15 research outputs found

    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

    Integrated programming and mathematics in schools - A solid fundation for a future engineering education?

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    The interest in programming in schools has the last decade increased, and many countries have introduced programming as part of the school curriculum. Teaching of programming to students in primary and secondary school is often focused on the computer sciences aspect of programming. The current study is a part of the recently initiated research project ‚ÄúProgramming for understanding mathematics‚ÄĚ which has a different emphasis; the project investigates how the mathematical competence of the students are affected by actively using programming in mathematics lessons. In this paper, a recognized analytical framework for analysing the cognitive demand of mathematical tasks is presented. We extend the framework to include the analysis of tasks that utilize programming, allowing us to distinguish between tasks that are demanding due to the mathematical content, but the programming aspect of the task is trivial, and tasks that are cognitive demanding due to complex programming, but the mathematics is simple. We use the extended framework to analyse tasks in four mathematics textbooks written for 16-17 year old students by two major publishers in Norway. The results show that the tasks provided in the textbooks mainly focus on elementary programming skills, and the tasks give limited experiences with cognitive demanding programming tasks

    Viscous fingering in fractured porous media

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    The effect of heterogeneities induced by highly permeable fracture networks on viscous miscible fingering in porous media is examined using high-resolution numerical simulations. We consider the planar injection of a less viscous fluid into a two-dimensional fractured porous medium which is saturated with a more viscous fluid. This problem contains two sets of fundamentally different preferential flow regimes; the first is caused by the viscous fingering and the second is due to the permeability contrasts between the fractures and rock matrix. We study the transition from a regime where the flow is dominated by the viscous instabilities, to a regime where the heterogeneities induced by the fractures define the flow paths. We find that fractures greatly affect the viscous fingering, even for small permeability differences between the rock matrix and the fractures. The interaction between the viscosity contrast and permeability contrast causes channeling of the less viscous fluid through the fractures and back to the rock. This channeling stabilizes the displacement front in the rock matrix, and the viscous fingering ceases for the higher permeability contrast. Several different fracture geometries are considered, and we observe a complex interplay between the geometries and unstable flow. While we find that the most important dimensionless number determining the effect of the fracture network is a weighted ratio of the permeability of the fractures and the permeability of the rock matrix, the exact point for the cross-over regime is highly dependent on the geometry of the fracture network.Comment: To reproduce simulations, see "R. L. Berge, I. Berre, E. Keilegavlen, and J. M. Nordbotten. Viscous fingering in fractured porous media. Zenodo. doi: 10.5281/zenodo.3249931, June 2019

    Finite volume discretization for poroelastic media with fractures modeled by contact mechanics

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    A fractured poroelastic body is considered where the opening of the fractures is governed by a nonpenetration law, whereas slip is described by a Coulomb‚Äźtype friction law. This physical model results in a nonlinear variational inequality problem. The variational inequality is rewritten as a complementary function, and a semismooth Newton method is used to solve the system of equations. For the discretization, we use a hybrid scheme where the displacements are given in terms of degrees of freedom per element, and an additional Lagrange multiplier representing the traction is added on the fracture faces. The novelty of our method comes from combining the Lagrange multiplier from the hybrid scheme with a finite volume discretization of the poroelastic Biot equation, which allows us to directly impose the inequality constraints on each subface. The convergence of the method is studied for several challenging geometries in 2D and 3D, showing that the convergence rates of the finite volume scheme do not deteriorate when it is coupled to the Lagrange multipliers. Our method is especially attractive for the poroelastic problem because it allows for a straightforward coupling between the matrix deformation, contact conditions, and fluid pressure.publishedVersio

    PorePy: an open-source software for simulation of multiphysics processes in fractured porous media

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    Development of models and dedicated numerical methods for dynamics in fractured rocks is an active research field, with research moving towards increasingly advanced process couplings and complex fracture networks. The inclusion of coupled processes in simulation models is challenged by the high aspect ratio of the fractures, the complex geometry of fracture networks, and the crucial impact of processes that completely change characteristics on the fracture-rock interface. This paper provides a general discussion of design principles for introducing fractures in simulators, and defines a framework for integrated modeling, discretization, and computer implementation. The framework is implemented in the open-source simulation software PorePy, which can serve as a flexible prototyping tool for multiphysics problems in fractured rocks. Based on a representation of the fractures and their intersections as lower-dimensional objects, we discuss data structures for mixed-dimensional grids, formulation of multiphysics problems, and discretizations that utilize existing software. We further present a Python implementation of these concepts in the PorePy open-source software tool, which is aimed at coupled simulation of flow and transport in three-dimensional fractured reservoirs as well as deformation of fractures and the reservoir in general. We present validation by benchmarks for flow, poroelasticity, and fracture deformation in porous media. The flexibility of the framework is then illustrated by simulations of non-linearly coupled flow and transport and of injection-driven deformation of fractures. All results can be reproduced by openly available simulation scripts.publishedVersio

    Numerical methods for coupled processes in fractured porous media

    Get PDF
    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

    Unstructured Voronoi grids conforming to lower-dimensional objects

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    We present a novel mixed-dimensional method for generating unstructured polyhedral grids that conform to prescribed geometric objects in arbitrary dimensions. Two types of conformity are introduced: (i) control-point alignment of cell centroids to accurately represent horizontal and multilateral wells or create volumetric representations of fracture networks, and (ii) boundary alignment of cell faces to accurately preserve lower-dimensional geological objects such as layers, fractures, faults, and/or pinchouts. The prescribed objects are in this case assumed to be lower-dimensional, and we create a grid hierarchy in which each lower-dimensional object is associated with a lower-dimensional grid. Further, the intersection of two objects is associated with a grid one dimension lower than the objects. Each grid is generated as a clipped Voronoi diagram, also called a perpendicular bisector (PEBI) grid, for a carefully chosen set of generating points. Moreover, each grid is generated in such a way that the cell faces of a higher-dimensional grid conform to the cells of all lower-dimensional grids. We also introduce a sufficient and necessary condition which makes it easy to check if the sites for a given perpendicular bisector grid will conform to the set of prescribed geometric objects.Unstructured Voronoi grids conforming to lower-dimensional objectsacceptedVersio

    Unstructured PEBI-grids Adapting to Geological Features in Subsurface Reservoirs

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    Extensive research has been done on the generation of unstructured grids in reservoir simulation, with the aim of better representing the geology. We introduce UPR (Unstructured PEBI-grids for Reservoirs), a free, open source module for the Matlab Reservoir Simulation Toolbox. This module automates the generation of grids that conform to structures in subsurface reservoirs. The module implements the new methods presented in this thesis. These methods generate PEBI-grids that conform to wells and faults. The grids honor faults exactly. By using our novel method for treating intersections, it handles several hard cases robustly, including, intersection of multiple faults, intersections of wells and faults, and faults intersecting at sharp angles. We have also generalized our method to three dimensions, and present how one can create unstructured 3D PEBI-grids that conform exactly to faults

    Reactivation of fractures in subsurface reservoirs - A numerical approach using a static-dynamic friction model

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    Fluid-induced slip of fractures is characterized by strong multiphysics couplings. Three physical processes are considered: Flow, rock deformation and fracture deformation. The fractures are represented as lower-dimensional objects embedded in a three-dimensional domain. Fluid is modeled as slightly compressible, and flow in both fractures and matrix is accounted for. The deformation of rock is inherently different from the deformation of fractures; thus, two different models are needed to describe the mechanical deformation of the rock. The medium surrounding the fractures is modeled as a linear elastic material, while the slip of fractures is modeled as a contact problem, governed by a static-dynamic friction model. We present an iterative scheme for solving the non-linear set of equations that arise from the models, and suggest how the step parameter in this scheme should depend on the shear modulus and mesh size
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