In the specified parameter range, k nw varies from 0.02 to 0.54 (Fig. 8a). For high log Ca values (between −1.25 and +0.5), k nw does not vary with log Ca and is only influenced by M , as indicated by the almost straight vertical borders between color bands. This finding agrees with Fig. 7, which shows that for log Ca ≥ −1.25, the nonwetting fluid relative permeability curve is not affected by Ca . However, as log Ca becomes lower than −1.25 (Ca value ≈ 0.06),k nw can be seen to vary on both the x axis (log M ) and the y axis (log Ca ), that is,k nw starts to be affected by the capillary force. The diagram agrees with our finding in section 3.2 that for logCa lower than −1.25, k nw decreases asCa decreases.
The k nw variation with M (x axis) is more pronounced for log M > 0 (M> 1) than for log M < 0 (Fig. 8a). This result is consistent with our findings in section 3.1 and is caused by the lubrication effect resulting from viscosity differences. Whenμ nw is higher than μ w, the resulting instability and viscous coupling effect enhances the relative permeability of the nonwetting fluid. The diagram clearly captures the contrast between k nw at log M < 0 and at log M > 0.
The range of k nw variation in low M and low Ca conditions is obvious in the magnified part of the diagram (Fig. 8b). The range of k nw values is 0.02 to 0.25, in a part of parameter space where both Ca and Minfluence k nw and thus the borders between color bands are curved. The diagram is useful for evaluatingk nw when the effects of viscosity gradient and capillary number are considered simultaneously, and the graph neatly summarizes all the main results of this study. The diagram shows that capillary force starts affecting the flow of the nonwetting fluid when log Ca is less than −1.25 and that the effect of changes in capillary number may be just as important as that of changes in viscosity ratio. The graph also shows that k nw is affected by M under all conditions, but most markedly when logM > 0. The graph yields a good estimate ofk nw at 20% saturation based on the M andCa parameters.