Motivation

In many practical cases the reservoir flow created by well or group of wells is getting aligned with a specific linear direction away from well.

This happens when well is placed in a channel or a narrow compartment. It also happens around fracture planes and conductive faults.

This type of flow is called linear fluid flow and corresponding PTA type library models provides a reference for linear fluid flow diagnostics.

Inputs & Outputs

Inputs		Outputs
$\begin{array}{l}q_t\end{array}$	total sandface rate	$\begin{array}{l}p(x)\end{array}$	reservoir pressure
$\begin{array}{l}p_i\end{array}$	initial formation pressure
$\begin{array}{l}d\end{array}$	reservoir channel width
$\begin{array}{l}\sigma\end{array}$	transmissibility, $\begin{array}{l}\sigma = \frac{k \, h}{\mu}\end{array}$
$\begin{array}{l}\chi\end{array}$	pressure diffusivity, $\begin{array}{l}\chi = \frac{k}{\mu} \, \frac{1}{\phi \, c_t}\end{array}$

Detailing

$\begin{array}{l}k\end{array}$	absolute permeability	$\begin{array}{l}c_t\end{array}$	total compressibility, $\begin{array}{l}c_t = c_r + c\end{array}$
$\begin{array}{l}h\end{array}$	effective thickness	$\begin{array}{l}{c_r}\end{array}$	pore compressibility
$\begin{array}{l}\mu\end{array}$	dynamic fluid viscosity	$\begin{array}{l}c\end{array}$	fluid compressibility
$\begin{array}{l}{\phi}\end{array}$	porosity

Linear fluid flow	Homogenous reservoir	Infinite boundary	Zero wellbore radius	Slightly compressible fluid flow	Constant rate production
$\begin{array}{l}p(t, {\bf r}) \rightarrow p(x)\end{array}$ $\begin{array}{l}{\bf r} \in ℝ^2 = \{ x, y\}\end{array}$	$\begin{array}{l}M(x, p)=M =\rm const\end{array}$ $\begin{array}{l}\phi(x, p)=\phi =\rm const\end{array}$ $\begin{array}{l}h(x)=h =\rm const\end{array}$	$\begin{array}{l}x \rightarrow \infty\end{array}$	$\begin{array}{l}r_w = 0\end{array}$	$\begin{array}{l}c_t(p) = c_r +c = \rm const\end{array}$	$\begin{array}{l}q_t = \rm const\end{array}$

Definition

(1)	$\begin{array}{l}\displaystyle \frac{\partial p}{\partial t} = \chi \, \frac{d^2 p}{dx^2} = 0\end{array}$

(2)	$\begin{array}{l}\displaystyle p(t, x \rightarrow L ) = p_i\end{array}$

(3)	$\begin{array}{l}\displaystyle \frac{\partial p(t, x )}{\partial x} \bigg\|_{x \rightarrow 0} = \frac{q_t}{\sigma \, d}\end{array}$

Solution

(4)	$\begin{array}{l}\displaystyle p(x) = p_i - \frac{q_t}{\sigma \, d} (x - L)\end{array}$

Derivation