Question 2 of 4: Methane / n-Butane Binary P-x Diagram

Problem 2: Methane / n-Butane Binary P-x Diagram (25 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. Isothermal ($\approx150^{\circ}$F) methane/n-butane equilibrium data: bubble-branch liquid composition $x_{C_1}$ and dew-branch vapor composition $y_{C_1}=1-y_{C_4}$ at five pressures (table above); reservoir at 1000 psia with overall composition $z_{C_1}=0.40$; injection gas $z_{C_1}=0.90$.

Find. (a) butane's own saturation pressure at $150^{\circ}$F; (b) the critical composition and pressure of the methane/butane mixture; (c) whether the 1000-psia reservoir fluid is oil or gas, and its liquid molar fraction; (d) whether the injection gas and reservoir fluid are first-contact miscible.

Approach. Plot $x_{C_1}$ (bubble/liquid branch) and $y_{C_1}=1-y_{C_4}$ (dew/vapor branch) against pressure to trace the closed two-phase loop (methane is supercritical at $150^{\circ}$F, so the loop does not reach $x=1$); read/extrapolate the low-pressure end for part (a) and the high-pressure convergence for part (b); apply the lever rule at the 1000-psia isobar for part (c); and check whether the straight mixing line between the two given compositions crosses the two-phase envelope for part (d).

1. Build and plot the P-x diagram. Converting the table's vapor-phase butane fraction to a vapor-phase methane fraction, $y_{C_1}=1-y_{C_4}$: $(150,0.03)$, $(200,0.23)$, $(500,0.61)$, $(1000,0.72)$, $(1500,0.68)$. Plotting $x_{C_1}$ (bubble curve, liquid) and $y_{C_1}$ (dew curve, vapor) against pressure (Fig. 3) traces the classic closed two-phase loop for a light/heavy binary at fixed sub-critical-for-the-heavy-component temperature: the bubble curve rises monotonically with pressure (more methane dissolves in the liquid), while the dew curve rises to a maximum near 1000 psia and then bends back (retrograde behaviour) as the two branches converge toward the mixture's critical point beyond the last tabulated row.
2. Part (a): butane's saturation pressure at $150^{\circ}$F. At $x_{C_1}=y_{C_1}=0$ (pure butane), both branches must meet at butane's own vapor pressure. Extrapolating the bubble branch's first two points $(150\text{ psia},0.002)\to(200\text{ psia},0.02)$ back to $x=0$ gives $P\approx144.4$ psia; extrapolating the dew branch's first two points $(150,0.03)\to(200,0.23)$ back to $y=0$ gives $P\approx142.5$ psia. The two independent extrapolations agree closely, so $$\boxed{P_{sat,\,\text{butane}}\ (150^{\circ}\text{F})\approx144\ \text{psia}}$$ (consistent with the lowest tabulated row, 150 psia, already sitting almost at the pure-butane point).
3. Part (b): critical point. The bubble and dew branches converge as pressure rises; extending the bubble curve's rising trend and the dew curve's now-falling trend from the last two rows (1000, 1500 psia) to where they intersect gives $$\boxed{P_c\approx1789\ \text{psia},\quad x_c\approx0.66\ \text{mole fraction methane}}$$ — the top of the closed loop in Fig. 3, beyond the range actually tabulated, read here by extrapolation exactly as the exam's "carefully plot...and identify" instruction intends.
4. Part (c): state and liquid fraction at 1000 psia, $z_{C_1}=0.40$. At 1000 psia the table gives the bubble (liquid) composition $x_{C_1}=0.31$ and dew (vapor) composition $y_{C_1}=0.72$. Since $x_{C_1}=0.31\lt z_{C_1}=0.40\lt y_{C_1}=0.72$, the overall composition point lies inside the two-phase envelope at this pressure — the in-situ fluid is genuinely two-phase, not single-phase. Applying the lever rule, $$L=\frac{y_{C_1}-z_{C_1}}{y_{C_1}-x_{C_1}}=\frac{0.72-0.40}{0.72-0.31}=\frac{0.32}{0.41}=\boxed{0.7805\ (78.05\%\text{ liquid, molar}),\qquad V=0.2195\ (21.95\%\text{ vapor}).}$$ Since liquid moles dominate by more than 3:1, the fluid is classified as an oil (with associated free gas), not a gas.
5. Part (d): first-contact miscibility. Two fluids are first-contact miscible only if every mixture along the straight composition line joining them remains single-phase at reservoir (P,T). Here the reservoir fluid's own composition ($z_{C_1}=0.40$) already sits inside the two-phase envelope at 1000 psia (Part c). Any blend formed by injecting the 90%-methane gas into this reservoir fluid moves along the mixing line from $z=0.40$ toward $z=0.90$, and since the starting point is itself two-phase, blends near the reservoir-fluid end of that line are unavoidably two-phase as well. $$\boxed{\text{Not first-contact miscible} \text{ — the mixing path must cross the two-phase region (}x_{C_1}=0.31\text{ to }y_{C_1}=0.72\text{ at 1000 psia) before any single-phase blend is reached.}}$$