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 wells (see linear fluid flow).

This happens when wells are placed in a channel or a narrow compartment.

It also happens around fracture planes and conductive faults. It also develops temporarily at early times of the transients in horizontal wells.

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	$\begin{array}{l}p(t,x)\end{array}$	reservoir pressure
$\begin{array}{l}p_i\end{array}$	initial formation pressure	$\begin{array}{l}p_{wf}(t)\end{array}$	well bottomhole 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

Physical Model

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

Mathematical Model

Definition

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

(2)	$\begin{array}{l}\displaystyle p(t = 0, x) = p_i\end{array}$

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

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

Solution

(5)	$\begin{array}{l}\displaystyle p(t,x) = p_i - \frac{q_t}{\sigma \, d} \bigg[ \sqrt{\frac{4 \chi t}{\pi}} \exp \bigg( -\frac{x^2}{4 \chi t} \bigg) - x \, \bigg[ 1- {\rm erf} \bigg(\frac{x}{\sqrt{4 \, \chi \, t}} \bigg) \bigg] \bigg]\end{array}$