GEOLOGY 724
Quiz #1 (TAKE HOME)
100 Points
NAME: ____________________________________________________________
Governing equations are written using several different types of notation: partial derivatives, vector notation, del notation and summation notation. It is important to be able to translate from one notation to another. All of the above-mentioned notations are used in the groundwater literature. Write the following equations in the notations indicated.
1.
 2 h
 x
2

 2 h
 y
2

 2 h
 z
2

0
(5 points) vector notation:
(5 points) del notation:
2.

 x
( K xx
 h
 x
)


 y
( K yy
 h
 y
)


 z
(5 points) vector notation:
(5 points) del notation:
(5 points) summation notation:
3.
 q
 x x 
 q y
 y

 q
 z z 
0
(5 points) vector notation:
(5 points) del notation:
( K zz
 h
 z
)

0

-2-
4.
Matching. Consider equations 1 and 2 above. Match each equation with the appropriate description. There may be two correct answers. Give both answers for full credit.
(5 points) a) homogeneous and isotropic
Eq. 1 ______ b) heterogeneous and isotropic b) homogeneous and anisotropic
Eq. 2 ______ d) heterogeneous and anisotropic e) Laplace’s equation
5.
Write the components of the hydraulic conductivity tensor in matrix form, using either letters or numbers as subscsripts.
(5 points)
K =
6.
In what situation are 6 of the components of the hydraulic conductivity tensor equal to zero?
(5 points)
7.
Equations 1, 2 and 3 on the previous page are all steady-state equations.
How can you tell by looking at the equations?
(5 points)

-3-
8.
When grad operates on a scaler, it produces a ______________ but when grad operates on a vector, we have a special kind of vector multiplication known as
______________ and this operation produces a _______________________.
(3 points)
9.
What is the effect on the vector q when the tensor K operates on the vector grad h?
(2 points)
10.
We can write Darcy’s Law in three dimensions as: q =

K grad h where q and grad h are __________________ because they have both __________________ and __________________. Whereas, h is a __________________ because it has
__________________ only. The parameter K is known as a __________________ and in its most general form must be represented by __________________ components.
The other “law” in addition to Darcy’s law that is used to derive the general equation describing groundwater flow is known as ____________________________________.
A word equation for the steady-state version of this law is: _______________________
______________________________________________________________________.
(10 points)
11.
Name (in words) the three components of a transient mathematical model.
(6 points)

-4-
12.
Perform the matrix multiplication indicated by equation A.10 on p. 208 of Wang and Anderson and write out expressions that define the components of the hydraulic conductivity tensor in a general global coordinate system (in two-dimensions).
Show all work.
(24 points)