1. Soil Physics 2010 Outline Announcements Where were we? More saturated flow

2. Soil Physics 2010 Announcements Homework is due now. If all homeworks are handed in now, I will post my answers right after class Reminder: Exam Friday Example exam is posted.

3. Soil Physics 2010 Where were we? Height Water pressure Water volume / unit time Length of flow Cross-sectional area of flow Proportionality coefficient: Hydraulic conductivity

4. Pressure = 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 water’s potential energy is the same all through the pool. Surface water has elevation; deep water has pressure. Depth Pressure Elevation Soil Physics 2010 Pressure + Elevation Potential Energy

5. Soil Physics 2010 Units in Darcy’s Law Unitless Velocity Usually we give the pressure term in units of length, so the gradient is unitless

6. Soil Physics 2010 Is this velocity how fast the water moves? Velocity No. Water flows only through the pores. Water flows through an area A  Water flows at mean velocity LL hh

7. Soil Physics 2010 Key implications of Darcy’s law For flow through a uniform medium, the hydraulic gradient is constant. The flow is linearly proportional to the gradient, as in Hooke’s law, Fick’s law, Fourier’s law, etc. K is a property of the medium.

8. Soil Physics 2010 More on Darcy’s law There is no flow without an energy (hydraulic) gradient Components of the gradient: elevation pressure velocity (?) For unit area, use What are the units of q?

9. Soil Physics 2010 More on Darcy’s law Elevation: potential energy z Pressure:“virtual” elevation p/gp/g Velocity: kinetic energy v 2 /2g The energy gradient has 3 components:

10. Soil Physics 2010 Total potential energy The potential energy of water can be expressed several different ways: energy per unit mass:  J kg -1 energy per unit volumepN m -2 = Pa energy per unit weighthm H 2 O  = p /  w h = p /  w g =  / g Basis symbolunits It is convenient to think of the energy in terms of h

11. Soil Physics 2010 Darcy in layered systems Steady-state flow Unit gradient overall L1=L2L1=L2 q 1 = ? q 2 = ? K 2 = 0.1 cm/s K 1 = 0.2 cm/s  h 1 = ?  h 2 = ? Atmosperic pressure at top & bottom

12. Soil Physics 2010 Continuity requires greater gradient for smaller K Darcy in layered systems K 2 = 0.1 cm/s K 1 = 0.2 cm/s L 1 = L 2, so L1=L2L1=L2

13. Soil Physics 2010 K 2 = 0.1 cm/s K 1 = 0.2 cm/s Darcy in layered systems

14. Soil Physics 2010 Darcy in artificial systems: 20cm 40cm 60cm D C B A E Given this system, with steady-state water flow, what are the values of the head components at each point?

15. Soil Physics 2010 20cm 40cm 60cm D C B A E pressure p = 0 at points A and E We know: Elevations can be read from the diagram elevation + pressure total head (energy) Steady-state flow →q is the same everywhere →linear energy gradient Darcy in artificial systems:

16. Soil Physics 2010 Construct a table: Elevation + pressure = Total ABCDEABCDE Pressure = 0 at A and E 0000 20cm 40cm 60cm D C B A E Darcy in artificial systems:

17. Soil Physics 2010 Take E as reference height 40 60 0 0000 20cm 40cm 60cm D C B A E Darcy in artificial systems: Elevation + pressure = Total ABCDEABCDE

18. Soil Physics 2010 40 60 0 0000 40 0 Elevation + pressure = Total 20cm 40cm 60cm D C B A E Darcy in artificial systems: Elevation + pressure = Total ABCDEABCDE

19. Soil Physics 2010 40 60 0 0000 40 33.3 26.6 20 0 Uniform medium: linear drop in head with distance 20cm 40cm 60cm D C B A E Darcy in artificial systems: Elevation + pressure = Total ABCDEABCDE so at 1/6 of  L, we’ve used 1/6 of  h  L=120 cm  h = 40cm 5/6 * 40 = 33.3 Elevation + pressure = Total

20. Soil Physics 2010 40 60 0 -26.7 -33.3 -40 0 40 33.3 26.7 20 0 Fill in the rest by difference 20cm 40cm 60cm D C B A E Darcy in artificial systems: Elevation + pressure = Total ABCDEABCDE

21. Soil Physics 2010 Darcy in artificial systems: Summary: You can use the pieces you know to assemble the whole puzzle. Every piece of information is needed: data and theory. 20cm 40cm 60cm D C B A E