Volumetric Dry-Gas Reservoir — p/Z Material Balance: 24-Pet-A3 Fundamental Reservoir Engineering, December 2015, Q4

Question 4: Volumetric Dry-Gas Reservoir — p/Z Material Balance (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Gas specific gravity, $\gamma_g$	0.65 (air = 1)
Initial pressure, $p_i$	4000 psia
Formation temperature, $T$	200°F = 660°R
Cumulative production at $p_1=3000$ psia, $G_{p1}$	500 MMSCF
Target pressure, $p_2$	2000 psia

Find. The original gas in place $G$, and the cumulative gas production $G_{p2}$ when $p$ drops to 2000 psia.

Check: no gas composition (impurities) is given, so the sweet-gas Standing pseudo-critical correlation applies directly; the $Z$-factors below use the Dranchuk–Abu-Kassem (DAK) analytical fit to the Standing–Katz chart rather than a manual chart read, for reproducible precision.

Approach. Compute pseudo-critical properties from $\gamma_g$, get $T_{pr}$ and $P_{pr}$ at each pressure, solve for $Z$ via the DAK correlation, then fit the two known $(G_p,\,p/Z)$ points to the straight-line volumetric gas material balance $p/Z=(p_i/Z_i)(1-G_p/G)$ to get $G$, and evaluate it again at $p_2$.

Pseudo-critical properties. $T_{pc}=168+325\gamma_g-12.5\gamma_g^2=168+325(0.65)-12.5(0.65)^2=374.0$°R; $P_{pc}=677+15.0\gamma_g-37.5\gamma_g^2=677+15(0.65)-37.5(0.65)^2=670.9$ psia. $T_{pr}=T/T_{pc}=660/374.0=\boxed{1.765}$ (constant — temperature does not change).
Z-factors at each pressure (DAK correlation). $P_{pr,i}=4000/670.9=5.96\Rightarrow Z_i=0.9350$; $P_{pr,1}=3000/670.9=4.47\Rightarrow Z_1=0.8925$; $P_{pr,2}=2000/670.9=2.98\Rightarrow Z_2=0.8860$.
p/Z values. $p_i/Z_i=4000/0.935=4278.3$ psia; $p_1/Z_1=3000/0.893=3361.2$ psia; $p_2/Z_2=2000/0.886=2257.4$ psia.
Solve for $G$ from the two known points, then $G_{p2}$. The straight line through $(0,\,p_i/Z_i)$ and $(G_{p1},\,p_1/Z_1)$ has $x$-intercept $G=\dfrac{(p_i/Z_i)\,G_{p1}}{(p_i/Z_i)-(p_1/Z_1)}=\dfrac{(4278.3)(500)}{4278.3-3361.2}=\boxed{2333\ \text{MMSCF}}$. Then $G_{p2}=G\left(1-\dfrac{p_2/Z_2}{p_i/Z_i}\right)=2333\left(1-\dfrac{2257.4}{4278.3}\right)=\boxed{1102\ \text{MMSCF}}$.

Fig. 2 — p/Z straight-line volumetric-gas material balance; the two measured points fix the line, whose $G_p$-axis intercept is the original gas in place.

Quantity	Value
$T_{pc}$, $P_{pc}$	374.0 °R, 670.9 psia
$Z_i$, $Z_1$, $Z_2$	0.9350, 0.8925, 0.8860
Original gas in place, $G$	2333 MMSCF (2.33 Bscf)
Cumulative production at $p=2000$ psia, $G_{p2}$	1102 MMSCF

Concept & Theory

Topic: Volumetric (no water influx) dry-gas material balance.

Key relations: $p/Z=(p_i/Z_i)(1-G_p/G)$ — a straight line in $G_p$; real-gas $Z$ from the Standing–Katz correlation via pseudo-reduced properties, $T_{pr}=T/T_{pc}$, $P_{pr}=p/P_{pc}$.

Why this works: With no aquifer support, all production is accounted for purely by gas expansion, so a plot of $p/Z$ against $G_p$ is exactly linear; any two points on it (plus the origin's $p_i/Z_i$) fully determine $G$ without needing a full pressure history.

Common pitfall: Extrapolating $p/Z$ to $G_p=G$ using raw pressure instead of $p/Z$ — the material balance is linear in $p/Z$, not in $p$ itself, because $Z$ itself varies with pressure.

Source: Ahmed, Reservoir Engineering Handbook, 5th ed., Ch. 5 (gas material balance); Lyons (ed.), Standard Handbook of Petroleum and Natural Gas Engineering, 3rd ed. (Standing–Katz $Z$-factor correlation, DAK fit).

Question 4 of 7: Volumetric Dry-Gas Reservoir — p/Z Material Balance

Question 4: Volumetric Dry-Gas Reservoir — p/Z Material Balance (20 marks)