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 a type library model provides a reference for linear fluid flow diagnostics.

Inputs & Outputs

Detailing

$\begin{array}{l}\sigma = \frac{k \, h}{\mu}\end{array}$	transmissibility
$\begin{array}{l}\chi = \frac{k}{\mu} \, \frac{1}{\phi \, c_t}\end{array}$	pressure diffusivity
$\begin{array}{l}k\end{array}$	absolute permeability
$\begin{array}{l}\phi\end{array}$	porosity
$\begin{array}{l}\mu\end{array}$	dynamic fluid viscosity
$\begin{array}{l}c_t = c_r + c\end{array}$	total compressibility
$\begin{array}{l}c_r\end{array}$	pore compressibility
$\begin{array}{l}c\end{array}$	fluid compressibility

Constant rate production	$\begin{array}{l}q_t = \rm const\end{array}$
Linear fluid flow	$\begin{array}{l}p(t, x)\end{array}$
Slightly compressible fluid flow	$\begin{array}{l}c_t(p) = c_r +c = \rm const\end{array}$
Homogeneous reservoir	$\begin{array}{l}M(x, p)=M =\rm const\end{array}$ $\begin{array}{l}\phi(x, p)=\phi =\rm const\end{array}$ $\begin{array}{l}h(x)=h =\rm const\end{array}$
Infinite boundary	$\begin{array}{l}x \rightarrow \infty\end{array}$

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