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TY - THES AU - Boisserée, Simon Melchior TI - Space-time adaptive methods for poroviscoelastic flow models PB - RWTH Aachen University VL - Dissertation CY - Aachen M1 - RWTH-2025-09929 SP - 1 Online-Ressource : Illustrationen PY - 2025 N1 - Veröffentlicht auf dem Publikationsserver der RWTH Aachen University 2026 N1 - Dissertation, RWTH Aachen University, 2025 AB - In this work we consider a system of equations that arises in geophysics when modeling subsurface flows of water, magma, or gases such as CO2 in a poroviscoelastic medium. We start by deriving the model equations in terms of porosity and effective pressure before transforming them into a nondimensional form. Furthermore, we state two simplified models directly arising from the full viscoelastic model, namely the viscous limit and low-porosity approximation. We demonstrate that the analysis for the full model and the low-porosity approximation can be unified due to a transformation of variables, implying that it only remains to show well-posedness for the viscous limit as a separate case. We define a notion of solution that includes discontinuous porosity distributions as they often occur in nature at the interface between sedimentary layers. We proceed by proving well-posedness of the viscoelastic model and the viscous limit. We mainly follow the idea of combining a fixed-point iteration for the porosity equation with regularity theory for the equation modeling the effective pressure. With this strategy, it turns out that in the viscous limit a result allowing very general discontinuous initial porosity distributions only follows if we restrict ourselves to one- and two-dimensional domains. However, by assuming that the initial porosity has a bounded total variation and by using a compactness argument for sequences of approximate solutions, we obtain existence of solutions also in arbitrary dimensions. This includes all practically relevant cases of discontinuous initial porosity distributions. For the viscoelastic model, we use a similar fixed-point argument as before. Here, we can prove existence and uniqueness of solutions for discontinuous initial porosity distributions, but only by assuming certain Hölder regularity of the shape of the discontinuities as well as Hölder smoothness of the initial functions away from the discontinuities. Using the same fixed-point formulation as before, we introduce an adaptive space-time method to solve the viscoelastic model numerically. This method can also be modified slightly to cover the viscous limit as well. Furthermore, we prove convergence and a posteriori error bounds for this approach. Then, we present some numerical tests for geophysically relevant problem settings. For these tests, we show numerically that our methods generate quasi-optimal approximations of the corresponding analytical solutions, independent of the presence of a discontinuity. As a geophysical application, we discuss the coupling of our method with a chemical transport problem since it provides new insights into the formation of chemical anomalies, which, for example, can lead to the formation of ore deposits in the Earth's subsurface. Then, we briefly discuss other possible space-time adaptive approaches to solving the model problem. Next, we compare our approach with a finite difference scheme, where it turns out that this method only reaches low convergence rates for discontinuous porosity distributions. Finally, we show some further advantages of space-time methods for solving related inverse problems. We present two different approaches that attempt to invert a simplified model problem. LB - PUB:(DE-HGF)11 DO - DOI:10.18154/RWTH-2025-09929 UR - https://publications.rwth-aachen.de/record/1022300 ER -