Final Review

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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)

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Numerical Derivatives

• finite difference approximation
• slope between points
• dy/dx Dy/Dx

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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)

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Partial Derivatives

• Functions of more than one variable
• Example h(x,y) x4 y3 xy

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Partial Derivatives

• Partial derivative of h with respect to x at a y
  location y0
• Notation ?h/?xyy0
• 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

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Partial Derivatives

• Example
• h(x,y) x4 y3 x2y xy
• ?h/?x 4x3 2xy y
• ?h/?xyy0 4x3 2xy0 y0

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WHY?
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Gradients

• del h (or grad h)
• Darcys Law

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Equipotentials/Velocity Vectors
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Capture Zones
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Watersheds
http//www.bsatroop257.org/Documents/Summer20Camp
/Topographic20map20of20Bartle.jpg
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Watersheds
http//www.bsatroop257.org/Documents/Summer20Camp
/Topographic20map20of20Bartle.jpg
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Capture Zones
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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

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Water (Mass) Balance

• Principal components
• Precipitation
• Evaporation
• Transpiration
• Runoff
• P E T Ro Change in Storage
• Units?

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Ground Water (Mass) Balance

• Principal components
• Recharge
• Inflow
• Transpiration
• Outflow
• R Qin T Qout Change in Storage

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Ground Water Basics

• Porosity
• Head
• Hydraulic Conductivity

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Porosity Basics

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

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Porosity Basics

• Can re-write that as
• Then incorporate
• Solid density rs
• Msolids/Vsolids
• Bulk density rb
• Msolids/Vtotal
• rb/rs Vsolids/Vtotal

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Ground Water Flow

• Pressure and pressure head
• Elevation head
• Total head
• Head gradient
• Discharge
• Darcys Law (hydraulic conductivity)
• Kozeny-Carman Equation

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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)

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Pressure and Pressure Head

• Pressure relative to atmospheric, so P 0 at
  water table
• P rghp
• r density
• g gravity
• hp depth

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

• Water wants to fall
• Potential energy

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Elevation Head (increases with height above datum)
Elevation
Elevation Head
Elevation datum
Head
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Total Head

• For our purposes
• Total head Pressure head Elevation head
• Water flows down a total head gradient

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P 0 ( Patm)
Pressure Head
Total Head (constant hydrostatic equilibrium)
Elevation
Elevation Head
Elevation datum
Head
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Head Gradient

• Change in head divided by distance in porous
  medium over which head change occurs
• A slope
• dh/dx unitless

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Discharge

• Q (volume per time L3T-1)
• q (volume per time per area L3T-1L-2 LT-1)

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Darcys 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
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Mean Pore Water Velocity

• Darcy velocity
• q -K ?h/?x
• Mean pore water velocity
• v q/ne

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Intrinsic Permeability
L2
L T-1
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More on gradients
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More on gradients

• Three point problems

h
412 m
h
400 m
100 m
h
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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
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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
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Types of Porous Media
Isotropic
Anisotropic
Heterogeneous
Homogeneous
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Hydraulic Conductivity Values
K (m/d)
8.6
0.86
Freeze and Cherry, 1979
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Layered media (horizontal conductivity)
Q4
Q3
Q2
Q1
Q Q1 Q2 Q3 Q4
K2
b2
K1
b1
Flow
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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
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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

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Transmissivity

• T Kb
• gpd/ft, ft2/d, m2/d

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Schematic
b2 (or h2)
T2 (or K2)
i 2
d2
k2
b1
T1
i 1
d1
k1
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Pumped Aquifer Heads
b2 (or h2)
i 2
d2
b1
i 1
d1
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Heads
h2 - h1
b2 (or h2)
h2
i 2
d2
h1
b1
i 1
d1
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Flows
h2
h2 - h1
b2 (or h2)
h1
i 2
d2
qv
b1
i 1
d1
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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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Derivation of 1-D Laplace Equation

• Inflows - Outflows 0
• (qxx- qxxDx)DyDz 0

Governing Equation
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Boundary Conditions

• Constant head h constant
• Constant flux dh/dx constant
• If dh/dx 0 then no flow
• Otherwise constant flow

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General Analytical Solution of 1-D Laplace
Equation
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Particular Analytical Solution of 1-D Laplace
Equation (BVP)
BCs - Derivative (constant flux) e.g., dh/dx0
0.01 - Constant head e.g., h100 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.
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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
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Finite Difference Solution of 1-D Laplace
Equation (ctd)
Combine 1st order derivative approximations to
get 2nd order derivative approximation.
Solve for h
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2-D Finite Difference Approximation
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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

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Derivation of Poisson Equation
(qxx- qxxDx)Dyb RDxDy 0
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General Analytical Solution of 1-D Poisson
Equation
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Water balance

• Qin RDxDy Qout 0
• qin bDy RDxDy qout bDy 0
• -K dh/dxin bDy RDxDy -K dh/dxout bDy 0
• -T dh/dxin Dy RDxDy -T dh/dxout Dy 0
• -T dh/dxin RDx T dh/dxout 0

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Dupuit Assumption

• Flow is horizontal
• Gradient slope of water table
• Equipotentials are vertical

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Dupuit Assumption
(qxx hxx - qxxDx hxDx)Dy RDxDy 0
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Transient Problems

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

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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 2002Dr. T.
  Brikowski, UTD

http//www.utdallas.edu/brikowi/Teaching/Geohydro
logy/LectureNotes/Regional_Flow/Storativity.html
66
1-D Transient GW Flow
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1-D Transient GW Flow Deriving the Governing PDE
qxxDx
qxx

• DVw -DxDy S Dh

(qxxDx - qxx)Dyb -SDxDy?h/?t
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(No Transcript)
69
Finite Difference Solution

• First order spatial derivative

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Second order spatial derivative
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Finite Difference Solution

• Temporal devivative

hx, t-Dt
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All together
73

• Stability criterion TDt/(SDx2) lt ½.

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Diffusion
75

• Ficks Law

• Heat/Diffusion Equation

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Temporal Derivative
Cx, t-Dt
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All together
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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 Ficks 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

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Convection-Dispersion Equation

• Key difference from diffusion here!
• Convective flux

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CDE
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Finite Difference Spatial
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Finite Difference Temporal
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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/Dx2 and vDt/2Dx

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

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Fitting the CDE
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Adsorption Isotherm

• Linear Cs Kd C

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Koc Values

• Kd Koc foc

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Organic Carbon Partitioning Coefficients for
Nonionizable Organic Compounds. Adapted from
USEPA, Soil Screening Guidance Technical
Background Document. http//www.epa.gov/superfund/
resources/soil/introtbd.htm
90
Retardation

• Incorporate adsorbed solute mass

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Sample problem

• A tanker truck collision has resulted in a spill
  of 5000 L of the insecticide diazinon 2000 m from
  the City of Miamis 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
  0.0001. 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.

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Retardation
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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(hz) rsgz r(hz) 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/(1.025 - 1) z
• h 40 z

98
Major Cations and Anions

• Cations
• 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
http//water.usgs.gov/pubs/wri/wri024045/htms/repo
rt2.htm
103
Redox Reactions

• O2 (disappear)
• NO3- (disappear)
• Fe/Mn (appear in solution)
• SO42- (disappear)
• CH4 (appear)