Browsing by Author "Flemisch, Bernd (PD Dr. rer. nat.)"

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An XFEM-based model for fluid flow in fractured porous media
(2015) Schwenck, Nicolas; Flemisch, Bernd (PD Dr. rer. nat.)
Many fields of applications for porous media flow include geometrically anisotropic inclusions and strongly discontinuous material coefficients which differ in orders of magnitude. If the extension of those heterogeneities is small in normal direction compared to the tangential directions, e.g., long and thin, those features are called fractures. Examples which include such fractured porous-media systems in earth sciences include reservoir engineering, groundwater-resource management, carbon capture and storage (CCS), radioactive-waste reposition, coal bed methane migration in mines, geothermal engineering and hydraulic fracturing. The analysis and prediction of flow in fractured porous-media systems is important for all the aforementioned applications. Experiments are usually too expensive and time consuming to satisfy the demand for fast but accurate decision making information. Many different conceptual and numerical models to treat fractured porous-media systems can be found in the literature. However, even in the time of large supercomputers with massive parallel computing power, the computational efficiency, and therefore the economic efficiency, plays a dominating role in the evaluation of simulation software. In this thesis an efficient method to simulate flow in fractured porous media systems is presented. Darcy flow in fractures and matrix is assumed. The presented method is suited best for flow regimes depending on both, the fractures and the surrounding rock matrix and is able to account for highly conductive but also almost impermeable fractures with respect to the surrounding matrix. The newly developed method is based on a co-dimension one conceptual model for the fracture network which is embedded in the surrounding matrix. The basis for this model reduction is given in Martin et al. (2005). Numerically the fracture network is resolved by its own grid and coupled to the independent matrix grid. The discretization on this matrix grid allows jumps in the solution across the geometrical position of the fractures within elements by discontinuous basis functions. This discretization method is known as eXtended Finite Element Method (XFEM). A similar approach was simultaneously developed in D’Angelo and Scotti (2012). The main novelty of this work is the extension of the aforementioned conceptual model, which only accounts for a single fracture ending on the boundary of the matrix domain, towards more complex fracture networks and suitable boundary conditions. This work can be structured into the development and implementation of three conceptual models (see 1–3 below) and their respective validation. It is followed by an evaluation of quality and efficiency with respect to established models (see 4 below). The implementation is carried out using DUNE, a toolbox for solving partial differential equations. 1. The first extension is the treatment of fractures, which end inside the domain. This includes the conceptual coupling at the fracture tips as well as the numerical treatment within the XFEM of the matrix elements, in which the fracture ends. The validation shows, that the proposed treatment is efficient and for most validation cases produces the desired accuracy. 2. In the second part, a conceptual model for intersecting fractures is developed and the implementation within the XFEM is presented. The validation shows that the proposed model and implementation can capture the complex physics of fracture crossings very accurately. 3. Of special interest are the boundary conditions for lower-dimensional fractures intersecting the matrix boundary. The established models very often use for simplicity constant values across lower-dimensional intersections. This is physically not always correct, because in reality lower-dimensional objects do not exist and if a gradient on the rock matrix boundary exists, there is also a gradient on the fracture boundary. Therefore, a sophisticated interpolation method is proposed. It is easy to apply because very often discrete measured data is given to the model as input anyway and the proposed interpolation of values at the boundary is separated from the flow problem inside the domain. The concepts and results of the crossing model (2) and the boundary-condition interpolation (3) are published in Schwenck et al. (2014). 4. To show the performance of the newly developed model including the three major aspects mentioned above, it is compared against several established models and implementations within the simulation framework DuMux. The results of this comparison are published in Schwenck et al. (2015). The model presented here combines the advantages of lower-dimensional models and non-matching grids while keeping the ability to represent the fracture geometry and its influence on the matrix flow field exactly. Therefore, it is an efficient alternative to established models.