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We present a numerical simulation study of advective-diffusive scalar transport in three-dimensional high-contrast discontinuous permeability fields, generated with a truncated pluri-Gaussian geostatistical approach. A range of permeability contrasts, correlation lengths, and Péclet numbers are studied to characterise the transition to non-Fickian transport behaviour. This is triggered by high permeability contrasts between different zones and is enhanced by the presence of connected higher permeability channels, which are characterised by high advective flow velocities. In this case, the overall transport behaviour deviate from the macroscopic advection-dispersion model based on a Fickian analogy. The numerical experiments are run with an Eulerian approach using a novel unified numerical framework based on the finite-volume library \of, for i) generating random pluri-Gaussian porous media, ii) solving the steady state Darcy-scale flow, iii) solving the advection diffusion equation, iv) computing post-processing quantities such as first order statistics, spatial probability density functions and breakthrough curves. We identify a hierarchy of non-Fickian transport triggering factors, the strength of permebility contrast being the pivotal driver. The Péclet number and the characteristic length at which facies transitions are observed as secondary factors. Transport remains Fickian when the facies conductivities differ by up to one order of magnitude. Greater permeability contrasts act strengthen the emergence of fast flow channels leading to non-Fickian transport.