Continuous Relative Permeability Model for Compositional Reservoir Simulation, Using the True Critical Point and Accounting for Miscibility

Compositional simulators conventionally use Li's correlation to approximate the critical temperature (T_c) of hydrocarbon mixtures, used to arbitrarily label them as ‘gas’ or ‘oil’ outside the two-phase (gas/oil) envelope. T_c also feeds into the calculation of hydrocarbon-to-water relative permeabilities, typically interpolated between oil-to-water and gas-to-water based on the parameter T_c/T. This approach is clearly inconsistent when crossing the phase envelope in case of incorrect phase labeling.

We here propose to replace Li's correlation by a rigorous calculation of T_c using Michelsen's (1984)'s algorithm when phase identification is required. Using the gas-oil interfacial tension (IFT) within the phase envelope, and a fictitious gas-oil IFT computed at the saturation pressure (p_sat) outside the envelope, the hydrocarbon-to-water relative permeability is then built by interpolation between oil-to-water and gas-to-water relative permeabilities. The derivatives of p_sat (hence the IFT) with respect to the primary variables are computed analytically, to ensure robust fully implicit simulations.

In order to address the observation that gas/oil relative permeability curves tend to straight lines when approaching to the critical point, a second level of interpolation with respect to the IFT is applied within the phase envelope between miscible and immiscible three-phase models. Continuity is, by construction, guaranteed at any possible phase-state transition.

The proposed relative permeability model is first tested standalone (i.e., on a single cell) with different hydrocarbon mixtures, by analysis of the dependent parameter (true or fictitious IFT) and the relative permeabilities at different p-T conditions; in particular, continuity at each relevant interface of the p-T diagram is illustrated.

The model is secondly implemented in our In-House Research Reservoir Simulator (IHRRS), and tested on a synthetic 2D cross-section undergoing near-critical gas injection. We observe that with conventional models based on Li's correlation, discontinuities in the relative permeability model when crossing the phase envelope occur, as well as spurious phase flipping. No such unphysical behavior is observed with the proposed approach, while requiring the same input data.

There is of course a computational cost involved in properly calculating T_c, which is partly offset by the improved model convergence; because it is a cell-by-cell calculation, the overhead however scales down very well with parallelization.