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Table of Contents

Lecture 1: Vadose zone flow and transport modeling: An overview. 3

Lecture 2: The HYDRUS-1D software for simulating one-dimensional variably-

saturated water flow and solute transport. 33Computer Session 1: HYDRUS-1D: Infiltration of water into a one-dimensional soil

profile. 37

Lecture 3: On the characterization and measurement of the hydraulic properties ofunsaturated porous media. 43

Lecture 4: Application of the finite element method to variably-saturated water flow and

solute transport. 59

Computer Session 2: HYDRUS-1D: Water flow and solute transport in a layered soil

profile. 65

Lecture 5: Inverse modeling. 77Computer Session 3: HYDRUS-1D: One- or multi-step outflow experiment. 89

Lecture 6a: Application of the finite element method to 2D variably-saturated water flow

and solute transport. 95

Lecture 6b: HYDRUS (2D/3D) software for simulating two- and three-dimensional

variably-saturated water flow and solute transport. 99

Computer Session 4: HYDRUS (2D/3D): Subsurface line source. 109

Computer Session 5: HYDRUS (2D/3D): Furrow infiltration with a solute pulse. 119

Computer Session 6: HYDRUS (2D/3D): Flow and transport in a transect to a stream.125

Computer Session 7: HYDRUS (2D/3D): Three-Dimensional Water Flow and Solute

Transport. 135

Lecture 7: Preferential and Nonequilibrium Flow and Transport. 143Computer Session 8: HYDRUS-1D: Nonequilibrium Flow and Transport. 155

Lecture 8: Coupled movement of water, vapor, and energy. 161Computer Session 9: HYDRUS-1D: Coupled movement of water, vapor, and energy.167

Lecture 9: Multicomponent biogeochemical transport modeling using the HYDRUS

computer software packages;Introduction to the HP1 code, which was obtained

by coupling HYDRUS-1D with the PHREEQC biogechemical code. 173Computer Session 10: Application of HP1 to a simple solute transport problem

involving cation exchange. 187

Lecture 10: Other applications and future plans in HYDRUS development. 195

References 207

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Contents-1

VadoseZoneFlowandTransportModeling:AnOverview

IntroductiontoHY

DRUS-1D,itsFunctionsandWindows

ComputerSession1:Infiltrationinto1DSoilProfile

UnsaturatedSoilH

ydraulicProperties,RETCandRosett

a

NumericalSolution

sfor1DVariably-SaturatedFlowand

SoluteTransport

ComputerSession2:TransientWaterFlowandSolute

TransportinaLayeredSoilProfile

ParameterEstimationandInverseModeling

ComputerSession3:InversemodelingOne-orMulti-step-

OutflowMethod

Contents-2

2D/3DNumericsforVariably-SaturatedFlowan

dTransport

IntroductiontoHYDRUS(2D/3D),itsStructure

and

Windows

Computer

Session4:InfiltrationfromSubsurfaceSource

Computer

Session5:FurrowIrrigationwithaSolutePulse

Computer

Session6a:WaterFlowtoaStream

Computer

Session6b:SolutePlumeMigratingtoaStream

Computer

Session7:3DWaterFlowandSolute

Transport

OtherTop

ics,OpenSession

Contents-3

PreferentialandN

onequilibriumFlowandTransport

ComputerSession

8:NonequilibriumFlowandTransport

CoupledMovementofWater,VaporandEnergy

ComputerSession

9:CoupledWater,VaporandEnergy

Transport

BiogeochemicalTransport-IntroductiontoHP1(couple

d

HYDRUS-1Dand

PHREEQC)andUNSATCHEM

ComputerSession

10:ApplicationofHP1toCation

Exchange

OtherApplication

s,FuturePlans

OpenSession

VadoseZoneFlowandTransport

Modeling

AnOverview

Jirk

aimnek1andRienvanGenuch

ten2

1DepartmentofEnvironmentalSciences

UniversityofCalifornia,Riverside,CA

2DepartmentofMechanicalEngineerin

g,

FederalUniversityofRiodeJaneiro,Br

azil

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Agricultural

Applications

Precipitation

Irrigation

Runoff

Evaporation

Transpiration

RootWaterUptake

CapillaryRise

DeepDrainage

Fertigation

Pesticides

Fumigants

Colloids

Pathogens

Industria

landEnvironmentalApp

lications

Control

Planes

SourceZone

Observationwells

IndustrialP

ollution

MunicipalP

ollution

LandfillCovers

WasteRepo

sitories

Radioactive

Waste

DisposalSit

es

Remediation

BrineRelea

ses

Contaminant

Plumes

Seepageof

Wastewater

from

LandTreatment

Systems

Environmental

Applications H

illel(2003

)

EcologicalApps

CarbonStorageand

Fluxes

HeatExchangeand

Fluxes

NutrientTransport

SoilRespiration

Microbiological

Processes

EffectsofClimate

Change

RiparianSystems

Stream-Aquifer

Interactions

GoverningEquations

Variably-S

aturatedWaterFlow(Richards

Equation)

h

Kh

Kh

S

t

z

z

()

()

=

HeatMov

ement

SoluteTr

ansport(Convection-DispersionEquation)

p

w

w

C

T

T

qT

C

CST

t

z

z

z

()

()

=

s

c

cD

qc

t

t

z

z

(

)

(

)

=

5


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HYDRUSGraphical

Interface

HYDRUSGraphical

Interface

Input,Output,Meshgen

HYDRUSM

odularStructure

HYDRUS

HYDRUSMainModule

MainModule

WaterFlow

SoluteTransport

HeatTransport

InverseOptimization

EquationSolvers

SoilHydraulicProperties

PedotransferFunctions

RootUptake

HYDRU

SSoftwarePackages

WaterFlow

:

WaterFlow

:

Richardse

quationforvariably-saturatedwaterflow

Variousmodelsofsoilhydraulicproperties

Hysteresis

Sinktermtoaccountforwateruptakebyplantroots

HeatTransport:

HeatTransport:

Conductionandconvectionwithflowingwater

SoluteTran

sport:

SoluteTran

sport:

Convective

-dispersivetransportintheliquidphase,diffu

sioninthe

gaseousph

ase

Nonlinearnonequilibriumreactionsbetweenthesolidan

dliquidphases

Linearequ

ilibriumreactionsbetweentheliquidandgaseousphases

Zero-orderproduction

First-orderdegradationreactions

Physicalnonequilibriumsolutetransport

HYDRUS-R

eferences

imnek,J.,M.ejna,H.

Saito,M.Sakai,andM.Th.vanGenuchten,T

he

HYDRUS

HYDRUS--1D1DSoftwarePackageforSimulatingtheOne

One--Dimensional

Dimensional

MovementofWater,Heat,andMultipleSolutesinVariably-SaturatedMedia,

Version4.0,HYDRUSSoftwareSeries1,DepartmentofEnvironmental

Sciences,UniversityofCa

liforniaRiverside,Riverside,CA,pp.315,200

8.

imnek,J.,M.Th.vanG

enuchten,andM.ejna,TheHYDRUS

HYDRUSSoftw

are

PackageforSimulatingTwo

Two--andThree

andThree--Dimensional

DimensionalMovementofWater,

Heat,andMultipleSolute

sinVariably-SaturatedMedia,TechnicalManual

TechnicalManual,

Version1.0,PCProgress,

Prague,CzechRepublic,pp.241,2007.

imnek,J.,M.ejna,andM.Th.vanGenuchten,TheHYDRUSSoftw

are

PackageforSimulatingTwo

Two--andThree

andThree--Dimensional

DimensionalMovementofWater,

Heat,andMultipleSolute

sinVariably-SaturatedMedia,UserManual

UserManual,

Version1.0,PCProgress,

Prague,CzechRepublic,pp.161,2007.

http://www.pc-pro

gress.com/en/Default.aspx

HYDRU

S-HistoryofDevelopm

ent

Israel:Neuman[1972]-UNSAT

U.ofArizona:D

avisandNeuman[1983]

Princeton

U.:

vanGenu

chten[1978]

Agr.Univ.inWageningen:

Feddesetal.[1978]

Vogel[1987]-SWMII

MIT:Celia

etal.[1990]

US

SL-SWMS-2D

im

neketal.[1992]

USSL-HYDRUS-2D(1.0)

imneketal.[1996]

USSL-CHAIN-2D

imnekandvan

Genuchten[1994]

IGWMC-HYD

RUS

HYD

RUS--2D2D(2.0)

imneketal.[1999]

UCR,PC-ProgressHYDR

US(2D/3D)

HYDR

US(2D/3D)

imneketal.[2007]

USSL-SWMS-3D

imneketal.[1995]

6


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CapillaryRise

Whenasmallcylindric

alglasscapillarytubeisinsertedinawater

reservoiropentoatmosphere,waterwillriseupwardinthetube.

Lap

lace

Lap

lace

Equa

tion:

Equa

tion:

2

cos

H

gR

=

surfacetension

contactangle

bulkdensityof

water

ggravitationalacceleration

Rcapillaryradiu

s

Hcapillaryrise

Retention

Curve

Soil-waterch

aracteristiccurve

-characterizestheenergystatusofthesoilwater

RetentionCu

rve

Soil-watercharacteristiccurve

Characterizestheenergy

statusofthesoilwater

0100

200

300

400

500

0

0

.1

0.2

0.3

0.4

0.5

WaterContent[-]

|Pressurehead|[cm]

Loam

Sand

Clay

WaterF

lowinSoils

Groundwaterflow(DarcysLaw)

(

)

s

s

P

z

dH

q

K

K

L

dz

+

=

=

Unsaturatedwaterflow(Darcy-Buckingh

amLaw)

(

)

()

()

h

z

dH

q

Kh

Kh

L

dz

+

=

=

H-sumofthematric(h)andgravitational(z)head

8


9/213

WaterFlow-

RichardsEquation

Thegoverningflowequationforone-dimensionalisothermal

Darcianflowinavaria

bly-saturatedisotropicrigidporous

medium:

-volumetricwatercontent[L3L-3]

h

-pressurehead[L]

K

-unsaturatedhydraulicconductivity[LT-1]

z

-vertical

coordinatepositiveupward[L]

t

-time[T]

S

-rootwateruptake[T-1]

()

()

()

()

h

h

=

Kh

+Kh

Sh

t

z

z

()

()

h

q

=

Sh

t

z

WaterF

low-RichardsEquation

()

()

()

A

A

ij

iz

i

j

h

h

=

Kh

K

+K

Sh

t

x

x

Thegovernin

gflowequationfortwo-dimensionalis

othermal

Darcianflow

inavariably-saturatedisotropicrigid

porous

medium:

-volumetricwatercontent[L3L-3]

h

-pressurehead[L]

K

-unsaturatedhydraulicconductivity[LT-1]

KijA-componentsofaanisotropytensor[-]

xi

-spatialcoordinates[L]

z

-verticalcoordinatepositiveupward[L]

t

-time[T]

S

-rootwateruptake[T-1]

RetentionCu

rve,

(h)

Soil-watercharacteristiccurve

Characterizestheenergy

statusofthesoilwater

0100

200

300

400

500

0

0

.1

0.2

0.3

0.4

0.5

WaterContent[-]

|Pressurehead|[cm]

Loam

Sand

Clay

Hydrau

licConductivity,K()

-characteriz

esresistanceofporousmediatowater

flow

-10-8-6-4-2024

0

0.1

0.2

0.3

0.4

0.5

WaterContent[-]

log(HydraulicConductivity)[cm/d]

Loam

Sand

Clay

9


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HydraulicConductivity,K()

Unsa

tura

tedhy

drau

liccon

duc

tiv

ity

decreasesasvo

lume

tricwa

terconten

t

decreases:

Cross-sec

tiona

lareaof

wa

ter

flow

decreases

Tortuos

ity

increases

Drag

forces

increase

Thus,

theunsa

tura

tedhyd

rau

liccon

duc

tiv

ity

isanon

linear

func

tionof

an

dh

.

Hydrau

licConductivity,K(h)

-characteriz

esresistanceofporousmediatowater

flow

-10-8-6-4-2024

0

1

2

3

4

5

log(|PressureHead|[cm])

log(HydraulicConductivity[cm/d])

Loam

Sand

Clay

RetentionCurve

BrooksandCorey[1964]:

vanGenuchten[1980]:

Kosugi[1996]:

s-saturatedwatercontent[-]

r-residualwaterconten

t[-]

n,h0,-empiricalpar

ameters[L-1],[-],[L],[-]

Se-effectivewatercontent[-]

11/

1

(1

)

e

n

n

S

h

=

+

-

-1/

|

|1

-1/

n

e

h

h

S

h

q=?

Tiledrains

Tiledrains

Bounda

ryConditions(System-Dependent)

Imperm

eablelayer

Soilsurface

Tiledrain

Groun

dwatertable

1DsoilprofileE

P

Drain

age

TheHYDRU

SSoftwarePackages

Variably-SaturatedFlow(RichardsEq.)

RootWaterUptake(waterandsalinitystress)


SolutesTransp

ort(decaychains,ADE)

-NonlinearS

orption

-ChemicalN

onequilibrium

-PhysicalNo

nequilibrium

HeatTranspor

t

PedotransferF

unctions(hydraulicproperties)

ParameterEstimation

InteractiveGraphics-BasedInterface

HYDRUS(2D/3D)andAdditionalModules

RootW

aterUptake

Feddesetal.[

1978]

01

Tp

=1mmd

-1

PressureHead,h

[L]

h1

h2

h3high

h3low

h4

Tp

=5mmd

-1

StressResponseFunction,[-]

01

Tp

=1mmd

-1

PressureHead,h

[L]

h1

h2

h3high

h3low

h4

Tp

=5mmd

-1

StressResponseFunction,[-]

(

)

()

(

)

()

(

)

(

)()

p

p

p

p

S

z,t=bzT

Sz,t=

hS

z,t

h

bzT

=

b

normalizedwateruptakedistribution[L-1]

Sp

potentialrootwateruptake[T-1]

S

actualrootwateruptake[T-1]

Tp

potentialtranspiration[LT-1]

stressresponsefunction[-]

x

z

R

S

Ta

Lt

b(x,z

)

x

z

R

S

Ta

Lt

b(x,z

)

13


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StressResponseFunctions

Waterandsolutestr

ess:

1

2

50

50

50

1

(,)

1 1

1

(,

)

1

1

p

p

p

hh

h

hh

hh

h

h h

h

=

+

+

=

+

+


h

pressurehead[L

]

hf

osmotichead[L]

h50

pressureheadat

whichwaterextractionrateisreducedby50%

[L]

h50

dittoforosmotic

head[L]

p1,p2

experimentalcon

stants[-](=3)

Transp

irationRates

(,)

(,)

(,)()

R

R

R

p

p

L

a

p

L

L

T

S

hzdz

T

Shzdz

T

ahzbzdz

=

=

=

b

normalizedwateruptakedistribution[L-1]


Sppotentialrootwateruptake[T-1]

S

actualrootwateruptake[T-1]

Tp

potentialtranspiration[LT-1]

Ta

actualtranspiration[LT-1]

SpatialRootD

istributionFunction

(

)

*

*

,

1

1

z

r

m

m

m

m

p

p

z

z

x

x

Z

X

z

x

bxz

Z

X

e

+

=

(

)

*

*

*

,,

1

1

1

y

x

z

m

m

m

p

p

p

x

x

y

y

z

z

X

Y

Z

m

m

m

x

y

z

bxyz

e

X

Y

Z

+

+

=

(Vrugtetal.,2001)

RootG

rowth

0

0

0

()

()

()

(

)

R

m

rt

m

L

t

Lft

L

ft

L

L

Le

=

=

+

LR

rootingdepth[L]

L0

initialrootingdepth[L]

Lm

maxim

umrootingdepth[L]

f

rootgr

owthcoefficient(Verhulst-Pearllogist

icfunction)

r

growth

rate[T-1]

14


15/213

TheHYDRU

SSoftwarePackages

Variably-Satur

atedFlow(RichardsEq.)


SolutesTransport(decaychains,ADE)

SolutesTransp

ort(decaychains,ADE)

--NonlinearS

orption

NonlinearS

orption

--ChemicalN

onequilibrium

ChemicalN

onequilibrium

--PhysicalNo

nequilibrium

PhysicalNo

nequilibrium

HeatTransport

PedotransferF

unctions(hydraulicproperties)

ParameterEst

imation



SoluteT

ransport-Convection-Dispersio

nEquation

One-dimensionalchemicaltransportduringtransientwater

flowinavariablysaturatedrigidporousmedium

c

-solutionconcentration[ML-3]

s

-adsorbedconcentration[MM-1]

-watercontent[L3L-3]

-soilbulkdensity[ML-3]

D

-dispersioncoefficient[L2T-1]

q

-volumetricflux[LT-1]

-rateconstantrepresentingreactions[ML-3T-1]

()

(

)

c

s

cD

qc

t

t

z

z

=

GeneralStructureoftheSystemofSolutes

HYDRU

SSoluteTransport

Radion

uclides:238Pu->234U->230Th->226Ra

Nitroge

n:(NH2)2CO->NH4+->NO2-->NO3-

Pesticid

es:aldicarb(oxime)->sulfone(sulfoneoxime)->sulfoxide(sulfoxide

oxime)

Chlorin

atedHydrocarbons:PCE->TCE->c-DCE->VC

->ethylene

Pharmaceuticals,hormones:Estrogen(17bEstradiol->Estrone->Estriol),

Testosterone

Explosives:TNT(->4HADNT->4ADNT->TAT),RDXH

MX

Transportofsingleions

Transportofmultipleions(sequentialfirst-orde

rdecay)

15


16/213

GoverningSo

luteTransportEquatio

ns

(

)

(

)

c

s

cD

qc

t

t

z

z

=

'

'

'

'

,

,

,

,

,

,

1

1

'

'

1

1

1

1

,

,

,

,

(

)

(

)

(

)

(2,)

w

g

k

k

k

k

k

k

k

k

wk

wk

k

sk

sk

k

gk

gk

k

w,k

k

s,k

k

g,k

k

wk

sk

gk

rk

s

c

s

ag

c

g

qc

+

+

=

D

+

aD

-

-

t

t

t

z

z

z

z

z

-

+

c-

+

s-

+

ag+

c

+

s

-

ag

+

+

+

a

Sc

k

n

w,s,g

subscriptscorrespondingwiththeliquid,solidand

gaseousphases,respectively

c,s,g

concentratio

ninliquid,solid,andgaseousphase,

respectively

GoverningSoluteTransportEq

uations

qi

volumetricflux[LT-1]

soilbulkdensity[ML-3]

a

aircontent[L3L-3]

S

sinkterminthewaterflowequation[T-1]

cr

concentrationofthesinkterm[ML-3]

Dw,D

g

dispersioncoefficientsfortheliquidandgaseousph

ase[L2T-1],

respectively

k

subscriptrepresentingthekthchainnumber

w,s,

g

first-orderrateconstantsforsolutesintheliquid,solid,and

gaseousphases[T-1],respectively

w,s,

g

zero-orderrateconstantsfortheliquid[ML-3T-1],solid[T-1],and

gaseousphases[ML-3T-1],respectively

w',

s',

g'first-orderrateconstantsforsolutesintheliquid,so

lidand

gaseousphases[T-1],respectively;theserateconstantsprovide

connectionsbetweentheindividualchainspecies.

ns

numberofsolutesinvolvedinthechainreaction

SoluteTrans

port-BoundaryConditions

First

First--type

type(orDirichlettype)boundaryconditions

Third

Third--type

type(Cauchytype)boundaryconditions

Second

Second--type

type(N

eumanntype)boundaryconditions

0

for(

)

ij

i

N

jc

D

n=

x,z

x

0

for(

)

ij

i

ii

ii

C

jc

-D

n

+qnc=qnc

x,z

x

0

(

)

(

)

for(

)

D

cx,z,t=

cx,z,t

x,z

SoluteTransportDispersionCo

efficient

Bear

Bear[1972]:

[1972]:

7/3 2 s

=

d

D=

|q|+D

Dd

-ionicormoleculardiffusio

ncoefficient

infreewater[L2T-1]

-tortuosityfactor[-]

-longitudinaldispersivity[L]

-watercontent[L3L-3]

q

-Darcysflux[LT-1]

MillingtonandQuirk[1961]:

s

-saturatedwatercontent[-]

16


17/213

SoluteTrans

port-DispersionCoefficien

t

Bear

Bear[1972]:

[1972]:

(

)ji

ij

T

ij

L

T

d

ij

qq

D=D|

q|

+D-D

+

D

|q|

Dd

-ionicormoleculardiffusioncoefficientinfreewater[L2T-1]

-tortuosityfactor

[-]

ij

-Kroneckerdelta

function(ij=1ifi=j,and

ij=0otherwise)

DL,DT-longitudinal

andtransversedispersivities[L]

2

2

2

2

(

)

x

z

xx

L

T

d

z

x

zz

L

T

d

x

z

xz

L

T

q

q

D

=D

+D

+

D

|q|

|q|

q

q

D

=D

+D

+

D

|q|

|q| q

q

D

=D-D

|q|

Disper

sivityasaFunctionofScale

Gelharetal.(1985)

TheHYDRUSSoftwarePackages




-NonlinearSorption

-ChemicalN

onequilibrium

-PhysicalNonequilibrium

HeatTransport

PedotransferFunctions(hydraulicproperties)

ParameterEstimation

InteractiveGr

aphics-BasedInterface


Convec

tion-DispersionEquation

Linea

rAdsorption

1

d

d

K

s

Kc

R

=

=

+

ij

i

i

j

Rc

c

D

qc

t

x

x

=

+

Kd

-distributioncoefficient[L3M-1]

R

-retardationfactor[-]

s

-solidphaseconcentration[MM-1]

c

-liquidphaseconcentration[ML-3]

17


18/213

Convection-DispersionEquation

TransientTr

ansport(2D)

Steady-State

Transport(1D)

ij

i

i

j

Rc

c

D

qc

t

x

x

=

+

22

22

R

R

c

c

c

R

D

v

t

z

z

c

c

c

D

v

t

z

z

=

=

DR

-retarded

dispersioncoefficient[L2T-1]

vR-retarded

velocity[LT-1]

Nonline

arEquilibriumAdsorp

tion

Equation

Model

Reference

s=kc+k

1

2

Linear

LapidusandAmundson[195

2]

Lindstrometal.[1967]

2

1

k

s=

kc

Freundlich

Freundlich[1909]

12

1kc

s=

+

k

c

Langmuir

Langmuir[1918]

3

3

12

1

k

k

kc

s=

+

k

c

Freundlich-Langmuir

Sips[1950]

1

3

2

4

1

1

kc

k

c

s=

+

+kc

+kc

DoubleLangmuir

ShapiroandFried[1959]

2

3

1

k

/k

c

s=

kc

ExtendedFreundlich

Sibbesen[1981]

1

2

3

1

kc

s=

+kc+k

c

Gunary

Gunary[191970]

2

1

3

k

s=k

-k

c

Fitter-Sutton

FitterandSutton[1975]

4

3

1

2

{1[1

]}k

k

s=k

-

+k

c

Barry

Barry[1992]

2

1

ln(

)

RT

s=

kc

k

Temkin

BacheandWilliams[1971]

1

2

exp(2

)

s=kc

-ks


vanGenuchtenetal.[1974]

s s

c

c

k

c

c

k

c

c

T

T

T

=

+

[

(

)exp{

(

)}]

1

2

2

modifiedKielland

LaiandJurinak[1971]

NonlinearEquilibriumAdsorption

HYDRUSassumesnonequilibriuminteractionsbetween

the

solution(c)andad

sorbed(s)concentrations,andequilibrium

interactionbetwee

nthesolution(c)andgaseous(g)

concentrationsofthesoluteinthesoilsystem.

Liquid-Solid:

ageneralizednonlinear

nonlinear

(Freundlich-Langmuir)empiricalequation

ks,,empirica

lconstants

1skc

s=

+c

TheHY

DRUSSoftwarePackages

Variab

ly-SaturatedFlow(RichardsEq.)

RootW

aterUptake(waterandsalinity

stress)


-Non

linearSorption

-Che

micalNonequilibrium

-Phy

sicalNonequilibrium

HeatT

ransport

PedotransferFunctions(hydraulicprop

erties)

ParameterEstimation


HYDR

US(2D/3D)andAdditionalModules

18


19/213

Non-EquilibriumAdsorptionEquations

Equation

Model

Reference

1

2

(

)

s=

kc+

k

-s

t

Linear

LapidusandAmundson[1952]

Oddsonetal.[1970]

2

1

(

)

k

s=

k

-s

c

t

Fr

eundlich

HornsbyandDavidson[1973]

vanGenuchtenetal.[1974]

12

1

s

kc

=

-s

t

+kc

La

ngmuir

Hendricks[1972]

3

3

12

1

k

k

s

kc

=

-s

t

+kc

Fr

eundlich-

La

ngmuir

imunekandvanGenuchten[1994]

1

(

)sinh

T

T

T

i

s

s

s

=

s

-s

k

t

s

s

FavaandEyring[42]

2

1

2

exp(

){

exp(2

)

}

s=

ks

kc

-ks

-s

t


1

2

k

k

s=

c

s

t

LeenheerandAhlrichs[1971]

Enfieldetal.[1976]

NonequilibriumTwo-SiteAdsorptionModel

se

Ty

pe-1siteswithinstantaneoussorption

sk

Ty

pe-2siteswithkineticsorption

f

fra

ctionofexchangesitesassumedtobeat

equilibrium

e

k

s=s+s

(1

)

(1

)

1

k

s

k

ks

s

s

kc

=

-

-

-

+

-f

f

s

s

t

+

c

es

s

=f

t

t

Two-SiteChemicalNonequilibriumTranspo

rt

,

[(1

)

]

k

k

k

d

sk

s=

-fKc-s-

s

t

(

)

(

)

[(1

)

]

d

k

d

d

l

s,e

c

+fK

c=

D

-qc-

t

z

z

-fK

c-s-

c-fK

c

Linearsorption:

Mo

bile

Co

llo

ids,

Cc

Stra

ine

dCo

llo

ids,

Sc

str

Attac

he

dC

ollo

ids,

Sc

att

Air-Wa

ter

Interface

Co

llo

ids,

c

kac

kdc

kstrk

aca

kdca

aca

s

str

s

Air

Wa

ter

So

lid

Colloid,Virus,andBacteriaTransport

19


20/213

Colloid,Virus,andBacteriaTransport

2

1

2

2

s

s

c

c

c

D

v

t

t

t

x

x

+

+

=

1

1

a

t

d

s

k

c

ks

t

=

max m

ax

max

1

t

s

s

s

s

s

=

=

kstr

-straining[T-1]

ka

-deposition(attachment)coefficient[T-1]

kd

-entrainment(detachment)coefficient[T-1]

-reductionofattachmentcoefficientduetoblockageofsorptionsites

2

str

str

s

k

c

t

=

0

c

str

c

d

x

x

d

+

=

Attachment/Detach

ment

Attachment/Detach

ment

Straining

Straining

TheHY

DRUSSoftwarePacka

ges


RootW

aterUptake(waterandsalinitystress)

Solutes

Transport(decaychains,ADE)

-NonlinearSorption

-ChemicalNonequilibrium


HeatTransport


ParameterEstimation

Interac

tiveGraphics-BasedInterface

HYDRUS(2D/3D)andAdditionalModu

les

Two-RegionPhy

sicalNonequilibriumTransp

ort

(

)

,

,

(

)(

)

m

m

d

m

m

m

m

m

im

m

wm

d

ms

m

c

+fk

c=

D

-qc

-

c-c

-

+fk

c

t

z

z

,

,

[

(1

)

]

(

)[

(

)

]

im

im

d

m

im

im

wim

d

simim

c

+

-f

k

=

c-c

-

+1-f

k

c

t

Two-RegionPhysicalNonequilibriumTransport

20


21/213

Interaction

AmongPhases

Liquid-Gas:

alinear

linearrelation(HenrysLa

w)

kg

emp

iricalconstantequalto(KHRTA)-1

KH

Hen

ry'sLawconstant

R

universalgasconstant

TA

absolutetemperature

g

g=kc

Volatilization

(

)

(

)

(

)

w

a

w

a

ij

ij

i

i

i

j

j

s

c

ag

c

g

+

+

=

D

+aD

-qc-qg+

t

t

t

x

x

x

g

g

kc

=

Steady-State(anewretardationfactorandeffectiv

ediffusion

coefficient):

2

2

1

w

a

g

i

i

g

w

a

d

ij

ij

g

i

j

i

E

E

ij

i

i

j

i

a

k

q

qk

K

c

a

c

c

+

=

D

D

k

-

t

xx

x

c

c

c

R

=D

-q

t

xx

x

+

+

+

TemperatureDependenceofTransportand

ReactionCoefficients

ar,aT

coefficientvaluesatareferenceabsolutetemperatu

re,

TrA,andabsolutetemperature,T

A,respectively

E

activationen

ergyofthereactionorprocess

(

)

exp

A

A r

T

r

A

A r

ET

-T

a=a

RTT

Mostofthediffusion

(Dw,Dg),distribution(ks,

kg),andreaction

rate(w,s,g,w

',s',

g',w,s,and

g)coefficientsare

stronglytemperature

dependent.HYDRUSassumesthatth

is

dependencycanbeexpressedbyanArrheniusequation

[StummandMorgan

,1981].

TheHYDRUSSoftwarePackages


RootW

aterUptake(waterandsalinitys

tress)

Solutes


-NonlinearSorption


-Phys

icalNonequilibrium

HeatTr

ansport

Pedotra

nsferFunctions(hydraulicproperties)

Parame

terEstimation


HYDRU

S(2D/3D)andAdditionalModu

les

21


22/213

GoverningH

eatTransportEquation

Sophocleous[197

9]:

ij()

apparentther

malconductivityofthesoil

C(),C

w

volumetriche

atcapacitiesoftheporousmediumandthe

liquidphase,respectively

deVries[1963]:

volumetricfraction

n,o,g,wsubscriptsrep

resentingsolidphase,organicmatter,gaseous

phase,andliq

uidphase,respectively.

()

n

n

o

o

w

g

C

=C

+C

+C

+Ca

()

()

ij

w

i

i

j

i

T

T

T

C

=

-Cq

t

x

x

x

Therma

lConductivity

0(

)thermalconductivityoftheporousmedium(solid

plu

swater)intheabsenceofflow

L,T

longitudinalandtransversethermaldispe

rsivities,

respectively

ChungandHorton[1987]

b1,b2,b3

empiricalparameters

0

()

(

)

(

)

i

j

ij

T

w

ij

L

T

w

ij

qq

=

C|q|

+

-

C

+

|q|

0.5

0

1

2

3

()

w

w

=b+b

+b

TheHYDRUS

SoftwarePackages


RootWaterUp

take(waterandsalinitystress)


-NonlinearSorption



HeatTransport


ParameterEstimation

InteractiveGra

phics-BasedInterface


PTFsbyCarselandParrish(1

988)

Averagevalues

ofselectedsoilwaterretentionparametersfor12major

soiltexturalgroups

22


23/213

Pedotransfer

Functions:Rosetta

Schaapetal.(2001)

Model

InputData

TXT

TexturalClass

SSC

Sand,Silt,Clay%

SSCBD

Same+BulkDensity

SSCBD+

33

SSCBD+

at33kPa

SSCBD+

33+

1500

Same+

at1500kPa

Pedotra

nsferFunctions:Rosetta

TexturalCla

ssAverages:Rosetta

Texturalclass

r

s

n

Ks

[L3L-3]

[L3L-3]

[cm-1]

[-]

[cmd-1]

Sand

0.053

0.375

0.035

3.18

643.

LoamySand

0.049

0.390

0.035

1.75

105.

SandyLoam

0.039

0.387

0.027

1.45

38.2

Loam

0.061

0.399

0.011

1.47

12.0

Silt

0.050

0.489

0.007

1.68

43.7

SiltyLoam

0.065

0.439

0.005

1.66

18.3

SandyClayLoam

0.063

0.384

0.021

1.33

13.2

ClayLoam

0.079

0.442

0.016

1.41

8.18

SiltyClayLoam

0.090

0.482

0.008

1.52

11.1

SandyClay

0.117

0.385

0.033

1.21

11.4

SiltyClay

0.111

0.481

0.016

1.32

9.61

Clay

0.098

0.459

0.015

1.25

14.8

SoilhydraulicparametersfortheanalyticalfunctionsofvanGenuchten(1980)forthe

twelvetexturalclassesofthe

USDAtexturaltriangleobtainedwiththeRosettaligh

t

program(Schaapetal.,2001

).

TheHY

DRUSSoftwarePacka

ges


RootW

aterUptake(waterstress)

Solutes


-NonlinearSorption



HeatTransport


ParameterEstimation

Interac



les

23


24/213

ParameterEstimationwithHYDRUS

ParameterEst

imation:

ParameterEst

imation:

-Soilhydraulicparameters

-Solute

transportandreactionparameters

-Heatt

ransportparameters

Sequence:

Sequence:-Independently

-Simultaneously

-Sequentially

Method:

Method: -Marquardt-Levenbergoptimization

Objective

FunctionforInversePro

blems

2

*

,

1

1

2

*

1

1

2

*

1

(,,

)

[

(,)-

(,,)]

[

()-

(,)]

[

-

]

q

qj

q

pj

b

m

n

j

ij

j

i

j

i

j

i

m

nj

ij

j

i

j

i

j

i

n

j

j

j

j

bqp

v

w

gxt

gxtb

v

w

p

p

b

b

b

v

=

=

=

=

=

=

1stterm:

1stterm:

de

viationsbetweenmeasuredandcalculatedspa

ce-time

va

riables

2ndterm:

2ndterm:

differencesbetweenindependentlymeasured,pj*,andpredicted,

pj,soilhydraulicproperties

3rdterm:

3rdterm:

pe

naltyfunctionfordeviationsbetweenpriorkn

owledgeofthe

soilhydraulicparameters,bj*,andtheirfinalestimates,bj.

Formulation

oftheInverseProblem

Theproblemcanbesimplifiedinto

theWeightedLeast

theWeightedLeast--SquaresProblem

SquaresProblem

wi-weightofapar

ticularmeasuredpoint

2

*

1

()

n

i

i

i

i

w

q

q

=

=

TheHY

DRUSSoftwarePacka

ges


RootW

aterUptake(waterstress)

Solutes


-NonlinearSorption



HeatTransport


ParameterEstimation

Interac



les

24


25/213


26/213

Cut-offWall

FiniteElementMesh

Cut-offWall

SolutePlume

PlumeMovementin

a

TransectwithStream

26


27/213

TheHYDRU

SSoftwarePackages


RootWaterU

ptake(waterstress)


-NonlinearSorption

-ChemicalN

onequilibrium


HeatTransport


ParameterEstimation

InteractiveGr

aphics-BasedInterface

HYDRUS(2D

/3D)andAdditionalModules

HYDRU

S(2D/3D)NewFeatu

res

Waterflowinadual-porositysystemallowingforpreferentialflow

infractures

ormacroporeswhilestoringwaterinthematrix.

Rootwateruptakewithcompensation.

Spatialroot

distributionfunctionsofVrugtetal.(2002

).

SoilhydraulicpropertymodelsofKosugi(1995)andD

urner(1994).

Transportofviruses,colloids,and/orbacteriausingan

attachment/detachmentmodel,straining,filtrationtheory,and

blockingfun

ctions.

Aconstructedwetlandmodule(onlyin2D).

ThehysteresismodelofLenhardetal.(1991)toelimin

atepumping

bykeepingt

rackofhistoricalreversalpoints.

Newprintm

anagementoptions.

Dynamic,sy

stem-dependentboundaryconditions.

Flowingpar

ticlesintwo-dimensionalapplications.

27


28/213

HYDRUS(2D

/3D)NewFeatures

CompletelynewGUI

basedonHi-End3Dgraphicslibraries.

MDIarchitecturem

ultipleprojectsandmultipleviews.

Neworganizationofg

eometricobjects.

Navigatorwindowwithanobjectexplorer.

Manynewfunctionsimprovingtheuser-friendliness,suchasdrag-

and-dropandcontextsensitivepop-upmenus.

Improvedinteractive

toolsforgraphicalinput.

SavingCross-SectionsandMesh-Linesforchartswithinagiven

project.

DisplayOptionsall

colors,linestyles,fontsandotherparameters

ofgraphicalobjectscanbecustomized.

Extendedprintoptions.

Extendedinformation

intheProjectManager(includingproject

preview).

Manyadditionalimprovements.

Geochem

icalModeling

HYDRUS

HYDRUS--1D1D(imneketal.,1998)

-variablysatura

tedwaterflow

-heattransport

-rootwateruptake

-solutetranspor

t

UNSATCHE

M

UNSATCHE

M(imneketal.,1996)

-carbondioxide

transport

-majorionchem

istry

-cationexcha

nge

-precipitation-dissolution(instantaneousandkinetic)

-complexatio

n

HYDRUS-1D

+UNSATCHEM

HYDRU

S-1D+UNSATCHEM

H4SiO4,

H3SiO4-,H2SiO42

-

3

Silicaspecies

6

PCO2,

H2CO3

*,

CO3

2-,HCO

3-,H+,

OH-,

H2O

7

CO2-H2O

species

5

Ca,

Mg,

Na,

K

4

Sorbeds

pecies

(exchangeable)

4

CaCO3,

CaSO4

2H2O,

Mg

CO3

3H2O,

Mg5(CO3)4(OH)2

4H2O,

Mg2Si3O7.5

(OH)

3H2O,CaMg(CO3)2

6

Precipita

ted

species

3

CaCO3o,

CaHCO3+,

CaSO4o,

MgCO3o,

MgHCO3+,

MgSO4o,

NaCO

3-,NaHCO3o,

NaSO4-,KSO4-

10

Complex

ed

species

2

Ca2+,

Mg2+,

Na+,

K+,

SO4

2-,Cl-,

NO3-

7

Aqueous

components

1Kineticreactio

ns:calciteprecipitation/dissolution,dolomitedissolution

Activitycoefficients:extendedDebye-Hckelequations,Pitzerexpressions

28


29/213

LysimeterStu

dy

Gonalvesetal.(2006)

HYDRUS

HYDRUS

--1D1D[[imimnekneketal.,1998]:

etal.,1998]:

VariablySaturatedWaterFlow

SoluteTransport

Heattransport

Rootwateruptake

PHREEQ

C

PHREEQ

C[ParkhurstandAppelo,1999]:

[ParkhurstandAppelo,1999]:

Availablechemicalreactions:

Aqueouscom

plexation

Redoxreactions

Ionexchange(Gains-Thomas)

Surfacecom

plexationdiffusedouble-layermodelandnon-

electrostatic

surfacecomplexationmodel

Precipitation/dissolution

Chemicalkinetics

Biologicalre

actions

HP1-CoupledHYDRUS-1DandPHREEQC

HYDRUS-1D

GUIforHP1

HP1examples

Transportofheavymetals(Zn2+,Pb2+,andCd

2+)

subjecttomultiplecationexchange

Transportwithmineraldissolutionofamorph

ousSiO2

andgibbsite(Al(OH)3)

Heavym

etaltransportinamediumwithapH

-

dependentcationexchangecomplex

Infiltrationofahyperalkalinesolutioninacla

ysample

(thisexampleconsiderskineticprecipitation-d

issolution

ofkaolinite,illite,quartz,calcite,dolomite,gypsum,

hydrota

lcite,andsepiolite)

Long-te

rmtransientflowandtransportofmajor

cations(Na+,K

+,Ca2+,andMg2+)andheavym

etals

(Cd2+,Z

n2+,andPb2+)inasoilprofile.

Kinetic

biodegradationofNTA(biomass,coba

lt)

29


30/213


31/213

Mathematica

lVerification

Question:Whena

llnonlinear,coupled,and/ortransient

processesareintro

ducedintothemodel,howdoweknow

thatthecomputer

codegivesaccuratenumericalresults?

ApproximateTests:

Massbalanceerrors

Self-consistency

(differentgridsandtimesteps)

Comparisonwithothercodes

Mathem

aticalVerification

Doestheco

mputercode(model)provideanaccurate

solutionofthegovernmentPDEsfordifferen

tinitialand

boundaryc

onditionswithintherangeofpossiblemodel

parametervalues?

Verificationofpartsofthecode:

Comparewithanalyticalsolutions(steady-stateflow)

Comparewithlinearizedsolutions(simplifiedconstitutiverelationships)

Steady-state

solutions

Homogeneou

smedia

Simplifiedin

itialandboundaryconditions

ModelValidation

Doesthemodel(i.e.,theequationsembeddedinthe

code)correctly

representtheactualprocesses?

Amodelisasimplified

representationoftherealsystemorproc

ess

Approximationsarise

becauseof

Incorporationofa

limitednumberofprocesses

Limitedunderstandingoftheactualprocess

Inabilitytotranslateobservedprocessesintousablemathematics

(howtoquantifythings?)

Inconsistencyofsm

all-scaleheterogeneitieswithnumericalg

rid

(effectiveparameters)

31


32/213

32


33/213

TheHYDRUS

-1DSoftwareforSimulating

One-Dimen

sionalVariably-Saturated

WaterFlowandSoluteTransport

Jirkaim

nek,RienvanGenuchten,

andMiroslavejna

DepartmentofEnv

ironmentalSciences,UniversityofCalifornia

Riverside,CA

FederalUniversityofRiodeJaneiro,RiodeJaneiro,Brazil

PC-Prog

ress,Ltd.,Prague,CzechRepublic

HYDR

US-1D-Functions

D

ataandProjectManagement

D

ataPre-Processing

-Inputparameter

-Transportdomaindesign

-Finiteelementgridgenerator

-Initialandboundarycondition

s

C

omputations

D

ataPost-Processing

-Graphicaloutput

-ASCIIoutput

HYDRUS-1D

-ModelStructure

HYDRUS1D

HYDRUS1D

-majormodule

POSITION

POSITION

-projectmanager

PROFILE

PROFILE

-flowdomaindesign

-finiteelementgenerator

-initialconditionsanddomain

properties

HYDRUS

HYDRUS

-waterflowandsolutetransport

calculations

HYDRU

S-1D-FortranApplic

ation

Water,Solute,andHeatMovement

Water,Solute,andHeatMovement:

-one-dim

ensionalporousmedia

RichardsE

quation

RichardsE

quation-saturated-unsaturatedwaterflow

-porousmedia:

-unsaturated

-partiallysaturated

-fullysaturated

-sinkterm-wateruptakebyplantroots

-waterstress

-salinitystress

SoilHydra

ulicProperties

SoilHydra

ulicProperties

-vanGen

uchten[1980]

-Brooksa

ndCorey[1964]

-modifiedvanGenuchtentypefunctions[VogelandCislerov

a,1989]

-dual-porositymodelofDurner[1994]

Hysteresis

Hysteresis

-Scottetal.[1983],

KoolandParker[1988]

RootGrow

th

RootGrow

th-logisticgrowthfunction

33


34/213

HYDRUS-1D

-FortranApplication

SoluteTransport

SoluteTransport-convection-dispersionequation

-liquid,solid,andgaseousphase

-nonlinearadso

rption[Freundlich-Langmuirequations]

-nonequilibirum

[two-sitesorptionmodel,mobile-immobilewater]

-HenrysLaw

-convectionand

dispersioninliquidphase,diffusioningaseousphase

-zero-orderpro

ductioninallthreephases

-first-orderdeg

radationinallthreephases

-chainreactions

HeatTransport

HeatTransport-co

nvection-dispersionequation

-heatconductio

n

-convection

NonuniformSoils

Scaling

ScalingProcedurefo

rHeterogeneousSoils

HYDRUS-1D-FortranApplication

ThreeStabilizingOptions

StabilizingOptionstoavoidoscillationinthen

umerical

solutionof

thesolutetransportequation:

-upstreamweighting

-artificialdispersion

-performanceindex

WaterFlowBoundaryConditions:

WaterFlowBoundaryConditions:

-prescribedheadandflux

-atmosph

ericconditions

-seepageface

-freedrainage

-deepdrainage

-horizont

aldrains

Flowand

Transport:

Flowand

Transport:

-verticaldirection

-horizontaldirection

-generallyinclineddirection

HYDRUS-1D

MajorModule

HYDRU

S-1D-MajorModule

Mainprog

ramunit

Mainprog

ramunitofthesystem

Controlse

xecutionoftheprogram

Determines

whichotheroptionalmodulesarenecessaryforap

articular

application

PrePre--processingunit

processingunit

-specificationofallparametersneededtosuccessfullyru

nHYDRUS

-smallcatalogofsoilhydraulicproperties

-Rose

ttapedotransferfunctionsbasedonNeuralNetworks

Post

Post--processingunit

proce

ssingunit

-simp

lex-ygraphsforgraphicalpresentationofsoilhydraulicproperties

andotheroutputresults

34


35/213

HYDRUS-1D

-MajorModule

Pre-processing

Post-processing

HYDRU

S-1D-Preprocessing

WaterFlo

wModels

SoluteTransp

ortModels

HYDRUS-1D

Post-processing

ProfileInformation

ObservationNodes

POSITION-ProjectManager

Manages

dataof

existingp

rojects

Locates

Opens

Copies

Deletes

Renames

--desiredprojectsor

selectedinpu

tand/or

outputdata

35


36/213

PROFILE-Dom

ainDesign,Mesh

Generator,DomainProperties

Discretization

Discretization

ofthesoilprofileintofiniteelements

InitialConditions

InitialConditionsforpressureheads,water

contents,temp

eratures,andconcentrations

RootUptakeD

istribution

RootUptakeD

istribution

MaterialLaye

rs

MaterialLaye

rs-parameterswhichdescribethe

propertiesoft

heflowdomain

-material

distribution

-scalingfactors

ObservationN

odes

ObservationN

odes

PROFILE

-DomainDesign,MeshGenerator,

DomainPr

operties

HYDRUS-1D

-3000

-2500

-2000

-1500

-1000

-5000

0.0

0.1

0.2

0.3

0.4

0.5

WaterContents[-]

-40

-200

20

40

60

50

100

150

200

250

300

Time[days]

potTop

potRoot

actTop

actRoot

actBot

CumulativeFluxes

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

50

100

1

50

200

250

300

T

ime[days]

AllFluxes

-300

-200

-1000

100

200

50

100

150

200

250

300

Time[days]

ObservationNodes

36


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Computer Session 1

Computer Session 1

The purpose of Computer Sessions 1, 2, and 3 is to give users hands-on experience

with the HYDRUS-1D software package (version 3.0). Three examples are given

to familiarize users with the major parts and modules of HYDRUS-1D (e.g., the

project manager, Profile and Graphics modules), and with the main concepts and

procedures of pre- and post-processing (e.g., domain design, finite element

discretization, initial and boundary conditions specification, and graphical display

of results).

The following three examples are considered in Computer Sessions 1, 2, and 3,

respectively:

I. Direct Problem: Infiltration into a one-dimensional soil profile (ComputerSession 1)

A. Water flow

B. Solute transport

C. Possible additional modifications

II. Direct Problem: Water flow and solute transport in a multilayered soil profile

(Computer Session 2)

III. Inverse Problem: One-step outflow method (Computer Session 3)

The first example represents the direct problem of infiltration into a 1-meter deep

loamy soil profile. The one-dimensional profile is discretized using 101 nodes.

Infiltration is run for one day. Ponded infiltration is initiated with a 1-cm constant

pressure head at the soil surface, while free drainage is used at the bottom of the

soil profile. The example is divided into three parts: (A) first, only water flow is

considered, after which (B) solute transport is added. Several other modifications

are suggested in part (C). These include (1) a longer simulation time, (2)

accounting for solute retardation, (3) using a two-layered soil profile, and (4)

implementing an alternative spatial discretization. Users in this example becomefamiliar with most dialog windows of the main module, and get an introduction

into using the external graphical Profile module with which one specifies initial

conditions, selects observation nodes, and so on.

37


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Computer Session 1

A. Infiltration of Water into a One-Dimensional SoilProfile

Project ManagerButton "New"

Name: Infiltr1

Description: Infiltration of water into soil profile

Button "OK"

Main ProcessesHeading: Infiltration of water into soil profile

Button "Next"

Geometry InformationButton "Next"

Time InformationFinal Time: 1

Initial Time Step: 0.0001

Minimum Time Step: 0.000001

Button "Next"

Print InformationNumber of Print Times: 12

Button "Select Print Times"

Button "Next"

Water Flow - Iteration CriteriaButton "Next"

Water Flow - Soil Hydraulic ModelButton "Next"

Water Flow - Soil Hydraulic ParametersCatalog of Soil Hydraulic Properties: Loam

Button "Next"

38


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Computer Session 1

Water Flow - Boundary ConditionsUpper Boundary Condition: Constant Pressure Head

Lower Boundary Condition: Free Drainage

Button "Next"

Soil Profile - Graphical EditorMenu: Conditions->Initial Conditions->Pressure Head

or Toolbar: red arrow

Button "Edit condition", select withMouse the first node and specify 1

cm pressure head.

Menu: Conditions->Observation Points

Button "Insert", Insert nodes at 20, 40, 60, 80, and 100 cm

Menu: File->Save Data

Menu: File->Exit

Soil Profile - SummaryButton "Next"

Execute HYDRUS

OUTPUT:Observation Points

Profile InformationWater Flow - Boundary Fluxes and Heads

Soil Hydraulic Properties

Run Time Information

Mass Balance Information

-100

-80

-60

-40

-20

0

20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Time [days]

N1

N2

N3

N4

N5

Observation Nodes: Pressure Heads

-100

-80

-60

-40

-20

0

-100 -80 -60 -40 -20 0 20

h [cm]

Profi le Info rmatio n: Pressure Head

Close Project

39


40/213

Computer Session 1

B. Infiltration of Water and Solute into

a One-Dimensional Soil Profile

Project ManagerClick on Infiltr1Button "Copy"

New Name: Infiltr2

Description: Infiltration of Water and Solute into Soil Profile

Button "OK", "Open"

Main ProcessesCheck "Solute Transport"

Button "OK"

Solute Transport - General InformationButton "Next"

Solute Transport - Transport ParametersDisp. = 1 cm

Button "Next"

Solute Transport - Reaction ParametersButton "Next"

Solute Transport - Boundary ConditionsUpper Boundary Condition: 1

Lower Boundary Condition: Zero Gradient

Button "Next"

Execute HYDRUS

OUTPUT:Observation Points

Profile Information

Solute Transport - Boundary Actual and Cumulative Fluxes

40


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Computer Session 1

C. Possible Modifications

1. Longer simulation time:

Project ManagerClick on Infiltr2

Button "Copy"

New Name: Infiltr3

Button "OK", "Open"

Time Information:Final Time: 2.5 d

Print Information

Button "Select Print Times"Button "Default"

Button "Next"

2. Retardation:Solute Transport - Reaction ParametersKd = 0.5

3. Two Soil Horizons:Geometry Information

Number of Soil Materials: 2

Water Flow - Soil Hydraulic Parameters1. line - Silt

Solute Transport - Reaction ParametersKd= 0

Soil Profile - Graphical EditorButton "Edit condition", select withMouse the lower 50 cm and specify

Material 2.

Menu: File->Save Data

Menu: File->Exit

41


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Computer Session 1

4. Different Spatial Discretization:

Soil Profile - Graphical EditorMenu: Conditions->Profile Discretization

or Toolbar: ladderButton "Insert Fixed", at 50 cm

Button "Density", at 50 cm 0.5, at the soil surface 0.3

Menu: Conditions->Initial Conditions->Pressure Head

or Toolbar: red arrow

Button "Edit condition", select withMouse the first node and specify 1

cm pressure head.

Menu: Conditions->Observation Points

Button "Insert", Insert nodes at 20, 40, 60, 80, and 100 cm

Menu: File->Save DataMenu: File->Exit

-100

-80

-60

-40

-20

0

-100 -80 -60 -40 -20 0 20

h [cm]

Prof i l e Infor mation : Pressure Head

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.0 0.5 1.0 1.5 2.0 2.5

Time [days]

N1

N2

N3

N4

N5

Observation Nodes: Water Content

42


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OntheCharac

terizationandMeasurement

oftheH

ydraulicPropertiesof

Unsatu

ratedPorousMedia

RienvanG

enuchten1andJirkaimnek

2

1Departm

entofMechanicalEngineering

FederalUniversityofRiodeJaneiro,Brazil

2Departm

entofEnvironmentalSciences

UniversityofCalifornia,Riverside,CA

RichardsE

quationforVariably-Satura

tedFlow

()

()

()

h

h

=

K

+K

t

z

z

SoilW

aterRetentionCurve,(h)

Hydr

aulicConductivityFunction,K(h)orK

()

-

volumetricwatercontent[L3L-3]

h-

pressurehead[L]

K-

unsaturatedhydraulicconductivity[LT-1]

z-

verticalcoordinatepositiveupward[L]

t-

time[T]

S-

rootwateruptake[T-1]

SoilWaterR

etentionCurve,

(h)

HydraulicConductivityFuncti

on,K()

43


44/213

SoilWaterH

ysteresis

Outlin

e

DirectMeasurementsofHydraulicProperties

StatisticalPore-SizeDistributionModels

Pedot

ransferFunctions

TheR

osettaPTFcode

TheU

NSODADatabase

StructuredMedia

Ongoing/FutureResearch

Unsaturated

HydraulicConductivity

Steady

Steady--StateMethods

StateMethods

UsingDarcysla

w:q=-K(h)(dh/dz-1)

Long-ColumnM

ethod

CentrifugeMeth

ods

DirectTransientMethods

DirectTransientMethods

HorizontalInfiltration(BruceandKlute,1956)

SorptivityMethods(Dirksen,1975)

One-Step/Multi-StepOutflowMethod(Passioura,1975)

Hot-AirMethod

(Aryaetal.,1975)

EvaporationMe

thod(Boelsetal.,1978)

ParameterOptimizationMethods

ParameterOptimizationMethods

(Kooletal.,1985;imnekandvanGenuchten,1996)

LaboratoryMetho

ds

Tempe

PressureCell

44


45/213

PeteShouses

TempeCellSetupatUSSalinityLaboratory

UnsaturatedHydraulicCondu

ctivity

DirectMethods

DirectMethods

InstantaneousProfileMethods(Watson,1966)

Unit-GradientMethods(Sissonetal.,1980)

Plane

-of-Zero-FluxMethod

Sorpt

ivityMethods(ClothierandWhite,1981)

ConstantHeadPermeameters(Reynoldsetal.,1983)

Tensi

onInfiltrometers(WhiteandPerroux,1988)

Param

eterOptimizationMethods

Param

eterOptimizationMethods

Russo

etal.(1991)

Abba

spouretal.(1996)

imneketal.(1998,2000)

...

FieldMeth

ods

Schematicof

TensionInfiltrometer

Thesupply

pressurehead

hwet=h2-h1

=

=

22/

(

)e

s

e

K

S

=K

S

+

+

Se

-effectivewaterconten

t

r,

s

-residualandsaturatedwatercontents

,n,m(=1-1/n),land

-emp

iricalparameters

Ks

-saturatedhydraulicconductivity

50


51/213

LognormalDistributionModel(Kosugi,1996):

(

)

0

ln

/

()

1

()

2

2

r

e

s

r

hh

h

S

h

erfc

=

=

(

)

2

0

ln

/

1

()

2

2

l

s

e

hh

Kh

KS

erfc

=

+

Se

-effectivewatercontent

r,

s

-residualan

dsaturatedwatercontents

h0,,andl-empiricalp

arameters

Ks

-saturatedh

ydraulicconductivity

SoilHydraulicPropertyModels

10-2

10-1

100

101

102

103

10

So

ilWa

ter

Pressure

Hea

d(-mm

)

10-7

10-6

10-5

10-4

10-3

10-2

10-

HydraulicConductivity(mm/sec)

Observed

BimodalHydraulicConductivity

Durner(1994):

Se

-effectivewatercontent

r,s

-residualand

saturatedwatercontents,respectively

k

-numberofoverlappingsubregions

wi

-weightingfactorsforthesub-curves

i,ni,mi(=1-1/ni),andl-empiricalparametersofthesub-curves.

1

()

1

()

(1

)i

i

k

r

e

i

m

n

i

s

r

i

h-

S

h=

=

w

-

+

h

=

2

1/

1

2

1

11-(1-

)

()

i

i

i

i

k

m

me

i

i

k

i

l

s

e

i

k

i

i

i

i

w

S

K

=

wS

K

w

=

=

=

Thehydrauliccharacteristics

contain4+2kunknownparameters:

r,

s,i,ni,

l,andK

s.

Ofthese,

r,

s,andK

shavea

clearphysicalmeaning,whereasi,niandlareessentially

empiricalparametersdetermi

ningtheshapeoftheretentionandhydraulicconductivity

functions[vanGenuchten,19

80].

SoilHydraulicPropertyModels

00

.10

.20

.30

.40

.50

.6-1

0

1

2

3

4

5

Log

(|PressureHead[cm]|)

WaterContent[-]

Total

Matrix

Fracture

-10-8-6-4-20

-1

0

1

2

3

4

5

Log(|Pressu

reHead[cm]|)

Log(Conductivity[cm/days])

Total

Matrix

Fracture

Exampleofcom

positeretention(left)andhydraulicconductivity(right)

functions(r=0.00,

s=0.50,1=0.01cm-1,n1=1.50,l=0.5,Ks=1cmd-1,w1=0.975,

w2=0.025,

2=1.00cm-1,n2=5.00).

Durner(19

94):

SoilHyd

raulicPropertyModels

51


52/213

PedotransferFunctions

Predictthehydraulicpropertiesfrom

moreeasily

measureddata:

SoilTexture(classorparticle-sizedistribution)

BulkDensity

Porosity

OrganicMatterContent

SoilStructure

ClayMineralogy

ChemicalProper

ties(EC,pH,SAR,)

TwoApproaches:

TwoApproaches:

-Predictspecificr

etentionvalues

-Predictsoilhydr

aulicparameters


988)

PTFsbyCarselandParrish(1988)


988)

Averagevalues

ofselectedsoilwaterretentionparametersfor12major

soiltexturalgroups

52


53/213

HierarchicalNeural-NetworkBootstrapApproach

Hierarchy:tryto

matchvariouslevelsofdataavailability

Neuralnetworks:toprovidethemostaccurate

predictions

Bootstrap:gener

ateconfidenceintervalsofthe

predictions

Predictionof:wa

terretention,Ks,andtheunsaturated

hydraulicconductivity

Rosetta(Schaapetal.,2001)

Rosetta(Schaapetal.,2001)

Pedotra

nsferFunctions:Rosetta

Schaapetal.(2001)

Model

InputData

TXT

TexturalClass

SSC

Sand,Silt,Clay

%

SSCB

D

Same+BulkDensity

SSCB

D+

33

SSCBD+

at3

3kPa

SSCB

D+

33+

1500

Same+

at1500kPa

Predicted

parameters+

uncertainties

Hierarchica

lModels

Inputdata

Pedotransfer

Functions:Rosetta

53


54/213

Pedotransfer

Functions:Rosetta

0

100

0

20

80

20

40

60

40

60

40

60

80

20

80

100

0

100

Clay

[%]

Silt[%]

San

d[%]

LL

0

100

0

20

80

20

40

60

40

60

40

60

80

20

80

100

0

100

Clay

[%]

Silt[%]

San

d[%

]

SlS

sL

L

siL

Si

sc

L

cL

sicL

siC

sC

C

RosettasCalibrationData

R

etention

(N=2134)

UnsaturatedCo

nductivity

(N=235

)

Waterretention

SaturatedConductivity

R2

RMSEw

R2

RMSEs

Model

Input

r

s

Log

Logn

cm3/cm3 LogKs

(-)

H1

TexturalClass

0.06

6

0.1

43

0.2

03

0.4

52

0.0

72

0.4

27

0.7

39

H2

SSC

0.08

6

0.1

78

0.2

38

0.4

73

0.0

70

0.4

61

0.7

17

H3

SSCBD

0.09

4

0.5

81

0.2

65

0.4

95

0.0

60

0.5

35

0.6

66

H4

SSCBD

33

0.12

1

0.6

05

0.4

17

0.5

99

0.0

41

0.6

40

0.5

86

H5

SSCBD

33

1500

0.38

7

0.6

00

0.5

77

0.7

60

0.0

39

0.6

47

0.5

81

Directfittodata

-

-

-

-

0.0

12

-

-

RosettasP

erformance

SSC:

Sand,silt,

claypercentages

BD:

Bulkdensity

33,1500

Watercon

tentat33and1500kPa

RosettasClass-AveragePTFs

54


55/213

Pore-ConnectivityParameterL

(

)

2

1/

1

1

m

L

m

s

e

e

K

KS

S

=

UNSODA2UnsaturatedSoilHydraulicDatabase

http://www.ussl.ars.usda.gov/models.htm

MS-ACCESS

FlexibleQu

eries

Graphicssu

pport

Downloadable

DripIrrigation(Skaggsetal.,2004)

DripIrrigation(Skaggsetal.,2004)

55


56/213

HYDRUS

-0.93

1.3

1.4

0.023

0.33

0.021

l

Ks(cmhr-1)

n

s

r

Hydraulicproperties

Hydraulicproperties

estimatedusingRo

setta

estimatedusingRo

setta

pedotransferfunction

pedotransferfunction

(sand,silt,andclay

,bulk

(sand,silt,andclay

,bulk

density,1/3and15

bar

density,1/3and15

bar

watercontent)

watercontent)

0

10

20

30

40

50

60

DISTANCE(cm)

0

10

20

30

40

50

60

DISTANCE(cm)

-70

-60

-50

-40

-30

-20

-100

DEPTH(cm)

Observed

Predicted

0.1

0.1

2

0.1

4

0.1

6

0.1

8

0.2

0.2

2

0.2

4

0.2

60.2

8

0.3

Vo

lume

tricWa

ter

Con

ten

t

Trial1:5hourirrigation,20L/mappliedwater

Time=5.5h

r

0.1

0.1

2

0.1

4

0.1

6

0.1

8

0.2

0.2

2

0.2

4

0.2

60.2

8

0.3

Vo

lume

tricWa

ter

Con

ten

t

0

10

20

30

40

50

60

DISTANC

E(cm)

-70

-60

-50

-40

-30

-20

-100

DEPTH(cm)

0

10

20

30

40

50

60

DISTANCE(cm)

Observed

Predicted


Time=28hr

0.1

0.1

2

0.1

4

0.1

6

0.1

8

0.2

0.2

2

0.2

4

0.2

60.2

8

0.3

Vo

lume

tricWa

ter

Con

ten

t

0

10

20

30

40

5

0

60

DISTANCE(cm)

0

1

0

20

30

40

50

60

DISTANCE(cm)

-70

-60

-50

-40

-30

-20

-100

DEPTH(m)

Observed

Predicted

Trial2:10

hourirrigation,40L/mappliedwater

Time=10.7

5hr

56


57/213

0.1

0.1

2

0.1

4

0.1

6

0.1

8

0.2

0.2

2

0.2

4

0.2

60.2

8

0.3

V

olume

tricWa

ter

Con

ten

t

0

10

20

30

40

50

60

DISTANCE(cm)

0

10

20

30

40

50

60

DISTANCE

(cm)

-70

-60

-50

-40

-30

-20

-100

DEPTH(cm)

Predicted

Obse

rved


Time=31hr

0.1

0.1

2

0.1

4

0.1

6

0.1

8

0.2

0.2

2

0.2

4

0.2

60.2

8

0.3

Vo

lume

tricWa

ter

Con

ten

t

0

10

20

30

40

50

60

DISTANCE(cm)

0

10

20

30

40

50

60

DISTANCE(cm)

-70

-60

-50

-40

-30

-20

-100

DEPTH(m)

Observed

Predicted

Trial3:15

hourirrigation,60L/mappliedwater

Time=16h

r

0.1

0.1

2

0.1

4

0.1

6

0.1

8

0.2

0.2

2

0.2

4

0.2

60.2

8

0.3

Vo

lume

tricWa

ter

Con

ten

t

0

10

20

30

40

50

60

DISTANCE(cm)

0

10

20

30

40

50

60

DISTANC

E(cm)

-70

-60

-50

-40

-30

-20

-100

DEPTH(cm)

Predicted

Observed


Time=39hr

0

-10

-20

-30

-40

-50

DEPTH

(cm)

00.10.20.3

WATERCONTENT

DISTANCE=

0

0

-10

-20

-30

-40

-50

DEPTH

(cm)

00.10.20.3

WATERCONTENT

DISTANCE=

10

0

10

20

30

40

DISTANCE(cm)

00

.10

.20

.3

DEPTH=-1

0

0

10

20

30

40

DISTANCE(cm)

00

.10

.20

.3

DEPTH=-2

0

Trial1:5h

ourirrigation,20L/mappliedwater

Time=5.5hr

57


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0

-10

-20

-30

-40

-50

DEP

TH

(cm)

00

.10

.20

.3

WATERCONTENT

DISTANCE

=0

0

-10

-20

-30

-40

-50

DEP

TH

(cm)

00

.10

.20

.3

WATERCONTENT

DISTANCE

=10

0

10

20

30

40

DISTANCE(cm)

00

.10

.20

.3

DEPTH=-1

0

0

10

20

30

40

DISTANCE(cm)

00

.10

.20

.3

DEPTH=-2

0

Trial1:5hourirrig

ation,20L/mappliedwater

Time=28hr

Genericversussite-specificPTFs

Effectsofsoilstructure

Effectsofchemistryandclaymineralogy

-NRCSsoilcharacterizationdatabase

Lab

oratoryversusfielddata

Continuedatamining(UNSODA)

Future

PlansRosetta

HydraulicPr

operties-Challenges

Hysteresis

Dryendeffects;residualsaturation,r

Dynamiceffec

ts;non-equilibriumflow

Airentrapment(sversusporosity)

Swellingsoils;

effectsofchemistry

Descriptionnearsaturation

Secondordercontinuityin(h)(n

1.0)

Structuredmedia;preferentialflow

ScaleIssues(u

pscaling;effectiveproperties)

RequiredAccuracy(fluxvsprofilecontrolledinf.)

...

58


59/213

ApplicationofFiniteElementMethodto

1DVariably-SaturatedWaterFlow

andS

oluteTransport

Jirkaimnek

1andRienvanGenuchten2

1DepartmentofEnvironmentalSciences

Universityo

fCalifornia,Riverside,CA

2DepartmentofMechanicalEngineering

FederalUnive

rsityofRiodeJaneiro,Brazil

ApplicationofFiniteElementMethod

to1DVariably-SaturatedFlow

RichardsEquation:

cos

h

q

=

K

+

-S

-S

t

x

x

x

=

Finalfinite

differencescheme:

1,

1

1

1

1/2

1/2

-

-

j

k

j

j

j

j

i

i

i

i

i

q

q

=

S

t

x

+

+

+

+

+

12/1

1

11

12/

1

12/1

++

+

++

+ +

+ +

=

=

j i

i

j i

j i

j i

j i

K

xh

h

K

q

K

xh

K

q

xi+1

xi

xi-1

ti

ti+1

x

xi

xi-1

j ih1

j ih

1j ih

+

1j ih+

11j ih+

11j ih+

+

t



RichardsEquation:

cos

h

q

=

K

+

-S

-S

t

x

x

x

=

Finalfinitedifferencescheme:

1,

1

1,

1

1,

1,

+1,

+1

1,

1

1,

1

1

-1

1/2

-1/2

1,

1,

1/2

-1/2

1

-

1

-

-

-

+

-

j

k

j

k

j

k

jk

j

k

j

j

k

j

k

i

i

i

i

i

i

i

i

j

k

j

k

ji

i

i

i

i

h

h

h

h

K

K

=

S

K

K

t

x

x

x

x

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

where:

+1

1

1

1

1

1

1

1

1

1

1

1

1

1

1/2

1/2

-

=

=

2

2

j

j

i

i

i

i

i

i

i

i

j

,k

j

,k

j

,k

j

,k

i

i

i

i-

j+

,k

j

,k

i+

i-

t=

t

t

x-

x

x=

x

x

-x

x

x-x

2

+

+

K

K

K

K

=

=

K

K

+

+

+

+

+

+

+

+


1DV

ariably-SaturatedFlow

MatrixForm:

Thesymmetrical

tridiagonalmatrix[Pw

]

hastheform:

+1,

+1

+1,k

[

{

={

}

}

]

j

k

j

w

w

P

h

F

d

e

e

d

e

e

d

e

.

.

.

.

.

.

e

d

e

e

d

e

e

d

=

P

N

N

N

N

N

N

N

N

w

1

1

1

2

2

2

3

3

3

2

2

2

1

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

]

[

ApplicationofFiniteElementMethodto1D

Variably-SaturatedFlow

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Themass

Themass--conservativemethod

conservativemeth

odproposedbyCeliaetal.[1990],inwhich

j+1,k+1isexpandedinatrun

catedTaylorserieswithrespecttohabou

t

theexpansionpointhj+1,k,is

usedinthetimedifferencescheme:

+1,

+1

+1,

+1

+1,

+1,

+1,

+1

+1,

+1,

+1,

1

1

1

+1,

-

-

- -

-

+

t

j

k

j

j

k

j

k

j

k

j

j

k

j

k

j

k

j

i

i

i

i

i

i

i

i

i

i

j

k

j

j+

,k

j+

,k

i

i

j

k

i

i

i

=

=

=

t

t

t

t

h

h

C

t

+

+

+

=



Thediagonal

entriesdiandabove-diagonalentrieseiofthematrix[Pw

],

andtheentrie

sfiofvector{Fw

},aregivenby:

1

11

1

1

11

k,1+

2

+

2

+

+

+

+

+

i

,k

ji-

,k

ji

i

,k

ji

,k

ji+

ji

i

xK

+

K

xK

+

K

Ctx

=d

i

,k

ji+

,k

ji

i

xK

K-

=e

2

+

11

1

+

+

x

S

K

K+

tx

h

Ctx

=f

ji

,k

ji

,k

ji+

j i

k

j i

,k

ji

k

ji

i

-

2-

)

-

(

-

11

11

,1+

1

,1+

+

+



ImplementationoftheUpperFluxBoundaryCondition:

ImplementationoftheUpperFluxBoundaryCondition:

q

=-

-S

t

x

ThemassbalanceequationinsteadofDarcy'slaw

isdiscretized.

Discretizationgives:

Expandingthetimederivativeonthelefthand[Celiaetal.,1990],an

d

usingthediscretizedformo

fDarcy'slawforqN-1/2leadsto:

S

x

q

q(

-=

t

jN

N

,k

j N-

j N

j N

k

j N

-)

-

2

-

1

12/1

1+

1+,

1+

+

1

11

1

,1+

1

2

+

+

N

,k

jN-

,k

+jN

k

jN

N

N

x

K

+

K

Ct

2

x=

d

q

-

S

x

K

K

-)

tx

h

C

t

2x

=f

+j N

jN

N

,k

jN-

k

jN

j i

k

j N

,k

jN

k

jN

N

N

1

1

11

,

1+

,1+

1

,1+

1

2

-

2+

-

(

2

-

+

+

qNistheprescribedsoilsurfaceboundaryflux



ComputationofNodalFluxes:

Computatio

nofNodalFluxes:

S

+

t

x

-

+

x

h-

h

K-=

q

x

+

x

x

+

xh

-

h

K-

x

+

xh

-

h

K-

=

q

+

xh

-

h

K-

=

q

jN

jN

+jN

N

N

+j-N

+jN

+j-N

+j N

i

i

i

-i

+j -i

+ji

+j -i

-i

i

+ji

+j+i

+j+i

+j i

i

+j

+j

+j+

+j

-

2

1

1

1

11

1

1

11

1

12/1

1

1

1

11

1

12/1

1

1

11

12/1

1

1

1

12

12/11

1

1



60


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IterativeProcess:

IterativeProcess:Picard

Picardlinearization

linearization

j-timestep

k-iteration

1)Firsttimestep:hj+1,1=hinit,j=1,k=1

2)Derivationofthesystemoflinearizedalgebraicequations

usinghj+1,k,q

j+1,k,K

j+1,k,C

j+1,k

3)Gaussianelimination-hj+1,k+1,q

j+1,k+1

4)ToleranceCriteria:

abs(hj+1,k+1-hj+1,k)Grid - Height: 0.05

Menu: Condition->Profile Discretization

Button"Number": 50

Button"Insert Fixed" at 3.95 cm

Button"Density": deselect "Use upper", upper density =0.1 at 3.95 cm

Menu: Condition->Initial Condition->Pressure Head

Button"Edit condition"

Select entire profile: Top value=-2, Bottom value=2.52

Deselect "Use top value for both"

Lowest node = -1000 cm

Menu: Condition->Material Distribution

Button"Edit condition"

Select the ceramic plate and specify "Material Index"=2

Ditto for "subregions"Observation Points?

Soil Profile - Summary

Execute HYDRUS

OUTPUT:Water Flow - Boundary Fluxes and Heads

Cumulative Bottom FluxSoil Hydraulic Properties

Inverse Solution Information

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.001 0.01 0.1 1 10 100

Time [hours]

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Computer Session 3

Exercises:

Multistep Outflow Experiments (SGP97 project):

Height of the soil sample:5.9 cm

Thickness of the ceramic: 0.5 cm

Conductivity of the ceramic, boundary conditions, and output data

depend on the sample (see the Excel file).

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0 500000 1e+006 1.5e+006

Time [sec]

Cum. Bottom Flux

-600

-500

-400

-300

-200

-100

0

0 500000 1e+006 1.5e+006

Time [sec]

Bottom Pressure Head

93


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94


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Application

ofFiniteElementMethod

to2DVariably-SaturatedWaterFlow

and

SoluteTransport

Jirka

imn

ek1andRienvanGenuchten

2

1Departm

entofEnvironmentalSciences

Universit

yofCalifornia,Riverside,CA

2DepartmentofMechanicalEngineering

FederalUniversityofRiodeJaneiro,Brazil

DiscretizationUsingFiniteEle

ments

Examplesofthe

unstructuredtriangularfiniteelementgridsforreg

ular(left)and

irregular(right)

two-dimensionaltransportdomains.


2DVariably-SaturatedFlow

TheTheGalerkin

Galerkinmethod:

method:

ApplyingGreen'sfirstidentityandreplacinghbyh0

A

A

ij

iz

n

i

jh

-

K

K

+K

+S

d

=

t

x

x

(

)

'

e

e

e

A

n

n

ij

e

j

i

A

A

ij

iz

i

n

j

e

A

n

iz

n

e

i

h

+KK

d

=

t

x

x

h

K

K

+

K

n

d

+

x

-KK

-S

d

x

e

representsthedomainoccupiedbyelemente

e

isaboundarysegmentofelemente

Inmatrixfor

m:

}

{

}

{

}

{

}

]{

[

}{

]

[

D-

B-

Q

=h

A

+

dt

d F

[

(

)

]

4

e

n

m

A

l

nm

l

ij

e

i

j

A

A

A

xx

m

n

xz

1.1 curso de hydrus 1d

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