- The document discusses equations for analyzing groundwater flow in confined and unconfined aquifers. - For confined aquifers, the continuity equation is integrated over the aquifer thickness to derive an equation using transmissivity. Examples are presented of steady horizontal and radial flow. - For unconfined aquifers, Dupuit assumptions are used and the continuity equation is solved for steady 1D flow using the water table elevation. Worked examples are provided for both confined and unconfined cases.

1. Groundwater Flow Equations
Groundwater Hydraulics
Daene C. McKinney

2. 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

3. 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
x
x
D

2
x
x
D

2
)( x
x
q
q x
x
D



r
r
2
)( x
x
q
q x
x
D



r
r
xD
x
yD
zD
Control
volume

4. Mass Flux
• Mass flux = Mass in - Mass out:
mass flux in mass flux out2
x
x
D

2
x
x
D

2
)( x
x
q
q x
x
D



r
r
2
)( x
x
q
q x
x
D



r
r
xD
x
yD
zD
rqx -
¶ rqx( )
¶x
Dx
2
é
ë
ê
ù
û
úDyDz - rqx +
¶ rqx( )
¶x
Dx
2
é
ë
ê
ù
û
úDyDz = -
¶ rqx( )
¶x
DV
Mass fluxMass in Mass out

5. Mass Flux
• Mass flux =
• Continuity: Mass flux = change of mass
• Fluid mass in the volume:
• Continuity
mass flux in mass flux out2
x
x
D

2
x
x
D

2
)( x
x
q
q x
x
D



r
r
2
)( x
x
q
q x
x
D



r
r
xD
x
yD
zD
-
¶ rqx( )
¶x
DV
m =frDV
Mass flux change of mass

6. Aquifer Storage
Water compressibility Aquifer compressibility
Chain rule
Now, put it back into the continuity equation

7. Continuity Equation
¶
¶x
Kx
¶h
¶x
æ
è
ç
ö
ø
÷+
¶
¶y
Ky
¶h
¶y
æ
è
ç
ö
ø
÷+
¶
¶z
Kz
¶h
¶z
æ
è
ç
ö
ø
÷ = S
¶h
¶t

8. Confined Aquifer Flow

9. 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)dz
0
b
ò
Ground surface
Bedrock
Confined aquifer
Qx
K
x
yz
h
Head in confined aquifer
Confining Layer
b
qx(x,y,t)=
1
b
qx(x,y,z,t)dz
0
b
ò Qx = bqx

10. 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

11. Continuity Equation
• Continuity equation
• Darcy’s Law
• Continuity
-
¶Qx
¶x
= S
¶h
¶t
Qx = -Tx
¶h
¶x
¶
¶x
Tx
¶h
¶x
æ
è
ç
ö
ø
÷ = S
¶h
¶t
Ground surface
Bedrock
Confined aquifer
Qx
K
x
yz
h
Head in confined aquifer
Confining Layer
b
1
r
¶
¶r
r
¶h
¶r
æ
è
ç
ö
ø
÷ =
S
T
¶h
¶t
Radial Coordinates

12. Example – Horizontal Flow
• Consider steady flow from left to right in a confined aquifer
• Find: Head in the aquifer, h(x)
¶
¶x
T
¶h
¶x
æ
è
ç
ö
ø
÷ = S
¶h
¶t
= 0
T
d2
h
¶x2
= 0
Ground surface
Bedrock
Confined aquifer
Qx
K
x
yz
hB
Confining Layer
b
hA
L
steady flow
h(x)

13. 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/d
x
yz
hB=80m
Confining Layer
b
hA=100m
L=1000m

14. Unconfined Aquifer Flow

15. 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
¶
¶x
Kh
¶h
¶x
æ
è
ç
ö
ø
÷ = Sy
¶h
¶t

16. Steady Flow in an Unconfined Aquifer
• 1-D flow
• Steady State,
• K = constant
• Find h(x)
¶
¶x
Kh
¶h
¶x
æ
è
ç
ö
ø
÷ = Sy
¶h
¶t
h
FlowhA
hB
Water Table
Ground Surface
Bedrock L
x

17. 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=150m
x
K=0.1cm/s

18. 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

19. Groundwater Flow Equations
Examples

20. Example – Horizontal Flow
• Consider steady flow from left to right in a confined aquifer
• Find: Head in the aquifer, h(x)
¶
¶x
T
¶h
¶x
æ
è
ç
ö
ø
÷ = S
¶h
¶t
= 0
T
d2
h
¶x2
= 0
h(x) = hA +
hB - hA
L
x
Ground surface
Bedrock
Confined aquifer
Qx
K
x
yz
hB
Confining Layer
b
hA
L
steady flow
Head in the aquifer
h(x)

21. 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
L
x =100- 0.02x m q = -K
hB - 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/d
x
yz
hB=80m
Confining Layer
b
hA=100m
L=1000m

22. Steady Flow in an Unconfined Aquifer
• 1-D flow
• Steady State,
• K = constant
¶
¶x
Kh
¶h
¶x
æ
è
ç
ö
ø
÷ = Sy
¶h
¶t
d
dx
Kh
dh
dx
æ
è
ç
ö
ø
÷ = 0
h2
(x) = hA
2
+(
hB
2
- hA
2
L
)x
h
FlowhA
hB
Water Table
Ground Surface
Bedrock L
x
Q = (-K
dh
dx
)h = -
K
2
dh2
dx
= -
K
2
hB
2
- hA
2
L
æ
è
ç
ö
ø
÷

23. 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=150m
x
Q = -
K
2
hB
2
- 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