Engineering lesson
Explore gas Z correlations over reduced pressure
A gas Z comparison evaluates specific correlations at matched reduced pressure and temperature. Reduced variables depend on the selected pseudocritical properties, which are themselves model inputs.
By PetroCalcHub Team · Updated October 24, 2026 · Free to read and use
Control the reduced-state basis
Pr = P/Ppc and Tr = T/Tpc require absolute pressure and temperature. Using different pseudocritical assumptions for two models confounds the correlation comparison.
The site's workflow compares its supported DAK and Hall–Yarborough implementations. The recorded sweep retains the same reduced-temperature basis and reports numerical checks.
Inspect roots and applicability
An implicit correlation requires a numerical root. A small equation residual shows the selected equation was solved, not that the correlation accurately represents a particular gas.
Near difficult states, inspect warnings and the documented domain rather than accepting a smooth plot as proof of validity.
Use independent measurements for accuracy
A model-to-model difference is a sensitivity result. A laboratory or independently supported reference is needed to assess accuracy.
The explorer uses recorded solver evaluations, not interpolation presented as a fresh root solve. Exact values for each selected state appear in the table.
Explore the example
Original recorded synthetic cases solved with the site's gas-z-comparison engine. The full inputs, outputs and diagnostic rows are in the downloadable JSON. These are discrete solved states; the lesson does not interpolate a new solve.
Select one of 5 recorded scenarios. Intermediate states are not interpolated.
Pseudo-reduced pressure (dimensionless) = 3. DAK Z (dimensionless): 0.776127593; Hall-Yarborough Z (dimensionless): 0.774828273.
- DAK Z (dimensionless)
- Hall-Yarborough Z (dimensionless)
Worked example
DAK and Hall-Yarborough Gas Z-Factor Solver
Fixed controls: Pseudo-reduced absolute temperature = 1.5 dimensionless.
Relation and unit basis
Solve rho_r * Z_DAK(rho_r, T_pr) = 0.27 * P_pr / T_pr; Z = 0.27 * P_pr / (rho_r * T_pr).; Solve the Hall-Yarborough residual F(y) = 0 for 0 < y < 1; Z = A * P_pr / y; t = 1/T_pr.; Z_DAK = 1 + c1*rho_r + c2*rho_r^2 - c3*rho_r^5 + c4; coefficients A1 to A11 are listed below.; F(y) = -A*P_pr + (y+y^2+y^3-y^4)/(1-y)^3 - B*y^2 + C*y^D; B = 14.76t-9.76t^2+4.58t^3; C = 90.7t-242.2t^2+42.4t^3; D = 2.18+2.82t.
Pseudo-reduced pressure is varied within the supported input range. This comparison deliberately supports only 0.2 <= P_pr <= 15 and 1.2 <= T_pr <= 3. The software envelope is narrower than the cited DAK range; near-critical, subcritical and higher-pressure extensions are not implemented. Bounds are enforced, not treated as extrapolation warnings.
1. Define the baseline
Pseudo-reduced pressure (dimensionless) = 3
Fixed controls: Pseudo-reduced absolute temperature = 1.5 dimensionless.
2. Select the relation
Solve rho_r * Z_DAK(rho_r, T_pr) = 0.27 * P_pr / T_pr; Z = 0.27 * P_pr / (rho_r * T_pr).; Solve the Hall-Yarborough residual F(y) = 0 for 0 < y < 1; Z = A * P_pr / y; t = 1/T_pr.; Z_DAK = 1 + c1*rho_r + c2*rho_r^2 - c3*rho_r^5 + c4; coefficients A1 to A11 are listed below.; F(y) = -A*P_pr + (y+y^2+y^3-y^4)/(1-y)^3 - B*y^2 + C*y^D; B = 14.76t-9.76t^2+4.58t^3; C = 90.7t-242.2t^2+42.4t^3; D = 2.18+2.82t.
Pseudo-reduced pressure is varied within the supported input range. This comparison deliberately supports only 0.2 <= P_pr <= 15 and 1.2 <= T_pr <= 3. The software envelope is narrower than the cited DAK range; near-critical, subcritical and higher-pressure extensions are not implemented. Bounds are enforced, not treated as extrapolation warnings.
3. Substitute the values
P_pr = 3 dimensionless; with the fixed controls above, the workflow evaluates DAK Z = 0.776127593 dimensionless; Hall-Yarborough Z = 0.774828273 dimensionless
Retain intermediate precision. The displayed values are rounded only after evaluation.
4. Read the result
DAK Z (dimensionless) = 0.776127593; Hall-Yarborough Z (dimensionless) = 0.774828273
Single gas-phase correlation evaluation only. No dew-point or phase-equilibrium test, composition adjustment, sour-gas correction, density in mass units, gas compressibility derivative or fluid flash is performed. Passing an envelope check does not establish that a supplied fluid is in one phase. Brackets, a bounded iteration count and final residual tests prevent an unverified number being returned. The chart shows correlation sensitivity within the selected interval, not measured error bars. Correlation difference and solver residual are different diagnostics and must not be presented as guaranteed accuracy.
Result and interpretation
DAK Z (dimensionless) = 0.776127593; Hall-Yarborough Z (dimensionless) = 0.774828273
Single gas-phase correlation evaluation only. No dew-point or phase-equilibrium test, composition adjustment, sour-gas correction, density in mass units, gas compressibility derivative or fluid flash is performed. Passing an envelope check does not establish that a supplied fluid is in one phase. Brackets, a bounded iteration count and final residual tests prevent an unverified number being returned. The chart shows correlation sensitivity within the selected interval, not measured error bars. Correlation difference and solver residual are different diagnostics and must not be presented as guaranteed accuracy.
Select among recorded reduced-pressure states; the workflow performs a fresh correlation solve when opened.
Exact scenario data
Original recorded synthetic cases solved with the site's gas-z-comparison engine. The full inputs, outputs and diagnostic rows are in the downloadable JSON. These are discrete solved states; the lesson does not interpolate a new solve.
| Pseudo-reduced pressure (dimensionless) | DAK Z (dimensionless) | Hall-Yarborough Z (dimensionless) |
|---|---|---|
| 0.5 | 0.9509373 | 0.9496855 |
| 1 | 0.9034006 | 0.9018179 |
| 3 | 0.7761276 | 0.7748283 |
| 5 | 0.8091314 | 0.8068394 |
| 10 | 1.13002 | 1.133788 |
Scroll the table to see all columns. The highlighted row is the worked baseline. Downloads retain full numerical precision.
Test your understanding
Work through the question before revealing its answer. Use the stated assumptions and include units in every numerical step.
1. Does a zero residual prove that Z is physically accurate?
Show the worked answer
No. It proves consistency with the numerical correlation equation, subject to its model and input assumptions.
2. Reproduce the baseline DAK Z (dimensionless) without rounding intermediate values.
Show the worked answer
P_pr = 3 dimensionless; with the fixed controls above, the workflow evaluates DAK Z = 0.776127593 dimensionless; Hall-Yarborough Z = 0.774828273 dimensionless. DAK Z (dimensionless) = 0.776127593. Single gas-phase correlation evaluation only. No dew-point or phase-equilibrium test, composition adjustment, sour-gas correction, density in mass units, gas compressibility derivative or fluid flash is performed. Passing an envelope check does not establish that a supplied fluid is in one phase. Brackets, a bounded iteration count and final residual tests prevent an unverified number being returned. The chart shows correlation sensitivity within the selected interval, not measured error bars. Correlation difference and solver residual are different diagnostics and must not be presented as guaranteed accuracy.
3. Compare the first and last recorded scenarios. Which input changes?
Show the worked answer
Pseudo-reduced pressure (dimensionless) changes from 0.5 to 10. Other controls remain as stated in the premise. DAK Z (dimensionless) changes from 0.950937323 to 1.13001963.
4. What would be needed to apply this example to a different physical system?
Show the worked answer
Replace the synthetic controls with compatible documented inputs and check the relation's assumptions. Pseudo-reduced pressure is varied within the supported input range. This comparison deliberately supports only 0.2 <= P_pr <= 15 and 1.2 <= T_pr <= 3. The software envelope is narrower than the cited DAK range; near-critical, subcritical and higher-pressure extensions are not implemented. Bounds are enforced, not treated as extrapolation warnings. Single gas-phase correlation evaluation only. No dew-point or phase-equilibrium test, composition adjustment, sour-gas correction, density in mass units, gas compressibility derivative or fluid flash is performed. Passing an envelope check does not establish that a supplied fluid is in one phase. Brackets, a bounded iteration count and final residual tests prevent an unverified number being returned. The chart shows correlation sensitivity within the selected interval, not measured error bars. Correlation difference and solver residual are different diagnostics and must not be presented as guaranteed accuracy.
Before using the result
- Match pseudocritical definitions.
- Inspect root diagnostics.
- Do not rank accuracy from model agreement alone.
Single gas-phase correlation evaluation only. No dew-point or phase-equilibrium test, composition adjustment, sour-gas correction, density in mass units, gas compressibility derivative or fluid flash is performed. Passing an envelope check does not establish that a supplied fluid is in one phase. Brackets, a bounded iteration count and final residual tests prevent an unverified number being returned. The chart shows correlation sensitivity within the selected interval, not measured error bars. Correlation difference and solver residual are different diagnostics and must not be presented as guaranteed accuracy.
Continue learning
Start with the broader workflow guide · Related method guide
Sources and method
The explanation and synthetic examples are original PetroCalcHub teaching material. The references below support the definitions and model context. They do not supply field validation for the synthetic scenarios.