Abstract
Homogeneous effective permeabilities in the near-well region are generally obtained using analytical solutions for transient flow. In contrast, this paper focuses on heterogeneous permeability obtained from steady flow solutions, although extensions to unsteady flow are introduced too. Exterior calculus and its discretized form have been used as a guide to derive the system of algebraic equations. Edge-based vector potentials describing 3-D steady and unsteady flow without mass balance error stabilize the solution. After a physics-oriented introduction, relatively simple analytic examples of forward and inverse discrete modeling demonstrate the applicability.
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References
Abraham R, Marsden J.E. and Manifolds Ratiu T. (1988). Tensor Analysis, and Applications, 2nd edition. Springer-Verlag, New York
Aguilera R., Artindale J.S., Cordell G.M., Ng M.C., Nicholl G.W., Runions G.A. (1991) Horizontal Wells Gulf Publishing Company
Aziz K. and Settari A. (1979). Petroleum Reservoir Simulation. Applied Science Publishers, Ltd., London
Barree, R.D., Conway, M.W. Beyond beta factors: a complete model for Darcy, Forchheimer, and trans-Forchheimer flow in porous media, SPE Annual Technical Conference and Exhibition, Houston, 26–29 Sept. 2004, paper SPE 89325 (2004)
Barton M.L. and Cendes Z.J. (1987). New vector finite elements for three dimensional magnetic fields computations. J. Appl. Phys. 61(8): 3919–3921
Bear J. (1988). Dynamics of Fluids in Porous Media. Dover Publications, Inc., New York
Bloch A.M. (2003) Nonholonomic Mechanics and Control Springer-Verlag New York, Inc
Bossavit A. (1998). Computational Electromagnetism: Variational Formulations, Complementarity, Edge Elements. Academic Press, San Diego
Bossavit A. (2003). Mixed-hybrid methods in magnetostatics complementarity in one stroke. IEEE Trans. Magn. 39: 1099–1102
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)
Burke W.L. (1985). Applied Differential Geometry. Cambridge University Press, Cambridge
Carlslaw H.S. and Jaeger J.C. (1959). Conduction of Heat in Solids. Oxford University Press, Oxford
Chavent G., Jaffré J. Mathematical Models and Finite Elements for Reservoir Simulators. North-Holland, Amsterdam (1986)
Darling R.W.R. (1994). Differential Forms and Connections. Cambridge University Press, Cambridge
Deschamps G.A. (1981). Electromagnetism and differential forms. Proc. IEEE 69: 676–696
Flanders H. (1963). Differential Forms, With Applications to the Physical Sciences. Academic Press, New York
Fokker, P.A., Verga, F., Egberts, P.J.P. New semianalytic technique to determine horizontal well productivity index in fractured reservoirs, SPE Reservoir Evaluation and Engineering (Apr.), 123–131 (2005)
Frankel T. (2004). The Geometry of Physics, An Introduction. Cambridge University Press, New York
Henle M. (1994). A Combinatorial Introduction to Topology. Dover Publications, New York
Hirani, A.N. Discrete Exterior Calculus, Ph.D. thesis, California institute of Technology, Pasadena (2003) (http://etd.caltech.edu/etd/available/etd-05202003-095403/, also see http://arxiv.org/abs/ math.DG/0508341)
Joshi, S.D. Horizontal Well Technology. PennWell Publishing Company, Tulsa, Oklahoma (1991)
Kaasschieter E.F. and Huijben A.J.M. (1992). Mixed-hybrid finite elements and streamline computations for the potential flow problem. Numer. Meth. for PDEs 8: 221–266
Kruseman, G.P., De Ridder, N.A. Analysis and Evaluation of Pumping Test Data. Publication 47, International Institute for Land Reclamation and Improvement, Wageningen, The Netherlands (1990)
Lindell I.V. (2004). Differential Forms in Electromagnetism. Wiley Interscience, Hoboken, NJ
Magnus, W., Schoenmaker, W. Introduction to electromagnetism. In: Ciarlet P.G. (ed.) Handbook of Numerical Analysis Vol. XIII, pp. 3–103, Special Volume Numerical Methods in Electromagnetics, W.H.A. Schilders and E.J.W. ter Maten guest editors, Elsevier (2005)
Misner C.W, Thorne K.S. and Wheeler J.A. (1973). Gravitation. Freeman and Co., San Francisco
Morse P.M and Feshbach H. (1953). Methods of Theoretical Physics. McGraw-Hill, New York
Neuzil C.E. (2003). Hydromechanical coupling in geologic processes. Hydrogeol. J. 11(3): 41–83
Nield D.A. (1983). Boundary condition for the Rayleigh-Darcy problem: limitation of the Brinkman equation. J. Fluid Mech. 128: 37–40
Penrose P. (2005). The Road to Reality, A Complete Guide to the Laws of the Universe. Alfred A. Knopf, New York
Russell T.F. (2000). Relationships among some conservative discretization methods. In: Chen, Z., Ewing, R.E. and Shi, Z.-C. (eds) Numerical Treatment of Multiphase Flows in Porous Media, pp 267–282. Springer-Verlag, Berlin
Spivak M. (1979). A Comprehensive Introduction to Differential Geometry. Publish or Perish, Inc., Berkeley, CA
Tarhasaari, T., Kettunen, L., Bossavit, A.: An interpretation of the Galerkin method as the realization of a discrete Hodge operator. IEEE Trans. Mag. 35(3), 1494–1497 (1999)
Leeuwen V.C.I. (2005). A Matlab Implementation of the Edge-Based Face Element Method for Forward and Inverse Modelling of the Near-Well Region. Mathematical Institute, University Utrecht, The Netherlands
Weintraub S.H. (1997). Differential Forms: A Complement to Vector Calculus. Academic Press, San Diego
Yeten, B., Durlofsky, L.J., Aziz, K. Optimization of smart well control, 2002 SPE International Horizontal Well Technology Conference. Calgary, SPE 7903, 1–7 (2002)
Zijl W. (2004). A direct method for the identification of the permeability field based on flux assimilation by a discrete analog of Darcy’s law. Transport in Porous Media 56(1): 87–112
Zijl W. (2005). Face-centered and volume-centered discrete analogs of the exterior differential equations governing porous media flow I: Theory. Transport in Porous Media 60: 109–122
Zijl W and Nawalany M. (2004). The edge-based face element method for three-dimensional stream function and flux calculations in porous media flow. Transport in Porous Media 55(3): 361–382
Zijl, W, Trykozko, A. Numerical homogenization of the absolute permeability tensor around wells. SPEJ (Dec.), 399–408 (2001)
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Zijl, W. Forward and inverse modeling of near-well flow using discrete edge-based vector potentials. Transp Porous Med 67, 115–133 (2007). https://doi.org/10.1007/s11242-006-0027-7
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DOI: https://doi.org/10.1007/s11242-006-0027-7