Solving 3D Subsurface Flow and Transport with Adaptive Multigrid
Publication: Journal of Hydrologic Engineering
Volume 5, Issue 1
Abstract
This paper presents an adaptive multigrid approach, combining multigrid methods and adaptive local grid refinement, to solve 3D density-dependent flow and transport problems in the subsurface. This approach is incorporated with the Galerkin finite-element method to solve the modified Richards' equation in the flow module and the Lagrangian-Eulerian finite-element method to solve the advection-dispersion equation in the transport module, which are coupled through density effects. With multigrid methods, the linear/linearized matrix equations can be solved with O(n) manipulations to save computer time. With adaptive local grid refinement, computational accuracy is improved by refining only rough regions of the problem domain and the computation is efficiently achieved because computer efforts are focused on those regions. In this work, rough regions are determined by examining the mesh Péclet number during each nonlinear iteration of the flow module and by checking the smoothness of the Lagrangian concentrations over global elements in the Lagrangian step of the transport module. Based on the detected rough regions, a modular setting of grid generation is employed to generate local zooming grids as well as to prepare the information needed for applying multigrid methods. Two examples are given to demonstrate the success of this approach.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Anderson, D. A., Tannehill, J. C., and Pletcher, R. H. (1984). Computational fluid mechanics, and heat transfer, McGraw-Hill, New York.
2.
Babuska, I., Zienkiewicz, O. C., Gago, J., and de A. Oliverira, E. R., eds. (1986). Accuracy estimates and adaptive refinements in finite element computations. Wiley, Chichester, England, U.K.
3.
Berger, M. J., and Oliger, J. (1984). “Adaptive mesh refinement for hyperbolic partial differential equations.” J. Computational Phys., 53, 484–512.
4.
Bornemann, F., Erdmann, B., and Kornhuber, R. (1993). “Adaptive multilevel methods in three space dimensions.” Int. J. Numer. Methods in Engrg., 36, 3187–3203.
5.
Brandt, A. ( 1977). “Multi-level adaptive solutions to boundary-value problems.” Mathematics of Computation, 31(138), 333–390.
6.
Brebbia, C. A., and Aliabadi, M. H., eds. ( 1993). Adaptive finite and boundary element methods. Computational Mechanics Publications, Southampton, U.K.
7.
Briggs, W. L. ( 1987). A multigrid tutorial. Society of Industrial and Applied Mathematics, Lancaster Press, Pa.
8.
Cheng, H. P., Cheng, J. R., and Yeh, G. T. (1996a). “A particle tracking technique for the Lagrangian-Eulerian finite element method in multi-dimensions.” Int. J. Numer. Methods in Engrg., 39, 1115–1136.
9.
Cheng, H. P., Cheng, J. R., and Yeh, G. T. (1998a). “A Lagrangian-Eulerian method with adaptively local zooming approach to solve three-dimensional advection-diffusion transport equations.” Int. J. Numer. Methods in Engrg., 41, 587–615.
10.
Cheng, H. P., Yeh, G. T., Xu, J., Li, M. H., and Carsel, R. (1998b). “A study of incorporating the multigrid method into the three-dimensional finite element discretization: A modular setting and application.” Int. J. Numer. Methods in Engrg., 41, 499–526.
11.
Cheng, J. R., Cheng, H. P., and Yeh, G. T. (1996b). “A Lagrangian-Eulerian method with adaptively local zooming and peak/valley capturing approach to solve two-dimensional advection-diffusion transport equations.” Int. J. Numer. Methods in Engrg., 39, 987–1016.
12.
Cheng, J. R., Shobl, R. O., Yeh, G. T., Lin, H. C., and Choi, W. H. (1998). “Modeling of 2D density-dependent flow and transport in the subsurface.”J. Hydrologic Engrg., ASCE, 3(4), 248–257.
13.
Cussler, E. L. (1984). Diffusion: Mass transfer in fluid systems. Cambridge University Press, Cambridge, U.K.
14.
Demkowicz, L., Oden, J. T., Rachowicz, W., and Hardy, O. (1989). “Toward a universal h-p adaptive finite element strategy, Part 1. Constrained approximation and data structure.” Comp. Methods in Appl. Mechanic and Engrg., 77, 79–112.
15.
Galeati, G., Gambolati, G., and Neumann, S. P. (1992). “Coupled and partially coupled Eulerian-Lagrangian model of freshwater-seawater mixing.” Water Resour. Res., 28(1), 149–165.
16.
Hackbusch, W. (1985). Multi-grid methods and applications. Springer, Berlin.
17.
Henry, H. R. (1964). “Effect of dispersion on salt encroachment in coastal aquifers, sea water in coastal aquifers.” U.S. Geological Survey Water Supply Paper 1613-C, H. H. Cooper et al., eds., Washington, D. C., 35–69.
18.
Herbert, A. W., Jackson, C. P., and Lever, D. A. (1988). “Coupled groundwater flow and solute transport with fluid density strongly dependent upon concentration.” Water Resour. Res., 24(10), 1781–1795.
19.
Huyakorn, P. S., Anderson, P. F., Mercer, J. W., and White, H. O. (1987). “Saltwater intrusion in aquifer: Development and testing of a three-dimensional finite element model.” Water Resour. Res., 23(2), 293–312.
20.
Johns, R. T., and Resele, G. (1997). “Solutions and scaling of one-dimensional groundwater-solute flow with large density variations.” Water Resour. Res., 33(6), 1327–1334.
21.
McCormick, S. F. ( 1989). “Multilevel adaptive methods for partial differential equations.” Frontiers in applied mathematics, Vol. 6, Society for Industrial and Applied Mathematics, Philadelphia, Pa.
22.
Park, C. (1996). Closed-form solutions for steady state density-dependent flow and transport in a vertical soil column.” Water Resour. Res., 32(5), 1317–1322.
23.
Press, W. H., Teukoiskey, S. A., Vetterling, W. T., and Flannery, B. P. (1992). Numerical recipes in FORTRAN. Cambridge University Press, Cambridge, U.K.
24.
Rivara, M. C., and Levin, C. (1992). “A 3-D refinement algorithm suitable for adaptive and multi-grid techniques.” Communications in Appl. Numer. Methods, 8, 281–290.
25.
Rüde, U. ( 1993). “Mathematically and computational techniques for multilevel adaptive methods.” Frontiers in applied mathematics, Vol. 13, Society for Industrial and Applied Mathematics, Philadelphia, Pa.
26.
Trompert, R. A., Verwer, J. G., and Blom, J. G. (1993). “Computing brine transport in porous media with an adaptive-grid method.” Int. J. Numer. Methods in Fluids, 16, 43–63.
27.
Tübingen, J. B. (1995). “Tetrahedral grid refinement.” Computing, 55, 355–378.
28.
Wesseling, P. (1992). An introduction to multigrid methods. Wiley, New York.
29.
Wille, S. ∅. (1997). “Adaptive linearization and grid iterations with the tri-tree multigrid refinement-recoarsement algorithm for the Navier-Stokes equations.” Int. J. Numer. Methods in Fluids, 24, 155–168.
30.
Xu, J. (1992). “Iterative methods by space decomposition and subspace correction.” SIAM Rev., 34(4), 581–613.
31.
Yeh, G. T. (1981). “On the computation of Darcian velocity and mass balance in the finite element modeling of groundwater flow.” Water Resour. Res., 17(5), 1529–1534.
32.
Yeh, G. T. (1987). “3DFEMWATER: A three-dimensional finite element model of water flow through saturated-unsaturated media.” ORNL-6386, Oak Ridge National Laboratory, Oak Ridge, Tenn.
33.
Yeh, G. T. (1990). “A Lagrangian-Eulerian method with zoomable hidden fine-mesh approach to solving advection-dispersion equations.” Water Resour. Res., 26(6), 1133–1144.
34.
Yeh, G. T., Chang, J. R., Gwo, J. P., Lin, H. C., Richards, D. R., and Martin, W. D. (1994). “3DSALT: A three-dimensional finite element model of density-dependent flow and transport through saturated-unsaturated media.” U.S. Army Corps of Engineers, Instruction Rep. HL-94-1, Vicksburg, Miss.
35.
Zhang, S. ( 1988). “Multi-level iterative techniques,” PhD thesis, Dept. of Mathematics, Pennsylvania State University, University Park, Pa.
Information & Authors
Information
Published In
Journal of Hydrologic Engineering
Volume 5 • Issue 1 • January 2000
Pages: 74 - 81
History
Received: Jul 7, 1998
Published online: Jan 1, 2000
Published in print: Jan 2000
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.