Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-12T02:54:57.724Z Has data issue: false hasContentIssue false
  • English
  • Français

A Moving Mesh Finite Difference Method for Non-Monotone Solutions of Non-Equilibrium Equations in Porous Media

Part of: Equations of mathematical physics and other areas of application Partial differential equations, initial value and time-dependent initial-boundary value problems Numerical methods Representations of solutions Flows in porous media; filtration; seepage

Published online by Cambridge University Press:  28 July 2017

Hong Zhang  and
Paul Andries Zegeling
Hong Zhang*
Affiliation:
Department of Mathematics, Utrecht University, P.O.Box 80.010, 3508TA Utrecht, The Netherlands
Paul Andries Zegeling*
Affiliation:
Department of Mathematics, Utrecht University, P.O.Box 80.010, 3508TA Utrecht, The Netherlands
*
*Corresponding author. Email addresses:H.Zhang4@uu.nl (H. Zhang), P.A.Zegeling@uu.nl (P. A. Zegeling)
*Corresponding author. Email addresses:H.Zhang4@uu.nl (H. Zhang), P.A.Zegeling@uu.nl (P. A. Zegeling)
Get access

Abstract

An adaptive moving mesh finite difference method is presented to solve two types of equations with dynamic capillary pressure effect in porous media. One is the non-equilibrium Richards Equation and the other is the modified Buckley-Leverett equation. The governing equations are discretized with an adaptive moving mesh finite difference method in the space direction and an implicit-explicit method in the time direction. In order to obtain high quality meshes, an adaptive monitor function with directional control is applied to redistribute the mesh grid in every time step, then a diffusive mechanism is used to smooth the monitor function. The behaviors of the central difference flux, the standard local Lax-Friedrich flux and the local Lax-Friedrich flux with reconstruction are investigated by solving a 1D modified Buckley-Leverett equation. With the moving mesh technique, good mesh quality and high numerical accuracy are obtained. A collection of one-dimensional and two-dimensional numerical experiments is presented to demonstrate the accuracy and effectiveness of the proposed method.

Keywords

Relaxation non-equilibrium Richards equationmodified Buckley-Leverett equationsaturation overshoottraveling wave analysismoving mesh finite difference method

MSC classification

Secondary: 35C07: Traveling wave solutions 35Q35: PDEs in connection with fluid mechanics 65M50: Mesh generation and refinement 74S20: Finite difference methods 76S05: Flows in porous media; filtration; seepage
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 4 , October 2017 , pp. 935 - 964
Copyright
Copyright © Global-Science 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.)

Footnotes

Communicated by Tao Zhou

References

[1] Hill, D., Parlange, J.-Y., Wetting front instability in layered soils, Soil Science Society of America Journal 36 (5) (1972) 697–702.CrossRefGoogle Scholar
[2] Selker, J., Parlange, J.-Y., Steenhuis, T., Fingered flow in two dimensions: 2. predicting finger moisture profile,Water Resources Research 28 (9) (1992) 2523–2528.CrossRefGoogle Scholar
[3] Glass, R. J., Steenhuis, T. S., Parlange, J.-Y., Mechanism for finger persistence in homogeneous, unsaturated, porous media: Theory and verification., Soil Science 148 (1) (1989) 60–70.CrossRefGoogle Scholar
[4] DiCarlo, D. A., Experimental measurements of saturation overshoot on infiltration, Water Resources Research 40 (4) (2004) W04215.CrossRefGoogle Scholar
[5] Stauffer, F., Time dependence of the relations between capillary pressure, water content and conductivity during drainage of porous media, in: IAHR symposium on scale effects in porous media, Thessaloniki, Greece, Vol. 29, 1978, pp. 3–35.Google Scholar
[6] Hassanizadeh, S. M., Gray, W. G., Thermodynamic basis of capillary pressure in porous media, Water Resources Research 29 (10) (1993) 3389–3405.CrossRefGoogle Scholar
[7] Kalaydjian, F.-M., et al., Dynamic capillary pressure curve for water/oil displacement in porous media: Theory vs. experiment, in: SPE Annual Technical Conference and Exhibition, Society of Petroleum Engineers, 1992.Google Scholar
[8] Eliassi, M., Glass, R. J., On the continuum-scale modeling of gravity-driven fingers in unsaturated porous media: The inadequacy of the Richards equationwith standard monotonic constitutive relations and hysteretic equations of state, Water Resources Research 37 (8) (2001) 2019–2035.CrossRefGoogle Scholar
[9] Eliassi, M., Glass, R. J., On the porous continuum-scale modeling of gravity-driven fingers in unsaturated materials: Numerical solution of a hypodiffusive governing equation that incorporates a hold-back-pile-up effect, Water resources research 39 (6) (2003) 1167.CrossRefGoogle Scholar
[10] Nieber, J., Sheshukov, A., Egorov, A., Dautov, R., Non-equilibrium model for gravity-driven fingering in water repellent soils: Formulation and 2D simulations (2003) 245–258.Google Scholar
[11] Chapwanya, M., Stockie, J. M., Numerical simulations of gravity-driven fingering in unsaturated porous media using a nonequilibrium model, Water Resources Research 46 (9) (2010) W09534.CrossRefGoogle Scholar
[12] Cueto-Felgueroso, L., Juanes, R., A phase field model of unsaturated flow, Water Resources Research 45 (10) (2009) W10409.CrossRefGoogle Scholar
[13] Hilfer, R., Doster, F., Zegeling, P., Nonmonotone saturation profiles for hydrostatic equilibrium in homogeneous porous media, Vadose Zone Journal 11 (3) (2012). http://vzj.geoscienceworld.org/content/11/3/vzj2012.0021.CrossRefGoogle Scholar
[14] Doster, F., Zegeling, P., Hilfer, R., Numerical solutions of a generalized theory for macroscopic capillarity, Physical Review E 81 (3) (2010) 036307.CrossRefGoogle ScholarPubMed
[15] DiCarlo, D. A., Mirzaei, M., Aminzadeh, B., Dehghanpour, H., Fractional flow approach to saturation overshoot, Transport in porous media 91 (3) (2012) 955–971.CrossRefGoogle Scholar
[16] van Duijn, C., Hassanizadeh, S., Pop, I., Zegeling, P., et al., Non-equilibrium models for two phase flow in porous media: the occurence of saturation overshoots, In: CAPM 2013 - Proceedings of the 5th International Conference on Applications of Porous Media (2013) 59–70.Google Scholar
[17] Zegeling, P. A., An adaptive grid method for a non-equilibrium PDE model from porous media, J. Math. Study 48 (2) (2015) 187–198.CrossRefGoogle Scholar
[18] Zhang, H., Zegeling, P. A., A numerical study of two-phase flow models with dynamic capillary pressure and hysteresis, Transport in Prous media 116 (2) (2017) 825–846.CrossRefGoogle Scholar
[19] Egorov, A. G., Dautov, R. Z., Nieber, J. L., Sheshukov, A. Y., Stability analysis of gravity-driven infiltrating flow, Water resources research 39 (9) (2003) 1266.CrossRefGoogle Scholar
[20] Nieber, J. L., Dautov, R. Z., Egorov, A. G., Sheshukov, A. Y., Dynamic capillary pressure mechanism for instability in gravity-driven flows; review and extension to very dry conditions, Transport in porous media 58 (1-2) (2005) 147–172.CrossRefGoogle Scholar
[21] Van Duijn, C., Peletier, L., Pop, I., A new class of entropy solutions of the Buckley-Leverett equation, SIAM Journal on Mathematical Analysis 39 (2) (2007) 507–536.CrossRefGoogle Scholar
[22] Mikelić, A., A global existence result for the equations describing unsaturated flow in porous mediawith dynamic capillary pressure, Journal of Differential Equations 248 (6) (2010) 1561–1577.CrossRefGoogle Scholar
[23] Spayd, K., Shearer, M., The Buckley-Leverett equationwith dynamic capillary pressure, SIAM Journal on Applied Mathematics 71 (4) (2011) 1088–1108.CrossRefGoogle Scholar
[24] Cao, X., Pop, I., Uniqueness of weak solutions for a pseudo-parabolic equation modeling two phase flow in porous media, Applied Mathematics Letters 46 (2015) 25–30.CrossRefGoogle Scholar
[25] Kao, C.-Y., Kurganov, A., Qu, Z., Wang, Y., A fast explicit operator splitting method for modified Buckley-Leverett equations, Journal of Scientific Computing 64 (3) (2015) 837–857.CrossRefGoogle Scholar
[26] Zhang, H., Zegeling, P. A., A numerical study of two-phase flow with dynamic capillary pressure using an adaptive moving mesh method, arXiv preprint arXiv:1604.04863.Google Scholar
[27] Peszynska, M., Yi, S.-Y., Numerical methods for unsaturated flow with dynamic capillary pressure in heterogeneous porous media, Int J Numer Anal Model 5 (Special Issue) (2008) 126–149.Google Scholar
[28] Wang, Y., Kao, C.-Y., Central schemes for the modified Buckley–Leverett equation, Journal of Computational Science 4 (1) (2013) 12–23.CrossRefGoogle Scholar
[29] Hong, J. M.-K., Wu, J., Yuan, J.-M., The generalized Buckley-Leverett and the regularized Buckley-Leverett equations, Journal of Mathematical Physics 53 (5) (2012) 053701.CrossRefGoogle Scholar
[30] de Moraes, G., Teixeira, R. d. S., Alves, L. S. d. B., Validity of parametric restrictions to the modified Buckley-Leverett equations, Journal of Porous Media 19 (9) (2016) 811–819.CrossRefGoogle Scholar
[31] Hu, G., Zegeling, P. A., Simulating finger phenomena in porous media with a moving finite element method, Journal of computational physics 230 (8) (2011) 3249–3263.CrossRefGoogle Scholar
[32] Dong, H., Qiao, Z., Sun, S., Tang, T., Adaptive moving grid methods for two-phase flow in porous media, Journal of Computational and Applied Mathematics 265 (2014) 139–150.CrossRefGoogle Scholar
[33] Huang, W., Ren, Y., Russell, R. D., Moving mesh partial differential equations (MMPDEs) based on the equidistribution principle, SIAM Journal on Numerical Analysis 31 (3) (1994) 709–730.CrossRefGoogle Scholar
[34] Van Dam, A., Zegeling, P., Balanced monitoring of flow phenomena in moving mesh methods, Communications in Computational Physics 7 (1) (2010) 138–170.Google Scholar
[35] Huang, W., Russell, R. D., Analysis of moving mesh partial differential equations with spatial smoothing, SIAM Journal on Numerical Analysis 34 (3) (1997) 1106–1126.CrossRefGoogle Scholar
[36] Hilfer, R., Steinle, R., Saturation overshoot and hysteresis for twophase flow in porous media, The European Physical Journal Special Topics 223 (11) (2014) 2323–2338.CrossRefGoogle Scholar
[37] Hassanizadeh, S. M., Celia, M. A., Dahle, H. K., Dynamic effect in the capillary pressure–saturation relationship and its impacts on unsaturated flow, Vadose Zone Journal 1 (1) (2002) 38–57.CrossRefGoogle Scholar
[38] Egorov, A., Dautov, R., Nieber, J., Sheshukov, A., Stability analysis of traveling wave solution for gravity-driven flow, Developments in Water Science 47 (2002) 121–128.CrossRefGoogle Scholar
[39] Van Duijn, C., Fan, Y., Peletier, L., Pop, I. S., Travelling wave solutions for degenerate pseudo-parabolic equations modelling two-phase flow in porous media, Nonlinear Analysis: Real World Applications 14 (3) (2013) 1361–1383.Google Scholar
[40] Weizhang, H., Russell, R. D., Adaptive moving mesh methods, Vol. 174, Springer Science & Business Media, 2010.Google Scholar
[41] Kissling, F., Rohde, C., The computation of nonclassical shock waves in porous media with a heterogeneous multiscale method: the multidimensional case, Multiscale Modeling & Simulation 13 (4) (2015) 1507–1541.CrossRefGoogle Scholar
[42] Zhang, Z.-R., Tang, T., An adaptive mesh redistribution algorithm for convection-dominated problems, Communications on Pure and Applied Analysis 1 (3) (2002) 341–357. doi:10.3934/cpaa.2002.1.341. URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=116 CrossRefGoogle Scholar
[43] Van der Vorst, H. A., Bi-cgstab: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM Journal on scientific and Statistical Computing 13 (2) (1992) 631–644.CrossRefGoogle Scholar
[44] Skalicky, T., Laspack 1.12. 2 (1995), URL http://dddas.org/mgnet/Codes/laspack/html/laspack.html.Google Scholar
[45] Miller, K., Miller, R. N., Moving finite elements. I, SIAM Journal on Numerical Analysis 18 (6) (1981) 1019–1032.CrossRefGoogle Scholar
[46] Budd, C. J., Williams, J., Moving mesh generation using the parabolic Monge-Ampere equation, SIAM Journal on Scientific Computing 31 (5) (2009) 3438–3465.CrossRefGoogle Scholar
[47] Huang, W., Ren, Y., Russell, R. D., Moving mesh methods based on moving mesh partial differential equations, Journal of Computational Physics 113 (2) (1994) 279–290.CrossRefGoogle Scholar
[48] Zegeling, P., De Boer, W., Tang, H., Robust and efficient adaptive moving mesh solution of the 2-D Euler equations, Contemporary Mathematics 383 (2005) 419–430.Google Scholar
[49] Huang, W., Practical aspects of formulation and solution of moving mesh partial differential equations, Journal of Computational Physics 171 (2) (2001) 753–775.CrossRefGoogle Scholar
[50] Verwer, J. G., Blom, J., Furzeland, R., Zegeling, P., A moving-grid method for one-dimensional PDEs based on the method of lines, Stichting Mathematisch Centrum, 1988.Google Scholar
[51] Beckett, G., Mackenzie, J., Ramage, A., Sloan, D., On the numerical solution of one-dimensional PDEs using adaptive methods based on equidistribution, Journal of Computational Physics 167 (2) (2001) 372–392.CrossRefGoogle Scholar
[52] Wang, H., Li, R., Tang, T., Efficient computation of dendritic growth with r-adaptive finite element methods, Journal of Computational Physics 227 (12) (2008) 5984–6000.CrossRefGoogle Scholar
[53] Karlsen, K.H., Brusdal, K., Dahle, H. K., Evje, S., Lie, K.-A., The corrected operator splitting approach applied to a nonlinear advection-diffusion problem, Computer methods in applied mechanics and engineering 167 (3) (1998) 239–260.CrossRefGoogle Scholar
[54] Bauters, T., DiCarlo, D., Steenhuis, T., Parlange, J.-Y., Soil water content dependent wetting front characteristics in sands, Journal of Hydrology 231 (2000) 244–254.CrossRefGoogle Scholar
[55] Schroth, M., Istok, J., Ahearn, S., Selker, J., Characterization of miller-similar silica sands for laboratory hydrologic studies, Soil Science Society of America Journal 60 (5) (1996) 1331–1339.CrossRefGoogle Scholar
[56] Brooks, R. H., Corey, A., Properties of porous media affecting fluid flow, Journal of the Irrigation and Drainage Division 92 (2) (1966) 61–90.CrossRefGoogle Scholar
[57] Sander, G., Glidewell, O., Norbury, J., Dynamic capillary pressure, hysteresis and gravity-driven fingering in porous media, in: Journal of Physics: Conference Series, Vol. 138, IOP Publishing, 2008, p. 012023.Google Scholar