  This work is motivated by the need to develop efficient tools for uncertainty
quantification in subsurface flows associated with radioactive waste disposal
studies. We consider single phase flow problems in random porous media
described by correlated lognormal distributions. We are interested in the error
introduced by a finite element discretisation of these problems. In contrast to
several recent works on the analysis of standard nodal finite element
discretisations, we consider here mass-conservative lowest order Raviart-Thomas
mixed finite elements. This is very important since local mass conservation is
highly desirable in realistic groundwater flow problems. Due to the limited
spatial regularity and the lack of uniform ellipticity and boundedness of the
operator the analysis is non-trivial in the presence of lognormal random
fields. We establish finite element error bounds for Darcy velocity and
pressure, as well as for a more accurate recovered pressure approximation. We
then apply the error bounds to prove convergence of the multilevel Monte Carlo
algorithm for estimating statistics of these quantities. Moreover, we prove
convergence for a class of bounded, linear functionals of the Darcy velocity.
An important special case is the approximation of the effective permeability in
a 2D flow cell. We perform numerical experiments to confirm the convergence
results.
