$30$ points$-$grad$*(\backslash$rho $\_($m$)$v$\_($m$))=($del$)/($delt$)(\backslash$phi $\backslash$rho $\_($m$)$S$\_($m$)),$ for $,$m$=$o$,$w

where $\backslash$rho $\_($m$)$ is the density of phase m$,$S$\_($m$)$ is the saturation of phase m$,$ and $\backslash$phi is the porosity of the rock. In Eq$.$

v$\_($m$)$ is the velocity vector $($expressed by Darcy's equation$)$ of phase m$.$ Note that, we have two continuity

equations: one for oil phase when m$=$o and other for water phase when m$=$w$.$ For this two$-$phase system,

the sum of the saturations should always add up to unity; i$.$e$.,$ S$\_($o$)+$S$\_($w$)=1.$

$($a$)(15$ pts$)$ Assuming that density of phase m$(\backslash$rho $\_($m$))$ is a unique function of p$,$ saturation of each phase is a

unique function of time, and that the effect of capillary pressure between oil and water can be neglected

so that p$\_($o$)=$p$\_($w$)=$p$.$ Show that Eq$.5\mathrm{.}1$ for each phase can be written as:

$-$c$\_($m$)$v$\_($m$)*$gradp$-$grad$*$v$\_($m$)=$S$\_($m$)\backslash$phi c$\_($r$)($delp$)/($delt$)+\backslash$phi S$\_($m$)$c$\_($m$)($delp$)/($delt$)+\backslash$phi $($delS$\_($m$))/($delt$),$ for $,$m$=$o$,$w

where c$\_($m$)$ is the isothermal compressibility of phase m $,$ and c$\_($r$)15$pts Writing Eq$.5\mathrm{.}2$ for each phase, and then adding both equations side by side, show that one can

have the following equation for the oil & water system:

$-$c$\_($w$)$v$\_($w$)*$gradp$-$grad$*$v$\_($w$)-$c$\_($o$)$v$\_($o$)*$gradp$-$grad$*$v$\_($o$)=\backslash$phi c$\_($t$)($delp$)/($delt$)$

where c$\_($t$)$ is the total compressibility of the rock and fluid system and is given by:

c$\_($t$)=$c$\_($r$)+($S$\_($w$)$c$\_($w$)+$S$\_($o$)$c$\_($o$))$