Modelling two-phase flow at the micro-scale using a volume-of-fluid method

We present a numerical scheme to model two-phase flow in porous media where capillary forces dominate over viscous effects. The volume-of-fluid method is employed to capture the fluid-fluid interface whose dynamics are described based on a finite volume discretization of the Navier--Stokes equations. Interfacial forces are calculated directly on reconstructed interface elements such that the total curvature is preserved. The computed interfacial forces are explicitly added to the Navier--Stokes equations using a sharp formulation which effectively eliminates spurious currents. The numerical model is validated in terms of physics, robustness, and mesh convergence, using an extensive hierarchy of static and dynamic test cases including wetting effects at the solid interface in two and three space dimensions. Next we provide an extensive study of viscous coupling effects in porous media flows, where the flow of one phase in the centre of a pore affects the flow of phases in layers or corners and vice versa. We perform two-phase flow simulations for different fluid configurations in non-circular capillary tubes to investigate viscous coupling effects as a function of viscosity ratio, contact angle, wetting phase saturation and wettability. We demonstrate the accuracy of our code in determining fluid velocities and capillary pressures, even for slow flows, where previous approaches fail. We specifically show the dependence of velocity profile and consequently flow conductivities on viscosity ratio and interface boundary condition, by modelling immiscible two-phase flow through an equilateral triangular capillary tube with sandwiched layers. We also demonstrate that imposing no-flow or free-slip interface boundary conditions at a clean fluid-fluid interface with zero interfacial shear viscosity, may lead to under- or over-estimation of flow conductance in layers compared to the physically correct continuity boundary condition at the interface. We use two-phase direct numerical simulation results in conjunction with basic arguments from fluid mechanics to present parametric models that estimate fluid conductivities as a function of the geometry and viscosity ratio. These scaling models, which take into account the flow coupling, can then be incorporated into pore-to-Darcy-scale flow models, for example two-phase pore-network models, to study the effects of viscous coupling on macroscopic flow properties such as relative permeabilities. These network models provide a much more computationally efficient framework for the simulation of flows to the centimetre or larger scales. Furthermore, in future work our methods can be used to assess local recovery and displacement mechanisms in multiphase flow using pore-scale images of different rocks.Open Acces