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Single-well Pressure Pulsation survey
Motivation
One of the most important objectives of the well testing is to assess the drainable oil reserves and reservoir properties around tested well.
This particularly becomes important in appraisal drilling as well testing is the only source of this information.
The Self-Pulse Test (SPT) is a single-well pressure test with periodic changes in flow rate and pressure (see Fig. 1).
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title | Expand on analytical approach |
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When flow rate is being intentionally varied in harmonic cycles with sandface amplitude
and cycling frequency LaTeX Math Inline |
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body | \omega = \frac{2 \, \pi}{T} |
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: LaTeX Math Block |
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q(t) = q_0 \, \sin ( \omega \, t ) |
then bottom-hole pressure will follow the same variation pattern:
LaTeX Math Block |
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p_{wf}(t) = p_0 \, \sin ( \omega \, [ t - t_{\Delta} ] ) |
with a bottom-hole pressure amplitude
and the time delay . It takes some time (3-5 cycles
) for pressure to develop a stabilized response to rate variations.
The pressure delay
and associated dimensionless phase shift LaTeX Math Inline |
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body | \Delta = \omega \, t_{\Delta} |
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represent the inertia effects from the adjoined reservoir and characterized by formation pressure diffusivity: LaTeX Math Block |
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\chi = \Big < \frac{k}{\mu} \Big > \frac{1}{\phi \, c_t} |
The diffusion nature of pressure dictates that amplitude of pressure variation is proportional to amplitude of sandface flowerate variation and the ratio
is related to formation transmissibility: LaTeX Math Block |
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\sigma = \Big < \frac{k}{\mu} \Big > \, h |
The exact solution of differential diffusion equation for vertical well with negligible well storage and infinite boundary homogeneous reservoir can be represented by a system of non-linear algebraic equations, relating field-measured parameters
LaTeX Math Inline |
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body | \big\{ \frac{q_0}{p_0}, \, t_{\Delta} \big\} |
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to formation properties LaTeX Math Inline |
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body | \{ \sigma, \, \chi \} |
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: LaTeX Math Block |
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X =r_w \, \sqrt{ \frac{\omega}{\chi} } |
LaTeX Math Block |
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anchor | defDelta |
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alignment | left |
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\Delta = \omega \, t_{\Delta} = \frac{\pi}{4} - arctg{ \frac{Ker_1 X \cdot Kei \, X - Ker_1 X \cdot Kei \, X }{Ker_1 X \cdot Kei \, X +Ker_1 X \cdot Kei \, X } } |
LaTeX Math Block |
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anchor | defSigma |
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alignment | left |
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\sigma =\frac{1}{2 \pi} \, \frac{q_0}{p_0} \, \sqrt{ \frac{Ker^2 X + Kei^2 X}{Ker_1^2 X + Kei_1^2 X} } |
The above equations assume that diffusivity
and dimensionless radius are found from LaTeX Math Block Reference |
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– LaTeX Math Block Reference |
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and then is substituted to LaTeX Math Block Reference |
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to calculate transmissibility .
In case of a low frequency pulsations:
LaTeX Math Block |
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\omega \ll 0.00225 \, \frac{ \chi }{ r_w^2} \quad \Longleftrightarrow \quad X \ll 0.15 |
the equations
LaTeX Math Block Reference |
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– LaTeX Math Block Reference |
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can be explicitly resolved in terms of transmissibility and diffusivity:
LaTeX Math Block |
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\chi = 0.25 \, \omega \, \gamma^2 \, r_w^2 \, \exp \frac{\pi}{2 \, {\rm tg} \Delta } |
LaTeX Math Block |
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\sigma = \frac{q_0}{8 \, p_0 \, \sin \Delta} |
where
LaTeX Math Inline |
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body | \Delta = \omega \, t_{\Delta} |
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.
The above analytical approach (either
LaTeX Math Block Reference |
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– LaTeX Math Block Reference |
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or LaTeX Math Block Reference |
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– LaTeX Math Block Reference |
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) is rarely helpful in practise. The field operations are very finnicky and difficult to follow the pre-desgined sequence of clean harmonic pulsations.
As result, the flowrate variation becomes a complex sum of harmonics:
LaTeX Math Block |
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q(t) = q_0 + \sum_{n=0}^\infty q_n \, \sin ( \omega_n \, t ) |
and the pressure response becomes complex as well:
LaTeX Math Block |
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p_{wf}(t) = p_0 + \sum_{n=0}^\infty p_n \, \sin ( \omega_n \, [ t - t_{\Delta_n} ] )
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The use of analytical formulas requires fourier transformation to identify the key harmonics from the raw data with a manual control from analyst and a certain amount of subjectivism on which harmonics to pick up for calculating the transmissibility and diffusivity.
Once the harmonics are identified one need to search for the
LaTeX Math Inline |
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body | \{ \sigma, \, \chi \} |
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best fit to a complex system of non-linear algebraic equations: LaTeX Math Block |
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X_n =r_w \, \sqrt{ \frac{\omega_n}{\chi} } |
LaTeX Math Block |
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anchor | defDelta |
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alignment | left |
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\Delta_n = \omega_n \, t_{\Delta} = \frac{\pi}{4} - arctg{ \frac{Ker_1 X_n \cdot Kei \, X_n - Ker_1 X_n \cdot Kei \, X_n }{Ker_1 X_n \cdot Kei \, X_n +Ker_1 X_n \cdot Kei \, X_n } } |
LaTeX Math Block |
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anchor | defSigma |
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alignment | left |
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\sigma =\frac{1}{2 \pi} \, \frac{q_n}{p_n} \, \sqrt{ \frac{Ker^2 X_n + Kei^2 X_n}{Ker_1^2 X_n + Kei_1^2 X_n} } |
In practice, the most efficient methodology to interpret the SPT data is via fitting numerical model to the raw pressure-rate data.
Still, formulas
LaTeX Math Block Reference |
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and LaTeX Math Block Reference |
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play important academic role and provide fast track estimations in SPT engineering...
title | Expand more on the pressure-rate time lag in SPT |
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Objectives
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Deliverables
The advantages of SPT deliverables over conventional single-well test is illustrated below.
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Drainage volume
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Inputs
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Procedure
The typical SPT procedure is brought on Fig. 2.
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It normally consists ion three consequent tests with three different cycling frequencies:
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The total duration of the test is 155 T.
Typically T = 3 hrs and total test duration is around 20 days.
Every pulse includes one choke-up and one choke-down so that full SPT survey require 60 choke operations during 40 days which is a lot of field activity for a given well.
It would be extremely difficult to perform this manually and usual practice is to arrange a programmable remote-controlled flow variation.
Interpretation
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References
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Овчинников М.Н., Куштанова Г.Г., Гаврилов А.Г. СРЕДСТВА КОНТРОЛЯ ГИДРОДИНАМИЧЕСКИХ ПОТОКОВ В СКВАЖИННЫХ УСЛОВИЯХ И РАСЧЕТЫ ФИЛЬТРАЦИОННЫХ ПАРАМЕТРОВ ПЛАСТОВ - Казань 2012
Gavrilov_Self_Pulse_Testing.pdf
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