# Relative Permeability of Near-Miscible Fluids in Compositional Simulators

- 179 Downloads

## 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.

## References

- Al-Wahaibi, Y.M., Grattoni, C.A., Muggeridge, A.H.: Drainage and imbibition relative permeabilities at near miscible conditions. J. Pet. Sci. Eng.
**53**, 239 (2006). https://doi.org/10.1016/j.petrol.2006.06.005 CrossRefGoogle Scholar - Alzayer, A.N.: Relative permeability of near-miscible fluids in compositional simulators. Master’s Thesis, Stanford University (2015)Google Scholar
- Aziz, K., Settari, A.: Petroleum Reservoir Simulation. Chapman & Hall, London (1979)Google Scholar
- Blom, S.M.P.: Relative permeability to near-miscible fluids. Delft University of Technology, TU Delft (1999)Google Scholar
- Blunt, M.J.: An empirical model for three-phase relative permeability. SPE J.
**5**(December), 435 (2000). https://doi.org/10.2118/67950-PA CrossRefGoogle Scholar - Cao, H., Tchelepi, H.A., Wallis, J.R., Yardumian, H.E.: Parallel scalable unstructured CPR-type linear solver for reservoir simulation. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (2005)Google Scholar
- Cao, H., Zaydullin R., Obi E., et al.: Nonlinear convergence for near-miscible problem: a mystery unveiled for natural variable simulator. In: SPE Reservoir Simulation Conference. Society of Petroleum Engineers (2017)Google Scholar
- Christie, M., Blunt, M.: Tenth SPE comparative solution project: a comparison of upscaling techniques. SPE Reserv. Eval. Eng.
**4**, 308 (2001). https://doi.org/10.2118/72469-PA CrossRefGoogle Scholar - Coats, K.: An equation of state compositional model. Soc. Pet. Eng. J.
**20**, 363 (1980). https://doi.org/10.2118/8284-PA CrossRefGoogle Scholar - Fayers, F., Foakes, A., Lin, C., Puckett, D.: An improved three phase flow model incorporating compositional variance. In: SPE/DOE Improved Oil Recovery Symposium. Society of Petroleum Engineers (2000)Google Scholar
- Fevang, O.I., Whitson, C.H.: Modeling gas condensate well deliverability. In: SPE Reservoir Engineering (November 1996)Google Scholar
- Geoquest, S.: Eclipse 100 reference manual. Schlumberger Geoquest
**18**, 1178 (2001)Google Scholar - Gosset, R., Heyen, G., Kalitventzeff, B.: An efficient algorithm to solve cubic equations of state. Fluid Phase Equilib.
**25**(01), 51 (1986)CrossRefGoogle Scholar - Iranshahr, A., Voskov, D.V., Tchelepi, H.A.: Gibbs energy analysis: Compositional tie-simplex space. Fluid Phase Equilib.
**321**, 49 (2012). https://doi.org/10.1016/j.fluid.2012.02.001 CrossRefGoogle Scholar - Iranshahr, A., Voskov, D., Tchelepi, H.A.: A negative-flash tie-simplex approach for multiphase reservoir simulation. SPE J.
**18**(6), 1–140 (2013)CrossRefGoogle Scholar - Jerauld, G.: General three-phase relative permeability model for Prudhoe bay. In: SPE Reservoir Engineering, vol. 12 (November 1997). https://doi.org/10.2118/36178-PA
- Lake, L.W.: Enhanced Oil Recovery. Prentice Hall Inc., Old Tappan (1989)Google Scholar
- Macleod, D.: On a relation between surface tension and density. Trans. Faraday Soc.
**19**, 38 (1923)CrossRefGoogle Scholar - Metcalfe, R., Yarborough, L.: The effect of phase equilibria on the CO\(_{2}\) displacement mechanism. Soc. Pet. Eng. J
**19**(8), 242 (1979)CrossRefGoogle Scholar - Orr, F.M.: Theory of Gas Injection Processes. Tie-Line Publications, Copenhagen (2007)Google Scholar
- Orr Jr., F., Johns, R., Dindoruk, B.: Development of miscibility in four-component CO\(_{2}\) floods. SPE Reserv. Eng.
**8**(02), 135 (1993)CrossRefGoogle Scholar - Sugden, S.: The variation of surface tension with temperature and some related functions. J. Chem. Soc.
**125**, 32 (1924)CrossRefGoogle Scholar - Tang, D., Zick, A.: A new limited compositional reservoir simulator. In: SPE Symposium on Reservoir Simulation (1993)Google Scholar
- Voskov, D.V., Tchelepi, H.A.: Compositional space parameterization: multicontact miscible displacements and extension to multiple phases. SPE J.
**14**(03), 441 (2009)CrossRefGoogle Scholar - Voskov, D.V., Tchelepi, H.A.: Comparison of nonlinear formulations for two-phase multi-component EoS based simulation. J. Pet. Sci. Eng.
**82**, 101 (2012)CrossRefGoogle Scholar - Voskov, D., Younis, R., Tchelepi, H.: General nonlinear solution strategies for multi-phase multi-component EoS based simulation. In: SPE Reservoir Simulation Symposium Proceedings, vol. 1, pp. 649–663 (2009)Google Scholar
- Wallis, J.: Incomplete Gaussian elimination as a preconditioning for generalized conjugate gradient acceleration. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (1983)Google Scholar
- Weinaug, C.F., Katz, D.L.: Surface tensions of methane-propane mixtures. Ind. Eng. Chem.
**35**, 239 (1943). https://doi.org/10.1021/ie50398a028. CrossRefGoogle Scholar - Whitson, C.H., Brulé, M.R.: Phase Behavior. Henry L. Doherty Memorial Fund of AIME, Society of Petroleum Engineers, Richardson (2000)Google Scholar
- Yuan, C.: A new relative permeability model for compositional simulation of two and three phase flow. Ph.D. Thesis, University of Texas (2010)Google Scholar
- Yuan, C., Pope, G.: A new method to model relative permeability in compositional simulators to avoid discontinuous changes caused by phase-identification problems. SPE J.
**17**, 1221 (2012). https://doi.org/10.2118/142093-PA CrossRefGoogle Scholar - Zhou, Y., Tchelepi, H., Mallison, B.: Automatic differentiation framework for compositional simulation on unstructured grids with multi-point discretization schemes. In: SPE Reservoir Simulation Symposium (SPE 141592) (2011)Google Scholar
- Zick, A.: A combined condensing/vaporizing mechanism in the displacement of oil by enriched gases. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (1986)Google Scholar