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On the local and global existence of solutions to 1d transport equations with nonlocal velocity
Compressible and viscous two-phase flow in porous media based on mixture theory formulation
1. | Department of Energy and Petroleum Engineering, University of Stavanger, Stavanger, NO-4068, Norway |
2. | School of Mathematics, South China University of Technology, Guangzhou, 510641, China |
The purpose of this work is to carry out investigations of a generalized two-phase model for porous media flow. The momentum balance equations account for fluid-rock resistance forces as well as fluid-fluid drag force effects, in addition, to internal viscosity through a Brinkmann type viscous term. We carry out detailed investigations of a one-dimensional version of the general model. Various a priori estimates are derived that give rise to an existence result. More precisely, we rely on the energy method and use compressibility in combination with the structure of the viscous term to obtain $ H^1 $-estimates as well upper and lower uniform bounds of mass variables. These a priori estimates imply existence of solutions in a suitable functional space for a global time $ T>0 $. We also derive discrete schemes both for the incompressible and compressible case to explore the role of the viscosity term (Brinkmann type) as well as the incompressible versus the compressible case. We demonstrate similarities and differences between a formulation that is based, respectively, on interstitial velocity and Darcy velocity in the viscous term. The investigations may suggest that interstitial velocity seems more natural to use in the formulation of momentum balance than Darcy velocity.
References:
[1] |
P.Ø. Andersen, Y. Qiao, D. C. Standnes and S. Evje,
Co-current spontaneous imbibition in porous media with the dynamics of viscous coupling and capillary back pressure, SPE J., 24 (2019), 158-177.
doi: 10.2118/190267-MS. |
[2] |
T. Arbogast,
The existence of weak solutions to single porosity and simple dual-porosity models of two-phase incompressible flow, Nonlinear Anal., 19 (1992), 1009-1031.
doi: 10.1016/0362-546X(92)90121-T. |
[3] |
J. Bear and Y. Bachmat, Introduction to Modeling of Transport Phenomena in Porous Media, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990.
doi: 10.1007/978-94-009-1926-6. |
[4] |
D. Bresch, X. D. Huang and J. Li,
Global weak solutions to one-dimensional non-conservative viscous compressible two-phase system, Comm. Math. Phys., 309 (2012), 737-755.
doi: 10.1007/s00220-011-1379-6. |
[5] |
C. Cances, T. O. Gallouet and L. Monsaingeon,
The gradient flow structure of immiscible incompressible two-phase flows in porous media, C. R. Acad. Sci. Paris Ser. I Math., 353 (2015), 985-989.
doi: 10.1016/j.crma.2015.09.021. |
[6] |
C. Cances and M. Pierre,
An existence result for multidimensional immiscible two-phase flows with discontinuous capillary pressure field, SIAM J. Math. Anal., 44 (2012), 966-992.
doi: 10.1137/11082943X. |
[7] |
X. Cao and I. S. Pop,
Degenerate two-phase porous media flow model with dynamic capillarity, J. Diff. Eqs., 260 (2016), 2418-2456.
doi: 10.1016/j.jde.2015.10.008. |
[8] |
Z. Chen,
Degenerate two-phase incompressible flow: Ⅰ. Existence, uniqueness and regularity of a weak solution, J. Diff. Eqs., 171 (2001), 203-232.
doi: 10.1006/jdeq.2000.3848. |
[9] |
G. M. Coclite, S. Mishra, N. H. Risebro and F. Weber,
Analysis and numerical approximation of Brinkman regularization of two-phase flows in porous media, Comput. Geosci., 18 (2014), 637-659.
doi: 10.1007/s10596-014-9410-6. |
[10] |
J. M. Delhaye, M. Giot and M. L. Riethmuller, Thermohydraulics of Two-Phase Systems for Industrial Design and Nuclear Engineering, Von Karman Institute, McGraw-Hill, New York, 1981. |
[11] |
D. A. Drew and S. L. Passman, Theory of Multicomponent Fluids, Springer, 1999.
doi: 10.1007/b97678. |
[12] |
C. J. van Duijn, Y. Fan, L. A. Peletier and I. S. Pop,
Travelling wave solutions for degenerate pseudo-parabolic equation modelling two-phase flow in porous media, Nonlinear Anal. Real World Applications, 14 (2013), 1361-1383.
doi: 10.1016/j.nonrwa.2012.10.002. |
[13] |
C. J. van Duijn, L. A. Peletier and I. S. Pop,
A new class of entropy solutions of the Buckley-Leverett equation, SIAM J. Math. Anal., 39 (2007), 507-536.
doi: 10.1137/05064518X. |
[14] |
S. Evje,
An integrative multiphase model for cancer cell migration under influence of physical cues from the microenvironment, Chem. Eng. Sci., 165 (2017), 240-259.
doi: 10.1016/j.ces.2017.02.045. |
[15] |
S. Evje and H. Y. Wen,
Analysis of a compressible two-fluid Stokes system with constant viscosity, J. Math. Fluid Mech., 17 (2015), 423-436.
doi: 10.1007/s00021-015-0215-8. |
[16] |
S. Evje and H. Y. Wen,
Stability of a compressible two-fluid hyperbolic-elliptic system arising in fluid mechanics, Nonlin. Anal.: Real World Applications, 31 (2016), 610-629.
doi: 10.1016/j.nonrwa.2016.03.011. |
[17] |
S. Evje, W. Wang and H. Y. Wen,
Global well-posedness and decay rates of strong solutions to a non-conservative compressible two-fluid model, Arch. Rat. Mech. Anal., 221 (2016), 1285-1316.
doi: 10.1007/s00205-016-0984-0. |
[18] |
S. Evje and H. Y. Wen,
A Stokes two-fluid model for cell migration that can account for physical cues in the microenvironment, SIAM J. Math. Anal., 50 (2018), 86-118.
doi: 10.1137/16M1078185. |
[19] |
C. Galusinski and M. Saad,
On a degenerate parabolic system for compressible immiscible two-phase flows in porous media, Adv. Diff. Eqs., 9 (2004), 1235-1278.
|
[20] |
C. Galusinski and M. Saad,
A nonlinear degenerate system modeling water-gas in porous media, Disc. Cont. Dyn. Syst., 9 (2008), 281-308.
doi: 10.3934/dcdsb.2008.9.281. |
[21] |
C. Galusinski and M. Saad,
Two compressible immiscible fluids in porous media, J. Diff. Eqs., 244 (2008), 1741-1783.
doi: 10.1016/j.jde.2008.01.013. |
[22] |
S. M. Hassanizadeh,
Derivation of basic equations of mass transport in porous media, Part 2. Generalized Darcy's and Fick's laws, Adv. Water Resour., 9 (1986), 207-222.
doi: 10.1016/0309-1708(86)90025-4. |
[23] |
S. M. Hassanizadeh and W. G. Gray,
Toward an improved description of the physics of two-phase flow, Adv. Water Resour., 16 (1993), 53-67.
doi: 10.1016/0309-1708(93)90029-F. |
[24] |
S. M. Hassanizadeh and W. G. Gray,
Thermodynamic basis of capillary pressure in porous media, Water Resour. Res., 28 (1993), 3389-3405.
doi: 10.1029/93WR01495. |
[25] |
R. Juanes,
Nonequilibrium effects in models of three-phase flow in porous media, Adv. Wat. Res., 31 (2008), 661-673.
doi: 10.1016/j.advwatres.2007.12.005. |
[26] |
Z. Khalil and M. Saad,
Degenerate two-phase compressible immiscible flow in porous media: The case where the density of each phase depends on its own pressure, Math. Comput. Simulation, 81 (2011), 2225-2233.
doi: 10.1016/j.matcom.2010.12.012. |
[27] |
M. Krotkiewski, I. Ligaarden, K.-A. Lie and D. W. Schmid,
On the importance of the Stokes-Brikman equations for computing effective permeability in carbonate-karst reservoirs, Comm. Comput. Phys., 10 (2011), 1315-1332.
|
[28] |
L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Course of Theoretical Physics, 6, Pergamon
Press, 1984. |
[29] |
G. Lemon, J. R. King, H. M. Byrne, O. E. Jensen and K. M. Shakesheff,
Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory, J. Math. Biol., 52 (2006), 571-594.
doi: 10.1007/s00285-005-0363-1. |
[30] |
M. Muskat, Physical Principles of Oil Production, McGraw-Hill, New York, 1949. |
[31] |
Y. Qiao, P.Ø. Andersen, S. Evje and D. C. Standnes,
A mixture theory approach to model co- and counter-current two-phase flow in porous media accounting for viscous coupling, Advances Wat. Res., 112 (2018), 170-188.
doi: 10.1016/j.advwatres.2017.12.016. |
[32] |
K. R. Rajagopal,
On a hierarchy of approximate models for flows of incompressible fluids through porous solids, Math. Mod. Met. Appl. Sci., 17 (2007), 215-252.
doi: 10.1142/S0218202507001899. |
[33] |
B. Saad and M. Saad,
Study of full implicit petroleum engineering finite volume scheme for compressible two phase flow in porous media, SIAM J. Numer. Anal., 51 (2013), 716-741.
doi: 10.1137/120869092. |
[34] |
M. M. Schuff, J. P. Gore and E. A. Nauman,
A mixture theory model of fluid and solute transport in the microvasculature of normal and malignant tissues. I. Theory, J. Math. Biol., 66 (2013), 1179-1207.
doi: 10.1007/s00285-012-0528-7. |
[35] |
D. C. Standnes, S. Evje and P.Ø. Andersen,
A novel relative permeability model based on mixture theory approach accounting for solid-fluid and fluid-fluid interactions, Tran. Por. Media, 119 (2017), 707-738.
doi: 10.1007/s11242-017-0907-z. |
[36] |
D. C. Standnes and P.Ø. Andersen,
Analysis of the impact of fluid viscosities on the rate of countercurrent spontaneous imbibition, Energy & Fuels, 31 (2017), 6928-6940.
|
[37] |
J. Urdal, J. O. Waldeland and S. Evje, Enhanced cancer cell invasion caused by fibroblasts when fluid flow is present, Biomech. Model. Mechanobiol., preprint, (2019).
doi: 10.1007/s10237-019-01128-2. |
[38] |
F. J. Valdes-Parada, J. A. Ochoa-Tapia and J. Alvarez-Ramirez,
On the effective viscosity for the Darcy–Brinkman equation, Physica A, 385 (2007), 69-79.
doi: 10.1016/j.physa.2007.06.012. |
[39] |
J. O. Waldeland and S. Evje,
A multiphase model for exploring cancer cell migration driven by autologous chemotaxis, Chem. Eng. Sci., 191 (2018), 268-287.
|
[40] |
J. O. Waldeland and S. Evje,
Competing tumor cell migration mechanisms caused by interstitial fluid flow, J. Biomech., 81 (2018), 22-35.
|
[41] |
L. Wang, L.-P. Wang, Z. Guo and J. Mi,
Volume-averaged macroscopic equation for fluid flow in moving porous media, Int. J. Heat Mass Tran., 82 (2015), 357-368.
doi: 10.1016/j.ijheatmasstransfer.2014.11.056. |
[42] |
Y. S. Wu, Multiphase Fluid Flow in Porous and Fractured Reservoirs, Elsevier, 2016. |
show all references
References:
[1] |
P.Ø. Andersen, Y. Qiao, D. C. Standnes and S. Evje,
Co-current spontaneous imbibition in porous media with the dynamics of viscous coupling and capillary back pressure, SPE J., 24 (2019), 158-177.
doi: 10.2118/190267-MS. |
[2] |
T. Arbogast,
The existence of weak solutions to single porosity and simple dual-porosity models of two-phase incompressible flow, Nonlinear Anal., 19 (1992), 1009-1031.
doi: 10.1016/0362-546X(92)90121-T. |
[3] |
J. Bear and Y. Bachmat, Introduction to Modeling of Transport Phenomena in Porous Media, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990.
doi: 10.1007/978-94-009-1926-6. |
[4] |
D. Bresch, X. D. Huang and J. Li,
Global weak solutions to one-dimensional non-conservative viscous compressible two-phase system, Comm. Math. Phys., 309 (2012), 737-755.
doi: 10.1007/s00220-011-1379-6. |
[5] |
C. Cances, T. O. Gallouet and L. Monsaingeon,
The gradient flow structure of immiscible incompressible two-phase flows in porous media, C. R. Acad. Sci. Paris Ser. I Math., 353 (2015), 985-989.
doi: 10.1016/j.crma.2015.09.021. |
[6] |
C. Cances and M. Pierre,
An existence result for multidimensional immiscible two-phase flows with discontinuous capillary pressure field, SIAM J. Math. Anal., 44 (2012), 966-992.
doi: 10.1137/11082943X. |
[7] |
X. Cao and I. S. Pop,
Degenerate two-phase porous media flow model with dynamic capillarity, J. Diff. Eqs., 260 (2016), 2418-2456.
doi: 10.1016/j.jde.2015.10.008. |
[8] |
Z. Chen,
Degenerate two-phase incompressible flow: Ⅰ. Existence, uniqueness and regularity of a weak solution, J. Diff. Eqs., 171 (2001), 203-232.
doi: 10.1006/jdeq.2000.3848. |
[9] |
G. M. Coclite, S. Mishra, N. H. Risebro and F. Weber,
Analysis and numerical approximation of Brinkman regularization of two-phase flows in porous media, Comput. Geosci., 18 (2014), 637-659.
doi: 10.1007/s10596-014-9410-6. |
[10] |
J. M. Delhaye, M. Giot and M. L. Riethmuller, Thermohydraulics of Two-Phase Systems for Industrial Design and Nuclear Engineering, Von Karman Institute, McGraw-Hill, New York, 1981. |
[11] |
D. A. Drew and S. L. Passman, Theory of Multicomponent Fluids, Springer, 1999.
doi: 10.1007/b97678. |
[12] |
C. J. van Duijn, Y. Fan, L. A. Peletier and I. S. Pop,
Travelling wave solutions for degenerate pseudo-parabolic equation modelling two-phase flow in porous media, Nonlinear Anal. Real World Applications, 14 (2013), 1361-1383.
doi: 10.1016/j.nonrwa.2012.10.002. |
[13] |
C. J. van Duijn, L. A. Peletier and I. S. Pop,
A new class of entropy solutions of the Buckley-Leverett equation, SIAM J. Math. Anal., 39 (2007), 507-536.
doi: 10.1137/05064518X. |
[14] |
S. Evje,
An integrative multiphase model for cancer cell migration under influence of physical cues from the microenvironment, Chem. Eng. Sci., 165 (2017), 240-259.
doi: 10.1016/j.ces.2017.02.045. |
[15] |
S. Evje and H. Y. Wen,
Analysis of a compressible two-fluid Stokes system with constant viscosity, J. Math. Fluid Mech., 17 (2015), 423-436.
doi: 10.1007/s00021-015-0215-8. |
[16] |
S. Evje and H. Y. Wen,
Stability of a compressible two-fluid hyperbolic-elliptic system arising in fluid mechanics, Nonlin. Anal.: Real World Applications, 31 (2016), 610-629.
doi: 10.1016/j.nonrwa.2016.03.011. |
[17] |
S. Evje, W. Wang and H. Y. Wen,
Global well-posedness and decay rates of strong solutions to a non-conservative compressible two-fluid model, Arch. Rat. Mech. Anal., 221 (2016), 1285-1316.
doi: 10.1007/s00205-016-0984-0. |
[18] |
S. Evje and H. Y. Wen,
A Stokes two-fluid model for cell migration that can account for physical cues in the microenvironment, SIAM J. Math. Anal., 50 (2018), 86-118.
doi: 10.1137/16M1078185. |
[19] |
C. Galusinski and M. Saad,
On a degenerate parabolic system for compressible immiscible two-phase flows in porous media, Adv. Diff. Eqs., 9 (2004), 1235-1278.
|
[20] |
C. Galusinski and M. Saad,
A nonlinear degenerate system modeling water-gas in porous media, Disc. Cont. Dyn. Syst., 9 (2008), 281-308.
doi: 10.3934/dcdsb.2008.9.281. |
[21] |
C. Galusinski and M. Saad,
Two compressible immiscible fluids in porous media, J. Diff. Eqs., 244 (2008), 1741-1783.
doi: 10.1016/j.jde.2008.01.013. |
[22] |
S. M. Hassanizadeh,
Derivation of basic equations of mass transport in porous media, Part 2. Generalized Darcy's and Fick's laws, Adv. Water Resour., 9 (1986), 207-222.
doi: 10.1016/0309-1708(86)90025-4. |
[23] |
S. M. Hassanizadeh and W. G. Gray,
Toward an improved description of the physics of two-phase flow, Adv. Water Resour., 16 (1993), 53-67.
doi: 10.1016/0309-1708(93)90029-F. |
[24] |
S. M. Hassanizadeh and W. G. Gray,
Thermodynamic basis of capillary pressure in porous media, Water Resour. Res., 28 (1993), 3389-3405.
doi: 10.1029/93WR01495. |
[25] |
R. Juanes,
Nonequilibrium effects in models of three-phase flow in porous media, Adv. Wat. Res., 31 (2008), 661-673.
doi: 10.1016/j.advwatres.2007.12.005. |
[26] |
Z. Khalil and M. Saad,
Degenerate two-phase compressible immiscible flow in porous media: The case where the density of each phase depends on its own pressure, Math. Comput. Simulation, 81 (2011), 2225-2233.
doi: 10.1016/j.matcom.2010.12.012. |
[27] |
M. Krotkiewski, I. Ligaarden, K.-A. Lie and D. W. Schmid,
On the importance of the Stokes-Brikman equations for computing effective permeability in carbonate-karst reservoirs, Comm. Comput. Phys., 10 (2011), 1315-1332.
|
[28] |
L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Course of Theoretical Physics, 6, Pergamon
Press, 1984. |
[29] |
G. Lemon, J. R. King, H. M. Byrne, O. E. Jensen and K. M. Shakesheff,
Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory, J. Math. Biol., 52 (2006), 571-594.
doi: 10.1007/s00285-005-0363-1. |
[30] |
M. Muskat, Physical Principles of Oil Production, McGraw-Hill, New York, 1949. |
[31] |
Y. Qiao, P.Ø. Andersen, S. Evje and D. C. Standnes,
A mixture theory approach to model co- and counter-current two-phase flow in porous media accounting for viscous coupling, Advances Wat. Res., 112 (2018), 170-188.
doi: 10.1016/j.advwatres.2017.12.016. |
[32] |
K. R. Rajagopal,
On a hierarchy of approximate models for flows of incompressible fluids through porous solids, Math. Mod. Met. Appl. Sci., 17 (2007), 215-252.
doi: 10.1142/S0218202507001899. |
[33] |
B. Saad and M. Saad,
Study of full implicit petroleum engineering finite volume scheme for compressible two phase flow in porous media, SIAM J. Numer. Anal., 51 (2013), 716-741.
doi: 10.1137/120869092. |
[34] |
M. M. Schuff, J. P. Gore and E. A. Nauman,
A mixture theory model of fluid and solute transport in the microvasculature of normal and malignant tissues. I. Theory, J. Math. Biol., 66 (2013), 1179-1207.
doi: 10.1007/s00285-012-0528-7. |
[35] |
D. C. Standnes, S. Evje and P.Ø. Andersen,
A novel relative permeability model based on mixture theory approach accounting for solid-fluid and fluid-fluid interactions, Tran. Por. Media, 119 (2017), 707-738.
doi: 10.1007/s11242-017-0907-z. |
[36] |
D. C. Standnes and P.Ø. Andersen,
Analysis of the impact of fluid viscosities on the rate of countercurrent spontaneous imbibition, Energy & Fuels, 31 (2017), 6928-6940.
|
[37] |
J. Urdal, J. O. Waldeland and S. Evje, Enhanced cancer cell invasion caused by fibroblasts when fluid flow is present, Biomech. Model. Mechanobiol., preprint, (2019).
doi: 10.1007/s10237-019-01128-2. |
[38] |
F. J. Valdes-Parada, J. A. Ochoa-Tapia and J. Alvarez-Ramirez,
On the effective viscosity for the Darcy–Brinkman equation, Physica A, 385 (2007), 69-79.
doi: 10.1016/j.physa.2007.06.012. |
[39] |
J. O. Waldeland and S. Evje,
A multiphase model for exploring cancer cell migration driven by autologous chemotaxis, Chem. Eng. Sci., 191 (2018), 268-287.
|
[40] |
J. O. Waldeland and S. Evje,
Competing tumor cell migration mechanisms caused by interstitial fluid flow, J. Biomech., 81 (2018), 22-35.
|
[41] |
L. Wang, L.-P. Wang, Z. Guo and J. Mi,
Volume-averaged macroscopic equation for fluid flow in moving porous media, Int. J. Heat Mass Tran., 82 (2015), 357-368.
doi: 10.1016/j.ijheatmasstransfer.2014.11.056. |
[42] |
Y. S. Wu, Multiphase Fluid Flow in Porous and Fractured Reservoirs, Elsevier, 2016. |







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