The objective of this work is to parallelize, using the Application Programming Interface (API) OpenMP (Open Multi-Processing) and Intel Xeon Phi coprocessor based on Intel Many Integrated Core (MIC) architecture, the numerical method used to solve the algebraic system resulting from the discretization of the differential partial equation (PDE) that describes the single-phase flow in an oil reservoir. The set of governing equations are the continuity equation and the Darcy's law. The Hydraulic Diffusivity Equation (HDE), for the unknown pressure, is obtained from these fundamental equations and it is discretized by means of the Finite Difference Method (FDM) along with a time implicit formulation. Different numerical tests were performed to study the computational efficiency of the parallelized versions of Conjugate Gradient, BiConjugate Gradient and BiConjugate Gradient Stabilized methods. Speedup results were considered to evaluate the performance of the parallel algorithms for the horizontal well simulation case. The methodology also included a sensibility analysis for different production scenarios including variations on the permeability, formation-valuefactor, well length and production rate.

Wellbore flow and its coupling with porous media has been an active field of research, mainly because wellbore hydraulics (especially in complex geometry wells, like horizontal or multilateral) plays an important role in the pressure and production profiles. An inherent difficulty is associated to the multiscale nature of the flow problem: the cell hosting the well is generally much larger than wellbore radius. As a consequence, average pressure in a cell is not a good approximation of the well pressure. Moreover, at the interface between well and reservoir there is a change in flow pattern, from porous media to pipe flow. In this work a well-reservoir coupling technique is developed and applied to the numerical simulation of horizontal wells in gas reservoirs. This model considers wellbore hydraulics and storage.We consider the 3D single-phase isothermal flow of a compressible fluid in porous media and 1D single-phase isothermal flow of a compressible fluid in the wellbore. In both regions, mass conservation, momentum balance and real gas equation of state are the governing equations. Conservative forms are used for continuity and momentum. Flow continuity and pressure equilibrium are considered at the interface between well and reservoir. The nonlinear partial differential equations can be rewritten to allow the coupling of the pressure field between the porous media (reservoir) and the free media (well) by means of an implicit formulation for the finite difference method presented. A hybrid grid technique was applied to improve the description of the flow close to the wellbore. In addition, pressure in reservoir and wellbore are solved simultaneously, while velocities in wellbore are updated at every iterative step. An iterative method to solve numerically the resulting linear system for the primitive variable pressure was developed and it includes the wellbore/reservoir coupling into a finite difference numerical model. This methodology is useful for the analysis of transient pressure and velocity behavior in both reservoir and wellbore including the change in flow pattern from porous media to pipe flow. Effects of permeability, gas specific gravity, formation damage and grid refinement are considered in different sensitivity analysis for several well-reservoir systems. Results for pressure and pressure derivative followed the expected behavior of the reservoir model chosen, both in early and long times. Effects of friction, convection and compressibility in wellbore hydraulics were successfully included in the numerical approach.

Resumo. Apresenta-se aqui um novo método de alta resolução para leis de conservação escalares que utiliza de uma metodologia do tipo REA ("Reconstruction", "Evolve", "Average"). O método combina a estratégia de [5] para as etapas de "Reconstruction" e "Average" com um método lagrangeano localmente conservativo (veja [3, 1]) para a etapa de "Evolve". Estabelece-se a relação entre o novo esquema e uma estratégia puramente euleriana. O novo esquema foi utilizado para aproximar numericamente leis de conservação escalares em uma dimensão espacial e produziu resultados bastante satisfatórios quando comparados com o esquema central de segunda ordem de [5]. O novo esquema foi aplicado na aproximação numérica das equações de Burgers e Buckley-Leverett, sendo estaúltima utilizada com o intuito de avaliar o desempenho do método em escoamentos bifásicos em meios porosos. IntroduçãoNa modelagem de fenômenos físicos por leis de conservação hiperbólica, freqüente-mente aparecem não-linearidades que dificultam o desenvolvimento de métodos que façam o cálculo preciso da solução numérica. Estas soluções não devem apresentar oscilações espúrias ou difusão numérica acentuada. Neste sentido, apresenta-se aqui uma metodologia do tipo REA: "Reconstruction", "Evolve" e "Average" que além das características desejáveis acima, não faz uso de soluções de problemas de Riemann eé baseada em uma estratégia euleriano-lagrangeana. A etapa de "Reconstruction" caracteriza-se por uma aproximação linear por partes. A metodologia foi aplicada em duas equações: a equação de Burgers, e a equação de Buckley-Leverett, estaúltima usada na modelagem de escoamentos bifásicos em reservatórios de petróleo. O novo Método Euleriano-LagrangeanoA seguiré apresentada a construção do novo esquema euleriano-lagrangeano, baseado no desenvolvimentos de [3,4]. O método utiliza evoluções para malhas deslo-1 smancuso@iprj.uerj.br 2 pereira@iprj.uerj.br 3 ev grazione@iprj.uerj.br

In reservoir simulation, the most known well-reservoir coupling technique is based on the Peaceman's equivalent radius, which is applied on the productivity index computation, that is used to relate flow rate, wellbore pressure, and the mesh block pressure. Original Peaceman's model considers the assumption of steady-state flow, leading to an artifact on the calculation of the wellbore pressure, called numerical wellbore storage. This artifact is more significant for coarser meshes and initial time instants, and it is also a function of fluid and rock properties. In this context, we have implemented and compared different Peaceman's technique extensions for productivity index calculation incorporating transient effects to prevent numerical wellbore storage. We have also considered some production scenarios for a vertical well and single-phase oil flow.

In this work, we perform a comparative study of some of the most well-known approaches for solving the system of algebraic equations, obtained by discretizing the governing equations using the Finite Volume Method, for a three-dimensional two-phase (water-oil and water-gas) flow in an oil reservoir. We consider that the flow is isothermal, the fluids immiscible, and we take into account the compressibility of the fluid and the porous matrix. We also use a model of well-reservoir coupling for specified flow rates of injection and production. The solution strategies considered are the Fully Implicit Method, the IMPES Method, the Sequential Method, and a Picard-Newton Method, which represents the main contribution of this work. To illustrate the accuracy of the methods, we considered a two-phase flow in slab geometry, two-phase flow in a five-spot arrangement well, and gas production in a reservoir. For the cases simulated here, the Picard-Newton Method was able to correctly reproduce the flow physics with accuracy comparable to the other three methods.