Modeling of Flow in Three-Dimensional, Multizone, Anisotropic Porous Media with Weakly Singular Integral Equation Method
Publication: Journal of Engineering Mechanics
Volume 135, Issue 8
Abstract
A symmetric Galerkin boundary element method is developed for modeling steady-state Darcy’s flow in three-dimensional porous media. The proposed technique is capable of treating a nonhomogeneous medium that consists of several regions possessing different permeabilities and may contain a surface of discontinuity such as impermeable seals. The key governing equations are established based on a pair of weakly singular weak-form integral equations for the fluid pressure and the fluid flux. The crucial feature of those integral equations are that they are completely regularized such that all involved kernels are only weakly singular and that they are applicable to a medium possessing generally anisotropic permeability. A final system of governing integral equations is obtained in a symmetric form and validity of all involved integrals only requires continuity of the pressure boundary data; as a consequence, continuous interpolations can be employed everywhere in the numerical approximation. To accurately capture the jump of the fluid pressure in the local region near the boundary of the discontinuity surface, special tip elements are employed. To further enhance accuracy and computational efficiency of the method, special integration quadrature is adopted to treat both weakly singular and nearly singular integrals and an interpolation strategy is utilized to evaluate the kernels for anisotropic permeability. As demonstrated by various numerical experiments, the current method yields highly accurate results with only weak dependence on mesh refinement.
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Acknowledgments
The writer gratefully acknowledges partial support provided by Chulalongkorn University in terms of grants for development of new faculty staff, Ratchadaphiseksomphot Endowment Fund.
References
Bonnet, M. (1995). “Regularized direct and indirect symmetric variational BIE formulations for three-dimensional elasticity.” Eng. Anal. Boundary Elem., 15, 93–102.
Brebbia, C. A., and Dominguez, J. (1989). Boundary elements: An introductory course, 2nd Ed., McGraw-Hill, New York.
Brezzi, F., Hughes, T. J. R., Marini, L. D., and Masud, A. (2005). “Mixed discontinuous Galerkin methods for Darcy flow.” J. Sci. Comput., 22, 119–145.
Deans, S. R. (1983). The radon transform and some of its applications, Wiley, New York.
Dominguez, J., Ariza, M. P., and Gallego, R. (2000). “Flux and traction boundary elements without hypersingular or strongly singular integrals.” Int. J. Numer. Methods Eng., 48, 111–135.
Frangi, A., and Maier, G. (2003). “The symmetric Galerkin BEM in linear and non-linear fracture mechanics of zonewise homogeneous solids.” Recent advance in boundary element and their solid mechanics applications, D. Beskos, ed., Springer, Berlin, 167–204.
Gel’fand, I. M., and Shilov, G. E. (1964). Generalized functions: Properties and operations, Vol. 1, Academic, New York.
Gray, L. J., Martha, L. F., and Ingraffea, A. R. (1990). “Hypersingular integrals in boundary element fracture analysis.” Int. J. Numer. Methods Eng., 29, 1135–1158.
Gray, L. J., and Paulino, G. H. (1997). “Symmetric Galerkin boundary integral formulation for interface and multi-zone problems.” Int. J. Numer. Methods Eng., 40, 3085–3101.
Hayami, K. (1992). “A projection transformation method for nearly singular surface boundary element integrals.” Lecture notes in engineering, Vol. 73, C. A. Brebbia, and S. A. Orszag, eds., Springer, Berlin.
Hayami, K., and Brebbia, C. A. (1988). “Quadrature methods for singular and nearly singular integrals in 3-D boundary element method.” Boundary elements X, Vol. 1, C. A. Brebbia, ed., Springer, Berlin, 237–264.
Hayami, K., and Matsumoto, H. (1994). “A numerical quadrature for nearly singular boundary element integrals.” Eng. Anal. Boundary Elem., 13, 143–154.
Helgason, S. (1999). The radon transform, 2nd Ed., Birkhäuser, Boston.
Hughes, T. J. R., Masud, A., and Wan, J. (2006). “A discontinuous-Galerkin finite element method for Darcy flow.” Comput. Methods Appl. Mech. Eng., 195, 3347–3381.
Jorge, A. B., Ribeiro, G. O., Cruse, T. A., and Fisher, T. S. (2001). “Self-regular boundary integral equation formulations for Laplace’s equation in 2-D.” Int. J. Numer. Methods Eng., 51, 1–29.
Li, H. B., and Han, G. M. (1985). “A new method for evaluating singular integral in stress analysis of solids by the direct boundary element method.” Int. J. Numer. Methods Eng., 21, 2071–2098.
Li, S., and Mear, M. E. (1998). “Singularity-reduced integral equations for displacement discontinuities in three-dimensional linear elastic media.” Int. J. Fract., 93, 87–114.
Li, S., Mear, M. E., and Xiao, L. (1998). “Symmetric weak-form integral equation method for three dimensional fracture analysis.” Comput. Methods Appl. Mech. Eng., 151, 435–459.
Liggett, J. A., and Liu, L. F. (1983). The boundary integral equation method for porous media flow, Allen and Unwin, London.
Martha, L. F., Gray, L. J., and Ingraffea, A. R. (1992). “Three-dimensional fracture simulation with a single-domain, direct boundary element formulation.” Int. J. Numer. Methods Eng., 35, 1907–1921.
Martin, P. A., and Rizzo, F. J. (1996). “Hypersingular integrals: How smooth must the density be?” Int. J. Numer. Methods Eng., 39, 687–704.
Masud, A., and Hughes, T. J. R. (2002). “A stabilized mixed finite element method for Darcy flow.” Comput. Methods Appl. Mech. Eng., 191, 4341–4370.
McLean, W. (2000). Strongly elliptic systems and boundary integral equations, Cambridge University Press, New York.
Nagarajan, A., Lutz, E. D., and Mukherjee, S. (1994). “A novel boundary elements method for linear elasticity with no numerical integration for 2D and line integrals for 3D problems.” J. Appl. Mech., 61, 264–269.
Rungamornrat, J. (2006). “Analysis of 3D cracks in anisotropic multi-material domain with weakly singular SGBEM.” Eng. Anal. Boundary Elem., 30, 834–846.
Rungamornrat, J., and Mear, M. E. (2008a). “Weakly-singular, weak-form integral equations for cracks in three-dimensional anisotropic media.” Int. J. Solids Struct., 45, 1283–1301.
Rungamornrat, J., and Mear, M. E. (2008b). “A weakly-singular SGBEM for analysis of cracks in 3D anisotropic media.” Comput. Methods Appl. Mech. Eng., 197, 4319–4332.
Rungamornrat, J., and Wheeler, M. F. (2006a). “Weakly-singular integral equations for steady-state flow in isotropic porous media.” Rep. No. 05-30, Institute for Computational Engineering and Sciences, Univ. of Texas at Austin, Austin, Tex.
Rungamornrat, J., and Wheeler, M. F. (2006b). “Weakly-singular integral equations for Darcy’s flow in anisotropic porous media.” Eng. Anal. Boundary Elem., 30, 237–246.
Shiah, Y. C., and Tan, C. L. (2004). “BEM treatment of three-dimensional anisotropic field problems by direct domain mapping.” Eng. Anal. Boundary Elem., 28, 43–52.
Sutradhar, A., and Paulino, G. H. (2004). “A simple boundary element method for problems of potential in non-homogeneous media.” Int. J. Numer. Methods Eng., 60, 2203–2230.
Xiao, L. (1998). “Symmetric weak-form integral equation method for three dimensional fracture analysis.” Ph.D. dissertation, Univ. of Texas at Austin, Austin, Tex.
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© 2009 ASCE.
History
Received: Dec 18, 2006
Accepted: Mar 24, 2009
Published online: Jul 15, 2009
Published in print: Aug 2009
Notes
Note. Associate Editor: Arif Masud
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