Advertisement

Abstract

The aim of this book is the development of a numerical method to model the stress field in geothermal reservoirs. For this purpose, a method of fundamental solutions to the quasistatic equations of poroelasticity as first given by Biot is developed. This chapter summarizes theoretical results about such a method as well as the results of numerical tests about the performance of the developed fundamental solution methods. Furthermore, some topics for further research on the method of fundamental solutions and its incorporation for modeling the stress field in geothermal reservoirs are provided.

Keywords

Source Point Radial Basis Function Fundamental Solution Heat Equation Collocation Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 24.
    Barnett, A.H., Betcke, T.: Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains. J. Comput. Phys. 227, 7003–7026 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 29.
    Bauer, F., Gutting, M., Lukas, M.A.: Evaluation of parameter choice methods for regularization of ill-posed problems in geomathematics. In: W. Freeden, Z. Nashed, T. Sonar (eds.) Handbook of Geomathematics, 2nd edn. Springer, New York (2015). Accepted for publicationGoogle Scholar
  3. 30.
    Bauer, F., Lukas, M.A.: Comparingparameter [sic] choice methods for regularization of ill-posed problems. Math. Comput. Simul. 81, 1795–1841 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 34.
    Biot, M.A.: Le Problème de la Consolidation des Matières Argileuses sous une Charge. Ann. Soc. Sci. Brux. B55, 110–113 (1935)Google Scholar
  5. 37.
    Biot, M.A.: General solutions of the equations of elasticity and consolidation for a porous material. J. Appl. Mech. 78, 91–96 (1956)MathSciNetGoogle Scholar
  6. 38.
    Biot, M.A.: Mechanics of deformation and acoustic propagation in porous media. J. Appl. Phys. 33, 1482–1498 (1962)MathSciNetCrossRefGoogle Scholar
  7. 42.
    Buhmann, M.D.: Radial basis functions: theory and implementations. In: Cambridge Monographs on Applied and Computational Mathematics, vol. 12. Cambridge University Press, Cambridge (2003)Google Scholar
  8. 57.
    Costabel, M.: Boundary integral operators for the heat equation. Integral Equ. Operat. Theory 13, 498–552 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 77.
    Fasshauer, G.E.: Solving partial differential equations by collocation with radial basis functions. In: A. Le Méhauté, C. Rabut, L.L. Schumaker (eds.) Surface Fitting and Multiresolution Methods, pp. 131–138. Vanderbilt University Press, Nashville (1997)Google Scholar
  10. 91.
    Freeden, W., Michel, V.: Multiscale Potential Theory with Applications to Geoscience. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2004)CrossRefzbMATHGoogle Scholar
  11. 115.
    Golberg, M.A.: The method of fundamental solutions for Poisson’s equation. Eng. Anal. Bound. Elem. 16, 205–213 (1995)CrossRefGoogle Scholar
  12. 135.
    Ingber, M.S., Chen, C.S., Tanski, J.A.: A mesh free approach using radial basis functions and parallel domain decomposition for solving three-dimensional diffusion equations. Int. J. Numer. Method. Eng. 60, 2183–2201 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 176.
    Liu, G., Xie, K., Zheng, R.: Model of nonlinear coupled thermo-hydro-elastodynamics response for a saturated poroelastic medium. Sci. China, Ser. E: Technol. Sci. 52, 2373–2383 (2009)Google Scholar
  14. 180.
    Lowitzsch, S.: Error estimates for matrix-valued radial basis function interpolation. J. Approx. Theory 137, 238–249 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 181.
    Lowitzsch, S.: Matrix-valued radial basis functions: stability estimates and applications. Adv. Comput. Math. 23, 299–315 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 198.
    Miehe, C., Méndez Diez, J., Göktepe, S., Schänzel, L.M.: Coupled thermoviscoplasticity of glassy polymers in the logarithmic strain space based on the free volume theory. Int. J. Solid. Struct. 48, 1799–1817 (2010)CrossRefGoogle Scholar
  17. 207.
    Nakao, S., Ishido, T.: Pressure-transient behavior during cold water injection into geothermal wells. Geothermics 27, 401–413 (1998)CrossRefGoogle Scholar
  18. 208.
    Nardini, D., Brebbia, C.A.: A new approach for free vibration analysis using boundary elements. In: C.A. Brebbia (ed.) Boundary Element Methods in Engineering: Proceedings of the Fourth International Seminar, Southhampton, Sept 1982, pp. 312–326. Springer, Berlin (1982)CrossRefGoogle Scholar
  19. 212.
    Neumaier, A.: Solving ill-conditioned and singular linear systems: a tutorial on regularization. SIAM Rev. 40, 636–666 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 219.
    Partridge, P.W., Sensale, B.: The method of fundamental solutions with dual reciprocity for diffusion and diffusion-convection using subdomains. Eng. Anal. Bound. Elem. 24, 633–641 (2000)CrossRefzbMATHGoogle Scholar
  21. 235.
    Reutskiy, S.Y.: The method of approximate fundamental solutions (MAFS) for Stefan problems. Eng. Anal. Bound. Elem. 36, 281–292 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 239.
    Runge, C.: Zur Theorie der eindeutigen analytischen Functionen. Acta Math. 6, 229–234 (1885)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 264.
    Smyrlis, Y.S.: Mathematical foundation of the MFS for certain elliptic systems in linear elasticity. Numer. Math. 112, 319–340 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 268.
    Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis, 2nd edn. Springer, New York (1993). Translated from German by R. Bartels, W. Gautschi, and C. WitzgallGoogle Scholar
  25. 281.
    Valtchev, S.S., Roberty, N.C.: A time-marching MFS scheme for heat conduction problems. Eng. Anal. Bound. Elem. 32, 480–493 (2008)CrossRefzbMATHGoogle Scholar
  26. 286.
    Wang, W., Kosakowski, G., Kolditz, O.: A parallel finite element scheme for thermo-hydro-mechanical (THM) coupled problems in porous media. Comput. Geosci. 35, 1631–1641 (2009)CrossRefGoogle Scholar
  27. 288.
    Wen, P.H., Liu, Y.W.: The fundamental solution of poroelastic plate saturated by fluid and its applications. Int. J. Numer. Anal. Method. Geomech. 34, 689–709 (2010)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Matthias Albert Augustin
    • 1
  1. 1.AG GeomathematikTechnische Universität KaiserslauternKaiserslauternGermany

Personalised recommendations