Motivation

In many practical cases the reservoir flow created by well is getting aligned with a radial direction towards or away from well.

This type of flow is called radial fluid flow and a type library model provides a reference for radial fluid flow diagnostics.

Inputs & Outputs

Detailing

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

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

Definition

(1)	$\begin{array}{l}\displaystyle \frac{\partial p}{\partial t} = \chi \, \left( \frac{\partial^2 p}{\partial r^2} + \frac{1}{r} \frac{\partial p}{\partial r} \right)\end{array}$

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

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

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

Solution

(5)	$\begin{array}{l}\displaystyle p(t,r) = p_i + \frac{q_t}{4 \pi \sigma} \, {\rm Ei} \bigg( - \frac{r^2}{4 \chi t} \bigg)\end{array}$

(6)	$\begin{array}{l}\displaystyle p_{wf}(t) = p_i + \frac{q_t}{4 \pi \sigma} \, \bigg[ - 2S + {\rm Ei} \bigg( - \frac{r_w^2}{4 \chi t} \bigg) \bigg]\end{array}$

Derivation