Skip to main content Accessibility help

Superconvergence of a fully conservative finite difference method on non-uniform staggered grids for simulating wormhole propagation with the Darcy–Brinkman–Forchheimer framework

  • Xiaoli Li (a1) and Hongxing Rui (a2)


In this paper, a finite difference scheme on non-uniform staggered grids is proposed for wormhole propagation with the Darcy–Brinkman–Forchheimer framework in porous media by introducing an auxiliary flux variable to guarantee full mass conservation. Error estimates for the pressure, velocity, porosity, concentration and auxiliary flux with second-order superconvergence in different discrete norms are established rigorously and carefully on non-uniform grids. We also obtain second-order superconvergence for some terms of the $H^{1}$ norm of the velocity on non-uniform grids. Finally, some numerical experiments are presented to verify the theoretical analysis and effectiveness of the proposed scheme.


Corresponding author

Email address for correspondence:


Hide All
Alazmi, B. & Vafai, K. 2000 Analysis of variants within the porous media transport models. Trans. ASME J. Heat Transfer 122, 303326.
Arbogast, T., Wheeler, M. F. & Yotov, I. 1997 Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal. 34, 828852.
Fredd, C. N. & Fogler, H. S. 1998 Influence of transport and reaction on wormhole formation in porous media. AIChE J. 44, 19331949.
Girault, V. & Lopez, H. 1996 Finite-element error estimates for the MAC scheme. IMA J. Numer. Anal. 16, 347379.
Girault, V. & Raviart, P.-A. 1979 An analysis of a mixed finite element method for the Navier–Stokes equations. Numer. Math. 33, 235271.
Girault, V. & Raviart, P.-A. 2012 Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms, vol. 5. Springer Science & Business Media.
Golfier, F., Zarcone, C., Bazin, B., Lenormand, R., Lasseux, D. & Quintard, M. 2002 On the ability of a Darcy-scale model to capture wormhole formation during the dissolution of a porous medium. J. Fluid Mech. 457, 213254.
Han, H. & Wu, X. 1998 A new mixed finite element formulation and the MAC method for the Stokes equations. SIAM J. Numer. Anal. 35, 560571.
Kanschat, G. 2008 Divergence-free discontinuous Galerkin schemes for the Stokes equations and the MAC scheme. Intl J. Numer. Meth. Fluids 56, 941950.
Kladias, N. & Prasad, V. 1991 Experimental verification of Darcy–Brinkman–Forchheimer flow model for natural convection in porous media. J. Thermophys. Heat Transfer 5, 560576.
Kou, J., Sun, S. & Wu, Y. 2016 Mixed finite element-based fully conservative methods for simulating wormhole propagation. Comput. Meth. Appl. Mech. Engng 298, 279302.
Li, X. & Rui, H. 2016 A two-grid block-centered finite difference method for nonlinear non-Fickian flow model. Appl. Maths Comput. 281, 300313.
Li, X. & Rui, H. 2017a Characteristic block-centered finite difference method for simulating incompressible wormhole propagation. Comput. Maths Applics. 73, 21712190.
Li, X. & Rui, H. 2017b A two-grid block-centered finite difference method for the nonlinear time-fractional parabolic equation. J. Sci. Comput. 72, 863891.
Li, X. & Rui, H. 2018a Block-centered finite difference method for simulating compressible wormhole propagation. J. Sci. Comput. 74, 11151145.
Li, X. & Rui, H. 2018b Superconvergence of characteristics marker and cell scheme for the Navier–Stokes equations on nonuniform grids. SIAM J. Numer. Anal. 56, 13131337.
Liu, M., Zhang, S., Mou, J. & Zhou, F. 2013 Wormhole propagation behavior under reservoir condition in carbonate acidizing. Transp. Porous Med. 96, 203220.
Liu, W., Cui, J. & Xin, J. 2018 A block-centered finite difference method for an unsteady asymptotic coupled model in fractured media aquifer system. J. Comput. Appl. Maths 337, 319340.
Minev, P. 2008 Remarks on the links between low-order DG methods and some finite-difference schemes for the Stokes problem. Intl J. Numer. Meth. Fluids 58, 307318.
Pan, H. & Rui, H. 2012 Mixed element method for two-dimensional Darcy–Forchheimer model. J. Sci. Comput. 52, 563587.
Panga, M. K., Ziauddin, M. & Balakotaiah, V. 2005 Two-scale continuum model for simulation of wormholes in carbonate acidization. AIChE J. 51, 32313248.
Perot, B. 2000 Conservation properties of unstructured staggered mesh schemes. J. Comput. Phys. 159, 5889.
Perot, J. B. 2011 Discrete conservation properties of unstructured mesh schemes. Annu. Rev. Fluid Mech. 43, 299318.
Raviart, P.-A. & Thomas, J.-M. 1977 A Mixed Finite Element Method for 2-nd Order Elliptic Problems. Springer.
Rees, D. A. S. 1999 Darcy–Brinkman free convection from a heated horizontal surface. Numer. Heat Transfer 35, 191204.
Rui, H. & Li, X. 2017 Stability and superconvergence of MAC scheme for Stokes equations on nonuniform grids. SIAM J. Numer. Anal. 55, 11351158.
Rui, H. & Pan, H. 2012 A block-centered finite difference method for the Darcy–Forchheimer model. SIAM J. Numer. Anal. 50, 26122631.
Rui, H. & Pan, H. 2013 Block-centered finite difference methods for parabolic equation with time-dependent coefficient. Japan J. Indust. Appl. Maths 30, 681699.
Rui, H., Zhao, D. & Pan, H. 2015 A block-centered finite difference method for Darcy–Forchheimer model with variable Forchheimer number. Numer. Meth. Part. Diff. Equ. 31, 16031622.
Vafai, K. & Kim, S. J. 1995 On the limitations of the Brinkman–Forchheimer-extended Darcy equation. Intl J. Heat Fluid Flow 16, 1115.
Weiser, A. & Wheeler, M. F. 1988 On convergence of block-centered finite differences for elliptic problems. SIAM J. Numer. Anal. 25, 351375.
Wu, Y., Salama, A. & Sun, S. 2015 Parallel simulation of wormhole propagation with the Darcy–Brinkman–Forchheimer framework. Comput. Geotech. 69, 564577.
Zhao, C., Hobbs, B., Hornby, P., Ord, A., Peng, S. & Liu, L. 2008 Theoretical and numerical analyses of chemical-dissolution front instability in fluid-saturated porous rocks. Intl J. Numer. Anal. Meth. Geomech. 32, 11071130.
Zhao, C., Hobbs, B. E. & Ord, A. 2014 Theoretical analyses of chemical dissolution-front instability in fluid-saturated porous media under non-isothermal conditions. Intl J. Numer. Anal. Meth. Geomech. 102, 799820.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed