Potential flow

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Lecture 24-26

• Potential flow

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Assumptions

• Steady state
• 1 or 2-d flow
• Homogeneous aquifer
• Isotropic
• No sources/sinks except wells
• For unconfined datum is base of aquifer

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Recall
If we define flow over the aquifer depth Q then
Call
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Recall flow equation
Which is Laplace equation
Solutions to this equation are called potential
functions Equipotential contours are given by
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Potential function F
F is called the discharge potential for confined
aquifers it is
And for unconfined aquifer it is
Where H is the saturated thickness
Now we can write
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Streamlines
Notice that the flow is perpendicular to
Equipotential contours. i,.e. flow // 0 Let us
define curves tangent to the flow vector and call
them streamlines
Relation between Fand ? Slope at p is
P
? ?1
Also along this streamline
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Comparing (1) and (2) we can see that
remember
In radial coordinates
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We can show that the discharge between two
streamlines is the numerical Difference between
their constant values ?1 ?2
What does this mean in terms of shape of stream
tube?
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Potential stream functions for uniform flow
Uniform flow in x-direction
Solving for the stream potential functions we
get
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Potential stream functions for uniform flow
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Potential stream functions for flow from an
injection well
Solving for the stream and equipotential functions
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A well in a uniform field
Using superposition we can write the stream
potential function for this case
For a pumping well
The corresponding potential function is
Notice that 1) values for constants are
arbitrary, 2) we can multiply RHS by any
constant 3) If we have more than one well we add
another well term
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Capture zone for a producing well
Confined aquifer, hydraulic gradient in
x-direction 0.01, b 10 m , K20 m/d Pumping
rate 200 m3/day
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Applications

• A single well to capture contaminants from a
  point source
• A pumping well to remove contaminated zone from
  the capture zone of a municipal well
• Two pumping wells to prevent contaminant
  migration in a certain direction

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Example 4-7 McWhorter Sunada

• Contaminated water from several waste-holding
  ponds seeps into an aquifer in which uniform flow
  exists. A well is to be constructed 300 m down
  gradient from the ponds in an attempt to
  intercept the contaminated groundwater. i0.002,
  T 0.13 m2/s. If the width of contaminant source
  area is 200 m, measured perpendicular to the
  direction of groundwater flow, calculate minimum
  rate of pumping to intercept the contaminated
  water. Neglect dispersion and discharge of
  contaminated water is small.