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

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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.

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References

  • Adachi, J., Siebrits, E., Peirce, A., Desroches, J.: Computer simulation of hydraulic fractures. Int. J. Rock Mech. Min. Sci. 44(5), 739–757 (2007)

    Article  Google Scholar 

  • 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 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12(2), 155–164 (1941)

    Article  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  Google Scholar 

  • Coussy, O.: Poromechanics. Wiley, New York (2004)

    MATH  Google Scholar 

  • Ehlers, W., Bluhm, J.: Porous Media: Theory, Experiments and Numerical Applications. Springer, Berlin (2013)

    MATH  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Guiducci, C., Collin, F., Radu, J.P., Pellegrino, A., Charlier, R.: Numerical modeling of hydro-mechanical fracture behaviour. NUMOG VIII, pp. 293–299 (2003)

  • Hanowski, K.K., Sander, O.: Simulation of deformation and flow in fractured. Poroelastic materials. ArXiv e-prints (2016)

  • 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

    Article  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Nordgren, R., et al.: Propagation of a vertical hydraulic fracture. Soc. Pet. Eng. J. 12(04), 306–314 (1972)

    Article  Google Scholar 

  • Oliver, D.S., Chen, Y.: Recent progress on reservoir history matching: a review. Comput. Geosci. 15(1), 185–221 (2011)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Perkins, T., Kern, L., et al.: Widths of hydraulic fractures. J. Pet. Technol. 13(09), 937–949 (1961)

    Article  Google Scholar 

  • Renner, J., Steeb, H.: Modeling of Fluid Transport in Geothermal Research, pp. 1443–1500. Springer, Berlin (2015)

    Google Scholar 

  • 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)

    Article  Google Scholar 

  • Rodrigues, J.: The Noordbergum effect and characterization of aquitards at the Rio Maior mining project. Ground Water 21, 200–207 (1983)

    Article  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Shen, B., Stephansson, O., Rinne, M.: Hydro-Mechanical Coupling, pp. 77–82. Springer, Dordrecht (2014)

    Google Scholar 

  • Sneddon, I.N., Elliot, H.A.: The opening of a Griffith crack under internal pressure. Q. Appl. Math. 4(3), 262–267 (1946)

    Article  MathSciNet  Google Scholar 

  • 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 

  • 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

    Article  MATH  Google Scholar 

  • Vinci, C.: Hydro-mechanical coupling in fractured rocks: modeling and numerical simulations. Ph.D. thesis. Ruhr-University Bochum (2014)

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Wang, H.F.: Theory of Linear Poroelasticity. Princeton University Press, Princeton (2000)

    Google Scholar 

  • 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

    Article  Google Scholar 

  • Yew, C.H., Weng, X.: Mechanics of Hydraulic Fracturing. Gulf Professional Publishing, Houston (2014)

    Google Scholar 

  • Zheltov, A.K.: Formation of vertical fractures by means of highly viscous liquid. In: 4th World Petroleum Congress. World Petroleum Congress (1955)

Download references

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”.

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  1. Institute of Applied Mechanics, University of Stuttgart, Pfaffenwaldring 7, 70569, Stuttgart, Germany

    Patrick Schmidt & Holger Steeb

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Schmidt, P., Steeb, H. Numerical aspects of hydro-mechanical coupling of fluid-filled fractures using hybrid-dimensional element formulations and non-conformal meshes. Int J Geomath 10, 14 (2019). https://doi.org/10.1007/s13137-019-0127-5

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