Fractional calculus has been introduced as an e cient tool for modeling physical phenomena, thanks to its memory and hereditary properties. For example, fractional models have been successfully used to describe anomalous di↵usion processes such as contaminant transport in soil, oil flow in porous media, and groundwater flow. These models capture important features of particle transport such as particles with velocity variations and long-rest periods. Mathematical modeling of physical phenomena requires the identification of pa- rameters and variables from available measurements. This is referred to as an inverse problem. In this work, we are interested in studying theoretically and numerically inverse problems for space Fractional Advection-Dispersion Equation (FADE), which is used to model solute transport in porous media. Identifying parameters for such an equa- tion is important to understand how chemical or biological contaminants are trans- ported throughout surface aquifer systems. For instance, an estimate of the di↵eren- tiation order in groundwater contaminant transport model can provide information about soil properties, such as the heterogeneity of the medium. Our main contribution is to propose a novel e cient algorithm based on modulat-ing functions to estimate the coe cients and the di↵erentiation order for space FADE, which can be extended to general fractional Partial Di↵erential Equation (PDE). We also show how the method can be applied to the source inverse problem. This work is divided into two parts: In part I, the proposed method is described and studied through an extensive numerical analysis. The local convergence of the proposed two-stage algorithm is proven for 1D space FADE. The properties of this method are studied along with its limitations. Then, the algorithm is generalized to the 2D FADE. In part II, we analyze direct and inverse source problems for a space FADE. The problem consists of recovering the source term using final observations. An analytic solution for the non-homogeneous case is derived and existence and uniqueness of the solution are established. In addition, the uniqueness and stability of the inverse problem is studied. Moreover, the modulating functions-based method is used to solve the problem and it is compared to a standard Tikhono-based optimization technique.

This work determines a suitable value for a dispersion coefficient to be used in the One-Dimensional Advection-Dispersion equation to model dispersion within sewer collection systems. Dispersion coefficients for sewer systems have only recently begun to be studied, and there is not yet an established value that is commonly accepted. The work described in this paper aimed, through observational study, to find a suitable value to be used. Salt tracers were placed in two separate reaches of sewer line. The first line studied was a straight, linear reach of sewer that included three manholes. The tracer was placed in the first manhole and the conductivity was measured at the two consecutive manholes downstream. These measurements were compared to a model developed using the 1D Advection-Dispersion Equation. The flow information and sewer network geometry was used in the model and the dispersion coefficient was adjusted to find a best fit. It was found that a value of 0.18 m2/s for the dispersion coefficient provided the best statistical match. The next reach of sewer that was studied was a reach with a 90 degree angle. This section was chosen to observe the effect that mixing has on dispersion, because of the change in direction of flow. The same procedure was applied, and an optimal dispersion coefficient of 0.22 m2/s was found. These values represent optimal dispersion coefficients under a specific set of conditions. It should not be assumed that they will provide accurate results in all circumstances, but are rather a base point for average flows under dry, stable conditions. Using these values inferences can begin to be made about dispersion characteristics throughout the entire sewer network. This can lead to specific engineering applications, and well as applications in other fields of study.

Les évaluations de vulnérabilité spécifiques pour les ressources d'eau souterraine sont des méthodes SIG qui établissent des indices qualitatifs spatiaux qui déterminent la sensibilité à l'infiltration de polluants superficiels. D'autre part, les fonctions de transfert, comme la Distribution de Temps de Séjour (DTS), sont utilisées pour prévoir le changement temporel de qualité de l'eau au forage, mais ils ne permettent pas de relier les concentrations aux pratiques observées sur le bassin versant. Basé sur une approche analytique, un modèle pour le transport souterrain des produits phytosanitaires liant la vulnérabilité spécifique SIG à la DTS a été développée. L’idée étant de relier les notions de vulnérabilité aux méthodes de prédiction des chroniques de la qualité des eaux. L'outil estime la qualité de l'eau à partir de l'ensemble de données des cartes de vulnérabilité. La validation de cette méthode de vulnérabilité spécifique est possible à partir des données de suivi de la qualité de l’eau au forage. Une formulation de paramètres équivalents a été proposée pour prendre en compte les caractéristiques hydrodynamiques des compartiments de sol (zone saturée et zone non saturée). Une validation théorique de l'approche est faite à l’aide de modèles souterrain à différence finie : HYDRUS et MODFLOW. Une application a été réalisée sur l'aquifère karstique du Val d'Orléans. Elle a permis de déterminer la DTS des concentrations en pesticides dans l'eau souterraine et a mis en évidence la présence non négligeable des métabolites du métazachlore dans le forage. En même temps, le modèle DTS, lié au SIG rend possible la localisation des zones contributrices dans le bassin versant. / The specific vulnerability estimations for the groundwater resources are GIS methods that establish spatial qualitative indices which determine the sensitivity of infiltration from surface contaminants. On the other hand, the transfer functions, using the Residence Time Distribution (RTD), are used to predict temporal water quality change in a borehole, but they do not integrate the spatial variability of the land use. Based on an analytic (advection / dispersion equation) approach, a simple GIS-linked RTD model for groundwater transport has been developed. The tool estimates the water quality from the vulnerability map dataset. This method enables to validate the specific vulnerability maps with the water quality monitoring at the borehole. It links the impacts of land use with the temporal evolution of the water quality. A equivalent formulation parameters is proposed to take into account the hydrodynamic characteristics of the soil compartments (unsaturated zone and Saturated Zone). A theoretical validation of the approach is made from finite-difference groundwater models: HYDRUS and MODFLOW. Also, an application of the RTD compilation was realized on the Val d’Orléans karstic aquifer. This last methodology allowed to determine the RTD of pesticides into the groundwater and highlighted the not insignificant presence of the metabolite of the metazachlor in the groundwater drilling. At the same time, the GIS-linked RTD model makes possible the localization of the contributing zones in the watershed.

The classical Fickian model for solute transport in porous media cannot correctly predict the spreading (the dispersion) of contaminant plumes in a heterogeneous subsoil unless its structure is completely characterized. Although the required precision is outside the reach of current field characterization methods, the classical Fickian model remains the most widely used model among practitioners. Two approaches can be adopted to solve the effect of physical heterogeneity on transport. First, upscaling methods allow one to compute “apparent” scale-dependent parameters to be used in the classical Fickian model. In the second approach, upscaled (non-Fickian) transport equations with scale-independent parameters are used. This research aims at comparing upscaling methods for Fickian transport parameters with non-Fickian upscaled transport equations, and evaluate their capabilities to predict solute transport in heterogeneous media. The models were tested using simplified numerical examples (perfectly stratified aquifers and bidimensional heterogeneous media). Hypothetical lognormal permeability fields were investigated, for different values of variance, correlation length and anisotropy ratio. Examples exhibiting discrete and multimodal permeability distributions were also investigated using both numerical examples and a physical laboratory experiment. It was found that non-Fickian transport equations involving fractional derivatives have higher upscaling capabilities regarding the prediction of contaminant plume migration and spreading, although their key parameters can only be inferred from inverse modelling of test data.

A cross-section averaged Advection-Dispersion equation (ADE) model was developed to simulate the transport of fertilizer in furrow irrigation. The advection and dispersion processes were solved separately by implementing the method of the characteristics with cubic spline interpolation (and natural boundary condition) and weighted finite difference scheme respectively. A zero-flux boundary condition during advance and an advective gradient at the downstream end of an open furrow were established. Local pseudo-steady state was assumed in order to apply Fischer's longitudinal dispersion equation under non-uniform and unsteady furrow flow conditions. Also, several parameters were used to evaluate the ADE model and fertigation performance.A field tracer experiment in two types of downstream-end furrow and two treatments was conducted and described. Infiltration and roughness parameters were calibrated by implementing a volume balance approach. The calibrated parameters were used as input data to run the surface irrigation model (SRFR). The roughness coefficient was 0.045 for wheel and 0.055 for non-wheel furrow treatment for bare soil. The root mean square error (RMSE) comparing the computed and observed infiltrated volume was in the range of 0.09-0.38 m3. The close match between simulated and observed data indicates an acceptable calibration. Pulses of fertilizer injected at the head end of four furrows each having unique management characteristics were simulated satisfactorily during the entire duration of the irrigation event. The constant value of the longitudinal dispersion coefficient was 1 m2 min-1 and yielded an acceptable space-time evolution of the pulses of tracer injected. Similar results for the dispersion coefficient were obtained with Fischer's equation in non-uniform and unsteady stream flow conditions in the furrow. An evaluation of several fertigation strategies for furrow systems indicated that fertigation by pulses could help reduce leaching and runoff losses in surface irrigation systems.

No description available.

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. / The contamination risk assessment and the choice of suitable cleanup techniques for pollutants in fractured rock depends on the quantification of the transport phenomena. Fracture geometry often described by the apertures is the major parameter that controls indirectly solute transport in fractured rock. The simplest approach is describing fractures as a pair of smooth parallel plates with constant aperture. However, there is a lack of information about the suitability for using a constant aperture for the conservative solute transport prediction in a single fracture of Serra Geral Aquifer (SGA), Brazil. The aim of this work is to evaluate the effect of aperture variability in a natural single rough-walled fracture of Serra Geral Aquifer on conservative solute transport. A natural core of fractured basalt was used for a hydraulic and tracer tests (laboratory scale). The aperture field was measured using the advanced, high-resolution and tridimensional technique X-ray computed tomography. The measured tracer concentration was validated by means of an analytical solution of the Advection-dispersion Equation (ADE). The ADE fit performance was measured against experimental breakthrough curves for four distinct kind of constant apertures. It was found that water flow and solute transport can take place through micrometric fractures, eventually leading the SGA contamination. Results show that the mass balance aperture is the only appropriate \"equivalent aperture\" for describing solute transport in a single rough-walled fracture. The results showed that ADE is not appropriate for modeling the complete behavior of experimental breakthrough curves because of the dimensional non-uniform velocity field. Therefore, the aperture heterogeneity must be considered in solute transport simulation.

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. / The contamination risk assessment and the choice of suitable cleanup techniques for pollutants in fractured rock depends on the quantification of the transport phenomena. Fracture geometry often described by the apertures is the major parameter that controls indirectly solute transport in fractured rock. The simplest approach is describing fractures as a pair of smooth parallel plates with constant aperture. However, there is a lack of information about the suitability for using a constant aperture for the conservative solute transport prediction in a single fracture of Serra Geral Aquifer (SGA), Brazil. The aim of this work is to evaluate the effect of aperture variability in a natural single rough-walled fracture of Serra Geral Aquifer on conservative solute transport. A natural core of fractured basalt was used for a hydraulic and tracer tests (laboratory scale). The aperture field was measured using the advanced, high-resolution and tridimensional technique X-ray computed tomography. The measured tracer concentration was validated by means of an analytical solution of the Advection-dispersion Equation (ADE). The ADE fit performance was measured against experimental breakthrough curves for four distinct kind of constant apertures. It was found that water flow and solute transport can take place through micrometric fractures, eventually leading the SGA contamination. Results show that the mass balance aperture is the only appropriate \"equivalent aperture\" for describing solute transport in a single rough-walled fracture. The results showed that ADE is not appropriate for modeling the complete behavior of experimental breakthrough curves because of the dimensional non-uniform velocity field. Therefore, the aperture heterogeneity must be considered in solute transport simulation.

Zlomkový kalkulus je matematická disciplína zabývající se vlastnostmi derivací a integrálů neceločíselných řádů (nazývaných zlomkové derivace a integrály, zkráceně diferintegrály) a metodami řešení diferenciálních rovnic obsahujících zlomkové derivace neznámé funkce (tzv. zlomkovými diferenciálními rovnicemi). V této práci představujeme standardní přístupy k definicím zlomkového kalkulu a důkazy některých základních vlastností diferintegrálů. Dále uvádíme krátký přehled metod řešení některých lineárních zlomkových diferenciálních rovnic a vymezujeme hranice jejich použitelnosti. Na závěr si všímáme některých fyzikálních aplikací zlomkového kalkulu.

Devido aos riscos de contaminação dos recursos naturais solo e água, ao alto custo, ao tempo e ao esforço humano nas investigações de campo, os modelos matemáticos, aliados às técnicas numéricas e aos avanços computacionais, constituem uma ferramenta importante na previsão do deslocamento de solutos, contribuindo assim, para o controle de alterações ambientais. No Brasil, a modelação de fluxo e transporte de solutos na zona não-saturada é voltada, quase que exclusivamente, aos problemas relacionados às atividades agrícolas. Entretanto, tão importante quanto a problemática dos produtos químicos nas atividades agrícolas é a questão de poluição e contaminação do solo e da água por chorume, gerado pelos resíduos sólidos domiciliares. Neste trabalho, é desenvolvido e validado um modelo computacional unidimensional para simulação de fluxo e transporte de solutos na zona não-saturada do solo. O modelo matemático é dado pela equação diferencial parcial não-linear de Richards, que rege o movimento de água no solo, e a equação diferencial parcial linear de advecção-dispersão, do transporte de solutos, acompanhadas das condições iniciais e de contorno. A equação de Richards é dada em função do potencial matricial da água e a equação de transporte de solutos estima a evolução temporal da concentração de solutos no perfil do solo. Devido à dificuldade de se obter soluções analíticas destas equações, são resolvidas numericamente pelo método de elementos finitos. As referidas equações são resolvidas utilizando-se malhas uniformes inicialmente. Com a finalidade de obter simulações mais eficientes, a um custo computacional reduzido, é empregada a adaptatividade com refinamento h na malha de elementos finitos. A função interpolação polinomial utilizada é de grau 2 ou maior que garante a conservação de massa. Na equação de Richards, a derivada temporal é aproximada por um quociente de diferença finita e é aplicado o esquema de Euler explícito e na equação de advecção-dispersão, é aproximada por um quociente de diferença finita, aplicando-se o esquema de Euler implícito, devido à linearidade da equação. O sistema operacional é o Linux Ubuntu 32 bits, o ambiente de programação é o PZ, escrito em linguagem de programação C++. Na validação do modelo, utilizam-se dados disponíveis na literatura. Os resultados são comparados, utilizando-se malhas uniformes e malhas adaptativas com refinamento h. Usando-se as malhas uniformes para o problema de Richards e de transporte de potássio, o tempo de execução é de 22 minutos e a memória utilizada de 6164 Kb. Com as malhas adaptadas, o tempo de execução é de 3 minutos e 27 segundos, consumindo 5876 Kb de memória. Houve, portanto, uma redução de 84,32% no tempo de execução, usando-se malhas adaptativas. A utilização da função interpolação polinomial de grau 2 ou maior e o refinamento h, permitem uma boa concordância do modelo na comparação com soluções disponíveis na literatura. / Due to the risks of contamination of soil and water resources, the high cost, time and human effort in the field investigations, the mathematical models, combined with numerical techniques and computational advances, are important tools in forecasting the movement of solutes thereby contributing to the control of environmental alteration. In Brazil, modeling of flow and solute transport in the unsaturated zone is focused, almost exclusively, on problems related to agricultural activities. However, as important as the problematical of chemicals products in agricultural activities is the issue of pollution and contamination of soil and water by leachate, generated by municipal solid wastes. In this work, an one-dimensional computational model for simulation of flow and solute transport in the unsaturated soil has been developed and validated. The mathematical model is given by the Richards\'s non-linear partial differential equation, which determines the movement of water in the soil, and the advection-dispersion linear partial differential equation, of the solute transport, together with initial and boundary conditions. The Richards equation is a function of the water pressure head and the solute transport equation estimate the temporal evolution of the solutes concentration in the soil profile. Due to the difficulty of obtaining analytical solutions of these equations, they are solved numerically using the finite element method. The governing equations are solved using initially a uniform mesh. In order to obtain more efficient simulations with low computational cost, adaptativity with h refinement on the finite element mesh is implemented. The interpolation function is of degree two or higher, assuring mass conservation. In Richards\' equation, the temporal derivative is approximated by Euler explicit finite difference. For the advection-dispersion equation, due to the linearity of the equation, an implicit finite difference scheme is used. The code is written in the programming language C++ based on the programming environment PZ using operating system Linux Ubuntu 32 bit. Model results are validated in comparison with data available in the literature. The results are evaluated using uniform meshes and with h refinement adaptive mesh. Using the uniform meshes for the problem of Richards and transport of potassium, the running time is 22 minutes and 6164 Kb of memory is used. With the adapted meshes, the execution time is 3 minutes and 27 seconds, consuming 5,876 Kb of memory. Therefore there was a reduction of 84.32% in execution time, using adaptive meshes. The interpolation function with degree two or higher and the h refinement, with reduction of the computation time, showed a good agreement in comparison with the literature.