Numerical simulation of two-phase flow in petroleum reservoirs using high-order CPR method coupled to a non-orthodox MPFA-D finite volume scheme

• Autor(a) principal: GALINDEZ RAMIREZ, Gustavo
• Data de Publicação: 2018
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Repositório Institucional da UFPE
• dARK ID: ark:/64986/0013000015rxt
• Texto Completo: https://repositorio.ufpe.br/handle/123456789/33064
• topic: Engenharia Mecânica; High order methods; CPR; MLP; MPFA-D; IMPES; Raviart-Thomas interpolation; The Piola trasformation; Twophase flows in porous media

The study and development of high resolution numerical approximations for the modeling and simulation of multiphase flows in petroleum reservoirs is still a challenge, from the computational viewpoint, due to the difficulties posed by some physical features such as heterogeneity and anisotropy of the medium, that are of paramount importance in this class of applications. Several methods have been proposed in the past that are based on FD (Finite-Difference), FV (Finite-Volume) or FE (Finite-Element). These methods, in their classical formulations, are of low order of approximation and suffer excessive smearing at saturation front introducing error into the numerical solution. These deficiencies can be mitigated or suppressed using highresolution methods such as the k-exact or ENO (Essentially non-Oscillatory) FV methods, which require large stencils to reconstruct high order polynomial within a control volume, resulting in an increase of the storage requirements and computational cost. On the other hand, over the last decades DG (Discontinuous Galerkin), SV (Spectral Volume), SD (Spectral Difference) and FR (Flux Reconstruction)/CPR (Correction Procedure via Reconstruction) methods were developed, which can achieve high order accuracy via a compact stencil consisting of the current cell and its immediate neighbors. In addition, the FR/CPR recovery simplified versions of nodal DG, SV and SD methods by choosing an adequate polynomial reconstruction function, whose coefficients are preprocessed and stored. The focus of this work is to investigate and to apply a very high resolution CPR method for the discretization of the saturation equation, which is generally advection-dominated and that results from the modeling of the 2-D Oil-Water displacement through porous formations. In order to suppress numerical oscillations (under/over shoots) near shocks that are typical in higher order schemes, and handing the high accuracy in smooth regions of the solution a hierarchical multi-dimensional limiting strategy (MLP) is used in the reconstruction stage. The integration in time is carried out using a third-order Runge-Kutta method. To solve the pressure equation a non-orthodox cell centered MPFA-D (Multipoint Flux Approximation-Diamond type) finite volume method is employed. In order to properly couple the MPFA-D method with the CPR formulation, it is necessary to obtain an adequate velocity reconstruction throughout the control volumes of the mesh. Because the cell-centered finite volume method naturally delivers fluxes across cell faces that belong to the primal grid, a reconstruction operator based on the lowest Raviart-Thomas interpolation functions and the Piola transformation is built, to get the complete knowledge of conservative velocity field throughout the domain. The reconstruction operator receives, as input, the density fluxes across control volume faces and returns the point-wise values of velocity anywhere within the cell. Finally, the coupling of the pressure-saturation system of equations is carried out using a classical IMPES (IMplicit Pressure Explicit Saturation) procedure. Some two-phase flow benchmark problems in one and two dimensions were analyzed and numerical and/or analytical comparisons have been used to verify the accuracy, efficiency and shock-capturing capability of the proposed methodology.