We present a discrete-domain approach to three-phase displacements and
hysteresis in porous media. In this method, constrained energy
minimization leads to evolution equations for local saturations that
describe a wide range of three-phase displacements, including pressure-
and saturation-controlled displacement with or without preservation of
one of the defending phases. Under action of global saturation
constraints, irreversible displacements lead to significant fluid
redistribution, as well as abrupt fluctuations of both the three-phase
saturation paths and the corresponding capillary pressures. These
features are a consequence of Haines jumps with cooperative behavior
that occur at pore scale in three-phase systems. The method is a fast
and convenient way to investigate hysteresis behavior of three-phase
displacement in porous media. As free energy is an extensive property,
the framework links pore and core scales and provide a means to achieve
upscaled three-phase displacements for higher-order hysteresis loops,
which rarely is obtained in time-consuming three-phase measurements or
pore-scale simulations.