Derivation of linear single-phase pressure diffusion @model

We start with the general from of (Single-phase pressure diffusion @model:1):

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

(2)	$\begin{array}{l}\displaystyle {\bf u} = - M \cdot ( \nabla p - \rho \, {\bf g})\end{array}$

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

where

$\begin{array}{l}p(t, {\bf r})\end{array}$	reservoir pressure	$\begin{array}{l}t\end{array}$	time
$\begin{array}{l}\rho({\bf r})\end{array}$	fluid density	$\begin{array}{l}{\bf r }\end{array}$	position vector
$\begin{array}{l}\phi({\bf r})\end{array}$	effective porosity	$\begin{array}{l}{\bf r }_k\end{array}$	position vector of the $\begin{array}{l}k\end{array}$ -th source
$\begin{array}{l}c_t({\bf r})\end{array}$	total compressibility	$\begin{array}{l}\delta ( \bf r )\end{array}$	Dirac delta function
$\begin{array}{l}M({\bf r})\end{array}$	reservoir fluid mobility $\begin{array}{l}M({\bf r}) = \frac{k({\bf r})}{\mu}\end{array}$	$\begin{array}{l}\nabla\end{array}$	gradient operator
$\begin{array}{l}k({\bf r})\end{array}$	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 under Darcy flow
$\begin{array}{l}q_k(t)\end{array}$	sandface flowrates of the $\begin{array}{l}k\end{array}$ -th source	$\begin{array}{l}\Gamma\end{array}$	reservoir boundary
$\begin{array}{l}q_\Gamma(t)\end{array}$	flow through the reservoir boundary $\begin{array}{l}\Gamma\end{array}$ , which is aquifer or gas cap

Let's neglect the non-linear term $\begin{array}{l}c \cdot ( {\bf u} \, \nabla p)\end{array}$ for low compressibility fluid $\begin{array}{l}c \sim 0\end{array}$ which is equivalent to assumption of nearly constant fluid density: $\begin{array}{l}\rho(p) = \rho = \rm const\end{array}$ .

Together with constant pore compressibility $\begin{array}{l}c_\phi = \rm const\end{array}$ this will lead to constant total compressibility $\begin{array}{l}c_t = c_\phi + c \approx \rm const\end{array}$ .

Assuming that permeability and fluid viscosity do not depend on pressure $\begin{array}{l}k(p) = k = \rm const\end{array}$ and $\begin{array}{l}\mu(p) = \mu = \rm const\end{array}$ one arrives to the differential equation with constant coefficients:

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

(5)	$\begin{array}{l}\displaystyle {\bf u} = - M \cdot ( \nabla p - \rho \, {\bf g})\end{array}$

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

or

(7)	$\begin{array}{l}\displaystyle \phi \cdot c_t \cdot \partial_t p + \nabla {\bf u} = 0\end{array}$

(8)	$\begin{array}{l}\displaystyle {\bf u} = - M \cdot ( \nabla p - \rho \, {\bf g})\end{array}$

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

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

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Derivation of linear single-phase pressure diffusion @model