  In this paper, we present a mixed Generalized Multiscale Finite Element
Method (GMsFEM) for solving flow in heterogeneous media. Our approach
constructs multiscale basis functions following a GMsFEM framework and couples
these basis functions using a mixed finite element method, which allows us to
obtain a mass conservative velocity field. To construct multiscale basis
functions for each coarse edge, we design a snapshot space that consists of
fine-scale velocity fields supported in a union of two coarse regions that
share the common interface. The snapshot vectors have zero Neumann boundary
conditions on the outer boundaries and we prescribe their values on the common
interface. We describe several spectral decompositions in the snapshot space
motivated by the analysis. In the paper, we also study oversampling approaches
that enhance the accuracy of mixed GMsFEM. A main idea of oversampling
techniques is to introduce a small dimensional snapshot space. We present
numerical results for two-phase flow and transport, without updating basis
functions in time. Our numerical results show that one can achieve good
accuracy with a few basis functions per coarse edge if one selects appropriate
offline spaces.
