1. Soil Physics 2010 Outline Announcements Richards’ equation Unsaturated flow

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

3. h  Soil Physics 2010 Question 1 Drying Wetting

4. h  0 0 h  0 0 Soil Physics 2010 Question 2 Different lines show different possibilities

5. Soil Physics 2010 Why different flow equations? Steady-stateTransient Saturated Unsaturated Darcy’s law Darcy’s law (with K(  )) N/A Richards’ equation Darcy’s law:  changes with time No K(  ) No  No  (  )

6. Soil Physics 2010 Equation of Continuity (Conservation of Mass) Steady-stateTransient Saturated Unsaturated Darcy’s law Darcy’s law (with K(  )) Richards’ equation Input – Output = Change in Storage =

7. Soil Physics 2010 Richards’ equation Given Darcy’s law: Let things change from place to place (say, in the x-direction) We also want conservation of mass So we substitute it in to the left-hand side

8. Soil Physics 2010 Richards’ equation But this doesn’t allow K to change with  So we permit that, and… voilà: Richards’ equation We can generalize it to 2 or 3 dimensions… … and add in anisotropy

9. Soil Physics 2010 Richards’ equation Remember that the potential gradient,, combines elevation, osmotic, pressure, and matric components (among others). Sometimes it’s convenient to separate out the elevation part: Vertical Horizontal Just remember that this  doesn’t include elevation!

10. Soil Physics 2010 Topp & Dane, Methods of soil analysis K(  ), averages by texture Coarse soils: Lower  Higher K s More abrupt drop At low  : Small  → big  K Huge range of K Huge uncertainty in K

11. Soil Physics 2010 K(  ) and K(  ) for 3 textures (Mualem-van Genuchten functions) K()K() K()K() ()() K(  ) has more hysteresis

12. Soil Physics 2010 How do we measure K(  ) in the lab? 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

13. 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

14. 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. (A spreadsheet’s Solver solves an inverse problem.)