Outline
• Announcements
• Richards’ equation
• Unsaturated flow
Soil Physics 2010
Announcements
• Homework 4 due March 3
• Excel Solver demo on course website
•
Soil Physics 2010
Quiz!
Question 1
Drying
h
Wetting
q
Soil Physics 2010
Question 2
Soil Physics 2010
Different lines show
different possibilities
h
h
0
0
0
q
0
y
Why different flow equations?
Steady-state
Transient
Saturated
Darcy’s law
N/A
Unsaturated
Darcy’s law
(with K(q))
Richards’
equation
q changes
with time
No K(q)
y
Darcy’s law: q  K A
L
Soil Physics 2010
No q
No q(y)
Equation of Continuity
(Conservation of Mass)
Steady-state
Saturated Darcy’s law
Unsaturated
Darcy’s law
(with K(q))
Transient
Richards’
equation
Input – Output = Change in Storage
q

x
Soil Physics 2010
=
q
t
q q


x t
Richards’ equation
Given Darcy’s law:
y
qK
L
Let things change
from place to place
q   y 
 K

x x  x 
(say, in the x-direction)
q q
We also want


conservation of mass
x t
So we substitute it in q     K y 
to the left-hand side t
x  x 
Soil Physics 2010
Richards’ equation
But this doesn’t allow
K to change with q
q
  y 
  K
t
x  x 
q
 
y 
   K q 
So we permit that, and…

t
x 
x 
voilà:
Richards’ equation
We can generalize it to 2 or 3 dimensions…
q
 
y   
y   
y 
   K x q 
  K y q 
  K z q 


t
x 
x  y 
y  z 
z 
Soil Physics 2010
… and add in anisotropy
Richards’ equation
Remember that the
q
 
y 
   K q 
t
x 
x 
y
potential gradient,
,
x
combines elevation, osmotic, pressure, and
matric components (among others).
Sometimes it’s
q
 
 y

   K q  
 1
convenient to
t
x 
 x

separate out the
elevation part: q     K q   y  0 


t
x 
 x

Vertical
Horizontal
Just remember that this y doesn’t include elevation!
Soil Physics 2010
K(q), averages by texture
Coarse soils:
Lower f
Higher Ks
More abrupt drop
Topp & Dane,
Methods of
soil analysis
At low q:
Small q → big K
Huge range of K
Huge uncertainty in K
Soil Physics 2010
K(q) and K(y) for 3 textures
(Mualem-van Genuchten functions)
K(q)
1.E+02
K(y)
1.E+02
1.E-02
1.E+00
1.E-01
1.E+00
1.E-02
1.E-04
1.E-04
1.E+00
0
0.1
0.2
0.3
0.4
0.5
0.6
1.E-02
1.E+01
1.E+02
1.E+03
1.E+04
Sand
clay
loam
1.E-06
1.E-06
1.E-08
1.E-08
Sand
1.E-10
1.E-10
clay
loam
1.E-12
1.E-12
1.E+04
1.E-14
1.E-14
Sand
1.E+03
clay
1.E-16
1.E-16
loam
1.E+02
1.E+01
1.E+00
0
1.E-01
Soil Physics 2010
1.E-02
0.1
0.2
q (y )
0.3
0.4
0.5
0.6
K(y) has more
hysteresis
How do we measure K(q) in the lab?
Ks is pretty easy.
K(q) is slow, and hard to control.
• Apply water at steady q < Ks
• Wait till outflow = inflow
• Measure q and/or y across a
“test interval”
Soil Physics 2010
•
•
•
•
•
Prevent evaporation
Water evenly, no disturbance
Tall column, or tension at bottom
Tensiometer can change flow
Measure q with gamma-rays
How do we measure K(q) in the lab?
K(q) is slow, and hard to control.
Other methods:
• Centrifuge
• Evaporation
• One-step
• Multi-step
As q decreases:
Soil Physics 2010
Slower
Harder to control
More uncertainty
How do we measure K(q) 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.)
Soil Physics 2010