Unstable displacement in a plane-parallel microchannel

Unstable immiscible water-oil displacement is considered in a Hele-Shaw cell at a constant pressure drop or at a constant flow rate. Unstable displacement in a Hele-Shaw cell with small gaps is characterized by valuable capillary forces, which lead to the displaced phase pinching. The capillary boundary between water and oil represents a curved cylindrical surface with two radii: a radius of 10 pm and a radius that is several orders of magnitude higher and corresponds to a varying shape of the unstable displacement boundary. The power dependence of the fractal dimension on the injected volume of the displacing fluid at a constant pressure drop is revealed, which does not change with the pressure drop increased by 2.5 times. An outburst of the displacing fluid at a constant pressure drop leads to a redistribution of local pressures, a change in structure, and a gradual increase in the velocity of “viscous fingers”. It is shown that the displacement process is characterized by four stages, which alternate almost stepwise in terms of the content of the displaced oil taken relative to the volume of injected water: 100 %, 25 %, 3.5 %, and 0.5 %. At the initial stage, displacement at a constant flow rate is more efficient, while for large volumes, it is more efficient at a constant pressure drop.