  In the present contribution, we construct a virtual element (VE)
discretization for the problem of miscible displacement of one incompressible
fluid by another, described by a time-dependent coupled system of nonlinear
partial differential equations. Our work represents a first study to
investigate the premises of virtual element methods (VEM) for complex fluid
flow problems. We combine the VEM discretization with a time stepping scheme
and develop a complete theoretical analysis of the method under the assumption
of a regular solution. The scheme is then tested both on a regular and on a
more realistic test case.
