Outline
• Announcements
• Where were we?
• More saturated flow
Soil Physics 2010
Announcements
• Homework is due now. If all
homeworks are handed in now, I will
post my answers right after class
• Reminder: Exam Friday
• Example exam is posted.
Soil Physics 2010
Where were we?
Q  KA
Water
pressure
h1  z1   h 2 
L
Cross-sectional
area of flow
Water volume /
unit time
Soil Physics 2010
Length
of flow
Proportionality
coefficient:
Hydraulic
conductivity
Height
z2 
Pressure = Elevation?
When you swim underwater,
your ears feel pressure
Depth
Why doesn’t the water at the
bottom of the pool – under lots of
pressure – shoot up to the top?
Soil Physics 2010
The water’s potential energy is
the same all through the pool.
Surface water has elevation;
deep water has pressure.
Pressure
+ Elevation
Potential Energy
Units in Darcy’s Law
Q  KA
3
L

T
Soil Physics 2010
L
2
L
h
L
LL
T
L
Velocity
Unitless
Usually we give the
pressure term in units
of length, so the
gradient is unitless
Is this velocity how fast the water moves?
3
L

T
L
2
L
T
L L
L
h
Q  KA
Velocity
No.
L
h
L
Water flows only through the pores.
Water flows through an area Af
Water flows at mean velocity u 
Soil Physics 2010
Q
Af

K h
f L
Key implications of Darcy’s law
For flow through a uniform medium,
the hydraulic gradient is constant.
The flow is linearly proportional to
the gradient, as in Hooke’s law, Fick’s
law, Fourier’s law, etc.
K is a property of the medium.
Soil Physics 2010
More on Darcy’s law
There is no flow without an
energy (hydraulic) gradient
Components of the gradient:
• elevation
• pressure
• velocity (?)
For unit area, use q  K
Soil Physics 2010
h
L
What are the units of q?
More on Darcy’s law
The energy gradient has 3 components:
Elevation:
potential energy
Pressure:
“virtual” elevation
Velocity:
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kinetic energy
z
p/rg
v2/2g
Total potential energy
The potential energy of water can be expressed
several different ways:
Basis
symbol units
energy per unit mass:
y
J kg-1
energy per unit volume
p
N m-2 = Pa
energy per unit weight
h
m H2O
It is convenient to think of
the energy in terms of h
Soil Physics 2010
y = p / rw
h = p / rw g = y / g
Darcy in layered systems
Steady-state flow
Unit gradient overall
K1= 0.2 cm/s
Atmosperic
pressure at top
& bottom
K2= 0.1 cm/s
L1
=
L2
h1 = ?
h2 = ?
q K
Soil Physics 2010
h
L
q1 = ? q2 = ?
Darcy in layered systems
q1  q 2
 h1
K1= 0.2 cm/s
K2= 0.1 cm/s
 h2
K1
 K2
L2
L1 L1
=
L2 L1 = L2, so
K 1  h1  K 2  h 2
Continuity requires greater gradient for smaller K
Soil Physics 2010
Darcy in layered systems
K 1  h1  K 2  h 2
0 . 2  h1  0 . 1  h 2
K1= 0.2 cm/s
K2= 0.1 cm/s
h1 
1
h1 
2
3
3
L1  L2 
L1  L2 
 h1   h 2  L1  L 2
Soil Physics 2010
Darcy in artificial systems:
Given this system, with steady-state water flow,
what are the values of the head components at each point?
40cm
20cm
C
B
D
A
60cm
E
Soil Physics 2010
Darcy in artificial systems:
We know:
pressure p = 0 at
points A and E
40cm
20cm
C
B
Elevations can be read
from the diagram
D
A
60cm
E
elevation
+ pressure
total head (energy)
Steady-state flow
→ q is the same everywhere
→ linear energy gradient
Soil Physics 2010
Darcy in artificial systems:
Construct a table:
40cm
20cm
C
B
Elevation + pressure = Total
A
D
0
B
A
60cm
E
C
D
E
0
Pressure = 0 at A and E
Soil Physics 2010
Darcy in artificial systems:
Elevation + pressure = Total
40cm
20cm
C
B
D
A
60cm
E
A
40
B
60
C
60
D
60
E
0
0
0
Take E as reference height
Soil Physics 2010
Darcy in artificial systems:
Elevation + pressure = Total
40cm
20cm
C
B
D
A
60cm
E
A
40
B
60
C
60
D
60
E
0
0
40
0
0
Elevation + pressure = Total
Soil Physics 2010
Darcy in artificial systems:
Elevation + pressure = Total
40cm
20cm
C
B
D
A
60cm
E
A
40
B
60
33.3
C
60
26.6
D
60
20
E
0
0
0
40
0
Uniform medium: linear drop in head with distance
so at 1/6 of L, we’ve used 1/6 of h
Soil Physics 2010
L=120 cm
h = 40cm
5/6 * 40 = 33.3
Darcy in artificial systems:
Elevation + pressure = Total
40cm
20cm
C
B
D
A
60cm
E
A
40
0
40
B
60
-26.7
33.3
C
60
-33.3
26.7
D
60
-40
20
E
0
0
0
Fill in the rest by difference
Soil Physics 2010
Darcy in artificial systems:
Summary:
40cm
20cm
C
B
D
60cm
• You can use the pieces you
know to assemble the
whole puzzle.
E
• Every piece of information
is needed: data and theory.
A
Soil Physics 2010