Derivation of Single-phase pressure diffusion @model

We start with reservoir pressure diffusion outside wellbore:

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

(2)	$\begin{array}{l}\displaystyle \int_{{\Gamma}} \, \rho(p) \, {\bf u} \, d {\bf A} = m_{\Gamma}(t)\end{array}$

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
$\begin{array}{l}m_k(t)\end{array}$	mass rate at $\begin{array}{l}k\end{array}$ -th well $\begin{array}{l}m_k(t) = \rho(p) \cdot q_k(t)\end{array}$
$\begin{array}{l}q_k(t)\end{array}$	sandface flowrate at $\begin{array}{l}k\end{array}$ -th well
$\begin{array}{l}\rho(p)\end{array}$	fluid density as function of reservoir fluid pressure $\begin{array}{l}p\end{array}$

Then use the following equality:

(3)

$\begin{array}{l}\displaystyle d(\rho \, \phi) = \rho \, d \phi + \phi \, d\rho = \rho \, \phi \, \left( \frac{d \phi }{\phi} + \frac{d \rho }{\rho} \right) = \rho \, \phi \, \left( \frac{1}{\phi} \frac{d \phi}{dp} \, dp + \frac{1}{\rho} \frac{d \rho}{dp} \, dp \right) = \rho \, \phi \, (c_{\phi} \, dp + c \, dp) = \rho \, \phi \, c_t \, dp\end{array}$

to arrive at:

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

(5)	$\begin{array}{l}\displaystyle \int_{{\Gamma}} \, {\bf u} \, d {\bf A} = q_{\Gamma}(t)\end{array}$

where

$\begin{array}{l}c_t = с_\phi+ c\end{array}$

total compressibility

Then use the following equality:

(6)	$\begin{array}{l}\displaystyle \nabla \, ( \rho \, {\bf u}) = \rho \, \nabla \, {\bf u} + (\nabla \rho, \, {\bf u}) = \rho \, \nabla \, {\bf u} + \frac{d\rho}{dp} \cdot (\nabla p, \, {\bf u}) = \rho \, \nabla \, {\bf u} + \rho \, c \cdot (\nabla p, \, {\bf u})\end{array}$

where $\begin{array}{l}\displaystyle c(p) = \frac{1}{\rho} \frac{d\rho}{dp}\end{array}$ is fluid compressibility.

Substituting (6) into (4) and reducing the density $\begin{array}{l}\rho(p)\end{array}$ one arrives to:

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

(8)	$\begin{array}{l}\displaystyle \int_{{\Gamma}} \, {\bf u} \, d {\bf A} = q_{\Gamma}(t)\end{array}$

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Derivation of Single-phase pressure diffusion @model