Flow and transport in fractured poroelastic media

Abstract

We study flow and transport in fractured poroelastic media using Stokes flow in the fractures and the Biot model in the porous media. The Stokes–Biot model is coupled with an advection–diffusion equation for modeling transport of chemical species within the fluid. The continuity of flux on the fracture-matrix interfaces is imposed via a Lagrange multiplier. The coupled system is discretized by a finite element method using Stokes elements, mixed Darcy elements, conforming displacement elements, and discontinuous Galerkin for transport. The stability and convergence of the coupled scheme are analyzed. Computational results verifying the theory as well as simulations of flow and transport in fractured poroelastic media are presented.

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References

  1. Acosta, G., Apel, T., Durán, R., Lombardi, A.: Error estimates for Raviart–Thomas interpolation of any order on anisotropic tetrahedra. Math. Comput. 80(273), 141–163 (2011)

    MathSciNet  MATH  Google Scholar 

  2. Aizinger, V., Dawson, C.N., Cockburn, B., Castillo, P.: The local discontinuous Galerkin method for contaminant transport. Adv. Water Resour. 24, 73–87 (2000)

    MATH  Google Scholar 

  3. Ambartsumyan, I., Ervin, V.J., Nguyen, T., Yotov, I.: A nonlinear Stokes–Biot model for the interaction of a non-Newtonian fluid with poroelastic media I: well-posedness of the model. arXiv:1803.00947 (2018a)

  4. Ambartsumyan, I., Khattatov, E., Yotov, I., Zunino, P.: A Lagrange multiplier method for a Stokes–Biot fluid-poroelastic structure interaction model. Numer. Math. 140(2), 513–553 (2018b)

    MathSciNet  MATH  Google Scholar 

  5. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2001/02)

    MathSciNet  MATH  Google Scholar 

  6. Badia, S., Quaini, A., Quarteroni, A.: Coupling Biot and Navier–Stokes equations for modelling fluid-poroelastic media interaction. J. Comput. Phys. 228(21), 7986–8014 (2009)

    MathSciNet  MATH  Google Scholar 

  7. Bazilevs, Y., Takizawa, K., Tezduyar, T.E.: Computational Fluid–Structure Interaction: Methods and Applications. Wiley, New York (2013)

    Google Scholar 

  8. Beavers, G., Joseph, D.: Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30(1), 197–207 (1967)

    Google Scholar 

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

    MATH  Google Scholar 

  10. Boffi, D., Brezzi, F., Fortin, M., et al.: Mixed Finite Element Methods and Applications, vol. 44. Springer, Berlin (2013)

    Google Scholar 

  11. Boon, W.M., Nordbotten, J.M., Yotov, I.: Robust discretization of flow in fractured porous media. SIAM J. Numer. Anal. 56(4), 2203–2233 (2018)

    MathSciNet  MATH  Google Scholar 

  12. Both, J., Kumar, K., Nordbotten, J.M., Radu, F.A.: Anderson accelerated fixed-stress splitting schemes for consolidation of unsaturated porous media. Comput. Math. Appl. https://doi.org/10.1016/j.camwa.2018.07.033 (2018)

    MathSciNet  Google Scholar 

  13. Brenner, S.C.: Poincaré–Friedrichs inequalities for piecewise \({H}^1\) functions. SIAM J. Numer. Anal. 41(1), 306–324 (2003)

    MathSciNet  MATH  Google Scholar 

  14. Bukac, M., Yotov, I., Zakerzadeh, R., Zunino, P.: Partitioning strategies for the interaction of a fluid with a poroelastic material based on a Nitsche’s coupling approach. Comput. Methods Appl. Mech. Eng. 292, 138–170 (2015)

    MathSciNet  MATH  Google Scholar 

  15. Bukac, M., Yotov, I., Zunino, P.: An operator splitting approach for the interaction between a fluid and a multilayered poroelastic structure. Numer. Methods Partial Differ. Equ. 31(4), 1054–1100 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Bukac, M., Yotov, I., Zunino, P.: Dimensional model reduction for flow through fractures in poroelastic media. ESAIM Math. Model. Numer. Anal. 51(4), 1429–1471 (2017)

    MathSciNet  MATH  Google Scholar 

  17. Bungartz, H.-J., Schäfer, M.: Fluid–Structure Interaction: Modelling, Simulation, Optimisation, vol. 53. Springer, Berlin (2006)

    Google Scholar 

  18. Ciarlet, P.: The finite element method for elliptic problems. Class. Appl. Math. 40, 1–511 (2002)

    MathSciNet  Google Scholar 

  19. Cockburn, B., Dawson, C.: Approximation of the velocity by coupling discontinuous Galerkin and mixed finite element methods for flow problems. Comput. Geosci. 6(3–4), 505–522 (2002). Locally conservative numerical methods for flow in porous media

    MathSciNet  MATH  Google Scholar 

  20. Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection–diffusion systems. SIAM J. Numer. Anal. 35(6), 2440–2463 (1998)

    MathSciNet  MATH  Google Scholar 

  21. D’Angelo, C., Scotti, A.: A mixed finite element method for Darcy flow in fractured porous media with non-matching grids. ESAIM Math. Model. Numer. Anal. 46(2), 465–489 (2012)

    MATH  Google Scholar 

  22. Dawson, C.: Conservative, shock-capturing transport methods with nonconservative velocity approximations. Comput. Geosci. 3(3–4), 205–227 (1999)

    MathSciNet  MATH  Google Scholar 

  23. Dawson, C., Sun, S., Wheeler, M.F.: Compatible algorithms for coupled flow and transport. Comput. Methods Appl. Mech. Eng. 193(23–26), 2565–2580 (2004)

    MathSciNet  MATH  Google Scholar 

  24. Discacciati, M., Miglio, E., Quarteroni, A.: Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 43(1–2), 57–74 (2002)

    MathSciNet  MATH  Google Scholar 

  25. Flemisch, B., Berre, I., Boon, W., Fumagalli, A., Schwenck, N., Scotti, A., Stefansson, I., Tatomir, A.: Benchmarks for single-phase flow in fractured porous media. Adv. Water Res. 111, 239–258 (2018)

    Google Scholar 

  26. Formaggia, L., Quarteroni, A., Veneziani, A.: Cardiovascular Mathematics: Modeling and Simulation of the Circulatory System, vol. 1. Springer, Berlin (2010)

    Google Scholar 

  27. Frih, N., Roberts, J.E., Saada, A.: Modeling fractures as interfaces: a model for Forchheimer fractures. Comput. Geosci. 12(1), 91–104 (2008)

    MathSciNet  MATH  Google Scholar 

  28. Frih, N., Martin, V., Roberts, J.E., Saada, A.: Modeling fractures as interfaces with nonmatching grids. Comput. Geosci. 16(4), 1043–1060 (2012)

    Google Scholar 

  29. Fumagalli, A., Scotti, A.: A reduced model for flow and transport in fractured porous media with non-matching grids. In: Numerical Mathematics and Advanced Applications 2011, pp. 499–507. Springer, Heidelberg (2013)

    Google Scholar 

  30. Fumagalli, A., Scotti, A.: Numerical modelling of multiphase subsurface flow in the presence of fractures. Commun. Appl. Ind. Math. 3(1), e–380, 23 (2012)

  31. Galdi, G.P., Rannacher, R. (eds.): Fundamental Trends in Fluid–Structure Interaction. Contemporary Challenges in Mathematical Fluid Dynamics and Its Applications, vol. 1. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2010)

    Google Scholar 

  32. Galvis, J., Sarkis, M.: Non-matching mortar discretization analysis for the coupling Stokes–Darcy equations. Electron. Trans. Numer. Anal. 26(20), 07 (2007)

    MathSciNet  MATH  Google Scholar 

  33. Ganis, B., Mear, M.E., Sakhaee-Pour, A., Wheeler, M.F., Wick, T.: Modeling fluid injection in fractures with a reservoir simulator coupled to a boundary element method. Comput. Geosci. 18(5), 613–624 (2014)

    MathSciNet  MATH  Google Scholar 

  34. Girault, V., Rivière, B.: DG approximation of coupled Navier–Stokes and Darcy equations by Beaver–Joseph–Saffman interface condition. SIAM J. Numer. Anal. 47(3), 2052–2089 (2009)

    MathSciNet  MATH  Google Scholar 

  35. 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(4), 587–645 (2015)

    MathSciNet  MATH  Google Scholar 

  36. Hecht, F.: New development in FreeFem++. J. Numer. Math. 20(3–4), 251–265 (2012)

    MathSciNet  MATH  Google Scholar 

  37. Kim, J., Tchelepi, H.A., Juanes, R.: Stability and convergence of sequential methods for coupled flow and geomechanics: drained and undrained splits. Comput. Methods Appl. Mech. Eng. 200(23–24), 2094–2116 (2011a)

    MathSciNet  MATH  Google Scholar 

  38. Kim, J., Tchelepi, H.A., 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(13–16), 1591–1606 (2011b)

    MathSciNet  MATH  Google Scholar 

  39. Kovacik, J.: Correlation between Young’s modulus and porosity in porous materials. J. Mater. Sci. Lett. 18(13), 1007–1010 (1999)

    Google Scholar 

  40. Layton, W.J., Schieweck, F., Yotov, I.: Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40(6), 2195–2218 (2003)

    MathSciNet  MATH  Google Scholar 

  41. Lee, S., Mikelić, A., Wheeler, M.F., Wick, T.: Phase-field modeling of proppant-filled fractures in a poroelastic medium. Comput. Methods Appl. Mech. Eng. 312, 509–541 (2016a)

    MathSciNet  Google Scholar 

  42. Lee, S., Wheeler, M.F., Wick, T.: Pressure and fluid-driven fracture propagation in porous media using an adaptive finite element phase field model. Comput. Methods Appl. Mech. Eng. 305, 111–132 (2016b)

    MathSciNet  MATH  Google Scholar 

  43. Lesinigo, M., D’Angelo, C., Quarteroni, A.: A multiscale Darcy–Brinkman model for fluid flow in fractured porous media. Numer. Math. 117(4), 717–752 (2011)

    MathSciNet  MATH  Google Scholar 

  44. Martin, V., Jaffre, J., Roberts, J.E.: Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26(5), 1667–1691 (2005)

    MathSciNet  MATH  Google Scholar 

  45. Mikelić, A., Wheeler, M.F.: Convergence of iterative coupling for coupled flow and geomechanics. Comput. Geosci. 17(3), 455–461 (2013)

    MathSciNet  MATH  Google Scholar 

  46. Mikelić, A., Wheeler, M.F., Wick, T.: Phase-field modeling of a fluid-driven fracture in a poroelastic medium. Comput. Geosci. 19(6), 1171–1195 (2015)

    MathSciNet  MATH  Google Scholar 

  47. Morales, F.A., Showalter, R.E.: The narrow fracture approximation by channeled flow. J. Math. Anal. Appl. 365(1), 320–331 (2010)

    MathSciNet  MATH  Google Scholar 

  48. Morales, F.A., Showalter, R.E.: A Darcy–Brinkman model of fractures in porous media. J. Math. Anal. Appl. 452(2), 1332–1358 (2017)

    MathSciNet  MATH  Google Scholar 

  49. Oden, J.T., Babuska, I., Baumann, C.E.: A discontinuous \(hp\) finite element method for diffusion problems. J. Comput. Phys. 146(2), 491–519 (1998)

    MathSciNet  MATH  Google Scholar 

  50. Peaceman, D.W.: Fundamentals of Numerical Reservoir Simulation. Elsevier, Amsterdam (1977)

    Google Scholar 

  51. Radu, F.A., Pop, I.S., Attinger, S.: Analysis of an Euler implicit-mixed finite element scheme for reactive solute transport in porous media. Numer. Methods Partial Differ. Equ. 26(2), 320–344 (2010)

    MathSciNet  MATH  Google Scholar 

  52. Richter, T.: Fluid–Structure Interactions: Models, Analysis and Finite Elements, vol. 118. Springer, Berlin (2017)

    Google Scholar 

  53. Rivière, B., Yotov, I.: Locally conservative coupling of Stokes and Darcy flows. SIAM J. Numer. Anal. 42(5), 1959–1977 (2005)

    MathSciNet  MATH  Google Scholar 

  54. Rivière, B., Wheeler, M.F., Girault, V.: Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. I. Comput. Geosci. 3(3–4), 337–360 (1999)

    MathSciNet  MATH  Google Scholar 

  55. Saffman, P.: On the boundary condition at the surface of a porous medium. Stud. Appl. Math. 50(2), 93–101 (1971)

    MathSciNet  MATH  Google Scholar 

  56. Scott, R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)

    MathSciNet  MATH  Google Scholar 

  57. Showalter, R.E.: Poroelastic filtration coupled to Stokes flow. In: Control Theory of Partial Differential Equations, vol. 242. Lecture Notes in Pure and Applied Mathematics, pp. 229–241. Chapman & Hall/CRC, Boca Raton, FL (2005)

  58. Sun, S., Rivière, B., Wheeler, M.F.: A combined mixed finite element and discontinuous Galerkin method for miscible displacement problem in porous media. In: Recent Progress in Computational and Applied PDEs (Zhangjiajie, 2001), pp. 323–351. Kluwer/Plenum, New York (2002)

    Google Scholar 

  59. Sun, S., Wheeler, M.F.: Discontinuous Galerkin methods for coupled flow and reactive transport problems. Appl. Numer. Math. 52(2–3), 273–298 (2005a)

    MathSciNet  MATH  Google Scholar 

  60. Sun, S., Wheeler, M.F.: Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media. SIAM J. Numer. Anal. 43(1), 195–219 (2005b)

    MathSciNet  MATH  Google Scholar 

  61. Vassilev, D., Yotov, I.: Coupling Stokes–Darcy flow with transport. SIAM J. Sci. Comput. 31(5), 3661–3684 (2009)

    MathSciNet  MATH  Google Scholar 

  62. Vassilev, D., Wang, C., Yotov, I.: Domain decomposition for coupled Stokes and Darcy flows. Comput. Methods Appl. Mech. Eng. 268, 264–283 (2014)

    MathSciNet  MATH  Google Scholar 

  63. Wheeler, M.F., Darlow, B.L.: Interior penalty Galerkin procedures for miscible displacement problems in porous media. In: Computational Methods in Nonlinear Mechanics (Proceedings of Second International Conference, University of Texas, Austin, TX, 1979), pp. 485–506. North-Holland, Amsterdam-New York (1980)

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Correspondence to Ivan Yotov.

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Partially supported by DOE Grant DE-FG02-04ER25618 and NSF Grants DMS 1418947 and DMS 1818775.

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Ambartsumyan, I., Khattatov, E., Nguyen, T. et al. Flow and transport in fractured poroelastic media. Int J Geomath 10, 11 (2019). https://doi.org/10.1007/s13137-019-0119-5

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Keywords

  • Fluid-poroelastic structure interaction
  • Stokes-Biot model
  • Coupled flow and transport
  • Fractured poroelastic media

Mathematics Subject Classification

  • 76S05
  • 76D07
  • 74F10
  • 65M60
  • 65M12