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Outline. Where were we? Hysteresis Constitutive relationships. Soil Physics 2010. Where were we? Wet region. Pore only drains if: Big enough Not isolated Air can get to it. h. Air entry Air access Structural pores. Wet. q. Soil Physics 2010. A model porous medium being drained.
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Outline • Where were we? • Hysteresis • Constitutive relationships Soil Physics 2010
Where were we? Wet region Pore only drains if: Big enough Not isolated Air can get to it h Air entry Air access Structural pores Wet q Soil Physics 2010
A model porous medium being drained Drainage allowed: Pore radius: Big Small Soil Physics 2010
A model porous medium being drained Drainage allowed: Pore radius: Big Small Soil Physics 2010
A model porous medium being drained Drainage allowed: Pore radius: Big Small Soil Physics 2010
A model porous medium being drained Drainage allowed: Pore radius: Big Small Soil Physics 2010
A model porous medium being drained Drainage allowed: Pore radius: Big Small Soil Physics 2010
Middle region 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 Middle h q Soil Physics 2010
Dry region 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 q → 0 as h → ∞ (for example, drying at 105° for 24 hrs) Dry h q 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. Soil Physics 2010
Ink-bottle pores q Confuses individual pore with whole soil h Soil Physics 2010
Converging & diverging Draining Soil Physics 2010
Converging & diverging 2 Wetting Soil Physics 2010
Ink-bottle pore Dh → r Soil Physics 2010
Ink-bottle pore r is too big for h! Dh → r Jump started where pore radius increases – where r too big for h 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 Soil Physics 2010
Ink-bottle pore hysteresis 1 pore: draining h wetting q Soil Physics 2010
Ink-bottle pores A cluster of larger pores surrounded by smaller pores Soil Physics 2010
Hysteresis h wetting draining q 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 q to er for TDR) • van Genuchten q-h relationship Soil Physics 2010
Empirical & phenomenological equations Brooks & Corey: h log h hb qsqat saturation qrq at 1.5 MPa (“residual”) hb bubbling pressure l fitting (“pore size distribution index”) log q log q hb: Lowest pressure at which air can flow through the soil Soil Physics 2010
Empirical & phenomenological equations van Genuchten: qs qr h hb qsq at saturation qrq at 1.5 MPa a1/hb n, m fitting. Often, m ≡ 1-(1/n) q 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 q to er for TDR) • van Genuchten q-h relationship • Physico-empirical equations • Brooks & Corey? Soil Physics 2010
Competition for pores • When both water and air are present (middle range): • Water is in the smaller pores • Air is in the larger pores h wetting draining q Soil Physics 2010
