(1)	$\begin{array}{l}\displaystyle \phi \cdot \partial_\tau \Psi - \nabla \cdot \left( k \cdot \vec \nabla \Psi \right) = 0\end{array}$

(2)	$\begin{array}{l}\displaystyle -\frac{k}{\mu} \int_{\Sigma} \, \frac{\partial p}{ d n} \, d \sigma = q(t)\end{array}$

where

$\begin{array}{l}p(t, {\bf r})\end{array}$	reservoir pressure	$\begin{array}{l}t\end{array}$	time
$\begin{array}{l}\phi({\bf r})\end{array}$	effective porosity	$\begin{array}{l}{\bf r }\end{array}$	position vector
$\begin{array}{l}c_t(p)\end{array}$	total compressibility	$\begin{array}{l}{\bf r }_k\end{array}$	position vector of the $\begin{array}{l}k\end{array}$ -th source
$\begin{array}{l}k\end{array}$	formation permeability to a given fluid	$\begin{array}{l}\nabla\end{array}$	gradient operator
$\begin{array}{l}\mu(p)\end{array}$	dynamic viscosity of a given fluid	$\begin{array}{l}\displaystyle \Psi(p) =2 \, \int_0^p \frac{p \, dp}{\mu(p) \, Z(p)}\end{array}$	Pseudo-Pressure
$\begin{array}{l}Z(p)\end{array}$	fluid compressibility factor	$\begin{array}{l}\displaystyle \tau(t) = \int_0^t \frac{dt}{\mu(p) \, c_t(p)}\end{array}$	Pseudo-Time
		$\begin{array}{l}q_k(t)\end{array}$	sandface flowrates to/from the well-reservoir contact

Derivation

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See also