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Numerical aspects of hydro-mechanical coupling of fluid-filled fractures using hybrid-dimensional element formulations and non-conformal meshes

  • Patrick SchmidtEmail author
  • Holger Steeb
Original Paper
  • 4 Downloads
Part of the following topical collections:
  1. Numerical methods for processes in fractured porous media

Abstract

In the field of porous and fractured media, subsurface flow provides insight into the characteristics of fluid storage and properties connected to underground matter and heat transport. Subsurface flow is precisely described by many diffusion based models in the literature. However, diffusion-based models lack to reproduce important hydro-mechanical coupling phenomena like inverse water-level fluctuations (Noordbergum effect). In theory, contemporary modeling approaches, such as direct numerical simulations (DNS) of surface-coupled fluid-solid (fracture) interactions or coarse-grained continuum approaches like Biot’s theory, are capable of capturing such phenomena. Nevertheless, during modeling processes of fractures with high aspect ratios, DNS methods with the explicit discretization of the fluid domain fail, and coarse-grained continuum approaches require a non-linear formulation for the fracture deformation since large deformation can be reached easily within fractures. Hence a hybrid-dimensional approach uses a parabolic velocity profile to avoid an explicit discretization of the fluid domain within the fracture. For fracture flow, the primary variable is the pressure field only, and the fracture domain is reduced by one dimension. The interaction between the fracture and the surrounding matrix domain, respectively, is realized by modified balance equations. The coupled system is numerically stiff when fluids are described with a low compressibility modulus. Two algorithms are proposed within this work, namely the weak coupling scheme, which uses an implicit staggered-iterative algorithm to find the residual state and the strong coupling scheme which directly couples both domains by implementing interface elements. In the course of this work, a consistent implementation scheme for the coupling of hybrid-dimensional elements with a surrounding bulk matrix is proposed and validated and tested throughout different numerical experiments.

Keywords

Fracture flow Hydromechanical coupling Deformation-induced flow Pressure diffusion 

Mathematics Subject Classification

86-08 

Notes

Acknowledgements

The authors gratefully acknowledge the funding provided by the German Federal Ministry of Education and Research (BMBF) for the GeomInt project, Grant Number 03A0004E, within the BMBF Geoscientific Research Program “Geo:N Geosciences for Sustainability”.

References

  1. Adachi, J., Siebrits, E., Peirce, A., Desroches, J.: Computer simulation of hydraulic fractures. Int. J. Rock Mech. Min. Sci. 44(5), 739–757 (2007)CrossRefGoogle Scholar
  2. Alnæs, M.S., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., Richardson, C., Ring, J., Rognes, M.E., Wells, G.N.: The FEniCS project version 1.5. Arch. Numer. Softw. 3(100), 9–23 (2015)Google Scholar
  3. Bastian, P., Heimann, F., Marnach, S.: Generic implementation of finite element methods in the distributed and unified numerics environment (DUNE). Kybernetika 46(2), 294–315 (2010)MathSciNetzbMATHGoogle Scholar
  4. Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12(2), 155–164 (1941)CrossRefGoogle Scholar
  5. Brenner, K., Hennicker, J., Masson, R., Samier, P.: Gradient discretization of hybrid-dimensional darcy flow in fractured porous media with discontinuous pressures at matrix—fracture interfaces. IMA J. Numer. Anal. 37(3), 1551–1585 (2017).  https://doi.org/10.1093/imanum/drw044 MathSciNetCrossRefzbMATHGoogle Scholar
  6. Castelletto, N., White, J.A., Tchelepi, H.A.: Accuracy and convergence properties of the fixed-stress iterative solution of two-way coupled poromechanics. Int. J. Numer. Anal. Methods Geomech. 39(14), 1593–1618 (2015).  https://doi.org/10.1002/nag.2400 CrossRefGoogle Scholar
  7. Coussy, O.: Poromechanics. Wiley, New York (2004)zbMATHGoogle Scholar
  8. Ehlers, W., Bluhm, J.: Porous Media: Theory, Experiments and Numerical Applications. Springer, Berlin (2013)zbMATHGoogle Scholar
  9. Geertsma, J., De Klerk, F.: A rapid method of predicting width and extent of hydraulically induced fractures. J. Pet. Technol. 21(12), 1–571 (1969)CrossRefGoogle Scholar
  10. Geuzaine, C., Remacle, J.F.: Gmsh: a 3-D finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng. 79(11), 1309–1331 (2009)MathSciNetCrossRefGoogle Scholar
  11. Girault, V., Wheeler, M.F., Ganis, B., Mear, M.E.: A lubrication fracture model in a poro-elastic medium. Math. Models Methods Appl. Sci. 25(04), 587–645 (2015).  https://doi.org/10.1142/S0218202515500141 MathSciNetCrossRefzbMATHGoogle Scholar
  12. Girault, V., Kumar, K., Wheeler, M.F.: Convergence of iterative coupling of geomechanics with flow in a fractured poroelastic medium. Comput. Geosci. 20(5), 997–1011 (2016).  https://doi.org/10.1007/s10596-016-9573-4 MathSciNetCrossRefzbMATHGoogle Scholar
  13. Guiducci, C., Collin, F., Radu, J.P., Pellegrino, A., Charlier, R.: Numerical modeling of hydro-mechanical fracture behaviour. NUMOG VIII, pp. 293–299 (2003)Google Scholar
  14. Hanowski, K.K., Sander, O.: Simulation of deformation and flow in fractured. Poroelastic materials. ArXiv e-prints (2016)Google Scholar
  15. Kim, J.M., Parizek, R.R.: Numerical simulation of the noordbergum effect resulting from groundwater pumping in a layered aquifer system. J. Hydrol. 202(1), 231–243 (1997).  https://doi.org/10.1016/S0022-1694(97)00067-X CrossRefGoogle Scholar
  16. Kim, J., Tchelepi, H., Juanes, R.: Stability and convergence of sequential methods for coupled flow and geomechanics: drained and undrained splits. Comput. Methods Appl. Mech. Eng. 200(23), 2094–2116 (2011).  https://doi.org/10.1016/j.cma.2011.02.011 MathSciNetCrossRefzbMATHGoogle Scholar
  17. Kim, J., Tchelepi, H., Juanes, R.: Stability and convergence of sequential methods for coupled flow and geomechanics: fixed-stress and fixed-strain splits. Comput. Methods Appl. Mech. Eng. 200(13), 1591–1606 (2011).  https://doi.org/10.1016/j.cma.2010.12.022 MathSciNetCrossRefzbMATHGoogle Scholar
  18. Martin, V., Jaffr, J., Roberts, J.: Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26(5), 1667–1691 (2005).  https://doi.org/10.1137/S1064827503429363 MathSciNetCrossRefzbMATHGoogle Scholar
  19. Nordgren, R., et al.: Propagation of a vertical hydraulic fracture. Soc. Pet. Eng. J. 12(04), 306–314 (1972)CrossRefGoogle Scholar
  20. Oliver, D.S., Chen, Y.: Recent progress on reservoir history matching: a review. Comput. Geosci. 15(1), 185–221 (2011)CrossRefGoogle Scholar
  21. Ortiz, R,A.E., Renner, J., Jung, R.: Hydromechanical analyses of the hydraulic stimulation of borehole basel 1. Geophy. J. Int. 185(3), 1266–1287 (2011)CrossRefGoogle Scholar
  22. Ortiz, R,A.E., Jung, R., Renner, J.: Two-dimensional numerical investigations on the termination of bilinear flow in fractures. Solid Earth 4(2), 331–345 (2013)CrossRefGoogle Scholar
  23. Peirce, A.P., Siebrits, E.: A dual mesh multigrid preconditioner for the efficient solution of hydraulically driven fracture problems. Int. J. Numer. Methods Eng. 63(13), 1797–1823 (2005).  https://doi.org/10.1002/nme.1330 MathSciNetCrossRefzbMATHGoogle Scholar
  24. Perkins, T., Kern, L., et al.: Widths of hydraulic fractures. J. Pet. Technol. 13(09), 937–949 (1961)CrossRefGoogle Scholar
  25. Renner, J., Steeb, H.: Modeling of Fluid Transport in Geothermal Research, pp. 1443–1500. Springer, Berlin (2015)Google Scholar
  26. Renshaw, C.E.: On the relationship between mechanical and hydraulic apertures in rough-walled fractures. J. Geophys. Res. Solid Earth 100(B12), 24629–24636 (1995)CrossRefGoogle Scholar
  27. Rodrigues, J.: The Noordbergum effect and characterization of aquitards at the Rio Maior mining project. Ground Water 21, 200–207 (1983)CrossRefGoogle Scholar
  28. Sandve, T., Berre, I., Nordbotten, J.: An efficient multi-point flux approximation method for discrete fracturematrix simulations. J. Comput. Phys. 231(9), 3784–3800 (2012).  https://doi.org/10.1016/j.jcp.2012.01.023 MathSciNetCrossRefzbMATHGoogle Scholar
  29. Segura, J.M., Carol, I.: On zero-thickness interface elements for diffusion problems. Int. J. Numer. Anal. Methods Geomech. 28(9), 947–962 (2004).  https://doi.org/10.1002/nag.358 CrossRefzbMATHGoogle Scholar
  30. Segura, J.M., Carol, I.: Coupled HM analysis using zero-thickness interface elements with double nodes. Part II: verification and application. Int. J. Numer. Anal. Methods Geomech. 32(18), 2103–2123 (2008)CrossRefGoogle Scholar
  31. Segura, J.M., Carol, I.: Coupled HM analysis using zero-thickness interface elements with double nodes. Part I: theoretical model. Int. J. Numer. Anal. Methods Geomech. 32(18), 2083–2101 (2008)CrossRefGoogle Scholar
  32. Settgast, R.R., Fu, P., Walsh, S.D., White, J.A., Annavarapu, C., Ryerson, F.J.: A fully coupled method for massively parallel simulation of hydraulically driven fractures in 3-dimensions. Int. J. Numer. Anal. Methods Geomech. 41(5), 627–653 (2017)CrossRefGoogle Scholar
  33. Shen, B., Stephansson, O., Rinne, M.: Hydro-Mechanical Coupling, pp. 77–82. Springer, Dordrecht (2014)Google Scholar
  34. Sneddon, I.N., Elliot, H.A.: The opening of a Griffith crack under internal pressure. Q. Appl. Math. 4(3), 262–267 (1946)MathSciNetCrossRefGoogle Scholar
  35. Taleghani, A.D.: Analysis of Hydraulic Fracture Propagation in Fractured Reservoirs: An Improved Model for the Interaction Between Induced and Natural Fractures. The University of Texas at Austin, Austin (2009)Google Scholar
  36. Tunc, X., Faille, I., Gallouët, T., Cacas, M.C., Havé, P.: A model for conductive faults with non-matching grids. Comput. Geosci. 16(2), 277–296 (2012).  https://doi.org/10.1007/s10596-011-9267-x CrossRefzbMATHGoogle Scholar
  37. Vinci, C.: Hydro-mechanical coupling in fractured rocks: modeling and numerical simulations. Ph.D. thesis. Ruhr-University Bochum (2014)Google Scholar
  38. Vinci, C., Renner, J., Steeb, H.: A hybrid-dimensional approach for an efficient numerical modeling of the hydro-mechanics of fractures. Water Resour. Res. 50(2), 1616–1635 (2014)CrossRefGoogle Scholar
  39. Vinci, C., Steeb, H., Renner, J.: The imprint of hydro-mechanics of fractures in periodic pumping tests. Geophys. J. Int. 202(3), 1613–1626 (2015)CrossRefGoogle Scholar
  40. Wang, H.F.: Theory of Linear Poroelasticity. Princeton University Press, Princeton (2000)Google Scholar
  41. Woodbury, A., Zhang, K.: Lanczos method for the solution of groundwater flow in discretely fractured porous media. Adv. Water Resour. 24(6), 621–630 (2001).  https://doi.org/10.1016/S0309-1708(00)00047-6 CrossRefGoogle Scholar
  42. Yew, C.H., Weng, X.: Mechanics of Hydraulic Fracturing. Gulf Professional Publishing, Houston (2014)Google Scholar
  43. Zheltov, A.K.: Formation of vertical fractures by means of highly viscous liquid. In: 4th World Petroleum Congress. World Petroleum Congress (1955)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Applied MechanicsUniversity of StuttgartStuttgartGermany

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