# American Institute of Mathematical Sciences

October  2013, 33(10): 4497-4530. doi: 10.3934/dcds.2013.33.4497

## Weak solutions of a gas-liquid drift-flux model with general slip law for wellbore operations

 1 University of Stavanger (UiS), 4036 Stavanger 2 University of Stavanger, NO-4036 Stavanger, Norway

Received  November 2012 Revised  January 2013 Published  April 2013

In this work we study a compressible gas-liquid models highly relevant for wellbore operations like drilling. The model is a drift-flux model and is composed of two continuity equations together with a mixture momentum equation. The model allows unequal gas and liquid velocities, dictated by a so-called slip law, which is important for modeling of flow scenarios involving for example counter-current flow. The model is considered in Lagrangian coordinates. The difference in fluid velocities gives rise to new terms in the mixture momentum equation that are challenging to deal with. First, a local (in time) existence result is obtained under suitable assumptions on initial data for a general slip relation. Second, a global in time existence result is proved for small initial data subject to a more specialized slip relation.
Citation: Steinar Evje, Huanyao Wen. Weak solutions of a gas-liquid drift-flux model with general slip law for wellbore operations. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4497-4530. doi: 10.3934/dcds.2013.33.4497
##### References:
 [1] C. S. Avelar, P. R. Ribeiro and K. Sepehrnoori, Deepwater gas kick simulation, J. Pet. Sci. Eng., 67 (2009), 13-22. doi: 10.1016/j.petrol.2009.03.001. [2] M. Baudin, C. Berthon, F. Coquel, R. Masson and Q. H. Tran, A relaxation method for two-phase flow models with hydrodynamic closure law, Numer. Math., 99 (2005), 411-440. doi: 10.1007/s00211-004-0558-1. [3] P. Chen and T. Zhang, A vacuum problem for multidimensional compressible Navier-Stokes equations with degenerate viscosity coefficients, Comm. Pure. Appl. Anal., 7 (2008), 987-1016. doi: 10.3934/cpaa.2008.7.987. [4] S. Evje, Weak solution for a gas-liquid model relevant for describing gas-kick oil wells, SIAM J. Math. Anal., 43 (2011), 1887-1922. doi: 10.1137/100813932. [5] S. Evje and K.-K. Fjelde, Hybrid flux-splitting schemes for a two-phase flow model, J. Comput. Phys., 175 (2002), 674-701. doi: 10.1006/jcph.2001.6962. [6] S. Evje and K.-K. Fjelde, On a rough AUSM scheme for a one dimensional two-phase model, Computers & Fluids, 32 (2003), 1497-1530. doi: 10.1016/S0045-7930(02)00113-5. [7] S. Evje and K. H. Karlsen, Global existence of weak solutions for a viscous two-phase model, J. Diff. Equations, 245 (2008), 2660-2703. doi: 10.1016/j.jde.2007.10.032. [8] S. Evje and K. H. Karlsen, Global weak solutions for a viscous liquid-gas model with singular pressure law, Comm. Pure Appl. Anal., 8 (2009), 1867-1894. doi: 10.3934/cpaa.2009.8.1867. [9] S. Evje, T. Flåtten and H. A. Friis, Global weak solutions for a viscous liquid-gas model with transition to single-phase gas flow and vacuum, Nonlinear Analysis TMA, 70 (2009), 3864-3886. doi: 10.1016/j.na.2008.07.043. [10] L. Fan, Q. Q. Liu and C. J. Zhu, Convergence rates to stationary solutions of a gas-liquid model with external forces, Nonlinearity, 25 (2012), 2875-2901. doi: 10.1088/0951-7715/25/10/2875. [11] I. Faille and E. Heintze, A rough finite volume scheme for modeling two-phase flow in a pipeline, Computers & Fluids, 28 (1999), 213-241. doi: 10.1016/S0045-7930(98)00023-1. [12] K.-K. Fjelde and K.-H. Karlsen, High-resolution hybrid primitive-conservative upwind schemes for the drift flux model, Computers & Fluids, 31 (2002), 335-367. doi: 10.1016/S0045-7930(01)00041-X. [13] T. Flåtten and Munkejord, The approximate Riemann solver of Roe applied to a drift-flux two-phase flow model, ESAIM: Math. Mod. Num. Anal., 40 (2006), 735-764. doi: 10.1051/m2an:2006032. [14] S. L. Gavrilyuk and J. Fabre, Lagrangian coordinates for a drift-flux model of a gas-liquid mixture, Int. J. Multiphase Flow, 22 (1996), 453-460. doi: 10.1016/0301-9322(95)00085-2. [15] D. Hoff, Construction of solutions for compressible, isentropic Navier-Stokes equations in one space dimension with nonsmooth initial data, Proc. Roy. Soc. Edinburgh Sect. A, 103 (1986), 301-315. doi: 10.1017/S0308210500018953. [16] M. Ishii, "Thermo-Fluid Dynamic Theory of Two-Phase Flow," Eyrolles, Paris, 1975. [17] S. Jiang, Z. Xin and P. Zhang, Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity, Meth. Appl. Anal., 12 (2005), 239-252. [18] T.-P. Liu, Z. Xin and T. Yang, Vacuum states for compressible flow, Discrete Continuous Dyn. Sys., 4 (1998), 1-32. [19] Q. Q. Liu and C. J. Zhu, Asymptotic behavior of a viscous liquid-gas model with mass-dependent viscosity and vacuum, J. Differential Equations, 252 (2012), 2492-2519. doi: 10.1016/j.jde.2011.10.018. [20] R. J. Lorentzen and K. K. Fjelde, Use of slopelimiter techniques in traditional numerical methods for multi-phase flow in pipelines and wells, Int. J. Numer. Meth. Fluids, 48 (2005), 723-745. doi: 10.1002/fld.952. [21] T. Luo, Z. Xin and T. Yang, Interface behavior of compressible Navier-Stokes equations with vacuum, SIAM J. Math. Anal., 31 (2000), 1175-1191. doi: 10.1137/S0036141097331044. [22] J. M. Masella, Q. H. Tran, D. Ferre and C. Pauchon, Transient simulation of two-phase flows in pipes, Int. J. of Multiphase Flow, 24 (1998), 739-755. doi: 10.1016/S0301-9322(98)00004-4. [23] S. T. Munkejord, S. Evje and T. Flåtten, The multi-staged centred scheme approach applied to a drift-flux two-phase flow model, Int. J. Num. Meth. Fluids, 52 (2006), 679-705. doi: 10.1002/fld.1200. [24] M. Okada, Free-boundary value problems for the equation of one-dimensional motion of viscous gas, Japan J. Indust. Appl. Math., 6 (1989), 161-177. doi: 10.1007/BF03167921. [25] M. Okada, S. Matusu-Necasova and T. Makino, Free-boundary problem for the equation of one-dimensional motion of compressible gas with density-dependent viscosity, Ann. Univ. Ferrara Sez VII(N.S.), 48 (2002), 1-20. [26] J. E. Romate, An approximate Riemann solver for a two-phase flow model with numerically given slip relation, Computers & Fluid, 27 (1998), 455-477. doi: 10.1016/S0045-7930(97)00067-4. [27] J. Schlegel, T. Hibiki and M. Ishii, Development of a comprehensive set of drift-flux constitutive models for pipes of various hydraulic diameters, Prog. Nuc. Energy, 52 (2010), 666-677. doi: 10.1016/j.pnucene.2010.03.007. [28] T. Yang, Z.-A. Yao and C.-J. Zhu, Compressible Navier-Stokes equations with density-dependent viscosity and vacuum, Comm. Partial Diff. Eq., 26 (2001), 965-981. doi: 10.1081/PDE-100002385. [29] T. Yang and H.-J. Zhao, A vacuum problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity, J. Differential Equations, 184 (2002), 163-184. doi: 10.1006/jdeq.2001.4140. [30] T. Yang and C.-J. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum, Comm. Math. Phys., 230 (2002), 329-363. doi: 10.1007/s00220-002-0703-6. [31] L. Yao, H.-L. Guo and Z.-H. Guo, A note on viscous liquid-gas two-phase flow model with mass-dependent viscosity and vacuum, Nonlinear Analysis: Real World Applications, 13 (2012), 2323-2342. doi: 10.1016/j.nonrwa.2012.02.001. [32] L. Yao and C.-J. Zhu, Free boundary value problem for a viscous two-phase model with mass-dependent viscosity, J. Differential Equations, 247 (2009), 2705-2739. doi: 10.1016/j.jde.2009.07.013. [33] L. Yao and C. J. Zhu, Existence and uniqueness of global weak solution to a two-phase flow model with vacuum, Math. Ann., 349 (2010), 903-928. doi: 10.1007/s00208-010-0544-0. [34] L. Yao, T. Zhang and C. J. Zhu, Existence and asymptotic behavior of global weak solutions to a 2d viscous liquid-gas two-phase flow model, SIAM J. Math. Anal., 42 (2010), 1874-1897. doi: 10.1137/100785302. [35] L. Yao, T. Zhang and C. J. Zhu, A blow-up criterion for a 2D viscous liquid-gas two-phase flow model, J. Differential Equations, 250 (2011), 3362-3378. doi: 10.1016/j.jde.2010.12.006. [36] T. Zhang and D.-Y. Fang, Global behavior of compressible Navier-Stokes equations with a degenerate viscosity coefficient, Arch. Rational Mech. Anal., 182 (2006), 223-253. doi: 10.1007/s00205-006-0425-6. [37] N. Zuber and J. A. Findlay, Average volumetric concentration in two-phase flow systems, J. Heat Transfer, 87 (1965), 453-468. doi: 10.1115/1.3689137.

show all references

##### References:
 [1] C. S. Avelar, P. R. Ribeiro and K. Sepehrnoori, Deepwater gas kick simulation, J. Pet. Sci. Eng., 67 (2009), 13-22. doi: 10.1016/j.petrol.2009.03.001. [2] M. Baudin, C. Berthon, F. Coquel, R. Masson and Q. H. Tran, A relaxation method for two-phase flow models with hydrodynamic closure law, Numer. Math., 99 (2005), 411-440. doi: 10.1007/s00211-004-0558-1. [3] P. Chen and T. Zhang, A vacuum problem for multidimensional compressible Navier-Stokes equations with degenerate viscosity coefficients, Comm. Pure. Appl. Anal., 7 (2008), 987-1016. doi: 10.3934/cpaa.2008.7.987. [4] S. Evje, Weak solution for a gas-liquid model relevant for describing gas-kick oil wells, SIAM J. Math. Anal., 43 (2011), 1887-1922. doi: 10.1137/100813932. [5] S. Evje and K.-K. Fjelde, Hybrid flux-splitting schemes for a two-phase flow model, J. Comput. Phys., 175 (2002), 674-701. doi: 10.1006/jcph.2001.6962. [6] S. Evje and K.-K. Fjelde, On a rough AUSM scheme for a one dimensional two-phase model, Computers & Fluids, 32 (2003), 1497-1530. doi: 10.1016/S0045-7930(02)00113-5. [7] S. Evje and K. H. Karlsen, Global existence of weak solutions for a viscous two-phase model, J. Diff. Equations, 245 (2008), 2660-2703. doi: 10.1016/j.jde.2007.10.032. [8] S. Evje and K. H. Karlsen, Global weak solutions for a viscous liquid-gas model with singular pressure law, Comm. Pure Appl. Anal., 8 (2009), 1867-1894. doi: 10.3934/cpaa.2009.8.1867. [9] S. Evje, T. Flåtten and H. A. Friis, Global weak solutions for a viscous liquid-gas model with transition to single-phase gas flow and vacuum, Nonlinear Analysis TMA, 70 (2009), 3864-3886. doi: 10.1016/j.na.2008.07.043. [10] L. Fan, Q. Q. Liu and C. J. Zhu, Convergence rates to stationary solutions of a gas-liquid model with external forces, Nonlinearity, 25 (2012), 2875-2901. doi: 10.1088/0951-7715/25/10/2875. [11] I. Faille and E. Heintze, A rough finite volume scheme for modeling two-phase flow in a pipeline, Computers & Fluids, 28 (1999), 213-241. doi: 10.1016/S0045-7930(98)00023-1. [12] K.-K. Fjelde and K.-H. Karlsen, High-resolution hybrid primitive-conservative upwind schemes for the drift flux model, Computers & Fluids, 31 (2002), 335-367. doi: 10.1016/S0045-7930(01)00041-X. [13] T. Flåtten and Munkejord, The approximate Riemann solver of Roe applied to a drift-flux two-phase flow model, ESAIM: Math. Mod. Num. Anal., 40 (2006), 735-764. doi: 10.1051/m2an:2006032. [14] S. L. Gavrilyuk and J. Fabre, Lagrangian coordinates for a drift-flux model of a gas-liquid mixture, Int. J. Multiphase Flow, 22 (1996), 453-460. doi: 10.1016/0301-9322(95)00085-2. [15] D. Hoff, Construction of solutions for compressible, isentropic Navier-Stokes equations in one space dimension with nonsmooth initial data, Proc. Roy. Soc. Edinburgh Sect. A, 103 (1986), 301-315. doi: 10.1017/S0308210500018953. [16] M. Ishii, "Thermo-Fluid Dynamic Theory of Two-Phase Flow," Eyrolles, Paris, 1975. [17] S. Jiang, Z. Xin and P. Zhang, Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity, Meth. Appl. Anal., 12 (2005), 239-252. [18] T.-P. Liu, Z. Xin and T. Yang, Vacuum states for compressible flow, Discrete Continuous Dyn. Sys., 4 (1998), 1-32. [19] Q. Q. Liu and C. J. Zhu, Asymptotic behavior of a viscous liquid-gas model with mass-dependent viscosity and vacuum, J. Differential Equations, 252 (2012), 2492-2519. doi: 10.1016/j.jde.2011.10.018. [20] R. J. Lorentzen and K. K. Fjelde, Use of slopelimiter techniques in traditional numerical methods for multi-phase flow in pipelines and wells, Int. J. Numer. Meth. Fluids, 48 (2005), 723-745. doi: 10.1002/fld.952. [21] T. Luo, Z. Xin and T. Yang, Interface behavior of compressible Navier-Stokes equations with vacuum, SIAM J. Math. Anal., 31 (2000), 1175-1191. doi: 10.1137/S0036141097331044. [22] J. M. Masella, Q. H. Tran, D. Ferre and C. Pauchon, Transient simulation of two-phase flows in pipes, Int. J. of Multiphase Flow, 24 (1998), 739-755. doi: 10.1016/S0301-9322(98)00004-4. [23] S. T. Munkejord, S. Evje and T. Flåtten, The multi-staged centred scheme approach applied to a drift-flux two-phase flow model, Int. J. Num. Meth. Fluids, 52 (2006), 679-705. doi: 10.1002/fld.1200. [24] M. Okada, Free-boundary value problems for the equation of one-dimensional motion of viscous gas, Japan J. Indust. Appl. Math., 6 (1989), 161-177. doi: 10.1007/BF03167921. [25] M. Okada, S. Matusu-Necasova and T. Makino, Free-boundary problem for the equation of one-dimensional motion of compressible gas with density-dependent viscosity, Ann. Univ. Ferrara Sez VII(N.S.), 48 (2002), 1-20. [26] J. E. Romate, An approximate Riemann solver for a two-phase flow model with numerically given slip relation, Computers & Fluid, 27 (1998), 455-477. doi: 10.1016/S0045-7930(97)00067-4. [27] J. Schlegel, T. Hibiki and M. Ishii, Development of a comprehensive set of drift-flux constitutive models for pipes of various hydraulic diameters, Prog. Nuc. Energy, 52 (2010), 666-677. doi: 10.1016/j.pnucene.2010.03.007. [28] T. Yang, Z.-A. Yao and C.-J. Zhu, Compressible Navier-Stokes equations with density-dependent viscosity and vacuum, Comm. Partial Diff. Eq., 26 (2001), 965-981. doi: 10.1081/PDE-100002385. [29] T. Yang and H.-J. Zhao, A vacuum problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity, J. Differential Equations, 184 (2002), 163-184. doi: 10.1006/jdeq.2001.4140. [30] T. Yang and C.-J. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum, Comm. Math. Phys., 230 (2002), 329-363. doi: 10.1007/s00220-002-0703-6. [31] L. Yao, H.-L. Guo and Z.-H. Guo, A note on viscous liquid-gas two-phase flow model with mass-dependent viscosity and vacuum, Nonlinear Analysis: Real World Applications, 13 (2012), 2323-2342. doi: 10.1016/j.nonrwa.2012.02.001. [32] L. Yao and C.-J. Zhu, Free boundary value problem for a viscous two-phase model with mass-dependent viscosity, J. Differential Equations, 247 (2009), 2705-2739. doi: 10.1016/j.jde.2009.07.013. [33] L. Yao and C. J. Zhu, Existence and uniqueness of global weak solution to a two-phase flow model with vacuum, Math. Ann., 349 (2010), 903-928. doi: 10.1007/s00208-010-0544-0. [34] L. Yao, T. Zhang and C. J. Zhu, Existence and asymptotic behavior of global weak solutions to a 2d viscous liquid-gas two-phase flow model, SIAM J. Math. Anal., 42 (2010), 1874-1897. doi: 10.1137/100785302. [35] L. Yao, T. Zhang and C. J. Zhu, A blow-up criterion for a 2D viscous liquid-gas two-phase flow model, J. Differential Equations, 250 (2011), 3362-3378. doi: 10.1016/j.jde.2010.12.006. [36] T. Zhang and D.-Y. Fang, Global behavior of compressible Navier-Stokes equations with a degenerate viscosity coefficient, Arch. Rational Mech. Anal., 182 (2006), 223-253. doi: 10.1007/s00205-006-0425-6. [37] N. Zuber and J. A. Findlay, Average volumetric concentration in two-phase flow systems, J. Heat Transfer, 87 (1965), 453-468. doi: 10.1115/1.3689137.
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