1. Soil Physics 2010 Outline Announcements Where were we? Measuring unsaturated flow Soil water diffusivity

2. Soil Physics 2010 Announcements Homework 4 due March 3 Excel Solver demo on course website

3. Soil Physics 2010 Where were we? K s is pretty easy. K(  ) is slow, and hard to control. Apply water at steady q < K s Wait till outflow = inflow Measure  and/or  across a “test interval” Prevent evaporation Water evenly, no disturbance Tall column, or tension at bottom Tensiometer can change flow Measure  with gamma-rays

4. Soil Physics 2010 How do we measure K(  ) in the lab? K(  ) is slow, and hard to control. Other methods: Centrifuge Evaporation One-step Multi-step As  decreases: Slower Harder to control More uncertainty

5. Soil Physics 2010 How do we measure K(  ) 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.)

6. Soil Physics 2010 Most estimation methods start with the WRC [  (  )], because it’s our best estimate of the pore size distribution What’s the equation for K(  )? Capillary tube approach Cut-and-rejoin approach 1950s – 1960s

7. Soil Physics 2010 Estimating K(  ) from  (  ) mm  draining r Known volume of pores with a known radius

8. How do you average conductivities? Network Z = 2 Z = ∞ 20 > Z > 2 Serial Parallel (like bundle of tubes) Soil Physics 2010

9. Beyond averages Network 2 < Z < 50 What value of conductor, if it replaced every other conductor in the system, would give an equivalent system conductivity? Effective Medium Theory: Soil Physics 2010

10. Examples (many, but not all, found in Hillel): …but none of them works What is the equation for K(  )? (Brooks & Corey, 1964)

11. Widely used models: take your pick Soil Physics 2010 van Genuchten’s  (  ) model combined with Mualem’s K(S) model: Burdine: Mualem:

12. Hydraulic diffusivity Soil Physics 2010 Different forms of Richards’ equation have different advantages and disadvantages. “Mixed form”: both  and h “h” form “  ” form

13. Soil Physics 2010 “The same equations have the same solutions” Richard Feynman So “differential water capacity” c(h), also called c(  ),

14. Soil Physics 2010 “The same equations have the same solutions” Richard Feynman “hydraulic diffusivity” So

15. h(  ), K(  ), c(  ) and D(  ) K()K() Soil Physics 2010

16. So what’s the point? “The same equations have the same solutions” Richard Feynman Heat flow equation Diffusion equation Hydraulic diffusivity equation if D(  ) constant in x Extremely well studied equations

17. Soil Physics 2010 Cost / benefit analysis for the hydraulic diffusivity equation: Cost: assumptions of No hysteresis Horizontal only c(  ) and K(  ) (and thereby D(  )) don’t change in x or t Benefit: an equation that Has only 1 variable (  ) that changes in x and t Has only 1 function (D) that needs to be measured or estimated Has centuries of mathematical history

18. Soil Physics 2010 What good is math history? A cool trick: the Boltzmann transformation notice that D has units of L 2 /t, which is characteristic of the diffusion equation. Introduce a new variable B ≡ x/t 1/2. Then Given,  varies only in B ODE, not PDE

19. Soil Physics 2010 1 function in 1 unknown Boltzmann variable B ≡ x/t 1/2 Horizontal infiltration Bruce & Klute setup Hydraulic diffusivity experiment  B

20. Soil Physics 2010 What use is it? From this (easy) experiment we get D(  ) From a water retention curve (also fairly easy) we get c(  ) Combining them, we get K(  ), which is way hard to measure.  B