  The contribution deals with the mathematical modelling of fluid flow in
porous media, in particular water flow in soils. The motivation is to describe
the competition between gravity and capillarity, or, in other words, between
transport and diffusion. The analysis is based on a mathematical model
developed by B. Detmann, C. Gavioli, and P. Krej\v{c}\'i, in which the effects
of gravity are included in a novel way. The model consists of a nonlinear
partial differential equation describing both the gravitational transport and
the capillary diffusion of water. Although analytical solutions can be obtained
for some special cases, only numerical solutions are available in more general
situations. The solving algorithm is based on a time discretisation and the
finite element method, and is written in Matlab. The results of the numerical
simulations are shown and the behaviour of the model is discussed.
