Accurate and simple liquid holdup prediction method for liquid-rich high-viscosity two-phase flows from horizontal to vertical

Liquid holdup governs the hydrostatic pressure gradient and the liquid inventory of gas–liquid pipelines, and the available correlations lose accuracy when the liquid is viscous, when the pipe is inclined, or both. This work develops and tests a correlation for liquid-rich, liquid-dominated flow, H_L ≥ 0.50, in which the liquid viscosity and the pipe inclination enter the drift-flux closures explicitly. Two databases of measured holdup were assembled: a low-viscosity benchmark of 601 points, turbulent at 99.8 % of them, and a high-viscosity database of 1069 points spanning 0.10–6.39 Pa s and horizontal to vertical flow, laminar throughout, of which 396 were held out. Twenty-three published correlations were evaluated on both. Those leading at low viscosity acquire negative biases of −5.5 to −23.6 % on the viscous data, because a distribution parameter calibrated on a turbulent profile cannot supply the slip a laminar one requires, and seventeen of the twenty-three are less accurate there than a constant-holdup baseline. The proposed correlation writes the distribution parameter as the no-slip gas fraction plus a coefficient nearly constant over the liquid-dominated domain, median 1.246. That coefficient follows a power law in viscosity, carrying the distribution parameter from 1.74 to 2.03 onto the fully developed laminar value, and falls toward the no-slip value as the pipe is raised to vertical. The correlation is a single explicit expression: non-iterative, bounded, monotone, and needing no flow-pattern input. It predicts the training set with a mean absolute percentage error of 4.71 % and the independent test set with 6.12 %, first in every comparison on the viscous data and ahead of a unified mechanistic model. Its holdup floor of 0.459–0.571 is reported as an applicability condition testable from the inputs.