On the prediction of three-phase relative permeabilities using two-phase constitutive relationships

The predictions of two recent and two classical mathematical models are compared with experimentally measured three-phase relative permeabilities. Experimentally determined constitutive relationships in two-phase systems were used as model input parameters to numerically predict relative permeabilities (k(r)) in three-phase systems. Then the estimated results were compared with experimental three-phase permeabilities measured along decreas-ing water saturation/decreasing oil saturation/increasing gas saturation (DDI) paths. The results of the individual models to each of the three fluids involved (water, oil, and gas) were analyzed in detail. The simulated results showed that the Total Differential (TD) compatible model overestimates significantly both the global mobilities as well as the relative phase permeabilities in the three-phase system. There was improvement in the prediction with the TD compatible model when experimental data were used to locally impose the global mobility and fractional water and gas fluxes in the ternary diagram. Globally, the best prediction of the measured k(r) values was obtained with the so-called mechanistic model. However, its numerical implementation requires a preliminary calibration of the relative phase permeabilities in a three-phase system against experimental data along one DDI path to quantify the required six characteristic coefficients. In contrast to the TD compatible model, which by construction does not exhibit any numerical instabilities, elliptic zones in the water-oil (NAPL)-gas ternary diagram were identified in the mechanistic model.