Wave Propagation in Drilling, Well Logging and Reservoir Applications

458 Pages·2014·4.672 MB·English

by  Wilson C. Chin

Wave Propagation in Drilling, Well
Logging and Reservoir
Applications
Scrivener Publishing
100 Cummings Center, Suite 541J
Beverly, MA 01915-6106
Publishers at Scrivener
Martin Scrivener ([email protected])
Phillip Carmical ([email protected])
Wave Propagation in
Drilling, Well Logging
and Reservoir
Applications
Wilson C. Chin, Ph.D., M.I.T.
Stratamagnetic Soft ware, LLC,
Houston, Texas
Copyright © 2014 by Scrivener Publishing LLC. All rights reserved.
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Cover design by Kris Hackerott
Library of Congr ess Cataloging-in-Publication Data:
ISBN 978-1-118-92589-8
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
To Jamie,
And to those who enjoy a good mystery or an exasperating puzzle, duti-
fully examine our photograph. Carefully peruse the bow, the way it’s held
and executed. “What eff ect does its position and velocity have on the sound
created? How do waves that refl ect from the ends behave at the contact
itself and how do they travel aft erwards? What changes when the environ-
ment changes?”

Contents
Preface xxi
Acknowledgements xxiii
1 Overview and Fundamental Ideas 1
1.1 Th e Classical Wave Equation 2
1.1.1 Fundamental properties 2
1.1.2 Refl ection properties 5
1.1.2.1 Example 1-1. Rigid end termination 5
1.1.2.2 Example 1-2. Stress-free end 6
1.1.2.3 Note on acoustics 6
1.2 Fundamental Representation 7
1.2.1 Taylor series 7
1.2.2 Fourier series 7
1.3 Separation of Variables and Eigenfunction Expansions 8
1.3.1 Example 1-3. String with pinned ends and general
initial conditions 9
1.3.2 Example 1-4. String with distributed forces 10
1.3.3 Example 1-5. Alternative boundary conditions 11
1.3.4 Example 1-6. Mixed boundary conditions 11
1.3.5 Example 1-7. Problems without initial conditions 13
1.3.5.1 Example 1-7a. Naive approach 13
1.3.5.2 Example 1-7b. Correct approach 14
1.3.5.3 Example 1-7c. Faster approach 14
1.3.6 Example 1-8. Dissipative wave solution 14
vii
viii Contents
1.4 Standing Versus Propagating Waves 16
1.4.1 Standing waves 16
1.4.2 Propagating waves 16
1.4.3 Combined standing and propagating waves 17
1.4.4 Characterizing propagating waves 17
1.5 Laplace Transforms 20
1.5.1 Wave equation derivation 20
1.5.2 Example 1-9. String falling under its own weight 21
1.5.3 Example 1-10. Semi-infi nite string with a general
end support 22
1.5.3.1 Example 1-10a. Rectangular pulse 25
1.5.3.2 Example 1-10b. Impulse response 25
1.5.3.3 Example 1-10c. Incident sinusoidal
wavetrain 26
1.6 Fourier Transforms 26
1.6.1 Example 1-11. Propagation of an initially static
disturbance 27
1.6.2 Example 1-12. Directional properties, special wave 28
1.7 External Forces Versus Boundary Conditions 30
1.7.1 Single point force 30
1.7.2 Properties of point loads 32
1.7.2.1 Example 1-13. Boundary conditions versus
forces 32
1.7.2.2 Couples or dipoles 33
1.7.2.3 Multiple forces and higher order moments 36
1.7.2.4 Symmetries and anti-symmetries 36
1.7.2.5 Impulse response 36
1.7.2.6 On the subtle meaning of impulse 39
1.7.2.7 Example 1-14. Incorrect use of impulse
response 39
1.7.2.8 Additional models 39
1.7.2.9 Other delta function properties 40
1.8 Point Force and Dipole Wave Excitation 42
1.8.1 Example 1-15. Finite string excited by a time-varying
concentrated point force 42
1.8.2 Example 1-16. Finite string excited by a time-varying
point dipole (i.e., a force couple) 44
1.8.3 Example 1-17. Splitting of an applied initial
disturbance 45