Characterizing the multiscale structure of fluctuations of transported quantities in a disordered medium

The transport of a scalar quantity in a disordered medium is a common problem in science and engineering. To understand the interplay between deterministic transport dynamics and stochasticity of the underlying microstructure, we analyse a simple model for unidirectional advection-diffusion-reaction over a random array of point sinks. The homogenized concentration distribution over a periodic array provides a leading-order approximation for a wide range of ergodic stationary random sink distributions of comparable mean density. However the fluctuations about this state depend strongly on the statistical properties of the array, the relative sizes of the scale-separation parameter (the ratio of mean inter-sink distance to domain size) and the physical parameters (expressed as dimensionless Péclet and Damköhler numbers). Using a combination of Monte-Carlo simulation and asymptotic analysis, we characterize the spatial variability and correlation statistics of the transported quantity and show how underlying regularity of the microstructure, particularly at low Péclet numbers, ensures a much smaller fluctuation magnitude than in the case of a uniformly random microstructure. Even when sink locations are almost uncorrelated to each other, we find that the concentration fluctuations correlate strongly over lengthscales comparable to the whole domain. Thus boundary conditions can determine the spatial profile of both the averaged leading-order distribution of the transported quantity and its fluctuations.