Goal oriented adaptivity for coupled flow and transport problems with applications in oil reservoir simulations. (English) Zbl 1173.76346
Summary: Goal oriented adaptivity has become an important and widely used technique for solving partial differential equations at a reduced computational cost. The technique is based on a computable estimate of the error in a linear functional of interest, and an adaptive strategy that improves the solution, for instance by refining the mesh locally according to the error estimator.
In this paper, we extend this theory to a multi-physical setting involving a coupled set of equations. Here adaptive algorithms become even more crucial since we do not only need estimators that indicate in which part of the domain the solution needs to be improved, but also which equation contributes most to the error, and thus, needs to be solved more accurately.
We derive a goal oriented error estimator for a system of coupled pressure and transport equations that serves as a basic model for oil reservoir simulation. We utilize a 2D model to present extensive numerical results, where we explain in detail how the adaptive algorithm works in practice.
In this paper, we extend this theory to a multi-physical setting involving a coupled set of equations. Here adaptive algorithms become even more crucial since we do not only need estimators that indicate in which part of the domain the solution needs to be improved, but also which equation contributes most to the error, and thus, needs to be solved more accurately.
We derive a goal oriented error estimator for a system of coupled pressure and transport equations that serves as a basic model for oil reservoir simulation. We utilize a 2D model to present extensive numerical results, where we explain in detail how the adaptive algorithm works in practice.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76S05 | Flows in porous media; filtration; seepage |
| 86A05 | Hydrology, hydrography, oceanography |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
Keywords:
finite element methods; adaptivity; duality; a posteriori error estimation; oil reservoir simulationReferences:
| [1] | Aarnes, J. E.; Gimse, T.; Lie, K.-A., An introduction to the numerics of flow in porous media using Matlab, Geometrical Modelling Numerical Simulation and Optimization, Industrial Mathematics at SINTEF (2005), Springer |
| [2] | J.E. Aarnes, V. Kippe, K.-A. Lie, A.B. Rustad, Modelling of multiscale structures in flow simulations for petroleum reservoirs, in: G. Hasle, K.-A. Lie, E. Quak (Eds.), Geometrical Modeling, Numerical Simulations and Optimization: Industrial Mathematics at SINTEF, Springer (in press).; J.E. Aarnes, V. Kippe, K.-A. Lie, A.B. Rustad, Modelling of multiscale structures in flow simulations for petroleum reservoirs, in: G. Hasle, K.-A. Lie, E. Quak (Eds.), Geometrical Modeling, Numerical Simulations and Optimization: Industrial Mathematics at SINTEF, Springer (in press). · Zbl 1178.76331 |
| [3] | Adams, R. A., Sobolev spaces, Pure and Applied Mathematics, vol. 65 (1975), Academic Press: Academic Press New York · Zbl 0186.19101 |
| [4] | Arbogast, T.; Bryant, S. L., A two-scale numerical subgrid technique for waterflood simulations, SPE J. (2002) |
| [5] | Arnold, D. N.; Brezzi, F., Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér., 19, 7-32 (1985) · Zbl 0567.65078 |
| [6] | Bangerth, W.; Rannacher, R., Adaptive Finite Element Methods for Differential Equations (2003), Birkhäuser · Zbl 1020.65058 |
| [7] | Bramble, J. H.; Xu, J., A local post-processing technique for improving the accuracy in mixed finite-element approximations, SIAM J. Numer. Anal., 26, 1267-1275 (1989) · Zbl 0688.65061 |
| [8] | Brenner, S. C.; Scott, L. R., The Mathematical Theory of Finite Element Methods (1994), Springer · Zbl 0804.65101 |
| [9] | Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods (1991), Springer · Zbl 0788.73002 |
| [10] | Dawson, C.; Sun, S.; Wheeler, M. F., Compatible algorithms for coupled flow and transport, Comput. Methods Appl. Mech. Engrg., 193, 2565-2580 (2004) · Zbl 1067.76565 |
| [11] | Eriksson, K.; Estep, D.; Hansbo, P.; Johnson, C., Introduction to adaptive methods for differential equations, Acta Numer., 1-54 (1995) |
| [12] | Eriksson, K.; Estep, D.; Hansbo, P.; Johnson, C., Computational Differential Equations (1996), Studentlitteratur · Zbl 0946.65049 |
| [13] | Eriksson, K.; Johnson, C., Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems, Math. Comp., 60, 167-188 (1993) · Zbl 0795.65074 |
| [14] | V. Heuveline, R. Rannacher, Adaptive FEM for eigenvalue problems with application in hydrodynamic stability analysis, in: Proceedings of the International Conference on Advances in Numerical Mathematics, September 16-17, 2005, Moscow, Special issue of J. Numer. Math. (in press).; V. Heuveline, R. Rannacher, Adaptive FEM for eigenvalue problems with application in hydrodynamic stability analysis, in: Proceedings of the International Conference on Advances in Numerical Mathematics, September 16-17, 2005, Moscow, Special issue of J. Numer. Math. (in press). · Zbl 1043.65122 |
| [15] | Johnson, C., Numerical Solution of Partial Differential Equations by the Finite Element Method (1987), Studentlitteratur · Zbl 0628.65098 |
| [16] | M.G. Larson and F. Bengzon, Adaptive finite element approximation of multiphysics problems, Preprint 2006, Commun. Numer. Methods Engrg. (submitted for publication).; M.G. Larson and F. Bengzon, Adaptive finite element approximation of multiphysics problems, Preprint 2006, Commun. Numer. Methods Engrg. (submitted for publication). · Zbl 1141.78328 |
| [17] | M.G. Larson, A. Målqvist, A posteriori error estimates in the energy norm for mixed finite element methods, Numer. Math. (submitted for publication).; M.G. Larson, A. Målqvist, A posteriori error estimates in the energy norm for mixed finite element methods, Numer. Math. (submitted for publication). · Zbl 1173.74431 |
| [18] | Larsson, S.; Thomée, V., Partial Differential Equations with Numerical Methods (2005), Springer · Zbl 1065.65002 |
| [19] | Lovadina, C.; Stenberg, R., Energy norm a posteriori error estimates for mixed finite element methods, Math. Comp., 75, 1659-1674 (2006) · Zbl 1119.65110 |
| [20] | Nédélec, J.-C., A new family of mixed finite elements in \(\textbf{R}^3\), Numer. Math., 50, 57-81 (1986) · Zbl 0625.65107 |
| [21] | Raviart, P.; Thomas, J., A mixed finite element method for second order elliptic problems, (Mathematical Aspects of the Finite Element Method, Lecture Notes in Mathematics, vol. 606 (1977), Springer), 292-315 · Zbl 0362.65089 |
| [22] | Stenberg, R., Postprocessing schemes for some mixed finite elements, RAIRO Modél. Math. Anal. Numér., 25, 151-167 (1991) · Zbl 0717.65081 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
