Int. J. Environ. Res., 3(2):317-326, Spring 2009ISSN: 1735-6865

317

*Corresponding author E-mail:ssabahi@ut.ac.ir

Received 19 March 2008; Revised 15 Dec. 2008; Accepted 25 Dec. 2008

Inverse Method to Estimate the Mass of Contamination Sourceby Comparing Analytical with Numerical Results

Ardestani, M. and Sabahi, M.S.*

Graduate Faculty of Environment, University of Tehran, P. O. Box 14155-6135, Tehran,Iran

ABSTRACT: The development of a conceptual model that best accounts for the different parametersinfluencing LNAPL fate and transport in groundwater is ultimate the key to a successful simulationof LNAPL concentration in groundwater. Characterization of hydrocarbon sources and identificationof areas with heavy LNAPL loadings from point and non-point sources is important for land useplanners and environmental regulators.Bistoon petrochemical site has discharged the light non-aqueous phase liquid (LNAPL) contamination for 10 months. Since the amount of contamination inthis source is not clear and the only available data is the amount of contamination in the observedwell, an attempt was made to solve this problem by using inverse method. The amount of sourcecontaminant was found through observed data in the well. In this method the analytical results withnumerical ones were compared. Upon computation of the contamination concentration, the mass ofcontaminants can be calculated by multiplying concentration by volume. The numerical methodused is finite difference which is an engine in Modflow program.

Key words: Contamination, NAPLs, Benzene, Modeling, Numerical method, Analytical method

INTRODUCTIONCrude oil, refined petroleum products, as well

as polycyclic aromatic hydrocarbons are ubiquitousin various environmental compartments(Onwurah, et al., 2007). Groundwatercontamination by hazardous substance iscommonly the result of accidental spill that occurduring production, storage and transportationactivities. The groundwater contaminationconstitutes of organic and inorganic contaminants.The most common classes of organic groundwaterinclude aromatic hydro-carbons, chlorinatedsolvents and pesticides (Figs. 1, 2, 3). Commoninorganic pollutant include nitrate (NO3 − ), arsenic(As + ), selenium (Se + ), and toxic heavy metalssuch as lead (Pb 2+ ), cadmium (Cd 4+ ), andchromium (Cr 6+ ). Petroleum hydrocarbonscomprise a diverse group of compounds. Gasoline,which is a very common groundwater pollutant, is

also a complex mixture. Gasoline constitutes aregenerally isopentane, p-xylene, n-propylbenzen,2,3-dimethylbutane, n-butane, n-pentane, andtoluene, which together make up over 50% of themixture. Shahidi Bonjar (2007) reported aboutadverse inhibitory effect of V-Guard and E-Guardgasoline additives against soil beneficialStreptomyces. He concluded that gasolineadditives contaminate soil and groundwater byfuel leaks and spills.

The most important constitute is benzene,which is highly soluble and as a result highly mobilein groundwater aquifers. The drinking waterstandard for benzene (5 Lg /µ ) is much morestringent than that of other mono-aromatichydrocarbon such as toluene (1000 Lg /µ ) andxylenes (10,000). Another important group ofpollutants are the poly-nuclear aromatic hydro-carbons (PAHs), which are commonly found nearcoal conversion facilities and petroleum refineries.

318

Ardestani, M. and Sabahi, M.S.

These hydrophobic pollutants are of majorconcern to both public and environmental healthbecause of their tendency to concentrate in foodchain (Mcelory et al., 1989) and acute toxicity(Heitkamp and Cerniglia, 1988). PAHs are theprinciple constituents are creosote, which is acomplex mixture of about 200 compounds alsocontaining phenolic and heterocyclic pollutants.Hydrocarbons that exist as a separate, immisciblephase comes in contact with water and/or air –the so-called Non-aqueous phase liquids (NAPLs).Non-aqueous phase liquids (NAPLs) aredifferences in the physical and chemical propertiesof water and NAPL result in the formation of aphysical interface between the liquids whichprevent the two fluids from mixing. Non-aqueousphase liquids are typically classified as either light

non-aqueous phase liquids (LNAPLs) which havedensities less than water, or dense non-aqueousphase liquids (DNAPLs) which have densitiesgreater than water. (Newell et al., 1997).

The development of a conceptual model thatbest accounts for the different parametersinfluencing LNAPL fate and transport ingroundwater is ultimately the key to a successfulsimulation of LNAPL concentration ingroundwater. Characterization of hydrocarbonsources and identification of areas with heavyLNAPL loadings from point and non-point sourcesis important for land use planners, environmentalregulators. This is essential for developing fateand transport models. Accurate quantification ofLNAPL leaching to groundwater is difficult dueto the complex interaction between land use

Fig. 1. Common aromatic hydrocarbon that contaminates soil and groundwater aquifers(Keith and Telliard, 1979)

320

Mass of Contamination Source

There are some evidences implying that due tosome problems, this plant has been the main sourceof contaminates for groundwater. Observationsshow that the period of leaching LNAPL ingroundwater from plant is 10 months. LaboratoryObservations also suggest that there are somekinds of LNAPL exist in the groundwater.However it has been mentioned already sincetoxicity of the benzene is higher than other mixture,this study focused on benzene as an indicator.Based on laboratory report benzene constitutes 2percent of the mixture therefore the value thatwill be mentioned for concentration is only forbenzene and not for the mixture. Most of the soilsin the study area are categorized as well drained.The ground water depth is around 80 meters inthe study area. Average annual precipitation is 448mm. Coefficient infiltration to the aquifer is 0.6.The groundwater flow is from north east to southwest (EIA Reports, 2005).

Fig. 4. depicts a pictorial representation of theproposed framework for modeling the impact ofland use on LNAPL contamination ofgroundwater. The framework is a simplificationto the itemized conception. The framework alsoincorporates the identification of the spatialdistribution of the on-ground LNAPL sources andcorresponding loadings, and the modeling of thegroundwater flow system and the LNAPL fateand transport processes. The modeling frameworkrelies on MODFLOW (Harbaugh and McDonald,1996) for the simulation of the groundwater flowmodel and on RT3D (Clement, 1998) in order tosimulate the LNAPL fate and transport processesin groundwater. Both, MODFLOW and RT3Dutilize the same finite-difference grid. There aredifferent mechanisms that control the fate andtransport of LNAPL in the soil and groundwaterzones. In addition, the models utilized in theframework of different levels of complexitiesamong them. However it is very important to keepin mind that these models and components to beexecuted in a sequential manner as depicted in(Fig.4). These linkages and procedures aresummarized as follows:

A- On ground LNAPL loading is computed andall surface losses of LNAPL are accounted for.Subsequently the net loading is considered as netinput of LNAPL to the soil zone.

LNAPL in Groundwater

On-Ground LNAPL Loading ModelLNAPL sources:

ReservoirSudden release

Plants pipe

Mass of LNAPL Leaching to Groundwater

Processing Code

SSM Package of RT3D

RT3D

LMT Package

Modflow

Velocity Distribution

Fig. 4. A schematic of the linkages between themodels of on-ground LNAPL loading, MODFLOW

and RT3D

B- This LNAPL input is subject to differenttransformations in the soil zone. The LNAPL thatleaches to the groundwater is dealt with spatiallywhere it is written into the sink and source mixing(SSM) input file (supported by RT3D). SSM filecontains all the information about theconcentrations (or mass distribution) of LNAPLthat enter the aquifer corresponding to the pointand non-point sources. Generally, the SSM file isimmense in terms of its size. This is due to thefact that at each time step can be specified foreach cell of the model domain the correspondingmass of LNAPL leaching to groundwater.Toassure efficiency in framework implementation, itis appropriate to develop the SSM fileautomatically. This feature would enhance the

Int. J. Environ. Res., 3(2):317-326, Spring 2009

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implementation of the modeling framework inhighly distributed fine resolution situations.C- Once the SSM file is developed automatically;RT3D can be executed after the development ofthe necessary input files.

Since this modeling framework is intended forregional analysis, fine time steps were avoided.Monthly time steps were considered for the RT3Dmodels while MODFLOW was developed understeady-state condit ions and consequentassumptions. A transient groundwater flow modelwould capture more adequately many of thedynamics of the drivers that ultimately dictate thetransport of LNAPL in groundwater and hencethe concentration distribution. Nevertheless, atransient model would imply that all input data tobe variable with time which was unavailable forthe study under consideration.The first step todevelop the modeling framework is to characterize

of the spatial variability of the on-ground LNAPLsources and to estimate the corresponding spatialand temporal distribution of the LNAPL loadings.Fig. 5.depicts Schematic view of part of thepetrochemical site. The main source of the LNAPLcontaminant related to the manholes and pipe linesthat were shown in (Fig. 5).

The partial differential equation that governsthe three-dimensional transport of a single chemicalconstituent in groundwater, considering advection,dispersion, fluid sinks/sources, equilibrium-controlled sorption, and first-order irreversible ratereactions is described as follow; (Zheng andBennett, 1995):

Where C is the dissolved concentration(ML 3− ); C is the adsorbed concentration

Pond

3#well

4#well

100#well

2#well

Manholes and pipelines

5#well

North

Flare

Reservation Tank

CPS

Source of Contamination

West site boundary

200#well

Fig. 5. Schematic view of part of the petrochemical site (Petrochemical report, 2006)

)()()()(

CCCq

CVxx

cD

xt

cR b

ss

iij

iji φ

ρλ

φ

θ+−+

∂

∂−

∂

∂

∂

∂=

∂

∂(1)

322

(MM 1− ); t is time (T); Dij is the hydrodynamicdispersion coefficient tensor (L 2 T 1− ); VVi is thepore water velocity (LT 1− ); qs is the volumetricflow rate per unit volume of aquifer and representsfluid sources and sinks (T 1− ); Cs is theconcentration of the fluid source or sink flux(ML 3− ); λ is the reaction rate constant (T 1− ); Ris the retardation factor (L0); bρ is the bulk densityof the porous medium (ML); and is the porosity(L0).As can be concluded from Eq. (1), the LNAPLfate and transport model requires the velocity ofthe groundwater flow. It is necessary to develop agroundwater flow model to obtain the velocity field.The following governing equation of the three-dimensional groundwater flow has to be solved andthe head distribution and subsequently the velocityare obtained and computed (Schwartz and Zhang,2003):

thsw

zhK

zyhK

yxhK

x szzyyxx ∂∂

=−∂∂

∂∂

+∂∂

∂∂

+∂∂

∂∂ )()()( (2)

Where Kxx, Kyy and Kzz are values of hydraulicconductivity (LT 1− ) along x, y, and z coordinateaxes; h is the hydraulic head, W is a flux term thataccounts for pumping (LT), recharge, or othersources and sinks; Ss is the specific storage (L);and t is time (T). The solution to Eq. (2) provides atransient prediction of hydraulic head in a three-dimensional domain for an (4)anisotropic hydraulic-conductivity field (Schwartz and Zhang,2003).Coefficients of hydrodynamic dispersion aregiven by the following equations:

DL = αLvi + D*

DT = αTvi + D*(3)(4)

Where DL and DT are the longitudinal andtransverse hydrodynamic dispersion coefficients,respectively; D* is the effective diffusioncoefficient (L 2 T 1− ); and αL and αT are thelongitudinal and transverse dispersive ties (L),respectively. In this model the effective diffusioncoefficient is omitted due to high amount of Pecletnumber. The Peclet number is defined as follows(Perkins and Johnson, 1963):

P = *0

Ddvx (4)

Where vx is the average linear velocity, d0 is themean grain diameter, and D* is the diffusion

coefficient. As the Peclet number increases,advective-dispersive processes (first term inequation 3 and 4) become increasingly moreimportant than diffusion.There are two ways tosolve equation 1: Numerical solution and Analyticalsolution. The analytical solution was used in thispaper is multidimensional transport from a finite,planar source of contaminant under transientconditions (Newell et al., 1996):

C(x,y,z,t)=(8

0C)exp[

x

xα2 (1 )41

c

c

vλα

+ ]

.erfc( )2

41

tvv

tvx

cx

c

xc

α

λα+−

.[erf( )2

2x

Yy

yα

+-erf

( )2

2x

Yy

yα

−] . [erf( )

2 xZz

zα+

-erf( )2 x

Zz

zα−

]

(6)

Where C= contaminant source, C0= initialcontaminant concentration at the source, x=distance down gradient of source, y= distance fromthe centerline of the source, z= vertical distancefrom the groundwater surface to the measurementpoint, Y= source withes, Z= source depth, xα =

longitudinal dipersivity, yα = horizontal transverse

dispersivity, zα = vertical transverse dispersivity,,λ = site specific first-order decay coefficient, t=time, vc= contaminant velocity in groundwater,erf(x)= error function and erfc(x)=complementary error function= 1- erf(x) Bysolving equation 6, the amount of contaminant inthe source zone can be estimated.The surficialgeology of the study area shows that there is onemajor geologic layer. This layer is composedmainly of stratified sand and gravel. Typically, thislayer is more than 80 m thick. Also the type ofboundary conditions and the flow pattern areassumed as shown in (Table 1 & Fig. 6).Table 1. Description of boundary conditions used in

developing of the groundwater flowSegments Flow

AB Constant Head BC No-Flow CD Constant Head DA No-Flow

Ardestani, M. and Sabahi, M.S.

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323

Fig. 6. Site domain, Bistoon technical report, 2005

The model domain was uniformly discretized intoa finite-difference grid of 67×57 m2 cells. Modelcalibration was conducted by altering thetransmissivity values until the simulatedpotentiometric heads matched closely the observedvalue.The LNAPL leakage sources in the studyare from the manholes as shown in Fig.5. Asmentioned earlier, RT3D was used in developingthe LNAPL fate and transport model using thesame finite-difference grid as in MODFLOW.RT3D interfaces directly with MODFLOW. Itretrieves the saturated thickness for each cell,fluxes across cell interfaces in three principledirections, and the locations of flow rates of thevarious sources and sinks. For temporaldiscretization, each year was divided into 12 stressperiods where each period corresponds to onemonth during which all inputs are constant.In RT3D‘Instantaneous Aerobic Decay of C6H6 Module’is selected. In this module instantaneous reaction

model is used to simulate aerobic degradation ofC6H6. This module simulates the instantaneousbiodegradation of fuel hydrocarbons under aerobicconditions. The overall aerobic reactionstoichiometry for a fuel hydrocarbon compoundcan be written as:

C6H6 + 7.5O2 6CO2 + 3H2O (7)

Equation (1) is solved both for benzene and oxygensimultaneously. At each time step, an instantaneousreaction algorithm is used to model the removalrates. According to this algorithm, eitherhydrocarbon or oxygen (which is limiting) will bereduced to zero within a grid cell, after a reactiontime step. The reaction algorithm is expressed by(Rifai et al., 1990):

H (t+1) = H (t) - FO(t)

O (t+1) = 0 when H (t) >(8)

FO(t)

Mass of Contamination Source

324

O (t+1) = O (t) - H (t).F,H (t+1) = 0 when O (t) > H (t). F

(9)

Where t refers to a particular time step and F isthe stoichiometric ratio.

One of important factors in groundwatermodeling is calibration. The purpose of calibratingthe mathematical model of LNAPL fate andtransport is to update the critical input parameterssuch that the simulated LNAPL concentrations arein close agreement with the field observedconcentrations (Zheng and Bennett, 1995).

The most critical and uncertain parametersshould be calibrated. Table 2 input parametersSince LNAPL contaminations occur after plantestablishment initial concentration was assumedzero. Therefore only the longitudinal disprsivity isused for calibration. (Fig.7). depicts the algorithmthat is used for solution in this study. The first stepto solve equation 6 is to determine parametersconstitute the equation. λ and vc are theunknown(Table 3). Prior to determine vc the value

φρ db

c

KvvR +== 1 (10)

Input A = C(x,y,z)

Analytical solutionmultidimensional transport

from a finite, planar, continuous source of

contaminant under transient conditions

Find C0 Although this value is

approximate

Numerical solution with C0Modflow

Finite difference method

Find C(x,y.z)C(x,y.z) = B

A = B Solution is complete

Change Model’s Parameters

TrueFalse

of retardation coefficient (R) should be known.Assuming the sorption can be described by linearpartitioning between the water and the aquifermatrix, the retardation coefficient is given byequation 10 (Freeze and Cherry 1979):

Fig. 7. Solution algorithm used in this study

Where bρ is the bulk density aquifer material,

dK is the distribution coefficient betweengroundwater and aquifer, and φ is the total porosity..The distribution coefficient ( ) can be estimatedon the soil adsorption coefficient for soil organiccarbon (Koc) and the fraction of soil organic carbon(ƒoc) as in equation 11:

Kd =ƒoc Koc (11)

dK

The value of Koc for benzene is 83.2 and thevalue is assigned for ƒoc is about 0.015 according

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Int. J. Environ. Res., 3(2):317-326, Spring 2009

to Bistoon EIA report (2003). It should bementioned that this value is in the range of reportsby Domenico and Schwartz (1998). By substitutingthe available data in equation 10 and 11, the resultswould be concluded: Kd is equal to 1.25 (Cm3/g)and. the value of R is about 7. Thus in this aquifer,benzene would travel 7 times slower thangroundwater velocity which vc is 0.3 (m/d). ë isboth chemical-specific and site-specific, and isdependent on such site-specific conditions aselectron-acceptor concentrations, soil chemistry,geology, water temperature, microbial populations,and concentrations of other constituents. Due tothe great amount of uncertainty and difficulty inquantifying these conditions, the first-order decayrate is generally selected as a conservativeliterature value or simply neglected. (Nevin et al.,1997). The amount of set to 0.

RESULTS & DISCUSSIONThe amount of the value of C0 is used for

numerical method obtained from the mean valueof C0 in well # 3 and C0 in well # 100 out ofanalytical method by using equation 6. Thereforetwo values for well # 3 and well # 100 are available.By averaging these values a C0 for numericalmethod will be reached. Using this C0 in Modflowand the result of the numerical will be comparedwith observation data. Because of assuming thatsome value is zero during the analytical method,the difference between numerical and observationwill be occurring. This process is performed untilthe difference between numerical results andobservation results is minus. (Fig. 8 & 9) depictthe final results of the solution. Since in numericalresults the decay coefficient is used by enteringthe amount of oxygen concentration the value ofnumerical results is estimated lower than analyticalresults. However in well # 100 the condition isdifferent. In this well the value of C0 for numericalresults is larger than real value. This is because inthis well the contamination takes a longer time thanwell # 3 to reach. Therefore all the concentrationvalues are lower than well # 3. By averaging thetwo C0 the value for well # 100 is larger than realone. However this difference is compensating withdecay coefficient. Then in well # 100 the differencebetween observation data and numerical result isnot as large as in well # 3.

Table 2. Data that is used for analytical andnumerical model

Data Ф 0.35 kd 1.25 (cm3/gr) λ Estimated 0 R 7 v 2.13 (m/d) vc 0.3 (m/d) ρb 1.7 (gr/cm3) foc 0.015 koc for benzene 83.2 Initial Concentration of Dissolved O2

4 (mg/L)

Sy Specific yield (Storage) 0.27 Y (source dimension) 2 (m) Z (source dimension) 1 (m)

Table 3. Final results after solution

Concentration Versus time well#3

02468

1012141618

90 120 180 240 300 360Day

Con

cent

ratio

n(µg

/L)

Numerical ResultsObservarion Results

Concentration Versus time well#100

01234567

90 120 160 250 300 330Day

Con

cent

ratio

n(µg

/L)

Numerical Results

Observation Results

Results αx 30 (m) αy 6.5 (m) αz 2 (m) Estimated C0 of benzene 900 (µg/m3) Estimated Mass 2700 µg

Fig. 8. Concentration of benzene versus time for well# 3

Fig. 9. Concentration of benzene versus time for well# 100

λ

326

CONCLUSIONIn normal case the concentration of source or

value of mass of contamination is known. Thereforethe model will be calibrated and predicts the valueof contamination in the future. However when themass of contaminant is unknown, a properjudgment for predicting the future is extremelydifficult. Thus in this work, analytical and numericalmethod were developed to estimate the initialconcentration of benzene. Benzene volatile andmobile in soil evaporates rapidly in water, andbiodegraded slowly in aerobic soil. Since the valueof ë in analytical solution is assumed zero a littledifference between the results of numerical datawith observation concentration data occurs. Innumerical method first the groundwater flowmodeled by using Modflow. Then the transport ofcontamination is solved by RT3D module, usingthe method of instantaneous aerobic decay ofbenzene. The value of C0 for analytical solution issupported by analytical solution. These values arecompared until the difference between numericalresults and observation data is minus and in thiscase the solution is completed.

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Ardestani, M. and Sabahi, M.S.