A fully coupled 3-D mixed finite element model of Biot consolidation.(English)Zbl 1305.76055
Summary: The numerical solution to the Biot equations of 3-D consolidation is still a challenging task because of the ill-conditioning of the resulting algebraic system and the instabilities that may affect the pore pressure solution. Recently new approaches have been advanced based on mixed formulations. In the present paper a fully coupled 3-D mixed finite element model is developed with the aim at alleviating the pore pressure numerical oscillations at the interface between materials with different permeabilities. A solution algorithm is implemented that takes advantage of the block structure of the discretized problem. The proposed model is verified against well-known analytical solutions and successfully experimented with in realistic applications of soil consolidation.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
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References:
| [1] | Terzaghi, K., Erdbaumechanik auf Bodenphysikalischer Grundlage (1925), F. Duticke: F. Duticke Vienna ·JFM 51.0655.07 |
| [2] | Biot, M. A., General theory of three-dimensional consolidation, J. Appl. Phys., 12, 155-164 (1941) ·JFM 67.0837.01 |
| [3] | Coussy, O., Mechanics of Porous Continua (1995), J. Wiley & Sons: J. Wiley & Sons New York, NY ·Zbl 0838.73001 |
| [4] | Pao, W. K.S.; Lewis, R. W.; Masters, I., A fully coupled hydro-thermo-poro-mechanical model for black oil reservoir simulation, Int. J. Numer. Anal. Meth. Geomech., 25, 1229-1256 (2001) ·Zbl 1016.74023 |
| [5] | Ferronato, M.; Gambolati, G.; Teatini, P.; Baù, D., Radioactive marker measurements in heterogeneous reservoirs: numerical study, Int. J. Geomech., 4, 79-92 (2004) |
| [6] | Yin, S.; Dusseault, M. B.; Rothenburg, L., Thermal reservoir modeling in petroleum geomechanics, Int. J. Numer. Anal. Meth. Geomech., 33, 449-485 (2009) ·Zbl 1273.74092 |
| [7] | Teatini, P.; Ferronato, M.; Gambolati, G.; Gonella, M., Groundwater pumping and land subsidence in the Emilia-Romagna coastland, Italy: modeling the past occurrence and the future trend, Water Resour. Res., 42 (2006) |
| [8] | Hudson, J.; Stephansson, O.; Andersson, J.; Tsang, C. F.; Ling, L., Coupled T-H-M issues related to radioactive waste repository design and performance, Int. J. Rock Mech. Mining Sci., 38, 143-161 (2001) |
| [9] | Roose, T.; Netti, P.; Munn, L.; Boucher, Y.; Jain, R., Solid stress generated by spheroid growth estimated using a linear poroelastic model, Microvasc. Res., 66, 204-212 (2003) |
| [10] | Swan, C.; Lakes, R.; Brand, R.; Stewart, K., Micromechanically based poroelastic modeling of fluid flow in Haversian bone, J. Biomech. Eng., 125, 25-37 (2003) |
| [11] | Ferronato, M.; Gambolati, G.; Teatini, P., Ill-conditioning of finite element poroelasticity equations, Int. J. Solids Struct., 38, 5995-6014 (2001) ·Zbl 1075.74643 |
| [12] | Bergamaschi, L.; Ferronato, M.; Gambolati, G., Mixed constraint preconditioners for the iterative solution to FE coupled consolidation equations, J. Comput. Phys., 227, 9885-9897 (2008) ·Zbl 1154.65015 |
| [13] | Ferronato, M.; Pini, G.; Gambolati, G., The role of preconditioning in the solution to FE coupled consolidation equations by Krylov subspace methods, Int. J. Numer. Anal. Meth. Geomech., 33, 405-423 (2009) ·Zbl 1272.74610 |
| [14] | Wheeler, M. F.; Gai, X., Iteratively coupled mixed and Galerkin finite element methods for poro-elasticity, Numer. Meth. Part. D. E., 23, 785-797 (2007) ·Zbl 1115.74054 |
| [15] | Idelson, S. R.; Heinrich, J. C.; Onate, E., Petrov-Galerkin methods for the transient advective-diffusive equation with sharp gradients, Int. J. Numer. Meth. Eng., 39, 1455-1473 (1996) ·Zbl 0869.76041 |
| [16] | Vermeer, P. A.; Verruijt, A., An accuracy condition for consolidation by finite elements, Int. J. Numer. Anal. Meth. Geomech., 5, 1-14 (1981) ·Zbl 0456.73060 |
| [17] | Reed, M. B., An investigation of numerical errors in the analysis of consolidation by finite elements, Int. J. Numer. Anal. Meth. Geomech., 8, 243-257 (1984) ·Zbl 0536.73089 |
| [18] | Sandhu, R. S.; Liu, H.; Singh, K. J., Numerical performance of some finite element schemes for analysis of seepage in porous elastic media, Int. J. Numer. Anal. Meth. Geomech., 1, 177-194 (1977) ·Zbl 0366.65049 |
| [19] | Murad, M. A.; Loula, A. F.D., On the stability and convergence of finite element approximations of Biot’s consolidation problem, Int. J. Numer. Meth. Eng., 37, 645-667 (1994) ·Zbl 0791.76047 |
| [20] | Murad, M. A.; Loula, A. F.D., Improved accuracy in finite element analysis of Biot’s consolidation problem, Comput. Meth. Appl. Mech. Eng., 95, 359-382 (1992) ·Zbl 0760.73068 |
| [21] | Tchonkova, M.; Peters, J.; Sture, S., A new mixed finite element method for poro-elasticity, Int. J. Numer. Anal. Meth. Geomech., 32, 579-606 (2008) ·Zbl 1273.74550 |
| [22] | Phillips, P. J.; Wheeler, M. F., A coupling of mixed and continuous Galerkin finite element methods for poroelasticity I: the continuous-in-time case, Comput. Geosci., 11, 131-144 (2007) ·Zbl 1117.74015 |
| [23] | Phillips, P. J.; Wheeler, M. F., A coupling of mixed and continuous Galerkin finite element methods for poroelasticity II: the discret-in-time case, Comput. Geosci., 11, 145-158 (2007) ·Zbl 1117.74016 |
| [24] | Phillips, P. J.; Wheeler, M. F., A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity, Comput. Geosci., 12, 417-435 (2008) ·Zbl 1155.74048 |
| [25] | Jha, B.; Juanes, R., A locally conservative finite element framework for the simulation of coupled flow and reservoir geomechanics, Acta Geotech., 2, 139-153 (2007) |
| [26] | R.W. Freund, N.M. Nachtingal, A new Krylov-subspace method for symmetric indefinite linear systems, in: Proceedings of the 14th IMACS World Congress on Computational and Applied Mathematics, 1994, pp. 1253-1256.; R.W. Freund, N.M. Nachtingal, A new Krylov-subspace method for symmetric indefinite linear systems, in: Proceedings of the 14th IMACS World Congress on Computational and Applied Mathematics, 1994, pp. 1253-1256. |
| [27] | Roberts, J. E.; Thomas, J. M., Mixed and hybrid methods, (Ciarlet, P. G.; Lions, J. L., Handbook of Numerical Analysis, vol. 2 (1991), North-Holland: North-Holland Amsterdam) ·Zbl 0875.65090 |
| [28] | Zienkiewicz, O. C.; Taylor, R. L., The Finite Element Method (2000), Butterworth-Heinemann: Butterworth-Heinemann Oxford, UK ·Zbl 0991.74002 |
| [29] | Toh, K. C.; Phoon, K. K., Comparison between iterative solution of symmetric and non-symmetric forms of Biot’s FEM equations using the generalized Jacobi preconditioner, Int. J. Numer. Anal. Meth. Geomech., 32, 1131-1146 (2008) ·Zbl 1273.74551 |
| [30] | Ferronato, M.; Bergamaschi, L.; Gambolati, G., Performance and robustness of block constraint preconditioners in finite element coupled consolidation models, Int. J. Numer. Meth. Eng., 81, 381-402 (2010) ·Zbl 1183.74271 |
| [31] | Wang, H. F., Theory of Linear Poroelasticity (2001), Princeton University Press: Princeton University Press Princeton, NJ |
| [32] | Wang, S. J.; Hsu, K. C., The application of the first-order second-moment method to analyze poroelastic problems in heterogeneous porous media, J. Hydrol., 369, 209-221 (2009) |
| [33] | Mandel, J., Consolidation des sols (étude mathématique), Géotechnique, 30, 287-289 (1953) |
| [34] | Abousleiman, Y.; Cheng, A. H.D.; Cui, L.; Detournay, E.; Roegiers, J. C., Mandel’s problem revisited, Géotechnique, 46, 187-195 (1996) |
| [35] | Cui, L.; Cheng, A. H.D.; Kaliakin, V. N.; Abousleiman, Y.; Roegiers, J. C., Finite element analyses of anisotropic poroelasticity: a generalized Mandel’s problem and an inclined borehole problem, Int. J. Numer. Anal. Meth. Geomech., 20, 381-401 (1996) ·Zbl 0859.73073 |
| [36] | L. Bergamaschi, S. Mantica, Efficient algebraic solvers for mixed finite element models of porous media flows, in: A.A. Aldama et al. (Eds.), Proceedings of 11th International Conference on Computational Methods in Water Resources, Computational Mechanics and Elsevier Applied Sciences, Southampton, London, UK, 1996, pp. 481-488.; L. Bergamaschi, S. Mantica, Efficient algebraic solvers for mixed finite element models of porous media flows, in: A.A. Aldama et al. (Eds.), Proceedings of 11th International Conference on Computational Methods in Water Resources, Computational Mechanics and Elsevier Applied Sciences, Southampton, London, UK, 1996, pp. 481-488. |
| [37] | Gambolati, G.; Gatto, P.; Freeze, R. A., Mathematical simulation of the subsidence of Venice: 2. Results, Water Resour. Res., 10, 563-577 (1974) |
| [38] | Gambolati, G.; Ricceri, G.; Bertoni, W.; Brighenti, G.; Vuillermin, E., Mathematical simulation of the subsidence of Ravenna, Water Resour. Res., 27, 2899-2918 (1991) |
| [39] | Verruijt, A., Elastic storage of aquifers, (De Wiest, R., Flow Through Porous Media (1969), Academic Press: Academic Press New York, NY), 331-376 |
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