1887

Abstract

Summary

We present a novel and efficient preconditioning technique to solve the non-symmetric system of equations associated with Lagrange multipliers-based discretization schemes, such as Mixed Hybrid Finite Element method (MHFE) and Mimetic Finite Difference method (MFD). These types of discretization have been gaining popularity lately and here we develop a fully dedicated preconditioner for them. Preconditioners are key to improve the efficiency of Krylov subspace methods, that provide a solution to the sequence of large-size, and often ill-conditioned, systems of equations originating from reservoir numerical simulations.

The mathematical model of flow in porous media is governed by a set of two coupled nonlinear equations: the momentum and mass balance equations, discretized using either the MHFE or the MFD, and the Finite Volume method (FV), respectively. Unknowns are located on elements (element pressure and saturation) and faces (face pressure and phase capillary pressure), the latter behaving as Lagrange multipliers. The problem is solved by adopting a fully implicit approach and linearization is provided by a Newton-Raphson method, which leads to a block-structured Jacobian matrix. An original numerical formulation of the mass balance equation, where the continuity of fluxes is strongly imposed with the aim of increasing the efficiency of the nonlinear iteration, has been investigated. The resulting block Jacobian is not symmetric, thus requiring special preconditioning tools for its efficient solution. The preconditioning approach exploits the Jacobian block structure to develop a multi-stage strategy that addresses separately the problem unknowns. A crucial point is the approximation of the resulting Schur complements, which is carried out at an algebraic level by applying proper restriction operators to the full matrix blocks. The selection of such restrictors is carried out with the aid of a domain decomposition technique algebraically enhanced by a dynamic minimal residual strategy. The proposed block preconditioner has been tested through an extensive experimentation on unstructured and highly heterogeneous reservoir systems, pointing out its robustness and computational efficiency.

Loading

Article metrics loading...

/content/papers/10.3997/2214-4609.202035072
2020-09-14
2021-07-27
Loading full text...

Full text loading...

References

  1. Abushaikha, A.S. and Terekhov, K.M.
    [2020] A fully implicit mimetic finite difference scheme for general purpose subsurface reservoir simulation with full tensor permeability. Journal of Computational Physics, 406, 109194.
    [Google Scholar]
  2. Abushaikha, A.S., Voskov, D.V. and Tchelepi, H.A.
    [2017] Fully implicit mixed-hybrid finite-element discretization for general purpose subsurface reservoir simulation. Journal of Computational Physics, 346, 514–538.
    [Google Scholar]
  3. Chavent, G. and Roberts, J.
    [1991] A unified physical presentation of mixed, mixed-hybrid finite elements and standard finite difference approximations for the determination of velocities in waterflow problems. Advances in Water Resources, 14(6), 329–348.
    [Google Scholar]
  4. Christie, M. and Blunt, M.
    [2001] Tenth SPE comparative solution project: A comparison of upscaling techniques. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers, Houston, Texas, 308–317.
    [Google Scholar]
  5. Ferronato, M.
    [2012] Preconditioning for sparse linear systems at the dawn of the 21st century: History, current developments, and future perspectives. ISRN Applied Mathematics, 2012, 127647.
    [Google Scholar]
  6. Ferronato, M., Franceschini, A., Janna, C., Castelletto, N. and Tchelepi, H.A.
    [2019] A general preconditioning framework for coupled multiphysics problems with application to contact- and poromechanics. Journal of Computational Physics, 398, 108887.
    [Google Scholar]
  7. Franceschini, A., Castelletto, N. and Ferronato, M.
    [2019] Block preconditioning for fault/fracture mechanics saddle-point problems. Computer Methods in Applied Mechanics and Engineering, 344, 376–401.
    [Google Scholar]
  8. Matringe, S.F., Juanes, R. and Tchelepi, H.A.
    [2007] Mixed-finite-element and related-control-volume discretizations for reservoir simulation on three-dimensional unstructured grids. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers, 106117.
    [Google Scholar]
  9. Moortgat, J. and Firoozabadi, A.
    [2016] Mixed-hybrid and vertex-discontinuous-Galerkin finite element modeling of multiphase compositional flow on 3D unstructured grids. Journal of Computational Physics, 315, 476–500.
    [Google Scholar]
  10. Puscas, M.A., Enchéry, G. and Desroziers, S.
    [2018] Application of the mixed multiscale finite element method to parallel simulations of two-phase flows in porous media. Oil & Gas Science and Technology – Revue d’IFP Energies nouvelles, 73, 38.
    [Google Scholar]
  11. Raviart, P.A. and Thomas, J.M.
    [1977] A mixed finite element method for 2-nd order elliptic problems. In: Galligani, I. and Magenes, E. (Eds.) Mathematical Aspects of Finite Element Methods. Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 292–315.
    [Google Scholar]
  12. Saad, Y. and Schultz, M.H.
    [1986] GMRES: A generalized minimal residual algorithm for solving non-symmetric linear systems. SIAM Journal on Scientific and Statistical Computing, 7(3), 856–869.
    [Google Scholar]
  13. van der Vorst, H.A.
    [1992] Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing, 13(2), 631–644.
    [Google Scholar]
  14. Wang, K., Liu, H., Luo, J. and Chen, Z.
    [2018] Efficient CPR-type preconditioner and its adaptive strategies for large-scale parallel reservoir simulations. Journal ofComputational and Applied Mathematics, 328, 443–468.
    [Google Scholar]
  15. Younes, A., Ackerer, P. and Delay, F.
    [2010] Mixed finite elements for solving 2-D diffusion-type equations. Reviews of Geophysics, 48(1), RG1004.
    [Google Scholar]
http://instance.metastore.ingenta.com/content/papers/10.3997/2214-4609.202035072
Loading
/content/papers/10.3997/2214-4609.202035072
Loading

Data & Media loading...

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error