A localized orthogonal decomposition method for semi-linear elliptic problems

In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. For this purpose we construct a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear fine-scale computations in small patches that have a diameter of order H |log H| where H is the coarse mesh size. Without any assumptions on the type of the oscillations in the coefficients, we give a rigorous proof for a linear convergence of the H1-error with respect to the coarse mesh size. To solve the arising equations, we propose an algorithm that is based on a damped Newton scheme in the multiscale space

On the optimization of iterative schemes for solving non-linear and/or coupled PDEs

In this thesis we study the optimization of iterative schemes as both linearization methods, and as splitting methods for solving non-linear and coupled partial differential equations (PDEs). We consider two equations that are describing processes in porous media; Richards’ equation, a possibly degenerate, non-linear and elliptic/parabolic equation that models flow of water in saturated/unsaturated porous media, and Biot’s equations, a coupled system of equations that models flow in deformable porous media. For Richards’ equation we compare the numerical properties of several linearization schemes, including the Newton-Raphson method, the modified Picard method and the L-scheme. Additionally, we prove convergence of the linearly and globally convergent L-scheme and discuss theoretically and practically how to choose its stabilization parameter optimally in the sense that convergence is obtained in the least amount of iterations. The second aim of the thesis is to effectively solve the quasi-static, linear Biot model. We consider the fixed-stress splitting scheme, which is a popular method for iteratively solving Biot’s equations. It is well-known that the convergence of the method is strongly dependent on the applied stabilization parameter. We propose a new approach to optimize this parameter, and show theoretically that it does not only depend on the mechanical properties and the coupling coefficient, but also on the fluid’s flow properties. The type of analysis presented in this thesis is not restricted to a particular spatial discretization, but we require it to be inf-sup stable. The convergence proof also applies to low-compressible or incompressible fluids, and low-permeable porous media. We perform illustrative numerical examples, including a well-known benchmark problem, Mandel’s problem. The results largely agree with the theoretical findings. Furthermore, we show numerically that for conditionally inf-sup stable discretizations, the performance of the fixed-stress splitting scheme behaves in a manner which contradicts the theory provided for inf-sup stable discretizations.Masteroppgave i anvendt og beregningsorientert matematikkMAMN-MABMAB39

Infiltration from the pedon to global grid scales: an overview and outlook for land surface modelling

Infiltration in soils is a key process that partitions precipitation at the land surface in surface runoff and water that enters the soil profile. We reviewed the basic principles of water infiltration in soils and we analyzed approaches commonly used in Land Surface Models (LSMs) to quantify infiltration as well as its numerical implementation and sensitivity to model parameters. We reviewed methods to upscale infiltration from the point to the field, hill slope, and grid cell scale of LSMs. Despite the progress that has been made, upscaling of local scale infiltration processes to the grid scale used in LSMs is still far from being treated rigorously. We still lack a consistent theoretical framework to predict effective fluxes and parameters that control infiltration in LSMs. Our analysis shows, that there is a large variety in approaches used to estimate soil hydraulic properties. Novel, highly resolved soil information at higher resolutions than the grid scale of LSMs may help in better quantifying subgrid variability of key infiltration parameters. Currently, only a few land surface models consider the impact of soil structure on soil hydraulic properties. Finally, we identified several processes not yet considered in LSMs that are known to strongly influence infiltration. Especially, the impact of soil structure on infiltration requires further research. In order to tackle the above challenges and integrate current knowledge on soil processes affecting infiltration processes on land surface models, we advocate a stronger exchange and scientific interaction between the soil and the land surface modelling communities

Modeling and application of soil moisture at varying spatial scales with parameter scaling

The dissertation focuses on characterization of subpixel variability within a satellite-based remotely sensed coarse-scale soil moisture footprint. The underlying heterogeneity of coarse-scale soil moisture footprint is masked by the area-integrated properties within the sensor footprint. Therefore, the soil moisture values derived from these measurements are an area average. The variability in soil moisture within the footprint is introduced by inherent spatial variability present in rainfall, and geophysical parameters (vegetation, topography, and soil). The geophysical parameters/variables typically interact in a complex fashion to make soil moisture evolution and dependent processes highly variable, and also, introduce nonlinearity across spatio-temporal scales. To study the variability and scaling characteristics of soil moisture, a quasi-distributed Soil-Vegetation-Atmosphere-Transfer (SVAT) modeling framework is developed to simulate the hydrological dynamics, i.e., the fluxes and the state variables within the satellite-based soil moisture footprint. The modeling framework is successfully tested and implemented in different hydroclimatic regions during the research. New multiscale data assimilation and Markov Chain Monte Carlo (MCMC) techniques in conjunction with the SVAT modeling framework are developed to quantify subpixel variability and assess multiscale soil moisture fields within the coarse-scale satellite footprint. Reasonable results demonstrate the potential to use these techniques to validate multiscale soil moisture data from future satellite mission e.g., Soil Moisture Active Passive (SMAP) mission of NASA. The results also highlight the physical controls of geophysical parameters on the soil moisture fields for various hydroclimatic regions. New algorithm that uses SVAT modeling framework is also proposed and its application demonstrated, to derive the stochastic soil hydraulic properties (i.e., saturated hydraulic conductivity) and surface features (i.e., surface roughness and volume scattering) related to radar remote sensing of soil moisture

Coherent gas flow patterns in heterogeneous permeability fields: Coherent gas flow patterns in heterogeneous permeability fields: from bench-scale to field-scale

Gas injection into saturated porous media has a high practical relevance. It is applied in groundwater remediation (air sparging), in CO2 sequestration into saline aquifers, and in enhanced oil recovery of petroleum reservoirs. This wide range of application necessitates a comprehensive understanding of gas flow patterns that may develop within the porous media and required modeling of multi-phase flow. There is an ongoing controversy in literature, if continuum models are able to describe the complex flow pattern observed in heterogeneous porous media, especially the channelized stochastic flow pattern. Based on Selker’s stochastic hypothesis, a gas channel is caused by a Brownian-motion process during gas injection. Therefore, the pore-scale heterogeneity will determine the shape of the single stochastic gas channels. On the other hand there are many studies on air sparging, which are based on continuum modeling. Up to date it is not clear under which conditions a continuum model can describe the essential features of the complex gas flow pattern. The aim of this study is to investigate the gas flow pattern on bench-scale and field scale using the continuum model TOUGH2. Based on a comprehensive data set of bench-scale experiments and field-scale experiments, we conduct for the first time a systematic study and evaluate the prediction ability of the continuum model. A second focus of this study is the development of a “real world”-continuum model, since on all scales – pore-scale, bench scale, field scale – heterogeneity is a key driver for the stochastic gas flow pattern. Therefore, we use different geostatistical programs to include stochastic conditioned and unconditioned parameter fields. Our main conclusion from bench-scale experiments is that a continuum model, which is calibrated by different independent measurements, has excellent prediction ability for the average flow behavior (e.g. the gas volume-injection rate relation). Moreover, we investigate the impact of both weak and strong heterogeneous parameter fields (permeability and capillary pressure) on gas flow pattern. The results show that a continuum model with weak stochastic heterogeneity cannot represent the essential features of the experimental gas flow pattern (e.g., the single stochastic gas channels). Contrary, applying a strong heterogeneity the continuum model can represent the channelized flow. This observation supports Stauffer’s statement that a so-called subscale continuum model with strong heterogeneity is able to describe the channelized flow behavior. On the other hand, we compare the theoretical integral gas volumes with our experiments and found that strong heterogeneity always yields too large gas volumes. At field-scale the 3D continuum model is used to design and optimize the direct gas injection technology. The field-scale study is based on the working hypotheses that the key parameters are the same as at bench-scale. Therefore, we assume that grain size and injection rate will determine whether coherent channelized flow or incoherent bubbly flow will develop at field-scale. The results of four different injection regimes were compared with the data of the corresponding field experiments. The main conclusion is that because of the buoyancy driven gas flow the vertical permeability has a crucial impact. Hence, the vertical and horizontal permeability should be implemented independently in numerical modeling by conditioned parameter fields

Three-dimensional electrical impedance tomography to monitor unsaturated moisture ingress in cement-based materials

The development of tools to monitor unsaturated moisture flow in cement-based material is of great importance, as most degradation processes in cement-based materials take place in the presence of moisture. In this paper, the feasibility of electrical impedance tomography (EIT) to monitor three-dimensional (3D) moisture flow in mortar containing fine aggregates is investigated. In the experiments, EIT measurements are taken during moisture ingress in mortar, using electrodes attached on the outer surface of specimens. For EIT, the so-called difference imaging scheme is adopted to reconstruct the change of the 3D electrical conductivity distribution within a specimen caused by the ingress of water into mortar. To study the ability of EIT to detect differences in the rate of ingress, the experiment is performed using plain water and with water containing a viscosity-modifying agent yielding a slower flow rate. To corroborate EIT, X-ray computed tomography (CT) and simulations of unsaturated moisture flow are carried out. While X-ray CT shows contrast with respect to background only in highly saturated regions, EIT shows the conductivity change also in the regions of low degree of saturation. The results of EIT compare well with simulations of unsaturated moisture flow. Moreover, the EIT reconstructions show a clear difference between the cases of water without and with the viscosity-modifying agent and demonstrate the ability of EIT to distinguish between different flow rates

Large-scale hydrological modelling :physical parameterisation for groundwater recharge

PhD ThesisThere is currently worldwide interest in the effect of human activity on tile global environment, especially the effect of greenhouse gases and land-use change on the global climate, and models are being developed to study both global change and the local effects of global change. The research reported here (funded by CNPq-Brazil) involves the development of GRASP:Groundwater Recharge modelling Approach with a Scaling up Procedure. GRASP has been integrated into the UP (Upscaled Physically-based) macromodel, developed under the UK NERC TIGER programme, which is designed for studying the effects of climate and land-use change on the availability and quality of water resources. The UP macromodel will be coupled to the UK Meteorological. Office's Unified (weather and climate) model to create a state-of-the-art coupled atmospheric/hydrological model. Several important requirements for the design of new large-scale hydrological models are identified in a wide ranging review on GCMs; (General Circulation Models) and physically -based hydrological modelling, and these requirements have been applied in the development of GRASP(and UP). The main requirements are a physical basis, proper treatment of spatial variability, and simplicity. Using the concept of partial analysis, two point-scale models, SM (Soil Moisture content approach) and TF (Transfer Function approach), are developed for recharge, both based on the one-dimensional Richards' equation. SM is a simple two-parameter model relating recharge to water storage in the unsaturated zone, and several unsuccessful attempts are made to link its parameters to physical propcrties. TF is a transfer function model, and is parameterised using the matric potential and unsaturated hydraulic conductivity functions using a new approach developed especially for GRASP. Both SM and TF are verified against numerical solutions of Richards' equation.CNPq: COPPE/UFRfJ

Transport processes in fractured porous media

112 stranThis habilitation thesis summarizes author's theoretical work related to development of the Flow123d simulator. This includes especially methods and algorithms for solving Darcy ow problems in saturated and unsaturated fractured porous media. A model with semi-discrete fractures called mixed dimension model is derived at the beginning. Then the abstract model for advection-di usion equation is applied to the Darcy ow. The mixed-hybrid formulation of the Darcy ow mixed dimension problem is presented followed by its discretization using Raviart-Thomas nite elements. An analytical solution to a test single fracture problem is supplied which allows veri cation of the model's implementation. Finally, the BDDC method is applied to obtain a scalable solver of the linear systems arising from the problem's discretization. Subsequently, new developments for the non-conforming mixed meshes are presented. Four methods with common strategy are used to introduce a coupling between equations living on the intersecting nite element meshes of di erent dimension. Further a family of e cient algorithms for computing mesh intersections is presented. Final chapter is devoted to the Richards' equation and modi cation of the mixed-hybrid scheme in order to satisfy discrete maximum principle. This is of particular importance for the Richards' equation where short time steps are often necessary which leads to strong oscillations for the schemes that violate DMP

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