1. 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).

When flow rate is being intentionally varied in harmonic cycles with sandface amplitude  and cycling frequency :

q(t) = q_0 \, \sin ( \omega  \, t )

then after a certain while (normally 3-5 cycles) the bottom-hole pressure becomes varying with the same frequency:

p_{wf}(t) = p_0 \, \sin ( \omega \, [ t - t_{\Delta} ] )

with a bottom-hole pressure amplitude  and the time delay .

The time shift   represents the inertia effects from the adjoined reservoir and characterized by formation pressure diffusivity:

\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:

\sigma = \Big <  \frac{k}{\mu}    \Big >  h

In case of a low frequency pulsations the relationships between field-measured parameters  and formation properties  is given by simple analytical formulas:

\sigma = \frac{q_0}{8 \, p_0 \, \sin \Delta}

\chi = 0.25 \, \omega \, \gamma^2 \, r_w^2 \, \exp \frac{\pi}{2 \, {\rm tg} \Delta }

where   is dimenshionless phase shift.

This only works for lengthy cyclings with sufficiently low frequency:

\omega \ll 0.00225 \, \frac{ \chi }{ r_w^2}

There are exact analytical formulas for arbitrary frequencies but they are rarely helpful in practise.

The field operations are very finnicky and difficult to follow the pre-desgined schematics with harmonic pulsations.

The use of analytical formulas requires fourier transformation to isolate appropriate harmonics from the raw data and this needs a manual control from analyst.

The most efficient methodology to interpret the practical SPT data is via fitting numerical model to the raw pressure-rate data.

Still, formulas  and  play important academic role and provide fast track estimations in SPT engineering.

The advantages of SPT over conventional single-well test is illustrated below.

BUS – Build-up Survey 

Conventional single-well testing is based on long-term monitoring of downhole pressure response to the step change in flow rate (usually shut-in or close-in).

The primary hard data deliverables are:
 

• formation pressure 
  
  
• skin-factor S
   
• average transmissibility in drainage area 
  
  
• time to reach the reservoir boundary 

The conditional deliverables from build-up survey would be:

Deliverables	Description	Non-BUS Input Parameters	Key Uncertainties
V_o = \frac{4 \, \sigma \, t_e \, (1-s_{wi})}{c_t} where   is total compressibility: c_t = c_r + (1-s_{wi}) \, c_o + s_{wi} \, c_w and are rock, oil and water compressibility.	Drainable oil reserves	The rock compressibility is defined from core lab study or empirical porosity correlations Fluid compressibility from PVT Initial water saturation from SCAL	Rock compressibility Initial water saturation
A_e = 4 \, \chi \, t_e where  is pressure diffusivity: \chi = \big< \frac{k}{\mu} \big> \, \frac{1}{\phi \, c_t} where is reservoir porosity, is fluid mobility: \big< \frac{k}{\mu} \big> = k_a \, \bigg[ \frac{k_{rw}}{\mu_w} + \frac{k_{ro}}{\mu_o} \bigg] is absolute permeability to air, are relative permeabilities to water and oil, are water and oil viscosities	Drainage area	Formation porosity Absolute permeability to air from core study Relative permeabilities from SCAL Fluid viscosities from PVT	Absolute permeability to air Relative permeabilities
h = \sigma \, \bigg< \frac{k}{\mu} \bigg>^{-1}	Effective reservoir thickness	Absolute permeability to air from core study Relative permeabilities from SCAL Fluid viscosities from PVT	Absolute permeability to air Relative permeabilities

As one can see, the drainage area and the reservoir thickness are conditioned by core data which may not be representative of the whole drainage area.

SPT – Self-Pulse Testing

The single-well self-pulse test is based on long-term monitoring of downhole pressure response to the periodic rate step change (usually shut-in or close-in).

If flowrate 

The primary hard data deliverables are:

• formation pressure 
  
  
• skin-factor S
   
• near  and far  zone transmissibility 
  
  
• near  and far  zone pressure diffusivity
  
  
• time to reach the reservoir boundary 

The SPT is correlating pressure variation with pre-designed flowrate variation sequence and tracks:

• pressure response amplitude which depends on formation transmissibility  

and

• time lag between flowrate variation and pressure response which depends on formation diffusivity .

This allows estimating effective formation thickness  directly from field survey without assumptions on core-based permeability (compare with ) and consequently leads to assessing the drainange area , fluid mobility  and absolute permeability  with lesser uncertainties than in BUS: 

h = \frac{\sigma}{\phi \, c_t \, \chi}

A_e = \frac{4 \, \sigma \, t_e}{c_t \, h}

\bigg< \frac{k}{\mu} \bigg>  = \chi \, \phi \, c_t

k_a =   \frac{\bigg< \frac{k}{\mu} \bigg>}{\bigg[ \frac{k_{rw}}{\mu_w} + \frac{k_{ro}}{\mu_o} \bigg]}

The absoluite permeability from SPT  is usually stacked up against core-based permeability  to validate the core samples and assess the effects of macroscopic features which are overlooked at core-plug size level.

Running SPT in two different cycling frequences allows assessing the near and far resevroir zones spearately.

The usual SPT workflow includes several cycling tests with different frequencies, the lower the frequency the longer the scanning range.

This captures variation of permeability and thickness when moving away from well location.

Together with deconvolution, the SPT is reproducing conventional PTA information and providing additional data on pressure diuffusivity.

This maybe used as estimation of permeability and thickness separately and their variation away from well location.

2. Objectives

• Assess reservoir volume around well
  
  
• Assess reservoir permeability and thickness variation around well

3. Deliverables

4. Inputs

5. Procedure

Test = Test 1 + Test 2 + Test 3

1. Test 1 = high freq pulsations (10 pulses with period T)
   
   
2. Test 2 = mid freq pulsations (10 pulses with with period 5T)
   
   
3. Test 3 = Low freq pulsations  (10 pulses with period 25 T)
   
   

So that total duration of the test is 310 T.

Typically T = 3 hrs and total test duration is around 40 days.

6. Interpretation

1. Numerical model
   
   
   1. Single well with circle boundary
      
      
   2. High density LGR
      
      
   3. High density time grid (seconds)
      
      
2. Automated pressure match in simulation software
   
   

References