Research Article Generalizing Source Geometry of...

Research ArticleGeneralizing Source Geometry of Site Contamination bySimulating and Analyzing Analytical Solution ofThree-Dimensional Solute Transport Model

Xingwei Wang1 Jiajun Chen1 Hao Wang2 and Jianfei Liu3

1 Key Laboratory for Water and Sediment Sciences of Ministry of Education School of Environment Beijing Normal UniversityBeijing 100875 China

2 CECEP LampT Environmental Technology Co Ltd Beijing 100085 China3 School of Civil Engineering Henan Polytechnic University Jiaozuo 454003 China

Correspondence should be addressed to Jiajun Chen jeffchenbnueducn

Received 28 February 2014 Accepted 13 May 2014 Published 13 August 2014

Academic Editor Oluwole Daniel Makinde

Copyright copy 2014 Xingwei Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Due to the uneven distribution of pollutions and blur edge of pollutant area there will exist uncertainty of source term shapein advective-diffusion equation model of contaminant transport How to generalize those irregular source terms and deal withthose uncertainties is very critical but rarely studied in previous research In this study the fate and transport of contaminant fromrectangular and elliptic source geometry were simulated based on a three-dimensional analytical solute transport model and thesource geometry generalization guideline was developed by comparing the migration of contaminant The result indicated that thevariation of source area size had no effect on pollution plumemigration when the plumemigrated as far as five times of source sidelength The migration of pollution plume became slower with the increase of aquifer thickness The contaminant concentrationwas decreasing with scale factor rising and the differences among various scale factors became smaller with the distance to fieldincreasing

1 Introduction

Concern about contamination of the subsurface environmenthas greatly stimulated research of solute transport phenom-ena in porous media A number of solute transport studiesaimed at solving the advective-diffusion equation (ADE) fornonreactive and reactive solutes subject to various initialand boundary conditions [1] Several analytical solutionsfor one- two- and three-dimensional ADEs have beendeveloped for predicting the transport of various contami-nants in the subsurface [2] For instance Ogata and Banks[3] Sauty [4] and Van Genuchten [5] formulated severalanalytical solutions for the one-dimensional ADE subjectto the first-type (Dirichlet) second-type (Neumann) andthird-type (Cauchy) boundary conditions respectively Batu[6 7] compiled analytical solutions to two-dimensional ADEwith various source boundary conditions The analyticalsolutions for three-dimensional ADE have been derived bySagar [8] Domenico [9] Leij et al [10] Batu [11] Sim and

Chrysikopoulos [12] and Park and Zhan [13] Analyticalsolutions for ADE play important roles in giving initialor approximate estimates of contaminant distributions insoil or aquifer systems [14] Although large amounts oftwo- or three-dimensional analytical models are available atpresent [15] thosemodels illustrated the surface condition fortransport from a regular source

The regular sources included point source line sourceand area source (rectangular or elliptic) For example Simand Chrysikopoulos [12] investigated the effect of aquiferboundary conditions and the source geometry on solutetransport in subsurface porous formations Porous mediawith either semi-infinite or finite thickness and source geom-etry with either a point source or an elliptic source wereexamined Park and Zhan [13] tested the sensitivity of the linesource solutions to source geometry dispersion coefficientsand distance to the source The results indicated that theconcentration at a near field point was sensitive to the sourcegeometry when the dispersion coefficients are anisotropic

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 646495 8 pageshttpdxdoiorg1011552014646495

2 Mathematical Problems in Engineering

and it was less sensitive to the source geometry when thedispersion coefficients were isotropic The concentration at afar field was found to be almost independent of the sourcegeometry Zhao et al [16] investigated the effects of differentdomain shapes in general and trapezoidal domain shape inparticular on the morphological evolution of NAPL dissolu-tion fronts in two-dimensional fluid-saturated porousmediaDomain shapes had a significant effect on the propagatingspeed and an increase in the divergent angle of a trapezoidaldomain could lead to a decrease in the propagating speedof the NAPL dissolution front The above-mentioned studiescommendably analyzed the fate and transport of contaminantunder various conditions from a regular source geometryHowever due to the complex condition in reality such asuneven distribution of pollutions and blur edge of pollutantarea the shape of pollution plume is not explicit in theinitial period of contaminated site investigation On thisbackground there will exist uncertainty of source term shapein ADE model of contaminant transport How to generalizethose irregular source terms and deal with those uncertaintiesis very critical and can influence the accuracy of finalcalculation result Nevertheless few previous studies havebeen conducted concerning those problems and then pro-vided decision-support for site remediation practice throughnumerical simulation methods

Therefore the objective of this study is to develop aguideline for source geometry generalization which differsfrom previous research in two aspects First it can answerthe questions of ldquohow to use the regular source geometry tosubstitute the irregular source geometryrdquo and ldquowhat regularsource geometry can be used for substitutionrdquo Second it canhelp technicians examine and predict the contaminant trans-port in subsurface flow systems in actual site remediationWe start from identifying contaminant from regular (ierectangular and elliptic) source geometry investigating itsapplication scope and condition from its analytical solutionsof ADE as well as the corresponding influence factors Thensimulating flow and transport of contaminants from tworegular source geometries under the same condition compareflow and transport of contaminant based on the simulationresults to develop source geometry generalization guidelineThe obtained results will provide useful information andtechnical support for estimating the distribution of contami-nant concentration in the initial period of contaminated siteinvestigation

2 Mathematical Model

This study considers the problem of contaminant transport insaturated homogeneous porousmedia accounting for three-dimensional hydrodynamic dispersion in a uniform flowfield and first-order decay of liquid phase and sorbed con-taminants with different decay rates Before formulating themodel several assumptionsmust bemade First it is assumedthat the upper and lower boundaries of homogeneous porousmedia are impermeable (no-flow boundary) if a water tableboundary exists the slope of the water table is small andthe aquifer is horizontal without curvature and parallel tothe lower boundary Second there exists one-dimensional

x

y

z

VH

y0

x0

minusy

Figure 1 The general geometry of the rectangular source

steady-state ground water flow along the 119909-axis and theaveraged actual flow velocity is 119906 Third cardinal directionof dispersion coefficient is consistent with the coordinateaxis Fourth there is no pollution in the study area at initialmoments and the continuous injection strength is 119868

119886

Based on the above assumptions the advective-diffusionequation can be formulated by the following partial differen-tial equation

120597119862

120597119905

= 119863119909

1205972

119862

1205971199092+ 119863119910

1205972

119862

1205971199102+ 119863119911

1205972

119862

1205971199112minus 119906

120597119862

120597119909

minus 120582119862 +

119868119886

119899

(1)

The corresponding initial and boundary conditions are asfollows

119862|119905=0= 0 minusinfin lt 119909 119910 lt infin 0 lt 119911 lt 119867

119862|119909rarrplusmninfin

= 0 minusinfin lt 119910 lt infin 0 lt 119911 lt 119867 119905 gt 0



120597119862

120597119911

10038161003816100381610038161003816100381610038161003816119911=119911table

=

120597119862

120597119911

10038161003816100381610038161003816100381610038161003816119911=119867

= 0 minusinfin lt 119909 119910 lt infin

119905 gt 0

(2)

where 119862 denotes the solute concentration 119863119909 119863119910 and 119863

119911

are the longitudinal lateral and vertical hydrodynamic dis-persion coefficients respectively 119906 represents the averagedactual steady-state pore water velocity 119905 is time 119909 119910 and 119911stands for the spatial coordinates in the longitudinal lateraland vertical directions respectively 120582 is the decay rate ofsolutes 119899 is the porosity of the porous medium 119868

119886is the

continuous injection strength119867 denotes aquifer thickness

21 Analytical Solution of Rectangular Source The generalgeometry of the rectangular source is show in Figure 1 Theorigin of the coordinate system is at the upper boundaryThepositive 119911-axis is downward The aquifer is assumed infinitein the 119909- and 119910-directions but finite in the 119911-direction witha thickness of 119867 The rectangular source of contaminationis located on top of an aquifer without considering itsthickness influence with 119909 isin [0 119909

0] 119910 isin [minus119910

0 1199100]

Mathematical Problems in Engineering 3

H

x

yV

z

minusy

a1

a2

Figure 2 The general geometry of the elliptic source

and 119911 isin [0119867] which is described mathematically by thefollowing expression

119868119886=

1198680119891 (119905) minus119909

0lt 119909lt 119909

0 minus1199100lt 119910 lt 119910

0 119911table lt 119911 lt 119867

0 otherwise(3)

Through using Greenrsquos function methods three-dimensionalanalytical solution for rectangular source can be obtained [13]as follows

119862 (119909 119910 119911 119905)

=

1

4119899119867

times int

119905

0

119868119886(119905 minus 120591) exp (minus120582119905)

times [erfc119909 minus 119906120591 minus 119909

0

2radic119863119909120591

minus erfc119909 minus 119906120591 + 119909

0

2radic119863119909120591

]

times[

[

[

erfc119910 minus 1199100

2radic119863119910120591

minus erfc119910 + 1199100

2radic119863119910120591

]

]

]

times [1 + 2

infin

sum

119899=1

cos119899120587 (119911 minus 119911table)

119867

exp(minus1198631199111198992

1205872

1198672

120591)]119889120591

(4)

where 1198680is a constant and 119891(119905) is a function of time erfc[119909]

denotes residual error function equal to 1minus(2radic120587) int1199090

119890minus1199112

119889119911

22 Analytical Solution of Elliptic Source Consider solutemovement from an elliptic source as sketched in Figure 2Theset of coordinate system is similar to the rectangular sourcethe source contaminant is located on top of an aquifer andthe solute may move from the source which has a negligiblethickness by diffusion or advection with 119909 isin [0 119886

1] 119910 isin

[minus1198862 1198862] and 119911 isin [0119867] the elliptic source geometry is

defined mathematically by the following expression

119868119886=

1198680119891 (119905)

1199092

1198862

1

+

1199102

1198862

2

le 1

0 otherwise(5)

The governing solute transport equations are solved analyti-cally by employing Laplace Fourier and finite Fourier cosinetransform techniques

119862 (119909 119910 119911 119905)

=

1

4119899119867radic120587119863119909

times int

119905

0

int

1198861

minus1198861

119868119886(119905 minus 120591) 119865

1(120591)

times int

120591

0

1198652(120591) 1198653(119909 minus 119902 119901) 119865

4(119901) 1198655(119901)

119889119901

radic119901

+

1198653(119909 minus 119902 120591)

radic120591

1198654(120591) 1198655(120591) 119889119901119889120591

1198651(119905) = exp [minus (119903

2+ 1205825) 119905]

1198652(119905) = 119868

1(2radic11990311199032119901 (119905 minus 119901))radic

11990311199032119901

119905 minus 119901

1198653(119909 119905)

= exp[ 1199061199092119863119909

minus

1199092

4119863119909119905

minus 119905 (1199031+ 120582 minus 119903

2minus 1205825+

1199062

4119863119909

)]

1198654(119905) = erfc [119896

2(119902 119910 119905)] minus erfc [119896

1(119902 119910 119905)]

1198961(119902 119910 119905) =

119910 + 119884

2radic119863119910119905

1198962(119902 119910 119905) =

119910 minus 119884

2radic119863119910119905

119884 = 1198862radic1 minus

1199022

1198862

1

1198655(119905) = 1 + 2

infin

sum

119899=1

cos119899120587 (119911 minus 119911table)

119867

exp(minus1198631199111198992

1205872

1198672

119905)

(6)

where 1199031and 1199032are the forward and reverse rate coefficients

120582 is decay rate of liquid phase solute 1205825is decay rate of sorbed

contaminant

3 Model Simulations and Discussion

Model simulations are performed for two different sourceconfigurations The integrals present in the analytical solu-tions are evaluated numerically by using globally adaptivequadrature algorithms The number of terms is selected sothat additional terms do not alter the summation more than


Table 1 Model simulation parameters for two source geometries

Elliptic source geometry Rectangular source geometryParameter Value Parameter Value120582 and 120582s 001 dayminus1 120582 and 120582s 001 dayminus1

119906 5 cmdayminus1 119906 5 cmdayminus1

119868119886

100mg cmminus2 dayminus1 119868119886

100mg cmminus2 dayminus1

119863119909

2000 cm2 dayminus1 119863119909

2000 cm2 dayminus1

119863119910

500 cm2 dayminus1 119863119910

500 cm2 dayminus1

119863119911

500 cm2 dayminus1 119863119911

500 cm2 dayminus1

1199031

002 dayminus1 119885table 500 cm1199032

001 dayminus1 119867 200 cm119885table 500 cm 119909

050 cm

119867 200 cm 1199100

25 cm1198861

100(120587)05 cm1198862

50(120587)05 cm

minus500

minus550

minus600

minus650

minus700

Z(c

m)

X (cm)

0 100 200 300 400 500 600 700 800 900 1000

040404

08

08

08

161

6 16

32

Figure 3 Contaminant concentration contours on the 119909119911-plane(119910 = 0) at 119905 = 10 119905 = 100 and 119905 = 500 days under conditionsfor rectangular source geometry

10119864 minus 7 In order to compare the fate and transport ofcontaminant from rectangular and elliptic source geometryit is assumed that aspect ratio (length-width ratio for rect-angular source and major axis-minor axis ratio for ellipticsource) of rectangular source geometry is the same as thatof elliptic source geometry besides that the area of twodisparate sources is equal which can guarantee the uniformtotal pollution load under the same continuous injectionstrength condition

31 Rectangular Source Type The groundwater table and thebottom of the finite thickness aquifer are assumed to belocated at 119911 = 119911table = 0 cm and 119911 = 119867 = 200 cmrespectively Other parameters are shown in Table 1 withthem concentrations can be calculated at any given time forcontinuous source with consideration of the first-order decayusing FORTRAN language

Figure 3 displays the fate and transport of contaminantfrom rectangular source on the 119909119911-plane (119910 = 0) at 119905 = 10119905 = 100 and 119905 = 500 days under conditions by continuousinjection where the black line stands for 119905 = 10 days thered line represents 119905 = 100 days and the blue line denotes119905 = 500 days As expected the pollution plume spreadsout with time and the plume has already reached the lowerboundary of aquifers at 119905 = 100 daysThe degree of spreadingalong 119909-axis is more than that along 119911-axis and themigrationspeed of contaminant at the near field is lower than that ofcontaminant at the far field The reason is due to the fact

minus500

minus550

minus600

minus650

minus700

Z(c

m)

X (cm)

0 100 200 300 400 500 600 700 800 900 1000

04

0816

1632

Figure 4 Contaminant concentration contours on the 119909119911-plane(119910 = 0) under larger source area unchanged source area andsmaller source area conditions at 119905 = 100 days for rectangular sourcegeometry

minus500

minus550

minus600

minus650

minus700minus750

Z(c

m)

X (cm)

0

0

0

100 200 300 400 500 600 700 800 900 1000

1000

1000

04

0404

08

0808

16

16

16

32326

4

Figure 5 Contaminant concentration contours on the 119909119911-plane(119910 = 0) under 119867 = 150 cm 119867 = 200 cm and 119867 = 250 cmconditions at 119905 = 100 days for rectangular source geometry

that there exists one-dimensional steady-state ground waterflow along the 119909-axis and the dispersion effect along 119909-axis(119863119909= 2000) is larger than that along 119911-axis (119863

119911= 500)

At initial time due to the higher concentration gradient andgravity the plume spread quickly both along 119909-axis and 119911-axis however this phenomenon is insignificant over time bycontinuous injectionThese results agree with the findings byCianci et al [17] investigating contaminant transport througha saturated porous medium in a semi-infinite domain Incomparison with the work by Leij et al [18] which illustratethat the relatively high maximum concentration occurs atthe surface This conclusion is obtained based on the surfacecondition transporting from a rectangular source which isalso indicated in this research

Figure 4 illustrates contaminant concentration contourson the 119909119911-plane (119910 = 0) under larger source area (red line)unchanged source area (black line) and smaller source areaconditions (blue line) at 119905 = 100 days The result of largerarea is obtained under condition that rectangular source areais double while the continuous injection strength reduces tohalf compared to the original source area Similarly smallerarea means that rectangular source area reduces to half whilethe continuous injection strength is double From the figurewe can draw a conclusion that the influence of source areasize decreases with the increase of distance to field when thepollution plume migrates to a certain distance the variationof source area size has no effect on plumemigration In orderto make this problem clear we conduct the following study

Figure 5 presents contaminant concentration contours onthe 119909119911-plane (119910 = 0) under 119867 = 150 cm (red line) 119867 =

200 cm (black line) and 119867 = 250 cm (blue line) conditionsat 119905 = 100 days which shows the influence of aquifer thick-ness on pollutant migration The pollution plume migratesslower with aquifer thickness increasing In order to analyze


00102030405060

0 100 200 300 400 500 600 700 800 900 1000x (cm)

Con

cent

ratio

n at

05H

SF = 25

SF = 30

SF = 10

SF = 15

SF = 20

(mg

cm2 )

(a)

0

02

04

06

08

0 100 200 300 400 500 600 700 800 900 1000x (cm)

SF = 25

SF = 30

SF = 10

SF = 15

SF = 20

Con

cent

ratio

n at

05H

(mg

cm2 )

(b)

Figure 6 Contaminant concentration at half aquifer thickness under various scale factors at 119905 = 100 days for rectangular source geometry

Table 2 The related calculation parameters for Figure 6

(a)

1199090(cm) 119910

0(cm) 119868

119886(mgcm2) 119867 (cm)

SF = 10 50 25 100 100SF = 15 50 25 100 150SF = 20 50 25 100 200SF = 25 50 25 100 250SF = 30 50 25 100 300

(b)

1199090(cm) 119910

0(cm) 119868

119886(mgcm2) 119867 (cm)

SF = 10 150 25 1003 300SF = 15 100 25 50 300SF = 20 75 25 2003 300SF = 25 60 25 2503 300SF = 30 50 25 100 300

the comprehensive influence of both source area and aquiferthickness we define a scale factor as follows

SF = 119867

119871119878

(7)

where SF denotes scale factor 119867 is aquifer thickness 119871119878

stands for width of rectangular source (or major axis ofelliptic source)

Figure 6 shows contaminant concentration at half aquiferthickness under various scale factors at 119905 = 100 days Thecalculation parameters are listed in Table 2 The results inFigure 6(a) are obtained with a fixed rectangular source areaunder various aquifer thicknesses conditions (as shown inTable 2(a)) From the vertical perspective the contaminantconcentration at half aquifer thickness is decreasing withscale factor rising besides differences of contaminant con-centration decrease among various scale factors From thehorizontal perspective differences of contaminant concen-tration among various scale factors become smaller with thedistance to field increasing Figure 6(b) is achieved with afixed aquifer thickness under various rectangular sources

minus500

minus550

minus600

minus650

minus700Z(c

m)

X (cm)

0 100 200 300 400 500 600 700 800 900 1000

32

16

16

16

0808

08

040404

Figure 7 Contaminant concentration contours on the 119909119911-plane(119910 = 0) under 119905 = 10 119905 = 100 and 119905 = 500 days under conditionsfor elliptic source geometry

conditions (as shown in Table 2(b)) The contaminant con-centration at half aquifer thickness is almost unchangeablewith scale factor increasing when the distance to field isgreater than 250 cm In other words the influence of sourcearea size can be neglected when the distance to field is greaterthan 250 cm

32 Elliptic Source Type Table 1 lists the related parametersfor model simulation of elliptic source Figure 7 illustratescontaminant concentration contours on the 119909119911-plane (119910 = 0)under 119905 = 10 days (black line) 119905 = 100 days (red line)and 119905 = 500 days (blue line) conditions by continuousinjection The pollution plume spreads similarly to that ofrectangular source geometry but the speed of contaminantmigration is slower than that of rectangular source type inboth transverse and longitudinal directions These resultsagree with the findings by Sim andChrysikopoulos [12] basedon a continuous source loading from elliptic source geometryin saturated homogeneous porous media

Figure 8 displays contaminant concentration contours onthe 119909119911-plane (119910 = 0) under various elliptic source areasat 119905 = 100 days (larger source area (red line) unchangedsource area (black line) and smaller source area conditions(blue line)) The influence of source area size on pollutionplume migration is insignificant It is consistent with that ofrectangular source type

Figure 9 illustrates contaminant concentration contourson the 119909119911-plane (119910 = 0) under 119867 = 150 cm (red line) 119867 =

200 cm (black line) and 119867 = 250 cm (blue line) conditions


minus500

minus550

minus600

minus650

minus700

Z(c

m)

X (cm)

0 100 200 300 400 500 600 700 800 900 1000

32 16

16

08

08

04

04

Figure 8 Contaminant concentration contours on the 119909119911-plane(119910 = 0) under larger source area unchanged source area andsmaller source area conditions at 119905 = 100 days for elliptic sourcegeometry

at 119905 = 100 days The influence of aquifer thickness for ellipticsource is the same as that of rectangular source the pollutionplume migrates fast with aquifer thickness decreasing Theinfluence of scale factor on pollution plume migration fromelliptic source geometry is the same as that of scale factorfrom rectangular source geometry under the two parameterconditions listed in Table 2

33 Comparative Analysis Figure 10 displays the variation ofCRCE defined as dimensionless coefficient indicating con-taminant concentration of rectangular source type dividedby contaminant concentration of elliptic source type undervarious scale factors along transverse direction The CRCEin the near field is smaller than that in the far field andespecially for 119909 = 100 cm the value is almost equalto 1 In other words the contaminant concentration fromrectangular source is nearly the same as that from ellipticsource at 119909 = 100 cmonwater table and the CRCE is slightlydecreasing with scale factor increasing The research studiedby Chrysikopoulos [19] also indicated that predictions ofcontaminant concentrations were sensitive to the sourcegeometry for short downstream distance However whenscale factor increases to a certain degree (ie SF = 70)the CRCE keeps constant In Figure 10(b) the situation isslightly different that the CRCE increases with scale factorrising However the CRCE keeps constant when scale factorincreased to SF = 90 The main reason is attributed to thefact that the same and continuous injection patternsmake thedifferences of contaminant concentration between two sourcegeometries in the near field insignificant on water table Sincecontaminant migration from rectangular source geometryis slightly faster than that from elliptic source geometryas previously studied so the difference between the twosource geometries in the far field is significant while it keepsconstant if scale factor reaches a certain value Assumingthat if the lower boundary is infinite (ie the scale factor isbig enough) both source geometries could be regarded aspoint source geometry In this context there would be nodifference on contaminant migration between the two sourcegeometries

From the above discussion a conclusion is obtainedthat the CRCE keeps constant under condition that thescale factor is equal to or greater than 9 Meaning that therectangular source geometry can substitute for the ellipticsource geometry under condition that scale factor is equal to

minus500

minus550

minus600

minus650

minus700

minus750

Z(c

m)

X (cm)

0

0

0

100 200 300 400 500 600 700 800 900 1000

1000

1000

3216

16

08 08

08 04

04

04

Figure 9 Contaminant concentration contours on the 119909119911-plane(119910 = 0) under 119867 = 150 cm 119867 = 200 cm and 119867 = 250 cmconditions at 119905 = 100 days for elliptic source geometry

or greater than 9 In other words we can use the data of con-taminant migration from rectangular source geometry andthe corresponding value of CRCE to analyze and estimatethe contaminant migration from elliptic source geometry

4 Summary

Three-dimensional analytical solutions for contaminanttransport in subsurface porous media from rectangularsource geometry and elliptic source geometry were inves-tigated accounting for three-dimensional hydrodynamicadvection-dispersion in a uniformflowfield first-order decayrates The fate and transport of contaminant were simulatedbased on a continuous source loading Several interestingsolutions were obtained as follows

(1) Themigration of pollution plume from the two sourcegeometries shows the same trend under the sameaspect ratio the equal source area size and theuniform total pollution load condition except that themigration of pollution plume from rectangular sourcegeometry is faster than that from elliptic sourcegeometry

(2) The influence of source area size on pollution plumemigration decreases with the distance to field increas-ing In particular when pollution plume migrates toa certain distance as far as five times of source sidelength (119871

119878) the variation of source area size has no

effect on pollution plume migration

(3) The migration of pollution plume becomes slowerwith the increase of aquifer thickness for both sourcegeometries Compared with elliptic source geometrythe phenomenon of rectangular source geometry ismore significant

(4) The contaminant concentration is decreasing withscale factor rising and the differences among variousscale factors get smaller with the distance to fieldincreasing

(5) When scale factor is equal to or greater than 9 theCRCE would keep constant meaning that we canuse the data of contaminant migration from rectan-gular source geometry and the corresponding valueof CRCE to analyze and estimate the contaminantmigration from elliptic source geometry


10

15

20

25

30

1 2 3 4 5 6 7 8 9 10SF

CRC

Eat

z=0

x = 100 cmx = 200 cmx = 500 cm

x = 800 cmx = 900 cm

(a)

10

15

20

25

30

35

1 2 3 4 5 6 7 8 9 10SF

x = 100 cmx = 200 cmx = 500 cm

x = 800 cmx = 900 cm

CRC

Eat

z=05H

(b)

Figure 10 The variation of CRCE with the increase of scale factor at various locations along transverse direction

Since calculations of three-dimensional analytical solu-tions for contaminant transport from rectangular sourcegeometry are simpler than those from elliptic source geom-etry (ie the rectangular source geometry requires littleinformation on contaminated site and little calculation) wecan use the data of contaminant migration from rectangularsource geometry to obtain the contaminant transport fromelliptic source geometry to simplify the actual engineeringproblem The results would provide useful information andtechnical support for estimating the distribution of contami-nant in the initial period of contaminated site investigation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 41272248) and National Scienceand Technology Support Program (no 2011BAC12B02) Theauthors would like to extend special thanks to the editor andthe anonymous reviewers for their constructive commentsand suggestions in improving the quality of this paper

References

[1] S K Yadav A Kumar and N Kumar ldquoHorizontal solute trans-port from a pulse type source along temporally and spatiallydependent flow analytical solutionrdquo Journal of Hydrology vol412-413 pp 193ndash199 2012

[2] J Mieles andH Zhan ldquoAnalytical solutions of one-dimensionalmultispecies reactive transport in a permeable reactive barrier-aquifer systemrdquo Journal of Contaminant Hydrology vol 134-135pp 54ndash68 2012

[3] A Ogata and R B Banks ldquoA solution of differential equationof longitudinal dispersion in porous mediardquo Geological SurveyProfessional Paper (United States) vol 411 pp A1ndashA7 1961

[4] J-P Sauty ldquoAn analysis of hydrodispersive transfer in aquifersrdquoWater Resources Research vol 16 no 1 pp 145ndash158 1980

[5] M T Van Genuchten ldquoAnalytical solutions for chemical trans-port with simultaneous adsorption zero-order production andfirst-order decayrdquo Journal of Hydrology vol 49 no 3-4 pp 213ndash233 1981

[6] V Batu ldquoA generalized two-dimensional analytical solution forhydrodynamic dispersion in bounded media with the first-typeboundary condition at the sourcerdquo Water Resources Researchvol 25 no 6 pp 1125ndash1132 1989

[7] V Batu ldquoA generalized two-dimensional analytical solute trans-port model in bounded media for flux-type finite multiplesourcesrdquoWater Resources Research vol 29 no 8 pp 2881ndash28921993

[8] B Sagar ldquoDispersion in three dimensions approximate analyticsolutionsrdquo Journal of the Hydraulics Division ASCE vol 108no 1 pp 47ndash62 1982

[9] P A Domenico ldquoAn analytical model for multidimensionaltransport of a decaying contaminant speciesrdquo Journal of Hydrol-ogy vol 91 no 1-2 pp 49ndash58 1987

[10] F J Leij T H Skaggs and M T van Genuchten ldquoAnalyti-cal solutions for solute transport in three-dimensional semi-infinite porous mediardquoWater Resources Research vol 27 no 10pp 2719ndash2733 1991

[11] V Batu ldquoA generalized three-dimensional analytical solutetransport model for multiple rectangular first-type sourcesrdquoJournal of Hydrology vol 174 no 1-2 pp 57ndash82 1996

[12] Y Sim andCV Chrysikopoulos ldquoAnalytical solutions for solutetransport in saturated porous media with semi-infinite or finitethicknessrdquo Advances in Water Resources vol 22 no 5 pp 507ndash519 1999

[13] E Park and H Zhan ldquoAnalytical solutions of contaminanttransport from finite one- two- and three-dimensional sourcesin a finite-thickness aquiferrdquo Journal of Contaminant Hydrologyvol 53 no 1-2 pp 41ndash61 2001

[14] J Chen Y Liu C Liang C Liu and C Lin ldquoExact analyticalsolutions for two-dimensional advection-dispersion equationin cylindrical coordinates subject to third-type inlet boundaryconditionrdquo Advances in Water Resources vol 34 no 3 pp 365ndash374 2011

[15] J Chen J Chen C Liu C Liang and C Lin ldquoAnalyticalsolutions to two-dimensional advection-dispersion equation incylindrical coordinates in finite domain subject to first- andthird-type inlet boundary conditionsrdquo Journal of Hydrology vol405 no 3-4 pp 522ndash531 2011


[16] C Zhao B E Hobbs and A Ord ldquoEffects of domain shapeson the morphological evolution of nonaqueous-phase-liquiddissolution fronts in fluid-saturated porous mediardquo Journal ofContaminant Hydrology vol 138-139 pp 123ndash140 2012

[17] R Cianci MMassabo and O Paladino ldquoAn analytical solutionof the advection dispersion equation in a bounded domain andits application to laboratory experimentsrdquo Journal of AppliedMathematics vol 2011 Article ID 493014 14 pages 2011

[18] F J Leij E Priesack and M G Schaap ldquoSolute transportmodeled with Greenrsquos functions with application to persistentsolute sourcesrdquo Journal of Contaminant Hydrology vol 41 no1-2 pp 155ndash173 2000

[19] C V Chrysikopoulos ldquoThree-dimensional analytical models ofcontaminant transport from nonaqueous phase liquid pool dis-solution in saturated subsurface formationsrdquo Water ResourcesResearch vol 31 no 4 pp 1137ndash1145 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


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International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


and it was less sensitive to the source geometry when thedispersion coefficients were isotropic The concentration at afar field was found to be almost independent of the sourcegeometry Zhao et al [16] investigated the effects of differentdomain shapes in general and trapezoidal domain shape inparticular on the morphological evolution of NAPL dissolu-tion fronts in two-dimensional fluid-saturated porousmediaDomain shapes had a significant effect on the propagatingspeed and an increase in the divergent angle of a trapezoidaldomain could lead to a decrease in the propagating speedof the NAPL dissolution front The above-mentioned studiescommendably analyzed the fate and transport of contaminantunder various conditions from a regular source geometryHowever due to the complex condition in reality such asuneven distribution of pollutions and blur edge of pollutantarea the shape of pollution plume is not explicit in theinitial period of contaminated site investigation On thisbackground there will exist uncertainty of source term shapein ADE model of contaminant transport How to generalizethose irregular source terms and deal with those uncertaintiesis very critical and can influence the accuracy of finalcalculation result Nevertheless few previous studies havebeen conducted concerning those problems and then pro-vided decision-support for site remediation practice throughnumerical simulation methods

Therefore the objective of this study is to develop aguideline for source geometry generalization which differsfrom previous research in two aspects First it can answerthe questions of ldquohow to use the regular source geometry tosubstitute the irregular source geometryrdquo and ldquowhat regularsource geometry can be used for substitutionrdquo Second it canhelp technicians examine and predict the contaminant trans-port in subsurface flow systems in actual site remediationWe start from identifying contaminant from regular (ierectangular and elliptic) source geometry investigating itsapplication scope and condition from its analytical solutionsof ADE as well as the corresponding influence factors Thensimulating flow and transport of contaminants from tworegular source geometries under the same condition compareflow and transport of contaminant based on the simulationresults to develop source geometry generalization guidelineThe obtained results will provide useful information andtechnical support for estimating the distribution of contami-nant concentration in the initial period of contaminated siteinvestigation

2 Mathematical Model

This study considers the problem of contaminant transport insaturated homogeneous porousmedia accounting for three-dimensional hydrodynamic dispersion in a uniform flowfield and first-order decay of liquid phase and sorbed con-taminants with different decay rates Before formulating themodel several assumptionsmust bemade First it is assumedthat the upper and lower boundaries of homogeneous porousmedia are impermeable (no-flow boundary) if a water tableboundary exists the slope of the water table is small andthe aquifer is horizontal without curvature and parallel tothe lower boundary Second there exists one-dimensional

x

y

z

VH

y0

x0

minusy

Figure 1 The general geometry of the rectangular source

steady-state ground water flow along the 119909-axis and theaveraged actual flow velocity is 119906 Third cardinal directionof dispersion coefficient is consistent with the coordinateaxis Fourth there is no pollution in the study area at initialmoments and the continuous injection strength is 119868

119886

Based on the above assumptions the advective-diffusionequation can be formulated by the following partial differen-tial equation

120597119862

120597119905

= 119863119909

1205972

119862

1205971199092+ 119863119910

1205972

119862

1205971199102+ 119863119911

1205972

119862

1205971199112minus 119906

120597119862

120597119909

minus 120582119862 +

119868119886

119899

(1)

The corresponding initial and boundary conditions are asfollows

119862|119905=0= 0 minusinfin lt 119909 119910 lt infin 0 lt 119911 lt 119867





120597119862

120597119911

10038161003816100381610038161003816100381610038161003816119911=119911table

=

120597119862

120597119911

10038161003816100381610038161003816100381610038161003816119911=119867

= 0 minusinfin lt 119909 119910 lt infin

119905 gt 0

(2)

where 119862 denotes the solute concentration 119863119909 119863119910 and 119863

119911

are the longitudinal lateral and vertical hydrodynamic dis-persion coefficients respectively 119906 represents the averagedactual steady-state pore water velocity 119905 is time 119909 119910 and 119911stands for the spatial coordinates in the longitudinal lateraland vertical directions respectively 120582 is the decay rate ofsolutes 119899 is the porosity of the porous medium 119868

119886is the

continuous injection strength119867 denotes aquifer thickness

21 Analytical Solution of Rectangular Source The generalgeometry of the rectangular source is show in Figure 1 Theorigin of the coordinate system is at the upper boundaryThepositive 119911-axis is downward The aquifer is assumed infinitein the 119909- and 119910-directions but finite in the 119911-direction witha thickness of 119867 The rectangular source of contaminationis located on top of an aquifer without considering itsthickness influence with 119909 isin [0 119909

0] 119910 isin [minus119910

0 1199100]


H

x

yV

z

minusy

a1

a2



119868119886=

1198680119891 (119905) minus119909

0lt 119909lt 119909

0 minus1199100lt 119910 lt 119910

0 119911table lt 119911 lt 119867

0 otherwise(3)


119862 (119909 119910 119911 119905)

=

1

4119899119867

times int

119905

0

119868119886(119905 minus 120591) exp (minus120582119905)


0

2radic119863119909120591

minus erfc119909 minus 119906120591 + 119909

0

2radic119863119909120591

]

times[

[

[

erfc119910 minus 1199100

2radic119863119910120591

minus erfc119910 + 1199100

2radic119863119910120591

]

]

]

times [1 + 2

infin

sum

119899=1

cos119899120587 (119911 minus 119911table)

119867

exp(minus1198631199111198992

1205872

1198672

120591)]119889120591

(4)



119890minus1199112

119889119911


1] 119910 isin



119868119886=

1198680119891 (119905)

1199092

1198862

1

+

1199102

1198862

2

le 1

0 otherwise(5)


119862 (119909 119910 119911 119905)

=

1

4119899119867radic120587119863119909

times int

119905

0

int

1198861

minus1198861

119868119886(119905 minus 120591) 119865

1(120591)

times int

120591

0

1198652(120591) 1198653(119909 minus 119902 119901) 119865

4(119901) 1198655(119901)

119889119901

radic119901

+

1198653(119909 minus 119902 120591)

radic120591

1198654(120591) 1198655(120591) 119889119901119889120591

1198651(119905) = exp [minus (119903

2+ 1205825) 119905]

1198652(119905) = 119868


11990311199032119901

119905 minus 119901

1198653(119909 119905)

= exp[ 1199061199092119863119909

minus

1199092

4119863119909119905

minus 119905 (1199031+ 120582 minus 119903

2minus 1205825+

1199062

4119863119909

)]

1198654(119905) = erfc [119896

2(119902 119910 119905)] minus erfc [119896

1(119902 119910 119905)]

1198961(119902 119910 119905) =

119910 + 119884

2radic119863119910119905

1198962(119902 119910 119905) =

119910 minus 119884

2radic119863119910119905

119884 = 1198862radic1 minus

1199022

1198862

1

1198655(119905) = 1 + 2

infin

sum

119899=1

cos119899120587 (119911 minus 119911table)

119867

exp(minus1198631199111198992

1205872

1198672

119905)

(6)



contaminant







119868119886



119863119909

2000 cm2 dayminus1 119863119909

2000 cm2 dayminus1

119863119910

500 cm2 dayminus1 119863119910

500 cm2 dayminus1

119863119911

500 cm2 dayminus1 119863119911

500 cm2 dayminus1

1199031



050 cm

119867 200 cm 1199100

25 cm1198861

100(120587)05 cm1198862

50(120587)05 cm

minus500

minus550

minus600

minus650

minus700

Z(c

m)

X (cm)

0 100 200 300 400 500 600 700 800 900 1000

040404

08

08

08

161

6 16

32





minus500

minus550

minus600

minus650

minus700

Z(c

m)

X (cm)

0 100 200 300 400 500 600 700 800 900 1000

04

0816

1632


minus500

minus550

minus600

minus650

minus700minus750

Z(c

m)

X (cm)

0

0

0

100 200 300 400 500 600 700 800 900 1000

1000

1000

04

0404

08

0808

16

16

16

32326

4



119911= 500)






00102030405060

0 100 200 300 400 500 600 700 800 900 1000x (cm)

Con

cent

ratio

n at

05H

SF = 25

SF = 30

SF = 10

SF = 15

SF = 20

(mg

cm2 )

(a)

0

02

04

06

08

0 100 200 300 400 500 600 700 800 900 1000x (cm)

SF = 25

SF = 30

SF = 10

SF = 15

SF = 20

Con

cent

ratio

n at

05H

(mg

cm2 )

(b)



(a)

1199090(cm) 119910

0(cm) 119868

119886(mgcm2) 119867 (cm)

SF = 10 50 25 100 100SF = 15 50 25 100 150SF = 20 50 25 100 200SF = 25 50 25 100 250SF = 30 50 25 100 300

(b)

1199090(cm) 119910

0(cm) 119868

119886(mgcm2) 119867 (cm)

SF = 10 150 25 1003 300SF = 15 100 25 50 300SF = 20 75 25 2003 300SF = 25 60 25 2503 300SF = 30 50 25 100 300


SF = 119867

119871119878

(7)




minus500

minus550

minus600

minus650

minus700Z(c

m)

X (cm)

0 100 200 300 400 500 600 700 800 900 1000

32

16

16

16

0808

08

040404








minus500

minus550

minus600

minus650

minus700

Z(c

m)

X (cm)

0 100 200 300 400 500 600 700 800 900 1000

32 16

16

08

08

04

04





minus500

minus550

minus600

minus650

minus700

minus750

Z(c

m)

X (cm)

0

0

0

100 200 300 400 500 600 700 800 900 1000

1000

1000

3216

16

08 08

08 04

04

04



4 Summary










10

15

20

25

30

1 2 3 4 5 6 7 8 9 10SF

CRC

Eat

z=0

x = 100 cmx = 200 cmx = 500 cm

x = 800 cmx = 900 cm

(a)

10

15

20

25

30

35

1 2 3 4 5 6 7 8 9 10SF

x = 100 cmx = 200 cmx = 500 cm

x = 800 cmx = 900 cm

CRC

Eat

z=05H

(b)





Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










H

x

yV

z

minusy

a1

a2



119868119886=

1198680119891 (119905) minus119909

0lt 119909lt 119909

0 minus1199100lt 119910 lt 119910

0 119911table lt 119911 lt 119867

0 otherwise(3)


119862 (119909 119910 119911 119905)

=

1

4119899119867

times int

119905

0

119868119886(119905 minus 120591) exp (minus120582119905)


0

2radic119863119909120591

minus erfc119909 minus 119906120591 + 119909

0

2radic119863119909120591

]

times[

[

[

erfc119910 minus 1199100

2radic119863119910120591

minus erfc119910 + 1199100

2radic119863119910120591

]

]

]

times [1 + 2

infin

sum

119899=1

cos119899120587 (119911 minus 119911table)

119867

exp(minus1198631199111198992

1205872

1198672

120591)]119889120591

(4)



119890minus1199112

119889119911


1] 119910 isin



119868119886=

1198680119891 (119905)

1199092

1198862

1

+

1199102

1198862

2

le 1

0 otherwise(5)


119862 (119909 119910 119911 119905)

=

1

4119899119867radic120587119863119909

times int

119905

0

int

1198861

minus1198861

119868119886(119905 minus 120591) 119865

1(120591)

times int

120591

0

1198652(120591) 1198653(119909 minus 119902 119901) 119865

4(119901) 1198655(119901)

119889119901

radic119901

+

1198653(119909 minus 119902 120591)

radic120591

1198654(120591) 1198655(120591) 119889119901119889120591

1198651(119905) = exp [minus (119903

2+ 1205825) 119905]

1198652(119905) = 119868


11990311199032119901

119905 minus 119901

1198653(119909 119905)

= exp[ 1199061199092119863119909

minus

1199092

4119863119909119905

minus 119905 (1199031+ 120582 minus 119903

2minus 1205825+

1199062

4119863119909

)]

1198654(119905) = erfc [119896

2(119902 119910 119905)] minus erfc [119896

1(119902 119910 119905)]

1198961(119902 119910 119905) =

119910 + 119884

2radic119863119910119905

1198962(119902 119910 119905) =

119910 minus 119884

2radic119863119910119905

119884 = 1198862radic1 minus

1199022

1198862

1

1198655(119905) = 1 + 2

infin

sum

119899=1

cos119899120587 (119911 minus 119911table)

119867

exp(minus1198631199111198992

1205872

1198672

119905)

(6)



contaminant







119868119886



119863119909

2000 cm2 dayminus1 119863119909

2000 cm2 dayminus1

119863119910

500 cm2 dayminus1 119863119910

500 cm2 dayminus1

119863119911

500 cm2 dayminus1 119863119911

500 cm2 dayminus1

1199031



050 cm

119867 200 cm 1199100

25 cm1198861

100(120587)05 cm1198862

50(120587)05 cm

minus500

minus550

minus600

minus650

minus700

Z(c

m)

X (cm)

0 100 200 300 400 500 600 700 800 900 1000

040404

08

08

08

161

6 16

32





minus500

minus550

minus600

minus650

minus700

Z(c

m)

X (cm)

0 100 200 300 400 500 600 700 800 900 1000

04

0816

1632


minus500

minus550

minus600

minus650

minus700minus750

Z(c

m)

X (cm)

0

0

0

100 200 300 400 500 600 700 800 900 1000

1000

1000

04

0404

08

0808

16

16

16

32326

4



119911= 500)






00102030405060

0 100 200 300 400 500 600 700 800 900 1000x (cm)

Con

cent

ratio

n at

05H

SF = 25

SF = 30

SF = 10

SF = 15

SF = 20

(mg

cm2 )

(a)

0

02

04

06

08

0 100 200 300 400 500 600 700 800 900 1000x (cm)

SF = 25

SF = 30

SF = 10

SF = 15

SF = 20

Con

cent

ratio

n at

05H

(mg

cm2 )

(b)



(a)

1199090(cm) 119910

0(cm) 119868

119886(mgcm2) 119867 (cm)

SF = 10 50 25 100 100SF = 15 50 25 100 150SF = 20 50 25 100 200SF = 25 50 25 100 250SF = 30 50 25 100 300

(b)

1199090(cm) 119910

0(cm) 119868

119886(mgcm2) 119867 (cm)

SF = 10 150 25 1003 300SF = 15 100 25 50 300SF = 20 75 25 2003 300SF = 25 60 25 2503 300SF = 30 50 25 100 300


SF = 119867

119871119878

(7)




minus500

minus550

minus600

minus650

minus700Z(c

m)

X (cm)

0 100 200 300 400 500 600 700 800 900 1000

32

16

16

16

0808

08

040404








minus500

minus550

minus600

minus650

minus700

Z(c

m)

X (cm)

0 100 200 300 400 500 600 700 800 900 1000

32 16

16

08

08

04

04





minus500

minus550

minus600

minus650

minus700

minus750

Z(c

m)

X (cm)

0

0

0

100 200 300 400 500 600 700 800 900 1000

1000

1000

3216

16

08 08

08 04

04

04



4 Summary










10

15

20

25

30

1 2 3 4 5 6 7 8 9 10SF

CRC

Eat

z=0

x = 100 cmx = 200 cmx = 500 cm

x = 800 cmx = 900 cm

(a)

10

15

20

25

30

35

1 2 3 4 5 6 7 8 9 10SF

x = 100 cmx = 200 cmx = 500 cm

x = 800 cmx = 900 cm

CRC

Eat

z=05H

(b)





Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119868119886



119863119909

2000 cm2 dayminus1 119863119909

2000 cm2 dayminus1

119863119910

500 cm2 dayminus1 119863119910

500 cm2 dayminus1

119863119911

500 cm2 dayminus1 119863119911

500 cm2 dayminus1

1199031



050 cm

119867 200 cm 1199100

25 cm1198861

100(120587)05 cm1198862

50(120587)05 cm

minus500

minus550

minus600

minus650

minus700

Z(c

m)

X (cm)

0 100 200 300 400 500 600 700 800 900 1000

040404

08

08

08

161

6 16

32





minus500

minus550

minus600

minus650

minus700

Z(c

m)

X (cm)

0 100 200 300 400 500 600 700 800 900 1000

04

0816

1632


minus500

minus550

minus600

minus650

minus700minus750

Z(c

m)

X (cm)

0

0

0

100 200 300 400 500 600 700 800 900 1000

1000

1000

04

0404

08

0808

16

16

16

32326

4



119911= 500)






00102030405060

0 100 200 300 400 500 600 700 800 900 1000x (cm)

Con

cent

ratio

n at

05H

SF = 25

SF = 30

SF = 10

SF = 15

SF = 20

(mg

cm2 )

(a)

0

02

04

06

08

0 100 200 300 400 500 600 700 800 900 1000x (cm)

SF = 25

SF = 30

SF = 10

SF = 15

SF = 20

Con

cent

ratio

n at

05H

(mg

cm2 )

(b)



(a)

1199090(cm) 119910

0(cm) 119868

119886(mgcm2) 119867 (cm)

SF = 10 50 25 100 100SF = 15 50 25 100 150SF = 20 50 25 100 200SF = 25 50 25 100 250SF = 30 50 25 100 300

(b)

1199090(cm) 119910

0(cm) 119868

119886(mgcm2) 119867 (cm)

SF = 10 150 25 1003 300SF = 15 100 25 50 300SF = 20 75 25 2003 300SF = 25 60 25 2503 300SF = 30 50 25 100 300


SF = 119867

119871119878

(7)




minus500

minus550

minus600

minus650

minus700Z(c

m)

X (cm)

0 100 200 300 400 500 600 700 800 900 1000

32

16

16

16

0808

08

040404








minus500

minus550

minus600

minus650

minus700

Z(c

m)

X (cm)

0 100 200 300 400 500 600 700 800 900 1000

32 16

16

08

08

04

04





minus500

minus550

minus600

minus650

minus700

minus750

Z(c

m)

X (cm)

0

0

0

100 200 300 400 500 600 700 800 900 1000

1000

1000

3216

16

08 08

08 04

04

04



4 Summary










10

15

20

25

30

1 2 3 4 5 6 7 8 9 10SF

CRC

Eat

z=0

x = 100 cmx = 200 cmx = 500 cm

x = 800 cmx = 900 cm

(a)

10

15

20

25

30

35

1 2 3 4 5 6 7 8 9 10SF

x = 100 cmx = 200 cmx = 500 cm

x = 800 cmx = 900 cm

CRC

Eat

z=05H

(b)





Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00102030405060

0 100 200 300 400 500 600 700 800 900 1000x (cm)

Con

cent

ratio

n at

05H

SF = 25

SF = 30

SF = 10

SF = 15

SF = 20

(mg

cm2 )

(a)

0

02

04

06

08

0 100 200 300 400 500 600 700 800 900 1000x (cm)

SF = 25

SF = 30

SF = 10

SF = 15

SF = 20

Con

cent

ratio

n at

05H

(mg

cm2 )

(b)



(a)

1199090(cm) 119910

0(cm) 119868

119886(mgcm2) 119867 (cm)

SF = 10 50 25 100 100SF = 15 50 25 100 150SF = 20 50 25 100 200SF = 25 50 25 100 250SF = 30 50 25 100 300

(b)

1199090(cm) 119910

0(cm) 119868

119886(mgcm2) 119867 (cm)

SF = 10 150 25 1003 300SF = 15 100 25 50 300SF = 20 75 25 2003 300SF = 25 60 25 2503 300SF = 30 50 25 100 300


SF = 119867

119871119878

(7)




minus500

minus550

minus600

minus650

minus700Z(c

m)

X (cm)

0 100 200 300 400 500 600 700 800 900 1000

32

16

16

16

0808

08

040404








minus500

minus550

minus600

minus650

minus700

Z(c

m)

X (cm)

0 100 200 300 400 500 600 700 800 900 1000

32 16

16

08

08

04

04





minus500

minus550

minus600

minus650

minus700

minus750

Z(c

m)

X (cm)

0

0

0

100 200 300 400 500 600 700 800 900 1000

1000

1000

3216

16

08 08

08 04

04

04



4 Summary










10

15

20

25

30

1 2 3 4 5 6 7 8 9 10SF

CRC

Eat

z=0

x = 100 cmx = 200 cmx = 500 cm

x = 800 cmx = 900 cm

(a)

10

15

20

25

30

35

1 2 3 4 5 6 7 8 9 10SF

x = 100 cmx = 200 cmx = 500 cm

x = 800 cmx = 900 cm

CRC

Eat

z=05H

(b)





Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus500

minus550

minus600

minus650

minus700

Z(c

m)

X (cm)

0 100 200 300 400 500 600 700 800 900 1000

32 16

16

08

08

04

04





minus500

minus550

minus600

minus650

minus700

minus750

Z(c

m)

X (cm)

0

0

0

100 200 300 400 500 600 700 800 900 1000

1000

1000

3216

16

08 08

08 04

04

04



4 Summary










10

15

20

25

30

1 2 3 4 5 6 7 8 9 10SF

CRC

Eat

z=0

x = 100 cmx = 200 cmx = 500 cm

x = 800 cmx = 900 cm

(a)

10

15

20

25

30

35

1 2 3 4 5 6 7 8 9 10SF

x = 100 cmx = 200 cmx = 500 cm

x = 800 cmx = 900 cm

CRC

Eat

z=05H

(b)





Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10

15

20

25

30

1 2 3 4 5 6 7 8 9 10SF

CRC

Eat

z=0

x = 100 cmx = 200 cmx = 500 cm

x = 800 cmx = 900 cm

(a)

10

15

20

25

30

35

1 2 3 4 5 6 7 8 9 10SF

x = 100 cmx = 200 cmx = 500 cm

x = 800 cmx = 900 cm

CRC

Eat

z=05H

(b)





Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Generalizing Source Geometry of...

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