Gravity segregation in steady-state horizontal flow in homogeneous reservoirs

The model of Stone for gravity segregation in steady, horizontal gas–liquid flow in homogeneous porous media is very useful and apparently general. For instance, it predicts that for a given reservoir and well pattern, the distance gas and liquid travel before complete segregation depends only on pressure gradient. This finding both implies an optimal injection strategy and suggests that override may be unavoidable if injection well pressure is limited. This model has lacked a sound theoretical foundation, however, and the range of its applicability was unknown. A proof is presented that this model applies to steady-state gas–liquid flow, and also foam flow, as long as the standard assumptions of fractional flow theory (incompressible flow, Newtonian mobilities, local equilibrium) apply. The key step in this proof is the substitution of the stream function for the vertical coordinate z in the governing partial differential equation. With this proof, the useful model of Stone can be applied with confidence to a wide range of situations. The horizontal position of the point of complete segregation of gas and water flow in the new proof matches that of Stone, but there is still no rigorous solution for the curves separating override, underride and mixed zones, or for the vertical height of the position of complete segregation.