Number of microstates and configurational entropy for steady-state two-phase flows in pore networks

Steady-state two-phase flow in porous media is a process whereby a wetting phase displaces a non-wetting phase within a pore network. It is a stationary, off equilibrium process -in the sense that it is maintained in dynamic equilibrium on the expense of energy supplied to the system. The efficiency of the process depends on its spontaneity, measurable by the rate of global entropy production. The latter has been proposed to comprise two components: the rate of mechanical energy dissipation at constant temperature (a thermal entropy component, Q/T, in the continuum mechanics scale) and a configurational entropy production component (a Boltzmann-type statistical-entropy component, klnW), due to the existence of a canonical ensemble of flow configurations, physically admissible to the externally imposed macrostate stationary conditions. Here, the number of microstates, lnW, in steady-state two-phase flows in pore networks is estimated in three stages: Combinatorics are implemented to evaluate the number of identified microstates per physically admissible internal flow arrangement compatible with the imposed stationary flow conditions. Then, “Stirling’s approximation limiting procedure” is applied to downscale the computational effort associated with the operations between large factorial numbers. Finally, the number of microstates is estimated by contriving a limiting procedure over the canonical ensemble of the physically admissible flow configurations. Counting the microstates is a prerequisite for estimating the process configurational entropy in order to implement the Maximum Entropy Production principle and justify the existence of optimum operating conditions.