An iterative MsCV method coupled to the high-resolution CPR approach via different solution smoothers for the simulation of oil-water flows in 2D petroleum reservoirs on unstructured grids

Resumo

The use of very high-order schemes to solve the transport problem, together with multiscale methods
applied to multi-phase flows in petroleum reservoirs has not been explored in literature. In this work, the Multiscale
Control-Volume (MsCV) method is used to solve the elliptical pressure problem while the hyperbolic saturation
equation is solved using the high-resolution nodal Correction Procedure via Reconstruction (CPR) approach. In
order to properly couple the previous set of equations, we use the Implicit Pressure Explicit Saturation (IMPES)
strategy and a velocity reconstruction operator based on the lowest order Raviart–Thomas shape functions. In
addition, a hierarchical Multidimensional Limiting Process (MLP) is employed in the reconstruction stage of CPR
approach to suppress numerical oscillations. To properly couple the MsCV method with the CPR approach an
adequate velocity reconstruction throughout all control volumes (CVs) is necessary to assure the accuracy of the
high-order method. Thus, the velocity field must present a proper degree of accuracy that, in general, is not handed
by multi-scale methods. To deal with this issue and to remove the high-frequency components of the error, we
have studied several smoothers. Hence, the aim of this paper is to describe the coupling of the MsCV method
with CPR for the first time in literature and to analyze the behavior and efficiency of different smoothers applied
to the elliptical problem in order to produce accurate velocity field and so proper two-phase flow results. Finally,
through two representative problems, which were solved to evaluate the accuracy, efficiency, and shock-capturing
capabilities of our new numerical methodology, we concluded that, for the same level o accuracy, our high-order
proposed methodology reduces the computational effort.