(Solved): Example 2.9. Show that Stone's Model 2 reduces to two-phase relative permeability data under the sa ...

Example 2.9. Show that Stone's Model 2 reduces to two-phase relative permeability data under the same conditions as Example 2.8. "roon $?{}^{\text{row }}{S}_{ww}=S_{\text{iw }}=k_{\text{rog }}{I}_g=0$. In other words, $k_{nocw}$ is the oil-phase relative permeability at $S_{iw}$. Stone's ${}^{26}$ Three-Phase Model 2. Stone's second model is a probability model based on channel flow considerations. ${}^{17}$ Like Stone's first model, the second model approximates $\mathrm{f}\left(S_w,S_g\right)$ in Eq. 2.145 using two sets of two-phase relative permeability data (Figs. 2.19 and 26 2.20) but does not require the knowledge of $S_{or}$ in a three-phase ${}_{5y_3}$ tem. The model equation is $\frac{k_{\text{ro }}}{k_{\text{rocw }}}=\left(k_{\text{row }}/k_{\text{rocw }}+k_{\text{rww }}\right)\left(k_{\text{rog }}/k_{\text{rocw }}+k_{\text{rg }}\right)?\left(k_{\text{rw }}+k_{\text{rg }}\right)\text{. }$ where $k_{ro}?0,k_{\text{row }}$ and $k_{rw}$ are obtained at $S_w$ by use of two-phase $0i1/w_{\text{wa. }}$ ter relative permeability data (Fig. 2.19), and $k_{\text{rog }}$ and $k_{\text{rg }}$ are obtained $S_g$ by use of two-phase oil/gas relative permeability data in the presence of irreducible water (Fig. 2.20). Again, Eq. 2.157 gives $k_{\text{rocw }}$. Both Stone's Models 1 and 2 reduce to the two-phase oil and water relative permeability data at $S_g=0$ and to the two-phase oil and ges relative permeability data at $S_w=S_{iw}$.