Definition and Counting of Configurational Microstates in Steady-State Two-Phase Flows in Pore Networks

Abstract

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 an off-equilibrium stationary process-in the sense that it is maintained in dynamic equilibrium at 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 the configurational entropy (a Boltzmann-Gibbs entropy component, klnW), due to the existence of a canonical ensemble of flow configurations, physically admissible to the externally imposed macrostate conditions. Here, we propose an analytical model to account the number of microstates, lnW, in two-phase flows in pore networks. Combinatorial analysis is implemented to evaluate the number of identified microstates per physically admissible internal flow arrangement, compatible with the imposed steady-state flow conditions. Then, Stirling's approximation is applied to downscale the large factorial numbers. Finally, the number of microstates is estimated by contriving an appropriate mixing scheme over the canonical ensemble of the physically admissible flow configurations. Indicative computations are furnished.

Ensemble distribution for immiscible two-phase flow in porous media

“…Since the late 80s, a theory of two-phase flow using a thermodynamic approach has been developed and employed by Hassanizadeh and Gray [5,6,9,11]. More recently, Valavanides and Daras [22] employ tools from statistical mechanics to describe flow. Hansen and Ramstad [16] proposed to develop a thermodynamical description of immiscible twophase flow in porous media based on the configurations of the fluid interfaces, an approach that is related to that of Valavanides and Daras.…”

Oil Fragmentation, Interfacial Surface Transport and Flow Structure Maps for Two-Phase Flow in Model Pore Networks. Predictions Based on Extensive, DeProF Model Simulations

“…The volume of V PAS is a measure of the degrees of freedom of the process and the variety of microstates it can attain; it is also related to the rate of entropy production (configurational entropy) within the process (Valavanides and Daras, 2016). The volume of the ensemble of interstitial flows is a measure of the internal degrees of freedom of the process; its size and shape depend on the externally imposed macroscopic flow conditions.…”

“…The existence of critical flow (optimum operating) conditions is therefore an inherent characteristic of steadystate two-phase flows in real porous media, remaining in latency until indicated by DeProF model predictions (Valavanides and Payatakes, 2003). Recently, in order to develop a theoretical framework based on statistical thermodynamics that would justify the existence of a locus optimum operating conditions, Valavanides and Daras (2016) proposed a model for accounting the number of the process microstates.…”

“…This equiprobability is a fundamental assumption of the DeProF model, related to the ergodic nature of the process (see '2.3. Discussion on Assumption of the Process Ergodicity' in Valavanides and Daras, 2016) To detect these flow configurations, the domain {S w , b, v} of all possible flow arrangement values is partitioned using sufficiently fine steps to obtain a 3D grid (Fig. 3).…”

Advances in Pore-Scale Simulation of Oil Reservoirs

“…Macroscopic flow can be decomposed into two prototype flows: The connected oil pathway flow (CPF) and the disconnected oil flow (DOF). The latter can be further decomposed into ganglion dynamics (GD) flow and drop traffic flow (DTF) [15]. If the interstitial velocity is large enough, the viscous force dominates the fluid flow.…”