An Error Estimate of a Modified Method of Characteristics Modeling Advective-Diffusive Transport in Randomly Heterogeneous Porous Media

Abstract

We analyze a stochastic modified method of characteristics (MMOC) modeling advective-diffusive transport in randomly heterogeneous porous media. Under the log-normal assumption of the porous media and the finite-dimensional noise assumption that leads to unbounded diffusivity, we prove an optimal-order error estimate for the stochastic MMOC scheme. Numerical experiments are presented to substantiate the numerical analysis.

Keywords

Uncertainty quantification / MMOC / advective-diffusive transport

References

[1] R.A. Adams and J. J.F. Fournier, Sobolev Spaces. Elsevier, San Diego, 2003.

[2] T. Arbogast and M. Wheeler, A characteristic-mixed finite element method for advectiondominated transport problems. SIAM J. Numer. Anal., 32 (1992), 404-424.

[3] I.Babuška, R. Tempone, and G.E. Zouraris, Galerkin finite element approximations of stochastic elliptic differential equations. SIAM J. Numer. Anal., 42 (2004), 800-825.

[4] J. Bear, Dynamics of Fluids in Porous Materials. American Elsevier, New York, 1972.

[5] M. Bieri, R. Andreev, and C. Schwab, Sparse tensor discretization of elliptic sPDEs. SIAM J. Sci. Comput., 31 (2009), 4281-4304.

[6] S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods. Springer Science and Business Media, 2007.

[7] G. Dagan, Flow and Transport in Porous Formations. Springer, New York, 1989.

[8] J. Douglas, C. Huang, and F. Pereira, The modified method of characteristics with adjusted advection. Numer. Math., 83 (1999), 353-369.

[9] J. Douglas and T. Russell, Numericalmethods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J. Numer. Anal., 19 (1982), 871-885.

[10] L. Evans, Partial Differential Ecuations. Graduate Studies in Mathematics, Volume 19, American Mathematical Society, Rhode Island, 1998.

[11] R. Ewing (ed.), The Mathematics of Reservoir Simulation, Research Frontiers in Applied Mathematics, 1, SIAM, Philadelphia, 1984.

[12] R.E. Ewing,T.F. Russell, and M.F. Wheeler, Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics, Comput. Meth. Appl. Mech. Eng., 47, (1984), 73-92.

[13] Z. Fang, J. Li, T. Tang, and T. Zhou, Efficient stochastic Galerkin methods for Maxwell’s equations with random inputs. J. Sci. Comput., 80 (2019), 248-267.

[14] L. Gao, H. Chen, and H.Wang, A uniform estimate for the characteristics-mixedmethod for transient advection-diffusion equations in multiple space dimensions. Appl. Math. Comp., 280 (2016), 86-102.

[15] H. Gao andW. Sun, Optimal error analysis of Crank-Nicolson lowest-order Galerkin-mixed finite elementmethod for incompressiblemiscible flow in porousmedia. NumerMeth. PDEs, 36 (2020), 1773-1789.

[16] L. Gelhar, Stochastic Subsurface Hydrology. Prentice-Hall, Englewood cliffs, NJ, 1993.

[17] C. J. Gittelson, Stochastic Galerkin discretization of the log-normal isotropic diffusion problem. Math. Models Meth. Appl. Sci., 20 (2010), 237-263.

[18] R. Ghanem and P. Spanos, Stochastic Finite Elements: A Spectral Approach. Springer-Verlag, 1991.

[19] H. Harbrecht, M. Peters, M. Siebenmorgen, On the quasi-Monte Carlo method with Halton points for elliptic PDEs with log-normal diffusion. Math. Comp., 86 (2017), 771-797.

[20] R. Helmig, Multiphase Flow and Transport Processes in the Subsurface. Springer, Berlin, 1997.

[21] W.Hackbusch, Integral Equations: Theory and Numerical Treatment. ISNMVol. 120, Birkhäuser, 1995.

[22] V. Hoang and C. Schwab, N-term Wiener chaos approximation rates for elliptic PDEs with lognormal Gaussian random inputs. Math. Model. Meth. Appl. Sci., 24 (2014), 797-826.

[23] H. Hu, Y. Chen, and Y. Huang, A characteristic finite element two-grid algorithm for a compressible miscible displacement problem. Adv. Comput. Math., 46 (2020), 15.

[24] B. Li and W. Sun, Unconditional convergence and optimal error estimates of a Galerkinmixed FEM for incompressible miscible flow in porous media. SIAM J. Numer. Anal., 51 (2013), 1959-1977.

[25] Y. Marzouk and D. Xiu, A stochastic collocation approach to Bayesian inference in inverse problems. Commun. Comput. Phys., 6 (2009), 826-847.

[26] J. Nordbotten and M. Celia, Geological Storage of CO₂: Modeling Approaches for Large-scale Simulation. JohnWiley and Sons, Hoboken, New Jersey, 2012.

[27] G. Pinder and M. Celia, Subsurface Hydrology. JohnWiley and Sons, New York, 2006.

[28] L. Roman and M. Sarkis, Stochastic Galerkin method for elliptic SPDEs: A white noise approach. Discontinuous and Continuous Dynamical Systems-Series B, 6 (2006), 941-955.

[29] J. Shen, T. Tang, and L. Wang, Spectral Methods: Algorithms, Analysis and Applications. Springer Ser. Comput. Math. Vol. 41, Springer, Heidelberg, 2011.

[30] X. Wan and B. Rozovskii, The Wick-Malliavin approximation of elliptic problems with lognormal random coefficients. SIAM J. Sci. Comput., 35 (2013), A2370-A2392.

[31] J. Wang, Z. Si, and W. Sun, A new error analysis of characteristics-mixed FEMs for miscible displacement in porous media. SIAM J. Numer. Anal., 52 (2014), 3000-3020.

[32] H. Wang, An optimal-order error estimate for an ELLAM scheme for two-dimensional advection-diffusion equations, SIAM J. Numer. Anal., 37 (2000), 1338-1368.

[33] H. Wang, H. Dahle, R. Ewing, M. Espedal, R. Sharpley, and S. Man, An ELLAMScheme for advection-diffusion equations in two dimensions. SIAMJ. Sci. Comput., 20 (1999), 2160-2194.

[34] H. Wang, R. Ewing, G. Qin, S. Lyons, M. Al-Lawatia, and S. Man, A family of Eulerian-Lagrangian localized adjoint methods for multi-dimensional advection-reaction equations. J. Comput. Phys., 15 (1999), 120-163.

[35] H. Wang, Y. Ren, J. Jia, and M. celia, A probabilistic collocation Eulerian-Lagrangian localized adjoint method on sparse grids for assessing CO2 leakage through wells in randomly heterogeneous porous media. Comput.Methods Appl.Mech. Engrg., 292 (2015), 35-53.

[36] H. Wang, W. Zhao, and R. Ewing, A numerical modeling of multicomponent compressible flows in porous media with multiple wells by an Eulerian-Lagrangian method. Comput. Visual. Sci., 8 (2005), 69-81.

[37] H.Wang and X. Zheng, An optimal-order error estimate of the lowest-order ELLAM-MFEM approximation to miscible displacement in three space dimensions. J. Comput. Appl. Math., 375 (2020), 112819.

[38] H. Wang and K. Wang, Uniform estimates of Eulerian-Lagrangian methods for transient convection-diffusion equations in multiple space dimensions. SIAM J. Numer. Anal., 48 (2010), 1444-1473.

[39] K. Wang and H. Wang, An optimal order estimate for MMOC-MFEM approximations to porous medium flow. Numer. Methods Partial Differential Eq., 25 (2009), 1283-1302.

[40] K. Wang and H. Wang, A Uniform estimate for the MMOC for two-dimensional advectiondiffusion equations. Numer. Meth. Partial Diff. Eq., 26 (2010), 1054-1069.

[41] K. Wang and H. Wang, Uniform estimates for a family of Eulerian-Lagrangian methods for time-dependent convection-diffusion equations with degenerate diffusion. IMA J. Numer. Anal., 31 (2011), 1006-1037.

[42] D. Xiu and J. Hesthaven, High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput., 27 (2005), 1118-1139.

[43] D. Xiu and G. Karniadakis, Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Methods Appl.Mech. Engrg., 191 (2002), 4927-4948.

[44] D. Xiu and J. Shen, Efficient stochastic Galerkin methods for random diffusion equations. J. Comput. Phys., 228 (2009), 266-281.

[45] D. Zhang, Stochastic Methods for Flow in Porous Media, Coping with Uncertainties. Academic Press, New York, 2002.

[46] T. Zhou and T. Tang, Note on coefficient matrices from stochastic Galerkin methods for random diffusion equations. J. Comput. Phys., 229 (2010), 8225-8230.