The Dykstra-Parsons model describes layer 1-D oil displacement by water in multilayered reservoirs. The main assumptions of the model are: piston-like displacement of oil by water, no crossflow between the layers, all layers are individually homogeneous, constant total injection rate, and injector-producer pressure drop for all layers is the same. Main drawbacks of Dykstra-Parsons method are that it does not take into account Buckley-Leverett displacement and the possibility of different oil-water relative permeability for each layer. A new analytical model for layer 1-D oil displacement by water in multilayered reservoir has been developed that incorporates Buckley-Leverett displacement and different oilwater relative permeability and water injection rate for each layer (layer injection rate varying with time). The new model employs an extensive iterative procedure, thus requiring a computer program. To verify the new model, calculations were performed for a two-layered reservoir and the results compared against that of numerical simulation. Cases were run, in which layer thickness, permeability, oil-water relative permeability and total water injection rate were varied. Main results for the cases studied are as follows. First, cumulative oil production up to 20 years based on the new model and simulation are in good agreement. Second, model water breakthrough times in the layer with the highest permeability-thickness product (kh) are in good agreement with simulation results. However, breakthrough times for the layer with the lowest kh may differ quite significantly from simulation results. This is probably due to the assumption in the model that in each layer the pressure gradient is uniform behind the front, ahead of the front, and throughout the layer after water breakthrough. Third, the main attractive feature of the new model is the ability to use different oil-water relative permeability for each layer. However, further research is recommended to improve calculation of layer water injection rate by a more accurate method of determining pressure gradients between injector and producer.

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Dans cette thèse, notre but est de donner un cadre numérique général pour approcher les solutions des problèmes du transport optimal (TO). L’idée générale est d’introduire une régularisation entropique du problème initial. Le problème régularisé correspond à minimiser une entropie relative par rapport à une mesure de référence donnée. En effet, cela équivaut à trouver la projection d’un couplage par rapport à la divergence de Kullback-Leibler. Cela nous permet d’utiliser l’algorithme de Bregman/Dykstra et de résoudre plusieurs problèmes variationnels liés au TO. Nous nous intéressons particulièrement à la résolution des problèmes du transport optimal multi-marges (TOMM) qui apparaissent dans le cadre de la dynamique des fluides (équations d’Euler incompressible à la Brenier) et de la physique quantique (la théorie de fonctionnelle de la densité ). Dans ces cas, nous montrons que la régularisation entropique joue un rôle plus important que de la simple stabilisation numérique. De plus, nous donnons des résultats concernant l’existence des transports optimaux (par exemple des transports fractals) pour le problème TOMM. / In this thesis we aim at giving a general numerical framework to approximate solutions to optimal transport (OT) problems. The general idea is to introduce an entropic regularization of the initialproblems. The regularized problem corresponds to the minimization of a relative entropy with respect a given reference measure. Indeed, this is equivalent to find the projection of the joint coupling with respect the Kullback-Leibler divergence. This allows us to make use the Bregman/Dykstra’s algorithm and solve several variational problems related to OT. We are especially interested in solving multi-marginal optimal transport problems (MMOT) arising in Physics such as in Fluid Dynamics (e.g. incompressible Euler equations à la Brenier) and in Quantum Physics (e.g. Density Functional Theory). In these cases we show that the entropic regularization plays a more important role than a simple numerical stabilization. Moreover, we also give some important results concerning existence and characterization of optimal transport maps (e.g. fractal maps) for MMOT .