Transport in Porous Media

, Volume 79, Issue 3, pp 469–485 | Cite as

Comparison of Two Mathematical Models for 3D Groundwater Flow: Block-Centered Heads and Edge-Based Stream Functions

  • G. A. MohammedEmail author
  • W. Zijl
  • O. Batelaan
  • F. De Smedt


Traditionally, groundwater flow models, as well as oil reservoir models, are based on the block-centered finite difference method. Well-known models based on this approach are MODFLOW (groundwater) and ECLIPSE (oil and gas). Such models are well proven and robust; their underlying principles are well understood by hydrologists and petroleum reservoir engineers. Nevertheless, the desire to improve the block-centered finite difference paradigm has always been alive, for instance, to be able to apply deformed grid blocks, or to model anisotropy that is not aligned along the coordinate axes. This article introduces the edge-based stream function as a potential alternative to the paradigmatic model, not only to mitigate the above mentioned limitations, but especially for its promise to inverse modeling. Computer programs have been developed for the discrete analog equations of the stream function method and the conventional method. The two methods are tested by using synthetic forward modeling problems of uniform and radial flow. The theoretical formulation and the numerical results show that the two methods are algebraically equivalent and yield the same flux output. However, for rectangular grid blocks and anisotropy aligned along the coordinate axes, the block-centered method is shown to be computationally more efficient than the edge-based stream function method. The major advantage of the stream function method is that it is linear in the resistivities, proving it an ideal candidate for direct inverse modeling. Moreover, any arbitrary specification of stream functions yields a solution that satisfies the mass balance.


Discrete analogs Block-centered method 3D stream function Edge-based vector potential 

List of Symbols

(Vectors are denoted by letters with right-pointing arrow above their names. Matrices and arrays are denoted as upper case symbols)


Surface area of grid block face f (L 2)


N V × N F matrix relating volumes to faces, discrete differential operator corresponding to div (–)


Hydraulic head gradient in the i direction (–)


Head gradient vector (–)


Column array of dimension N F representing head differences including those caused by boundary conditions (L)


Component of hydraulic conductivity tensor along the i and j directions (L/T)


N V × N V matrix containing hydraulic conductivities and grid metrics (L 2/T)


Hydraulic conductivity tensor (L/T)


Number of boundary nodes, boundary faces, and boundary edges (–)


Number of grid blocks in the x, y, and z directions (–)


Number of grid volumes, faces, and edges (–)


Flux density in the i direction (L 3/T/L 2)


Flux density vector (L 3/T/L 2)


Column array of dimension N F representing normal components of flux densities \({\vec{q}}\) integrated over faces (L 3/T)


N F × N E matrix relating faces to edges, discrete differential operator corresponding to curl or rot (–)

\({\dot {s}_0}\)

Storage per unit volume (1/T)

\({\dot {{S}}_{\rm o}}\)

Column array of dimension N V representing storage (L 3/T)


Cartesian coordinate in the i direction (L)

Greek Symbols


Hydraulic head (L)


Column array of dimension N V representing heads at the center of grid volumes (L)


Component of resistivity tensor along the i and j directions (T/L)


Resistivity tensor (T/L)


N V × N V matrix containing resistivities and grid metrics (T/L 2)


Column array of dimension N F representing oriented boundary heads, nonzero only on boundary faces (L)


Angular coordinate (rad)


Vector potential (L 3/T)


Column array of dimension N E representing 3D stream function, i.e., vector potentials integrated along edges (L 3/T)

Additional Symbols


Cross product


Dot product

 = (∂/∂x, ∂/∂y, ∂/∂z), del (nabla) operator in 3D Cartesian coordinate system (1/L)


Divergence, \({\nabla \cdot \vec{a}=(\partial a_x /\partial x+\partial a_y /\partial y+\partial a_z /\partial z)}\)


Curl, \({\nabla \times \vec{a}=\left( {\partial a_z /\partial \tilde{y}-\partial a_y /\partial z,\partial a_x /\partial \tilde{z}-\partial a_z /\partial x,\partial a_y /\partial \tilde{x}-\partial a_x /\partial y}\right)}\)


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aziz K., Settari A.: Petroleum Reservoir Simulation. Applied Science Publishers, Ltd., London (1979)Google Scholar
  2. Barrett R., Berry M., Chan T.F., Demmel J., Donato J., Dongarra J., Eijkhout V., Pozo R., Romine C., Van der Vorst H.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, Philadelphia (1994)Google Scholar
  3. Barton M.L., Cendes Z.J.: New vector finite elements for three-dimensional magnetic field computation. J. Appl. Phys. 61(8), 3919–3921 (1987). doi: 10.1063/1.338584 CrossRefGoogle Scholar
  4. Bear J.: Dynamics of Fluids in Porous Media. Dover Publications, Inc., New York (1972)Google Scholar
  5. Bossavit A.: Computational Electromagnetism. Academic Press, San Diego (1998)Google Scholar
  6. Bossavit A.: Generalized finite differences in computational electromagnetism. Prog. Electromagn. Res. 32, 45–64 (2001). doi: 10.2528/PIER00080102 CrossRefGoogle Scholar
  7. Bossavit, A.: Discretization of electromagnetic problems: The “Generalized finite differences” approach. In: Ciarlet, P. G. (ed.) Handbook of Numerical Analysis, vol. XIII, pp. 105–191, Special volume numerical methods in electromagnetics, W.H.A. Schilders and E.J.W. ter Maten (guest editors), Elsevier (2005)Google Scholar
  8. Brouwer, G.K., Fokker, P.A., Wilschut, F., Zijl, W.: A direct inverse model to determine permeability fields from pressure and flow rate measurements. Math. Geosci. (2008, accepted)Google Scholar
  9. Flanders H.: Differential Forms with Applications to the Physical Sciences. Courier Dover Publications, New York (1989)Google Scholar
  10. Frankel T.: The Geometry of Physics, An Introduction. Cambridge University Press, New York (2004)Google Scholar
  11. Freeze R.A., Cherry J.A.: Groundwater. Prentice-Hall, Inc., New Jersey (1979)Google Scholar
  12. Göckeler M., Schücker T.: Differential Geometry, Gauge Theories and Gravity. Cambridge University Press, New York (1987)Google Scholar
  13. Ivancevic V.G., Ivancevic T.T.: Natural Biodynamics. World Scientific Publishing Co. Pte. Ltd, Singapore (2005)Google Scholar
  14. Kaasschieter E.F., Huijben A.J.M.: Mixed-hybrid finite elements and streamline computations for the potential flow problem. Numer. Methods Partial Differ. Equ. 8, 221–226 (1992)CrossRefGoogle Scholar
  15. Parker J.W., Ferraro R.D., Liewer P.C.: Comparing 3D finite element formulations modeling scattering from a conducting sphere. IEEE Trans. Magn. 29(2), 1646–1649 (1993). doi: 10.1109/20.250721 CrossRefGoogle Scholar
  16. Perot J.B., Subramania V.: Discrete calculus methods for diffusion. J. Comput. Phys. 224, 59–81 (2007). doi: 10.1016/ CrossRefGoogle Scholar
  17. Schwartz F.W., Zhang H.: Fundamentals of Ground Water. Wiley, New York (2003)Google Scholar
  18. Shewchuk J.R.: An Introduction to the Conjugate Gradient Method Without the Agonizing Pain. Carnegie Mellon University, Pittsburgh (1994)Google Scholar
  19. Smith L., Wheatcraft S.W.: Groundwater flow. In: Maidment, D.R.(eds) Handbook of Hydrology, pp. 6.1–6.58. McGraw-Hill, Inc., New York (2003)Google Scholar
  20. Tóth J.: Hydraulic continuity in large sedimentary basins. Hydrogeol. J. 3(4), 4–16 (1995). doi: 10.1007/s100400050250 CrossRefGoogle Scholar
  21. Trykozko A., Brouwer G., Zijl W.: Downscaling: a complement to homogenization. Int. J. Numer. Anal. Model. 5(suppl.), 157–170 (2008)Google Scholar
  22. Weintraub S.H.: Differential Forms a Complement to Vector Calculus. Academic Press, Inc., San Diego (1997)Google Scholar
  23. Zijl W.: A direct method for the identification of the permeability field based on flux assimilation by a discrete analog of Darcy’s law. Transp. Porous Med. 56, 87–112 (2004). doi: 10.1023/B:TIPM.0000018405.22085.99 CrossRefGoogle Scholar
  24. Zijl W.: Face-centered and volume-centered discrete analogs of the exterior differential equations governing porous medium flow I: theory, II examples. Transp. Porous Med. 60, 109–133 (2005). doi: 10.1007/s11242-004-4044-0 CrossRefGoogle Scholar
  25. Zijl W.: Forward and inverse modeling of near-well flow using discrete edge-based vector potentials. Transp. Porous Med. 67, 115–133 (2007). doi: 10.1007/s11242-006-0027-7 CrossRefGoogle Scholar
  26. Zijl W., Nawalany M.: Natural Groundwater Flow. Lewis Publishers, Boca Raton (1993)Google Scholar
  27. Zijl W., Nawalany M.: The edge-based face element method for 3D-stream function and flux calculations in porous media flow. Transp. Porous Med. 55, 361–382 (2004). doi: 10.1023/B:TIPM.0000013258.79763.a2 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • G. A. Mohammed
    • 1
    Email author
  • W. Zijl
    • 1
  • O. Batelaan
    • 1
    • 2
  • F. De Smedt
    • 1
  1. 1.Department of Hydrology and Hydraulic EngineeringVrije Universiteit BrusselBrusselsBelgium
  2. 2.Department of Earth and Environmental SciencesKatholieke Universiteit LeuvenLeuvenBelgium

Personalised recommendations