Analytical solution to steady state, viscosity-gravity driven horizontal flow

Reservoir injection of low density fluids often results in phase segregation at some distance from the well. Similarly, gravity effects can be observed in coreflooding experiments, e.g., drainage by N2 or CO2. It is important to model these processes as they have substantial impact, for example, on recovery factor and history matching results. We formulate equations for steady state immiscible two-phase flow where capillary forces are negligible while gravity and viscous effects are significant. Both phases are injected simultaneously at the inlet boundary with a given fractional flow. A solution is derived using the method of characteristics allowing to predict pressure and saturation in two-dimensional space. It consists of linear boundaries separating three regions of constant saturation describing the transition between a mixed wetting/nonwetting zone and a segregated zone, in which the lighter phase is above the heavier. The solution is compared to numerical simulations and to an analytical formula presented by Stone [1982] and Jenkins [1984]. It is found that when segregation occurs relatively near the inlet (e.g., for low flow rates) the new solution is more accurate than the existing formula, while otherwise its accuracy is reduced.