The flow of two immiscible fluids through a porous medium depends
on the complex interplay between gravity, capillarity, and viscous
forces. The interaction between these forces and the geometry of
the medium gives rise to a variety of complex flow regimes that are
difficult to describe using continuum models. Although a number of
pore-scale models have been employed, a careful investigation of
the macroscopic effects of pore-scale processes requires methods
based on conservation principles in order to reduce the number of
modeling assumptions. In this work we perform direct numerical simulations
of drainage by solving Navier-Stokes equations in the pore space
and employing the Volume Of Fluid (VOF) method to track the evolution
of the fluid-fluid interface. After demonstrating that the method
is able to deal with large viscosity contrasts and model the transition
from stable flow to viscous fingering, we focus on the macroscopic
capillary pressure and we compare different definitions of this quantity
under quasi-static and dynamic conditions. We show that the difference
between the intrinsic phase-average pressures, which is commonly
used as definition of Darcy-scale capillary pressure, is subject
to several limitations and it is not accurate in presence of viscous
effects or trapping. In contrast, a definition based on the variation
of the total surface energy provides an accurate estimate of the
macroscopic capillary pressure. This definition, which links the
capillary pressure to its physical origin, allows a better separation
of viscous effects and does not depend on the presence of trapped
fluid clusters.