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This paper presents a theoretical method for the upscaling of the time-harmonic Maxwell equations. We use the eddy current approximation of the Maxwell equations to describe the fields in heterogeneous materials. The magnetic permeability of the media is assumed to have random heterogeneities given by a Gaussian random field. The upscaling is based on the coarse graining method which applies projections and Green's function formalism in Fourier space to scale the electric field. An upscaled Maxwell equation is derived which includes an effective magnetic permeability tensor. The effective permeability explicitly depends on the given scale for the upscaling. The scale-dependent permeability is calculated by a second-order perturbative expansion, and we discuss the future verification and the application of the results.

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In this study we present a numerical analysis for the self-averaging of the longitudinal dispersion coefficient for transport in heterogeneous media. This is done by investigating the mean-square sample-to-sample fluctuations of the dispersion for finite times and finite numbers of modes for a quasi-periodic random field using analytical arguments as well as numerical simulations. We consider transport of point-like injections in a quasi-periodic random field with a Gaussian correlation function. In particular, we focus on the asymptotic and pre-asymptotic behaviour of the fluctuations with the aid of a probability density function for the dispersion, and we verify the logarithmic growth of the sample-tosample fluctuations as earlier reported in [Eberhard, 2004]. We also comment on the choice of the relevant parameters to generate quasi-periodic realizations with respect to the self-averaging of transport in statistically homogeneous Gaussian velocity fields.