Island Recharge Problem

1
Island Recharge Problem Analytical
Solutions (with R 0.00305 ft/day)
h(x) R (L2 x2) / 2T
21.96
1D confined inverse solution with h(0) 20,
gives R
0.00278 ft/day.
2
2D Confined Numerical Solution
Gauss-Seidel Iteration Formula
3
Island Recharge Problem 4 X 7 Grid (a 4000 ft)
4
Island Recharge Problem Solutions (with R
0.00305 ft/day)
21.96
Too low error 0.13 ft
Same conceptual model
5
Sensitivity of solution to grid spacing
1. The approximation to the derivative
improves as grid spacing ? zero
as ? x ? 0
2. The calculation of the water budget
improves with smaller grid spacing.
6
L
?y/2
For full area (L x 2L) IN 8784E 2 ft3/day
?x/2
2L
7
Unconfined version of the Island Recharge Problem
R
groundwater divide
ocean
b
ocean
x 0
x L
x - L
8
Derivation of 2D unconfined equation with the
Dupuit Assumptions

1. No vertical flow
2. No seepage face

h
Impermeable rock
9
Unconfined version of the Island Recharge Problem
R
groundwater divide
ocean
b
ocean
datum
x 0
x L
x - L
Let b 100 ft K 100 ft/d so that at xL,
Kb 10,000 ft2/day
1. Could define an unconfined transmissivity T
u Kh
10
General Approaches for Unconfined Aquifers
1. Define an unconfined transmissivity Tu Kh
Update Tx Kxh and Ty Kyh during the solution.
MODFLOW uses this approach.
11
Unconfined version of the Island Recharge Problem
R
groundwater divide
ocean
b
ocean
datum
x 0
x L
x - L
Let b 100 ft K 100 ft/d so that at xL,
Kb 10,000 ft2/day
12
This equation is linear in h2.
Let v h2
Poisson Equation
Solve the finite difference equations for v and
then solve for h as ? v
13
Unconfined Poisson Equation
Confined Poisson Equation
14
Unconfined aquifer
Poisson eqn solved for v
h SQRT(v)
15
Outflow terms for the Water Balance of the
Unconfined Problem with ?x ?y
Qx -K ?v /2
Also Qy -K ?v /2
16
Fluxes calculated using v
Qx K ?v /2
17
Island Recharge Problem Analytical
Solutions (with R 0.00305 ft/day)
21.96
?
18
Unconfined version of the Island Recharge Problem
R
groundwater divide
ocean
b
ocean
datum
x 0
x L
x - L
Let b 100 ft K 100 ft/d so that at xL,
Kb 10,000 ft2/day
19
1D solution for unconfined aquifer
Governing Eqn.
Boundary conditions
20
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
21
Island Recharge Problem
R
groundwater divide
ocean
b
ocean
datum
x 0
x L
x - L
Let b 100 ft K 100 ft/d so that at xL,
Kb 10,000 ft2/day
At groundwater divide confined aquifer h
ho unconfined aquifer hb ho
22
h(x) R (L2 x2) / 2T
confined
at x 0 h ho
ho R L2 / 2T
R 2 Kb ho / L2
b
hL
? (b ho)2 R L2 /K b2
0 L
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.