Analysis of counter-current imbibition with gravity in weakly water-wet systems

Counter-current imbibition in one dimension is analyzed, in which a wetting phase (water) displaces a non-wetting phase against gravity. An approximate analytical approach is used to derive an expression for the saturation profile in the case where the mobility of the displaced phase at the inlet is finite. This approach is applicable to waterflooding in hydrocarbon reservoirs, flow in geothermal systems, or the displacement of non-aqueous phase liquid (NAPL) or air by water. Solutions are developed for both gravity-dominated and capillary-dominated cases. In the capillary-dominated limit the predicted recoveries compare very well with experimental data from the literature, over all time scales. The use of this expression for field-scale dual-porosity modeling of flow in fractured systems is briefly discussed.