(1)	$\begin{array}{l}\displaystyle {\rm F}_{\Gamma}(p, {\bf u}) = 0\end{array}$

where

$\begin{array}{l}\Gamma\end{array}$	reservoir boundary
$\begin{array}{l}p\end{array}$	reservoir pressure
$\begin{array}{l}{ \bf u }\end{array}$	fluid velocity
$\begin{array}{l}{\rm F}_{\Gamma}(p, {\bf u})\end{array}$	some function

(2)	$\begin{array}{l}\displaystyle {\rm F}_{\Gamma}(p, {\bf u}) = \big[ a \cdot (p({\bf r}) - p_0) + \epsilon \cdot {\bf n} \cdot M \, (\nabla p - \rho \, {\bf g}) \big]_{{\bf r} \in \Gamma} = 0\end{array}$

where

$\begin{array}{l}p(t, {\bf r})\end{array}$	reservoir pressure	$\begin{array}{l}t\end{array}$	time
$\begin{array}{l}\rho({\bf r},p)\end{array}$	fluid density	$\begin{array}{l}{\bf r }\end{array}$	position vector
$\begin{array}{l}M = k / \mu\end{array}$	phase mobility	$\begin{array}{l}\nabla\end{array}$	gradient operator
$\begin{array}{l}k\end{array}$	formation permeability to a given fluid	$\begin{array}{l}{ \bf g }\end{array}$	gravity vector
$\begin{array}{l}\mu\end{array}$	dynamic viscosity of a given fluid	$\begin{array}{l}{ \bf u }\end{array}$	fluid velocity
$\begin{array}{l}{ \bf n }\end{array}$	external normal to the reservoir boundary $\begin{array}{l}\Gamma\end{array}$	$\begin{array}{l}\epsilon \in \{ 0,1 \}\end{array}$	a binary value

The two extreme cases of (2) are:

Constant Pressure	No flow
$\begin{array}{l}p({\bf r})_{{\bf r} \in \Gamma} = p_0 = \rm const\end{array}$	$\begin{array}{l}{\bf n} \cdot {\bf u} \, \Big\|_{{\bf} in \Gamma} = {\bf n} \cdot M \, (\nabla p - \rho \, {\bf g}) \, \Big\|_{{\bf} in \Gamma} = 0\end{array}$

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