Motivation

The pipeline and wellbore flow simulations require a model of static fluid pressure $\begin{array}{l}p\end{array}$ variation as a function of elevation $\begin{array}{l}z\end{array}$ :

(1)	$\begin{array}{l}\displaystyle p = p(z)\end{array}$

Output

$\begin{array}{l}p(z)\end{array}$

Fluid pressure $\begin{array}{l}p\end{array}$

Input

$\begin{array}{l}z\end{array}$	Elevation	$\begin{array}{l}\rho_0\end{array}$	Fluid density at Logging reference point $\begin{array}{l}z_0\end{array}$
$\begin{array}{l}z_0\end{array}$	Logging reference point (usually at surface)	$\begin{array}{l}c_0\end{array}$	Fluid Compressibility at Logging reference point $\begin{array}{l}z_0\end{array}$
$\begin{array}{l}g\end{array}$	Standard gravity constant

The static balance equation for fluid column is:

(2)	$\begin{array}{l}\displaystyle \frac{dp}{dz}= \rho(p) \cdot g\end{array}$

Full-range model

(3)	$\begin{array}{l}\displaystyle p(z) = p_0 + \rho_0 \cdot g \cdot (z-z_0)\end{array}$

(4)	$\begin{array}{l}\displaystyle p(z) = p_0 \cdot \exp \left[ - \frac{\rho_0 \, g}{p_0} \cdot (z-z_0) \right]\end{array}$

(5)	$\begin{array}{l}\displaystyle \frac{1+ c_0 \, p(z)}{1 + c_0 \, p_0} = \exp \left[ \frac{ с_0 \cdot \rho_0 \cdot g \cdot (z-z_0)}{1+c_0 \, p_0} \right]\end{array}$

(6)	$\begin{array}{l}\displaystyle p_2 - p_1 = \rho_0 \cdot g \cdot (z_2-z_1)\end{array}$

(7)	$\begin{array}{l}\displaystyle p_2 - p_1 = p_1 \cdot \left( \exp \left[ \frac{ \rho_0 \cdot g \cdot (z_2-z_1)} {p_0} \right] - 1 \right)\end{array}$

(8)	$\begin{array}{l}\displaystyle p_2 - p_1 = \frac{ (1+c_0 \, p_1)}{c_0} \cdot \left( \exp \left[ \frac{ с_0 \cdot \rho_0 \cdot g \cdot (z_2-z_1)}{1+c_0 \, p_0} \right] - 1 \right)\end{array}$