Modelling of chloride transport in non-saturated concrete : from microscale to macroscale

Corrosion of steel bars embedded in concrete has a great influence on structural performance and durability of reinforced concrete. Chloride penetration is considered to be a primary cause of concrete deterioration in a vast majority of structures. Therefore, modelling of chloride penetration into concrete has become an area of great interest. The present work focuses on modelling of chloride transport in concrete. The differential macroscopic equations which govern the problem were derived from the equations at the microscopic scale by comparing the porous network with a single equivalent pore whose properties are the same as the average properties of the real porous network. The resulting transport model, which accounts for diffusion, migration, advection, chloride binding and chloride precipitation, consists of three coupled differential equations. The first equation models the transport of chloride ions, while the other two model the flow of the pore water and the heat transfer. In order to calibrate the model, the material parameters to determine experimentally were identified. The differential equations were solved by means of the finite element method. The classical Galerkin method was employed for the pore solution flow and the heat transfer equations, while the streamline upwind Petrov Galerkin method was adopted for the transport equation in order to avoid spatial instabilities for advection dominated problems. The finite element codes are implemented in Matlab® . To retrieve a good understanding of the influence of each variable and parameter, a detailed sensitivity analysis of the model was carried out. In order to determine the diffusive and hygroscopic properties of the studied concretes, as well as their chloride binding capacity, an experimental analysis was performed. The model was successfully compared with experimental data obtained from an offshore oil platform located in Brazil. Moreover, apart from the main objectives, numerous results were obtained throughout this work. For instance, several diffusion coefficients and the relation between them are discussed. It is shown how the electric field set up between the ionic species depends on the gradient of the species’ concentrations. Furthermore, the capillary hysteresis effects are illustrated by a proposed model, which leads to the determination of several microstructure properties, such as the pore size distribution and the tortuosity-connectivity of the porous network. El fenómeno de corrosión del acero de refuerzo embebido en el hormigón ha tenido gran influencia en estructuras de hormigón armado, tanto en su funcionalidad estructural como en aspectos de durabilidad. La penetración de cloruros en el interior del hormigón esta considerada como el factor principal en el deterioro de la gran mayoría de estructuras. Por lo tanto, la modelización numérica de dicho fenómeno ha generado gran interés. El presente trabajo de investigación se centra en la modelización del transporte de cloruros en el interior del hormigón. Las ecuaciones diferenciales que gobiernan los fenómenos a nivel macroscópico se deducen de ecuaciones planteadas a nivel microscópico. Esto se obtiene comparando la red porosa con un poro equivalente, el cual mantiene las mismas propiedades de la red porosa real. El modelo está constituido por tres ecuaciones diferenciales acopladas que consideran el transporte de cloruros, el flujo de la solución de poro y la transferencia de calor. Con estas ecuaciones se tienen en cuenta los fenómenos de difusión, migración, advección, combinación y precipitación de cloruros. El análisis llevado a cabo en este trabajo ha definido los parámetros necesarios para calibrar el modelo. De acuerdo con ellas, se seleccionaron los ensayos experimentales a realizar. Las ecuaciones diferenciales se resolvieron mediante el método de elementos finitos. El método clásico de Galerkin se empleó para solucionar las ecuaciones de flujo de la solución de poro y de la transferencia de calor, mientras que el método streamline upwind Petrov-Galerkin se utilizó para resolver la ecuación de transporte de cloruros con la finalidad de evitar inestabilidades espaciales en problemas con advección dominante. El código de elementos finitos está implementado en Matlab® . Con el objetivo de facilitar la comprensión del grado de influencia de cada variable y parámetro, se realizó un análisis de sensibilidad detallado del modelo. Se llevó a cabo una campaña experimental sobre los hormigones estudiados, con el objeto de obtener sus propiedades difusivas, químicas e higroscópicas. El modelo se contrastó con datos experimentales obtenidos en una plataforma petrolera localizada en Brasil. Las simulaciones numéricas corroboraron los datos experimentales. Además, durante el desarrollo de la investigación se obtuvieron resultados paralelos a los planteados inicialmente. Por ejemplo, el análisis de diferentes coeficientes de difusión y la relación entre ellos. Así como también se observó que el campo eléctrico establecido entre las especies iónicas disueltas en la solución de poro depende del gradiente de concentración de las mismas. Los efectos de histéresis capilar son expresados por el modelo propuesto, el cual conduce a la determinación de una serie de propiedades microscópicas, tales como la distribución del tamaño de poro, además de la tortuosidad y conectividad de la red porosa.

Modelling of chloride transport in non-saturated concrete : from microscale to macroscale

A model for chloride transport in concrete is proposed. The model accounts for transport several transport mechanisms such as diffusion, advection, migration, etc. This work shows the chloride transport equations at the macroscopic scale in non-saturated concrete. The equations involve diffusion, migration, capillary suction, chloride combination and precipitation mechanisms. The material is assumed to be infinitely rigid, though the porosity can change under influence of chloride binding and precipitation. The involved microscopic and macroscopic properties of the materials are measured by standardized methods. The variables which must be imposed on the boundaries are temperature, relative humidity and chloride concentration. The output data of the model are the free, bound, precipitated and total chloride ion concentrations, as well as the pore solution content and the porosity. The proposed equations are solved by means of the finite element method (FEM) implemented in MATLAB (classical Galerkin formulation and the streamline upwind Petrov-Galerkin (SUPG) method to avoid spatial instabilities for advection dominated flows).

Simulación numérica de un problema de contaminación atmosférica

La ecuación en derivadas parciales de advección difusión con reacción química es la base de los modelos de dispersión de contaminantes en la atmósfera, y los diferentes métodos numéricos empleados para su resolución han sido objeto de amplios estudios a lo largo de su desarrollo. En esta Tesis se presenta la implementación de un nuevo método conservativo para la resolución de la parte advectiva de la ecuación en derivadas parciales que modela la dispersión de contaminantes dentro del modelo mesoescalar de transporte químico CHIMERE. Este método está basado en una técnica de volúmenes finitos junto con una interpolación racional. La ventaja de este método es la conservación exacta de la masa transportada debido al empleo de la ley de conservación de masas. Para ello emplea una formulación de flujo basado en el cálculo de la integral ponderada dentro de cada celda definida para la discretización del espacio en el método de volúmenes finitos. Los resultados numéricos obtenidos en las simulaciones realizadas (implementando el modelo conservativo para la advección en el modelo CHIMERE) se han comparado con los datos observados de concentración de contaminantes registrados en la red de estaciones de seguimiento y medición distribuidas por la Península Ibérica. Los datos estadísticos de medición del error, la media normalizada y la media absoluta normalizada del error, presentan valores que están dentro de los rangos propuestos por la EPA para considerar el modelo preciso. Además, se introduce un nuevo método para resolver la parte advectivadifusiva de la ecuación en derivadas parciales que modeliza la dispersión de contaminantes en la atmósfera. Se ha empleado un método de diferencias finitas de alto orden para resolver la parte difusiva de la ecuación de transporte de contaminantes junto con el método racional conservativo para la parte advectiva en una y dos dimensiones. Los resultados obtenidos de la aplicación del método a diferentes situaciones incluyendo casos académicos y reales han sido comparados con la solución analítica de la ecuación de advección-difusión, demostrando que el nuevo método proporciona un resultado preciso para aproximar la solución. Por último, se ha desarrollado un modelo completo que contempla los fenómenos advectivo y difusivo con reacción química, usando los métodos anteriores junto con una técnica de diferenciación regresiva (BDF2). Esta técnica consiste en un método implícito multipaso de diferenciación regresiva de segundo orden, que nos permite resolver los problemas rígidos típicos de la química atmosférica, modelizados a través de sistemas de ecuaciones diferenciales ordinarias. Este método hace uso de la técnica iterativa Gauss- Seidel para obtener la solución de la parte implícita de la fórmula BDF2. El empleo de la técnica de Gauss-Seidel en lugar de otras técnicas comúnmente empleadas, como la iteración por el método de Newton, nos proporciona rapidez de cálculo y bajo consumo de memoria, ideal para obtener modelos operativos para la resolución de la cinética química atmosférica. ABSTRACT Extensive research has been performed to solve the atmospheric chemicaladvection- diffusion equation and different numerical methods have been proposed. This Thesis presents the implementation of an exactly conservative method for the advection equation in the European scale Eulerian chemistry transport model CHIMERE based on a rational interpolation and a finite volume algorithm. The advantage of the method is that the cell-integrated average is predicted via a flux formulation, thus the mass is exactly conserved. Numerical results are compared with a set of observation registered at some monitoring sites in Spain. The mean normalized bias and the mean normalized absolute error present values that are inside the range to consider an accurate model performance. In addition, it has been introduced a new method to solve the advectiondiffusion equation. It is based on a high-order accurate finite difference method to solve de diffusion equation together with a rational interpolation and a finite volume to solve the advection equation in one dimension and two dimensions. Numerical results obtained from solving several problems include academic and real atmospheric problems have been compared with the analytical solution of the advection-diffusion equation, showing that the new method give an efficient algorithm for solving such problems. Finally, a complete model has been developed to solve the atmospheric chemical-advection-diffusion equation, adding the conservative method for the advection equation, the high-order finite difference method for the diffusion equation and a second-order backward differentiation formula (BDF2) to solve the atmospheric chemical kinetics. The BDF2 is an implicit, second order multistep backward differentiation formula used to solve the stiff systems of ordinary differential equations (ODEs) from atmospheric chemistry. The Gauss-Seidel iteration is used for approximately solving the implicitly defined BDF solution, giving a faster tool than the more commonly used iterative modified Newton technique. This method implies low start-up costs and a low memory demand due to the use of Gauss-Seidel iteration.

Design of a Semi-analytical Method to Describe Plasma-Wall Interactions

Electric probes are objects immersed in the plasma with sharp boundaries which collect of emit charged particles. Consequently, the nearby plasma evolves under abrupt imposed and/or naturally emerging conditions. There could be localized currents, different time scales for plasma species evolution, charge separation and absorbing-emitting walls. The traditional numerical schemes based on differences often transform these disparate boundary conditions into computational singularities. This is the case of models using advection-diffusion differential equations with source-sink terms (also called Fokker-Planck equations). These equations are used in both, fluid and kinetic descriptions, to obtain the distribution functions or the density for each plasma species close to the boundaries. We present a resolution method grounded on an integral advancing scheme by using approximate Green's functions, also called short-time propagators. All the integrals, as a path integration process, are numerically calculated, what states a robust grid-free computational integral method, which is unconditionally stable for any time step. Hence, the sharp boundary conditions, as the current emission from a wall, can be treated during the short-time regime providing solutions that works as if they were known for each time step analytically. The form of the propagator (typically a multivariate Gaussian) is not unique and it can be adjusted during the advancing scheme to preserve the conserved quantities of the problem. The effects of the electric or magnetic fields can be incorporated into the iterative algorithm. The method allows smooth transitions of the evolving solutions even when abrupt discontinuities are present. In this work it is proposed a procedure to incorporate, for the very first time, the boundary conditions in the numerical integral scheme. This numerical scheme is applied to model the plasma bulk interaction with a charge-emitting electrode, dealing with fluid diffusion equations combined with Poisson equation self-consistently. It has been checked the stability of this computational method under any number of iterations, even for advancing in time electrons and ions having different time scales. This work establishes the basis to deal in future work with problems related to plasma thrusters or emissive probes in electromagnetic fields.

Método multigrid algébrico: reutilização das estruturas multigrid no transporte de contaminantes

A necessidade de obter solução de grandes sistemas lineares resultantes de processos de discretização de equações diferenciais parciais provenientes da modelagem de diferentes fenômenos físicos conduz à busca de técnicas numéricas escaláveis. Métodos multigrid são classificados como algoritmos escaláveis.Um estimador de erros deve estar associado à solução numérica do problema discreto de modo a propiciar a adequada avaliação da solução obtida pelo processo de aproximação. Nesse contexto, a presente tese caracteriza-se pela proposta de reutilização das estruturas matriciais hierárquicas de operadores de transferência e restrição dos métodos multigrid algébricos para acelerar o tempo de solução dos sistemas lineares associados à equação do transporte de contaminantes em meio poroso saturado. Adicionalmente, caracteriza-se pela implementação das estimativas residuais para os problemas que envolvem dados constantes ou não constantes, os regimes de pequena ou grande advecção e pela proposta de utilização das estimativas residuais associadas ao termo de fonte e à condição inicial para construir procedimentos adaptativos para os dados do problema. O desenvolvimento dos códigos do método de elementos finitos, do estimador residual e dos procedimentos adaptativos foram baseados no projeto FEniCS, utilizando a linguagem de programação PYTHONR e desenvolvidos na plataforma Eclipse. A implementação dos métodos multigrid algébricos com reutilização considera a biblioteca PyAMG. Baseado na reutilização das estruturas hierárquicas, os métodos multigrid com reutilização com parâmetro fixo e automática são propostos, e esses conceitos são estendidos para os métodos iterativos não-estacionários tais como GMRES e BICGSTAB. Os resultados numéricos mostraram que o estimador residual captura o comportamento do erro real da solução numérica, e fornece algoritmos adaptativos para os dados cuja malha retornada produz uma solução numérica similar à uma malha uniforme com mais elementos. Adicionalmente, os métodos com reutilização são mais rápidos que os métodos que não empregam o processo de reutilização de estruturas. Além disso, a eficiência dos métodos com reutilização também pode ser observada na solução do problema auxiliar, o qual é necessário para obtenção das estimativas residuais para o regime de grande advecção. Esses resultados englobam tanto os métodos multigrid algébricos do tipo SA quanto os métodos pré-condicionados por métodos multigrid algébrico SA, e envolvem o transporte de contaminantes em regime de pequena e grande advecção, malhas estruturadas e não estruturadas, problemas bidimensionais, problemas tridimensionais e domínios com diferentes escalas.

Migração de solutos em basalto fraturado: quantificação experimental em laboratório e validação matemática

A avaliação do risco a contaminação e a escolha de técnicas de remediação de poluentes em aquíferos fraturados depende da quantificação dos fenômenos envolvidos no transporte de solutos. A geometria da fratura, usualmente caracterizada pela abertura, é o principal parâmetro que indiretamente controla o transporte nos aquíferos fraturados. A simplificação mais comum desse problema é assumir que as fraturas são um par de placas planas e paralelas, isto é, com uma abertura constante. No entanto, por causa do limitado número de trabalhos experimentais, não está esclarecida a adequabilidade do uso de uma abertura constante para simular o transporte conservativo em fraturas do Aquífero Serra Geral (ASG), Brasil. O objetivo deste trabalho é avaliar a influência da abertura de uma fratura natural do Aquífero Serra Geral sob o transporte conservativo de solutos. Uma amostra natural de basalto fraturado foi usada em um experimento hidráulico e de transporte de um traçador conservativo (escala de laboratório). O campo de abertura foi medido usando a técnica avançada, de alta resolução e tridimensional, chamada microtomografia computadorizada de raios-X. A concentração de traçador medida foi utilizada para validar uma solução analítica unidimensional da Equação de Advecção-dispersão (ADE). O desemprenho do ajuste da ADE às curvas de passagem experimentais foi avaliado para quatro diferentes tipos de aberturas constantes. Os resultados mostraram que o escoamento de água e o transporte de contaminantes pode ocorrer através de fraturas micrométricas, ocasionando, eventualmente, a contaminação do ASG. A abertura de balanço de massa é a única que pode ser chamada propriamente de \"abertura equivalente\". O uso de aberturas constantes na ADE não permitiu representar completamente o formato das curvas de passagem porque o campo de velocidade não é uniforme e intrinsicamente bidimensional. Portanto, na simulação do transporte deve-se incorporar a heterogeneidade da abertura da fratura.

Age determination of sediments of Site LO09-14 (Table 1)

A composite record (LO09-14) of three sediment cores from the subpolar North Atlantic (Reykjanes Ridge) was investigated in order to assess surface ocean variability during the last 11 kyr. The core site is today partly under the influence of the Irminger Current (IC), a branch of the North Atlantic Drift continuing northwestward around Iceland. However, it is also proximal to the Sub-Arctic Front (SAF) that may cause extra dynamic hydrographic conditions. We used statistical methods applied to the fossil assemblages of diatoms to reconstruct quantitative sea surface temperatures (SSTs). Our investigations give evidence for different regional signatures of Holocene surface oceanographic changes in the North Atlantic. Core LO09-14 reveal relatively low and highly variable SSTs during the early Holocene, indicating a weak IC and increased advection of subpolar water over the site. A mid-Holocene thermal optimum with a strong IC occurs from 7.5 to 5 kyr and is followed by cooler and more stable late Holocene surface conditions. Several intervals throughout the Holocene are dominated by the diatom species Rhizosolenia borealis, which we suggest indicates proximity to a strongly defined convergence front, most likely the SAF. Several coolings, reflecting southeastward advection of cold and ice-bearing waters, occur at 10.4, 9.8, 8.3, 7.9, 6.4, 4.7, 4.3 and 2.8 kyr. The cooling events recorded in the LO09-14 SSTs correlate well with both other surface records from the area and the NADW reductions observed at ODP Site 980 indicating a surface-deepwater linkage through the Holocene.

Stable isotopes, Mg/Ca-ratios, and biogenic opal of five sediment cores from the subarctic northwest Pacific

Based on models and proxy data it has been proposed that salinity-driven stratification weakened in the subarctic North Pacific during the last deglaciation, which potentially contributed to the deglacial rise in atmospheric carbon dioxide. We present high-resolution subsurface temperature (TMg/Ca) and subsurface salinity-approximating (d18Oivc-sw) records across the last 20,000 years from the subarctic North Pacific and its marginal seas, derived from combined stable oxygen isotopes and Mg/Ca ratios of the planktonic foraminiferal species Neogloboquadrina pachyderma (sin.). Our results indicate regionally differing changes of subsurface conditions. During the Heinrich Stadial 1 and the Younger Dryas cold phases our sites were subject to reduced thermal stratification, brine rejection due to sea-ice formation, and increased advection of low-salinity water from the Alaskan Stream. In contrast, the Bølling-Allerød warm phase was characterized by strengthened thermal stratification, stronger sea-ice melting, and influence of surface waters that were less diluted by the Alaskan Stream. From direct comparison with alkenone-based sea surface temperature estimates (SSTUk'37), we suggest deglacial thermocline changes that were closely related to changes in seasonal contrasts and stratification of the mixed layer. The modern upper-ocean conditions seem to have developed only since the early Holocene.

Sedimentology of the Reykjanes Ridge, Southeast Greenland

Marine records from the Reykjanes Ridge indicate ice sheet variations and abrupt climate changes. One of these records, ice-rafted detritus (IRD), serves as a proxy for iceberg discharges that probably indicates ice sheet fluctuations. The IRD records suggest that iceberg discharge 68,000-10,000 yr B.P. happened more frequently than the 7000- to 10,000-yr spacing of the Heinrich events. An IRD peak 67,000 to 63,000 yr B.P. further suggests that the Middle Weichselian glaciation started about 12,000 yr earlier in the North Atlantic than in the Norwegian Sea. Several later IRD events, in contrast, correlate with Norwegian Sea IRD-rich layers and imply coeval ice sheet advances in the North Atlantic and the Norwegian Sea. Coccoliths in a core from the Reykjanes Ridge show distinct peaks in species that record occasional inflow of warm surface water during the last glaciation, as previously reported from the eastern Labrador Sea. High abundances of coccoliths, together with a decrease ofNeogloboquadrina pachydermasin. and relatively low delta18O values, imply enhanced advection of the North Atlantic Current 69,000-67,000 yr B.P., 56,000-54,000 yr B.P., 35,000-33,000 yr B.P., and 26,000-23,000 yr B.P. This advection provided a regional moisture source for extension of ice sheets onto the shelf. In contrast, most of the IRD events are characterized by cold polar surface water masses indicating rapid variations in ocean surface conditions.

(Table 1) Age determination of sediment core MD02-2589

Sortable silt mean grain sizes together with oxygen and carbon isotopic data produced on the benthic foraminiferal species Fontbotia wuellerstorfi are used to construct high-resolution records of near-bottom flow vigour and deep water ventilation at a core site MD02-2589 located at 2660 m water depth on the southern Agulhas Plateau. The results suggest that during glacial periods (marine oxygen isotope stages 2 and 6, MIS 2 and MIS 6, respectively), there was a persistent contribution of a well-ventilated water mass within the Atlantic to Indian oceanic gateway with a d13C signature similar to present-day Northern Component Water (NCW), e.g., North Atlantic Deep Water (NADW). The records of chemical ventilation and near-bottom flow vigor reflect changes in the advection of northern source waters and meridional variability in the location of the Antarctic Circumpolar Current and its associated fronts. We suggest that during Termination II (TII), changes in chemical ventilation are largely decoupled from near-bottom physical flow speeds. A mid-TII climate optimum is associated with a low-flow speed plateau concurrent with a period of increased ventilation shown in the benthic d13C of other Southern Ocean records but not in our benthic d13C of MD02-2589. The climate optimum is followed by a period of southern cooling around 128 ka coincident with a stronger influence of NCW to interglacial levels at around 124 ka. All proxy records show a near synchronous and rapid shift during the transition from MIS 5a-4 (73 ka). This large event is attributed to a rapid decrease in NADW influence and replacement over the Agulhas Plateau by southern source waters.

Age, pollen and spores of ODP Site 175-1082

The present-day condition of bipolar glaciation characterized by rapid and large climate fluctuations began at the end of the Pliocene with the intensification of the Northern Hemisphere continental glaciations. The global cooling steps of the late Pliocene have been documented in numerous studies of Ocean Drilling Program (ODP) sites from the Northern Hemisphere. However, the interactions between oceans and between land and ocean during these cooling steps are poorly known. In particular, data from the Southern Hemisphere are lacking. Therefore I investigated the pollen of ODP Site 1082 in the southeast Atlantic Ocean in order to obtain a high-resolution record of vegetation change in Namibia between 3.4 and 1.8 Ma. Four phases of vegetation development are inferred that are connected to global climate change. (1) Before 3 Ma, extensive, rather open grass-rich savannahs with mopane trees existed in Namibia, but the extension of desert and semidesert vegetation was still restricted. (2) Increase of winter rainfall dependent Renosterveld-like vegetation occurred between 3.1 and 2.2 Ma connected to strong advection of polar waters along the Namibian coast and a northward shift of the Polar Front Zone in the Southern Ocean. (3) Climatically induced fluctuations became stronger between 2.7 and 2.2 Ma and semiarid areas extended during glacial periods probably as the result of an increased pole-equator thermal gradient and consequently globally enhanced atmospheric circulation. (4) Aridification and climatic variability further increased after 2.2 Ma, when the Polar Front Zone migrated southward and the influence of Atlantic moisture brought by the westerlies to southern Africa declined. It is concluded that the positions of the frontal systems in the Southern Ocean which determine the locations of the high-pressure cells over the South Atlantic and the southern Indian Ocean have a strong influence on the climate of southern Africa in contrast to the climate of northwest and central Africa, which is dominated by the Saharan low-pressure cell.