GLY 6826
Assignment 1 Finitedifference modeling using a spreadsheet
60 points Due Sept 7, 2011
In this assignment, you will work through some basics of finitedifference numerical
modeling and start modeling using a spreadsheet program.
Purpose
: To provide insight on how finitedifference approximations work and to
introduce basic concepts of iterations and convergence criteria.
Please submit your answers online through the Sakai web site. Include an Excel
workbook with your spreadsheets.
Part I. Finite Difference Approximation of 1D steady state flow (25 pts)
Consider a one dimensional, homogeneous “aquifer” with steadystate fluid flow.
1.
(1 pt) What is the appropriate governing equation for fluid flow?
To work up the finite difference approximation for the hydraulic head at a point “i” (so
we’ll refer to the hydraulic head as h
i
), consider 2 locations in the aquifer that are equally
spaced on either side of i (call them i1 and i+1) and the distance between each (i and i+1
and i and i1) is
∆
x. The first derivative of h with respect to distance, dh/dx, is replaced
by the difference,
∆
h/
∆
x .
2.
A. (1 pt) Write an expression for
∆
h/
∆
x at point i1/2, located half way between i
1 and i
B.(1 pt) Write an expression for
∆
h/
∆
x at a point i+1/2 half way between i and
i+1.
C.
(2 pt) Using your solutions from #2 and #3, write an expression for the second
derivative of h with respect to x at point i.
D.
(2 pt) Substitute this expression into equation 1 and solve for h
i
.
Consider a steady state 1D flow problem representing a crosssection through an aquifer
with a fullypenetrating lake or river on one side (head of 100 m) and a fullypenetrating
lake or river on the other side (head of 200 m). Assume the aquifer is homogeneous. The
length of the aquifer is 10 km, and it is discretized (divided) into 11 Excel cells (
∆
x=1000
m).
For this exercise, we’ll consider the “node” where the head to be determined is
1
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View Full Documentlocated at the center of each Excel cell. You know the head on the left and right hand
side, and you have figured out (in #5) how to solve for head at any point.
3.
(2 pts) Before you begin the numerical approximation:
This problem has a simple
analytical solution. What is the head at each cell?
100
?
?
?
?
?
?
?
?
?
200
4.
(2 pts) Type in your expression from #5 into all of the interior nodes (translating
into an Excel formula). What happens and why?
There are two methods to solve this in Excel. Both require “iterations”. The first method
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 Fall '11
 Screaton
 Numerical Analysis, pts, lake or river

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