1 / 103

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:

gunda
Download Presentation

Final Review

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related