Soil Physics 2010

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 0 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 0 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

Is it 5/12, or –5/12?

12

5

24

520

The equation subtracts 0, which delays the peak.So to advance a point, we subtract a negative 0.

Soil Physics 2010

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 2Egbert 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?

31zT 41

35At what z isA0 e-z/d ≤ 4?

What is z if 10 e-z/8.4 = 4?

e-z/8.4 = 0.4

-z/8.4 = ln(0.4)

z = -8.4 ln(0.4)

At least7.7 cm deep

z

Question 3 (extra credit)

Why do soil physicistsdisproportionately hail fromUtah, 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 2010

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.

Back to Mass Transport

Why is solute movement interesting or important?

Isn’t it the same as the water movement?

How (and why) would solutes move differently than water?

Pollutants, fertilizers, tracers, lawsuits…

Mostly, but instead of wanting to know the mean velocity, now we’re interested in the arrival time distribution

Precipitation & dissolution, oxidation & reduction, sorption, decay, diffusion…

Soil Physics 2010

Arrival time distribution?

Soil Physics 2010

Monday I called this a breakthrough curve

t0

t1

t2

C/C0

t

x t3

Often, time of first arrival is of greatest interest

Diffusion with Convection

Sir Geoffrey Taylor examined a “slug” of dye traveling in a tube of flowing water (early 1950s).

vThe slug moved at the mean water velocity, and spread out, but remained symmetrical.

This seemed remarkable to Taylor.

t0 t1 t2 t3

Soil Physics 2010

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

What was going on?

2

2

0 1R

rvrv

tv

xtx

C

C

00

1,

R tube radiusv0 velocity at centerr distance from center,

0 < r < R

C dye concentrationC0 input dye concentrationx distance from dye inlett time

the breakthrough curve should be

Soil Physics 2010 t

C

C0

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

mf D

vRD

192

20

2

Df dispersion coefficientDm diffusion coefficientR tube radiusv0 velocity at centerSoil Physics 2010

You don’t need to know this one

So what?

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

The dye slug appeared to be diffusing, but much faster than by diffusion alone.

v

t0 t1 t2 t3

Soil Physics 2010

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 Convection-Dispersion Equation (CDE), widely used to describe dispersion.

v

t0 t1 t2 t3

Soil Physics 2010

Convection-Dispersion Equation

x

Cv

x

CD

t

CR f

2

2

Diffusion equation

Convection - like the continuity

equation,

Retardation(sorption and such)

x

Cv

t

C

v

t0 t1 t2 t3

Soil Physics 2010

Dispersion coefficient

Df = *v is called

“dispersivity”

A scaling issue

Lon

gitu

din

al D

isp

ersi

vity

(

m)

Scale (m)

Dispers

ivity in

creasin

g with

scale?

after Gelhar, 1992

If increases with x (or t), then the CDE needs at least one more parameter.

Soil Physics 2010

“The all-too-frequent ‘patch’ solution, which

allows the dispersivity [] to change with

travel distance or time, is mathematically incorrect, and contradicts the fundamental assumptions [of] the CDE.”

-- Brian Berkowitz

So the CDE is patched for both velocity and scale

“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

Streamtube

CDE

Soil Physics 2010

(not taught in 577)

CTRW

CPA

MIM

• Each region has a different mean velocity

• Different velocities within each region

• Could presumably have diffusion within and between streamtubes

Streamtube model

Soil Physics 2010

Without mixing between streamtubes, the breakthrough curve (a.k.a. arrival time distribution) could be calculated directly from the velocity distribution.

Streamtube model

Engineers call this a transfer function –

like Taylor’s parabolic velocity profile for a tube.

tv

xtx

C

C

00

1,

Soil Physics 2010

Flow parallel to layering approximates the case of no mixing between streamtubes

Where streamtubes work well

Soil Physics 2010

Summary:

CDE says dispersion is like diffusion

Streamtubes say dispersion is about different flow velocities

Neither upscales reliably