American Institute of Mathematical Sciences

Advanced
September  2019, 14(3): 489-536. doi: 10.3934/nhm.2019020

Compressible and viscous two-phase flow in porous media based on mixture theory formulation

Yangyang Qiao 1, , Huanyao Wen 2, and  Steinar Evje 1,,

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

* Corresponding author: Steinar Evje

Received  June 2018 Revised  March 2019 Published  May 2019

Fund Project: Wen was supported by the National Natural Science Foundation of China (Grant No. 11671150, 11722104) and by GDUPS (2016).

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.

Keywords: Darcy's equation, mixture theory, two-phase flow, Brinkman's equation, viscous coupling, compressible flow, incompressible flow, Navier-Stokes.
Mathematics Subject Classification: 65M06, 76T10, 76N10.
Citation: Yangyang Qiao, Huanyao Wen, Steinar Evje. Compressible and viscous two-phase flow in porous media based on mixture theory formulation. Networks & Heterogeneous Media, 2019, 14 (3) : 489-536. doi: 10.3934/nhm.2019020
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[11]

D. A. Drew and S. L. Passman, Theory of Multicomponent Fluids, Springer, 1999. doi: 10.1007/b97678.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[28]

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Course of Theoretical Physics, 6, Pergamon Press, 1984.  Google Scholar

[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.  Google Scholar

[30]

M. Muskat, Physical Principles of Oil Production, McGraw-Hill, New York, 1949. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[40]

J. O. Waldeland and S. Evje, Competing tumor cell migration mechanisms caused by interstitial fluid flow, J. Biomech., 81 (2018), 22-35.   Google Scholar

[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.  Google Scholar

[42]

Y. S. Wu, Multiphase Fluid Flow in Porous and Fractured Reservoirs, Elsevier, 2016. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[11]

D. A. Drew and S. L. Passman, Theory of Multicomponent Fluids, Springer, 1999. doi: 10.1007/b97678.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[28]

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Course of Theoretical Physics, 6, Pergamon Press, 1984.  Google Scholar

[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.  Google Scholar

[30]

M. Muskat, Physical Principles of Oil Production, McGraw-Hill, New York, 1949. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[40]

J. O. Waldeland and S. Evje, Competing tumor cell migration mechanisms caused by interstitial fluid flow, J. Biomech., 81 (2018), 22-35.   Google Scholar

[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.  Google Scholar

[42]

Y. S. Wu, Multiphase Fluid Flow in Porous and Fractured Reservoirs, Elsevier, 2016. Google Scholar

Figure 1.  Water fractional flow function $ \hat{f}_w(s_w) $ as given by (4.127) for the incompressible model obtained by using the parameters specified in Table 1 (left figure) and initial water saturation (3.84) profile (right), both similar to that used in [9]
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Figure 2.  Upper row: Results produced by the discrete scheme described in Appendix D (incompressible model). Three kinds of curves are plotted including the case without viscous effect, i.e., $ \varepsilon_w = \varepsilon_o = 0 $, the one based on using Darcy velocity $ U_i $ $ (i = w, o) $ in the viscous term, and the one with interstitial velocity $ u_i $ $ (i = w, o) $ in the viscous term. The left figure shows results with $ \varepsilon = \varepsilon_w = \varepsilon_o = 10^7 $ whereas the right figure shows results with $ \varepsilon = \varepsilon_w = \varepsilon_o = 10^6 $. Lower row: The results of two corresponding cases with $ \varepsilon_w = \varepsilon_o = 10^7 $ and $ \varepsilon_w = \varepsilon_o = 10^6 $ after a dimensionless time, 0.65, produced by the numerical scheme described in [9] to solve the model (1.13). From these computations we see that the solution is sensitive to whether the interstitial velocity $ u_i $ or the Darcy velocity $ U_i $ appear in the viscous term. In particular, the use of Darcy velocity seems to generate considerably more oscillatory behavior behind the "water bank" formed at the front
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Figure 3.  Simulation results with smaller viscous parameters after 10 days of water flooding. Three kinds of curves are compared: zero viscous effect, Darcy velocity $ U_i $ in viscous term and interstitial velocity $ u_i $ in viscous term. It shows that the viscous constant water level gradually vanishes when $ \varepsilon $ is as low as $ 10^3 $ and $ 10^2 $
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Figure 4.  The results after 10 days with initial data are shown in Fig. 1 with interstitial velocity in viscous term. Four curves are compared: the one with large values of $ \varepsilon_w $ and $ \varepsilon_o $, $ 10^6 $; the second one with large $ \varepsilon_o $, $ 10^6 $ and small $ \varepsilon_w $, $ 10^4 $; the third one with large $ \varepsilon_w $, $ 10^6 $ and small $ \varepsilon_o $, $ 10^4 $ and the last one with small values of $ \varepsilon_w $ and $ \varepsilon_o $, $ 10^4 $. It shows that the displaced oil influences significantly on the constant level of water which displaces oil
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Figure 5.  Initial water saturation profile from Coclite et al. [9]
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Figure 6.  Left: The results from Coclite et al. [9] based on Darcy velocity in viscous term. Right: Numerical scheme (after 8 days) which uses interstitial velocity in viscous term with different viscous values $ \varepsilon = 0, 10^3, 10^4, 10^5 $ and $ \varepsilon_w = \varepsilon_o = \varepsilon $
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Figure 7.  Comparison between the compressible model and the incompressible model for water-oil flow with $ \varepsilon_w = \varepsilon_o = \varepsilon = 10^7, 10^6 $. After the same period of 10 days, water flow in the compressible model is delayed compared with water profiles in the incompressible model, for both situations with interstitial velocity and Darcy velocity in viscous terms
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Figure 8.  The water pressure evolution in the compressible model for the case with Darcy velocity in viscous term (left figure) and the case with interstitial velocity in viscous term (right figure). Water pressure increases with time in the water displacing part of the reservoir layer which leads to a compression effect where the magnitude of the viscous terms increase and thereby slows down the displacement of the water front
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Figure 9.  Left: Comparison of saturation profiles for water injection and gas injection, respectively, after the same time period (10 days) in the compressible model using interstitial velocity in viscous term ($ \varepsilon_w = \varepsilon_o = \varepsilon = 10^7 $). Right: The gas saturation profile shown at different times
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Table 1.  Input parameters of reservoir and fluid properties used for for the below simulations. Note that $ P_{wL} $ is the boundary pressure at left for the incompressible model whereas for the compressible model it represents the initial pressure distribution
Parameter Dimensional Value Parameter Dimensional Value
$ \, L $ $ 100 $ $ \text{m} $ $ \, I_w $ $ 1.5 $
$ \, \phi $ $ 1 $ $ \, I_o $ $ 1.5 $
$ \, A $ $ 1 $ $ \text{m}^2 $ $ \, I $ $ 0 $ (Pa$ \cdot $s)$ ^{-1} $
$ \, \tilde{\rho}_{w0} $ $ 1 $ $ \text{g}/\text{cm}^3 $ $ \, \alpha $ $ 0 $
$ \, \tilde{\rho}_{o0} $ $ 1 $ $ \text{g}/\text{cm}^3 $ $ \, \beta $ $ 0 $
$ \, C_{w} $ $ 10^6 $ $ \text{m}^2/\text{s}^2 $ $ \, \varepsilon_w $ $ 10^7, 10^6, 10^5, 10^4, 10^3, 10^2 $ $ \text{cP} $
$ \, C_{o} $ $ 5\cdot10^5 $ $ \text{m}^2/\text{s}^2 $ $ \, \varepsilon_o $ $ 10^7, 10^6, 10^5, 10^4, 10^3, 10^2 $ $ \text{cP} $
$ \, \mu_w $ $ 1 $ $ \text{cP} $ $ \, K $ $ 1000 $ $ \text{mD} $
$ \, \mu_o $ $ 1 $ $ \text{cP} $ $ \, k_{rw}^{max}=1/I_w $ $ 0.667 $
$ \, Q $ $ 8.004 $ $ \text{m}^3/\text{day} $ $ \, k_{ro}^{max}=1/I_o $ $ 0.667 $
$ \, P_{wL} $ $ 10^6 $ $ \text{Pa} $ $ \, T $ $ 10 $ $ \text{days} $
$ \, N_x $ $ 2001 $ $ \, \triangle t $ $ 8640 $ $ \text{s} $
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Parameter Dimensional Value Parameter Dimensional Value
$ \, L $ $ 100 $ $ \text{m} $ $ \, I_w $ $ 1.5 $
$ \, \phi $ $ 1 $ $ \, I_o $ $ 1.5 $
$ \, A $ $ 1 $ $ \text{m}^2 $ $ \, I $ $ 0 $ (Pa$ \cdot $s)$ ^{-1} $
$ \, \tilde{\rho}_{w0} $ $ 1 $ $ \text{g}/\text{cm}^3 $ $ \, \alpha $ $ 0 $
$ \, \tilde{\rho}_{o0} $ $ 1 $ $ \text{g}/\text{cm}^3 $ $ \, \beta $ $ 0 $
$ \, C_{w} $ $ 10^6 $ $ \text{m}^2/\text{s}^2 $ $ \, \varepsilon_w $ $ 10^7, 10^6, 10^5, 10^4, 10^3, 10^2 $ $ \text{cP} $
$ \, C_{o} $ $ 5\cdot10^5 $ $ \text{m}^2/\text{s}^2 $ $ \, \varepsilon_o $ $ 10^7, 10^6, 10^5, 10^4, 10^3, 10^2 $ $ \text{cP} $
$ \, \mu_w $ $ 1 $ $ \text{cP} $ $ \, K $ $ 1000 $ $ \text{mD} $
$ \, \mu_o $ $ 1 $ $ \text{cP} $ $ \, k_{rw}^{max}=1/I_w $ $ 0.667 $
$ \, Q $ $ 8.004 $ $ \text{m}^3/\text{day} $ $ \, k_{ro}^{max}=1/I_o $ $ 0.667 $
$ \, P_{wL} $ $ 10^6 $ $ \text{Pa} $ $ \, T $ $ 10 $ $ \text{days} $
$ \, N_x $ $ 2001 $ $ \, \triangle t $ $ 8640 $ $ \text{s} $
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