  In this study, we investigate the complexity of two-phase flow (air/water) in
a heterogeneous soil sample by using complex network theory, where the supposed
porous media is non-deformable media, under the time-dependent gas pressure.
Based on the different similarity measurements (i.e., correlation, Euclidean
metrics) over the emerged patterns from the evolution of saturation of
non-wetting phase of a multi-heterogeneous soil sample, the emerged complex
networks are recognized. Understanding of the properties of complex networks
(such degree distribution, mean path length, clustering coefficient) can be
supposed as a way to analysis of variation of saturation profiles structures
(as the solution of finite element method on the coupled PDEs) where complexity
is coming from the changeable connection and links between assumed nodes. Also,
the path of evolution of the supposed system will be illustrated on the state
space of networks either in correlation and Euclidean measurements. The results
of analysis showed in a closed system the designed complex networks approach to
small world network where the mean path length and clustering coefficient are
low and high, respectively. As another result, the evolution of macro -states
of system (such mean velocity of air or pressure) can be scaled with
characteristics of structure complexity of saturation. In other part, we tried
to find a phase transition criterion based on the variation of non-wetting
phase velocity profiles over a network which had been constructed over
correlation distance.
