Accessing Preferential Foam Flow Paths in 2D Micromodel Using a Graph-Based 2-Parameter Model

Abstract

This study uses experimental data of pore-scale foam flow inside a high-complexity network to fit a graph-based model describing preferential flow paths using microstructural characteristics of the porous medium. Two experiments, with equal gas fractions but varying injection rates, are modeled in parallel. Proposed paths are solution paths to the k-Shortest Paths with Limited Overlap (k-SPwLO) problem, applied to a graph representation of the porous medium with edge weights representing local throat properties. A 1-parameter model based on throat radius only is tested before integrating a second parameter, describing the alignment of the pores surrounding the throat with respect to the injection pressure gradient. The preferential paths observed in both experiments differ in quantity and with respect to the specific porous zones used. As such, the best fit preferential path models for either experiment show different dependencies on the microstructural parameters. The optimized model for the high injection rate experiment markedly shows a dependence on the pore alignment with pressure gradient as well as throat size, whereas the lower injection rate experiment was best fitted to a model that only includes the throat radius. Overall, the graph-based framework was able to capture many high-flow zones in various model parameter combinations, perhaps as consequence of the relatively spiked throat size distribution of the model.

Appendices

Appendix 1: Porous Network and Extracted Graph

Regarding the structural parameters used for the weights of the edges connecting the artificial inlet/outlet nodes to the rest of the network, the throat size for each edge was trivially given by the throat that connected the network to the inlet/outlet zone. However, the angle µ that shows the pore-to-pore alignment with pressure gradient was set equal to 0 (perfect alignment with gradient) for all nodes connected to either to inlet or outlet nodes. Indeed, as the center of mass of the inlet and outlet nodes are positioned in the central axis, taking the alignment angle with this position would create erroneous bias against pores outwards from the central axis (Fig. 21).

Fig. 21

Top: Micromodel structure with porous area seen in black and obstacles in white. Fluid injection can be performed on the full width of the model. Bottom: Extracted graph with added inlet and outlet nodes that share edges with all the adjacent nodes to the inlet or outlet of the model

Appendix 2: Full Model Experimental Match of Proposed Paths

Here, we give two examples of full model flowmaps overlaid with the paths for the best models for each experiment.

In Fig. 22, we show the best model for Exp. 1: $\left( {\alpha ,\beta } \right) = \left( {2.2,0.6} \right)$ with a high intensity threshold on the pore activity classification. Flow intensity is shown in grayscale within the porous network.

Fig. 22

Experiment 1 high intensity threshold best model proposed paths. The color code is given in Fig. 8. Flow is from top to bottom

In Fig. 23, we show the best model for Exp. 2: $\left( {\alpha ,\beta } \right) = \left( {1.2,0} \right)$ with a high intensity threshold on the pore activity classification.

Fig. 23

Experiment 2 high intensity threshold best model proposed paths. The color code is given in Fig. 8. Flow is from top to bottom

Appendix 3: Comparison Between Models Minimizing Path Distance and Number of Pores

In Fig. 24, we show the first 5 proposed paths in two types of models used in Fig. 13: the model minimizing the total physical distance (red crosses in Fig. 13) and the model minimizing the number of pores (green plusses in Fig. 13). The velocity intensity of Exp. 1 with flow from top to bottom is shown in grayscale within the porous area for comparison.

Fig. 24

Comparison of proposed paths for different models used in Fig. 13. The paths in light green are proposed by the model minimizing physical distance, whereas the paths in light blue are proposed by the model minimizing the number of pores. The areas in dark blue are common to both sets of paths