(1)	$\begin{array}{l}\displaystyle t^*(t) = \mu_(p_{ref}) \cdot c_t(p_{ref}) \cdot \tau(t) = \mu_(p_{ref}) \cdot c_t(p_{ref}) \cdot \int_0^t \frac{dt}{\mu(p) \, c_t(p)}\end{array}$

where

$\begin{array}{l}p_{ref}\end{array}$	reference pressure
$\begin{array}{l}\tau(t)\end{array}$	pseudo-time at time moment $\begin{array}{l}t\end{array}$
$\begin{array}{l}\mu(p)\end{array}$	dynamic fluid viscosity
$\begin{array}{l}c_t(p)\end{array}$	total compressibility
$\begin{array}{l}p(t)\end{array}$	bottom-hole pressure or field average formation pressure as function of time (see below)

It is widely used in Pressure Diffusion @model and transient data analysis (PTA / RTA ) of strongly compressible fluids.

The choice of reference pressure $\begin{array}{l}p_{ref}\end{array}$ is made by engineer depending on common sense and objectives of the study.

The usual practise is to select $\begin{array}{l}p_{ref} = p(t=0)\end{array}$ as the bottom-hole pressure at initial time moment.

Selection of $\begin{array}{l}p(t)\end{array}$ depends on objectives.

In Pressure Build Up (PBU) survey the $\begin{array}{l}p(t)\end{array}$ is calculated as sandface bottom-hole pressure: $\begin{array}{l}p(t) = p_{wf}(t)\end{array}$ .

In Drawdown survey (DD) survey the $\begin{array}{l}p(t)\end{array}$ is calculated as field-average formation pressure: $\begin{array}{l}p(t) = p_r(t)\end{array}$ .

In case the Normalized Pseudo-Time is used to linearize the Pressure Diffusion equation the choice of reference pressure is not going to affect the solution.

Normalized Pseudo-Time does represent a time in terms of physical property and has the same dimension unlike Pseudo-Time.

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See also