Phase-field modelling of a liquid/liquid immiscible displacement through a network of capillaries

•GPU algorithm for numerical modelling of multi-phase flows.•Direct calculation of capillary pressure for a complex interface in a micro-matrix.•Displacement flows in a single capillary and in uniform matrices occur qualitatively similarly.•Integral parameters quickly converge upon increase of the matrix size.•Capillary pressure is given by the difference of the flow rates for the two-phase and single-phase flows. The liquid/liquid displacement through a 2D uniform network of capillaries is numerically modelled with the use of the phase-field approach. The detailed structure of the flow fields within the uniform matrices of different sizes (with the different number of pores) is examined with the aim to reveal the asymptotic behaviour, pertinent for a sufficiently large matrix, that could be used for representation of a porous medium. The integral characteristics of the flow that do not depend on the matrix size can be used for calculation of the parameters of a macroscopic (Darcy) approach. We demonstrate that qualitatively the displacement occurs very similarly in the matrices with the different number of structural elements. In particular, we show that the capillary pressure remains nearly constant during the displacement run until the break-through time (with some minor variations related to a particular shape of a matrix). Upon increase of the matrix size the magnitude of the capillary pressure (and all other integral characteristics of the two-phase flow) quickly converges to the limiting value (so that the limiting results are already reached for a matrix with 6×6 elements), giving the direct procedure for calculation of the capillary pressure in a porous medium.