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978-3-8439-0996-9, Reihe Mathematik
Analysis and Numerics for Nonclassical Wave Fronts in Porous Media
134 Seiten, Dissertation Universität Stuttgart (2013), Softcover, A5
In this thesis, we consider the infiltration of a wetting, incompressible fluid into a porous medium and the formation of saturation overshoots. Saturation overshoots are peaks in the saturation profile close to the infiltrating fluid front. If higher order effects are neglected, such two-phase flow problems in porous media can be modelled as a hyperbolic conservation law with non-convex flux function as macroscale model. It is well known, that there can be multiple weak solutions. To ensure uniqueness, we use on the microscale either a kinetic relation or an extended system. The extended system takes a dynamic capillary pressure into account, modeled by the rate-dependent approach of Hassanizadeh and Gray (1993). We present a new multidimensional mass-conserving numerical method to solve the infiltration problems on the macroscale. The method belongs to the class of Heterogeneous Multiscale Methods in the sense of E and Engquist (2003) and is stable. A key part of the approximation is a novel numerical flux function for the multidimensional setting, which captures undercompressive waves and generalizes the one-dimensional approach of Boutin et al. (2008). Furthermore, we improve the overall computational complexity, using a data-based approach. Finally, we validate the numerical method and test it on several infiltration problems.
In the last part of this thesis, we consider conservation laws with spatially discontinuous flux, resulting from spatial heterogeneities. These conservation laws are perturbed by diffusion and dispersion terms, coming from the rate-dependent (dynamic) capillary pressure. We investigate the singular limit of a scalar model problem as the diffusion and dispersion parameters tend to zero, showing strong convergence towards a weak solution of the limit conservation law. Finally, we apply the Heterogeneous Multiscale Method to heterogeneous porous media and perform numerical simulations for various exemplary situations.