Abstract
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.
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Abbreviations
- :
-
(Vectors are denoted by letters with right-pointing arrow above their names. Matrices and arrays are denoted as upper case symbols)
- A f :
-
Surface area of grid block face f (L 2)
- D:
-
N V × N F matrix relating volumes to faces, discrete differential operator corresponding to div (–)
- h i :
-
Hydraulic head gradient in the i direction (–)
- \({\vec{h}}\) :
-
Head gradient vector (–)
- H:
-
Column array of dimension N F representing head differences including those caused by boundary conditions (L)
- k ij :
-
Component of hydraulic conductivity tensor along the i and j directions (L/T)
- K:
-
N V × N V matrix containing hydraulic conductivities and grid metrics (L 2/T)
- \({\vec{\vec{k}}}\) :
-
Hydraulic conductivity tensor (L/T)
- N BN, N BF, N BE :
-
Number of boundary nodes, boundary faces, and boundary edges (–)
- N X , N Y , N Z :
-
Number of grid blocks in the x, y, and z directions (–)
- N V, N F, N E :
-
Number of grid volumes, faces, and edges (–)
- q i :
-
Flux density in the i direction (L 3/T/L 2)
- \({\vec{q}}\) :
-
Flux density vector (L 3/T/L 2)
- Q:
-
Column array of dimension N F representing normal components of flux densities \({\vec{q}}\) integrated over faces (L 3/T)
- R:
-
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)
- x i :
-
Cartesian coordinate in the i direction (L)
- \({\phi}\) :
-
Hydraulic head (L)
- Φ:
-
Column array of dimension N V representing heads at the center of grid volumes (L)
- γ ij :
-
Component of resistivity tensor along the i and j directions (T/L)
- \({\vec{\vec{\gamma}}}\) :
-
Resistivity tensor (T/L)
- Γ:
-
N V × N V matrix containing resistivities and grid metrics (T/L 2)
- Π:
-
Column array of dimension N F representing oriented boundary heads, nonzero only on boundary faces (L)
- θ :
-
Angular coordinate (rad)
- \({\vec{\psi}}\) :
-
Vector potential (L 3/T)
- Ψ:
-
Column array of dimension N E representing 3D stream function, i.e., vector potentials integrated along edges (L 3/T)
- × :
-
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, \({\nabla \times \vec{a}=\left( {\partial a_z /\partial \tilde{y}-\partial a_y /\partial z,\partial a_x /\partial \tilde{z}-\partial a_z /\partial x,\partial a_y /\partial \tilde{x}-\partial a_x /\partial y}\right)}\)
References
Aziz K., Settari A.: Petroleum Reservoir Simulation. Applied Science Publishers, Ltd., London (1979)
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)
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
Bear J.: Dynamics of Fluids in Porous Media. Dover Publications, Inc., New York (1972)
Bossavit A.: Computational Electromagnetism. Academic Press, San Diego (1998)
Bossavit A.: Generalized finite differences in computational electromagnetism. Prog. Electromagn. Res. 32, 45–64 (2001). doi:10.2528/PIER00080102
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)
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)
Flanders H.: Differential Forms with Applications to the Physical Sciences. Courier Dover Publications, New York (1989)
Frankel T.: The Geometry of Physics, An Introduction. Cambridge University Press, New York (2004)
Freeze R.A., Cherry J.A.: Groundwater. Prentice-Hall, Inc., New Jersey (1979)
Göckeler M., Schücker T.: Differential Geometry, Gauge Theories and Gravity. Cambridge University Press, New York (1987)
Ivancevic V.G., Ivancevic T.T.: Natural Biodynamics. World Scientific Publishing Co. Pte. Ltd, Singapore (2005)
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)
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
Perot J.B., Subramania V.: Discrete calculus methods for diffusion. J. Comput. Phys. 224, 59–81 (2007). doi:10.1016/j.jcp.2006.12.022
Schwartz F.W., Zhang H.: Fundamentals of Ground Water. Wiley, New York (2003)
Shewchuk J.R.: An Introduction to the Conjugate Gradient Method Without the Agonizing Pain. Carnegie Mellon University, Pittsburgh (1994)
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)
Tóth J.: Hydraulic continuity in large sedimentary basins. Hydrogeol. J. 3(4), 4–16 (1995). doi:10.1007/s100400050250
Trykozko A., Brouwer G., Zijl W.: Downscaling: a complement to homogenization. Int. J. Numer. Anal. Model. 5(suppl.), 157–170 (2008)
Weintraub S.H.: Differential Forms a Complement to Vector Calculus. Academic Press, Inc., San Diego (1997)
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
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
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
Zijl W., Nawalany M.: Natural Groundwater Flow. Lewis Publishers, Boca Raton (1993)
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
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Mohammed, G.A., Zijl, W., Batelaan, O. et al. Comparison of Two Mathematical Models for 3D Groundwater Flow: Block-Centered Heads and Edge-Based Stream Functions. Transp Porous Med 79, 469–485 (2009). https://doi.org/10.1007/s11242-009-9336-y
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DOI: https://doi.org/10.1007/s11242-009-9336-y