A moving finite element framework for fast infiltration in nonlinear poroelastic media. (English) Zbl 1460.65126

Summary: Poroelasticity theory can be used to analyse the coupled interaction between fluid flow and porous media (matrix) deformation. The classical theory of linear poroelasticity captures this coupling by combining Terzaghi’s effective stress with a linear continuity equation. Linear poroelasticity is a good model for very small deformations; however, it becomes less accurate for moderate to large deformations. On the other hand, the theory of large-deformation poroelasticity combines Terzaghi’s effective stress with a nonlinear continuity equation. In this paper, we present a finite element solver for linear and nonlinear poroelasticity problems on triangular meshes based on the displacement-pressure two-field model. We then compare the predictions of linear poroelasticity with those of large-deformation poroelasticity in the context of a two-dimensional model problem where flow through elastic, saturated porous media, under applied mechanical oscillations, is considered. In addition, the impact of introducing a deformation-dependent permeability according to the Kozeny-Carman equation is explored. We computationally show that the errors in the displacement and pressure fields that are obtained using the linear poroelasticity are primarily due to the lack of the kinematic nonlinearity. Furthermore, the error in the pressure field is amplified by incorporating a constant permeability rather than a deformation-dependent permeability.


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
Full Text: DOI


[1] Almeida, ES; Spilker, RL, Finite element formulations for hyperelastic transversely isotropic biphasic soft tissues, Comput. Methods Appl. Mech. Engrg., 151, 513-538 (1998) · Zbl 0920.73350
[2] Anand, L., Moderate deformations in extension-torsion of incompressible isotropic elastic materials, J. Mech. Phys. Solids, 34, 3, 293-304 (1986)
[3] Auton, L., MacMinn, C.: From Arteries to Boreholes: Steady-State Response of a Poroelastic Cylinder to Fluid Injection. In: Proc. R. Soc. A, vol. 473, pp. 20160753 (2017) · Zbl 1404.76251
[4] Bear, J.; Corapcioglu, M., Mathematical model for Regional Land Subsidence Due to Pumping: 1. Integrated Aquifer Subsidence Equations Based on Vertical Displacement Only, Water Resour Res., 17, 937-946 (1981)
[5] Biot, MA, General Theory of Three-Dimensional Consolidation, J. Appl. Phys., 12, 155-164 (1941) · JFM 67.0837.01
[6] Biot, MA, General Solutions of the Equations of Elasticity and Consolidation for a Porous Material, J. Appl. Mech., 78, 91-96 (1956) · Zbl 0074.19101
[7] Borja, R.I., Choo, J., White, J.A.: Rock Moisture Dynamics, Preferential Flow, and the Stability of Hillside Lopes. In: Multi-Hazard Approaches to Civil Infrastructure Engineering, pp. 443-464. Springer (2016)
[8] Borregales, M., Kumar, K., Nordbotten, J.M., Radu, F.A.: Iterative solvers for Biot model under small and large deformation arXiv:1905.12996v1 [math.NA] (2019)
[9] Borregales, M.; Radu, FA; Kumar, K.; Nordbotten, JM, Robust iterative schemes for non-linear poromechanics, Comput. Geosci., 22, 1021-1038 (2018) · Zbl 1402.65109
[10] Both, JW; Borregales, M.; Nordbotten, JM; Kumar, K.; Radu, FA, Robust fixed stress splitting for Biot’s equations in heterogeneous media, Appl. Math. Lett., 68, 101-108 (2017) · Zbl 1383.74025
[11] Braess, D.: Finite elements: Theory, fast solvers, and applications in solid mechanics. Cambridge University Press (2001) · Zbl 0976.65099
[12] Brun, M.K., Ahmed, E., Berre, I., Nordbotten, J.M., Radu, F.A.: Monolithic and splitting solution schemes for fully coupled quasi-static thermo-poroelasticity with nonlinear convective transport. arXiv:1902.05783 (2019) · Zbl 1451.74204
[13] Callari, C.; Abati, A., Finite element methods for unsaturated porous solids and their application to dam engineering problems, Comput. Struct., 87, 485-501 (2009)
[14] Celia, MA, Geological storage of captured carbon dioxide as a large-scale carbon mitigation option, Water Resour. Res., 53, 3527-3533 (2017)
[15] Cowin, SC, Bone poroelasticity, J. Biomech., 32, 217-238 (1999)
[16] Dziuk, G.; Elliott, C., Finite elements on evolving surfaces, IMA J. Numer. Anal., 27, 262-292 (2007) · Zbl 1120.65102
[17] Ferronato, M.; Gambolati, G.; Janna, C.; Teatini, P., Numerical modelling of regional faults in land subsidence prediction above gas/oil reservoirs, Int. J. Numer. Anal. Meth. Geomech., 32, 633-657 (2008) · Zbl 1273.74532
[18] Franceschini, G.; Bigoni, D.; Regitnig, P.; Holzapfel, G., Brain tissue deforms similarly to filled elastomers and follows consolidation theory, J. Mech. Phys. Solids, 54, 2592-2620 (2006) · Zbl 1162.74303
[19] Fu, G., A high-order HDG method for the Biot’s consolidation model, Comput. Math. Appl., 77, 1, 237-252 (2019) · Zbl 1442.65257
[20] Gambolati, G.; Teatini, P.; Baú, D.; Ferronato, M., Importance of poroelastic coupling in dynamically active aquifers of the Po river basin, Italy, Water Resour. Res., 36, 9, 2443-2459 (2000)
[21] Gawin, D.; Baggio, P.; Schrefler, BA, Coupled heat, water and gas flow in deformable porous media, Internat. J. Numer. Methods Fluids, 20, 969-987 (1995) · Zbl 0854.76052
[22] Haga, JB; Osnes, H.; Langtangen, HP, On the causes of pressure oscillations in low-permeable and low-compressible porous media, Int. J. Numer. Anal. Meth. Geomech., 36, 12, 1507-1522 (2012)
[23] Hencky, H., The law of elasticity for isotropic and quasi-isotropic substances by finite deformations, J. Rheol., 2, 2, 169-176 (1931)
[24] Hu, X.; Mu, L.; Ye, X., Weak Galerkin method for the Biot’s consolidation model, Comput. Math. Appl., 75, 2017-2030 (2018) · Zbl 1409.76063
[25] Hu, X.; Rodrigo, C.; Gaspar, FJ; Zikatanov, LT, A nonconforming finite element method for the Biot’s consolidation model in poroelasticity, J. Comput. Appl. Math., 310, 143-154 (2017) · Zbl 1381.76175
[26] Juanes, R.; Jha, B.; Hager, B.; Shaw, J.; Plesch, A.; Astiz, L.; Dieterich, J.; Frohlich, C., Were the May 2012 Emilia-Romagna earthquakes induced? A coupled flow-geomechanics modeling assessment, Geophys. Res. Lett., 43, 13, 6891-6897 (2016)
[27] List, F.; Radu, FA, A study on iterative methods for solving Richards’ equation, Comput. Geosci., 20, 341-353 (2016) · Zbl 1396.65143
[28] MacMinn, CW; Dufresne, ER; Wettlaufer, JS, Large Deformations of a Soft Porous Material, Phys. Rev. Appl., 5, 044020 (2016)
[29] Markert, B.: Porous Media Viscoelasticity with Application to Polymeric Foams. Ph.D. Thesis, Institut für Mechanik Lehrstuhl II. Universität Stuttgart (2005)
[30] Moeendarbary, E.; Valon, L.; Fritzsche, M.; Harris, AR; Moulding, DA; Thrasher, AJ; Stride, E.; Mahadevan, L.; Charras, GT, The cytoplasm of living cells behaves as a poroelastic material, Nat. Mater., 12, 253-261 (2013)
[31] Murad, MA; Loula, AF, On stability and convergence of finite element approximations of Biot’s consolidation problem, Int. J. Numer. Meth. Engng., 37, 645-667 (1994) · Zbl 0791.76047
[32] Na, S.; Sun, W., Computational thermo-hydro-mechanics for multiphase freezing and thawing porous media in the finite deformation range, Comput. Methods Appl. Mech. Engrg., 318, 667-700 (2017) · Zbl 1439.74114
[33] Nield, D.A., Bejan, A.: Convection in Porous Media. Springer International Publishing AG (2017) · Zbl 1375.76004
[34] Rahrah, M., Lopez-Peña, L.A., Vermolen, F., Meulenbroek, B.: Network-inspired versus Kozeny-Carman based permeability-porosity relations applied to Biot’s poroelasticity model. arXiv:2004.09373(2020)
[35] Rahrah, M.; Vermolen, F., Monte Carlo Assessment of the Impact of Oscillatory and Pulsating Boundary Conditions on the Flow Through Porous Media, Transp. Porous Med., 123, 1, 125-146 (2018)
[36] Rahrah, M., Vermolen, F.: Uncertainty Quantification in Injection and Soil Characteristics for Biot’s Poroelasticity Model. In: European Conference on Numerical Mathematics and Advanced Applications ENUMATH 2017, pp. 645-652. Springer (2019) · Zbl 1425.74486
[37] Rutqvist, J., The Geomechanics of CO2 Storage in Deep Sedimentary Formations, Geotech. Geol. Eng., 30, 3, 525-551 (2012)
[38] Sahimi, M., Flow and Transport in Porous Media and Fractured Rock (1995), Weinheim: VCH, Weinheim · Zbl 0849.76001
[39] Schrefler, B., Computer modelling in environmental geomechanics, Comput. Struct., 79, 2209-2223 (2001)
[40] Soga, K.; Alonso, E.; Yerro, A.; Kumar, K.; Bandara, S., Trends in large-deformation analysis of landslide mass movements with particular emphasis on the material point method, Géotechnique, 66, 3, 248-273 (2016)
[41] Støverud, KH; Darcis, M.; Helmig, R.; Hassanizadeh, SM, Modeling Concentration Distribution and Deformation During Convection-Enhanced Drug Delivery into Brain Tissue, Transp. Porous Med., 92, 1, 119-143 (2012)
[42] Sun, M.; Rui, H., A coupling of weak Galerkin and mixed finite element methods for poroelasticity, Comput. Math. Appl., 73, 804-823 (2017) · Zbl 1369.76028
[43] Sun, W.; Chen, Q.; Ostien, JT, Modeling the hydro-mechanical responses of strip and circular punch loadings on water-saturated collapsible geomaterials, Acta Geotech., 9, 903-934 (2014)
[44] Szulczewski, M.L., MacMinn, C.W., Herzog, H.J., Juanes, R.: Lifetime of Carbon Capture and Storage as a Climate-Change Mitigation Technology. In: Proc. Natl. Acad. Sci. U.S.A., vol. 109, pp. 5185 - 5189 (2012)
[45] 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, 1, W01406 (2006)
[46] Terzaghi, K.: Theoretical soil mechanics. Chapman And Hall Limited, London (1951)
[47] Tsai, TL; Chang, KC; Huang, LH, Body force effect on consolidation of porous elastic media due to pumping, J. Chin. Inst. Eng., 29, 1, 75-82 (2006)
[48] Wang, H.F.: Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology. Princeton University Press (2000)
[49] Wang, SJ; Hsu, KC, Dynamics of deformation and water flow in heterogeneous porous media and its impact on soil properties, Hydrol. Process., 23, 3569-3582 (2009)
[50] Wheeler, M.; Xue, G.; Yotov, I., Coupling multipoint flux mixed finite element methods with continuous Galerkin methods for poroelasticity, Comput. Geosci., 18, 57-75 (2014) · Zbl 1395.65093
[51] White, J., Chiaramonte, L., Foxall, S.E.W., Hao, Y., Ramirez, A., McNab, W.: Geomechanical behavior of the reservoir and caprock system at the In Salah CO_2 storage project. In: Proc. Natl. Acad. Sci., vol. 111, pp. 8747-8752 (2014)
[52] White, JA; Borja, RI, Stabilized low-order finite elements for coupled solid-deformation/fluid-diffusion and their application to fault zone transients, Comput. Methods Appl. Mech. Engrg., 197, 4353-4366 (2008) · Zbl 1194.74480
[53] You, L.; Liu, H., A two-phase flow and transport model for the cathode of PEM fuel cells, Int. J. Heat Mass Transfer, 45, 2277-2287 (2002) · Zbl 0993.76556
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.