Analytical and variational numerical methods for unstable miscible displacement flows in porous media. (English) Zbl 1375.76188
Summary: The miscible displacement of one fluid by another in a porous medium has received considerable attention in subsurface, environmental and petroleum engineering applications. When a fluid of higher mobility displaces another of lower mobility, unstable patterns – referred to as viscous fingering – may arise. Their physical and mathematical study has been the object of numerous investigations over the past century. The objective of this paper is to present a review of these contributions with particular emphasis on variational methods. These algorithms are tailored to real field applications thanks to their advanced features: handling of general complex geometries, robustness in the presence of rough tensor coefficients, low sensitivity to mesh orientation in advection dominated scenarios, and provable convergence with fully unstructured grids. This paper is dedicated to the memory of Dr. Jim Douglas Jr., for his seminal contributions to miscible displacement and variational numerical methods.
MSC:
| 76S05 | Flows in porous media; filtration; seepage |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 76M30 | Variational methods applied to problems in fluid mechanics |
| 76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |
Keywords:
miscible displacement; viscous fingering; flow instabilities; Hele-Shaw; variational methods; numerical simulation; Galerkin methodsSoftware:
C-AMSReferences:
| [1] | Chen, Z.; Huan, G.; Ma, Y., Computational Methods for Multiphase Flows in Porous Media, vol. 2 (2006), SIAM · Zbl 1092.76001 |
| [2] | Peaceman, D. W., Improved treatment of dispersion in numerical calculation of multidimensional miscible displacement, Soc. Pet. Eng. J., 6, 213-216 (1966) |
| [3] | Fabrie, P.; Langlais, M., Mathematical analysis of miscible displacement in porous medium, SIAM J. Math. Anal., 23, 6, 1375-1392 (1992) · Zbl 0772.76070 |
| [4] | Tan, C.; Homsy, G., Stability of miscible displacements in porous media: rectilinear flow, Phys. Fluids, 29, 3549-3556 (1986) · Zbl 0608.76087 |
| [5] | Tan, C.; Homsy, G., Simulation of nonlinear viscous fingering in miscible displacement, Phys. Fluids, 31, 1330-1338 (1988) |
| [6] | Tan, C.; Homsy, G., Viscous fingering with permeability heterogeneity, Phys. Fluids A, Fluid Dyn., 4, 1099-1101 (1992) |
| [7] | Tan, C.; Homsy, G., Stability of miscible displacements in porous media: radial source flow, Phys. Fluids, 30, 1239-1245 (1987) · Zbl 0623.76035 |
| [8] | Manickam, O.; Homsy, G., Simulation of viscous fingering in miscible displacements with nonmonotonic viscosity profiles, Phys. Fluids, 6, 95-107 (1994) · Zbl 0922.76168 |
| [9] | Homsy, G. M., Viscous fingering in porous media, Annu. Rev. Fluid Mech., 19, 271-311 (1987) |
| [10] | Zimmerman, W.; Homsy, G., Three-dimensional viscous fingering: a numerical study, Phys. Fluids A, Fluid Dyn., 4, 1901-1914 (1992) · Zbl 0763.76022 |
| [11] | Zimmerman, W.; Homsy, G., Viscous fingering in miscible displacements: unification of effects of viscosity contrast, anisotropic dispersion, and velocity dependence of dispersion on nonlinear finger propagation, Phys. Fluids A, Fluid Dyn., 4, 2348-2359 (1992) |
| [12] | Zimmerman, W.; Homsy, G., Nonlinear viscous fingering in miscible displacement with anisotropic dispersion, Phys. Fluids A, Fluid Dyn., 3, 1859-1872 (1991) · Zbl 0745.76090 |
| [13] | Araktingi, U. G.; Orr, F. M., Viscous fingering in heterogeneous porous media, SPE Adv. Technol. Ser., 1, 71-80 (1993) |
| [14] | Chen, C.-Y.; Meiburg, E., Miscible porous media displacements in the quarter five-spot configuration. Part 1. The homogeneous case, J. Fluid Mech., 371, 233-268 (1998) · Zbl 0946.76093 |
| [15] | Chen, C.-Y.; Meiburg, E., Miscible porous media displacements in the quarter five-spot configuration. Part 2. Effect of heterogeneities, J. Fluid Mech., 371, 269-299 (1998) · Zbl 0946.76093 |
| [16] | Meiburg, E.; Chen, C.-Y., High-accuracy implicit finite-difference simulations of homogeneous and heterogeneous miscible-porous-medium flows, SPE J., 5, 129-137 (2000) |
| [17] | Riaz, A.; Tchelepi, H. A., Linear stability analysis of immiscible two-phase flow in porous media with capillary dispersion and density variation, Phys. Fluids, 16, 4727-4737 (2004) · Zbl 1187.76445 |
| [18] | Riaz, A.; Tchelepi, H. A., Numerical simulation of immiscible two-phase flow in porous media, Phys. Fluids, 18, Article 014104 pp. (2006) |
| [19] | Tchelepi, H. A.; Orr, F. M.; Rakotomalala, N.; Salin, D.; Woumeni, R., Dispersion, permeability heterogeneity, and viscous fingering: acoustic experimental observations and particle tracking simulations, Phys. Fluids A, Fluid Dyn., 5, 1558-1574 (1993) |
| [20] | Saffman, P. G.; Taylor, G., The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid, (Proceedings of the Royal Society of A: Mathematical, Physical and Engineering Sciences, vol. 245 (1958), The Royal Society), 312-329 · Zbl 0086.41603 |
| [21] | Saffman, P. G., Viscous fingering in Hele-Shaw cells, J. Fluid Mech., 173, 73-94 (1986) · Zbl 0623.76115 |
| [22] | Chang, S.; Slattery, J. C., A linear stability analysis for miscible displacements, Transp. Porous Media, 1, 179-199 (1986) |
| [23] | Raghavan, R.; Marsden, S. S., The stability of immiscible liquid layers in a porous medium, J. Fluid Mech., 48, 143-159 (1971) · Zbl 0213.54302 |
| [24] | Shikaze, S. G.; Sudicky, E.; Schwartz, F., Density-dependent solute transport in discretely-fractured geologic media: is prediction possible?, J. Contam. Hydrol., 34, 273-291 (1998) |
| [25] | Schincariol, R. A.; Schwartz, F. W.; Mendoza, C. A., Instabilities in variable density flows: stability and sensitivity analyses for homogeneous and heterogeneous media, Water Resour. Res., 33, 31-41 (1997) |
| [26] | Zhang, H.; Schwartz, F. W., Multispecies contaminant plumes in variable density flow systems, Water Resour. Res., 31, 837-847 (1995) |
| [27] | Schincariol, R. A.; Schwartz, F. W., An experimental investigation of variable density flow and mixing in homogeneous and heterogeneous media, Water Resour. Res., 26, 2317-2329 (1990) |
| [28] | Cuthiell, D.; Kissel, G.; Jackson, C.; Frauenfeld, T.; Fisher, D.; Rispler, K., Viscous fingering effects in solvent displacement of heavy oil, J. Can. Pet. Technol., 45, Article 07 pp. (2006) |
| [29] | Sharma, J.; Inwood, S. B.; Kovscek, A., Experiments and analysis of multiscale viscous fingering during forced imbibition, SPE J., 17, 1-142 (2012) |
| [30] | Holm, L., Miscible Displacement, Petroleum Engineering Hand Book, 1-45 (1987), Society of Petroleum Engineers: Society of Petroleum Engineers Richardson, TX |
| [31] | Bear, J.; Bachmat, Y., Introduction to Modeling of Transport Phenomena in Porous Media, vol. 4 (2012), Springer Science & Business Media |
| [32] | Nield, D. A.; Bejan, A., Convection in Porous Media (2006), Springer Science & Business Media · Zbl 1256.76004 |
| [33] | Kolditz, O.; Ratke, R.; Diersch, H.-J. G.; Zielke, W., Coupled groundwater flow and transport: 1. Verification of variable density flow and transport models, Adv. Water Resour., 21, 27-46 (1998) |
| [34] | Diersch, H.-J.; Kolditz, O., Variable-density flow and transport in porous media: approaches and challenges, Adv. Water Resour., 25, 899-944 (2002) |
| [35] | Hidalgo, J. J.; Carrera, J., Effect of dispersion on the onset of convection during \(CO_2\) sequestration, J. Fluid Mech., 640, 441-452 (2009) · Zbl 1183.76712 |
| [36] | Pau, G. S.H.; Bell, J. B.; Pruess, K.; Almgren, A. S.; Lijewski, M. J.; Zhang, K.; Yang, C.; Gu, Y., High-resolution simulation and characterization of density-driven flow in \(CO_2\) storage in saline aquifers, Adv. Water Resour., 33, 443-455 (2010) |
| [37] | Peaceman, D. W., Fundamentals of Numerical Reservoir Simulation (1977) · Zbl 0204.28001 |
| [38] | Douglas, J.; Roberts, J. E., Numerical methods for a model for compressible miscible displacement in porous media, Math. Comput., 41, 441-459 (1983) · Zbl 0537.76062 |
| [39] | Choquet, C., Parabolic and degenerate parabolic models for pressure-driven transport problems, Math. Models Methods Appl. Sci., 20, 543-566 (2010) · Zbl 1195.35190 |
| [40] | Amirat, Y.; Hamdache, K.; Ziani, A., Mathematical analysis for compressible miscible displacement models in porous media, Math. Models Methods Appl. Sci., 6, 729-747 (1996) · Zbl 0859.35087 |
| [41] | Huang, H.; Wattenbarger, R.; Gai, X.; Brown, W. P.; Hehmeyer, O. J.; Wang, J.; Long, T. A., Using a Fully Coupled Flow and Geomechanical Simulator to Model Injection Into Heavy Oil Reservoirs, vol. 71 (2013), Wiley Online Library · Zbl 1430.76443 |
| [42] | Farina, A.; Fasano, A.; Mikelić, A., On the equations governing the flow of mechanically incompressible, but thermally expansible, viscous fluids, Math. Models Methods Appl. Sci., 18, 813-857 (2008) · Zbl 1200.35213 |
| [43] | Antontsev, S. N.; Kazhiktov, A.; Monakhov, V. N., Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, vol. 22 (1989), Elsevier |
| [44] | Aganović, I.; Mikelić, A., Homogenization of nonstationary flow of a two-constituant mixture through a porous medium, Asymptot. Anal., 6, 173-189 (1992) · Zbl 0763.76077 |
| [45] | Mikelić, A.; Aganović, I., Homogenization of stationary flow of miscible fluids in a domain with a grained boundary, SIAM J. Math. Anal., 19, 287-294 (1988) · Zbl 0645.76099 |
| [46] | Mikelić, A., Mathematical theory of stationary miscible filtration, J. Differ. Equ., 90, 186-202 (1991) · Zbl 0735.35100 |
| [47] | Mikelić, A., Regularity and uniqueness results for two-phase miscible flows in porous media, (Flow in Porous Media (1993)), 139-154 · Zbl 0805.76081 |
| [48] | Balhoff, M.; Mikelić, A.; Wheeler, M. F., Polynomial filtration laws for low Reynolds number flows through porous media, Transp. Porous Media, 81, 35-60 (2010) |
| [49] | Firdaouss, M.; Guermond, J. L.; Le Quéré, P., Nonlinear corrections to Darcy’s law at low Reynolds numbers, J. Fluid Mech., 343, 331-350 (1997) · Zbl 0897.76091 |
| [50] | Marušić-Paloka, E.; Mikelić, A., The derivation of a nonlinear filtration law including the inertia effects via homogenization, Nonlinear Anal., 42, 97-137 (2000) · Zbl 0965.35131 |
| [51] | Mei, C. C.; Auriault, J. L., The effect of weak inertia on flow through a porous medium, J. Fluid Mech., 222, 647-663 (1991) · Zbl 0718.76099 |
| [52] | E, W.; Engquist, B.; Li, X.; Ren, W.; Vanden-Eijnden, E., Heterogeneous multiscale methods: a review, Commun. Comput. Phys., 2, 367-450 (2007) · Zbl 1164.65496 |
| [53] | Yue, X.; E, W., Numerical methods for multiscale transport equations and application to two-phase porous media flow, J. Comput. Phys., 210, 656-675 (2005) · Zbl 1089.76049 |
| [54] | Li, R. N.; Yue, X. L., Application and validation of an upscaling method for unsaturated water flow processes in heterogeneous soils, Vadose Zone J., 14 (2015) |
| [55] | Henning, P.; Ohlberger, M.; Schweizer, B., Adaptive heterogeneous multiscale methods for immiscible two-phase flow in porous media, Comput. Geosci., 19, 99-114 (2015) · Zbl 1326.76060 |
| [56] | Efendiev, Y.; Hou, T. Y., Multiscale Finite Element Methods: Theory and Applications, vol. 4 (2009), Springer Science & Business Media · Zbl 1163.65080 |
| [57] | Efendiev, Y.; Ginting, V.; Hou, T.; Ewing, R., Accurate multiscale finite element methods for two-phase flow simulations, J. Comput. Phys., 220, 155-174 (2006) · Zbl 1158.76349 |
| [58] | Lunati, I.; Jenny, P., Multiscale finite-volume method for density-driven flow in porous media, Comput. Geosci., 12, 337-350 (2008) · Zbl 1259.76051 |
| [59] | Ginting, V., Computational upscaled modeling of heterogeneous porous media flow utilizing finite volume method (2004), Texas A & M University, Ph.D. thesis |
| [60] | Durlofsky, L. J., Coarse scale models of two phase flow in heterogeneous reservoirs: volume averaged equations and their relationship to existing upscaling techniques, Comput. Geosci., 2, 73-92 (1998) · Zbl 0943.76085 |
| [61] | Efendiev, Y.; Durlofsky, L. J.; Lee, S. H., Modeling of subgrid effects in coarse-scale simulations of transport in heterogeneous porous media, Water Resour. Res., 36, 2031-2041 (2000) |
| [62] | Drazin, P., Introduction to Hydrodynamic Stability, vol. 32 (2002), Cambridge University Press · Zbl 0997.76001 |
| [63] | Drazin, P. G.; Reid, W. H., Hydrodynamic Stability (2004), Cambridge University Press · Zbl 0449.76027 |
| [64] | I_Udovich, V. I., The Linearization Method in Hydrodynamical Stability Theory, vol. 74 (1989), American Mathematical Soc. · Zbl 0727.76039 |
| [65] | Wloka, J., Partial Differential Equations (1987), Cambridge University Press · Zbl 0623.35006 |
| [66] | Coskuner, G.; Bentsen, R. G., An extended theory to predict the onset of viscous instabilities for miscible displacements in porous media, Transp. Porous Media, 5, 473-490 (1990) |
| [67] | Lajeunesse, E.; Martin, J.; Rakotomalala, N.; Salin, D., 3D instability of miscible displacements in a Hele-Shaw cell, Phys. Rev. Lett., 79, 5254 (1997) |
| [68] | Lajeunesse, E.; Martin, J.; Rakotomalala, N.; Salin, D.; Yortsos, Y., Miscible displacement in a Hele-Shaw cell at high rates, J. Fluid Mech., 398, 299-319 (1999) · Zbl 0942.76508 |
| [69] | d’Olce, M.; Martin, J.; Rakotomalala, N.; Salin, D.; Talon, L., Pearl and mushroom instability patterns in two miscible fluids’ core annular flows, Phys. Fluids, 20, Article 024104 pp. (2008) · Zbl 1182.76207 |
| [70] | Riaz, A.; Meiburg, E., Radial source flows in porous media: linear stability analysis of axial and helical perturbations in miscible displacements, Phys. Fluids, 15, 938-946 (2003) · Zbl 1186.76444 |
| [71] | Yortsos, Y., Stability of displacement processes in porous media in radial flow geometries, Phys. Fluids, 30, 2928-2935 (1987) · Zbl 0635.76095 |
| [72] | Christie, M.; Bond, D., Detailed simulation of unstable processes in miscible flooding, SPE Reserv. Eng., 2, 514-522 (1987) |
| [73] | DeGregoria, A., A predictive Monte Carlo simulation of two-fluid flow through porous media at finite mobility ratio, Phys. Fluids, 28, 2933-2935 (1985) · Zbl 0573.76093 |
| [74] | De Wit, A.; Homsy, G., Viscous fingering in periodically heterogeneous porous media. I. Formulation and linear instability, J. Chem. Phys., 107, 9609-9618 (1997) |
| [75] | De Wit, A.; Homsy, G., Viscous fingering in periodically heterogeneous porous media. II. Numerical simulations, J. Chem. Phys., 107, 9619-9628 (1997) |
| [76] | Yortsos, Y. C.; Zeybek, M., Dispersion driven instability in miscible displacement in porous media, Phys. Fluids, 31, 3511-3518 (1988) |
| [77] | Hickernell, F.; Yortsos, Y., Linear stability of miscible displacement processes in porous media in the absence of dispersion, Stud. Appl. Math., 74, 93-115 (1986) · Zbl 0603.76091 |
| [78] | Balasubramaniam, R.; Rashidnia, N.; Maxworthy, T.; Kuang, J., Instability of miscible interfaces in a cylindrical tube, Phys. Fluids, 17, Article 052103 pp. (2005) · Zbl 1187.76035 |
| [79] | Scoffoni, J.; Lajeunesse, E.; Homsy, G., Interface instabilities during displacements of two miscible fluids in a vertical pipe, Phys. Fluids, 13, 553-556 (2001) · Zbl 1184.76487 |
| [80] | Taylor, G. I., Dispersion of soluble matter in solvent flowing slowly through a tube, (Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 219 (1953), The Royal Society), 186-203 |
| [81] | Aris, R., On the dispersion of a solute in a fluid flowing through a tube, (Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 235 (1956), The Royal Society), 67-77 |
| [82] | Caflisch, R.; Rubinstein, J., Lectures on the Mathematical Theory of Multiphase-Flow (1984), Courant Institute of Mathematical Sciences: Courant Institute of Mathematical Sciences New York |
| [83] | Mercer, G.; Roberts, A., A centre manifold description of contaminant dispersion in channels with varying flow properties, SIAM J. Appl. Math., 50, 1547-1565 (1990) · Zbl 0711.76086 |
| [84] | Rosencrans, S., Taylor dispersion in curved channels, SIAM J. Appl. Math., 57, 1216-1241 (1997) · Zbl 1023.35018 |
| [85] | Mikelić, A.; Devigne, V.; Van Duijn, C. J., Rigorous upscaling of the reactive flow through a pore, under dominant Péclet and Damkohler numbers, SIAM J. Math. Anal., 38, 1262-1287 (2006) · Zbl 1120.35007 |
| [86] | Mikelić, A.; Rosier, C., Rigorous upscaling of the infinite adsorption rate reactive flow under dominant Péclet number through a pore, Ann. Univ. Ferrara, 53, 333-359 (2007) · Zbl 1181.35013 |
| [87] | Van Duijn, C. J.; Mikelić, A.; Pop, I. S.; Rosier, C., Effective dispersion equations for reactive flows with dominant Péclet and Damkohler numbers, Adv. Chem. Eng., 34, 1-45 (2008) |
| [88] | Choquet, C.; Mikelić, A., Laplace transform approach to the rigorous upscaling of the infinite adsorption rate reactive flow under dominant Péclet number through a pore, Appl. Anal., 87, 1373-1395 (2008) · Zbl 1154.35304 |
| [89] | Choquet, C.; Mikelić, A., Rigorous upscaling of the reactive flow with finite kinetics and under dominant Péclet number, Contin. Mech. Thermodyn., 21, 125-140 (2009) · Zbl 1170.76057 |
| [90] | Rubinstein, J.; Mauri, R., Dispersion and convection in periodic porous media, SIAM J. Appl. Math., 46, 1018-1023 (1986) · Zbl 0614.76098 |
| [91] | Taylor, G. I., The dispersion of matter in turbulent flow through a pipe, (Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 223 (1954), The Royal Society), 446-468 |
| [92] | Karniadakis, G. E.; Beskok, A.; Aluru, N., Microflows and Nanoflows: Fundamentals and Simulation, vol. 29 (2006), Springer Science & Business Media |
| [93] | Scheidegger, A., Typical solutions of the differential equations of statistical theories of flow through porous media, Eos, 39, 929-932 (1958) |
| [94] | Camacho, J., Thermodynamics of Taylor dispersion: constitutive equations, Phys. Rev. E, 47, 1049 (1993) |
| [95] | Camacho, J., Purely global model for Taylor dispersion, Phys. Rev. E, 48, 310 (1993) |
| [96] | Camacho, J., Thermodynamic functions for Taylor dispersion, Phys. Rev. E, 48, 1844 (1993) |
| [97] | Berentsen, C.; Verlaan, M.; Van Kruijsdijk, C., Upscaling and reversibility of Taylor dispersion in heterogeneous porous media, Phys. Rev. E, 71, Article 046308 pp. (2005) |
| [98] | Chang, H.-C.; Balakotaiah, V., Hyperbolic homogenized models for thermal and solutal dispersion, SIAM J. Appl. Math., 63, 1231-1258 (2003) · Zbl 1037.76050 |
| [99] | Chakraborty, S.; Balakotaiah, V., Spatially averaged multi-scale models for chemical reactors, Adv. Chem. Eng., 30, 205-297 (2005) |
| [100] | Mikelić, A.; Van Duijn, C. J., Rigorous derivation of a hyperbolic model for Taylor dispersion, Math. Models Methods Appl. Sci., 21, 1095-1120 (2011) · Zbl 1223.35036 |
| [101] | Carbonell, R.; Whitaker, S., Dispersion in pulsed systems—II: theoretical developments for passive dispersion in porous media, Chem. Eng. Sci., 38, 1795-1802 (1983) |
| [102] | Paine, M.; Carbonell, R.; Whitaker, S., Dispersion in pulsed systems—I: heterogeneous reaction and reversible adsorption in capillary tubes, Chem. Eng. Sci., 38, 1781-1793 (1983) |
| [103] | Quintard, M.; Whitaker, S., Transport in ordered and disordered porous media: volume-averaged equations, closure problems, and comparison with experiment, Chem. Eng. Sci., 48, 2537-2564 (1993) |
| [104] | Mei, C. C., Method of homogenization applied to dispersion in porous media, Transp. Porous Media, 9, 261-274 (1992) |
| [105] | Auriault, J.-L.; Adler, P., Taylor dispersion in porous media: analysis by multiple scale expansions, Adv. Water Resour., 18, 217-226 (1995) |
| [106] | Salles, J.; Thovert, J.-F.; Delannay, R.; Prevors, L.; Auriault, J.-L.; Adler, P., Taylor dispersion in porous media. Determination of the dispersion tensor, Phys. Fluids A, Fluid Dyn., 5, 2348-2376 (1993) · Zbl 0796.76089 |
| [107] | Edwards, D.; Shapiro, M.; Brenner, H.; Shapira, M., Dispersion of inert solutes in spatially periodic, two-dimensional model porous media, Transp. Porous Media, 6, 337-358 (1991) |
| [108] | Perrins, W.; McKenzie, D.; McPhedran, R., Transport properties of regular arrays of cylinders, (Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 369 (1979), The Royal Society), 207-225 |
| [109] | Koch, D. L.; Cox, R. G.; Brenner, H.; Brady, J. F., The effect of order on dispersion in porous media, J. Fluid Mech., 200, 173-188 (1989) · Zbl 0659.76102 |
| [110] | Allaire, G.; Raphael, A.-L., Homogenization of a convection-diffusion model with reaction in a porous medium, C. R. Math., 344, 523-528 (2007) · Zbl 1114.35007 |
| [111] | Allaire, G.; Mikelić, A.; Piatnitski, A., Homogenization approach to the dispersion theory for reactive transport through porous media, SIAM J. Math. Anal., 42, 125-144 (2010) · Zbl 1213.35057 |
| [112] | Allaire, G.; Brizzi, R.; Mikelić, A.; Piatnitski, A., Two-scale expansion with drift approach to the Taylor dispersion for reactive transport through porous media, Chem. Eng. Sci., 65, 2292-2300 (2010) |
| [113] | Papanicolaou, G. C., Diffusion in random media, (Surveys in Applied Mathematics (1995), Springer), 205-253 · Zbl 0846.60081 |
| [114] | Van Duijn, C. J.; Knabner, P., Travelling waves in the transport of reactive solutes through porous media: adsorption and binary ion exchange - Part 1, Transp. Porous Media, 8, 167-194 (1992) |
| [115] | Knabner, P.; van Duijn, C. J.; Hengst, S., An analysis of crystal dissolution fronts in flows through porous media. Part 1: compatible boundary conditions, Adv. Water Resour., 18, 171-185 (1995) |
| [116] | Marušić-Paloka, E.; Piatnitski, A. L., Homogenization of a nonlinear convection-diffusion equation with rapidly oscillating coefficients and strong convection, J. Lond. Math. Soc., 72, 391-409 (2005) · Zbl 1106.35006 |
| [117] | Amaziane, B.; Bourgeat, A.; Jurak, M., Effective macrodiffusion in solute transport through heterogeneous porous media, Multiscale Model. Simul., 5, 184-204 (2006) · Zbl 1187.76746 |
| [118] | Bourgeat, A.; Jurak, M.; Piatnitski, A., Averaging a transport equation with small diffusion and oscillating velocity, Math. Methods Appl. Sci., 26, 95-117 (2003) · Zbl 1013.76069 |
| [119] | Gelhar, L. W.; Welty, C.; Rehfeldt, K. R., A critical review of data on field-scale dispersion in aquifers, Water Resour. Res., 28, 1955-1974 (1992) |
| [120] | Goode, D. J.; Konikow, L. F., Apparent dispersion in transient groundwater flow, Water Resour. Res., 26, 2339-2351 (1990) |
| [121] | Horne, R. N.; Rodriguez, F., Dispersion in tracer flow in fractured geothermal systems, Geophys. Res. Lett., 10, 289-292 (1983) |
| [122] | Petitjeans, P.; Chen, C.-Y.; Meiburg, E.; Maxworthy, T., Miscible quarter five-spot displacements in a Hele-Shaw cell and the role of flow-induced dispersion, Phys. Fluids, 11, 1705-1716 (1999) · Zbl 1147.76474 |
| [123] | Fernandez, J.; Kurowski, P.; Petitjeans, P.; Meiburg, E., Density-driven unstable flows of miscible fluids in a Hele-Shaw cell, J. Fluid Mech., 451, 239-260 (2002) · Zbl 1156.76311 |
| [124] | Graf, F.; Meiburg, E.; Haertel, C., Density-driven instabilities of miscible fluids in a Hele-Shaw cell: linear stability analysis of the three-dimensional Stokes equations, J. Fluid Mech., 451, 261-282 (2002) · Zbl 1050.76024 |
| [125] | Goyal, N.; Meiburg, E., Unstable density stratification of miscible fluids in a vertical Hele-Shaw cell: influence of variable viscosity on the linear stability, J. Fluid Mech., 516, 211-238 (2004) · Zbl 1131.76321 |
| [126] | Vanaparthy, S. H.; Meiburg, E.; Wilhelm, D., Density-driven instabilities of miscible fluids in a capillary tube: linear stability analysis, J. Fluid Mech., 497, 99-121 (2003) · Zbl 1074.76025 |
| [127] | Kaviany, M., Principles of Heat Transfer in Porous Media (2012), Springer Science & Business Media |
| [128] | Ene, H. I.; Poliševski, D., Thermal Flows in Porous Media, vol. 1 (2012), Springer Science & Business Media · Zbl 1017.35016 |
| [129] | Ene, H. I.; Poliševski, D., Steady convection in a porous layer with translational flow, Acta Mech., 84, 13-18 (1990) |
| [130] | Ene, H. I., Effects of anisotropy on the free convection from a vertical plate embedded in a porous medium, Transp. Porous Media, 6, 183-194 (1991) |
| [131] | Ene, H. I.; Sanchez-Palencia, E., On thermal equation for flow in porous media, Int. J. Eng. Sci., 20, 623-630 (1982) · Zbl 0491.76098 |
| [132] | Fayers, F.; Jouaux, F.; Tchelepi, H., An improved macroscopic model for viscous fingering and its validation for 2D and 3D flows. 1. Non-gravity flows, In Situ, 18, 43-78 (1994) |
| [133] | Ennis-King, J. P.; Paterson, L., Role of convective mixing in the long-term storage of carbon dioxide in deep saline formations, SPE J., 10, 349-356 (2005) |
| [134] | Ennis-King, J.; Preston, I.; Paterson, L., Onset of convection in anisotropic porous media subject to a rapid change in boundary conditions, Phys. Fluids, 17, Article 084107 pp. (2005) · Zbl 1187.76141 |
| [135] | Hesse, M. A., Mathematical modeling and multi scale simulation of \(CO_2\) storage in saline aquifers (2008), Stanford University, Ph.D. thesis |
| [136] | Lindeberg, E.; Wessel-Berg, D., Vertical convection in an aquifer column under a gas cap of \(CO_2\), Energy Convers. Manag., 38, S229-S234 (1997) |
| [137] | Xu, X.; Chen, S.; Zhang, D., Convective stability analysis of the long-term storage of carbon dioxide in deep saline aquifers, Adv. Water Resour., 29, 397-407 (2006) |
| [138] | Riaz, A.; Hesse, M.; Tchelepi, H. A.; Orr, F. M., Onset of convection in a gravitationally unstable diffusive boundary layer in porous media, J. Fluid Mech., 548, 87-111 (2006) |
| [139] | Gounot, J.; Caltagirone, J. P., Stabilité et convection naturelle au sein d’une couche poreuse non homogène, Int. J. Heat Mass Transf., 32, 1131-1140 (1989) · Zbl 0673.76097 |
| [140] | Hidalgo, J. J.; Carrera, J.; Medina, A., Role of salt sources in density-dependent flow, Water Resour. Res., 45, Article W05503 pp. (2009) |
| [141] | Pau, G. S.; Bell, J. B.; Pruess, K.; Almgren, A. S.; Lijewski, M. J.; Zhang, K., High-resolution simulation and characterization of density-driven flow in \(CO_2\) storage in saline aquifers, Adv. Water Resour., 33, 443-455 (2010) |
| [142] | Pau, G. S.H.; Almgren, A. S.; Bell, J. B.; Lijewski, M. J., A parallel second-order adaptive mesh algorithm for incompressible flow in porous media, Philos. Trans. R. Soc. Lond. A, 367, 4633-4654 (2009) · Zbl 1192.76056 |
| [143] | Koval, E. J., A method for predicting the performance of unstable miscible displacement in heterogeneous media, Soc. Pet. Eng. J., 3, 145-154 (1963) |
| [144] | Todd, M.; O’dell, P.; Hirasaki, G., Methods for increased accuracy in numerical reservoir simulators, Soc. Pet. Eng. J., 12, 515-530 (1972) |
| [145] | Wooding, R. A., Growth of fingers at an unstable diffusing interface in a porous medium or Hele-Shaw cell, J. Fluid Mech., 39, Article 03 pp. (1969) |
| [146] | Menon, G.; Otto, F., Dynamic scaling in miscible viscous fingering, Commun. Math. Phys., 257, 303-317 (2005) · Zbl 1135.76323 |
| [147] | Manickam, O.; Homsy, G., Fingering instabilities in vertical miscible displacement flows in porous media, J. Fluid Mech., 288, 75-102 (1995) · Zbl 0853.76024 |
| [148] | DiCarlo, D. A., Experimental measurements of saturation overshoot on infiltration, Water Resour. Res., 40 (2004) |
| [149] | Van Duijn, C. J.; Peletier, L. A.; Pop, I. S., A new class of entropy solutions of the Buckley-Leverett equation, SIAM J. Math. Anal., 39, 507-536 (2007) · Zbl 1149.35054 |
| [150] | Menon, G.; Otto, F., Fast communication: diffusive slowdown in miscible viscous fingering, Commun. Math. Sci., 4, 267-273 (2006) · Zbl 1184.35260 |
| [151] | Yortsos, Y. C.; Salin, D., On the selection principle for viscous fingering in porous media, J. Fluid Mech., 557, 225-236 (2006) · Zbl 1094.76021 |
| [152] | Yortsos, Y., A theoretical analysis of vertical flow equilibrium, Transp. Porous Media, 18, 107-129 (1995) |
| [153] | Russell, T. F.; Wheeler, M. F., Finite element and finite difference methods for continuous flows in porous media, (The Mathematics of Reservoir Simulation, vol. 1 (1983)), 35-106 · Zbl 0572.76089 |
| [154] | Ewing, R. E.; Russell, T. F., Efficient time-stepping methods for miscible displacement problems in porous media, SIAM J. Numer. Anal., 19, 1-67 (1982) · Zbl 0498.76084 |
| [155] | Ewing, R. E.; Wheeler, M. F., Galerkin methods for miscible displacement problems in porous media, SIAM J. Numer. Anal., 17, 351-365 (1980) · Zbl 0458.76092 |
| [156] | Bell, J. B.; Shubin, G. R.; Wheeler, M. F., Analysis of a new method for computing the flow of miscible fluids in a porous medium, SIAM J. Numer. Anal., 22, 1041-1050 (1985) · Zbl 0586.76168 |
| [157] | Huang, H.; Scovazzi, G., A high-order, fully coupled, upwind, compact discontinuous Galerkin method for modeling of viscous fingering in compressible porous media, Comput. Methods Appl. Mech. Eng., 263, 169-187 (2013) · Zbl 1286.76145 |
| [158] | Scovazzi, G.; Huang, H.; Collis, S.; Yin, J., A fully-coupled upwind discontinuous Galerkin method for incompressible porous media flows: high-order computations of viscous fingering instabilities in complex geometry, J. Comput. Phys., 252, 86-108 (2013) · Zbl 1349.76264 |
| [159] | Shubin, G.; Bell, J., An analysis of the grid orientation effect in numerical simulation of miscible displacement, Comput. Methods Appl. Mech. Eng., 47, 47-71 (1984) · Zbl 0535.76117 |
| [160] | Potempa, T. C., Finite element methods for convection dominated transport problems (1982), Rice University, Ph.D. thesis |
| [161] | Hughes, T. J.; Engel, G.; Mazzei, L.; Larson, M. G., The continuous Galerkin method is locally conservative, J. Comput. Phys., 163, 467-488 (2000) · Zbl 0969.65104 |
| [162] | Dawson, C.; Sun, S.; Wheeler, M. F., Compatible algorithms for coupled flow and transport, Comput. Methods Appl. Mech. Eng., 193, 2565-2580 (2004) · Zbl 1067.76565 |
| [164] | Hou, T. Y.; Wu, X.-H., A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 134, 169-189 (1997) · Zbl 0880.73065 |
| [165] | Ayuso de Dios, B.; Holst, M.; Zhu, Y.; Zikatanov, L., Multilevel preconditioners for discontinuous, Galerkin approximations of elliptic problems, with jump coefficients, Math. Comput., 83, 1083-1120 (2014) · Zbl 1296.65153 |
| [166] | Lee, S.; Lee, Y.-J.; Wheeler, M. F., A locally conservative enriched Galerkin approximation and efficient solver for elliptic and parabolic problems, SIAM J. Sci. Comput., 38, A1404-A1429 (2016) · Zbl 1337.65128 |
| [167] | Wang, Y.; Hajibeygi, H.; Tchelepi, H. A., Algebraic multiscale solver for flow in heterogeneous porous media, J. Comput. Phys., 259, 284-303 (2014) · Zbl 1349.76835 |
| [168] | Tene, M.; Wang, Y.; Hajibeygi, H., Adaptive algebraic multiscale solver for compressible flow in heterogeneous porous media, J. Comput. Phys., 300, 679-694 (2015) · Zbl 1349.76272 |
| [169] | Tene, M.; Al Kobaisi, M.; Hajibeygi, H., Algebraic multiscale solver for flow in heterogeneous fractured porous media, (SPE Reservoir Simulation Symposium (2015), Society of Petroleum Engineers) |
| [170] | Natvig, J. R.; Lie, K.-A., Fast computation of multiphase flow in porous media by implicit discontinuous Galerkin schemes with optimal ordering of elements, J. Comput. Phys., 227, 10108-10124 (2008) · Zbl 1218.76029 |
| [171] | Riviere, B.; Wheeler, M., A posteriori error estimates and mesh adaptation strategy for discontinuous Galerkin methods applied to diffusion problems, Comput. Math. Appl., 46, 141-163 (2003) · Zbl 1059.65098 |
| [172] | Riviere, B., Discontinuous Galerkin finite element methods for solving the miscible displacement problem in porous media (2000), The University of Texas at Austin, Ph.D. thesis |
| [173] | Sun, S.; Wheeler, M. F., A posteriori error estimation and dynamic adaptivity for symmetric discontinuous Galerkin approximations of reactive transport problems, Comput. Methods Appl. Mech. Eng., 195, 632-652 (2006) · Zbl 1091.76040 |
| [174] | Lee, S.; Wheeler, M. F., Adaptive enriched Galerkin methods for miscible displacement problems with entropy residual stabilization, J. Comput. Phys., 331, 19-37 (2017) · Zbl 1378.76048 |
| [175] | Aziz, K.; Settari, A., Petroleum Reservoir Simulation (1979), Chapman & Hall |
| [176] | Weiser, A.; Wheeler, M. F., On convergence of block-centered finite differences for elliptic problems, SIAM J. Numer. Anal., 25, 351-375 (1988) · Zbl 0644.65062 |
| [177] | Samarskii, A. A., Local one dimensional difference schemes on non-uniform nets, USSR Comput. Math. Math. Phys., 3, 572-619 (1963) · Zbl 0273.65079 |
| [178] | Gerschgorin, S., Fehlerabschätzung für das differenzenverfahren zur lösung partieller differentialgleichungen, Z. Angew. Math. Mech., 10, 373-382 (1930) · JFM 56.0467.03 |
| [179] | Bramble, J.; Hubbard, B.; Thomee, V., Convergence estimates for essentially positive type discrete Dirichlet problems, Math. Comput., 23, 695-709 (1969) · Zbl 0217.21902 |
| [180] | Nitsche, J.; Nitsche, J. C., Error estimates for the numerical solution of elliptic differential equations, Arch. Ration. Mech. Anal., 5, 293-306 (1960) · Zbl 0097.33103 |
| [181] | Kellogg, R., An error estimate for elliptic difference equations on a convex polygon, SIAM J. Numer. Anal., 3, 79-90 (1966) · Zbl 0143.17602 |
| [182] | Bramble, J. H., On the convergence of difference approximations for second order uniformly elliptic operators, (Numerical Solution of Field Problems in Continuum Physics, vol. 2 (1970)), 201 · Zbl 0234.65086 |
| [183] | Kreiss, H.-O.; Manteuffel, T. A.; Swartz, B.; Wendroff, B.; White, A., Supra-convergent schemes on irregular grids, Math. Comput., 47, 537-554 (1986) · Zbl 0619.65055 |
| [184] | Forsyth, P.; Sammon, P., Quadratic convergence for cell-centered grids, Appl. Numer. Math., 4, 377-394 (1988) · Zbl 0651.65086 |
| [185] | Wheeler, M. F.; Xue, G.; Yotov, I., A multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra, Numer. Math., 121, 165-204 (2012) · Zbl 1277.65100 |
| [186] | Aavatsmark, I., Comparison of monotonicity for some multipoint flux approximation methods, (Finite Volumes for Complex Applications, vol. 5 (2008)), 19-34 · Zbl 1422.65301 |
| [187] | Heinemann, Z.; Brand, C.; Munka, M.; Chen, Y., Modeling reservoir geometry with irregular grids, (SPE Symposium on Reservoir Simulation (1989), Society of Petroleum Engineers) |
| [188] | LeVeque, R. J., Finite Volume Methods for Hyperbolic Problems, vol. 31 (2002), Cambridge University Press · Zbl 1010.65040 |
| [189] | Cai, Z.; Douglas, J.; Park, M., Development and analysis of higher order finite volume methods over rectangles for elliptic equations, Adv. Comput. Math., 19, 3-33 (2003) · Zbl 1020.65087 |
| [190] | Chen, L., A new class of high order finite volume methods for second order elliptic equations, SIAM J. Numer. Anal., 47, 4021-4043 (2010) · Zbl 1261.65109 |
| [191] | Chen, Z.; Wu, J.; Xu, Y., Higher-order finite volume methods for elliptic boundary value problems, Adv. Comput. Math., 37, 191-253 (2012) · Zbl 1266.65180 |
| [192] | Eymard, R.; Gallouët, T.; Herbin, R., Finite volume methods, Handb. Numer. Anal., 7, 713-1018 (2000) · Zbl 0981.65095 |
| [193] | Babuška, I., The finite element method with Lagrangian multipliers, Numer. Math., 20, 179-192 (1973) · Zbl 0258.65108 |
| [194] | Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Fr. Autom. Inform. Rech. Opér., Anal. Numér., 8, 129-151 (1974) · Zbl 0338.90047 |
| [195] | Raviart, P.-A.; Thomas, J.-M., A mixed finite element method for 2-nd order elliptic problems, (Mathematical Aspects of Finite Element Methods (1977), Springer), 292-315 · Zbl 0362.65089 |
| [196] | Nédélec, J.-C., Mixed finite elements in \(R^3\), Numer. Math., 35, 315-341 (1980) · Zbl 0419.65069 |
| [197] | Brezzi, F.; Douglas, J.; Marini, L. D., Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47, 217-235 (1985) · Zbl 0599.65072 |
| [198] | Brezzi, F.; Douglas, J.; Fortin, M.; Marini, L. D., Efficient rectangular mixed finite elements in two and three space variables, Modél. Math. Anal. Numér., 21, 581-604 (1987) · Zbl 0689.65065 |
| [199] | Brezzi, F.; Douglas, J.; Durán, R.; Fortin, M., Mixed finite elements for second order elliptic problems in three variables, Numer. Math., 51, 237-250 (1987) · Zbl 0631.65107 |
| [200] | Chen, Z.; Douglas, J., Prismatic mixed finite elements for second order elliptic problems, Calcolo, 26, 135-148 (1989) · Zbl 0711.65089 |
| [201] | Warren, J.; Price, H.; Skiba, F.; Varga, R., Miscible displacement: the liquid-liquid case, (Fall Meeting of the Society of Petroleum Engineers of AIME (1961), Society of Petroleum Engineers) |
| [202] | Price, H.; Cavendish, J.; Varga, R., Numerical methods of higher-order accuracy for diffusion-convection equations, Soc. Pet. Eng. J., 8, 293-303 (1968) |
| [203] | Settari, A.; Price, H.; Dupont, T., Development and application of variational methods for simulation of miscible displacement in porous media, Soc. Pet. Eng. J., 17, 228-246 (1977) |
| [204] | Young, L. C., A finite-element method for reservoir simulation, Soc. Pet. Eng. J., 21, 115-128 (1981) |
| [205] | Ewing, R. E.; Wheeler, M. F., Galerkin methods for miscible displacement problems with point sources and sinks-unit mobility ratio case, (Gross, K. I., Mathematical Methods in Energy Research (1984), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia), 40-58 · Zbl 0551.76079 |
| [206] | Douglas, J.; Wheeler, M. F.; Darlow, B. L.; Kendall, R. P., Special issue on oil reservoir simulation self-adaptive finite element simulation of miscible displacement in porous media, Comput. Methods Appl. Mech. Eng., 47, 131-159 (1984) · Zbl 0545.76128 |
| [207] | Sammon, P. H., Numerical approximations for a miscible displacement process in porous media, SIAM J. Numer. Anal., 23, 508-542 (1986) · Zbl 0608.76084 |
| [208] | Masud, A.; Hughes, T. J., A stabilized mixed finite element method for Darcy flow, Comput. Methods Appl. Mech. Eng., 191, 4341-4370 (2002) · Zbl 1015.76047 |
| [209] | Douglas, J.; Dupont, T., Interior penalty procedures for elliptic and parabolic Galerkin methods, (Computing Methods in Applied Sciences (1976), Springer), 207-216 |
| [210] | Wheeler, M. F.; Darlow, B. L., Interior penalty Galerkin procedures for miscible displacement problems in porous media, (Computational Methods in Nonlinear Mechanics (1980)), 485-506 · Zbl 0444.76081 |
| [211] | Douglas, J.; Wheeler, M. F.; Darlow, B. L.; Kendall, R. P., Self-adaptive finite element simulation of miscible displacement in porous media, Comput. Methods Appl. Mech. Eng., 47, 131-159 (1984) · Zbl 0545.76128 |
| [212] | Arnold, D. N.; Brezzi, F.; Cockburn, B.; Marini, L. D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39, 1749-1779 (2002) · Zbl 1008.65080 |
| [213] | Riviere, B.; Wheeler, M. F.; Girault, V., A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems, SIAM J. Numer. Anal., 39, 902-931 (2001) · Zbl 1010.65045 |
| [214] | Sun, S.; Wheeler, M. F., Discontinuous Galerkin methods for coupled flow and reactive transport problems, Appl. Numer. Math., 52, 273-298 (2005) · Zbl 1079.76584 |
| [215] | Sun, S.; Wheeler, M. F., Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media, SIAM J. Numer. Anal., 43, 195-219 (2005) · Zbl 1086.76043 |
| [216] | Brezzi, F.; Manzini, G.; Marini, D.; Pietra, P.; Russo, A., Discontinuous finite elements for diffusion problems, (Atti Convegno in Onore di F. Brioschi. Atti Convegno in Onore di F. Brioschi, Milan, 1997, vol. 16 (1999)), 197-217 |
| [217] | Hughes, T. J.R.; Masud, A.; Wan, J., A stabilized mixed discontinuous Galerkin method for Darcy flow, Comput. Methods Appl. Mech. Eng., 195, 3347-3381 (2006) · Zbl 1120.76040 |
| [218] | Brezzi, F.; Hughes, T. J.R.; Marini, L. D.; Masud, A., Mixed discontinuous Galerkin methods for Darcy flow, J. Sci. Comput., 22, 119-145 (2005) · Zbl 1103.76031 |
| [219] | Brezzi, F.; Hughes, T. J.; Marini, L. D.; Masud, A., Mixed discontinuous Galerkin methods for Darcy flow, J. Sci. Comput., 22, 119-145 (2005) · Zbl 1103.76031 |
| [220] | Hughes, T. J.R.; Scovazzi, G.; Franca, L. P., Multiscale and stabilized methods, (Stein, E.; de Borst, R.; Hughes, T. J.R., Encyclopedia of Computational Mechanics (2004), John Wiley & Sons) · Zbl 1190.76001 |
| [221] | Brezzi, F.; Cockburn, B.; Marini, L. D.; Süli, E., Stabilization mechanisms in discontinuous Galerkin finite element methods, Comput. Methods Appl. Mech. Eng., 195, 3293-3310 (2006) · Zbl 1125.65102 |
| [222] | Peraire, J.; Persson, P.-O., The compact discontinuous Galerkin (CDG) method for elliptic problems, SIAM J. Sci. Comput., 30, 1806-1824 (2008) · Zbl 1167.65436 |
| [223] | Cockburn, B.; Shu, C.-W., The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35, 2440-2463 (1998) · Zbl 0927.65118 |
| [224] | Sun, S.; Liu, J., A locally conservative finite element method based on piecewise constant enrichment of the continuous Galerkin method, SIAM J. Sci. Comput., 31, 2528-2548 (2009) · Zbl 1198.65197 |
| [225] | LeVeque, R. J., Numerical Methods for Conservation Laws, vol. 132 (1992), Springer · Zbl 0847.65053 |
| [226] | Godunov, S. K., A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Mat. Sb., 89, 271-306 (1959) · Zbl 0171.46204 |
| [227] | Van Leer, B., Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method, J. Comput. Phys., 32, 101-136 (1979) · Zbl 1364.65223 |
| [228] | Bell, J. B.; Dawson, C. N.; Shubin, G. R., An unsplit higher order Godunov method for scalar conservation laws in multiple dimensions, J. Comput. Phys., 74, 1, 1-24 (1988) · Zbl 0684.65088 |
| [229] | Douglas, J., Simulation of miscible displacement in porous media by a modified method of characteristic procedure, (Numerical Analysis (1982), Springer: Springer Berlin/Heidelberg), 64-70 · Zbl 0476.76100 |
| [230] | Russell, T. F., Finite elements with characteristics for two-component incompressible miscible displacement, (SPE Reservoir Simulation Symposium (1982), Society of Petroleum Engineers) |
| [231] | Ewing, R.; Russell, T.; Wheeler, M. F., Simulation of miscible displacement using mixed methods and a modified method of characteristics, (SPE Reservoir Simulation Symposium (1983), Society of Petroleum Engineers) |
| [232] | Russell, T. F.; Wheeler, M. F.; Chiang, C., Large-scale simulation of miscible displacement by mixed and characteristic finite element methods, (Mathematical and Computational Methods in Seismic Exploration and Reservoir Modeling (1986)), 85-107 |
| [233] | Moissis, D., Simulation of viscous fingering during miscible displacement in nonuniform porous media (1988), Rice University, Ph.D. thesis |
| [234] | Moissis, D.; Wheeler, M.; Miller, C., Simulation of miscible viscous fingering using a modified method of characteristics: effects of gravity and heterogeneity, SPE Adv. Technol. Ser., 1, 62-70 (1993) |
| [235] | Douglas, J.; Russell, T. F., Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal., 19, 871-885 (1982) · Zbl 0492.65051 |
| [236] | Ewing, R. E.; Russell, T. F.; Wheeler, M. F., Special issue on oil reservoir simulation convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics, Comput. Methods Appl. Mech. Eng., 47, 73-92 (1984) · Zbl 0545.76131 |
| [237] | Dawson, C. N.; Russell, T. F.; Wheeler, M. F., Some improved error estimates for the modified method of characteristics, SIAM J. Numer. Anal., 26, 1487-1512 (1989) · Zbl 0693.65062 |
| [238] | Arbogast, T.; Wheeler, M. F., A characteristics-mixed finite element method for advection-dominated transport problems, SIAM J. Numer. Anal., 32, 404-424 (1995) · Zbl 0823.76044 |
| [239] | Arbogast, T.; Huang, C.-S., A fully mass and volume conserving implementation of a characteristic method for transport problems, SIAM J. Sci. Comput., 28, 2001-2022 (2006) · Zbl 1293.76097 |
| [240] | Hughes, T. J., Finite element methods for convection dominated flows, (Proceedings of the Annual Meeting of the American Society of Mechanical Engineers (1979), ASME) · Zbl 0418.00017 |
| [241] | Loula, A. F.; Garcia, E. L.; Coutinho, A. L., Miscible displacement simulation by finite element methods in distributed memory machines, Comput. Methods Appl. Mech. Eng., 174, 339-354 (1999) · Zbl 0961.76044 |
| [242] | Coutinho, A. L.; Alves, J. L., Parallel finite element simulation of miscible displacements in porous media, SPE J., 1, 487-500 (1996) |
| [243] | Coutinho, A. L.; Alves, J. L., Finite element simulation of nonlinear viscous fingering in miscible displacements with anisotropic dispersion and nonmonotonic viscosity profiles, Comput. Mech., 23, 108-116 (1999) · Zbl 0962.76554 |
| [244] | Coutinho, A.; Dias, C.; Alves, J.; Landau, L.; Loula, A.; Malta, S.; Castro, R.; Garcia, E., Stabilized methods and post-processing techniques for miscible displacements, Comput. Methods Appl. Mech. Eng., 193, 1421-1436 (2004) · Zbl 1079.76580 |
| [245] | Brooks, A. N.; Hughes, T. J., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 32, 199-259 (1982) · Zbl 0497.76041 |
| [246] | Lasaint, P.; Raviart, P.-A., On a finite element method for solving the neutron transport equation, (Mathematical Aspects of Finite Elements in Partial Differential Equations. Mathematical Aspects of Finite Elements in Partial Differential Equations, Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974 (1974), Academic Press: Academic Press New York), 89-123, Publication No. 33 · Zbl 0341.65076 |
| [247] | Brezzi, F.; Marini, L. D.; Süli, E., Discontinuous Galerkin methods for first-order hyperbolic problems, Math. Models Methods Appl. Sci., 14, 1893-1903 (2004) · Zbl 1070.65117 |
| [248] | Riviere, B.; Wheeler, M. F., Discontinuous Galerkin methods for flow and transport problems in porous media, Commun. Numer. Methods Eng., 18, 63-68 (2002) · Zbl 0996.76056 |
| [249] | Scovazzi, G.; Gerstenberger, A.; Collis, S. S., A discontinuous Galerkin method for gravity-driven viscous fingering instabilities in porous media, J. Comput. Phys. (2012) · Zbl 1286.76152 |
| [250] | Gerstenberger, A.; Scovazzi, G.; Collis, S. S., Computing gravity-driven viscous fingering flows in complex subsurface geometries: a high-order discontinuous Galerkin approach, Comput. Geosci. (2012) · Zbl 1382.86003 |
| [251] | Guermond, J.-L.; Pasquetti, R.; Popov, B., Entropy viscosity method for nonlinear conservation laws, J. Comput. Phys., 230, 4248-4267 (2011) · Zbl 1220.65134 |
| [252] | Bonito, A.; Guermond, J.-L.; Popov, B., Stability analysis of explicit entropy viscosity methods for non-linear scalar conservation equations, Math. Comput., 83, 1039-1062 (2014) · Zbl 1291.65277 |
| [253] | Delshad, M.; Kong, X.; Tavakoli, R.; Hosseini, S. A.; Wheeler, M. F., Modeling and simulation of carbon sequestration at Cranfield incorporating new physical models, Int. J. Greenh. Gas Control, 18, 463-473 (2013) |
| [254] | Rassenfoss, S., A Tricky Tradeoff-Can Adding a Little Solvent Yield a Lot More Heavy Crude?, vol. 64(06) (2012), Society of Petroleum Engineers |
| [255] | Boone, T. T.; Wattenbarger, C. C.; Clingman, S.; Dickson, J. L., An integrated technology development plan for solvent-based recovery of heavy oil, (SPE Heavy Oil Conference and Exhibition (2011), Society of Petroleum Engineers) |
| [256] | Jha, R. K.; Kumar, M.; Benson, I.; Hanzlik, E., New insights into steam/solvent-coinjection-process mechanism, SPE J., 18, 867-877 (2013) |
| [258] | McDuff, D.; Shuchart, C. E.; Jackson, S.; Postl, D.; Brown, J. S., Understanding wormholes in carbonates: unprecedented experimental scale and 3-D visualization, (SPE Annual Technical Conference and Exhibition (2010), Society of Petroleum Engineers) |
| [259] | Shuchart, C.; Alvarez, J.; McDuff, D.; Chang, D.; Troshko, A., Advances in ExxonMobil’s carbonate stimulation methodology-from wormholes to long-term productivity, (International Petroleum Exhibition and Conference (2010)), 1-4 |
| [260] | Valsecchi, P.; McDuff, D.; Chang, D.-L.; Huang, H.; Burdette, J.; Long, T. A.; Karmonik, C., Simulation and visualization of near-well flow, (SPE International Production and Operations Conference & Exhibition (2012), Society of Petroleum Engineers), 15 |
| [261] | Alghamdi, A.; Nasr-El-Din, M. A.; Hill, A. D.; Nasr-El-Din, H. A., Diversion and propagation of viscoelastic surfactant based acid in carbonate cores, (SPE International Symposium on Oilfield Chemistry (2009), Society of Petroleum Engineers) |
| [262] | Melendez, M. G.; Pournik, M.; Zhu, D.; Hill, A. D., The effects of acid contact time and the resulting weakening of the rock surfaces on acid fracture conductivity, (European Formation Damage Conference (2007), Society of Petroleum Engineers) |
| [263] | Saneifar, M.; Nasr-El-Din, H. A.; Nasralla, R. A.; Fahes, M. M.; Hill, D., Effect of spent acids on the wettability of sandstones and carbonates at high temperature and pressure, (SPE European Formation Damage Conference (2011), Society of Petroleum Engineers) |
| [264] | Saneifar, M.; Fahes, M. M.; Hill, D.; Zoghbi, B.; Sheikh Ali, N., An experimental investigation of carbonate rock wettability after stimulation fluids invasion, (SPE EUROPEC/EAGE Annual Conference and Exhibition (2011), Society of Petroleum Engineers) |
| [265] | Morgenthaler, L. N.; Zhu, D.; Mou, J.; Hill, A. D., Effect of reservoir mineralogy and texture on acid response in heterogeneous sandstones, (SPE Annual Technical Conference and Exhibition (2006), Society of Petroleum Engineers) |
| [266] | Economides, M. J.; Hill, A. D.; Ehlig-Economides, C.; Zhu, D., Petroleum Production Systems (2012), Pearson Education |
| [267] | Stalkup, F., Miscible Displacement, Monograph (Society of Petroleum Engineers of AIME) (1983), Henry L. Doherty Memorial Fund of AIME, Society of Petroleum Engineers of AIME |
| [268] | Tchelepi, H. A.; Orr, F. M., Interaction of viscous fingering, permeability heterogeneity, and gravity segregation in three dimensions, SPE Reserv. Eng., 9, 266-271 (1994) |
| [269] | Tchelepi, H., Viscous fingering, gravity segregation and permeability heterogeneity in two-dimensional and three dimensional flows (1994), Stanford University, Ph.D. thesis |
| [270] | Christie, M.; Muggeridge, A.; Barley, J., 3D simulation of viscous fingering and WAG schemes, SPE Reserv. Eng., 8, Article 01 pp. (1993) |
| [272] | Kinzelbach, W., The random walk method in pollutant transport simulation, (Groundwater Flow and Quality Modelling (1988), Springer), 227-245 |
| [273] | Chang, Y.-B.; Lim, M.; Pope, G.; Sepehrnoori, K., \(CO_2\) flow patterns under multiphase flow: heterogeneous field-scale conditions, SPE Reserv. Eng., 9, 208-216 (1994) |
| [274] | Mohanty, K.; Johnson, S., Interpretation of laboratory gasfloods with multidimensional compositional modeling, SPE Reserv. Eng., 8, 59-66 (1993) |
| [275] | Douglas, J.; Ewing, R. E.; Wheeler, M. F., The approximation of the pressure by a mixed method in the simulation of miscible displacement, ESAIM: Math. Model. Numer. Anal., 17, 17-33 (1983) · Zbl 0516.76094 |
| [276] | Darlow, B. L.; Ewing, R. E.; Wheeler, M. F., Mixed finite element method for miscible displacement problems in porous media, Soc. Pet. Eng. J., 24, 391-398 (1984) |
| [277] | Sun, S.; Riviere, B.; Wheeler, M. F., A combined mixed finite element and discontinuous Galerkin method for miscible displacement problem in porous media, (Recent Progress in Computational and Applied PDEs (2002), Springer), 323-351 · Zbl 1060.76072 |
| [278] | Moortgat, J.; Sun, S.; Firoozabadi, A., Compositional modeling of three-phase flow with gravity using higher-order finite element methods, Water Resour. Res., 47 (2011) |
| [279] | Hoteit, H.; Firoozabadi, A., Multicomponent fluid flow by discontinuous Galerkin and mixed methods in unfractured and fractured media, Adv. Water Resour., 41 (2005) |
| [280] | Wheeler, M. F., Numerical Simulation in Oil Recovery (1988), Springer-Verlag New York Inc.: Springer-Verlag New York Inc. New York, NY |
| [281] | Li, J.; Riviere, B., Numerical solutions of the incompressible miscible displacement equations in heterogeneous media, Comput. Methods Appl. Mech. Eng., 292, 107-121 (2015) · Zbl 1423.76254 |
| [282] | Li, J.; Riviere, B., High order discontinuous Galerkin method for simulating miscible flooding in porous media, Comput. Geosci., 19, 1251-1268 (2015) · Zbl 1391.76732 |
| [283] | Riaz, A.; Tchelepi, H. A., Influence of relative permeability on the stability characteristics of immiscible flow in porous media, Transp. Porous Media, 64, 315-338 (2006) |
| [284] | Wheeler, M. F.; Dawson, C., An operator-splitting method for advection-diffusion-reaction problems, (The Mathematics of Finite Elements and Applications VI (1987)), 463-482 · Zbl 0678.76095 |
| [285] | The Hanford Site (2016) |
| [286] | Arbogast, T.; Wheeler, M. F., A parallel numerical model for subsurface contaminant transport with biodegradation kinetics, Math. Finite Elem. Appl., 199-213 (1994) · Zbl 0828.76047 |
| [287] | Sun, S.; Wheeler, M. F., Anisotropic and dynamic mesh adaptation for discontinuous Galerkin methods applied to reactive transport, Comput. Methods Appl. Mech. Eng., 195, 3382-3405 (2006) · Zbl 1175.76096 |
| [288] | Sun, S.; Wheeler, M. F., Discontinuous Galerkin methods for simulating bioreactive transport of viruses in porous media, Adv. Water Resour., 30, 1696-1710 (2007) |
| [289] | Chiang, C. Y.; Dawson, C. N.; Wheeler, M. F., Modeling of in-situ biorestoration of organic compounds in groundwater, Transp. Porous Media, 6, 667-702 (1991) |
| [290] | Dawson, C. N.; Duijn, C. J.V.; Wheeler, M. F., Characteristic-Galerkin methods for contaminant transport with nonequilibrium adsorption kinetics, SIAM J. Numer. Anal., 31, 982-999 (1994) · Zbl 0808.76046 |
| [291] | Blyton, C. A.; Gala, D. P.; Sharma, M. M., A comprehensive study of proppant transport in a hydraulic fracture, (SPE Annual Technical Conference and Exhibition (2015), Society of Petroleum Engineers), 21 |
| [292] | Malhotra, S.; Lehman, E. R.; Sharma, M. M., Proppant placement using alternate-slug fracturing, SPE J., 19, 974-985 (2014) |
| [293] | Lee, S.; Mikelić, A.; Wheeler, M. F.; Wick, T., Phase-field modeling of proppant-filled fractures in a poroelastic medium, Comput. Methods Appl. Mech. Eng. (2016) · Zbl 1439.74359 |
| [294] | Malhotra, S.; Sharma, M. M.; Lehman, E. R., Experimental study of the growth of mixing zone in miscible viscous fingering, Phys. Fluids, 27, Article 014105 pp. (2015) |
| [295] | Zingan, V.; Guermond, J.-L.; Morel, J.; Popov, B., Implementation of the entropy viscosity method with the discontinuous Galerkin method, Comput. Methods Appl. Mech. Eng., 253, 479-490 (2013) · Zbl 1297.76109 |
| [296] | Bonito, A.; Guermond, J.-L.; Lee, S., Numerical simulations of bouncing jets, Int. J. Numer. Methods Fluids, 80, 53-75 (2016) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
