  We present a theoretical and experimental investigation of slow drainage in
porous media with a gradient in the grains size (and hence in the typical
pores' throats), in an external gravitational field. We mathematically show
that such structural gradient and external force have a similar effect on the
obtained drainage patterns, when they stabilise the invasion front. With the
help of a newly introduced experimental set-up, based on the 3D-print of
transparent porous matrices, we illustrate this equivalence, and extend it to
the case where the front is unstable. We also present some invasion-percolation
simulations of the same phenomena, which are inline with our theoretical and
experimental results. In particular, we show that the width of stable drainage
fronts mainly scales with the spatial gradient of the average pore invasion
threshold and with the local distribution of this (disordered) threshold. The
scaling exponent results from percolation theory and is -0.57 for 2D systems.
Overall, we propose a unifying theory for the up-scaling of dual fluid flows in
most classical scenarii.
