1. Thermodynamic Phase Change & Saturation Plateau Pressure
When a supercritical dense-phase \(\text{CO}_2\) pipeline at initial conditions (\(P_0, T_0\)) experiences a catastrophic rupture, the fluid decompresses isentropically until intersecting the vapor-liquid phase boundary. Pure \(\text{CO}_2\) possesses a critical temperature of \(31.0^\circ\text{C}\) and critical pressure of \(73.8\ \text{bar}\). The saturation pressure (\(P_{\text{sat}}\)) forms a persistent pressure plateau:
\[ P_{\text{plat}} = P_{\text{sat}}(T_0) + \Delta P_{\text{impurities}} \]
Light impurities with higher volatility (\(\text{CH}_4, \text{N}_2, \text{H}_2\)) expand the two-phase envelope upwards, shifting the decompression plateau to significantly higher pressures:
\[ \Delta P_{\text{impurities}} \approx y_{\text{CH}_4}(4.2) + y_{\text{N}_2}(7.8) + y_{\text{H}_2}(11.5) \quad [\text{bar}] \]
2. Isentropic Decompression Wave Velocity (\(W\))
In single-phase dense fluid, acoustic wave speed is high (\(c_0 \approx 500\text{--}750\ \text{m/s}\)). Across the two-phase mixture region, sound speed collapses to the homogeneous equilibrium mixture (HEM) speed:
\[ c_{\text{HEM}} = \sqrt{\left(\frac{\partial P}{\partial \rho}\right)_s} \approx 80\text{--}145\ \text{m/s} \]
The decompression wave speed \(W(P)\) dictates how fast pressure drops ahead of the advancing crack tip.
3. Battelle Two-Curve Model (BTCM) Crack Velocity (\(V_{\text{crack}}\))
The Maxey / Battelle formulation determines the velocity of a running ductile axial shear fracture in line pipe steel under internal hoop stress \(\sigma_h = \frac{P \cdot D_{\text{mean}}}{2 t}\):
\[ V_{\text{crack}}(P) = C_1 \cdot \frac{\bar{\sigma}}{\sqrt{C_v / A_c}} \left(\frac{\sigma_h(P)}{\sigma_a} - 1\right)^{1/6} \]
where \(\bar{\sigma}\) is flow stress (\(\approx 1.15 \cdot \text{SMYS}\)), \(C_v\) is Upper Shelf Charpy V-notch energy, and \(\sigma_a = 0.333 \bar{\sigma}\) is the crack arrest stress threshold below which ductile tearing terminates.
4. Required Charpy Energy for Crack Arrest (\(C_{v,\text{req}}\))
For high-pressure dense \(\text{CO}_2\) pipelines, the uncorrected BTCM significantly under-predicts required toughness. Per ISO 27913 and EPRG recommendations, the Leis correction factor is applied for API 5L X65–X80 steels:
\[ C_{v,\text{req}} = F_{\text{Leis}} \cdot C_{v,\text{BTCM}} \qquad (F_{\text{Leis}} \approx 1.40\text{--}1.95) \]
If \(C_{v,\text{actual}} < C_{v,\text{req}}\), running ductile fracture cannot be arrested by steel toughness alone, and mechanical crack arrestors must be installed at intervals \(L_{\text{arrestor}} \le 250\text{--}350\ \text{m}\).