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Soil Physics 2010

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

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Soil Physics 2010

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

  2. 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. Soil Physics 2010

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

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

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

  6. Is this velocity how fast the water moves? Dh DL No. Water flows only through the pores. Water flows through an area Af Water flows at mean velocity Velocity Soil Physics 2010

  7. 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. Soil Physics 2010

  8. 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? Soil Physics 2010

  9. More on Darcy’s law z Elevation: potential energy p/rg Pressure: “virtual” elevation v2/2g Velocity: kinetic energy The energy gradient has 3 components: Soil Physics 2010

  10. Total potential energy The potential energy of water can be expressed several different ways: Basis symbol units energy per unit mass: y J kg-1 energy per unit volume p N m-2 = Pa energy per unit weight h m H2O It is convenient to think of the energy in terms of h y = p / rw h = p / rwg = y / g Soil Physics 2010

  11. Darcy in layered systems Dh1 = ? K1= 0.2 cm/s Atmosperic pressure at top & bottom Dh2 = ? K2= 0.1 cm/s q1 = ? q2 = ? Steady-state flow Unit gradient overall L1 = L2 Soil Physics 2010

  12. Darcy in layered systems K1= 0.2 cm/s L1 = L2 L1 = L2, so K2= 0.1 cm/s Continuity requires greater gradient for smaller K Soil Physics 2010

  13. Darcy in layered systems K1= 0.2 cm/s K2= 0.1 cm/s Soil Physics 2010

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

  15. Darcy in artificial systems: 40cm C D B 20cm A 60cm 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 Soil Physics 2010

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

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

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

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

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

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

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