Trajectories as Training Images to Simulate Advective‐Diffusive, Non‐Fickian Transport

We propose a spatial Markov model to simulate transport in three‐dimensional complex porous media flows. Our methodology is inspired by the concept of training images from geostatistics. Instead of using a training image we use highly resolved training trajectories obtained by high‐resolution particle tracking, from which we sample increments in our random walk model. To reflect higher‐order processes, subsequent increments are correlated. The approach can be split into three steps. First, we subdivide ( cut ) the training trajectories to form an archive of trajectory segments. Next, we recursively sample segments, where subsequent samples are chosen conditioned to the previous one to ensure continuity and smoothness of velocity ( conditional copy ). Finally, we merge ( paste ) consecutive segments together to generate simulated trajectories of arbitrary length. This training trajectory approach aims to overcome three common shortcomings of spatial Markov models: (1) We simulate finite‐Péclet transport in three dimensions without commonly made simplifications (e.g., dimensionality reduction, and neglecting diffusion). (2) We do not parameterize dependence via a high‐dimensional transition matrix. (3) We simulate transport at the resolution of the (highly resolved) training trajectories, which can be important for processes such as mixing and reaction. To validate our methodology, we apply it to simulate transport within a three‐dimensional sandstone sample and compare predictions of a broad range of benchmark metrics against measurements from direct numerical simulations. We demonstrate that the training trajectories approach accurately represents three‐dimensional particle motion, suggesting that this method can capture the governing dependence structure and simulate transport processes in full complexity.