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 :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? 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), T 0 = 31 °C, A 0 = 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 T 0 = 31 °C, A 0 = 10 °C, and d = 8.4 cm. How deep should Egbert make his summer home? 31 z T At what z is A 0 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 least 7.7 cm deep z

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 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 t0t0 t1t1 t2t2 C/C0C/C0 t x t3t3 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). v The slug moved at the mean water velocity, and spread out, but remained symmetrical. This seemed remarkable to Taylor. t0t0 t1t1 t2t2 t3t3 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? Rtube radius v 0 velocity at center rdistance from center, 0 < r < R Cdye concentration C 0 input dye concentration xdistance from dye inlet ttime the breakthrough curve should be Soil Physics 2010 t CC0CC0

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 D f dispersion coefficient D m diffusion coefficient Rtube radius v 0 velocity at center Soil 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 t0t0 t1t1 t2t2 t3t3 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 t0t0 t1t1 t2t2 t3t3 Soil Physics 2010

Convection-Dispersion Equation Diffusion equation Convection - like the continuity equation, Retardation (sorption and such) v t0t0 t1t1 t2t2 t3t3 Soil Physics 2010 Dispersion coefficient D f =  * v  is called “dispersivity”

A scaling issue Longitudinal Dispersivity  (m) Scale (m) Dispersivity  increasing 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. 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