USFKAD is an encoded expert system for the eigenfunction expansion of solutions to the wave, diffusion, and Laplace equations: both homogeneous and nonhomogenous; one, two, or three dimensions; Cartesian, cylindrical, or spherical coordinates; Dirichlet, Neumann, Robin, or singular boundary conditions; in time, frequency, or Laplace domain. The user follows a menu to enter his/her choices and the output is a LaTeX file containing the formula for the solution together with the transcendental equation for the eigenvalues (if necessary) and the projection formulas for the coefficients. The file is suitable for insertion into a book or journal article, and as a teaching aid. Virtually all cases are covered, including the Mellin, spherical harmonic, Bessel, modified Bessel, spherical Bessel, Dini, Hankel, Weber, MacDonald, and Kantorovich-Lebedev expansions, mixed spectrum, and rigid body modes.

L’objectif de ce travail est le développement d’un modèle et d’une méthode numérique adaptés à la simulation duruissellement d’eau de pluie sur des surfaces agricoles. Pour cela, nous utilisons un système d’équations aux dérivéespartielles pour les eaux peu profondes : le système de Saint Venant. La pluie et l’infiltration y sont pris en compte parl’ajout de termes source et le couplage avec un modèle d’infiltration. Le système étant hyperbolique et conservatifnous choisissons d’utiliser un schéma aux volumes finis avec reconstruction hydrostatique. Avec cette reconstructionnous obtenons un schéma permettant de traiter les équilibres stationnaires et les interfaces sec/mouillé dues auxévénements pluvieux. Tout d’abord, nous effectuons une comparaison systématique de différents flux numériques,de différentes reconstructions d’ordre deux (MUSCL et ENO modifiées) et de différentes méthodes de traitementdes frottements sur des solutions analytiques unidimensionnelles. Ces comparaisons nous permettent de choisirune méthode numérique adaptée à la simulation du ruissellement qui en outre capture les phénomènes de typeroll-waves. Ensuite nous nous intéressons au traitement de la pluie et de l’infiltration à l’aide du modèle de Green-Ampt. Cette méthode généralisée en dimension deux est validée sur des résultats expérimentaux obtenus à l’INRAd’Orléans et à l’IRD. La méthode numérique choisie et validée a été implémentée dans FullSWOF_2D un logiciellibre écrit en C++ pour la simulation de ruissellement d’eau de pluie sur des surfaces agricoles. / In this work, we aim at developping both a model and a numerical method adapted to the simulation of rainfallrunoff on agricultural fields. The model used is a system of partial differential equations for shallow water flows :namely the system of Saint Venant. Rain and infiltration are treated as source terms and with a coupled model ofinfiltration. Because of the hyperbolicity and conservativity properties, we are using a finite volume scheme with thehydrostatic reconstruction. This reconstruction allows us to deal with stationary equilibria and wet/dry transitionsdu to rainfall events. First, a systematic comparison on analytic solutions is achieved between various numericalfluxes, two second order reconstructions (a MUSCL and a modified ENO reconstructions) and different numericaltreatment of the friction term. This allows us to choose a numerical method adapted to runoff simulation. Moreoverit is able to catch roll waves phenomenons. Then we are interested in the rain and infiltration treatment. Infiltrationis modeled thanks to the Green Ampt model. This method is thereafter generalized to two dimensions and validatedon experimental results obtained by the INRA of Orléans and the IRD. The chosen numerical method has beenimplemented in an C++ open-source software for rainfall runoff on agricultural fields simulations : FullSWOF_2D.

The exploitation of unconventional reservoirs goes hand in hand with the practice of hydraulic fracturing and, with an ever increasing demand in energy, this practice is set to experience significant growth in the coming years. Sophisticated analytic models are needed to accurately describe fluid flow in a hydraulic fracture and the problem has been approached from different directions in the past 3 decades - starting with the use of line-source functions for the infinite conductivity case, followed by the application of Laplace Transforms and the Boundary-Element Method for the finite-conductivity case. This topic remains an active area of research and, for the more complicated physical scenarios such as multiple transverse fractures in ultra-tight reservoirs, answers are presently being sought. Fractal theory has been successfully applied to pressure transient testing, albeit with an emphasis on the effects of natural fractures in pressure-rate behavior. In this work, we begin by performing a rigorous analytical and numerical study of the Fractal Diffusivity Equation and we show that it is more fundamental than the classic linear and radial diffusivity equations. Subsequently, we combine the Fractal Diffusivity Equation with the Trilinear Flow Model, culminating in a new semi-analytic solution for flow in a finite-conductivity vertical fracture which we name the "Fractal-Fracture Solution". This new solution is instantaneous and has an overall accuracy of 99.7%, thus making it comparable to the Trilinear Pseudoradial Solution for practical purposes. It may be used for pressure transient testing and reservoir characterization of hydrocarbon reservoirs being produced by a vertically fractured well. Additionally, this is the first time that fractal theory is used in fluid flow in porous media to address a problem not related to reservoir heterogeneity. Ultimately, this work is a demonstration of the untapped potential of fractal theory; our approach is very flexible and we believe that the same methodology may be extended to develop new reservoir flow solutions for pressing problems that the industry currently faces.