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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 = & &
GEOL473 Hydrogeology
Dr. Schradh Saenton Topic 3: Groundwater Flow and Well Hydraulics
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=
GEOL473 Hydrogeology
Dr. Schradh Saenton Topic 3: Groundwater Flow and Well Hydraulics
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
GEOL473 Hydrogeology
Dr. Schradh Saenton Topic 3: Groundwater Flow and Well Hydraulics
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
GEOL473 Hydrogeology
Dr. Schradh Saenton Topic 3: Groundwater Flow and Well Hydraulics
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
GEOL473 Hydrogeology
Dr. Schradh Saenton Topic 3: Groundwater Flow and Well Hydraulics
3-6
Pumpinginanaquiferwithdifferenttransmissivity(T)andstorativity(S)
HighS LowS
IMPERMEABLE ROCK
Confined Aquifer
GroundWell
Potentiometric Surface t = 0
IMPERMEABLE ROCKConfined Aquifer
GroundWell
Potentiometric Surface t = 0
HighT LowT
IMPERMEABLE ROCK
Confined Aquifer
GroundWell
Potentiometric Surface t = 0
IMPERMEABLE ROCKConfined Aquifer
GroundWell
Potentiometric Surface t = 0
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]
GEOL473 Hydrogeology
Dr. Schradh Saenton Topic 3: Groundwater Flow and Well Hydraulics
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
GEOL473 Hydrogeology
Dr. Schradh Saenton Topic 3: Groundwater Flow and Well Hydraulics
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
GEOL473 Hydrogeology
Dr. Schradh Saenton Topic 3: Groundwater Flow and Well Hydraulics
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
GEOL473 Hydrogeology
Dr. Schradh Saenton Topic 3: Groundwater Flow and Well Hydraulics
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 = +
GEOL473 Hydrogeology
Dr. Schradh Saenton Topic 3: Groundwater Flow and Well Hydraulics
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)
GEOL473 Hydrogeology
Dr. Schradh Saenton Topic 3: Groundwater Flow and Well Hydraulics
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
GEOL473 Hydrogeology
Dr. Schradh Saenton Topic 3: Groundwater Flow and Well Hydraulics
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.
GEOL473 Hydrogeology
Dr. Schradh Saenton Topic 3: Groundwater Flow and Well Hydraulics
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!!
GEOL473 Hydrogeology
Dr. Schradh Saenton Topic 3: Groundwater Flow and Well Hydraulics
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).
GEOL473 Hydrogeology
Dr. Schradh Saenton Topic 3: Groundwater Flow and Well Hydraulics
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
GEOL473 Hydrogeology
Dr. Schradh Saenton Topic 3: Groundwater Flow and Well Hydraulics
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
GEOL473 Hydrogeology
Dr. Schradh Saenton Topic 3: Groundwater Flow and Well Hydraulics
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
GEOL473 Hydrogeology
Dr. Schradh Saenton Topic 3: Groundwater Flow and Well Hydraulics
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
GEOL473 Hydrogeology
Dr. Schradh Saenton Topic 3: Groundwater Flow and Well Hydraulics
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)
GEOL473 Hydrogeology
Dr. Schradh Saenton Topic 3: Groundwater Flow and Well Hydraulics
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.
GEOL473 Hydrogeology
Dr. Schradh Saenton Topic 3: Groundwater Flow and Well Hydraulics
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
GEOL473 Hydrogeology
Dr. Schradh Saenton Topic 3: Groundwater Flow and Well Hydraulics
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