Analytical%20and%20Numerical%20Solutions%20are%20affected%20by:

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Island Recharge Problem

• Analytical and Numerical Solutions are affected
  by
• Differences in conceptual model (confined vs
  unconfined)
• Dimensionality (1D vs 2D)
• Numerical Solutions also affected by
• Grid Spacing (e.g., 4000 ft vs 1000 ft)

Now, lets add a pumping well!
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Q
R ?x ?y
?y
?x
Point source (L3/T)
Distributed source (L3/T)
R Q/?x ?y
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P - R
Distributed sink
In a finite difference model, all sinks
(including pumping) are represented as
distributed sinks.
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Island Recharge Problem
ocean
L
y
2L
well
ocean
ocean
Only ¼ of the pumping well is in the upper right
hand quadrant. Qwell QT / 4
x
ocean
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Bottom 4 rows
Assume pumping from the well where Qwell 0.1
IN IN is the inflow to the top right hand
quadrant.
Pumping is treated as a diffuse sink.
Well
P Qwell / (?x ? y)/4 Qwell / (a2/4)
4 Qwell / a2
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Also
where QT 0.1 (4)(IN) IN is the inflow
to the upper right hand quadrant.
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Write a new finite difference expression
P 0 except at the pumping well where P
4Qwell/a2
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Bottom 4 rows
Pumping is treated as a diffuse sink.
Well
Head computed by the FD model is the average head
in the cell, not the head in the well.
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Q
P ?x ?y
?y
?x
Point sink (L3/T)
Distributed sink (L3/T)
P Q/?x ?y
Finite difference models simulate all
sources/sinks as distributed sources/sinks
finite element models simulate all sources/sinks
as point sources/sinks.
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Use eqn. 5.1 (Thiem equation for confined
aquifers) or equation 5.7 (unconfined version of
the Thiem equation) in AW to calculate an
approximate value for the head in the pumping
well in a finite difference model.
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unconfined aquifer
confined aquifer
Thiem equation for steady state flow to a
pumping well.
Figure from Hornberger et al. 1998
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Use eqn. 5.1 (confined aquifer) or 5.7
(unconfined aquifer) in AW to calculate an
approximate value for the head in the pumping
well in a finite difference model.
Sink node (i, j)
r a
re
(i1, j)
hi,j is the average head in the cell.
re is the radial distance from the node where
head is equal to the average head in the cell,
hi,j
Using the Thiem eqn., we find that re 0.208 a
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Solution by Iteration

1. Gauss-Seidel Iteration
2. SOR (Successive Over Relaxation)

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G-S
where c error or residual
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SOR Formula
Relaxation factor 1 Gauss-Seidel lt 1
under-relaxation gt1 over-relaxation
where, for example,
(Gauss-Seidel Formula for Laplace Equation)
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SOR solution for confined Island Recharge Problem
The Gauss-Seidel formula for the confined Poisson
equation
where
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