surface head controlled by overflow

bottom head controlled by overflow

reference measurement line

- - (the top of the strong box)

bottom base I I V r l .X Centrifuge platform - - -

tt conductivity probe

FIG. 2. The configuration of centrifugal models 1 and 2 (in parentheses). L1 = 120 mm; L2 = 45 mm; L3 = 13 mm (6.5 mm); H1 = 43 mm (-77 mm, i.e., below the reference measurement line); H2 = 7 mm (-95 mm); Ah = 36 mm (18 mm).

XU A A cell D I

E cellB ---.-- Average A A A - - - A A A - - ; w a d ..... P)

- cell C A ~ A - - k * M ~ ~ : 210 - A A A - - - , - ~ o g p n

O A -;.we$ g e a - - , -w o g L l n 0

3 -I/. o g e " o n 0 0

150 1 I I I

1 800 1900 2000 2100 2200

Time (s) FIG. 3. Typical load cell response during centrifuge flight; R is

the regression coefficient.

(BTC, i.e., C,(L, t) where L is the soil column length and t is time) to be determined.

Results and conclusions

Steady flow The flow rate was measured with a flux tank containing

four load cells. As the effluent water discharged to this tank, the load on these four cells increased. By inspecting the change (with time) of the total load on the cells, we can determine whether the flow is steady or not: a steady flow should give a linear change of the total load. The typical test result shown in Fig. 3 indicates a nearly perfect linear increase of the total load, confirming a steady flow through model.

Modelling of models The two models designed for the purpose of modelling

of models have a scale number of 2, i.e., N = 2. Suppose that model 2 is used to simulate model 1. If the correct scal- ing is to be achieved, CaIn2(tm2, Lm2) = Caml(tml, where tm2 = tlnI/4, and Lm, = Lll11/2 are preset. Three sets of tests

e Model 1 (N = 10) o. 0. 0 Model 2 (N = 20)

0. 0.

O .

-1

600 800 1000 1200 1400 1600 1800 2000

tllll (min) FIG. 4. Modelling of models tests: set 1 (MT1 and MT2).

were performed (see Table 1). For the first set of tests, MTI and MT2, BTC curves were obtained for model I at an acceleration of log and model 2 at an acceleration of 20g. By scaling the time of model 2's BTC (the data [C,;, ti]) according to t,,,, = t,,,, X 4, the transformed data [C,,, 4 X ti] from model 2 can be plotted together with the BTC of model 1 (Fig. 4). The result shows that the two BTC curves are reasonably close to each other, which suggests that model 2 could in fact model model 1, i.e., C,n12(tn12, Lm2) = Cani~(tml* 'nil).

The main area of disagreement between the BTCs in Fig. 4 is in the receding limb. This is due to the small error in the pulse time of the applied solute. The solute mass under each curve (Mo in Table 1) differs by about 5%. The other two sets of tests were also carried out at different cen- trifugal accelerations as listed in Table I . The results from these two sets of tests also show that modelling model 1 by model 2 is satisfactory (Figs. 5 and 6). The grain PCclet numbers in all the tests were less than unity, as required by the scaling analysis; the highest value was about 0.8. The BTCs obtained from the tests all exhibited tailing, a sig- nificant feature of the nonequilibrium transport process,

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L1 ET AL. 475

I 0 Model 1 (N = 30) I o Model 2 (N = 60)

t",, (mi4 FIG. 5 . Modelling of models tests: set 2 (MT3 and MT4).

indicating that physical nonequilibrium transport occurred dur- ing the tests. A reference experiment with pure silt was con- ducted prior to the tests, in which a symmetric BTC was obtained without tailing (Li 1993). This confirms that the physical nonequilibrium transport in the centrifuge models was due to the local soil inhomogeneity.

To estimate the error of the physical modelling approach, the first three temporal moments of the BTCs for both mod- els were calculated. These three moments physically rep- resent the mass conservation, mean movement, and vari- ance and are defined as follows (e.g., Barry and Parker1987):

Mi = j: t i c ( t ) d t

where C(t) is the BTC concentration and Mi (i = 0, 1, 2) is the ith-order temporal moment. The zeroth order moments for models 1 and 2 in each set of tests (Table 1) are not exactly the same. As mentioned, this is likely due to small experi- mental differences in the timing of the step pulse input of

0 ~ o $ e l 1 (N = 50)

o Model 2 (N = 100)

t,,,, (mi4 FIG. 6. Modelling of models tests: set 3 (MT5 and MT6).

solute. It is, therefore, necessary to transform the raw moments to Dirac temporal moments, i.e., the moments that would result from a Dirac pulse of solute at the column entrance at t = 0. The transformed moments are more rep- resentative of the transport mechanism, since the effects of the boundary condition (i.e., the different pulse input of solute) are excluded. We shall describe the calculation of these transformed moments briefly following Valocchi (1985).

The solute concentration at the column exit for a non- Dirac source at the column entrance is the convolution of the Dirac response function with the entrance boundary condi- tion. The Laplace-transformed exit concentration F(p) is thus

E51 ?( p) = "b PM P) where p is the Laplace transform variable, g(p) is the trans- formed Dirac response function, and &(p) is the Laplace transform of the boundary condition. The temporal moments calculated from [5] are

- d i i ( p ) l + [ , i! ] d i - j b ( p ) d ; i ( p ) +(-1)i b ( 0 1 7

d p ' ,,=, , ( i - j ) ! dpi-j

where M,, and Mgi are, respectively, the ith order temporal moment of the input pulse and the Dirac response. Clearly, g(0) = 1 while Ib(0) = coTo = Mo is the total mass of the solute added to the column, where To is the step pulse time. For the step pulse input applied in the tests, we have

Therefore, we can calculate the first three orders of M,i from [6] and [7]:

M,, = g(0) = 1

The variance (i.e., the second central moment o" can then be calculated from Mg2 as

The results listed in Table 1 show that these transformed moments for the two models are very close to each other. The M,, results represent the mean travel time of solute through the soil, and the dispersion induced by the soil profile is quantified by o'. Along with Figs. 4-6, these moments sug-

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476 CAN GEOTECH J VOL 31, 1994 , TABLE 2. Typical grain PCclet numbers of the model and

prototype for different soils (D* = lo-' m2/s)

Coarse sand Fine sand Silt

Grain size rl (m) lo-' 0-5

K ( d s ) 10-4 10-5 lo-" Fluid head gradient 0.0 1 0.0 1 0.01 P, (prototype) 1 0.0 1 0.000 1 P, ( 1 00g model) 100 1 0.0 1

~cknowled'pents b

This research was supported in part by the Australian Research Council. The first author thanks the Australian International Development Assistance Bureau (AIDAB) for financial support under Australian Developing Country Scholarship Scheme (ADCSS). The authors gratefully acknowledge the excellent technical support of Mr. Don Herley during the course of the centrifugal tests.

gest that the transport processes in these two models are similar. Recall that the scale-factor number between the two physical models is only 2. Modelling of models with a larger scale factor number is limited by the capability of the cen- trifuge apparatus. However, the similarity between the mod- els in the present tests is strong enough to indicate that there exists internal consistency of the models. If the test results are to be used to predict the prototype transport process, then the scale factor number will relate to the accel- eration level. For example, the scale factor number between MT6 and the prototype is 100; therefore, we can predict the prototype transport process at the time scale of a year from MT6, the test time being about an hour.

Conclclding remarks The results from modelling of models indicate that centri-

fugal modelling of the physical nonequilibrium transport process with relatively slow flow in locally inhomogeneous soils may be feasible. Such feasibility is valid under certain scaling conditions. The primary scaling condition is that the grain PCclet number (P,) must be less than unity (Hensley and Schofield, 1991): V,,,d/D:~ < 1. This will put limits on the prototype distance (L,) and velocity (V,) that can be simu- lated. Table 2 lists typical grain PCclet numbers that are calculated for different soil materials assuming a hydraulic gradient of 0.01, which we take as a representative field value for the purpose of illustration. According to these val- ues, exact physical modelling is possible if the soil is fine sand or silt (clay), whereas it would not be practical for coarse sand. More precisely, given a transport process in the prototype in which the transport velocity (V,) and distance (L,) are fixed, the limitation of V,,, < D":ld implies that the maximum possible scale number is N,,, < D'i:ldV,. If the model is set up with a scale number higher than N,,,,, then the dispersion coefficient will increase with velocity in the model. This will result in an overestimation of the dispersive solute movement in the prototype, and so would provide an upper bound on solute travel time through the prototype profile. Such a bound could be useful if a hazardous solute was being considered.

The feasibility of centrifugal modelling also depends on the scaling of the sorption rate a. If the flow velocity in the model is high beyond the low-velocity range in which a = V'JD, rx will scale as ~:!:lh' (Li 1993). In such a case, scal- ing of the local heterogeneity length (h) is required for exact centrifugal modelling. This would be impractical due to the lack of information to characterize and reconstruct natural inhomogeneous soils.

The test results also indicate that the condition of P.. less than unity seems more restrictive on the velocity scaling than that due to the scaling of a, which has not been defined quantitatively. In other words, if the condition of P, < 1 can be satisfied in the centrifuge model, then the scaling of rx may also be achievable.

Arulanandan, K., Thompson, P.Y., Kutter, B.L., Meegoda, N.J., Muraleetharan, K.K., and Yogachandran, C. 1988. Centrifuge modelling of transport processes for pollutants in soils. ASCE Journal of Geotechnical Engineering, 114(2): 185 -205.

Barrow, N.J. 1989. Suitability of sorption-desorption models to simulate partitioning and movement of ions in soils. I11

Inorganic contaminants in the vadose zone. Edited by B. Bar- Yosef and N.J. Barrow. Springer-Verlag, Berlin. pp. 18-3 1.

Barry, D.A., and Parker, J.C. 1987. Approximations for solute transport through porous media with flow transverse to lay- ering. Transport in Porous Media, 2(1): 65-82.

Barry, D.A., and Sposito, G. 1989. Analytical solution of a convection-dispersion model with' time-dependent transport coefficients. Water Resources Research, 25(12): 2407-2416.

Bear, J. 1972. Dynamics of fluids in porous media. Elsevier Science Publishing Co., Inc.,?New York.

Brusseau, M.L. 1992. Transport of rate-limited sorbing solutes in heterogeneous porous media: Application of a one-dimensional multifactor nonideality model to field data. Water Resources Research, 28(9): 2485-2497.

Brusseau, M.L., and Rao, P.S.C. 1990. Modelling solute transport in structured soils: A review. Geoderma, 46: 169-192.

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Cooke, B., and Mitchell, R.J. 1991. Physical modelling of a dis- solved contaminant in an unsaturated sand. Canadian Geotechni- cal Journal. 28: 829-833.

Head, K.H. 1984. Soil laboratory testing (vols. 1 and 2): Soil classification and conlpaction tests. ELE International Limited, London.

Hensley, P.J. 1987. Geotechnical centrifuge modelling of haz- ardous waste migration. It2 Proceedings of a Workshop on Land Disposal of Hazardous Waste, Eng~neering and Environ- mental Issues, Cambridge University, Cambridge, U.K., 9-11 Sept. pp. 139-151.

Hensley, P.J., and Schofield, A.N. 1991. Accelerated physical modelling of hazardous waste transport. GCotechnique, 41(3): 447- 466.

Li, L. 1993. Physical modelling of nonreactive chemical transport in locally stratified soils. M.Eng. Sci. dissertation, University of Western Australia, Nedlands.

Li, L., Barry, D.A. Hensley, P.J., and Bajracharya, K. 1993. Non- reactive chemical transport in structured soil: The potential for centrifugal modelling. In Geotechnical management of waste and contamination. Edited by R. Fell, T. Phillips, and C. Gerrard. A.A. Balkema, The Netherlands. pp. 425-43 1.

Murali, V., and Aylmore, L.A.G. 1980. No-flow equilibrium and adsorption dynamics during ionic transport in soils. Nature (London), 283(1): 467-469.

Patricia, M.T., and Donald, R.N. 1984. Scale factors i n soil sci- ence. Soil Science Society of America Journal, 48(5): 953-960.

Valocchi, A.J. 1985. Validity of the local equilibrium assump- tion for modelling sorbing solute transport through homoge- neous soils. Water Resources Research, 21(6): 808-820.

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L1 ET AL.

List of symbols solute concentration[^^-3] dimensionless concentration, clc, input solute concentration characteristic grain size, d,o dispersion coefficient molecular diffusion coefficient gravitational acceleration contrifugal acceleration, Ng characteristic length of the local flow variations soil column length zeroth-order moment, total injected mass per unit flow rate i th-order temporal moment ith-order Dirac temporal moment scale-factor number column Ptclet number, VLID grain Ptclet number, VdlD* scale factor scale of 01

time

5

pulse time of sblute input \

pore water velocity in the mobile region distance dimensionless distance, xlL nondimensional time, VtIL water content in the mobile region water content in the immobile region ratio of mobile region volume to total volume, 0,l (0 , + 0,) nonequil ibrium sorpt ion ra te (or mass transfer coefficient) Damkohler number, aLIV0, second central moment or variance

Subscripts: a mobile region b immobile region m model m l model 1 m2 model 2 p prototype

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