Laplace-transform analytic-element method for transient porous-media flow



A unified theory of the Laplace-transform analytic-element method (LT-AEM) for solving transient porous-media flow problems is presented. LT-AEM applies the analytic-element method (AEM) to the modified Helmholtz equation, the Laplace-transformed diffusion equation. LT-AEM uses superposition and boundary collocation with Laplace-space convolution to compute flexible semi-analytic solutions from a small collection of fundamental elements. The elements discussed are derived using eigenfunction expansions of element shapes in their natural coordinates. A new formulation for a constant-strength line source is presented in terms of elliptical coordinates and complex-parameter Mathieu functions. Examples are given illustrating how leaky and damped-wave hydrologic problems can be solved with little modification using existing LT-AEM techniques.


Analytic element Diffusion equation Elliptical coordinates Laplace transform Mathieu functions Modified Helmholtz equation Transient line source 



Analytic-element method


Laplace-transform analytic-element method


  1. 1.
    Kraemer SR (2007) Analytic element ground water modeling as a research program (1980 to 2006). Ground Water 45(4): 402–408CrossRefGoogle Scholar
  2. 2.
    Strack ODL, Haitjema HM (1981) Modeling double aquifer flow using a comprehensive potential and distributed singularities 1. Solution for homogeneous permeability. Water Resour Res 17(5): 1535–1549Google Scholar
  3. 3.
    Boyd JP (2000) Chebyshev and Fourier spectral methods. 2nd ed. Dover Publications, New YorkGoogle Scholar
  4. 4.
    Kuhlman KL (2008) Laplace transform analytic element method. VDM Verlag, SaarbrückenGoogle Scholar
  5. 5.
    Fitts CR (1991) Modeling three-dimensional flow about ellipsoidal inhomogeneities with application to flow to a gravel-packed well and flow through lens-shaped inhomogeneities. Water Resour Res 27(5): 815–824CrossRefADSGoogle Scholar
  6. 6.
    Furman A, Neuman SP (2003) Laplace-transform analytic element solution of transient flow in porous-media. Adv Water Res 26(12): 1229–1237CrossRefGoogle Scholar
  7. 7.
    Bakker M, Strack ODL (2003) Analytic elements for multiaquifer flow. J Hydrol 271(1–4): 119–129CrossRefGoogle Scholar
  8. 8.
    Bakker M, Nieber JL (2004) Two-dimensional steady unsaturated flow through embedded elliptical layers. Water Resour Res 40(12): W12406CrossRefADSGoogle Scholar
  9. 9.
    Hunt RJ (2006) Ground water modeling applications using the analytic element method. Ground Water 44(1): 5–15CrossRefGoogle Scholar
  10. 10.
    Strack ODL (1999) Principles of the analytic element method. J Hydrol 226(3–4): 128–138CrossRefGoogle Scholar
  11. 11.
    Strack ODL (2003) Theory and applications of the analytic element method. Rev Geophys 41(2): 1005–1021CrossRefADSGoogle Scholar
  12. 12.
    Strack ODL (1989) Groundwater mechanics. Prentice-Hall, Englewood CliffsGoogle Scholar
  13. 13.
    Haitjema HM (1995) Analytic element modeling of groundwater flow. Academic Press, LondonGoogle Scholar
  14. 14.
    Haitjema HM, Strack ODL (1985) An initial study of thermal energy storage in unconfined aquifers. Technical report PNL-5818 UC-94e, Pacific Northwest LaboratoriesGoogle Scholar
  15. 15.
    Zaadnoordijk WJ, Strack ODL (1993) Area sinks in the analytic element method for transient groundwater flow. Water Resour Res 29(12): 4121–4129CrossRefADSGoogle Scholar
  16. 16.
    Butler JJ, Liu W (1993) Pumping tests in nonuniform aquifers: the radially asymmetric case. Water Resour Res 29(2): 259–269CrossRefADSGoogle Scholar
  17. 17.
    Bakker M (2004) Transient analytic elements for periodic Dupuit–Forchheimer flow. Adv Water Resour 27(1): 3–12CrossRefADSGoogle Scholar
  18. 18.
    Strack ODL (2006) The development of new analytic elements for transient flow and multiaquifer flow. Ground Water 44(1): 91–98CrossRefGoogle Scholar
  19. 19.
    Bakker M, Nieber JL (2004) Analytic element modeling of cylindrical drains and cylindrical inhomogeneities in steady two-dimensional unsaturated flow. Vadose Zone J 3(3): 1038–1049CrossRefGoogle Scholar
  20. 20.
    Kuhlman KL, Warrick AW (2008) Quasilinear infiltration from an elliptical cavity. Adv Water Resour 31(8): 1057–1065CrossRefADSGoogle Scholar
  21. 21.
    Churchill RV (1972) Operational Mathematics 3rd edn. McGraw-Hill, New YorkMATHGoogle Scholar
  22. 22.
    Kuhlman KL, Neuman SP (2006) Recent advances in Laplace transform analytic element method (LT-AEM) theory and application to transient groundwater flow. In: Computational methods in water resources, vol XVIGoogle Scholar
  23. 23.
    Duffin RJ (1971) Yukawan potential theory. J Math Anal Appl 35(1): 105–130MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Graff KF (1991) Wave motion in elastic solids. Dover Publications, New YorkGoogle Scholar
  25. 25.
    Moore RK (1964) Wave and diffusion analogies. McGraw-Hill, New YorkGoogle Scholar
  26. 26.
    Özişik NM (1993) Heat conduction. 2nd ed. Wiley-Interscience, New YorkGoogle Scholar
  27. 27.
    Abramowitz M, Stegun IA (eds) (1964) Handbook of mathematical functions with formulas, graphs and mathematical tables. Number 55 in Applied Mathematics Series. National Bureau of StandardsGoogle Scholar
  28. 28.
    Bakker M (2004) Modeling groundwater flow to elliptical lakes and through multi-aquifer elliptical inhomogeneities. Adv Water Resour 27(5): 497–506ADSGoogle Scholar
  29. 29.
    Smith BF, Bjorstad PE, Gropp W (1996) Domain decomposition. Cambridge University Press, CambridgeMATHGoogle Scholar
  30. 30.
    Andrews LC (1998) Special functions of mathematics for engineers. 2nd ed. SPIE Press, BellinghamGoogle Scholar
  31. 31.
    McLachlan NW (1947) Theory and application of Mathieu functions. Oxford University Press, OxfordMATHGoogle Scholar
  32. 32.
    Arscott FM (1981) Ordinary and partial differential equations. In: Everitt WN, Sleeman BD (eds) Lecture notes in mathematics, vol 846. The land beyond Bessel: a survey of higher special functions. Springer, HeidelbergGoogle Scholar
  33. 33.
    Morse PM, Feshbach H (1953) Methods of theoretical physics. Vols 1 and 2. McGraw-Hill, New YorkGoogle Scholar
  34. 34.
    Oleksy Cz (1996) A convergence acceleration method for Fourier series. Comput Phys Commun 96(1): 17–26MATHCrossRefMathSciNetADSGoogle Scholar
  35. 35.
    Lanczos C (1988) Applied analysis. Dover Publications, New YorkGoogle Scholar
  36. 36.
    Cohen AM (2007) Numerical methods for Laplace transform inversion. Springer, HeidelbergMATHGoogle Scholar
  37. 37.
    Davies B (2002) Integral transforms and their application. 3rd ed. Springer, HeidelbergGoogle Scholar
  38. 38.
    Abate J, Valkò PP (2003) Multi-precision Laplace transform inversion. Int J Numer Methods Eng 60(5): 979–993CrossRefGoogle Scholar
  39. 39.
    Janković I, Barnes R (1999) High-order line elements in modeling two-dimensional groundwater flow. J Hydrol 226(3–4): 211–223CrossRefGoogle Scholar
  40. 40.
    Anderson, E, Bai Z, Dongarra J, Greenbaum A, McKenney A, Du Croz J, Hammarling S, Demmel J, Bischof C, Sorensen D (1990) LAPACK: a portable linear algebra library for high-performance computers. In: Proceedings of the 1990 ACM/IEEE conference on supercomputing. IEEE Computer Society, pp 2–11Google Scholar
  41. 41.
    Golub GH, van Loan CF (1996) Matrix computations. 3rd ed. Johns Hopkins University Press, BaltimoreMATHGoogle Scholar
  42. 42.
    Lawson CL, Hanson RJ (1974) Solving least squares problems. SIAM, PhiladelphiaMATHGoogle Scholar
  43. 43.
    Hantush MS (1960) Modification of the theory of leaky aquifers. J Geophys Res 65(11): 3713–3725CrossRefADSGoogle Scholar
  44. 44.
    Lee T-C (1999) Applied mathematics in hydrogeology. CRC Press, Boca RatonGoogle Scholar
  45. 45.
    Theis CV (1935) The relation between lowering of the piezometric surface and the rate and duration of discharge of a well using ground-water storage. Trans Am Geophys Union 16(2): 519–524Google Scholar
  46. 46.
    Bear J (1988) Dynamics of fluids in porous media. Dover Publications, New YorkGoogle Scholar
  47. 47.
    Nield DA, Bejan A (2006) Convection in porous media. 3rd ed. Springer, HeidelbergGoogle Scholar
  48. 48.
    Löfqvist T, Rehbinder G (1993) Transient flow towards a well in an aquifer including the effect of fluid inertia. Appl Sci Res 51(3): 611–623MATHCrossRefGoogle Scholar
  49. 49.
    Vásquez JL (2007) The porous medium equation: mathematical theory. Oxford University Press, OxfordGoogle Scholar
  50. 50.
    Strack ODL, Janković I (1999) A multi-quadric area-sink for analytic element modeling of groundwater flow. J Hydrol 226(3–4): 299–196Google Scholar
  51. 51.
    Bakker M (2008) Derivation and relative performance of strings of line elements for modeling (un)confined and semi-confined flow. Adv Water Resour 31(6): 906–914CrossRefADSGoogle Scholar
  52. 52.
    Suribhatla RM, Bakker M, Bandilla K, Janković I (2004) Steady two-dimensional groundwater flow through many elliptical inhomogeneities. Water Resour Res 40(4): W04202CrossRefADSGoogle Scholar
  53. 53.
    Moon P, Spencer DE (1961) Field theory handbook: including coordinate systems differential equations and their solutions. Springer-Verlag, HeidelbergMATHGoogle Scholar
  54. 54.
    Alhargan FA (2000) Algorithm 804: subroutines for the computation of Mathieu functions of integer order. ACM Trans Math Software 26(3): 408–414CrossRefMathSciNetGoogle Scholar
  55. 55.
    Gutiérrez Vega JC, Rodriguez Dagnino RM, Meneses Nava AM, Chávez Cerda S (2003) Mathieu functions, a visual approach. Am J Phys 71(3): 233–242CrossRefADSGoogle Scholar
  56. 56.
    Tranter CJ (1951) Heat conduction in the region bounded internally by an elliptical cylinder and an analogous problem in atmospheric diffusion. Q J Mech Appl Math 4(4): 461–465MATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    Kucûk F, Brigham WE (1979) Transient flow in elliptical systems. Soc Petroleum Eng J 267: 401–410Google Scholar
  58. 58.
    Riley MF (1991) Finite conductivity fractures in elliptical coordinates. PhD thesis, Stanford UniversityGoogle Scholar
  59. 59.
    Erricolo D (2003) Acceleration of the convergence of series containing Mathieu functions using Shanks transformations. IEEE Antennas Wirel Propag Lett 2: 58–61CrossRefADSGoogle Scholar
  60. 60.
    de Hoog FR, Knight JH, Stokes AN (1982) An improved method for numerical inversion of Laplace transforms. SIAM J Stat Comput 3(3): 357–366MATHCrossRefGoogle Scholar
  61. 61.
    Chaos-Cador L, Ley-Koo E (2002) Mathieu functions revisited: matrix evaluation and generating functions. Rev Mex Fis 48(1): 67–75MathSciNetGoogle Scholar
  62. 62.
    Delft Numerical Analysis Group (1973) On the computation of Mathieu functions. J Eng Math 7(1): 39–61CrossRefGoogle Scholar
  63. 63.
    Green DJ, Michaelson S (1965) Series solution of certain Sturm-Liouville eigenvalue problems. Comput J 7(4): 322–336MATHMathSciNetGoogle Scholar
  64. 64.
    Stamnes JJ, Spjelkavik B (1995) New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders. Pure Appl Opt 4(3): 251–262CrossRefADSGoogle Scholar
  65. 65.
    Blanch G, Clemm DS (1969) Mathieu’s equation for complex parameters: tables of characteristic values. Technical report, Aerospace Research Laboratories, US Air ForceGoogle Scholar
  66. 66.
    Alhargan FA (2000) Algorithms for the computation of all Mathieu functions of integer orders. ACM Trans Math Software 26(3): 390–407CrossRefMathSciNetGoogle Scholar
  67. 67.
    Hunter C, Guerrieri B (1981) The eigenvalues of Mathieu’s equation and their branch points. Stud Appl Math 64: 113–141MATHMathSciNetGoogle Scholar
  68. 68.
    Arscott FM (1964) Periodic differential equations. Macmillan, New YorkMATHGoogle Scholar
  69. 69.
    Arscott FM, Darai A (1981) Curvilinear co-ordinate systems in which the Helmholtz equation separates. IMA J Appl Math 27(1): 33–70MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of Hydrology and Water ResourcesUniversity of ArizonaTucsonUSA

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