1. Soil Physics 2010 Outline Where were we? Hysteresis Constitutive relationships

2. Soil Physics 2010 Where were we? Wet region  h Wet Pore only drains if: Big enough Not isolated Air can get to it Air entry Air access Structural pores

3. A model porous medium being drained Drainage allowed: Pore radius: Big Small Soil Physics 2010

4. Pore radius: Big Small Drainage allowed: Soil Physics 2010 A model porous medium being drained

5. Pore radius: Big Small Drainage allowed: Soil Physics 2010 A model porous medium being drained

6. Pore radius: Big Small Drainage allowed: Soil Physics 2010 A model porous medium being drained

7. Pore radius: Big Small Drainage allowed: Soil Physics 2010 A model porous medium being drained

8. Soil Physics 2010 Middle region  h Middle Air and water are both continuous Best conditions for life Reasonable reflection of pore size distribution Mixed textural & structural pores at wetter part Textural pores at drier part Hysteresis

9. Soil Physics 2010 Dry region  h Dry Most water is in films sorbed to solid surface Water retention mostly determined by surface area Little or no hysteresis (if at equilibrium) Water flow in films is very slow  → 0 as h → ∞ (for example, drying at 105° for 24 hrs)

10. Soil Physics 2010 Hysteresis Thermostats Wind turbine Domain concept “Ink bottle” pores Individual pores Pore regions Sir James Albert Ewing: The macroscopic, continuous hysteresis that we see is the result of many microscopic hysteretic events, each of which is abrupt rather than smooth.

11. Soil Physics 2010 Ink-bottle pores h  Confuses individual pore with whole soil

12. Soil Physics 2010 Converging & diverging Draining

13. Soil Physics 2010 Wetting Converging & diverging 2

14.  h → r Soil Physics 2010 Ink-bottle pore

15.  h → r Soil Physics 2010 Ink-bottle pore r is too big for h! Jump started where pore radius increases – where r too big for h

16. Soil Physics 2010 Ink-bottle pore r is too big for h! Jump started where pore radius decreases – where r is too small for h

17. Soil Physics 2010 Ink-bottle pore hysteresis h  draining wetting 1 pore:

18. Soil Physics 2010 Ink-bottle pore s A cluster of larger pores surrounded by smaller pores

19. Soil Physics 2010 Hysteresis h  draining wetting

20. Soil Physics 2010 2 kinds of equations Physically-based and/or derived equations Darcy’s law Capillary rise Newton’s laws Terzaghi’s effective stress equation Empirical, phenomenological, and/or statistical equations Topp’s equation (relating  to  r for TDR) van Genuchten  -h relationship

21. Soil Physics 2010 Empirical & phenomenological equations Brooks & Corey:  s  at saturation  r  at 1.5 MPa (“residual”) h b bubbling pressure fitting (“pore size distribution index”) log  log h log  h h b : Lowest pressure at which air can flow through the soil hbhb

22. Soil Physics 2010 Empirical & phenomenological equations  s  at saturation  r  at 1.5 MPa  1 / h b n, m fitting. Often, m ≡ 1-(1/n) van Genuchten: ss hbhb rr  h

23. Soil Physics 2010 3 kinds of equations Physically-based and/or derived equations Darcy’s law Capillary rise Newton’s laws Empirical, phenomenological, and/or statistical equations Topp’s equation (relating  to  r for TDR) van Genuchten  -h relationship Physico-empirical equations Brooks & Corey?

24. Soil Physics 2010 Competition for pores h  draining wetting When both water and air are present (middle range): Water is in the smaller pores Air is in the larger pores