85 research outputs found
Mathematical modelling and analysis of soil and plant root interactions
The influence of plants on soil water transport is a relevant factor in a number
of ecological contexts. Examples include: the resistance of crops to drought, the
prevention of floods and the protection of soils from erosion. There exists strong
experimental evidence that interactions between soil and plant roots change a soil’s
hydraulic properties. Nevertheless, it remains a challenge to anticipate the impact
of specific root traits on the infiltration of water through soil.
In an attempt to address the issue above, this thesis presents modifications of
Richards’ equation—the classic model for water transport through soil—to incorporate some effects that root systems are known to have on soil hydraulic properties.
First, a model is developed that incorporates the phenomenon of root-oriented
preferential flow. Using the finite element method and Bayesian optimisation, a
pipeline is developed to calibrate the model against experimental data. Moreover, it
is shown how existing root architectural models can be used in conjunction with our
model to investigate the influence that root system traits have on infiltration and
water uptake. Results suggest that this modification of Richards’ equation leads
to improved agreement of simulations with reference pore water pressure profiles,
which are derived from experimental data regarding the hydraulic conductivity of
vegetated soils.
Following this, the developed model is used to obtain simulations of various
infiltration scenarios. These reveal that, up to a critical point, increasing preferential
flow strength reduces water loss from the rooted zone. Furthermore, evidence is
provided to suggest that root systems with a reduced gravitropic response allow a
greater retention of water in the rooted zone following precipitation and, hence, are
among the most effective at delaying the onset of water deficits.
In another case, an alternative modification is proposed whereby Richards’ equation is coupled with an equation for water transport through roots. This model
accounts for root water uptake and hydraulic lift through a Neumann boundary
condition at the root-soil interface. By using the methods of Rothe and Galerkin,
existence of a solution to this coupled model is then established. Uniqueness is
shown by Kruzkov’s variable doubling method, but applied only in time.UK Engineering and Physical Sciences Research Council (EPSRC) grant
EP/L016508/01Scottish Funding Counci
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
Bioprocess Systems Engineering Applications in Pharmaceutical Manufacturing
Biopharmaceutical and pharmaceutical manufacturing are strongly influenced by the process analytical technology initiative (PAT) and quality by design (QbD) methodologies, which are designed to enhance the understanding of more integrated processes. The major aim of this effort can be summarized as developing a mechanistic understanding of a wide range of process steps, including the development of technologies to perform online measurements and real-time control and optimization. Furthermore, minimization of the number of empirical experiments and the model-assisted exploration of the process design space are targeted. Even if tremendous progress has been achieved so far, there is still work to be carried out in order to realize the full potential of the process systems engineering toolbox. Within this reprint, an overview of cutting-edge developments of process systems engineering for biopharmaceutical and pharmaceutical manufacturing processes is given, including model-based process design, Digital Twins, computer-aided process understanding, process development and optimization, and monitoring and control of bioprocesses. The biopharmaceutical processes addressed focus on the manufacturing of biopharmaceuticals, mainly by Chinese hamster ovary (CHO) cells, as well as adeno-associated virus production and generation of cell spheroids for cell therapies
Adaptive numerical techniques for problems related to flow in porous media
The solution of partial differential equations modelling water infiltration into
soil poses many challenges. The multi-scale and nonlinear nature of soil
makes the design of robust and accurate numerical schemes particularly difficult. In addition, error estimation is complicated by low solution regularity.
In this thesis, we investigate the mathematical and numerical aspects of the
approximation of problems related to subsurface flow by the finite element
method. We begin with a variational inequality as a simplified model (albeit
of significant interest and complexity in its own right) of a seepage problem.
The so-called Signorini problem includes many of the key difficulties, namely
nonlinear boundary conditions and lack of dual regularity. We derive rigorous and computable a posteriori error estimates using duality arguments
that require careful analysis of primal and dual problems. Crucial in this argument is the design of a novel nonlinear bound-preserving interpolant that
respects various inequalities related to the weak form of the problem. These
estimates are used to implement a mesh adaptive routine. We then study
a physically realistic seepage problem complete with nonlinear coefficients
and mixed boundary conditions and inequality constraints. This time, we
apply the dual-weighted residual framework of a posteriori error estimation
and derive error estimates that are used to optimise the computational mesh
for a quantity of interest. The estimates are tested on realistic groundwater
scenarios that utilise field data. We conclude with a numerical study of a
time-dependent and nonlinear model of two-dimensional subsurface flow. We
introduce a method to regularise the nonlinearity in the soil porosity function
and derive a posteriori error estimates that account for this approximation in
linear elliptic and parabolic cases. We show that in the nonlinear parabolic
case, this regularisation mitigates the commonly observed failure of nonlinear
solvers for Richards’ equation
Computational Modelling of Concrete and Concrete Structures
Computational Modelling of Concrete and Concrete Structures contains the contributions to the EURO-C 2022 conference (Vienna, Austria, 23-26 May 2022). The papers review and discuss research advancements and assess the applicability and robustness of methods and models for the analysis and design of concrete, fibre-reinforced and prestressed concrete structures, as well as masonry structures. Recent developments include methods of machine learning, novel discretisation methods, probabilistic models, and consideration of a growing number of micro-structural aspects in multi-scale and multi-physics settings. In addition, trends towards the material scale with new fibres and 3D printable concretes, and life-cycle oriented models for ageing and durability of existing and new concrete infrastructure are clearly visible. Overall computational robustness of numerical predictions and mathematical rigour have further increased, accompanied by careful model validation based on respective experimental programmes. The book will serve as an important reference for both academics and professionals, stimulating new research directions in the field of computational modelling of concrete and its application to the analysis of concrete structures. EURO-C 2022 is the eighth edition of the EURO-C conference series after Innsbruck 1994, Bad Gastein 1998, St. Johann im Pongau 2003, Mayrhofen 2006, Schladming 2010, St. Anton am Arlberg 2014, and Bad Hofgastein 2018. The overarching focus of the conferences is on computational methods and numerical models for the analysis of concrete and concrete structures
Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018
This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions
Two Phase Flow, Phase Change and Numerical Modeling
The heat transfer and analysis on laser beam, evaporator coils, shell-and-tube condenser, two phase flow, nanofluids, complex fluids, and on phase change are significant issues in a design of wide range of industrial processes and devices. This book includes 25 advanced and revised contributions, and it covers mainly (1) numerical modeling of heat transfer, (2) two phase flow, (3) nanofluids, and (4) phase change. The first section introduces numerical modeling of heat transfer on particles in binary gas-solid fluidization bed, solidification phenomena, thermal approaches to laser damage, and temperature and velocity distribution. The second section covers density wave instability phenomena, gas and spray-water quenching, spray cooling, wettability effect, liquid film thickness, and thermosyphon loop. The third section includes nanofluids for heat transfer, nanofluids in minichannels, potential and engineering strategies on nanofluids, and heat transfer at nanoscale. The forth section presents time-dependent melting and deformation processes of phase change material (PCM), thermal energy storage tanks using PCM, phase change in deep CO2 injector, and thermal storage device of solar hot water system. The advanced idea and information described here will be fruitful for the readers to find a sustainable solution in an industrialized society
Iterative schemes for surfactant transport in porous media
In this work, we consider the transport of a surfactant in variably saturated porous media. The water flow is modelled by the Richards equations and it is fully coupled with the transport equation for the surfactant. Three linearization techniques are discussed: the Newton method, the modified Picard, and the L-scheme. Based on these, monolithic and splitting schemes are proposed and their convergence is analyzed. The performance of these schemes is illustrated on five numerical examples. For these examples, the number of iterations and the condition numbers of the linear systems emerging in each iteration are presented.publishedVersio
Numerical Modeling of Permafrost in Heterogeneous Media
Interest in numerical modeling of permafrost has increased over the past decade due to accelerating rates of permafrost thaw. Discontinuous permafrost regions are particularly susceptible to climate change since small increases in mean annual temperature may lead to significant permafrost thaw, landscape change, changes in hydrologic connectivity, and greenhouse gas (GHG) emissions. The influence of local heterogeneities on the short- and long-term evolution of permafrost bodies is poorly understood. In order to numerically simulate the freeze-thaw processes in heterogenous media, a robust numerical model is desirable to overcome potential instabilities induced by heterogeneity in soil thermal properties. Here, such a model is developed, supplemented by a careful evaluation of the impact of heterogeneity upon the soil freezing curve, and applied to investigate the influence of local heterogeneity upon discontinuous permafrost evolution.
Numerical models of permafrost evolution in porous media typically rely upon a smooth continuous relation between pore ice saturation and sub-freezing temperature, rather than the abrupt phase change that occurs in pure media. Soil scientists have known for decades that this function, known as the soil freezing curve (SFC), is related to the soil-water-characteristic-curve (SWCC) for unfrozen soils due to the analogous capillary and sorptive effects experienced during both soil freezing and drying. Herein we demonstrate that other factors beyond the SFC-SWCC relationship can influence the potential range over which pore water phase change occurs. In particular, we provide a theoretical extension for the functional form of the SFC based upon the presence of spatial heterogeneity in both soil thermal conductivity and the freezing point depression of water. We infer the functional form of the SFC from many abrupt-interface 1-D numerical simulations of heterogeneous systems with prescribed statistical distributions of water and soil properties. The proposed SFC paradigm extension has the appealing features that it (1) is determinable from measurable soil and water properties, (2) collapses into an abrupt phase transition for homogeneous media, (3) describes a wide range of heterogeneity within a single functional expression, and (4) replicates the observed hysteretic behavior of freeze-thaw cycles in soils.
SFCs are used in all the permafrost models that use a continuum phase-change criterion. Here, an efficient enthalpy-based continuum numerical approach is introduced for solving heat transfer problems with non-isothermal phase change. In order to simulate permafrost over time spans of several years, a robust and efficient model is required. In the present setting, the heat transfer problem is converted to a minimization problem, in which we minimize a potential function that characterizes the governing heat transfer PDE within a time discrete framework. The use of the trust region minimization algorithm proves desirable due to the highly nonlinear energy functional which also involves non-convex terms induced by phase change. Results obtained show satisfactory agreement with existing analytical solutions. Moreover, the grid and timestep convergence studies conducted to examine the dependence of the solution on mesh and timestep sizes indicate robust convergence rates. This is the first application of trust region energy minimization algorithm in permafrost simulation.
The two-dimensional enthalpy-based numerical model with continuum phase-change is applied to study the effect of heterogeneity in the soil freezing point on conduction-driven talik formation. This model is rigorously verified against Lunardini's solution (Lunardini 1981), which is an analytical solution of the Stefan Problem with a non-isothermal phase-change criterion. Stochastic realizations of spatially correlated distributed soil thermal parameter fields are generated using the geostatistical software library (GSLIB) for a variety of correlation lengths and variances in material properties. These are used as input to the 2-D permafrost model under fully saturated conditions. The simulation results indicate that local heterogeneities have conditional effects on the formation of unfrozen zones and, eventually, talik. This influence is exacerbated under the presence of advective heat transfer, where small perturbations to the liquid water saturation can lead to preferential flow conduits. This work is extended by conducting sensitivity analysis to study the relative dependence of talik formation on different sources of heterogeneity (e.g. soil density and boundary conditions)
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