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Groundwater Flow Equations
Groundwater Hydraulics
Daene C. McKinney
Summary
• General Groundwater Flow– Control Volume Analysis– General Continuity Equation
• Confined Aquifer Flow– Continuity Equation– Integrate over vertical dimension– Transmissivity– Continuity– Examples
• Unconfined Aquifer Flow– Darcy Law– Continuity Equation– Examples
Control Volume• Control volume of dimensions Dx, Dy, Dz
• Completely saturated with a fluid of density r
x
yz
Mass flux in Mass flux out
2
xx
D
2
xx
D
2
)( x
x
qq x
x
D
rr
2
)( x
x
qq x
x
D
rr
xD
x
yD
zDControl
volume
Mass Flux
• Mass flux = Mass in - Mass out:
mass flux in mass flux out2
xx
D
2
xx
D
2
)( x
x
qq x
x
D
rr
2
)( x
x
qq x
x
D
rr
xD
x
yD
zD
rqx -¶ rqx( )¶x
Dx
2
é
ëê
ù
ûúDyDz - rqx +
¶ rqx( )¶x
Dx
2
é
ëê
ù
ûúDyDz = -
¶ rqx( )¶x
DV
Mass fluxMass in Mass out
Mass Flux
• Mass flux =
• Continuity: Mass flux = change of mass
• Fluid mass in the volume:
• Continuity
mass flux in mass flux out2
xx
D
2
xx
D
2
)( x
x
qq x
x
D
rr
2
)( x
x
qq x
x
D
rr
xD
x
yD
zD
-¶ rqx( )¶x
DV
m = frDV
Mass flux change of mass
Aquifer Storage
Water compressibility Aquifer compressibility
Chain rule
Now, put it back into the continuity equation
Continuity Equation
¶
¶xKx
¶h
¶x
æ
è ç
ö
ø ÷ +
¶
¶yKy
¶h
¶y
æ
è ç
ö
ø ÷ +
¶
¶zKz
¶h
¶z
æ
è ç
ö
ø ÷ = S
¶h
¶t
Confined Aquifer Flow
Horizontal Aquifer Flow
• Most aquifers are thin compared to horizontal extent– Flow is horizontal, qx and qy
– No vertical flow, qz = 0
– Average properties over aquifer thickness (b)
h(x,y,t)= 1
b h(x,y,z,t)dz0
bò
Ground surface
Bedrock
Confined aquiferQx
K
x
yz
h
Head in confined aquifer
Confining Layer
b
qx(x,y,t)= 1
b qx(x,y,z,t)dz0
bò
Qx = bqx
Aquifer Transmissivity
• Transmissivity (T) – Discharge through thickness of
aquifer per unit width per unit head gradient
– Product of conductivity and thickness
Hydraulic
gradient = 1 m/m
b
1 m
1 m
1 m
Transmissivity, T, volume
of water flowing an area 1
m x b under hydraulic
gradient of 1 m/m
Conductivity, K, volume of water
flowing an area 1 m x 1 m under
hydraulic gradient of 1 m/m
Continuity Equation
• Continuity equation
• Darcy’s Law
• Continuity
-¶Qx
¶x= S
¶h
¶t
Qx = -Tx¶h
¶x
¶
¶xTx
¶h
¶x
æ
èç
ö
ø÷ = S
¶h
¶t
Ground surface
Bedrock
Confined aquiferQx
K
x
yz
h
Head in confined aquifer
Confining Layer
b
1
r
¶
¶rr
¶h
¶r
æ
èç
ö
ø÷ =
S
T
¶h
¶t
Radial Coordinates
Example – Horizontal Flow
• Consider steady flow from left to right in a confined aquifer
• Find: Head in the aquifer, h(x)
¶
¶xT
¶h
¶x
æ
èç
ö
ø÷ = S
¶h
¶t= 0
Td2h
¶x2= 0
Ground surface
Bedrock
Confined aquifer
Qx
K
xyz
hB
Confining Layer
b
hA
L
steady flow
h(x)
Example – Horizontal Flow• L = 1000 m, hA = 100 m, hB = 80 m, K = 20 m/d, f = 0.35
• Find: head, specific discharge, and average velocity
Ground surface
Bedrock
Confined aquifer
Qx
K=2-m/dxyz
hB=80m
Confining Layer
b
hA=100m
L=1000m
Unconfined Aquifer Flow
Flow in an Unconfined Aquifer
• Dupuit approximations – Slope of the water table is small
– Velocities are horizontal
Ground surface
Bedrock
Unconfined aquifer
Water table
Dx
Qx
K
h
x
yz
Qx = qxh = (-K¶h
¶x)h
-¶Qx
¶x= Sy
¶h
¶t
¶
¶xKh
¶h
¶x
æ
è ç
ö
ø ÷ = Sy
¶h
¶t
Steady Flow in an Unconfined Aquifer
• 1-D flow
• Steady State,
• K = constant
• Find h(x)
¶
¶xKh
¶h
¶x
æ
è ç
ö
ø ÷ = Sy
¶h
¶t
h
FlowhA
hB
Water Table
Ground Surface
Bedrock Lx
Steady Flow in an Unconfined Aquifer
• K = 10-1 cm/sec
• L = 150 m
• hA = 6.5 m
• hB = 4 m
• x = 150 m
• Find h(x), Q
h
FlowhA=6.5m
hB=4m
Water Table
Ground Surface
Bedrock L=150mx
K=0.1cm/s
Summary
• General Groundwater Flow– Control Volume Analysis– General Continuity Equation
• Confined Aquifer Flow– Continuity Equation– Integrate over vertical dimension– Transmissivity– Continuity– Examples
• Unconfined Aquifer Flow– Darcy Law– Continuity Equation– Examples
Groundwater Flow EquationsExamples
Example – Horizontal Flow
• Consider steady flow from left to right in a confined aquifer
• Find: Head in the aquifer, h(x)
¶
¶xT
¶h
¶x
æ
èç
ö
ø÷ = S
¶h
¶t= 0
Td2h
¶x2= 0
h(x) = hA + hB - hA
Lx
Ground surface
Bedrock
Confined aquifer
Qx
K
xyz
hB
Confining Layer
b
hA
L
steady flow
Head in the aquifer
h(x)
Example – Horizontal Flow• L = 1000 m, hA = 100 m, hB = 80 m, K = 20 m/d, f = 0.35
• Find: head, specific discharge, and average velocity
h(x) = hA +hB - hA
Lx =100 - 0.02x m
q = -KhB - hA
L
= -(20 m /d)80 -100
1000
= 0.4 m /day
v =q
f
=1.14 m /day
Ground surface
Bedrock
Confined aquifer
Qx
K=2-m/dxyz
hB=80m
Confining Layer
b
hA=100m
L=1000m
Steady Flow in an Unconfined Aquifer
• 1-D flow
• Steady State,
• K = constant
¶
¶xKh
¶h
¶x
æ
è ç
ö
ø ÷ = Sy
¶h
¶t
d
dxKh
dh
dx
æ
è ç
ö
ø ÷ = 0
h2(x) = hA2 + (
hB2 - hA
2
L)x
h
FlowhA
hB
Water Table
Ground Surface
Bedrock Lx
Q = (-Kdh
dx)h = -
K
2
dh2
dx= -
K
2
hB2 - hA
2
L
æ
è ç
ö
ø ÷
Steady Flow in an Unconfined Aquifer
• K = 10-1 cm/sec
• L = 150 m
• hA = 6.5 m
• hB = 4 m
• x = 150 m
• Find Q
h
FlowhA=6.5m
hB=4m
Water Table
Ground Surface
Bedrock L=150mx
Q = -K
2
hB2 - hA
2
L
æ
è ç
ö
ø ÷ = -
86.4 m /d
2
6.52 - 42
150
æ
è
ç ç
ö
ø
÷ ÷
= 7.56 m3 /d /m
K=0.1cm/s