Fluid Flow in Petroleum Reservoirs
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Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016 Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures Fall 2016 Petroleum Engineering 620 Instructor: Dr. Tom BLASINGAME TA: Alex VALDES-PEREZ Texas A&M University Office: Richardson 821A Office: Richardson 821 College of Engineering [email protected] [email protected] MW 07:55 pm-09:10 pm RICH 319 (in-class lectures) Required Texts/Resources: (a. Book must be purchased. b. Out of Print/Public Domain. Some texts may also be available at www.scribd.com) 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 (1937). 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: (a. Local bookstores. b. Special order or TAMU library. Some texts also available at www.scribd.com) 1.a Calculus, 4th edition: Frank Ayres and Elliot Mendelson, Schaum's Outline Series (1999) (Remedial text) 2.a Differential Equations, 2nd edition: Richard Bronson, Schaum's Outline Series (1994) (Remedial text) 3.a Laplace Transforms, M.R. Spiegel, Schaum's Outline Series (1965) (Remedial text) 4.a Numerical Analysis, F. Scheid, Schaum's Outline Series, McGraw-Hill Book Co, New York (1968). (Remedial text) 5.b The Mathematics of Diffusion, 2nd edition, J. Crank, Oxford Science Publications (1975). (important/historical) 6.b Table of Integrals, Series, and Products, I.S. Gradshteyn and I.M. Ryzhik, Academic Press (1980). (very important/historical) 7.b Methods of Numerical Integration, P.F. Davis and P. Rabinowitz, Academic Press, New York (1989). (perhaps useful for research) 8.b An Atlas of Functions, J. Spanier and K. Oldham, Hemisphere Publishing (1987). (perhaps useful for research) 9.b Adv. Math. Methods for Eng. and Scientists, 2nd edition, C.M. Bender and S.A. Orsag, McGraw-Hill (1978). (excellent text) 10.b Asymptotic Approximations of Integrals, R. Wong, Academic Press (1989). (perhaps useful for research) 11.b 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_16C/ 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] Assigned Problems ...... 90 percent Class Participation (subjective, based on opinion of the instructor) ...... 10 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 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. (Page 2 of 24) Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016 Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures Fall 2016 Work Requirements: (layout/format/etc.) You must show ALL work — as appropriate, YOU MUST: — Show all details in your calculations (no skipped steps) — all portions of all analysis relations must be shown. — Show all units in each calculation. 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. 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) 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. (Page 3 of 24) Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016
Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Course Outline/Topics Fall 2016 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. (Page 4 of 24) Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016 Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Tentative Course Schedule Fall 2016 Month Date Day Topic Lecture or Potential Assignment Topic Aug 29 M Course Introduction Syllabus Aug 31 W Review of Functions Lec_01_Mod1_ML_01_Rev_of_Fcns.pdf Sep 05 M Approximation of Functions Lec_02_Mod1_ML_02_Fcn_Approx.pdf Sep 07 W 1st Order Ordinary Differential Equations Lec_03_Mod1_ML_03_1st_Order_ode.pdf Sep 12 M 2nd Order Ordinary Differential Equations Lec_04_Mod1_ML_04_2nd_Order_ode.pdf Sep 14 W The Laplace Transform Lec_05_Mod1_ML_05_LaplaceTrans.pdf Sep 19 M Introduction to Special Functions Lec_06_Mod1_ML_06_SpecialFcns.pdf Sep 21 W Porosity and Permeability Concepts Lec_07_Mod2_PtrPhy_01_PorPerm.pdf Sep 26 M Correlation of Petrophysical Data Lec_08_Mod2_PtrPhy_02_DataCorel.pdf Sep 28 W Development of Permeability/Darcy's Law Lec_09_Mod2_PtrPhy_03_Perm_Dev.pdf Oct 03 M Capillary Pressure Lec_10_Mod2_PtrPhy_04_Cap_Pres.pdf Oct 05 W Relative Permeability Lec_11_Mod2_PtrPhy_05_Rel_Perm.pdf Oct 10 M Electrical Properties of Reservoir Rocks Lec_12_Mod2_PtrPhy_06_Elec_Prop.pdf Oct 12 W Single-Phase, Steady-State Flow Lec_13_Mod3_FunFld_01_SSDarcy.pdf Oct 17 M Non-Laminar Flow in Porous Media Lec_14_Mod3_FunFld_02_SSNonDarcy.pdf Oct 19 W Material Balance Concepts Lec_15_Mod3_FunFld_03_MatBal.pdf Oct 24 M Pseudosteady-State Flow (Circular Res.) Lec_16_Mod3_FunFld_04_PSS_Flow.pdf Oct 26 W Liquid Flow Diffusivity Equation Lec_17_Mod3_FunFld_05_DifEq_Liq.pdf Oct 31 M Gas Flow Diffusivity Equation Lec_18_Mod3_FunFld_06_DifEq_Gas.pdf Nov 02 W Multiphase Flow Diffusivity Equation Lec_19_Mod3_FunFld_07_DifEq_MlPhs.pdf Nov 07 M Dimensionless Variables/Radial Flow Lec_20_Mod4_ResFlw_01_DimLssVar.pdf Nov 09 W Solutions — Radial Flow Diffusivity Eq. Lec_21_Mod4_ResFlw_02_RadFlwSln.pdf Nov 14 M Solutions — Radial Flow Diffusivity Eq. Lec_21_Mod4_ResFlw_02_RadFlwSln.pdf Nov 16 W Solutions — Linear Flow Diffusivity Eq. Lec_22_Mod4_ResFlw_03_LinFlwSln.pdf
Nov 21 M Solutions — Fractured Well (High FcD) Lec_23_Mod4_ResFlw_04_FracWellSln.pdf Nov 23 W Solutions — Dual Porosity Reservoirs Lec_24_Mod4_ResFlw_05_NatFrcResSln.pdf Nov 24 Th Thanksgiving Holiday (no class) — Nov 28 M Direct Solution — Gas Diffusivity Equation Lec_25_Mod4_ResFlw_06_DrtSlnGas.pdf Nov 30 W Convolution Lec_26_Mod4_ResFlw_07_Convolution.pdf Dec 05 M Wellbore Storage Lec_27_Mod4_ResFlw_08_WellboreStrg.pdf Dec 07 W Extra Class —
Dec 12 M Any/all remaining assignments due. — (http://registrar.tamu.edu/general/finalschedu le.aspx#0-Fall2016) Dec 15 R Final grades due GRADUATING students. — (http://registrar.tamu.edu/general/calendar.aspx) Dec 19 M Final grades for all students Fall 2016 term. — (http://registrar.tamu.edu/general/calendar.aspx) Comments: 1. All class sessions will also be recorded and put on the eCampus system as well as the instructor's website. 2. We will meet will also host "as needed" math remediation sessions on Friday evenings, these sessions will also be recorded. 3. This lecture schedule is tentative and is subject to change (as a legal disclaimer). (Page 5 of 24) Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016 Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Required University Statements — Required by Texas A&M University Fall 2016 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." (Page 6 of 24) Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016 Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Assignment Coversheet — Required by University Policy Fall 2016
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Petroleum Engineering Number — Course Title Assignment Number— Assignment Title Assignment Date — Due Date 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: "On my honor, as an Aggie, I have neither given nor received unauthorized aid on this academic work."
______(Print your name)
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______(Page 7 of 24) Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016 Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Appendix — Extended Description of Course Objectives Fall 2016 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): 1 f '(x) ( f (x 2x) 8 f (x x) 8 f (x x) f (x 2x)) (x)4 12x Second Derivative, f''(x): 1 f ''(x) ( f (x 2x) 16 f (x x) 30 f (x) 16 f (x x) f (x 2x)) (x)4 12(x)2 Third Derivative, f'''(x): 1 f '''(x) ( f (x 3x) 8 f (x 2x) 13 f (x x) 13 f (x x) 8 f (x 2x) 8(x)3 f (x 3x)) (x)4 Fourth Derivative, fiv(x): 1 f iv (x) ( f (x 3x) 12 f (x 2x) 39 f (x x) 6(x)4 56 f (x) 39 f (x x) 12 f (x 2x) f (x 3x)) (x)4 (Page 8 of 24) Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016 Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Appendix — Extended Description of Course Objectives Fall 2016
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): xn n1 x (x)2 I(x) f (x)dx [ f (x ) f (x )] x f (x ) [ f '(x ) f '(x )] 2 0 n i 12 n 0 x0 i1 x x where x n 0 n — Simpson's rule: (with correction) (be able to develop — see Hornbeck): xn n1 n2 4 x (x) iv I(x) f (x)dx [ f (x0) f (xn) 4 f (xi ) 2 f (xi )] (xn x0) f (x) 3 180 x0 i1 i2 iodd ieven x x x x where n must be even. Also x n 0 and x n 0 . n 2 — Gaussian quadrature: (weights and abscissas from Abramowitz and Stegun: Handbook of Mathematical Functions, Table 25.4, pgs. 916-919): xn n xn x0 xn x0 xn x0 f (x) dx wi f (zi ) where zi ( )xi ( ) 2 2 2 x0 i1 — Laguerre quadrature: (weights and abscissas from Abramowitz and Stegun: Handbook of Mathematical Functions, Table 25.9, pgs. 923): n n x x e f (x) dx wi f (xi ) or g(x) dx wi e ig(xi ) 0 i1 0 i1 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. (Page 9 of 24) Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016 Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Appendix — Extended Description of Course Objectives Fall 2016
Course Objectives (Continued) Module 1: Advanced Mathematics Relevant to Problems in Engineering (continued) 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: dy 1. Given a0 a1 y r(t) , we must rearrange to yield the following form: dt dy 1 [r(t) a1 y] dt a0
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: d 2 y dy 1. Given a0 a1 a2 y r(t) , we must rearrange to yield the following form: dt 2 dt d 2 y 1 dy d 2 y 1 dy [r(t) a1 a2 y] or [r(t) a1v a2 y] , where v dt 2 a0 dt dt 2 a0 dt 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 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: st 1 x x f (s) L( f (t)) e f (t)dt or e f ( )dx (using x=st) s s 0 0 Definition of the Inverse Laplace Transform: (Mellin Inversion Integral) yi 1 f (t) L1( f (s)) est f (s)ds 2i yi — 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 The unit step function is given by: u(t a) 0 t a ) a - t ( 0 u(t a) 1 t a u And its Laplace transform is: - 1 1 f (u) eas a s t (Page 10 of 24) Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016 Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Appendix — Extended Description of Course Objectives Fall 2016
Course Objectives (Continued) Module 1: Advanced Mathematics Relevant to Problems in Engineering (continued) — Be able to develop and apply the Laplace transform formulas for the discrete data functions shown below. + Step Data Function: n 1 sti1 f (s) ( fi fi 1)e where (t0=0 and f0=0) s i1 + Piecewise Linear Data Function: (Roumboutsos and Stewart Method) n1 1 1 1 f (u) m (1 est1 ) m (esti1 esti ) m estn1 2 1 2 i 2 n s s i2 s
where the slope terms (mi's) are taken as backward differences given by fi fi1 mi ti ti1 + Piecewise Log-Linear Data Function: (Blasingame Method) a1 a2 a2 f (s) (v1, st1) (v2, st2) (v2, st1) sv1 sv2 sv2 an1 an1 an an ... (vn1, stn1) (vn1, stn2) (vn) (vn, stn1) svn1 svn1 svn svn 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 f (s) . + Laplace Transform of a Generic Time Dependent Derivative: d n L( f (t)) s n f (s) s n1 f (t 0) s n2 f '(t 0) ... sf n2 (t 0) f n1(t 0) dt n where n2 n1 c0 f (t 0), c1 f '(t 0), c2 f ''(t 0)...cn2 f (t 0), cn1 f (t 0) — 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 1 x f (s) est f (t)dt w f ( k ) s k s 0 k 1 (Page 11 of 24) Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016 Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Appendix — Extended Description of Course Objectives Fall 2016
Course Objectives (Continued) Module 1: Advanced Mathematics Relevant to Problems in Engineering (continued) — 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 st 1 xk f (s) e f (t)dt wk f ( ) s s 0 k 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 n ln(2) (2n)! (1)k ln(2) f (t) f ( [n k]) Gaver t (n 1)! (n k)!k! t k 0 The Gaver-Stehfest formula for numerical Laplace transform inversion is n ln(2) ln(2) f (t) V f ( i) GaverStehfest t i t i1 and the Stehfest extrapolation coefficients are given n n Min(i, ) n i 2 k (2k)! V (1) 2 2 i n i1 ( k)!k!(k 1)!(i k)!(2k i)! k [ ] 2 2 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) 2 2 d y dy 2 2 z z (z n )y 0 and has the solution y c1Jn(z) c2Yn(z) dz2 dz
+ 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) 2 2 d y dy 2 2 z z (z n )y 0 and has the solution y c1In(z) c2Kn(z) dz2 dz Be able to use the Bessel functions in numerical problem solving efforts and theoretical developments; especially recurrence relations, integral definitions, and Laplace transforms. (Page 12 of 24) Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016 Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Appendix — Extended Description of Course Objectives Fall 2016
Course Objectives (Continued) 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). q 1 p 2 v k r avg p where k p is considered to be a "geometry" factor. Ax x 8 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. 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). Be able to recognize, develop, and apply the Taylor series (finite-difference) formulas for numerical differentiation of an analytic function. 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: 1 pc 2 ow cos( ) r 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 k *3 ow dSw* 2 n p2 0 c 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. (Page 13 of 24) Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016 Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Appendix — Extended Description of Course Objectives Fall 2016
Course Objectives (Continued) 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: 2 1 3 ow 1 k 10.66 * dSw* where * (1 Swi ) 2 n p2 0 c 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: 3 2 1 3 2 1 k * ow [ ] k 10.66 * ow [ ] n 2 2 or n 2 2 (field units) pd pd 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 pc 1 Sw * pD (1 SwD ) where pD and SwD 1 Sw pd 1 Swi 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 Sw* 1 1 1 dSw * dSw * 2 p2 pc S * c k (S *)2 0 and k (1 S *)2 w rw w 1 rn w 1 1 1 dSw * dSw * p2 p2 0 c 0 c 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: o (32/ ) o 2 (12/ ) krw krwSw * and krn krn (1 Sw*) [1 Sw * ] 1 where the Brooks and Corey capillary pressure model is given by pc pd Sw * (Page 14 of 24) Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016 Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Appendix — Extended Description of Course Objectives Fall 2016
Course Objectives (Continued) Module 2: Petrophysical Properties (continued) o o and krw and krn 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: (32 / ) krwD (1 SwD ) — Dimensionless non-wetting phase relative permeability: 2 (12 / ) krnD SwD [1 (1 SwD ) ] — Dimensionless relative permeability ratio function: 2 krnD SwD (12/ ) [(1 SwD ) 1] k 2 rwD (1 SwD ) — Dimensionless saturation functions: 1 Sw * * Sw Swi SwD 1 Sw and Sw 1 SwD 1 Swi 1 Swi 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. (Page 15 of 24) Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016 Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Appendix — Extended Description of Course Objectives Fall 2016
Course Objectives (Continued) 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. c p c(p)2 2 p t k t — "Small and Constant Compressibility" Case: Base relation for all developments in reservoir engineering and well testing. c p 2 p t k t 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
2 ct p p 2 p p p p p p (ct )n k t k ta where the "pseudopressure" function, pp, is given by: p p B k 1 p ( ) dp or p (B) dp p k n B p n B pbase pbase and the "pseudotime" function, ta, is given by: t 1 ta (ct )n dt ( p)ct ( p) 0 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: k ko g [ p ] ( ) [ p ] ( ) oBo t Bo g Bg t Bg 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: k () c p [ p] [ p] t t z k z t Be able to develop the diffusivity equation for single-phase gas flow in terms of the following: pseudopressure, pressure-squared, and pressure. (Page 16 of 24) Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016 Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Appendix — Extended Description of Course Objectives Fall 2016
Course Objectives (Continued) Module 3: Fundamentals of Flow in Porous Media (continued) — "Pseudopressure" Formulation: p 2 ct p pg z p p pg where p pg ( )n dp k t p z pbase — "Pressure-Squared" Formulation: 2 2 2 2 ct 2 c ( p ) [ln( z)]( p ) ( p ) if z constant then 2 ( p2 ) t ( p2) p2 k t k t — "Pressure" Formulation: 2 z 2 c p p c p p [ln( )](p) t if constant then 2 p t p p k t z k t 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: (ovo) (o) (wvw) (w) t t Gas Continuity Equation: vo vw ( g vg )tot [ pg vg Rso gsc Rsw gsc ] [( g )tot ] Bo Bw t Be able to write Darcy's law velocity relations for each phase. The general form is given by: ki vi pi where i = oil, gas, and water. i 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: ko kw ovo osc po wvw wsc pw oBo wBw Gas Flux Equation:
kg ko kw (gvg )tot gsc[ pg Rso po Rsw pw] g Bg oBo wBw 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: So Sw (o ) oSo osc (w) wSw wsc Bo Bw (Page 17 of 24) Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016 Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Appendix — Extended Description of Course Objectives Fall 2016
Course Objectives (Continued) Module 3: Fundamentals of Flow in Porous Media (continued) Gas Mass Equation:
Rso Rsw Sg So Sw (g )tot g Sg So gsc Sw gsc gsc[ Rso Rsw ] Bo Bw Bg Bo Bw
Assuming no capillary pressure forces (p po pg pw) , be able to develop the generalized diffusivity relations for each phase. (Martin Eqs. 1-3) "Oil" Equation: "Water" Equation: k S k S [ o p] ( o ) [ w p] ( w ) oBo t Bo wBw t Bw "Gas" Equation:
kg ko kw Sg So Sw [( Rso Rsw )p] [( Rso Rsw )] g Bg oBo wBw t Bg Bo Bw
2 NEGLECTING the Sop,Swp, and pp p terms — be able to develop the diffusivity relations for each phase as shown by Martin (Eqs. 7-9) "Oil" Equation: "Water" Equation: k S k S o 2 p ( o ) w 2 p ( w ) o Bo t Bo wBw t Bw "Gas" Equation:
kg ko kw 2 Sg So Sw ( Rso Rsw ) p [ ( Rso Rsw )] g Bg oBo wBw t Bg Bo Bw 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: dSo So dBo o dSw Sw dBw w ct ct dp Bo dp t dp Bw dp t Total Compressibility:
So dBo So Bg dRso Sw dBw SwBg dRsw S g dBg ct Bo dp Bo dp Bw dp Bw dp Bg dp or,
1 dBo Bg dRso 1 dBw Bg dRsw 1 dBg ct [ ]So [ ]Sw [ ]S g Bo dp Bo dp Bw dp Bw dp Bg dp or finally, ct coSo cwSw cg S g where, (Page 18 of 24) Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016 Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Appendix — Extended Description of Course Objectives Fall 2016
Course Objectives (Continued) Module 4: Reservoir Flow Solutions dB 1 dBo Bg dRso 1 dBw Bg dRsw 1 g co ,cw , and cg Bo dp Bo dp Bw dp Bw dp Bg dp Total Pressure Equation: k 2 ct ko g kw p where t t t o g w 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: 2 pD 1 pD pD 2 r r t rD D D D — Dimensionless Initial and Boundary Conditions: + Dimensionless Initial Condition pD (rD ,tD 0) 0 (uniform pressure in reservoir) + Dimensionless Inner Boundary Condition p [r D ] 1 D rD 1 (constant rate at the well) rD + Dimensionless Outer Boundary Conditions a. "Infinite-Acting" Reservoir pD (rD , tD ) 0 b. "No-Flow" Boundary p [r D ] 0 D rD reD (No flux across the reservoir boundary) rD c. Constant Pressure Boundary pD(reD,tD ) 0 (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: 2 1 pD pD 1 pD pD 1 d dpD [rD ] [rD ] upD r r r 2 r r t r dr dr D D D rD D D D D D D (Page 19 of 24) Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016 Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Appendix — Extended Description of Course Objectives Fall 2016
Course Objectives (Continued) Module 4: Reservoir Flow Solutions (continued) General Solution:
pD (rD,u) Al0( urD ) BK0( urD ) Derivative of the General Solution: dpD A ul1( urD ) B uK1( urD) drD 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 K0( urD ) 1 pD (rD,u) K0( urD ) u uK1( u ) u Constant Rate-Constant Pressure Relation: (from van Everdingen and Hurst) 1 1 q (u) D 2 u pD(u) 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: 2 d pD 1 pD [1 ] 0 (infinite-acting reservoir case only) 2 D d D D "Log Approximation" Solution for the Diffusivity Equation: 1 1 4 1 1 p (r ,u) K ( ur ) ln[ ] D D u 0 D 2u 2 u (=0.577216…Euler's constant) e rD 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 pD (rD,tD 0) 0 (uniform pressure in reservoir) + Dimensionless Inner Boundary Condition p [r D ] 1 D rD 1 (constant rate at the well) rD + Dimensionless Outer Boundary Conditions a. Prescribed Flux at the Boundary p [r D ] q (t ) D rD reD Dext D rD (Page 20 of 24) Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016 Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Appendix — Extended Description of Course Objectives Fall 2016
Course Objectives (Continued) Module 4: Reservoir Flow Solutions (continued) b. Constant Pressure at the Boundary pD(rD reD,tD ) 0 (No flux across the reservoir boundary) — Particular Solutions in the Laplace Domain: + "Infinite-acting" reservoir behavior
1 K0( urD ) pD (rD,u) u uK1( u ) Or the line source approximation 1 pD (rD,u) K0 ( urD ) (where uK ( u ) 1 , for u 0 ) u 1 + Bounded circular reservoir — "no-flow" at the outer boundary (i.e., qDext (tD ) 0 )
1 K0 ( urD )l1( ureD ) K1( ureD )l0( urD ) pD (rD,u) (constant rate at the well) u uK1( u )l1( ureD ) ul1( u )K1( ureD ) + Bounded circular reservoir — "constant-pressure" at the outer boundary
1 K0( urD )l0( ureD) K0( ureD )l0( urD ) pD (rD,u) (constant rate at the well) u uK1( u )l0( ureD ) ul1( u )K0( ureD) + Bounded circular reservoir — "prescribed flux" at the outer boundary
1 K0( urD )l1( ureD ) K1( ureD)l0( urD) pD (rD,u) u uK1( u )l1( ureD) ul1( u )K1( ureD ) 1 u K0( urD ) ul1( u ) l0( urD ) uK1( u ) qDext (u)[ ] u ureD uK1( u )l1( ureD ) ul1( u )K1( ureD ) 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) 2 1 rD pD (tD ,rD ) E1( ) 2 4tD or the so-called "log approximation" 1 4 tD pD (tD,rD ) ln( ) 2 2 e rD — Bounded circular reservoir — "no-flow" at the outer boundary 2 2 2 2 2 1 r 1 r 2t r r 1 r p (t , r ) E ( D ) E ( eD ) D exp( eD ) ( D )exp( eD ) D D D 2 1 4t 2 1 4t 2 4t 2 4 4t D D reD D 2reD D
and its "well testing" derivative function, pD'=d/dtD[pD(rD,tD)] is given by (Page 21 of 24) Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016 Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Appendix — Extended Description of Course Objectives Fall 2016
Course Objectives (Continued) Module 4: Reservoir Flow Solutions (continued) 2 2 2 2 2 1 r 2t r 1 r r r p' (t , r ) exp( D ) D exp( eD ) ( D eD )exp( eD ) D D D 2 4t 2 4t 2t 4 8 4t D reD D D D — Bounded circular reservoir — "constant pressure" at the outer boundary 2 2 2 1 r 1 r 1 r D eD 2 2 eD pD (tD, rD ) E1( ) E1( ) (reD rD ) exp( ) 2 4tD 2 4tD 8tD 4tD
and its "well testing" derivative function, pD'=d/dtD[pD(rD,tD)] is given by 2 2 2 2 1 rD 1 reD 1 2 2 reD reD p'D (tD, rD) exp( ) exp( ) (reD rD )( 1) exp( ) 2 4tD 2 4tD 8tD 4tD 4tD 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) 1 1 p (| xD | 1, yD 0,u) p [(xD x'wD ),u]dx'wD D,cfracs 2 D,cls 1 where the cls subscript means Continuous Line Source — "Infinite-acting" reservoir behavior (line source solution) u (1xD ) u (1xD ) 1 1 pD,cfracs,inf (| xD |1, yD 0,u) [ K0 (z)dz K0 (z)dz] 2u u 0 0 — Bounded circular reservoir — "no-flow" at the outer boundary
pD,cfracs,nfb (| xD |1, yD 0,u) pD,cfracs,inf (| xD |1, yD 0,u) u (1xD ) u (1 xD ) 1 1 K1( ureD ) [ I0(z)dz I0 (z)dz] 2u u I ( ur ) 1 eD 0 0 — Bounded circular reservoir — "constant pressure" at the outer boundary
pD,cfracs,cpb(| xD |1, yD 0,u) pD,cfracs,inf (| xD |1, yD 0,u) u (1xD ) u (1 xD ) 1 1 K0( ureD ) [ I0(z)dz I0(z)dz] 2u u I ( ur ) 0 eD 0 0 (Page 22 of 24) Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016 Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Appendix — Extended Description of Course Objectives Fall 2016
Course Objectives (Continued) Module 4: Reservoir Flow Solutions (continued) 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": (1 )u f (u) (1 )u Solutions in the Laplace domain:
1 K0( uf (u)rD ) 1 1 4 1 1 pD (rD,u) K0( uf (u)rD ) ln( ) u uf (u)K ( uf (u)) u 2u 2 2 uf (u) 1 e rD — Line source solution in the real domain: 1 4 t 1 1 p (t , r ) ln( D ) E ( t ) E ( t ) S D D D 2 2 2 1 (1 ) D 2 1 (1 ) D e rD Be able to develop the Laplace and real domain results given by Warren and Root for pseudosteady-state matrix flow. These relations are 1 1 1 p'D (tD, rD) exp( tD ) exp( tD) 2 2 (1 ) 2 (1 ) 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. Where we assume that the β(y) function can be re-cast as a t unique function of time (i.e., β(y) can be written as β(t)). y y 2 y (y) g(t )d Using β(t) requires assumptions as to flow regimes--we t will demonstrate this assuming pseudosteady-state flow. 0 Taking the Laplace transform of this relation gives 2 y(u) [u y(u) y(t 0)]g(u) 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: 2 ppD 1 ppD ct ppD ppD ct (t ) [ (tD ) ] 2 r r c t D t c rD D D i ti D D i ti where k r 1 kh t 0.0002637 t ppD ( p pi pp ) D 2 rD 141.2 qB ictirw rw (Page 23 of 24) Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016 Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Appendix — Extended Description of Course Objectives Fall 2016
Course Objectives (Continued) Module 4: Reservoir Flow Solutions (continued) and the pseudopressure function is given by: p p 1 i zi p p p i Bgi dp dp Bg pi z pbase pbase — Substituting the convolution formulation into the right-hand-side of the real gas diffusivity equation gives 2 tD 1 p pD p pD 1 p pD p pD [r ] g(t )d D 2 D rD rD rD r rD rD D 0 2 1 d dp pD (u) d p pD (u) 1 dp pD (u) [rD ] ug(u) p pD (u) (Laplace domain relation) r dr dr 2 r dr D D D drD D D
Be familiar with and be able to develop the g(u) term. The g(tD) term is defined by: tD p pD p pD (tD ) g(tD )d tD 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: n pwD (tD ) (qDj qDj1) psD,cr (tD tDj1) (discrete rate changes) j1 tD pwD (tD ) q'D ( ) psD,cr (tD )d (continuous rate changes) 0 — Variable-Pressure Drop Case: n ( pi pwf , j ) qtD (tD ) qDcp (tD tDj1) (discrete rate changes) ( p p ) j1 i r Be able to derive the general convolution identity in the Laplace domain from the integral form of the variable-rate convolution identity. pwD (u) qqD (u) psD,cr (u) 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: 1 1 qD,cp (u) u2 psD,cr (u) — Real domain result: tD tD qD,cp ( ) psD,cr (tD )d tD or psD,cr ( )qD,cp (tD )d tD 0 0 (Page 24 of 24) Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs Syllabus and Administrative Procedures — Fall 2016 Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Appendix — Extended Description of Course Objectives Fall 2016
Course Objectives (Continued) Module 4: Reservoir Flow Solutions (continued) 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: dpwf dptf (qsf q )B 24Cs[ ] dt dt — Pressure Relations (for small times/wellbore storage domination): qB pwf pi t (for small times, i.e., wellbore storage domination) 24Cs or tD pwD (for small times, i.e., wellbore storage domination) CD — Laplace Domain Identity: 1 pwD (u) 1 2 (valid for all times) u CD psD (u)
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.