Motivation

In many practical cases the Radial Flow Pressure Diffusion is evolving towards pressure stabilization and can be efficiently analyzed using the steady state flow model.

Inputs & Outputs

Inputs		Outputs
$\begin{array}{l}q_t\end{array}$	total sandface rate	$\begin{array}{l}p(r)\end{array}$	reservoir pressure
$\begin{array}{l}{p_i}\end{array}$	initial formation pressure	$\begin{array}{l}p_{wf}\end{array}$	well bottomhole pressure
$\begin{array}{l}\sigma\end{array}$	transmissibility, $\begin{array}{l}\sigma = \frac{k \, h}{\mu}\end{array}$
$\begin{array}{l}S\end{array}$	skin-factor
$\begin{array}{l}r_w\end{array}$	wellbore radius
$\begin{array}{l}r_e\end{array}$	drainage radius

Detailing

Radial fluid flow	Homogenous reservoir	Finite reservoir flow boundary	Slightly compressible fluid flow	Constant rate	Constant skin
$\begin{array}{l}p(t, {\bf r}) \rightarrow p(r)\end{array}$ $\begin{array}{l}{\bf r} \in ℝ^2 = \{ x, y\}\end{array}$	$\begin{array}{l}M(r, p)=M =\rm const\end{array}$ $\begin{array}{l}\phi(r, p)=\phi =\rm const\end{array}$ $\begin{array}{l}h(r)=h =\rm const\end{array}$ $\begin{array}{l}c_r(r)=c_r =\rm const\end{array}$	$\begin{array}{l}r_w \leq r \leq r_e < \infty\end{array}$	$\begin{array}{l}c_t(r,p) = \rm const\end{array}$	$\begin{array}{l}q_t = \rm const\end{array}$	$\begin{array}{l}S = \rm const\end{array}$

Definition

(1)	$\begin{array}{l}\displaystyle r_{wf} < r \leq r_e\end{array}$

(2)	$\begin{array}{l}\displaystyle p(t, r ) = p(r) \Leftrightarrow \frac{\partial p}{\partial t} = 0\end{array}$

(3)	$\begin{array}{l}\displaystyle \frac{\partial^2 p}{\partial r^2} + \frac{1}{r} \frac{\partial p}{\partial r} =0\end{array}$

(4)	$\begin{array}{l}\displaystyle p(r_e ) = p_i\end{array}$

(5)	$\begin{array}{l}\displaystyle \left[ r\frac{\partial p(r)}{\partial r} \right]_{r \rightarrow r_w} = \frac{q_t}{2 \pi \sigma}\end{array}$

(6)	$\begin{array}{l}\displaystyle p_{wf}= p(r_w ) - S \cdot r_w \, \frac{\partial p}{\partial r} \Bigg\|_{r=r_w}\end{array}$

Solution

(7)	$\begin{array}{l}\displaystyle p(r) = p_i + \frac{q_t}{2 \pi \sigma} \, \ln \frac{r}{r_e} = p(r_w) + \frac{q_t}{2 \pi \sigma} \, \ln \frac{r}{r_w} , \quad r_{wf} < r \leq r_e\end{array}$

(8)	$\begin{array}{l}\displaystyle p_{wf} = p_i - \frac{q_t}{2 \pi \sigma} \, \bigg[ S + \ln \frac{r_e}{r_w} \bigg]\end{array}$

Derivation

Equation (7) shows how the basic diffusion model parameters impact the relation between drawdown $\begin{array}{l}\Delta p = p_i - p_{wf}\end{array}$ and total sandface flowrate $\begin{array}{l}q_t\end{array}$ and plays important methodological role as they are used in many algorithms and express-methods of Pressure Testing.

Productivity Index Analysis

The Total Sandface Productivity Index for low-compressibility fluid and low-compressibility rocks does not depend on formation pressure, bottomhole pressure and the flowrate and can be expressed as:

(9)	$\begin{array}{l}\displaystyle J_t = \frac{q_t}{p_i - p_{wf}(t)} =\frac{2 \pi \sigma}{\ln \frac{r_e}{r_w} + S} = {\rm const}\end{array}$

The Field-average Productivity Index for low-compressibility fluid and low-compressibility rocks does not depend on formation pressure, bottomhole pressure and the flowrate and can be expressed as:

(10)	$\begin{array}{l}\displaystyle J_t = \frac{q_t}{p_r(t) - p_{wf}(t)} =\frac{2 \pi \sigma}{\ln \frac{r_e}{r_w} + 0.5 +S} = {\rm const}\end{array}$