Well Models for Mimetic Finite Difference Methods and Improved Representation of Wells inMultiscale Methods

Abstract

In reservoir simulation, the modeling and the representation of wells are critical factors. The standard approach for well modeling is to couple the well to the reservoir through the use of a well index, which relates the well pressure and flow rate to grid cell quantities. Well models for the recent mimetic finite difference methods (FDMs) are an unexplored field, but are necessary in order to use these methods for reservoir simulations. In this thesis, we develop numerical well indices for mimetic FDMs by extending the well-known Peaceman radial-flow well model. The performance of the new well indices is tested on both homogeneous and heterogeneous 2D reservoir models with uniform Cartesian grids. The results are compared against a two-point flux approximation using Peaceman's well index and a reference solution obtained on a near-well radial grid. The tests show that it is critical to use specially adapted well indices for mimetic FDMs. Furthermore, we consider improvements in the representation of wells in the also recently developed multiscale mixed finite element method (MsMFEM). This method uses a coarse partition of an underlying fine subgrid for simulations, while subscale heterogeneities and wells are incorporated through the use of locally defined basis functions. These basis functions are computed by solving a number of local flow problems on the fine grid by a subgrid solver. Mimetic FDMs have been shown to be particularly versatile as subgrid solvers. In MsMFEM the wells are represented by well basis functions and the well model in the subgrid solver. The modeling of the flow near the wells is of great importance in order to produce an accurate global flow scenario. In this thesis, we show that the accuracy of MsMFEM can be improved by an overlap technique that extends the support of the well basis functions. Tests performed on both homogeneous and heterogeneous 2D reservoir models with uniform, square, coarse grids show that the most efficient way of representing a well in MsMFEM is to make a coarse grid partition that places the well in the center of the coarse well-block. In cases where this is not possible, the overlap technique is shown to be a successful remedy.