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The concept of d3f:

Numerical simulation of density driven flows with d3f
The finite-volume simulator d3f was developed by Simulation in Technology for variable-density flow and transport in porous media modeled by equations (1)-(3), [Fein 1998]. The building blocks of the code include unstructured grid generation, local-grid refinement, robust multigrid solvers, and parallelization techniques as implemented in the basic library uG [Bastian, Wittum 1994], [Bastian, Wittum et al 2001], [Lang, Wittum 2005]. Fully implicit, backward Euler discretization is applied in time and central differencing in space, leading to a non-monotonic, second-order accurate discretization scheme. An internal control of numerical accuracy is given by Peclet number and Courant number specification, invoking adaptive grid-refinement and time stepping. The discretized non-linear system of equations (1)-(3) is solved via Newton linearization, using a multigrid solver accelerated by BiCGStab. This solution strategy has been found to be highly robust and efficient even for large and complex problems [Fein et al., 1998]. A main feature of d3f is combination of unstructured grids allowing dynamic adaptive grid refinement with parallelization and multigrid methods.

For the simulations needed here, we propose to simulate the fully coupled equations, i.e. without Boussinesq approximation. The Boussinesq approximation is generally inappropriate, when significant density changes arise from solute concentrations and the contours of equipotential pressures are no longer orthogonal to the velocity vectors. A consistent velocity approximation for density-driven flow is implemented in d3f [see, Knabner and Frolkovic, 1996; Frolkovic, 1998]. This consistency for the derivatives of the velocity vector in the generalized Darcy law uses same-order approximations for the pressure gradients and the gravity term, thus avoiding spurious velocities. d3f uses unstructured grids and allows hierarchical local adaptive refinement als well as standard uniform refinement of the grid.

With d3f numerous density driven flow problems have been computed. First we computed the standard test, the Elder problem in 2d and 3d. We soon found out that these problems exihibiting an unstable layering were not really understood. Klaus Johannsen presented a thorough investigation of the nonlinear behaviour of this problem, using d3f,[Johannsen, 2003] He was able to get a new understanding of the problem, but furthermore of the substantial difference between Boussinesq approximation and a ful model of density driven flows. For validation, a special physical experiment was set up and carefully compared with computed results, [Johannsen et al., 2002]. Then we used d3f to compute numerous real life cases like saltwater intrusion into the Chinese island Wei Zhou, flow and transport areound saltdomes like Gorleben, Höfer, Asse and long-time flow studies for the WIPP nucelear waste disposal in New Mexico using massively parallel computers.

All these problems were groundwater flow problems on the scale of local aquifers, either with known coefficients or using the geostatistics feature of d3f. Also novel methods for solving inverse problems have been introduced and applied, [Schulz&Wittum, 1998], [Wittum, Schulz et al,2003]. We further developed a suite of gridding tools in particular for anisotropic geological domains, [Feuchter et al 2001], [Fuchs & Wittum, 2003]. Special solvers for heterogenous anisotropic problems have been developed by Wagner and Wittum, [Wagner 2000], [Wagner 2001], [Wagner & Wittum 1997].


The concept of r3t:

The software package r3t can realise numerical simulations of very large systems of coupled partial and ordinary differential equations that arise from the modelling of radioactive contaminant transport in porous media. It can solve precisely nontrivial mathematical problems like advection-dominated system with different retardation of transport for each component and with nonlinear Freundlich sorption and/or precipitation. Additionally, long time simulations on complex 3D geological domains with locally refined unstructured grids can be realised.

The software package r3t (radionuclides, reactions, retardation and transport) can help to numerically simulate the spreading of radioactive contaminants in flowing groundwater. The general mathematical model includes not only transport by advection, diffusion and dispersion, but also reactions due to decay, sorption (kinetic and in equilibrium, linear and nonlinear), immobilisation and precipitation of contanminants. Numerical approximation by finite volume methods follows mass conservation rules, it is formally second order accurate in time and space, and it producess numerical solutions without oscillations. The package r3t is based in UG toolbox (Unstructured Grids) and numerical simulations can be realised using unstructured 2D/3D grids that can be refined and unrefined locally during computations.

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