Flow Rate Dependency of Steady-State Two-Phase Flows in Pore Networks: Universal, Relative Permeability Scaling Function and System-Characteristic Invariants

Abstract

The phenomenology of steady-state two-phase flow in porous media is conventionally recorded by the relative permeability diagrams in terms of saturation. Yet, theoretical, numerical and laboratory studies of flow in artificial pore network models and natural porous media have revealed a significant dependency on the flow rates—especially when the flow regime is capillary to capillary/viscous and part of the disconnected non-wetting phase remains mobile. These studies suggest that relative permeability models should incorporate the functional dependence on flow intensities. In the present work, a systematic dependence of the pressure gradient and the relative permeabilities on flow rate intensity is revealed. It is based on extensive simulations of steady-state, fully developed, two-phase flows within a typical 3D model pore network, implementing the DeProF mechanistic–stochastic model algorithm. Simulations were performed across flow conditions spanning 5 orders of magnitude, both in the capillary number, Ca, and the flow rate ratio, r, and for different favorable /unfavorable viscosity ratio fluid systems. The systematic, flow rate dependency of the relative permeabilities can be described analytically by a universal scaling function along the entire domain of the independent variables of the process, Ca and r. This universal scaling comprises a kernel function of the capillary number, Ca, that describes the asymmetric effects of capillarity across the entire flow regime—from capillarity-dominated to mixed capillarity/viscosity- to viscosity-dominated flows. It is shown that the kernel function, as well as the locus of the cross-over relative permeability values, are single-variable functions of the capillary number; they are both identified as viscosity ratio invariants of the system. Both invariants can be correlated with the structure of the pore network, through a function of Ca. Consequently, the correlation is associated with the wettability characteristics of the system. Among the potential applications, the proposed, universal, flow rate dependency scaling laws are the improvement of core analysis and dynamic rock-typing protocols, as well as integration into field-scale simulators or associated machine learning interventions for improved specificity/accuracy.