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Numerical Studies of Three-dimensional Stochastic Darcy’s Equation and Stochastic Advection-Diffusion-Dispersion Equation

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Abstract

Solute transport in randomly heterogeneous porous media is commonly described by stochastic flow and advection-dispersion equations with a random hydraulic conductivity field. The statistical distribution of conductivity of engineered and naturally occurring porous material can vary, depending on its origin. We describe solutions of a three-dimensional stochastic advection-dispersion equation using a probabilistic collocation method (PCM) on sparse grids for several distributions of hydraulic conductivity. Three random distributions of log hydraulic conductivity are considered: uniform, Gaussian, and truncated Gaussian (beta). Log hydraulic conductivity is represented by a Karhunen-Loève (K-L) decomposition as a second-order random process with an exponential covariance function. The convergence of PCM has been demonstrated. It appears that the accuracy in both the mean and the standard deviation of PCM solutions can be improved by using the Jacobi-chaos representing the truncated Gaussian distribution rather than the Hermite-chaos for the Gaussian distribution. The effect of type of distribution and parameters such as the variance and correlation length of log hydraulic conductivity and dispersion coefficient on leading moments of the advection velocity and solute concentration was investigated.

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References

  1. Graham, W.D., McLaughlin, D.: Stochastic analysis of nonstationary subsurface solute transport: 1. Unconditional moments. Water Resour. Res. 25(2), 215–232 (1989)

    Article  Google Scholar 

  2. Dagan, G.: Flow and Transport in Porous Formations. Springer, New York (1989)

    Google Scholar 

  3. Gelhar, L.W.: Stochastic Subsurface Hydrology. Prentice–Hall, Englewood Cliffs (1993)

    Google Scholar 

  4. Zhang, D., Neuman, S.P.: Eulerian-Lagrangian analysis of transport conditioned on hydraulic data, 1. analytical-numerical approach. Water Resour. Res. 31, 39–51 (1995)

    Article  Google Scholar 

  5. Xiu, D., Hesthaven, J.S.: High order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118–1139 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Ghanem, R.G., Spanos, P.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)

    MATH  Google Scholar 

  7. Xiu, D., Karniadakis, G.E.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Xiu, D.: Generalized (Wiener-Askey) polynomial chaos. Ph.D. thesis, Brown University, Division of Applied Mathematics (2004)

  9. Xiu, D., Karniadakis, G.: Supersensitivity due to uncertain boundary conditions. Int. J. Numer. Methods Eng. 61(12), 2114–2138 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bernardi, C., Maday, Y.: Properties of some weighted Sobolev spaces and application to spectral approximations. SIAM J. Numer. Anal. 26, 769–829 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  11. Canuto, C., Quarteroni, A.: Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comput. 38, 67–86 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  12. Smolyak, S.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Sov. Math. Dokl. 4, 240–243 (1963)

    Google Scholar 

  13. Tatang, M., McRae, G.: Direct treatment of uncertainty in models of reaction and transport. Tech. rep., MIT Tech. Rep. (1994)

  14. Novak, E., Ritter, K.: High dimensional integration of smooth functions over cubes. Numer. Math. 75, 79–97 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  15. Novak, E., Ritter, K.: Simple cubature formulas with high polynomial exactness. Constr. Approx. 15, 499–522 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  16. Barthelmann, V., Novak, E., Ritter, K.: High dimensional polynomial interpolation on sparse grids. Adv. Comput. Math. 12, 273–288 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Gerstner, T., Griebel, M.: Dimension-adaptive tensor-product quadrature. Computing 71(1), 65–87 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Ganapathysubramanian, B., Zabaras, N.: Sparse grid collocation schemes for stochastic natural convection problems. J. Comput. Phys. 225(1), 652–685 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. Griebel, M.: Adaptive sparse grid multilevel methods for elliptic PDEs based on finite differences. Computing 61(2), 151–180 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. Ma, X., Zabaras, N.: An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations. J. Comput. Phys. 228, 3084–3113 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  21. Foo, J.Y.: Multi-element probabilistic collocation in high dimensions: Applications to systems biology and physical systems. Ph.D. thesis, Brown University, Division of Applied Mathematics (2008)

  22. Li, H., Zhang, D.: Probabilistic collocation method for flow in porous media: Comparisons with other stochastic methods. Water Resour. Res. 43, W9409 (2007). doi:10.1029/2006WR005673

    Article  Google Scholar 

  23. Lin, G., Tartakovsky, A.: An efficient, high-order probabilistic collocation method on sparse grids for three-dimensional flow and solute transport in randomly heterogeneous porous media. Adv. Water Resour. 32(5), 712–722 (2009)

    Article  Google Scholar 

  24. Morales-Casique, E., Neuman, S.P., Guadagnini, A.: Nonlocal and localized analyses of nonreactive solute transport in bounded randomly heterogeneous porous media: computational analysis. Adv. Water Resour. 8(29), 1238–1255 (2006)

    Article  Google Scholar 

  25. Jarman, K.D., Russell, T.F.: Eulerian moment equations for 2-D stochastic immiscible flow. Multiscale Model. Simul. 4(1), 598–608 (2003)

    Article  MathSciNet  Google Scholar 

  26. Emmanuel, S., Berkowitz, B.: Mixing-induced precipitation and porosity evolution in porous media. Adv. Water Res. 28, 337–344 (2005)

    Article  Google Scholar 

  27. Dur, A.: On the optimality of the discrete Karhunen-loève expansion. SIAM J. Control Optim. 36(6), 1937–1939 (1998)

    Article  MathSciNet  Google Scholar 

  28. Loève, M.: Probability Theory, 4th edn. Springer, New York (1977)

    MATH  Google Scholar 

  29. Frauenfelder, P., Schwab, C., Todor, R.A.: Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Eng. 194, 205–228 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  30. Strait, P.: Sample function regularity for Gaussian processes with the parameter in a Hilbert space. Pac. J. Math. 19(1), 159–173 (1966)

    MATH  MathSciNet  Google Scholar 

  31. Huang, S., Quek, S., Phoon, K.: Convergence study of the truncated Karhunen-Loeve expansion for simulation of stochastic processes. Int. J. Numer. Methods Eng. 52(9), 1029–1043 (2001)

    Article  MATH  Google Scholar 

  32. Cools, R.: Monomial cubature rules since ‘Stroud’: a compilation. J. Comput. Appl. Math. 48, 309–326 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  33. Cools, R.: Monomial cubature rules since ‘Stroud’: a compilation—part 2. J. Comput. Appl. Math. 112, 21–27 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  34. Gautschi, W.: On generating orthogonal polynomials. SIAM J. Sci. Stat. Comput. 3(3), 289–317 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  35. Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  36. Babus̆ka, I., Chatzipantelidis, P.: On solving elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 191, 4093–4122 (2002)

    Article  MathSciNet  Google Scholar 

  37. Wan, X., Xiu, D., Karniadakis, G.E.: Stochastic solutions for the two-dimensional advection-diffusion equation. SIAM J. Sci. Comput. 26, 578–590 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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  1. Computational Mathematics, Pacific Northwest National Laboratory, 902 Battelle Blvd., Box 999, Richland, WA, 99352, USA

    G. Lin & A. M. Tartakovsky

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  1. G. Lin
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  2. A. M. Tartakovsky
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Lin, G., Tartakovsky, A.M. Numerical Studies of Three-dimensional Stochastic Darcy’s Equation and Stochastic Advection-Diffusion-Dispersion Equation. J Sci Comput 43, 92–117 (2010). https://doi.org/10.1007/s10915-010-9346-5

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  • DOI: https://doi.org/10.1007/s10915-010-9346-5

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