1. 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, let’s add a pumping well!

2. Q
R x y y x
Point source
(L3/T)
Distributed source
(L3/T)
R = Q/x y

3. P = - R
Distributed sink
In a finite difference model, all sinks (including pumping)
are represented as distributed sinks.

4. 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

5. 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

6. Also:
where: QT = 0.1 [ (4)(IN) ]
IN is the inflow to the upper
right hand quadrant.

7. Write a new finite difference expression
P = 0 except at the pumping well where P = 4Qwell/a2

8. 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.

9. 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.

10. Use eqn. 5.1 (Thiem equation for confined aquifers)
or equation 5.7 (unconfined version of the Thiem equation)
in A&W to calculate an approximate value for the head
in the pumping well in a finite difference model.

11. unconfined aquifer confined aquifer Thiem equation for steady state
flow to a pumping
well.
Figure from Hornberger et al. 1998

12. Use eqn. 5.1 (confined aquifer) or 5.7 (unconfined aquifer)
in A&W to calculate an approximate value for the head
in the pumping well in a finite difference model.
Sink node (i, j)
r = a re
(i+1, 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 = a

13. Solution by Iteration Gauss-Seidel Iteration
SOR (Successive Over Relaxation)

14.   where c = error or residual In SOR (successive over-relaxation):
G-S
where c = error or residual
In SOR (successive over-relaxation):
SOR
relaxation factor 

15. (Gauss-Seidel Formula for Laplace Equation)
SOR Formula
Relaxation factor
= 1 Gauss-Seidel
< 1 under-relaxation
>1 over-relaxation
where, for example,
(Gauss-Seidel Formula for Laplace Equation)

16. Spreadsheet SOR solution for confined Island Recharge Problem
The Gauss-Seidel formula for
the confined Poisson equation
where
Spreadsheet