Physical modelling of a dissolved contaminant in an unsaturated sand

B. COOKE AND R. J. MITCHELL Queen's University, Kingston, Ont., Canada K7L 3N6

Received April 3, 1991

Accepted July 19, 1991

Computer solutions to the problem of dissolved contaminant movements in saturated soils are readily available. However, for partially saturated soils, extreme nonlinearity of some variables and difficulties in determining values for other variables make computer solutions predicting the movement of dissolved contaminants highly unreliable. Physical modelling in a geotechnical centrifuge is presented as a viable alternative to numerical modelling. Scaling laws for physical modelling are presented in this paper, and dimensional analysis is used to identify potential difficulties in application of model test results to prototype situations. Preliminary results of experimental work are presented. These results indicate that centrifuge modelling may be a viable approach to the evaluation of dissolved contaminant transport in partially saturated soils.

Key words: contaminant transport, centrifuge, unsaturated.

Des solutions d'ordinateur pour le problkme de mouvements de contaminants dans des sols satures sont aisement disponibles. Cependant, pour les sols partiellement satures, des non-linearitis extremes de certaines variables et des difficultes a determiner des valeurs pour d'autres variables, rendent trhs peu fiables la prediction des mouvements des contaminants par des solutions d'ordinateurs. La modelisation physique dans un centrifuge de geotechnique est presentee comme alternative viable a la modelisation numerique. Les lois d'echelle pour la modelisation physique sont presentees dans cet article, et l'analyse dimensionnelle est utilisee pour identifier les difficultes d'application des resultats de l'essai sur modttle aux conditions du prototype. Les resultats prkliminaires du travail experimental sont prksentes. Ces resultats indiquent que la modelisation par centrifuge peut @tre une approche viable pour Cvaluer le transport de contaminant dissout dans les sols partiellement saturks.

Mots clgs : transport de contaminant, centrifuge, non-sature. [Traduit par la redaction]

Can. Geotech. J. 28, 829-833 (1991)

Introduction a(Oc> a J(Oc) a(vC)

The usual point of departure for studies of the transport 131 - = - [ D ~ ] - - of dissolved contaminants by groundwater movement is the at az az equation of advective-dispersive transport, which in the simple, one-dimensional form for a nonreactive soluble is

ac a2c ac [ I ] 8- = OD- - V-

at az2 az where Cis the solute concentration, t is time, z is the vertical position coordinate, O is the volumetric water content (equal to porosity in a saturated soil), D is the coefficient of hydrodynamic dispersion, which combines the effects of both molecular diffusion and mechanical dispersion, and v is the specific discharge, or Darcy flux as defined by Darcy's Law:

where K is the hydraulic conductivity and H i s the hydraulic head.

Values may be determined experimentally, with varying degrees of difficulty, for all the parameters needed to solve the advective-dispersive transport equation for fully saturated soils. Numerical solutions usually involve finite difference techniques, and a number of software packages are commercially available for solving [I] under various combinations of boundary conditions, initial conditions, soil types, and contaminant types.

However, for transport through partially saturated soils the analysis is much more complex, since 8, D, and v can no longer be assumed to remain constant. Equation [l] is rewritten for unsaturated soils as Printed in Canada / Imprime au Canada

Although the variations of D and O have long been recognized (Porter et al. 1958), a satisfactory means of describing these variations in a general mathematical model has yet to be formulated (Allen 1985). The variation of v in the adaptation of Darcy's Law to the unsaturated case may be expressed as

where Z is the elevation head and 9 is the suction head. The variation of K with O is complex, difficult to define, and may range over a number of orders of magnitude. As well, $ varies with O in a nonlinear and nonunique manner.

The mathematical problems caused by nonlinearities, combined with the difficulty in defining values for most of the hydraulic parameters, render [3] practically unsolvable by computer modelling, except, perhaps, for the case of shallow soil profiles, which are of interest to agricultural engineers and hydrologists. Although there have been a number of theoretical models for flow and transport under unsaturated conditions described in the literature (e.g., van Genuchten and Wierenga 1976; Rose et a[. 1982; Bresler and Dagan 1983), it is generally agreed (Russo 1989; Butters et al. 1989) that none have been satisfactorily validated under field conditions.

Because of the aforementioned difficulties in developing satisfactory numerical models, research work is currently underway to develop equipment, and to test techniques, to allow the accurate physical modelling of two- and three-

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830 CAN. GEOTECH. J . VOL. 28, 1991

phase flow phenomena in a geotechnical centrifuge. This paper reports on the scaling considerations that have been addressed in developing the research study and on some pre- liminary experimental results.

Scaling considerations The essence of centrifuge modelling is the simulation of

gravitational mass forces, which occur in the real (prototype) situation, by centripetal mass forces created by centrifugal acceleration on the scaled model. The stress distribution in a prototype soil profile, for example, can be accurately reproduced in a centrifuge by subjecting a 1 /N scale model to a centrifugal acceleration of Ng. Because of the stress dependence of many soil properties, centrifugal models are more suitable for geotechnical studies than laboratory models under normal gravity.

For centrifuge studies involving flow of fluids, it has been established (Goodings 1984) that the N-fold increase in gravity in a centrifuge model results in an N-fold increase in velocity of Darcy flow through soil over that in the proto- type. Combined with the reduction in dimensions by a fac- tor of N, this results in a reduction by a factor N~ in the time required for flow phenomena to occur in the centrifuge model. In the current study, modelling done at 100 x g can simulate, in 1 h, advective transport which takes over a year in the prototype situation. This is an obvious advantage of accelerated modelling, but it must be recognized that disper- sive transport and chemical changes may not be adequately modelled.

Several researchers have recently begun to apply centrifuge modelling techniques to the problem of contaminant transport. Gronow et al. (1988) have modelled leachate migration through saturated soils. Arulanandan et al. (1988) have established scaling laws for both saturated and unsaturated contaminant transport; these authors list eight dimensionless variables, noted below, which may be used to describe contaminant transport in porous media:

C nl = - (concentration number)

P f

Pfvl = Re (Reynolds number) n2 = - P

vt a 3 = - (advection number)

L

(diffusion number) X4 = - L~

,OfgL1 (capillary effects number) ng = - T f

S n6 = - (adsorption number)

Pf

gt2 P, = - (dynamic effects number)

1 vl

ns = - = Pe (Peclet number) Dm

where C is solute concentration, p is fluid viscosity, Dm is the coefficient of molecular diffusion, S is mass of adsorbed solute per unit volume, v is the average interstitial fluid

velocity, T f is thk fluid-particle surface tension, p f is fluid density, g is gravitational acceleration, 1 is characteristic microscopic length, L is characteristic macroscopic length, and t is time.

The adsorption number (a6) is related to the mass transfer aspect of chemical reactions in the transport pro- cess, but none of the above n terms accounts for chemical reaction rates. This oversight may be corrected by the introduction of a dimensionless number incorporating k , the reaction rate constant. Assuming first-order reactions, the dimensions of k are T - ' , so one possible definition of a "chemical effects number" is

n9 = kt (chemical effects number)

By using the same soil and contaminant in both the model and the prototype, equality between model and prototype is achieved for five of these numbers. The remaining num- bers are the Reynolds number, the Peclet number, the "dynamic effects number," and the "chemical effects number."

It has been shown (Bear 1972) that if Re < 1, flow in porous media is laminar, so even without equality of Reynolds number between model and prototype, Darcy's Law remains valid and flow phenomena may be accurately modelled. For the relatively fine-grained soils used in this study, flow velocities were low enough that the condition of Re < 1 was easilv met.

The Peclect numbe; relates the two factors that constitute the coefficient of hydrodynamic dispersion, namely mechanical dispersion and molecular diffusion. The mechanical dispersion term is a function of flow velocity and path length, and it is not possible to scale it from model to prototype while maintaining all the other scaling relation- ships. However, as long as Pe < 1, molecular diffusion dominates the dispersion process and equality of Peclet numbers is not necessary (Arulanandan et al. 1988). Again, for the soil used in this study, the velocity is low enough that this condition obtains, and accurate modelling of the dispersion phenomena may be assumed.

The dynamic effects number is important in problems where inertial forces are significant, such as the study of earthquake or pile-driving effects. Similitude of this num- ber can be ignored for laminar flow problems in porous media (Zelikson 1984).

The chemical effects number is important in problems where chemical reactions may affect the rate of contami- nant transport, either through chemical alteration of the solute itself or through processes that change the nature of the porous medium, such as dissolution. The inequality of this number in model and prototype suggests that centrifuge modelling of transport of reactive solutes would be inap- propriate in some cases and in other cases would require careful evaluation of ~arametr ic tests to relate model test results to a prototype situation. In the current study, this difficulty was avoided by use of a nonreactive tracer.

Since this study deals with partially saturated soils, it is important to note that among the dimensionless numbers which are equal in model and prototype is the "capillary effects number," thus indicating that the moisture-content profile of the model unsaturated soil column will be geometrically similar to that of the prototype. This is a result of using the same soil in the model as in the prototype, which means equality of pore sizes and therefore equality of matric

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COOKE AND MITCHELL 831

FIG. 1. Soil column and tank assembly mounted in centrifuge strongbox.

suction causing capillary rise. However, the unit weight of water in the model is N times as great owing to the increased gravity, so the water column supported by this suction will only be 1 /N as high as in the prototype, thus maintaining geometric similarity.

Procedure The 3 m radius, 30 g-tonne geotechnical centrifuge that

has been in operation at the Civil Engineering Department of Queen's University since 1985 was adapted for geo- environmental research early in 1988 by constructing a 0.25 x 0.4 x 0.5 m (on radius) strongbox to carry various apparatuses that were designed for hydraulic conductivity and contaminant transport studies.

The soil used in this study was a natural fine silty sand (72% sand, 25% silt, 3% clay) from Shirley's Bay, near Ottawa. It had an average saturated hydraulic conductivity at the void ratio used in these tests of 3.7 x m/s.

The apparatus used in this study is shown in Fig. 1 and is described in detail by Mitchell and Cooke (1991). The water tanks are mounted on springs and connected to direct current displacement transducers (DCDTs). Spring constants are matched to the gravitational forces being simulated in any given test run such that each water tank moves up (or down) as water leaves (or enters) that tank, thus maintain- ing a constant water elevation. The movement of each tank is detected to f 0.01 mm by the DCDTs, and the measure- ment of mass transfer is accurate, by calibration, to 2 or 3 g of water. The 100 mm diameter lucite soil column con- tainer was sectioned to allow examination of each 20 mm section following a model test.

The model soil columns were prepared by compacting the soil into the lucite mold using energy input per unit volume equivalent to the standard Proctor test. Since the mold was composed of 20 mm rings, the height could be adjusted to suit the requirements of the particular test by addition or removal of rings. To represent a soil profile with a nominal prototype height of 15 m, 30 and 16 cm columns were used for 50 x g and 100 x g models, respectively. Each soil column was saturated from the bottom under a hydraulic gradient of between 2 and 3, then placed into the centrifuge strongbox for consolidation and permeability testing.

The soil column was connected to the water supply and receiving tanks, and then the centrifuge was brought up to the target speed (to produce an acceleration of either 50 or 100 xg). The upper water level was set approximately at the elevation of the top of the soil column to model a pro- totype gradient of unity. The bottom water level, represent- ing the groundwater table, was set at an elevation between 1 and 2 cm from the bottom of the soil column for both the permeability test and the drainage test (described below). Water was permeated through at this constant gradient until the hydraulic conductivity stabilized.

The centrifuge was then stopped and the supply (top) tank disconnected. The soil column was again accelerated and allowed to drain for 30 000 h prototype time (3 h at 100 x g or 12 h at 50 x g). While this drainage test was in progress, a 10% NaCl solution was added to the top of the column, using an in-flight dispensing apparatus. The volume of solu- tion added was equivalent to a prototype spill of 2000 L; that is, 2 cm3 of tiacer was added in the 100 x g tests, and 16 cm3 was added in the 50 x g tests. This addition took place after 7500 h drainage for series A tests and after 15 000 h for series B tests. Thus the tracer was allowed to infiltrate for 22 500 h for series A and 15 000 h for series B tests.

At the completion of the test, the soil column was parti- tioned into 20 mm slices. Each slice was dried to determine moisture content and analyzed for the quantity of the salt tracer by conductivity testing.

The volume of water draining into the receiving tank was monitored continuously throughout the test, and the final volume was checked by weighing the water collected in the tank at the end of the test.

Verification of results A standard method of verifying results in centrifuge

modelling is through "modelling of models," whereby the same experiment is carried out at more than one scale. If the results are similar at two different scales, then it is gen- erally assumed they are valid for any scale, including pro- totype. In the current work, all experiments were run at scale factors of 50 and 100. To investigate the existence of real- time phenomena such as chemical effects, much larger scale variations would have to be used.

Another means of verifying results, of course, is through comparison with numerical models. The soil column drainage was modelled numerically using RICHARDS, an implicit finite difference program written for this research, based on the solution to the Richards equation (Richards 1931), which is outlined by Whisler and Watson (1968). Input data for the program was obtained from standard pressure extraction tests run to establish the soil mois-

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832 CAN. GEOTECH. J . VOL. 28, 1991

- 50 g MODELS 100 g MODELS

............ COMPUTER MODEL

PROTONPE TIME, x lo3 (h)

FIG. 2. Results of column drainage test.

- 50 g MODELS 100 g MODELS

........ COMPUTER MODEL

HEIGHT ABOVE WL (rn)

FIG. 3. Moisture-content profiles. WL, water level.

ture characteristic curve. This curve was fitted to the van Genuchten (1980) model by nonlinear least squares anal- ysis, using the program SOIL (El-Kadi 1987).

Test results Typical results of the column drainage aspect of the tests

for both 50 x g and 100 x g models are presented in Fig. 2, as well as the computer solution from RICHARDS. To effect meaningful comparisons, results are presented in dimen- sionless form as the outflow volume, Q, divided by the equi- librium outflow volume, Q,. The value of Q, was deter- mined by numerical integration between the saturated water content and the equilibrium profile as determined by the pressure extraction tests. Excellent agreement between the 50 x g model results and the computer simulation is evident in Fig. 2. The 100 x g model results differ slightly but are also in good agreement.

0 . 0 1 I I I I I 0 1 2 3 4 5

PROTOTYPE DEPTH ( m )

FIG. 4. Cumulative tracer mass, series A test.

0.0P I I I I

0 1 2 3 4 5

PROTOTYPE DEPTH (rn)

FIG. 5. Cumulative tracer mass, series B test

The moisture profile at the completion of testing is shown in Fig. 3 , again compared with the computer prediction. Again, there is good agreement between the 50 x g and 100 x g models and the physical model data support the computed curves.

The variation between 50 x g and 100 x g results is con- sidered to be due to the fact that a 20 mm slice from the soil column represents 2 m in the 100 x g tests and 1 m in the 50 x g tests, thus the sampling frequency is twice as high with respect to prototype results in the 50 x g models.

It should also be noted that the data in Fig. 3 do not cor- respond to the static soil moisture characteristic curve, which would have taken several days to establish at 50 xg.

Figures 4 and 5 show the cumulation of tracer (salt) mass with depth for the two sets of models, presented as a pro-

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COOKE A N D MITCHELL 833

A SERIES A

A SERIES B

PROTOTYPE DEPTH (m)

FIG. 6. Cumulative tracer mass, 500 x g tests.

portion of the total mass added. Figure 4 gives results for the series A tests, in which the tracer infiltrated for 22 500 h, and Fig. 5 gives the series B results, with 15 000 h infiltra- tion. In both cases there is excellent agreement between the 50 x g and 100 x g models. A computer model is not avail- able for comparison with this data.

Figure 6 compares the cumulative infiltration for the 50 x g series A and B models, demonstrating the advance of the tracer with time. It is clear from Figs. 4-6 that the modelling of models technique provides a level of confidence in accepting these tracer test results as being representative of prototype behaviour.

Summary and conclusions

Previous research has established that fluid flow through saturated and unsaturated soil may be accurately modelled in a geotechnical centrifuge. The current centrifugal model- ling research confirms the theoretical work of Arulanandan et al. (1988), which indicates that dissolved contaminant movement through unsaturated soil may be accurately rep- resented by centrifuge modelling. These modelling tech- niques may prove useful in representing specific contami- nant events and in the validation of numerical models developed to simulate such events on a computer. Additional centrifuge modelling, using a wider range of scale factors, is needed to determine whether the contaminant movement may be altered by chemical or other real-time effects not accounted for in this study.

Acknowledgements

Financial support for the modelling work reported in this paper was provided by the Ontario Ministry of the Envi- ronment, project No. 505G. The support of the Natural

Sciences and Engineering Research Council of Canada (NSERC) in providing funds to construct and operate the Queen's University geotechnical centrifuge centre is also gratefully acknowledged. Mr. Cooke also acknowledges the support of NSERC through a 1967 Science and Engineer- ing Scholarship.

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