Mathematical development and verification of a non-orthogonal finite volume model for groundwater flow applications

Advances in Water Resources

Volumn 30, Issue 1, 2007, Pages 29-42

  Loudyi, D ^a   Falconer, R A ^a   Lin, B ^a  

^a CARDIFF UNIVERSITY   (United Kingdom)

Author keywords

Finite difference comparison; Finite volume; Gradient approximation; Groundwater flow; Non orthogonal grid

Indexed keywords

AQUIFERS; FINITE DIFFERENCE METHOD; FINITE VOLUME METHOD; HYDRAULICS; LEAST SQUARES APPROXIMATIONS; MATHEMATICAL MODELS;

GRADIENT APPROXIMATION; HYDRAULIC EQUATIONS; NON-ORTHOGONAL GRID;

GROUNDWATER FLOW;

AQUIFERS; FINITE DIFFERENCE METHOD; FINITE VOLUME METHOD; GROUNDWATER FLOW; HYDRAULICS; LEAST SQUARES APPROXIMATIONS; MATHEMATICAL MODELS;

AQUIFER; DISCHARGE; FINITE DIFFERENCE METHOD; FINITE VOLUME METHOD; GROUNDWATER FLOW; HYDRAULIC HEAD;

EURASIA; EUROPE; TAWE RIVER; UNITED KINGDOM; WALES; WESTERN EUROPE;

EID: 33750947856     PISSN: 03091708     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.advwatres.2006.02.010     Document Type: Article

References (53)

1. Aavatsmark I. An introduction to multipoint flux approximations for quadrilateral grids. Comput Geosci 6 3-4 (2002) 405-432
2. Amtec. Tecplot mesh generator, version 1.0. User's manual. Amtec Engineering, Inc., Bellevue, Washington, USA, 1999. Available from: http://www.tecplot.com.
3. Andersen PF. A manual of instructional problems for the USGS MODFLOW model. EPA/600/R-93/010, 1993.
4. Anderson M.P., and Woessner W.W. Applied groundwater modeling: simulation of flow and advective transport (1992), Academic Press, San Diego
5. 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
6. Barrash W., and Dougherty M.E. Modeling axially symmetric and nonsymmetric flow to a well with MODFLOW, and application to Goddard2 well test, Boise, Idaho. Ground Water 35 4 (1997) 602-611
7. Bear J. Dynamics of fluids in porous media (1972), American Elsevier, New York
8. Bear J. Hydraulics of groundwater (1979), McGraw-Hill, New York
9. Cai Z., Jones J.E., McCormick S.F., and Russell T.F. Control-volume mixed finite element methods. Comput Geosci 1 3-4 (1997) 289-315
10. Carrera J, Melloni G. The simulation of solute transport: an approach free of numerical dispersion. SAND86-7095, Sandia National Laboratories, Albuquerque, NM, 1987.
11. Celia M.A., Russell T.F., Herrera I., and Ewing R.E. Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation. Adv Water Resour 13 4 (1990) 187-206
12. Chow P., Cross M., and Pericleous K. A natural extension of the conventional finite volume method into polygonal unstructured meshes for CFD application. Appl Math Model 20 2 (1996) 170-183
13. Croft TN. Unstructured mesh-finite volume algorithms for swirling, turbulent, reacting flows, Ph.D. Thesis, University of Greenwich, UK, 1998.
14. Di Giammarco P., Todini E., and Lamberti P. A Conservative finite elements approach to overland flow: the control volume finite element formulation. J Hydrol 175 1-4 (1996) 267-291
15. Digimap. University of Edinburgh, 2004. Available from: http://www.edina.ac.uk/digimap/.
16. Elmahi I., Benkhaldoun F., Vilsmeier R., Gloth O., Patschull A., and Hǎnel D. Finite volume simulation of a droplet flame ignition on unstructured meshes. J Comput Appl Math 103 1 (1999) 187-205
17. Faille I. A control volume method to solve elliptic equation on a two-dimensional irregular mesh. Comput Meth Appl Mech Eng 100 (1992) 275-290
18. Faust CR, Sims PN, Spalding CP, Andersen PF, Lester BH, Shupe MG, et al. FTWORK: groundwater flow and solute transport in three dimensions. In: Documentation versions 2.8, GeoTrans, Sterling, VA, 1993.
19. Ferguson W.J., and Turner I.W. A control volume finite element numerical simulation of the drying of spruce. J Comput Phys 125 1 (1996) 59-70
20. Gottardi G., and Venutelli M. One-dimensional moving finite-element model of solute transport. Ground Water 32 4 (1994) 645-649
21. Haitjema H.M., Kelson V.A., and de Lange W. Selecting MODFLOW cell sizes for accurate flow fields. Ground Water 39 6 (2001) 931-938
22. Heberton CI, Russell TF, Konikow LF, Hornberger GZ. A three-dimensional finite volume Eulerian-Lagrangian localized adjoint method (ELLAM) for solute-transport modeling. US Geological Survey Water-Resources Investigations Report 00-4087, 2000.
23. Hirsch C. Numerical computation of internal and external flows. Fundamentals of numerical discretisation vol. 1 (1988), John Wiley & Sons, Chichester
24. Hyman J.M., and Shashkov M. Mimetic discretizations for Maxwell's equations. J Comput Phys 151 2 (1999) 881-909
25. Hyman J., Morel J., Shashkov M., and Steinberg S. Mimetic finite difference methods for diffusion equations. Comput Geosci 6 3-4 (2002) 333-352
26. Jayantha P.A., and Turner I.W. A comparison of gradient approximations for use in finite-volume computational models for two-dimensional diffusion equations. Numer Heat Transfer 40 5 (2001) 376-390 Part B
27. Jayantha P.A., and Turner I.W. A second-order finite volume technique for simulating transport in anisotropic media. Int J Numer Meth Heat Fluid Flow 13 1 (2003) 31-56
28. Jayantha P.A., and Turner I.W. Second order control-volume finite-element least-squares strategy for simulating diffusion in strongly anisotropic media. J Comput Math 23 1 (2005) 1-16
29. Kershaw D.S. Differencing of the diffusion equation in lagrangian hydrodynamic codes. J Comput Phys 39 (1981) 375-395
30. Kirpup SM. Fortran codes for computing the discrete Helmholtz integral operators: user guide. Report MCS-96-06, Department of Mathematics and Computer Sciences, University of Salford, UK, 1996.
31. Klausen R.A., and Russell T.F. Relationships among some locally conservative discretization methods which handle discontinuous coefficients. Comput Geosci 8 3-4 (2004) 341-377
32. Konikow LF, Goode DJ, Hornberger GZ. A three-dimensional method-of-characteristics solute-transport model (MOC3D). US Geological Survey Water-Resources Investigations Report 96-4267, 1996.
33. Leake SA, Claar DV. Procedures and computer programs for telescopic mesh refinement using MODFLOW. US Geological Survey Open-File Report 99-238, 1999.
34. Lin HC, Richards DR, Yeh GT, Cheng HP, Jones NL. FEMWATER: a three-dimensional finite element computer model for simulating density dependent flow and transport, in variably saturated media. Report CHL-96-12, US Army Corps of Engineer, Vicksburg, MS, 1997.
35. Liu F., and Jameson A. Multigrid Navier-Stokes calculations for three-dimensional cascades. AIAA J 31 10 (1993) 1785-1791
36. Loudyi D. A 2D Finite volume model for groundwater flow simulations: integrating non-orthogonal grid capability into MODFLOW, Ph.D. Thesis, Cardiff University, UK, 2005.
37. McDonald MG, Harbaugh AW. A modular three-dimensional finite-difference ground-water flow model. US Geological Survey Techniques of Water-Resources Investigations, Book 6, 1988 [chapter A1].
38. Meerschaert M.M., and Tadjeran C. Finite difference approximations for fractional advection-dispersion flow equations. J Comput Appl Math 172 1 (2004) 65-77
39. Mehl S., and Hill M.C. Three-dimensional local grid refinement method for block-centered finite-difference groundwater models using iteratively coupled shared nodes: a new method of interpolation and analysis of errors. Adv Water Resour 27 9 (2004) 899-912
40. Morel J.E., Dendy Jr., Hall M.L., and White S.W. A cell-centered Lagrangian-mesh diffusion differencing scheme. J Comput Phys 103 2 (1992) 286-299
41. Morel J.E., Roberts R.M., and Shashkov M.J. A local support-operators diffusion discretization scheme for quadrilateral r-z meshes. J Comput Phys 144 1 (1998) 17-51
42. Neuman S.P. Adaptive Eulerian-Lagrangian finite element method for advection-dispersion. Int J Numer Meth Eng 20 2 (1984) 321-337
43. Osnes H., and Langtangen H.P. An efficient probabilistic finite element method for stochastic groundwater flow. Adv Water Resour 22 2 (1998) 185-195
44. Patankar S.V. Numerical heat transfer and fluid flow (1980), Hemisphere, Washington, DC
45. Spitz F.J., Nicholson R.S., and Daryll A.P. A nested rediscretization method to improve pathline resolution by eliminating weak sinks representing wells. Ground Water 39 5 (2001) 778-785
46. Tannehill J.C., Anderson D.A., and Pletcher R.H. Computational fluid mechanics and heat transfer (1997), Taylor & Francis, London
47. Townley LR. AQUIFEM-N: a multi-layered finite element aquifer flow model, user's manual and description. CSIRO Division of Water Resources, Perth, Western Australia, 1990.
48. Turkel E. Accuracy of schemes with nonuniform meshes for compressible fluid flows. Appl Numer Math 2 6 (1985) 529-550
49. Turner I.W., and Ferguson W.J. An unstructured mesh cell-centered control volume method for simulating heat and mass transfer in porous media: application to softwood drying, Part I: The isotropic model. Appl Math Model 19 11 (1995) 654-667
50. US Environmental Protection Agency (USEPA). A technical guide to ground-water model selection at sites contaminated with radioactive substances. EPA/402/R-94/012, Office of Solid Waste and Emergency Response, US Environmental Protection Agency, Washington, DC, 1994.
51. Voss CI. A finite element simulation model for saturated-unsaturated, fluid-density-dependent ground-water flow with energy transport or chemically-reactive single-species solute transport. US Geological Survey Water Resources Investigation Report 84-4369, 1984.
52. Wasantha Lal A.M. Weighted implicit finite-volume model for overland flow. J Hydraul Eng 124 9 (1998) 941-950
53. Zheng C, Wang PP. MT3DMS: A modular three-dimensional multispecies transport model for simulation of advection, dispersion, and chemical reactions of contaminants in groundwater systems: Documentation and user's guide. Contract Report SERDP-99-1, US Army Engineer Research and Development Center, Vicksburg, MS, 1999.