Derivation of pseudo-linear pressure diffusion @model

(1)	$\begin{array}{l}\displaystyle \frac{\partial (\rho \phi)}{\partial t} + \nabla \, ( \rho \, {\bf u}) = \rho \cdot \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k)\end{array}$

(2)	$\begin{array}{l}\displaystyle \int_{\Sigma_k} \, {\bf u} \, d {\bf A} = q_k(t)\end{array}$

where

$\begin{array}{l}\Sigma_k\end{array}$	well-reservoir contact of the $\begin{array}{l}k\end{array}$ -th well
$\begin{array}{l}d {\bf \Sigma}\end{array}$	normal vector of differential area on the well-reservoir contact, pointing inside wellbore

and use

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to arrive at:

(3)	$\begin{array}{l}\displaystyle \rho \, c_t \cdot \frac{\partial (p)}{\partial t} + \nabla \, ( \rho \, {\bf u}) = 0\end{array}$

(4)	$\begin{array}{l}\displaystyle \int_{\Sigma_k} \, {\bf u} \, d {\bf A} = q_k(t)\end{array}$

where

$\begin{array}{l}c_t\end{array}$

Let's assume Darcy flow and ignore gravity forces:

(5)	$\begin{array}{l}\displaystyle {\bf u} = \frac{k}{\mu} \nabla \, p\end{array}$

so that diffusion equation becomes:

(6)	$\begin{array}{l}\displaystyle \rho \, c_t \cdot \frac{\partial p}{\partial t} + \nabla \, ( k \cdot \frac{\rho}{\mu} \, \nabla \, p) = 0\end{array}$

(7)	$\begin{array}{l}\displaystyle \frac{k}{\mu} \cdot \int_{\Sigma_k} \, {\bf p} \, d {\bf A} = q_k(t)\end{array}$

Let's express the density via Z-factor:

(8)	$\begin{array}{l}\displaystyle \rho = \frac{M}{RT} \, \frac{p}{Z(p)}\end{array}$

where

and assuming the fluid temperature $\begin{array}{l}T\end{array}$ does not change over time and space during the modelling period:

(9)	$\begin{array}{l}\displaystyle \rho \, c_t \, \mu \cdot \frac{p}{\mu \, Z} \cdot \frac{\partial p}{\partial t} + \nabla \, ( k \cdot \frac{p}{\mu \, Z} \, \nabla \, p) = 0\end{array}$

(10)	$\begin{array}{l}\displaystyle \frac{k}{\mu} \cdot \int_{\Sigma_k} \, {\bf p} \, d {\bf A} = q_k(t)\end{array}$

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