Question

Consider 1D oil--water flow with $k_{ro}^o=1,\ k_{rw}^o=0.2,\ S_{wr}=0.2,\ S_{or}=0.4$. Oil and water viscosities are $5$ cP and $1$ cP. With \[ k_{ro}=k_{ro}^o(1-S_w^*),\quad k_{rw}=k_{rw}^o(S_w^*),\quad S_w^*=\frac{S_w-S_{wr}}{1-S_{or}-S_{wr}}, \] the total relative mobility at $S_w=0.4$ is _________ cP$^{-1}$ (rounded to one decimal place).

Hint:

At mid effective saturation ($S_w^*\!=\!0.5$) with linear Corey-type curves, $k_{ro}$ and $k_{rw}$ contribute equally; divide by viscosities and add to get total mobility.

Answer 1

- Compute effective saturation: \[ S_w^*=\frac{0.4-0.2}{1-0.4-0.2}=\frac{0.2}{0.4}=0.5. \] - Relative permeabilities: \[ k_{ro}=1(1-0.5)=0.5,\qquad k_{rw}=0.2(0.5)=0.1. \] - Mobilities: \[ \lambda_o=\frac{k_{ro}}{\mu_o}=\frac{0.5}{5}=0.1,\qquad \lambda_w=\frac{k_{rw}}{\mu_w}=\frac{0.1}{1}=0.1. \] - Total relative mobility: \ $\lambda_t=\lambda_o+\lambda_w=0.1+0.1=\mathbf{0.2\ \mathrm{cP^{-1}}}$ (one decimal place: $0.2$).