Petroleum Engineering 603
EXAM 1
Date: October 5, 2004
Version 1A
Name:_________________________________________
You may use one sheet of handwritten notes, attach to exam when turned in.
You may use a calculator.
Do not unstaple the examination booklet.
Time allotted for the examination is 120 minutes.
When you have completed the examination, read and sign the statement below, then turn in the examination booklet.
STATEMENT OF ACADEMIC INTEGRITY
I pledge that I have neither given nor received aid in completing this examination. I have followed the strictures of the Texas A&M University Aggie Code of Honor during this examination.
Signature: ____________________________________
1

Beginning with the Continuity Equation as shown below, derive the Diffusivity
Equation for general reservoir flow, considering a single phase, constant compressibility liquid, and using density as the dependent variable. Note all assumptions in your derivation, especially those relating to the linearity of your final derived Diffusivity
Equation. Include derivation of pore volume compressibility (c f
) as a function of fluid density.
  
 u = -

(

 t

)
2

3

Consider a single phase, horizontal, 1-D linear reservoir initially at constant pressure.
For times larger than zero, pressure is held constant at the right boundary at the initial pressure. For times larger than zero, fluid is produced at the left boundary at constant rate. a) Show the partial differential equation that describes fluid flow for this specific reservoir geometry, along with initial and boundary conditions expressed in equation form. b)
Sketch a time sequence of pressure profiles [curves of p(x,t) vs. x, for increasing time]. Show the trend of increasing time, and identify all reservoir flow regimes.
4

The following sketch shows the dimensions of two adjacent reservoir simulation gridblocks.
Calculate water production rate (q w
), using upstream relative permeability (k rw
) for data below:
Gridblock 1 Gridblock 2 p
S w
Z
1990.0
0.60
-5610.0
1950.0 psia
0.80
-5615.0 ft, subsea k rw
P cow
B w

 w w
/144 k
0.08
20.0
1.024
0.3350
0.48
100.0
0.30
23.0 psia
1.020 rcf/scf
0.3360 cp
0.48 psia/ft
100.0 md
100 ft 100 ft
100 ft
1 2
20 ft
5

6

Derive the first order, forward difference, Taylor’s Series approximation of p’’(x) using points p i
, p i+1
and p i+2
. Your derivation should include the order of the largest term of the truncation error. p’’(x)  (p i
- 2*p i+1
+ p i+2
)/(  x 2 )
7

8

Wattenbarger’s Eq. 160 from Chapter 1 was used to calculate the analytical solution for dimensionless pressure profile. Write a VBA function that evaluates Eq. 160 as a truncated series. Your function should continue to add terms to the series until the magnitude of the last
(and final) term added is less than 1.0x10
-6 . Assume that the module for your function contains
“Option Explicit” at the top. Consider the elements of programming style recommended in
PETE 603. u(x,t) = 1 + 2

 n=1
(- 1 ) n

 n -
1
2


 cos



 n -
1
2





1
2


2
 2 t
9

10