Transcript Saturated flow - Soil Physics, Iowa State University

Outline
• Announcements
• Todd on aggregates
• Saturated flow of water
Soil Physics 2010
Announcements
• No office hours today
• Reminder: Homework due Feb. 10
• Reminder: Exam Feb. 12
• Example exam is now posted.
Don’t panic! I covered material in a
different order that year, and the class
was not dual-listed at the 400-level.
Soil Physics 2010
Todd on Aggregates
Soil Physics 2010
Saturated flow
When is the soil saturated?
Near the surface under heavy rain
Confined aquifer
Under a lake or stream
Recharge basin
Go deep enough…
It turns out that the saturated flow equations are
just special (simpler) cases of flow more generally.
Soil Physics 2010
The city of Dijon
1800
Soil Physics 2010
Ancient capitol fallen
on hard times
France, 1800s
Political turmoil:
Paris
• 1804 - 1815 First Empire - Napoleon
Dijon
• 1815 - 1830 Restoration - Louis
XVIII - Charles X
• 1830 - 1848 Bourgeois Monarchy Louis-Philippe
• 1848 – 1851 Second Republic
• 1852 - 1870 Second Empire Napoleon III
Soil Physics 2010
Lyon
Henry Darcy
Fellow students at
L’École des Ponts et Chaussées*:
Cauchy
Chézy
Coliolis
Dupuit
Fresnel
Navier
Pitot
St. Venant
Henry graduated 12th in
his class. He had been 1st,
but he got violently upset
with a chemistry professor
over a question about
* The school of Bridges and Roads
cooking.
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Most famous publication
Soil Physics 2010
Darcy’s experiment
reported in Appendix D of
Les Fontaines Publiques…
Soil Physics 2010
Henry Darcy
• Public water supply in Dijon: clean,
reliable, gravity flow 12 km to 142
public fountains
• 4 km railroad tunnel allowed the main
Paris-Lyon track to pass through Dijon
• Several large bridges near Dijon
• Improvements to the Pitot tube
• Extensive studies on:
Flow through open channels
“One does not get the main
Flow through pipes
square in town named in one’s
Road construction
• Near the end of his life, a few
studies on flow through sand
Soil Physics 2010
honor on the basis of some
column experiments.”
-- Allan Freeze, 1994
Henry Darcy
“As much as possible, one should favor the free drawing of water
because it is necessary for public health. A city that cares for the
interest of the poor class should not limit their water, just as
daytime and light are not limited.”
Soil Physics 2010
Darcy’s experiment
• Column filled with sand
to different depths
• Different water pressures 2.5 m
applied across column
• Discharge Q measured
(volume of water / time)
• Experiment repeated with
different sands
1 m3
Soil Physics 2010
Darcy’s 3 observations:
1: Q  cross-sectional Area
2: Q  drop in (pressure + elevation)
3: Q  1 / flow distance
Soil Physics 2010
State of the art
For slow flow through a pipe,
 R Dp
Q
8h L
4
(Poiseuille’s law)
Q discharge
R radius
h viscosity
Dp pressure drop
L length
R Dp
u
8h L
2
Mean velocity is
Soil Physics 2010
Darcy’s 1st observation
Darcy’s experiments used a vertical column, but a horizontal column is simpler.
1: Q  cross-sectional Area
2: Q  drop in (pressure + elevation)
3: Q  1 / flow distance
Soil Physics 2010
Assume zero resistance in pipes
Darcy’s 2nd observation
1: Q  cross-sectional Area
2: Q  drop in (pressure + elevation)
3: Q  1 / flow distance
Soil Physics 2010
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 energy is the same all
through the pool. Surface
water has elevation; deep
water has pressure.
Pressure
+ Elevation
Energy
Darcy’s 3rd observation
1: Q  cross-sectional Area
2: Q  drop in (pressure + elevation)
3: Q  1 / flow distance
L
Soil Physics 2010
Darcy’s Law as he wrote it
Water
pressure
Height

h1  z1   h2  z2 
Q  KA
L
Cross-sectional
area of flow
Water volume /
unit time
Soil Physics 2010
Length
of flow
Proportionality
coefficient:
Hydraulic
conductivity
Units in Darcy’s Law

h1  z1   h2  z2 
Q  KA
L
L
L 2 LL
 L
T T
L
3
Velocity
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Unitless
Key implications of Darcy’s law: 1
 h2  h1 
Q   KA

 L 
For flow through a uniform medium,
the hydraulic gradient is constant.
Taking the derivative of Darcy’s law,
q   Kh
gives the Laplace equation,
 h0
2
Soil Physics 2010
Key implications of Darcy’s law: 2
The flow is linearly proportional to the gradient.
This puts Darcy’s law into the same class as several other equations:
1
  
E
q   Kh
j  V
qh  T
f   DC
Soil Physics 2010
Hooke’s law (elasticity)
Darcy’s law (hydraulic conductivity)
Ohm’s law (electrical conductivity)
Fourier’s law (heat conduction)
Fick’s law (diffusion)
Key implications of Darcy’s law: 3
K is a property of the medium*.
The hydraulic conductivity K is not changed
by whether the water flows up instead of down,
or by having a greater or smaller gradient,
or by the pressures or elevations themselves.
In fact, K can often be predicted with reasonable
accuracy, given some other information about the
medium. For example, the porosity and the grain
size or pore size distribution allow a fair estimate.
*
Soil Physics 2010
Also of the fluid – see later