Petroleum Engineering 620 — Fluid Flow in Petroleum ...

Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures

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Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Course Introduction

Fall 2018

Instructor Information:

Petroleum Engineering 620 Instructor: Dr. Tom BLASINGAME TAs: Mr. Jorge GARCIA/Mr. Rui KOU Fluid Flow in Petroleum Reservoirs Richardson 821A Richardson 821 Texas A&M University +1.979.845.2292 [email protected] MW 07:55 pm-09:10 pm RICH 319 [email protected] [email protected] (in-class lectures) (Please always use e-mail to contact me) (Please always use e-mail to contact the TAs)

Goals of PETE 620:

In simple terms, the goal of PETE 620 (Fluid Flow in Petroleum Reservoirs) is to take the student from "sand grains to the classic solutions used in reservoir engineering." The path begins with a review of mathematics because many of us need to re-familiarize ourselves with algebra, calculus, differential equations, numerical methods, special functions, and other related topics/subjects. This review is intended to address most of the skills/topics which will be required in the course.

We then proceed to study the (mostly) empirical aspects of geology and petrophysics (rock properties), and then on to the fundamental building blocks of reservoir engineering: permeability, capillary pressure, relative permeability, and the electrical properties of reservoir rocks. After this we work through the "flow relations" (steady-state Darcy and non-Darcy flow), then Material Balance (needed for conservation of mass), pseudosteady-state flow, and finally the "diffusion" (or diffusivity) equations. At this point it is worth noting that we will have addressed the "fundamental" aspects of these building blocks — the assumptions, the limitations, and the need for advances in concepts for flow in porous media; and perhaps most importantly, the inherent non-linearities that exist for the "flow equations" used in Petroleum Reservoir Engineering.

In the last (and most important) portion of the course we consider the classic reservoir solutions for radial and linear flow, flow in fractured wells, flow in naturally-fractured (or dual porosity) systems, and the definition and use of convolution in reservoir engineering applications.

Assignments for PETE 620:

The assignments in this course vary from fundamental developments to solutions which could be used in reservoir engineering practice. Students are expected to demonstrate mastery of all fundamental concepts covered in the course. In addition, the instructor wishes to provide students with concepts, problems, and/or applications which will be useful for research.

The tentative assignment topics for the Fall 2018 are given as follows:

● (Math) Laplace Transforms ● Material Balance (liquid or gas) ● Solutions for Fractured Wells ● (Math) Special Functions ● Pseudosteady-State Flow (liquid or gas) ● Dual Porosity Reservoirs ● Correlation of Petrophysical Data ● Radial Flow Solutions ● Convolution/Deconvolution ● Capillary Pressure/Relative Permeability ● Linear Flow Solutions ● Wellbore Storage

Philosophy about Life:

● Most Appropriate Quote: Opportunity is missed by most people because it is dressed in overalls and looks like work... (hard work is the only path…) Thomas A. Edison, American Inventor (1847-1931)

● Important Rules for Life: — Never own anything that eats while you sleep… — Always work harder than those you work for… — Never own anything that needs repainting… — If you have to herd cats, then be a rat… — Never own anything that you can't drive a nail in... — Never say no, and there's no limit to where you can go…

Brief Bio: Thomas A. Blasingame, Ph.D., P.E.

● Professor, Department of Petroleum Engineering at Texas A&M University in College Station Texas ● Holds a joint appointment in the Department of Geology and Geophysics at Texas A&M University ● Holder of the Robert L. Whiting Professorship in Petroleum Engineering ● B.S., M.S., and Ph.D. degrees in Petroleum Engineering from Texas A&M University. ● Teaching/Research activities: ● Technical Contributions:

— Petrophysics — Pressure transient test analysis — Reservoir engineering — Analysis of production data — Analysis/interpretation of well performance — Reservoir management — Exploitation of unconventional reservoirs — Diagnostic characterization of reservoir performance — Technical mathematics. — General reservoir engineering

● Student counts to date (Aug 2018): 66 M.S. (thesis), 34 M.Eng. (report, non-thesis), and 14 Ph.D. students.

Guidance:

● Orientation — This is graduate school, the (only) person you are competing against is in the mirror. ● Work Quality — My highest commandment is that you submit your best work; and ONLY your best work. ● Focus — This is an essential course in reservoir engineering, results are used throughout the discipline. ● Timeliness — This material is very "dense" — do not underestimate your workload and timing. ● Course Materials — The material will teach itself, but you must put your energy and enthusiasm into the course. ● Connections — I am here to help; I will answer any/all relevant correspondence within 24 hours (unless I am totally offline).


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Fall 2018

Instructor Information:

Petroleum Engineering 620 Instructor: Dr. Tom BLASINGAME TAs: Mr. Jorge GARCIA/Mr. Rui KOU Fluid Flow in Petroleum Reservoirs Richardson 821A Richardson 821 Texas A&M University +1.979.845.2292 [email protected] MW 07:55 pm-09:10 pm RICH 319 [email protected] [email protected] (in-class lectures) (Please always use e-mail to contact me) (Please always use e-mail to contact the TAs)

Required Texts/Resources: (a. Book must be purchased. b. Out of Print/Public Domain.)

1.a Advanced Mathematics for Engineers and Scientists, M.R. Spiegel, Schaum's Series (1971). [The 1st edition, the 1971 text.] 2.a Conduction of Heat in Solids, 2nd edition, H. Carslaw and J. Jaeger, Oxford Science Publications (1959). 3.b Handbook of Mathematical Functions, M. Abramowitz and I. Stegun, Dover Pub. (1972). 4.b Table of Laplace Transforms, G.E. Roberts and H. Kaufman, W.B. Saunder, Co. (1964). 5.b Numerical Methods, R.W. Hornbeck, Quantum Publishers, Inc., New York (1975). 6.b Approximations for Digital Computers: Hastings, C., Jr., et al, Princeton U. Press, Princeton, New Jersey (1955). 7.b Handbook for Computing Elementary Functions: L.A. Lyusternik, et al, Pergamon Press (1965). 8.b The Flow of Homogeneous Fluids Through Porous Media: M. Muskat, McGraw-Hill, New York (1946). 9.b Advanced Calculus for Applications: F. B. Hildebrand, Prentice-Hall (1962). 10.b Reservoir Sandstones: R.R. Berg, Prentice Hall (1985).

Optional Texts/Resources: (Try local bookstores or online vendors, or the TAMU library.)

1. Calculus, 4th edition: Frank Ayres and Elliot Mendelson, Schaum's Outline Series (1999) (Remedial text) 2. Differential Equations, 2nd edition: Richard Bronson, Schaum's Outline Series (1994) (Remedial text) 3. Laplace Transforms, M.R. Spiegel, Schaum's Outline Series (1965) (Remedial text) 4. Numerical Analysis, F. Scheid, Schaum's Outline Series, McGraw-Hill Book Co, New York (1968). (Remedial text) 5. The Mathematics of Diffusion, 2nd edition, J. Crank, Oxford Science Publications (1975). (important/historical) 6. Table of Integrals, Series, and Products, I.S. Gradshteyn and I.M. Ryzhik, Academic Press (1980). (very important/historical) 7. Methods of Numerical Integration, P.F. Davis and P. Rabinowitz, Academic Press, New York (1989). (perhaps useful for research) 8. An Atlas of Functions, J. Spanier and K. Oldham, Hemisphere Publishing (1987). (perhaps useful for research) 9. Adv. Math. Methods for Eng. and Scientists, 2nd edition, C.M. Bender and S.A. Orsag, McGraw-Hill (1978). (excellent text) 10. Asymptotic Approximations of Integrals, R. Wong, Academic Press (1989). (perhaps useful for research) 11. Asymptotics and Special Functions, F.W.J. Olver, Academic Press (1974). (perhaps useful for research)

Course and Reference Materials:

The course materials for this course are located at:

http://www.pe.tamu.edu/blasingame/data/P620_18C/

Basis for Grade: [Grade Cutoffs (Percentages) → A: > 90 B: 89.99 to 80 C: 79.99 to 70 D: 69.99 to 60 F: < 59.99]

(Due 19 Nov 2018) Reading Portfolio [30 reviews for assigned publications] ...................................................................... 30 percent (Due 03 Dec 2018) Homework Portfolio [12 problems from texts, journal articles, etc.] .......................................................... 50 percent (Due 10 Dec 2018) Final Examination [specialized problems taken from the literature and/or texts] ................................... 20 percent

Total = 100 percent

Policies and Procedures:

1. Students are expected to attend class every session. Resident (not Distance Learning students) are REQUIRED to attend class every session. Distance Learning students are expected review lecture materials within 24 hours of the lecture being given. This is not a casual requirement, penalties can and will be assigned for missing class.

2. Policy on Grading a. All work in this course is graded on the basis of answers only — any partial credit is at the discretion of the instructor. b. All work requiring calculations shall be properly and completely documented for credit. c. All grading shall be done by the instructor, or under his direction and supervision, and the decision of the instructor is final.

3. Policy on Regrading a. Only in very rare cases will assignments/exams be considered for re-grading — partial credit (if any) is not subject to appeal. b. Work which, while possibly correct, but cannot be followed, will be considered incorrect. c. Grades assigned to homework problems will not be considered for regrading. d. If regrading is necessary, the student is to submit a letter to the instructor explaining the situation that requires consideration

for regrading, the material to be regraded must be attached to this letter. The letter and attached material must be received within one week from the date returned by the instructor.

4. The grade for a late assignment is zero. Homework will be considered late if it is not turned in at the start of class on the due date. If a student comes to class after homework has been turned in and after class has begun, the student's homework will be considered late and given a grade of zero. Late or not, all assignments must be turned in. A course grade of Incomplete will be given if any assignment is missing, and this grade will be changed only after all required work has been submitted.

5. Each student should review the University Regulations concerning attendance, grades, and scholastic dishonesty. In particular, anyone caught cheating on an examination or collaborating on an assignment where collaboration is not specifically authorized by the instructor will be removed from the class roster and given an F (failure grade) in the course.


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Fall 2018

Work Requirements: (layout/format/etc.)

You must show ALL work — as appropriate, YOU MUST: — WORK: You must show all details in your calculations (no skipped steps) — all portions of all analysis relations must be shown. — UNITS: You must show all units in all calculations.

Work layout: (as appropriate for a given problem) — NEATNESS: You will be graded on the neatness of your work. — LABELS: All work, trends, and features on every plot MUST be appropriately labeled — no exceptions.

■ Work: All work must be fully labeled and documented — equations, relations, calculations, etc. ■ Trends: This includes the slope, intercept, and the information used to construct a given trend. ■ Features: Any description of features/points of interest on a given trend (times, pressures, etc.).

— 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.

Plots/Plotting: (as required) — SYMBOLS: Use symbols for "data" (if "data" are presented — e.g., reference solutions given as discrete data points). — LINES: Use lines to represent models. — COLORS: Use black for all axes and gridlines. Use primary colors (red, green, blue), avoid pastel colors. — etc.: Please do NOT use a border or "frame" around your plots.

Typing: ALL WORK SUBMITTED IN THIS COURSE MUST BE TYPED, NO HANDWRITTEN WORK IS PERMITTED.

Scanning: >300 dpi COLOR scan from a printer/scanner — DO NOT SUBMIT PHOTOS (photos will not be accepted).

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 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 specifically required to perform your own work.

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.

Student Obligation:

You must prepare your work as instructed above, or you will be assessed SEVERE grading penalties.

e-mail Protocols

In order to manage your correspondence, I require that you use the following protocol.

Subject Line: [YYYYMMDD] (YOURLASTNAME) Subject (date) (your last name) (Subject of your e-mail) Body:

Dr. BLASINGAME:

I would like to enquire about the following: * Question 1 ... (be clear and concise) * Question 2 ... (be clear and concise) * Question 3 ... (be clear and concise)

I thank you for your assistance.

YourFirstName YOURLASTNAME (contact information) E: (TAMU) E: (personal) T: (a phone contact) (I will NEVER call you without first sending an e-mail or text)

Comments: DO NOT FORWARD/REPLY TO EMAILS FROM ECAMPUS — SEND A NEW NOTE. The subject line is used to file e-mail (this is why this specific subject line is required). Every effort will be made to answer every e-mail, but PLEASE avoid trivial enquiries (consult the syllabus for "administrative" issues). I am more than happy to address questions by e-mail — i.e., issues/errors/etc. and/or need help with something relevant to the course. Courier New 10pt Bold font is required.

Computational Tools:

In this course you are NOT required to work in a particular computational environment. However; you should be/must be proficient at whatever computational tool(s) you use for work in this course. Example products/computational environments include

Visual Basic (VB) via MS Excel. MATLAB (http://www.mathworks.com/products/matlab/). Mathematica (https://www.wolfram.com/mathematica/). Programming Languages: C++, FORTRAN, Pascal, machine language, the Univac, an abacus, etc.

Please note that YOU are RESPONSIBLE for your computer-aided solutions. Depending on the assignment you may be asked for a copy of your source code and should provide relevant commentary/documentation in your source code sufficient for your work to be traced. You will also be asked for an outline/workflow for any/all computational solutions.


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Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs

Course Outline/Topics Fall 2018

Course Description:

Graduate Catalog: Analysis of fluid flow in bounded and unbounded reservoirs, wellbore storage, phase redistribution, finite and infinite conductivity vertical fractures, dual-porosity systems.

Translation: Development of skills required to derive "classic" problems in reservoir engineering and well testing from the fundamental principles of mathematics and physics. Emphasis is placed on a mastery of fundamental calculus, analytical and numerical solutions of 1st and 2nd order ordinary and partial differential equations, as well as extensions to non-linear partial differential equations that arise for the flow of fluids in porous media.

Course Outline/Topics:

Advanced Mathematics Relevant to Problems in Engineering: (used throughout assignments)

Approximation of Functions Taylor Series Expansions and Chebyshev Economizations Numerical Differentiation and Integration of Analytic Functions and Applications Least Squares

First-Order Ordinary Differential Equations Second-Order Ordinary Differential Equations The Laplace Transform Fundamentals of the Laplace Transform Properties of the Laplace Transform Applications of the Laplace Transform to Solve Linear Ordinary Differential Equations Numerical Laplace Transform and Inversion

Special Functions

Petrophysical Properties:

Porosity and Permeability Concepts Correlation of Petrophysical Data Concept of Permeability — Darcy's Law Capillary Pressure Relative Permeability Electrical Properties of Reservoir Rocks

Fundamentals of Flow in Porous Media:

Steady-State Flow Concepts: Laminar Flow Steady-State Flow Concepts: Non-Laminar Flow Material Balance Concepts Pseudosteady-State Flow in a Circular Reservoir Development of the Diffusivity Equation for Liquid Flow Development of the Diffusivity Equations for Gas Flow Development of the Diffusivity Equation for Multiphase Flow

Classical Reservoir Flow Solutions:

Dimensionless Variables and the Dimensionless Radial Flow Diffusivity Equation Solutions of the Radial Flow Diffusivity Equation — Infinite-Acting Reservoir Case Laplace Transform (Radial Flow) Solutions — Bounded Circular Reservoir Cases Real Domain (Radial Flow) Solutions — Bounded Circular Reservoir Cases Linear Flow Solutions: Infinite and Finite-Acting Reservoir Cases Solutions for a Fractured Well — High Fracture Conductivity Cases Dual Porosity Reservoirs — Pseudosteady-State Interporosity Flow Behavior Direct Solution of the Gas Diffusivity Equation Using Laplace Transform Methods Convolution and Concepts and Applications in Wellbore Storage Distortion

Advanced Reservoir Flow Solutions: (Possible Coverage)

Multilayered Reservoir Solutions Dual Permeability Reservoir Solutions Horizontal Well Solutions Radial Composite Reservoir Solutions Models for Flow Impediment (Skin Factor)

Applications/Extensions of Reservoir Flow Solutions: (Possible Coverage)

Oil and Gas Well Flow Solutions for Analysis, Interpretation, and Prediction of Well Performance Low Permeability/Heterogeneous Reservoir Behavior Macro-Level Thermodynamics (coupling PVT behavior with Reservoir Flow Solutions) External Drive Mechanisms (Water Influx/Water Drive, Well Interference, etc.). Hydraulic Fracturing/Solutions for Fractured Well Behavior Analytical/Numerical Solutions of Various Reservoir Flow Problems. Applied Reservoir Engineering Solutions — Material Balance, Flow Solutions, etc.


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Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Tentative Course Schedule

Fall 2018

Course Schedule:

Month Date Day Topic Lecture or Potential Assignment Topic

Aug 27 M Course Introduction Syllabus

Aug 29 W Review of Functions Lec_01_Mod1_ML_01_Rev_of_Fcns.pdf

Sep 03 M Approximation of Functions Lec_02_Mod1_ML_02_Fcn_Approx.pdf

Sep 05 W 1st Order Ordinary Differential Equations Lec_03_Mod1_ML_03_1st_Order_ode.pdf

Sep 10 M 2nd Order Ordinary Differential Equations Lec_04_Mod1_ML_04_2nd_Order_ode.pdf

Sep 12 W The Laplace Transform Lec_05_Mod1_ML_05_LaplaceTrans.pdf

Sep 17 M Introduction to Special Functions Lec_06_Mod1_ML_06_SpecialFcns.pdf

Sep 19 W Porosity and Permeability Concepts Lec_07_Mod2_PtrPhy_01_PorPerm.pdf

Sep 24 M Correlation of Petrophysical Data Lec_08_Mod2_PtrPhy_02_DataCorel.pdf

Sep 26 W Development of Permeability/Darcy's Law Lec_09_Mod2_PtrPhy_03_Perm_Dev.pdf

Oct 01 M Capillary Pressure Lec_10_Mod2_PtrPhy_04_Cap_Pres.pdf

Oct 03 W Relative Permeability Lec_11_Mod2_PtrPhy_05_Rel_Perm.pdf

Oct 08 M Electrical Properties of Reservoir Rocks Lec_12_Mod2_PtrPhy_06_Elec_Prop.pdf

Oct 10 W Single-Phase, Steady-State Flow Lec_13_Mod3_FunFld_01_SSDarcy.pdf

Oct 15 M Non-Laminar Flow in Porous Media Lec_14_Mod3_FunFld_02_SSNonDarcy.pdf

Oct 17 W Material Balance Concepts Lec_15_Mod3_FunFld_03_MatBal.pdf

Oct 22 M Pseudosteady-State Flow (Circular Res.) Lec_16_Mod3_FunFld_04_PSS_Flow.pdf

Oct 24 W Liquid Flow Diffusivity Equation Lec_17_Mod3_FunFld_05_DifEq_Liq.pdf

Oct 29 M Gas Flow Diffusivity Equation Lec_18_Mod3_FunFld_06_DifEq_Gas.pdf

Oct 31 W Multiphase Flow Diffusivity Equation Lec_19_Mod3_FunFld_07_DifEq_MlPhs.pdf

Nov 05 M Dimensionless Variables/Radial Flow Lec_20_Mod4_ResFlw_01_DimLssVar.pdf

Nov 07 W Solutions — Radial Flow Diffusivity Eq. Lec_21_Mod4_ResFlw_02_RadFlwSln.pdf

Nov 12 M Solutions — Radial Flow Diffusivity Eq. Lec_21_Mod4_ResFlw_02_RadFlwSln.pdf

Nov 14 W Solutions — Linear Flow Diffusivity Eq. Lec_22_Mod4_ResFlw_03_LinFlwSln.pdf

Nov 19 M Solutions — Fractured Well (High FcD) Lec_23_Mod4_ResFlw_04_FracWellSln.pdf

Nov 21 W Solutions — Dual Porosity Reservoirs Lec_24_Mod4_ResFlw_05_NatFrcResSln.pdf

Nov 22 Th Thanksgiving Holiday (no class) —

Nov 26 M Direct Solution — Gas Diffusivity Equation Lec_25_Mod4_ResFlw_06_DrtSlnGas.pdf

Nov 28 W Convolution Lec_26_Mod4_ResFlw_07_Convolution.pdf

Dec 03 M Wellbore Storage Lec_27_Mod4_ResFlw_08_WellboreStrg.pdf

Dec 05 W Extra Class —

Dec 10 M Any/all remaining assignments due. (http://registrar.tamu.edu/Courses,-Registration,-Scheduling/Final-Examination-Schedules#5-December10(Monday))

Dec 13 R Final grades due GRADUATING students. (http://registrar.tamu.edu/Catalogs,-Policies-Procedures/Academic-Calendar)

Dec 17 M Final grades for ALL students Fall 2018 term (http://registrar.tamu.edu/Catalogs,-Policies-Procedures/Academic-Calendar)

Comments:

1. All class sessions will also be recorded and put on the eCampus system as well as the instructor's website. 2. Friday class sessions will also be used from time-to-time (substitute lecture dates, math recitations, etc.). 3. This lecture schedule is tentative and is subject to change (as a legal disclaimer).


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____________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Reading Portfolio Assignment

Monday 19 November 2018 [by 5:00 p.m. (i.e., 16:59:59 US CST)]

Assignment Coversheet

[This sheet (or the sheet provided for a given assignment) must be included with EACH 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.

Aggie Honor Code "An Aggie does not lie, cheat, or steal or tolerate those who do."

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:


_______________________________ (Print your name) _______________________________ (Your signature)

____________________________________________________________________________________________________________________________________________________________________________________________________________________________________

(Include this page in your submission of the Reading Portfolio)


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Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Reading Portfolio Assignment (15 points/30 papers for review)

Fall 2018

Reading Portfolio: Orientation

Based on the "Reading Portfolio Inventory" given on the next page, you are to create a "Reading Portfolio." The Reading Portfolio is assigned as follows:

● Due date: Monday 19 November 2018 [by 5:00 p.m. (i.e., 16:59:59 US CST)] ● Submission: Submit as a SINGLE .pdf file named: P620_18C_ReadPort_YOURLASTNAME.pdf ● Recipient: Send to [email protected]

It is suggested that the student create an MS Word document as the working document for their Reading Portfolio — however; ONLY a single .pdf file will be accepted as the Reading Portfolio submission. Students ARE permitted to communicate — HOWEVER, students are not permitted to submit shared or copied work. Shared or copied work will result in a zero (0) score for the Reading Portfolio assignment.

Failure to submit the Reading Portfolio will result a zero (0) score for this portion of the student's grade. ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Reading Portfolio — 1-Page Review

Fall 2018

(Students must use EXACTLY this template format and 9-point "Times New Roman" font must be used for text)

Article Being Reviewed: (give full reference citation)

(example)

Archie, G.E. (1950) "Introduction to Petrophysics of Reservoir Rocks," Bull., AAPG 34, 943-961.

Review Questions: (All questions must be answered) (… at least 2 bullet points are required for each question)

● What is/are the problem(s) solved?

— Bullet Point 1 — Bullet Point 2 — …

● What are the assumptions and limitations of the solutions/results?


● What are the practical applications of the solutions/results?


● (optional) How could the solutions/results be extended or improved?


● (optional) Are there applications other than those given by the author(s) where the solution(s) could be applied?


____________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Guidance on Reading Portfolio Assignment: Submit a SINGLE .pdf file to [email protected] by 16:59:59 US CST on 19 November 2018. ● It is suggested that you use MS Word to create your Reading Portfolio, then create a .pdf from this file. ● The purpose of this assignment is to prepare you for the "homework portfolio" as well as possible questions for the final exam. ● Poor/fair quality work submissions will NOT be accepted. ● Your submission file must be named: P620_18C_ReadPort_YOURLASTNAME.pdf

Note: ● Students ARE permitted to communicate — HOWEVER, students ARE NOT permitted to submit shared or copied work. ● Sharing or copying work will result in a zero (0) score for the Reading Portfolio assignment.



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Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Reading Portfolio Inventory — Inventory

Fall 2018

Reading Portfolio: Inventory

The "Reading Portfolio Inventory" is assigned as follows:

1. Archie, G.E. (1950) "Introduction to Petrophysics of Reservoir Rocks," Bull., AAPG 34, 943-961.

2. Bentsen, R. G. (1984) "A Functional Relationship for Macroscopic Capillary Pressure," Society of Petroleum Engineers.

3. Berg, R.R. (1985) "Reservoir Sandstones," Prentice Hall, Englewood Cliffs, New Jersey.

4. Byrnes P.A., Cluff R.M., Webb J.C. (2009) "Analysis of Critical Permeability, Capillary and Electrical Properties for Mesaverde Tight Gas Sandstones from Western US Basins," DOE Technical report (DE-FC26-05NT42660) submitted to DOE and NETL (355 pages).

5. Byrnes, A. P., and Castle, J. W. (2000) Comparison of Core Petrophysical Properties between Low-Permeability Sandstone Reser-voirs: Eastern U.S. Medina Group and Western U.S. Mesaverde Group and Frontier Formation. Society of Petroleum Engineers. doi:10.2118/60304-MS (https://doi.org/10.2118/60304-MS)

6. Camacho-V, R. G. (1987) Well Performance under Solution Gas Drive, Ph.D. Dissertation, University of Tulsa.

7. Carslaw, H. and Jaeger, J. (1959) Conduction of Heat in Solids, 2nd edition, Oxford Science Publications. [Chapters 12 and 13]

8. Fancher, G. H., Lewis, J. A., and Barnes, K. B. (1933) Some physical characteristics of oil sands: Pennsylvania State College, Min. Ind. Exp. Sta. Bull. 12, p. 141.

9. Gates, J.I. and Templaar-Lietz, W. (1950) "Relative Permeabilities of California Cores by the Capillary Pressure Method," API Drilling and Production Practices, 285-302.

10. Gaver, D.P., Jr. (1966) "Observing Stochastic Processes and Approximate Transform Inversion," Operations Research, vol. 14, No. 3, 444-459.

11. Geertsma, J. (1974) Estimating the Coefficient of Inertial Resistance in Fluid Flow through Porous Media. Society of Petroleum Engineers. doi:10.2118/4706-PA (https://doi.org/10.2118/4706-PA)

12. Gomez, C.T., Dvorkin, J. and Vanorio, T. (2010) "Laboratory measurements of porosity, permeability, resistivity, and velocity on Fontainebleau sandstones." GEOPHYSICS, 75(6), E191-E204. (https://doi.org/10.1190/1.3493633)

13. Gringarten, A. C., Ramey, H. J., and Raghavan, R. (1974) Unsteady-State Pressure Distributions Created by a Well With a Single Infinite-Conductivity Vertical Fracture. Society of Petroleum Engineers. doi:10.2118/4051-PA (https://doi.org/10.2118/4051-PA)

14. Hastings, C., Jr., et al (1955) Approximations for Digital Computers, Princeton U. Press, Princeton, New Jersey.

15. Hornbeck, R.W. (1975) Numerical Methods, Quantum Publishers, Inc., New York. [Chapters 2-9 (inclusive)]

16. Klinkenberg, L.J.: (1941) "The Permeability of Porous Media to Liquids and Gases," API Drilling and Production Practices, 200-213.

17. Muskat, M. (1946) The Flow of Homogeneous Fluids Through Porous Media: McGraw-Hill, New York.

18. Nelson, P.H. (1994) "Permeability-Porosity Relationships in Porous Rocks," The Log Analyst (May-June 1994), 38-62.

19. Nelson, P.H. (2009) Pore throat sizes in sandstone, tight gas sandstones and shales: AAPG Bulletin, v. 93/3, p. 329-340.

20. Pirson, S.J. (1950) Elements of Oil Reservoir Engineering, McGraw-Hill, New York.

21. Russell, D. G., Goodrich, J. H., Perry, G. E., and Bruskotter, J. F. (1966) Methods for Predicting Gas Well Performance. Society of Petroleum Engineers. doi:10.2118/1242-PA (https://doi.org/10.2118/1242-PA)

22. Spiegel, M.R. (1971) Advanced Mathematics for Engineers and Scientists, Schaum's Series (1971). [Chapter 1,2,4,9,10, and 12]

23. Standing, M.B.: "Notes on Relative Permeability Relationships," Course Notes, Trondheim, Norway (1975).

24. Thom, W.T., Jr. (1929) Petroleum and Coal — Keys to the Future, Princeton U. Press, Princeton, New Jersey. [Chapters 7-12 (inclusive)]

25. Thomas, L. K., Katz, D. L., and Tek, M. R. (1968) Threshold Pressure Phenomena in Porous Media. Society of Petroleum Engineers. doi:10.2118/1816-PA (https://doi.org/10.2118/1816-PA)

26. Van Everdingen, A. F., and Hurst, W. (1949) The Application of the Laplace Transformation to Flow Problems in Reservoirs. Society of Petroleum Engineers. doi:10.2118/949305-G (https://doi.org/10.2118/949305-G)

27. ver Wiebe, W.A. (1951) "How Oil is Found," Edwards, Ann Arbor, MI.

28. Warren, J. E. and Root, P. J. (1963) The Behavior of Naturally Fractured Reservoirs. Society of Petroleum Engineers. doi:10.2118/426-PA (https://doi.org/10.2118/426-PA)

29. Waxman, M. H., and Smits, L. J. M. (1968) Electrical Conductivities in Oil-Bearing Shaly Sands. Society of Petroleum Engineers. doi:10.2118/1863-A (https://doi.org/10.2118/1863-A)

30. Wyllie, M.R.J., and Spangler, M.B. (1951) "The Application of Electrical Resistivity Measurements to the Problem of Fluid Flow in Porous Media," Gulf Research and Development Co., Pittsburgh, PA.



Fall 2018 (version: 20181118)

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Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Reading Portfolio Inventory — Topics Table

Fall 2018

Reading Portfolio: Topics Table

The "Reading Portfolio Topics Table" is provided below — the student is free to complete the reading portfolio at their own pace, this table is provided for guidance purposes only.

Topic Reference Article

Course Introduction 24_Thm_Ptr_y_Col_(1929).pdf 27_VWe_Oil_Fnd_(1953).pdf

Review of Functions (22_Spiegel_Text)

Approximation of Functions (22_Spiegel_Text) 15_Hnb_Num_Mth_(1975).pdf

1st Order Ordinary Differential Equations (22_Spiegel_Text)

2nd Order Ordinary Differential Equations (22_Spiegel_Text)

The Laplace Transform (22_Spiegel_Text) 10_OpRes_1966_(Gaver).pdf

Introduction to Special Functions (22_Spiegel_Text) 14_Hst_Apr_Cmp_(1955).pdf

Porosity and Permeability Concepts 01_AAPG_Bull_34_1950_(Archie).pdf 03_Brg_Snd_Res_(1986).pdf 18_SPWLA_v35_1994_(Nelson).pdf

Correlation of Petrophysical Data 04_DOE_2009_(Byrnes).pdf 05_SPE_060304_(Byrnes).pdf 12_GEOP_v_75_(Gomez).pdf 19_AAPG_Bull_93_2009_(Nelson].pdf

Development of Permeability/Darcy's Law 08_Penn_State_(1933)_(Fancher).pdf 16_API_(Klinkenberg).pdf

Capillary Pressure 23_Standing_(UTrondheim_1975).pdf 25_SPE_001816_(Thomas).pdf

Relative Permeability 02_SPE_013831_(Bentsen).pdf 09_API_1950_(Gates).pdf

Electrical Properties of Reservoir Rocks 29_SPE_001863_(Waxman_y_Smits).pdf 30_Wyl_Apl_Elc_Res_(1951).pdf

Single-Phase, Steady-State Flow 20_Psn_Elm_Rsr_Eng_(1950).pdf

Non-Laminar Flow in Porous Media 11_SPE_004706_(Geertsma).pdf

Material Balance Concepts 17_Txt_Msk_Flw_Fld_(1946).pdf

Pseudosteady-State Flow (Circular Res.) 06_Camacho_(Dis_UTulsa_1987).pdf

Liquid Flow Diffusivity Equation —

Gas Flow Diffusivity Equation 21_SPE_001242_(Russell).pdf

Multiphase Flow Diffusivity Equation —

Dimensionless Variables/Radial Flow —

Solutions — Radial Flow Diffusivity Eq. 26_SPE_949305-G_(van_Everdingen_y_Hurst).pdf

Solutions — Radial Flow Diffusivity Eq. (07_Carslaw_and_Jaeger_Text)

Solutions — Linear Flow Diffusivity Eq. (07_Carslaw_and_Jaeger_Text)

Solutions — Fractured Well (High FcD) 13_SPE_004051_(Gringarten).pdf

Solutions — Dual Porosity Reservoirs 28_SPE_000426_(Warren_y_Root).pdf

Direct Solution — Gas Diffusivity Equation —

Convolution —

Wellbore Storage —



Fall 2018 (version: 20181118)

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____________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Homework Portfolio Assignment

Monday 03 December 2018 [by 5:00 p.m. (i.e., 16:59:59 US CST)]















____________________________________________________________________________________________________________________________________________________________________________________________________________________________________

(Include this page in your submission of the Homework Portfolio)


Fall 2018 (version: 20181118)

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Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Homework Portfolio Assignment (50 points/12 problems)

Fall 2018

Homework Portfolio: Orientation

The purpose of a "homework portfolio" is to ensure that the student has developed competent skills in the course objectives. The portfolio MUST be submitted, and will be assigned a numeric grade (0 to 100) at the discretion of the instructor. Each student must work each problem completely and independently (i.e., each student will create their own solution work showing all solutions steps — no detail(s) should be omitted). Students ARE permitted to communicate — HOWEVER, students are not permitted to submit shared or copied work. Shared or copied work will result in a zero (0) score for the Homework Portfolio assignment.

The Homework Portfolio is assigned as follows:

● Due date: Monday 03 December 2018 [by 5:00 p.m. (i.e., 16:59:59 US CST)] ● Submission: Submit as a SINGLE .pdf file named: P620_18C_HwkPort_YOURLASTNAME.pdf ● Recipient: Send to [email protected] (use TAMU "FILEX" system for very large files)

Failure to submit the Homework Portfolio will result a zero (0) score for this portion of the student's grade.

Homework Portfolio: Inventory of Problems

Problem 1 — Advanced Mathematics for Engineers and Scientists, M.R. Spiegel, Schaum's Series (1971)

(1 pt) 1.1 Solved Problem 1.15 Derivative of a Power Law Function (1 pt) 1.2 Solved Problem 1.18 Derivative of Logarithms (1 pt) 1.3 Solved Problem 1.23 Integral of a Power Law Function (1 pt) 1.4 Solved Problem 1.28 Simple Numerical Integration (1 pt) 1.5 Solved Problem 1.40 Integration Using Taylor Series

Problem 2 — Numerical Methods, R.W. Hornbeck, Quantum Publishers, Inc., New York (1975)

(1 pt) 2.1 Section 7.1 Derivation of Normal Equations (Eqs: 7.1-7.9) (you must provide all details) (2 pts) 2.2 Illustrative Problem 7.2 Least Squares Fitting — Polynomial and a Sine Wave (you must provide all details and a solution plot)


(1 pt) 3.1 Solved Problem 2.18 Derivation of the Integrating Factor (1 pt) 3.2 Solved Problem 2.25 Solution of an ODE of order higher than one using the Integrating Factor (1 pt) 3.3 Solved Problem 2.46 Solution of Solved Problem 2.44 by the Taylor series method (1 pt) 3.4 Solved Problem 2.47 Solution of Solved Problem 2.44 by Picard's method (1 pt) 3.5 Solved Problem 2.48 Solution of Solved Problem 2.44 by the Runge-Kutta method


(1 pt) 4.1 Solved Problem 4.39 Inverse Laplace Transform of a Rational Function (using Partial Fractions) (1 pt) 4.2 Solved Problem 10.1 Method of Frobenius — Series Solutions of Bessel's Differential Equation) (1 pt) 4.3 Solved Problem 12.24 Solution of the Radial Flow Equation using Separation of Variables

Problem 5 — Capillary Pressure, Permeability, and Relative Permeability (from the course notes)

(1 pt) 5.1 Derive the capillary pressure relation for a single tube. (1 pt) 5.2 Derive the permeability relation for porous media using the "bundle of capillary tubes" model (from PETE 428 Course Notes). (1 pt) 5.3 Derive the "field units" form of the Purcell-Burdine permeability equation. (2 pts) 5.4 Derive the Brooks-Corey-Burdine equation for relative permeability.

Problem 6 — Steady-State and Pseudosteady-State Flow Solutions (from the course notes)

(1 pt) 6.1 Derive the steady-state laminar flow relation for the horizontal, radial flow of a gas in porous media (p2 form). (1 pt) 6.2 Derive the pseudosteady-state flow relation for (liquid case). (1 pt) 6.3 Derive the pseudosteady-state flow relation for p(r,t) (liquid case). (1 pt) 6.4 Derive the constant pressure flow relation for "boundary-dominated" flow conditions (liquid case). (not in course notes) (1 pt) 6.5 Derive the linear pressure flow relation for "boundary-dominated" flow conditions (liquid case). (not in course notes)

Problem 7 — Diffusivity Equations (from the course notes)

(1 pt) 7.1 Derive the "diffusivity" equation for the flow of a slightly compressible liquid in porous media — "pressure" form. (1 pt) 7.2 Derive the "diffusivity" equation for the flow of a gas in porous media — "p2" form. (2 pts) 7.3 Derive the "diffusivity" equations (and compressibility and saturation relations) for multiphase flow in porous media (cf ≠ 0).

Problem 8 — Solutions for Radial Flow (from the course notes)

(1 pt) 8.1 Derive the Boltzmann transform solution for the infinite-acting reservoir case. (2 pts) 8.2 Derive the Laplace transform solutions for the infinite-acting reservoir case ("cylindrical source" and "line source" solutions). (2 pts) 8.3 Derive the Laplace transform solution for the "no-flow" at the outer boundary condition. (2 pts) 8.4 Derive the "real domain" solution for the "no-flow" at the outer boundary condition.

)( wfpp

Guidance on Homework Portfolio Assignment: Submit a SINGLE .pdf file to [email protected] by 16:59:59 US CST on 03 December 2018. ● You must type your Homework Portfolio. ● Your work may include output from computer programs such as MS Excel, but this work must be clearly shown and documented. ● It is suggested that you create a "work file" in MS Word or MS PowerPoint to capture your work, then create a .pdf from this file. ● The standard of submission must be near "publication quality," poor/fair quality work submissions will NOT be accepted. ● Your submission file must be named P620_18C_HwkPort_YOURLASTNAME.pdf.

Note: ● Students ARE permitted to communicate — HOWEVER, students ARE NOT permitted to submit shared or copied work. ● Unless otherwise stated, students ARE NOT permitted to use symbolic/numeric software (e.g., Mathematica, Theorist, MATLAB). ● Using software (unless specifically permitted), and/or sharing or copying work will result in a zero (0) score for the Homework Portfolio.



Fall 2018 (version: 20181118)

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Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Homework Portfolio Assignment (50 points/12 problems)

Fall 2018

Homework Portfolio: Inventory of Problems (continued)

Problem 9 — Solutions for Fractured Wells (from the course notes)

(1 pt) 9.1 Derive the Laplace domain solutions for linear flow in an "infinite-acting" reservoir. (from "Linear Flow Solutions" lecture) (1 pt) 9.2 Derive the Laplace domain solution for a well with a vertical fracture in an "infinite-acting" reservoir as given by Ozkan. (2 pts) 9.3 Using the real domain solution (Eq. 19 in the notes), derive the Gringarten-Ramey-Raghavan solution given by Eq. 36.

Problem 10 — Solutions for Naturally-Fractured Reservoirs (from the course notes)

(1 pt) 10.1 Derive Eq. 46 in the course notes. (1 pt) 10.2 Derive Eq. 53 and Eq. 55 in the course notes.

Problem 11 — Convolution (from the course notes)

(1 pt) 11.1 Derive Eq. 11 (course notes), dimensionless convolution result, discrete step variable-rate case (in the real domain). (1 pt) 11.2 Derive Eq. 25 (course notes), dimensionless convolution result, discrete step variable-pressure drop case (in the real domain). (1 pt) 11.3 Derive Eq. 45 (course notes), dimensionless constant rate-constant pressure convolution identity (in the Laplace domain).

Problem 12 — Wellbore Storage (from the course notes)

(1 pt) 12.1 Derive Eq. 26 (course notes), dimensionless wellbore storage flowrate relation due to wellbore storage. (1 pt) 12.2 Derive Eq. 35 (course notes), dimensionless wellbore storage pressure identity in the Laplace domain. (2 pts) 12.3 Make an exhaustive attempt to analytically invert Eq. 40 (course notes) into the real domain (this will be extremely difficult).

Guidance on Homework Portfolio Assignment: Submit a SINGLE .pdf file to [email protected] by 16:59:59 US CST on 03 December 2018. ● You must type your Homework Portfolio. ● Your work may include output from computer programs such as MS Excel, but this work must be clearly shown and documented. ● It is suggested that you create a "work file" in MS Word or MS PowerPoint to capture your work, then create a .pdf from this file. ● The standard of submission must be near "publication quality," poor/fair quality work submissions will NOT be accepted. ● Your submission file must be named P620_18C_HwkPort_YOURLASTNAME.pdf.

Note: ● Students ARE permitted to communicate — HOWEVER, students ARE NOT permitted to submit shared or copied work. ● Unless otherwise stated, students ARE NOT permitted to use symbolic/numeric software (e.g., Mathematica, Theorist, MATLAB). ● Using software (unless specifically permitted), and/or sharing or copying work will result in a zero (0) score for the Homework Portfolio.



Fall 2018 (version: 20181118)

(Page 13 of 33)

____________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Final Examination

Monday 10 December 2018 [by 5:00 p.m. (i.e., 16:59:59 US CST)]















____________________________________________________________________________________________________________________________________________________________________________________________________________________________________

(Include this page in your submission of the Final Examination)


Fall 2018 (version: 20181118)

(Page 14 of 33)

Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs (Orientation) Final Examination (25 points)

Fall 2018

Final Examination: Orientation

Students ARE NOT permitted to communicate or collaborate — this is an Exam, any sharing/copying will result in a zero score. Students ARE NOT permitted to use symbolic/numeric software (e.g., Mathematica, Theorist, etc.).

The Final Examination is assigned as follows:

● Due date: Monday 10 Dec 2018 [by 5:00 p.m. (i.e., 16:59:59 US CST)] ● Submission: Submit as a SINGLE .pdf file named: P620_18C_Exam_YOURLASTNAME.pdf ● Recipient: Send to [email protected] (use TAMU "FILEX" system for very large files) ● References: Van Everdingen, A. F., and Hurst, W. (1949) The Application of the Laplace Transformation to Flow Problems in

Reservoirs. Society of Petroleum Engineers. doi:10.2118/949305-G (doi.org/10.2118/949305-G)

Failure to submit the Final Exam will result a zero (0) score for this portion of the student's grade.

The inventory of examination problems is given below:

(20 pts) Numerical Inversion of Laplace Transform Solutions (Van Everdingen and Hurst) using the Stehfest Algorithm

You are to prepare a computational module to numerically invert the Laplace transform solutions as follows from the Van Everdingen and Hurst reference cited above:

● Eq. VI-4 (Laplace domain solution for a well produced at a constant flowrate, infinite-acting homogenous reservoir) ● Eq. VI-27 (Laplace domain solution for a well produced at a constant pressure, infinite-acting homogenous reservoir) ● Eq. VII-4 (Laplace domain solution for a well produced at a constant pressure, bounded circular reservoir) ● Eq. VII-11 (Laplace domain solution for a well produced at a constant flowrate, bounded circular reservoir) ● Eq. VIII-10 (Laplace domain solution for a well with wellbore storage effects, infinite-acting homogenous reservoir)*

* Tabulated solutions for this case (wellbore storage effects) can be found in the reference:

Agarwal, R.G., Al-Hussainy, R., and Ramey, H.J. Jr. (1970) An Investigation of Wellbore Storage and Skin Effect in Unsteady Liquid Flow: I. Analytical Treatment, Society of Petroleum Engineers. (doi:10.2118/2466-PA)

For each component problem you are to: ● Compute the numerical inversion solution and its derivative using the Stehfest algorithm and provide a sample table. ● Plot the numerical inversion solution and its derivative as appropriate, Cartesian, semilog, log-log. ● Plot any existing real-domain solutions as data for comparison on the plots with the numerical inversion solutions.

[Note — models are plotted as lines, data are plotted as symbols. (publication-quality plots are required)]

Guidance on Final Examination: Submit a SINGLE .pdf file to [email protected] by 16:59:59 US CST on 10 December 2018. ● You must type your Final Examination. ● Your work may include output from computer programs such as MS Excel, but this work must be clearly shown and documented. ● It is suggested that you create a "work file" in MS Word or MS PowerPoint to capture your work, then create a .pdf from this file. ● The standard of submission must be near "publication quality," poor/fair quality work submissions will NOT be accepted. ● Your submission file must be named P620_18C_Exam_YOURLASTNAME.pdf.

Note: ● Students ARE NOT permitted to communicate or collaborate — this is an Exam, any sharing/copying will result in a zero score. ● Students ARE NOT permitted to use symbolic/numeric software (e.g., Mathematica, Theorist, etc.). ● Last warning — this is an Exam — sharing or copying work will result in a zero (0) score for this Exam.

(Include this page in your submission of the Final Examination)


Fall 2018 (version: 20181118)

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Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Required University Statements — Required by Texas A&M University

Fall 2018

Americans with Disabilities Act (ADA) Statement: (Last Revision: 05 November 2015)

The Americans with Disabilities Act (ADA) is a federal anti-discrimination statute that provides comprehensive civil rights protection for persons with disabilities. Among other things, this legislation requires that all students with disabilities be guaranteed a learning environment that provides for reasonable accommodation of their disabilities. If you believe you have a disability requiring an accommodation, please contact Disability Services, currently located in the Disability Services building at the Student Services at White Creek complex on west campus or call 979-845-1637. For additional information, visit http://disability.tamu.edu.

Aggie Honor Code: (http://student-rules.tamu.edu/aggiecode)

"An Aggie does not lie, cheat or steal, or tolerate those who do."

Definitions of Academic Misconduct:

1. CHEATING: Intentionally using or attempting to use unauthorized materials, information, notes, study aids or other devices or materials in any academic exercise.

2. FABRICATION: Making up data or results, and recording or reporting them; submitting fabricated docu-ments.

3. FALSIFICATION: Manipulating research materials, equipment or processes, or changing or omitting data or results such that the research is not accurately represented in the research record.

4. MULTIPLE SUBMISSION: Submitting substantial portions of the same work (including oral reports) for credit more than once without authorization from the instructor of the class for which the student submits the work.

5. PLAGIARISM: The appropriation of another person's ideas, processes, results, or words without giving ap-propriate credit.

6. COMPLICITY: Intentionally or knowingly helping, or attempting to help, another to commit an act of aca-demic dishonesty.

7. ABUSE AND MISUSE OF ACCESS AND UNAUTHORIZED ACCESS: Students may not abuse or misuse computer access or gain unauthorized access to information in any academic exercise. See Student Rule 22: http://student-rules.tamu.edu/

8. VIOLATION OF DEPARTMENTAL OR COLLEGE RULES: Students may not violate any announced departmental or college rule relating to academic matters.

9. UNIVERSITY RULES ON RESEARCH: Students involved in conducting research and/or scholarly activities at Texas A&M University must also adhere to standards set forth in the University Rules.

For additional information please see:

http://student-rules.tamu.edu/.

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 permission.

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 own, even if you should have the permission of that person. Plagiarism is one of the worst academic sins, for the plagiarist 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."


Fall 2018 (version: 20181118)

(Page 16 of 33)

Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Appendix — Extended Description of Course Objectives

Fall 2018

Learning Objectives

The student should be able to demonstrate mastery of objectives in the following areas:

Module 1 — Advanced Mathematics Relevant to Problems in Engineering Module 2 — Petrophysical Properties Module 3 — Fundamentals of Flow in Porous Media Module 4 — Reservoir Flow Solutions Module 5 — Applications/Extensions of Reservoir Flow Solutions

Considering these modular topics, we have the following catalog of course objectives:

Module 1: Advanced Mathematics Relevant to Problems in Engineering

Fundamental Topics in Mathematics:

Work fundamental problems in algebra and trigonometry, including partial fractions and the factoring of equations.

Perform elementary and advanced calculus: analytical integration and differentiation of elementary functions (polynomials, exponentials, and logarithms), trigonometric functions (sin, cos, tan, sinh, cosh, tanh, and combinations), and special functions (Error, Gamma, Exponential Integral, and Bessel functions).

Derive the Taylor series expansions and Chebyshev economizations for a given function.

Derive and apply formulas for the numerical differentiation and integration of a function using Taylor series expansions. Specifically, be able to derive the forward, backward, and central "finite-difference" relations for differentiation, as well as the "Trapezoidal" and "Simpson's" Rules for integration.

Apply the Gaussian and Laguerre quadrature formulas for numerical integration.

Numerical Differentiation and Integration of Analytic Functions:

Be able to recognize, develop, and apply the Taylor series (finite-difference) formulas for numerical differentiation of an analytic function.

— The O(x)4 derivatives are expressed as:

First Derivative, f'(x):

4)())2()(8)(8)2((12

1)(' xxxfxxfxxfxxf

xxf

Second Derivative, f''(x):

42

)())2()(16)(30)(16)2(()(12

1)('' xxxfxxfxfxxfxxf

xxf

Third Derivative, f'''(x):

4

3

)())3(

)2(8)(13)(13)2(8)3(()(8

1)('''

xxxf

xxfxxfxxfxxfxxfx

xf

Fourth Derivative, fiv(x):

4

4

)())3()2(12)(39)(56

)(39)2(12)3(()(6

1)(

xxxfxxfxxfxf

xxfxxfxxfx

xf iv


Fall 2018 (version: 20181118)

(Page 17 of 33)


Fall 2018

Course Objectives (Continued)

Module 1: Advanced Mathematics Relevant to Problems in Engineering (continued)

Be able to recognize and apply the following formulas and methodologies for numerical integration.

— Trapezoidal rule: (with correction) (be able to develop — see Hornbeck):

)](')('[12

)()()]()([

2)()( 0

21

10

0

xfxfx

xfxxfxfx

dxxfxI n

n

iin

nx

x

where n

xxx n 0

— Simpson's rule: (with correction) (be able to develop — see Hornbeck):

)()(180

)(])(2)(4)()([

3)()( 0

42

eveni2

1

oddi1

0

0

xfxxx

xfxfxfxfx

dxxfxI ivn

n

ii

n

iin

nx

x

where n must be even. Also n

xxx n 0 and

20xx

x n .

— Gaussian quadrature: (weights and abscissas from Abramowitz and Stegun: Handbook of Mathematical Functions, Table 25.4, pgs. 916-919):

n

iii

nnx

x

zfwxx

dxxf

1

0

0

)(2

)( where )2

()2

( 00 xxx

xxz n

in

i

— Laguerre quadrature: (weights and abscissas from Abramowitz and Stegun: Handbook of Mathematical Functions, Table 25.9, pgs. 923):

n

iii

x xfwdxxfe

10

)( )( or )( )(

10

n

ii

xi xigewdxxg

Solution of First and Second Order Ordinary Differential Equations:

First Order Ordinary Differential Equations:

— Classify the order of a differential equation (order of the highest derivative). — Verify a given solution of a differential equation via substitution of a given solution into the original

differential equation. — Solve first order ordinary differential equations using the method of separation of variables (or separable

equations). — Derive the method of integrating factors for a first order ordinary differential equation. — Apply the Euler and Runge-Kutta methods to numerically solve first order ordinary differential equations.

Solution of First Order Ordinary Differential Equations:

— Be able to derive the method of integrating factors for a first order ordinary differential equation. — Be able to determine the solution of a first order ordinary differential equation using the method of

integrating factors.

Second Order Ordinary Differential Equations: — Develop the homogeneous (or complementary) solution of a 2nd order ordinary differential equation

(ODE) using y=emx as a trial solution. — Develop the particular solution of a 2nd order ordinary differential equation (ODE) using the method of

undetermined coefficients.


Fall 2018 (version: 20181118)

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Fall 2018



Application of the Runge-Kutta Method:

— Be able to apply the Runge-Kutta methods to numerically solve 1st order ordinary differential equations given a general 1st order relation of the form:

1. Given )(10 tryadt

dya , we must rearrange to yield the following form:

])([1

10

yatradt

dy

2. We also require the "initial" conditions: ti and yi=y(ti), where ti is usually set equal to zero (but does not have to be set to zero).

— Be able to apply the Runge-Kutta methods to numerically solve 2nd order ordinary differential equations given a general 2nd order relation of the form:

1. Given )(212

2

0 tryadt

dya

dt

yda , we must rearrange to yield the following form:

])([1

2102

2ya

dt

dyatr

adt

yd or ])([

121

02

2yavatr

adt

yd , where

dt

dyv

2. For 2nd order equations, we again require "initial" conditions, but now we include a first derivative term. In this case we require: ti, yi=y(ti), and vi=v(ti) where again, ti is usually set equal to zero (but does not have to be set to zero).

The Laplace Transform:

Fundamentals of the Laplace Transform:

— Be able to state the definition of the Laplace transformation and its inverse.

Definition of the Laplace Transform:

dttfetfLsf st )())(()(

0

or dxs

xfe

sx )(

1

0

(using x=st)

Definition of the Inverse Laplace Transform: (Mellin Inversion Integral)

dssfei

sfLtf

iy

iy

st )(2

1))(()( 1

— Be able to prove that the Laplace transform is a linear operator. — Be able to derive the Laplace transforms given on page 98 of the Spiegel text. — Be familiar with, and be able to derive, the operational theorems for the Laplace transform as given on

pages 101-102 of the Spiegel text.

Properties of the Laplace Transform:

— Be familiar with the "unit step" function shown below

-1

0

1

u(t

-a)

t a

The unit step function is given by:

atatu

atatu

1)(

0)(

And its Laplace transform is:

ases

uf 1

)(


Fall 2018 (version: 20181118)

(Page 19 of 33)


Fall 2018



— Be able to develop and apply the Laplace transform formulas for the discrete data functions shown below.

+ Step Data Function:

n

i

istii eff

ssf

1

11)(1

)( where (t0=0 and f0=0)

+ Piecewise Linear Data Function: (Roumboutsos and Stewart Method)

1

2

12

12

1121

)(1

)1(1

)(n

i

nstnistist

ist em

seem

sem

suf

where the slope terms (mi's) are taken as backward differences given by

1

1

ii

iii tt

ffm

+ Piecewise Log-Linear Data Function: (Blasingame Method)

),(),(),()( 1222

2222

1111 stv

s

astv

s

astv

s

asf

vvv

),()(),(),(... 12111

1111

nn

nvn

nnvn

nnnvn

nnnvn stv

s

av

s

astv

s

astv

s

a

The slope and intercept terms ('s and 's) are shown graphically in the attached notes. Also, Γ(x) is the Gamma function and γ(a,x) is the first incomplete Gamma function.

Applications of the Laplace Transform to Solve Linear Ordinary Differential Equations:

— Be able to develop the Laplace transform of a given differential equation and its initial condition(s). This requires the Laplace transform of each time-derivative, then substitution into the differential form, the

result is an algebraic expression in terms of s and )(sf .

+ Laplace Transform of a Generic Time Dependent Derivative:

)0()0(...)0(')0()())(( 1221 tftsftfstfssfstfdt

dL nnnnn

n

n

where

)0(),0()...0(''),0('),0( 11

22210

tfctfctfctfctfc nn

nn

— Be able to resolve the algebra resulting from the Laplace transform of a given differential equation and its initial condition(s) into a closed and hopefully, invertible form.

— Be able to invert the closed form Laplace transform solution of a given differential equation using the fundamental properties of Laplace transforms, Laplace transform tables, partial fractions.

Numerical Laplace Transform and Inversion:

— Be able to use the Gauss-Laguerre integration formula for numerical Laplace transformation. The Laguerre quadrature weights, wk, and abscissas, xk, can be obtained from Abramowitz and Stegun.

n

k

kk

sts

xfw

sdttfesf

10

)(1

)()(


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— Be familiar with the development of the Gaver formula for numerical Laplace transformation, and note its similarity to the Widder inversion formula given in the Cost (AIAA Journal) paper.

n

k

kk

sts

xfw

sdttfesf

10

)(1

)()(

— Be able to use the Gaver and Gaver-Stehfest numerical inversion algorithms for the inversion of Laplace transforms.

+ The Gaver formula for numerical Laplace transform inversion is

])[)2ln(

(!)!(

)1(

)!1(

)!2()2ln()(

0

knt

fkknn

n

ttf

n

k

k

Gaver

The Gaver-Stehfest formula for numerical Laplace transform inversion is

))2ln(

()2ln(

)(

1

it

fVt

tfn

iiStehfestGaver

and the Stehfest extrapolation coefficients are given

)2

,(

]2

1[

2

)!2()!()!1(!)!2

(

)!2(2)1(

niMin

ik

in

iikkikkk

n

kn

kV

Introduction to Special Functions:

Special Functions in Petroleum Engineering Applications

— Be familiar with and be able to compute the following special functions which have applications in petroleum engineering:

+ Exponential Integral (Ei (x) and E1 (x)= -Ei (-x)) + Gamma and Incomplete Gamma Functions ((x), and (a,x), (a,x) and B(z,w)) + Error and Complimentary Error Functions (erf(x) and erfc(x)) + Bessel Functions: J0(x), J1(x), Y0(x), and Y1(x) + Modified Bessel Functions: I0(x), I1(x), K0(x), and K1(x), as well as the integrals of I0(x) and K0(x).

Bessel Functions

— Be familiar with the following Bessel functions:

+ Bessel Functions: Jn(x) and Yn(x), where Bessel's differential equation is given as: (Abramowitz and Stegun; Chapter 9, Eq. 9.1.1)

0)( 222

22 ynz

dz

dyz

dz

ydz and has the solution )()( 21 zYczJcy nn

+ Modified Bessel Functions: In(x) and Kn(x), where Bessel's "modified" differential equation is given as: (Abramowitz and Stegun; Chapter 9, Eq. 9.6.1)

0)( 222

22 ynz

dz

dyz

dz

ydz and has the solution )()( 21 zKczIcy nn

Be able to use the Bessel functions in numerical problem solving efforts and theoretical developments; especially recurrence relations, integral definitions, and Laplace transforms.


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Module 2: Petrophysical Properties

Introduction to Porosity and Permeability Concepts:

Be able to recognize and classify rock types as clastics (sandstones) and carbonates (limestones, chalks, dolstones) and be familiar with the characteristics of porosity that these rocks exhibit.

Be able to distinguish between effective and total porosity and be familiar with the meanings of primary (or depositional) porosity and secondary (or post-depositional) porosity.

Be familiar with factors which affect porosity. In particular, the shapes, arrangements, and distributions of grain particles and the effect of cementation, vugs, and fractures on porosity.

Be familiar with the concept of permeability for porous rocks and be aware of the correlative relations for porosity and permeability.

Be familiar with "friction factor"-"Reynolds Number" plotting concept put forth by Cornell and Katz for flow through porous media. Be aware that this plotting concept validates Darcy's law empirically (the unit slope line on the left portion of the plot, laminar flow).

Development of a Semi-Empirical Concept of Permeability: Darcy's Law:

Be able to develop a velocity/pressure gradient relation for modeling the flow of fluids in pipes (i.e., the Poiseuille equation).

x

pk

A

qv p

xavg

1

where 8

2rk p is considered to be a "geometry" factor.

Be familiar with the general assumptions and limitations of the Poiseuille equation.

Be able to derive the "units" of a Darcy (1 Darcy = 9.86923x10-9 cm2).

Be able to derive the field units form of Darcy's law.

Introduction to Capillary Pressure and Relative Permeability:

Be familiar with the concept of "capillary pressure" for tubes as well as for porous media--and be able to derive the capillary pressure relation for fluid rise in a tube:

rp owc

1)cos(2

Be familiar with and be able to derive the permeability and relative permeability relations for porous media using the "bundle of capillary tubes" model as provided by Nakornthap and Evans. The permeability result is given by:

*1

2*

1

02

23 dSw

pnk

c

ow

Be familiar with the concept of "relative permeability" and the factors which should and should not affect this function. Also, be familiar with the laboratory techniques for measuring relative permeability.


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Module 2: Petrophysical Properties (continued)

Development of the Brooks-Corey-Burdine Equation for Permeability and the Development of a Type Curve Analysis Approach for Capillary Pressure Data:

Be able to derive the "field units" form of the Purcell-Burdine permeability equation (k in md, ow in dyne/cm and, pc in psia). The Purcell-Burdine permeability equation as provided by Nakornthap and Evans is given in terms of absolute (i.e., metric) units. The "field units" result is given by:

*1

2*66.10

1

02

23 dSw

pnk

c

ow where )1(* wiS

Be familiar with and be able to derive the Brooks-Corey-Burdine equation for permeability based on the Purcell-Burdine permeability equation (as given above). This result is given by:

]2

[1

*2

23

dow

pnk or ]

2[

1*66.10

223

dow

pnk (field units)

Be able to discuss the possible applications for the Brooks-Corey-Burdine permeability equation.

Be familiar with and be able to derive a type curve matching approach for capillary pressure data based on the Brooks-Corey model for capillary pressure and saturation given below.

1

)1(

wDD Sp where d

cD p

pp and *1

1

1Sw

S

SwS

wiwD

Electrical Properties of Reservoir Rocks: Be familiar with the definition of the formation resistivity factor, F, as well as the effects of reservoir and

fluid properties on this parameter. Be familiar with and be able to use the Archie and Humble equations to estimate porosity given the formation

resistivity factor, F. Be familiar with the definition of the resistivity index, I, as well as the effects of reservoir and fluid properties

on this parameter and also be familiar with the Archie result for water saturation, Sw. Be familiar with the "shaly sand" models given by Waxman and Smits for relating the resistivity index with

saturation and for relating formation factor with porosity.

Development of a Type Curve Analysis Approach for Relative Permeability Data

Be familiar with and be able to derive the Burdine relative permeability equations (this derivation is provided in detail by Nakornthap and Evans). These relations are

*1

*1

*)(1

02

*

02

2

wc

wS

wc

wrw

dSp

dSp

Sk

and

*1

*1

*)1(1

02

1

*2

2

wc

wS

wc

wrn

dSp

dSp

Sk

Be familiar with and be able to derive the Brooks-Corey-Burdine equations for relative permeability based on the combination of the Burdine relative permeability equations (shown above) and the Brooks and Corey capillary pressure model. These results are given by:

)/23(* worwrw Skk and ]*1[*)1( )/21(2 ww

ornrn SSkk

where the Brooks and Corey capillary pressure model is given by 1

*

wdc Spp


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Module 2: Petrophysical Properties (continued)

and orwk and o

rnk are the "endpoint" relative permeability values.

Be familiar with and be able to derive a type curve matching approach for relative permeability data based on the Brooks-Corey-Burdine relative permeability models. The "dimensionless" variables for this development are given below. — Dimensionless wetting phase relative permeability:

)/23()1( wDrwD Sk

— Dimensionless non-wetting phase relative permeability:

])1(1[ )/21(2 wDwDrnD SSk

— Dimensionless relative permeability ratio function:

]1)1[()1(

)/21(2

2

wDwD

wD

rwD

rnD SS

S

k

k

— Dimensionless saturation functions:

*11

1w

wi

wwD S

S

SS

and wDwi

wiww S

S

SSS

11

*

Module 3: Fundamentals of Flow in Porous Media

Steady-State Flow Concepts: Laminar Flow

Derive the concept of permeability (Darcy's Law) using the analogy of the Poiseuille equation for the flow of fluids in capillaries. Be able to derive the "units" of a "Darcy" (1 Darcy = 9.86923x10-9 cm2), and be able to derive Darcy's Law in "field" and "SI" units.

Derive the single-phase, steady-state flow relations for the laminar flow of gases and compressible liquids using Darcy's Law — in terms of pressure, pressure-squared, and pseudopressure, as appropriate.

Derive the steady-state "skin factor" relations for radial flow.

Steady-State Flow Concepts: Non-Laminar Flow

Demonstrate familiarity with the concept of "gas slippage" as defined by Klinkenberg.

Derive the single-phase, steady-state flow relations for the non-laminar flow of gases and compressible liquids using the Forchheimer equation (quadratic in velocity) — in terms of pressure, pressure-squared, and pseudopressure, as appropriate.

Material Balance Concepts:

Be able to identify/apply material balance relations for gas and compressible liquid systems.

Be familiar with and be able to apply the "Havlena-Odeh" formulations of the oil and gas material balance equations.

Pseudosteady-State Flow Concepts:

Demonstrate familiarity with and be able to derive the single-phase, pseudosteady-state flow relations for the laminar flow of compressible liquids in a radial flow system (given the radial diffusivity equation as a starting point).

Sketch the pressure distributions during steady-state and pseudosteady-state flow conditions in a radial system.


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Module 3: Fundamentals of Flow in Porous Media (continued)

Development of Diffusivity Equation: Pressure and Pseudopressure Forms, General and Radial Flow Geometries:

Be able to describe in words and in terms of mathematical expressions the mass continuity relation for flow through porous media.

Be able to develop the "diffusivity" equations for the flow of a slightly compressible liquid in porous media--"pressure" form, general flow geometry.

— "Gradient-Squared" Case: General form for a slightly compressible liquid.

t

p

k

cppc t

)( 22

— "Small and Constant Compressibility" Case: Base relation for all developments in reservoir engineering and well testing.

t

p

k

cp t

2

Be able to derive the pseudopressure/pseudotime forms of the diffusivity equation for cases where fluid density and viscosity are functions of pressure for a general flow geometry.

"Pseudopressure-Time" Form "Pseudopressure-Pseudotime" Form

t

ptp

p

k

cp

2

a

pntp t

pc

kp

)(2

where the "pseudopressure" function, pp, is given by:

dpB

k

k

Bp

p

basep

np )(

or dp

BBp

p

basep

np 1

)(

and the "pseudotime" function, ta, is given by:

dtpcp

ctt

tnta

0)()(

1)(

Development of Diffusivity Equations for the Flow of a Real Gas: Pressure and Pressure-Squared and Pseudopressure Forms:

Be familiar with and be able to derive the single-phase diffusivity equations in terms of formation volume factors (Bo or Bg) for both the oil and gas cases. These results are given as:

Single-Phase Oil Equation: Single-Phase Gas Equation:

)(][o

poo

oBtB

k

)(][

gp

gg

g

BtB

k

Be able to develop the general form of the diffusivity equation for single-phase gas flow in terms of pressure (and p/z) — starting from the density formulation. These relations are given by:

Density Formulation: General Form: Single-Phase Gas Equation:

t

pk

)(

][

tz

p

k

cp

zt

][

Be able to develop the diffusivity equation for single-phase gas flow in terms of the following: pseudopressure, pressure-squared, and pressure.


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— "Pseudopressure" Formulation:

t

p

k

cp

pgtpg

2 where dp

z

p

p

zp

p

basep

npg

)(

— "Pressure-Squared" Formulation:

)()()][ln()( 2222

22 ptk

cpz

pp t

if constantz then )()( 222 p

tk

cp t

— "Pressure" Formulation:

t

p

k

cp

p

z

pp t

22 ))]([ln( if constant

z

p

then

t

p

k

cp t

2

Development of Diffusivity Equations for the Multiphase Flow:

Be able to develop the continuity relations for the oil, gas, and water phases in terms of the fluid densities. Assume that the gas phase includes gas liberated from the oil and water phases.

Oil Continuity Equation: Water Continuity Equation:

)()( ooo tv

)()( www tv

Gas Continuity Equation:

])[(][)( totggscsww

wgscso

o

oggtotgg t

RB

vR

B

vvpv

Be able to write Darcy's law velocity relations for each phase. The general form is given by:

ii

ii p

kv

where i = oil, gas, and water.

Be able to develop the mass flux relations for the oil, gas, and water phases in terms of the fluid formation volume factors. Again, assume that the gas phase includes gas liberated from the oil and water phases.

Oil Flux Equation: Water Flux Equation:

ooo

ooscoo p

B

kv

w

ww

wwscww p

B

kv

Gas Flux Equation:

][)( www

wswo

oo

osog

gg

ggsctotgg p

B

kRp

B

kRp

B

kv

Be able to develop the mass relations for the oil, gas, and water phases in terms of the fluid formation volume factors. As before, assume that the gas phase includes gas liberated from the oil and water phases.

Oil Mass Equation: Water Mass Equation:

o

ooscooo B

SS )(

w

wwscwww B

SS )(


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Gas Mass Equation:

][ )(w

wsw

o

oso

g

ggscgsc

w

swwgsc

o

sooggtotg B

SR

B

SR

B

S

B

RS

B

RSS

Assuming no capillary pressure forces )( wgo pppp , be able to develop the generalized

diffusivity relations for each phase. (Martin Eqs. 1-3)

"Oil" Equation: "Water" Equation:

)(][o

o

oo

oB

S

tp

B

k

)(][

w

w

ww

wB

S

tp

B

k

"Gas" Equation:

)]([])[(w

wsw

o

oso

g

g

ww

wsw

oo

oso

gg

g

B

SR

B

SR

B

S

tp

B

kR

B

kR

B

k

NEGLECTING the ,, pSpS wo and 2ppp terms — be able to develop the diffusivity relations

for each phase as shown by Martin (Eqs. 7-9)

"Oil" Equation: "Water" Equation:

)(2

o

o

oo

oB

S

tp

B

k

)(2

w

w

ww

wB

S

tp

B

k

"Gas" Equation:

)]( [)( 2

w

wsw

o

oso

g

g

ww

wsw

oo

oso

gg

g

B

SR

B

SR

B

S

tp

B

kR

B

kR

B

k

Development of Diffusivity Equations for the Multiphase Flow — Martin's Saturation Equations and the Concept of Total Compressibility:

Be familiar with and be able to derive the Martin relations for total compressibility and the associated saturation-pressure relations (Eqs. 10 and 11).

Oil Saturation Equation: Water Saturation Equation:

tt

oo

o

oo cdp

dB

B

S

dp

dS

tt

ww

w

ww cdp

dB

B

S

dp

dS

Total Compressibility:

dp

dB

B

S

dp

dR

B

BS

dp

dB

B

S

dp

dR

B

BS

dp

dB

B

Sc

g

g

gsw

w

gww

w

wso

o

goo

o

ot

or,

gg

gw

sw

w

gw

wo

so

o

go

ot S

dp

dB

BS

dp

dR

B

B

dp

dB

BS

dp

dR

B

B

dp

dB

Bc ]

1[]

1[]

1[

or finally,

ggwwoot ScScScc

where,


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Module 4: Reservoir Flow Solutions

,1

,1

dp

dR

B

B

dp

dB

Bc

dp

dR

B

B

dp

dB

Bc sw

w

gw

ww

so

o

go

oo and

dp

dB

Bc

g

gg

1

Total Pressure Equation:

t

cp

t

t

2 where w

w

g

g

o

ot

kkk

Dimensionless Variables and the Dimensionless Radial Flow Diffusivity Equation:

Be able to develop the dimensionless form of the single-phase radial flow diffusivity equation as well as the appropriate dimensionless forms of the initial and boundary conditions, including the developments of dimensionless radius, pressure, and time.

— The Dimensionless Diffusivity Equation:

D

D

D

D

DD

Dt

p

r

p

rr

p

12

2

— Dimensionless Initial and Boundary Conditions:

+ Dimensionless Initial Condition

0)0,( DDD trp (uniform pressure in reservoir)

+ Dimensionless Inner Boundary Condition

1][ 1

DrD

DD r

pr (constant rate at the well)

+ Dimensionless Outer Boundary Conditions

a. "Infinite-Acting" Reservoir

0),( DDD trp

b. "No-Flow" Boundary

0][

eDrDrD

DD r

pr (No flux across the reservoir boundary)

c. Constant Pressure Boundary

0),( DeDD trp (Constant pressure at the reservoir boundary)

Be able to derive the conversion factors for dimensionless pressure and time, for both SI and "field" units.

Solutions of the Radial Flow Diffusivity Equation Using the Laplace Transform:

Be able to recognize that the Laplace transform of the dimensionless form of the single-phase radial flow diffusivity equation is the modified Bessel differential equation. Also, be able to write the general solution for this transformed differential equation.

Dimensionless Diffusivity Equation: Laplace Transform of Diffusivity Equation:

D

D

D

D

DD

D

D

DD

DD t

p

r

p

rr

p

r

pr

rr

1

][1

2

2 D

D

DD

DDpu

dr

pdr

dr

d

r][

1


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Module 4: Reservoir Flow Solutions (continued)

General Solution:

)()(),( 00 DDDD ruBKruAlurp

Derivative of the General Solution:

)()( 11 DDD

D ruKuBruluAdr

pd

Be able to develop the particular solution (in Laplace domain) for the constant rate and constant pressure inner boundary conditions and the infinite-acting reservoir outer boundary condition. Also, be able to use the van Everdingen and Hurst result to convert the constant rate case to the constant wellbore pressure case.

Constant Rate Solution: (infinite-acting reservoir)

)(1

)(

)(1),( 0

1

0D

DDD ruK

uuKu

ruK

uurp

Constant Rate-Constant Pressure Relation: (from van Everdingen and Hurst)

)(

11)(

2 upuuq

DD

Be able to develop the real domain (time) solution for the constant rate inner boundary condition and the infinite-acting reservoir outer boundary condition using both the Laplace transform and the Boltzmann transform approaches. Also be able to develop the "log-approximation" for this solution.

Boltzmann Transform of the Diffusivity Equation:

0]1

1[2

2

D

D

D

D p

Dd

pd

(infinite-acting reservoir case only)

"Log Approximation" Solution for the Diffusivity Equation:

]114

ln[2

1)(

1),(

20 ureuruK

uurp

DDDD (=0.577216…Euler's constant)

Laplace Transform Solutions of the Radial Flow Diffusivity Equation for a Bounded Circular Reservoir:

Be able to derive the particular solutions (in Laplace domain) for a well produced at a constant flow rate in a homogeneous reservoir for the following initial condition, subject to the following initial and outer boundary conditions:

— Dimensionless Initial and Boundary Conditions:

+ Dimensionless Initial Condition

0)0,( DDD trp (uniform pressure in reservoir)

+ Dimensionless Inner Boundary Condition

1][ 1

DrD

DD r

pr (constant rate at the well)

+ Dimensionless Outer Boundary Conditions

a. Prescribed Flux at the Boundary

)(][ DDexteDrDrD

DD tq

r

pr


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b. Constant Pressure at the Boundary

0),( DeDDD trrp

— Particular Solutions in the Laplace Domain:

+ "Infinite-acting" reservoir behavior

)(

)(1),(

1

0

uKu

ruK

uurp D

DD

Or the line source approximation

)(1

),( 0 DDD ruKu

urp (where 1)(1 uKu , for 0u )

+ Bounded circular reservoir — "no-flow" at the outer boundary (i.e., 0)( DDext tq )

)()()()(

)()()()(1),(

1111

0110

eDeD

DeDeDDDD

ruKuluruluKu

rulruKrulruK

uurp

(constant rate at the well)

+ Bounded circular reservoir — "constant-pressure" at the outer boundary

)()()()(

)()()()(1),(

0101

0000

eDeD

DeDeDDDD

ruKuluruluKu

rulruKrulruK

uurp

(constant rate at the well)

+ Bounded circular reservoir — "prescribed flux" at the outer boundary

)()()()(

)()()()(1),(

1111

0110

eDeD

DeDeDDDD

ruKuluruluKu

rulruKrulruK

uurp

)()()()(

)()()()(])[(

1

1111

1010

eDeD

DD

eDDext

ruKuluruluKu

uKurululuruK

ru

uuq

u

Real Domain Solutions of the Radial Flow Diffusivity Equation for a Bounded Circular Reservoir:

Be able to derive the following particular solutions in the real domain from the appropriate Laplace transform solutions for an unfractured well produced at a constant flow rate in a homogeneous reservoir for the following outer boundary conditions:

— "Infinite-acting" reservoir behavior (line source solution)

)4

(2

1),(

2

1D

DDDD t

rErtp

or the so-called "log approximation"

)4

ln(2

1),(

2D

DDDD

r

t

ertp

— Bounded circular reservoir — "no-flow" at the outer boundary

)4

exp()4

1

2()

4exp(

2)

4(

2

1)

4(

2

1),(

2

2

22

2

2

1

2

1D

eD

eD

D

D

eD

eD

D

D

eD

D

DDDD t

r

r

r

t

r

r

t

t

rE

t

rErtp

and its "well testing" derivative function, pD'=d/dtD[pD(rD,tD)] is given by


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)4

exp()84

(2

1)

4exp(

2)

4exp(

2

1),('

2222

2

2

D

eDeDD

DD

eD

eD

D

D

DDDD t

rrr

tt

r

r

t

t

rrtp

— Bounded circular reservoir — "constant pressure" at the outer boundary

)4

exp()(8

1)

4(

2

1)

4(

2

1),(

222

2

1

2

1D

eDDeD

DD

eD

D

DDDD t

rrr

tt

rE

t

rErtp

and its "well testing" derivative function, pD'=d/dtD[pD(rD,tD)] is given by

)4

exp()14

)((8

1)

4exp(

2

1)

4exp(

2

1),('

2222

22

D

eD

D

eDDeD

DD

eD

D

DDDD t

r

t

rrr

tt

r

t

rrtp

Solutions for the Behavior of a Fractured Well in a Bounded Circular Reservoir: Infinite and Finite-Acting Reservoir Cases:

Be familiar with the concept of a well with a uniform flux or infinite conductivity vertical fracture in a homogeneous reservoir. Note that the uniform flux condition implies that the rate of fluid entering the fracture is constant at any point along the fracture. On the other hand, for the infinite conductivity case, we assume that there is no pressure drop in the fracture as fluid flows from the fracture tip to the well.

Be able to derive the following real and Laplace domain (line source) solutions for a well with a uniform flux or infinite conductivity vertical fracture in a homogeneous reservoir.

— General Result: (cfracs subscript means Continuous Fracture Source)

wDwDDclsDDDcfracsD dxuxxpuyxp ']),'[(2

1),0,1|(|

1

1,,

where the cls subscript means Continuous Line Source

— "Infinite-acting" reservoir behavior (line source solution)

])()([1

2

1),0,1|(|

)1(

0

0

)1(

0

0inf,, dzzKdzzKuu

uyxpDxuDxu

DDcfracsD

— Bounded circular reservoir — "no-flow" at the outer boundary

),0,1|(|),0,1|(| inf,,,, uyxpuyxp DDcfracsDDDnfbcfracsD

])()([)(

)(1

2

1)1(

0

0

)1(

0

01

1 dzzIdzzIruI

ruK

uu

DxuDxu

eD

eD

— Bounded circular reservoir — "constant pressure" at the outer boundary

),0,1|(|),0,1|(| inf,,,, uyxpuyxp DDcfracsDDDcpbcfracsD

])()([)(

)(1

2

1)1(

0

0

)1(

0

00

0 dzzIdzzIruI

ruK

uu

DxuDxu

eD

eD


Fall 2018 (version: 20181118)

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Fall 2018



Dual Porosity Reservoirs — Warren and Root Approach — Pseudosteady-State Matrix Behavior:

Be familiar with the "fracture" and "matrix" models developed by Warren and Root.

Be able to develop the Laplace and real domain results given by Warren and Root for pseudosteady-state matrix flow. These relations are

— Laplace domain results:

+ Warren and Root "Interporosity Flow Function":

u

uuf

)1(

)1()(

Solutions in the Laplace domain:

))(

114ln(

2

1))((

1

))(()(

))((1),(

2201

0uufreu

ruufKuuufKuuf

ruufK

uurp

DD

DDD

— Line source solution in the real domain:

StEtEr

t

ertp DD

D

DDDD

)

)1((

2

1)

)1((

2

1)

4ln(

2

1),( 112

Be able to develop the Laplace and real domain results given by Warren and Root for pseudosteady-state matrix flow. These relations are

))1(

exp(2

1)

)1(exp(

2

1

2

1),(' DDDDD ttrtp

Direct Solution of the Gas Diffusivity Equation Using Laplace Transform Methods:

Be familiar with the convolution form of a non-linear partial differential equation (with a non-linear right-hand-side term), as shown below.

dtgy

t

yyy

t

)()(

0

2

Where we assume that the β(y) function can be re-cast as a unique function of time (i.e., β(y) can be written as β(t)). Using β(t) requires assumptions as to flow regimes--we will demonstrate this assuming pseudosteady-state flow.

Taking the Laplace transform of this relation gives

)()]0()([)(2 ugtyuyuuy

Be able to develop the generalized Laplace domain formulation of the non-linear radial gas diffusivity equation using the β(t) approach.

— The real gas diffusivity equation (in radial coordinates) is given in dimensionless form by:

D

pDD

D

pD

tii

t

D

pD

DD

pD

t

pt

t

p

c

c

r

p

rr

p

)(

12

2

])([

tii

tD c

ct

where

)(2.141

1ppipD pp

qB

khp

t

rc

kt

wtiiD 2

0002637.0

w

D r

rr


Fall 2018 (version: 20181118)

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Fall 2018



and the pseudopressure function is given by:

dpz

p

p

zdp

BBp

p

basepi

iip

basepg

giip

1

— Substituting the convolution formulation into the right-hand-side of the real gas diffusivity equation gives

dtgp

r

p

rr

p

r

pr

rr D

DtpD

D

pD

DD

pD

D

pDD

DD)(

1][

1

02

2

)()()(1)(

])(

[1

2

2

upuugdr

upd

rdr

upd

dr

updr

dr

d

r pDD

pD

DD

pD

D

pDD

DD (Laplace domain relation)

Be familiar with and be able to develop the g(u) term. The g(tD) term is defined by:

dtgp

t

pt D

DtpD

D

pDD )()(

0

Convolution:

Be familiar with and be able to derive the convolution sums and integrals for the variable-rate and variable pressure drop cases.

— Variable-Rate Case:

)()()( 1,1

1

DjDcrsD

n

jDjDjDwD ttpqqtp (discrete rate changes)

dtpqtp D

Dt

crsDDDwD )()(')(

0

, (continuous rate changes)

— Variable-Pressure Drop Case:

)( )(

)()( 1

1

,

DjDDcp

n

j ri

jwfiDtD ttq

pp

pptq (discrete rate changes)

Be able to derive the general convolution identity in the Laplace domain from the integral form of the variable-rate convolution identity.

)()()( , upuqup crsDqDwD

Be able to derive the real and Laplace domain identities for relating the constant pressure and constant rate cases: (from van Everdingen and Hurst)

— Laplace domain result:

)(

11)(

,2, upuuq

crsDcpD

— Real domain result:

DDcrsD

Dt

cpD tdtpq )()( ,

0

, or DDcpD

Dt

crsD tdtqp )()( ,

0

,


Fall 2018 (version: 20181118)

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Fall 2018



Concepts and Applications in Wellbore Storage Distortion:

Be familiar with and, based on physical principles, be able to derive the relations to model the phenomena of "wellbore storage." In particular, you should be able to derive the following:

— General Rate Relation:

][24)(dt

dp

dt

dpCBq

tfwfsqsf

— Pressure Relations (for small times/wellbore storage domination):

tC

qBpp

siwf 24 (for small times, i.e., wellbore storage domination)

or

D

DwD C

tp (for small times, i.e., wellbore storage domination)

— Laplace Domain Identity:

DsD

wDCu

up

up2

)(

11

)(

(valid for all times)

Module 4: Reservoir Flow Solutions — Under Consideration

Multilayered Reservoir Solutions

Dual Permeability Reservoir Solutions

Horizontal Well Solutions

Radial Composite Reservoir Solutions

Various Models for Flow Impediment (Skin Factor) Module 5: Applications/Extensions of Reservoir Flow Solutions — Under Consideration

Oil and Gas Well Flow Solutions for Analysis, Interpretation, and Prediction of Well Performance.

Low Permeability/Heterogeneous Reservoir Behavior.

Macro-Level Thermodynamics (coupling PVT behavior with Reservoir Flow Solutions).

External Drive Mechanisms (Water Influx/Water Drive, Well Interference, etc.).

Hydraulic Fracturing/Solutions for Fractured Well Behavior.

Analytical/Numerical Solutions of Various Reservoir Flow Problems.

Applied Reservoir Engineering Solutions — Material Balance, Flow Solutions, etc. ■

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