Numerical methods for coupled processes in fractured porous media

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

Modelling and quantification of structural uncertainties in petroleum reservoirs assisted by a hybrid cartesian cut cell/enriched multipoint flux approximation approach

Efficient and profitable oil production is subject to make reliable predictions about reservoir performance. However, restricted knowledge about reservoir distributed properties and reservoir structure calls for History Matching in which the reservoir model is calibrated to emulate the field observed history. Such an inverse problem yields multiple history-matched models which might result in different predictions of reservoir performance. Uncertainty Quantification restricts the raised model uncertainties and boosts the model reliability for the forecasts of future reservoir behaviour. Conventional approaches of Uncertainty Quantification ignore large scale uncertainties related to reservoir structure, while structural uncertainties can influence the reservoir forecasts more intensely compared with petrophysical uncertainty. What makes the quantification of structural uncertainty impracticable is the need for global regridding at each step of History Matching process. To resolve this obstacle, we develop an efficient methodology based on Cartesian Cut Cell Method which decouples the model from its representation onto the grid and allows uncertain structures to be varied as a part of History Matching process. Reduced numerical accuracy due to cell degeneracies in the vicinity of geological structures is adequately compensated with an enhanced scheme of class Locally Conservative Flux Continuous Methods (Extended Enriched Multipoint Flux Approximation Method abbreviated to extended EMPFA). The robustness and consistency of proposed Hybrid Cartesian Cut Cell/extended EMPFA approach are demonstrated in terms of true representation of geological structures influence on flow behaviour. In this research, the general framework of Uncertainty Quantification is extended and well-equipped by proposed approach to tackle uncertainties of different structures such as reservoir horizons, bedding layers, faults and pinchouts. Significant improvements in the quality of reservoir recovery forecasts and reservoir volume estimation are presented for synthetic models of uncertain structures. Also this thesis provides a comparative study of structural uncertainty influence on reservoir forecasts among various geological structures