  The subsurface flow is usually subject to uncertain porous media structures.
In most cases, however, we only have partial knowledge about the porous media
properties. A common approach is to model the uncertain parameters of porous
media as random fields, then the statistical moments (e.g. expectation) of the
Quantity of Interest(QoI) can be evaluated by the Monte Carlo method. In this
study, we develop a full multigrid-multilevel Monte Carlo (FMG-MLMC) method to
speed up the evaluation of random parameters effects on single-phase porous
flows. In general, MLMC method applies a series of discretization with
increasing resolution and computes the QoI on each of them. The effective
variance reduction is the success of the method. We exploit the similar
hierarchies of MLMC and multigrid methods and obtain the solution on coarse
mesh $Q^c_l$ as a byproduct of the full multigrid solution on fine mesh $Q^f_l$
on each level $l$. In the cases considered in this work, the computational
saving due to the coarse mesh samples saving is $20\%$ asymptotically. Besides,
a comparison of Monte Carlo and Quasi-Monte Carlo (QMC) methods reveals a
smaller estimator variance and a faster convergence rate of the latter approach
in this study.
