numerical study of soil heterogeneity effects on contaminant transport in unsaturated soil (model...

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech. (2011)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nag.1100

Numerical study of soil heterogeneity effects on contaminanttransport in unsaturated soil (model development and validation)

M. Mousavi Nezhad1,2, A. A. Javadi1,*,†, A. Al-Tabbaa2 and F. Abbasi3

1Department of Engineering, University of Exeter, Exeter, Devon EX4 4QF, UK2Engineering Department, Cambridge University, Cambridge, CB2 1 PZ, UK

3Agricultural Research Institute AERI, PO Box 31585-845, Karaj, Iran

SUMMARY

The movement of chemicals through soil to groundwater is a major cause of degradation of water resources.In many cases, serious human and stock health implications are associated with this form of pollution. Thestudy of the effects of different factors involved in transport phenomena can provide valuable information tofind the best remediation approaches. Numerical models are increasingly being used for predicting oranalyzing solute transport processes in soils and groundwater. This article presents the development of astochastic finite element model for the simulation of contaminant transport through soils with the main focusbeing on the incorporation of the effects of soil heterogeneity in the model. The governing equations ofcontaminant transport are presented. The mathematical framework and the numerical implementation ofthe model are described. The comparison of the results obtained from the developed stochastic model withthose obtained from a deterministic method and some experimental results shows that the stochastic model iscapable of predicting the transport of solutes in unsaturated soil with higher accuracy than deterministic one.The importance of the consideration of the effects of soil heterogeneity on contaminant fate is highlightedthrough a sensitivity analysis regarding the variance of saturated hydraulic conductivity as an index of soilheterogeneity. Copyright © 2011 John Wiley & Sons, Ltd.

Received 14 February 2011; Revised 20 June 2011; Accepted 26 August 2011

KEY WORDS: stochastic finite element; contaminant transport; soil heterogeneity; unsaturated soil

1. INTRODUCTION

In recent years, interest in understanding the mechanisms and prediction of contaminant transportthrough soils has dramatically increased. This is due to growing evidence and public concern that thequality of the subsurface environment is being adversely affected by industrial, municipal, andagricultural activities. Contamination of groundwater is an issue of major concern in residential areas,which may occur because of spillages of hazardous chemicals, dumping of toxic waste, landfills,waste water, or industrial discharges [1]. Hazardous waste disposal is increasingly becoming one ofthe most serious problems confronting health and the environment. The movement of chemicalsthrough the soil to the groundwater represents a degradation of these water resources. Hence, themanagement of contaminated lands for control of groundwater quality will be a crucial requirementfor sustainability. The effective management of contaminated land and the selection of appropriateand efficient remedial technologies are strongly dependent on the accuracy of predictive models forthe simulation of flow and solute transport in the soil. The simulation of hydraulic phenomena in soilis a complex problem because of the complex and heterogeneous nature of soils and the participationof a large number of factors involved. Therefore, many efforts have been made to understand and

*Correspondence to: A. A. Javadi, Department of Engineering, University of Exeter, Exeter, Devon EX4 4QF, UK.†E-mail: [email protected]

Copyright © 2011 John Wiley & Sons, Ltd.

M. MOUSAVI NEZHAD ET AL.

simplify these procedures for modeling purposes. These simplified models describe small-scaleconceptualizations of various physical, chemical [2–8], and biological mechanisms [9–11] that affectunsaturated flow and contaminant transport in soils. Recent studies have shown that current modelsand methods do not adequately describe the leaching of nutrients through soil, often underestimatingthe risk of groundwater contamination by surface-applied chemicals and overestimating theconcentration of resident solutes [12, 13]. One of the most challenging problems in modeling ofsolute transport in soils is how to effectively characterize and quantify the uncertainties and thepotential fluctuations in hydraulic parameters of the soil (resulting from natural heterogeneity of soil)to achieve a reliable prediction. Despite the evidence from field-scale observations and experimentalstudy about the significant effects of soil heterogeneity on contaminant transport [14–19], theseeffects have been not included in most the existing numerical models.

Recently, stochastic methods have been developed and probabilistic frameworks have been used forconsidering soil heterogeneity in the simulation of flow and contaminant transport [20]. Monte Carlomethod (MCM) is a powerful technique for considering uncertainties in a system. This method hasbeen widely used for the prediction of flow and contaminant transport in soil [21–23]. In thismethod, different realizations of the participating parameters in the domain under consideration aregenerated, and the analysis is repeated for each set of parameters. Therefore, depending on thenumber of the contributing parameters in the problem and the size of the domain underconsideration, the MCM can be computationally intensive and time consuming. Analytical-basedstochastic methods [24–26] are another type of modeling techniques that can be used to deal withthe probabilistic nature of hydrological processes in the soil. Analytical methods can provideconceptual understanding and insight into the effects of different factors involved in a process.However, analytical-based probabilistic methods cannot be readily used for solving complexproblems, which reduces their applicability in practical conditions. Despite the development of fewanalytical or numerical models (usually integrated with the MCM) for the incorporation of soilheterogeneity in contaminant transport, they are neither computationally efficient nor practical forcomplex problems. Hence, the focus of this article is the development of a computationally efficientstochastic numerical model that is able to simulate and predict the in situ problems with complexboundary conditions.

In what follows, the main governing phenomena of contaminant transport including advection,mechanical dispersion, and molecular diffusion are presented. The procedure of the extraction of thestochastic partial differential equations of water flow and solute transport from perturbation-spectralmethod and computation of their numerical solution are explained. The stochastic contaminanttransport equation is solved numerically using the finite element method, subject to prescribed initialand boundary conditions. The model is validated through the comparison of its results withexperimental results. The effects of soil heterogeneity on the mechanisms of advection and thedispersion of contaminant are studied through the simulation of a laboratory and field-scaleexperiments. A sensitivity analysis is conducted to study how soil heterogeneity affects thedimensions and concentration of contaminant plume and to investigate the relationship betweenspatial variability of soil saturated hydraulic conductivity and contaminant transport phenomena.

2. CONCEPTUAL MODEL

Contaminant transport in soil occurs through the advection, dispersion, and diffusion mechanisms.Advection is the transport of material caused by the net flow of the fluid in which that material issuspended. Whenever a fluid is in motion, all contaminants in the flowing fluid, including bothmolecules and particles, are advected along with the fluid [27]. The process by which contaminantsare transported by the random thermal motion of contaminant molecules is called diffusion [28].Mechanical dispersion is a mixing or spreading process caused by small-scale fluctuation ingroundwater velocity along the tortuous flow paths within individual pores [13]. The transportmechanisms are therefore dependent on the hydraulic properties of soil, and the mathematical partialdifferential governing equations of these phenomena are function of hydraulic parameters of soil.The hydraulic properties of soil change randomly in the space and cannot be uniquely defined at

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2011)DOI: 10.1002/nag

SOIL HETEROGENEITY EFFECTS ON CONTAMINANT TRANSPORT

entire large-scale aquifer. It is impractical, however, to measure these properties at all points or even ata relatively large number of points. However, from a finite number of observations, these propertiesmay be modeled as random variables or, with a higher level of sophistication, as random processeswith the actual medium properties viewed as a particular realization of these processes. Hence, thesimulation of water flow and contaminant transport in soil necessitates the solution of two partialdifferential equations with random coefficients.

Stochastic finite element (SFE) method is an approach that can be used to find the solution of partialdifferential equations with random coefficients. SFE method is a generalization of the deterministicfinite element (DFE) method to incorporate the random fluctuations in the material and geometricproperties of the model. The main steps of SFE method include (i) the selection of appropriateprobabilistic models for the random fields used to simulate the uncertainties in the system and in theboundary conditions, (ii) the discretization of these random fields, that is, the replacement of therandom fields with an equivalent set of random variables, (iii) the formulation of the FE system ofequations using standard methods and their solution, and (iv) the estimation of probabilisticcharacteristics of the system response. In this work, an analytical spectral-based stochastic method isused to incorporate random parameters in advective-dispersive solute transport governing equationand to develop a probabilistic model for these phenomena in heterogeneous soil. Velocity iscalculated by Darcy’s equation. Eulerian advection and dispersion equations used in this model arefunctions of the hydraulic properties of the soil. Hydraulic conductivity and volumetric watercontent vary randomly through the domain because of natural heterogeneity. It is assumed that theselocal properties are realizations of three-dimensional, spatially correlated random fields. Theserandom parameters are defined by combination of a mean value and possible perturbations aroundthe mean. A log-normal distribution is assumed for saturated hydraulic conductivity. The localdeterministic governing equation of solute transport is averaged over the ensemble of possible soilproperty realizations, and an appropriate large-scale probabilistic model for solute transport isdeveloped. An exponential covariance function is considered for random variables, and crosscorrelations between local soil properties and output fluctuations are evaluated by spectral approach.Then, the expressions for the effective solute transport parameters and macrodispersion aredeveloped. Also, the variance of solute distribution is evaluated by the spectral method, which is anindex of reliability of the model.

The extracted analytical stochastic partial differential equations are solved by the finite elementmethod. In this work, an SFE model is developed to solve the governing equations. The stochasticpartial differential equations of transport are solved using a Galerkin finite element technique and afinite difference operator for discretization in time. Figure 1 shows the general structure of thedeveloped SFE model.

3. CLASSICAL MATHEMATICAL MODEL

The governing equation for advective-dispersive solute transport in soil can be written as [13]

@ nSð Þ@t

¼ @

@xinDij

@S

@xj

� �� @ qiSð Þ

@xiþ Fs (1)

where n is the effective porosity of soil, S is solute concentrations [M] [L]�3, qi is specific groundwaterdischarge [L] [T]�1that is evaluated using the Darcy equation, Dij is the coefficient of dispersivitytensor [L]2[T]�1, and Fs is the sink/source of contaminant [M] [L]�3[T]�1. Subscripts i and jrepresent the directions parallel and perpendicular to the mean flow direction, respectively. Dij isevaluated as [29]

Dij ¼ aij � qi þ Dm if i ¼ jDij ¼ Dm if i 6¼ j

(2)

where Dm is the coefficient of molecular diffusion [L]2[T]�1 and aijis solute dispersivity [L][T]�1.


Start

Input data (geometry, initial and boundary conditions)

Increment time

Implement finite diference scheme

Iteration loop

Tim

e increment

loop

NO

YES

Implement boundary condition

Convergence

C

K

Configure element

andmatrices

Assemble global

, , andC Kmatrices

Evaluate the expected value of

random parameters

Evaluate and

Output results ( , and )

STOP

YES

NO

Time <Time

2c

2c

Evaluate velocity

Configure element

andmatrices

M

M U

2h

2h

H

U

C

Figure 1. General structure of developed SFE model.




4. STOCHASTIC MODEL

Assuming that the natural logarithm of saturated hydraulic conductivity lnKs, specific moisturecapacitance C, scaling parameter a, local specific dischargeqi, concentration S, capillary tension headc, and soil moisture content θ are realizations of stationary random fields yields

lnKs ¼ F þ f (3)

C ¼ Γ þ g (4)

a ¼ Aþ a (5)

qi ¼ �qi þ q′i i ¼ 1;2;3 (6)

S ¼ �Sþ S′ (7)

θ ¼ �θþ θ′ (8)

c ¼ H þ h (9)

where F; Γ; A; �qi; �S; �θ, and H are the mean values and f ; g; a; q′i; S′; θ′, and hare the fluctuations.

The mean values are deterministic, smooth spatial functions, and the fluctuations are realizations ofthree-dimensional zero mean second-order stationary random fields.

To obtain the model for transient mean solute transport in soils, we averaged Equation (1) over theensemble of realizations of the random fields a and f. Taking the expected value of Equation (1) yields [24]

@E nS½ �@t

¼ @

@xiEij

@E S½ �@xj

� �� @E Sqið Þ½ �

@xi

¼ @

@xiEij

@�S

@xj

� �� @E Sqið Þ½ �

@xi

(10)

where Eij is equal to nDij.Substituting Equations (3)-(9) into Equation (10), the expected value of the last term on the right-

hand side of Equation (10) can be written as

E Sqið Þ½ � ¼ �S�qi þ E S′q′i� �

(11)

Using Fick’s law yields [24]

E S′q′i� � ¼ �E ij

@�S

@xj(12)

where E ij is the effective bulk macrodispersion coefficient tensor.A macrodispersivity tensor is defined as

Aij ¼ E ij

qj j (13)

where |q| represents the magnitude of mean specific discharge, and the direction of q is given by [24]



f ¼ arctgK i′i′Ji′

K j′j′Jj′

!(14)

where i′ and j′ represent the horizontal and vertical directions in Cartesian coordinate system and Jrepresent the mean hydraulic gradient.

Substituting Equation (13) into Equation (12) yields

E S′q′i� � ¼ �Aij qj j @

�S

@xj(15)

Substituting Equations (11) and (13) into Equation (10) and assuming that the fluctuations θ′ and S′are small (neglecting the second order terms) yields

@�θ�S@t

¼ @

@xiEij þ Aij qj j� @�S

@xj

� �� @ �S�qið Þ

@xi(16)

Equation (16) is the mean solute transport equation. The term Aij|q|, the bulk macrodispersioncoefficient, takes into account the dispersion due to the spatial variability of qi. The macrodispersioncoefficient used in this work is evaluated as [30]

Aii ¼ s2f liljT22 þ 2x2T23 þ x4T33�

pg2b(17)

Aij ¼s2f liljJ

2j x4T33�

pg2bJ2i(18)

Ajj ¼s2f liljJjx

2 T23 þ x2T33� pg2bJi

(19)

where lis correlation scale of random parameters involved in the governing equations of the transportprocess [L] and

g2 ¼ qj j2K2mJ

2i b

2 (20)

b ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2i sinfð Þ2 þ l2j cosfð Þ2

q(21)

x2 ¼ l2i sinfð Þ2l2j þ cosfð Þ2 (22)

where Km= exp{F�AH�E[ah]}.If a and f are uncorrelated, then

b2 ¼ 1þ z2H (23)

If a and f are perfectly correlated, then

b2 ¼ 1� zH (24)



where z2 is given by

z2 ¼ s2a=s2f (25)

and

Ji ¼ Ji′ cosfþ Jj′ sinf (26)

Jj ¼ �Ji′ sinfþ Jj′ cosf (27)

The integrals T22, T23, and T33 are evaluated numerically, and they are given by the followingexpressions [30]

T22 ¼ 4Z0

p=2

1=8 a2 � c2�

p� 8c2�

ln c2=a2�

=a2 � c2� �

cosfð Þ4df (28)

T23 ¼ 4Z0

p=2 Z10

r5 1� 2 sinfð Þ2 cosfð Þ2h i

= a2r2 þ c2�

1þ r2� 2h i

drdf (29)

T33 ¼ 4Z0

p=2

1=8 a2 � c2�

p� 8c2= a2 � c2� �

ln c2=a2� � �

sinfð Þ4df (30)

where

a2 ¼ cosfð Þ2 þ x2 sinfð Þ2 (31)

and

c2 ¼ A2L2j b2 cosfð Þ2 (32)

Lj ¼ Jj þ @H

@xj(33)

For the simulation of transient problems, the mean solute transport equation is coupled with themean flow equation. The mean flow equation can be obtained by averaging the classical water flowgoverning equation over the ensemble of possible realizations of the stochastic processes f, a, and gas [31]

� @Y@t

¼ C@H

@t¼ @

@xi′K i′j′

@ H þ zð Þ@xi′

� �þ Q (34)

where Y is the mean soil moisture content [L]3[L]�3 and C is the effective specific moisture capacity[L]�1, which is defined as [32]



C ¼ � @

@HE θ Hð Þ½ � � E gh½ �ð Þ (35)

where E[] indicates the expected value (mean) operator of a random variable, which is defined as theintegral of the random variable with respect to its probability measure. K i′j′ is the effective hydraulicconductivity tensor, which is defined as [33]

K i′j′ ¼ Km exps2e2þ ti′Ji′

� �no sum on i□

K i′j′ ¼ 0 i′ 6¼ j′(36)

where

Km ¼ exp F � AH � E ah½ �f g (37)

Ji′ ¼@ H þ zð Þ

@xi′(38)

s2e ¼ s2f þ A2E h2� �þ H2s2a � 2AE fh½ � � 2HE fa½ � þ 2AHE ah½ � (39)

ti′ ¼ E f@h

@xi′

� �� HE a

@h

@xi′

� �(40)

wheres2f is the variance of natural logarithm of saturated hydraulic conductivity and lnKs ands2a are thevariance of a.

The prediction of a concentration distribution consists of the ensemble mean value and aquantification of the deviation around the mean (sc). An analytical expression for the evaluation ofconcentration variation proposed by Vomvoris and Gelhar [34] was used in this work, which is given as

sc2 ¼ Tijs2f l2j GjGi (41)

where for the general case with anisotropic ln K, Tij is given by [34]

Tii ¼ 2

3g2Kerirj� � (42)

Tjj ¼ 16g2

12r2ið Þ þ

1R2

� �þ 1R

0:5þ 1R2

� �ln

Rþ 2R2ð ÞR� 2R2ð Þ

� �� (43)

where

ri ¼ljli

(44)

rj ¼ljlj

(45)

R ¼ 1� ri2 (46)



5. NUMERICAL SOLUTION

Mean capillary tension head is the primary unknown variable in Equation (34). An approximatesolution of this variable over each element is defined in terms of its nodal values and associatednodal shape function as

H ¼XMm¼1

NmHm (47)

where H is the approximated value of capillary tension head, Nm is the shape function at node m, Hm isthe mean capillary tension head value at node m, and M is the number of nodes in the domain.

Replacing the primary unknown variable with its shape function approximation, Equation (34) iswritten as

C@H

@t� @

@xi′K i′j′

@ H þ z� @xi′

" #� Q ¼ RΩ (48)

Using the Galerkin weighted residual approach to minimize the residual error represented by thisapproximation, the discretized global finite element equation for water flow takes the form [31]

p½ � H:

n oþ K� �

Hf g � Tf g ¼ 0f g (49)

where

p½ � ¼XM1

ZΩ

NCN�

dΩ (50)

K� � ¼XM

1

ZΩ

rNKrN�

dΩ (51)

Tf g ¼XM1

ZΩ

N@K z

@zÞdΩ

�(52)

where {T}is the force vector including gravitational force vector and groundwater sink/source vectorover the problem domain.

Numerical evaluation of the transient mean flow equation is completed using time discretization ofEquation (49) by application of a fully implicit mid-interval backward difference algorithm. Applyingthe finite difference scheme [35] to Equation (49) will result in the final form of equation system forevaluation of pressure heads in the problem domain as [31]

p½ � þ ’Δt K� ��

Hf gtþΔt ¼p½ � � 1� ’ð ÞΔt K� ��

Hf gt þ Δt 1� ’ð Þ Tf gt þ ’ Tf gtþΔt

� (53)

where ’ takes on the value 1 for the backward finite difference scheme.The governing differential equations for contaminant transport as defined in the previous section

have one variable �S. This variable is primary unknown. The primary unknown can be approximatedusing the shape function approach as



�S ¼XMm¼1

Nm�Sm (54)

where Nm is the shape function and �Sm is the nodal mean solute concentration. Replacing the primaryunknown with shape function approximation of the previous equation, Equation (16) can be written as

@�θ�S@t

� @

@xi½ Eij þ Aijq� @�S

@xj� þ @ �S�qi

� �@xi

¼ RΩ (55)

Using the Galerkin weighted residual approach to minimize the residual error represented by thisapproximation, the discretized global finite element equation for solute takes the form

Wd�Sdt

þ U�Sþ Fs ¼ 0 (56)

where

W ¼XM1

ZΩ

θ�SΔt

� �NNdΩ (57)

U ¼XM1

ZΩ

�q�SNrN þ E þ Aqð Þ�S rNrNð Þ½ �dΩ (58)

F ¼XM1

ZΩ

N2 �q�S� E þ qAð Þr�Sð ÞdΓΩ (59)

where ΓΩ represents the boundary of domain.Applying a finite difference scheme [35] to Equation (56) will result in

W θ�Sð ÞtþΔt � θ�Sð ÞtΔt

þ U 1� ’ð Þ�St þ ’�StþΔt

h iþ F

tþΔt

s ¼ 0 (60)

where Δt is the time step, ’ is a parameter equal to 1 for the backward finite difference scheme, and tand t+Δt stand for time levels. Equation (60) will give the distribution of the contaminantconcentrations at various points within the soil and at different times, taking into account the effectsof soil heterogeneity on contaminant transport.

6. MODEL VERIFICATION AND NUMERICAL STUDY

The developed SFE model has been verified against the results obtained from the MCM for steady-state flow and transient contaminant transport, which were documented elsewhere [36]. Here, it isused for the simulation of a laboratory-scale solute transport experiment [14] to numerically studythe effect of soil heterogeneity on dispersion mechanism. Also, the model is verified for the case oftransient flow and contaminant transport by the simulation of a field-scale solute transport problem[37, 38].



6.1. Solute transport in stratified soil

The stratified form of soil configuration is the most commonly used simplified form for modeling ofheterogeneity of soil. Soil stratification parallel and inclined to the main flow direction is applicableto natural ground conditions with horizontal groundwater flow. Hence, the study of solute transportin stratified soil even in the laboratory-scale can be useful in the understanding of transportmechanisms that usually occur in real field conditions. In this example, a laboratory-scaleexperiment performed by Al-Tabbaa et al. [14] was simulated using the SFE model. The experimentwas also simulated deterministically using the same model by setting the spatial variability ofhydraulic parameters to zero. The test was carried out in a rectangular laboratory-scale Perspex tank0.45m long, 0.38m wide, and 0.25m high, along which the main flow was imposed. One inlet andthree outlet flow ports were used as shown schematically in Figure 2. These ports were screenedwith fine wire mesh to eliminate possible clogging by the soil and were adjusted such that the flowin and out of the tank could be fixed at predetermined rates [14]. A peristaltic flow pump, located atthe inflow position, was used to provide the inflow water and contaminant solution into the soiltank. One-dimensional flow condition was imposed. A sodium chloride solution containing 8 g/L(0.137M) concentration was used as the nonreactive contaminant. This is a typical landfill leachateconcentration of Cl�1 and typical collective concentration of Na+ and other similar ions such as K+.The concentration of sodium chloride was measured at the outflow position using a conductivitymeter. All experiments were conducted at an ambient temperature of approximately 24�C [14]. Twogradings of sand were used: fine to medium (FMS) and coarse (CS), as shown in the particle sizedistribution curves in Figure 3. The sands were placed at a constant porosity of 40% in theexperiments. Two different sand stratification configurations parallel to mean flow direction weretested here, named as PF1 and PF2. In PF1 configuration, a CS layer was located between two FMSlayers, and in PF2, two layers of a CS soil were located at the top and bottom of a layer of FMSsoil. The bulk longitudinal and transverse dispersion coefficients of CS layer are equal to 6.6�10�6

and 6.0�10�7 m2/s, respectively, and they are equal to 4.2�10�6 m2/s and 4.0�10�7 m2/s for theFMS layer. The saturated hydraulic conductivity of each layer was measured experimentally, and onthe basis of the statistical assessment of Ks, the variance of the sample s2f was considered equal to 7with correlation scale l equal to 0.08m.

Uncontaminated water was initially permeated through the tank to ensure steady-state conditionswere achieved before the saline solution was introduced and to establish the required outflowconditions at each port to maintain constant velocity. The discharge velocity used in all theexperiments was approximately 4.5 � 10�5 m/s. The dimensions of the problem domain and thefinite element mesh generated for simulation are presented in Figure 4.

Figures 5 and 6 show concentration isochrones for the tests PF1 and PF2, respectively. The resultsobtained using the SFE model are in better agreement with the experimental results than those obtainedusing the deterministic method. Because of the effect of heterogeneity in the vertical direction,transverse dispersion increases; hence, more concentration is transferred vertically from the highconcentration layer, CS, to the low concentration layers, FMS. Hence, concentration in the FMSlayers increases. As the mean flow is in the direction parallel to the layering and the soil propertiesin horizontal direction are comparatively uniform, longitudinal dispersion increases very slightly. Inthe CS, the solute concentration obtained from the SFE method is less than that obtained by DFE. Itis shown that the effect of soil heterogeneity on the transverse dispersion of solute is dominant to itseffects on the longitudinal dispersion in these configurations that mean flow is parallel to layering.

inflow outflow

0.45 m 0.38 m

0.25

m

Figure 2. Schematic diagram of the tank used in the experiment [14].


Figure 3. Problem definition [14].

0.45 m

0.25

m

Pollution source

Figure 4. Problem definition.


6.2. Field-scale transient flow and solute transport

The developed SFE model was also applied to a case study to show reliability and applicability of themodel to field-scale problems. The case study involves a field-scale experiment, conducted by Abbasiet al. [37,38] at the Maricopa Agricultural Center, Phoenix, AZ, to investigate the distribution of soilmoisture and solute concentration in the soil profile below and adjacent to the agricultural irrigationfurrows. The soil in the field site was bare sandy loam. The experiment was carried out on 115-m-long furrows under free-draining condition, spaced 1m apart (Figure 7). The experiment includedthree furrows, one monitored nonwheel furrow in the middle and two guard wheel furrows, one ateach side. The experiment was run with two irrigation events 10 days apart; the first irrigation lasted275min and the second irrigation 140min. Two sets of neutron probe access tubes were installed at5 and 110m along the monitored furrow. Hereafter, we refer to these locations as the inlet andoutlet sites, respectively. Each set included five 2.2-m-long neutron probe access tubes, installed intwo rows at different locations in a cross section perpendicular to the furrow, spaced 0.5m apart toavoid mutual interference of the readings (Figure 8).


0

0.2

0.4

0.6

0.8

1

1.2

Top (FMS) Middle (CS) Bottom (FMS)

Position in tank (sand type)Rel

ativ

e m

ean

so

lute

co

nce

ntr

atio

n (

C/C

o)

45mins

75mins

105mins

30mins

Figure 5. Concentration isochrones for the parallel sand stratification configurations in test PF1 (blue line,experimental result; dashed red line, stochastic result; solid red line, deterministic result).

0

0.2

0.4

0.6

0.8

1

1.2

Top (CS) Middle (FMS) Bottom (CS)

Position in tank (sand type)Rel

ativ

e m

ean

so

lute

co

nce

ntr

atio

n (

C/C

o)

45mins

75mins

105mins

30mins

Figure 6. Concentration isochrones for the parallel sand stratification configurations in test PF2 (blue line,experimental result; dashed red line, stochastic result; solid red line, deterministic result).

5m

Inflow E

nd

105 m5m

115 m

Soil and water sampling station

Neutron probe access tube site

Figure 7. Plan view of the furrow irrigation field experiments, not to scale [39].


To determine gravimetric water content, we collected soil samples 12 h and 5 days after theirrigation at three different locations: at 5 and 110m (the inlet and outlet sites, respectively). Thesamples were collected from one side of the monitored furrows at three locations (top, side, andbottom of the furrows; e.g. at locations 1, 2, and 3 in Figure 8) in a cross section perpendicular tothe furrow axis at similar depths as used for the neutron probe measurements. Water flow depths inthe furrows were measured at the inlet and outlet sites every few minutes as soon as water reached a


50 cm 25 cm

25 cm25 cm 25 cm

1

2 3

45

Figure 8. Position of neuron probe access tubes at different locations in the furrow cross section. Numbersrelate to access tubes installed in two different rows; the first row includes tubes 2 and 4 along the sides and

the second row includes tubes 1, 3, and 5 [39].


particular station. Geometries of the furrows were measured with a profilometer before and after theirrigation at the inlet and outlet sites. These measured geometries served as the upper boundaries forthe numerical calculations. The values of the saturated θsand residual θrsoil water contents wereconsidered as 0.411 and 0.106, respectively. Measured soil water contents before the experimentswere used as initial conditions within the flow domain, and initial Bromide concentration wasassumed to be zero through the entire domain. Time- and space-dependent flow depths (surfaceponding, h(x,t) in Figure 9) were specified as the upper boundary condition in the furrow duringirrigation, whereas averages of measured pan evaporation rates from the nearest weather station(approximately 150m away from the experimental field), and estimated reference evapotranspirationrates obtained using the Penman–Monteith method (as reported by Abbasi et al. [37]) were used asatmospheric boundary conditions. As indicated earlier, in each experiment, flow depths at the inletand outlet sites were frequently measured. No-flux boundary conditions were applied to both sidesof the flow domain. Bromide in the form of CaBr2 was injected at a constant rate of 6.3 g Br/Lduring the entire irrigation. A Cauchy (solute flux) condition was used for the upper boundarycondition for solute transport, whereas free-drainage conditions for both water and solute wereapplied to the lower boundary of the domain (Figure 9). The parameters used in the SFE model andHYDROUS-2D are summarized at Table I.

Measured and predicted (using the SFE model developed in this study and HYDROUS2-D model[38]) bromide concentrations at the inlet and outlet sites of the experiment are presented inFigure 10. The results are given using one-dimensional curves to provide a better visual comparisonbetween the measured and calculated distributions. The results are given at two different times(5 days after the start of the first and the second irrigation) and for three different locations in the

0

-1000 100

Furrow Width (cm)

No flux boundaries

Free-draining boundary

h(x,t)

Atmospheric boundary condition

So

il P

rofi

le D

epth

(cm

) h(x,t) is flow depth

Figure 9. Boundary conditions used for numerical modeling [37].


Table I. Parameter values used for the numerical simulations.

Site Ks(m/s) s2f aL(m) aT(m) lz(m) lx(m) A(1/m) s2a(1/m2)

Simultaneouslyoptimization

Inlet 1.39� 10-3 2.22� 10-1 4.4� 10-2 3.9Outlet 1.59� 10-3 9.1� 10-2 1.0� 10-4 3.9

Two-step optimization Inlet 7.6� 10-4 2.005� 10-3 4.34� 10-2 3.9Outlet 1.78� 10-3 1.74� 10-2 4.0� 10-4 3.9

Stochastic finite element Inlet 1.5� 10-3 0.6 2.0� 10-1 2.0� 10-2 0.2 1 3.9 1.3Outlet 1.5� 10-3 0.6 2.0� 10-1 2.0� 10-2 0.2 1 3.9 1.3

c) Outlet site: 5 days after the first irrigation

-80

-55

-30

-5

0 2 4 6 8

Bromide concentration (g l-3)

So

il P

rofi

le D

epth

(cm

)

-80

-55

-30

-5

0 2 4 6 8


So

il P

rofi

le D

epth

(cm

)

-80

-55

-30

-5

0 2 4 6 8


So

il P

rofi

le D

epth

(cm

)

-80

-55

-30

-5

0 3 6 9


So

il P

rofi

le D

epth

(c

m)

Bottom of Furrow Side of Furrow Top of Furrow

a) Inlet site: 5 days after the first irrigation

-80

-55

-30

-5

0 3 6 9Bromide concentration (g l-3)

So

il P

rofi

le D

epth

(c

m)

-80

-55

-30

-5

0 3 6 9


So

il P

rofi

le D

epth

(c

m)

-80

-55

-30

-5

0 2 4 6


So

il P

rofi

le D

epth

(c

m)

-80

-55

-30

-5

0 2 4 6Bromide concentration (g l-3)

So

il P

rofi

le D

epth

(c

m)

-80

-55

-30

-5

0 2 4 6


So

il P

rofi

le D

epth

(c

m)

b) Inlet site: 5 days after the second irrigation

-80

-55

-30

-5

0 1 2 3


So

il P

rofi

le D

epth

(cm

)

-80

-55

-30

-5

0 1 2 3


So

il P

rofi

le D

epth

(cm

)

-80

-55

-30

-5

0 1 2 3


So

il P

rofi

le D

epth

(cm

)

d) Outlet site: 5 days after the second irrigation

Figure 10. Measured and predicted (using SFE and HYDRUS2-D models) bromide concentration for the in-let and outlet sites (measured, blue points; simultaneous, solid black lines; two-step optimization, dashed

lines; SFE, solid red lines).




furrow cross section (bottom, side, and top of the furrow) up to a depth of 100 cm below the groundsurface. The results are plotted versus depth (instead of versus lateral distance) because considerablymore data were available along the depth. The black solid and dashed lines show simulation resultsobtained by HYDRUS-2D in combination with simultaneous and two-step optimization approaches,respectively [3]. The solid red lines show results obtained using the SFE model. From thecomparison of the results, it is concluded that SFE method produced better agreement with theobserved concentrations, than HYDROUS-2D. HYDROUS-2D is a deterministic numerical code,and effects of spatial variability of soil hydraulic parameters are not considered in this model.Saturated hydraulic conductivity used in this model was obtained by inverse estimation using twodifferent simultaneous and two-step optimization approaches, and different values were found foreach inlet and outlet section of the problem that shows spatial variability of the hydraulic parametersof domain.

The comparison of the two solute concentration profiles is difficult because it is impossible to detectif the differences in the profiles are due to the different flow fields and or due to the differentapproaches used to solve the contaminant transport equation. To study the effect of the inclusion ofmacrodispersion as a transport mechanism, we solved the transient unsaturated flow equation usinga stochastic approach and the transport equation twice, once using a deterministic approach for thetransport part and the second time using a stochastic approach. Figure 11 shows the mean soluteconcentration through the domain after 2 h, 2 days, and 5 days after the start of the first irrigation.

The results obtained using the SFE model show more uniform distribution of Br concentration at thecontaminated plume than DFE. The deterministic model overestimates the solute concentrationspecially around of the solute source. This means that the effect of including the variability of thesoil properties in the transport equation through the macrodispersivity tensor is to increase the lateraland longitudinal spreading of the contaminant. Therefore, greater amount of solute disperses throughthe domain and it is distributed in a larger area. This can be seen in Figures 11(a) and 11(b) that, forexample, the area with concentration between 2 and 3mg/L is larger for the stochastic case incomparison with the deterministic one. The source of solute dispersion is the fluctuations of water

(a) Deterministic

Furrow width (m)

2 hours after irrigation 2days after irrigation 5days after irrigation

Fu

rro

w d

epth

(m)

(b) Stochastic

Figure 11. Br concentration distributions at t = 2 h, 2 days, and 5 days after the start of first irrigation for thecases of (a) DFE and (b) SFE.



velocity. Hence, in heterogeneous soils, a higher amount of fluctuation is expected for water velocityand consequently, there is more potential for solute to disperse through the domain.

Figure 12 shows the variance of Bromide concentration through the domain for 2 h, 2 days, and5 days after the start of the first irrigation. The variance is higher along the border of thecontaminant plume and the area with high concentration gradient but, the coefficient of variation isstill low. The amount of the variance decreases with time as solute concentration gradient decreases.The direct relationship between the variance of concentration with concentration gradient is shownin Equation (41). In areas where the solute concentration is faced with a sudden change in thedomain, more uncertainty is expected in transport mechanisms. After 2 days when the concentrationdistribution is relatively uniform, the variance is lower than those after 2 h and 5 days. After 5 days,which solute is going out of the domain from the free drainage boundary the variance is high aroundthe bottom and left boundaries of the contaminated domain where the concentration is considerablydecreasing and approaching zero. This means that the uncertainty in the predicted bromideconcentration is greater along the borders of the contaminant plume and the area with higherconcentration gradient compared with the other regions of the domain, but the predicted values arestill reliable because the coefficient of variation is low.

6.3. Sensitivity analysis

The model is also used to study numerically the effects of soil heterogeneity on transport mechanismsand contaminant fate. To study the effects of soil heterogeneity on the mechanism of dispersion, weattempted to eliminate the effects of variations in the flow field conditions because of soilheterogeneity on solute transport and by keeping a constant flow field condition in the domain.Hence, the water flow equation was solved only with one value for the saturated hydraulicconductivity equal to 0.6, whereas the solute transport equation was solved for three different valuesfor saturated hydraulic conductivity equal to 0.3, 0.6, and 7. Figure 13 shows the bromideconcentration distribution under the furrow (described in previous section) at inlet and outlet sitesfor three different values for the variance of saturated hydraulic conductivity. It is shown that thesolute disperses in the wider area with more uniform concentration. The maximum concentration is5.61 for the case with the variance equal to 7 and 3.84 for the case with variance equal to 0.3. It canbe concluded that ignoring the soil heterogeneity may lead to the overestimation of the soluteconcentration and the underestimation of the dimension of contaminated plume. The variance ofsaturated hydraulic conductivity can be considered as an index of heterogeneity of soil. As thevariance increases, the solute spreads faster in the domain. The heterogeneity of soil causes thefluctuation of water velocity to increase. As the solute dispersion is caused by the changes inthe water velocity, soil heterogeneity provides greater potential for solute dispersion, so higheramount of solute spreads because of higher heterogeneity.

To study the effects of soil heterogeneity on advection mechanism, we set the coefficient ofmacrodispersivity to zero and solved the solute transport equation three times using the velocityobtained from solving water flow equation using three different values for s2f equal to 0.3, 0.6, and

Furrow width (m)

Fu

rro

w d

epth

(m)

Figure 12. Br concentration variance (�10�3) at t = 2 h, 2 days, and 5 days after the start of first irrigation.


a) After 2 hours from the start of the irrigation in inlet site

c) After 2 hours from the start of the irrigation in outlet site

d) After 5 days from the start of the irrigation in outlet site

Furrow width (m)

2f 0.3= 2

f 0.6= 2f 7=

Fu

rro

w d

epth

(m)

Fu

rro

w d

epth

(m)

b) After 5 days from the start of the irrigation in inlet site

Figure 13. Effects of soil heterogeneity on solute distribution due to dispersion.


7. As the groundwater velocity is dependent on flow field conditions (wetting or drying condition), 2 hafter the start of the first irrigation is chosen when flow occurs under a wetting condition (flow fieldresults are not presented here). Figure 14 shows bromide concentration distribution under the furrowfor three different flow field conditions obtained using different variances of saturated hydraulicconductivity after 2 h from the start of the first irrigation. The results show that for this case, thespeed of solute movement due to advection in vertical direction decreases when the variance s2fincreases. This is due to the reduction of water flow velocity in vertical direction due to soilheterogeneity. The experimental study shows that the configuration of the soil under the furrow ishorizontally stratified. Field observations show the water flow in the soil with horizontal


Furrow width (m)

2f 0.3= 2

f 0.6= 2f 7=

Fu

rro

w d

epth

(m)

Figure 14. Effects of soil heterogeneity on solute distribution due to advection for 2 h after the start of firstirrigation.


stratification formation moves slower in the direction perpendicular to the stratification in the wettingcondition [30], which is in accordance with the result presented here.

It can be seen that soil heterogeneity plays a significant role in transport and distribution of soluteconcentrations. This analysis indicates the importance of consideration of the effect soilheterogeneity and random spatial variability of hydraulic parameters of soil in modeling thetransport of contaminants in soils.

7. CONCLUSION

One of the most challenging problems in modeling of solute transport in soils is how to effectivelycharacterize and quantify the effects of soil heterogeneity on the transport processes. Recent studieshave shown that the current models of contaminant transport analysis are not able to adequatelydescribe the effects of soil heterogeneity. Furthermore, the effect of soil heterogeneity on the fateand transport of contaminants is not included in most of the existing numerical models forcontaminant transport. This article presented a coupled transient SFE model for predicting the flowof water and contaminant transport in unsaturated soils. The model is capable of simulating thephenomena governing contaminant transport in soils including advection, dispersion, and diffusion.The mathematical framework and the numerical implementation of the model were presented. Themodel was validated by application to two test cases with the aim of studying the effectsheterogeneity of soil on contaminant transport. The SFE model performed well in predictingtransport of contaminants through the soil. The comparison of the results of the SFE and DFEmodels with the experimental results shows that the SFE model is capable of predicting the solutetransport with higher accuracy than DFE.

An interesting feature of SFE model is that only limited numbers of stochastic properties (e.g. mean,variance, correlation scale) of soil hydraulic parameters are required to evaluate the outputs of themodel, which can be measured or estimated with site investigation. Hence, the variance of hydraulicparameters is used as an index of amount of soil heterogeneity. Sensitivity analysis of the modeloutput with respect to the variance of soil hydraulic parameters makes it possible to quantitativelystudy the effects of soil heterogeneity on the solute transport processes. It can be concluded thatdispensing with the soil heterogeneity in the solute transport modeling may cause the overestimationof the solute concentration and the underestimation of the dimensions of contaminated plume. Soilheterogeneity causes an enhancement in the solute spread in all directions in the contaminatedplume, whereas in the case of water flow in direction parallel to soil layering, the enhancement intransverse direction of mean flow is higher than the one in longitudinal direction. The resultshighlight the importance of consideration of spatial variability in saturated hydraulic conductivity ofsoil in the simulation and prediction of contaminant fate.



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