Final Review

1. Final Review

2. Ground Water Basics • Porosity • Head • Hydraulic Conductivity • Transmissivity

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

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

5. Porosity Basics • Volumetric water content (q) • Equals porosity for saturated system

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

7. Multiple Choice:Water flows…? • Uphill • Downhill • Something else

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

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

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

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

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

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

14. Potential/Potential Diagrams • Total potential = elevation potential + pressure potential • Pressure potential depends on depth below a free surface • Elevation potential depends on height relative to a reference (slope is 1)

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

16. Discharge • Q (volume per time) Specific Discharge/Flux/Darcy Velocity • q (volume per time per unit area) • L3 T-1 L-2→ L T-1

17. Darcy’s Law • Q = -K dh/dx A where K is the hydraulic conductivity and A is the cross-sectional flow area 1803 - 1858 www.ngwa.org/ ngwef/darcy.html

18. Darcy’s Law • Q = K dh/dl A • Specific discharge or Darcy ‘velocity’: qx = -Kx∂h/∂x … q = -K gradh • Mean pore water velocity: v = q/ne

19. Intrinsic Permeability L2 L T-1

20. Kozeny-Carman Equation

21. Transmissivity • T = Kb

22. Darcy’s Law • Q = -K dh/dl A • Q, q • K, T

23. Mass Balance/Conservation Equation • I = inputs • P = production • O = outputs • L = losses • A = accumulation

24. qx|x Dz qx|x+Dx Dx Dy Derivation of 1-D Laplace Equation • Inflows - Outflows = 0 • (q|x - q|x+Dx)DyDz = 0 • q|x – (q|x +Dx dq/dx) = 0 • dq/dx = 0 (Continuity Equation) (Constitutive equation)

25. General Analytical Solution of 1-D Laplace Equation

26. Particular Analytical Solution of 1-D Laplace Equation (BVP) BCs: - Derivative (constant flux): e.g., dh/dx|0 = 0.01 - Constant head: e.g., h|100 = 10 m After 1st integration of Laplace Equation we have: After 2nd integration of Laplace Equation we have: Incorporate derivative, gives A. Incorporate constant head, gives B.

27. Finite Difference Solution of 1-D Laplace Equation Need finite difference approximation for 2nd order derivative. Start with 1st order. Look the other direction and estimate at x – Dx/2:

28. Finite Difference Solution of 1-D Laplace Equation (ctd) Combine 1st order derivative approximations to get 2nd order derivative approximation. Set equal to zero and solve for h:

29. 2-D Finite Difference Approximation

30. Matrix Notation/Solutions • Ax=b • A-1b=x

31. Toth Problems • Governing Equation • Boundary Conditions

32. Recognizing Boundary Conditions • Parallel: • Constant Head • Constant (non-zero) Flux • Perpendicular: No flow • Other: • Sloping constant head

33. Internal ‘Boundary’ Conditions • Constant head • Wells • Streams • Lakes • No flow • Flow barriers • Other

34. Poisson Equation • Add/remove water from system so that inflow and outflow are different • R can be recharge, ET, well pumping, etc. • R can be a function of space and time • Units of R: L T-1

35. Poisson Equation (qx|x+Dx - qx|x)Dyb -RDxDy = 0

36. Dupuit Assumption • Flow is horizontal • Gradient = slope of water table • Equipotentials are vertical

37. Dupuit Assumption (qx|x+Dx hx|x+Dx- qx|x hx|x)Dy - RDxDy = 0

38. Capture Zones

39. Water Balance and Model Types

40. Water Balance • Given: • Recharge rate • Transmissivity • Find and compare: • Inflow • Outflow

41. Water Balance • Given: • Recharge rate • Flux BC • Transmissivity • Find and compare: • Inflow • Outflow

42. 2Dy Y 1Dy 0 0 1Dx 2Dx Effective outflow boundary Block-centered model Only the area inside the boundary (i.e. [(imax -1)Dx] [(jmax -1)Dy] in general) contributes water to what is measured at the effective outflow boundary. In our case this was 23000  11000, as we observed. For large imax and jmax, subtracting 1 makes little difference. X

43. Effective outflow boundary Mesh-centered model 2Dy An alternative is to use a mesh-centered model. This will require an extra row and column of nodes and the constant heads will not be exactly on the boundary. Y 1Dy 0 0 1Dx 2Dx X

44. Dupuit Assumption Water Balance Effective outflow area h1 (h1 + h2)/2 h2

45. Geostatistics

46. Basic definitions • Variance: • Standard Deviation:

47. Basic definitions • Number of pairs

48. Basic definitions • Number of pairs:

49. h Basic definitions • Lag (h) • Separation distance (and possibly direction)

50. h Basic definitions • Variance: • Variogram: