Advertisement

Mathematical Geology

, Volume 36, Issue 2, pp 217–238 | Cite as

Stochastic Modeling of Variably Saturated Transient Flow in Fractal Porous Media

  • Luis Guarracino
  • Juan E. Santos
Article

Abstract

This work presents the application of a Monte Carlo simulation method to perform an statistical analysis of transient variably saturated flow in an hypothetical random porous media. For each realization of the stochastic soil parameters entering as coefficients in Richards' flow equation, the pressure head and the flow field are computed using a mixed finite element procedure for the spatial discretization combined with a backward Euler and a modified Picard iteration in time. The hybridization of the mixed method provides a novel way for evaluating hydraulic conductivity on interelement boundaries. The proposed methodology can handle both large variability and fractal structure in the hydraulic parameters. The saturated conductivity Ks and the shape parameter αvg in the van Genuchten model are treated as stochastic fractal functions known as fractional Brownian motion (fBm) or fractional Gaussian noise (fGn). The statistical moments of the pressure head, water content, and flow components are obtained by averaging realizations of the fractal parameters in Monte Carlo fashion. A numerical example showing the application of the proposed methodology to characterize groundwater flow in highly heterogeneous soils is presented.

unsteady unsaturated flow finite elements heterogeneous soils 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

references

  1. Allen, M. B., and Murphy, C., 1985, A finite element collocation method for variably saturated flows in porous media: Numer. Methods Partial Differ. Equations, v. 3, p. 229-239.Google Scholar
  2. Arnold, D. N., and Brezzi, F., 1985, Mixed and nonconforming finitenelement methods: Implementation, postprocessing and error estimates: R.A.I.R.O. Modélisation, Mathématique et Analyse Numérique, v. 19, p. 7-32.Google Scholar
  3. Beckie, R., Wood, E. F., and Aldama, A. A., 1993, Mixed finite element simulation of saturated groundwater flow using a multigrid accelerated domain decomposition technique: Water Resour. Res., v. 29, p. 3145-3157.Google Scholar
  4. Bergamaschi, L., and Putti, M., 1999, Mixed finite elements and Newton-type linearization for the solution of Richards' equation: Int. J. Numer. Methods Eng., v. 45, p. 1025-1046.Google Scholar
  5. Carsel, R. F., and Parrish, R. S., 1988, Developing join probability distributions of soil water characteristics: Water Resour. Res., v. 24, p. 755-769.Google Scholar
  6. Celia, M. A., Bouloutas, E. T., and Zarba, R. L., 1990, A general mass-conservative numerical solution for the unsaturated flow equation: Water Resour. Res., v. 26, p. 1483-1496.Google Scholar
  7. Chounet, L. M., Hilhorst, D., Jouron, C., Kelanemer, Y., and Nicolas, P., 1999, Simulation of water flow and heat transfer in soils by means of a mixed finite element method: Adv. Water Resour., v. 22, p. 445-460.Google Scholar
  8. Dagan, G., 1989, Flow and transport in porous formations: Springer, New York, 465 p.Google Scholar
  9. Douglas, J., Jr., and Roberts, J. E., 1985, Global estimates for mixed methods for second order elliptic equations: Math. Comput., v. 44, p. 39-52.Google Scholar
  10. Ferrante, M., and Yeh, T. C. J., 1999, Head and flux variability in heterogeneous unsaturated soils under transient flow conditions: Water Resour. Res., v. 35, p. 1471-1479.Google Scholar
  11. Guarracino, L., 2001, Modelado numérico del flujo de aguas subterráneas y transporte de solutos en medios porosos heterogéneos: Unpublished Doctoral Dissertation, Universidad Nacional de La Plata, Argentina, 118 p.Google Scholar
  12. Harter, T., and Yeh, T. C. J., 1998, Flow in unsaturated random porous media, nonlinear numerical analysis and comparison to analytical stochastic models: Adv. Water Resour., v. 22, p. 257-272.Google Scholar
  13. Hassan, A., Cushman, J. H., and Delleur, J. W., 1998, A Monte Carlo assessment of Eulerian flow and transport perturbation models: Water Resour. Res., v. 34, p. 1143-1163.Google Scholar
  14. Kemblowski, M. W., and Chang, C. M., 1993, Infiltration in soils with fractal permeability distribution: Ground Water, v. 31, p. 187-192.Google Scholar
  15. Lessoff, S. C., Idelman, P., and Dagan, G., 2000, A note on the influence of a constant velocity boundary condition on flow and transport in heterogeneous formations: Water Resour. Res., v. 36, p. 3095-3101.Google Scholar
  16. Moltz, F. J., and Boman, G. K., 1995, Futher evidence of fractal structure in hydraulic conductivity distributions: Geophys. Res. Lett., v. 22, p. 2545-2548.Google Scholar
  17. Moltz, F. J., Liu, H. H., and Szulga, J., 1997, Fractional Brownian motion and fractional Gaussian noise in subsurface hydrology: A review, presentation of fundamental properties and extensions: Water Resour. Res., v. 33, p. 2273-2286.Google Scholar
  18. Mosé, R., Siegel, P., Ackerer, P., and Chavent, G., 1994, Application of the mixed hybrid finite element approximation in a groundwater flow model: Luxury or necessity?: Water Resour. Res., v. 30, p. 3001-3012.Google Scholar
  19. Naff, R. L., Haley, D. F., and Sudicky, E. A., 1998, High-resolution Monte Carlo simulation of flow and conservative transport in heterogeneous porous media 1. Methodology and flow results: Water Resour. Res., v. 34, p. 663-677.Google Scholar
  20. Nedelec, J., 1980, Mixed finite elements in R3: Numer. Math., v. 35, p. 315-341.Google Scholar
  21. Neuman, S. P., 1994, Generalized scaling of permeabilities: Validation and effect of support scale: Geophys. Res. Lett., v. 21, p. 349-352.Google Scholar
  22. Raviart, P. A., and Thomas, J. M., 1977, A mixed finite element method for second order elliptic problems: Mathematical aspects of the finite element method: Lect. Notes Math., v. 606, p. 292-315.Google Scholar
  23. Richards, L., 1931, Capillary conduction of liquids through porous mediums: Physics, v. 1, p. 318-333.Google Scholar
  24. Ross, P. J., and Parlange, J. Y., 1994, Comparing exact and numerical solutions for Richards' equation for one-dimensional infiltration and drainage: Soil Sci., v. 157, p. 341-344.Google Scholar
  25. Russo, D., and Bouton, M., 1992, Statistical analysis of spatial variability in unsaturated flow parameters: Water Resour. Res., v. 28, p. 1911-1925.Google Scholar
  26. Russo, D., Russo, I., and Laufer, A., 1997, On the spatial variability of parameters of the unsaturated hydraulic conductivity: Water Resour. Res., v. 33, p. 945-956.Google Scholar
  27. Salandin, P., and Fiorotto, V., 1998, Solute transport in highly heterogeneous aquifers: Water Resour. Res., v. 34, p. 949-961.Google Scholar
  28. Strivastava, R., and Yeh, T. C. J., 1991, Analytical solutions for one-dimensional, transient infiltration toward the water table in homogeneous and layered soils: Water Resour. Res., v. 27, p. 753-762.Google Scholar
  29. Tartakovsky, D. M., Neuman, S. P., and Lu, Z., 1999, Conditional stochastic averaging of steady state unsaturated flow by means of Kirchhoff transformation: Water Resour. Res., v. 35, p. 731-745.Google Scholar
  30. Ünlü, K., Nielsen, D. R., and Biggar, J. W., 1990, Stochastic analysis of unsaturated flow: One-dimensional Monte Carlo simulation and comparisons with spectral perturbation analysis and field observations: Water Resour. Res., v. 26, p. 2207-2218.Google Scholar
  31. van Genutchen, M. T., 1980, A closed-form equation for predicting the hydraulic conductivity of unsaturated soils: Soil Sci. Soc. Am. J., v. 44, p. 892-898.Google Scholar
  32. Voss, R. F., 1988, Fractals in nature: From characterization to simulation, in Peitgen, H., and Saupe, D., eds., The science of fractal images: Springer, New York, p. 21-69.Google Scholar

Copyright information

© International Association for Mathematical Geology 2004

Authors and Affiliations

  • Luis Guarracino
    • 1
  • Juan E. Santos
    • 1
    • 2
  1. 1.CONICET, Facultad de Ciencias Astronómicas y GeofísicasUniversidad Nacional de La PlataLa PlataArgentina
  2. 2.Department of MathematicsPurdue UniversityWest Lafayette

Personalised recommendations