Reappraisal of Upscaling Descriptors for Transient Two-Phase Flows in Fibrous Media

Abstract

Transient two-phase flows within fibrous media are considered at local scale. Upscaling these flows constitute a key procedure towards a tractable description in an industrial context. However, the task remains challenging as a time-dependent behaviour is observed within a geometrically complex structure with interplay of various physical phenomena (capillary effects, viscous dissipation, etc.). The usual upscaling strategies encountered in both soil sciences and composite materials communities are reviewed, compared, and finally adapted to reach a method that is relevant to describe fibrous media imbibition. Using finite element flow simulations on statistical representative volume elements, the proposed approach first considers several definitions for saturation in order to characterise the flow dynamics as well as the characteristic length associated with the transient behaviour. Next, two methods are proposed to assess a resulting capillary pressure, demonstrating the importance to properly define the capillary pressure acting on the interface. The first one considers the mean pressure jump at the interface, while the second one uses a machine-learning technique, namely Gaussian process regression, to retrieve the mean curvature of the interface. Those methods are found to be both consistent and in agreement with the results from the literature. Finally, a novel approach that stochastically describes the position of the flow front through a presence distribution is detailed. The spread of the front can be compared to the saturation length, and its value has been found to be small enough to be neglected at upper scale, justifying the use of sharp interface models for similar porous media and flow settings.

Appendix A

The numerical method that has been employed to simulate transient two-phase flows has been first validated on a test case that is here quickly presented. It should be noticed that it is not necessarily straightforward to identify a test case in the literature that is representative of the impregnation flows under consideration. We have here followed the micromodel experiment proposed by Kunz et al. (2016) which consists in the imbibition of a fibrous-like geometry with consideration of the capillary effects. The study by Kunz et al. provides both experimental and numerical results and has been also considered in other studies as a reference case to assess the validity of other numerical methods. The geometry and boundary conditions associated with this experiment are indicated in Fig. 21 and the fluid properties in Table 2.

Table 2 Material properties associated with the Kunz et al. (2016) test case flow

Fig. 21

Test case proposed by Kunz et al. (2016) for transient two-phase flows with consideration of capillary effects: geometry and boundary conditions

Experimental (Kunz et al. 2016) and numerical (Kunz et al. 2016; Konangi et al. 2021) results are compared in Fig. 22. For that purpose, four reference states $S_0 , S_1 , S_2 , S_3$ observed during the experimental drainage are considered. The corresponding times are indicated for each case, assuming $S_0$ as the initial time. For each numerical work, the comparable flow states are represented.

Fig. 22

Comparison between similar configurations: experimental and numerical results from Kunz et al. (2016) and Konangi et al. (2021) and present study

It can be seen that all the numerical strategies satisfactorily predict the flow path as well as the ganglia position and size. However, the time associated with each state does not match between the simulations and the experiment. Kunz et al. attribute this discrepancy to an additional flow resistance and to a stick-slip phenomenon (Kunz et al. 2016). However, both Konangi et al. and Kunz et al. simulations are in agreement, regarding the time predictions. A factor of 10 can be observed between our simulations and the other two. This can be explained as our model is two-dimensional, contrary to the others: the geometry height $H = 0.1$ mm is not taken into account. This has an significative impact on the characteristic velocity of the flow, as a pressure drop is prescribed at the volume inlet/outlet. Assuming the flow rate Q to be constant in the height direction, the 3D and 2D cases can be related as:

$$\begin{aligned} Q_{\mathrm{3D}} = \int _S\varvec{v}\cdot d\varvec{S} = H\int _{\ell } \varvec{v} \cdot d\varvec{\ell }= H Q_{\mathrm{2D}} \end{aligned}$$

(18)

where S (resp. $\ell$) is the section in 3D (resp. 2D). As the unit system is in millimetres here, the discrepancy of one order of magnitude between both cases can be explained.

In a more quantitative way, the values of saturation associated with each state can be compared (Table 3). All the values are in satisfactorily agreement, showing that the simulations predict with accuracy the residual phase content.

Table 3 Liquid saturation $S_L$ for the four reference states

Our modelling strategy of transient two-phase flows within porous structures has been here validated from a micromodel experiment (Kunz et al. 2016). The results have been compared to three reference results, including both numerical and experimental works. Our observations are consistent with the other numerical studies, and the main discrepancies have been explained. As regards the experimental results, the proposed strategy predicts with accuracy the flow path as well as the ganglia position.

However, some differences are observed between the numerical approaches and experimental observations. Those can be attributed to modelling errors, but also to the experimental procedure. Indeed, with a numerical method akin to Konangi et al., Ambekar et al. obtain very accurate time predictions for similar geometries (Ambekar et al. 2021). In the end, despite the strong assumptions of our model (incompressibility, Stokes equations, simplified behaviour of the vapour phase, etc.), the validity of our strategy has been here highlighted through this test case study.