  We consider a two-dimensional model of a porous medium where circular grains
are uniformly distributed in a squared container. We assume that such medium is
partially filled with water and that the stationary interface separating the
water phase from the air phase is described by the balance of capillarity and
gravity. Taking the unity as the average distance between grains, we identify
four asymptotic regimes that depend on the Bond number and the size of the
container. We analyse, in probabilistic terms, the possible global interfaces
that can form in each of these regimes. In summary, we show that in the regimes
where gravity dominates the probability of configurations of grains allowing
solutions close to the horizontal solution is close to one. Moreover, in such
regimes where the size of the container is sufficiently large we can describe
deviations from the horizontal in probabilistic terms.
  On the other hand, when capillarity dominates while the size of the container
is sufficiently large, we find that the probability of finding interfaces close
to the graph of a given smooth curve without self-intersections is close to
one.
