Motivation

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

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, $\begin{array}{l}\sigma = \frac{k \, h}{\mu}\end{array}$
$\begin{array}{l}\chi\end{array}$	pressure diffusivity, $\begin{array}{l}\chi = \frac{k}{\mu} \, \frac{1}{\phi \, c_t}\end{array}$
$\begin{array}{l}S\end{array}$	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
$\begin{array}{l}p(t, {\bf r})\end{array}$ $\begin{array}{l}{\bf r} \in ℝ^2 = \{ x, y\}\end{array}$	$\begin{array}{l}M(r, p)=M =\rm const\end{array}$ $\begin{array}{l}\phi(r, p)=\phi =\rm const\end{array}$ $\begin{array}{l}h(r)=h =\rm const\end{array}$ $\begin{array}{l}c_r(r)=c_r =\rm const\end{array}$	$\begin{array}{l}r_w \leq r \leq r_e < \infty\end{array}$	$\begin{array}{l}c_t(r,p) = \rm const\end{array}$	$\begin{array}{l}q_t = \rm const\end{array}$	$\begin{array}{l}S = \rm const\end{array}$

Mathematical Model

Definition

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

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

(3)	$\begin{array}{l}\displaystyle \left[ \frac{\partial p}{\partial r} \right]_{r=r_e} = 0\end{array}$

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

(5)	$\begin{array}{l}\displaystyle p_{wf}(t)= p(t, r_w) - S \cdot \left[ r \frac{\partial p(t,r)}{\partial r} \right]_{r=r_w} = p(t, r_w) - \frac{q_t}{2 \pi \sigma} S\end{array}$

Disclaimer

It is important to note that equations (1)– (4) do not constitute a complete CVP as it does not specify the initial condition.

Solution

(6)	$\begin{array}{l}\displaystyle p(t,r) = p_i - \frac{{\rm w \,} q_t }{V_e \, \phi \, c_t} \, t + \frac{{\rm w \,} q_t }{4\pi \sigma} \left[ 2 \ln \frac{r}{r_e} - \frac{r^2}{r_e^2} + 1 \right] , \quad r_{wf} < r \leq r_e, \quad {\rm w }= 1 - \frac{r_w^2}{r_e^2}\end{array}$

(7)	$\begin{array}{l}\displaystyle p_e(t) = p_i - \frac{{\rm w \,} q_t}{V_e \phi c_t}t\end{array}$

(8)	$\begin{array}{l}\displaystyle p_{wf}(t) = p_e(t) - \frac{q_t}{2 \pi \sigma} \, \left[ {\rm w\, } \ln \frac{r_e}{r_w} + 0.5 + S \right]\end{array}$

Derivation

Approximations

(9)	$\begin{array}{l}\displaystyle {\rm w} = 1 - \frac{r_w^2}{r_e^2} \approx 1\end{array}$

(10)	$\begin{array}{l}\displaystyle p(t,r) \approx p_i - \frac{q_t}{V_e \, \phi \, c_t} \, t + \frac{q_t}{4\pi \sigma} \bigg[ 2 \ln \frac{r}{r_e} - \frac{r^2}{r_e^2} + 1 \bigg] , \quad r_{wf} < r \leq r_e\end{array}$

(11)	$\begin{array}{l}\displaystyle p_{wf}(t) \approx p_e(t) - \frac{q_t}{2 \pi \sigma} \, \left[ \ln \frac{r_e}{r_w} + 0.5 + S \right]\end{array}$

(12)	$\begin{array}{l}\displaystyle p_e(t) \approx p_i - \frac{q_t}{V_e \phi c_t}t\end{array}$

Applications

Equation (6) 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:

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

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:

(14)	$\begin{array}{l}\displaystyle J_t = \frac{q_t}{p_r(t) - p_{wf}(t)} =\frac{2 \pi \sigma}{\ln \frac{r_e}{r_w} + 0.75 +S} = {\rm const}\end{array}$

Page tree

Motivation

Inputs & Outputs

Physical Model

Mathematical Model

Approximations

Applications

See Also

Page tree

Pseudo-Steady State Radial Flow Pressure Diffusion @model

Motivation

Inputs & Outputs

Physical Model

Mathematical Model

Approximations

Applications

See Also