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Shvidler, M. I.; Karasaki, Kenzi

doi:10.1023/a:1021129325701

Cited by 35 publications

(9 citation statements)

References 3 publications

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“…Because the methods of Euclidean geometry, which ordinarily deals with regular sets, are purely suited for describing objects such as in nature, stochastic models are taken into account [4]. Another possible way of describing a complex structure of the pore space is to use fractal theory of sets of fractional dimensionality [5]- [8].…”

Section: Introductionmentioning

confidence: 99%

Fractional hydrodynamic equations for fractal media

2005

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We use the fractional integrals in order to describe dynamical processes in the fractal media. We consider the "fractional" continuous medium model for the fractal media and derive the fractional generalization of the equations of balance of mass density, momentum density, and internal energy. The fractional generalization of Navier-Stokes and Euler equations are considered. We derive the equilibrium equation for fractal media. The sound waves in the continuous medium model for fractional media are considered.

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Section: Introductionmentioning

confidence: 99%

Fractional hydrodynamic equations for fractal media

2005

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“…[9,10] and references therein). Most approaches start by employing the Reynolds decomposition, A = A + A , to represent the random quantities in (3.6) as the sum of their ensemble means A and zero-mean fluctuations about the mean A .…”

Section: Cumulative Distribution Function Methodsmentioning

confidence: 99%

“…Here, we employ the so-called LED closure, which replaces the random Green's function G with its 'mean-field approximation' G. The latter is given by a solution of (3.9) with the average velocity ṽ used in place of its random counterpartṽ. Assuming that the CDF gradient∇F(ỹ, t) varies slowly in space and time, we obtain a closed-form expression for the cross-correlation term Q (appendix A), 10) where the Einstein notation is used to indicate summation over repeated indices. …”

Section: Cumulative Distribution Function Methodsmentioning

confidence: 99%

Cumulative distribution function solutions of advection–reaction equations with uncertain parameters

2014

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We derive deterministic cumulative distribution function (CDF) equations that govern the evolution of CDFs of state variables whose dynamics are described by the first-order hyperbolic conservation laws with uncertain coefficients that parametrize the advective flux and reactive terms. The CDF equations are subjected to uniquely specified boundary conditions in the phase space, thus obviating one of the major challenges encountered by more commonly used probability density function equations. The computational burden of solving CDF equations is insensitive to the magnitude of the correlation lengths of random input parameters. This is in contrast to both Monte Carlo simulations (MCSs) and direct numerical algorithms, whose computational cost increases as correlation lengths of the input parameters decrease. The CDF equations are, however, not exact because they require a closure approximation. To verify the accuracy and robustness of the large-eddydiffusivity closure, we conduct a set of numerical experiments which compare the CDFs computed with the CDF equations with those obtained via MCSs. This comparison demonstrates that the CDF equations remain accurate over a wide range of statistical properties of the two input parameters, such as their correlation lengths and variance of the coefficient that parametrizes the advective flux.

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“…If the field ( ) − V x y is differentiable, the convergence of the convolution in (19) depends on the behavior of the integrand at 0 = y and → ∞ y . An investigation of the asymptotic behaviors of the function ( ) G y (Shvidler, 1966;1985) showed that if scale heterogeneity is finite and y is very small, the principal part of the Green`s function is ( )~1/ 4 G y y πσ * . Here…”

Section: Global Symmetrymentioning

confidence: 99%

“…In some cases they are more-or-less hypothetical (Saffman, 1971;Dagan, 1979). Approximate nonlocal equations have been developed in the framework of perturbation methods (Lomakin, 1970;Shvidler, 1985;Keller, 2001). Neuman and Orr (1993) considered steady flow in a bounded domain and found the averaged exact system of two equations.…”

Section: Introductionmentioning

confidence: 99%

Exact Averaging of Stochastic Equations for Flow in Porous Media

Shvidler

Karasaki

2007

Transp Porous Med

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It is well known that at present, exact averaging of the equations for flow and transport in random porous media have been proposed for limited special fields. Moreover, approximate averaging methods-for example, the convergence behavior and the accuracy of truncated perturbation series-are not well studied, and in addition, calculation of high-order perturbations is very complicated. These problems have for a long time stimulated attempts to find the answer to the question: Are there in existence some, exact, and sufficiently general forms of averaged equations? Here, we present an approach for finding the general exactly averaged system of basic equations for steady flow with sources in unbounded stochastically homogeneous fields. We do this by using (1) the existence and some general properties of Green's functions for the appropriate stochastic problem, and (2) some information about the random field of conductivity. This approach enables us to find the form of the averaged equations without directly solving the stochastic equations or using the usual assumption regarding any small parameters. In the common case of a stochastically homogeneous conductivity field we present the exactly averaged new basic nonlocal equation with a unique kernel-vector. We show that in the case of some type of global symmetry (isotropy, transversal isotropy, or orthotropy), we can for threedimensional and two-dimensional flow in the same way derive the exact averaged nonlocal equations with a unique kernel-tensor. When global symmetry does not exist, the nonlocal equation with a kernel-tensor involves complications and leads to an ill-posed problem.

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Cited by 35 publications

References 3 publications

Fractional hydrodynamic equations for fractal media

Fractional hydrodynamic equations for fractal media

Cumulative distribution function solutions of advection–reaction equations with uncertain parameters

Exact Averaging of Stochastic Equations for Flow in Porous Media

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