Outline
• Announcements
• Where were we?
• Measuring unsaturated flow
• Soil water diffusivity
Soil Physics 2010
Announcements
• Homework 4 due March 3
• Excel Solver demo on course website
Soil Physics 2010
Where were we?
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. Requires way more computer
resources than a simple statistical fit.
(The Excel Solver solves an inverse problem.)
Soil Physics 2010
What’s the equation for K(q)?
Most estimation methods start with the
WRC [q (y)], because it’s our best
estimate of the pore size distribution
Capillary tube approach
Cut-and-rejoin approach
1950s – 1960s
Soil Physics 2010
Estimating K(q)
from q(y)
Known volume
of pores with a
known radius
ym
r
Soil Physics 2010
q
How do you average conductivities?
Network
Parallel
Serial
(like bundle of tubes)
Z=∞
20 > Z > 2
N
K Arith   f i K i
i 1
Soil Physics 2010
K Geom


   f i K i 
 i

1
Z=2
n
K harm 
1
N
fi

i 1 K i
Beyond averages
Effective Medium Theory:
Network
What value of conductor,
if it replaced every other
conductor in the system,
would give an equivalent
system conductivity?
N

2 < Z < 50
N

i 1
Soil Physics 2010
i 1
Ki  K
fi
0
*
Ki  2K
Ki  K
fi

0
Ki  Z 1 K *
2

*

*
What is the equation for K(q)?
Examples (many, but not all, found in Hillel):
K q   K sq
q 
K q   K s  
 
K S   Ks S
K y  
n
n
22.5
y
n
Ks
K y  
n
1  ay

K y  
Ks
y 
1   
y c 
…but none of them works
(Brooks & Corey, 1964)
Soil Physics 2010
Ks
n
Widely used models: take your pick
q
Burdine:
K S   K s S
2
y

y
0
0
2
r
dq
2
r
dq
q
Mualem:
K S   K s
van Genuchten’s q(y)
model combined with
Mualem’s K(S) model:
Soil Physics 2010
 y  2 dq 
 0 r

S 

2
  y r dq 
 0


2
K S   K s S 1  1  S


2
m
1m
Hydraulic diffusivity
Different forms of Richards’ equation have
different advantages and disadvantages.
q
 
h 



K
q
t x 
x 
“Mixed form”:
both q and h
q
 
q 

D q  

t x 
x 
“q ” form
h  
h 


c h  
K
h
t x 
x 
“h” form
Soil Physics 2010
“The same equations have the same solutions”
Richard Feynman
h  
h 


c h  
K
h
t x 
x 
q
c(h), also called c(q), 
h
“differential
water capacity”
h q h q


So c h 
t h t t
Soil Physics 2010
“The same equations have the same solutions”
Richard Feynman
q
 
q 



D
q


t x 
x 
K q 
K q 
D q  

q
cq 
h
“hydraulic
diffusivity”
q
h q
h
So D q 
 K q 
 K q 
x
q x
x
Soil Physics 2010
h(q), K(q), c(q) and D(q)
1.E+01
1.E+04
Sand
1.E+03
0
clay
0.1
0.2
0.3
0.4
0.5
0.6
1.E-01
loam
1.E+02
1.E-03
1.E+01
Sand
clay
1.E+00
0
1.E-01
0.1
hq 
0.2
0.3
0.4
0.5
0.6
1.E-05
loam
q
c q  
h
1.E-07
1.E+02
1.E-02
1.E+02
1.E+00
0
0.1
0.2
0.3
0.4
0.5
0.6
1.E+00
1.E-02
0
1.E-04
0.1
0.2
0.3
0.4
0.5
0.6
1.E-02
1.E-06
Sand
1.E-08
clay
loam
1.E-10
Sand
1.E-04
clay
K q 
D q  
cq 
loam
1.E-12
1.E-14
Soil
1.E-16Physics 2010
K(q)
1.E-06
1.E-08
So what’s the point? “The same equations have
the same solutions”
Richard Feynman
C
C
D 2
t
x
Diffusion
equation
T
T
 DT 2
t
x
Heat flow
equation
2
2
Extremely
well studied
equations
q
 
q 
q



D
q
 Dq  2


t x 
x 
x
2
Soil Physics 2010
Hydraulic
diffusivity
equation
if D(q) constant in x
Cost / benefit analysis for the
hydraulic diffusivity equation:
Cost: assumptions of
• No hysteresis
• Horizontal only
• c(q) and K(q) (and thereby D(q))
don’t change in x or t
Benefit: an equation that
q
q
 Dq  2
t
x
• Has only 1 variable (q) that changes in x
and t
• Has only 1 function (D) that needs to be
measured or estimated
• Has centuries of mathematical history
Soil Physics 2010
2
What good is math history?
A cool trick: the Boltzmann transformation

q

q
Given
 Dq  2 ,
t
x
2
notice that D has units of L2/t, which is
characteristic of the diffusion equation.
Introduce a new variable B
B dq
dq
 D q  2
2 dB
dB
2
Soil Physics 2010
≡ x/t1/2. Then
• q varies only in B
• ODE, not PDE
1 function in 1 unknown
Boltzmann variable B
B dq
dq
 D q  2
2 dB
dB
2
≡ x/t1/2
Horizontal infiltration
Bruce & Klute setup
Hydraulic diffusivity experiment
q
Soil Physics 2010
B
What use is it?
• From this (easy) experiment we
get D(q)
• From a water retention curve
(also fairly easy) we get c(q)
• Combining them, we get K(q),
which is way hard to measure.
q
Soil Physics 2010
B