Visual Analysis of Displacement Processes in Porous Media using Spatio-Temporal Flow Graphs

TL;DR

The paper addresses the challenge of understanding two-phase displacement in porous media across ensembles by developing a time map that condenses image series and a spatio-temporal flow graph that encodes displacement timing and topology. This approach enables interactive, ensemble-focused visualization and robust analysis across arbitrary geometries, revealing that global metrics like the capillary number $Ca$ can be misleading due to local constraints. Key contributions include the time map construction, flow-front segmentation with per-front metrics, a robust graph generation procedure, and an interactive framework that links graph topology to experimental context. The work demonstrates qualitative and actionable insights into breakthrough behavior and finger dynamics, and outlines clear directions for extending to simulations, graph-based metrics, and 3D data integration.

Abstract

We developed a new approach comprised of different visualizations for the comparative spatio-temporal analysis of displacement processes in porous media. We aim to analyze and compare ensemble datasets from experiments to gain insight into the influence of different parameters on fluid flow. To capture the displacement of a defending fluid by an invading fluid, we first condense an input image series to a single time map. From this map, we generate a spatio-temporal flow graph covering the whole process. This graph is further simplified to only reflect topological changes in the movement of the invading fluid. Our interactive tools allow the visual analysis of these processes by visualizing the graph structure and the context of the experimental setup, as well as by providing charts for multiple metrics. We apply our approach to analyze and compare ensemble datasets jointly with domain experts, where we vary either fluid properties or the solid structure of the porous medium. We finally report the generated insights from the domain experts and discuss our contribution's advantages, generality, and limitations.

Figures (14)

• Figure 1: Illustration of porous medium terminology (adapted from Frey2021).
• Figure 2: Overview of our datasets with different solid geometries: \ref{['fig:datasets-circular']} circular, \ref{['fig:datasets-octagonal']} octagonal, and \ref{['fig:datasets-triangular']} triangular. The invading fluid (gray/black) is entering from the right and displaces the defending fluid (white).
• Figure 3: Time map with discrete time steps mapped to a periodic color map. \ref{['fig:displacement-overview']} Time map for the whole domain (quantized). \ref{['fig:displacement-original']} Zoom-in on a region (red box) where small velocities lead to sub-pixel flow front propagation. \ref{['fig:displacement-quantized']} Quantization "resolves" this issue and removes noise.
• Figure 4: Edge creation during graph generation. \ref{['fig:edges-merge']} If two flow fronts of the same time frame arrive at a junction, they merge into a single flow front. This is reflected in the graph by a node with two incoming edges (orange). \ref{['fig:edges-deadend']} If at a junction two flow fronts of different time frames meet, the flow already progressed through the first flow front, and the later arriving fluid goes into a "dead end". This is reflected as a sink node (red).
• Figure 5: Velocity calculation at the graph edges. \ref{['fig:hausdorff-issue']} At splits with wide angle, the large distance between the centers of mass is not representative when calculating velocity. \ref{['fig:hausdorff-solution']} Modified Hausdorff distances for the calculation of the velocity for the specified edge (black arrow). Only minimum distances from the interface $\Gamma$ are considered in forward (orange arrows) and reverse time (green arrows), respectively.