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

Abstract

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.

Flexible and Efficient Simulation of Spatio-Temporal Processes with Advection

“…Why would we like to consider all these different processes for spatio-temporal phenomena? Just like for temporal processes, understanding processes with a certain large frequency spectral decay, versus boundedness at small frequencies, allows us to understand different types of phenomena, see for example examples such as cyclones and more complex advection and transport [3,21,12,32,23,6,30]. Now to realistically capture that phenomena that dissipate, the author of [9, p. 396] proposed covariance functions with elliptical contours.…”

“…The fundamental way 3-dimensional space and 2+1dimensional space-time are different is that isotropy is a natural null model in the former scenario, but not at all in the latter scenario. A number of applied studies contain features that require new methodology, such as cyclones and more complex advection and transport [3,21,12,32,23,6,30]. These do require models able to capture more than a strict frozen field model.…”

Upscaling of solute plumes in periodic porous media through a trajectory based spatial Markov model

“…In this work, we pursue this objective with the aim of opening new pathways for the application of SMM approaches to transport in porous media at laboratory and field scales. To achieve this goal, our work starts from that of Sund et al (2017), Sherman et al (2019), andMost et al (2019) where a trajectory-based SMM (here labeled tSMM) was formulated. The methodology relies on a set of numerically simulated Lagrangian trajectories obtained for a single unit cell of the porous medium, which is then used to predict transport across much larger distances.…”

Upscaling of Solute Plumes in Periodic Porous Media Through a Trajectory‐Based Spatial Markov Model

“…In this work, we pursue this objective with the aim of opening new pathways for the application of SMM approaches to transport in porous media at laboratory and field scales. To achieve this goal, our work starts from that of Sund et al (2017), Sherman et al (2019), and Most et al (2019) where a trajectory‐based SMM (here labeled tSMM) was formulated. The methodology relies on a set of numerically simulated Lagrangian trajectories obtained for a single unit cell of the porous medium, which is then used to predict transport across much larger distances.…”