Outline
• Announcements
• Heitman’s comment:
“I like your slides. I can almost
picture the confused looks.”
• Back to solute movement
Soil Physics 2010
Announcements
• Homework due now
• Review sessions this week:
• 11:00 – 1:00 today in 1581
• 11:00 – 1:00 Friday in G217
•
Quiz!
(virtually)
Soil Physics 2010
Question 1
When it’s 10:06 a.m. here in Iowa, it is
5:06 a.m. in Hawai’i.
What is the phase shift j0 for calling
Henry in Honolulu?
Soil Physics 2010
Question 1
When it’s 10:06 a.m. here in Iowa, it is 5:06 a.m. in Hawai’i.
What is the phase shift j0 for calling Henry in Honolulu?
That’s a 5 hour time difference.
The earth’s rotation is a daily phenomenon,
so the appropriate period is one day (24 hr).
The time difference between Ames and
Honolulu is therefore
5 5p
j 0  2p
Is it 5p /12, or –5p /12?
Soil Physics 2010
24

12
The equation subtracts j0, which delays the peak.
So to advance a point, we subtract a negative j0.
Question 2
Egbert the earthworm can’t tolerate
temperatures > 37 °C.
In central Iowa, where she currently lives,
he* estimates that during July (her least
favorite month), T0 = 31 °C, A0 = 10 °C, and
d = 8.4 cm.
How deep should Egbert make his summer
home?
* earthworms are hermaphrodites
Soil Physics 2010
Question 2
Egbert the earthworm can’t tolerate temperatures > 35 °C.
She estimates that T0 = 31 °C, A0 = 10 °C, and d = 8.4 cm.
How deep should Egbert make his summer home?
At what z is
A0 e-z/d ≤ 4?
What is z if
10 e-z/8.4 = 4?
T
z
e-z/8.4 = 0.4
-z/8.4 = ln(0.4)
z = -8.4 ln(0.4)
Soil Physics 2010
31
At least z
7.7 cm deep
35
41
Question 3 (extra credit)
Why do soil physicists
disproportionately hail from
Utah, Israel, and the Netherlands?
Soil Physics 2010
Question 3 (extra credit)
Why do soil physicists disproportionately hail from
Utah, Israel, and the Netherlands?
Soil physics frequently involves
managing water, especially too much or
too little.
Utah and Israel have too little; the
Netherlands has too much.
A kid growing up in those places knows
that soil physicists are the true heroes.
Soil Physics 2010
Back to Mass Transport
Why is solute movement interesting or important?
Pollutants, fertilizers, tracers, lawsuits…
Isn’t it the same as the water movement?
Mostly, but instead of wanting to know the
mean velocity, now we’re interested in the
arrival time distribution
How (and why) would solutes move differently
than water?
Precipitation & dissolution, oxidation &
reduction, sorption, decay, diffusion…
Soil Physics 2010
Arrival time distribution?
Monday I called this a
breakthrough curve
t0
C/C0
Often, time of first arrival
is of greatest interest
t1
t2
x
Soil Physics 2010
t3
t
Diffusion with Convection
Sir Geoffrey Taylor examined a “slug” of dye
traveling in a tube of flowing water (early 1950s).
t0
t1
t2
v
The slug moved at the mean water velocity,
and spread out, but remained symmetrical.
This seemed remarkable to Taylor.
Soil Physics 2010
t3
Why was this remarkable?
Taylor knew that water flowing through a tube
has a parabolic velocity profile. Water in the
center flows at twice the mean water velocity.
The velocity profile is not symmetrical, but the
dye slug was symmetrical.
Soil Physics 2010
Diffusion with Convection
In fact, given a parabolic
velocity profile

r2 
vr   v0 1  2 
 R 
R
v0
r
tube radius
velocity at center
distance from center,
0<r<R
the breakthrough curve should be
C
x
x, t   1 
C0
v0t
C
C0
Soil Physics 2010
t
C
C0
x
t
dye concentration
input dye concentration
distance from dye inlet
time
What was going on?
Diffusion with Convection
Taylor realized that dye was diffusing radially,
between the fast water in the center, and the
slow water at the tube wall.
This made the dye disperse lengthwise
(longitudinally) in the tube in a way that
looked like diffusion, but was way faster
than diffusion.
Specifically, Taylor found
2 2
0
Rv
Df 
192Dm
Soil Physics 2010
You don’t need to
know this one
Df
Dm
R
v0
dispersion coefficient
diffusion coefficient
tube radius
velocity at center
So what?
The dye slug appeared to be diffusing, but
much faster than by diffusion alone.
In other words, a solute spreads out
much faster in flowing water
than in still water.
It also spreads faster when the
water is flowing faster
t0
Soil Physics 2010
t1
t2
v
t3
Last bundle-of-tubes of the semester
Fast-forward through a few decades…
If a porous medium can be approximated as a
bundle of tubes…
…then dispersion in a porous medium is
like dispersion in a bundle of tubes.
This is the premise of the ConvectionDispersion Equation (CDE), widely used to
describe dispersion.
t0
Soil Physics 2010
t1
t2
v
t3
Convection-Dispersion Equation
C
C
C
R
 Df

v
2
t
x
x
2
Diffusion
equation
Retardation
(sorption
and such)
Soil Physics 2010
Dispersion
coefficient
Df = a *v
Convection
- like the continuity
equation,
C
C
 v
t
x
a is called
t0
“dispersivity”
t1
v
t2
t3
Longitudinal Dispersivity a (m)
A scaling issue
If a increases with x (or t), then the
CDE needs at least one more parameter.
Soil Physics 2010
Scale (m)
after Gelhar, 1992
So the CDE is patched for both velocity and scale
“The all-too-frequent ‘patch’ solution, which
allows the dispersivity [a] to change with
travel distance or time, is mathematically
incorrect, and contradicts the fundamental
assumptions [of] the CDE.”
-- Brian Berkowitz
“I don’t believe in it anymore”
-- T.C. “Jim” Yeh
(regarding use of the CDE at the field scale)
Soil Physics 2010
Alternatives to the CDE
CDE
Streamtube
CTRW
CPA
MIM (not taught in 577)
Soil Physics 2010
Streamtube model
• Each region has a different mean velocity
• Different velocities within each region
• Could presumably have diffusion within and
between streamtubes
Soil Physics 2010
Streamtube model
Without mixing between streamtubes, the
breakthrough curve (a.k.a. arrival time
distribution) could be calculated directly from
the velocity distribution.
Engineers call this a
transfer function –
like Taylor’s parabolic
velocity profile for a tube.
Soil Physics 2010
C
x
x, t   1 
C0
v0t
Where streamtubes work well
Flow parallel to layering approximates the case of
no mixing between streamtubes
Summary:
CDE says dispersion is like diffusion
Streamtubes say dispersion is about
different flow velocities
Soil Physics 2010
Neither upscales reliably