(1)	$\begin{array}{l}\displaystyle \phi \cdot \partial_\tau \Psi - \nabla \cdot \left( k \cdot \vec \nabla \Psi \right) = 0\end{array}$

(2)	$\begin{array}{l}\displaystyle -\frac{k}{\mu} \int_{\Sigma} \, \nabla p \, d {\bf A} = q(t)\end{array}$

where

$\begin{array}{l}p(t, {\bf r})\end{array}$	reservoir pressure	$\begin{array}{l}t\end{array}$	time
$\begin{array}{l}\phi({\bf r})\end{array}$	effective porosity	$\begin{array}{l}{\bf r }\end{array}$	position vector
$\begin{array}{l}c_t(p)\end{array}$	total compressibility	$\begin{array}{l}\nabla\end{array}$	gradient operator
$\begin{array}{l}k\end{array}$	formation permeability to a given fluid	$\begin{array}{l}d {\bf A}\end{array}$	normal surface element of well-reservoir contact
$\begin{array}{l}\mu(p)\end{array}$	dynamic viscosity of a given fluid	$\begin{array}{l}\displaystyle \Psi(p) =2 \, \int_0^p \frac{p \, dp}{\mu(p) \, Z(p)}\end{array}$	Pseudo-Pressure
$\begin{array}{l}Z(p)\end{array}$	fluid compressibility factor	$\begin{array}{l}\displaystyle \tau(t) = \int_0^t \frac{dt}{\mu(p) \, c_t(p)}\end{array}$	Pseudo-Time
		$\begin{array}{l}q(t)\end{array}$	sandface flowrates to/from the well-reservoir contact

Derivation

Derivation of pseudo-linear pressure diffusion @model

In some practical cases the complex $\begin{array}{l}c_t \, \mu\end{array}$ can be considered as constant in time which makes

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a linear differential equation.

But during the early transition times the pressure drop is usually high and the complex $\begin{array}{l}c_t \, \mu\end{array}$ can not be considered as constant in time which leads to distortion of pressure transient diagnostics at early times.

In this case one can use Pseudo-Time, calculated by means of the bottom-hole pressure $\begin{array}{l}p_{BHP}(t)\end{array}$ :

(3)	$\begin{array}{l}\displaystyle \tau(t) = \int_0^t \frac{dt}{\mu(p_{BHP}(t)) \, c_t(p_{BHP}(t))}\end{array}$

to correct early-time transient behaviour.

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See also

Page tree

Pseudo-linear pressure diffusion @model

See also