Motivation

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

This type of reservoir fluid flow is called radial fluid flow and corresponding pressure diffusion models provide a diagnostic basis for pressure-rate base reservoir flow analysis.

The radial flow can be infinite acting or boundary dominated or transiting from one to another.

Although the actual reservoir fluid flow may not have an axial symmetry around the well-reservoir contact or around reservoir inhomogeneities (like boundary and faults and composite areas) but still in many practical cases the long-term correlation between the flowrate and bottom-hole pressure response can be approximated by a radial flow pressure 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

$\begin{array}{l}k\end{array}$	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

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

Mathematical Model

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

Applications

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. It also called Dupuis

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

Page tree

Motivation

Inputs & Outputs

Physical Model

Mathematical Model

Applications

See Also

Page tree

Steady State Radial Flow Pressure Diffusion @model

Motivation

Inputs & Outputs

Physical Model

Mathematical Model

Applications

See Also