Difference between revisions of "Hall Plot"

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The interpretation technique is fitting the data points with the straight lines to estimate injectivity change.
 
The interpretation technique is fitting the data points with the straight lines to estimate injectivity change.
 +
 +
[[Hall Plot]] is available online at [https://ep.pengtools.com/daily/measures/water-injectors E&P Portal].
 +
  
 
[[File:Hall Plot.png|link=https://ep.pengtools.com/waterflooding/hall-plot]]
 
[[File:Hall Plot.png|link=https://ep.pengtools.com/waterflooding/hall-plot]]
  
−
<center>[[Hall Plot]] in the [https://ep.pengtools.com/waterflooding/hall-plot E&P Portal]</center>
+
<center>[[Hall Plot]] in the [https://ep.pengtools.com/daily/measures/water-injectors E&P Portal]</center>
  
 
==Math & Physics==
 
==Math & Physics==
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Hall integrated both sides with respect to time <ref name=Buell1990/>:
 
Hall integrated both sides with respect to time <ref name=Buell1990/>:
 
:<math> \sum q_{inj} \Delta t = \frac{kh}{141.2 B \mu} \times J_D \times \sum (P_{wf} - \bar{P}) \Delta t </math>
 
:<math> \sum q_{inj} \Delta t = \frac{kh}{141.2 B \mu} \times J_D \times \sum (P_{wf} - \bar{P}) \Delta t </math>
−
The Cartesian graph of cumulative injection versus cumulative <math>(P_{wf} - \bar{P})</math> yields a straight line with slope:
+
The Cartesian graph of cumulative <math>(P_{wf} - \bar{P})</math> versus cumulative injection yields a straight line with slope:
  
−
:<math> m = \frac{kh}{141.2 B \mu} \times J_D </math>
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:<math> m = \frac{141.2 B \mu}{khJ_D} </math>
  
 
and [[Injectivity index]] is:
 
and [[Injectivity index]] is:
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===Hall Plot Application with Wellhead Pressure===
 
===Hall Plot Application with Wellhead Pressure===
  
−
Since bottomhole pressure and reservoir pressure are not readily available, the [[Hall Plot]] can be used with tubing head pressure (<math>THP</math>):
+
Since bottomhole and reservoir pressures are not readily available, the [[Hall Plot]] can be used with tubing head pressure.
 +
 
 +
Given:
 +
:<math> P_{wf} = P_{tub} + \Delta P_{friction} + \Delta P_{gravity} </math>
  
−
===Application with Tubing Head Pressure===
+
Rearranging above:
 +
:<math> \sum P_{tub} \Delta t = \frac{141.2 B \mu}{kh J_D} \times \sum q_{inj} \Delta t + \sum ( \bar{P} - \Delta P_{friction} - \Delta P_{gravity} ) \Delta t </math>
  
−
[your derivation here]
+
Under the assumption that the reservoir pressure, friction and gravity terms are constant, a plot of cumulative tubing head pressure versus cumulative injection will yield a straight line. The slope of this line represents the '''apparent injectivity''', which includes the effects of friction, gravity, and reservoir pressure.
  
−
Rearranging:
+
==Discussion==
−
:<math> \sum P_{wf} \Delta t = \frac{141.2 B \mu}{kh J_D} \times \sum q_{inj} \Delta t + \sum \bar{P} \Delta t</math>
 
  
 +
The [[Hall Plot]] is primarily used for qualitative analysis of injectivity changes. A change in slope may indicate:
  
−
Given:
+
*Increasing slope indicates formation damage, scaling, increasing skin, or external radius decrease.
−
:<math> P_{wf} = P_{wh} + \Delta P_{friction} + \Delta P_{gravity} </math>
+
*Decreasing slope indicates stimulation, fracturing, increasing water relative permeability (e.g., with surfactant), or increasing well radius (e.g., with acid).
 +
*Constant slope indicates constant skin, reservoir pressure, wellbore radius, external radius and permeability-thickness (kh).
 +
 
 +
Early in the life of a waterflood, the [[Hall Plot]] may exhibit a concave upward appearance. This is caused by the expansion of the external drainage radius and increase in reservoir pressure <ref name=DW/>.
 +
 
 +
The tubing head pressure plot serves as a first-step diagnostic. Any slope changes observed should be verified with a cumulative <math>P_{wf} - \bar{P}</math> versus cumulative <math>q_{inj}</math> plot for confirmation.
 +
 
 +
Multiple wells can be plotted on the same Hall Plot for comparison. This allows for easy identification of wells that are underperforming or responding to stimulation.
 +
 
 +
[[File:Multiple_wells_Hall_Plot.png|Hall Plot example for multiple wells]]
 +
 
 +
<center>[[Hall Plot]] for multiple wells in the [https://ep.pengtools.com/daily/measures/water-injectors E&P Portal]</center>
  
 
== Nomenclature ==
 
== Nomenclature ==
  
−
==See Also==
+
:<math> B </math> = formation volume factor, bbl/stb
−
[[Injectivity index]]
+
:<math> I </math> = injectivity index, stb/d/psia
 +
:<math> J_D </math> = dimensionless injectivity index, dimensionless
 +
:<math> kh </math> = permeability times thickness, md·ft
 +
:<math> m </math> = slope of Hall Plot, psia·d/bbl
 +
:<math> P_{wf} </math> = bottomhole flowing pressure, psia
 +
:<math> \bar{P} </math> = average reservoir pressure, psia
 +
:<math> P_{tub} </math> = tubing head pressure, psia
 +
:<math> q_{inj} </math> = injection rate, stb/d
 +
:<math> r_w </math> = wellbore radius, ft
 +
:<math> r_e </math> = external drainage radius, ft
 +
:<math> S </math> = skin factor, dimensionless
 +
:<math> \Delta P_{friction} </math> = frictional pressure drop, psia
 +
:<math> \Delta P_{gravity} </math> = hydrostatic pressure drop, psia
 +
 
 +
===Greek symbols===
 +
 
 +
:<math> \mu </math> = viscosity, cp
 +
:<math> \Delta t </math> = time step, days/months
  
−
[[Category:E&P Portal]]
+
== References ==
  
−
{{#seo:
 
−
|title=Hall Plot
 
−
|titlemode= replace
 
−
|keywords=Hall plot, injection well analysis, waterflood performance, injectivity analysis, formation damage detection
 
−
|description=Hall Plot method for analyzing waterflood injection well performance, detecting formation damage, and evaluating stimulation effectiveness.
 
−
}}
 
 
<references>
 
<references>
  
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}}</ref>
 
}}</ref>
  
 +
<ref name=DW>
 +
{{cite book
 +
|last1= Wolcott |first1=Don
 +
|title=Applied Waterflood Field Development
 +
|date=2009
 +
|publisher=Energy Tribune Publishing Inc
 +
|place=Houston
 +
|url=https://www.amazon.com/Applied-Waterflood-Field-Development-Wolcott/dp/0578023946/ref=sr_1_1?ie=UTF8&qid=1481788841&sr=8-1&keywords=Don+wolcott
 +
}}</ref>
  
 
</references>
 
</references>
 +
 +
==See Also==
 +
* [[Injectivity index]]
 +
* [[JD]]
 +
* [[Prats effective well radius]]
 +
 +
[[Category:E&P Portal]]
 +
 +
{{#seo:
 +
|title=Hall Plot
 +
|titlemode=replace
 +
|keywords=Hall plot, injection well analysis, waterflood performance, injectivity analysis, formation damage detection, injectivity index, waterflood
 +
|description=Hall Plot method for analyzing waterflood injection well performance, detecting formation damage, and evaluating stimulation effectiveness. Includes application with tubing head pressure and interpretation guidelines.
 +
}}

Latest revision as of 18:47, 28 August 2026

Brief

Hall Plot is the engineering technique to analyze performance of injection wells published in 1963 by Howard Hall [1].

Hall Plot is applied to determine the change in the Injectivity index of the wells.

Hall Plot uses readily available Well flowing data: injection rate and wellhead pressures.

The interpretation technique is fitting the data points with the straight lines to estimate injectivity change.

Hall Plot is available online at E&P Portal.


Hall Plot.png

Hall Plot in the E&P Portal

Math & Physics

From Darcy's law for an unfractured injection well located in the center of a circular drainage area, the injection rate in pseudo-steady state is:

{q_{inj}} = \frac{kh}{141.2 B \mu} \times J_D \times (P_{wf} - \bar{P})

Hall integrated both sides with respect to time [2]:

\sum q_{inj} \Delta t = \frac{kh}{141.2 B \mu} \times J_D \times \sum (P_{wf} - \bar{P}) \Delta t

The Cartesian graph of cumulative (P_{wf} - \bar{P}) versus cumulative injection yields a straight line with slope:

m = \frac{141.2 B \mu}{khJ_D}

and Injectivity index is:

I = \frac{1}{m}

Hall Plot Application with Wellhead Pressure

Since bottomhole and reservoir pressures are not readily available, the Hall Plot can be used with tubing head pressure.

Given:

P_{wf} = P_{tub} + \Delta P_{friction} + \Delta P_{gravity}

Rearranging above:

\sum P_{tub} \Delta t = \frac{141.2 B \mu}{kh J_D} \times \sum q_{inj} \Delta t + \sum ( \bar{P} - \Delta P_{friction} - \Delta P_{gravity} ) \Delta t

Under the assumption that the reservoir pressure, friction and gravity terms are constant, a plot of cumulative tubing head pressure versus cumulative injection will yield a straight line. The slope of this line represents the apparent injectivity, which includes the effects of friction, gravity, and reservoir pressure.

Discussion

The Hall Plot is primarily used for qualitative analysis of injectivity changes. A change in slope may indicate:

  • Increasing slope indicates formation damage, scaling, increasing skin, or external radius decrease.
  • Decreasing slope indicates stimulation, fracturing, increasing water relative permeability (e.g., with surfactant), or increasing well radius (e.g., with acid).
  • Constant slope indicates constant skin, reservoir pressure, wellbore radius, external radius and permeability-thickness (kh).

Early in the life of a waterflood, the Hall Plot may exhibit a concave upward appearance. This is caused by the expansion of the external drainage radius and increase in reservoir pressure [3].

The tubing head pressure plot serves as a first-step diagnostic. Any slope changes observed should be verified with a cumulative P_{wf} - \bar{P} versus cumulative q_{inj} plot for confirmation.

Multiple wells can be plotted on the same Hall Plot for comparison. This allows for easy identification of wells that are underperforming or responding to stimulation.

Hall Plot example for multiple wells

Hall Plot for multiple wells in the E&P Portal

Nomenclature

B = formation volume factor, bbl/stb
I = injectivity index, stb/d/psia
J_D = dimensionless injectivity index, dimensionless
kh = permeability times thickness, md·ft
m = slope of Hall Plot, psia·d/bbl
P_{wf} = bottomhole flowing pressure, psia
\bar{P} = average reservoir pressure, psia
P_{tub} = tubing head pressure, psia
q_{inj} = injection rate, stb/d
r_w = wellbore radius, ft
r_e = external drainage radius, ft
S = skin factor, dimensionless
\Delta P_{friction} = frictional pressure drop, psia
\Delta P_{gravity} = hydrostatic pressure drop, psia

Greek symbols

\mu = viscosity, cp
\Delta t = time step, days/months

References

  1. ↑ Hall, H.N. (1963). "How To Analyze Waterflood Injection Well Performance". World Oil. 
  2. ↑ Buell, R.S.; Kazemi, H.; Poettmann, F.H. (1990). "Analyzing Injectivity of Polymer Solutions With the Hall Plot". SPE Reservoir Engineering. 5 (1). doi:10.2118/16963-PA. 
  3. ↑ Wolcott, Don (2009). Applied Waterflood Field Development. Houston: Energy Tribune Publishing Inc. 

See Also

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