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Final Review

Final Review. Exam cumulative: incorporate complete midterm review. Calculus Review. Derivative of a polynomial. In differential Calculus, we consider the slopes of curves rather than straight lines For polynomial y = ax n + bx p + cx q + …, derivative with respect to x is:

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Final Review

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  1. Final Review • Exam cumulative: incorporate complete midterm review

  2. Calculus Review

  3. Derivative of a polynomial • In differential Calculus, we consider the slopes of curves rather than straight lines • For polynomial y = axn + bxp + cxq + …, derivative with respect to x is: • dy/dx = a n x(n-1) + b p x(p-1) + c q x(q-1) + …

  4. Example y = axn + bxp + cxq + … dy/dx = a n x(n-1) + b p x(p-1) + c q x(q-1) + …

  5. Numerical Derivatives • ‘finite difference’ approximation • slope between points • dy/dx ≈Dy/Dx

  6. Derivative of Sine and Cosine • sin(0) = 0 • period of both sine and cosine is 2p • d(sin(x))/dx = cos(x) • d(cos(x))/dx = -sin(x)

  7. Partial Derivatives • Functions of more than one variable • Example: h(x,y) = x4 + y3 + xy

  8. Partial Derivatives • Partial derivative of h with respect to x at a y location y0 • Notation ∂h/∂x|y=y0 • Treat ys as constants • If these constants stand alone, they drop out of the result • If they are in multiplicative terms involving x, they are retained as constants

  9. Partial Derivatives • Example: • h(x,y) = x4 + y3 + x2y+ xy • ∂h/∂x = 4x3 + 2xy + y • ∂h/∂x|y=y0 = 4x3 + 2xy0+ y0

  10. WHY?

  11. Gradients • del h (or grad h) • Darcy’s Law:

  12. Equipotentials/Velocity Vectors

  13. Capture Zones

  14. Watersheds http://www.bsatroop257.org/Documents/Summer%20Camp/Topographic%20map%20of%20Bartle.jpg

  15. Watersheds http://www.bsatroop257.org/Documents/Summer%20Camp/Topographic%20map%20of%20Bartle.jpg

  16. Capture Zones

  17. Water (Mass) Balance • In – Out = Change in Storage • Totally general • Usually for a particular time interval • Many ways to break up components • Different reservoirs can be considered

  18. Water (Mass) Balance • Principal components: • Precipitation • Evaporation • Transpiration • Runoff • P – E – T – Ro = Change in Storage • Units?

  19. Ground Water (Mass) Balance • Principal components: • Recharge • Inflow • Transpiration • Outflow • R + Qin – T – Qout = Change in Storage

  20. Ground Water Basics • Porosity • Head • Hydraulic Conductivity

  21. Porosity Basics • Porosity n (or f) • Volume of pores is also the total volume – the solids volume

  22. Porosity Basics • Can re-write that as: • Then incorporate: • Solid density: rs = Msolids/Vsolids • Bulk density: rb = Msolids/Vtotal • rb/rs = Vsolids/Vtotal

  23. Ground Water Flow • Pressure and pressure head • Elevation head • Total head • Head gradient • Discharge • Darcy’s Law (hydraulic conductivity) • Kozeny-Carman Equation

  24. Pressure • Pressure is force per unit area • Newton: F = ma • Fforce (‘Newtons’ N or kg ms-2) • m mass (kg) • a acceleration (ms-2) • P = F/Area (Nm-2 or kg ms-2m-2 = kg s-2m-1 = Pa)

  25. Pressure and Pressure Head • Pressure relative to atmospheric, so P = 0 at water table • P = rghp • r density • g gravity • hpdepth

  26. P = 0 (= Patm) Pressure Head Pressure Head (increases with depth below surface) Elevation Head

  27. Elevation Head • Water wants to fall • Potential energy

  28. Elevation Head (increases with height above datum) Elevation Elevation Head Elevation datum Head

  29. Total Head • For our purposes: • Total head = Pressure head + Elevation head • Water flows down a total head gradient

  30. P = 0 (= Patm) Pressure Head Total Head (constant: hydrostatic equilibrium) Elevation Elevation Head Elevation datum Head

  31. Head Gradient • Change in head divided by distance in porous medium over which head change occurs • A slope • dh/dx [unitless]

  32. Discharge • Q (volume per time: L3T-1) • q (volume per time per area: L3T-1L-2 = LT-1)

  33. Darcy’s Law • q = -K dh/dx • Darcy ‘velocity’ • Q = K dh/dx A • where K is the hydraulic conductivity and A is the cross-sectional flow area • Transmissivity T = Kb • b = aquifer thickness • Q = T dh/dx L • L = width of flow field 1803 - 1858 www.ngwa.org/ ngwef/darcy.html

  34. Mean Pore Water Velocity • Darcy ‘velocity’: q = -K ∂h/∂x • Mean pore water velocity: v = q/ne

  35. Intrinsic Permeability L2 L T-1

  36. More on gradients

  37. More on gradients • Three point problems: h 412 m h 400 m 100 m h

  38. More on gradients h = 10m • Three point problems: • (2 equal heads) 412 m h = 10m 400 m CD • Gradient = (10m-9m)/CD • CD? • Scale from map • Compute 100 m h = 9m

  39. More on gradients h = 11m • Three point problems: • (3 unequal heads) Best guess for h = 10m 412 m h = 10m 400 m • Gradient = (10m-9m)/CD • CD? • Scale from map • Compute CD 100 m h = 9m

  40. Types of Porous Media Isotropic Anisotropic Heterogeneous Homogeneous

  41. Hydraulic Conductivity Values K (m/d) 8.6 0.86 Freeze and Cherry, 1979

  42. Layered media (horizontal conductivity) Q4 Q3 Q2 Q1 Q = Q1 + Q2 + Q3 + Q4 K2 b2 Flow K1 b1

  43. Layered media(vertical conductivity) R4 Q4 Flow R3 Q3 K2 b2 R2 K1 b1 Q2 Controls flow R1 Q1 Q ≈ Q1 ≈ Q2 ≈ Q3 ≈ Q4 The overall resistance is controlled by the largest resistance: The hydraulic resistance is b/K R = R1 + R2 + R3 + R4

  44. Aquifers • Lithologic unit or collection of units capable of yielding water to wells • Confined aquifer bounded by confining beds • Unconfined or water table aquifer bounded by water table • Perched aquifers

  45. Transmissivity • T = Kb gpd/ft, ft2/d, m2/d

  46. Schematic T2 (or K2) b2 (or h2) i = 2 k2 d2 T1 b1 i = 1 k1 d1

  47. T2 (or K2) k2 T1 k1 Pumped Aquifer Heads b2 (or h2) i = 2 d2 b1 i = 1 d1

  48. T2 (or K2) k2 T1 k1 Heads h2 - h1 b2 (or h2) h2 i = 2 d2 h1 b1 i = 1 d1

  49. T2 (or K2) k2 T1 k1 Flows h2 h2 - h1 b2 (or h2) h1 i = 2 d2 qv b1 i = 1 d1

  50. Terminology • Derive governing equation: • Mass balance, pass to differential equation • Take derivative: • dx2/dx = 2x • PDE = Partial Differential Equation • CDE or ADE = Convection or Advection Diffusion or Dispersion Equation • Analytical solution: • exact mathematical solution, usually from integration • Numerical solution: • Derivatives are approximated by finite differences

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