Problem $1$: Gas fractional flow Equation

$(5$ Marks$)$

Work out the derivation procedure for the fractional flow curve for gas $(f_g)$ Eq$.(32).$ You need to provide a detailed procedure similar to the one used to derive fractional flow curve for water.

Construct the fractional flow curve for a reservoir where oil is displaced by carbon dioxide gas and the following fluids properties are known:

${}_o=1\mathrm{.}81$mPa.$s$

${}_g=0\mathrm{.}035$mPa.$s$

$S_{or}=0\mathrm{.}24$

$S_{gi}=0\mathrm{.}05$

And the measured relative permeability curves are shown in the figure below

$3.$ What will be the effect of having oil viscosity of $1\mathrm{.}0$mPa.$s$ and $5\mathrm{.}0$mPa.$s$ on the constructed above $($in $2)$ fractional flow curve. All three fractional flow curves for the three different oil viscosities have to be plotted on the same figure.