# Relative Permeability of Near-Miscible Fluids in Compositional Simulators

## Abstract

Miscible gas injection is one of the most effective enhanced oil recovery techniques. There are several challenges in accurately modeling this process, which occurs in the near-miscible region. The adjustment of relative permeability for near-miscible processes is the main focus of this work. The dependence of relative permeability on phase identification can lead to significant complications while simulating near-miscible displacements. We present an analysis of how existing methods incorporate compositional dependence in relative permeability functions. The sensitivity of the different methods to the choice of reference points is presented with guidelines to limit the modification of the relative permeabilities to physically reasonable values. We distinguish between the two objectives of reflecting near-miscible behavior and ensuring smooth transitions across phase changes. We highlight an important link that combines the two objectives in a more general framework. We make use of Gibbs free energy as a compositional indicator in the generalized framework. The new approach was implemented in an automatic differentiation general purpose research simulator and tested on a set of near-miscible gas-injection problems. We show that including compositional dependencies in the relative permeability near the critical point impacts the simulation results with significant improvements in nonlinear convergence.

## Keywords

Relative permeability Gas injection Miscible Compositional Simulation Surface tension Gibbs free energy## List of symbols

- \(k_{\text {rp}}^{\mathrm{Cor}}\)
Corrected relative permeability of phase

*p*- \(k_{\text {rp}}^{\mathrm{Imm}}\)
Immiscible relative permeability of phase

*p*- \(k_{\text {rp}}^{\mathrm{Mis}}\)
Miscible relative permeability of phase

*p*- \(k_{\text {rp-ep}}\)
End-point relative permeability of phase

*p*- \(F_k\)
Interpolation parameter

- \(\sigma \)
Surface tension (dynes/cm)

- \(\sigma _0\)
Reference surface tension (dynes/cm)

*n*Exponent

- \(N_{\text {cap}}\)
Capillary number

*u*Superficial velocity (m/s)

- \(\mu \)
Viscosity (cp)

- \(\alpha \)
Rock-dependent constant from Fevang and Whitson (1996)

- \(N_{\text {c}}\)
Number of components

- \(P_i\)
Parachor of component

*i*—empirical constant [\((\hbox {dyne/cm})^{1/4}\,(\hbox {m}^3\hbox {/mol})\)]- \(x_i\)
Liquid molar fraction of component

*i*- \(y_i\)
Vapor molar fraction of component

*i*- \(\rho ^m_{\text {L}}\)
Liquid molar density (g–Mole/cc)

- \(\rho ^m_{\text {V}}\)
Vapor molar density (g–Mole/cc)

- \(S_i\)
Saturation of phase

*i*- \(\xi _i\)
Parachor-weighted molar density of cell

*i*[\((\hbox {dyne/cm})^{1/4}\)]- \(\xi _{p0}\)
Reference parachor-weighted molar density of phase

*p*[\((\hbox {dyne/cm})^{1/4}\)]- \(f_{pi}\)
Fugacity of component

*i*in phase*p*(bars)- \(g_p^*\)
Normalized Gibbs free energy of phase

*p*- \(g_i^*\)
Normalized Gibbs free energy of cell

*i*- \(g_0^*\)
Reference normalized Gibbs free energy

- \(x_{\mathrm{D}}\)
Dimensionless distance

## Notes

### Acknowledgements

We would like to thank Chengwu Yuan for his support and Curtis Whitson for his feedback. We also would like to thank Saudi Aramco and the Stanford University Petroleum Research Institute for Reservoir Simulation (SUPRI-B) for financial support.

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