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Petroleum Engineering 620 Fluid Flow in Petroleum Reservoirs

Syllabus and Administrative Procedures — Fall 2011

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

Fall 2011

Petroleum Engineering 620 Instructor: Dr. Tom Blasingame Texas A&M University Office: Richardson 821/TBA College of Engineering TL: +1.979.845.2292 TR 19:00-21:00 RICH 313 (for in-class lectures) EM: [email protected]

Required Texts/Resources: (*Book must be purchased. #Out of Print/Public Domain — Electronic file to be made available by instructor.)

*1. Advanced Mathematics for Engineers and Scientists, M.R. Spiegel, Schaum's Series (1971). *2. Conduction of Heat in Solids, 2nd edition, H. Carslaw and J. Jaeger, Oxford Science Publications (1959). #3. Handbook of Mathematical Functions, M. Abramowitz and I. Stegun, Dover Pub. (1972). #4. Table of Laplace Transforms, G.E. Roberts and H. Kaufman, W.B. Saunder, Co. (1964). #5. Numerical Methods, R.W. Hornbeck, Quantum Publishers, Inc., New York (1975). #6. Approximations for Digital Computers: Hastings, C., Jr., et al, Princeton U. Press, Princeton, New Jersey (1955). #7. Handbook for Computing Elementary Functions: L.A. Lyusternik, et al, Pergamon Press, (1965).

Optional Texts/Resources: (+Special order at MSC Bookstore or check TAMU library. #Local bookstores)

#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_11C/

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]

Homework Assignments ............................................................................................................................................................... 90 percent Class Participation (subjective, based on opinion of the instructor) .......................................................................................... 10 percent Total = 100 percent

Policies and Procedures:

1. Students are expected to keep pace in the course.

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.

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.



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Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs

Course Outline/Topics Fall 2011

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 2011

Month Date Day Method Topic Lecture Aug 30 T Class Review of Functions P620_11C_Lec_01_Mod1_ML_01_Rev_of_Fcns.pdf

Sep 01 R Class Approximation of Functions P620_11C_Lec_02_Mod1_ML_02_Fcn_Approx.pdf

Sep 06 T Vid/NZ 1st Order Ordinary Differential Equations P620_11C_Lec_03_Mod1_ML_03_1st_Order_ode.pdf

Sep 08 R Vid/NZ 2nd Order Ordinary Differential Equations P620_11C_Lec_04_Mod1_ML_04_2nd_Order_ode.pdf

Sep 13 T Class The Laplace Transform P620_11C_Lec_05_Mod1_ML_05_LaplaceTrans.pdf

Sep 15 R Class Introduction to Special Functions P620_11C_Lec_06_Mod1_ML_06_SpecialFcns.pdf

Sep 20 T Vid/NZ Porosity and Permeability Concepts P620_11C_Lec_07_Mod2_PtrPhy_01_PorPerm.pdf

Sep 22 R Vid/NZ Correlation of Petrophysical Data P620_11C_Lec_08_Mod2_PtrPhy_02_DataCorel.pdf

Sep 27 T Vid/Trvl Development of Permeability/Darcy's Law P620_11C_Lec_09_Mod2_PtrPhy_03_Perm_Dev.pdf

Sep 29 R Vid/Trvl Capillary Pressure P620_11C_Lec_10_Mod2_PtrPhy_04_Cap_Pres.pdf

Oct 04 T Vid/SPE Relative Permeability P620_11C_Lec_11_Mod2_PtrPhy_05_Rel_Perm.pdf

Oct 06 R Vid/SPE Electrical Properties of Reservoir Rocks P620_11C_Lec_12_Mod2_PtrPhy_06_Elec_Prop.pdf

Oct 11 T Vid/NZ Single-Phase, Steady-State Flow P620_11C_Lec_13_Mod3_FunFld_01_SSDarcy.pdf

Oct 13 R Vid/NZ Non-Laminar Flow in Porous Media P620_11C_Lec_14_Mod3_FunFld_02_SSNonDarcy.pdf

Oct 18 T Class Material Balance Concepts P620_11C_Lec_15_Mod3_FunFld_03_MatBal.pdf

Oct 20 R Class Pseudosteady-State Flow (Circular Res.) P620_11C_Lec_16_Mod3_FunFld_04_PSS_Flow.pdf

Oct 25 T Vid/NZ Liquid Flow Diffusivity Equation P620_11C_Lec_17_Mod3_FunFld_05_DifEq_Liq.pdf

Oct 27 R Class Gas Flow Diffusivity Equation P620_11C_Lec_18_Mod3_FunFld_06_DifEq_Gas.pdf

Nov 01 T Vid/SPE Multiphase Flow Diffusivity Equation P620_11C_Lec_19_Mod3_FunFld_07_DifEq_MlPhs.pdf

Nov 03 R Vid/SPE Dimensionless Variables/Radial Flow P620_11C_Lec_20_Mod4_ResFlw_01_DimLssVar.pdf

Nov 08 T Vid/NZ Solutions — Radial Flow Diffusivity Eq. P620_11C_Lec_21_Mod4_ResFlw_02_RadFlwSln.pdf

Nov 10 R Vid/NZ Solutions — Radial Flow Diffusivity Eq. P620_11C_Lec_21_Mod4_ResFlw_02_RadFlwSln.pdf

Nov 15 T Class Solutions — Linear Flow Diffusivity Eq. P620_11C_Lec_22_Mod4_ResFlw_03_LinFlwSln.pdf

Nov 17 R Class Solutions — Fractured Well (High FcD) P620_11C_Lec_23_Mod4_ResFlw_04_FracWellSln.pdf

Nov 22 T Vid/NZ Solutions — Dual Porosity Reservoirs P620_11C_Lec_24_Mod4_ResFlw_05_NatFrcResSln.pdf

Nov 24 R — Thanksgiving Holiday (no class) —

Nov 29 T Class Direct Solution — Gas Diffusivity Equation P620_11C_Lec_25_Mod4_ResFlw_06_DrtSlnGas.pdf

Dec 01 R Class Convolution P620_11C_Lec_26_Mod4_ResFlw_07_Convolution.pdf

Dec 06 T Class Wellbore Storage P620_11C_Lec_27_Mod4_ResFlw_08_WellboreStrg.pdf

Dec 12 T — Any/all remaining assignments due.

(http://registrar.tamu.edu/general/calendar.aspx) —

Dec 15 R — Final grades for all students GRADUATING in the Fall 2011 term.

—

Dec 19 M — Final grades for all students Fall 2011 term. — Notes:

1. Class = Lecture in classroom (RICH 313) 2. Vid/NZ = Lecture video from home in New Zealand (should be easily available by e-mail ([email protected])). 3. Vid/SPE = Lecture video during 2 SPE conferences (likely to have limited e-mail availability during those days). 4. Vid/Trvl = Lecture video during travel days (limited or no e-mail availability during those specific days).



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

Fall 2011

Americans with Disabilities Act (ADA) Statement:

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 the Department of Student Life, Services for Students with Disabilities in Room B118 of Cain Hall, or call 845-1637.

Aggie Honor Code: (http://www.tamu.edu/aggiehonor/)

"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."



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Petroleum Engineering 620 — Fluid Flow in Petroleum Reservoirs Assignment Coversheet — Required by University Policy

Fall 2011 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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) _______________________________ (Your signature)

____________________________________________________________________________________________________________________________________________________________________________________________________________________________________



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

Fall 2011

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



(Page 7 of 23)


Fall 2011 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.



(Page 8 of 23)




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

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

)(



(Page 9 of 23)




— 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

)()(



(Page 10 of 23)




— 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.



(Page 11 of 23)



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.

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:

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.



(Page 12 of 23)



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



(Page 13 of 23)



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.



(Page 14 of 23)



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.



(Page 15 of 23)




— "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 )(



(Page 16 of 23)




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,



(Page 17 of 23)



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



(Page 18 of 23)



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



(Page 19 of 23)




b. Constant Pressure at the Boundary

0),( DeDDD trrp (No flux across the reservoir boundary)

— 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



(Page 20 of 23)




)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



(Page 21 of 23)




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

and the pseudopressure function is given by



(Page 22 of 23)




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

,



(Page 23 of 23)




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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