Foam Displacements with Multiple Steady States
- William Richard Rossen (U. of Texas at Austin) | Johannes Bruining (Delft U. of Technology)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- March 2007
- Document Type
- Journal Paper
- 5 - 18
- 2007. Society of Petroleum Engineers
- 3 Production and Well Operations, 4.1.5 Processing Equipment, 4.3.4 Scale, 5.8.6 Naturally Fractured Reservoir, 5.4 Enhanced Recovery, 2.5.2 Fracturing Materials (Fluids, Proppant), 5.2.1 Phase Behavior and PVT Measurements, 3.2.4 Acidising, 5.4.6 Thermal Methods, 5.5.3 Scaling Methods, 5.4.2 Gas Injection Methods, 1.6.9 Coring, Fishing, 5.3.2 Multiphase Flow
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A number of experimental and theoretical studies suggest that the fractional-flow function for foam can be either multivalued in water saturation or else can comprise distinct fractional-flow curves for two or more foam regimes, with jumps between them where each regime reaches its limiting condition. We construct fractional-flow solutions for these cases. When such a foam is employed in a surfactant-alternating-gas (SAG) process, the usual "tangency?? condition is modified and the foam can be considerably weaker than the foam formed at what appears to be the point of tangency of the multivalued fractional-flow function. If the capillary-pressure function Pc (Sw ) differs between foam regimes, that difference can substantially change the nature of the displacement. This alters the effective fractional-flow function and hence the global solution of the equations. It is therefore important to determine how capillary pressure varies in foam displacements, by direct measurement in situ if possible. Special care is needed in numerical simulation of processes using fixed grids if capillary pressure depends directly on foam regime. Using gridblocks that are too large can weaken the effect of capillary pressure that would enforce the correct shock on the small scale. Using an upscaled fractional-flow function appears to eliminate this problem, however.
Foams are injected into geological formations for gas diversion in improved oil recovery (IOR) (Schramm 1994; Rossen 1996), acid diversion in matrix acid well stimulation (Gdanski 1993), and environmental remediation (Hirasaki et al. 2000). In IOR and environmental remediation, it is often useful to inject gas and surfactant solution in alternating slugs, a process called SAG injection. SAG injection holds several advantages over continuous coinjection of gas and liquid, as described elsewhere (Shi and Rossen 1998; Shan and Rossen 2004).
Method of Characteristics and Fractional-Flow Theory. Many problems involving conservation equations can be formulated in the so-called hyperbolic framework. The ensuing equations can then be solved using the method of characteristics (see, for example, Smoller (1980), after page 266). The solution consists of rarefactions or spreading waves, constant states, and shocks. Additional conditions are also required to obtain a unique solution. Numerical solutions of the equations can, in effect, pick out the wrong uniqueness conditions and give inaccurate results. A complete analysis requires the traveling-wave representation of a shock. Bruining and Van Duijn (2000, 2007) present an example in which the conditions on the traveling wave are essential to identifying the correct solution of the macroscopic equations. Bruining et al. (2002, 2004) give a regularization procedure for constructing such a traveling wave solution for an application of steam injection. The conditions on the traveling wave must be kept in mind when using a graphical procedure for finding the solution with the method of characteristics, a technique introduced by Buckley and Leverett (1941). It is widely used for petroleum engineering applications [see also Pope (1980), who generalized fractional-flow theory to deal with more complex problems]. In this formulation, the uniqueness condition is called the Welge tangent condition (1952).
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Barenblatt, G.I., Entov, V.M., and Ryzhik, V.M. 1991. Theory of FluidsThrough Natural Rocks. Dordrecht: Academic Publishers.
Bedrikovetsky, P., Marchesin, D., and Ballin, P.R. 1996. Mathematical Model for ImmiscibleDisplacement Honouring Hysteresis. Paper SPE 36132 presented at the SPELatin American and Caribbean Petroleum Engineering Conference, Port-of-Spain,Trinidad and Tobago, 23-26 April. DOI:http://www.spe.org/elibrary/servlet/spepreview?id=36132-MS.
Bedrikovetsky, P.G., Potsch, K.T., Polyanin, A.D., and Zhurov, A.I. 1995. Upscaling of the Waterflood ReservoirProperties on the Core Level: Laboratory Study, Macro, and Micro Modeling.Paper SPE 29870 presented at the SPE Middle East Oil Show, Bahrain, 11-14March. DOI: http://www.spe.org/elibrary/servlet/spepreview?id=29870-MS.
Bruining, J. and van Duijn, C.J. 2000. Uniqueness Conditions in aHyperbolic Model for Oil Recovery by Steamdrive. ComputationalGeosciences B 4 (1): 65-98. DOI: http://dx.doi.org/10.1023/A:1011555715400.
Bruining, J. and van Duijn, C.J. 2007. Traveling Waves in a FiniteCondensation Rate Model for Steam Injection. Computational Geosciences10 (4): 373-387.
Bruining, J., Marchesin, D., and Schecter, S. 2004. Steam Condensation Wavesin Water-Saturated Porous Rock. Qualitative Theory of Dynamical Systems5: 81-106.
Buckley, S.E. and Leverett, M.C. 1941. Mechanism of Fluid Displacement inSands. Trans., AIME 146: 107.
Dafermos, C.M. 2000. Hyperbolic Conservation Laws in ContinuumPhysics. Vol. 325. Grundlehren der Mathematischen Wissenschaften. Berlin:Springer-Verlag.
van Dyke, M. 1975. Perturbation Methods in Fluid Mechanics. Ch.5. Stanford: The Parabolic Press.
Gauglitz, P.A., Friedmann, F., Kam, S.I., and Rossen, W.R. 2002. FoamGeneration in Porous Media. Chem. Eng. Sci. 57: 4037-4052.
Gdanski, R.D. 1993. Experience and Research Show Best Designs forFoam-Diverted Acidizing. Oil Gas J. 9: 85.
Glimm, J. 1988. The Interaction of Nonlinear Hyperbolic Waves. Comm. PureAppl. Math. 41: 569-590.
Godlewski, E. and Raviart, P.A. 1996. Numerical Approximation ofHyperbolic Systems of Conservation Laws. Applied Mathematical Series118. Berlin: Springer.
Guzman, R.E. and Fayers, F.J. 1997. Solutions to the Three-PhaseBuckley-Leverett Problem. SPEJ 2 (3): 301-311. SPE-35156-PA.DOI: http://www.spe.org/elibrary/servlet/spepreview?id=35156-PA.
Hirasaki, G.J. et al. 2000. Field Demonstration of the Surfactant/FoamProcess for Remediation of a Heterogeneous Aquifer Contaminated With DNAPL.InNAPL Removal: Surfactants, Foams, and Microemulsions,ed.S. Fiorenza, C.A. Miller, C.L. Oubre, and C.H. Ward. Boca Raton,Florida: Lewis Publishers.
Hundsdorfer, W. and Verwer, J.G. 2003. Numerical Solution of TimeDependent Advection-Diffusion Reaction Equations. Springer Series inComputational Mathematics. Berlin: Springer.
Isaacson, E., Marchesin, D., and Plohr, B. 1990. Transitional Waves forConservation Laws. SIAM J. Math. Anal. 21: 837-866.
Kam, S.I. and Rossen, W.R. 2003. A Model for Foam Generation inHomogeneous Media. SPEJ 8 (4): 417-425. SPE-87334-PA. DOI:http://www.spe.org/elibrary/servlet/spepreview?id=87334-PA.
Khatib, Z.I., Hirasaki, G.J., and Falls, A.H. 1988. Effects of Capillary Pressure onCoalescence and Phase Mobilities in Foams Flowing Through Porous Media.SPERE 3 (3): 919-926. SPE-15442-PA. DOI:http://www.spe.org/elibrary/servlet/spepreview?id=15442-PA.
Kibodeaux, K.R. and Rossen, W.R. 1997. Coreflood Study ofSurfactant-Alternating-Gas Foam Processes: Implications for Field Design.Paper SPE 38318 presented at the SPE Western Regional Meeting, Long Beach,California, 25-27 June. DOI:http://www.spe.org/elibrary/servlet/spepreview?id=38318-MS.
Kovscek, A.R., Patzek, T.W., and Radke, C.J. 1995. A Mechanistic PopulationBalance Model for Transient and Steady-State Foam Flow in Boise Sandstone.Chem. Eng. Sci. 50: 3783-3799.
Kruzkov, S. 1970. First Order Quasilinear Equations in Several IndependentVariables. Mat Sb. 123: 228-255; English translation in Math.USSR, Sbornik 10: 217-243.
Lake, L.W. 1989. Enhanced Oil Recovery. Englewood Cliffs, New Jersey:Prentice Hall.
Martinsen, H.A. and Vassenden, F. 1999. Foam-Assisted Water Alternating Gas(FAWAG) Process on Snorre. Paper presented at the European IOR Symposium,Brighton, U.K., 18-20 August.
Medeiros, H.B., Marchesin, D., and Paes Leme, P.J. 1998. Hysteresis inTwo-Phase Flow: A Simple Mathematical Model. Comp. Appl. Math 17:81-99.
Oleinik, O.A. 1963a. Discontinuous Solutions of Non-Linear DifferentialEquations. Am. Math. Soc. Trans. 26: 95-172.
Oleinik, O.A. 1963b. Uniqueness and Stability of the Generalized Solution ofthe Cauchy Problem for a Quasi-Linear Equation. Am. Math Soc. Trans.33: 285-290.
Persoff, P., Radke, C.J., Pruess, K., Benson, S.M., and Witherspoon, P.A.1991. A Laboratory Investigationof Foam Flow in Sandstone at Elevated Pressure. SPERE 6 (3):365-372. SPE-18781-PA. DOI:http://www.spe.org/elibrary/servlet/spepreview?id=18781-PA.
Pope, G.A. 1980. The Applicationof Fractional-Flow Theory to Enhanced Oil Recovery. SPEJ 20(3): 191-205. SPE-7660-PA. DOI:http://www.spe.org/elibrary/servlet/spepreview?id=7660-PA.
Ransohoff, T.C. and Radke, C.J. 1988. Mechanisms of Foam Generation inGlass-Bead Packs. SPERE 3 (2): 573-585. SPE-15441-PA. DOI:http://www.spe.org/elibrary/servlet/spepreview?id=15441-PA.
Rossen, W.R. 1996. Foams in Enhanced Oil Recovery. InFoams: Theory,Measurements and Applications, ed.R.K. Prud'homme and S. Khan. NewYork: Marcel Dekker.
Rossen, W.R. and Bruining, J. 2004. Foam Displacements With MultipleSteady States. Paper SPE 89397 presented at the SPE/DOE Symposium onImproved Oil Recovery, Tulsa, 17-21 April. DOI:http://www.spe.org/elibrary/servlet/spepreview?id=89397-MS.
Rossen, W.R. and Gauglitz, P.A. 1990. Percolation Theory of Creation andMobilization of Foam in Porous Media. AIChE J. 36: 1176-1188.
Rossen, W.R., Zeilinger, S.C., Shi, J.-X., and Lim, M.T. 1999. Simplified Mechanistic Simulation ofFoam Processes in Porous Media. SPEJ 4 (3): 279-287.SPE-57678-PA. DOI:http://www.spe.org/elibrary/servlet/spepreview?id=57678-PA.
Schramm, L.L., ed. 1994. Foams: Fundamentals and Applications in thePetroleum Industry. ACS Advances in Chemistry Series. No. 242.Washington, DC: American Chem. Soc.
Serre, D. 1999. Systems of Conservation Laws, Vol 1:Hyperbolicity, Entropies and Shock Waves. Cambridge: Cambridge U.Press.
Shan, D. and Rossen, W.R. 2004. Optimal Injection Strategies for FoamIOR. SPEJ 9 (2): 132-150. SPE-88811-PA. DOI:http://www.spe.org/elibrary/servlet/spepreview?id=88811-PA.
Shi, J.-X., and Rossen, W.R. 1998. Improved Surfactant-Alternating-GasFoam Process To Control Gravity Override. Paper SPE 39653 prepared forpresentation at the SPE/DOE Improved Oil Recovery Symposium, Tulsa, 19-22April. DOI: http://www.spe.org/elibrary/servlet/spepreview?id=39653-MS.
Smoller, J. 1980. Shock Waves and Reaction-Diffusion Equations. NewYork: Springer-Verlag.
Tanzil, D., Hirasaki, G.J., and Miller, C.A. 2002. Conditions for Foam Generation inHomogeneous Porous Media. Paper SPE 75176 presented at the SPE/DOE ImprovedOil Recovery Symposium, Tulsa, 13-17 April. DOI:http://www.spe.org/elibrary/servlet/spepreview?id=75176-MS.
van der Net, A. and Zitha, P.L.J. 2005. A New Method for the Determination ofFoam Capillary Pressures in Porous Media. Paper SPE 96988 presented at theSPE Annual Technical Conference and Exhibition, Dallas, 9-12 October. DOI:http://www.spe.org/elibrary/servlet/spepreview?id=96988-MS.
Wassmuth, F.R., Green, K.A., and Randall, L. 2001. Details of In-Situ Foam PropagationExposed With Magnetic Resonance Imaging. SPEREE 4 (2):135-145. SPE-71300-PA. DOI:http://www.spe.org/elibrary/servlet/spepreview?id=71300-PA.
Welge, H.J. 1952. A Simplified Method for Computing Oil Recovery by Gas orWater Drive. Trans., AIME 195, 91-98.
Xu, Q. and Rossen, W.R. 2004. Experimental Study of Gas Injectionin Surfactant-Alternating-Gas Foam Process. SPEREE 7(6):438-448. SPE-84183-PA. DOI:http://www.spe.org/elibrary/servlet/spepreview?id=84183-PA.
Zeilinger, S.C. 1996. A Modeling and Experimental Study of Foam in AcidDiversion and Enhanced Oil Recovery. PhD dissertation. Austin, Texas: U. ofTexas at Austin.
Zhou, Z.H. and Rossen, W.R. 1995. Applying Fractional-Flow Theory toFoam Processes at the ‘Limiting Capillary Pressure'. SPE AdvancedTechnology 3: 154-162. SPE-24180-PA.http://www.spe.org/elibrary/servlet/spepreview?id=24180-PA.