Derivation of Linear pressure diffusion @model

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

and neglect the non-linear term $\begin{array}{l}c \cdot ( {\bf u} \, \nabla p)\end{array}$ for low compressibility fluid $\begin{array}{l}c \sim 0\end{array}$ 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:

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

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

or

(5)	$\begin{array}{l}\displaystyle \phi \, c_t \cdot \partial_t p + = \nabla \left( \frac{k}{\mu} \cdot ( \nabla p - \rho \, {\bf g}) \right) + \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k)\end{array}$

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Derivation of Linear pressure diffusion @model