147 research outputs found
Analytical and Numerical Aspects of Porous Media Flow
The Brinkman equations model fluid flow through porous media and are particularly interesting in regimes where viscous shear effects cannot be neglected. Two model parameters in the momentum balance function as weights for the terms related to inter-particle friction and bulk resistance. If these are not in balance, then standard finite element methods might suffer from instabilities or error estimates might deteriorate. In particular the limit case, where the Brinkman problem reduces to a Darcy problem, demands for special attention. This thesis proposes a low-order finite element method which is uniformly stable with respect to the flow regimes captured by the Brinkman model, including the Darcy limit. To that end, linear equal-order approximations are combined with a pressure stabilization technique, a grad-div stabilization, and a penalty-free non-symmetric Nitsche method. The combination of these ingredients allows to develop a robust method, which is proven to be well-posed for the whole family of problems in two spatial dimensions, even if any Brinkman parameter vanishes. An a priori error analysis reveals optimal convergence in the considered norm. A convergence study based on problems with known analytic solutions confirms the robust first order convergence for reasonable ranges of numerical (stabilization) parameters. Further, numerical investigations that partly extend the theoretical framework are considered, revealing strengths and weaknesses of the approach. An application motivated by the optimization of geothermal energy production completes the thesis. Here, the proposed method is included in a multi-physics discrete model, appropriate to describe the thermo-hydraulics in hot, sedimentary, essentially horizontal aquifers. An immersed boundary method is adopted in order to allow a flexible, automatic optimization without regenerating the computational mesh. Utilizing the developed computational framework, the optimized multi-well arrangements with respect to the net energy gain are presented and discussed for different geothermal and hydrogeological setups. The results show that taking into account heterogeneous permeability structures and variable aquifer temperatures might drastically affect the optimal configuration of the wells
Mathematical analysis, scaling and simulation of flow and transport during immiscible two-phase flow
Fluid flow and transport in fractured geological formations is of fundamental
socio-economic importance, with applications ranging from oil
recovery from the largest remaining hydrocarbon reserves to bioremediation
techniques. Two mechanisms are particularly relevant for flow
and transport, namely spontaneous imbibition (SI) and hydrodynamic
dispersion. This thesis investigates the influence of SI and dispersion
on flow and transport during immiscible two-phase flow.
We make four main contributions. Firstly, we derive general, exact analytic
solutions for SI that are valid for arbitrary petrophysical properties.
This should finalize the decades-long search for analytical solutions
for SI. Secondly, we derive the first non-dimensional time for SI that
incorporates the influence of all parameters present in the two-phase
Darcy formulation - a problem that was open for more than 90 years.
Thirdly, we show how the growth of the dispersive zone depends on the
flow regime and on adsorption. To that end we derive the first known
set of analytical solutions for transport that fully accounts for the effects
of capillarity, viscous forces and dispersion. Finally, we provide numerical
tools to investigate the influence of heterogeneity by extending the
higher order finite-element finite-volume method on unstructured grids
to the case of transport and two-phase flow
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