205 research outputs found
A posteriori error estimation and modeling of unsaturated flow in fractured porous media
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 suïŹiciently 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 porĂžst medium med sprekker, (2) a posteriori feilestimater for blandet-dimensjonale elliptiske ligninger, og (3) bidrag til Ă„pen kildekode for komplekse multifysikk-prosesser i porĂžse medier.
I det fÞrste bidraget anvender vi en Discrete-Fracture Matrix (DFM) (Diskret-Sprekk Matrise) metode til Ä sette opp en modell hvor Richard's ligning modellerer vann-flyt i matrisen, og sprekkene representeres som lavere-dimensjonale Äpne kanaler, som naturlig virker som kapillÊrbarrierer til vann-flyten. Derfor vil vann i matrisen kun fÄ tilgang til sprekken nÄr kapillÊrbarrieren blir brutt. NÄr det inntreffer, antar vi at vannet i sprekken flyter nedover uten motstand, og at hydrostatisk ekvilibrium derfor inntreffer Þyeblikkelig. Slike antakelser kan rettferdiggjÞres for sprekker med tilstrekkelig stor apertur (Äpning), hvor kapillÊrkrefter 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 porÞse medier med sprekker. Her anvender vi teorien for "funksjonal a posteriori feilestimater" til Ä finne Þvre skranker for primÊr og dual variablene, samt Þvre og nedre skranker for primÊr-dual paret. Dessuten viser vi at vi kan forbedre standardresultatene fra "funksjonal a posteriori feilestimater" ved Ä foreslÄ fire mÄte Ä estimere residualfeilen basert pÄ bevaringsegenskapene til diskretiseringen. De fire forskjellige bevaringsegenskapene er; ingen bevaringsegenskap, under- domene bevaring, lokal bevaring og punktvis bevaring. Dette fÞrer til skarpere skranker som er mulige Ä beregne nÄr 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 Äpen kildekode. FÞrst presenterer vi imple- menteringen av en multipunktfluks-basert modul for flyt i umettet deformerbart porÞst medium som er kompatibelt med "Matlab Reservoir Simulation Toolbox"(MRST). I tillegg presenterer vi et nytt Python-basert rammeverk for simulering av multifysikkprosesser i porÞse medier med sprekker, som heter PorePy. Dette rammeverket er designet for Ä hÄndtere geometrier med blandede dimensjoner og er derfor optimalt for DFM modeller. De to fÞrste bidragene i avhandlingen (nevnt over) er implementert i PorePy.Doktorgradsavhandlin
Simulation of rock salt dissolution and its impact on land subsidence
Extensive land subsidence can occur due to subsurface dissolution of evaporites such as halite and gypsum. This paper explores techniques to simulate the salt dissolution forming an intrastratal karst, which is embedded in a sequence of carbonates, marls, anhydrite and gypsum. A numerical model is developed to simulate laminar flow in a subhorizontal void, which corresponds to an opening intrastratal karst. The numerical model is based on the laminar steady-state Stokes flow equation, and the advection dispersion transport equation coupled with the dissolution equation. The flow equation is solved using the nonconforming Crouzeix-Raviart (CR) finite element approximation for the Stokes equation. For the transport equation, a combination between discontinuous Galerkin method and multipoint flux approximation method is proposed. The numerical effect of the dissolution is considered by using a dynamic mesh variation that increases the size of the mesh based on the amount of dissolved salt. The numerical method is applied to a 2D geological cross section representing a Horst and Graben structure in the Tabular Jura of northwestern Switzerland. The model simulates salt dissolution within the geological section and predicts the amount of vertical dissolution as an indicator of potential subsidence that could occur. Simulation results showed that the highest dissolution amount is observed near the normal fault zones, and, therefore, the highest subsidence rates are expected above normal fault zones
Postprocessing of Non-Conservative Flux for Compatibility with Transport in Heterogeneous Media
A conservative flux postprocessing algorithm is presented for both
steady-state and dynamic flow models. The postprocessed flux is shown to have
the same convergence order as the original flux. An arbitrary flux
approximation is projected into a conservative subspace by adding a piecewise
constant correction that is minimized in a weighted norm. The application
of a weighted norm appears to yield better results for heterogeneous media than
the standard norm which has been considered in earlier works. We also
study the effect of different flux calculations on the domain boundary. In
particular we consider the continuous Galerkin finite element method for
solving Darcy flow and couple it with a discontinuous Galerkin finite element
method for an advective transport problem.Comment: 34 pages, 17 figures, 11 table
Reactive Flows in Deformable, Complex Media
Many processes of highest actuality in the real life are described through systems of equations posed in complex domains. Of particular interest is the situation when the domain is variable, undergoing deformations that depend on the unknown quantities of the model. Such kind of problems are encountered as mathematical models in the subsurface, or biological systems. Such models include various processes at different scales, and the key issue is to integrate the domain deformation in the multi-scale context. Having this as the background theme, this workshop focused on novel techniques and ideas in the analysis, the numerical discretization and the upscaling of such problems, as well as on applications of major societal relevance today
Discontinuous Finite Element Methods for Interface Problems: Robust A Priori and A Posteriori Error Estimates
For elliptic interface problems in two and three dimensions, this paper studies a
priori and residual-based a posteriori error estimations for the CrouzeixâRaviart nonconforming and
the discontinuous Galerkin finite element approximations. It is shown that both the a priori and the
a posteriori error estimates are robust with respect to the diffusion coefficient, i.e., constants in the
error bounds are independent of the jump of the diffusion coefficient. The a priori estimates are also
optimal with respect to local regularity of the solution. Moreover, we obtained these estimates with
no assumption on the distribution of the diffusion coefficient
A linear domain decomposition method for partially saturated flow in porous media
The Richards equation is a nonlinear parabolic equation that is commonly used
for modelling saturated/unsaturated flow in porous media. We assume that the
medium occupies a bounded Lipschitz domain partitioned into two disjoint
subdomains separated by a fixed interface . This leads to two problems
defined on the subdomains which are coupled through conditions expressing flux
and pressure continuity at . After an Euler implicit discretisation of
the resulting nonlinear subproblems a linear iterative (-type) domain
decomposition scheme is proposed. The convergence of the scheme is proved
rigorously. In the last part we present numerical results that are in line with
the theoretical finding, in particular the unconditional convergence of the
scheme. We further compare the scheme to other approaches not making use of a
domain decomposition. Namely, we compare to a Newton and a Picard scheme. We
show that the proposed scheme is more stable than the Newton scheme while
remaining comparable in computational time, even if no parallelisation is being
adopted. Finally we present a parametric study that can be used to optimize the
proposed scheme.Comment: 34 pages, 13 figures, 7 table
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