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

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

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	$\begin{array}{l}d {\bf \Sigma}\end{array}$	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 Pseudo-Time being proportional to freagular time:

(3)	$\begin{array}{l}\displaystyle \tau(t) = \frac{t}{\mu \, c_t}\end{array}$

and one can write the diffusion equation as:

(4)	$\begin{array}{l}\displaystyle \phi \, c_t \, \mu \cdot \frac{\partial \Psi}{\partial \tau} - \nabla \cdot \left( k \cdot \vec \nabla \Psi \right) = 0\end{array}$

which is a treat it as a differential equation with linear coefficients.

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 $\begin{array}{l}p_{BHP}(t)\end{array}$ :

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

to correct early-time transient behaviour.

In case of the ideal gas equation of state, the Z-factor has a unit value: $\begin{array}{l}Z(p) = 1\end{array}$ , viscosity does not depend on pressure $\begin{array}{l}\mu(p) = \mu\end{array}$ and total compressibility is fully defined by fluid compressibility $\begin{array}{l}\displaystyle c_t = c_r + c \sim \frac{1}{p}\end{array}$ which simplifies the expression for Pseudo-Pressure and Pseudo-Time as to:

(6)	$\begin{array}{l}\displaystyle \Psi(p) = \frac{p^2}{\mu}\end{array}$

(7)	$\begin{array}{l}\displaystyle \tau(t) = \frac{1}{\mu} \int_0^t p_{BHP}(t) dt\end{array}$

Page tree

See also

Page tree

Pseudo-linear pressure diffusion @model

See also