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
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

15. Watersheds

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
F force (‘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
hp depth

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

27. Elevation Head
Water wants to fall
Potential energy

28. (increases with height above datum)
Elevation Head
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. (constant: hydrostatic equilibrium)
P = 0 (= Patm)
Pressure Head
(constant: hydrostatic equilibrium)
Total Head
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 Q = K dh/dx A Transmissivity T = Kb
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
ngwef/darcy.html

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

35. Intrinsic Permeability
L T-1 L2

36. More on gradients

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

38. More on gradients Three point problems: Gradient = (10m-9m)/CD CD?
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 Three point problems: Gradient = (10m-9m)/CD CD?
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 ValuesK
(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)Q4 R4
Flow Q3 R3 K2 b2 R2 K1 b1 Q2
Controls flow Q1 R1
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. Pumped Aquifer Heads k1 T1 k2 T2 (or K2) b2 (or h2) i = 2 d2 b1 i = 1

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

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

50. Terminology Derive governing equation: Take derivative:
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

51. Derivation of 1-D Laplace Equation
Inflows - Outflows = 0
(qx|x- qx|x+Dx)DyDz = 0 Dx Dy
qx|x
qx|x+Dx Dz
Governing Equation

52. Boundary Conditions Constant head: h = constant
Constant flux: dh/dx = constant
If dh/dx = 0 then no flow
Otherwise constant flow

53. General Analytical Solution of 1-D Laplace Equation

54. 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.

55. 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:

56. Finite Difference Solution of 1-D Laplace Equation (ctd)
Combine 1st order derivative approximations to get 2nd order derivative approximation.
Solve for h:

57. 2-D Finite Difference Approximation

58. 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
Units of R: L T-1

59. Derivation of Poisson Equation
(qx|x- qx|x+Dx)Dyb + RDxDy =0

60. General Analytical Solution of 1-D Poisson Equation

61. Water balance Qin + RDxDy – Qout = 0 qin bDy + RDxDy – qout bDy = 0
-K dh/dx|in bDy + RDxDy – -K dh/dx|out bDy = 0
-T dh/dx|in Dy + RDxDy – -T dh/dx|out Dy = 0
-T dh/dx|in + RDx +T dh/dx|out = 0

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

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

64. Transient Problems Transient GW flow Diffusion
Convection-Dispersion Equation
All transient problems require specifying initial conditions (in addition to boundary conditions)

65. Storage Coefficient/Storativity
S is storage coefficient or storativity: The amount of water stored or released per unit area of aquifer given unit head change
Typical values of S (dimensionless) are 10-5 – 10-3
Measuring storativity: derived from observations of multi-well tests
GEOS 4310/5310 Lecture Notes, Fall 2002 Dr. T. Brikowski, UTD

66. 1-D Transient GW Flow

67. 1-D Transient GW Flow: Deriving the Governing PDEDx b
qx|x
qx|x+Dx
DVw = DxDy S Dh
(qx|x - qx|x+Dx)Dyb = SDxDyh/t

69. Finite Difference Solution
First order spatial derivative:
h|x
h|x+Dx x
x +Dx
C/x|x+Dx/2
Estimate here

70. Second order spatial derivative
h|x
h|x+Dx x
x +Dx
h|x-Dx
x -Dx
h/x|x+Dx/2
Estimate here
h/x|x-Dx/2
2h/x2|x

71. Finite Difference Solution
Temporal devivative
h|x, t-Dt
h|x, t x
x +Dx
C/t|t-Dt/2
Estimate here
t-Dt t
x -Dx

72. All together:

73. Stability criterion (Mesh Ratio): TDt/(S(Dx)2) < ½.

74. Diffusion
x + Dx Dy Dz x
jx|x

75. Fick’s Law:
Heat/Diffusion Equation:

76. Temporal Derivative C|x, t-Dt C|x, t x x +Dx C/t|t-Dt/2
Estimate here
t-Dt t
x -Dx

77. All together:

78. Boundary conditions Specify either
Concentrations at the boundaries, or
Chemical flux at the boundaries (usually zero)
Fixed concentration boundary concept is simple.
Chemical flux boundary is slightly more difficult. We go back to Fick’s law:
Notice that if ∂C/∂x = 0, then there is no flux. The finite difference expression we developed for ∂C/∂x is
Setting this to 0 is equivalent to

79. Convection-Dispersion Equation
Key difference from diffusion here!
Convective flux

80. CDE

81. Finite Difference: Spatial
C|x
C|x+Dx x
x +Dx
C|x-Dx
x -Dx
C/x|x+Dx/2
Estimate here
C/x|x-Dx/2
2C/x2|x

82. Finite Difference: Temporal
C|x, t x
x +Dx
C/t|t-Dt/2
Estimate here
t-Dt t
x -Dx

83. Centered Finite Difference
For first order spatial derivative:
Worked for estimating second order derivative (estimate ended up at x).
Need centered derivative approximation

84. All together:
prone to numerical instabilities depending on the values of the factors DDt/(Dx)2 and vDt/2Dx (CFL)

85. Boundary conditions Specify either
Concentrations at the boundaries, or
Chemical flux at the boundaries
Fixed concentration boundary concept is simple
Chemical flux boundary is slightly more difficult. We go back to the flux
Notice that if ∂C/∂x = 0, then there is no dispersive flux but there can still be a convective flux. This would apply at the end of a soil column for example; the water carrying the chemical still flows out of the column but there is no more dispersion. One of the finite difference expressions we developed for ∂C/∂x is
Setting this to 0 is equivalent to

86. Fitting the CDE

87. Adsorption Isotherm
Linear: Cs = Kd C C Cs

88. Koc Values
Kd = Koc foc

89. Organic Carbon Partitioning Coefficients for Nonionizable Organic Compounds. Adapted from USEPA, Soil Screening Guidance: Technical Background Document.
Compound
mean Koc (L/kg)
Acenaphthene
5,028
1,4-Dichlorobenzene(p)
687
Methoxychlor
80,000
Aldrin
48,686
1,1-Dichloroethane 54
Methyl bromide 9
Anthracene
24,362
1,2-Dichloroethane 44
Methyl chloride 6
Benz(a)anthracene
459,882
1,1-Dichloroethylene 65
Methylene chloride 10
Benzene 66
trans-1,2-Dichloroethylene 38
Naphthalene
1,231
Benzo(a)pyrene
1,166,733
1,2-Dichloropropane 47
Nitrobenzene
141
Bis(2-chloroethyl)ether 76
1,3-Dichloropropene 27
Pentachlorobenzene
36,114
Bis(2-ethylhexyl)phthalate
114,337
Dieldrin
25,604
Pyrene
70,808
Bromoform
126
Diethylphthalate 84
Styrene
912
Butyl benzyl phthalate
14,055
Di-n-butylphthalate
1,580
1,1,2,2-Tetrachloroethane 79
Carbon tetrachloride
158
Endosulfan
2,040
Tetrachloroethylene
272
Chlordane
51,798
Endrin
11,422
Toluene
145
Chlorobenzene
260
Ethylbenzene
207
Toxaphene
95,816
Chloroform 57
Fluoranthene
49,433
1,2,4-Trichlorobenzene
1,783
DDD
45,800
Fluorene
8,906
1,1,1-Trichloroethane
139
DDE
86,405
Heptachlor
10,070
1,1,2-Trichloroethane 77
DDT
792,158
Hexachlorobenzene
Trichloroethylene 97
Dibenz(a,h)anthracene
2,029,435
a-HCH (a-BHC)
1,835
o-Xylene
241
1,2-Dichlorobenzene(o)
390
b-HCH (b-BHC)
2,241
m-Xylene
204
g-HCH (Lindane)
1,477
p-Xylene
313

90. Retardation
Incorporate adsorbed solute mass

91. Sample problem:
A tanker truck collision has resulted in a spill of 5000 L of the insecticide diazinon 2000 m from the City of Miami’s water supply wells. Use a rule of thumb to estimate the dispersivity for the plume that is carrying the contaminant from the spill site to the wells.

92. Sample problem:
The transmissivity determined from aquifer tests is 100,000 m2 d-1 and the aquifer thickness is 20 m. The head in wells 1000 m apart along the flow path is 3.1 and 3 m. What is the gradient? What is the mean pore water velocity and what is the dispersion coefficient?

93. Sample problem:
You look up the Koc value of diazinon (290 ml/g). The aquifer material you tested has an foc of What is the Kd? If the porosity is 50% and the bulk density is 1.5 Kg L-1, what is R?
Assume retarded piston flow and estimate the arrival time of the insecticide at the well field using the appropriate data from the preceding problems.

94. Retardation

95. Aquifer Tests
Theis
Matching aquifer test data to the Theis type curve has resulted in the match point coordinates 1/u = 10, W(u) = 1, t = 83.9 minutes, and s =0.217 m. The pumping rate is 1 m3 min-1 and the observation well is 100 m away from the pumping well. Compute the aquifer transmissivity and storativity. Be sure to specify the units.
Hints:
T = Q/(4ps) W(u)
S = 4Ttu/r2.

96. Ghyben-Herzberg h z z Pf = Ps rg(h+z) = rsgz r(h+z) = rsz
rh = (rs-r) z→ h r/(rs-r) = z

97. Ghyben-Herzberg Seawater: 1.025 g cm-3 h r/(rs-r) = z
h 1/( ) = z
h 40 = z

98. Major Cations and Anions
Ca2+, Mg2+, Na+, K+
Anions:
Cl-, SO42-, HCO3-

99. Chemical Concentration Conversions
Usually given ML-3 (e.g. mg L-1 mg L-1)
Convert to mol L-1:
Convert to mol (+/-) L-1:

100. Charge Balance

101. Piper Diagrams
Convert to % mol (+/-)

102. Stiff Diagrams

103. Redox Reactions O2 (disappear) NO3- (disappear)
Fe/Mn (appear in solution)
SO42- (disappear)
CH4 (appear)