soil physics 2010 outline announcements richards’ equation unsaturated flow
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Soil Physics 2010
Outline
• Announcements
• Richards’ equation
• Unsaturated flow
Soil Physics 2010
Announcements
• Homework 4 due March 3
• Excel Solver demo on course website
• Quiz!
h
Soil Physics 2010
Question 1
Drying
Wetting
h
0
0
h
0
0
Soil Physics 2010
Question 2 Different lines show different possibilities
Soil Physics 2010
Why different flow equations?
Steady-state Transient
Saturated
Unsaturated
Darcy’s law
Darcy’s law (with K())
N/A
Richards’ equation
Darcy’s law:L
AKq
changes with time
No K()
No No ()
Soil Physics 2010
Equation of Continuity(Conservation of Mass)
Steady-state Transient
Saturated
Unsaturated
Darcy’s law
Darcy’s law (with K())
Richards’ equation
Input – Output = Change in Storage
x
q
=t
tx
q
Soil Physics 2010
Richards’ equation
LKq
Given Darcy’s law:
xK
xx
q
Let things change from place to place (say, in the x-direction)
tx
q
We also want
conservation of mass
xK
xt
So we substitute it in
to the left-hand side
Soil Physics 2010
Richards’ equation
xK
xt
But this doesn’t allow
K to change with
xK
xt
So we permit that, and…
voilà: Richards’ equation
zK
zyK
yxK
xt zyx
We can generalize it to 2 or 3 dimensions…
… and add in anisotropy
Soil Physics 2010
Richards’ equation
xK
xt
Remember that the
potential gradient, ,
combines elevation, osmotic, pressure, and matric components (among others).
x
Sometimes it’s convenient to separate out the elevation part:
1 x
Kxt
Vertical
0 x
Kxt
Horizontal
Just remember that this doesn’t include elevation!
Soil Physics 2010
Topp & Dane, Methods of soil analysis
K(), averages by texture
Coarse soils:Lower Higher Ks
More abrupt drop
At low :Small → big KHuge range of KHuge uncertainty in K
Soil Physics 2010
K() and K() for 3 textures(Mualem-van Genuchten functions)
1.E-16
1.E-14
1.E-12
1.E-10
1.E-08
1.E-06
1.E-04
1.E-02
1.E+00
1.E+02
0 0.1 0.2 0.3 0.4 0.5 0.6
Sand
clay
loam
K()
1.E-16
1.E-14
1.E-12
1.E-10
1.E-08
1.E-06
1.E-04
1.E-02
1.E+00
1.E+02
1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04
Sand
clay
loam
K()
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
0 0.1 0.2 0.3 0.4 0.5 0.6
Sand
clay
loam
()
K() has more hysteresis
Soil Physics 2010
How do we measure K() in the lab?Ks is pretty easy.K() is slow, and hard to control.
• Apply water at steady q < Ks
• Wait till outflow = inflow
• Measure and/or across a “test interval”• Prevent evaporation
• Water evenly, no disturbance
• Tall column, or tension at bottom
• Tensiometer can change flow
• Measure with gamma-rays
Soil Physics 2010
How do we measure K() in the lab?
K() is slow, and hard to control.
Other methods:
• Centrifuge
• Evaporation
• One-step
• Multi-step
As decreases:SlowerHarder to controlMore uncertainty
Soil Physics 2010
How do we measure K() in the field?
• Instantaneous profile• Various others• Best solved with Inverse methods
The “forward problem”:Given the parameters and boundary conditions, simulate what happened (or will happen).
The “inverse problem:Given the data and the boundary conditions, estimate the parameter values. (A spreadsheet’s Solver solves an inverse problem.)