Analytical Solutions for Spontaneous Imbibition: Fractional-Flow Theory and Experimental Analysis
- Karen S. Schmid (The Boston Consulting Group) | Nayef Alyafei (Texas A&M University at Qatar) | Sebastian Geiger (Heriot-Watt University) | Martin J. Blunt (Imperial College London)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- December 2016
- Document Type
- Journal Paper
- 2,308 - 2,316
- 2016.Society of Petroleum Engineers
- Fractured Reservoirs, Analytical solution, Spontaneous Imbibition, Capillary flow, Reservoir Engineering
- 7 in the last 30 days
- 662 since 2007
- Show more detail
- View rights & permissions
|SPE Member Price:||USD 12.00|
|SPE Non-Member Price:||USD 35.00|
We present analytical solutions for capillary-controlled displacement in one dimension by use of fractional-flow theory. We show how to construct solutions with a spreadsheet that can be used for the analysis of experiments as well as matrix-block-scale recovery in field settings. The solutions can be understood as the capillary analog to the classical Buckley-Leverett solution (Buckley and Leverett 1942) for viscous-dominated flow, and are valid for cocurrent and countercurrent spontaneous imbibition (SI), as well as for arbitrary capillary pressure and relative permeability curves. They can be used to study the influence of wettability, predicting saturation profiles and production rates characteristic for water-wet and mixed-wet conditions. We compare our results with in-situ measurements of saturation profiles for SI in a water-wet medium. We show that the characteristic shape of the saturation profile is consistent with the expected form of the relative permeabilities. We discuss how measurements of imbibition profiles, in combination with other measurements, could be used to determine relative permeability and capillary pressure.
|File Size||995 KB||Number of Pages||9|
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. http://dx.doi.org/10.2118/16323-PA.
Barenblatt, G. I., Entov, V. M., and Ryzhik, V. M. 1990. Theory of Fluid Flows Through Natural Rocks. Dordrecht, The Netherlands: Springer.
Behbahani, H. and Blunt, M. J. 2005. Analysis of Imbibition in Mixed-Wet Rocks Using Pore-Scale Modeling. SPE J. 10 (4): 466–474. SPE-90132-PA. http://dx.doi.org/10.2118/90132-PA.
Bjørnara, T. I. and Mathias, S. A. 2013. A Pseudospectral Approach to the McWhorter and Sunada Equation for Two-Phase Flow in Porous Media with Capillary Pressure. Computat. Geosci. 17 (6): 889–897. http://dx.doi.org/10.1007/s10596-013-9360-4.
Bourbiaux, B. and Kalaydjian, F. 1990. Experimental Study of Cocurrent and Countercurrent Flows in Natural Porous Media. SPE Res Eng 5 (3): 361–368. SPE-18283-PA. http://dx.doi.org/10.2118/18283-PA.
Buckley, S. E. and Leverett, M. C. 1942. Mechanism of Fluid Displacement in Sands. In Transactions of the Society of Petroleum Engineers, Vol. 146, Part 1, SPE-942107-G, 107–116. http://dx.doi.org/10.2118/942107-G.
Chen, Q., Gingras, M. K., and Balcom, B. J. 2003. A Magnetic Resonance Study of Pore Filling Processes During Spontaneous Imbibition in Berea Sandstone. J. Chem. Phys. 119: 9609. http://dx.doi.org/10.1063/1.1615757.
Cil, M. and Reis, J. C. 1996. A Multi-Dimensional, Analytical Model for Countercurrent Water Imbibition into Gas-Saturated Matrix Blocks. J. Pet. Sci. Eng. 16 (1–3): 61–69. http://dx.doi.org/10.1016/0920-4105(95)00055-0.
Dake, L. P. 1983. Fundamentals of Reservoir Engineering, Vol. 8. Amsterdam: Elsevier Science.
Di Donato, G., Lu, H., Tavassoli, Z. et al. 2007. Multirate-Transfer Dual-Porosity Modeling of Gravity Drainage and Imbibition. SPE J. 12 (1): 77–88. SPE-93144-PA. http://dx.doi.org/10.2118/93144-PA.
Geiger, S., Dentz, M., and Neuweiler, I. 2013. A Novel Multi-Rate Dual-Porosity Model for Improved Simulation of Fractured and Multiporosity Reservoirs. SPE J. 18 (4): 670–684. SPE-148130-PA. http://dx.doi.org/10.2118/148130-PA.
Helmig, R. 1997. Multiphase Flow and Transport Processes in the Subsurface. Berlin: Springer.
Kashchiev, D. and Firoozabadi, A. 2002. Analytical Solutions for 1D Countercurrent Imbibition in Water-Wet Media. SPE J. 8 (4): 401–408. SPE-87333-PA. http://dx.doi.org/10.2118/87333-PA.
Le Guen, S. S. and Kovscek, A. R. 2006. Nonequilibrium Effects During Spontaneous Imbibition. Transport Porous Med. 63 (1): 127–146. http://dx.doi.org/10.1007/s11242-005-3327-4.
Maier, C. and Geiger, S. 2013. Combining Unstructured Grids, Discrete Fracture Representation and Dual-Porosity Models for Improved Simulation of Naturally Fractured Reservoirs. Presented at the SPE Reservoir Characterization and Simulation Conference and Exhibition, Abu Dhabi, 16–18 September. SPE-166049-MS. http://dx.doi.org/10.2118/166049-MS.
Maier, C., Schmid, K. S., Elfeel, M. A. et al. 2013. Multi-Rate Mass-Transfer Dual-Porosity Modelling Using the Exact Analytical Solution for Spontaneous Imbibition. Presented at the EAGE Annual Conference and Exhibition Incorporating SPE Europec, London, 10–13 June. SPE-164926-MS. http://dx.doi.org/10.2118/164926-MS.
Mason, G. and Morrow, N. R. 2013. Developments in Spontaneous Imbibition and Possibilities for Future Work. J. Pet. Sci. Eng. 110 (October): 268–293. http://dx.doi.org/10.1016/j.petrol.2013.08.018.
McWhorter, D. B. and Sunada, D. K. 1990. Exact Integral Solutions for Two-Phase Flow. Water Resour. Res. 26 (3): 399–413. http://dx.doi.org/10.1029/WR026i003p00399.
McWhorter, D. B. and Sunada, D. K. 1992. Exact Integral Solutions for Two-Phase Flow: Reply. Water Resour. Res. 28 (5): 1479. http://dx.doi.org/10.1029/92WR00474.
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. http://dx.doi.org/10.2118/38443-PA.
Ryazanov, A. V., van Dijke, M. I. J., and Sorbie, K. S. 2009. Two-Phase Pore-Network Modelling: Existence of Oil Layers During Water Invasion. Transport Porous Med. 80 (1): 79–99. http://dx.doi.org/10.1007/s11242-009-9345-x.
Ryazanov, A. V., van Dijke, M. I. J., and Sorbie, K. S. S. 2010. Pore-Network Prediction of Residual Oil Saturation Based on Oil Layer Drainage in Mixed-wet Systems. Presented at the SPE Improved Oil Recovery Symposium, Tulsa, 24–28 April. SPE-129919-MS. http://dx.doi.org/10.2118/129919-MS.
Schmid, K. S. and Geiger, S. 2012. Universal Scaling of Spontaneous Imbibition for Water-Wet Systems. Water Resour. Res. 48 (3): W03507. http://dx.doi.org/10.1029/2011WR011566.
Schmid, K. S. and Geiger, S. 2013. Universal Scaling of Spontaneous Imbibition for Arbitrary Petrophysical Properties: Water-Wet and Mixed-Wet States and Handy’s Conjecture. J. Pet. Sci. Eng. 101 (January): 44–61. http://dx.doi.org/10.1016/j.petrol.2012.11.015.
Schmid, K. S., Geiger, S., and Sorbie, K. S. 2011. Semianalytical Solutions for Co- and Countercurrent Imbibition and Dispersion of Solutes in Immiscible Two-Phase Flow. Water Resour. Res. 47 (2): W02550. http://dx.doi.org/10.1029/2010WR009686.
Tao, T. 2006. Nonlinear Dispersive Equations: Local and Global Analysis. Providence, Rhode Island: American Mathematical Society.
Yortsos, Y. C. and Fokas, A. S. 1983. An Analytical Solution for Linear Waterflood Including the Effects of Capillary Pressure. SPE J. 23 (1): 115–124. SPE-9407-PA. http://dx.doi.org/10.2118/9407-PA.
Zhou, D., Jia, L., Kamath, J. et al. 2002. Scaling of Counter-Current Imbibition Processes in Low-Permeability Porous Media. J. Pet. Sci. Eng. 33 (1–3): 61–74. http://dx.doi.org/10.1016/S0920-4105(01)00176-0.
Zhou, X., Morrow, N. R., and Ma, S. 2000. Interrelationship of Wettability, Initial Water Saturation, Aging Time, and Oil Recovery by Spontaneous Imbibition and Waterflooding. SPE J. 5 (2): 199–207. SPE-62507-PA. http://dx.doi.org/10.2118/62507-PA.