hidraULICA

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