Unconditionally convergent nonlinear solver for hyperbolic conservation laws with S-shaped flux functions

Jenny, Patrick; Tchelepi, Hamdi A.; Lee, Seong H.

doi:10.1016/j.jcp.2009.06.032

Cited by 102 publications

(72 citation statements)

References 9 publications

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“…Nevertheless, it is safe to state that reservoir simulation using standard FIM (i.e., Implicit-PPU) is far from being unconditionally convergent [8,18]. In practice, nonlinear convergence problems in the course of a reservoir-simulation are a major concern because they can easily cause severe restrictions on the timestep size that can be used.…”

Section: Introductionmentioning

confidence: 99%

Hybrid upwind discretization of nonlinear two-phase flow with gravity

Lee

Efendiev

Tchelepi

2015

Advances in Water Resources

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Section: Introductionmentioning

confidence: 99%

Hybrid upwind discretization of nonlinear two-phase flow with gravity

Lee

Efendiev

Tchelepi

2015

Advances in Water Resources

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“…These include the development of improved numerical schemes for simulations of multidimensional wave-oriented upwind schemes [33], an exponential integrator for advection-dominated reactive flow [73], and an unconditionally convergent nonlinear solver for hyperbolic conservation laws [45]. An open-source MATLAB implementation offering a flexible discretization which can be used in more complex structures has also been developed [56].…”

mentioning

confidence: 99%

“…The implementation of these methods is described in detail, and the associated code is made available as part of the widely used open source finite element library deal.II [6,7] through the extensively documented tutorial program Step-43 [20]. We note that for more complex models than the ones considered here, an additional point that needs to be thoroughly addressed is the design of nonlinear solvers (see, for example, [45]), but we will not consider this here.…”

mentioning

confidence: 99%

An $h$-Adaptive Operator Splitting Method for Two-Phase Flow in 3D Heterogeneous Porous Media

Chueh¹,

Djilali²,

Bangerth³

2013

SIAM J. Sci. Comput.

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Abstract. The simulation of multiphase flow in porous media is a ubiquitous problem in a wide variety of fields, such as fuel cell modeling, oil reservoir simulation, magma dynamics, and tumor modeling. However, it is computationally expensive. This paper presents an interconnected set of algorithms which we show can accelerate computations by more than two orders of magnitude compared to traditional techniques, yet retains the high accuracy necessary for practical applications. Specifically, we base our approach on a new adaptive operator splitting technique driven by an a posteriori criterion to separate the flow from the transport equations, adaptive meshing to reduce the size of the discretized problem, efficient block preconditioned solver techniques for fast solution of the discrete equations, and a recently developed artificial diffusion strategy to stabilize the numerical solution of the transport equation. We demonstrate the accuracy and efficiency of our approach using numerical experiments in one, two, and three dimensions using a program that is made available as part of a large open source library.

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“…The second part of the nonlinear solver is the limitation of variables to avoid oscillatory behavior of the nonlinear Newton loop, usually associated with relative permeability function. One solution was proposed by Jenny et al (2009) and based on the fact that the saturation should not pass the inflection point of the phase flux function. However, computation of the inflection point is a challenging problem in itself for general fractional flow curves, since the viscosity is a nonlinear function of pressure and composition.…”

Section: Nonlinear Solvermentioning

confidence: 99%

“…However, computation of the inflection point is a challenging problem in itself for general fractional flow curves, since the viscosity is a nonlinear function of pressure and composition. An efficient method for general purpose compositional simulation is to use under-relaxation of saturation in the neighborhood of the inflection point (Jenny et al 2009) to prevent oscillations in the Newton process due to non-convexity in flux function.…”

Section: Nonlinear Solvermentioning

confidence: 99%

An Extended Natural Variable Formulation for Compositional Simulation Based on Tie-Line Parameterization

Voskov

2011

Transp Porous Med

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A nonlinear formulation based on extension of natural variables set is proposed for modeling compositional two-phase flow in porous media. The focus here is on numerical general-purpose simulation using the fully implicit method. In the formulation, the phase fraction and the saturation change "continuously" in the immiscible region of the compositional space (i.e., sub-critical region). Inside the two-phase region, these variables are identical to the saturation and phase-fraction of the standard approach. In the single-phase regions, however, these saturation-like and phase-fraction-like variables can become negative, or larger that unity. We demonstrate that when this variable set is used, the equation-of-state (EoS)-based thermodynamic equilibrium computations are resolved completely within the global Newton loop. That is, need not to separate things into phase stability and flash computations. Compared to the standard natural variables approach, the number of global Newton iterations grows only slightly, but overall, the new approach leads to more efficient simulations. Moreover, the continuous variation of both the saturation and phase fraction across phase boundaries results in improved behavior of the nonlinear (Newton) solver. Two different strategies are used to deal with the densities. The first scheme honors the nonlinear dependence of the overall density on phase fractions and saturation, and the second employs a linearized relation for the overall density. Both schemes are compared with the standard natural variables formulation using several challenging compositional problems.

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Unconditionally convergent nonlinear solver for hyperbolic conservation laws with S-shaped flux functions

Cited by 102 publications

References 9 publications

Hybrid upwind discretization of nonlinear two-phase flow with gravity

Hybrid upwind discretization of nonlinear two-phase flow with gravity

An $h$-Adaptive Operator Splitting Method for Two-Phase Flow in 3D Heterogeneous Porous Media

An Extended Natural Variable Formulation for Compositional Simulation Based on Tie-Line Parameterization

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