gw flow & well hydraulics

GEOL473 Hydrogeology

Dr. Schradh Saenton Topic 3: Groundwater Flow and Well Hydraulics

3-1

3. Groundwater Flow Equations & Well Hydraulics

GroundwaterFlowEquationsGWfloweqn.=mathematicalexpressionusedtodescribethebehaviorofgroundwater

flowinporousmedia.Darcyslaw:

Inonedimension: x XQ h

q KA x

= =

Inthreedimension: 0 0

0 0

0 0

hx x x

hy y y

hz z z

q K

q q K

q K

= = v

Massbalanceequation:m& =massflowrateenteringorexitingasurface[M/T]

GWfloweqn=Darcyslaw+MassBalanceeqn.

InOut=Accumulation[M/T]

Control VolumeControl Volume

x y

z

x y

z

xm& xm& x xm +& x xm +&

in out

mm m

t = & &



3-2

[ ]( )x wm q x y z= & [ ]( )x x wm q x x y z+ = + & [ ]( )y wm q y x z= & [ ]( )y y wm q y y x z+ = + & [ ]( )z wm q z x y= & [ ]( )z z wm q z z x y+ = + &

Substituteporosity wVV

= andvolumeV x y z= intotheaboveexpressionDividethroughby x y z ,wewillhave

Assumethatdensitydoesntchangemuchwithinthecontrolvolume,wecantakedensityoutofthelefthandside,thenwewillhave

Takelimit , , 0x y z ,wecantransformthedifferenceintothedifferentialform.

Recallthedefinitionofderivative:0

( ) ( )limx

df f x x f xdx x

+ = .

[ ] [ ]in out x x x y y y z z zm m m m m m m m mt + + + = = + + & & & & & & & &

[ ] [ ]

[ ] [ ]

( )( ) ( )

( ) ( )

( ) ( )

w ww x w x

w y w y

w z w x

Vq x y z q x x y z

t

q y x z q y y x z

q z x y q z z x y

= + + +

+ +

[ ] [ ]

[ ] [ ]

( )( ) ( )

( ) ( )

( ) ( )

ww x w x

w y w y

w z w x

x y z q x y z q x x y zt

q y x z q y y x z

q z x y q z z x y

= + + +

+ +

[ ] [ ]

[ ] [ ]

( ) ( )( )

( ) ( )

( ) ( )

w x w xw

w y w y

w z w x

q x q x x

t x

q y q y y

y

q z q z z

z

+ = + +

+ +

( ) ( )1 ( ) ( ) ( ) ( )y xw x x z xw

q y y q yq x x q x q z z q zt t x y z

+ + + + =

w wm V=



3-3

Fromdarcyslaw:

x x

hq K

x= y y

hq K

y= z z

hq K

z=

Finally,wehavederivedthegeneralgroundwaterflowequation.Forconfinedaquifer: ( )w w sg g S + = orspecificstorage

S x y z

h h h hS K K K

t x x y y z z = + +

Forunconfinedaquifer: ( )w w Sg g b + = (S=storativity=Sy+bSS)

x y z

h h h hS T T T

t x x y y z z = + +

Note: T=Kb=transmissivity[L2/T] b=aquiferssaturatedthickness

1 yw x zw

qq qt t x y z

+ =

0 0 0

( ) ( )1 ( ) ( ) ( ) ( )lim lim limy xw x x z xx y z

w

q y y q yq x x q x q z z q zt t x y z

+ + + + =

( )w w x y zh h h hg g K K Kt x x y y z z + = + +

1 ww

w

hg

t t =

COMPRESSIBILITY OF WATER 1

ww

hg

t t =

COMPRESSIBILITY OF AQUIFER



3-4

ExampleIII1

Solvegroundwaterflowequationinthefollowing2Dproblembyassumingsteady

stateandaquiferisisotropicandhomogeneous.

Solvethisproblembyhand,wewillhavethesolutionfor ( , )h x y asfollows:

0

(2 1)(2 1)

0 2 (2 1)20

cos cosh4( , )

2 (2 1) cosh

m ym xs s

m ym s

cs csh x y y

m

++

+=

= + + !!!!!

***Theaboveexampleispresentedhereonlyforillustrationandtoshowthatgroundwater

flowequationisnoteasilysolvedbyhandtoobtainclosedformsolution.***

Groundwaterflowtoapumpingwell

Boundaryconditions

Beforepumping Whilepumping



3-5

Coneofdepression

=dewateredzoneinanaquifer(unconfinedaquifer)

=aconeofdepressedpotentiometricsurface(confinedaquifer)

Drawdown(s)[L]

=depressedwaterlevel(orpotentiometricsurface)

Radiusofinfluence

=distancefrompumpingwellwheredrawdownisessentiallyzero

IMPERMEABLE ROCK

Confined Aquifer

Ground

Well

IMPERMEABLE ROCK

UnConfined Aquifer

Ground

Well



3-6

Pumpinginanaquiferwithdifferenttransmissivity(T)andstorativity(S)

HighS LowS

IMPERMEABLE ROCK

Confined Aquifer

GroundWell

Potentiometric Surface t = 0

IMPERMEABLE ROCKConfined Aquifer

GroundWell


HighT LowT

IMPERMEABLE ROCK

Confined Aquifer

GroundWell


IMPERMEABLE ROCKConfined Aquifer

GroundWell


HighSWewillgetmorewaterforthesameheaddrop.HighTWaterflowsmoreeasily.

Radialflowtowell

GWfloweqnincylindricalcoordinatecanbeusedto

describeradialflowtowell.

t1t2

Pumping Well

2

2

1S h h hT t r r r = +

( , )h h r t=Topview

t2>t1

Radiusofinfluence

h=hydraulichead[L]

r=radialdisrance[L]

t=time[T]



3-7

Whathappensduringpumping?

Whenthewellisbeingpumped,adrawdownofheadiscreatedaroundthewell

formingaconeofdepression.Insomecases,ifaquiferisbeingpumpedlongenough,

drawdowncanreachequilibrium(i.e.headdoesntchangewithtimeanymore).

r1r2

Pumping Well

Observation Wells

Time

Dra

wdo

wn

(m)

IMPERMEABLE ROCK

Confined Aquifer

GroundPumping Well

Potentiometric Surface

OW1 OW2



3-8

Equilibriumflowtowell

- Occurswhenaquiferispumpedforaverylongtime.

- Waterlevel(orpotentiometricsurface)doesnotchangewithtime.

- WecanusedarcyslawtocalculateKORTifweknowQandhydraulicheadsat

twolocations(i.e.calledpumpingtest)

1.Confinedaquifer

2

2 1 1

2

2 1 1

ln2 ( )

ln2 ( )

Q rT

h h r

Q rK

b h h r

= =

2.Unconfinedaquifer

22 22 1 1

ln( )Q r

Kh h r

=

ExampleIII2

Awellinconfinedaquiferispumpedatarateof220gallon/min.Measurementof

drawdownintwoobservationwellsshowsthatafter1270minofpumping,nofurther

drawdownisoccurring.Well#1islocatedat26ftfrompumpingwellandhasahydraulic

headof29.34ftabovethetopofaquifer.Well#2islocatedat73ftfromthepumpingwell

andhasahydraulicheadof32.56ftabovethetopofaquifer.UseThiemequationtofind

aquifertransmissivityifaquiferthicknessis25m.

h2 h1

TheimEquation



3-9

r1=26ft h1=29.34ft

r2=73ft h2=32.56ft

Steadystate(equilibrium)pumpinginanunconfinedaquifer

(Exampleofmodelsimulation)

NoPumpingCondition

3

3

min

ftday

1ft 1440min220

7.48gal 1day

42,400

galQ =

=

3

2

2 1 1

ftday

ln2 ( )

42400 73 ftln

2 (32.56ft 29.34ft)

Q rT

h h r

= = 26 ft

22164 ft day

=

Crosssection

Plan

view

[ ]2ft2164 78ft 27.7ft/dayd

T Kb K K = = =

Convert Q from gal/mintoft3/day



3-10

Withpumpingatthecenteroftheaquifer

Nonequilibriumpumping(ortransient)

Drawdown(s)isafunctionoftimeanddistance

Confinedaquifer

UseTheissolution Aquiferishomogeneous,

isotropic,andisofinfinite

extent

Wellcompletelypenetrates(andgetwaterfrom)the

entireaquifer

Transmissivityisconstant Waterisremovedfromstorageanddischargeinstantaneously.

Crosssection

Plan

view

2

2

1S h h hT t r r r = +



3-11

TheisSolution

0 ( )4Q

s h h W uT= =

where2

4r S

uTt

= ,and2 3

( ) 0.5772 ln2 2! 3 3!

x

u

e u uW u dx u u

x

= = + + L

s drawdown[L]

h0 initialheadinawellatdistancer[L]

h headatdistancerattimet[L]

t timesincepumpingbegins[T]

r distancefrompumpingwell[L]

Q pumpingrate[L3/T]

T transmissivity[L2/T]

S storativity[]

b aquiferssaturatedthickness[L]

W(u) wellfunction[]

u auxiliaryparameter[]

ExampleIII3

Awellislocatedinconfinedaquiferwithahydraulicconductivityof14.9m/dayand

astorativityof0.0051.Theaquiferis20.1mthickandispumpedatarateof2725m3/day.

Whatisthedrawdownatadistanceof7.0mfromthewellafter1dayofpumping?

2m m

day day14.9 (20.1m) 299.49T Kb = = =

[ ] [ ][ ]2

224

mday

7m 0.00512.086 10

4 4 299.49 1dayr S

uTt

= = =

FindthevalueofW(u)fromthetable(appendix1)



3-12

Fromtable+linearinterpolationW(u)=7.9398.Thereforethedrawdownatdistanceof7.0mafter1dayofpumpingis

3

2

mday

0 mday

2725( ) 7.9398 5.75m

4 4 299.49Q

s h h W uT

= = = =

Unconfinedaquifer

UseNeumansolution Aquiferishomogeneousandisof

infiniteextent

Initiallywaterispumpedfromstorage(Ss)

Later,waterisbeingdrainedduetogravity(Sy)

Assumedrawdownisnegligiblecomparedtosaturatedthickness

RadialK(orKr)canbedifferentfromverticalK(orKv) Neumansolutionisvalidonlywhendrawdownisverysmallcomparedtoaquifers

thicknessor s b



3-13

Neumansolution

0 ( , , )4 A BQ

s h h W u uT= =

Where2

4Ar S

uTt

= ,2

4y

B

r Su

Tt= and

2

2v

h

r Kb K

=

( , , )A BW u u canbeobtainedfromtable(appendix6A,6B)

s drawdown[L]

h0 initialheadinawellatdistancer[L]

h HEADATDISTANCErATTIMEt[L]

t timesincepumpingbegins[T]

r distancefrompumpingwell[L]

Q pumpingrate[L3/T]

Kv verticalhydraulicconductivity[L/T]

Kh horizontalhydraulicconductivity[L/T]

S storativityatearlytime(inthiscase,S=bSs)[]

Sy specificyield[]

b initialaquiferssaturatedthickness[L]

Atearlypumpingtime,useuA

Whereatlatertime,useuB.



3-14

ExampleIII4

Awellislocatedinanunconfinedaquiferwithaverticalandhorizontalhydraulic

conductivitiesof1.26and15.8m/day,respectively.Thevalueofspecificstorageis0.00025

m1,andspecificyieldis0.12.Theaquifersinitialsaturatedthicknessis20.1mandis

pumpedatarateof275m3/day.Whatisthedrawdownatadistanceof7.0mfromthewell

after1and50dayofpumping?

r=7m Q=275m3/day

Kv=1.26m/day Kh=15.3m/day

b=20.1m Sy=0.12

S=bSs=(0.00025m1)(20.1m)=0.005 T=Khb=(15.8m/day)(20.1m)=317.58m2/day

[ ][ ]

2 m2day

22 mday

7m 1.260.01

20.1m 15.3v

h

r Kb K

= = [seetableinappendix6A,6B]

After1day(earlytime)

[ ] [ ][ ]2

224

mday

7m 0.0051.93 10

4 4 317.58 1dayAr S

uTt

= = = 1 5185Au = ( ) 3.46W u =

Drawdown:3

2

mday

mday

275( ) 3.46 0.238m

4 4 317.58Q

s W uT

= = =

After50day(latetime)

[ ] [ ][ ]2

225

mday

7m 0.129.26 10

4 4 317.58 50dayy

B

r Su

Tt= = = 1 10802Bu = ( ) 8.672W u =

Drawdown:3

2

mday

mday

275( ) 8.672 0.598m

4 4 317.58Q

s W uT

= = =

Check:s=0.238,0.597b=20.1.OK!!



3-15

Summary:CharacteristicsofAnalyses

Equilibrium

(orsteadystate)

NonEquilibrium

(ortransient)

1. ObtainedmostaccurateT,Kvalues

2. Usefulwhenlongtermpumpinghas

beenestablished

3. Equationsusefulindesigninga

pumptest(estimatemaximum

drawdown)

4. Cannotobtaininformationon

storage

5. Thiemsolution

1. Candeterminestorativity(in

pumpingtest)

2. Getresultsatearlytime

3. Analysisismorecomplicated

4. Theissolutionconfined5. Neumansolutionunconfined6. Neumansolutionisapplicableonly

forsb

Principleofsuperposition

Iftherearemorethanonepumpingwells,drawdownatobservationwellcanbedeterminedusingprincipleofsuperposition(drawdowncanbeaddedorsubtracted)

Normally,thismethodisvalidonlyforconfinedaquifer.However,insomecases,itmaybeapplicableinunconfinedaquiferifdrawdownisnegligiblecomparedto

saturatedthickness(sb).



3-16

ExampleIII5

Twowellsinaconfinedaquifer(b=20m,S=

0.0075,K=1.75m/day)arepumping

simultaneouslyattheratesof200and400m3/day,

respectively.Calculatedrawdownatanobservation

wellatt=4day.

Usingprincipleofsuperposition: 1 2total pw pws s s= + 1. calculatedrawdownatOWfromPW1

T=Kb=(1.75m/day)(20m)=35m2/day[ ] [ ]

[ ]222

mday

97m 0.00750.126

4 4 35 4dayr S

uTt

= = = ( ) 1.667W u =

3

2

mday

1 mday

200( ) 1.667 0.758m

4 4 35pwQ

s W uT

= = =

2. calculatedrawdownatOWfromPW2

[ ] [ ][ ]2

22

mday

175m 0.00750.410

4 4 35 4dayr S

uTt

= = = ( ) 0.688W u =

3

2

mday

2 mday

400( ) 0.688 0.626m

4 4 35pwQ

s W uT

= = =

ExampleIII6

FromExampleIII5,if,insteadof

pumpingwater fromPW2,waterisinjectedat

therateof+400m3/daywhilePW1isstill

pumpingattherateof200m3/day.Calculate

drawdowninobservationwellattimet=4day.

|

PW1Q1 = -200 m3/day

PW2Q2 = -400 m3/day

Observation Well

r 2= 17

5 m

r1 = 97 m

1 2

0.758 0.626

1.38m

total PW PWs s s= += +=

|

PW1Q1 = -200 m3/day

INJW2Q2 = +400 m3/day

Observation Well

r 2= 17

5 m

r1 = 97 m



3-17

Frompreviousexample,

1pws =drawdownduetopumpingfromPW1(waterleveldecreases)

=+0.758m

2injws =drawupduetoinjectionofwaterinPW2(waterlevelincreases)

=0.626m

Fromprincipleofsuperposition,drawdownatobservationwellafterday4is

1 2 0.758 0.626 0.13mtotal PW INJWs s s= + = = .Thus,waterleveldecreasesonly0.13minthiscase.

MethodofImages

Whenanaquiferisconnectedtoriverorimpermeablebarrier,pumpingwellwillbe

affectedbytheseboundaries.Onecanusemethodofimagetocalculatecorrect

drawdownattheobservationwell.

Ifboundaryisriver(watersupply)animagewellwillbeaninjectionwell.Ifboundaryisimpermeablerockanimagewellwillbeapumpingwell.

Onceanimagewelliscreated,onecanuseprincipleofsuperpositiontodetermine

accuratedrawdownfromthepumping/injectingwells.

EXAMPLEIII7



3-18

Ifboundaryisriver,imagewellofapumpingwellisaninjectionwell.Ontheother

hand,ifthewellisaninjectionwell,imagewillbeaninjectionwell.Ifboundaryisan

impermeablerock,imageofthepumpingwellisapumpingwellwhereimageofaninjection

wellisaninjectionwell.

RiverBoundary ImpermeableBoundary

Well Image Well Image

pumping injection pumping Pumping

injection Injection injection injection

ExampleIII7

AnaquiferisbeginpumpedfromPW1(pumpingwell#1).Ifthisaquiferisconnected

toanimpermeable(noflow)boundary,suggestthemethodhowtocalculatedrawdownin

anobservationwell(OW).

PW1

|OW

r1

IMPE

RM

EAB

LE B

OU

ND

AR

Y

Sincetheboundaryisimpermeable,animagewellshouldbeapumpingwelland

drawdownatOWcanbecalculatedfromPW1andPW1imgasshownbelow.

PW1

|OW

r1

IMP

ER

ME

ABL

E B

OU

ND

AR

Yd d

PW1-IMG

r2



3-19

Principleofsuperposition:

[ ]

1 1

1 1

1 1

( ) ( )4 4

( ) ( )4

total PW PW IMG

PW PW IMG

PW PW IMG

s s s

Q QW u W u

T TQ

W u W uT

= += +

= +

where21

1 4PWr S

uTt

= and22

1 4PW IMGr S

uTt

=

Seawaterintrusion

Inacoastalaquifer,freshwaterusuallyoverliesontopofsalinegroundwaterduetodensitydifference

Salinegroundwatercanmovelandwardiftoomuchpumpingoccurs.

confinedaquifer

unconfinedaquifer



3-20

Boundarybetweenfreshandsalinegroundwaterisnotsharp!Thereisazoneofdiffusionwheretheconcentration(ofsalt)gradientexists.

GhybenBerzbergPrinciple

Inanunconfinedaquifer,therelationshipbetweenwatertableanddepthtosaline

groundwaterisdescribedbythefollowingexpression:

( , ) ( , )freshsaline fresh

z x y h x y

=

where

( , )z x y depthofsaltwaterinterfacebelowsealevel(L)

( , )h x y elevationofwatertableabovesealevel(L)

fresh densityoffreshgroundwater(M/L3) saline densityofsalinegroundwater(M/L3)



3-21

ExampleIII8

Ifdensityoffreshwaterandsalinegroundwaterare1000and1025kg/m3,respectively,

whatistheratioof ( , ) ( , )z x y h x y ?

1000( , ) ( , )

1025 1000( , )

40( , )

z x y h x y

z x yh x y

= =

Thus,depthtowhichfreshgroundwaterextendsbelowsealevelisapproximately40times

theheightofwatertableabovesealevel.



3-22

Exercise

1. Awellthatpumpsataconstantrateof78,000ft3/dayhasachievedequilibriumsothatthereisnochangeinthedrawdownwithtime.Thewelltapsaconfinedaquiferthatis18ftthick.Anobservationwell125ftawayhasaheadof277ftabovesealevel;anotherobservationwell385ftawayhasaheadof291ft.ComputethevalueofaquifertransmissivityusingThiemequation.

[Ans:997.5ft2/day]2. Awellthatpumpsataconstantrateof78,000ft3/dayhasachievedequilibriumsothat

thereisnochangeinthedrawdownwithtime.Thewelltapsanunconfinedaquiferthatconsistsofsandoverlyingimpermeablebedrockatanelevationof260ftabovesealevel.Anobservationwell125ftawayhasaheadof277ftabovesealevel;anotherobservationwell385ftawayhasaheadof291ft.ComputethevalueofhydraulicconductivityusingThiemequation.

[Ans:41.6ft/day]3. Acommunityisinstallinganewwellinaregionallyconfinedaquiferwithatransmissivity

of1589ft2/dayandastorativityof0.0005.Theplannedpumpingrateis325gal/min.Thereareseveralnearbywellstappingthesameaquifer,andtheprojectmanagerneedstoknowifthenewwellwillcausesignificantinterferencewiththesewells.Computethetheoreticaldrawdowncausedbythenewwellafter30daysofcontinuouspumpingatthefollowingdiances:50,150,250,500,1000,3000,6000,and10,000ft.

[Ans:35.56,28.70,25.58,21.14,16.85,6.87,5.87,and3.18ft]4. Awellthatisscreenedinaconfinedaquiferistobepumpedatarateof165,000ft3/day

for30days.Iftheaquifertansmissivityis5320ft2/day,andthestorativityis0.0007,whatisthedrawdownatdistancesof50,150,250,500,1000,3000,5000,and10,000ft?

5. Awellisbeingpumpedfromanunconfinedaquiferthathasinitialsaturatedthicknessof

30m.Thisaquiferhassimilarverticalandhorizontalconductivities(i.e.,Kv=Kh=10m/day)withSs=0.0001m1andSy=0.2.Calculatedrawdownatobservationwell,locatedat5.477mawayfromthepumpingwell,attimet=1day(earlytime)andt=50day(latetime).UseQ=100m3/day.

6. Awell(PW1)ispumpingwaterfrom

confinedaquifer,thatisclosetoanimpermeablerock,atarateof100m3/day.Calculatedrawdownatobservationwell(OW)attimet=5d.GivenS=0.0005,T=500m2/day.

PW1

|OW

r1 = 100 m

IMPE

RM

EABL

E BO

UN

DA

RY150 m

Q1 = -100 m3/day

60



3-23

7. Awell(PW1)ispumpingwaterfromconfinedaquifer,thatisclosetoafullypenetratedriver,atarateof150m3/day.Calculatedrawdownatobservationwell(OW)attimet=10d.GivenS=0.0005,T=500m2/day.

PW1

|OW

r1 = 75 m

RIV

ER

120 mQ1 = -150 m3/day

75

gw flow & well hydraulics

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