Numerical methods for coupled processes in fractured porous media
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Numerical simulations have become essential in the planning and execution of operations in the subsurface, whether this is geothermal energy production or storage, carbon sequestration, petroleum production, or wastewater disposal. As the computational power increases, more complex models become feasible, not only in the form of more complicated physics, but also in the details of geometric constraints such as fractures, faults and wells. These features are often of interest as they can have a profound effect on different physical processes in the porous medium. This thesis focuses on modeling and simulations of fluid flow, transport and deformation of fractured porous media. The physical processes are formulated in a mixed-dimensional discrete fracture matrix model, where the rock matrix, fractures, and fracture intersections form a hierarchy of subdomains of different dimensions that are coupled through interface laws. A new discretization scheme for solving the deformation of a poroelastic rock coupled to a Coulomb friction law governing fracture deformation is presented. The novelty of this scheme comes from combining an existing finite-volume discretization for poroelasticity with a hybrid formulation that adds Lagrange multipliers on the fracture surface. This allows us to formulate the inequalities as complementary functions and solve the corresponding non-linear system using a semi-smooth Newton method. The mixed-dimensional framework is used to investigate non-linear coupled flow and transport. Here, we study how highly permeable fractures affect the viscous fingering in a porous medium and show that there is a complex interplay between the unstable viscous fingers and the fractures. The computer code of the above contributions of the thesis work has been implemented in the open-source framework PorePy. The introduction of fractures is a challenge to the discretization and the implementation of the governing equations, and the aim of this framework is to enable researchers to overcome many of the technical difficulties inherent to fractures, allowing them to easily develop models for fractured porous media. One of the large challenges for the mixed-dimensional discrete fracture matrix models is to create meshes that conform to the fractures, and we present a novel algorithm for constructing conforming Voronoi meshes. The proposed algorithm creates a mesh hierarchy, where the faces of the rock matrix mesh conform to the cells of the fractures, and the faces of the fracture mesh conform to the cells of the fracture intersections. The flexibility of the mixed-dimensional framework is exemplified by the wide range of applications and models studied within this thesis. While these physical processes might be fairly well known in a porous medium without fractures, the results of this thesis improves our understanding as well as the models and solution strategies for fractured porous media.
Paper A: Reactivation of Fractures in Subsurface Reservoirs – A numerical approach using a static-dynamic friction model. Runar L. Berge, Inga Berre, Eirik Keilegavlen. In: Numerical Mathematics and Advanced Applications ENUMATH 2017, pp 653-660. The article is not available in BORA due to publisher restrictions. The published version is available at: https://doi.org/10.1007/978-3-319-96415-7_60Paper B: Finite volume discretization for poroelastic media with fractures modeled by contact mechanics. Runar L. Berge, Inga Berre, Eirik Keilegavlen, Jan-Martin Nordbotten, Barbara Wohlmuth. International Journal for Numerical Methods in Engineering, in press. The accepted version is available in the main thesis. The published version is available at: https://doi.org/10.1002/nme.6238Paper C: Viscous fingering in fractured porous media. Runar L. Berge, Inga Berre, Eirik Keilegavlen, Jan-Martin Nordbotten. The article is not available in BORA.Paper D: PorePy: An Open-Source Software for Simulation of Multiphysics Processes in Fractured Porous Media. Eirik Keilegavlen, Runar L. Berge, Alessio Fumagalli, Michele Starnoni, Ivar Stefansson, Jhabriel Varela, Inga Berre. The article is available in the main thesis. The article is also available at: https://arxiv.org/abs/1908.09869Paper E: Unstructured Voronoi grids conforming to lower dimensional objects. Runar L. Berge, Øystein S. Klemetsdal, Knut-Andreas Lie. Computational Geosciences, 2019, 23(1):169–188. The article is not available in BORA due to publisher restrictions. The published version is available at: https://doi.org/10.1007/s10596-018-9790-0
PublisherThe University of Bergen
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