Modeling cadmium accumulation in radish, carrot, spinach and cabbage

Applied Mathematical Modelling 31 (2007) 1652–1661 www.elsevier.com/locate/apm

Modeling cadmium accumulation in radish, carrot, spinach and cabbage P. Verma a

a,*

, K.V. George a, H.V. Singh a, R.N. Singh

b

National Environmental Engineering Research Institute, Nehru Marg, Nagpur 440 020, India b National Geophysical Research Institute, Uppal Road, Hydrabad, India Received 1 September 2005; received in revised form 1 April 2006; accepted 11 May 2006 Available online 27 July 2006

Abstract Heavy metals like cadmium and arsenic have serious health consequences and ecosystem impacts. Due to various factors including the disposal of municipal and industrial wastes, application of fertilizers, atmospheric deposition and discharge of wastewater on land, has resulted in increase in the concentration of heavy metals in the soil. Crops and vegetables grown on such soil accumulate heavy metals, which leads to phyto-toxicity. For understanding and managing precious natural resources, mathematical models are increasingly being used. This paper describes a dynamic macroscopic numerical model for heavy metal transport and its uptake by vegetables in the root zone. The model is applied for simulating cadmium uptake by radish (Raphanus sativus), carrot (Daucos carota), spinach (Spinacia oleracea) and cabbage (Brassica oleracea) by using measured field data. The governing non-linear partial differential equations are solved numerically by an implicit finite difference method using Picard’s iterative technique and the source code is written in MATLAB. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Modeling; Cadmium uptake; Numerical simulation; Phyto-toxicity

1. Introduction Soil normally contains a low concentration of heavy metals such as copper (Cu) and zinc (Zn), which are the essential micronutrients for the optimum growth of the plants. Metals like cadmium (Cd), arsenic (Ar), chromium (Cr), lead (Pb), nickel (Ni) and selenium (Se) are usually not found in agricultural soil and are toxic to plants [1]. Disposal of municipal and industrial wastes, application of phosphorous and nitrogen fertilizer and atmospheric deposition leads to a high concentration of specific heavy metals in the soil, which in turn are taken up by the plants and enter into the food chain resulting in phyto-toxicity [2]. Several researchers have studied the heavy metal accumulation in crops, the factors responsible for it, and its effect on human and animal health. In Japan the study on cadmium poisoning in humans revealed that it is due to rice grown on fields, which have *

Corresponding author. Tel.: +91 712 2249886 88x413; fax: +91 712 2249881. E-mail address: [email protected] (P. Verma).

0307-904X/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2006.05.008

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been irrigated by river water contaminated from mining operations [3]. A reduction in plant growth was observed due to the presence of elevated levels of heavy metals like copper, lead and mercury [4]. The application of fly ash and co-recycling of sewage sludge in soil increases its pH, which results in a reduction of metal uptake by crops [5]. Metal uptake and its translocation studies was carried out by various researchers [6–8] for different vegetables and crops under varying soil conditions, which is helpful in understanding the uptake and the transport mechanism. The major factors influencing the uptake of heavy metal by crops are soil pH [9], the amount of heavy metal present in the soil [10], and the oxidation and reduction potential of the soil affecting the bioavailability [11]. Other minor factors such as cation exchange capacity [12], content of oxides of iron and manganese in soil [13], interactive effects of zinc, copper, nickel, manganese, selenium and phosphorus [14] and soil temperature [15] are not nearly as consistent and clear cut as the major factors mentioned above. Several mathematical models exist for simulating solute and nutrients uptake by crops for steady state moisture and solute conditions [16–18]. These models were extended for heavy metal uptake by crops under steady state condition of moisture and heavy metal, however they ignore the influencing factors mentioned above [19,20]. Moreover their application is limited because they cannot simulate field conditions that are dynamic in nature. Simulation of heavy metals uptake by crops first requires the modeling of water uptake by root. A literature review classifies the study of water uptake by plant/crop into two categories. The first category (microscopic approach) considers radial flow of water to a single root [21–23] and helps in understanding the root extraction process. The second category (macroscopic approach) treats the root system as a single unit that extracts the moisture from each differential volume of the root zone [24–26] and can be incorporated as a sink term in the water balance equation. Various analytical expressions exist for predicting root growth with time [27,28,26]. Some of these equations assume that root density changes with depth and not with time and others assume the root distribution, to be a function of time and soil water pressure head. The root growth equation is described in terms of root elongation minus the death rate of roots. Once the movement of water in the unsaturated zone is simulated, the solute movement can be modeled. Solute movement in an unsaturated zone has been modeled by various researchers [29–31]. This study is based on the theoretical mathematical model proposed by Rao et al. [32] for simulating heavy metal uptake by plants and is improved by taking soil pH into consideration. The root zone extends maximum up to 2–3 m in length for crops and the soil properties remains the same for such shallow depth, the porous medium is taken to be homogenous and isotropic in the formulation. 2. Model description 2.1. Governing equation for water flow and its uptake by crops in the unsaturated zone The one-dimensional mass balance equation for the flow of water in an unsaturated zone with a sink term can be written as [33]:    oh o oh cðhÞ ¼ kðhÞ  1  Sðz; tÞ: ot oz oz

ð1Þ

Here h is the suction head, c(h) is the soil moisture capacity defined as oh/oh, k(h), the suction head dependent hydraulic conductivity, and z, the soil depth taken to be positive upwards. S(z, t) is the sink term for water extraction, which is a spatial and temporal function expressed as the volume of water per unit volume of soil per unit time. The solution of Eq. (1) requires soil moisture retention characteristics and the unsaturated hydraulic conductivity. Since volumetric water content and unsaturated hydraulic conductivity are non-linear functions of suction head, explicit expressions developed by van Genucthen [34] and van Genucthen et al. [35], based on the experimental data set, are adopted in this model. These expressions can be written respectively as: hðhÞ  hr 1 ¼ ; ð1 þ ðahÞn Þm hs  hr

ð2Þ

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and h i2 n1 n m 1  ðahÞ f1 þ ðahÞ g kðhÞ ¼ k s

n

m

½1 þ ðahÞ  2

:

ð3Þ

Here h(h) represents the water retention curve, defining the water content as a function of the soil water pressure, hs and hr are the saturation and residual moisture content, respectively. a, n and m are the curve shape parameters and ks is the saturated hydraulic conductivity of the soil. The first derivative of Eq. (2) gives the expression for soil moisture capacity as: cðhÞ ¼

oh ð1  nÞan hn1 ðhs  hr Þ ¼ : n mþ1 oh ½1 þ ðahÞ 

ð4Þ

The root water extraction function is a sink term in Eq. (1), which represents the volume of water withdrawn by plant roots. From many expressions for such functions available in the literature [36–38], the linear root water uptake model suggested by Prasad [39], is adopted in this model, since it has been verified for several soil moisture conditions:   2T p ðtÞ Zr Sðz; tÞ ¼ bðhÞ 1  : ð5Þ Z max Z max Here Tp is the potential transpiration rate, Zr the rooting depth, Zmax the maximum rooting depth and b(h) the suction head dependent reduction factor. At any time the value of S(z, t) gives the uptake rate at the top of the root and varies linearly to zero at the bottom of the root. In the water uptake equation (5), the water extraction rate is proportional to the root depth, Zr, which is a function of time. Empirical equations have been developed by many researchers to predict the root growth with time [40]. Some of these empirical equations require detailed information on the root to predict the growth of the root at a microscopic rate and are impractical to use since they require extensive data, which is again a limitation. Among various expressions, the one suggested by Borg and Grimes [28] is adopted since it has been developed after an extensive regression analysis using 150-field observations:    DAP Z r ¼ 0:5Z max 1 þ sin 3:03  1:47 : ð6Þ DTM Here DAP and DTM are respectively the days after planting and days to maturity of the plant under consideration. This empirical relation does not take into account the root distribution with depth. 2.2. Governing equation for heavy metal transport and its uptake by crops The governing one-dimensional equation for heavy metal transport through soil can be written as [32]:   o o o oM w ðqM s Þ þ ðhM w Þ ¼ hD  qM w þ S p : ð7Þ ot ot oz oz Here q is the bulk density of the soil, Ms, the mass of the heavy metal adsorbed in the soil column, h, the volumetric moisture content, Mw, the concentration of heavy metal in water phase and D is the hydrodynamic dispersion coefficient, which is a function of pore velocity V. q is the Darcy velocity and Sp the heavy metal uptake by plant roots. The pore water velocity and hydrodynamic dispersion coefficient can be written respectively as V ¼ q=n; D ¼ aV þ b:

ð8Þ ð9Þ

Here n is the porosity of the soil, a is the longitudinal dispersivity and b is the molecular diffusion coefficient. The Darcy velocity, q is given by   oh 1 : ð10Þ q ¼ kðhÞ oz

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Generally one heavy metal in soil is accompanied by others. Since most of these heavy metals are divalent cations they compete for sorption sites [41], but the results are not clear about which heavy metal has more affinity as compared to others for plant uptake and results cannot be generalized. Therefore, a competitive effect of other heavy metals has not been included in this study. The heavy metal in soil solution is assumed to be governed by linear isothermal adsorption and is proportional to the soil heavy metal concentration [42]: M w ¼ kd M s;

ð11Þ

here kd is the partition coefficient. The uptake of heavy metals depends upon various factors including soil and plant characteristics. The mathematical models are based on transportation of the ions in the plant roots by mass flow and diffusion along the concentration gradients. The ions are absorbed by roots at a rate that depends on the concentration of the ions at the root surface and are assumed to follow a Michaelis–Menton type relationship. The expression is further modified for soil pH by taking into account a pH factor in the uptake term in the form as S p ¼ F pH

I max M w : Km þ Mw

ð12Þ

Here Imax is the uptake rate at infinite concentration, Km is the Michaelis and Menton constant, and FpH, the soil pH factor. Data selected from results published by the Council of Agricultural Science and Technology [43], illustrates how Cd accumulation in Swiss chard and oat grain varies with soil pH. The mean Cd concentration in the Swiss chard reduced from 40.5 mg/kg to 3.01 mg/kg by varying pH from 5.2 to 6.1. Similar results were reported by Chaney et al. [44], which showed that a reduction in the Cd concentration of soybean leaves from 33 mg/kg to 5 mg/kg when the pH of the soil on which the plants were grown was increased by liming from pH 5.3 to 7.0. 3. Solution of the governing equations Eqs. (1)–(12) form a complete set of governing equations, which are non-linear partial differential equations. Thus an analytical solution is not possible and hence a numerical method is applied to obtain the solution by prescribing initial and boundary conditions for flow and heavy metal transport equations. The governing partial differential equations when solved using finite difference method results in a set of non-linear equations, which is solved by Picard’s iterative technique. The governing equation (1), along with equations (3)–(6) is solved first to obtain the spatial and temporal variation of suction head (h) in soil column. The moisture content h(h) and flux (q) in the soil column are obtained by substituting the computed suction head (h) in Eqs. (2) and (10). Eqs. (7) is then solved along with Eqs. (11) and (12) to obtain the heavy metal adsorbed in the soil and water phase and heavy metal uptake by plant root system. Fig. 1 shows the flow chart for the computation scheme. The source code for the solution of partial differential equation is written in MATLAB. 4. Result and discussion The above model is applied to simulate cadmium uptake in vegetables by using the experimental data generated by Page [6]. He studied cadmium accumulation in vegetables like radish (Raphanus sativus), carrot (Daucos carota), spinach (Spinacia oleracea) and cabbage (Brassica oleracea) grown on metal contaminated soil. Table 1 shows the soil and model parameters and Table 2 summarizes the water requirement, maximum rooting depth, crop maturity period, moisture content and cadmium levels in the radish, carrot, spinach and cabbage. The governing equation for flow is solved first using the following initial and boundary condition, which can be written as:  2  oh ¼ 0 initially at t = 0, oz2 t¼0 h(0, t) = 0 cm at groundwater level, h(L, t) = 30 cm at ground surface.

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Water flow & Uptake by Root c(h)

∂h ∂ ⎡ ⎛ ∂h ⎞ ⎤ = k ( h )⎜ − 1⎟ ⎥ − S ( z , t ) ∂t ∂z ⎣⎢ ⎝ ∂z ⎠⎦

Discritization using Implicit FDM ( Δ z, Δt ) Non-linear set of algebraic Equation

Ai hit−+11 + Bi hit +1 + C i hit++11 + Di = 0 Input Tp (t ), α (h),

Assume h and Calculate the Coefficients A (h, k), B (c, h, k), C (h, k), D(c, h, k)

Zmax, DAP, DTM n, a, θ s, θ r

Recalculate A (h, k), B (c, h, k), C (h, k), D (c, h, k)

IC & BC for h Linear Set of algebraic Equation

Ai hit−+11 + Bi hit +1 + C i hit++11 + Di = 0 Solution of algebraic Equation Compute h ’

Output Calculate q, θ ,v

Replace h with computed value of h’

YES

Suction Head

NO

| h – h’ |< ε

Water flow & Uptake by Root ∂ ∂M w ∂ ∂ − qM w ) + S p ( ρM s) + (θ M w ) = (θ D ∂z ∂z ∂t ∂t

Discritization using Implicit FDM ( Δz, Δt ) Non-linear set of algebraic Equation t +1

t +1

t +1

A ' i M w i −1 + B ' i M w i + C ' i M w i + 1 + D ' i = 0 Input a, b, kd, I max, km, ρ & IC & BC for Mw

Assume Mw and Calculate the Coefficients A’ (Mw), B ’ (Mw), C’ (Mw), D’ (Mw) Recalculate A (Mw), B (Mw), C (Mw), D(Mw)

Linear Set of algebraic Equation t +1

t +1

t +1

A ' i M w i −1 + B ' i M w i + C ' i M w i + 1 + D ' i = 0 Solution of algebraic Equation Compute Mw’

Calculate Ms & Heavy Metal uptake by crops

Output

Mw

YES

Replace Mw with computed value of Mw ’

| Mw – Mw ’ |< ε

Fig. 1. Flow chart for computation scheme.

NO

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Table 1 Soil and model parameters Parameter

Symbol

Value

Unit

Saturated moisture content Residual moisture content Saturated hydraulic conductivity Bulk density of soil Height of unsaturated zone Dispersion coefficient Diffusion coefficient Partition coefficient Space step Time step Empirical shape factor Empirical shape factor Empirical shape factor Soil pH factor Suction head dependent reduction factor

hs hr ks q L a b kd Dz Dt n m a FpH a(h)

0.41 0.05 0.67 1.67 300 0.008 0.002 0.3 10 1 1.52 1.0 0.000542 1.0 1.0

– – cm/h g/cm3 cm cm cm2/s – cm h – – cm1 – –

Table 2 Vegetable parameters Parameter

Carrot

Radish

Spinach

Cabbage

Maximum rooting depth, Zmax (m) Crop maturity period (d) Water requirement (cm) Average wet weight of vegetable (kg) Moisture content (%) Average dry weight of vegetable (kg) Cadmium accumulation (mg/kg) (dry weight basis) Total cadmium accumulated in vegetable (lg)

0.4 120 45 0.120 86.0 0.0168 0.980 16.464

0.4 45 45 0.045 94.5 0.002475 0.310 0.7672

0.2 50 30 0.050 92.1 0.00395 3.090 12.2055

0.3 90 50 0.700 91.9 0.0567 0.093 5.2731

Here L is the length measured from water table to ground surface. For the simulation, an unsaturated soil depth of 3 m is considered, which is irrigated with wastewater having cadmium concentration of 0.005 g/ml. The model is run for a time span equal to the maturity period of the vegetable under consideration. The upper boundary is subjected to a suction head of 30 cm, which provides sufficient water for the vegetable to grow under unstressed conditions. Therefore the value of reduction factor b(h) is taken as 1 in the simulation. For each vegetable (radish, carrot, spinach and cabbage) under consideration, a number of model runs were carried out by varying the transpiration rate until the computed cumulative water uptake equals the total water requirement of the crop (Table 2). It can be inferred from Table 2 and the model results (Fig. 2) that though carrot and radish have the same maximum rooting depth, the later achieves it faster because of a lesser maturity period. The daily water uptake for different vegetables is depicted in Fig. 3 in which radish shows the highest maximum daily water uptake, followed by spinach, cabbage and carrot. Fig. 4 shows the cumulative water uptake by the different vegetables. After calibrating the cumulative water uptake using transpiration rate, the governing equation for heavy metal transport in the unsaturated zone is solved by using the following initial and boundary conditions prescribed in terms of cadmium concentration: Mw(z, 0) = 0 g/ml initially at t = 0, Mw(0, t) = 0 g/ml at groundwater level, Mw(L, t) = 0.005 g/ml at ground surface.

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P. Verma et al. / Applied Mathematical Modelling 31 (2007) 1652–1661 40 Carrot Radish

Root Length (cm)

30

Cabbage

20 Spinach

10

0

0

20

40

60

80

100

120

Time (days)

Fig. 2. Root growth with time.

2.5

2

Water Uptake (cm)

Radish

1.5 Spinach

Cabbage

1

0.5 Carrot

0

0

20

40

60

80

100

120

Time (days)

Fig. 3. Daily water uptake.

Similarly the maximum metal uptake rate (Imax) and Michaelis and Menton Constant (Km) are estimated in a similar way by comparing the computed cumulative metal uptake by the crop root with the measured total cadmium in the vegetable (Table 2). A sensitivity analysis for the plant parameters revealed that Imax is more sensitive than Km and therefore Imax has been first adjusted followed by Km. The simulation is carried out by initially maintaining a constant value for Km (which is less than the metal concentration in wastewater) and varying the maximum metal uptake rate. A number of simulations were carried out until the cumulative metal uptake by the roots nearly equals the measured metal content in the vegetable. Further, the Km value is varied

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60

50

Cabbage

Carrot

Water Uptake (cm)

Radish 40

Spinach

30

20

10

0 0

20

40

60

80

100

120

Time (days)

Fig. 4. Cumulative water uptake.

20

Carrot

Cadmium Uptake (ug)

16

12 Spinach 8 Cabbage 4

Radish 0 0

20

40

60

80

100

120

Time (days)

Fig. 5. Cumulative cadmium uptake.

Table 3 Uptake parameters for vegetables Parameters

Carrot

Radish

Spinach

Cabbage

Transpiration rate (cm/h) Maximum uptake rate, Imax (g/ml/h) Michaelis Menton constant, km (g/ml)

0.120 0.00008 0.00138

0.320 0.00004 0.02439

0.195 0.000067 0.0015

0.180 0.000043 0.004

until the cumulative metal uptake equals the metal content in the vegetable. Fig. 5 shows the cumulative heavy metal uptake and Table 3 shows the uptake parameters for carrot, radish, spinach and cabbage.

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5. Conclusion Verma et al. [45] carried out simulation for water movement and its uptake by plant roots in unsaturated zone. This paper is the extension of the earlier work and takes into account heavy metal movement in unsaturated zone and its uptake by plant roots. The governing equations developed in the mathematical model were solved using finite difference techniques to obtain the cadmium accumulation in radish, carrot, spinach and cabbage. The model can be used for predicting metal accumulation in different vegetables under dynamic field conditions using appropriate water extraction and metal uptake functions. This will be helpful in deciding the vegetable type to be grown, quantity of liming to be done for reducing the metal accumulation without hampering the vegetable growth. The model can be extended to two or three dimensions but will require detail information on complex root geometry, which at present is difficult and expensive to obtain. Presently such type of data available is very less; therefore a one-dimensional formulation is found to be more appropriate. This model can be further modified by taking into account complex root geometry and incorporating different processes such as degradation, chemical transformation, etc. for non-conservative pollutants in the rhizo-sphere. Acknowledgements The authors are thankful to Anamika Srivastava for valuable suggestions and the Director, NEERI, for kind permission to publish this paper. The authors also want to thank the anonymous reviewer for suggesting important changes for improving the manuscript. References [1] R. Tucker, D.H. Hardy, C.E. Stokes, Heavy Metals in North Carolina Soils, Occurrence and Significance, N.C. Department of Agriculture and Consumer Services, Agronomic division, Raleigh, 2003. Available from: . [2] W.R. Berti, L.W. Jacobs, Chemistry and phytotoxicity of soil trace elements from repeated sewage sludge applications, J. Environ. Qual. 25 (1996) 1025–1032. [3] Y. Takijima, F. Katsumi, K. Takezawa, Cadmium contamination of soils and rice plants caused by zinc mining. II Soil conditions of contaminated paddy fields which influence heavy metal contents of rice, Soil Sci. Plant Nutr. 19 (1973) 173–182. [4] R.H. Merry, K.G. Tiller, A.M. Alston, The effects of contamination of soil with copper, lead, mercury and arsenic on the growth and composition of plants. I Effects of season, genotype, soil temperature and fertilizers, Plant Soil. 91 (1986) 115–128. [5] A.L. Adriano, A.L. Page, Co-recycling of sewage sludge and fly ash: heavy metal uptake by crops, in: International Conference on Heavy Metal in the Environment, Commission of the European communities, WHO, Amsterdam, 1981, pp. 185–188. [6] A.L. Page, Cadmium in soil and its accumulation by food corps, in: International Conference on Heavy Metal in the Environment, Commission of the European communities, WHO, Amsterdam, 1981, pp. 206–213. [7] G. Ide, B. Becker, Relationship between the arsenic concentration in soil and in the cropped vegetablesInternational Conference on Heavy Metal in the Environment, Hamburg, vol. 2, CEP Consultants Ltd, Norwich, UK, 1995, pp. 302–304. [8] B. Szteke, R. Jedrzejczak, The variability of heavy metal contents in plant and soil samples from fields of one farmInternational Conference on Heavy Metal in the Environment, Hamburg, vol. 2, CEP Consultants Ltd., Norwich, UK, 1995, pp. 228–231. [9] CAST (Council of Agricultural Science and Technology), Application of sewage sludge to cropland: appraisal of potential hazards of the heavy metals to plants and animals, Council for agricultural science and technology, No. 64, Ames, Iowa, 1976, pp. 63. [10] J.M.A.S. Valadares, M. Gal, U. Mingelgrin, A.L. Page, Some heavy metals in soils treated with sewage sludge, their effects on yield and their uptake by plants, J. Environ. Qual. 12 (1983) 49–57. [11] F.T. Bingham, A.L. Page, R.J. Mahler, T.J. Ganje, Cadmium availability to rice in sludge amended soil under ‘flood’ and ‘nonflood’ culture, Soil Sci. Soc. Am. J. 40 (1976) 715–719. [12] R.J. Mahler, F.T. Bingham, A.L. Page, Cadmium-enriched sewage sludge application to acid and calcareous soils: 1. Effect on yield and cadmium uptake by lettuce and Swiss chard, J. Environ. Qual. 7 (1978) 274–281. [13] E.A. Forbes, A.M. Posner, J.P. Quirk, The specific adsorption of divalent Cd, Co, Cu, Pb, and Zn on goethite, Soil Sci. 27 (1976) 154– 160. [14] J.J. Street, B.R. Sabey, W.L. Lindsay, Influence of pH, phosphorus, cadmium, sewage sludge and incubation time on the solubility and plant uptake of cadmium, J. Environ. Qual. 7 (1978) 286–290. [15] P.M. Giordano, D.A. Mays, A.D. Behel, Soil temperature effects on the uptake of cadmium and zinc by vegetables grown on sludge amended soil, J. Environ. Qual. 8 (1979) 232–236. [16] P.H. Nye, F.H.C. Marriott, A theoretical study of the distribution of substances around roots resulting from simultaneous diffusion and mass flow, Plant Soil. 30 (1969) 459–472. [17] N. Classen, S.A. Barber, Simulation model for nutrients uptake from soil by a growing plant root system, Agrono. J. 68 (1976) 961– 964.

P. Verma et al. / Applied Mathematical Modelling 31 (2007) 1652–1661

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[18] E. Hoffland, H.S. Bolemhof, P.A. Leffelaar, G.R. Findenegg, J.A. Nelemans, Simulation of nutrients uptake by a growing root system considering increasing root density and inter root competition, Plant Soil. 124 (1990) 149–155. [19] G.L. Mullins, L.E. Sommers, S.A. Barber, Modeling the plant uptake of cadmium and zinc from soils treated with sewage sludge, Soil Sci. Soc. Am. J. 50 (1986) 1245–1250. [20] S.E. Jorgensen, Modeling the contamination of agricultural products by lead and cadmium, in: A. Marani (Ed.), Advances in Environmental Modeling, 1988, pp. 343–350. [21] F.J. Molz, I. Remson, A.A. Fungaroli, R.L. Drake, Soil moisture availability for transpiration, Water Resour. Res. 4 (1968) 1161– 1169. [22] D. Hillel, V. Beek, H. Talpaz, A microscopic scale model of soil water uptake and salt movement to plant roots, Soil. Sci. 120 (1975) 385–399. [23] H.R. Rowse, W.K. Mason, H.M. Taylor, A microcomputer simulation model of soil water extraction by soyabeans, Soil. Sci. 130 (1983) 218–224. [24] F.J. Molz, I. Remson, Extraction term models of soil moisture use by transpiring plants, Water Resour. Res. 6 (1970) 1346–1356. [25] S.R. Singh, A. Kumar, Analysis of soil water dynamics, Agric. Engg. ISAE. 22 (2) (1985) 50–74. [26] M.A. Marino, J.C. Tracy, Flow of water through root-soil environment, Irrig. Drain. Eng. 114 (1988) 558–604. [27] D. Hillel, H. Talpaz, Simulation of root growth and its effect on the pattern of soil water uptake by a non uniform root system, Soil Sci. 121 (1976) 307–312. [28] H. Borg, D.W. Grimes, Depth developments of roots with time, an empirical description, Trans. ASAE 29 (1) (1986) 194–197. [29] W.J. Grenney, C.L. Caupp, R.C. Sims, E.E. Short, A mathematical model for the fate of hazardous substances in soil: Model description and experimental results, Haz. Waste Mater. 4 (1987) 223–237. [30] N. Rajmohan, L. Elango, Modelling the movement of chloride and nitrogen in the unsaturated zone, in: L. Elango, R. Jayakumar (Eds.), Modelling in Hydrogeology, Allied Publishers Limited, New Delhi, 2001, pp. 209–225. [31] S.J. Van der Hoven, D.K. Solomon, G.R. Moline, Modeling unsaturated flow and transport in the saprolite of fractured sedimentary rock: effects of periodic wetting and drying, Water Resour. Res. 39 (7) (2003) SBH6 1–SBH6 11. [32] S. Rao, S. Mathur, Modeling heavy metal (cadmium) uptake by soil plant root system, J. Irrig. Drain. Eng. 120 (1994) 89–96. [33] S. Mathur, Settlement of soil due to water uptake by plant roots, Int. J. Numer. Methods Geomech. 23 (1999) 1349–1357. [34] M.TH.V. Genucthen, A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. Am. J. 44 (1980) 892–898. [35] M.TH.V. Genucthen, D.R. Nielsen, On describing and predicting the hydraulic properties of unsaturated soils, Ann. Geophys. 3 (1985) 615–628. [36] M.N. Nimha, R.J. Hanks, Model for estimating soil water plant and atmospheric interrelations, 1. Description and sensitivity, Soil Sci. Soc. Am. Proc. 37 (1973) 522–527. [37] P.A.C. Raats, Distribution of salts in the root zone, J. Hydrol. 99 (1973) 297–306. [38] J.C. Hoogland, R.A. Fedes, C. Belmans, Root water uptake model depending on soil water pressure head and maximum extraction rate, Acta Hort. 119 (1981) 123–131. [39] R. Prasad, A linear water uptake model, J. Hydrol. 99 (1988) 297–306. [40] F.J. Molz, Models of water transport in the soil-plant system: a review, Water Resour. Res. 17 (1981) 1245–1260. [41] T.J. Christensen, Cadmium soil sorption at low concentration. VI: A model for zinc competition, Water Air Soil Pollut. 34 (1987) 305–314. [42] T.J. Christensen, Cadmium soil sorption at low concentration. III: Prediction and observation of mobility, Water Air Soil Pollut. 26 (1985) 255–264. [43] CAST (Council of Agricultural Science and Technology), Effect of sewage sludge on the cadmium and zinc content of crops, Council for agricultural science and technology, No. 83, Ames, Iowa, 1980, pp. 77. [44] R.L. Chaney, M.C. White, P.W. Simon, Plant uptake of heavy metals from sewage sludge applied to land, in: Proceedings of Second National Conference on Municipal Sludge Management, 1975, pp. 169–174. [45] P. Verma, K.V. George, H.V. Singh, T.P. Mathew, R.N. Singh, Simulating water movement and its uptake by plants roots in unsaturated zone, Int. J. Environ. Stud. 61 (1) (2004) 39–48.