1. Final Review

2. Ground Water Basics Porosity Head Hydraulic Conductivity Transmissivity

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

4. Porosity Basics Can re-write that as: Then incorporate: Solid density:  s = M solids /V solids Bulk density:  b = M solids /V total  b  s = V solids /V total

5. Porosity Basics Volumetric water content (  ) –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 =  gh p –  density –g gravity –h p depth

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

10. Elevation Head Water wants to fall Potential energy

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

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

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

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) L 3 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 www.ngwa.org/ ngwef/darcy.html 1803 - 1858

18. Darcy’s Law Q = K dh/dl A Specific discharge or Darcy ‘velocity’: q x = -K x ∂h/∂x … q = -K grad h Mean pore water velocity: v = q/n e

19. Intrinsic Permeability L T -1 L2L2

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. Derivation of 1-D Laplace Equation Inflows - Outflows = 0 (q| x - q| x+  x )  y  z = 0 q| x – (q| x +  x dq/dx) = 0 dq/dx = 0 (Continuity Equation) xx yy qx|xqx|x q x | x+  x zz (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 1 st integration of Laplace Equation we have: Incorporate derivative, gives A. After 2 nd integration of Laplace Equation we have: Incorporate constant head, gives B.

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

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

29. 2-D Finite Difference Approximation

30. Matrix Notation/Solutions Ax=b A -1 b=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 (q x | x+  x - q x | x )  yb -R  x  y = 0

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

37. Dupuit Assumption (q x | x+  x h x | x+  x - q x | x h x | x )  y - R  x  y = 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. X 0 2x2x1x1x 2y2y 1y1y 0 Y Effective outflow boundary Only the area inside the boundary (i.e. [(i max -1)  x] [(j max -1)  y] 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 i max and j max, subtracting 1 makes little difference. Block-centered model

43. X 0 2x2x1x1x 2y2y 1y1y 0 Y Effective outflow boundary 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. Mesh-centered model

44. Dupuit Assumption Water Balance h1h1 h2h2 Effective outflow area (h 1 + h 2 )/2

45. Geostatistics

46. Basic definitions Variance: Standard Deviation:

47. Basic definitions Number of pairs

48. Basic definitions Number of pairs:

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

50. Basic definitions Variance: Variogram: h

51. The variogram Captures the intuitive notion that samples taken close together are more likely to be similar that sample taken far apart

52. Common Variogram Models

54. Basic definitions Kriging: BLUE

55. Kriging Estimates

56. Random Numbers; Pure Nugget # # One variable definition: # to start the variogram modeling user interface. # data(K): 'rand.csv', x=1, y=2, v=3;

57. Unconditioned Simulation Specify mean and neighborhood Specify variogram Simulation should honor variogram.cmd file/mask map # Unconditional Gaussian simulation on a mask # (local neighborhoods, simple kriging) # defines empty variable: data(dummy): dummy, sk_mean=100, max=20, min=10, force; variogram(dummy): 10 Sph(10); mask: 'gridascii.prn'; method: gs; # Gaussian simulation instead of kriging predictions(dummy): 'gs.out';

58. Unconditional Simulation

59. Simulated Field/Known Variogram

60. Conditional Gaussian Simulation Specify data Fit and specify variogram Simulation should honor variogram and be responsive to values at ‘conditioning’ points # Gaussian simulation, conditional upon data # (local neighborhoods, simple kriging) data(SC): 'SC_rand.csv', x=1, y=2, v=3,average,max=20, sk_mean=1400; method: gs; variogram(SC): 400000Nug(0)+3.5e+006 Gau(0.035); #Gridded Output mask: 'ga_SC.prn'; predictions(SC): 'SC_pred.prn';

61. Kriging Specify data Fit and specify variogram Simulation should honor variogram and return exact values at sampling points Optimal estimate too far from sample data is mean # # Kriging # (local neighbourhoods, simple and ordinary kriging) # data(SC): 'SC_rand.csv', x=1, y=2, v=3,average,max=20, sk_mean=1400; variogram(SC): 400000Nug(0)+3.5e+006 Gau(0.035); #Gridded Output mask: 'ga_SC.prn'; predictions(SC): 'SC_Krpred.prn';

62. Gaussian Simulation/Kriging

63. Gaussian Kriging

64. Transient Ground Water Flow

65. Transient Flow Equation  Vw =  x  y S  h (q x | x - q x | x +  x)  yb + (q y | y - q y | y +  y)  xb = S  x  y  h/  t

66. Finite Difference x -  x h| x, t x x +  x  h/  t| t-  t/2 Estimate here t-  t t h| x, t-  t

67. CFL Condition The stability criterion (for 1-D) is: T/S  t/  x 2  ½

68. Quasi-3D Models Leakance and head-dependent boundaries

69. Assumptions: Flow is 2-D horizontal in ‘aquifer’ layers Flow is vertical in ‘confining’ layers There is a significant difference in hydraulic conductivity between aquifers and confining layers Aquifer layers are connected by leakage across confining layers

70. Schematic i = 1 i = 2 d1d1 b1b1 d2d2 b 2 (or h 2 ) k1k1 T1T1 k2k2 T 2 (or K 2 )

71. Pumped Aquifer Heads i = 1 i = 2 d1d1 b1b1 d2d2 b 2 (or h 2 )k1k1 T1T1 k2k2 T 2 (or K 2 )

72. Heads i = 1 i = 2 d1d1 b1b1 d2d2 b 2 (or h 2 )k1k1 T1T1 k2k2 T 2 (or K 2 ) h1h1 h2h2 h 2 - h 1

73. Flows i = 1 i = 2 d1d1 b1b1 d2d2 b 2 (or h 2 )k1k1 T1T1 k2k2 T 2 (or K 2 ) h1h1 h2h2 h 2 - h 1 qvqv

74. Leakance Leakage coefficient, resistance (inverse) Leakance From below: From above:

75. Equations Fully 3-D Confined Unconfined

76. Poisson Equation

77. Finite Elements  : basis functions

78. Finite Elements  : hat functions

79. Fracture/Conduit Flow

80. Basic Fluid Dynamics

81. Momentum p = mu

82. Viscosity Resistance to flow; momentum diffusion Low viscosity: Air High viscosity: Honey Kinematic viscosity:

83. Reynolds Number The Reynolds Number (Re) is a non-dimensional number that reflects the balance between viscous and inertial forces and hence relates to flow instability (i.e., the onset of turbulence) Re = v L/ L is a characteristic length in the system Dominance of viscous force leads to laminar flow (low velocity, high viscosity, confined fluid) Dominance of inertial force leads to turbulent flow (high velocity, low viscosity, unconfined fluid)

84. Re << 1 (Stokes Flow) Tritton, D.J. Physical Fluid Dynamics, 2 nd Ed. Oxford University Press, Oxford. 519 pp.

85. Separation

86. Eddies, Cylinder Wakes, Vortex Streets Re = 30 Re = 40 Re = 47 Re = 55 Re = 67 Re = 100 Re = 41 Tritton, D.J. Physical Fluid Dynamics, 2 nd Ed. Oxford University Press, Oxford. 519 pp.

87. L Flow u a x y z Poiseuille Flow

88. In a slit or pipe, the velocities at the walls are 0 (no-slip boundaries) and the velocity reaches its maximum in the middle The velocity profile in a slit is parabolic and given by: x = 0x = a/2 u(x) G can be due to gravitational acceleration (G =  g in a vertical slit) or the linear pressure gradient (P in – P out )/L

89. Poiseuille Flow Maximum Average x = 0x = a/2 u(x)

90. Kirchoff’s Current Law Kirchoff’s law states that the total current flowing into a junction is equal to the total current leaving the junction. I2I2I2I2 I3I3I3I3 node I 1 I 1 flows into the node I 2 I 2 flows out of the node I 3 I 3 flows out of the node I 1 = I 2 + I 3 I 1 = I 2 + I 3 Gustav Kirchoff was an 18th century German mathematician I1I1I1I1

91. Ohm’s law relates the flow of current to the electrical resistance and the voltage drop V = IR (or I = V/R) where: –I = Current –V = Voltage drop –R = Resistance Ohm’s Law is analogous to Darcy’s law

92. Poiseuille's law can related to Darcy’s law and subsequently to Ohm's law for electrical circuits.Ohm's law Cubic law: A = a *unit depth

93. Fracture Network L 23 -216 lu - L 12 Q 12 Q 34 Q 56 PP  P 12  P 23  P 34 Q 23 Q 45  P 45  P 56 L 45 36 lu

94. Matrix Form

95. Back Solution Have conductivities and, from the matrix solution, the gradients –Compute flows Also have end pressures –Compute intermediate pressures from  Ps

96. a Hydrologic-Electric Analogy Poiseuille's law corresponds to the Kirchoff/Ohm’s Law for electrical circuits, where pressure drop Δp is replaced by voltage V and flow rate by current I I 12 I 23 I 56 I 45 ΔP 12 ΔP 23 ΔP 34 ΔP 45 ΔP 56 I 23 I 45 I 34 0.66 0.11 0.11 1.0 0.14 0.14 1.8 0.18 0.19 4.1 0.27 0.28 7.2 0.36 0.37 43.0 0.87 0.92 Re Q (lu 3 /ts) Kirchoff’s LBM Q = 0.11 lu 3 /ts Kirchoff LBM

97. Eddies Re = 9 3.3 mm x 2.7 mm 3 mm 2 mm Bai, T., and Gross, M.R., 1999, J Geophysical Res, 104, 1163-1177 Serpa, CY, 2005, Unpublished MS Thesis, FIU Flow