The computation of nonclassical shock waves with a heterogeneous
multiscale method

Kissling, Frederike; Rohde, Christian

doi:10.3934/nhm.2010.5.661

Cited by 20 publications

(9 citation statements)

References 19 publications

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“…The appearances and disappearances of instabilities and overshoots during two-phase flow in porous media are shown theoretically by means of travelling wave analysis by van Duijn et al [47]. Fingering is also discussed in detail by Rohde and Kissling [48], Kissling et al [49] and DiCarlo [50]. As the dynamic saturation front in a porous domain could vary within the domain and K r relationships are primarily point relationships, it becomes important to know the location dependent K r -S w relationship.…”

Section: Introductionmentioning

confidence: 99%

Scale dependency of dynamic relative permeability–satuartion curves in relation with fluid viscosity and dynamic capillary pressure effect

et al. 2016

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Capillary pressure-saturation-relative permeability relationships (P c -S w -K r ) are functions of importance in modeling and simulations of the hydrodynamics of two-phase flow in porous media. These relationships are found to be affected by porous medium and fluid properties but the manner in which they are affected is a topic of intense discussion. For example, reported trends in fluid viscosity and boundary conditions effects have been found to be contrary to each other in different studies. In this work, we determine the dependency of dynamic K r -S w relationships (averaged data) on domain scale in addition to investigating the effects of fluid viscosity and boundary pressure using silicone oil (i.e. 200 and 1000 cSt) and water as the respective non-wetting and wetting fluids with a view to eliminating some of the uncertainties reported in the literature. Water relative permeability, K rw , was found to increase with increasing wetting phase saturation but decreases with the increase in viscosity ratio. On the other hand, the oil relative permeability, K rnw , was found to increase with the increasing non-wetting phase saturation in addition to the increase in viscosity ratio. Also, it was found that with the increasing boundary pressure K rw decreases while K rnw increases. The influence of scale on relative permeability was slightly indicated in the non-wetting phase with K rnw decreasing as domain size increases. Effect of measurement location on dynamic relative permeability was explored which is rarely found in the literature. Comparison was also made between K r -S w relationships obtained under static and dynamic condition. Finally, mobility ratio (m) and dynamic coefficient (s) were plotted as a function of water saturation (S w ), which showed that m decreases as s increases at a given saturation, or vice versa.

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Section: Introductionmentioning

confidence: 99%

Scale dependency of dynamic relative permeability–satuartion curves in relation with fluid viscosity and dynamic capillary pressure effect

et al. 2016

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“…Most notably, the construction ensures mass conservation (Theorem 4.5). The whole approach can be understood as a multidimensional generalization of the one-dimensional work in [6,25]. While a complete error analysis of the HMM appears to be out of reach, we show in Theorem 4.6 that the HMM is at least L ∞ -stable and preserves non-negativity of the saturation.…”

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confidence: 85%

“…From the solution, the approximate plateau value S ij = S ε ij is determined. OUTPUT: Plateau value S ij For solving (6.2) and for the approximation of the plateau value, we use a semi-implicit Euler discretization and refer to [23,25] and the discussions therein. Alternatively, one can use the methods discussed in [1,11,16,20].…”

Section: Computationalmentioning

confidence: 99%

The Computation of Nonclassical Shock Waves in Porous Media with a Heterogeneous Multiscale Method: The Multidimensional Case

Kissling¹,

Rohde²

2015

Multiscale Model. Simul.

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We consider two-phase flow problems in porous media with overshooting waves. If higher order effects are neglected, such problems are governed by a hyperbolic conservation law with non-convex flux function as macroscale model. It is well known, that there can be multiple weak solutions. To ensure uniqueness, we use on the microscale either a kinetic relation or in a second approach an extended system, taking rate-dependent capillary pressure effects into account. We present a new multidimensional mass-conserving numerical method to solve the macroscale model. The method belongs to the class of Heterogeneous Multiscale Methods in the sense of E&Engquist (2003) and is L ∞-stable. A key part of the approximation is a novel numerical flux function for the multidimensional setting, which captures undercompressive waves and generalizes the one-dimensional approach of Boutin et al. (2008). Furthermore, we improve the overall computational complexity, using a data-based approach. Finally, we validate the numerical method and test it on several infiltration problems.

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“…The second model we consider in our experiments stems from the work , which is an extension of to the multidimensional case. Therein, a two‐phase flow problem in porous media on a 2D domain is investigated.…”

Section: Applicationsmentioning

confidence: 99%

Surrogate modeling of multiscale models using kernel methods

Wirtz

Karajan²,

Haasdonk

2014

Int. J. Numer. Meth. Engng

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SUMMARYThis work investigates the possibilities of acceleration and approximation of multiscale systems using kernel methods. The key element is to learn the interface between the different scales using a fast surrogate for the microscale model, which is given by multivariate kernel expansions. The expansions are computed using statistically representative samples of input and output of the microscale model. We apply both support vector machines and a vectorial kernel greedy algorithm as learning methods. We demonstrate the applicability of the resulting surrogate models using two multiscale models from different engineering disciplines. We consider, first, a human spine model coupling a macroscale multibody system with a microscale intervertebral spine disc model and, second, a model for simulation of saturation overshoots in porous media involving nonclassical shock waves. Copyright © 2014 John Wiley & Sons, Ltd.

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The computation of nonclassical shock waves with a heterogeneous multiscale method

Cited by 20 publications

References 19 publications

Scale dependency of dynamic relative permeability–satuartion curves in relation with fluid viscosity and dynamic capillary pressure effect

Scale dependency of dynamic relative permeability–satuartion curves in relation with fluid viscosity and dynamic capillary pressure effect

The Computation of Nonclassical Shock Waves in Porous Media with a Heterogeneous Multiscale Method: The Multidimensional Case

Surrogate modeling of multiscale models using kernel methods

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