Soil Physics 2010

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

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

3. Question 1 Drying h q Wetting Soil Physics 2010

4. Question 2 Different lines show different possibilities h h 0 0 0 0 q y Soil Physics 2010

5. Why different flow equations? No Dq No q(y) No K(q) q changes with time Steady-state Transient Saturated Darcy’s law N/A Richards’ equation Darcy’s law (with K(q)) Unsaturated Darcy’s law: Soil Physics 2010

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

7. 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 Soil Physics 2010

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

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

10. K(q), averages by texture • Coarse soils: • Lower f • Higher Ks • More abrupt drop Topp & Dane, Methods of soil analysis • At low q: • Small Dq → big DK • Huge range of K • Huge uncertainty in K Soil Physics 2010

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

12. How do we measure K(q) in the lab? Ks is pretty easy. K(q) is slow, and hard to control. • Apply water at steady q < Ks • Wait till outflow = inflow • Measure q and/or y across a “test interval” • Prevent evaporation • Water evenly, no disturbance • Tall column, or tension at bottom • Tensiometer can change flow • Measure q with gamma-rays Soil Physics 2010

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

14. How do we measure K(q) 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.) Soil Physics 2010