soil physics 2010
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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?. - PowerPoint PPT PresentationTRANSCRIPT
Soil Physics 2010
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?
L
zhzhKAQ 2211
HeightWater
pressure
Water volume / unit
time
Length of flow
Cross-sectional area of flow
Proportionality coefficient:Hydraulic
conductivity
Pressure = Elevation?
When you swim underwater, your ears feel pressure
Why doesn’t the water at the bottom of the pool – under lots of pressure – shoot up to the top?
The water’s potential energy is the same all through the pool. Surface water has elevation; deep water has pressure.
Dep
th
Pressure
Elevat
ion
Soil Physics 2010
Pressure+ Elevation
Potential Energy
Soil Physics 2010
Units in Darcy’s Law
L
hKAQ
L
LLL
T
L
T
L 2
3
UnitlessVelocity
Usually we give the pressure term in
units of length, so the gradient is
unitless
Soil Physics 2010
Is this velocity how fast the water moves?
L
LLL
T
L
T
L 2
3
Velocity
No.
Water flows only through the pores.
Water flows through an area AWater flows at mean velocity
L
hKAQ
Lh
L
hK
A
Qu
Soil Physics 2010
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, useL
hKq
What are the units of q?
Soil Physics 2010
More on Darcy’s law
Elevation: potential energy z
Pressure: “virtual” elevation p/gVelocity: kinetic energy v2/2g
The energy gradient has 3 components:
Soil Physics 2010
Total potential energy
The potential energy of water can be expressed several different ways:
energy per unit mass: J kg-1
energy per unit volume p N m-2 = Pa
energy per unit weight h m H2O
= p / w
h = p / w g = / g
Basis symbol units
It is convenient to think of the energy in terms
of h
Soil Physics 2010
Darcy in layered systems
Steady-state flowUnit gradient overall
L1 =L2
q1 = ? q2 = ?
K2= 0.1 cm/s
K1= 0.2 cm/s
L
hKq
h1 = ?
h2 = ?Atmosperic
pressure at top & bottom
Soil Physics 2010
Continuity requires greater gradient for smaller K
Darcy in layered systems21 qq
2
22
1
11 L
hK
L
hK
K2= 0.1 cm/s
K1= 0.2 cm/s
2211 hKhK L1 = L2, so
L1 =L2
Soil Physics 2010
K2= 0.1 cm/s
K1= 0.2 cm/s
Darcy in layered systems
2121 LLhh
2211 hKhK
21 1.0 2.0 hh
211 3
1LLh
211 3
2LLh
Soil Physics 2010
Darcy in artificial systems:
20cm
40cm
60cm
DCB
A
E
Given this system, with steady-state water flow,what are the values of the head components at each point?
Soil Physics 2010
20cm
40cm
60cm
DCBA
E
pressure p = 0 at points A and EWe know:
Elevations can be read from the diagram
elevation+ pressuretotal head (energy)
Steady-state flow→ q is the same everywhere
→ linear energy gradient
Darcy in artificial systems:
Soil Physics 2010
Construct a table: Elevation + pressure = Total
A
B
C
D
E
Pressure = 0 at A and E
0
0
20cm
40cm
60cm
DCBA
E
Darcy in artificial systems:
Soil Physics 2010
Take E as reference height
40
60
60
60
0
0
0
20cm
40cm
60cm
DCBA
E
Darcy in artificial systems:
Elevation + pressure = Total
A
B
C
D
E
Elevation + pressure = Total
Soil Physics 2010
40
60
60
60
0
0
0
40
0
Elevation + pressure = Total
20cm
40cm
60cm
DCBA
E
Darcy in artificial systems:
Elevation + pressure = Total
A
B
C
D
E
Elevation + pressure = Total
Soil Physics 2010
40
60
60
60
0
0
0
40
33.3
26.6
20
0
Uniform medium: linear drop in head with distance
20cm
40cm
60cm
DCBA
E
Darcy in artificial systems:
Elevation + pressure = Total
A
B
C
D
E
so at 1/6 of L, we’ve used 1/6 of hL=120 cm h = 40cm 5/6 * 40 = 33.3
Elevation + pressure = Total
Soil Physics 2010
40
60
60
60
0
0
-26.7
-33.3
-40
0
40
33.3
26.7
20
0
Fill in the rest by difference
20cm
40cm
60cm
DCBA
E
Darcy in artificial systems:
Elevation + pressure = Total
A
B
C
D
E
Elevation + pressure = Total
Soil Physics 2010
Darcy in artificial systems:
Summary:
• You can use the pieces you know to assemble the whole puzzle.
• Every piece of information is needed: data and theory.
20cm
40cm
60cm
DCBA
E