Groundwater reactive transport models are an essential tool for the analysis of coupled geochemical processes in Earth Systems. On a molecular level, chemical reactions are the result of collision and subsequent recombination of the molecules present in the system. One way to deal with reactive transport is using a continuum mechanics approach; for example, in a well-mixed system, reactions are a function of reactant concentrations. An alternative is to use a Lagrangian approach, by tracking all molecules and specifying reactions in terms of collision theory. A computationally efficient family of methods to solve the reactive transport problem is based on Particle Tracking (PT). While the number of particles in most realistic applications is in the order of 10^6 í¯€-10^9, the number of molecules even in diluted systems might be in the order of fractions of the Avogadro number. Thus, each particle actually represents a group of potentially reactive molecules. The use of a low number of particles may result not only in loss of accuracy, but also may lead to an improper reproduction of the mixing process, limited by diffusion. Recent works have used this effect as a proxy to model incomplete mixing in porous media. In this work, we propose using a Kernel Density Estimation (KDE) of the concentrations that allows getting the expected results for a well-mixed solution with a limited number of particles. The idea consists of treating each particle as a sample drawn from the pool of molecules that it represents; this way, the actual location of a tracked particle is seen as a sample drawn from the density function of the location of molecules represented by that given particle, rigorously represented by a kernel density function. The probability of reaction can be obtained by combining the kernels associated to two potentially reactive particles. We demonstrate that the observed deviation in the reaction vs time curves in numerical experiments reported in the literature could be attributed to a deficient estimation of the concentrations based on particles reconstruction, and not to the occurrence of true incomplete mixing. We further explore the evolution of the kernel size with time, linking it to the diffusion process. Our results show that KDEs are powerful tools to improve reactive transport simulations, and indicates that incomplete mixing in diluted systems should be modeled based on alternative mechanistic models and not on a limited number of particles.

The novel coronavirus has already spread to almost every country in the world and has infected over 3 million people. To understand the transmission mechanism of this highly contagious virus, it is necessary to study the potential factors, including meteorological conditions. Here, we present a machine learning approach to study the effect of temperature, humidity and wind speed on the number of infected people in the three most populous autonomous communities in Spain. We find that there is a moderate inverse correlation between temperature and the daily number of infections. This correlation manifests for temperatures recorded up to 6 days before the onset, which corresponds well to the known mean incubation period of COVID-19. We also show that the correlation for humidity and wind speed is not significant.

Random walk particle tracking methods are a computationally efficient family of methods to solve reactive transport problems. While the number of particles in most realistic applications is in the order of 10^6 - 10^9, the number of reactive molecules even in diluted systems might be in the order of fractions of the Avogadro number. Thus, each particle actually represents a group of potentially reactive molecules. The use of a low number of particles may result not only in loss of accuracy, but also may lead to an improper reproduction of the mixing process, limited by diffusion. Recent works have used this effect as a proxy to model incomplete mixing in porous media. The main contribution of this thesis is to propose a reactive transport model using a Kernel Density Estimation (KDE) of the concentrations that allows getting the expected results for a well-mixed solution with a limited number of particles. The idea consists of treating each particle as a sample drawn from the pool of molecules that it represents; this way, the actual location of a tracked particle is seen as a sample drawn from the density function of the location of molecules represented by that given particle, rigorously represented by a kernel density function. The probability of reaction can be obtained by combining the kernels associated with two potentially reactive particles. We demonstrate that the observed deviation in the reaction vs time curves in numerical experiments reported in the literature could be attributed to the statistical method used to reconstruct concentrations (fixed particle support) from discrete particle distributions, and not to the occurrence of true incomplete mixing. We further explore the evolution of the kernel size with time, linking it to the diffusion process. Our results show that KDEs are powerful tools to improve computational efficiency and robustness in reactive transport simulations, and indicates that incomplete mixing in diluted systems should be modeled based on alternative mechanistic models and not on a limited number of particles. Motivated by this potential, we extend the KDE model to simulate nonlinear adsorption which is a relevant process in many fields, such as product manufacturing or pollution remediation in porous materials. We show that the proposed model is able to reproduce the results of the Langmuir and Freundlich isotherms and to combine the features of these two classical adsorption models. In the Langmuir model, it is enough to add a finite number of sorption sites of homogeneous sorption properties, and to set the process as the combination of the forward and the backward reactions, each one of them with a pre-specified reaction rate. To model the Freundlich isotherm instead, typical of low to intermediate range of solute concentrations, there is a need to assign a different equilibrium constant to each specified sorption site, provided they are all drawn from a truncated power-law distribution. Both nonlinear models can be combined in a single framework to obtain a typical observed behavior for a wide range of concentration values. This approach opens up a new way to predict and control an adsorption-based process using a particle-based method with a finite number of particles. Finally, by classifying the particles to mobile and immobile states and employing transition probabilities between these two states, we take into account the porosity of the diluted system in the KDE model. The state of a particle is an attribute that defines the domain at which the particle is present at a given time within the porous medium. The transition probabilities are controlled by two parameters which implicitly determine the porosity. Simulations results show a good agreement with the analytical solutions of complete and incomplete mixing solutions, independent of the number of particles. Random walk particle tracking methods son una familia de métodos computacionalmente eficientes para resolver problemas de transporte reactivo. Mientras que el número de partículas en las aplicaciones más realistas es del orden de 10 ^ 6 - 10 ^ 9, el número de moléculas reactivas incluso en sistemas diluidos podría ser del orden de las fracciones del número de Avogadro. Así, cada partícula representa en realidad un grupo de moléculas potencialmente reactivas. El uso de un número bajo de partículas puede resultar no sólo en pérdida de precisión, sino que también puede conducir a una reproducción inadecuada del proceso de mezcla, limitado por la difusión. Trabajos recientes han utilizado este efecto como una aproximación para modelar la mezcla incompleta en medios porosos. La principal contribución de esta tesis es proponer un modelo de transporte reactivo utilizando una KDE (Kernel Density Estimation) de las concentraciones que permite obtener los resultados esperados para una solución bien mezclada con un número limitado de partículas. La idea consiste en tratar cada partícula como una muestra extraída del conjunto de moléculas que representa; de esta manera, la localización real de una partícula seguida se ve como una muestra extraída de la función de densidad de la localización de moléculas representadas por esa partícula dada, rigurosamente representada por una función de densidad de núcleo. La probabilidad de reacción puede obtenerse combinando los granos asociados con dos partículas potencialmente reactivas. Demostramos que la desviación observada en las curvas de reacción frente a tiempo en experimentos numéricos reportados en la literatura podría atribuirse al método estadístico utilizado para reconstruir las concentraciones (soporte de partículas fijas) a partir de distribuciones de partículas discretas y no a la aparición de mezclas verdaderamente incompletas. Nuestros resultados muestran que los KDE son potentes herramientas para mejorar la eficiencia computacional y la robustez en simulaciones de transporte reactivo, e indica que la mezcla incompleta en sistemas diluidos debe modelarse basándose en modelos mecánicos alternativos y no en un número limitado de partículas. Motivados por este potencial, ampliamos el modelo KDE para simular la adsorción no lineal, que es un proceso relevante en muchos campos, como la fabricación de productos o la remediación de la contaminación en materiales porosos. Se muestra que el modelo propuesto es capaz de reproducir los resultados de las isotermas de Langmuir y Freundlich y de combinar las características de estos dos modelos clásicos de adsorción. En el modelo de Langmuir, basta con añadir un número finito de sitios de sorción de propiedades de sorción homogéneas y establecer el proceso como la combinación de las reacciones hacia adelante y hacia atrás, cada una de ellas con una reacción preespecificada tarifa. Para modelar la isoterma de Freundlich en su lugar, típica de las concentraciones de soluto de bajo a intermedio, es necesario asignar una constante de equilibrio diferente a cada sitio de sorción especificado, siempre que se extraigan de una distribución truncada de la ley de potencia. Ambos modelos no lineales se pueden combinar en un marco único para obtener un comportamiento observado típico para una amplia gama de valores de concentración. Por último, al clasificar las partículas en estados móviles e inmóviles y emplear probabilidades de transición entre estos dos estados, se tiene en cuenta la porosidad del sistema diluido en el modelo KDE. Las probabilidades de transición son controladas por dos parámetros que determinan implícitamente la porosidad. Los resultados de las simulaciones muestran un buen acuerdo con las soluciones analíticas de soluciones de mezcla completas e incompletas, independientemente del número de partículas.