Derivation of Linear pressure diffusion @model

We start with reservoir pressure diffusion outside wellbore:

(1)	$\begin{array}{l}\displaystyle \frac{\partial (\rho \phi)}{\partial t} + \nabla \, ( \rho \, {\bf u}) = 0\end{array}$

(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

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

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

where

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

total compressibility

We start with (Single-phase pressure diffusion @model:1):

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

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

and neglect the non-linear term $\begin{array}{l}c \cdot ( {\bf u} \, \nabla p)\end{array}$ for low compressibility fluid $\begin{array}{l}c \sim 0\end{array}$ or equivalently to a constant fluid density: $\begin{array}{l}\rho(p) = \rho = \rm const\end{array}$ .

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

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:

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

(9)	$\begin{array}{l}\displaystyle {\bf u} = - \frac{k}{\mu} \cdot ( \nabla p - \rho \, {\bf g})\end{array}$

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Derivation of Linear pressure diffusion @model