456 research outputs found

    A study on iterative methods for solving Richards` equation

    Full text link
    This work concerns linearization methods for efficiently solving the Richards` equation,a degenerate elliptic-parabolic equation which models flow in saturated/unsaturated porous media.The discretization of Richards` equation is based on backward Euler in time and Galerkin finite el-ements in space. The most valuable linearization schemes for Richards` equation, i.e. the Newtonmethod, the Picard method, the Picard/Newton method and theLscheme are presented and theirperformance is comparatively studied. The convergence, the computational time and the conditionnumbers for the underlying linear systems are recorded. The convergence of theLscheme is theo-retically proved and the convergence of the other methods is discussed. A new scheme is proposed,theLscheme/Newton method which is more robust and quadratically convergent. The linearizationmethods are tested on illustrative numerical examples

    Numerical Treatment of State-Dependent Permeability in Multiphysics Problems

    Get PDF
    Constitutive laws relating fluid potentials and fluxes in a nonlinear manner are common in several porous media applications, including biological and reactive flows, poromechanics, and fracture deformation. Compared to the standard, linear Darcy's law, such enhanced flux relations increase both the degree of nonlinearity, and, in the case of multiphysics simulations, coupling strength between processes. While incorporating the nonlinearities into simulation models is thus paramount for computational efficiency, correct linearization, as is needed for incorporation in Newton's method, is challenging from a practical perspective. The standard approach is therefore to ignore nonlinearities in the permeability during linearization. For finite volume methods, which are popular in porous media applications, complete linearization is feasible only for the simplest flux discretization, namely the two-point flux approximation. We introduce an approximated linearization scheme for finite volume methods that is exact for the two-point scheme and can be applied to more advanced and accurate discretizations, exemplified herein by a multi-point flux stencil. We test the new method for both nonlinear porous media flow and several multiphysics simulations. Our results show that the new linearization consistently outperforms the standard approach. Moreover our scheme achieves asymptotic second order convergence of the Newton iterations, in contrast to the linear convergence obtained with the standard approach.publishedVersio

    A linear domain decomposition method for two-phase flow in porous media

    Full text link
    This article is a follow up of our submitted paper [11] in which a decomposition of the Richards equation along two soil layers was discussed. A decomposed problem was formulated and a decoupling and linearisation technique was presented to solve the problem in each time step in a fixed point type iteration. This article extends these ideas to the case of two-phase in porous media and the convergence of the proposed domain decomposition method is rigorously shown.Comment: 8 page

    An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow

    Get PDF
    International audienceIn this paper we derive an a posteriori error estimate for the numerical approximation of the solution of a system modeling the flow of two incompressible and immiscible fluids in a porous medium. We take into account the capillary pressure, which leads to a coupled system of two equations: parabolic and elliptic. The parabolic equation may become degenerate, i.e., the nonlinear diffusion coefficient may vanish over regions that are not known a priori. We first show that, under appropriate assumptions, the energy-type-norm differences between the exact and the approximate nonwetting phase saturations, the global pressures, and the Kirchhoff transforms of the nonwetting phase saturations can be bounded by the dual norm of the residuals. We then bound the dual norm of the residuals by fully computable a posteriori estimators. Our analysis covers a large class of conforming, vertex-centered finite volume-type discretizations with fully implicit time stepping. As an example, we focus here on two approaches: a ''mathematical'' scheme derived from the weak formulation, and a phase-by-phase upstream weighting ''engineering'' scheme. Finally, we show how the different error components, namely the space discretization error, the time discretization error, the linearization error, the algebraic solver error, and the quadrature error can be distinguished and used for making the calculations efficient

    Development of robust and efficient solution strategies for coupled problems

    Get PDF
    Det er mange modeller i moderne vitenskap hvor sammenkoblingen mellom forskjellige fysiske prosesser er svært viktig. Disse finner man for eksempel i forbindelse med lagring av karbondioksid i undervannsreservoarer, flyt i kroppsvev, kreftsvulstvekst og geotermisk energiutvinning. Denne avhandlingen har to fokusområder som er knyttet til sammenkoblede modeller. Det første er å utvikle pålitelige og effektive tilnærmingsmetoder, og det andre er utviklingen av en ny modell som tar for seg flyt i et porøst medium som består av to forskjellige materialer. For tilnærmingsmetodene har det vært et spesielt fokus på splittemetoder. Dette er metoder hvor hver av de sammenkoblede modellene håndteres separat, og så itererer man mellom dem. Dette gjøres i hovedsak fordi man kan utnytte tilgjengelig teori og programvare for å løse hver undermodell svært effektivt. Ulempen er at man kan ende opp med løsningsalgoritmer for den sammenkoblede modellen som er trege, eller ikke kommer frem til noen løsning i det hele tatt. I denne avhandlingen har tre forskjellige metoder for å forbedre splittemetoder blitt utviklet for tre forskjellige sammenkoblede modeller. Den første modellen beskriver flyt gjennom deformerbart porøst medium og er kjent som Biot ligningene. For å anvende en splittemetode på denne modellen har et stabiliseringsledd blitt tilført. Dette sikrer at metoden konvergerer (kommer frem til en løsning), men dersom man ikke skalerer stabiliseringsleddet riktig kan det ta veldig lang tid. Derfor har et intervall hvor den optimale skaleringen av stabiliseringsleddet befinner seg blitt identifisert, og utfra dette presenteres det en måte å praktisk velge den riktige skaleringen på. Den andre modellen er en fasefeltmodell for sprekkpropagering. Denne modellen løses vanligvis med en splittemetode som er veldig treg, men konvergent. For å forbedre dette har en ny akselerasjonsmetode har blitt utviklet. Denne anvendes som et postprosesseringssteg til den klassiske splittemetoden, og utnytter både overrelaksering og Anderson akselerasjon. Disse to forskjellige akselerasjonsmetodene har kompatible styrker i at overrelaksering akselererer når man er langt fra løsningen (som er tilfellet når sprekken propagerer), og Anderson akselerasjon fungerer bra når man er nærme løsningen. For å veksle mellom de to metodene har et kriterium basert på residualfeilen blitt brukt. Resultatet er en pålitelig akselerasjonsmetode som alltid akselererer og ofte er svært effektiv. Det siste modellen kalles Cahn-Larché ligningene og er også en fasefeltmodell, men denne beskriver elastisitet i et medium bestående av to elastiske materialer som kan bevege seg basert på overflatespenningen mellom dem. Dette problemet er spesielt utfordrende å løse da det verken er lineært eller konvekst. For å håndtere dette har en ny måte å behandle tidsavhengigheten til det underliggende koblede problemet på blitt utviklet. Dette leder til et diskret system som er ekvivalent med et konvekst minimeringsproblem, som derfor er velegnet til å løses med de fleste numeriske optimeringsmetoder, også splittemetoder. Den nye modellen som har blitt utviklet er en utvidelse av Cahn-Larché ligningene og har fått navnet Cahn-Hilliard-Biot. Dette er fordi ligningene utgjør en fasefelt modell som beskriver flyt i et deformerbart porøst medium med to poroelastiske materialer. Disse kan forflytte seg basert på overflatespenning, elastisk spenning, og poretrykk, og det er tenkt at modellen kan anvendes i forbindelse med kreftsvulstmodellering.There are many applications where the study of coupled physical processes is of great importance. These range from the life sciences with flow in deformable human tissue to structural engineering with fracture propagation in elastic solids. In this doctoral dissertation, there is a twofold focus on coupled problems. Firstly, robust and efficient solution strategies, with a focus on iterative decoupling methods, have been applied to several coupled systems of equations. Secondly, a new thermodynamically consistent coupled system of equations is proposed. Solution strategies are developed for three different coupled problems; the quasi-static linearized Biot equations that couples flow through porous materials and elastic deformation of the solid medium, variational phase-field models for brittle fracture that couple a phase-field equation for fracture evolution with linearized elasticity, and the Cahn-Larché equations that model elastic effects in a two-phase elastic material and couples an extended Cahn-Hilliard phase-field equation and linearized elasticity. Finally, the new system of equations that is proposed models flow through a two-phase deformable porous material where the solid phase evolution is governed by interfacial forces as well as effects from both the fluid and elastic properties of the material. In the work that concerns the quasi-static linearized Biot equations, the focus is on the fixed-stress splitting scheme, which is a popular method for sequentially solving the flow and elasticity subsystems of the full model. Using such a method is beneficial as it allows for the use of readily available solvers for the subproblems; however, a stabilizing term is required for the scheme to converge. It is well known that the convergence properties of the method strongly depend on how this term is chosen, and here, the optimal choice of it is addressed both theoretically and practically. An interval where the optimal stabilization parameter lies is provided, depending on the material parameters. In addition, two different ways of optimizing the parameter are proposed. The first is a brute-force method that relies on the mesh independence of the scheme's optimal stabilization parameter, and the second is valid for low-permeable media and utilizes an equivalence between the fixed-stress splitting scheme and the modified Richardson iteration. Regarding the variational phase-field model for brittle fracture propagation, the focus is on improving the convergence properties of the most commonly used solution strategy with an acceleration method. This solution strategy relies on a staggered scheme that alternates between solving the elasticity and phase-field subproblems in an iterative way. This is known to be a robust method compared to the monolithic Newton method. However, the staggered scheme often requires many iterations to converge to satisfactory precision. The contribution of this work is to accelerate the solver through a new acceleration method that combines Anderson acceleration and over-relaxation, dynamically switching back and forth between them depending on a criterion that takes the residual evolution into account. The acceleration scheme takes advantage of the strengths of both Anderson acceleration and over-relaxation, and the fact that they are complementary when applied to this problem, resulting in a significant speed-up of the convergence. Moreover, the method is applied as a post-processing technique to the increments of the solver, and can thus be implemented with minor modifications to readily available software. The final contribution toward solution strategies for coupled problems focuses on the Cahn-Larché equations. This is a model for linearized elasticity in a medium with two elastic phases that evolve with respect to interfacial forces and elastic effects. The system couples linearized elasticity and an extended Cahn-Hilliard phase-field equation. There are several challenging features with regards to solution strategies for this system including nonlinear coupling terms, and the fourth-order term that comes from the Cahn-Hilliard subsystem. Moreover, the system is nonlinear and non-convex with respect to both the phase-field and the displacement. In this work, a new semi-implicit time discretization that extends the standard convex-concave splitting method applied to the double-well potential from the Cahn-Hilliard subsystem is proposed. The extension includes special treatment for the elastic energy, and it is shown that the resulting discrete system is equivalent to a convex minimization problem. Furthermore, an alternating minimization solver is proposed for the fully discrete system, together with a convergence proof that includes convergence rates. Through numerical experiments, it becomes evident that the newly proposed discretization method leads to a system that is far better conditioned for linearization methods than standard time discretizations. Finally, a new model for flow through a two-phase deformable porous material is proposed. The two poroelastic phases have distinct material properties, and their interface evolves according to a generalized Ginzburg–Landau energy functional. As a result, a model that extends the Cahn-Larché equations to poroelasticity is proposed, and essential coupling terms for several applications are highlighted. These include solid tumor growth, biogrout, and wood growth. Moreover, the coupled set of equations is shown to be a generalized gradient flow. This implies that the system is thermodynamically consistent and makes a toolbox of analysis and solvers available for further study of the model.Doktorgradsavhandlin
    corecore