agricultural soil typically found in Victoria, Australia [11]. ..... G. Allinson, F. Stagnitti, S.A. Salzman, K.J. Dover, J.P. Venner and L.A. Thwaites, A comparison of the.
MATHEMATICAL AND COMPUTER MODELLING PERGAMON
Mathematical and Computer Modelling 34 (2001) 433-440 www.elsevier, nl/locate/mcm
M o d e l l i n g Solute Transport in S t r u c t u r e d Soils: P e r f o r m a n c e Evaluation of t h e A D R and T R M M o d e l s F. STAGNITTI* School of Ecology and Environment, Deakin University P.O. Box 423, Warrnambool 3280, Australia frankst©deakin, edu. au
LING LI AND A. BARRY Department of Civil and Environmental Engineering University of Edinburgh, Scotland, U.K. G. ALLINSON School of Ecology and Environment, Deakin University P.O. Box 423, Warrnambool 3280, Australia
J.-Y. P A R L A N G E A N D W. S T E E N H U I S Department of Agricultural and Biological Engineering Cornell University, Ithaca, NY, U.S.A. E. LAKSHMANAN Department of Geology, Anna University Chennai-600 025, India
(Received August 1999; revised and accepted April 2000) A b s t r a c t - - T h e movement of chemicals through the soil to the groundwater or discharged to surface waters represents a degradation of these resources. In many cases, serious human and stock health implications are associated with this form of pollution. The chemicals of interest include nutrients, pesticides, salts, and industrial wastes. Recent studies have shown that current models and methods do not adequately describe the leaching of nutrients through soil, often underestimating the risk of groundwater contamination by surface-applied chemicals, and overestimating the concentration of resident solutes. This inaccuracy results primarily from ignoring soil structure and nonequilibrium between soil constituents, water, and solutes. A multiple sample percolation system (MSPS), consisting of 25 individual collection wells, was constructed to study the effects of localized soil heterogeneities on the transport of nutrients (NO3, Cl-, PO 3-) in the vadose zone of an agricultural soil predominantly dominated by clay. Very significant variations in drainage patterns across a small spatial scale were observed (one-way ANOVA, p < 0.001) indicating considerable heterogeneity in water flow patterns and nutrient leaching. Using data collected from the multiple sample percolation experiments, this paper compares the performance of two mathematical models for predicting solute transport, the advective-dispersion model with a reaction term (ADR), and a two-region preferential flow model (TRM) suitable for modelling nonequilibrium transport. These results have implications *Author to whom all correspondence should be addressed. The first author acknowledges funding support from the Australian Research Council's Large Grant Scheme (Grant Nos. A89701825 and A10014154); Department of Industry, Science and Tourism, International Collaboration (Grant #16), the Australia-India Council, and a travel fellowship from the Australian Academy of Science to the United Kingdom. The last author acknowledges funding support from the Indian Council of Scientific and Industrial Research. 0895-7177/01/$ - see front matter (~) 2001 Elsevier Science Ltd. All rights reserved. PII: S0895-7177(01)00074-7
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434
F. STAGNITTIet al. for modelling solute transport and predicting nutrient loading on a larger scale. @ 200i Elsevier Science Ltd. All rights reserved. Keywords--Advective-dispersion equation, Preferential flow, Numerical methods, Solute transport, Nutrients, Nitrate, Chloride, Phosphate, Pollution..
1. I N T R O D U C T I O N The movement of chemicals through the soil to the groundwater or discharged to surface waters represents a degradation of these resources. In many cases, serious human and stock health implications are associated with this form of pollution. The chemicals of interest include nutrients, pesticides, salts, and industrial wastes. In the case of nutrients, leaching losses also represent a decline in soil fertility with economic consequences, and in the case of nitrate leaching, reduction of productivity due to soil acidification [1,2]. In the case of pesticides and industrial wastes, small but highly toxic chemicals can be transported to groundwater and remain there for hundreds and even thousands of years [3-7]. Monitoring and modelling solute transport is complicated by nonrandom spatial and temporal variations of physical, chemical, and biological components of soils. Laboratory experiments based on repacked, sterile, homogenized soil cores bear little resemblance to physical reality and have often been shown to underestimate the solute loss and risk of contamination to groundwater reserves [8]. Hence, a better knowledge of the factors causing nonequilibrium and heterogeneous solute transport is required for sustainable ecological management of nutrients in agricultural soils [9]. Multiple sample percolation systems are an excellent means to obtain accurate values of the solute and water flux in vadose zone experiments and have been shown to effectively represent the impacts of soil heterogeneity on these fluxes [10]. Using this apparatus, an experiment was recently conducted to investigate the impacts of soil structure on nutrient transport in an agricultural soil typically found in Victoria, Australia [11]. The data from this experiment is used here to compare and contrast two approaches to modelling solute transport in the unsaturated zone. The models considered in this paper are the advective-dispersion equation with reaction (ADR) and a two-region preferential flow model (TRM).
2. M A T E R I A L S
AND METHODS
Free-draining, undisturbed soil columns offer the best means to study nutrient transport under field conditions because they preserve the natural structure of the soil. A large undisturbed soil core (42.5cm x 42.5cm wide x 40cm deep) was extracted from a farm located in the Western District of Victoria, Australia. A nmltiple sample percolation system was constructed to sample moisture and chemicals leaching from the soil core. The multiple sample percolation system consists of a metal-alloy base-plate that is shaped into 25 equal sized collection wells (funnels) o1' mini-catchments. Leachate solutions were analysed for NO3, CI-, and PO 3-. The daily leachate concentrations collected from each of the 25 individual collection wells were aggregated to calculate a total daily concentration for the entire soil core for each ion for each day of the duration of the experiment. The daily total concentrations were used to fit concentration breakthrough curves for the T R M and ADR models. The full experimental details are described in [8].
3. T H E O R Y 3.1. E q u i l i b r i u m T r a n s p o r t M o d e l ( A D R ) The most challenging problem confronting mathematical modelling of solute transport in field soils is how to effectively characterize and quantify the geometric, hydraulic, and chemical properties of porous media. Recently, Stagnitti et al. [10] and de Rooij and Stagnitti [12] proposed the
Modelling Solute Transport
435
use of the beta distribution to describe the statistical variation in solute and moisture patterns. However, this approach provides little insight into the complex physical and chemical processes that govern adsorption and transport in soils. The issue of how to deal with heterogeneity in the soil remains unresolved. To date, therefore, it is not surprising that very few attempts have specifically included soil structure in modelling solute transport [13]. To reduce the complexity involved in modelling solute transport, many models rely on the simplifying assumption of homogeneous soil structure and instantaneous sorption, sometimes referred to as the LEA (linear equilibrium adsorption) assumption. The general equation governing contaminant transport under saturated, steady flow conditions, and with chemical reaction, has the form of the classical advection-dispersion-reaction equation [14] n °c
-
Ot = t)-g~z2
-
v °c
Oz
- 7c,
(1)
where R is the retardation factor (which is equal to 1 + pk/O and where p is the density of water, k is the distribution coefficient of absorption, and 0 is the water content), "7 is the reaction rate coefficient, D is the dispersion coefficient, and V is the pore water velocity. Analytical solutions to equation (1) for most practical circumstances have been compiled by Parker and van Genuchten [14], van Genuchten and Alves [15], and Barry and Sposito [16]. 3.2. N o n e q u i l i b r i u m T r a n s p o r t M o d e l - - T h e
Two Region Model (TRM)
The ADR has a simple form because it describes an ideal process, i.e., equilibrium transport. However, the LEA assumption is normally not valid in field studies. Nonideal transport (nonequilibrium transport) as observed in many experiments is more the norm than the exception. The causes of nonequilibrium transport in soils are soil heterogeneity and chemical nonequilibrium sorption [17]. Nonequilibrium transport is mainly due to physical and chemical phenomena reflecting the heterogeneous properties of soils. Also, biological forces are increasingly being recognized as a source of nonequilibrium [13]. In this paper, we examine the nonequilibrium transport caused by possible preferential flow through the soil column. Based on the bicontinuum conceptualization, Coats and Smith [18] developed the two-region model (TRM) to describe nonequilibrium solute transport in aggregated soils. The governing equations are R OCm
1 -/30C~m
Ot + T Ot 1 -/30Cim /3
at
_ D O2Cm
Oz 2
OCm
K,,
- a3(c,, - c~,,),
(2a)
0--7 '
(2b)
where the subscripts m and i m denote regions in which mobile and immobile solute transport may occur,/3 is the ratio of the mobile region to the entire pore volume, i.e.,/3 = 0m/(0m + 0im), 0 is water content, Vm is the flow velocity in the mobile region and the velocity in the immobile region is zero by definition (so the averaged flow velocity is V,~/3), and a~ is the rate coefficient (in the nondimensional form, c~ = w L / V m , and L is the column length). The bicontinuum concept physically represents the soil structure in aggregated soils. The region within the aggregates is the immobile region where water and solutes are stagnant except for lateral diffusion. The region between the aggregates is the mobile region where water and solute move due to adveetion and dispersion. The lateral diffusion has been simplified by using the first-order equation (2b). Although this model was originally developed for solute transport in aggregated soils, it is often used t o m o d e l other nonequilibrium transport processes. For example, Coats and Smith [18], Herr et aL [19], and Li et al. [20] have reported applications of this model to solute transport in stratified soils. Herein, we report the use of the T R M to investigate solute transport in an undisturbed soil column and compare it with the ADR.
F. STAGNITTIet al.
436
4. R E S U L T S A N D D I S C U S S I O N 4.1. Effects of Spatial Variability
in Leaching
T a b l e 1 presents t h e d a i l y average v o l u m e of l e a c h a t e collected for each well of t h e m u l t i p l e s a m p l e p e r c o l a t i o n s y s t e m . T h e d a t a has been r a n k e d in o r d e r of m a g n i t u d e of e l u t e d leachate. Wells 1 to 5, 6, 10, 11, 15, 16, 20, a n d 21 to 25 were l o c a t e d on t h e edge of t h e p e r c o l a t i o n s y s t e m , a n d t h e s e s u r r o u n d e d t h e inner wells n u m b e r e d 7 to 9, 12 t o 14, a n d 17 t o 19. T h e t a b l e i l l u s t r a t e s s u b s t a n t i a l h e t e r o g e n e i t y in p e r c o l a t i o n r a t e s from t h e core. T h e e l u t e d v o l u m e s for each well were found to be s t a t i s t i c a l l y v e r y highly v a r i a b l e (one-way A N O V A , p < 0.001). Table 1. Average daily volume of soil moisture collected by each MSPS well. The data are the arithmetic mean and standard errors determined over an 18 day period. Low Flow Wells
Medium Flow Wells
Rapid Flow Wells
Well No.
Mean
St. Err.
Well No.
Mean
St. Err.
Well No.
Mean
St. Err.
8 3 1 2 6 11 4 17
11.9 26.5 31.4 55.5 57.5 60.0 66.8 84.3
5.38 5.39 3.23 8.94 11.75 6.19 9.85 16.93
14 18 12 13 16 10 19 9
84.8 87.0 106.7 117.9 120.6 123.1 127.7 131.3
7.49 6.71 7.08 8.57 10.69 19.92 13.86 12.49
21 7 24 22 23 5 25 20 15
133.0 135.0 142.7 197.2 199.2 204.3 241.4 307.1 333.5
36.52 9.06 25.32 37.77 35.41 33.25 44.49 43.61 49.38
F i g u r e 1 p r e s e n t s t h e results from one of t h e e x p e r i m e n t s c o n d u c t e d on t h e u n d i s t u r b e d , freed r a i n i n g , field soil. T h e c u m u l a t i v e solute m a s s a n d v o l u m e of s o i l - m o i s t u r e e l u t e d from t h e core were m o n i t o r e d over an 18 d a y period. T h e i r r i g a t i o n r a t e a p p l i e d to t h e surface of t h e core was u n i f o r m in t i m e a n d areal coverage. P e r c o l a t i o n of soil-water from t h e core was n e a r l y u n i f o r m as i n d i c a t e d on t h e figure w i t h a close c o r r e s p o n d e n c e to a o n e - t o - o n e line. T h e r e l a t i v e m a s s of N O 3 leaching from t h e core was v e r y closely m a t c h e d to t h e relative a m o u n t of C I - , i n d i c a t i n g t h a t C1- m a y be a s u i t a b l e p r o x y for N O ~ in solute t r a n s p o r t studies. As m u c h as 50% of t h e a p p l i e d m a s s of C1- a n d NO 3 leached from t h e core in 30% of t h e a p p l i e d i r r i g a t i o n w i t h i n five d a y s after a p p l i c a t i o n , i n d i c a t i n g v e r y r a p i d t r a n s p o r t . However, w i t h i n t h e s a m e t i m e p e r i o d , less t h a n 15% of t h e a p p l i e d P O 3 - leached from t h e core, i n d i c a t i n g s t r o n g a d s o r p t i o n . 4.2. Modelling
Results
S o l u t e b r e a k t h r o u g h curves ( B T C ) are p r e s e n t e d in F i g u r e s 2 to 4. T h e r a t e of e v a p o t r a n s p i r a t i o n was d e t e r m i n e d from a w a t e r b u d g e t over t h e 18 days. T h e average r a t e was a p p r o x i m a t e l y 25%. Using t h i s value, t h e initial solute c o n c e n t r a t i o n s in t h e i r r i g a t i o n were d e t e r m i n e d t o be C0_ci = 6 1 8 6 . 1 0 m g / L , C0_NO3 = 273.65 m g / L , a n d C 0 _ P O 4 : 4724.10 m g / L . T h e d u r a t i o n t i m e for i r r i g a t i o n of t h e solutes was T = 0.484 days ( i n p u t pulse). For each s i m u l a t i o n , it was a s s u m e d t h a t t h e initial (resident) c o n c e n t r a t i o n s of solutes in t h e soil were negligible; i.e., Ci = 0 m g / L . Using t h e s e values, t h e A D R a n d T R M were fitted to t h e C1- d a t a , a n d B T C s are shown in F i g u r e s 2a a n d b. T h e fitting was achieved using s i m u l a t e d a n n e a l i n g [21]. Sireu l a t e d a n n e a l i n g is a b e t t e r t e c h n i q u e for finding global o p t i m a t h a n t r a d i t i o n a l g r a d i e n t - t y p e a p p r o a c h e s such as n o n l i n e a r least squares m e t h o d s [22]. A s s u m i n g t h a t C1- b e h a v e s as a cons e r v a t i v e element, t h e n 7 = 0 (i.e., no reaction) a n d R = 1 (i.e., no a d s o r p t i o n ) . T h e o p t i m a l values for t h e fitted p a r a m e t e r s in t h e A D R were V = 0.051 m / d a n d D = 0 . 0 0 6 m 2 / d (see Figure 2a). T h e m o d e l failed t o m a t c h t h e d a t a at t h e peak. T h e lack of a g r e e m e n t is d u e t o t h e failure of t h e A D R t o m o d e l t h e n o n e q u i l i b r i u m effects caused by t h e o b s e r v e d h e t e r o g e n e i t y in
Modelling Solute T r a n s p o r t
437
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Chloride ......... Nitrate
- . . . . 50%
,
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o;o, ur
---
~
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___.__
11
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Time (Days) Figure 1. Relative c u m u l a t i v e m a s s of solutes a n d v o l u m e of eluted soil-water collected for all wells p l o t t e d as a function of time.
percolation rates. The fitted velocity and dispersion coefficients, however, are within the experimental constraints. The T R M was fitted to the data (see Figure 2b), and the optimal values for the model parameters were V m = 0.057m/d, D = 0.001 m2/d, ~ = 0.47, and a = 1.72. The predicted and actual BTCs are in excellent agreement, suggesting that nonequilibrium t r a n s p o r t of C1- has occurred. The results also verify that little to no retardation of chloride in the core occurred. The fitted parameter values are also within experimental constraints and they have physical meaning. In particular, the averaged flow velocity, calculated according to Vm~, is in accordance with the experimental data. The fitted value for/3 is close to 0.5, which indicates t h a t nearly 50% of the pore volume is actively responsible for C1- transport. This too is supported by experimental observations. The predicted B T C for NO 3 using the A D R is plotted with the actual B T C in Figure 3a. In this case, knowledge gained from fitting the B T C s for C1- is used to hold the values for V and D constant. The values for V and D determined from fitting the A D R to the C1- d a t a set are used for the NO 3 d a t a set. However, in this case, the possibility of adsorption and reaction of NO 3 is possible and therefore permitted; i.e., R and 7 become "free" parameters and their values are determined by optimization. The optimal values for R and ~ were found to be 1.06 and 0.035 d -1, respectively, indicating a small adsorption and significant reaction rate. These p a r a m e t e r values also make sense physically, e.g., resulting from mineralization and denitrification in the soil. The fit is surprisingly good even though the ADR does not explicitly include assumptions of variable flow domains. In a similar fashion, the values for the parameters Vm, D, /77, and ~ in the T R M were fixed to be the same as those determined for the C1- BTC. In this case, however, the only free p a r a m e t e r is R. The optimal value for R was found to be 1.27, and the predicted B T C is presented in Figure 3b. The T R M overpredicted the concentration peak. This is not surprising, as the T R M in its present form does not contain a reaction term. The better performance of A D R in this case most likely results from the extra freedom of having two free parameters rather t h a n one. Indeed, if the T R M included reaction, then the performance for nitrate prediction will undoubtedly improve. The phosphate experimental data showed strong adsorption and reaction (see Figure 1). For this reason, we did not fit the T R M to this d a t a set. Figure 4 presents the predicte d B T C using the A D R with V and D fixed to the same values as determined for the C1- B T C and R and 7 fitted by optimization. The optimal values for R and 7 were 8.117 and
F, STAGNITTI et aL
438
0.0~ 0.0;
O tI
0.0~
0.0~
@
O.OZ r~
0.0,~ 0.00.01 0
1'o
0
20
Time (Days) (a)
0.08" 0"07t
•
o.o6r 0'05I o.o4r
oos r oo2 r =
~ 0.01t
0;
;
,;
20
Time (Days) (b) Figure 2. Comparison of breakthrough curves for chloride using (a) ADR and (b) TRM. Experimental data are circles, and model predictions are solid lines.
? = 1.781d -1, respectively, indicating, as expected, very strong adsorption and quick reaction. The ADR predicted the data reasonably well. Again, the good performance of the ADR here does not necessarily imply that nonequilibrium transport is negligible; rather it may be due to the extra freedom in the fitting process. Also, the reaction rate appears to be too fast to be physically realistic. 5. C O N C L U S I O N S Soil structm'e has an important role in distributing moisture and solutes. Considerable smallscale heterogeneity in the volume and solute concentrations in the percolate was observed. The performance of two models, the ADR and TRM, were contrasted and compared. The ADR model performed reasonably well even though the experimental data exhibited considerable heterogeneity in the percolation rate and solute concentrations. There are currently no analytical solutions for the T R M with a reaction term. Consequently, the TRM in its present form can only be expected to perform well for solutes that have negligible reaction times (e.g., conservative tracers). We are presently developing a numerical solution for the TRM that incorporates a reaction term. This solution will also be coupled to an annealing optimization algorithm. We anticipate that this model will significantly improve the predictions of nonequilibrium solute transport in field soils.
439
Modelling Solute Transport
0.045
~Y o.04[ 0::r
~00:t ~°°I 0.01
i 0=[ o
,
.~
1'0
~
20
Time (Days)
(a)
0.07
"'" ' '
m
~
k
0,06
i
0.05 0.O4 0.03 0,02 0.01 10 15 Time (Days)
20
(b) Figure 3. Comparison of breakthrough curves for nitrate using (a) ADR and (b) TRM. Experimental d a t a are circles, and model predictions are solid lines.
e
~3
0 OoOe
en 5
• 10
15
2O
Time (Days) Figure 4. Breakthrough curve for phosphate predicted using TRM. Experimental data are circles, and model predictions are solid lines.
440
F. STAGNITTI et al.
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Stagnitti, Environmental fate of pesticides used in Australian viticulture. III. Fate of dithianon from vine to wine, Toxological and Environmental Chemistry 70, 385-400 (1999). 5. M. Allinson, B. Williams, G. Allinson, M. Graymore and F. Stagnitti, Environmental fate of pesticides used in Australian viticulture. IV. Aqueous stability of dithianon, Toxological and Environmental Chemistry 70, 401-414 (1999). 6. M. Craymore, G. Allinson, M. Allinson, F. Stagnitti, Y. Shibata and M. Morita, Environmental fate of pesticides used in Australian viticulture. V. Behaviour of atrazine in the soils of the South Australian Riverland, Toxological and Environmental Chemistry 70, 427-439 (1999). 7. Y. Kelsall, M. Allinson, G. Allinson, N. Turoczy, F. Stagnitti, M. Nishikawa and M. Morita, Leaching of copper, chromium and arsenic in a soil of the South West Victoria, Australia, Toxologieal and Environmental Chemistry 70, 375-384 (1999). 8. F. Stagnitti, J. Sherwood, G. Allinson, L. Evans, M. Allinson, L. Li and I. Phillips, An investigation of localised soil heterogeneities on solute transport using a multisegment percolation system, New Zealand Journal of Agricultural Research 41, 603-612 (1998). 9. F. Stagnitti, J.-Y. Parlange, T.S. Steenhuis, J. Boll , B. Pivertz and D.A. Barry, Transport of moisture and solutes in the unsaturated zone by preferential flow, In Environmental Hydrology, Water Science and Technology Library, Volume 15, Chapter 7, (Edited by V.P. Singh), pp. 193 224, Kluwer Academic, Dordrecht, (1995). 10. F. Stagnitti, L. Li, G. Allinson, I. Phillips, D. Lockington, A. Zeiliguer, M. Allinson, J. Lloyd-Smith and M. Xie, A mathematical model for estimating the extent of solute- and water-flux heterogeneity in multiple sample percolation experiments, Journal of Hydrology 215, 59-69 (1999). 11. F. Stagnitti, G. Allinson, J. Sherwood, M. Craymore, M. Allinson, N. Turoczy, L. Li and I. Phillips, Preferential leaching of nitrate, chloride and phosphate in an Australian clay soil, Toxological and Environmental Chemistry 70, 415-425 (1999). 12. G.H. de Rooij and F. Stagnitti, Spatial variability of solute leaching: Experimental validation of a quantitative parameterization, Soil Science Society o/America Journal 64 (2), 499-504 (2000). 13. F. Stagnitti, A model of the effects of non-uniform soil-water distribution on the subsurface migration of bacteria: Implications for land disposal of sewage, Mathl. Comput. Modelling 29 (4), 41-52 (1999). 14. J.C. Parker and M.T. van Genuchten, Determining transport parameters from laboratory and field tracer experiments, Virginia Agricultural Experimental Station, Bulletin 84-3, Virginia Polytechnical Institute and State University, Blacksburg, VA. 15. M.T. van Genuchten and W.J. Alves, Analytical solutions to the one-dimensional convection-dispersion solute transport equation, USDA Technical Bulletin, p. 1661. 16. D.A. Barry and G. Sposito, Application of the convection-dispersion model to solute transport in finite soil columns, Soil Science Society of America Journal 52, 3 9 (1988). 17. M.L. Brusseau and P.S.C. Rao, Modelling solute transport in structured soils: A review, Geoderma 46, 169-192 (1990). 18. K.H. Coats and B.D. Smith, Dead end pore volume and dispersion in porous media, Society for Petroleum Engineering Journal 4, 73-84 (1964). 19. M.G. Herr, C. Schafer and K. Spitz, Experimental studies of mass transport in porous media with local heterogeneities, Journal of Contaminant Hydrology 4, 127-137 (1989). 20. L. Li, D.A. Barry, P.J. Culligan-Hensley and K. Bajracharya, Mass transfer in soils with local stratification on hydraulic conductivity, Water Resources Research 30, 2891-2900 (1994). 21. L. Li, D.A. Barry and J. Morris, Parallel simulated annealing using CILK language: Application in estimating transport parameters for groundwater contaminants. In Proceedings of ModSim 97, Hobart, Tasmania, Australia, (1997). 22. L. Li, J. Morris, D.A. Barry and F. Stagnitti, CXTANNEAL: An improved program for estimating transport parameters, Environmental Modelling and Software 14 (6), 607 611 (1999).
