Advertisement

Modeling Deep Geothermal Reservoirs: Recent Advances and Future Perspectives

  • Matthias Augustin
  • Mathias Bauer
  • Christian Blick
  • Sarah Eberle
  • Willi Freeden
  • Christian Gerhards Maxim Ilyasov
  • René Kahnt
  • Matthias Klug
  • Sandra Möhringer
  • Thomas Neu
  • Helga Nutz
  • Isabel Michel née Ostermann
  • Alessandro Punzi
Reference work entry

Abstract

Modeling geothermal reservoirs is a key issue of a successful geothermal energy development. After over 40 years of study, many models have been proposed and applied to hundreds of sites worldwide. Nevertheless, with increasing computational capabilities, new efficient methods become available. The aim of this paper is to present recent progress on potential methods and seismic (post-)processing, as well as fluid and thermal flow simulations for porous and fractured subsurface systems. Commonly used procedures in industrial energy exploration and production such as forward modeling, seismic migration, and inversion methods together with continuum and discrete flow models for reservoir monitoring and management are explained, and some numerical examples are presented. The paper ends with the description of future fields of studies and points out opportunities, perspectives, and challenges.

Notes

Acknowledgements

The work of the Geomathematics Group Kaiserslautern and G.E.O.S Ingenieurgesellschaft mbH, Freiberg, is supported by the “Verbundprojekt GEOFÜND: Charakterisierung und Weiterentwicklung integrativer Untersuchungsmethoden zur Quantifizierung des Fündigkeitsrisikos” (PI: W. Freeden) Federal Ministry for Economic Affairs and Energy (BMWi) Germany. M. Augustin has been supported by a fellowship of the German National Academic Foundation (Studienstiftung des deutschen Volkes). C. Gerhards has been supported by a fellowship within the Postdoc-program of the German Academic Exchange Service (DAAD). S. Eberle is thankful for the support by the Rhineland-Palatinate Center of Excellence for Climate Change Impacts. M. Ilyasov, S. Möhringer, H. Nutz, I. Ostermann, and A. Punzi thank for the support by the Rhineland-Palatinate excellence research center “Center for Mathematical and Computational Modeling (CM)2” and the University of Kaiserslautern within the scope of the project “EGMS” (PI: W. Freeden).

References

  1. Addis MA (1997) The stress-depletion response of reservoirs. In: SPE annual technical conference and exhibition, San Antonio, 5–8 Oct 1997Google Scholar
  2. Adler PM, Thovert JF (1999) Theory and applications in porous media. Fractures and fracture networks, vol 15. Kluwer Academic, DordrechtGoogle Scholar
  3. Aitken M (2010) Why we still don’t understand the social aspects of wind power: a critique of key assumptions with the literature. Energy Policy 38:1834–1841CrossRefGoogle Scholar
  4. Altmann J, Dorner A, Schoenball M, Müller BIR, Müller T (2008) Modellierung von porendruckinduzierten Änderungen des Spannungsfeldes in Reservoiren. In: Kongressband, Geothermiekongress 2008, KarlsruheGoogle Scholar
  5. Arbogast T (1989) Analysis of the simulation of single phase flow through a naturally fractured reservoir. SIAM J Numer Anal 26:12–29MathSciNetzbMATHCrossRefGoogle Scholar
  6. Arbogast T, Douglas J, Hornung U (1990) Derivation of the double porosity model of single phase flow via homogenization theory. SIAM J Math Anal 21:823–836MathSciNetzbMATHCrossRefGoogle Scholar
  7. Assteerawatt A (2008) Flow and transport modelling of fractured aquifers based on a geostatistical approach. PhD thesis, Institute of Hydraulic Engineering, University of StuttgartGoogle Scholar
  8. Augustin M (2012) On the role of poroelasticity for modeling of stress fields in geothermal reservoirs. Int J Geomath 3:67–93MathSciNetzbMATHCrossRefGoogle Scholar
  9. Augustin M (2014) A method of fundamental solutions in poroelasticity to model the stress field in geothermal reservoirs. PhD thesis, Geomathematics Group, University of KaiserslauternGoogle Scholar
  10. Augustin M, Freeden W, Gerhards C, Möhringer S, Ostermann I (2012) Mathematische Methoden in der Geothermie. Math Semesterber 59:1–28MathSciNetzbMATHCrossRefGoogle Scholar
  11. Auradou H (2009) Influence of wall roughness on the geometrical, mechanical and transport properties of single fractures. J Phys D Appl Phys 42:214015CrossRefGoogle Scholar
  12. Auriault J-L (1973) Contribution à l’étude de la consolidation des sols. PhD thesis, L’Université scientifique et médicale de GrenobleGoogle Scholar
  13. Axelsson G, Gunnlaugsson E (2000) Long-term monitoring of high- and low-enthalpy fields under exploitation. In: World geothermal congress 2000, pre-congress course, KokonoeGoogle Scholar
  14. Baisch S, Carbon D, Dannwolf U, Delacou B, Delvaux M, Dunand F, Jung R, Koller M, Martin C, Sartori M, Secanell R, Vorös R (2009) Deep heat mining Basel – seismic risk analysis. SERIANEX. Technical report, study prepared for the Departement für Wirtschaft, Soziales und Umwelt des Kantons Basel-Stadt, Amt für Umwelt und EnergieGoogle Scholar
  15. Barenblatt GI, Zheltov IP, Kochina IN (1960) Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. PMM Sov Appl Math Mech 24:852–864zbMATHGoogle Scholar
  16. Barnett AH, Betcke T (2008) Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains. J Comput Phys 227:7003–7026MathSciNetzbMATHCrossRefGoogle Scholar
  17. Bauer M, Freeden W, Jacobi H, Neu T (2014a) Energiewirtschaft 2014. Springer Spektrum, WiesbadenCrossRefGoogle Scholar
  18. Bauer M, Freeden W, Jacobi H, Neu T (2014b) Handbuch Tiefe Geothermie. Springer Spektrum, Berlin/HeidelbergCrossRefGoogle Scholar
  19. Baysal E, Kosloff DD, Sherwood JWC (1983) Reverse time migration. Geophysics 48: 1514–1524CrossRefGoogle Scholar
  20. Baysal E, Kosloff DD, Sherwood JWC (1984) A two-way nonreflecting wave equation. Geophysics 49:132–141CrossRefGoogle Scholar
  21. Bear J (1972) Dynamics of fluids in porous media. Elsevier, New YorkzbMATHGoogle Scholar
  22. Bear J, Tsang CF, de Marsily G (1993) Flow and contaminant transport in fractured rock. Academic, San DiegoGoogle Scholar
  23. Berkowitz B (1995) Analysis of fracture network connectivity using percolation theory. Math Geol 27:467–483CrossRefGoogle Scholar
  24. Berkowitz B (2002) Characterizing flow and transport in fractured geological media: a review. Adv Water Resour 25:852–864CrossRefGoogle Scholar
  25. Billette F, Brandsberg-Dahl S (2005) The 2004 BP velocity benchmark. In: 67th annual international meeting EAGE, Madrid. Expanded abstracts. EAGEGoogle Scholar
  26. Biondi BL (2006) Three-dimensional seismic imaging. Society of Exploration Geophysicists, TulsaGoogle Scholar
  27. Biot MA (1935) Le problème de la consolidation des matières argileuses sous une charge. Ann Soc Sci Brux B55:110–113Google Scholar
  28. Biot MA (1941) General theory of three-dimensional consolidation. J Appl Phys 12:151–164zbMATHGoogle Scholar
  29. Biot MA (1955) Theory of elasticity and consolidation for a porous anisotropic solid. J Appl Phys 26:182–185MathSciNetzbMATHCrossRefGoogle Scholar
  30. Blakely RJ (1996) Potential theory in gravity & magnetic application. Cambridge University Press, CambridgeGoogle Scholar
  31. Blank L (1996) Numerical treatment of differential equations of fractional order. Technical report, numercial analysis report, University of ManchesterGoogle Scholar
  32. Bleistein N (1987) On the imaging of reflectors in the Earth. Geophysics 49:931–942CrossRefGoogle Scholar
  33. Bleistein N, Cohen JK, Stockwell JW (2000) Mathematics of multidimensional seismic imaging, migration, and inversion. Springer, New YorkzbMATHGoogle Scholar
  34. Bödvarsson G (1964) Physical characteristics of natural heat sources in Iceland. In: Proceedings of the United Nations conference on new sources of energy, vol 2. United NationsGoogle Scholar
  35. Bollhöfer M, Grote MJ, Schenk O (2008) Algebraic multilevel preconditioner for the Helmholtz equation in heterogeneous media. SIAM J Sci Comput 31:3781–3805MathSciNetzbMATHCrossRefGoogle Scholar
  36. Bonomi E, Pieroni E (1998) Energy-tuned absorbing boundary conditions. In: 4th SIAM international conference on mathematical and numerical aspects of wave propagation, Colorado School of MinesGoogle Scholar
  37. Bording RP, Liner CL (1994) Theory of 2.5-D reverse time migration. In: Proceedings, 64th annual international meeting: society of exploration geophysicists, Los AngelesGoogle Scholar
  38. Brouwer GK, Lokhorst A, Orlic B (2005) Geothermal heat and abandoned gas reservoirs in the Netherlands. In: Proceedings world geothermal congress 2005, AntalyaGoogle Scholar
  39. Browder FE (1962) Approximation by solutions of partial differential equations. Am J Math 84:134–160MathSciNetzbMATHCrossRefGoogle Scholar
  40. Brown SR (1987) Fluid flow through rock joints: the effect of surface roughness. J Geophys Res 92:1337–1347CrossRefGoogle Scholar
  41. Buhmann MD (2003) Radial basis functions: theory and implementations. Cambridge monographs on applied and computational mathematics, vol 12. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  42. Buske S (1994) Kirchhoff-Migration von Einzelschußdaten. Master thesis, Institut für Meteorologie und Geophysik der Johann Wolfgang Goethe Universität Frankfurt am MainGoogle Scholar
  43. Chen M, Bai M, Roegiers JC (1999) Permeability tensors of anisotropic fracture networks. Math Geol 31:355–373CrossRefGoogle Scholar
  44. Chen Z, Huan G, Ma Y (2006) Computational methods for multiphase flows in porous media. Computational science & engineering, vol 2. SIAM, PhiladelphiaGoogle Scholar
  45. Cheng H-P, Yeh G-T (1998) Development and demonstrative application of a 3-D numerical model of subsurface flow, heat transfer, and reactive chemical transport: 3DHYDROGEOCHEM. J Contam Hydrol 34:47–83CrossRefGoogle Scholar
  46. Claerbout J (2009) Basic Earth imaging. Stanford University, StanfordGoogle Scholar
  47. Darcy HPG (1856) Les Fontaines Publiques de la Ville de Dijon. Victor Dalmont, ParisGoogle Scholar
  48. de Boer R (2000) Theory of porous media – highlights in historical development and current state. Springer, BerlinzbMATHCrossRefGoogle Scholar
  49. Deng F, McMechan GA (2007) 3-D true amplitude prestack depth migration. In: Proceedings, SEG annual meeting, San AntonioCrossRefGoogle Scholar
  50. Dershowitz WS, La Pointe PR, Doe TW (2004) Advances in discrete fracture network modeling. In: Proceedings, US EPA/NGWA fractured rock conference, Portland, pp 882–894Google Scholar
  51. Diersch H-J (1985) Modellierung und numerische Simulation geohydrodynamischer Transportprozesse. PhD thesis, Akademie der Wissenschaften der DDRGoogle Scholar
  52. Diersch H-J (2000) Numerische Modellierung ober- und unterirdischer Strömungs- und Transportprozesse. In: Martin H, Pohl M (eds) Technische Hydromechanik 4 – Hydraulische und numerische Modelle. Verlag Bauwesen, BerlinGoogle Scholar
  53. Dietrich P, Helmig R, Sauter M, Hötzl H, Köngeter J, Teutsch G (2005) Flow and transport in fractured porous media. Springer, BerlinCrossRefGoogle Scholar
  54. Du X, Bancroft JC (2004) 2-D wave equation modeling and migration by a new finite difference scheme based on the Galerkin method. Technical report, CREWESCrossRefGoogle Scholar
  55. Durst P, Vuataz FD (2000) Fluid-rock interactions in hot dry rock reservoirs: a review of the HDR sites and detailed investigations of the Soultz-sous-Forets system. In: Proceedings of the world geothermal congess 2000, Kyushu-TohokuGoogle Scholar
  56. Eberle S (2014) Forest fire determination: theory and numerical aspects. PhD thesis, Geomathematics Group, University of KaiserslauternGoogle Scholar
  57. Eberle S, Freeden W, Matthes U (2015) Forest fire spreading. In Freeden W, Nashed B, Sonar T (Eds) Handbook of Geomathematics, 2nd Edn. SpringerGoogle Scholar
  58. Eker E, Akin S (2006) Lattice Boltzmann simulation of fluid flow in synthetic fractures. Transp Porous Media 65:363–384CrossRefGoogle Scholar
  59. Ene HI, Poliševski D (1987) Thermal flow in porous media. D. Reidel, DordrechtzbMATHCrossRefGoogle Scholar
  60. Engelder T, Fischer MP (1994) Influence of poroelastic behaviour on the magnitude of minimum horizontal stress, S h, in overpressured parts of sedimentary basins. Geology 22:949–952CrossRefGoogle Scholar
  61. Engl W, Hanke M, Neubauer A (1996) Regularization of inverse problems. Kluwer Academic, DordrechtzbMATHCrossRefGoogle Scholar
  62. Ernstson K, Alt W (2013) Gravity and geomagnetic methods in geothermal exploration: understanding and misunderstanding. World Min 65:115–122Google Scholar
  63. Evans KF, Cornet FH, Hashida T, Hayashi K, Ito T, Matsuki K, Wallroth T (1999) Stress and rock mechanics issues of relevance to HDR/HWR engineered geothermal systems: review of developments during the past 15 years. Geothermics 28:455–474CrossRefGoogle Scholar
  64. Expertengruppe “Seismisches Risiko bei hydrothermaler Geothermie” (2010) Das seismische Ereignis bei Landau vom 15. August 2009, Abschlussbericht. Technical report, on behalf of the Ministerium für Umwelt, Landwirtschaft, Ernährung, Weinbau und Forsten des Landes Rheinland-PfalzGoogle Scholar
  65. Fehlinger T (2009) Multiscale formulations for the disturbing potential and the deflections of the vertical in locally reflected physical geodesy. PhD thesis, Geomathematics Group, University of KaiserslauternGoogle Scholar
  66. Fisher N, Lewis T, Embleton B (1993) Statistical analysis of spherical data. Cambridge University Press, CambridgezbMATHGoogle Scholar
  67. Fomin S, Hashida T, Shimizu A, Matsuki K, Sakaguchi K (2003) Fractal concept in numerical simulation of hydraulic fracturing of the hot dry rock geothermal reservoir. Hydrol Process 17:2975–2989CrossRefGoogle Scholar
  68. Ford NJ, Simpson A (2001) The numerical solution of fractional differential equations: speed versus accuracy. Numer Algorithms 26:333–346MathSciNetzbMATHCrossRefGoogle Scholar
  69. Foulger G, Natland J, Presnall D, Anderson D (2005) Plates, plumes, and paradigms. Geological Society of America, BoulderGoogle Scholar
  70. Freeden C (2013) The role and the potential of communication by analysing the social acceptance of the German deep geothermal energy market. Master thesis, University of PlymouthGoogle Scholar
  71. Freeden W (1980) On the approximation of external gravitational potential with closed systems of (trial) functions. Bull Geod 54:1–20MathSciNetCrossRefGoogle Scholar
  72. Freeden W (1981) On approximation by harmonic splines. Manuscr Geod 6:193–244zbMATHGoogle Scholar
  73. Freeden W (1983) Least squares approximation by linear combination of (multi-)poles. Report 344, Departement of Geodetic Science and Surveying, The Ohio State University, ColumbusGoogle Scholar
  74. Freeden W (1999) Multiscale modelling of spaceborne geodata. Teubner, StuttgartzbMATHGoogle Scholar
  75. Freeden W (2011) Metaharmonic lattice point theory. CRC/Taylor & Francis, Boca RatonzbMATHGoogle Scholar
  76. Freeden W (2015) Geomathematics: Its Role, Its Aim, and Its Potential. In: Freeden W, Nashed Z, Sonar T (Eds) Handbook of Geomathematics, 2nd Edn. SpringerCrossRefGoogle Scholar
  77. Freeden W, Blick C (2013) Signal decorrelation by means of multiscale methods. World Min 65:304–317Google Scholar
  78. Freeden W, Gerhards C (2010) Poloidal and toroidal field modeling in terms of locally supported vector wavelets. Math Geosci 42:817–838MathSciNetzbMATHCrossRefGoogle Scholar
  79. Freeden W, Gerhards C (2013) Geomathematically oriented potential theory. Chapman & Hall/CRC, Boca RatonzbMATHGoogle Scholar
  80. Freeden W, Gutting M (2013) Special functions of mathematical (geo-)physics. Birkhäuser, BaselzbMATHCrossRefGoogle Scholar
  81. Freeden W, Kersten H (1981) A constructive approximation theorem for the oblique derivative problem in potential theory. Math Methods Appl Sci 3:104–114MathSciNetzbMATHCrossRefGoogle Scholar
  82. Freeden W, Mayer C (2003) Wavelets generated by layer potentials. Appl Comput Harm Anal 14:195–237MathSciNetzbMATHCrossRefGoogle Scholar
  83. Freeden W, Michel V (2004) Multiscale potential theory with applications to geoscience. Birkhäuser, BostonzbMATHCrossRefGoogle Scholar
  84. Freeden W, Nutz H (2011) Satellite gravity gradiometry as tensorial inverse problem. Int J Geomath 2:123–146MathSciNetzbMATHCrossRefGoogle Scholar
  85. Freeden W, Nutz H (2014) Mathematische Methoden. In: Bauer M, Freeden W, Jacobi H, Neu T (eds) Handbuch Tiefe Geothermie. Springer, Heidelberg, pp 125–222Google Scholar
  86. Freeden W, Reuter R (1990) A constructive method for solving the displacement boundary-value problem of elastostatics by use of global basis systems. Math Methods Appl Sci 12:105–128MathSciNetzbMATHCrossRefGoogle Scholar
  87. Freeden W, Schreiner M (2006) Local multiscale modelling of geoid undulations from deflections of the vertical. J Geodesy 79:641–651zbMATHCrossRefGoogle Scholar
  88. Freeden W, Schreiner M (2009) Spherical functions of mathematical geosciences: a scalar, vectorial, and tensorial setup. Springer, BerlinzbMATHGoogle Scholar
  89. Freeden W, Wolf K (2009) Klassische Erdschwerefeldbestimmung aus der Sicht moderner Geomathematik. Math Semesterber 56:53–77MathSciNetCrossRefGoogle Scholar
  90. Freeden W, Gervens T, Schreiner M (1998) Constructive approximation on the sphere (with applications to geomathematics). Oxford Science Publications/Clarendon, OxfordzbMATHGoogle Scholar
  91. Freeden W, Mayer C, Schreiner M (2003) Tree algorithms in wavelet approximations by Helmholtz potential operators. Numer Funct Anal Optim 24:747–782MathSciNetzbMATHCrossRefGoogle Scholar
  92. Freeden W, Fehlinger T, Klug M, Mathar D, Wolf K (2009) Classical globally reflected gravity field determination in modern locally oriented multiscale framework. J Geodesy 83:1171–1191CrossRefGoogle Scholar
  93. Gehringer M, Loksha V (2012) Handbook on planning and financing geothermal power generation. ESMAP (Energy Sector Management Assistence Programm), main findings and recommendations, The International Bank for Reconstruction and Development, WashingtonGoogle Scholar
  94. Georgsson LS, Friedleifsson IB (2009) Geothermal energy in the world from energy perspective. In: Short course IV on exploration for geothermal resources, Lake Naivasha, pp 1–22Google Scholar
  95. Geothermal Energy Association (2011) Annual US geothermal power production and development report. Technical reportGoogle Scholar
  96. Gerhards C (2011) Spherical multiscale methods in terms of locally supported wavelets: theory and application to geomagnetic modeling. PhD thesis, Geomathematics Group, University of KaiserslauternGoogle Scholar
  97. Gerhards C (2012) Locally supported wavelets for the separation of spherical vector fields with respect to their sources. Int J Wavel Multires Inf Proc 10:1250034MathSciNetzbMATHCrossRefGoogle Scholar
  98. Gerhards C (2014) A multiscale power spectrum for the analysis of the lithospheric magnetic field. Int J Geomath. 5:63–79MathSciNetzbMATHCrossRefGoogle Scholar
  99. Ghassemi A (2003) A thermoelastic hydraulic fracture design tool for geothermal reservoir development. Technical report, Department of Geology & Geological Engineering, University of North DakotaCrossRefGoogle Scholar
  100. Ghassemi A, Tarasovs S (2004) Three-dimensional modeling of injection induced thermal stresses with an example from Coso. In: Proceedings, 29th workshop on geothermal reservoir engineering, Stanford University, StanfordGoogle Scholar
  101. Ghassemi A, Zhang Q (2004) Poro-thermoelastic mechanisms in wellbore stability and reservoir stimulation. In: Proceedings, 29th workshop on geothermal reservoir engineering, Stanford University, StanfordGoogle Scholar
  102. Ghassemi A, Tarasovs S, Cheng AHD (2003) An integral equation solution for three-dimensional heat extraction from planar fracture in hot dry rock. Int J Numer Anal Methods Geomech 27:989–1004zbMATHCrossRefGoogle Scholar
  103. Golberg MA, Chen CS (1998) The method of fundamental solutions for potential, Helmholtz and diffusion problems. In: Golberg MA (ed) Boundary integral methods – numerical and mathematical aspects. Computational mechanics publications. WIT, Southhampton, pp 103–176Google Scholar
  104. Gorenflo R, Mainardi F (1997) Fractional calculus: integral and differential equations of fractional order. In: Carpinteri A, Mainardi F (eds) Fractals and fractional calculus in continuum mechanics. Springer, Wien, pp 223–276CrossRefGoogle Scholar
  105. Hammons TJ (2011) Geothermal power generation: global perspectives, technology, direct uses, plants, drilling and sustainability worldwide. In: Electricity infrastructures in the global marketplace. InTech, pp 195–234Google Scholar
  106. Haney MM, Bartel LC, Aldridge DF, Symons NP (2005) Insight into the output of reverse-time migration: what do the amplitudes mean? In: Proceedings, SEG annual meeting, HoustonGoogle Scholar
  107. Helmig R, Niessner J, Flemisch B, Wolff M, Fritz J (2014) Efficient modeling of flow and transport in porous media using multi-physics and multi-scale approaches. In: Freeden W, Nashed Z, Sonar T (eds) Handbook of geomathematics, 2nd edn. Springer, New YorkGoogle Scholar
  108. Heuer N, Küpper T, Windelberg D (1991) Mathematical model of a hot dry rock system. Geophys J Int 105:659–664CrossRefGoogle Scholar
  109. Hicks TW, Pine RJ, Willis-Richards J, Xu S, Jupe AJ, Rodrigues NEV (1996) A hydro-thermo-mechanical numerical model for HDR geothermal reservoir evaluation. Int J Rock Mech Min Sci 33:499–511CrossRefGoogle Scholar
  110. Hillis RR (2000) Pore pressure/stress coupling and its implications for seismicity. Explor Geophys 31:448–454CrossRefGoogle Scholar
  111. Hillis RR (2001) Coupled changes in pore pressure and stress in oil fields and sedimentary basins. Pet Geosci 7:419–425CrossRefGoogle Scholar
  112. Hillis RR (2003) Pore pressure/stress coupling and its implications for rock failure. In: Vanrensbergen P, Hillis RR, Maltman AJ, Morley CK (eds) Subsurface sediment mobilization. Geological Society of London, London, pp 359–368Google Scholar
  113. Ilyasov M (2011) A tree algorithm for Helmholtz potential wavelets on non-smooth surfaces: theoretical background and application to seismic data postprocessing. PhD thesis, Geomathematics Group, University of KaiserslauternGoogle Scholar
  114. International Energy Agency (2010) Annual report. Technical reportGoogle Scholar
  115. Itasca Consulting Group Inc (2000) UDEC user’s guide. MinnesotaGoogle Scholar
  116. Jackson JD (1998) Classical electrodynamics. Wiley, New YorkzbMATHGoogle Scholar
  117. Jacobs F, Meyer H (1992) Geophysik – Signale aus der Erde. Teubner, StuttgartCrossRefGoogle Scholar
  118. Jaeger JC, Cook NGW, Zimmerman RW (2007) Fundamentals of rock mechanics. Blackwell, MaldenGoogle Scholar
  119. Jia X, Hu T (2006) Element-free precise integration method and its application in seismic modelling and imaging. Geophys J Int 166:349–372CrossRefGoogle Scholar
  120. Jing L, Hudson JA (2002) Numerical methods in rock mechanics. Int J Rock Mech Min Sci 39:409–427CrossRefGoogle Scholar
  121. Jing Z, Willis-Richards J, Watanabe K, Hashida T (2000) A three-dimensional stochastic rock mechanics model of engineered geothermal systems in fractured crystalline rock. J Geophys Res 105:23663–23679CrossRefGoogle Scholar
  122. Jing Z, Watanabe K, Willis-Richards J, Hashida T (2002) A 3-D water/rock chemical interaction model for prediction of HDR/HWR geothermal reservoir performance. Geothermics 31:1–28CrossRefGoogle Scholar
  123. Johansson BT, Lesnic D (2008) A method of fundamental solutions for transient heat conduction. Eng Anal Bound Elem 32:697–703zbMATHCrossRefGoogle Scholar
  124. Johansson BT, Lesnic D, Reeve T (2011) A method of fundamental solutions for two-dimensional heat conduction. Int J Comput Math 88:1697–1713MathSciNetzbMATHCrossRefGoogle Scholar
  125. John V, Schmeyer E (2008) Finite element methods for time-dependent convection-diffusion-reaction equations with small diffusion. Comput Methods Appl Mech Eng 198:475–494MathSciNetzbMATHCrossRefGoogle Scholar
  126. John V, Kaya S, Layton W (2006) A two-level variational multiscale method for convection-dominated convection-diffusion equations. Comput Methods Appl Mech Eng 195:4594–4603MathSciNetzbMATHCrossRefGoogle Scholar
  127. Jung R (2007) Stand und Aussichten der Tiefengeothermie in Deutschland. Erdöl, Erdgas, Kohle 123:1–7Google Scholar
  128. Katsurada M (1989) A mathematical study of the charge simulation method II. J Fac Sci Univ Tokyo Sect IA Math 36:135–162MathSciNetzbMATHGoogle Scholar
  129. Katsurada M, Okamoto H (1996) The collocation points of the fundamental solution method for the potential problem. Comput Math Appl 31:123–137MathSciNetzbMATHCrossRefGoogle Scholar
  130. Kazemi H (1969) Pressure transient analysis of naturally fractured reservoirs with uniform fracture distribution. Soc Petrol Eng J 246:451–461CrossRefGoogle Scholar
  131. Kazemi H, Merril LS, Porterfield KL, Zeman PR (1976) Numerical simulation of water-oil flow in naturally fractured reservoirs. In: Proceedings, SPE-AIME 4th symposium on numerical simulation of reservoir performance, Los AngelesGoogle Scholar
  132. Kim I, Lindquist WB, Durham WB (2003) Fracture flow simulation using a finite-difference lattice Boltzmann method. Phys Rev E 67:046708CrossRefGoogle Scholar
  133. Kimura S, Masuda Y, Hayashi K (1992) Efficient numerical method based on double porosity model to analyze heat and fluid flows in fractured rock formations. JSME Int J Ser 2 35:395–399Google Scholar
  134. Kühn M (2009) Modelling feed-back of chemical reactions on flow fields in hydrothermal systems. Surv Geophys 30:233–251CrossRefGoogle Scholar
  135. Kühn M, Stöfen H (2005) A reactive flow model of the geothermal reservoir Waiwera, New Zealand. Hydrogeol J 13:606–626CrossRefGoogle Scholar
  136. Kupradze VD (1964) A method for the approximate solution of limiting problems in mathematical physics. USSR Comput Math Math Phys 4:199–205zbMATHCrossRefGoogle Scholar
  137. Lai M, Krempl E, Ruben D (2010) Introduction to continuum mechanics. Butterworth-Heinemann, BurlingtonGoogle Scholar
  138. Landau LD, Pitaevskii LP, Lifshitz EM, Kosevich AM (1986) Theory of elasticity. Theoretical physics, vol 7, 3rd edn. Butterworth-Heinemann, OxfordGoogle Scholar
  139. Lang U (1995) Simulation regionaler Strömungs- und Transportvorgänge in Karstaquifern mit Hilfe des Doppelkontinuum-Ansatzes: Methodenentwicklung und Parameteridentifikation. PhD thesis, University of StuttgartGoogle Scholar
  140. Lang U, Helmig R (1995) Numerical modeling in fractured media – identification of measured field data. In: Herbert M, Kovar K (eds) Groundwater quality: remediation and protection. IAHS and University Karlova, Prague, pp 203–212Google Scholar
  141. Lee J, Choi SU, Cho W (1999) A comparative study of dual-porosity model and discrete fracture network model. KSCE J Civ Eng 3:171–180CrossRefGoogle Scholar
  142. Li X (2008a) Convergence of the method of fundamental solutions for Poisson’s equation on the unit sphere. Adv Comput Math 28:269–282MathSciNetzbMATHCrossRefGoogle Scholar
  143. Li X (2008b) Rate of convergence of the method of fundamental solutions and hyperinterpolation for modified Helmholtz equations on the unit ball. Adv Comput Math 29:393–413MathSciNetzbMATHCrossRefGoogle Scholar
  144. Lomize GM (1951) Seepage in fissured rocks. State Press, MoscowGoogle Scholar
  145. Long J, Remer J, Wilson C, Witherspoon P (1982) Porous media equivalents for networks of discontinuous fractures. Water Resour Res 18:645–658CrossRefGoogle Scholar
  146. Luchko Y (2009) Maximum principle for the generalized time-fractional diffusion equation. J Math Anal Appl 351:218–223MathSciNetzbMATHCrossRefGoogle Scholar
  147. Luchko Y (2010) Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation. Comput Math Appl 59:1766–1772MathSciNetzbMATHCrossRefGoogle Scholar
  148. Luchko Y (2015) Fractional diffusion and wave propagation. In: Freeden W, Nashed M, Sonar T (Eds) Handbook of Geomathematics, 2nd Edn. SpringerGoogle Scholar
  149. Luchko Y, Punzi A (2011) Modeling anomalous heat transport in geothermal reservoirs via fractional diffusion equations. Int J Geomath 1:257–276MathSciNetzbMATHCrossRefGoogle Scholar
  150. Martin GS, Marfurt KJ, Larsen S (2002) Marmousi-2: an updated model for the investigation of AVO in structurally complex areas. In: Proceedings, SEG annual meeting, Salt Lake CityGoogle Scholar
  151. Maryška J, Severýn O, Vohralík M (2004) Numerical simulation of fracture flow in mixed-hybrid FEM stochastic discrete fracture network model. Comput Geosci 8:217–234MathSciNetzbMATHCrossRefGoogle Scholar
  152. Masahi M, King P, Nurafza P (2007) Fast estimation of connectivity in fractured reservoirs using percolation theory. SPE J 12:167–178CrossRefGoogle Scholar
  153. Mayer C (2007) A wavelet approach to the Stokes problem. Habilitation thesis, Geomathematics Group, University of KaiserslauternGoogle Scholar
  154. Mayer C, Freeden W (2015) Stokes problem, layer potentials and regularizations, multiscale applications. In: Freeden W, Nashed Z, Sonar T (Eds) Handbook of Geomathematics, 2nd Edn. SpringerGoogle Scholar
  155. McLean W, Mustapha K (2009) Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation. Numer Algorithms 52:69–88MathSciNetzbMATHCrossRefGoogle Scholar
  156. Menke W (1984) Geophysical data analysis: discrete inverse theory. Academic, OrlandozbMATHGoogle Scholar
  157. Michel V (2002) A multiscale approximation for operator equations in separable Hilbert spaces – case study: reconstruction and description of the Earth’s interior. Habilitation thesis, Geomathematics Group, University of KaiserslauternGoogle Scholar
  158. Michel V, Fokas AS (2008) A unified approach to various techniques for the non-uniqueness of the inverse gravimetric problem and wavelet based methods. Inverse Probl 24:045019MathSciNetzbMATHCrossRefGoogle Scholar
  159. Min KB, Jing L, Stephansson O (2004) Determining the equivalent permeability tensor for fractured rock masses using a stochastic REV approach: method and application to the field data from Sellafield, UK. Hydrogeol J 12:497–510CrossRefGoogle Scholar
  160. MIT (Massachusetts Institute of Technology) (2006) The future of geothermal energy. http://mitei.mit.edu/publications/reports-studies/future-geothermal-energy
  161. Mo H, Bai M, Lin D, Roegiers JC (1998) Study of flow and transport in fracture network using percolation theory. Appl Math Model 22:277–291CrossRefGoogle Scholar
  162. Moeck I, Kwiatek G, Zimmermann G (2009) The in-situ stress field as a key issue for geothermal field development – a case study from the NE German Basin. In: Proceedings, 71st EAGE conference & exhibition, AmsterdamGoogle Scholar
  163. Möhringer S (2014) Decorrelation of gravimetric data. PhD thesis, Geomathematics Group, University of KaiserslauternGoogle Scholar
  164. Mongillo M (2011) International efforts to promote global sustainable geothermal development. In: GIA annual report executive summary, Singapore, pp 1–19Google Scholar
  165. Morgan WJ (1971) Convective plumes in the lower mantle. Nature 230:42–43CrossRefGoogle Scholar
  166. Müller C (1998) Analysis of spherical symmetries in euclidean spaces. Applied mathematical sciences, vol 129. Springer, BerlinGoogle Scholar
  167. Müller C, Kersten H (1980) Zwei Klassen vollständiger Funktionensysteme zur Behandlung der Randwertaufgaben der Schwingungsgleichung \(\bigtriangleup U + k^{2}U = 0\). Math Method Appl Sci 2:48–67zbMATHCrossRefGoogle Scholar
  168. Nakao S, Ishido T (1998) Pressure-transient behavior during cold water injection into geothermal wells. Geothermics 27:401–413CrossRefGoogle Scholar
  169. Neuman S (2005) Trends, prospects and challenges in quantifying flow and transport through fractured rocks. Hydrogeol J 13:124–147CrossRefGoogle Scholar
  170. Neuman S, Depner J (1988) Use of variable-scale pressure test data to estimate the log hydraulic conductivity covariance and dispersivity of fractured granites near Oracle, Arizona. J Hydrol 102:475–501CrossRefGoogle Scholar
  171. Nolet G (2008) Seismic tomography: imaging the interior of the Earth and Sun. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  172. Oden M, Niemi A (2006) From well-test data to input to stochastic continuum models: effect of the variable support scale of the hydraulic data. Hydrogeol J 14:1409–1422CrossRefGoogle Scholar
  173. Ödner H (1998) One-dimensional transient flow in a finite fractured aquifer system. Hydrol Sci J 43:243–265CrossRefGoogle Scholar
  174. Ostermann I (2011a) Modeling heat transport in deep geothermal systems by radial basis functions. PhD thesis, Geomathematics Group, University of KaiserslauternGoogle Scholar
  175. Ostermann I (2011b) Three-dimensional modeling of heat transport in deep hydrothermal reservoirs. Int J Geomath 2:37–68MathSciNetzbMATHCrossRefGoogle Scholar
  176. O’Sullivan MJ, Pruess K, Lippmann MJ (2001) State of the art of geothermal reservoir simulation. Geothermics 30:395–429CrossRefGoogle Scholar
  177. Ouenes A (2000) Practical application of fuzzy logic and neural networks to fractured reservoir characterization. Comput Geosci 26:953–962CrossRefGoogle Scholar
  178. Peters RR, Klavetter EA (1988) A continuum model for water movement in an unsaturated fractured rock mass. Water Resour Res 24:416–430CrossRefGoogle Scholar
  179. Phillips PJ (2005) Finite element method in linear poroelasticity: theoretical and computational results. PhD thesis, University of Texas, AustinGoogle Scholar
  180. Phillips PJ, Wheeler MF (2007) A coupling of mixed and continuous Galerkin finite element methods for poroelasticity I: the continuous in time case. Comput Geosci 11:131–144MathSciNetzbMATHCrossRefGoogle Scholar
  181. Phillips WS, Rutledge JT, House LS, Fehler MC (2002) Induced microearthquake patterns in hydrocarbon and geothermal reservoirs: six case studies. Pure Appl Geophys 159:345–369CrossRefGoogle Scholar
  182. Podvin P, Lecomte I (1991) Finite difference computation of traveltimes in very contrasted velocity models: a massively parallel approach and its associated tools. Geophys J Int 105:271–284CrossRefGoogle Scholar
  183. Popov M (1982) A new method of computation of wave fields using Gaussian beams. Wave Motion 4:85–97MathSciNetzbMATHCrossRefGoogle Scholar
  184. Pruess K (1990) Modelling of geothermal reservoirs: fundamental processes, computer simulation and field applications. Geothermics 19:3–15CrossRefGoogle Scholar
  185. Pruess K, Narasimhan TN (1985) A practical method for modeling fluid and heat flow in fractured porous media. Soc Pet Eng J 25:14–26CrossRefGoogle Scholar
  186. Pruess K, Wang JSY, Tsang YW (1986) Effective continuum approximation for modeling fluid and heat flow in fractured porous tuff. Technical report, Sandia National Laboratories Report SAND86-7000, AlbuquerqueGoogle Scholar
  187. Reichenberger V, Jakobs H, Bastian P, Helmig R (2006) A mixed-dimensional finite volume method for two-phase flow in fractured porous media. Adv Water Resour 29:1020–1036CrossRefGoogle Scholar
  188. Renaut R, Fröhlich J (1996) A pseudospectral Chebychev method for 2D wave equation with domain stretching and absorbing boundary conditions. J Comput Phys 124:324–336MathSciNetzbMATHCrossRefGoogle Scholar
  189. Renner J, Steeb H (2015) Modeling of fluid transport in geothermal research. In: Freeden W, Nashed Z, Sonar T (Eds) Handbook of Geomathematics, 2nd Edn. SpringerGoogle Scholar
  190. Rice JR, Cleary MP (1976) Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents. Rev Geophys Space Phys 14:227–241CrossRefGoogle Scholar
  191. Ritter JRR, Christensen UR (2007) Mantle plumes: a multidisciplinary approach. Springer, BerlinCrossRefGoogle Scholar
  192. Runge C (1885) Zur Theorie der eindeutigen analytischen Funktionen. Acta Math 6:229–234MathSciNetCrossRefGoogle Scholar
  193. Rutqvist J, Stephansson O (2003) The role of hydromechanical coupling in fractured rock engineering. Hydrogeol J 11:7–40CrossRefGoogle Scholar
  194. Saemundsson K (2009) Geothermal systems in global perspective. In: Short course IV on exploration for geothermal resources, Lake NaivashaGoogle Scholar
  195. Sahimi M (1995) Flow and transport in porous media and fractured rock: from classical methods to modern approaches. VCH, WeinheimzbMATHGoogle Scholar
  196. Sanyal SK (2005) Classification of geothermal systems – a possible scheme. In: Proceedings, 30th workshop on geothermal reservoir engineering, Stanford University, Stanford, SGP-TR-176, pp 85–92Google Scholar
  197. Sanyal SK, Butler SJ, Swenson D, Hardeman B (2000) Review of the state-of-the-art of numerical simulation of enhanced geothermal systems. In: Proceedings, world geothermal congress, Kyushu-TohokuGoogle Scholar
  198. Schanz M (2001) Application of 3D time domain boundary element formulation to wave propagation in poroelastic solids. Eng Anal Bound Elem 25:363–376zbMATHCrossRefGoogle Scholar
  199. Schubert G, Turcotte DL, Olson P (2001) Mantle convection in the Earth and Planets. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  200. Schulz R (2009) Aufbau eines geothermischen Informationssystems für Deutschland. Technical report, Leibniz-Institut für Angewandte Geophysik, HannoverGoogle Scholar
  201. Semtchenok NM, Popov MM, Verdel AR (2009) Gaussian beam tomography. In: Extended abstracts, 71st EAGE conference & exhibition, AmsterdamGoogle Scholar
  202. Showalter RE (2000) Diffusion in poro-elastic media. J Math Anal Appl 251:310–340MathSciNetzbMATHCrossRefGoogle Scholar
  203. Smyrlis Y-S (2009a) Applicability and applications of the method of fundamental solutions. Math Comput 78:1399–1434MathSciNetzbMATHCrossRefGoogle Scholar
  204. Smyrlis Y-S (2009b) Mathematical foundation of the MFS for certain elliptic systems in linear elasticity. Numer Math 112:319–340MathSciNetzbMATHCrossRefGoogle Scholar
  205. Smyrlis Y-S, Karageorghis A (2009) Efficient implementation of the MFS: the three scenarios. J Comput Appl Math 227:83–92MathSciNetzbMATHCrossRefGoogle Scholar
  206. Snieder R (2002) The perturbation method in elastic wave scattering and inverse scattering in pure and applied science. In: General theory of elastic wave. Academic, San Diego, pp 528–542Google Scholar
  207. Snow DT (1965) A parallel plate model of fractured permeable media. PhD thesis, University of California, BerkeleyGoogle Scholar
  208. Stothoff S, Or D (2000) A discrete-fracture boundary integral approach to simulating coupled energy and moisture transport in a fractured porous medium. In: Faybishenko B, Witherspoon PA, Benson SM (eds) Dynamics of fluids in fractured rocks, concepts and recent advances. AGU geophysical monograph, vol 122. American Geophysical Union, Washington, DC, pp 269–279Google Scholar
  209. Sudicky EA, McLaren RG (1992) The Laplace transform Galerkin technique for large-scale simulation of mass transport in discretely fractured porous formations. Water Resour Res 28:499–514CrossRefGoogle Scholar
  210. Symes WW (2003) Kinematics of reverse time S-G migration. Technical report, Rice UniversityGoogle Scholar
  211. Symes WW (2007) Reverse time migration with optimal checkpointing. Geophysics 72:SM213–SM221CrossRefGoogle Scholar
  212. Takenaka H, Wang Y, Furumura T (1999) An efficient approach of the pseudospectral method for modelling of geometrical symmetric seismic wavefields. Earth Planets Space 51:73–79CrossRefGoogle Scholar
  213. Tang DH, Frind EO, Sudicky EA (1981) Contaminant transport in fractured porous media: analytical solution for a single fracture. Water Resour Res 17:555–564CrossRefGoogle Scholar
  214. Tran NH, Rahman SS (2006) Modelling discrete fracture networks using neuro-fractal-stochastic simulation. J Eng Appl Sci 1:154–160Google Scholar
  215. Travis BJ (1984) TRACR3D: a model of flow and transport in porous/fractured media. Technical report, Los Alamos National Laboratory LA-9667-MS, Los AlamosGoogle Scholar
  216. Trefftz E (1926) Ein Gegenstück zum Ritzschen Verfahren. In: Proceedings of the 2nd international congress for applied mechanics, ZürichGoogle Scholar
  217. Tsang Y, Tsang C (1987) Chanel flow model through fractured media. Water Resour Res 23:467–479CrossRefGoogle Scholar
  218. Tsang Y, Tsang C (1989) Flow chaneling in a single fracture as a two-dimensional strongly heterogeneous permeable medium. Water Resour Res 25:2076–2080CrossRefGoogle Scholar
  219. Tsang Y, Tsang C, Hale F, Dverstorp B (1996) Tracer transport in a stochastic continuum model of fractured media. Water Resour Res 32:3077–3092CrossRefGoogle Scholar
  220. Turcotte DL, Schubert G (2001) Geodynamics. Cambridge University Press, CambridgeGoogle Scholar
  221. Vidale J (1988) Finite-difference calculation of travel times. Bull Seismol Soc Am 78:2062–2076Google Scholar
  222. Walsh J (1929) The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions. Bull Am Math Soc 35:499–544MathSciNetzbMATHCrossRefGoogle Scholar
  223. Warren JE, Root PJ (1963) The behaviour of naturally fractured reservoirs. Soc Pet Eng J 228:245–255CrossRefGoogle Scholar
  224. Watanabe K, Takahashi T (1995) Fractal geometry characterization of geothermal reservoir fracture networks. J Geophys Res 100:521–528CrossRefGoogle Scholar
  225. Welding L (2007) GLITNIR geothermal research. In: United States geothermal energy market report, pp 1–37Google Scholar
  226. Wendland H (2005) Scattered data approximation. Cambridge monographs on applied and computational mathematics, vol 17. Cambridge University Press, CambridgeGoogle Scholar
  227. Wilson JT (1963) A possible origin of the Hawaiian island. Can J Phys 41:863–868CrossRefGoogle Scholar
  228. Wolf K (2009) Multiscale modeling of classical boundary value problems in physical geodesy by locally supported wavelets. PhD thesis, Geomathematics Group, University of KaiserslauternGoogle Scholar
  229. Wu YS (2000) On the effective continuum method for modeling multiphase flow, multicomponent transport and heat transfer in fractured rock. In: Faybishenko B, Witherspoon PA, Benson SM (eds) Dynamics of fluids in fractured rocks, concepts and recent advances. American Geophysical Union, Washington, DC, pp 299–312CrossRefGoogle Scholar
  230. Wu YS, Pruess K (2005) A physically based numerical approach for modeling fracture-matrix interaction in fractured reservoirs. In: Proceedings, world geothermal congress 2005, AntalyaGoogle Scholar
  231. Wu YS, Qin G (2009) A generalized numerical approach for modeling multiphase flow and transport in fractured porous media. Commun Comput Phys 6:85–108MathSciNetCrossRefGoogle Scholar
  232. Wu X, Pope GA, Shook GM, Srinivasan S (2005) A semi-analytical model to calculate energy production in single fracture geothermal reservoirs. Geotherm Resour Counc Trans 29:665–669Google Scholar
  233. Wu RS, Xie XB, Wu XY (2006) One-way and one-return approximations (de Wolf approximation) for fast elastic wave modeling in complex media. Adv Geophys 48:265–322CrossRefGoogle Scholar
  234. Xie XB, Wu RS (2006) A depth migration method based on the full-wave reverse time calculation and local one-way propagation. In: Proceedings, SEG annual meeting, New OrleansCrossRefGoogle Scholar
  235. Yilmaz O (1987) Seismic data analysis: processing, inversion, and interpretation of seismic data. Society of Exploration Geophysicists, TulsaGoogle Scholar
  236. Yin S (2008) Geomechanics-reservoir modeling by displacement discontinuity-finite element method. PhD thesis, University of Waterloo, OntarioGoogle Scholar
  237. Zhao C, Hobbs BE, Baxter K, Mühlhaus HB, Ord A (1999) A numerical study of pore-fluid, thermal and mass flow in fluid-saturated porous rock basins. Eng Comput 16:202–214zbMATHCrossRefGoogle Scholar
  238. Zhou XX, Ghassemi A (2009) Three-dimensional poroelastic simulation of hydraulic and natural fractures using the displacement discontinuity method. In: Proceedings of the 34th workshop on geothermal reservoir engineering, StanfordGoogle Scholar
  239. Zubkov VV, Koshelev VF, Lin’kov AM (2007) Numerical modeling of hydraulic fracture initiation and development. J Min Sci 43:40–56CrossRefGoogle Scholar
  240. Zyvoloski G (1983) Finite element methods for geothermal reservoir simulation. Int J Numer Anal Methods Geomech 7:75–86zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Matthias Augustin
    • 1
  • Mathias Bauer
    • 2
  • Christian Blick
    • 1
  • Sarah Eberle
    • 1
  • Willi Freeden
    • 1
  • Christian Gerhards Maxim Ilyasov
    • 1
  • René Kahnt
    • 3
  • Matthias Klug
    • 1
  • Sandra Möhringer
    • 1
  • Thomas Neu
    • 4
  • Helga Nutz
    • 1
  • Isabel Michel née Ostermann
    • 5
  • Alessandro Punzi
    • 1
  1. 1.Geomathematics GroupUniversity of KaiserslauternKaiserslauternGermany
  2. 2.CBM GmbHBexbachGermany
  3. 3.G.E.O.S. Ingenieurgesellschaft mbHFreibergGermany
  4. 4.Tiefe Geothermie Saar GmbHSaarbrückenGermany
  5. 5.Fraunhofer ITWMKaiserslauternGermany

Personalised recommendations