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.
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.
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