1,727 research outputs found
Adaptive asynchronous time-stepping, stopping criteria, and a posteriori error estimates for fixed-stress iterative schemes for coupled poromechanics problems
In this paper we develop adaptive iterative coupling schemes for the Biot
system modeling coupled poromechanics problems. We particularly consider the
space-time formulation of the fixed-stress iterative scheme, in which we first
solve the problem of flow over the whole space-time interval, then exploiting
the space-time information for solving the mechanics. Two common
discretizations of this algorithm are then introduced based on two coupled
mixed finite element methods in-space and the backward Euler scheme in-time.
Therefrom, adaptive fixed-stress algorithms are build on conforming
reconstructions of the pressure and displacement together with equilibrated
flux and stresses reconstructions. These ingredients are used to derive a
posteriori error estimates for the fixed-stress algorithms, distinguishing the
different error components, namely the spatial discretization, the temporal
discretization, and the fixed-stress iteration components. Precisely, at the
iteration of the adaptive algorithm, we prove that our estimate gives
a guaranteed and fully computable upper bound on the energy-type error
measuring the difference between the exact and approximate pressure and
displacement. These error components are efficiently used to design adaptive
asynchronous time-stepping and adaptive stopping criteria for the fixed-stress
algorithms. Numerical experiments illustrate the efficiency of our estimates
and the performance of the adaptive iterative coupling algorithms
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 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
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