Advanced numerical methods for transport and reaction in porous media Peter Frolkovič University of Heidelberg
Content R 3 T a software package for numerical simulation of radioactive contaminant transport in groundwater Motivation accuracy Advanced numerical methods: advection and nonlinear retardation advection and flux-based level set method advection and operator splitting 2
R 3 T: Retardation, Reaction, Radionuclides, Transport Issues mathematical computational Peter Frolkovič, Michael Lampe, Gabriel Wittum: r3t - software package for numerical simulations of radioactive contaminant transport in groundwater; WiR, Preprint 08/2005 3
R 3 T Mathematical issues decay reaction system diffusion-dispersion advection adsorption precipitation dissolution... 4
R 3 T Mathematical issues decay reaction system diffusion-dispersion advection adsorption precipitation dissolution... } - dominated case 4
R 3 T Computational issues parallel computations complex 3D unstructured grids advection described by density driven flow: 5
Motivation Importance of accuracy...-dominated case: one has to solve the pure cases well operator-splitting method: simple coupling of the well-solved pure cases to be compared: experiments <=> PDE <=> numerical solution experiments <=> numerical solution 6
Motivation Importance of accuracy (Elder example) one finger solution at t=20y C. Oldenburg, K. Pruess: Dispersive transport dynamics in a strongly coupled groundwater-brine flow system; Wat. Res. Res., 4 (1995), p. 289-302 7
Motivation Importance of accuracy (Elder example) half finger solution at t=20y P. Frolkovič, H. Schepper: Numerical modelling of convection dominated transport coupled with density driven flow in porous media; Adv. Wat. Res., 1 (2001), p. 63-72 8
Motivation Decay reactions t c i = λ i c i + k λ kc k Diffusion-dispersion t c i = (D i c i ) exact solutions of reactions 2nd order accurate implicit control-volume method multigrid solvers on locally adapted multilevel grids standard operator splitting method 2D test example of 40 PDEs + 120 ODEs 9
Motivation Advection t c = V (x) c Challenges: hyperbolic α T h parabolic with dispersion (Peclet number ) second order in time and space?! unstructured grids?! stability?! consistency?! non-physical oscillations?! 10
Advection + retardation Advection with nonlinear retardation t (R i (c)c i ) = V c i Analytical solutions for 1D hyperbolic equation correct speed of shocks... t θ(c) = x c t θ = x c(θ) θ(c) = c + c p c(θ) =? Peter Frolkovič, Jozef Kačur: Semi-analytical solutions for contaminant transport with nonlinear sorption in 1D; Computational Geosciences, 3 (2006), p. 279-290 11
Advection + retardation 1D hyperbolic problem: shocks, rarefaction waves,... click on pictures to open web page R(c) = 2 R(c) = 1 + c 1 2 12
Advection + retardation 2D: t (R(c)c) = V (x) c R(c) = 2 R(c) = 1 + c 1 2 13
Advection Linear advection equation (conservation laws) ( ) t c = V (x)c Finite volume method Ω i c n+1 i = Ω i c n i t n V ij c n+1/2 ij Computation of concentration in fluxes c n+1/2 ij :=? 14
Advection Linear advection equation (conservation laws) ( ) t c = V (x)c Finite volume method Ω i c n+1 i = Ω i c n i t n V ij c n+1/2 ij Godunov method (1st order upwind) c n+1/2 ij = c n i, V ij > 0 exact solution of local one-dimensional Riemann s problem the CFL restriction on time step t n 15
Advection Linear advection equation (conservation laws) ( ) t c = V (x)c Finite volume method Ω i c n+1 i = Ω i c n i t n V ij c n+1/2 ij Flux-based method of characteristics [Frolkovic, CVS 2002] c n+1/2 ij = k Λ i inflow ω k c n k exact discrete mass conservation, no oscillations,... no CFL restriction on time step t n 16
Advection - level set equation Linear advection equation (level set formulation) t c = V c, V = 0 Finite volume method Ω i c n+1 i = Ω i c n i t n V ij c n+1/2 ij Flux-based level set method c n+1/2 ij = c n i + t tc n i + h ij c n i, V ij > 0 finite Taylor expansion and t c n i = V i c n i 2nd order accurate on unstructured grids 17
Advection - level set equation Peter Frolkovič, Karol Mikula: High-resolution flux-based level set method; SIAM J. Sci. Comp., (2007), to appear 18
Advection - level set equation Peter Frolkovič, Karol Mikula: High-resolution flux-based level set method; SIAM J. Sci. Comp., (2007), to appear 18
Advection - level set equation Peter Frolkovič, Karol Mikula: High-resolution flux-based level set method; SIAM J. Sci. Comp., (2007), to appear 19
Advection - level set equation Rotation of Gaussian function t = 1 I N EOC min max 1 64 320-1.0E-2.788 128 640 2.61-1.4E-4.956 256 1280 2.84-1.3E-8.994 512 2560 2.69-5.7E-9.999 0.8 0.6 0.4 0.2 0 0.2 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.5 0 Peter Frolkovič, Christian Wehner: Flux-based level set method on rectangular grids and computation of first arrival time functions; submitted 20
Advection - level set equation Single-vortex benchmark 21
Advection + operator splitting Time splitting error serious difficulty for systems with retardation example of numerical results after two large time steps: 22
Advection + operator splitting Time splitting error serious difficulty for systems with retardation example of numerical results after two large time steps: standard operator splitting => large splitting error 22
Advection + operator splitting Time splitting error serious difficulty for systems with retardation example of numerical results after two large time steps: standard operator splitting => large splitting error Peter Frolkovič: Flux-based methods of characteristics for coupled transport equations in porous media; Comp. Visual. Sci., 6 (2004), p. 173-184 22
Advection + operator splitting Coupling of R3T with PHREEQ-C: equilibrium equations for pure phases and exchangers 23
Conclusions Importance of accuracy good solution of the pure cases, not only the -dominated cases necessary condition for succesful operator splitting method Godunov method for advection and nonlinear retardation flux-based level set method for the pure advection case small time splitting error for coupling with advection 24