Advertisement

Numerical Methods in Theories of Porous Materials

  • B. A. Schrefler
Conference paper
Part of the International Centre for Mechanical Sciences book series (CISM, volume 400)

Abstract

The numerical solution of a model for the mechanical behaviour of saturated-unsaturated porous media is presented. Averaging theories are used to derive the necessary balance equations. Constitutive equations and thermodynamic equations for the model closure are introduced. The governing equations are then discretised in space, by the finite element method, and in time, by finite differences. The solution is shown by a direct approach and domain decomposition together with a multifrontal technique. The numerical properties are discussed. Examples of thermoelastic consolidation of partially saturated soil and dynamic strain localisation of fully saturated sands are illustrated.

Keywords

Porous Medium Shear Band Porous Material Capillary Pressure Effective Plastic Strain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Hassanizadeh M. and Gray W. G., General conservation equations for multi-phase system: 1. Averaging technique, Adv. Water Res. 2, 131–144, 1979Google Scholar
  2. 2.
    Hassanizadeh M. and Gray W. G., General conservation equations for multi-phase system: 2. Mass, momenta, energy and entropy equations, Adv. Water Res. 2, 191–201, 1979Google Scholar
  3. 3.
    Hassanizadeh M. and Gray W. G., General conservation equations for multi-phase system: 3. Constitutive theory for porous media flow, Adv. Water Res. 3, 25–40, 1980Google Scholar
  4. 4.
    Lewis R.W., Schrefler B.A., The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media, John Wiley, Chichester, 1998MATHGoogle Scholar
  5. 5.
    Hassanizedh, S.M. and Gray, W.G., Mechanics and thermodynamics of multiphase flow in porous media including interphase transport, Adv. Water Resour., 13 (1990), 169–186.Google Scholar
  6. 6.
    Gray W. G. and Hassanizadeh M., Unsaturated flow theory including interfacial phenomena, Water Resour. Res. 27, (8), 1855–1863, 1991Google Scholar
  7. 7.
    Gray, W.G., and Hassanizadeh, S.M., Paradoxes and realities in unsaturated flow theory, Water Resour. Res., Vol. 27 (1991), 1847–1854.Google Scholar
  8. 8.
    Baggio P., Bonacina C., Schrefler B. A., Some considerations on modelling heat and mass transfer in porous media, Transport in Porous Media, 28 (1997), 233–281CrossRefGoogle Scholar
  9. 9.
    Zienkiewicz O.C., Chan A.H.C., Pastor M., Paul D.K. and Shiomi T., Static and dynamic behaviour of soils: a rational approach to quantitative solutions. I–Fully saturated problems, Proc. R. Soc.Lond., A429, 285–309, 1990MATHCrossRefGoogle Scholar
  10. 10.
    Zienkiewicz, O.C., Xie, Y.M., Schrefler, B.A., Ledesma, A. and Bicanic, N. (1990) Static and Dynamic behaviour of soils: a rational apprach to quantitative solutions. II. Semi-saturated problems, Proc. R. Soc. Lond. A 429, 311–21MATHCrossRefGoogle Scholar
  11. 11.
    Biot, M.A. (1956) General solution of the equation of elasticity and consolidation for a porous material, J. Appl. mech., 23, 91–96.MATHMathSciNetGoogle Scholar
  12. 12.
    Biot, M.A. (1941) General theory of three-dimensional consolidation, J. Appl. Phys., 12, 155–64.MATHCrossRefGoogle Scholar
  13. 13.
    Meroi E., Schrefler B.A., Zienkiewicz O.C., Large strain static and dynamic semisaturated soil behaviour, Int. J. Num. Anal. Meth. Geomechanics, 19, 2, 81–106, 1995CrossRefGoogle Scholar
  14. 14.
    Carter, J.P., Booker, J.R. and Small, J.C. (1979) The analysis of finite elastoplastic consolidation. Int. J. Num. Anal. Meth. Geomech., 2, 107–29.CrossRefGoogle Scholar
  15. 15.
    Zienkiewicz, O.C. and Shiomi, T. (1984) Dynamic behaviour of saturated porous media: The generalised Boit formulation and its numerical solution, Int. J. Num. Anal. Meth. Geomech., 8, 71–96.MATHCrossRefGoogle Scholar
  16. 16.
    Li, X., Zienkiewicz, O.C. and Xie, Y.M. (1990) A numerical model for immiscible two-phase fluid flow in porous medium and its time domain solution, Int. J. Num. Methods Engng., 30, 1195–1212.MATHCrossRefGoogle Scholar
  17. 17.
    Schrefler, B.A. and Zhan, X.Y. (1993) A fully coupled model for water flow and airflow in deformable porous media, Wat. Resour. Research, 29, 155–67.CrossRefGoogle Scholar
  18. 18.
    Gawin, D., Baggio, P. and Schrefler, B.A. (1995) Coupled heat, water and gas flow in deformable porous media, Int. J. Num. Meth. Fluids, 20, 969–87.MATHCrossRefGoogle Scholar
  19. 19.
    Zienkiewicz O.C., Taylor R.L., The Finite Element Method, McGraw-Hill Book Company, London, 1991Google Scholar
  20. 20.
    Babuska, I., Error bounds for finite element methods, Num. Math. 16 (1971), 322–333.MATHMathSciNetGoogle Scholar
  21. 21.
    Babuska, I., The finite element method with Lagrange multipliers, Num. Math. 20 (1973), 179–192.MATHMathSciNetGoogle Scholar
  22. 22.
    Brezzi, F., On the existence, uniqueness and approximations of saddle point problems arising from Lagrange multipliers, R.A.I.R.D. 8 (1974), 129–151.MATHMathSciNetGoogle Scholar
  23. 23.
    Richtmyer, R.D., Morton, K.W., Difference Methods for Initial-Value Problems, Interscience, New York, 1967.MATHGoogle Scholar
  24. 24.
    Ortega, J.M., Rheinboldt, W.C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, San Diego, 1970.MATHGoogle Scholar
  25. 25.
    Morris, J.L., Computational Methods in Elementary Numerical Analysis, Wiley, Chichester, 1983.MATHGoogle Scholar
  26. 26.
    Allen III, M.B., Herrera, I., Pinder, G.F., Numerical Modelling in Science and Engineering, Wiley, New York, 1988.Google Scholar
  27. 27.
    Farhat, C., Sobh, N., A consistency analysis of a class of concurrent transient implicit/explicit algorithms, Comput. Methods Appl. Mech. Engrg., 84 (1990), 147–162.MATHCrossRefGoogle Scholar
  28. 28.
    Schrefler, B.A., and Simoni, L., Capillary effects and compaction of gas reservoirs, Proc. 10th Int. Conf. Finite Elements in Fluids, M. Hafez and J.C. Heinrich, Eds., University of Arizona, Tucson, 1998, 70–75.Google Scholar
  29. 29.
    Wang, X., Gawin, D., Schrefler, B.A, A parallel algorithm for thermo-hydromechanical analysis of deforming porous media, Computational Mechanics, 19 (1996), 99–109.MATHCrossRefGoogle Scholar
  30. 30.
    Wang, X., and Schrefler, B.A., A multifrontal parallel algorithm for coupled thermohydro-mechanical analysis of deforming porous media, Int. J. Num. Eng., 43, (1998), 1069–1083.MATHCrossRefGoogle Scholar
  31. 31.
    Irons, B. M., A frontal solution program for finite element analysis, Int. J. num. Meth. Engng, 2, 5–32 (1970).MATHCrossRefGoogle Scholar
  32. 32.
    Hood, P., Frontal solution program for unsymmetric matrices, Int. J. num. Meth. Engng, 10, 379–399(1976).Google Scholar
  33. 33.
    Yeo, M. F., A more efficient front solution: allocating assembly locations by longevity consideration, Int. J. num. Meth. Engng, 7, 570–573 (1973).CrossRefGoogle Scholar
  34. 34.
    Abbas, S. F., Some novel applications of the frontal concept, Int. J. num. Meth. Engng, 15, 519–536 (1980).MATHCrossRefGoogle Scholar
  35. 35.
    Schrefler, B.A., Simoni. L. and Turska, E., Standard staggered and staggered Newton schemes in thermo-hydro-mechanical problems, Comput. Meth. Appl. mech. Engng., 144 (1997), 93–109.MATHGoogle Scholar
  36. 36.
    Geist, G. A., Sunderam, V., Network based concurrent computing on the PVM system, Oak Ridge National Laboratory, ORNL/TM-11760, 1991.Google Scholar
  37. 37.
    Geist, G. A., Beguelin, A., Dongarra, J., Jiang, W., Manchek, R., Sunderam, V., PVM 3 User’s guide and reference manual, Oak Ridge National Laboratory, ORNL/TM12187, 1994.Google Scholar
  38. 38.
    Butler, R., Lush, E., User’s guide to the P4 Programming System, Technical Report, TM-ANL/92/17, Argonne National Laboratory, 1992.Google Scholar
  39. 39.
    Gens, A., Jouanna, P., and Schrefler, B.A. (eds), Modern Issues in Non-Saturated Soils, CISM Courses ans Lectures, 357, Springer Verlag, Wien, 1995.Google Scholar
  40. 40.
    Zienkiewicz O.C., Chan A.H.C., Pastor M., Schrefler B.A., Shiomi T., Computational Soil Dynamics applied to Earthquake Engineering, John Wiley and Sons, 1999.Google Scholar
  41. 41.
    Brooks R.N., Corey A.T., Properties of porous media affecting fluid flow, J. Irrig. Drain. Div. Am. Soc. Civ. Eng., 92 (IR2), 61–68, 1966.Google Scholar
  42. 42.
    Schrefler, B.A., Zhang, H.W., Pastor, M., and Zienkiewicz, O.C., Strain localisation modelling and pore pressure in saturated sand samples, Computational Mechanics, 22, 266–280, 1998.MATHCrossRefGoogle Scholar
  43. 43.
    Desrues J., Mokni M., Drained, Undrained Biaxial Test Data on Hostun RF Sand, Data base Alert96, ALERT Web server. http://geo.hmg. inpg.fr/alert//people/DESRUES_Jacques/DESRUES_Jacques_Comput er_Pgm.5/Google Scholar
  44. 44.
    Desrues J:, Mokni M., Viggiani G., Experimental strain localisation in undrained biaxial tests on sand, McNU’97 Symposium, Northwestern University, USA 30 June-2 July 1997.Google Scholar
  45. 45.
    Desrues J:, Mokni M., Viggiani G., Experimental strain localisation in undrained biaxial tests on sand, to be published in Mechanics of Cohesive-Frictional Materials and Structures (1999).Google Scholar
  46. 46.
    Sluys, L.J., Wave propagation, localization and dispersion in softening solids, Ph.D.Thesis, Civil Engineering Department of Delf University of Technology, 1992.Google Scholar
  47. 47.
    Loret, B., and Prevost, J.H., Dynamic strain localisation in fluid-saturated porous media J. Eng. Mech. 11, 907–922, 1991.CrossRefGoogle Scholar
  48. 48.
    Needleman A., Material rate dependence and mesh sensitivity on localisation problems, Comp. Meth. Appl. Mech. Eng. 67, 69–86, 1988.MATHCrossRefGoogle Scholar
  49. 49.
    Schrefler B. A., Sanavia L. and Majorana C. E., A multiphase medium model for localisation and post-localisation simulation in geomaterial. Mechanics of Cohesive-Frictional Materials and Structures, 1, 95–114, 1996.CrossRefGoogle Scholar
  50. 50.
    Schrefler, B.A., Sanavia, L., and Majorana, C.E., A multiphase medium model for localisation and post-localisation simulation in geomaterials. Mechanics of Cohesive-Frictional Materials and Structures, 1 (1996), 95–114.CrossRefGoogle Scholar
  51. 51.
    Pastor, M., Zienkiewicz, O.C., and Chan, A.H.C., Generalized plasticity and the modelling of soil behaviour, Int. J. Num. Anal. Meths. Geomech., 14 (1990), 151–190.MATHCrossRefGoogle Scholar
  52. 52.
    Pastor M. and Zienkiewicz O. C., A generalized plasticity hierarchical model for sand under monotonic and cyclic loading, in G.N. Pande and W.F. Van Impe(eds.), Numerical Models in Geomechanics, Jackson and Son, London, 131–150, 1986Google Scholar
  53. 53.
    Mokni M., Relations entre déformations en masse et déformations localisées dans les matériaux granulaires, Ph. D. Thesis, Institut de Mécanique de Grenoble, France, 1992.Google Scholar
  54. 54.
    Gawin D., Sanavia L., Schrefler B.A., Cavitation modelling in saturated geomaterials with application to dynamic strain localisation, Int. J. Num. Meth. In fluids, 27, 109–125, 1998.Google Scholar
  55. 55.
    Safai N.M., Pinder G.F., `Vertical and horizontal land deformation in a desaturating porous medium’, Adv. Water Resour. 2, 19–25, 1979.Google Scholar
  56. 56.
    Schrefler, B.A., Mechasnics of saturated-unsaturated porous materials and quantitative solution,Sectional Lecture, XIX ICTAM, 1996, Kyoto, Japan, from Theoretical and Applied Mechanics, T. Tatsumi, E. Watanabe and T. Kambe, Eds. Elsevier Science E.V., 1997, 481–49Google Scholar

Copyright information

© Springer-Verlag Wien 1999

Authors and Affiliations

  • B. A. Schrefler
    • 1
  1. 1.University of PaduaPaduaItaly

Personalised recommendations