Abstract
Poroelastic models arise in reservoir modeling and many other important applications. Under certain assumptions, they involve a time-dependent coupled system consisting of Navier–Lamé equations for the displacements, Darcy’s flow equation for the fluid velocity and a divergence constraint equation. Stability for infinite time of the continuous problem and, second and third order accurate, time discretized equations are shown. Methods to handle the lack of regularity at initial times are discussed and illustrated numerically. After discretization, at each time step this leads to a block matrix system in saddle point form. Mixed space discretization methods and a regularization method to stabilize the system and avoid locking in the pressure variable are presented. A certain block matrix preconditioner is shown to cluster the eigenvalues of the preconditioned matrix about the unit value but needs inner iterations for certain matrix blocks. The strong clustering leads to very few outer iterations. Various approaches to construct preconditioners are presented and compared. The sensitivity of the number of outer iterations to the stopping accuracy of the inner iterations is illustrated numerically.
Similar content being viewed by others
References
Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941)
Biot, M.A.: Theory of elasticity and consolidation for porous anisotropic media. J. Appl. Phys. 26, 182–185 (1955)
Ewing, R. (ed.): The Mathematics of Reservoir Simulations, Frontiers in Appl. Math. vol. 1. SIAM, Philadelphia (1984)
Discacciati, M., Quarteroni, A.: Navier–Stokes/Darcy coupling: modelling, analysis and numerical approximation. Rev. Math. Comput. 22, 315–426 (2009)
Lipnikov, K.: Numerical Methods for the Biot Model in Poroelasticity. Ph. D. Thesis, University of Houston (2002)
Kuznetsov, Yu., Prokopenko, A.: A new multilevel algebraic preconditioner for the diffusion equation in heterogeneous media. Numer. Linear Algebra Appl. 17, 759–769 (2010)
Phillips, P.J.: Finite Element Methods in Linear Poroelasticity: Theoretical and Computational Results. Ph.D. Thesis, The University of Texas at Austin (2005)
Axelsson, O., Blaheta, R.: Preconditioning of matrices partitioned in two by two block form: eigenvalue estimates and Schwarz DD for mixed FEM. Numer. Linear Algebra Appl. 17, 787–810 (2010)
Axelsson, O.: Preconditioners for regularized saddle point matrices. J. Numer. Math. 19, 91–112 (2011)
Torquato, S.: Random Heterogeneous Materials. Microstructure and Macroscopic Properties. Springer, New York (2002)
Axelsson, O.: On iterative solvers in structural mechanics, separate displacement orderings and mixed variable methods. Math. Comput. Simul. 50(1–4), 11–30 (1999)
Phillips, P.J., Wheeler, M.F.: Overcoming the problem of locking in linear elasticity and poroelasticity: an heuristic approach. Comput. Geosci. 13, 5–12 (2009)
Haga, J.B., Osnes, H., Langtangen, H.P.: On the causes of pressure oscillations in low-permeable porous media. Int. J. Numer. Anal. Methods Geomech. (on line) (2011)
Detournay, E., Cheng, A.H.-D.: Fundamentals of poroelasticity. In: Fairhurst, C. (ed.) Chapter 5 in Comprehensive Rock Engineering Principles, Practice and Projects, Vol. II, Analysis and Design Method. Pergamon Press, pp. 113–171 (1993)
Biot, M.A., Willis, D.G.: The elastic coefficients of the theory of consolidation. J. Appl. Mech. 24, 594–601 (1957)
Terzaghi, K.: Theoretical Soil Mechanics. Wiley, New York (1943)
Axelsson, O., Gololobov, S.V.: Stability and error estimates for the \( \theta \)-method for strongly monotone and infinitely stiff evolution equations. Numer. Math. 89, 31–48 (2001)
Showalter, R.E.: Diffusion in poro-elastic media. J. Math. Anal. Appl. 251, 310–340 (2000)
Helfrisch, H.P.: Fehlerabschätzungen für das Galerkinverfahren zur Lösung von Evolutionsgleichungen. Manuscripta Math. 13, 219–235 (1974)
Rannacher, R.: Finite element solution of diffusion problems with irregular data. Numer. Math. 43, 309–327 (1984)
Murad, M.A., Thomée, V., Loula, A.F.D.: Asymptotic behaviour of semidiscrete finite element approximations of Biot’s consolidation problem. SIAM J. Numer. Anal. 33, 1065–1083 (1996)
Murad, M.A.: AFD Loula. On stability and convergence of finite element approximations of Biot’s consolidation problem. Int. J. Numer. Method Eng. 27, 645–667 (1994)
Gear, C.W., Petzold, L.R.: ODE methods for the solution of differential/algebraic systems. SIAM J. Numer. Anal. 23, 837–852 (1986)
Butcher, J.: Numerical Methods for Ordinary Differential Equations. Wiley, Chichester (2003)
Axelsson, O.: A class of \( A-\)stable methods. BIT 9, 185–199 (1969)
Axelsson, O.: On the efficiency of a class of \(A-\)stable methods. BIT 14, 279–287 (1974)
Axelsson, O., Blaheta, R., Sysala, S., Ahmad, B.: On the solution of high order stable time integration methods. J. Boundary Value Probl. Springer open 108 (2013). doi:10.1186/1687-2770-2013-108
Babuska, I.: Error-bounds for the finite element method. Numer. Math. 16, 322–333 (1971)
Brezzi, F.: On the existence, uniqueness and approximation of saddle point problems, arising from Lagrangian multipliers. R.A.I.R.O. Anal. Numer. 8, 129–151 (1974)
Raviart, R.A., Thomas, J.M.: Mixed finite element method for second order elliptic problems. Math. Aspects Finite Element Method Lect. Notes Math. 606, 292–315 (1977)
Axelsson, O., Barker, V.A., Neytcheva, M., Polman, B.: Solving the Stokes problem on a massively parallel computer. Math. Model. Anal. 4, 1–22 (2000)
Sagae, M., Tanabe, K.: Upper and lower bounds for the arithmetic–geometric–harmonic means of positive definite matrices. Linear Multilinear Algebra 37, 279–282 (1994)
Axelsson, O.: Iterative Solution Methods. Cambridge University Press, Cambridge (1994)
Haga, J.B., Osnes, H., Langtangen, H.P.: Efficient block preconditioners for the coupled equations of pressure and deformation in highly discontinuous media. Int. J. Numer. Anal. Methods Geomech. 35(13), 1466–1482 (2011)
Saad, Y.: A flexible inner-outer preconditioned GMRES-algorithm. SIAM J. Sci. Comput. 14, 461–469 (1993)
Axelsson, O., Blaheta, R., Neytcheva, M.: Preconditioning for boundary value problems using elementwise Schur complements. SIAM J. Matrix Anal. Appl. 31, 767–789 (2009)
Haga, J.B., Langtangen, H.P., Osnes, H.: A parallel block preconditioner for large scale poroelasticity with highly heterogeneous material parameters. Comput. Geosci. 16(3), 723–734 (2012)
Arbenz, P., Turan, E.: Preconditioning for large scale micro finite element analyses of 3D poroelasticity. In: Manninen, P., Öster, P. (eds.) Applied Parallel and Scientific Computing (PARA 2012), Lecture Notes in Computer Science 7782, pp. 361–374. Springer, Heidelberg (2013)
Ferronato, M., Castelletto, N., Gambolati, G.: A fully coupled 3-D mixed finite element model of Biot consolidation. J. Comput. Phys. 229, 4813–4830 (2010)
Acknowledgments
This work was supported by the European Regional Development Fund in the IT4Innovations Centre of Excellence project (CZ.1.05/1.1.00/02.0070).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated ByGabriel Wittum.
This paper is dedicated to the memory of Richard (Dick) E. Ewing for his friendly manner and concern for other peoples research. The first author was fortunate to meet Dick already when he worked at Mobil Oil, Texas. Later Dick invited him to visit University of Wyoming at Laramie, Wyoming which enabled collaborative research with one of Dick’s coworkers at that time, Panayot S. Vassilevsky. Later meetings with Dick at College Station, Texas, University of Nijmegen, The Netherlands and at conferences in Bulgaria were also always very useful.
Rights and permissions
About this article
Cite this article
Axelsson, O., Blaheta, R. & Byczanski, P. Stable discretization of poroelasticity problems and efficient preconditioners for arising saddle point type matrices. Comput. Visual Sci. 15, 191–207 (2012). https://doi.org/10.1007/s00791-013-0209-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00791-013-0209-0