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 corresponding pressure diffusion models provide a diagnostic basis for pressure-rate base reservoir flow analysis.

Although the actual flow may not have an axial symmetry around the well-reservoir contact or reservoir inhomogeneities (like boundary and faults and composite areas) but still the dominant part of wellbore and reservoir pressure variation is usually radial-flow or linear-flow and the two represent the basis for Pressure diffusion analysis.

Inputs & Outputs

Inputs		Outputs
$\begin{array}{l}q_t\end{array}$	total sandface rate	$\begin{array}{l}p(t,r)\end{array}$	reservoir pressure
$\begin{array}{l}{p_i}\end{array}$	initial formation pressure	$\begin{array}{l}{p_{wf}(t)}\end{array}$	well bottomhole pressure
$\begin{array}{l}\sigma\end{array}$	transmissibility
$\begin{array}{l}\chi\end{array}$	pressure diffusivity

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

Physical Model

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

Mathematical Model

Definition

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

(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