  Relative permeability is commonly used to model immiscible fluid flow through
porous materials. In this work we derive the relative permeability relationship
from conservation of energy, assuming that the system to be non-ergodic at
large length scales and relying on averaging in both space and time to
homogenize the behavior. Explicit criteria are obtained to define stationary
conditions: (1) there can be no net change for extensive measures of the system
state over the time averaging interval; (2) the net energy inputs into the
system are zero, meaning that the net rate of work done on the system must
balance with the heat removed; and (3) there is no net work performed due to
the contribution of internal energy fluctuations. Results are then evaluated
based on direct numerical simulation. Dynamic connectivity is observed during
steady-state flow, which is quantitatively assessed based the Euler
characteristic. We show that even during steady-state flow at low capillary
number ($\mathsf{Ca}\sim1\times10^5$), typical flow processes will explore
multiple connectivity states. The residence time for each connectivity state is
captured based on the time-and-space average. The distribution for energy
fluctuations is shown to be multi-modal and non-Gaussian when terms are
considered independently. However, we demonstrate that their sum is zero. Given
an appropriate choice of the thermodynamic driving force, we show that the
conventional relative permeability relationship is sufficient to model the
energy dissipation in systems with complex pore-scale dynamics that routinely
alter the structure of fluid connected pathways.
