A numerical modeling of multicomponent compressible flows in porous media with multiple wells by an Eulerian-Lagrangian method

Wang, Hong; Zhang, Weidong; Ewing, E.

doi:10.1007/s00791-005-0153-8

Cited by 11 publications

(6 citation statements)

References 20 publications

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“…By design, it automatically preserves the local mass constraint of tracer, but violates the local volume constraint of the bulk fluid due to numerical approximation of trace-back regions. Many ELLAM schemes are based on similar principles (see, e.g., [11,14,27,28,26]).…”

Section: Introductionmentioning

confidence: 99%

Stability, Monotonicity, Maximum and Minimum Principles, and Implementation of the Volume Corrected Characteristic Method

Arbogast¹,

Wang²

2011

SIAM J. Sci. Comput.

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We consider the volume corrected characteristics-mixed method (VCCMM) for tracer transport problems. The volume correction adjustment maintains the local volume conservation of bulk fluids and the numerical convergence of the method. We discuss some details of implementation by considering the scheme from an algebraic point of view. We show that the volume correction adjustment is important for stability and necessary for the monotonicity and the maximum and minimum principles of the method. We also derive a relatively weaker stability property for the uncorrected characteristic-mixed method (CMM). Some numerical experiments of a quarter "fivespot" pattern of wells are given to verify our theoretical results and compare the concentration errors of VCCMM and CMM due to random perturbations set up in the computation of the algorithm. More numerical tests, including one related to long-time nuclear waste storage, are given to compare VCCMM with CMM and Godunov's method, showing that VCCMM exhibits no overshoots or undershoots and less numerical diffusion.

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Section: Introductionmentioning

confidence: 99%

Stability, Monotonicity, Maximum and Minimum Principles, and Implementation of the Volume Corrected Characteristic Method

Arbogast¹,

Wang²

2011

SIAM J. Sci. Comput.

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“…Eulerian-Lagrangian schemes have been developed to approximate the advectiondiffusion equation (12), using Lagrangian characteristic methods for the transport and a fixed Eulerian grid for the diffusion. Included are the Eulerian-Lagrangian localized adjoint methods (ELLAM) [5,7,23,25,24] and the characteristics-mixed method (CMM) [1,3] and its two-phase variant [11], which are ELLAM schemes but emphasize their development in terms of the local mass constraint.…”

Section: Introductionmentioning

confidence: 99%

Convergence of a Fully Conservative Volume Corrected Characteristic Method for Transport Problems

Arbogast¹,

Wang²

2010

SIAM J. Numer. Anal.

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We consider the convergence of a volume corrected characteristics-mixed method (VCCMM) for advection-diffusion systems. It is known that, without volume correction, the method is first order convergent, provided there is a nondegenerate diffusion term. We consider the advective part of the system and give some properties of the weak solution. With these properties we prove that the volume corrected method, with no diffusion term, gives a lower order L 1-convergence rate of O(h/ √ Δt + h + (Δt) r), where r is related to the accuracy of the characteristic tracing. This result compares favorably to Godunov's method, but avoids the CFL constraint, so large time steps can be taken in practice. In fact, Godunov's method converges at O(h 1/2), which is our result for Δt = Ch, where now C is not limited. However, the optimal choice, Δt = Ch 2/(2r+1) , gives a better rate, O(h 2r/(2r+1)), than Godunov's method, e.g., O(h 2/3) if r = 1. With a nondegenerate diffusion term, we obtain an L 2-error estimate for the problem. We also prove the existence of, and give an error estimate for, a perturbed velocity field for which the volume is locally conserved. Finally, some convergence tests are given to verify the optimal convergence rate for r = 1.

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“…Note that the analytical tracking obtained from the mixed formulation is incorrect within the well elements. For example, in the center of an injection well, the magnitude of the velocity with the mixed approximation is zero, although it should have the largest value (see [15] for details). With triangular discretization, this problem can be easily circumvented by using a very fine mesh in the well region.…”

Section: Efficiency Of the Ellam In Highly Heterogeneous Domains Inclmentioning

confidence: 99%

Efficiency of the Eulerian–Lagrangian localized adjoint method for solving advection–dispersion equations on highly heterogeneous media

Ramasomanana

Younès

2011

Numerical Methods in Fluids

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SUMMARY The Eulerian–Lagrangian Localized Adjoint Method (ELLAM) is well adapted for advection‐dominated transport problems. The method can use large time steps because the advection term is accurately approximated without any CFL restriction. A new formulation of ELLAM was developed by Younes et al. (Adv. Water Resour. 2006; 29:1056–1074) for unstructured triangular meshes. This formulation uses only strategic numerical integration points and thus requires a very limited number of integration points (usually 1 per element). To avoid numerical dispersion due to interpolation when several time steps are used, the method continuously tracks characteristics, and only changes due to dispersion are interpolated at each time step. In this work, we study the applicability and efficiency of this formulation for highly heterogeneous domains. The results show that (i) this formulation remains efficient even for highly heterogeneous domains, (ii) special care must be taken for the approximation of the dispersive term when large time steps are used, (iii) interpolation at intermediate times can be used to reduce memory requirement for problems with a large number of elements and small time steps, and (iv) the method can be used successfully for heterogeneous problems including injection and pumping wells, and remains much more efficient than the discontinuous Galerkin finite element method even when small time steps are required. Copyright © 2011 John Wiley & Sons, Ltd.

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A numerical modeling of multicomponent compressible flows in porous media with multiple wells by an Eulerian-Lagrangian method

Cited by 11 publications

References 20 publications

Stability, Monotonicity, Maximum and Minimum Principles, and Implementation of the Volume Corrected Characteristic Method

Stability, Monotonicity, Maximum and Minimum Principles, and Implementation of the Volume Corrected Characteristic Method

Convergence of a Fully Conservative Volume Corrected Characteristic Method for Transport Problems

Efficiency of the Eulerian–Lagrangian localized adjoint method for solving advection–dispersion equations on highly heterogeneous media

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