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Laplace Transform Analytic Element Method for Transient Groundwater Flow Simulation Laplace Transform Analytic Element Method for Transient Groundwater Flow Simulation Item Type text; Electronic Dissertation Authors Kuhlman, Kristopher Lee Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 29/09/2021 19:08:37 Link to Item http://hdl.handle.net/10150/193735 LAPLACE TRANSFORM ANALYTIC ELEMENT METHOD FOR TRANSIENT GROUNDWATER FLOW SIMULATION by Kristopher L. Kuhlman A Dissertation Submitted to the Faculty of the DEPARTMENT OF HYDROLOGY &WATER RESOURCES In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY WITH A MAJOR IN HYDROLOGY In the Graduate College THE UNIVERSITY OF ARIZONA 2 0 0 8 2 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE As members of the Dissertation Committee, we certify that we have read the dis- sertation prepared by Kristopher L. Kuhlman entitled “Laplace transform analytic element method for transient groundwater flow simulation” and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy Date: April 24, 2008 Shlomo P. Neuman Date: April 24, 2008 Arthur W. Warrick Date: April 24, 2008 Paul A. Hsieh Date: April 24, 2008 Ty P. A. Ferre´ Date: April 24, 2008 Barry D. Ganapol Date: April 24, 2008 Cholik Chan Final approval and acceptance of this dissertation is contingent upon the can- didate’s submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. Date: April 24, 2008 Dissertation Director: Shlomo P. Neuman 3 STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the Univer- sity Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permis- sion, provided that accurate acknowledgment of source is made. Requests for per- mission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. SIGNED: Kristopher L. Kuhlman 4 ACKNOWLEDGMENTS This research was supported by the United States Geological Survey National In- stitutes for Water Resources Grant Program (award 200AZ68G) and by the C.W. & Modene Neely fellowship through the National Water Research Institute, in Foun- tain Valley, California. I thank my advisor, Shlomo Neuman, who conceptualized the LT-AEM, for being my mentor and teacher these last six years. He has both inspired and chal- lenged me to do things I would not have otherwise imagined possible. I thank Alex Furman, who made the idea of the LT-AEM happen, for setting me in the right direction at the beginning. Art Warrick proposed the idea behind the work in Appendix F after my oral comprehensive exam. Ty Ferre´ has given me a glimpse into the world of teaching, which I hope to pursue in my career. Each of my major and minor committee members have both directly and indirectly given me advice and insight into problems and ideas I have encountered in graduate school. I feel fortunate to interact with such a group of people, whom I consider to be my advi- sors. I thank my current and past colleagues in the department, including Bwalya Malama, Junfeng Zhu, Andreas Englert, Andrew Hinnell, Raghu Suribhatla, and Liang Xue, for many interesting discussions and projects over the years. I would not have applied to the HWR graduate program, were it not for the help and encouragement of my boss, Dennis Williams, and various co-workers at Geoscience Support Services, in Los Angeles. I worked there as a consultant for nearly four years, where they fostered my interest in groundwater-related things. This practical experience has been the foundation for everything I have learned in graduate school. Obviously, my ability to have done any of this comes from my supportive fam- ily. My wife, Sarah, and mother-in-law, Sue, have selflessly proof-read countless drafts, papers, abstracts, and applications. My mom and dad set me in the right direction, supported me along the way, and basically made me who I am; they were the first teachers I ever had. My dad would be the proudest of what I am accomplishing here. 5 To my dad. 6 TABLE OF CONTENTS LIST OF FIGURES ................................... 10 LIST OF TABLES .................................... 14 ABSTRACT ....................................... 15 CHAPTER 1. BACKGROUND ............................. 16 1.1. Motivation .................................. 16 1.2. AEM introduction .............................. 17 1.2.1. Transient AEM ........................... 19 1.2.2. Laplace-transform based methods ................ 21 1.3. Dissertation overview ........................... 23 CHAPTER 2. LT-AEM FOUNDATION ....................... 25 2.1. Governing equation ............................. 25 2.2. Superposition ................................ 27 2.3. Element derivation using eigenfunction expansion ........... 28 2.3.1. Geometric considerations ..................... 31 2.3.2. Sturm-Liouville ........................... 32 2.4. Convolution ................................. 35 2.4.1. Duhamel’s theorem ......................... 35 2.4.2. Convolution example ....................... 36 2.4.3. Time behaviors for aquifer tests .................. 38 2.5. Boundary matching ............................. 39 2.5.1. Simple illustrative example .................... 39 2.5.2. Boundary conditions ........................ 41 2.5.3. Detailed boundary matching example .............. 45 2.6. Solution for coefficients ........................... 47 2.6.1. Fixed-point iteration ........................ 48 2.6.2. Direct matrix approach ...................... 50 2.6.3. Computation of least-squares solution .............. 55 2.7. Solution for head or flux .......................... 57 CHAPTER 3. DERIVATION OF ELEMENTS ..................... 59 3.1. Circular elements .............................. 59 3.1.1. Well as a circle of small radius (no storage) ........... 63 3.1.2. Wellbore storage .......................... 66 3.2. Elliptical elements .............................. 69 7 TABLE OF CONTENTS—Continued 3.2.1. Elliptical special functions ..................... 71 3.2.2. Elliptical PDE solution ....................... 73 3.2.3. Specified flux line source ..................... 76 3.2.4. Uniform head ellipse ........................ 79 3.2.5. Elliptical source in unsaturated media .............. 80 3.3. Cartesian elements ............................. 81 3.4. Three-dimensional elements ........................ 83 3.4.1. Cylindrical coordinates ...................... 85 3.4.2. Rotational coordinates ....................... 89 3.4.3. 3D summary ............................ 94 CHAPTER 4. DISTRIBUTED SOURCES ........................ 95 4.1. Inhomogeneous sources .......................... 96 4.1.1. Decomposition of potential .................... 96 4.2. Homogeneous sources ........................... 99 4.2.1. Leaky aquifer source term ..................... 100 4.2.2. Layered system solution ...................... 107 4.2.3. Boulton’s delayed yield source term ............... 111 4.2.4. Source term from Darcy’s law ................... 115 CHAPTER 5. NUMERICAL INVERSE LAPLACE TRANSFORM ........... 118 5.1. General algorithm .............................. 118 5.2. Parallelization ................................ 120 5.3. Specific methods .............................. 121 5.3.1. Post-Widder ............................. 122 5.3.2. Schapery ............................... 125 5.3.3. Fourier series ............................ 126 5.3.4. Mobius¨ mapping .......................... 130 CHAPTER 6. LT-AEM INVERSE APPLICATIONS ................. 135 6.1. Boise aquifer test .............................. 135 6.1.1. LT-AEM model ........................... 137 6.1.2. Homogeneous model results ................... 137 6.1.3. Inhomogeneous model results .................. 139 6.1.4. Unconfined vs confined ...................... 141 6.2. Synthetic inverse problem ......................... 146 6.2.1. Synthetic problem description .................. 146 6.2.2. SCEM inverse approach ...................... 146 6.2.3. SCEM results ............................ 148 8 TABLE OF CONTENTS—Continued CHAPTER 7. CONCLUSIONS ............................. 152 APPENDIX A. LAPLACE TRANSFORM ....................... 158 A.1. Forward transform ............................. 158 A.1.1. Two-sided Laplace transform ................... 159 A.1.2. Fourier transform .......................... 160 A.2. Inverse transform .............................. 161 A.3. General properties ............................. 162 A.4. Some time behaviors ............................ 163 APPENDIX B. VECTOR COORDINATE CHANGE .................. 164 B.1. Metric coefficients .............................. 164 B.2. Vector transformation ............................ 165 B.3. Example transformation .......................... 165 APPENDIX
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