Cocurrent Spontaneous Imbibition in Porous Media With the Dynamics of Viscous Coupling and Capillary Backpressure
- Pål Ø. Andersen (University of Stavanger) | Yangyang Qiao (University of Stavanger) | Dag Chun Standnes (University of Stavanger) | Steinar Evje (University of Stavanger)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- February 2019
- Document Type
- Journal Paper
- 158 - 177
- 2019.Society of Petroleum Engineers
- Dynamic relative permeability, Capillary Back Pressure, Viscous coupling based on mixture theory, Co- and counter-current flow, Co-current spontaneous imbibition
- 9 in the last 30 days
- 248 since 2007
- Show more detail
- View rights & permissions
|SPE Member Price:||USD 12.00|
|SPE Non-Member Price:||USD 35.00|
This paper presents a numerical study of water displacing oil using combined cocurrent/countercurrent spontaneous imbibition (SI) of water displacing oil from a water-wet matrix block exposed to water on one side and oil on the other. Countercurrent flows can induce a stronger viscous coupling than during cocurrent flows, leading to deceleration of the phases. Even as water displaces oil cocurrently, the saturation gradient in the block induces countercurrent capillary diffusion. The extent of countercurrent flow may dominate the domain of the matrix block near the water-exposed surfaces while cocurrent imbibition may dominate the domain near the oil-exposed surfaces, implying that one unique effective relative permeability curve for each phase does not adequately represent the system. Because relative permeabilities are routinely measured cocurrently, it is an open question whether the imbibition rates in the reservoir (depending on a variety of flow regimes and parameters) will in fact be correctly predicted. We present a generalized model of two-phase flow dependent on momentum equations from mixture theory that can account dynamically for viscous coupling between the phases and the porous media because of fluid/rock interaction (friction) and fluid/fluid interaction (drag). These momentum equations effectively replace and generalize Darcy’s law. The model is parameterized using experimental data from the literature.
We consider a water-wet matrix block in one dimension that is exposed to oil on one side and water on the other side. This setup favors cocurrent SI. We also account for the fact that oil produced countercurrently into water must overcome the so-called capillary backpressure, which represents a resistance for oil to be produced as droplets. This parameter can thus influence the extent of countercurrent production and hence viscous coupling. This complex mixture of flow regimes implies that it is not straightforward to model the system by a single set of relative permeabilities, but rather relies on a generalized momentum-equation model that couples the two phases. In particular, directly applying cocurrently measured relative permeability curves gives significantly different predictions than the generalized model. It is seen that at high water/oil-mobility ratios, viscous coupling can lower the imbibition rate and shift the production from less countercurrent to more cocurrent compared with conventional modeling. Although the viscous-coupling effects are triggered by countercurrent flow, reducing or eliminating countercurrent production by means of the capillary backpressure does not eliminate the effects of viscous coupling that take place inside the core, which effectively lower the mobility of the system. It was further seen that viscous coupling can increase the remaining oil saturation in standard cocurrent-imbibition setups.
|File Size||1 MB||Number of Pages||20|
Ambrosi, D. and Preziosi, L. 2002. On the Closure of Mass Balance Models for Tumor Growth. Math. Mod. Meth. Appl. Sci. 12 (5): 737–754. https://doi.org/10.1142/S0218202502001878.
Andersen, P. Ø., Evje, S., and Kleppe, H. 2014. A Model for Spontaneous Imbibition as a Mechanism for Oil Recovery in Fractured Reservoirs. Transport Porous Med. 101 (2): 299–331. https://doi.org/10.1007/s11242-013-0246-7.
Andersen, P. Ø., Bratteka°s, B., Walrond, K. et al. 2017a. Numerical Interpretation of Laboratory Spontaneous Imbibition—Incorporation of the Capillary Back Pressure and How it Affects SCAL. Presented at the Abu Dhabi International Petroleum Exhibition & Conference, Abu Dhabi, 13–16 November. SPE-188625-MS. https://doi.org/10.2118/188625-MS.
Andersen, P. Ø., Evje, S., and Hiorth, A. 2017b. Modeling of Spontaneous-Imbibition Experiments With Porous Disk—On the Validity of Exponential Prediction. SPE J. 22 (5): 1326–1337. SPE-186094-PA. https://doi.org/10.2118/186094-PA.
Andersen, P. Ø., Skjæveland, S. M., and Standnes, D. C. 2017c. A Novel Bounded Capillary Pressure Correlation With Application to Both Mixed and Strongly Wetted Porous Media. Presented at the Abu Dhabi International Petroleum Exhibition & Conference, Abu Dhabi, 13–16 November. SPE-188291-MS. https://doi.org/10.2118/188291-MS.
Andersen, P. Ø., Standnes, D. C., and Skjæveland, S. M. 2017d. Waterflooding Oil-Saturated Core Samples—Analytical Solutions for Steady-State Capillary End Effects and Correction of Residual Saturation. J. Pet. Sci. Eng. 157 (August): 364–379. https://doi.org/10.1016/j.petrol.2017.07.027.
Anderson, W. G. 1987. Wettability Literature Survey Part 5: The Effects of Wettability on Relative Permeability. J Pet Technol 39 (11): 1453–1468. SPE-16323-PA. https://doi.org/10.2118/16323-PA.
Armstrong, R. T.,McClure, J. E., Berill,M. A. et al. 2017. Flow Regimes During Immiscible Displacement. Petrophysics 58 (1): 10–18. SPWLA-2017-v58n1a1.
Babchin, A., Yuan, J., and Nasr, T. 1998. Generalized Phase Mobilities in Gravity Drainage Processes. Presented at the Annual Technical Meeting, Calgary, Alberta, Canada, 8–10 June. PETSOC-98-09. https://doi.org/10.2118/98-09.
Barenblatt, G. I., Patzek, T. W., and Silin, D. B. 2003. The Mathematical Model of Nonequilibrium Effects in Water-Oil Displacement. SPE J. 8 (4): 409–416. SPE-87329-PA. https://doi.org/10.2118/87329-PA.
Bear, J. 1988. Dynamics of Fluids in Porous Media. Chelmsford, Massachusetts: Courier Corporation.
Bentsen, R. G. 2001. The Physical Origin of Interfacial Coupling in Two-Phase Flow Through Porous Media. Transport Porous Med. 44 (1): 109–122. https://doi.org/10.1023/A:1010791526239.
Bentsen, R. G. and Manai, A. A. 1992. Measurement of Concurrent and Countercurrent Relative Permeability Curves Using the Steady-State Method. AOSTRA J. Res. 7: 160–181.
Bourbiaux, B. J. and Kalaydjian, F. J. 1990. Experimental Study of Concurrent and Countercurrent Flows in Natural Porous Media. SPE Res Eval & Eng 5 (3): 361–368. SPE-18283-PA. https://doi.org/10.2118/18283-PA.
Bowen, R. M. 1980. Incompressible Porous Media Models by Use of the Theory of Mixtures. Int. J. Eng. Sci. 18 (9): 1129–1148. https://doi.org/10.1016/0020-7225(80)90114-7.
Bradford, S. A. and Leij, F. J. 1997. Estimating Interfacial Areas for Multi-Fluid Soil Systems. J. Contam. Hydrol. 27 (1–2): 83–105. https://doi.org/10.1016/S0169-7722(96)00048-4.
Byrne, H. M. and Preziosi, L. 2003. Modelling Solid Tumour Growth Using the Theory of Mixtures. Math. Med. Biol. 20 (4): 341–366. https://doi.org/10.1093/imammb20.4.341.
Dullien, F. A. L. and Dong, M. 1996. Experimental Determination of the Flow Transport Coefficients in the Coupled Equations of Two-Phase Flow in Porous Media. Transport Porous Med. 25 (1): 97–120. https://doi.org/10.1007/BF00141264.
Ehrlich, R. 1993. Viscous Coupling in Two-Phase Flow in Porous Media and its Effect on Relative Permeabilities. Transport Porous Med. 11 (3): 201–218. https://doi.org/10.1007/BF00614812.
Evje, S. 2017. An Integrative Multiphase Model for Cancer Cell Migration Under Influence of Physical Cues From the Microenvironment. Chem. Eng. Sci. 165 (29 June): 240–259. https://doi.org/10.1016/j.ces.2017.02.045.
Geffen, T. M., Owens, W. W., Parrish, D. R. et al. 1951. Experimental Investigation of Factors Affecting Laboratory Relative Permeability Measurements. J Pet Technol 3 (4): 99–110. SPE-951099-G. https://doi.org/10.2118/951099-G.
Goode, P. A. and Ramakrishnan, T. S. 1993. Momentum Transfer Across Fluid-Fluid Interfaces in Porous Media: A Network Model. AIChE J. 39 (7): 1124–1134. https://doi.org/10.1002/aic.690390705.
Haugen, Å., Fernø, M. A., Mason, G. et al. 2014. Capillary Pressure and Relative Permeability Estimated From a Single Spontaneous Imbibition Test. J. Pet. Sci. Eng. 115 (March): 66–77. https://doi.org/10.1016/j.petrol.2014.02.001.
Kalaydjian, F. 1990. Origin and Quantification of Coupling Between Relative Permeabilities for Two-Phase Flows in Porous Media. Transport Porous Med. 5 (3): 215–229. https://doi.org/10.1007/BF00140013.
Langaas, K. 1998. Viscous Coupling and Two-Phase Flow in Porous Media. Oral presentation given at ECMOR VI—6th European Conference on the Mathematics of Oil Recovery, Peebles, Scotland, 8–11 September. https://doi.org/10.3997/2214-4609.201406624.
Lefebvre du Prey, E. J. 1973. Factors Affecting Liquid-Liquid Relative Permeabilities of a Consolidated Porous Medium. SPE J. 13 (1): 39–47. SPE-3039-PA. https://doi.org/10.2118/3039-PA.
Li, H., Pan, C., and Miller, C. T. 2005. Pore-Scale Investigation of Viscous Coupling Effects for Two-Phase Flow in Porous Media. Phys. Rev. 72 (2): 026705. https://doi.org/10.1103/PhysRevE.72.026705.
Li, Y., Ruth, D., Mason, G. et al. 2006. Pressures Acting in Counter-Current Spontaneous Imbibition. J. Pet. Sci. Eng. 52 (1–4): 87–99. https://doi.org/10.1016/j.petrol.2006.03.005.
Lohne, A. 2013. User’s Manual for BugSim—An MEOR Simulator, Version 1.2. Technical report, International Research Institute of Stavanger, Stavanger, Norway.
Mason, G. and Morrow, N. R. 2013. Developments in Spontaneous Imbibition and Possibilities for Future Work. J. Pet. Sci. Eng. 110 (October): 268–293. https://doi.org/10.1016/j.petrol.2013.08.018.
Mason, G., Fischer, H., Morrow, N. R. et al. 2010. Correlation for the Effect of Fluid Viscosities on Counter-Current Spontaneous Imbibition. J. Pet. Sci. Eng. 72 (1–2): 195–205. https://doi.org/10.1016/j.petrol.2010.03.017.
Meng, Q., Liu, H., and Wang, J. 2015. Entrapment of the Non-Wetting Phase During Co-Current Spontaneous Imbibition. Energy Fuels 29 (2): 686–694. https://doi.org/10.1021/ef5025164.
Meng, Q., Liu, H., and Wang, J. 2017. Effect of Viscosity on Oil Production by Concurrent and Countercurrent Imbibition From Cores With Two Ends Open. SPE Res Eval & Eng 20 (2): 251–259. SPE-183635-PA. https://doi.org/10.2118/183635-PA.
Muskat, M., Wyckoff, R. D., Botset, H. G. et al. 1937. Flow of Gas-Liquid Mixtures Through Sands. Trans. AIME 123 (1): 69–96. SPE-937069-G. https://doi.org/10.2118/937069-G.
Nejad, K. S., Berg, E. A., and Ringen, J. K. 2011. Effect of Oil Viscosity on Water/Oil Relative Permeability. Oral presentation of paper SCA2011-12 given at the International Symposium of the Society of Core Analysts, Austin, Texas, 18–21 September.
Odeh, A. S. 1959. Effect of Viscosity Ratio on Relative Permeability. SPE-1189-G.
Pooladi-Darvish, M. and Firoozabadi, A. 2000. Cocurrent and Countercurrent Imbibition in a Water-Wet Matrix Block. SPE J. 5 (1): 3–11. SPE-38443-PA. https://doi.org/10.2118/38443-PA.
Prosperetti, A. and Tryggvason, G. eds. 2009. Computational Methods for Multiphase Flow. Cambridge, UK: Cambridge University Press.
Qiao, Y., Andersen, P. Ø., Evje, S. et al. 2018. A Mixture Theory Approach to Model Co- and Counter-Current Two-Phase Flow in Porous Media Accounting for Viscous Coupling. Adv. Water Resour. 112 (February): 170–188. https://doi.org/10.1016/j.advwatres.2017.12.016.
Rajagopal, K. R. and Tao, L. 1995. Mechanics of Mixtures, Vol. 35. Singapore: World Scientific.
Rapoport, L. A. and Leas, W. J. 1953. Properties of Linear Waterfloods. J Pet Technol 5 (5): 139–148. SPE-213-G. https://doi.org/10.2118/213-G.
Reeves, P. C. and Celia, M. A. 1996. A Functional Relationship Between Capillary Pressure, Saturation, and Interfacial Area as Revealed by a Pore-Scale Network Model. Water Resour. Res. 32 (8): 2345–2358. https://doi.org/10.1029/96WR01105.
Schuff, M. M., Gore, J. P., and Nauman, E. A. 2013. A Mixture Theory Model of Fluid and Solute Transport in the Microvasculature of Normal and Malignant Tissues. I. Theory. J. Math. Biol. 66 (6): 1179–1207. https://doi.org/10.1007/s00285-012-0528-7.
Standnes, D. C. 2004. Experimental Study of the Impact of Boundary Conditions on Oil Recovery by Co-Current and Counter-Current Spontaneous Imbibition. Energy Fuels 18 (1): 271–282. https://doi.org/10.1021/ef030142p.
Standnes, D. C. and Andersen, P. Ø. 2017. Analysis of the Impact of Fluid Viscosities on the Rate of Countercurrent Spontaneous Imbibition. Energy Fuels 31 (7): 6928–6940. https://doi.org/10.1021/acs.energyfuels.7b00863.
Standnes, D. C., Evje, S., and Andersen, P. Ø. 2016. A Novel Relative Permeability Model—A Two-Fluid Approach Accounting for Solid-Fluid and Fluid/Fluid Interactions. Oral presentation of paper SCA2016-060 given at the International Symposium of the Society of Core Analysts, Snowmass, Colorado, 21–26 August.
Standnes, D. C., Evje, S., and Andersen, P. Ø. 2017. A Novel Relative Permeability Model Based on Mixture Theory Approach Accounting for Solid-Fluid and Fluid/Fluid interactions. Transport Porous Med. 119 (3): 707–738. https://doi.org/10.1007/s11242-017-0907-z.
Terez, I. E. and Firoozabadi, A. 1999. Water Injection in Water-Wet Fractured Porous Media: Experiments and a New Model With Modified Buckley-Leverett Theory. SPE J. 4 (2): 134–141. SPE-56854-PA. https://doi.org/10.2118/56854-PA.
Wang, J., Dong, M., and Asghari, K. 2006. Effect of Oil Viscosity on Heavy Oil-Water Relative Permeability Curves. Presented at the SPE/DOE Symposium on Improved Oil Recovery, Tulsa, 22–26 April. SPE-99763-MS. https://doi.org/10.2118/99763-MS.
Wu, W.-T., Aubry, N., Antaki, J. F. et al. 2017. Flow of a Fluid-Solid Mixture: Normal Stress Differences and Slip Boundary Condition. Int. J. Non-Linear Mech. 90 (April): 39–49. https://doi.org/10.1016/j.ijnonlinmec.2017.01.004.
Xie, C., Raeini, A. Q., Wang, Y. et al. 2017. An Improved Pore-Network Model Including Viscous Coupling Effects Using Direct Simulation by the Lattice Boltzmann Method. Adv. Water Resour. 100 (February): 26–34. https://doi.org/10.1016/j.advwatres.2016.11.017.
Yuster, S. T. 1951. Theoretical Considerations of Multiphase Flow in Idealized Capillary Systems. Presented at the 3rd World Petroleum Congress, The Hague, 28 May–6 June. WPC-4129.