Wellbore Pressure-Temperature Correlation

In stagnant wellbore fluid conditions the pressure and temperature variations in time are linearly correlated:

(1)	$\begin{array}{l}\displaystyle p(t) = a + b \cdot T(t)\end{array}$

and the regression coefficient is related to along-hole pressure gradient $\begin{array}{l}\nabla p\end{array}$ and along-hole temperature gradient $\begin{array}{l}\nabla T\end{array}$ as:

(2)	$\begin{array}{l}\displaystyle b = \frac{d p}{ d T} = \frac{\nabla p }{ \nabla T }\end{array}$

Derivation

In stationary conditions:

(3)	$\begin{array}{l}\displaystyle T(z) = T_0 + \nabla T \cdot (z-z_0)\end{array}$

(4)	$\begin{array}{l}\displaystyle p(z) = p_0 + \nabla p \cdot (z-z_0)\end{array}$

Assume very slow quasi-isothermal variations of fluid column $\begin{array}{l}z = z(t)\end{array}$ then:

(5)	$\begin{array}{l}\displaystyle T(z(t)) = T_0 + \nabla T \cdot (z(t)-z_0) \rightarrow dT(t) = \nabla T \cdot dz(t)\end{array}$

(6)	$\begin{array}{l}\displaystyle p(z(t)) = p_0 + \nabla p \cdot (z(t)-z_0)\rightarrow dp(t) = \nabla p \cdot dz(t)\end{array}$

Dividing (6) by (5) leads to (2).

The analysis of wellbore pressure-temperature correlation helps in QC of gauge metrology and check for fluid stagnancy.

Non-linearity in $\begin{array}{l}dp \ {\rm vs} \ dT\end{array}$ is a sign of a poor gauge calibration or fluid movement.

In practical applications, the pressure and temperature gradients can be calculated form data readings of a pair of calibrated downhole pressure gauges:

$\begin{array}{l}\displaystyle {\rm Gauge_{up}} = \{ p_{\rm up}(t), \ T_{\rm up}(t) \}\end{array}$

$\begin{array}{l}\displaystyle {\rm Gauge_{down}} = \{ p_{\rm down}(t), \ T_{\rm down}(t) \}\end{array}$

with $\begin{array}{l}\Delta l_z\end{array}$ true vertical spacing between the sensors.

This allows calculating the pressure and temperature gradients along the hole as:

(7)	$\begin{array}{l}\displaystyle \nabla p(t) = \frac{p_{\rm down}(t)-p_{\rm up}(t)}{\Delta l_z} = \frac{\Delta p(t)}{\Delta l_z}\end{array}$

(8)	$\begin{array}{l}\displaystyle \nabla T(t) = \frac{T_{\rm down}(t)-T_{\rm up}(t)}{\Delta l_z} = \frac{\Delta T(t)}{\Delta l_z}\end{array}$

Building a linear regression with sliding window $\begin{array}{l}\delta_t\end{array}$ :

(9)	$\begin{array}{l}\displaystyle [ \Delta p(t) ]_{\delta_t} = \frac{dp}{dT} \cdot [ \Delta T(t) ]_{\delta_t}\end{array}$

provides a practical estimate of $\begin{array}{l}\displaystyle \frac{dp}{dT}\end{array}$ as function of time $\begin{array}{l}t\end{array}$ .

For stagnant vertical water column $\begin{array}{l}\rho_w\end{array}$ = 10³ kg/m³ with standard geothermal gradient $\begin{array}{l}\nabla T\end{array}$ = 0.025 K/m the derivative should be around $\begin{array}{l}\displaystyle \frac{dp}{dT} = \frac{\rho_w \, g}{\nabla T}\end{array}$ = 3.9 · 10⁵ Pa/K.

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Wellbore Pressure-Temperature Correlation