Derivation of pseudo-linear pressure diffusion @model

We start with reservoir pressure diffusion outside wellbore:

(1)	$\begin{array}{l}\displaystyle \frac{\partial (\rho \phi)}{\partial t} + \nabla \, ( \rho \, {\bf u}) = 0\end{array}$

(2)	$\begin{array}{l}\displaystyle \int_{\Sigma_k} \, {\bf u} \, d {\bf A} = q_k(t)\end{array}$

where

$\begin{array}{l}\Sigma_k\end{array}$	well-reservoir contact of the $\begin{array}{l}k\end{array}$ -th well
$\begin{array}{l}d {\bf \Sigma}\end{array}$	normal vector of differential area on the well-reservoir contact, pointing inside wellbore

Then use the following equality:

(3)

$\begin{array}{l}\displaystyle d(\rho \, \phi) = \rho \, d \phi + \phi \, d\rho = \rho \, \phi \, \left( \frac{d \phi }{\phi} + \frac{d \rho }{\rho} \right) = \rho \, \phi \, \left( \frac{1}{\phi} \frac{d \phi}{dp} \, dp + \frac{1}{\rho} \frac{d \rho}{dp} \, dp \right) = \rho \, \phi \, (c_{\phi} \, dp + c \, dp) = \rho \, \phi \, c_t \, dp\end{array}$

to arrive at:

(4)	$\begin{array}{l}\displaystyle \rho \, \phi \, c_t \cdot \frac{\partial p}{\partial t} + \nabla \, ( \rho \, {\bf u}) = 0\end{array}$

(5)	$\begin{array}{l}\displaystyle \int_{\Sigma_k} \, {\bf u} \, d {\bf A} = q_k(t)\end{array}$

where

$\begin{array}{l}c_t = с_\phi+ c\end{array}$

total compressibility

Let's assume Darcy flow with constant permeability $\begin{array}{l}\displaystyle \frac{dk}{dp} = 0\end{array}$ and ignore gravity forces:

(6)	$\begin{array}{l}\displaystyle {\bf u} = \frac{k}{\mu} \nabla \, p\end{array}$

so that diffusion equation becomes:

(7)	$\begin{array}{l}\displaystyle \rho \, \phi \, c_t \cdot \frac{\partial p}{\partial t} + \nabla \, ( k \cdot \frac{\rho}{\mu} \, \nabla \, p) = 0\end{array}$

(8)	$\begin{array}{l}\displaystyle \frac{k}{\mu} \cdot \int_{\Sigma_k} \, {\bf \nabla } \, p \cdot d {\bf A} = q_k(t)\end{array}$

Let's express the density via Z-factor:

(9)	$\begin{array}{l}\displaystyle \rho = \frac{M}{RT} \, \frac{p}{Z(p)}\end{array}$

where

$\begin{array}{l}T\end{array}$	fluid temperature
$\begin{array}{l}M\end{array}$	molar mass of a fluid
$\begin{array}{l}R\end{array}$	gas constant

and assuming the fluid temperature $\begin{array}{l}T\end{array}$ does not change over time and space during the modelling period:

(10)	$\begin{array}{l}\displaystyle \phi \, c_t \, \mu \cdot \frac{p}{\mu \, Z} \cdot \frac{\partial p}{\partial t} + \nabla \, ( k \cdot \frac{p}{\mu \, Z} \, \nabla \, p) = 0\end{array}$

(11)	$\begin{array}{l}\displaystyle \frac{k}{\mu} \cdot \int_{\Sigma_k} \, {\bf \nabla } \, p \cdot d {\bf A} = q_k(t)\end{array}$

or

(12)	$\begin{array}{l}\displaystyle \phi \, c_t \, \mu \cdot \frac{\partial \Psi}{\partial t} + \nabla \, ( k \cdot \nabla \, \Psi) = 0\end{array}$

(13)	$\begin{array}{l}\displaystyle \frac{k}{\mu} \cdot \int_{\Sigma_k} \, {\bf \nabla } \, p \cdot d {\bf A} = q_k(t)\end{array}$

where

$\begin{array}{l}\displaystyle \Psi(p) =2 \, \int_0^p \frac{p \, dp}{\mu(p) \, Z(p)}\end{array}$

Pseudo-Pressure

In some practical cases the complex $\begin{array}{l}c_t \, \mu\end{array}$ can be considered as constant in time which makes (12) 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 $\begin{array}{l}p_{BHP}(t)\end{array}$ :

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

to correct early-time transient behaviour which turn equation (12) into:

(15)	$\begin{array}{l}\displaystyle \phi \cdot \frac{\partial \Psi}{\partial \tau} + \nabla \, ( k \cdot \nabla \, \Psi) = 0\end{array}$

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Derivation of pseudo-linear pressure diffusion @model