Numerical methods for multiscale transport equations and application to two-phase porous media flow

Yue, Xingye; Weinan, E

doi:10.1016/j.jcp.2005.05.009

Cited by 20 publications

(11 citation statements)

References 22 publications

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“…An alternative derivation of the model for miscible flows in porous media can be obtained using the heterogeneous multiscale method (HMM) for numerical homogenization [52]. The heterogeneous multiscale method was used for the numerical homogenization of the two-phase immiscible flows from Darcy's level to the reservoir level (see [53,54,55] and references therein), but not for the upscaling from the pore level. In principle, the general strategy of the HMM numerical homogenization could be used, jointly with the rigorous homogenization results presented in this section, to handle more complicate chemistry and non-periodic porous media.…”

Section: Modeling Single Phase Mixtures Through Upscalingmentioning

confidence: 99%

“…We also note that the authors of [55] study the immiscible two-phase flows with only one single component, and the dispersion is not treated. Similarly, in [53] the effect of capillary pressure was neglected. Therefore, in our assessment, the HMM approach to miscible flows is a promising approach that may require further developments.…”

Section: Modeling Single Phase Mixtures Through Upscalingmentioning

confidence: 99%

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Analytical and variational numerical methods for unstable miscible displacement flows in porous media

Scovazzi

Wheeler

Mikelić

et al. 2017

Journal of Computational Physics

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International audienceThe miscible displacement of one fluid by another in a porous medium has received considerable attention in sub-surface, environmental and petroleum engineering applications. When a fluid of higher mobility displaces another of lower mobility, unstable patterns-referred to as viscous fingering-may arise. Their physical and mathematical study has been the object of numerous investigations over the past century. The objective of this paper is to present a review of these contributions with particular emphasis on variational methods. These algorithms are tailored to real field applications thanks to their advanced features: handling of general complex geometries, robustness in the presence of rough tensor coefficients, low sensitivity to mesh orientation in advection dominated scenarios, and provable convergence with fully unstructured grids. This paper is dedicated to the memory of Dr. Jim Douglas Jr., for his seminal contributions to miscible displacement and variational numerical methods

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Section: Modeling Single Phase Mixtures Through Upscalingmentioning

confidence: 99%

Section: Modeling Single Phase Mixtures Through Upscalingmentioning

confidence: 99%

Analytical and variational numerical methods for unstable miscible displacement flows in porous media

Scovazzi

Wheeler

Mikelić

et al. 2017

Journal of Computational Physics

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“…This way, we related the required non-dimensional source term for the energy equation to the non-dimensional source term in the equations for the vapor and liquid mass fractions S l!v . Inserting the detailed expressions for S nuc and S e c from (4) and (12), respectively, we may write…”

Section: Non-dimensional Formulationmentioning

confidence: 99%

Efficient second‐order time integration for single‐species aerosol formation and evolution

Winkelmann

Nordlund

Kuczaj

et al. 2013

Numerical Methods in Fluids

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SUMMARY The dynamics of a single‐species aerosol composed of droplets in air is described in terms of nucleation, evaporation, condensation, and coagulation processes. We present a comprehensive overview of the Euler–Euler formulation, which gives rise to a model in which fast nucleation that initiates aerosol droplets co‐exists with comparably slow condensation. The latter process is responsible for the subsequent growth of the droplets. To accurately represent the dynamical consequences of the fast nucleation process, while retaining numerical efficiency, a new second‐order time‐integration method for the nucleation, evaporation, and condensation processes is proposed and analyzed. The new time‐integration method takes the form of a ‘corrected Euler forward’ method. It includes rapid nucleation bursts and their possible cessation within a time step Δt. If the current nucleation burst persists for longer than the next time step, it is included fully, whereas cessation of the nucleation burst within the next Δt implies corrections to the effective rates in the algorithm. The identification of these two situations corresponds to the physical mechanism by which nucleation of a supersaturated vapor is halted because of the progressing condensation onto the already formed droplets. The resulting time‐integration method is shown to be second‐order accurate in time, whereas the computational costs per time step were found to be increased by less than 25% compared with the Euler forward method. The new method is also applied in combination with advective transport of the aerosol forming vapor to investigate a front of rapid nucleation. Adopting robust first‐order upwinding for the spatial discretization, we arrive at a flexible method that shows an overall first‐order convergence in Δt. For the full, spatially dependent system motivated by an aerosol of water droplets in air, the computational benefits of the new time‐integration method over the Euler forward scheme, are a factor of about 10 improvements in accuracy at a given Δt and a similar factor in computing time when keeping the same level of accuracy. Copyright © 2013 John Wiley & Sons, Ltd.

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“…However, in spite that, these traditional methods are widely used and perform very well in some situations; they are incapable of dealing with the large‐scale dynamic problems of the heterogeneous porous media on account of the large degrees of freedom (DOFs) introduced by resolving all small characteristic scales on the fine‐scale model and the low computational efficiency attributed to the computer memory and CPU time. Recently, a number of multiscale methods have been developed to circumvent these difficulties , such as the hierarchical multiscale method , the variational multiscale finite element method (MsFEM) , the modified MsFEM , the multiscale FVM , and the finite volume MsFEM , and so on. As one of the popular multiscale methods, the MsFEM, which can be dated back to the work presented by Babuska et al .…”

Section: Introductionmentioning

confidence: 99%

A coupling extended multiscale finite element method for dynamic analysis of heterogeneous saturated porous media

Zhang

Zheng

2015

Int. J. Numer. Meth. Engng

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A coupling extended multiscale finite element method (CEMsFEM) is developed for the dynamic analysis of heterogeneous saturated porous media. The coupling numerical base functions are constructed by a unified method with an equivalent stiffness matrix. To improve the computational accuracy, an additional coupling term that could reflect the interaction of the deformations among different directions is introduced into the numerical base functions. In addition, a kind of multi-node coarse element is adopted to describe the complex high-order deformation on the boundary of the coarse element for the two-dimensional dynamic problem. The coarse element tests show that the coupling numerical base functions could not only take account of the interaction of the solid skeleton and the pore fluid but also consider the effect of the inertial force in the dynamic problems. On the other hand, based on the static balance condition of the coarse element, an improved downscaling technique is proposed to directly obtain the satisfying microscopic solutions in the CEMsFEM. Both one-dimensional and two-dimensional numerical examples of the heterogeneous saturated porous media are carried out, and the results verify the validity and the efficiency of the CEMsFEM by comparing with the conventional finite element method.

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Numerical methods for multiscale transport equations and application to two-phase porous media flow

Cited by 20 publications

References 22 publications

Analytical and variational numerical methods for unstable miscible displacement flows in porous media

Analytical and variational numerical methods for unstable miscible displacement flows in porous media

Efficient second‐order time integration for single‐species aerosol formation and evolution

A coupling extended multiscale finite element method for dynamic analysis of heterogeneous saturated porous media

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