Given $S_{wi}$ and $S_{or}$ as $0\mathrm{.}3$ and $0\mathrm{.}3$ respectively and viscosity of water is $1cP.$

Given the expression for relative permeability as follows,

$k_{ro}=0\mathrm{.}8(1-S_{wD}{)}^2,k_{rw}=0\mathrm{.}2S_{wD}^2$ and $S_{wD}=\frac{S_w-S_{wi}}{1-S_{or}-S_{wi}}$

$($i$)$ Tabulate the fractional flow of water $($up to three decimal places$)$ displacing oil of

viscosity $40cP,$ at water saturation of $0\mathrm{.}4$

$($ii$)$ Calculate $f_w^{\text{'}}($a derivative of the fractional flow of water w$.$r$.$t$.$ saturation of water$)$

$($iii$)$ Calculate the average upstream water saturation when the injected front reaches the

outlet?

$($iv$)$ Calculate the volume of oil produced $($in the unit of pore volumes$)$ when the injected

front reaches the outlet

$($v$)$ Calculate the water saturation of the front when the injected fluid reaches the outlet?

$($vi$)$ When the water saturation at the outlet becomes $0\mathrm{.}54,$ what would be the average

water saturation of the core

$($vii$)$ When the water saturation at the outlet becomes $0\mathrm{.}54,$ what would be the total pore

volumes of oil produced till that time?