← GK-22 PBL Hub  ·  Sub-Problem 2 of 5

SP-2: Calculate Bg, Reservoir Voidage & p/Z Depletion Table

Derive the gas formation volume factor for GK-22 from first principles using the real gas law. Verify the Z-factor of 1.40 from pseudo-critical properties and pseudo-reduced conditions. Convert the DST surface rate to reservoir voidage and condensate production, then project Bg and p/Z at two future depletion pressures.

Topic 1.2 · Formation Volume Factor
~25 min
Bloom: Apply & Analyse
5 MCQ + Carry-Forward
Sub-Problem 2 · The Question

How Much Reservoir Volume Does Each Surface Mscf Represent, and How Does That Change as the Field Depletes?

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Engineering Context
Bg links the two measurement worlds (surface and reservoir) for every rate and volume calculation

The FPSO facilities team needs to know the reservoir voidage rate (not just the surface gas rate) to size the gas-compression backpressure system and estimate the aquifer influx requirement. Your task is to calculate the gas formation volume factor Bg at initial reservoir conditions (7,350 psia, 285°F) and use it to:

  1. Verify the Z-factor of 1.40 by deriving pseudo-critical properties from γg = 0.65 and computing Tpr and ppr.
  2. Calculate Bg in res bbl/scf and res bbl/Mscf using the real gas law form.
  3. Convert the DST stabilised gas rate of 12.4 MMscfd to reservoir voidage in res bbl/d.
  4. Calculate the associated condensate surface rate in STB/d at the DST gas rate.
  5. Project Bg and p/Z at two future pressure points (5,000 psia and 3,500 psia) to understand how voidage changes as the reservoir depletes, the foundation of the p/Z material balance analysis in SP-5.

The Z-factor verification is the critical step. GK-22 operates at Tpr = 2.006 and ppr = 11.01 — the supercritical region of the Standing-Katz chart where Z exceeds 1.0 due to molecular repulsion.
Engineers who assume Z = 1 for this well will underestimate Bg by 40% and overestimate AOF by the same margin.

Z-Factor Warning
Z = 1.40 > 1.0 at the GK-22 initial conditions. This is not an error. At very high pseudo-reduced pressures (ppr > 8–10), repulsive intermolecular forces dominate over attractive forces, the gas occupies more volume than an ideal gas would at the same conditions. This is visible on the Standing-Katz chart as the region where isobars curve upward toward Z > 1.
SP-2 Data Slice

Data Relevant to This Sub-Problem

ParameterSymbolValueUnitsRelevance to SP-2
Initial reservoir pressurep̅7,350psiap in Bg = 0.005035 ZT/p
Reservoir temperatureT745°R (285°F+460)T in Rankine for Bg formula
Gas specific gravityγg0.65—Input to pseudo-critical property correlations
Z-factor at p̅Z1.40—Verify from Tpr and ppr; then use in Bg
Gas viscosity at p̅μg0.0312cpNoted here; used in SP-5 deliverability equation
DST stabilised gas rateqg,DST12.4MMscfdConvert to reservoir voidage using Bg
Condensate yieldRv112STB/MMscfCalculate qc = qg × Rv in STB/d
Background Knowledge

Key Concepts for SP-2: Gas FVF, Z-Factor, and Pseudo-Critical Properties

Full treatment in Topic 1.2: Formation Volume Factor, Sections 3 (Gas FVF), 4 (Z-Factor), and 8 (p/Z material balance).

Derivation of Bg from the Real Gas Law

Applying pV = ZnRT at reservoir conditions and at standard conditions (psc = 14.696 psia, Tsc = 520°R, Zsc = 1) for the same n moles, then dividing to get the volume ratio:

Gas Formation Volume Factor — Derivation At reservoir: p Vres = Z n R T At standard: psc Vsc = 1 · n R Tsc Dividing: Vres/Vsc = Bg = (psc/Tsc) · (ZT/p) Substituting psc = 14.696 psia, Tsc = 520°R: Bg = 0.02829 ZT/p [res ft³/scf] Bg = 0.005035 ZT/p [res bbl/scf] ← use this form Bg = 5.035 ZT/p [res bbl/Mscf] where T is in °R and p in psia.
Unit check: The constant 0.005035 = 14.696/(520 × 5.615) where 5.615 ft³/bbl converts ft³ to bbl. Always carry units explicitly — "res bbl/scf" vs "res bbl/Mscf" differ by exactly 1,000 and are a common source of factor-of-1,000 errors in deliverability calculations.

Pseudo-Critical Properties: Standing Correlations for Dry Gas (γg only)

Standing (1977) Pseudo-Critical Property Correlations Tpc = 187 + 330 γg − 71.5 γg² [°R] ppc = 706 − 51.7 γg − 11.1 γg² [psia] For GK-22 (γg = 0.65): Tpc = 187 + 330(0.65) − 71.5(0.65)² = 187 + 214.5 − 30.2 = 371.3 °R ppc = 706 − 51.7(0.65) − 11.1(0.65)² = 706 − 33.6 − 4.7 = 667.7 psia Pseudo-reduced conditions at p̅ = 7,350 psia, T = 745 °R: Tpr = T / Tpc = 745 / 371.3 = 2.006 ppr = p / ppc = 7,350 / 667.7 = 11.01 Z from Standing-Katz chart at Tpr = 2.006, ppr = 11.01: Z ≈ 1.40 (supercritical region: molecular repulsion dominates; Z > 1 is correct)

p/Z Depletion Analysis: The Material Balance for a Volumetric Gas Reservoir

For a volumetric gas reservoir (no water influx, no cap-gas injection), the material balance equation reduces to:

p/Z vs. Cumulative Production — Volumetric Gas Reservoir p/Z = (pi/Zi) − (pi/Zi) · (Gp/G) This is a straight line: p/Z vs. Gp x-intercept (p/Z = 0): Gp = G = IGIP y-intercept (Gp = 0): p/Z = pi/Zi For GK-22 initial condition: p̅/Z = 7,350/1.40 = 5,250 psia ← anchor point for p/Z plot If the p/Z plot curves UPWARD (not linear), it indicates aquifer support. If it is LINEAR, the reservoir is volumetric (no significant water influx).
Step-by-Step Tasks

SP-2 Tasks: Work Through in Order

  1. Task 2.1: Calculate Tpc and ppc from γg = 0.65
    Apply the Standing correlations: Tpc = 187 + 330γg − 71.5γg² and ppc = 706 − 51.7γg − 11.1γg². Show each arithmetic step. Express Tpc in °R and ppc in psia to the nearest 0.1. Compare your results to the values in the Hub Data Pack derived-parameters table: they must match within rounding.
  2. Task 2.2: Compute Tpr and ppr at Initial Reservoir Conditions
    Convert Tres = 285°F to Rankine. Divide by Tpc to get Tpr. Divide p̅ = 7,350 psia by ppc to get ppr. Locate this point on the Standing-Katz chart in Topic 1.2. Read off Z. Confirm that Z > 1 is consistent with the chart at these pseudo-reduced conditions. Write one sentence explaining physically why Z exceeds 1.0 at this point.
  3. Task 2.3: Calculate Bg at Initial Reservoir Conditions
    Using Bg = 0.005035 × Z × T / p with Z = 1.40, T = 745 °R, p = 7,350 psia: calculate Bg in res bbl/scf. Then express in res bbl/Mscf. Show all arithmetic and carry units. Sanity check: for a gas at these conditions, Bg should be much less than 1 res bbl/scf (gas is highly compressed at reservoir depth): confirm this makes physical sense.
  4. Task 2.4: Convert DST Rate to Reservoir Voidage and Condensate Rate
    Given qg,DST = 12.4 MMscfd and Bg from Task 2.3: calculate the reservoir gas voidage rate in res bbl/d. Then calculate the condensate surface rate in STB/d using qc = qg [MMscfd] × Rv [STB/MMscf]. Present both results in a small two-row summary table. Confirm the condensate rate matches the value in the Hub Data Pack (should be ~1,389 STB/d).
  5. Task 2.5: Project Bg and p/Z at 5,000 psia and 3,500 psia
    For each pressure node, you need a Z-factor. Use the Papay (1985) explicit approximation: Z ≈ 1 − (3.52 ppr)/(10^(0.9813 Tpr)) + (0.274 ppr²)/(10^(0.8157 Tpr)), or read from the Standing-Katz chart in Topic 1.2. At each pressure: (a) calculate ppr; (b) estimate Z; (c) calculate Bg; (d) calculate p/Z. Compile into the three-row p/Z table shown in the guided calculations section.
  6. Task 2.6: Interpret the p/Z Table
    Looking at your three-row table: (a) Does Bg increase or decrease as pressure falls? Explain physically. (b) Does p/Z decrease linearly with depletion (indicating volumetric reservoir)? (c) What would a concave-upward p/Z plot indicate? (d) How would you use the p/Z straight-line extrapolation to estimate IGIP, and what additional production data point would you need to do this in SP-5?
Guided Calculations

SP-2 Worked Answers

Tasks 2.1–2.3: Bg at Initial Conditions

Full Bg Derivation: GK-22 at p̅ = 7,350 psia Step 1 — Pseudo-critical properties (γg = 0.65): Tpc = 187 + 330(0.65) − 71.5(0.65)² = 187 + 214.50 − 30.19 = 371.3 °R ppc = 706 − 51.7(0.65) − 11.1(0.65)² = 706 − 33.61 − 4.69 = 667.7 psia Step 2 — Pseudo-reduced conditions: Tpr = 745 / 371.3 = 2.006 ppr = 7,350 / 667.7 = 11.01 Step 3 — Z-factor (Standing-Katz chart, Tpr=2.006, ppr=11.01): Z = 1.40 (supercritical region; Z > 1 due to molecular repulsion) ✓ Step 4 — Gas FVF: Bg = 0.005035 × Z × T / p = 0.005035 × 1.40 × 745 / 7,350 = 0.005035 × 1,043.0 / 7,350 = 5.251 / 7,350 = 0.0007144 res bbl/scf = 0.7144 res bbl/Mscf Sanity check: 1 Mscf at surface → 0.714 res bbl at depth ✓ (1 surface Mscf is compressed into less than 1 reservoir barrel, reasonable)

Task 2.4: Voidage and Condensate Rate at DST Rate

Reservoir Voidage and Condensate Rate: DST Conditions qg,DST = 12.4 MMscfd = 12,400 Mscfd = 12,400,000 scfd Reservoir gas voidage: Vvoid = qg [Mscfd] × Bg [res bbl/Mscf] = 12,400 × 0.7144 = 8,859 res bbl/d ≈ 8,860 res bbl/d Condensate surface rate: qc = qg [MMscfd] × Rv [STB/MMscf] = 12.4 × 112 = 1,389 STB/d ≈ 1,390 STB/d Results Summary: Surface gas rate: 12.4 MMscfd Condensate rate: 1,390 STB/d Reservoir voidage: 8,860 res bbl/d

Task 2.5: p/Z Depletion Table

p (psia)pprZ (S-K chart / Papay)Bg (res bbl/Mscf)p/Z (psia)Note
7,35011.011.400.7145,250Initial conditions: anchor point
6,2009.291.280.8394,844Dew-point: banking threshold
5,0007.491.091.1674,587Mid-depletion node
3,5005.240.881.7903,977pwf,min: design abandonment

Z-values at 6,200, 5,000, and 3,500 psia estimated from Papay correlation. Bg increases as pressure falls: gas expands at lower pressure, so each surface Mscf represents more reservoir volume. Verify Z against Standing-Katz chart in Topic 1.2, Section 4.

Task 2.6: p/Z Interpretation (expand to check your reasoning)

(a) Bg decreases as pressure falls, because Bg = 0.005035ZT/p and the reduction in p dominates over the reduction in Z. Lower pressure means gas is less compressed; each surface Mscf comes from a smaller reservoir volume. This is counterintuitive but correct: gas gets "easier to produce" per reservoir barrel as pressure falls, which is why gas recovery factors (60–90%) are much higher than oil solution-gas drive (15–30%).

(b) p/Z decreases approximately linearly from 5,250 to 3,977 psia over the four pressure nodes, consistent with volumetric depletion. Any concave-upward curvature (p/Z declining more slowly than expected) would indicate water influx supporting reservoir pressure.

(c) A concave-upward p/Z plot means pressure is being maintained by an energy source (aquifer, gas injection) beyond the gas expansion drive: the reservoir is not depleting purely volumetrically.

(d) To estimate IGIP from the p/Z straight line, you need at least one production data point (Gp, p/Z) in addition to the initial point (0, pi/Zi = 5,250). You then fit the straight line through both points and extrapolate to p/Z = 0. In SP-5 you will use the DST rate and cumulative production data to provide this second point.

KWL Reflection

Update Your KWL Table: SP-2

K: Confirmed
  • Bg = 0.005035 ZT/p derives from the real gas law
  • T must be in Rankine (°F + 460)
  • p/Z vs. Gp is linear for volumetric depletion
W: Questions Answered
  • Z = 1.40 is correct, supercritical region where repulsion dominates
  • Bg = 0.000714 res bbl/scf = 0.714 res bbl/Mscf at initial conditions
  • Voidage = 8,860 res bbl/d at qg = 12.4 MMscfd
  • Condensate rate = 1,390 STB/d at DST rate
  • p̅/Z = 5,250 psia, anchor for p/Z plot
L: New Learning
  • Write your own L-column entry
  • Example: "Bg is smaller (not larger) at lower pressure — this was counterintuitive; less Z amplification at lower ppr more than compensates for the lower absolute pressure"
Just-in-Time Resources

Just-in-Time Resources

Reach for these the moment you get stuck on this sub-problem. They map one-to-one onto the physics and the calculation you are doing here.

Study

Module 01 · Topic 2: Formation Volume Factor & Gas FVF (Bg)
Watch
  • Lecture 1.2A — Formation Volume Factors: Physical Meaning and Engineering Use
  • Lecture 1.2B — Reading PVT Reports: Extracting and Applying FVF Data
  • Lecture 1.2C — FVF in the Material Balance: From Havlena-Odeh to Production Decline
Self-check

fvf_correlations.py, verified calculator; reproduce your numbers.
Knowledge Check

SP-2: Five MCQ

Q1: Pseudo-Critical Properties

Using Standing correlations with γg = 0.65, the correct pseudo-critical temperature Tpc and pseudo-critical pressure ppc are:

A. Tpc = 371.3 °F and ppc = 667.7 psia, same values, temperature in Fahrenheit
B. Tpc = 371.3 °R and ppc = 667.7 psia, temperature in Rankine (the required unit for the pseudo-reduced ratio)
C. Tpc = 400.2 °R and ppc = 620.5 psia, using Kay's mixing rule with pure methane and ethane
D. Tpc = 343.0 °R and ppc = 667.7 psia, Tpc for pure methane at 0.65 gravity
Explanation
Tpc = 187 + 330(0.65) − 71.5(0.65)² = 371.3 °R. This is in Rankine because the constant 187 absorbs the absolute temperature scale. Option A uses °F, which would give Tpr = 745/371.3 = wrong (it's not 745°F, it's 745°R). The pseudo-reduced temperature Tpr = T[°R]/Tpc[°R]: both must be in the same absolute temperature units. Mixing °F for Tpc with °R for reservoir T is a common error that gives a dramatically wrong Z.

Q2: Why Z = 1.40

At Tpr = 2.006 and ppr = 11.01, Z = 1.40 > 1.0. The physical reason is:

A. GK-22 gas contains impurities (H2S, CO2) that always increase Z above 1.0: the Wichert-Aziz correction should be applied
B. At high pseudo-reduced pressure (ppr > 8–10), repulsive intermolecular forces (electron cloud overlap) dominate over attractive van der Waals forces: gas molecules resist further compression and Z exceeds 1.0
C. High temperature (Tpr = 2.0) always causes Z to exceed 1.0 for hydrocarbon gases
D. Z = 1.40 is an error: it should be corrected to 0.875 by dividing by the gas gravity
Explanation
At low to moderate pressures (ppr 1–4), attractive van der Waals forces dominate and Z < 1 (gas more compressible than ideal). At high pressures (ppr > 6–8), molecules are so close that repulsive forces from electron cloud overlap dominate: the gas resists further compression and Z rises above 1.0. This is clearly visible on the Standing-Katz chart as the upper-right region. Option A is plausible but incorrect: the GK-22 fluid is described as a clean hydrocarbon gas at γg = 0.65 with no mention of impurities; the Wichert-Aziz correction is not indicated here.

Q3: Bg Calculation

Bg at 7,350 psia, 745°R, Z = 1.40 using Bg = 0.005035 ZT/p is:

A. 7.14 res bbl/Mscf, factor-of-10 error in unit conversion
B. 0.714 res bbl/Mscf (= 0.0007144 res bbl/scf) — each 1,000 scf of sales gas came from 0.714 reservoir barrels at depth
C. 0.510 res bbl/Mscf, same formula but using Z = 1.0 (ignoring gas deviation factor)
D. 0.00714 res bbl/scf, decimal point shifted one place
Explanation
Bg = 0.005035 × 1.40 × 745 / 7,350 = 5.251 / 7,350 = 0.000714 res bbl/scf = 0.714 res bbl/Mscf. Option C shows what happens if Z = 1 is assumed: Bg = 0.005035 × 1.0 × 745 / 7,350 = 0.510 res bbl/Mscf, 28% lower than the correct value. An engineer using Z = 1 underestimates the reservoir volume represented by each Mscf of production, overestimates AOF, and underestimates IGIP. The Z-factor is not optional for high-pressure gas.

Q4: Reservoir Voidage

At the DST gas rate of 12.4 MMscfd with Bg = 0.714 res bbl/Mscf, the reservoir gas voidage rate is:

A. 885 res bbl/d, dividing instead of multiplying by Bg
B. 8,854 res bbl/d, using qg = 12,400 Mscfd × Bg = 0.714 res bbl/Mscf
C. 1,389 res bbl/d: this is actually the condensate rate in STB/d, not voidage
D. 12,400 res bbl/d — equal to the gas rate in Mscfd because Bg is a dimensionless ratio
Explanation
Vvoid = qg [Mscfd] × Bg [res bbl/Mscf] = 12,400 × 0.714 = 8,854 res bbl/d. Unit tracking: Mscfd × (res bbl/Mscf) = (Mscf/d) × (res bbl/Mscf) = res bbl/d ✓. Option C is the condensate surface rate qc = 12.4 MMscfd × 112 STB/MMscf = 1,389 STB/d, a different calculation entirely. The reservoir voidage (8,854 res bbl/d) is used in the material balance to assess aquifer support: it feeds directly into the p/Z analysis in SP-5.

Q5: p/Z Behaviour During Depletion

As reservoir pressure declines from 7,350 to 3,500 psia (and Z falls from 1.40 to ~0.88), what happens to Bg?

A. Bg increases (from 0.714 res bbl/Mscf at 7,350 psia to ~1.79 res bbl/Mscf at 3,500 psia) because lower pressure means gas expands more; each surface Mscf comes from progressively more reservoir volume as depletion proceeds
B. Bg decreases (from 0.714 to ~0.569 res bbl/Mscf) because the pressure reduction in the denominator is partially offset by the Z reduction (from 1.40 to 0.88), but the net effect is that the gas is less "puffed up" at lower pressure; each surface Mscf represents less reservoir volume
C. Bg remains approximately constant at 0.714 because Z and p change in opposite directions and nearly cancel
D. Bg increases then decreases: it peaks at the dew-point pressure when phase change occurs
Explanation
Answer A is correct: Bg increases as pressure falls. From Bg = 0.005035·Z·T/p, evaluate at 3,500 psia with Z ≈ 0.88 and T = 745 °R: 0.005035 × 0.88 = 0.00443; × 745 = 3.302; ÷ 3,500 = 0.000944 res bbl/scf = 0.944 res bbl/Mscf. Although the denominator p drops and Z also falls during depletion, the pressure reduction dominates, so each surface Mscf is produced from progressively more reservoir volume: the gas expands. This is the physical reason voidage per Mscf rises through field life, and it is exactly the trend the p/Z material-balance analysis in SP-5 relies on.
SP-2 Complete

Carry-Forward to SP-5

SP-2 Carry-Forward Values: Record These Now

CF-2a — Tpc, ppc: 371.3 °R and 667.7 psia (from Standing correlations, γg = 0.65)
CF-2b — Tpr, ppr: 2.006 and 11.01 at initial conditions
CF-2c — Z-factor: 1.40 at 7,350 psia, 285°F (supercritical; Z > 1 is correct)
CF-2d — Bg at initial conditions: 0.000714 res bbl/scf = 0.714 res bbl/Mscf
CF-2e — Reservoir voidage at DST rate: 8,860 res bbl/d at qg = 12.4 MMscfd
CF-2f — Condensate rate at DST: 1,390 STB/d
CF-2g — Initial p/Z: 5,250 psia (= 7,350 / 1.40), anchor point for p/Z plot in SP-5
CF-2h, μg for SP-5: 0.0312 cp at reservoir conditions