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An efficient finite element procedure for analyzing three-phase porous media based on the relaxed Picard method. (English) Zbl 1352.76061
Summary: Effective simulation of the solid-liquid-gas coupling effect in unsaturated porous media is of great significance in many diverse areas. Because of the strongly nonlinear characteristics of the fully coupled formulations for the three-phase porous media, an effective numerical solution scheme, such as the finite element method with an efficient iterative algorithm, has to be employed. In this paper, an efficient finite element procedure based on the adaptive relaxed Picard method is developed for analyzing the coupled solid-liquid-gas interactions in porous media. The coupled model and the finite element analysis procedure are implemented into a computer code PorousH2M, and the proposed procedure is validated through comparing the numerical simulations with the experimental benchmarks. It is shown that the adaptive relaxed Picard method has salient advantage over the traditional one with respect to both the efficiency and the robustness, especially for the case of relatively large time step sizes. Compared with the Newton-Raphson scheme, the Picard method successfully avoids the unphysical ‘spurious unloading’ phenomenon under the plastic deformation condition, although the latter shows a better convergence rate. The proposed procedure provides an important reference for analyzing the fully coupled problems related to the multi-phase, multi-field coupling in porous media.
MSC:
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
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76S05 Flows in porous media; filtration; seepage
76T30 Three or more component flows
Software:
PorousH2M
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[1] Oettl, Numerical simulation of geotechnical problems based on a multi-phase finite element approach, Computers and Geotechnics 31 (8) pp 643– (2004) · doi:10.1016/j.compgeo.2004.10.002
[2] Hu, Modeling of coupled deformation, water flow and gas transport in soil slopes subjected to rain infiltration, Science China Technological Sciences 54 (10) pp 2561– (2011) · Zbl 1239.76065 · doi:10.1007/s11431-011-4504-z
[3] Khoei, Numerical modeling of multiphase fluid flow in deforming porous media: a comparison between two- and three-phase models for seismic analysis of earth and rockfill dams, Computers and Geotechnics 38 (2) pp 142– (2011) · doi:10.1016/j.compgeo.2010.10.010
[4] Li, A numerical model for immiscible two-phase fluid flow in a porous medium and its time domain solution, International Journal for Numerical Methods in Engineering 30 (6) pp 1195– (1990) · Zbl 0714.76098 · doi:10.1002/nme.1620300608
[5] Xikui, Multiphase flow in deforming porous media and finite element solutions, Computers & Structures 45 (2) pp 211– (1992) · Zbl 0769.76031 · doi:10.1016/0045-7949(92)90405-O
[6] Schrefler, A fully coupled model for water-flow and air-flow in deformable porous-media, Water Resources Research 29 (1) pp 155– (1993) · doi:10.1029/92WR01737
[7] Borja, Critical state plasticity. Part VII: triggering a shear band in variably saturated porous media, Computer Methods in Applied Mechanics and Engineering 261 pp 66– (2013) · Zbl 1286.74027 · doi:10.1016/j.cma.2013.03.008
[8] Schrefler, A fully coupled dynamic model for two-phase fluid flow in deformable porous media, Computer Methods in Applied Mechanics and Engineering 190 (24-25) pp 3223– (2001) · Zbl 0977.74019 · doi:10.1016/S0045-7825(00)00390-X
[9] Coussy, Poromechanics (2004)
[10] Boer, Trends in Continuum Mechanics of Porous Media (2005) · Zbl 1085.74002 · doi:10.1007/1-4020-3144-0
[11] Lewis, The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media (1998) · Zbl 0935.74004
[12] Wei, A continuum theory of porous media saturated by multiple immiscible fluids: II. Lagrangian description and variational structure, International Journal of Engineering Science 40 (16) pp 1835– (2002) · Zbl 1211.76124 · doi:10.1016/S0020-7225(02)00069-1
[13] Coussy, From mixture theory to Biot’s approach for porous media, International Journal of Solids and Structures 35 (34-35) pp 4619– (1998) · Zbl 0932.74014 · doi:10.1016/S0020-7683(98)00087-0
[14] Park, Are upwind techniques in multi-phase flow models necessary?, Journal of Computational Physics 230 (22) pp 8304– (2011) · Zbl 1408.76370 · doi:10.1016/j.jcp.2011.07.030
[15] Ataie-Ashtiani, Comparison of numerical formulations for two-phase flow in porous media, Geotechnical and Geological Engineering 28 (4) pp 373– (2010) · doi:10.1007/s10706-009-9298-4
[16] Bishop, The Principles of Effective Stress (1960)
[17] Kohgo, Theoretical aspects of constitutive modelling for unsaturated soils, Soils and Foundations 33 (4) pp 49– (1993) · doi:10.3208/sandf1972.33.4_49
[18] Borja, Cam-Clay plasticity. Part V: a mathematical framework for three-phase deformation and strain localization analyses of partially saturated porous media, Computer Methods in Applied Mechanics and Engineering 193 (48) pp 5301– (2004) · Zbl 1112.74430 · doi:10.1016/j.cma.2003.12.067
[19] Khalili, Effective stress in unsaturated soils: review with new evidence, International Journal of Geomechanics 4 (2) pp 115– (2004) · doi:10.1061/(ASCE)1532-3641(2004)4:2(115)
[20] Kim, Rigorous coupling of geomechanics and multiphase flow with strong capillarity, Spe Journal 18 (06) pp 1123– (2013) · doi:10.2118/141268-PA
[21] Van Genuchten, A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Science Society of America Journal 44 (5) pp 892– (1980) · doi:10.2136/sssaj1980.03615995004400050002x
[22] Wei, Formulation of capillary hysteresis with internal state variables, Water Resources Research 42 (7) pp W07405– (2006) · doi:10.1029/2005WR004594
[23] Pedroso, A novel approach for modelling soil-water characteristic curves with hysteresis, Computers and Geotechnics 37 (3) pp 374– (2010) · doi:10.1016/j.compgeo.2009.12.004
[24] Zienkiewicz, Volume 1, in: The Finite Element Method (1994)
[25] Kolditz, Numerical simulation of two-phase flow in deformable porous media: application to carbon dioxide storage in the subsurface, Mathematics and Computers in Simulation 82 (10) pp 1919– (2012) · doi:10.1016/j.matcom.2012.06.010
[26] Callari, Finite element methods for unsaturated porous solids and their application to dam engineering problems, Computers & Structures 87 (7-8) pp 485– (2009) · doi:10.1016/j.compstruc.2008.12.012
[27] Paniconi, A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems, Water Resources Research 30 (12) pp 3357– (1994) · doi:10.1029/94WR02046
[28] Huyakorn, A three - finite - model for simulating water flow in variably saturated porous media, Water Resources Research 22 (13) pp 1790– (1986) · doi:10.1029/WR022i013p01790
[29] Durbin, Adaptive underrelaxation of Picard iterations in ground water models, Ground Water 45 (5) pp 648– (2007) · doi:10.1111/j.1745-6584.2007.00329.x
[30] Liang, Reliability and Robustness of Engineering Software II pp 35– (1991) · doi:10.1007/978-94-011-3026-4_3
[31] Ribó, GiD Reference Manual, Pre and Post Processing System for FEM Calculations (2011)
[32] Liakopoulos, Transient Flow Through Unsaturated Porous Media (1964)
[33] Dangla, Drainage and drying of deformable porous materials: one dimensional case study, Iutam Symposium on Mechanics of Granular and Porous Materials 53 pp 427– (1997) · doi:10.1007/978-94-011-5520-5_38
[34] Akai, Finite element analysis of saturated-unsaturated seepage in soil, Proceedings of JSCE 264 pp 87– (1977)
[35] Meng-xi, Saturated-unsaturated unsteady seepage numerical analysis, Journal of Hydraulic Engineering 12 pp 38– (1999)
[36] Rong, Numerical analysis of saturated-unsaturated seepage problem of rock slope under rainfall infiltration, Rock and Soil Mechanics-Wuhan 26 (10) pp 1545– (2005)
[37] Potts, Finite Element Analysis in Geotechnical Engineering: Theory (1999) · doi:10.1680/feaiget.27534
[38] Crisfield, Non-linear Finite Element Analysis of Solids and Structures: Essentials (1991) · Zbl 0809.73005
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