Comparison of cell- and vertex-centered discretization methods for flow in a two-dimensional discrete-fracture-matrix system

Advances in Water Resources

Volumn 32, Issue 12, 2009, Pages 1740-1755

  Haegland H ^a   Assteerawatt, A ^b   Dahle, H K ^c   Eigestad, G T ^d   Helmig, R ^e  

^a Centre for Integrated Petroleum Research   (Norway)

^b RISK MANAGEMENT SOLUTIONS   (United Kingdom)

^c UNIVERSITY OF BERGEN   (Norway)

^d Perecon AS   (Norway)

^e UNIVERSITY OF STUTTGART   (Germany)

Author keywords

Box method; Discrete fracture model; Elliptic equation; Multi point flux approximation; Porous media; Streamlines

Indexed keywords

ADVECTIVE TRANSPORT; CHARACTERIZATION TECHNIQUES; COMPUTER EFFICIENCY; COMPUTER POWER; CONDITION NUMBERS; DIMENSIONAL MODEL; DISCRETE-FRACTURE MODEL; DISCRETE-FRACTURE MODELS; DISCRETIZATION METHOD; ELLIPTIC EQUATIONS; FRACTURE-MATRIX SYSTEMS; MATRIX; POROUS MEDIA; POROUS ROCKS; POST PROCESSING; PRESSURE EQUATIONS; STREAMLINE SIMULATION; VELOCITY FIELD;

DISCRETE EVENT SIMULATION; FLOW SIMULATION; MATHEMATICAL MODELS; NUMBER THEORY; POROUS MATERIALS; SIMULATORS; VOLUME MEASUREMENT;

FRACTURE;

ACCURACY ASSESSMENT; ADVECTION; COMPUTER SIMULATION; FLOW FIELD; FLOW MODELING; FLOW VELOCITY; FRACTURE; MATRIX; POROUS MEDIUM; TWO-DIMENSIONAL FLOW;

EID: 71549163516     PISSN: 03091708     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.advwatres.2009.09.006     Document Type: Article

References (88)

1. Aavatsmark I. An introduction to multipoint flux approximations for quadrilateral grids. Comput Geosci 6 (2002) 405-432
2. Aavatsmark I., Barkve T., Bøe Ø., and Mannseth T. Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media. J Comput Phys 127 (1996) 2-14
3. Aavatsmark I., Barkve T., Bøe Ø., and Mannseth T. Discretization on unstructured grids for inhomogeneous, anisotropic media. Part I: Derivation of the methods. SIAM J Sci Comput 19 5 (1998) 1700-1716
4. Aavatsmark I., and Eigestad G.T. Numerical convergence of the MPFA O-method and U-method for general quadrilateral grids. Int J Numer Methods Fluids 51 (2006) 939-961
5. Aavatsmark I, Eigestad GT, Heimsund BO, Mallison BT, Nordbotten J, Øian E. A new finite-volume approach to efficient discretization on challenging grids. SPE 106435. In: SPE reservoir simulation symposium, Houston, Texas; 26-28 February 2007.
6. Aavatsmark I., Eigestad G.T., and Klausen R.A. Numerical convergence of the MPFA O-method for general quadrilateral grids in two and three dimensions. In: Arnold D.N., Bochev P.B., Lehoucq R.B., Nicolaides R.A., and Shashkov M. (Eds). Compatible spatial discretizations. IMA volume series (2006), Springer, New York 1-21
7. Aavatsmark I., Eigestad G.T., Mallison B.T., and Nordbotten J.M. A compact multipoint flux approximation method with improved robustness. Numer Methods Part DE 24 5 (2008) 1329-1360
8. Adler P.M., and Thovert J.-F. Fractures and fracture networks. Theory and applications of transport in porous media vol. 15 (1999), Kluwer Academic Publishing, Boston
9. Arbogast T., Dawson C.N., Keenan P.T., Wheeler M.F., and Yotov I. Enhanced cell-centered finite differences for elliptic equations on general geometry. SIAM J Sci Comput 19 2 (1998) 404-425
10. Arbogast T., Wheeler M.F., and Yotov I. Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite-differences. SIAM J Numer Anal 34 2 (1997) 828-852
11. Arnold D.N., Boffi D., and Falk R.S. Quadrilateral H(div) finite elements. SIAM J Numer Anal 42 6 (2005) 2429-2451
12. Assteerawatt A, Helmig R, Hægland H, Bárdossy A, Dahle HK. Simulation of flow and transport processes in a discrete fracture-matrix system. I. Geostatistical generation of fractures on an aquifer analogue scale, in preparation.
13. Baliga B.R., and Patankar S.V. A new finite-element formulation for convection-diffusion problems. Numer Heat Transfer 3 (1980) 393-409
14. Bank R.E., and Rose D.J. Some error estimates for the box method. SIAM J Numer Anal 24 4 (1987) 777-787
15. Bank R.E., and Scott L.R. On the conditioning of finite element equations with highly refined meshes. SIAM J Numer Anal 26 6 (1989) 1383-1394
16. Barker J.A. Transport in fractured rock. In: Downing R.A., and Wilkinson W.B. (Eds). Applied groundwater hydrology (1991), Oxford Science Publications, Clarendon Press, Oxford 199-216
17. Bear J. Modeling flow and contaminant transport in fractured rocks. In: Bear J., Tsang C.-F., and de Marsily G. (Eds). Flow and contaminant transport in fractured rock (1993), Academic Press, San Diego 1-37
18. Belayneh M., Geiger S., and Matthäi S.K. Numerical simulation of water injection into layered fractured carbonate reservoir analogs. AAPG Bull 90 10 (2006) 1473-1493
19. Berg S.S., and Øian E. Hierarchical approach for simulating fluid flow in normal fault zones. Petrol Geosci 13 (2007) 25-35
20. Berkowitz B. Characterizing flow and transport in fractured geological media: a review. Adv Water Resour 25 (2002) 861-884
21. Bogdanov I.I., Mourzenko V.V., and Thovert J.-F. Effective permeability of fractured porous media in steady state flow. Water Resour Res 39 1 (2003)
22. Bonnet E., Bour O., Odling N.E., Davy P., Main I., Cowie P., et al. Scaling of fracture systems in geological media. Rev Geophys 39 3 (2001) 347-383
23. Brezzi F., and Fortin M. Mixed and hybrid finite element methods. Springer series in computational mathematics vol. 15 (1991), Springer-Verlag, New York
24. Cai Z., Jones J.E., McCormick S.F., and Russell T.F. Control-volume mixed finite element methods. Comput Geosci 1 (1997) 289-315
25. Cai Z., Mandel J., and McCormick S. The finite volume element method for diffusion equations on general triangulations. SIAM J Numer Anal 28 2 (1991) 392-402
26. Caillabet Y., Fabrie P., Landereau P., Noetinger B., and Quintard M. Implementation of a finite-volume method for the determination of effective parameters in fissured porous media. Numer Methods Part DE 16 (2000) 237-263
27. Castillo P. A review of the local discontinuous Galerkin (LDG) method applied to elliptic problems. Appl Numer Math 56 (2006) 1307-1313
28. Chen Q.-Y., Wan J., Yang Y., and Mifflin R.T. Enriched multi-point flux approximation for general grids. J Comput Phys 227 (2008) 1701-1721
29. Chen Z. On the control volume finite element methods and their application to multiphase flow. Netw Heterogeneous Media 1 4 (2006) 689-706
30. Ciarlet P.G. The finite element method for elliptic problems (1980), North-Holland, Amsterdam
31. Committee on Fracture Characterization and Fluid Flow, US National Committee for Rock Mechanics, Geotechnical Board, Board on Energy and Environmental Systems, Commission on Engineering and Technical Systems, and National Research Council. Rock fractures and fluid flow: contemporary understanding and applications. National Academy Press, Washington, DC; 1996.
32. Cordes C., and Kinzelbach W. Continuous groundwater velocity fields and path lines in linear, bilinear, and trilinear finite elements. Water Resour Res 28 11 (1992) 2903-2911
33. Datta-Gupta A., and King M.J. Streamline simulation: theory and practice (2007), SPE Text book
34. Davis TA. UMFPACK, Unsymmetric MultiFrontal method; 2008. .
35. In: Dietrich P., Helmig R., Sauter M., Hötzl H., Köngeter J., and Teutsch G. (Eds). Flow and transport in fractured porous media (2005), Springer, Berlin
36. Edwards M.G., and Pal M. Positive-definite q-families of continuous sub-cell Darcy-flux CVD(MPFA) finite volume schemes and the mixed finite element method. Int J Numer Methods Fluids 57 (2008) 355-387
37. Edwards M.G., and Rogers C.F. Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput Geosci 2 (1998) 259-290
38. Eigestad G.T., Aavatsmark I., and Espedal M. Symmetry and M-matrix issues for the O-method on an unstructured grid. Comput Geosci 6 (2002) 381-404
39. Eigestad G.T., and Klausen R.A. On the convergence of the multi-point flux approximation O-method: numerical experiments for discontinuous permeability. Numer Methods Part DE 21 6 (2005) 1079-1098
40. Ewing R.E., Lin T., and Lin Y. On the accuracy of the finite volume element method based on piecewise linear polynomials. SIAM J Numer Anal 39 6 (2002) 1865-1888
41. Eymard R., Gallouët T., and Herbin R. Finite volume methods. In: Ciarlet P.G., and Lions J.L. (Eds). Handbook of numerical analysis vol. VII (2000), North-Holland, Amsterdam
42. Friis H.A., Edwards M.G., and Mykkeltveit J. Symmetric positive definite flux-continuous full-tensor finite-volume schemes on unstructured cell-centered triangular grids. SIAM J Sci Comput 31 2 (2008) 1192-1220
43. Fuchs A. Optimierte Delaunay-Triangulierungen zur Vernetzung getrimmter NURBS-Körper. PhD thesis, Stuttgart University, 1999.
44. Gebauer S, Neunhäuserer L, Kornhuber R, Ochs S, Hinkelmann R, Helmig R. Equidimensional modelling of flow and transport processes in fractured porous systems I. In: Proceedings of the XIV international conference on computational methods in water resources, vol. 1, 23-28 June, Delft, The Netherlands; 2002.
45. Golub G.H., and Van Loan C.F. Matrix computations. Johns Hopkins studies in the mathematical sciences. 3rd ed. vol. 3 (1996), Johns Hopkins University Press, Baltimore
46. Granet S, Fabrie P, Lemonnier P, Quintard M. A single-phase flow simulation of fractured reservoir using a discrete representation of fractures. In: Proceedings of the 6th European conference on the mathematics of oil recovery, Peebles, Scotland; 1998.
47. Granet S., Fabrie P., Lemonnier P., and Quintard M. A two-phase flow simulation of a fractured reservoir using a new fissure element method. J Petrol Sci Eng 32 (2001) 35-52
48. Hægland H. Streamline methods with application to flow and transport in fractured media. PhD thesis, Dept. of Mathematics, University of Bergen, Norway; 2009. .
49. Higham N.J. Accuracy and stability of numerical algorithms. 2nd ed. (2002), Society for Industrial and Applied Mathematics, Philadelphia
50. Holm R., Kaufmann R., Heimsund B.-O., Øian E., and Espedal M.S. Meshing of domains with complex internal geometries. Numer Linear Algebra Appl 13 9 (2006) 717-731
51. Hoteit H., and Firoozabadi A. Multicomponent fluid flow by discontinuous Galerkin and mixed methods in unfractured and fractured media. Water Resour Res 41 (2005) W11412 10.1029/2005WR004339
52. Hoteit H., and Firoozabadi A. An efficient numerical model for incompressible two-phase flow in fractured media. Adv Water Resour 31 (2008) 891-905
53. Huyakorn P.S., and Pinder G.F. Computational methods in subsurface flow (1983), Academic Press, New York
54. Hyman J., Shashkov M., and Steinberg S. The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials. J Comput Phys 132 (1997) 130-148
55. Juanes R., Samper J., and Molinero J. A general and efficient formulation of fractures and boundary conditions in the finite element method. Int J Numer Methods Eng 54 (2002) 1751-1774
56. Karimi-Fard M., Durlofsky L.J., and Aziz K. An efficient discrete fracture model applicable for general purpose reservoir simulators. SPE J June (2004) 227-236
57. Keilegavlen E, Aavatsmark I. Monotonicity for MPFA methods on triangular grids. Comput Geosci, submitted for publication.
58. Kim J.-G., and Deo M.D. Finite element discrete-fracture model for multiphase flow in porous media. AIChE J 46 6 (2000) 1120-1130
59. Klausen R.A., Radu F.A., and Eigestad G.T. Convergence of MPFA on triangulations and Richards' equation. Int J Numer Methods Fluids (2008) 10.1002/fld.1787
60. Klausen R.A., and Winther R. Convergence of multipoint flux approximations on quadrilateral grids. Numer Methods Part DE (2005) 10.1002/num.20158
61. Klausen R.A., and Winther R. Robust convergence of multipoint flux approximation on rough grids. Numer Math 104 (2006) 317-337
62. In: Krizek M., Neittaanmäki P., Glowinski R., and Korotov S. (Eds). Conjugate gradient algorithms and finite element methods (2004), Springer
63. Li R., Chen Z., and Wu W. Generalized difference methods for differential equations (2000), Marcel Dekker, New York
64. Ma J., Couples G.D., and Harris S.D. A mixed finite element technique based on implicit discretization of faults for permeability upscaling in fault damage zones. Water Resour Res 42 (2006) W08413 10.1029/2005WR004686
65. Manzocchi T., Walsh J.J., Nell P., and Yielding G. Fault transmissibility multipliers for flow simulation models. Petrol Geosci 5 (1999) 53-63
66. Martin V., Jaffré J., and Roberts J.E. Modeling fractures and barriers as interfaces for flow in porous media. SIAM J Sci Comput 26 5 (2005) 1667-1691
67. Morton K.W., and Süli E. Finite volume methods and their analysis. IMA J Numer Anal 11 (1991) 241-260
68. Neuman S.P. Trends, prospects and challenges in quantifying flow and transport through fractured rocks. Hydrogeol J 13 (2005) 124-147
69. Neunhäuserer L. Diskretisierungsansätze zur Modellierung von Strömungs- und Transportprozessen in geklüftet-porösen Medien. PhD thesis, University of Stuttgart, 2003.
70. Neunhäuserer L, Gebauer S, Ochs S, Hinkelmann R, Kornhuber R, Helmig R. Equidimensional modelling of flow and transport processes in fractured porous systems II. In: Proceedings of the XIV international conference on computational methods in water resources, vol. 1, 23-28 June, Delft, The Netherlands; 2002.
71. Nordbotten J.M., Aavatsmark I., and Eigestad G.T. Monotonicity of control volume methods. Numer Math 106 (2007) 255-288
72. Ochs SO, Hinkelmann R, Neunhäuserer N, Suess M, Helmig R, Gebauer S, et al. Adaptive methods for the equidimensional modelling of flow and transport processes in fractured aquifers. In: 5th international conference on hydro-science & engineering, Warsaw, Poland; 2002.
73. Pal M., Edwards M.G., and Lamb A.R. Convergence study of a family of flux-continuous finite-volume schemes for the general tensor pressure equation. Int J Numer Methods Fluids 51 (2006) 1177-1203
74. Pinder C.F., Huyakorn P.S., and Sudicky E.A. Simulation of flow and transport in fractured porous media. In: Bear J., Tsang C.-F., and de Marsily G. (Eds). Flow and contaminant transport in fractured rock (1993), Academic Press, San Diego 395-435
75. Pruess K., Wang J.S.Y., and Tsang Y.W. On thermohydrologic conditions near high-level nuclear wastes emplaced in partially saturated fractured tuff. 1. Simulation studies with explicit consideration of fracture effects. Water Resour Res 26 6 (1990) 1235-1248
76. Reichenberger V., Jakobs H., Bastian P., and Helmig R. A mixed-dimensional finite volume method for two-phase flow in fractured porous media. Adv Water Resour 29 7 (2006) 1020-1036
77. Reichenberger V, Jakobs H, Bastian P, Helmig R, Niessner J. Complex gas-water processes in discrete fracture-matrix systems. Upscaling, mass-conservative discretization and efficient multilevel solution. Mitteilungsheft Nr. 130, Institut für Wasserbau, University of Stuttgart. ISBN:3-933761-33-6, 2004.
78. Sahimi M. Flow and transport in porous media and fractured rock: from classical methods to modern approaches (1995), VCH, Weinheim
79. Shashkov M., and Steinberg S. Conservative finite-difference methods on general grids, symbolic and numeric computation series (1996), CRC Press, Baca Raton
80. Sheng Z., and Yuan G. A nine point scheme for the approximation of diffusion operators on distorted quadrilateral meshes. SIAM J Sci Comput 30 3 (2008) 1341-1361
81. Silberhorn-Hemminger A. Modellierung von Kluftaquifersystemen: Geostatistische Analyse und deterministisch-stochastische Kluftgenerierung. PhD thesis, University of Stuttgart, 2002.
82. Stewart G.W., and Sun J.-G. Matrix pertubation theory. Computer science and scientific computing (1990), Academic Press, Boston
83. Strang G., and Fix G.J. An analysis of the finite element method. Prentice-Hall series in automatic computation (1973), Prentice-Hall, Englewood Cliffs, NJ
84. Teimoori A., Chen Z., Rahman S.S., and Tran T. Effective permeability calculation using boundary element method in naturally fractured reservoirs. Petrol Sci Technol 23 (2005) 693-709
85. Trefethen L.N., and Bau D. Numerical linear algebra (1997), SIAM, Philadelphia, Penn
86. Verma S, Aziz K. A control-volume scheme for flexible grids in reservoir simulation, SPE 37999. In: SPE reservoir simulation symposium, Dallas, Texas; 8-11 June 1997.
87. Weisstein EW, Plane-plane intersection. From math world - a wolfram web resource. .
88. Zienkiewicz O.C., Taylor R.L., and Zhu J.Z. The finite element method: its basis and fundamentals. 6th ed. (2005), Elsevier, Amsterdam