Geology 724
Mid-Term Exam – Spring 2001
(150 points)
Answer all questions in the blue book.
A. Matching (15 points)
1. div q
a.  K h
2. grad h
b. qx + qy + qz
3. q
c.
4. qx y  z
d.  K h
5. 2 h
e.   q
 2h  2h  2h


x 2 y 2 z 2
f.  Kx Ix A (where Ix is hydraulic gradient and A is area)
g.
h
h
h
1
2
3
x
y
z
(where 1,2, 3 are unit vectors)
B. True/False. Indicate whether the statement is true (T) or false (F). (10)
1. The hydraulic conductivity tensor is a symmetric tensor.
2. Boundary conditions always influence the solution of a steady-state finite difference
model.
3. Initial conditions for a transient problem are usually the same as the initial guesses
for the steady-state version of the problem.
2
C. Give a short answer to each of the following questions (25 points).
1. Under what conditions does the hydraulic conductivity tensor reduce to three
principal components: Kx, Ky, Kz?
2. If the full form of the hydraulic conductivity tensor must be used, what is the correct
expression of Darcy’s law for qx ?
3. The steady state mass balance equation for flow through a representative elementary
volume (REV) is div q = 0, which is shorthand for a sum of three terms. What are
the units of each term? What does each term physically represent?
4. When calculating head using a Gauss-Seidel iteration formula, the solution is
monitored by checking the residual. How is the residual calculated?
h
more accurate than
x
h
either the forward or backward difference approximation of
?
x
5. Why is the central difference approximation for the term
D. Equations (40 points)
1. Consider the one-dimensional transport equation given below, where DL , v, and R
are constants.
DL
 2c
c
c
v
R
2
x
x
t
a. Write a finite difference approximation for this equation that could be solved
explicitly.
(5)
b. If you were to re-arrange the finite difference equation in your answer to part a to
put it in a form (the so-called SOR form) that identifies the combination of
parameters that controls the stability of the solution, you would find that:
 = 2 DL t / R (x)2
Using this equation, write the stability criterion for this solution.
(5)
3
2. Consider the 2D Laplace equation:
 2h
 2h

0
x 2
z 2
a. Describe the groundwater flow system that is represented by this governing
equation, where h is head and z is the vertical axis. (That is, what is an appropriate
conceptual model for this governing equation?)
(5)
b. Write a finite difference expression for this equation assuming that x  z. (5)
c. Put your finite difference equation in a form suitable for Gauss-Seidel iteration. (5)
d. Now write the equation(s) needed for iteration by SOR.
(5)
e. Suppose you are constructing a spreadsheet model to solve the equation in part c
(Gauss-Seidel formula) and that the cell C4 is an interior node in the spreadsheet
model. Write the equation you would input to the spreadsheet. (5)
f. Assume that cell C8 is a side boundary node in a spreadsheet model that is set up
for a mesh-centered grid that has 5 rows (a – E) . Cell C8 is on a groundwater divide
(no flow boundary) located on the right hand side of the problem domain. Write the
appropriate equation for this cell for a model based on the equation in part c. (5)
3. Consider a one-dimensional problem represented by a mesh-centered grid with 6
nodes as shown below. Node 1 is a specified head node and node 6 lies on a
specified flow boundary, with flux equal to q entering the problem domain.
     
1
2
3
4
5
6
x
The finite difference equation for i = 2 is Bh2 + Ch3 = F2 – Ah1 and the entry for i = 3 is
Ah2 + Bh3 + Ch4 = F3. Write the entry for i =6 assuming that A  C.
(5)
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D. System Conceptualization (60 points)
1. The diagram below shows a cross section through an unconfined aquifer. There is
constant recharge across the water table, R, and seepage from an underlying
bedrock unit, B. While the exact location of the water table is unknown, it is known
to occur in the interval from z = H to z = Z. We wish to consider two-dimensional
flow in this cross section (i.e., we will not use the Dupuit assumptions but you may
assume that there is no seepage face).
R
z
Z
Groundwater
Divide
?
?
h=H
0
x
0
B
L
a. Assume that the aquifer is heterogeneous and anisotropic and that the flow system is
at steady state. Write the mathematical model for this system. When writing the
side and bottom boundary conditions indicate mathematically the location where
each equation applies (i.e., do not show locations by labeling a sketch). Also write
the boundary condition for the top boundary. (20)
b. Write the equations that you would use to develop a water balance computation for
this problem. (10)
c. Suppose that we want to simulate the transient response of the system to an increase
in the recharge rate, R. Write the appropriate transient governing equation. (5)
d. Your equation in part c should have a storage parameter. What is this storage
parameter called and how is it defined? (5)
5
2. Consider the transient problem in part D.1c above, but now suppose you decide to
simulate the problem in two dimensions in areal view (the x-y plane) using Dupuit
assumptions.
y
Hydraulic no flow boundary (streamline)
Groundwater
Divide
River
Hydraulic no flow boundary (streamline)
x
a. Write the governing equation for this problem.
(5)
b. Describe how you would define initial conditions for this problem.
(5)
c. Suppose we overlay a mesh-centered grid over the problem domain. The grid has 5
rows and 9 columns, including the nodes along the boundaries. How many nodes
are “active” nodes, i.e., nodes having unknown head values that are calculated
during the solution?
(5)
3. Recall the confined version of the Island Recharge Problem we considered in class.
Now suppose that there is a drought on the island and the recharge rate, R, drops to
zero. How would you modify your steady-state water budget computation when
solving the transient problem? Write a generic form of the equation(s) you would
introduce into your spreadsheet model to compute the storage term used in the water
balance and indicate whether the equation(s) would go on the inflow or outflow side
of the water balance. (5)