×

zbMATH — the first resource for mathematics

Two-phase flows in karstic geometry. (English) Zbl 1309.76204
Summary: Multiphase flow phenomena are ubiquitous. Common examples include coupled atmosphere and ocean system (air and water), oil reservoir (water, oil, and gas), and cloud and fog (water vapor, water, and air). Multiphase flows also play an important role in many engineering and environmental science applications.
In some applications such as flows in unconfined karst aquifers, karst oil reservoir, proton membrane exchange fuel cell, multiphase flows in conduits, and in porous media must be considered together. Geometric configurations that contain both conduit (or vug) and porous media are termed karstic geometry. Despite the importance of the subject, little work has been performed on multiphase flows in karstic geometry.
In this paper, we present a family of phase-field (diffusive interface) models for two-phase flow in karstic geometry. These models together with the associated interface boundary conditions are derived utilizing Onsager’s extremum principle. The models derived enjoy physically important energy laws. A uniquely solvable numerical scheme that preserves the associated energy law is presented as well.

MSC:
76T10 Liquid-gas two-phase flows, bubbly flows
76S05 Flows in porous media; filtration; seepage
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Brennen, Fundamentals of Multiphase Flow (2005) · doi:10.1017/CBO9780511807169
[2] Joseph, Fundamentals of Two-Fluid Dynamics. Part I, Interdisciplinary Applied Mathematics 3 (1993)
[3] Joseph, Fundamentals of two-fluid dynamics. Part II, Interdisciplinary Applied Mathematics 4 (1993)
[4] Katz, Changes in the isotopic and chemical composition of ground water resulting from a recharge pulse from a sinking stream, Journal of Hydrology 211 (14) pp 178– (1998) · doi:10.1016/S0022-1694(98)00236-4
[5] Kuniansky E Geological Survey Karst Interest Group Proceedings, U.S. Geological Survey Scientific Investigations Report 2008-5023 2008
[6] Kincaid T Exploring the Secrets of Wakulla Springs 2004
[7] Matusick, Comparative study of groundwater vulnerability in a karst aquifer in central florida, Geophysical Research Abstracts 9 pp 1– (2007)
[8] Taylor C Greene E Quantitative Approaches in Characterizing Karst Aquifers: U.S. Geological Survey Karst Interest Group Proceedings 2001
[9] Li, Contaminant sequestration in karstic aquifers: Experiments and quantification, Water Resources Research 44 (2) pp 2429– (2008) · doi:10.1029/2006WR005797
[10] Beavers, Boundary conditions at a naturally permeable wall, Journal of Fluid Mechanics 30 pp 197– (1967) · doi:10.1017/S0022112067001375
[11] Discacciati, Mathematical and numerical models for coupling surface and groundwater flows, Applied Numerical Mathematics 43 (1-2) pp 57– (2002) · Zbl 1023.76048 · doi:10.1016/S0168-9274(02)00125-3
[12] Discacciati, Numerical Mathematics and Advanced Applications 320 (2003)
[13] Discacciati, Navier-Stokes/Darcy coupling: modeling, analysis, and numerical approximation, Revista Matematica Complutense 22 (2) pp 315– (2009)
[14] Layton, Coupling fluid flow with porous media flow, SIAM Journal on Numerical Analysis 40 (6) pp 2195– (2002) · Zbl 1037.76014 · doi:10.1137/S0036142901392766
[15] Cao, Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition, Communications in Mathematical Sciences 8 (1) pp 1– (2010) · Zbl 1189.35244 · doi:10.4310/CMS.2010.v8.n1.a2
[16] Wu, A multiple-continuum model for simulating single-phase and multiphase flow in naturally fractured vuggy reservoirs, Journal of Petroleum Science and Engineering 78 (1) pp 13– (2011) · doi:10.1016/j.petrol.2011.05.004
[17] Gluyas, Petroleum Geology (2004)
[18] Tuber, Visualization of water buildup in the cathode of a transparent PEM fuel cell, Journal of Power Sources 124 (2) pp 403– (2003) · doi:10.1016/S0378-7753(03)00797-3
[19] Formaggia, Cardiovascular Mathematics: Modeling and simulation of the circulatory system (2010)
[20] Muskat, Two fluid systems in porous media. the encroachment of water into an oil sand, Physics 5 (9) pp 250– (1934) · JFM 60.1388.01 · doi:10.1063/1.1745259
[21] Saffman, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid, Proceedings of the Royal Society of London Series A 245 pp 312– (1958) · Zbl 0086.41603 · doi:10.1098/rspa.1958.0085
[22] Howison, A note on the two-phase Hele-Shaw problem, Journal of Fluid Mechanics 409 pp 243– (2000) · Zbl 0962.76028 · doi:10.1017/S0022112099007740
[23] Castro, Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves, Annals of Mathematics (2) 175 (2) pp 909– (2012) · Zbl 1267.76033 · doi:10.4007/annals.2012.175.2.9
[24] Bear, Dynamics of Fluids in Porous Media (1988) · Zbl 1191.76002
[25] Batchelor, An Introduction to Fluid Dynamics (2000) · Zbl 0958.76001 · doi:10.1017/CBO9780511800955
[26] Lighthill, Waves in Fluids (1978)
[27] Lamb, Hydrodynamics (1932)
[28] Abels, Diffuse Interface Models for Two-Phase Flows of Viscous Incompressible Fluids 36 (2007) · Zbl 1124.35060
[29] Kang, A boundary condition capturing method for multiphase incompressible flow, Journal of Scientific Computing 15 (3) pp 323– (2000) · Zbl 1049.76046 · doi:10.1023/A:1011178417620
[30] Sussman, An adaptive level set approach for incompressible two-phase flows, Journal of Computational Physics 148 (1) pp 81– (1999) · Zbl 0930.76068 · doi:10.1006/jcph.1998.6106
[31] Sussman, A sharp interface method for incompressible two-phase flows, Journal of Computational Physics 221 (2) pp 469– (2007) · Zbl 1194.76219 · doi:10.1016/j.jcp.2006.06.020
[32] Raessi, Consistent mass and momentum transport for simulating incompressible interfacial flows with large density ratios using the level set method, Computers & Fluids 63 pp 70– (2012) · Zbl 1365.76238 · doi:10.1016/j.compfluid.2012.04.002
[33] Jemison, A coupled level set-moment of fluid method for incompressible two-phase flows, Journal of Scientific Computing 54 (2-3) pp 454– (2013) · Zbl 1352.76091 · doi:10.1007/s10915-012-9614-7
[34] Anderson, Annu. Rev. Fluid Mech., vol.30 30, in: Annual review of fluid mechanics (1998)
[35] Gurtin, Two-phase binary fluids and immiscible fluids described by an order parameter, Mathematical Models and Methods in Applied Sciences 6 (6) pp 815– (1996) · Zbl 0857.76008 · doi:10.1142/S0218202596000341
[36] Lowengrub, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, Proceedings of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences 454 (1978) pp 2617– (1998) · Zbl 0927.76007 · doi:10.1098/rspa.1998.0273
[37] Lee, Modeling pinchoff and reconnection in a Hele-Shaw cell. I. The models and their calibration, Physics of Fluids 14 (2) pp 492– (2002) · Zbl 1184.76316 · doi:10.1063/1.1425843
[38] Rowlinson, Translation of J. D. van der Waals’ ”The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density”, Journal of Statistical Physics 20 (2) pp 197– (1979) · Zbl 1245.82006 · doi:10.1007/BF01011513
[39] Rayleigh, On the theory of surface forces.ii. compressible fluids, Philosophical Magazine Series 5 33 (201) pp 209– (1892) · JFM 24.1102.02 · doi:10.1080/14786449208621456
[40] Liu, A phase field model for the mixture of two incompressible fluids and its approximation by a fourier-spectral method, Physica D: Nonlinear Phenomena 179 (34) pp 211– (2003) · Zbl 1092.76069 · doi:10.1016/S0167-2789(03)00030-7
[41] Qian, A variational approach to moving contact line hydrodynamics, Journal of Fluid Mechanics 564 pp 333– (2006) · Zbl 1178.76296 · doi:10.1017/S0022112006001935
[42] Qian, Molecular scale contact line hydrodynamics of immiscible flows, Physical Review E 68 pp 016– (2003) · doi:10.1103/PhysRevE.68.016306
[43] Onsager, Reciprocal relations in irreversible processes. i., Physical Review 37 pp 405– (1931) · JFM 57.1168.10 · doi:10.1103/PhysRev.37.405
[44] Onsager, Reciprocal relations in irreversible processes. ii, Physical Review 38 pp 2265– (1931) · Zbl 0004.18303 · doi:10.1103/PhysRev.38.2265
[45] Onsager, Fluctuations and irreversible processes, Physical Review 91 pp 1505– (1953) · Zbl 0053.15106 · doi:10.1103/PhysRev.91.1505
[46] Helmholtz H Verhandlungen des naturhistorisch-medizinischen vereins zu heidelberg The collected works of Lars Onsager, Band V: 1-7. Reprinted in Helmholtz, H. (1882), Wissenschaftliche Abhandlungen 1 Johann Ambrosius Barth Leipzig
[47] Eck, On a phase-field model for electrowetting, Interfaces and Free Boundaries 11 (2) pp 259– (2009) · Zbl 1167.35553 · doi:10.4171/IFB/211
[48] Fontelos, On a phase-field model for electrowetting and other electrokinetic phenomena, SIAM Journal on Mathematical Analysis 43 (1) pp 527– (2011) · Zbl 1228.35082 · doi:10.1137/090779668
[49] Abels, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities, Mathematical Models and Methods in Applied Science 22 (3) pp 1150013(40 page– (2012) · Zbl 1242.76342 · doi:10.1142/S0218202511500138
[50] Doi, Variational principle for the kirkwood theory for the dynamics of polymer solutions and suspensions, The Journal of Chemical Physics 79 (10) pp 5080– (1983) · doi:10.1063/1.445604
[51] Le Bars, Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification, Journal of Fluid Mechanics 550 pp 149– (2006) · Zbl 1097.76066 · doi:10.1017/S0022112005007998
[52] Nield, Convection in Porous Media (1999) · Zbl 0924.76001 · doi:10.1007/978-1-4757-3033-3
[53] E, Phase separation in incompressible systems, Physical Review E 55 pp R3844– (1997) · Zbl 1110.82304 · doi:10.1103/PhysRevE.55.R3844
[54] Lee, Modeling pinchoff and reconnection in a Hele-Shaw cell. II. Analysis and simulation in the nonlinear regime, Physics of Fluids 14 (2) pp 514– (2002) · Zbl 1184.76317 · doi:10.1063/1.1425844
[55] Wise, Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn-Hilliard-Hele-Shaw system of equations, Journal of Scientific Computing 44 (1) pp 38– (2010) · Zbl 1203.76153 · doi:10.1007/s10915-010-9363-4
[56] Feng, Analysis of a Darcy-Cahn-Hilliard diffuse interface model for the Hele-Shaw flow and its fully discrete finite element approximation, SIAM Journal on Numerical Analysis 50 (3) pp 1320– (2012) · Zbl 1426.76258 · doi:10.1137/110827119
[57] Wang, Well-posedness of the Hele-Shaw-Cahn-Hilliard system, Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (3) pp 367– (2013) · Zbl 1291.35240 · doi:10.1016/j.anihpc.2012.06.003
[58] Wang, Long-time behavior for the Hele-Shaw-Cahn-Hilliard system, Asymptotic Analysis 78 (4) pp 217– (2012)
[59] Collins, An efficient, energy stable scheme for the cahn-hilliard-brinkman system, Communications in Computational Physics 13 pp 929– (2013) · Zbl 1373.76161 · doi:10.4208/cicp.171211.130412a
[60] Delllsola, Boundary conditions at fluid-permeable interfaces in porous media: A variational approach, International Journal of Solids and Structures 46 (17) pp 3150– (2009) · Zbl 1167.74393 · doi:10.1016/j.ijsolstr.2009.04.008
[61] Jones, Low reynolds number flow past a porous spherical shell, Mathematical Proceedings of the Cambridge Philosophical Society 1 73 pp 231– (1973) · Zbl 0262.76061 · doi:10.1017/S0305004100047642
[62] Chen, A parallel Robin-Robin domain decomposition method for the Stokes-Darcy system, SIAM Journal on Numerical Analysis 49 (3) pp 1064– (2011) · Zbl 1414.76017 · doi:10.1137/080740556
[63] Chen, Asymptotic analysis of the differences between the Stokes-Darcy system with different interface conditions and the Stokes-Brinkman system, Journal of Mathematical Analysis and Applications 368 (2) pp 658– (2010) · Zbl 1352.35093 · doi:10.1016/j.jmaa.2010.02.022
[64] Cao, Finite element approximations for Stokes-Darcy flow with Beavers-Joseph interface conditions, SIAM Journal on Numerical Analysis 47 (6) pp 4239– (2010) · Zbl 1252.76040 · doi:10.1137/080731542
[65] Chen, Efficient and long-time accurate second-order methods for Stokes-Darcy systems, SIAM Journal on Numerical Analysis 51 (5) pp 2563– (2013) · Zbl 1282.76094 · doi:10.1137/120897705
[66] Shen, A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscosities, SIAM Journal of Scientific Computing 32 (3) pp 1159– (2010) · Zbl 1410.76464 · doi:10.1137/09075860X
[67] Shen, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap 22, in: Multiscale modeling and analysis for materials simulation (2012)
[68] Lamorgese, Phase field approach to multiphase flow modeling, Milan Journal of Mathematics 79 (2) pp 597– (2011) · Zbl 1237.82031 · doi:10.1007/s00032-011-0171-6
[69] Boyer, A theoretical and numerical model for the study of incompressible mixture flows, Computers & Fluids 31 (1) pp 41– (2002) · Zbl 1057.76060 · doi:10.1016/S0045-7930(00)00031-1
[70] Ding, Diffuse interface model for incompressible two-phase flows with large density ratios, Journal of Computational Physics 226 (2) pp 2078– (2007) · Zbl 1388.76403 · doi:10.1016/j.jcp.2007.06.028
[71] Discacciati, Robin-Robin domain decomposition methods for the Stokes-Darcy coupling, SIAM Journal on Numerical Analysis 45 (3) pp 1246– (2007) · Zbl 1139.76030 · doi:10.1137/06065091X
[72] Rivière, Locally conservative coupling of Stokes and Darcy flows, SIAM Journal on Numerical Analysis 42 (5) pp 1959– (2005) · Zbl 1084.35063 · doi:10.1137/S0036142903427640
[73] Cai, Numerical solution to a mixed Navier-Stokes/Darcy model by the two-grid approach, SIAM Journal on Numerical Analysis 47 (5) pp 3325– (2009) · Zbl 1213.76131 · doi:10.1137/080721868
[74] Dawson, Analysis of discontinuous finite element methods for ground water/surface water coupling, SIAM Journal on Numerical Analysis 44 (4) pp 1375– (2006) · Zbl 1124.76025 · doi:10.1137/050639405
[75] Jäger, On the interface boundary condition of Beavers, Joseph, and Saffman, SIAM Journal on Applied Mathematics 60 (4) pp 1111– (2000) · Zbl 0969.76088 · doi:10.1137/S003613999833678X
[76] Landau, Course of Theoretical Physics. Vol. 5: Statistical Physics (1968)
[77] Modica, The gradient theory of phase transitions and the minimal interface criterion, Archive for Rational Mechanics and Analysis 98 (2) pp 123– (1987) · Zbl 0616.76004 · doi:10.1007/BF00251230
[78] Modica, Un esempio di gamma convergenza, Bollettino Della Unione Matematica Italiana 14 (1) pp 285– (1977)
[79] Fischer, Novel surface modes in spinodal decomposition, Physical Review Letters 79 pp 893– (1997) · doi:10.1103/PhysRevLett.79.893
[80] Racke, The Cahn-Hilliard equation with dynamic boundary conditions, Advances in Differential Equations 8 (1) pp 83– (2003) · Zbl 1035.35050
[81] Wu, Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions, Journal of Differential Equations 204 (2) pp 511– (2004) · Zbl 1068.35018 · doi:10.1016/j.jde.2004.05.004
[82] Duvaut, Inequalities in Mechanics and Physics (1976) · doi:10.1007/978-3-642-66165-5
[83] Girault, DG approximation of coupled Navier-Stokes and Darcy equations by Beaver-Joseph-Saffman interface condition, SIAM Journal on Numerical Analysis 47 (3) pp 2052– (2009) · Zbl 1406.76082 · doi:10.1137/070686081
[84] Kanschat, A strongly conservative finite element method for the coupling of Stokes and Darcy flow, Journal of Computational Physics 229 (17) pp 5933– (2010) · Zbl 1425.76068 · doi:10.1016/j.jcp.2010.04.021
[85] Eyre, Computational and mathematical models of microstructural evolution (San Francisco, CA, 1998), Mater. Res. Soc. Sympos. Proc 529 (1998)
[86] Wang, Unconditionally stable schemes for equations of thin film epitaxy, Discrete and Continuous Dynamical Systems 28 (1) pp 405– (2010) · Zbl 1201.65166 · doi:10.3934/dcds.2010.28.405
[87] Wise, An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM Journal on Numerical Analysis 47 (3) pp 2269– (2009) · Zbl 1201.35027 · doi:10.1137/080738143
[88] Girault, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms (1986) · Zbl 0585.65077 · doi:10.1007/978-3-642-61623-5
[89] Evans, Partial Differential Equations (2010) · Zbl 1194.35001 · doi:10.1090/gsm/019
[90] Zeidler, Nonlinear Functional Analysis and Its Application III.: Variational Methods and Optimization (1985) · Zbl 0583.47051 · doi:10.1007/978-1-4612-5020-3
[91] Grisvard, Elliptic Problems in Nonsmooth Domains 24 (1985) · Zbl 0695.35060
[92] Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities, Archive for Rational Mechanics and Analysis 194 (2) pp 463– (2009) · Zbl 1254.76158 · doi:10.1007/s00205-008-0160-2
[93] Abels, Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (6) pp 2403– (2009) · Zbl 1181.35343 · doi:10.1016/j.anihpc.2009.06.002
[94] Lowengrub, Analysis of a mixture model of tumor growth, European Journal of Applied Mathematics 24 (05) pp 691– (2013) · Zbl 1292.35153 · doi:10.1017/S0956792513000144
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.