Abstract
Combining a geological model with a geomechanical model, it generally turns out that the geomechanical model is built from units that are at least a 100 times larger in volume than the units of the geological model. To counter this mismatch in scales, the geological data model's heterogeneous fine-scale Young's moduli and Poisson's ratios have to be “upscaled” to one “equivalent homogeneous” coarse-scale rigidity. This coarse-scale rigidity relates the volume-averaged displacement, strain, stress, and energy to each other, in such a way that the equilibrium equation, Hooke's law, and the energy equation preserve their fine-scale form on the coarse scale. Under the simplifying assumption of spatial periodicity of the heterogeneous fine-scale rigidity, homogenization theory can be applied. However, even then the spatial variability is generally so complex that exact solutions cannot be found. Therefore, numerical approximation methods have to be applied. Here the node-based finite element method for the displacement as primary variable has been used. Three numerical examples showing the upper bound character of this finite element method are presented.
Similar content being viewed by others
REFERENCES
Arbogast, T., Wheeler, M. F., and Yotov, I., 1997, Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences: SIAM J. Numer. Anal., v. 34, no. 2, p. 828–852.
Auriault, J.-L., 1983, Effective macroscopic description for heat conduction in periodic composites: Int. J. Heat Mass Transfer, v. 26, no. 6, p. 861–869.
Bakhvalov, N., and Panansenko, G., 1994, Homogenization: Averaging processes in periodic media: Kluwer, Dordrecht, 354 p.
Bensoussan, A., Lions, J.-L., and Papanicolaou, G., 1978, Asymptotic analysis for periodic structures: North Holland, Amsterdam, 200 p.
Bossavit, A., 1998, Computational electromagnetism, variational formulations, complementarity, edge elements: Academic Press, San Diego, 321 p.
Davis, R. O., and Selvadurai, A. P. S., 1996, Elasticity and geomechanics: Cambridge University Press, New York, 371 p.
De Haan, H., Giesen, M., and Scheffers, B., 2000, Comparison of velocity models for 2D time-depth conversion of the Chalk Group: Inf.-TNO-NITG Ed. Geo-Energy, v. 5, May, p. 1-4.
De Witte, F. C., and Hendriks, M. A. N., 1998, Diana-finite element analysis, user's manual release 7, linear static analysis: TNO Building and Construction Research, Delft, 217 p.
Dogru, A. H., 2000, Megacell reservoir simulation: J. Pet. Technol., May, p. 54-60.
Duvaut, G., and Lions, J.-L., 1976, Inequalities in mechanics and physics: Springer, Berlin, 397 p.
Helbig, K., 1994, Foundations of aniosotropy for exploration seismics: Pergamon, New York, 486 p.
Hornung, U., 1997, Homogenization and porous media: Springer, Berlin, 381 p.
Lanczos, C., 1970, The variational principles of mechanics: Dover, New York, 319 p.
Lefik, M., and Schrefler, B. A., 1994, Application of the homogenisation method to the analysis of superconducting coils: Fusion Eng. Des., v. 24, p. 231–255.
Morse, Ph. M., and Feshbach, H., 1953, Methods of theoretical Physics: McGraw-Hill, New York, 1978 p.
Pellegrino, C., Galvanetto, U., and Schrefler, B. A., 1999, Numerical homogenization of periodic composite materials with non-linear material components: Int. J. Numer. Methods Eng., v. 46, p. 1609–1637.
Penman, J., 1988, Dual and complementary variational techniques for the calculation of electromagnetic fields: Adv. Electron. Electron Phys., v. 70, p. 315–364.
Rijpsma, G., and Zijl, W., 1998, Upscaling of Hooke's law for imperfectly layered rocks: Math. Geol., v. 30, no. 8, p. 943–969.
Sanchez-Palencia, E., 1980, Non-homogeneous media and vibration theory, in Lecture notes in Physics, Vol. 127: Springer, Berlin, 415 p.
Schrefler, B. A., Lefik, M., and Galvanetto, U., 1997, Correctors in a beam model for unidirectional composites: Mech. Computational Mater. Struct., v. 4, p. 159–190.
Strang, G., and Fix, G. J., 1973, An analysis of the finite element method: Prentice-Hall, Englewood Cliffs, NJ, 306 p.
Trykozko, A., Zijl, W., and Bossavit, A., 2001, Nodal and mixed finite elements for the numerical homogenization of 3D permeability: Computational Geosciences, v. 5, p. 61–84.
Van Dyke, M., 1975, Perturbation methods in fluid mechanics: Parabolic Press, Stanford, 271 p.
Zijl, W., and Trykozko, A., 2001-a, Numerical homogenization of the absolute permeability using the conformal-nodal and mixed-hybrid finite element method: Transp. Porous Media., v. 44, p. 33–52.
Zijl, W., and Trykozko, A., 2001-b, Numerical homogenization of the absolute permeability tensor around a well: SPE J., p. 399-408.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Zijl, W., Hendriks, M.A.N. & 't Hart, C.M.P. Numerical Homogenization of the Rigidity Tensor in Hooke's Law Using the Node-Based Finite Element Method. Mathematical Geology 34, 291–322 (2002). https://doi.org/10.1023/A:1014894923280
Issue Date:
DOI: https://doi.org/10.1023/A:1014894923280