由於鑿井過程或完井工作的影響，水井附近的水層組成有顯著改變時，則此水層可視為徑向雙層系統。於受壓含水層做定水頭試驗，本文對地下水流方程式及定水頭邊界條件，推導得一個新的井緣流量閉合解；再者，也提出一個有效率的數值方法估算此閉合解的數值。估算得的流量值與用數值逆轉在拉普拉斯域解的結果，並列於表內，結果顯示這兩個解相吻合。這個新解不但可以用於估算徑向雙層地下水水井邊緣的流量，而且可評估水井膚層對井緣流量的影響。此外，於受壓單層含水層系統，存在兩種不同型式的井緣流量解析解，本文證明這兩個解與本文所推導得的解是相等的。本文所提的數值方法估算此閉合解的結果，精度至小數第五位，此數值可符合工程應用所需的精度。

An aquifer may be considered as a radial two-layer system if the formation properties near the well are significantly changed due to the well construction and/or well development. A closed-form solution for transient flow rate across the wellbore is derived under a constant-head test from a radial two-layer ground-water flow equation, which is subject to the boundary condition of keeping a constant head at the well radius in a confined aquifer. An efficient numerical approach is proposed to evaluate this solution. Values of dimensionless transient flow rate versus dimensionless time are provided in tabular forms and compared with those obtained by a numerical inversion technique for the Laplace-domain solution. The results show that the values of these two solutions are in good agreement. This newly derived solution can be used not only for predicting the transient flow rate across the wellbore but also for identifying the effects of a finite-thickness skin on the estimation of transient flow rates in a radial two-layer ground-water system. Furthermore, two other closed-form solutions for describing the same flow rate in a single-layer confined aquifer but in different forms can also be seen in the literature. This paper proofs the equality of these three solutions. The results estimated by the proposed approach are to five decimal places, which may meet the need of accuracy in engineering applications.

Page
CHINESE ABSTRACT ……………………………………………… i
ABSTRACT ……………………………………………………… ii
ACKNOWLEDGMENTS ……………………………………………… iv
TABLE OF CONTENTS …………………………………………… v
LIST OF TABLES ……………………………………………… vii
LIST OF FIGURES ……………………………………………… viii
NOTATION ………………………………………………………… ix
CHAPTER 1 INTRODUCTION ……………………………………… 1
CHAPTER 2 MATHEMATICAL DERIVATIONS ……………………… 7
2.1 Radial Two-layer Flow Equations …………… 7
2.2 Solutions for Distributions of Hydraulic Head in the skin and Formation …………………………………… 8
2.3 Closed-form Solutions for Flow Rate across the Wellbore ……………………………………………… 9
2.4 Dimensionless Variables ……………………… 10
2.5 Single-layer Solution for Flow Rate across the Wellbore ……………… 12
2.6 Behavior of Integrand of (20) ……………… 13
CHAPTER 3 NUMERICAL EVALUATIONS ………………………… 15
3.1 Bessel Functions ………………………………… 15
3.2 Shanks Method …………………………………… 16
3.3 Gaussian Quadrature …………………………… 17
3.4 Numerical Inversion …………………………… 18
3.5 Numerical Integration ………………………… 20
3.5.1 Integration of single-layer solution
……………………………… 20
3.5.2 Integration of two-layer solution
………………………………… 23
CHAPTER 4 RESULTS ……………………………………………… 24
CHAPTER 5 CONCLUSSIONS ……………………………………… 27
LIST OF REFERENCES …………………………………………… 29
APPENDICES ……………………………………………………… 33
APPENDEX A DERIVATIONS OF (8) AND (9) …………34
APPENDEX B DERIVATIONS OF (13) ……………… 37
APPENDIX C FORMULAS OF I0(u), I1(u), K0(u), K1(u), J0(u), J1(u), Y0(u), AND Y1(u) ……………………………… 40
作者簡歷
著作目錄

Abramowitz, M., and I. A. Stegun, Handbook of mathematical functions with formulas, graphs and mathematical tables, National Bureau of Standards, Washington, Dover, Inc., 1964.
Batu, V., Aquifer hydraulics: A comprehensive guide to hydrogeologic data analysis, John Wiley and Sons, Inc., New York, 1998.
Burden, R. L., and J. D. Faires, Numerical Analysis, 6th Ed., Cole Publishing Company, Brooks, 1989.
Butler, J. J., Jr., Pumping tests in nonuniform aquifers — The radially symmetric case, J. Hydrol., 101, 15-30, 1988.
Carslaw, H. S., and J. C. Jaeger, Some two-dimensional problems in conduction of heat with circular symmetry, Some Problems in Conduction of Heat, 46, 361-388, 1939.
Carslaw, H. S., and J. C. Jaeger, Conduction of heat in solids, Second Ed., Clarendon Press, Oxford, 1959.
Chang, C. C., and C. S. Chen, Analysis of constant-head for a two-layer radially symmetric nonuniform model, Proceeding of the Third Groundwater Resources and Water Quality Protection Conference, National Central University, Chung-Li, Taiwan, 1999.
Comrie, L. J., Chamber’s six-figure mathematical tables, London, W. & R., Chambers, 1949.
Crump, K. S., Numerical inversion of Laplace transforms using a Fourier series approximation, Journal of the Association for Computing Machinery, 23(1), 89-96, 1976.
de Hoog, F. R., J. H. Knight, and A. N. Stokes, An improved method for numerical inversion of Laplace transforms, Society for Industrial and Applied Mathematics J. Sci. Stat. Comput., 3(3), 357-366, 1982.
Gerald, C. F., and P. O., Wheatley, Applied Numerical Analysis, 5th Ed., Addison-Wesley, California, 1989.
Goldstein, S., Some two-dimensional diffusion problems with circular symmetry, London Math. Society, Proc. Soc.,Ⅱ, 51-88, 1932.
Hantush, M. S., Flow of ground water in sands of nonuniform thickness; Park 1. Flow in a wedge-shaped aquifer, Journal of Geophysical Research, 67(2), 703-709, 1962.
Harvard Problem Report, A Function describing the conduction of heat in a solid medium bounded internally by a cylindrical surface, Computation Laboratory of Harvard University, No. 76, 1950.
Hildebrand, F. B., Advanced Calculus for Applications, Second Ed., Prentice-Hall, Inc., New Jersey, 1976.
Huang, C. P., H. D. Yeh, and S. Y. Yang, Applications of accelerate methods on the evaluation of two types of drawdown solutions, Proceedings of the 11th hydraulic engineering conference, National Taiwan University, Taipei, Taiwan, D147-152, 5-6 July 2000.
Ingeroll, L. R., F. T. W. Adler, H. J. Plass, and A. G. Ingersoll, Theory of earth heat exchangers for the heat pump, Journal section, Heating, Piping, and Air Conditioning, 113-122, 1950.
Ingersoll, L. R., O. J. Zobel, and A. C. Ingersoll, Heat conduction with engineering, geological, and other applications, Second Ed., University Wisconsin Press, Madison, 1954.
International Mathematics and Statistics Library, Inc., IMSL User's Manual, Vol. 2, IMSL, Inc., Houston, TX, 1987.
Jacob, C. E., and S. W. Lohman, Nonsteady flow to a well of constant drawdown in an extensive aquifer, Transactions, American Geophysical Union, 33(4), 559-569, 1952.
Jaeger, J. C., Heat flow in the region bounded internally by a circular cylinder, Proceedings of the Royal Society Edinburgh, Section A, 61, 223-228, 1942.
Jaeger, J. C., Numerical values for the temperature in radial heat flow, J. Math. Phys., 34, 316-321, 1956.
Jaeger, J. C., and M. Clarke, A short table of I(o, i; x), Proceedings of the Royal Society Edinburgh, Section A, 61, 229-230, 1942.
Lohman, S. W., Ground-Water Hydraulics, Geological Survey Professional Paper; 708, Washington: United States Government Printing Office, 1972.
Markle, J. M., R. K. Rowe, and K. S. Novakowski, A model for the constant-heat pumping test conducted in vertically fractured media, International Journal for Numerical and Analytical Methods in Geomechanics, 19, 457-473, 1995.
Novakowski, K. S., A composite analytical model for analysis of pumping tests affected by wellbore storage and finite thickness skin, Water Resour. Res., 25(9), 1937-1946, 1989.
Novakowski, K. S., Interpretation of the transient flow rate obtained from constant-head tests conducted in situ in clays, Can. Geotech. J., 30, 600-606, 1993.
Peng, H. Y., H. D. Yeh, and S. Y. Yang, Improved numerical evaluation of the radial groundwater flow equation, Advances in Water Resources, 2002. (Accepted)
Reed, J. E., Type Curves for Selected Problems of Flow to Wells in Confined Aquifers; Book 3 Applications of Hydraulics, United States Department of the Interior, 1980.
Shanks, D., Non-linear transformations of divergent and slowly convergent sequences, J. Math. Phys., 34, 1-42, 1955.
Smith, L. P., Heat flow in an infinite solid bounded internally by a cylinder, J. Applied Physics, 8(6), 45-49, 1937.
Spiegel, M. R., Laplace Transforms, Schaum Publishing Co., New York, 1965.
Stehfest, H., Numerical inversion of Laplace transforms, Commun. ACM, 13(1), 47-49, 1970.
Streltsova, T. D., Well Testing in Heterogeneous Formations, John Wiley and Sons, Inc., New York, 45-49, 1988.
Streltsova, T. D., and R. M. McKinley, Effect of flow time duration on buildup pattern for reservoirs with heterogeneous properties, Soc. Pet. Eng. J., 294-306, 1984.
Talbot, A., The accurate numerical inversion of Laplace transforms, J. Inst. Math. Appl., 23, 97-120, 1979.
Watson, G. N., A Treatise on the Theory of Bessel Functions, Second Ed., Cambridge University Press, 1958.
Wynn, P., On a device for computing the em(Sn) transformation, Math. Tables Other Aids Comp., 10, 91-96, 1956.