Soil Physics 2010

1. Outline • Announcements • Todd on aggregates • Saturated flow of water Soil Physics 2010

2. Announcements • No office hours today • Reminder: Homework due Feb. 10 • Reminder: Exam Feb. 12 • Example exam is now posted. Don’t panic! I covered material in a different order that year, and the class was not dual-listed at the 400-level. Soil Physics 2010

3. Todd on Aggregates Soil Physics 2010

4. Saturated flow When is the soil saturated? Near the surface under heavy rain Confined aquifer Under a lake or stream Recharge basin Go deep enough… It turns out that the saturated flow equations are just special (simpler) cases of flow more generally. Soil Physics 2010

5. The city of Dijon 1800 Ancient capitol fallen on hard times Soil Physics 2010

6. France, 1800s Paris Political turmoil: • 1804 - 1815 First Empire - Napoleon • 1815 - 1830 Restoration - Louis XVIII - Charles X • 1830 - 1848 Bourgeois Monarchy - Louis-Philippe • 1848 – 1851 Second Republic • 1852 - 1870 Second Empire - Napoleon III Dijon Lyon Soil Physics 2010

7. Henry Darcy Fellow students at L’École des Ponts et Chaussées*: Cauchy Chézy Coliolis Dupuit Fresnel Navier Pitot St. Venant Henry graduated 12th in his class. He had been 1st, but he got violently upset with a chemistry professor over a question about cooking. * The school of Bridges and Roads Soil Physics 2010

8. Most famous publication Soil Physics 2010

9. Darcy’s experiment reported in Appendix D of Les Fontaines Publiques… Soil Physics 2010

10. Henry Darcy • Public water supply in Dijon: clean, reliable, gravity flow 12 km to 142 public fountains • 4 km railroad tunnel allowed the main Paris-Lyon track to pass through Dijon • Several large bridges near Dijon • Improvements to the Pitot tube • Extensive studies on: • Flow through open channels • Flow through pipes • Road construction “One does not get the main square in town named in one’s honor on the basis of some column experiments.” -- Allan Freeze, 1994 • Near the end of his life, a few studies on flow through sand Soil Physics 2010

11. Henry Darcy “As much as possible, one should favor the free drawing of water because it is necessary for public health. A city that cares for the interest of the poor class should not limit their water, just as daytime and light are not limited.” Soil Physics 2010

12. Darcy’s experiment • Column filled with sand to different depths • Different water pressures applied across column • Discharge Q measured (volume of water / time) • Experiment repeated with different sands 2.5 m 1 m3 Soil Physics 2010

13. Darcy’s 3 observations: 1: Q cross-sectional Area 2: Q drop in (pressure + elevation) 3: Q 1 / flow distance Soil Physics 2010

14. State of the art For slow flow through a pipe, Q discharge R radius h viscosity Dp pressure drop L length (Poiseuille’s law) Mean velocity is Soil Physics 2010

15. Darcy’s 1st observation Darcy’s experiments used a vertical column, but a horizontal column is simpler. 1: Q cross-sectional Area 2: Q drop in (pressure + elevation) 3: Q 1 / flow distance Assume zero resistance in pipes Soil Physics 2010

16. Darcy’s 2nd observation 1: Q cross-sectional Area 2: Q drop in (pressure + elevation) 3: Q1 / flow distance Soil Physics 2010

17. Pressure = Elevation? Pressure Depth Elevation When you swim underwater, your ears feel pressure Why doesn’t the water at the bottom of the pool – under lots of pressure – shoot up to the top? The energy is the same all through the pool. Surface water has elevation; deep water has pressure. Pressure + Elevation Energy Soil Physics 2010

18. Darcy’s 3rd observation 1: Q cross-sectional Area 2: Q drop in (pressure + elevation) 3: Q 1 / flow distance L Soil Physics 2010

19. Darcy’s Law as he wrote it Water pressure Height Cross-sectional area of flow Length of flow Water volume / unit time Proportionality coefficient: Hydraulic conductivity Soil Physics 2010

20. Units in Darcy’s Law Velocity Unitless Soil Physics 2010

21. Key implications of Darcy’s law: 1 Taking the derivative of Darcy’s law, gives the Laplace equation, For flow through a uniform medium, the hydraulic gradient is constant. Soil Physics 2010

22. Key implications of Darcy’s law: 2 Hooke’s law (elasticity) Darcy’s law (hydraulic conductivity) Ohm’s law (electrical conductivity) Fourier’s law (heat conduction) Fick’s law (diffusion) The flow is linearly proportional to the gradient. This puts Darcy’s law into the same class as several other equations: Soil Physics 2010

23. Key implications of Darcy’s law: 3 K is a property of the medium*. The hydraulic conductivity K is not changed by whether the water flows up instead of down, or by having a greater or smaller gradient, or by the pressures or elevations themselves. In fact, K can often be predicted with reasonable accuracy, given some other information about the medium. For example, the porosity and the grain size or pore size distribution allow a fair estimate. *Also of the fluid – see later Soil Physics 2010