Upscaling for stationary transport in heterogeneous porous media

This paper focuses on upscaling of the transport equation for heterogeneous porous media with random flow. We develop an upscaling by the coarse graining method which considers the local flow field being a stationary random field and which is based on filtering procedures in Fourier space. The coarse graining method is used to obtain an upscaled dispersion tensor which depends on the given length scale of the upscaling. We give explicit results for the scale-dependent dispersion coefficient in lowest-order perturbation theory. For a global upscaling the upscaled value agrees with the well-known macrodispersion coefficient. Furthermore, we develop a numerical upscaling scheme based on coarse graining which allows us to test the theoretical results. The numerical coarse graining also allows us to compute the upscaled dispersion for higher variances. The theoretical results are in agreement with the numerical simulations.