Computational upscaled modeling of heterogeneous porous media flow utilizing finite volume method
In this dissertation we develop and analyze numerical method to solve general elliptic boundary value problems with many scales. The numerical method presented is intended to capture the small scales eﬀect on the large scale solution without resolving the small scale details, which is done through the construction of a multiscale map. The multiscale method is more eﬀective when the coarse element size is larger than the small scale length. To guarantee a numerical conservation, a ﬁnite volume element method is used to construct the global problem. Analysis of the multiscale method is separately done for cases of linear and nonlinear coeﬃcients. For linear coeﬃcients, the multiscale ﬁnite volume element method is viewed as a perturbation of multiscale ﬁnite element method. The analysis uses substantially the existing ﬁnite element results and techniques. The multiscale method for nonlinear coeﬃcients will be analyzed in the ﬁnite element sense. A class of correctors corresponding to the multiscale method will be discussed. In turn, the analysis will rely on approximation properties of this correctors. Several numerical experiments verifying the theoretical results will be given. Finally we will present several applications of the multiscale method in the ﬂow in porous media. Problems that we will consider are multiphase immiscible ﬂow, multicomponent miscible ﬂow, and soil inﬁltration in saturated/unsaturated ﬂow.