<p> Petroleum Engineering 620 — Fluid Flow in Reservoirs Homework 6 — The Laplace Transform Summer/Fall 2000 You are to work the following problems in complete detail—you must show all steps. Laplace Transform Theorems:  Solved Problems in Chapter 4 (Laplace Transforms) of the Schaum's Outline text, Advanced Mathematics for Engineers and Scientists, by Spiegel.  Application of Theorems: 4.27,4.28,4.30,4.33,4.34,4.36  Solved Problems in the Schaum's Outline text, Differential Equations, (2nd edition, by Bronson).  Chapter 14: The Laplace Transform: 14.5,14.6,14.7,14.8,14.9,14.21,14.22, 14.23,14.24  Chapter 15: Inverse Laplace Transform:15.8,15.9,15.10,15.17,15.18 Solution of Differential Equations Using Laplace Transforms:  Solved Problems in the Schaum's Outline text, Differential Equations, (2nd edition, by Bronson).  Chapter 17: Sol. by Laplace Transform: 17.1,17.2,17.3,17.4,17.5,17.9 Numerical Inversion of Laplace Transforms: Problems using the Gaver-Stehfest Algorithm: You are to solve the following cases taken from Solved Problems in Chapter 4 of the Schaum's Outline text, Advanced Mathematics for Engineers and Scientists, by Spiegel.  Problem 4.12b for 0.01<t<10 (use n=12 for the Stehfest algorithm)  Problem 4.13a for 0.01<t<10 (use n=12 for the Stehfest algorithm)  Problem 4.13b for 0.01<t<10 (use n=12 for the Stehfest algorithm) You are to use the Gaver-Stehfest algorithm to calculate the required results, and you are to then compare the exact and computed results by plotting these results on Car- tesian, semilog, and log-log scales (scale each graph appropriately to illustrate the relevant behavior for a particular function). The Gaver-Stehfest formula for numerical Laplace transform inversion is given by: ln(2) n ln(2) Gaverf -Stehfest(n,t) = ∑ Vi f i t i=1 t And the Stehfest extrapolation coefficients are given as: Min i,n 2 n n +i k 2 (2k)! Vi = (-1)2 ∑ n-k !k!(k -1)!(i-k)!(2k-i)! k= i+1 2 2 You are free to use any programming device you wish—most students will probably prefer to use MS Excel to compute the numerical inversion, and the easiest mechanism for Excel is to have the coefficients already computed. The Gaver- Stehfest coeffi-cients are given in a table attached to this assignment. A sample 2 Fortran program and data file are also provided for those students who wish to use Fortran. The data and software (sample Fortran program and data file) for this assignment are located at: http://pumpjack.tamu.edu/~t-blasingame/ 3</p><p>Petroleum Engineering 620 — Fluid Flow in Reservoirs Homework 6 — The Laplace Transform Summer/Fall 2000 Numerical Inversion of Laplace Transforms: (continued) Paper Reviews:  You are to provide a critical and detailed review (at least 1 page) for the following paper(s): (these papers are included in the course notes)  Cost, T.L.: "Approximate Laplace Transform Inversion in Viscoelastic Stress Analysis," AIAA Journal, (Dec. 1964), 2157-2166.  Davies, B. and Martin, B.: "Numerical Inversion of the Laplace Transform: a Survey and Comparison of Methods," Journal of Computational Physics, 33 (1979), 1-32.  Gaver, D.P., Jr.: "Observing Stochastic Processes and Approximate Transform Inversion," Operations Research, vol. 14, No. 3 (1966), 444-459.  Stehfest, H.: "Numerical Inversion of Laplace Transforms," Communications of the ACM (January 1970), 13, No. 1, 47-49. (Algorithm 368 with correction (October 1970), 13, No. 10) For each paper you are to address the following questions: (Type or write neatly)  Problem:  What is/are the problem(s) solved?  What are the underlying physical principles used in the solution(s)?  Assumptions and Limitations:  What are the assumptions and limitations of the solutions/results?  How serious are these assumptions and limitations?  Practical Applications:  What are the practical applications of the solutions/results?  If there are no obvious "practical" applications, then how could the solutions/ results be used in practice?  Discussion:  Discuss the author(s)'s view of the solutions/results.  Discuss your own view of the solutions/results.  Recommendations/Extensions:  How could the solutions/results be extended or improved?  Are there applications other than those given by the author(s) where the solution(s) or the concepts used in the solution(s) could be applied? Petroleum Engineering 620 — Fluid Flow in Reservoirs Homework 6 — The Laplace Transform Summer/Fall 2000</p><p>Coefficients for the Gaver-Stehfest Algorithm V(I,2) V(I,4) V(I,6) V(I,8) V(I,10) V(I,12) V(I,14) V(I,16) V(I,18) V(1,J) 2 -2 1 -3.333333E-01 8.333333E-02 -1.6666666E-02 2.77777777777E-03 -3.968253968253968E- 4.960317460317460E-05 04 V(2,J) -2 26 -49 4.833333E+01 -3.208333E+01 1.6016666E+01 -6.40277777777E+00 2.133730158730159E+00 -6.095734126984128E- 01 V(3,J) --- -48 366 -9.060000E+02 1.279000E+03 -1.2470000E+03 9.24050000000E+02 - 2.745940476190476E+02 5.510166666666667E+02 V(4,J) --- 24 -858 5.464666E+03 -1.562366E+04 2.7554333E+04 -3.45979277777E+04 3.350016111111111E+04 - 2.630695674603174E+04 V(5,J) ------810 -1.437666E+04 8.424416E+04 -2.6328083E+05 5.40321111111E+05 - 9.572572013888889E+05 8.126651111111111E+05 V(6,J) ------270 1.873000E+04 -2.369575E+05 1.3241387E+06 -4.39834636666E+06 1.007618376666667E+07 - 1.735869484583333E+07 V(7,J) ------1.194666E+04 3.759116E+05 -3.8917055E+06 2.10875917777E+07 - 1.824212226472222E+08 7.324138297777778E+07 V(8,J) ------2.986666E+03 -3.400716E+05 7.0532863E+06 -6.39449130444E+07 3.390596320730159E+08 - 1.218533288309127E+09 V(9,J) ------1.640625E+05 -8.0053365E+06 1.27597579550E+08 - 5.491680025283035E+09 1.052539536278571E+09 V(10,J) ------3.281250E+04 5.5528305E+06 -1.70137188083E+08 2.259013328583333E+09 - 1.736213111520684E+10 V(11,J) ------2.1555072E+06 1.50327467033E+08 - 3.945509690352738E+10 3.399701984433333E+09 V(12,J) ------3.5925120E+05 -8.45921615000E+07 3.582450461700000E+09 - 6.526651698517500E+10 V(13,J) ------2.74788847666E+07 - 7.873006832822083E+10 2.591494081366667E+09 V(14,J) ------3.92555496666E+06 1.227049828766667E+09 - 6.855644419612083E+10 V(15,J) ------4.198434347505357E+10 3.427345554285714E+08 V(16,J) ------4.284181942857143E+07 - 1.716093471183929E+10 V(17,J) ------4.204550039102679E+09 V(18,J) ------4.671722265669643E+08</p><p>4</p>