1. Island Recharge Problem
Analytical Solutions
(with R = ft/day)
Solution Type
Head at center of island (ft)
1D confined*
1D unconfined
2D confined
20.00
21.96
*1D confined inverse solution with h(0)= 20, gives R =
ft/day.

2. Unconfined version of the Island Recharge Problem
groundwater
divide h
ocean
ocean b
datum
x = - L
x = 0
x = L
Let b = 100 ft & K = 100 ft/d
so that
at x=L, Kb= 10,000 ft2/day
Can define an
unconfined “transmissivity”:
Tu = Kh

3. grid spacing decreases. x,  y IN 4000 6710E 2 1000 8243E 2 L
y/2
For full area (L x 2L)
IN = 8784E 2 ft3/day
x/2 2L
IN increases as
grid spacing decreases.
x,  y IN
E 2
E 2
E 2
E 2

4. Let v = h2
Solve the finite difference equations for v
and then solve for h as  v

5. Spreadsheets

6. Water Balance of the Unconfined Problem
Outflow terms for the
Water Balance of the Unconfined Problem
Qx = -K v /2
Qy = -K v /2

7. h (x) = [R (L2 - x2 )/K] + (hL)2
1D approximation for unconfined case
Governing Eqn.
Let v = h2
at x =0
Boundary conditions
h(L) = hL = 100
100
Analytical solution for 1D “unconfined” version of the problem
h2(x) = R (L2 - x2 )/K + (hL)2
h (x) = [R (L2 - x2 )/K] + (hL)2

8. h (x) = [R (L2 - x2 )/K] + (hL)2
Analytical solution for 1D “confined” version of the problem
h(x) = R (L2 – x2) / 2T
Analytical solution for 1D “unconfined” version of the problem
h (x) = [R (L2 - x2 )/K] + (hL)2

9. h (x) = [R (L2 - x2 )/K] + (hL)2
h(x) = R (L2 – x2) / 2T
“confined”
h = ho at x = 0
ho = R L2 / 2T
R = 2 Kb ho / L2
h (x) = [R (L2 - x2 )/K] + (hL)2
“unconfined”
at x = 0; h = b + ho
&
hL = b
 (b + ho)2 = [R L2 /K] + b2
R = (2 Kb ho / L2) + (ho2 K / L2)
 To maintain the same head (ho) at the groundwater divide as in the confined system, the 1D unconfined system requires that recharge rate, R, be augmented by the term shown in blue.

10. Bottom 4 rows Assume pumping from the well where: Qwell = 0.1 IN
& IN is the
inflow to the top right hand
quadrant.
Well
-R = Qwell / {(x  y)/4} = Qwell / (a2/4)
= 4 Qwell / a2

11. Q
R x y y x
Point source
(L3/T)
Distributed source
(L3/T)
Finite difference models simulate all sources as
distributed sources; finite element models simulate
all sources as point sources.

12. Gauss-Seidel Formula for Laplace Equation
SOR Formula
Relaxation factor
= 1 Gauss-Seidel
< 1 under-relaxation
>1 over-relaxation