Name: _____________________________
Assigned: 13 November 2012
Due:
12 December 2012 [by Midnight (23:59:59 US CST)]
Send a SINGLE .pdf file (no .doc, .xls, .??? — no exceptions) to: t-blasingame@tamu.edu, dl_coordinators@pe.tamu.edu
Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs
Course Examination [Blasingame]
Assigned: 13 November 2012 — Due: 12 December 2012 [by Midnight (23:59:59 US CST)]
Assignment Coversheet
(This sheet must be included with your work submission)
Required Academic Integrity Statement: (Texas A&M University Policy Statement)
Academic Integrity Statement
All syllabi shall contain a section that states the Aggie Honor Code and refers the student to the Honor Council Rules and
Procedures on the web.
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Upon accepting admission to Texas A&M University, a student immediately assumes a commitment to uphold the Honor
Code, to accept responsibility for learning and to follow the philosophy and rules of the Honor System. Students will be
required to state their commitment on examinations, research papers, and other academic work. Ignorance of the rules does
not exclude any member of the Texas A&M University community from the requirements or the processes of the Honor
System. For additional information please visit: www.tamu.edu/aggiehonor/
On all course work, assignments, and examinations at Texas A&M University, the following Honor Pledge shall be preprinted
and signed by the student:
"On my honor, as an Aggie,
I have neither given nor received unauthorized aid on this academic work."
Aggie Code of Honor:
An Aggie does not lie, cheat, or steal or tolerate those who do.
Required Academic Integrity Statement:
"On my honor, as an Aggie, I have neither given nor received
unauthorized aid on this academic work."
_______________________________ (Print your name)
_______________________________ (Your signature)
Coursework Copyright Statement: (Texas A&M University Policy Statement)
The handouts used in this course are copyrighted. By "handouts," this means all materials generated for this class, which
include but are not limited to syllabi, quizzes, exams, lab problems, in-class materials, review sheets, and additional problem
sets. Because these materials are copyrighted, you do not have the right to copy them, unless you are expressly granted
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As commonly defined, plagiarism consists of passing off as one’s own the ideas, words, writings, etc., that belong to another.
In accordance with this definition, you are committing plagiarism if you copy the work of another person and turn it in as your
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destroys the trust among colleagues without which research cannot be safely communicated.
If you have any questions about plagiarism and/or copying, please consult the latest issue of the Texas A&M University
Student Rules, under the section "Scholastic Dishonesty."
____________________________________________________________________________________________________________________________________________________________
Notes:
(Page 1 of 8)
— You MUST submit ALL assignments in a SINGLE .pdf file — P620_12C_CourseExam_YOURLASTNAME.pdf
— You MUST ensure that all pages are correctly ordered and scanned legibly. (poorly constructed/assembled work will not be graded)
— Neatness, proper structure, clarity of your work is critical. No scratch-outs, use a good pencil or type you work.
— This is an EXAMINATION, no collaboration is allowed, submit only your best work!
Name: _____________________________
Assigned: 13 November 2012
Due:
12 December 2012 [by Midnight (23:59:59 US CST)]
Send a SINGLE .pdf file (no .doc, .xls, .??? — no exceptions) to: t-blasingame@tamu.edu, dl_coordinators@pe.tamu.edu
Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs
Course Examination [Blasingame]
Assigned: 13 November 2012 — Due: 12 December 2012 [by Midnight (23:59:59 US CST)]
Exam Guidelines:
Deadline/Due Date:
● Due date: 12 December 2012 (23:59:59 US CST)
● Submit as a SINGLE .pdf file named: P620_12C_CourseExam_YOURLASTNAME.pdf
● Send to t-blasingame@tamu.edu and dl_coordinators@pe.tamu.edu
Rules:
1. You are to work independently.
[Collaborative efforts are not permitted (i.e., joint work). You are permitted to discuss problems — however; sharing of
solutions, plots, programs, etc. is not permitted and will result in a failure grade (F) for this course.]
2. You must show all work for credit.
[Unsupported work will not be given credit.]
3. You are given specific instructions for each problem, follow these guidelines exactly.
4. Be as neat and organized as possible.
[Work that can not be followed will be graded as incorrect. This rule is specifically directed towards work that is illegible,
poorly constructed, and/or is simply incomplete.]
5. You must provide an OUTLINE for each problem solution, where this outline should include:
— A restatement of each problem in your own words/terms.
— The assumptions/limitations made in order to solve the problem (if appropriate).
— A nomenclature section where major variables are defined (if appropriate).
— References or other citations (if appropriate).
Format for Individual Problems:
1. For derivation problems, YOU MUST USE THE FOLLOWING FORMAT:
— Type or write neatly, USE LINED PAPER IF WORK IS HANDWRITTEN.
— Use a dark lead pencil — DO NOT USE A PEN.
— You are to ONLY write (or print) on the FRONT of any particular page.
— Number the pages of your work in the upper right corner of each page as follows:
Problem #/page #/total pages (for a particular problem)
2. For computer-aided solutions, YOU MUST PROVIDE THE FOLLOWING: (if applicable)
— A complete development of the equations and algorithms used in the problem.
— A written summary description of the problem (2 paragraphs minimum).
— A copy of the computer code, the input file, and an abbreviated output file.
— For data plots, you are to:
■ Scale plots APPROPRIATELY, so that regions of interest/data features are clearly identified.
■ IF you provide a summary table of data, then such tables MUST be abbreviated to 1 page.
3. Work Layout: (as appropriate)
— NEATNESS:
You will be graded on the neatness of your work.
— LABELS:
All plots, trends, features MUST be appropriately labeled — no exceptions.
— LINES:
Use appropriate drafting care in construction of lines, trends, arrows, etc.
— SKETCHING: Take great care in any sketches you create/use in your work.
Scholastic Dishonesty:
THE STUDENT IS HEREBY WARNED THAT ANY/ALL ACTS OF SCHOLASTIC DISHONESTY WILL RESULT IN AN "F" GRADE
FOR ALL ASSIGNMENTS IN THIS PORTION OF THE COURSE. As a definition, "scholastic dishonesty" will include any or all of the
following acts:
● Unauthorized collaborations — you are explicitly forbidden from working together.
● Using work of others — you are explicitly forbidden from using the work of others — "others" is defined as students in this
course, as well as any other person. You are required to perform your own work.
This is a zero-tolerance policy, there will be no exceptions.
Work Standard:
Simply put, the expectation of the instructor (Blasingame) is that "perfection is the standard" — in other words, your work will be
judged against a perfect standard. If your submission is not your very best work, then don't submit it. You have an OBLIGATION to
submit only your very best work.
____________________________________________________________________________________________________________________________________________________________
Notes:
(Page 2 of 8)
— You MUST submit ALL assignments in a SINGLE .pdf file — P620_12C_CourseExam_YOURLASTNAME.pdf
— You MUST ensure that all pages are correctly ordered and scanned legibly. (poorly constructed/assembled work will not be graded)
— Neatness, proper structure, clarity of your work is critical. No scratch-outs, use a good pencil or type you work.
— This is an EXAMINATION, no collaboration is allowed, submit only your best work!
Name: _____________________________
Assigned: 13 November 2012
Due:
12 December 2012 [by Midnight (23:59:59 US CST)]
Send a SINGLE .pdf file (no .doc, .xls, .??? — no exceptions) to: t-blasingame@tamu.edu, dl_coordinators@pe.tamu.edu
Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs
Course Examination [Blasingame]
Assigned: 13 November 2012 — Due: 12 December 2012 [by Midnight (23:59:59 US CST)]
Exam Guidelines: (Continued)
Before you start:
● Read the examination in its entirety, and then read it again, using a highlighter to mark points of interest/etc.
● Assess the time you believe you will need — PLAN your work.
Value: (each problem is worth 20 points, including the bonus problem)
For a "B" grade you must complete the following problems:
Problem 1: Solutions for Radial Flow
Problem 2: Derivation of the Diffusivity Equation for Multiphase Flow
Problem 3: Solution of Bessel's Differential Equation
For an "A" grade you must complete the following ADDITIONAL problems:
Problem 4: Derivation of the Pressure-Squared (p2) Form of the Gas Pseudosteady-State Flow Equation
Problem 5: Least Squares
____________________________________________________________________________________________________________________________________________________________
Notes:
(Page 3 of 8)
— You MUST submit ALL assignments in a SINGLE .pdf file — P620_12C_CourseExam_YOURLASTNAME.pdf
— You MUST ensure that all pages are correctly ordered and scanned legibly. (poorly constructed/assembled work will not be graded)
— Neatness, proper structure, clarity of your work is critical. No scratch-outs, use a good pencil or type you work.
— This is an EXAMINATION, no collaboration is allowed, submit only your best work!
Name: _____________________________
Assigned: 13 November 2012
Due:
12 December 2012 [by Midnight (23:59:59 US CST)]
Send a SINGLE .pdf file (no .doc, .xls, .??? — no exceptions) to: t-blasingame@tamu.edu, dl_coordinators@pe.tamu.edu
Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs
Course Examination [Blasingame]
Assigned: 13 November 2012 — Due: 12 December 2012 [by Midnight (23:59:59 US CST)]
Problem 1: Solutions for Radial Flow
Given:
In this assignment you are to derive several solutions of the radial flow diffusivity equation, which is given as:
1   pD   2 pD
1 pD pD


......................................................................................... (1.1)
rD

2
rD rD  rD 
r
tD
rD
D rD
The initial and boundary conditions considered in this assignment are:
● Initial Condition:
[Uniform Initial Pressure Distribution] ........................................................... (1.2)
pD (rD , tD  0)  0
● Inner Boundary Condition:
 pD 
 1
rD

 rD  rD  1
[q = Constant (production case)] .................................................................... (1.3)
● Outer Boundary Condition 1:
["Infinite-Acting" Reservoir → No Reservoir Boundary]............................... (1.4)
pD (rD  , tD )  0
● Outer Boundary Condition 2:
 pD 
0
rD

 rD  rD  reD
["No Flow" at the Outer Boundary] ................................................................ (1.5)
Required:
1.1. REAL DOMAIN Solutions for a Well Produced at a Constant Rate in an "Infinite-Acting" Reservoir
1.1.1 You are to develop the solution for Eq. 1.1 subject to Eqs. 1.2-1.4 using the Boltzmann transform.
1.1.2 You are to develop the solution for Eq. 1.1 subject to Eqs. 1.2-1.4 using the Laplace transform.
1.2. REAL DOMAIN Solutions for a Well Produced at a Constant Rate in a Circular, "No Flow" Reservoir
1.2.1 You are to develop the solution for Eq. 1.1 subject to Eqs. 1.2, 1.3, and 1.5 — you are to obtain the following
form: (see course notes for help)
2
2
 r2   r2
 r2 
1  rD  1  reD 
t
1
E1
 E1
 2 D exp  eD    D   exp  eD 
2
2  4t D  2  4t D 
4
 4t D   2r 2
 4t D 
reD





  eD



.......................................................................................................................................................... (1.6)
You are to develop the solution for Eq. 1.1 subject to Eqs. 1.2, 1.3, and 1.5 — you are to obtain the following
form: (see van Everdingen and Hurst; and Matthews and Russell references for help)
pD (rD , reD , t D ) 
1.2.2
2
 rD2

reD
1
4
4
2
 tD  
ln[ rD ] 
[3reD
 4reD
ln[ reD ]  2reD
- 1]

2
2
2
2
4
 (reD  1)
(reD  1) 
4(reD  1)
.......................................................................................................................................................... (1.7)
pD (rD , reD , t D ) 
2
References: (included as attachments)
● van Everdingen, A.F. and Hurst W.: "The Application of the Laplace Transformation to Flow Problems in
Reservoirs," Trans., AIME (1949), 186, 305-324.
● Horner, D.R.: "Pressure Build-Up in Wells," Proc., Third World Pet. Cong., E.J. Brill, Leiden (1951) II, 503.
● Matthews, C. S. and Russell, D. G.: Pressure Buildup and Flow Tests in Wells, Monograph Series, Society of
Petroleum Engineers of AIME, Dallas (1967) 1. (Appendix A — Solutions for Radial Flow of Fluids of Small and
Constant Compressibility)
____________________________________________________________________________________________________________________________________________________________
Notes:
(Page 4 of 8)
— You MUST submit ALL assignments in a SINGLE .pdf file — P620_12C_CourseExam_YOURLASTNAME.pdf
— You MUST ensure that all pages are correctly ordered and scanned legibly. (poorly constructed/assembled work will not be graded)
— Neatness, proper structure, clarity of your work is critical. No scratch-outs, use a good pencil or type you work.
— This is an EXAMINATION, no collaboration is allowed, submit only your best work!
Name: _____________________________
Assigned: 13 November 2012
Due:
12 December 2012 [by Midnight (23:59:59 US CST)]
Send a SINGLE .pdf file (no .doc, .xls, .??? — no exceptions) to: t-blasingame@tamu.edu, dl_coordinators@pe.tamu.edu
Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs
Course Examination [Blasingame]
Assigned: 13 November 2012 — Due: 12 December 2012 [by Midnight (23:59:59 US CST)]
Problem 2: Derivation of the Diffusivity Equation for Multiphase Flow
Given:
This work is taken from materials provided in the PETE 620 lecture notes:
Fundamental Flow Lecture 7 — The Diffusivity Equation for the Multiphase Flow
Required:
You are to re-derive, in complete detail, the "diffusivity equations" for the oil, water, and gas phases; as well as the
"multiphase" diffusivity equation and the total mobility and total compressibility equations. You must include the
formation compressibility (cf) formulation starting from the continuity relations (no short-cuts). You must show all work
for credit, and you must perform the specific tasks.
2.1 You are to state all assumptions.
[Blasingame notes]
2.2 Develop the mass and mass flux equations for each phase.
[Blasingame notes]
2.3 Develop generalized diffusivity equation for each phase.
[Martin Eqs. 1-3]
2.4 Develop the "expanded" generalized diffusivity equation for each phase.
2.5 Apply the "Martin" criteria — neglect Sop, Swp, and
pp=p2.
2.6 Develop the "Martin" saturation-mobility-compressibility relations.
[Martin Eqs. 4-6]
[Martin Eqs. 7-9]
[Martin Eqs. 10-11]
2.7 Write the "Martin" total compressibility relation.
[No Eq. number, Martin pg. 310]
2.8 Write the "Martin" total mobility relation.
[No Eq. number, Martin pg. 310]
2.9 Write the "Martin" 3-phase diffusivity equation (and give all assumptions).
[Martin Eqs. 12]
As noted above, you MUST include the formation compressibility (cf) formulation starting from the continuity relations
(i.e., Eqs. 28, 29, and 30 in the Blasingame notes).
References: (attached)
● Martin, J.C.: "Simplified Equations of Flow in Gas Drive Reservoirs and the Theoretical Foundation of Multiphase
Pressure Buildup Analysis," Trans., AIME 216 (1959) 309-311.
● Blasingame, T.A.: "Derivation of the Perrine-Martin Diffusivity Equations for Individual Phases," Petroleum
Engineering 620 Course Notes, Texas A&M University (1993).
● Derivation of the Simplified Diffusivity Equation for Multiphase Flow ("Martin" Equations) taken from Lee, W.J. and
Wattenbarger, R.A. Gas Reservoir Engineering, SPE (1996)).
____________________________________________________________________________________________________________________________________________________________
Notes:
(Page 5 of 8)
— You MUST submit ALL assignments in a SINGLE .pdf file — P620_12C_CourseExam_YOURLASTNAME.pdf
— You MUST ensure that all pages are correctly ordered and scanned legibly. (poorly constructed/assembled work will not be graded)
— Neatness, proper structure, clarity of your work is critical. No scratch-outs, use a good pencil or type you work.
— This is an EXAMINATION, no collaboration is allowed, submit only your best work!
Name: _____________________________
Assigned: 13 November 2012
Due:
12 December 2012 [by Midnight (23:59:59 US CST)]
Send a SINGLE .pdf file (no .doc, .xls, .??? — no exceptions) to: t-blasingame@tamu.edu, dl_coordinators@pe.tamu.edu
Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs
Course Examination [Blasingame]
Assigned: 13 November 2012 — Due: 12 December 2012 [by Midnight (23:59:59 US CST)]
Problem 3: Solution of Bessel's Differential Equation
3.1. Solution of Bessel's differential equation using the "Method of Frobenius:"
Given:
Bessel's differential equation is defined as:
x 2 y''  x y'  ( x 2  n 2 ) y  0 .................................................................................................................................... (3.1)
Required:
You are to use the "Method of Frobenius" to solve for the general solution of Bessel's differential equation (Eq. 3.1).
You must show ALL work, ALL steps, ALL assumptions, etc — solutions which are incomplete will receive 0 (zero)
credit.
Hint:
The "Method of Frobenius" is used to develop the general solution of Bessel's differential equation in the text, Advanced
Mathematics for Engineers and Scientists, [M.R. Spiegel, Schaum's Series (1971), see pg. 228 (Chapter 10)]. As this
assignment is a Solved Problem (10.1) — perfection is expected!
References:
● Spiegel, M.R.: Advanced Mathematics for Engineers and Scientists, Schaum's Outline Series (1971).
3.2. Solution of Bessel's differential equation using the Runge-Kutta Method:
Given:
Bessel's differential equation as defined below in "Runge-Kutta" form:
t2
d2y
dt 2
t
dy
d2y
dy
v
 (t 2  z 2 ) y  0 or

 y , where v 
and z  0. ..................................................... (3.2)
2
dt
t
dt
dt
Required:
You are to use the Runge-Kutta Method to numerically solve Bessel's modified differential equation (in the form given
by Eq. 3.2):
— Subject to the following initial conditions: at t=0, y(0)=1, y'(0)=v(0)=0.
— Use t=0.01.
— Exact solution: y(t) = J0(t)
You are to plot the solution computed using the Runge-Kutta Method and the exact solution [y(t) = J0(t)] for the range
0<t<10, where your Runge-Kutta solution is plotted as a red line and the J0(t) is plotted as a blue line.
References:
● Spiegel, M.R.: Advanced Mathematics for Engineers and Scientists, Schaum's Outline Series (1971).
● Blasingame, T.A.: "Math Lecture 4b - Application of the Runge-Kutta Method," Petroleum Engineering 620 — Fluid
Flow in Petroleum Reservoirs, Texas A&M University (1995).
____________________________________________________________________________________________________________________________________________________________
Notes:
(Page 6 of 8)
— You MUST submit ALL assignments in a SINGLE .pdf file — P620_12C_CourseExam_YOURLASTNAME.pdf
— You MUST ensure that all pages are correctly ordered and scanned legibly. (poorly constructed/assembled work will not be graded)
— Neatness, proper structure, clarity of your work is critical. No scratch-outs, use a good pencil or type you work.
— This is an EXAMINATION, no collaboration is allowed, submit only your best work!
Name: _____________________________
Assigned: 13 November 2012
Due:
12 December 2012 [by Midnight (23:59:59 US CST)]
Send a SINGLE .pdf file (no .doc, .xls, .??? — no exceptions) to: t-blasingame@tamu.edu, dl_coordinators@pe.tamu.edu
Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs
Course Examination [Blasingame]
Assigned: 13 November 2012 — Due: 12 December 2012 [by Midnight (23:59:59 US CST)]
Problem 4: Derivation of the Pressure-Squared (p2) Form of the Gas Pseudosteady-State Flow Equation
Given:
The pressure-squared (p2) form of the gas pseudosteady-state flow equation is given as:
2 ) ........................................................................................................................................................ (4.1)
q g  C ( p 2  pwf
Starting with the generalized pressure-squared (p2) form of the gas diffusivity equation:
2 ( p 2 ) 

p
2
ln (z)( p 2 ) 2   ct
 2
( p ) ............................................................................................................ (4.2)
k t
Assuming (z) = constant, yields:
2 ( p 2 ) 
 ct 
( p 2 ) ................................................................................................................................................ (4.3)
k t
Assuming horizontal radial flow behavior, we obtain:
1    ( p 2 )   ct  2
( p ) ...................................................................................................................................... (4.4)
r

r r 
r 
k t


The "high pressure" gas material balance equation is given as:
p
1  ce ( p)( pi  p) 
z
 ................................................................... (4.5)
pi pi 1 
1

(Wp  Winj ) Bw  We 
Gp  Ginj  Wp Rsw  5.615
zi zi G 
Bg

Assuming the volumetric, dry gas condition only, the gas material balance equation becomes:

p pi

z zi

 Gp 
pi
d  p
2
2 1
1 
 and
 z    z G q g for a circle G  h (re  rw ) B
G
dt
 
i
gi


.......................................................... (4.6)
Required:
You are to derive the pressure-squared (p2) form of gas the pseudosteady-state flow equation given by Eq. 4.1 in complete
detail. You should start with Eq. 4.4, and use Eq. 4.6 as the material balance relation to represent the pseudosteady-state
flow condition.
Failure to show all steps will result in a zero (0) grade for this problem. You must also clearly state all assumptions, as
well as any presumed conditions or states.
Hints:
4.1. You should review:
● Fundamental Flow Lecture 4 — Pseudosteady-State Flow in a Circular Reservoir. This is the file:
P620_11B_Lec_16_Mod3_FunFld_04_PSS_Flow.pdf
4.2. You will need the derivative of Eq. 4.6 with respect to time, but in terms of p2 — not p. Using the chain rule, we obtain
the following starting point:
d  p  d  p  dp dp 2

dt  z  dp  z  dp 2 dt
also,
d  p  p  1 1 dz  p
dp 2
  
 c g and
 2 p ........................................... (4.7)



dp  z  z  p z dp  z
dp
These hints should get you to the same starting point as the liquid case (Eq. 13 in the handwritten notes portion of
Fundamental Flow Lecture 4) — obviously, the "c"-term must be correctly defined, and you must work the derivation in p2.
As a guiding comment, this derivation may not lead to a compact form (i.e., a simple, closed-form relation)
____________________________________________________________________________________________________________________________________________________________
Notes:
(Page 7 of 8)
— You MUST submit ALL assignments in a SINGLE .pdf file — P620_12C_CourseExam_YOURLASTNAME.pdf
— You MUST ensure that all pages are correctly ordered and scanned legibly. (poorly constructed/assembled work will not be graded)
— Neatness, proper structure, clarity of your work is critical. No scratch-outs, use a good pencil or type you work.
— This is an EXAMINATION, no collaboration is allowed, submit only your best work!
Name: _____________________________
Assigned: 13 November 2012
Due:
12 December 2012 [by Midnight (23:59:59 US CST)]
Send a SINGLE .pdf file (no .doc, .xls, .??? — no exceptions) to: t-blasingame@tamu.edu, dl_coordinators@pe.tamu.edu
Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs
Course Examination [Blasingame]
Assigned: 13 November 2012 — Due: 12 December 2012 [by Midnight (23:59:59 US CST)]
Problem 5: Least Squares
Given:
5.1. In the Hornbeck text (Numerical Methods), you are to begin with Eq. 7.2 and re-derive ALL STEPS such that you
obtain the matrix of "normal equations" (i.e., Eq. 7.9). This is essentially an exercise in copying and completing any
missing or incomplete steps — perfection is expected!
5.2. In the Hornbeck text (Numerical Methods), you are to solve Illustrative Problem 7.2 (all parts). You are to plot your
results — plotting the data as symbols and your functional approximations (models) as lines. As this is also a
"rework" of an existing problem — perfection is expected!
References:
● Hornbeck, R.W.: Numerical Methods, Quantum Publishers, Inc., New York (1975).
____________________________________________________________________________________________________________________________________________________________
Notes:
(Page 8 of 8)
— You MUST submit ALL assignments in a SINGLE .pdf file — P620_12C_CourseExam_YOURLASTNAME.pdf
— You MUST ensure that all pages are correctly ordered and scanned legibly. (poorly constructed/assembled work will not be graded)
— Neatness, proper structure, clarity of your work is critical. No scratch-outs, use a good pencil or type you work.
— This is an EXAMINATION, no collaboration is allowed, submit only your best work!