Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-26T08:01:34.141Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalizing the modified Buckley–Leverett equation with TCAT capillary pressure

Published online by Cambridge University Press:  03 July 2017

K. SPAYD
K. SPAYD*
Affiliation:
Department of Mathematics, Gettysburg College, Gettysburg, PA, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The Buckley–Leverett partial differential equation has long been used to model two-phase flow in porous media. In recent years, the PDE has been modified to include a rate-dependent capillary pressure constitutive equation, known as dynamic capillary pressure. Previous traveling wave analysis of the modified Buckley–Leverett equation uncovered non-classical solutions involving undercompressive shocks. More recently, thermodynamically constrained averaging theory (TCAT) has generalized the capillary pressure equation by including additional dependence on fluid properties. In this paper, the model and traveling wave analysis are updated to incorporate TCAT capillary pressure as a generalization of dynamic capillary pressure. Solutions of the corresponding Riemann problem are similar to previous results except in the physically relevant situation in which both phases are pure fluids. The results presented here shed new light on the nature of the interface between one pure fluid displacing another pure fluid, in accordance with TCAT.

Keywords

35K7035L6535L6776S05
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Issue 2 , April 2018 , pp. 338 - 351
Copyright
Copyright © Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Armstrong, R. T., Porter, M. L. & Wildenschild, D. (2012) Linking pore-scale interfactial curvature to column-scale capillary pressure. Adv. Water Resour. 46, 55–62.CrossRefGoogle Scholar
[2] Buckley, S. E. & Leverett, M. C. (1942) Mechanism of fluid displacement in sands. Soc. Pet. Eng. 146, 107–116.Google Scholar
[3] Cuesta, C. M., van Duijn, C. J. & Hulshof, J. (2000) Infiltration in porous media with dynamic capillary pressure: Travelling waves. Eur. J. Appl. Math. 11, 381–397.CrossRefGoogle Scholar
[4] Cuesta, C. M. & Hulshof, J. (2003) A model problem for groundwater flow with dynamic capillary pressure: Stability of traveling waves. Nonlinear Anal. 52, 1199–1218.CrossRefGoogle Scholar
[5] Cuesta, C. M., van Duijn, C. J. & Pop, I. S. (2006) Non-classical shocks for Buckley-Leverett: Degenerate pseudo-parabolic regularisation. In: Progress in Industrial Mathematics at ECMI 2004, Berlin, Springer, pp. 569–573.CrossRefGoogle Scholar
[6] Cueto-Felgueroso, L. & Juanes, R. (2008) Nonlocal interface dynamics and pattern formation in gravity-driven unsaturated flow through porous media. Phys. Rev. Lett. 101, 244504.CrossRefGoogle ScholarPubMed
[7] Das, D. B. & Mirzaei, M. (2012) Dynamic effects in capillary pressure relationships for two-phase flow in porous media: Experiments and numerical analyses. Am. Inst. Chem. Eng. J. 58 (12), 3891–3903.CrossRefGoogle Scholar
[8] DiCarlo, D. A. (2004) Experimental measurements of saturation overshoot on infiltration. Water Resour. Res. 43, W04215.1–W04215.9.Google Scholar
[9] DiCarlo, D. A. (2005) Modeling observed saturation overshoot with continuum additions to standard unsaturated theory. Adv. Water Resour. 28, 1021–1027.CrossRefGoogle Scholar
[10] DiCarlo, D. A. (2007) Capillary pressure overshoot as a function of imbibition flux and initial water content. Water Resour. Res. 40, W08402.1–W08402.7.Google Scholar
[11] DiCarlo, D. A. & Blunt, M. J. (2000) Determination of finger shape using the dynamic capillary pressure. Water Resour. Res. 36 (9), 2781–2785.CrossRefGoogle Scholar
[12] DiCarlo, D. A., Juanes, R., LaForce, T. & Witelski, T. P. (2008) Nonmonotonic traveling wave solutions of infiltration into porous media. Water Resour. Res. 44, W02406.1–W02406.17.CrossRefGoogle Scholar
[13] Evans, L. C. (1998) Partial Differential Equations, American Mathematical Society, Providence.Google Scholar
[14] Gray, W. G. & Miller, C. T. (2011) TCAT analysis of capillary pressure in non-equilibrium, two-fluid-phase, porous medium systems. Adv. Water Resour. 34, 770–778.CrossRefGoogle ScholarPubMed
[15] Gray, W. G. & Miller, C. T. (2014) Introduction to the Thermodynamically Constrained Averaging Theory for Porous Medium Systems, Springer, New York.CrossRefGoogle Scholar
[16] Gray, W. G., & Miller, C. T. & Schrefler, B. A. (2013) Averaging theory for description of environmental problems: What have we learned? Adv. Water Resour. 51, 123–138.CrossRefGoogle ScholarPubMed
[17] Hassanizadeh, S. M. & Gray, W. G. (1990) Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries. Adv. Water Resour. 13, 169–186.CrossRefGoogle Scholar
[18] Hassanizadeh, S. M. & Gray, W. G. (1993) Thermodynamic basis of capillary pressure in porous media. Water Resour. Res. 29, 3389–3405.CrossRefGoogle Scholar
[19] Hayes, B. & Shearer, M. (1999) Undercompressive shocks and Riemann problems for scalar conservation laws with nonconvex fluxes. Proc. R. Soc. Edinburgh Sect. A 129, 733–754.CrossRefGoogle Scholar
[20] Jacobs, D., McKinney, B. & Shearer, M. (1995) Traveling wave solutions of the modified Korteweg-deVries-Burgers equation. J. Differ. Equ. 116 (2), 448–467.CrossRefGoogle Scholar
[21] Kalisch, H., Mitrovic, D. & Nordbotten, J. M. (2015) Non-standard shocks in the Buckley-Leverett equation. J. Math. Anal. Appl. 428, 882–895.CrossRefGoogle Scholar
[22] Lax, P. D. (1957) Hyperbolic systems of conservation laws II. Commun. Pure Appl. Math. 10, 537–566.CrossRefGoogle Scholar
[23] Lovoll, G., Jankov, M., Maloy, K. J., Toussaint, R., Schmittbuhl, J., Schafer, G. & Meheust, Y. (2011) Influence of viscous fingering on dynamic saturation-pressure curves in porous media. Transp. Porous Media 86, 305–324.CrossRefGoogle Scholar
[24] Nieber, J. L., Dautov, R. Z., Egorov, A. G. & Sheshukov, A. Y. (2005) Dynamic capillary pressure mechanism for instability in gravity-driven flows; review and extension to very dry conditions. Transp. Porous Media 58, 147–172.CrossRefGoogle Scholar
[25] Peaceman, D. (1977) Fundamentals of Numerical Reservoir Simulation, Elsevier Scientific, Amsterdam, New York.Google Scholar
[26] Schecter, S. (1990) Simultaneous equilibrium and heteroclinic bifurcation of planar vector fields via the Melnikov integral. Nonlinearity 3, 79–99.CrossRefGoogle Scholar
[27] Shubao, T., Gang, L., Shunli, H. & Limin, Y. (2012) Dynamic effect of capillary pressure in low permeability reservoirs. Pet. Explor. Dev. 39 (3), 405–411.Google Scholar
[28] Spayd, K. & Shearer, M. (2011) The Buckley-Leverett equation with dynamic capillary pressure. SIAM J. Appl. Math. 71 (4), 1088–1108.CrossRefGoogle Scholar
[29] Xiong, Y. (2014) Flow of water in porous media with saturation overshoot: A review. J. Hydrol. 510, 353–362.CrossRefGoogle Scholar
[30] van Duijn, C. J., Peletier, L. A. & Pop, I. S. (2007) A new class of entropy solutions of the Buckley-Leverett equation. SIAM J. Math. Anal. 39 (2), 507–536.CrossRefGoogle Scholar
[31] Vladimirov, I. G. & Klimenko, A. Y. (2010) Tracing diffusion in porous media with fractal properties. Multiscale Modeling and Simulation 8 (4), 1178–1211.CrossRefGoogle Scholar