Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T12:29:14.308Z Has data issue: false hasContentIssue false
  • English
  • Français

Anisotropic Mesh Adaptation for 3D Anisotropic Diffusion Problems with Application to Fractured Reservoir Simulation

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  12 September 2017

Xianping Li  and
Weizhang Huang
Xianping Li*
Affiliation:
Department of Mathematics and Statistics, University of Missouri-Kansas City, Kansas City, MO 64110, U.S.A
Weizhang Huang*
Affiliation:
Department of Mathematics, University of Kansas, Lawrence, KS 66045, U.S.A
*
*Corresponding author. Email addresses:[email protected] (X. P. Li), [email protected] (W. Z. Huang)
*Corresponding author. Email addresses:[email protected] (X. P. Li), [email protected] (W. Z. Huang)
Get access

Abstract

Anisotropic mesh adaptation is studied for linear finite element solution of 3D anisotropic diffusion problems. The 𝕄-uniform mesh approach is used, where an anisotropic adaptive mesh is generated as a uniform one in the metric specified by a tensor. In addition to mesh adaptation, preservation of the maximum principle is also studied. Some new sufficient conditions for maximum principle preservation are developed, and a mesh quality measure is defined to server as a good indicator. Four different metric tensors are investigated: one is the identity matrix, one focuses on minimizing an error bound, another one on preservation of the maximum principle, while the fourth combines both. Numerical examples show that these metric tensors serve their purposes. Particularly, the fourth leads to meshes that improve the satisfaction of the maximum principle by the finite element solution while concentrating elements in regions where the error is large. Application of the anisotropic mesh adaptation to fractured reservoir simulation in petroleum engineering is also investigated, where unphysical solutions can occur and mesh adaptation can help improving the satisfaction of the maximum principle.

Keywords

Finite element methodanisotropic mesh adaptationthree dimensionalanisotropic diffusiondiscrete maximum principlepetroleum engineering

MSC classification

Secondary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65M50: Mesh generation and refinement
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 4 , November 2017 , pp. 913 - 940
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aavatsmark, I., Barkve, T., Bøe, Ø. and Mannseth, T., Discretization on unstructured grids for inhomogeneous, anisotropic media. I. Derivation of the methods, SIAM J. Sci. Comput., 19 (1998), pp. 1700–1716.Google Scholar
[2] Aavatsmark, I., Barkve, T., Bøe, Ø. and Mannseth, T., Discretization on unstructured grids for inhomogeneous, anisotropic media. II. Discussion and numerical results, SIAM J. Sci. Comput., 19 (1998), pp. 1717–1736.CrossRefGoogle Scholar
[3] Ait-Ali-Yahia, D., Baruzzi, G., Habashi, W. G., Fortin, M., Dompierre, J. and Vallet, M. G., Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver-independent CFD. Part II: Structured grids, Int. J. Numer. Meth. Fluids, 39 (2002), pp. 657–673.Google Scholar
[4] Bogdanov, I. I., Mourzenko, V. V. and Thovert, J. F., Effective permeability of fractured porous media in steady state flow, Water Res. Research, 39(39) (2003), pp. 257–263.CrossRefGoogle Scholar
[5] Borouchaki, H., George, P. L., Hecht, P., Laug, P. and Saletl, E., Delaunay mesh generation governed by metric specification: Part I. Algorithms, Fin. Elem. Anal. Des., 25 (1997), pp. 61–83.CrossRefGoogle Scholar
[6] Borouchaki, H., George, P. L. and Mohammadi, B., Delaunay mesh generation governed by metric specification: Part II. Applications, Fin. Elem. Anal. Des., 25 (1997), pp. 85–109.Google Scholar
[7] Bossen, F. J. and Heckbert, P. S.. A pliant method for anisotropic mesh generation, In Proceedings 5th International Meshing roundtable, Sandia National Laboratories, 1996, pp. 63–74.Google Scholar
[8] Brandts, J., Korotov, S., and Křížek, M., The discrete maximum principle for linear simplicial finite element approximations of a reaction-diffusion problem, Linear Algebra Appl., 429 (2008), pp. 2344–2357.Google Scholar
[9] Castro-Díaz, M. J., Hecht, F., Mohammadi, B. and Pironneau, O., Anisotropic unstructured mesh adaption for flow simulations, Int. J. Numer. Meth. Fluids, 25 (1997), pp. 475–491.Google Scholar
[10] Chan, T. F. and Shen, J., Non-texture inpainting by curvature driven diffusions (CDD), J. Vis. Commun. Image Rep, 12 (2000), pp. 436–449.Google Scholar
[11] Chan, T. F., Shen, J., and Vese, L., Variational PDE models in image processing, Not. AMS J., 50 (2003), pp. 14–26.Google Scholar
[12] Ciarlet, P. G. and Raviart, P. A., Maximum principle and uniform convergence for the finite element method, Comput. Methods Appl. Mech. Engrg., 2 (1973), pp. 17–31.Google Scholar
[13] Cipolla, C. L., Lolon, E. P., Erdle, J. C. and Rubin, B., Reservoir modeling in shale-gas reservoirs, SPE 125530, presented at the SPE Eastern Regional Meeting held in Charleston, West Virginia, 2009.Google Scholar
[14] Cipolla, C. L., Lolon, E. P., Erdle, J. C. and Tathed, V., Modeling well performance in shale-gas reservoirs, SPE 125532, presented at the SPE/EAGE Reservoir Characterization and Simulation Conference held in Abu Dhabi, UAE, 2009.Google Scholar
[15] Crumpton, P. I., Shaw, G. J. and Ware, A. F., Discretisation and multigrid solution of elliptic equations with mixed derivative terms and strongly discontinuous coefficients, J. Comput. Phys., 116 (1995), pp. 343–358.CrossRefGoogle Scholar
[16] Dobrzynski, C. and Frey, P., Anisotropic Tetrahedral Remesher/Moving Mesh Generation software, available at: http://www.math.u-bordeaux1.fr/~dobrzyns/logiciels/mmg3d.php, 2012.Google Scholar
[17] Dompierre, J., Vallet, M. G., Bourgault, Y., Fortin, M. and Habashi, W. G.. Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver-independent CFD. Part III: Unstructured meshes, Int. J. Numer. Meth. Fluids, 39 (2002), pp. 675–702.Google Scholar
[18] Drǎgǎnescu, A., Dupont, T. F. and Scott, L. R., Failure of the discrete maximum principle for an elliptic finite element problem, Math. Comp., 74(249) (2004), pp. 1–23.CrossRefGoogle Scholar
[19] Freeman, C. M., Moridis, G., Ilk, D., and Blasingame, T. A., A numerical study of performance for tight gas and shale gas reservoir systems, J. Petrol. Sci. Eng., 108 (2013), pp. 22–39.Google Scholar
[20] Garimella, R. V. and Shephard, M. S., Boundary layer meshing for viscous flows in complex domain, International Meshing Roundtable, 2000, pp. 107–118.Google Scholar
[21] Günter, S. and Lackner, K., A mixed implicit-explicit finite difference scheme for heat transport in magnetised plasmas, J. Comput. Phys., 228 (2009), pp. 282–293.Google Scholar
[22] Günter, S., Lackner, K. and Tichmann, C., Finite element and higher order difference formulations for modelling heat transport in magnetised plasmas, J. Comput. Phys., 226 (2007), pp. 2306–2316.CrossRefGoogle Scholar
[23] Günter, S., Yu, Q., Kruger, J. and Lackner, K., Modelling of heat transport in magnetised plasmas using non-aligned coordinates, J. Comput. Phys., 209 (2005), pp. 354–370.Google Scholar
[24] Habashi, W. G., Dompierre, J., Bourgault, Y., D. AIT-ALI-YAHIA, Fortin, M. and Vallet, M. G., Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver-independent CFD. Part I: General principles, Int. J. Numer. Meth. Fluids, 32 (2000), pp. 725–744.Google Scholar
[25] Hecht, F., Bidimensional Anisotropic Mesh Generator, Inria Report, 1998.Google Scholar
[26] Hecht, F.. New development in Freefem++, J. Numer. Math., 20(3-4) (2012), pp. 251–265.Google Scholar
[27] Huang, W., Variational mesh adaptation: isotropy and equidistribution, J. Comput. Phys., 174 (2001), pp. 903–924.Google Scholar
[28] Huang, W., Measuring mesh qualities and application to variational mesh adaptation, SIAM. J. Sci. Comput., 26 (2005), pp. 1643–1666.Google Scholar
[29] Huang, W., Metric tensors for anisotropic mesh generation, J. Comput. Phys., 204 (2005), pp. 633–665.Google Scholar
[30] Huang, W., Mathematical principles of anisotropic mesh adaptation, Comm. Comput. Phys., 1 (2006), pp. 276–310.Google Scholar
[31] Huang, W., Discrete maximum principle and a Delaunay-type mesh condition for linear finite element approximations of two-dimensional anisotropic diffusion problems, Numer. Math. Theor. Meth. Appl., 4 (2011), pp. 319–334.Google Scholar
[32] Huang, W., Kamenski, L. and Li, X., Anisotropic mesh adaptation for variational problems using error estimation based on hierarchical bases, Canadian Applied Mathematics Quarterly, 17(3) (2009), pp. 501–522.Google Scholar
[33] Karátson, J., Korotov, S. and Křížek, M., On discrete maximum principles for nonlinear elliptic problems, Math. Comput. Simulat., 76 (2007), pp. 99–108.Google Scholar
[34] Karras, D. A. and Mertzios, G. B., New PDE-based methods for image enhancement using SOM and Bayesian inference in various discretization schemes, Meas. Sci. Technol., 20 (2009), 104012.CrossRefGoogle Scholar
[35] Knupp, P. M., Algebraic mesh quality metrics for unstructured initial meshes, Fin. Elem. Anal. Des., 39 (2003), pp. 217–241.Google Scholar
[36] Knupp, P. M., Introducing the target-matrix paradigm for mesh optimization via node-movement, Eng. Comput., 28 (2012), pp. 419–429.Google Scholar
[37] Křížek, M. and Lin, Q., On diagonal dominance of stiffness matrices in 3d, East-West J. Numer. Math., 3 (1995), pp. 59–69.Google Scholar
[38] Kuzmin, D., Shashkov, M. J. and Svyatskiy, D., A constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems, J. Comput. Phys., 228 (2009), pp. 3448–3463.CrossRefGoogle Scholar
[39] Letniowski, F. W., Three-dimensional Delaunay triangulations for finite element approximations to a second-order diffusion operator, SIAM J. Sci. Stat. Comput., 13 (1992), pp. 765–770.Google Scholar
[40] Li, X. and Huang, W., An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems, J. Comput. Phys., 229 (2010), pp. 8072–8094.CrossRefGoogle Scholar
[41] Li, X. and Huang, W., Maximum principle for the finite element solution of time-dependent anisotropic diffusion problems, Numer. Meth. PDEs, 29 (2013), pp. 1963–1985.Google Scholar
[42] Li, X., Svyatskiy, D. and Shashkov, M., Mesh adaptation and discrete maximum principle for 2D anisotropic diffusion problems, LANL technical report, (2007), LA-UR 10-01227.Google Scholar
[43] Lipnikov, K., Shashkov, M., Svyatskiy, D. and Vassilevski, Y., Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes, J. Comput. Phys., 227 (2007), pp. 492–512.Google Scholar
[44] Liska, R. and Shashkov, M., Enforcing the discrete maximum principle for linear finite element solutions of second-order elliptic problems, Commun. Comput. Phys., 3 (2008), pp. 852–877.Google Scholar
[45] Loseille, A., Metric-orthogonal anisotropic mesh generation, Procedia Engineering, 82 (2014), pp. 403–415.Google Scholar
[46] Lu, C., Huang, W. and Qiu, J., Maximum principle in linear finite element approximations of anisotropic diffusion-convection-reaction problems, Numer. Math., 127 (2014), pp. 515–537.Google Scholar
[47] Lunati, I. and Lee, S. H., A dual-tube model for gas dynamics in fractured nanoporous shale formations, J. Fluid Mech., 757 (2014), pp. 943–971.Google Scholar
[48] Marcum, D. and Alauzet, F., Aligned metric-based anisotropic solution adaptivemesh generation, Procedia Engineering, 82 (2014), pp. 428–444.Google Scholar
[49] Mi, L., Jiang, H. and Li, J., The impact of diffusion type on multiscale discrete fracture model numerical simulation for shale gas, Journal of Natural Gas Science and Engineering, 20 (2014), pp. 74–81.Google Scholar
[50] Mlacnik, M. J. and Durlofsky, L. J., Unstructured grid optimization for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients, J. Comput. Phys., 216 (2006), pp. 337–361.Google Scholar
[51] Nishikawa, K. and Wakatani, M., Plasma Physics, Springer-Verlag Berlin Heidelberg, New York, 2000.Google Scholar
[52] Olorode, O. M., Freeman, C. M., Moridis, G. J. and Blasingame, T. A., High-resolution numerical modeling of complex and irregular fracture patterns in shale gas and tight gas reservoirs, Spe Reservoir Evaluation & Engineering, 16(4) (2013), pp. 443–455.Google Scholar
[53] Peraire, J., Vahdati, M., Morgan, K. and Zienkiewicz, O. C., Adaptive remeshing for compressible flow computations, J. Comput. Phys., 72 (1987), pp. 449–466.CrossRefGoogle Scholar
[54] Perona, P. and Malik, J., Scale-space and edge detection using anisotropic diffusion, IEEE T. Pattern Anal., 12 (1990), pp. 629–639.CrossRefGoogle Scholar
[55] Le Potier, C., A nonlinear finite volume scheme satisfying maximum and minimum principles for diffusion operators, Int. J. Finite Vol., 6(2) (2009), pp. 1–20.Google Scholar
[56] Rubin, B., Accurate simulation of non-darcy flow in stimulated fractured shale reservoir, SPE 132093, presented at the SPE Western Regional Meeting held in Anaheim, California, 2010.Google Scholar
[57] Sharma, P. and Hammett, G. W., Preserving monotonicity in anisotropic diffusion, J. Comput. Phys., 227 (2007), pp. 123–142.Google Scholar
[58] Silin, D. and Kneafsey, T., Shale gas: nanometer-scale observations and well modelling, Journal of Canadian Petroleum Technology, 51(6) (2012), pp. 464–475.Google Scholar
[59] Vu, M., Pouya, A. and Seyedi, D. M., Modelling of steady-state fluid flow in 3d fractured isotropic porous media: application to effective permeability calculation, Int. J. Numer. Anal. Meth. Geomech, 37 (2013), pp. 2257–2277.Google Scholar
[60] Wang, Q., Chen, X., Jha, A. N. and Rogers, H., Natural gas from shale formation âĂŞ the evolution, evidences and challenges of shale gas revolution in united states, Renewable and Sustainable Energy Reviews, 30 (2014), pp. 1–28.Google Scholar
[61] Wang, J. and Zhang, R., Maximum principle for P1-conforming finite element approximations of quasi-linear second order elliptic equations, SIAM J. Numer. Anal., 50 (2012), pp. 626–642.Google Scholar
[62] Weickert, J., Anisotropic diffusion in image processing, Teubner-Verlag, Stuttgart, Germany, 16(1) (1998), pp. 272.Google Scholar
[63] Xu, J. and Zikatanov, L., A monotone finite element scheme for convection-diffusion equations, Math. Comp., 69 (1999), pp. 1429–1446.Google Scholar
[64] Yamakawa, S. and Shimada, K., High quality anisotropic tetrahedral mesh generation via ellipsoidal bubble packing, IMR, 2000, PP. 263–273.Google Scholar
[65] Yuan, G. and Sheng, Z., Monotone finite volume schemes for diffusion equations on polygonal meshes, J. Comput. Phys., 227 (2008), pp. 6288–6312.Google Scholar
[66] Zhang, Y., Zhang, X., and Shu, C. W., Maximum-principle-satisfying second order discontinuous Galerkin schemes for convection-diffusion equations on triangular meshes, J. Comput. Phys., 234 (2013), pp. 295–316.Google Scholar