17 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

    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

    A posteriori error estimation and modeling of unsaturated flow in fractured porous media

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    This doctoral thesis focuses on three topics: (1) modeling of unsaturated flow in fractured porous media, (2) a posteriori error estimation for mixed-dimensional elliptic equations, and (3) contributions to open-source software for complex multiphysics processes in porous media. In our first contribution, following a Discrete-Fracture Matrix (DFM) approach, we propose a model where Richards' equation governs the water flow in the matrix, whereas fractures are represented as lower-dimensional open channels, naturally providing a capillary barrier to the water flow. Therefore, water in the matrix is only allowed to imbibe the fracture if the capillary barrier is overcome. When this occurs, we assume that the water inside the fracture flows downwards without resistance and, therefore, is instantaneously at hydrostatic equilibrium. This assumption can be justifiable for fractures with sufficiently large apertures, where capillary forces play no role. Mathematically, our model can be classified as a coupled PDE-ODE system of equations with variational inequalities, in which each fracture is considered a potential seepage face. Our second contribution deals with error estimation for mixed-dimensional (mD) elliptic equations, which, in particular, model single-phase flow in fractured porous media. Here, based on the theory of functional a posteriori error estimates, we derive guaranteed upper bounds for the mD primal and mD dual variables, and two-sided bounds for the mD primal-dual pair. Moreover, we improve the standard results of the functional approach by proposing four ways of estimating the residual errors based on the conservation properties of the approximations, that is, (1) no conservation, (2) subdomain conservation, (3) local conservation, and (4) pointwise conservation. This results in sharper and fully-computable bounds when mass is conserved either locally or exactly. To our knowledge, to date, no error estimates have been available for fracture networks, including fracture intersections and floating subdomains. Our last contribution is related to the development of open-source software. First, we present the implementation of a new multipoint finite-volume-based module for unsaturated poroelasticity, compatible with the Matlab Reservoir Simulation Toolbox (MRST). Second, we present a new Python-based simulation framework for multiphysics processes in fractured porous media, named PorePy. PorePy, by design, is particularly well-suited for handling mixed-dimensional geometries, and thus optimal for DFM models. The first two contributions discussed above were implemented in PorePy.Denne avhandlingen tar for seg tre emner: (1) modellering av flyt i umettet porøst medium med sprekker, (2) a posteriori feilestimater for blandet-dimensjonale elliptiske ligninger, og (3) bidrag til åpen kildekode for komplekse multifysikk-prosesser i porøse medier. I det første bidraget anvender vi en Discrete-Fracture Matrix (DFM) (Diskret-Sprekk Matrise) metode til å sette opp en modell hvor Richard's ligning modellerer vann-flyt i matrisen, og sprekkene representeres som lavere-dimensjonale åpne kanaler, som naturlig virker som kapillærbarrierer til vann-flyten. Derfor vil vann i matrisen kun få tilgang til sprekken når kapillærbarrieren blir brutt. Når det inntreffer, antar vi at vannet i sprekken flyter nedover uten motstand, og at hydrostatisk ekvilibrium derfor inntreffer øyeblikkelig. Slike antakelser kan rettferdiggjøres for sprekker med tilstrekkelig stor apertur (åpning), hvor kapillærkrefter ikke har noen innvirkning. Fra et matematisk standpunkt kan modellen klassifiseres som en sammenkoblet PDE-ODE med variasjonelle ulikheter hvor hver sprekk behandles som en filtreringsfase. Det andre bidraget tar for seg feilestimater for blandet-dimensjonale elliptiske ligninger, som modellerer en-fase flyt i porøse medier med sprekker. Her anvender vi teorien for "funksjonal a posteriori feilestimater" til å finne øvre skranker for primær og dual variablene, samt øvre og nedre skranker for primær-dual paret. Dessuten viser vi at vi kan forbedre standardresultatene fra "funksjonal a posteriori feilestimater" ved å foreslå fire måte å estimere residualfeilen basert på bevaringsegenskapene til diskretiseringen. De fire forskjellige bevaringsegenskapene er; ingen bevaringsegenskap, under- domene bevaring, lokal bevaring og punktvis bevaring. Dette fører til skarpere skranker som er mulige å beregne når masse er bevart enten lokalt, eller eksakt. Vi kjenner ikke til andre tilgjengelige feilestimater for sprekknettverk som inkluderer snitt av sprekker og sprekkrender som ligger innenfor domenets rand. Det siste bidraget omhandler utvikling av åpen kildekode. Først presenterer vi imple- menteringen av en multipunktfluks-basert modul for flyt i umettet deformerbart porøst medium som er kompatibelt med "Matlab Reservoir Simulation Toolbox"(MRST). I tillegg presenterer vi et nytt Python-basert rammeverk for simulering av multifysikkprosesser i porøse medier med sprekker, som heter PorePy. Dette rammeverket er designet for å håndtere geometrier med blandede dimensjoner og er derfor optimalt for DFM modeller. De to første bidragene i avhandlingen (nevnt over) er implementert i PorePy.Doktorgradsavhandlin

    Two-phase flow properties upscaling in heterogeneous porous media

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    The groundwater specialists and the reservoir engineers share the same interest in simulating multiphase flow in soil with heterogeneous intrinsic properties. They also both face the challenge of going from a well-modeled micrometer scale to the reservoir scale with a controlled loss of information. This upscaling process is indeed worthy to make simulation over an entire reservoir manageable and stochastically repeatable. Two upscaling steps can be defined: one from the micrometer scale to the Darcy scale, and another from the Darcy scale to the reservoir scale. In this thesis, a new second upscaling multiscale algorithm Finite Volume Mixed Hybrid Multiscale Methods (Fv-MHMM) is investigated. Extension to a two-phase flow system is done by weakly and sequentially coupling saturation and pressure via IMPES-like method
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