Block-Centered Finite Difference Method for Simulating Compressible Wormhole Propagation

In this paper, the block-centered finite difference method is introduced and analyzed to solve the compressible wormhole propagation. The coupled analysis approach to deal with the fully coupling relation of multivariables is employed. By this, stability analysis and error estimates for the pressure, velocity, porosity, concentration and its flux in different discrete norms are established rigorously and carefully on non-uniform grids. Finally, some numerical experiments are presented to verify the theoretical analysis and effectiveness of the given scheme.

[1]  Ricardo Durin.  Superconvergence for rectangular mixed finite elements , 2005 .

[2]  Hongxing Rui,et al.  Block-centered finite difference methods for parabolic equation with time-dependent coefficient , 2013, Japan Journal of Industrial and Applied Mathematics.

[3]  Alison Ord,et al.  Theoretical and numerical analyses of chemical‐dissolution front instability in fluid‐saturated porous rocks , 2008 .

[4]  Zhengguang Liu,et al.  A parallel CGS block-centered finite difference method for a nonlinear time-fractional parabolic equation , 2016 .

[5]  Xiaoli Li,et al.  Characteristic block-centred finite difference methods for nonlinear convection-dominated diffusion equation , 2017, Int. J. Comput. Math..

[6]  Michel Quintard,et al.  On the ability of a Darcy-scale model to capture wormhole formation during the dissolution of a porous medium , 2002, Journal of Fluid Mechanics.

[7]  Ricardo G. Durán,et al.  Superconvergence for rectangular mixed finite elements , 1990 .

[8]  M. Wheeler,et al.  Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Cell-Centered Finite Differences , 1997 .

[9]  Wei Liu,et al.  A Two-Grid Block-Centered Finite Difference Method For Darcy-Forchheimer Flow in Porous Media , 2015, SIAM J. Numer. Anal..

[10]  Shuyu Sun,et al.  Mixed finite element-based fully conservative methods for simulating wormhole propagation , 2016 .

[11]  Xiaoli Li,et al.  Characteristic block-centered finite difference method for simulating incompressible wormhole propagation , 2017, Comput. Math. Appl..

[12]  J. Nédélec.  Mixed finite elements in ℝ3 , 1980 .

[13]  Qinghua Zhang,et al.  Error analysis of the semi-discrete local discontinuous Galerkin method for compressible miscible displacement problem in porous media , 2015, Appl. Math. Comput..

[14]  H. Scott Fogler,et al.  Influence of Transport and Reaction on Wormhole Formation in Porous Media , 1998 .

[15]  P. Raviart,et al.  A mixed finite element method for 2-nd order elliptic problems , 1977 .

[16]  Xiaoli Li,et al.  A two-grid block-centered finite difference method for nonlinear non-Fickian flow model , 2016, Appl. Math. Comput..

[17]  Jean E. Roberts,et al.  Numerical methods for a model for compressible miscible displacement in porous media , 1983 .

[18]  A. Weiser,et al.  On convergence of block-centered finite differences for elliptic-problems , 1988 .

[19]  Shuyu Sun,et al.  Parallel simulation of wormhole propagation with the Darcy-Brinkman-Forchheimer framework , 2015 .

[20]  Mary F. Wheeler,et al.  A Two-Grid Finite Difference Scheme for Nonlinear Parabolic Equations , 1998 .

[21]  Murtaza Ziauddin,et al.  Two‐scale continuum model for simulation of wormholes in carbonate acidization , 2005 .

[22]  Hongxing Rui,et al.  A Block-Centered Finite Difference Method for the Darcy-Forchheimer Model , 2012, SIAM J. Numer. Anal..

[23]  S. Mauran,et al.  Application of the Carman–Kozeny Correlation to a High‐Porosity and Anisotropic Consolidated Medium: The Compressed Expanded Natural Graphite , 2001 .

[24]  Xiaoli Li,et al.  A Two-Grid Block-Centered Finite Difference Method for the Nonlinear Time-Fractional Parabolic Equation , 2017, Journal of Scientific Computing.

[25]  Shicheng Zhang,et al.  Wormhole Propagation Behavior Under Reservoir Condition in Carbonate Acidizing , 2012, Transport in Porous Media.