Soil Physics 2010

1. Outline • Announcements • Heitman’s comment: “I like your slides. I can almost picture the confused looks.” • Back to solute movement Soil Physics 2010

2. 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

3. 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

4. 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 Is it 5p /12, or –5p /12? The equation subtractsj0, which delays the peak. So to advance a point, we subtract a negative j0. Soil Physics 2010

5. 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

6. Question 2 z 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? 41 T 31 35 z What is z if 10 e-z/8.4 = 4? At least 7.7 cm deep e-z/8.4 = 0.4 -z/8.4 = ln(0.4) z = -8.4 ln(0.4) Soil Physics 2010

7. Question 3 (extra credit) Why do soil physicists disproportionately hail from Utah, Israel, and the Netherlands? Soil Physics 2010

8. 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

9. 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

10. C/C0 t0 t t1 t2 t3 Arrival time distribution? Monday I called this a breakthrough curve Often, time of first arrival is of greatest interest x Soil Physics 2010

11. t1 t0 t3 t2 v Diffusion with Convection Sir Geoffrey Taylor examined a “slug” of dye traveling in a tube of flowing water (early 1950s). The slug moved at the mean water velocity, and spread out, but remained symmetrical. This seemed remarkable to Taylor. Soil Physics 2010

12. 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

13. C C0 t Diffusion with Convection In fact, given a parabolic velocity profile R tube radius v0 velocity at center r distance from center, 0 < r < R the breakthrough curveshould be C dye concentration C0 input dye concentration x distance from dye inlet t time What was going on? Soil Physics 2010

14. 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 Df dispersion coefficient Dm diffusion coefficient R tube radius v0 velocity at center You don’t need to know this one Soil Physics 2010

15. t1 t0 t3 t2 v 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 Soil Physics 2010

16. t1 t0 t3 t2 Last bundle-of-tubes of the semester v 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. Soil Physics 2010

17. t1 t0 t3 t2 Convection - like the continuity equation, Convection-Dispersion Equation Dispersion coefficient Diffusion equation Retardation (sorption and such) v Df = a *v a is called “dispersivity” Soil Physics 2010

18. A scaling issue If aincreases with x (or t), then the CDE needs at least one more parameter. Dispersivity a increasing with scale? Longitudinal Dispersivity a (m) Scale (m) Soil Physics 2010 after Gelhar, 1992

19. 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

20. Alternatives to the CDE CTRW CPA MIM (not taught in 577) CDE Streamtube Soil Physics 2010

21. 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

22. 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

23. 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 • Neither upscales reliably Soil Physics 2010