Figure 1 A schematic overview of the different scaling regimes
for the mass M of the non-wetting cluster. The solid lines are the
boundaries
of the various regimes. In these regimes the M scales differently with
the system size L. The correlation length xg
scales differently with the bond number B.
When xs <xg
the local structure of the invading cluster is governed by the
interplay
of capillarity and the fractal properties of the pore space. Only parts
of the backbone of the pore structure can be invaded. Therefore, the
obtained
fractal dimension for small systems L<xs
is much lower, 1.40, than the one for ordinary IP, 1.82. By invasion in
a fractal pore space the structure of the invader cluster is much more
volatile, see figure 2.
Figure 2 – The invading cluster at breakthrough in a L=100 system: a) B=0 and b) B=100. In this picture I) refers to pores filled nw-fluid, II) to pores filled with non-trapped w-fluid, III) to pores which contain trapped w-fluid and IV) to solid pores
On larger length scales, xs
<L<xg, the
fractality
of the pore space is no longer important and the cluster grows as in
ordinary
IP. When L>xg,
gravity
becomes important and xg
scales with the bond number B as xg~B^(-0.57),
as in ordinary IP, while the fractal dimension becomes equal to the
euclidean
one.
When xg< xs
gravity is already important on length scales where the fractality of
the
medium has to be considered too. On small scales L<xg,
where only capillarity and fractality play a role the cluster structure
is again characterized by the fractal dimension of 1.40. On larger
length
scales, xg <L<
xs,
gravity promotes a more efficient invasion of the pore space and
different
fractal dimension is found, 1.52, see figure 2. The length scale xg
no longer follows ordinary IP scaling: xg
~B^(-0.69). When L>xs
the
fractal dimension of the invading cluster equals the euclidean one and
xg ~B ^(-0.69).
H.P. Huinink, M.A.J. Michels, The influence of buoyancy on drainage of a fractal porous medium; Physical Review E 66, 064301 (2002).