A coupled finite element-global random walk approach to advection-dominated transport in porous media with random hydraulic conductivity

Solute transport through heterogeneous porous media considered in environmental and industrial problems is often characterized by high Peclet numbers. In this paper we develop a new numerical approach to advection-dominated transport consisting of coupling an accurate mass-conservative mixed finite element method (MFEM), used to solve Darcy flows, with a particle method, stable and free of numerical diffusion, for non-reactive transport simulations. The latter is the efficient global random walk (GRW) algorithm, which performs the simultaneous tracking of arbitrarily large collections of particles on regular lattices at computational costs comparable to those of single-trajectory simulations using traditional particle tracking (PT). MFEM saturated flow solutions are computed for spatially heterogeneous hydraulic conductivities parameterized as samples of random fields. The coupling is achieved by projecting the velocity field from the MFEM basis onto the regular GRW lattice. Preliminary results show that MFEM-GRW is tens of times faster than the full MFEM flow and transport simulation.

[1]  Uwe Jaekel,et al.  Estimation of Macrodispersion by Different Approximation Methods for Flow and Transport in Randomly Heterogeneous Media , 2001 .

[2]  Peter Knabner,et al.  Simulation of carrier-facilitated transport of phenanthrene in a layered soil profile. , 2002, Journal of contaminant hydrology.

[3]  Nicolae Suciu,et al.  Evaluation of the first‐order approximations for transport in heterogeneous media , 2006 .

[4]  Harry Vereecken,et al.  Generalized random walk algorithm for the numerical modeling of complex diffusion processes , 2003 .

[5]  Sabine Attinger,et al.  Analysis of an Euler implicit‐mixed finite element scheme for reactive solute transport in porous media , 2009 .

[6]  Uwe Jaekel,et al.  Renormalization group analysis of macrodispersion in a directed random flow , 1997 .

[7]  Gilles Porel,et al.  Transport in a 2-D saturated porous medium: A new method for particle tracking , 1996 .

[8]  P. Kloeden,et al.  Numerical Solutions of Stochastic Differential Equations , 1995 .

[9]  Sabine Attinger,et al.  Accuracy of numerical simulations of contaminant transport in heterogeneous aquifers: A comparative study , 2011 .

[10]  Frederick Delay,et al.  Simulating Solute Transport in Porous or Fractured Formations Using Random Walk Particle Tracking: A Review , 2005 .

[11]  H Vereecken,et al.  Persistent memory of diffusing particles. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Peter Knabner,et al.  Order of Convergence Estimates for an Euler Implicit, Mixed Finite Element Discretization of Richards' Equation , 2004, SIAM J. Numer. Anal..

[13]  Jan Vanderborght,et al.  Numerical investigations on ergodicity of solute transport in heterogeneous aquifers , 2006 .

[14]  Nicolae Suciu,et al.  On the self-averaging of dispersion for transport in quasi-periodic random media , 2007 .

[15]  Haifeng Wang,et al.  Numerical implementation of mixing and molecular transport in LES/PDF studies of turbulent reacting flows , 2011, J. Comput. Phys..

[16]  Peter Knabner,et al.  Optimal order convergence of a modified BDM1 mixed finite element scheme for reactive transport in porous media , 2012 .

[17]  Sabine Attinger,et al.  A Mixed Hybrid Finite Element Discretization Scheme for Reactive Transport in Porous Media , 2008 .

[18]  Nicolae Suciu,et al.  G LOBAL RANDOM WALK ALGORITHM FOR DIFFUSION PROCESSES , 2011 .

[19]  Nicolae Suciu,et al.  Evaluation of overshooting errors in particle methods for diffusion by biased global random walk , 2006, Journal of Numerical Analysis and Approximation Theory.

[20]  Nicolae Suciu,et al.  Spatially inhomogeneous transition probabilities as memory effects for diffusion in statistically homogeneous random velocity fields. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.