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The isochronal test<ref name="r1" /> is a series of [[Single-point tests for gas wells|single-point tests]] developed to estimate stabilized deliverability characteristics without actually flowing the well for the time required to achieve stabilized conditions at each different rate. This article discusses the implementation and analysis of the isochroncal testing for [[Deliverability testing of gas wells|gas well deliverability tests]]. Both the Rawlins and Schellhardt and Houpeurt analysis techniques are presented in terms of pseudopressures.  
The isochronal test<ref name="r1">Cullender, M.H. 1955. The Isochronal Performance Method of Determining the Flow Characteristics of Gas Wells. In Petroleum Transactions, 204, 137-142. AIME.</ref> is a series of [[Single-point_tests_for_gas_wells|single-point tests]] developed to estimate stabilized deliverability characteristics without actually flowing the well for the time required to achieve stabilized conditions at each different rate. This article discusses the implementation and analysis of the isochroncal testing for [[Deliverability_testing_of_gas_wells|gas well deliverability tests]]. Both the Rawlins and Schellhardt and Houpeurt analysis techniques are presented in terms of pseudopressures.


==Isochronal test procedure and analysis==
== Isochronal test procedure and analysis ==
The isochronal test is conducted by alternately producing the well then shutting it in and allowing it to build to the average reservoir pressure before the beginning of the next production period. Pressures are measured at several time increments during each flow period. The times at which the pressures are measured should be the same relative to the beginning of each flow period. Because less time is required to build to essentially initial pressure after short flow periods than to reach stabilized flow at each rate in a flow-after-flow test, the isochronal test is more practical for low-permeability formations. A final stabilized flow point often is obtained at the end of the test. '''Fig. 1''' illustrates an isochronal test.  
 
The isochronal test is conducted by alternately producing the well then shutting it in and allowing it to build to the average reservoir pressure before the beginning of the next production period. Pressures are measured at several time increments during each flow period. The times at which the pressures are measured should be the same relative to the beginning of each flow period. Because less time is required to build to essentially initial pressure after short flow periods than to reach stabilized flow at each rate in a flow-after-flow test, the isochronal test is more practical for low-permeability formations. A final stabilized flow point often is obtained at the end of the test. '''Fig. 1''' illustrates an isochronal test.


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The isochronal test is based on the principle that the radius of drainage established during each flow period depends only on the length of time for which the well is flowed and not the flow rate. Consequently, the pressures measured at the same time periods during each different rate correspond to the same transient radius of drainage. Under these conditions, isochronal test data can be analyzed using the same theory as a flow-after-flow test, even though stabilized flow is not attained. In theory, a stabilized deliverability curve can be obtained from transient data if a single, stabilized rate and the corresponding BHP have been measured and are available.  
The isochronal test is based on the principle that the radius of drainage established during each flow period depends only on the length of time for which the well is flowed and not the flow rate. Consequently, the pressures measured at the same time periods during each different rate correspond to the same transient radius of drainage. Under these conditions, isochronal test data can be analyzed using the same theory as a flow-after-flow test, even though stabilized flow is not attained. In theory, a stabilized deliverability curve can be obtained from transient data if a single, stabilized rate and the corresponding BHP have been measured and are available.


The transient flow regime is modeled by
The transient flow regime is modeled by


[[File:Vol5 page 0851 eq 004.png]]....................(1)
[[File:Vol5 page 0851 eq 004.png|RTENOTITLE]]....................(1)


where ''p''<sub>''s''</sub> is the stabilized BHP measured before the test. The transient equation can be rewritten in a form similar to the stabilized equation for a circular drainage area. To start this process, write
where ''p''<sub>''s''</sub> is the stabilized BHP measured before the test. The transient equation can be rewritten in a form similar to the stabilized equation for a circular drainage area. To start this process, write


[[File:Vol5 page 0851 eq 005.png]]....................(2)
[[File:Vol5 page 0851 eq 005.png|RTENOTITLE]]....................(2)


Further, a transient radius of drainage is defined as
Further, a transient radius of drainage is defined as


[[File:Vol5 page 0852 eq 001.png]]....................(3)
[[File:Vol5 page 0852 eq 001.png|RTENOTITLE]]....................(3)


By substituting '''Eq. 3''' into '''Eq. 2''' and rearranging, the transient solution becomes
By substituting '''Eq. 3''' into '''Eq. 2''' and rearranging, the transient solution becomes


[[File:Vol5 page 0852 eq 002.png]]....................(4)
[[File:Vol5 page 0852 eq 002.png|RTENOTITLE]]....................(4)


which is valid at any ''fixed'' time because ''r''<sub>''d''</sub> is a function of time and not of flow rate. ''r''<sub>''d''</sub> has no rigorous physical significance. It is simply the radius that forces the transient equation to resemble the pseudosteady-state equation. In addition, do not confuse ''r''<sub>''d''</sub> with ''r''<sub>''i''</sub>, which is the transient radius of investigation given by '''Eq. 5'''.  
which is valid at any ''fixed'' time because ''r''<sub>''d''</sub> is a function of time and not of flow rate. ''r''<sub>''d''</sub> has no rigorous physical significance. It is simply the radius that forces the transient equation to resemble the pseudosteady-state equation. In addition, do not confuse ''r''<sub>''d''</sub> with ''r''<sub>''i''</sub>, which is the transient radius of investigation given by '''Eq. 5'''.


[[File:Vol5 page 0843 eq 001.png]]....................(5)
[[File:Vol5 page 0843 eq 001.png|RTENOTITLE]]....................(5)


Similar to Houpeurt’s equations, rewrite '''Eq. 4''' as
Similar to Houpeurt’s equations, rewrite '''Eq. 4''' as


[[File:Vol5 page 0852 eq 003.png]]....................(6)
[[File:Vol5 page 0852 eq 003.png|RTENOTITLE]]....................(6)


where
where


[[File:Vol5 page 0852 eq 004.png]]....................(7)
[[File:Vol5 page 0852 eq 004.png|RTENOTITLE]]....................(7)


and [[File:Vol5 page 0852 eq 005.png]]....................(8)
and [[File:Vol5 page 0852 eq 005.png|RTENOTITLE]]....................(8)


''b'' is not a function of time and will remain constant. Similarly, the intercept ''a''<sub>''t''</sub> is constant for each fixed time line or isochron.  
''b'' is not a function of time and will remain constant. Similarly, the intercept ''a''<sub>''t''</sub> is constant for each fixed time line or isochron.


The theory of isochronal test analysis implies that the transient pressure drawdowns corresponding to the same elapsed time during each different flow period will plot as straight lines with the same slope ''b''. The intercept a t for each line will increase with increasing time. Therefore, draw a line with the same slope, ''b'', through the final, stabilized data point, and use the coordinates of the stabilized point and the slope to calculate a stabilized intercept, ''a'', independent of time, where (for radial flow) the stabilized flow coefficient is defined by
The theory of isochronal test analysis implies that the transient pressure drawdowns corresponding to the same elapsed time during each different flow period will plot as straight lines with the same slope ''b''. The intercept a t for each line will increase with increasing time. Therefore, draw a line with the same slope, ''b'', through the final, stabilized data point, and use the coordinates of the stabilized point and the slope to calculate a stabilized intercept, ''a'', independent of time, where (for radial flow) the stabilized flow coefficient is defined by


[[File:Vol5 page 0853 eq 001.png]]....................(9)
[[File:Vol5 page 0853 eq 001.png|RTENOTITLE]]....................(9)
 
== Rawlins-Schellhardt analysis ==


==Rawlins-Schellhardt analysis==
In logarithmic form, the empirical equation introduced by Rawlins and Schellhardt for analysis of flow-after-flow test data is
In logarithmic form, the empirical equation introduced by Rawlins and Schellhardt for analysis of flow-after-flow test data is


[[File:Vol5 page 0853 eq 002.png]]....................(10)
[[File:Vol5 page 0853 eq 002.png|RTENOTITLE]]....................(10)


For isochronal tests, plot transient data measured at different flow rates but taken at the same time increments relative to the beginning of each flow period. The lines drawn through data points corresponding to the same fixed flow time prove to be parallel, so the value of n is constant and independent of time. However, the intercept, log (''C''), is a function of time, so a different intercept must be calculated for each isochronal line. This "transient" intercept is log (''C''<sub>''t''</sub>). In terms of this transient intercept, '''Eq. 11''' becomes
For isochronal tests, plot transient data measured at different flow rates but taken at the same time increments relative to the beginning of each flow period. The lines drawn through data points corresponding to the same fixed flow time prove to be parallel, so the value of n is constant and independent of time. However, the intercept, log (''C''), is a function of time, so a different intercept must be calculated for each isochronal line. This "transient" intercept is log (''C''<sub>''t''</sub>). In terms of this transient intercept, '''Eq. 11''' becomes


[[File:Vol5 page 0844 eq 001.png]]....................(11)
[[File:Vol5 page 0844 eq 001.png|RTENOTITLE]]....................(11)


[[File:Vol5 page 0853 eq 003.png]]....................(12)
[[File:Vol5 page 0853 eq 003.png|RTENOTITLE]]....................(12)


[[File:Vol5 page 0781 inline 001.png]] is replaced by ''p''<sub>''s''</sub> in the modified equation.  
[[File:Vol5 page 0781 inline 001.png|RTENOTITLE]] is replaced by ''p''<sub>''s''</sub> in the modified equation.


The conventional Rawlins-Schellhardt method of isochronal test analysis is to plot
The conventional Rawlins-Schellhardt method of isochronal test analysis is to plot


[[File:Vol5 page 0853 eq 004.png]]
[[File:Vol5 page 0853 eq 004.png|RTENOTITLE]]


for each time, giving a straight line of slope 1/''n'' and an intercept of
for each time, giving a straight line of slope 1/''n'' and an intercept of


[[File:Vol5 page 0853 eq 005.png]]
[[File:Vol5 page 0853 eq 005.png|RTENOTITLE]]
 
== Houpeurt analysis ==


==Houpeurt analysis==
Recall that the Houpeurt equation for analyzing flow-after-flow tests is
Recall that the Houpeurt equation for analyzing flow-after-flow tests is


[[File:Vol5 page 0853 eq 006.png]]....................(13)
[[File:Vol5 page 0853 eq 006.png|RTENOTITLE]]....................(13)


'''Eq. 13''' assumes stabilized flow conditions; however, in isochronal testing, measured transient data are being recorded. Consequently, for each isochronal (or fixed time) line, the equation for transient flow conditions is
'''Eq. 13''' assumes stabilized flow conditions; however, in isochronal testing, measured transient data are being recorded. Consequently, for each isochronal (or fixed time) line, the equation for transient flow conditions is


[[File:Vol5 page 0853 eq 007.png]]....................(14)
[[File:Vol5 page 0853 eq 007.png|RTENOTITLE]]....................(14)


where
where


[[File:Vol5 page 0854 eq 001.png]]....................(7)
[[File:Vol5 page 0854 eq 001.png|RTENOTITLE]]....................(7)


and [[File:Vol5 page 0854 eq 002.png]]....................(8)
and [[File:Vol5 page 0854 eq 002.png|RTENOTITLE]]....................(8)


The form of '''Eq. 14''' suggests that a plot of Δ''p''<sub>''p''</sub>/''q'' = [''p''<sub>''p''</sub>(''p''<sub>''s''</sub>) – ''p''<sub>''p''</sub>(''p''<sub>''wf,s''</sub>)]/''q'' vs. ''q'' will yield a straight line with slope ''b'' and intercept ''a''<sub>''t''</sub>. This theory can then be extended to the stabilized point and calculate a stabilized intercept, ''a'', using the coordinates of the stabilized point. The slope ''b'' remains the same.  
The form of '''Eq. 14''' suggests that a plot of Δ''p''<sub>''p''</sub>/''q'' = [''p''<sub>''p''</sub>(''p''<sub>''s''</sub>) – ''p''<sub>''p''</sub>(''p''<sub>''wf,s''</sub>)]/''q'' vs. ''q'' will yield a straight line with slope ''b'' and intercept ''a''<sub>''t''</sub>. This theory can then be extended to the stabilized point and calculate a stabilized intercept, ''a'', using the coordinates of the stabilized point. The slope ''b'' remains the same.


==Example: Analysis of isochronal tests==
== Example: Analysis of isochronal tests ==
Estimate the [[Glossary:Absolute open flow|absolute open flow]] (AOF) of this well<ref name="r1" /> using both the Rawlins and Schellhardt and the Houpeurt analyses. '''Table 1''' summarizes the isochronal test data. Assume ''p''<sub>''b''</sub> = 14.65 psia.


[[File:Vol5 page 0854 eq 003.png]]
Estimate the [[Glossary:Absolute_open_flow|absolute open flow]] (AOF) of this well<ref name="r1">Cullender, M.H. 1955. The Isochronal Performance Method of Determining the Flow Characteristics of Gas Wells. In Petroleum Transactions, 204, 137-142. AIME.</ref> using both the Rawlins and Schellhardt and the Houpeurt analyses. '''Table 1''' summarizes the isochronal test data. Assume ''p''<sub>''b''</sub> = 14.65 psia.


===Solution===
[[File:Vol5 page 0854 eq 003.png|RTENOTITLE]]
''Rawlins-Schellhardt analysis technique''. First, plot Δ''p''<sub>''p''</sub> = ''p''<sub>''p''</sub>(''p''<sub>''s''</sub>) – ''p''<sub>''p''</sub>(''p''<sub>''wf''</sub> ) vs. ''q'' on log-log coordinates ('''Fig 2''') and include the single stabilized, extended flow point. '''Table 2''' gives the plotting functions.  
 
=== Solution ===
 
''Rawlins-Schellhardt analysis technique''. First, plot Δ''p''<sub>''p''</sub> = ''p''<sub>''p''</sub>(''p''<sub>''s''</sub>) – ''p''<sub>''p''</sub>(''p''<sub>''wf''</sub> ) vs. ''q'' on log-log coordinates ('''Fig 2''') and include the single stabilized, extended flow point. '''Table 2''' gives the plotting functions.


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Calculate the deliverability exponent, ''n'', for each line or isochron using least-squares regression analysis. Note that, because the first data point for each isochron does not align with the data points at the last three flow rates ('''Fig. 2'''), the first data point is ignored in all subsequent calculations.  
Calculate the deliverability exponent, ''n'', for each line or isochron using least-squares regression analysis. Note that, because the first data point for each isochron does not align with the data points at the last three flow rates ('''Fig. 2'''), the first data point is ignored in all subsequent calculations.


'''Table 3''' summarizes the deliverability exponents determined with a least-squares regression analysis for each isochron. The arithmetic average of the n values in '''Table 3''' is 0.89.  
'''Table 3''' summarizes the deliverability exponents determined with a least-squares regression analysis for each isochron. The arithmetic average of the n values in '''Table 3''' is 0.89.


<gallery widths=300px heights=200px>
<gallery widths="300px" heights="200px">
File:Vol5 Page 0857 Image 0001.png|'''Table 3'''
File:Vol5 Page 0857 Image 0001.png|'''Table 3'''
</gallery>
</gallery>


Because 0.5 ≤ [[File:Vol5 page 0854 inline 001.png]] ≤ 1.0, AOF can be calculated or determined graphically using '''Fig. 3'''. AOF will be calculated in this example. First, determine the stabilized performance coefficient using the coordinates of the stabilized, extended flow point and ''n'' = [[File:Vol5 page 0854 inline 001.png]]:
Because 0.5 ≤ [[File:Vol5 page 0854 inline 001.png|RTENOTITLE]] ≤ 1.0, AOF can be calculated or determined graphically using '''Fig. 3'''. AOF will be calculated in this example. First, determine the stabilized performance coefficient using the coordinates of the stabilized, extended flow point and ''n'' = [[File:Vol5 page 0854 inline 001.png|RTENOTITLE]]:


[[File:Vol5 page 0854 eq 004.png]]
[[File:Vol5 page 0854 eq 004.png|RTENOTITLE]]


Then calculate the AOF potential:
Then calculate the AOF potential:


[[File:Vol5 page 0854 eq 005.png]]
[[File:Vol5 page 0854 eq 005.png|RTENOTITLE]]


To determine the AOF graphically, first calculate the pseudopressure at ''p''<sub>''b''</sub> and compute
To determine the AOF graphically, first calculate the pseudopressure at ''p''<sub>''b''</sub> and compute


[[File:Vol5 page 0854 eq 006.png]]
[[File:Vol5 page 0854 eq 006.png|RTENOTITLE]]


Then, draw a line of slope 1/[[File:Vol5 page 0854 inline 001.png]] through the stabilized flow point, extrapolate the line to the flow rate at Δ''p''<sub>''p''</sub> = ''p''<sub>''p''</sub>(''p''<sub>''s''</sub>) − ''p''<sub>''p''</sub>(''p''<sub>''b''</sub>), and read the AOF directly from the graph. The result is ''q''<sub>AOF</sub> = 4.04 MMscf/D.  
Then, draw a line of slope 1/[[File:Vol5 page 0854 inline 001.png|RTENOTITLE]] through the stabilized flow point, extrapolate the line to the flow rate at Δ''p''<sub>''p''</sub> = ''p''<sub>''p''</sub>(''p''<sub>''s''</sub>) − ''p''<sub>''p''</sub>(''p''<sub>''b''</sub>), and read the AOF directly from the graph. The result is ''q''<sub>AOF</sub> = 4.04 MMscf/D.


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''Houpeurt analysis technique''. Plot Δ''p''<sub>''p''</sub>/''q'' = [''p''<sub>''p''</sub>(''p''<sub>''s''</sub>) – ''p''<sub>''p''</sub>(''p''<sub>''wf''</sub> )]/''q'' vs. ''q'' on Cartesian graph paper ('''Fig. 4'''). '''Table 4''' gives the plotting functions. Construct best-fit lines through the isochronal data points for each time. Note that, for each flow time, the point corresponding to the lowest rate does fit on the same straight line, so all four data points will be used for the analysis of each isochron.  
''Houpeurt analysis technique''. Plot Δ''p''<sub>''p''</sub>/''q'' = [''p''<sub>''p''</sub>(''p''<sub>''s''</sub>) – ''p''<sub>''p''</sub>(''p''<sub>''wf''</sub> )]/''q'' vs. ''q'' on Cartesian graph paper ('''Fig. 4'''). '''Table 4''' gives the plotting functions. Construct best-fit lines through the isochronal data points for each time. Note that, for each flow time, the point corresponding to the lowest rate does fit on the same straight line, so all four data points will be used for the analysis of each isochron.


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Next, determine the slope ''b'' of each line or isochron. Values of ''b'' from least-squares regression analysis are summarized in '''Table 5'''. The arithmetic average value of the slopes in '''Table 5''' is 2.074 × 10<sup>4</sup> psia<sup>2</sup>/cp/(MMscf/D)<sup>2</sup>.  
Next, determine the slope ''b'' of each line or isochron. Values of ''b'' from least-squares regression analysis are summarized in '''Table 5'''. The arithmetic average value of the slopes in '''Table 5''' is 2.074 × 10<sup>4</sup> psia<sup>2</sup>/cp/(MMscf/D)<sup>2</sup>.


<gallery widths=300px heights=200px>
<gallery widths="300px" heights="200px">
File:Vol5 Page 0857 Image 0003.png|'''Table 5'''
File:Vol5 Page 0857 Image 0003.png|'''Table 5'''
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Calculate the stabilized isochronal deliverability line intercept using Δ''p''<sub>''p''</sub>/''q'' = 2.113 × 10<sup>6</sup> psia<sup>2</sup>/cp/(MMscf/D) at the extended, stabilized point.
Calculate the stabilized isochronal deliverability line intercept using Δ''p''<sub>''p''</sub>/''q'' = 2.113 × 10<sup>6</sup> psia<sup>2</sup>/cp/(MMscf/D) at the extended, stabilized point.


[[File:Vol5 page 0855 eq 001.png]]
[[File:Vol5 page 0855 eq 001.png|RTENOTITLE]]


Calculate the AOF potential using the average value of ''b'' and the stabilized value of ''a''.
Calculate the AOF potential using the average value of ''b'' and the stabilized value of ''a''.


[[File:Vol5 page 0855 eq 002.png]]
[[File:Vol5 page 0855 eq 002.png|RTENOTITLE]]


'''Fig. 5''' illustrates the results.  
'''Fig. 5''' illustrates the results.


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== Nomenclature ==
== Nomenclature ==
{|
{|
|''a''<sub>''t''</sub>
|=
|[[File:Vol5 page 0879 inline 002.png]], transient deliverability coefficient, psia<sup>2</sup>-cp/MMscf-D
|-
|-
|''a''<sub>''f''</sub>  
| ''a''<sub>''t''</sub>
|=  
| =
|[[File:Vol5 page 0879 inline 002.png]], depth of investigation along major axis in fractured well, ft  
| [[File:Vol5 page 0879 inline 002.png|RTENOTITLE]], transient deliverability coefficient, psia<sup>2</sup>-cp/MMscf-D
|-
| ''a''<sub>''f''</sub>
| =
| [[File:Vol5 page 0879 inline 002.png|RTENOTITLE]], depth of investigation along major axis in fractured well, ft
|-
|-
|''A''  
| ''A''
|=  
| =
|''πa''<sub>''f''</sub>''b''<sub>''f''</sub> , area of investigation in fractured well, ft<sup>2</sup>  
| ''πa''<sub>''f''</sub>''b''<sub>''f''</sub> , area of investigation in fractured well, ft<sup>2</sup>
|-
|-
|''b''  
| ''b''
|=  
| =
|[[File:Vol5 page 0880 inline 001.png]] (gas flow equation)  
| [[File:Vol5 page 0880 inline 001.png|RTENOTITLE]] (gas flow equation)
|-
|-
|''b''<sub>''f''</sub>  
| ''b''<sub>''f''</sub>
|=  
| =
|[[File:Vol5 page 0880 inline 002.png]], depth of investigation of along minor axis in fractured well, ft  
| [[File:Vol5 page 0880 inline 002.png|RTENOTITLE]], depth of investigation of along minor axis in fractured well, ft
|-
|-
|''c''  
| ''c''
|=  
| =
|compressibility, psi<sup>–1</sup>  
| compressibility, psi<sup>–1</sup>
|-
|-
|''c''<sub>''f''</sub>  
| ''c''<sub>''f''</sub>
|=  
| =
|formation compressibility, psi<sup>–1</sup>  
| formation compressibility, psi<sup>–1</sup>
|-
|-
|''c''<sub>''g''</sub>  
| ''c''<sub>''g''</sub>
|=  
| =
|gas compressibility, psi<sup>–1</sup>  
| gas compressibility, psi<sup>–1</sup>
|-
|-
|''c''<sub>''o''</sub>  
| ''c''<sub>''o''</sub>
|=  
| =
|oil compressibility, psi<sup>–1</sup>  
| oil compressibility, psi<sup>–1</sup>
|-
|-
|''c''<sub>''t''</sub>  
| ''c''<sub>''t''</sub>
|=  
| =
|''S''<sub>''o''</sub>''c''<sub>''o''</sub> + ''S''<sub>''w''</sub>''c''<sub>''w''</sub> + ''S''<sub>''g''</sub>''c''<sub>''g''</sub> + ''c''<sub>''f''</sub> = total compressibility, psi<sup>–1</sup>  
| ''S''<sub>''o''</sub>''c''<sub>''o''</sub> + ''S''<sub>''w''</sub>''c''<sub>''w''</sub> + ''S''<sub>''g''</sub>''c''<sub>''g''</sub> + ''c''<sub>''f''</sub> = total compressibility, psi<sup>–1</sup>
|-
|-
|''c''<sub>''w''</sub>  
| ''c''<sub>''w''</sub>
|=  
| =
|water compressibility, psi<sup>–1</sup>  
| water compressibility, psi<sup>–1</sup>
|-
|-
|[[File:Vol5 page 0789 inline 002.png]]  
| [[File:Vol5 page 0789 inline 002.png|RTENOTITLE]]
|=  
| =
|total compressibility evaluated at average drainage area pressure, psi<sup>–1</sup>  
| total compressibility evaluated at average drainage area pressure, psi<sup>–1</sup>
|-
|-
|''C''  
| ''C''
|=  
| =
|performance coefficient in gas-well deliverability equation, or wellbore storage coefficient, bbl/psi  
| performance coefficient in gas-well deliverability equation, or wellbore storage coefficient, bbl/psi
|-
|-
|''D''  
| ''D''
|=  
| =
|non-Darcy flow constant, D/Mscf  
| non-Darcy flow constant, D/Mscf
|-
|-
|''h''  
| ''h''
|=  
| =
|net formation thickness, ft  
| net formation thickness, ft
|-
|-
|''k''<sub>''g''</sub>  
| ''k''<sub>''g''</sub>
|=  
| =
|permeability to gas, md  
| permeability to gas, md
|-
|-
|''L''<sub>''f''</sub>  
| ''L''<sub>''f''</sub>
|=  
| =
|fracture half length, ft  
| fracture half length, ft
|-
|-
|''m''  
| ''m''
|=  
| =
|162.2 ''qBμ''/''kh'' = slope of middle-time line, psi/cycle  
| 162.2 ''qBμ''/''kh'' = slope of middle-time line, psi/cycle
|-
|-
|''n''  
| ''n''
|=  
| =
|inverse slope of the line on a log-log plot of the change in pressure squared or pseudopressure vs. gas flow rate  
| inverse slope of the line on a log-log plot of the change in pressure squared or pseudopressure vs. gas flow rate
|-
|-
|''p''<sub>''p''</sub>  
| ''p''<sub>''p''</sub>
|=  
| =
|pseudopressure, psia<sup>2</sup>/cp  
| pseudopressure, psia<sup>2</sup>/cp
|-
|-
|''p''<sub>''s''</sub>  
| ''p''<sub>''s''</sub>
|=  
| =
|stabilized shut-in BHP measured just before start of a deliverability test, psia  
| stabilized shut-in BHP measured just before start of a deliverability test, psia
|-
|-
|''p''<sub>''w''</sub>  
| ''p''<sub>''w''</sub>
|=  
| =
|BHP in wellbore, psi  
| BHP in wellbore, psi
|-
|-
|''p''<sub>''wf''</sub>  
| ''p''<sub>''wf''</sub>
|=  
| =
|flowing BHP, psi  
| flowing BHP, psi
|-
|-
|''p''<sub>''ws''</sub>  
| ''p''<sub>''ws''</sub>
|=  
| =
|shut-in BHP, psi  
| shut-in BHP, psi
|-
|-
|''q''  
| ''q''
|=  
| =
|flow rate at surface, STB/D  
| flow rate at surface, STB/D
|-
|-
|''r''<sub>''d''</sub>  
| ''r''<sub>''d''</sub>
|=  
| =
|effective drainage radius, ft  
| effective drainage radius, ft
|-
|-
|''r''<sub>''e''</sub>  
| ''r''<sub>''e''</sub>
|=  
| =
|external drainage radius, ft  
| external drainage radius, ft
|-
|-
|''r''<sub>''w''</sub>  
| ''r''<sub>''w''</sub>
|=  
| =
|wellbore radius, ft  
| wellbore radius, ft
|-
|-
|''s''  
| ''s''
|=  
| =
|skin factor, dimensionless  
| skin factor, dimensionless
|-
|-
|''S''<sub>''g''</sub>  
| ''S''<sub>''g''</sub>
|=  
| =
|gas saturation, fraction of pore volume  
| gas saturation, fraction of pore volume
|-
|-
|''S''<sub>''o''</sub>  
| ''S''<sub>''o''</sub>
|=  
| =
|oil saturation, fraction of pore volume  
| oil saturation, fraction of pore volume
|-
|-
|''S''<sub>''w''</sub>  
| ''S''<sub>''w''</sub>
|=  
| =
|water saturation, fraction of pore volume  
| water saturation, fraction of pore volume
|-
|-
|''t''  
| ''t''
|=  
| =
|elapsed time, hours  
| elapsed time, hours
|-
|-
|''T''  
| ''T''
|=  
| =
|reservoir temperature, °R  
| reservoir temperature, °R
|-
|-
|Δ''p''<sub>''p''</sub>  
| Δ''p''<sub>''p''</sub>
|=  
| =
|pseudopressure change since start of test, psia<sup>2</sup>/cp  
| pseudopressure change since start of test, psia<sup>2</sup>/cp
|-
|-
|''μ''  
| ''μ''
|=  
| =
|viscosity, cp  
| viscosity, cp
|-
|-
|[[File:Vol5 page 0844 inline 002.png]]  
| [[File:Vol5 page 0844 inline 002.png|RTENOTITLE]]
|=  
| =
|gas viscosity evaluated at average pressure, cp  
| gas viscosity evaluated at average pressure, cp
|-
|-
|''ϕ''  
| ''ϕ''
|=  
| =
|porosity, dimensionless  
| porosity, dimensionless
|}
|}


==References==
== References ==
<references>
 
<ref name="r1">Cullender, M.H. 1955. The Isochronal Performance Method of Determining the Flow Characteristics of Gas Wells. In ''Petroleum Transactions'', 204, 137-142. AIME. </ref>
<references />
</references>
 
== Noteworthy papers in OnePetro ==


==Noteworthy papers in OnePetro==
Use this section to list papers in OnePetro that a reader who wants to learn more should definitely read
Use this section to list papers in OnePetro that a reader who wants to learn more should definitely read


==External links==
== External links ==
 
Use this section to provide links to relevant material on websites other than PetroWiki and OnePetro
Use this section to provide links to relevant material on websites other than PetroWiki and OnePetro


==See also==
== See also ==
[[Deliverability testing of gas wells]]
 
[[Deliverability_testing_of_gas_wells|Deliverability testing of gas wells]]
 
[[Modified_isochronal_tests_for_gas_wells|Modified isochronal tests for gas wells]]


[[Modified isochronal tests for gas wells]]
[[Flow-after-flow_tests_for_gas_wells|Flow-after-flow tests for gas wells]]


[[Flow-after-flow tests for gas wells]]
[[Single-point_tests_for_gas_wells|Single-point tests for gas wells]]


[[Single-point tests for gas wells]]
[[Flow_equations_for_gas_and_multiphase_flow|Flow equations for gas and multiphase flow]]


[[Flow equations for gas and multiphase flow]]
[[PEH:Fluid_Flow_Through_Permeable_Media]]


[[PEH:Fluid Flow Through Permeable Media]]
[[Category:5.6 Formation evaluation and management]]

Latest revision as of 13:44, 12 June 2015

The isochronal test[1] is a series of single-point tests developed to estimate stabilized deliverability characteristics without actually flowing the well for the time required to achieve stabilized conditions at each different rate. This article discusses the implementation and analysis of the isochroncal testing for gas well deliverability tests. Both the Rawlins and Schellhardt and Houpeurt analysis techniques are presented in terms of pseudopressures.

Isochronal test procedure and analysis

The isochronal test is conducted by alternately producing the well then shutting it in and allowing it to build to the average reservoir pressure before the beginning of the next production period. Pressures are measured at several time increments during each flow period. The times at which the pressures are measured should be the same relative to the beginning of each flow period. Because less time is required to build to essentially initial pressure after short flow periods than to reach stabilized flow at each rate in a flow-after-flow test, the isochronal test is more practical for low-permeability formations. A final stabilized flow point often is obtained at the end of the test. Fig. 1 illustrates an isochronal test.

  • Fig. 1 – Pressure and flow rate history of a typical isochronal test.

    Fig. 1 – Pressure and flow rate history of a typical isochronal test.

The isochronal test is based on the principle that the radius of drainage established during each flow period depends only on the length of time for which the well is flowed and not the flow rate. Consequently, the pressures measured at the same time periods during each different rate correspond to the same transient radius of drainage. Under these conditions, isochronal test data can be analyzed using the same theory as a flow-after-flow test, even though stabilized flow is not attained. In theory, a stabilized deliverability curve can be obtained from transient data if a single, stabilized rate and the corresponding BHP have been measured and are available.

The transient flow regime is modeled by

RTENOTITLE....................(1)

where ps is the stabilized BHP measured before the test. The transient equation can be rewritten in a form similar to the stabilized equation for a circular drainage area. To start this process, write

RTENOTITLE....................(2)

Further, a transient radius of drainage is defined as

RTENOTITLE....................(3)

By substituting Eq. 3 into Eq. 2 and rearranging, the transient solution becomes

RTENOTITLE....................(4)

which is valid at any fixed time because rd is a function of time and not of flow rate. rd has no rigorous physical significance. It is simply the radius that forces the transient equation to resemble the pseudosteady-state equation. In addition, do not confuse rd with ri, which is the transient radius of investigation given by Eq. 5.

RTENOTITLE....................(5)

Similar to Houpeurt’s equations, rewrite Eq. 4 as

RTENOTITLE....................(6)

where

RTENOTITLE....................(7)

and RTENOTITLE....................(8)

b is not a function of time and will remain constant. Similarly, the intercept at is constant for each fixed time line or isochron.

The theory of isochronal test analysis implies that the transient pressure drawdowns corresponding to the same elapsed time during each different flow period will plot as straight lines with the same slope b. The intercept a t for each line will increase with increasing time. Therefore, draw a line with the same slope, b, through the final, stabilized data point, and use the coordinates of the stabilized point and the slope to calculate a stabilized intercept, a, independent of time, where (for radial flow) the stabilized flow coefficient is defined by

RTENOTITLE....................(9)

Rawlins-Schellhardt analysis

In logarithmic form, the empirical equation introduced by Rawlins and Schellhardt for analysis of flow-after-flow test data is

RTENOTITLE....................(10)

For isochronal tests, plot transient data measured at different flow rates but taken at the same time increments relative to the beginning of each flow period. The lines drawn through data points corresponding to the same fixed flow time prove to be parallel, so the value of n is constant and independent of time. However, the intercept, log (C), is a function of time, so a different intercept must be calculated for each isochronal line. This "transient" intercept is log (Ct). In terms of this transient intercept, Eq. 11 becomes

RTENOTITLE....................(11)

RTENOTITLE....................(12)

RTENOTITLE is replaced by ps in the modified equation.

The conventional Rawlins-Schellhardt method of isochronal test analysis is to plot

RTENOTITLE

for each time, giving a straight line of slope 1/n and an intercept of

RTENOTITLE

Houpeurt analysis

Recall that the Houpeurt equation for analyzing flow-after-flow tests is

RTENOTITLE....................(13)

Eq. 13 assumes stabilized flow conditions; however, in isochronal testing, measured transient data are being recorded. Consequently, for each isochronal (or fixed time) line, the equation for transient flow conditions is

RTENOTITLE....................(14)

where

RTENOTITLE....................(7)

and RTENOTITLE....................(8)

The form of Eq. 14 suggests that a plot of Δpp/q = [pp(ps) – pp(pwf,s)]/q vs. q will yield a straight line with slope b and intercept at. This theory can then be extended to the stabilized point and calculate a stabilized intercept, a, using the coordinates of the stabilized point. The slope b remains the same.

Example: Analysis of isochronal tests

Estimate the absolute open flow (AOF) of this well[1] using both the Rawlins and Schellhardt and the Houpeurt analyses. Table 1 summarizes the isochronal test data. Assume pb = 14.65 psia.

RTENOTITLE

Solution

Rawlins-Schellhardt analysis technique. First, plot Δpp = pp(ps) – pp(pwf ) vs. q on log-log coordinates (Fig 2) and include the single stabilized, extended flow point. Table 2 gives the plotting functions.

  • Fig. 2 – Rawlins-Schellhardt analysis.

    Fig. 2 – Rawlins-Schellhardt analysis.

  • Table 1

    Table 1

  • Table 2

    Table 2

Calculate the deliverability exponent, n, for each line or isochron using least-squares regression analysis. Note that, because the first data point for each isochron does not align with the data points at the last three flow rates (Fig. 2), the first data point is ignored in all subsequent calculations.

Table 3 summarizes the deliverability exponents determined with a least-squares regression analysis for each isochron. The arithmetic average of the n values in Table 3 is 0.89.

  • Table 3

    Table 3

Because 0.5 ≤ RTENOTITLE ≤ 1.0, AOF can be calculated or determined graphically using Fig. 3. AOF will be calculated in this example. First, determine the stabilized performance coefficient using the coordinates of the stabilized, extended flow point and n = RTENOTITLE:

RTENOTITLE

Then calculate the AOF potential:

RTENOTITLE

To determine the AOF graphically, first calculate the pseudopressure at pb and compute

RTENOTITLE

Then, draw a line of slope 1/RTENOTITLE through the stabilized flow point, extrapolate the line to the flow rate at Δpp = pp(ps) − pp(pb), and read the AOF directly from the graph. The result is qAOF = 4.04 MMscf/D.

  • Fig. 3 – Rawlins-Schellhardt analysis result.

    Fig. 3 – Rawlins-Schellhardt analysis result.

Houpeurt analysis technique. Plot Δpp/q = [pp(ps) – pp(pwf )]/q vs. q on Cartesian graph paper (Fig. 4). Table 4 gives the plotting functions. Construct best-fit lines through the isochronal data points for each time. Note that, for each flow time, the point corresponding to the lowest rate does fit on the same straight line, so all four data points will be used for the analysis of each isochron.

  • Fig. 4 – Houpeurt analysis of isochronal test data.

    Fig. 4 – Houpeurt analysis of isochronal test data.

  • Table 4

    Table 4

Next, determine the slope b of each line or isochron. Values of b from least-squares regression analysis are summarized in Table 5. The arithmetic average value of the slopes in Table 5 is 2.074 × 104 psia2/cp/(MMscf/D)2.

  • Table 5

    Table 5

Calculate the stabilized isochronal deliverability line intercept using Δpp/q = 2.113 × 106 psia2/cp/(MMscf/D) at the extended, stabilized point.

RTENOTITLE

Calculate the AOF potential using the average value of b and the stabilized value of a.

RTENOTITLE

Fig. 5 illustrates the results.

  • Fig. 5 – Houpeurt analysis of isochronal test data result.

    Fig. 5 – Houpeurt analysis of isochronal test data result.

Nomenclature

at = RTENOTITLE, transient deliverability coefficient, psia2-cp/MMscf-D
af = RTENOTITLE, depth of investigation along major axis in fractured well, ft
A = πafbf , area of investigation in fractured well, ft2
b = RTENOTITLE (gas flow equation)
bf = RTENOTITLE, depth of investigation of along minor axis in fractured well, ft
c = compressibility, psi–1
cf = formation compressibility, psi–1
cg = gas compressibility, psi–1
co = oil compressibility, psi–1
ct = Soco + Swcw + Sgcg + cf = total compressibility, psi–1
cw = water compressibility, psi–1
RTENOTITLE = total compressibility evaluated at average drainage area pressure, psi–1
C = performance coefficient in gas-well deliverability equation, or wellbore storage coefficient, bbl/psi
D = non-Darcy flow constant, D/Mscf
h = net formation thickness, ft
kg = permeability to gas, md
Lf = fracture half length, ft
m = 162.2 qBμ/kh = slope of middle-time line, psi/cycle
n = inverse slope of the line on a log-log plot of the change in pressure squared or pseudopressure vs. gas flow rate
pp = pseudopressure, psia2/cp
ps = stabilized shut-in BHP measured just before start of a deliverability test, psia
pw = BHP in wellbore, psi
pwf = flowing BHP, psi
pws = shut-in BHP, psi
q = flow rate at surface, STB/D
rd = effective drainage radius, ft
re = external drainage radius, ft
rw = wellbore radius, ft
s = skin factor, dimensionless
Sg = gas saturation, fraction of pore volume
So = oil saturation, fraction of pore volume
Sw = water saturation, fraction of pore volume
t = elapsed time, hours
T = reservoir temperature, °R
Δpp = pseudopressure change since start of test, psia2/cp
μ = viscosity, cp
RTENOTITLE = gas viscosity evaluated at average pressure, cp
ϕ = porosity, dimensionless

References

  1. ↑ 1.0 1.1 Cullender, M.H. 1955. The Isochronal Performance Method of Determining the Flow Characteristics of Gas Wells. In Petroleum Transactions, 204, 137-142. AIME.

Noteworthy papers in OnePetro

Use this section to list papers in OnePetro that a reader who wants to learn more should definitely read

External links

Use this section to provide links to relevant material on websites other than PetroWiki and OnePetro

See also

Deliverability testing of gas wells

Modified isochronal tests for gas wells

Flow-after-flow tests for gas wells

Single-point tests for gas wells

Flow equations for gas and multiphase flow

PEH:Fluid_Flow_Through_Permeable_Media

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