# Rheology of Liquid Foam flowing through Porous Media

A Salamé, R Choudhary, A-M Tang, S Vincent-Bonnieu, V Langlois, O Pitois

► To cite this version:

A Salamé, R Choudhary, A-M Tang, S Vincent-Bonnieu, V Langlois, et al., Rheology of Liquid Foam flowing through Porous Media. Physics of Fluids, 2025, 37, 10.1063/5.0268270. http://dx.doi.org/10.1063/5.0268270

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Submitted on 7 Sep 2025

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# Rheology of Liquid Foam flowing through Porous Media

A. Salamé,¹ R. Choudhary,¹ A.-M. Tang,¹ S. Vincent-Bonnieu,² V. Langlois,¹ and O. Pitois¹

¹)Laboratoire Navier, Université Gustave Eiffel, ENPC, Institut Polytechnique de Paris, CNRS, Champs-sur-Marne 77420, France

²)European Space Agency, Keplerlaan 1, 2200 AG Noordwijk, The Netherlands

(*Electronic mail: Olivier.Pitois@univ-eiffel.fr)

(Dated: 10 April 2025)

Liquid foams are widely used in porous media, like soils. Recent studies have explored foam flow in two-and three-dimensional porous media, but the effects of confinement on foam rheology remain poorly understood. We investigated foam flow through granular packing with stable bubble sizes and reduced compressibility across different liquid fractions. Our analysis shows that confined foam flow cannot be directly extrapolated from bulk foam properties. The apparent yield stress of confined foam increases with degree of confinement, governed by the bubble-to-grain size ratio. This increase can be significant, especially for foams with high liquid content. The increase aligns with a smooth transition between yield stress values of bulk foams and those under strong confinement. This suggests that the geometric effect of bubbles passing through small constrictions is not significant. Instead, liquid redistribution within pore spaces at grain surfaces and contact areas is crucial. This effect is modeled and agrees with our data and literature. Finally, stopping bubble coarsening is shown to be essential to accurately understand foam rheology in porous media.
<figure><img src="image_2.png" /><figcaption>FIG. 1. Presentation of the setup used to study liquid foam flow through a granular packing. Starting bottom left: large bubbles are generated from nitrogen saturated with perfluorohexane vapor (the latter is used to strongly reduce the rate of bubble coarsening), and then these bubbles are fragmented by passing through a so-called 'bubble breaker', made by stacking several nylon filters, resulting in foam with bubbles of radius $R_b$ and a liquid fraction of approximately 5%. This foam is further mixed in-line with a chosen quantity of foaming solution to increase the liquid fraction to the target value, $\phi_l$. Finally, the foam flows at a rate $Q_f(P_0)$ through the measurement cell, which consists of a column of length $L$ filled with compacted glass beads of size $R_g$. Note that due to gas compressibility, $\phi_l$, $R_b$ and all the flow rates depend on $\Delta P$, where their corresponding values are given relative to the reference pressure $P_0$. A pressure sensor embedded in the head of the column measures the driving pressure, $\Delta P$. Bubble size measurements $R_b$ are made at both the inlet and outlet of the column to quantify any change in bubble size during flow.</figcaption></figure>

# I. INTRODUCTION

Liquid foams, i.e., dispersions of gas bubbles in a liquid phase, belong to the class of yield stress fluids: they behave like a liquid above a stress threshold and as a solid below it¹. These materials are widely valued in the industrial and engineering fields² due to their unique viscoplastic properties combined with their lightness. Such properties are particularly useful in applications such as enhanced oil recovery, where foam reduces gas mobility and redirects the flow to less accessible zones, thus improving fluid displacement efficiency³,⁴. More recently, foams have been applied to soil remediation, both to clean contaminants and to contain them around the source
areas⁵,⁶.

Substantial experimental work has investigated the flow properties of foams in various porous media, including two-dimensional micromodels⁷, smooth capillaries⁸, periodically constricted tubes⁹,¹⁰, grain packings¹¹, and consolidated soils¹². In these studies, pressure measurements were used to characterize the rheological behavior of confined foams, highlighting the influence of liquid content and bubble size on foam stability, apparent viscosity, and dynamic resistance to flow. Jones et al.¹³ explored the dynamics of foam flow in porous media with a particular focus on how coarsening (i.e., the increase in average bubble size over time due to gas exchange between bubbles) affects foam behavior and mobility control. They found that the coarsening of the foam increases the pressure needed to reinitiate the flow after stopping for a while, a result attributed to the larger curvatures of the lamella that have had time to position themselves in areas where they provide the greatest resistance¹⁴. Using granular packings, Omirbekov et al.¹⁵,¹⁶ observed shear thinning behavior in foams, effectively modeled by the Herschel-Bulkley framework, where the apparent viscosity of the foam decreases with increasing shear rate:

$$ \mu_{\text{app}} = \frac{\sigma_y}{\dot{\delta}} + k \dot{\delta}^{n-1}, \qquad (1) $$

where $\mu_{\text{app}}$ is the apparent viscosity, $\dot{\delta}$ is the shear rate, $\sigma_y$, $k$ and $n$ are fitting parameters, representing the yield stress, the consistency index (or plastic viscosity), and the flow behavior index, respectively. Note that for small shear rates, i.e. $\dot{\delta} \ll (\sigma_y/k)^{1/n}$, the foam viscosity is approximately given by $\mu_{\text{app}} \approx \frac{\sigma_y}{\dot{\delta}}$ and the corresponding shear stress is $\sigma \approx \sigma_y$. Omirbekov et al. also noted that the foam yield stress deduced from their experiment varies with grain size: larger grains induce a lower yield stress¹⁵. In fact, the relationship between grain size and yield stress remains unclear, as yield stress depends on bubble size, which in turn changes over time due to coarsening. On the other hand, the rheology of bulk foam is well described by a Herschel-Bulkley framework and the dependence of the yield stress on bubble radius, $R_b$, and the liquid volume fraction within the foam, $\phi_l$, is expressed as¹:

$$ \sigma_y = \frac{\gamma}{R_b} \alpha (\phi_l^* - \phi_l)^2, \qquad (2) $$

where $\gamma$ is the gas/liquid surface tension, $\alpha$ is a dimensionless constant of the order of unity, and $\phi_l^* \approx 0.36$ is the critical liquid fraction¹.

A new factor has now emerged among the anticipated effects for confined foam. In a study focused on the size of liquid bridges mediated by volume exclusion of the bubbles at contact points between grains, it was shown that the liquid mobilized at these locations is drawn from the
foam filling the remaining pore space, thereby reducing its liquid content. More specifically, the effective liquid content of the foam filling the pores strongly depends on the bubble-to-grain size ratio¹⁷. Quantifying the impact of this effect on the apparent viscosity of the flowing foam would be valuable, as it has been completely overlooked until now.

Actually, current understanding of the effects of confinement on foam rheology in porous media is limited and lacks comprehensive physical and theoretical explanations. A key experimental challenge in foam flow through porous media lies in obtaining data that accurately reflect foam's rheological properties without interference from instability, gas compressibility and coarsening. In this work, we experimentally examine the influence of confinement on liquid foam flow through granular packing. At controlled liquid volume fractions and reduced compressibility effects, we present results on the rheology of confined foam with stabilized bubble size, analyzing how the bubble-to-grain size ratio affects foam properties in comparison with bulk foam. Then, we investigate the impact of foam coarsening on apparent rheological properties, emphasizing why suppressing coarsening is essential for accurately understanding foam flow in porous media.

## II. MATERIALS AND METHODS

### A. Foam production

Due to our thorough understanding of its physical properties¹⁸,¹⁹, we have chosen to work with the surfactant tetradecyltrimethylammonium bromide (TTAB) at 10 g/L in Milli-Q water as foaming solution, resulting in a gas/liquid surface tension $γ = 37$ mN/m. To generate the liquid foam, a flow of nitrogen gas $N_2$ at volume flow rate $Q_g(P_0)$ is regulated using a mass flow controller (Bronkhorst F-201CV-5K0-AAD-33-V) with a maximum flow rate of 500 mL/min. Note that the gas volume flow rates mentioned hereafter correspond to standard conditions of pressure and temperature. Since the effect of temperature does not play a role in our study (room temperature at 22°C), we will refer only to the reference pressure $P_0 = 1$ bar. Exiting the controller, the gas is bubbled into a small bottle partially filled with perfluorohexane ($C_6F_{14}$) and maintained in a water bath at 10°C, in order to control the partial pressure of perfluorohexane vapor in the bottle (Fig. 1). The role of $C_6F_{14}$ is to significantly reduce the coarsening rate of the generated foam²⁰, but note that the bottle can be easily bypassed to allow only nitrogen to pass through when necessary. This nitrogen-vapor mixture is then directed at the same flow rate $Q_g(P_0)$ to a column filled with the
<figure><img src="image_3.png" /><figcaption>FIG. 2. (a) Volume-weighted bubble size distributions of radii $R_i$ for various flow rates $Q_g(P_0)$ and with three different target bubble sizes $R_b = \langle R_i^3 \rangle / \langle R_i^2 \rangle$: 60, 95 and 180 µm. These distributions were achieved using specific configurations of nylon filters in the bubble breaker device as follows: for $R_b \approx 60$ µm we used 9 filters with 10 µm openings at $Q_g(P_0) = 5$ mL/min, 5 filters with 10 µm openings at $Q_g(P_0)=50$ mL/min, and 4 filters with 10 µm openings at $Q_g(P_0)=100$ mL/min; for $R_b \approx 95$ µm we used 5 filters with 10 µm openings at $Q_g(P_0)=5$ mL/min, 3 filters with 10 µm openings at $Q_g(P_0) = 50$ mL/min, and 3 filters with 10 µm openings at $Q_g(P_0)=100$ mL/min; for $R_b \approx 180$ µm we used 2 filters with 20 µm openings at $Q_g(P_0)=5$ mL/min, 2 filters with 30 µm openings at $Q_g(P_0)=50$ mL/min, and 2 filters with 30 µm openings at $Q_g(P_0) = 100$ mL/min. (b) Corresponding bubble size distributions of radii $R_i$ normalized by the corresponding average value $R_b$.</figcaption></figure>

foaming solution. Note that for low flow rates ($Q_g(P_0) \le 5$ mL/min), the bottle of $C_6F_{14}$ is placed before a mass flow controller with a maximum flow rate of 10 mL/min (Bronkhorst F-201CV-020-AAD-33-V). In this configuration, the vapor in the bottle is quickly pressurized, enabling the foam to be pushed downstream efficiently. If pressurization were due solely to nitrogen arriving from the outlet of this controller, foam production would take significantly longer. However, this setup requires that the perfluorohexane vapor passes through the controller's measurement chamber, necessitating a correction factor for the presumed flow rate. Regardless of configuration, gas flow generates large bubbles (with a radius of approximately 1 mm) that accumulate as foam in the upper part of the column, where they drain during their residence time before overflowing. At this stage, the foam composed of large bubbles is mixed in-line with additional foaming solution delivered by a syringe pump at flow rate $0.05Q_g(P_0)$, achieving a liquid fraction of approximately 5%. This liquid fraction was found to be optimal for fragmenting the large bubbles into bubbles
of the desired smaller size.

The large bubbles thus produced are directed toward a custom-built bubble breaker device (see Fig. 1), containing a set of *N* nylon filter sheets (Millipore), each with uniform rectangular openings of 10, 20 or 30 µm. The number of sheets, *N*, varies between 2 at high gas flow rates and 9 at low flow rates. These filters are cut into 10 mm diameter disks and incrementally layered between 5 mm thick supports (Fig. 1). To verify the size of the formed bubbles, a sample of the resulting fragmented bubbles is collected and confined between two glass plates, with a set gap maintained by spacers (Elcometer), allowing accurate measurement of bubble size distribution thanks to classical image processing. The bubble breaker device enables the production of three primary bubble sizes for our experiments, with Sauter mean radius $R_b = \langle R_i^3 \rangle / \langle R_i^2 \rangle \approx 60$ µm, 95 µm and 180 µm, where $R_i$ represents individual bubble radii. Volume weighted bubble size distributions are shown in Fig. 2a. Normalizing these distributions by $R_b$ demonstrates the consistent foam production across all cases, as evidenced by the similar normalized distributions (Fig. 2b), which is essential for our analysis. Thus, for each gas flow rate $Q_g(P_0)$, we optimize the configuration of sheets to consistently produce the same bubble size distribution. This setup enables the continuous generation of foam with bubble size of $R_b \pm 10$ µm, at flow rates $Q_g(P_0) = 2.5, 5, 10, 50$ and $100$ mL/min. The generated foam is either directed immediately to the measurement cell to achieve a 5% liquid fraction, or co-injected with the foaming solution at a flow rate $q_a$, controlled by a syringe pump, to increase the foam's liquid fraction $\phi_l$ up to 32% before entry into the measurement cell. Note that the liquid volume fraction of the foam, with respect to the reference pressure $P_0$, is determined by adjusting both the total liquid flow rate $Q_l = 0.05Q_g(P_0) + q_a$ and the gas flow rate $Q_g(P_0)$, which gives: $\phi_l = Q_l/(Q_l + Q_g(P_0))$. Therefore, the final flow rate of foam entering the measurement cell is $Q_f(P_0) = (1+0.05)Q_g(P_0) + q_a$ (see Fig. 1).

## B. Flow cell

The measurement cell consists of a cylindrical glass column with an inner diameter of ≈ 25 mm (associated to the cross-sectional area S), filled with compacted glass beads of radius $R_g = 400, 750, 1000, 1500$ or $2500$ µm (where $\Delta R_g/R_g \approx 7\%$). The beads are secured between two metallic grids with 200 µm openings to maintain stability during foam flow. For each bubble and grain size, we choose a bead packing height, among $L \approx 2, 4$ and $8$ cm, in order to minimize compressibility effects (a shorter column is chosen when higher pressures are expected) and to guarantee that the
<figure><img src="image_4.png" /><figcaption>FIG. 3. (a) Typical measurement sequences of the raw pressure drop, $\Delta P_r$ (see the corresponding plateaus), at a constant gas flow rate, $Q_g(P_0)$. The measurement is performed using two different sequences: (red) starting with a large liquid fraction, $\phi_l = \phi_{l1}$, we decrease it to $\phi_{l2}$, and then return to the initial value $\phi_{l1}$; (blue) starting with a small $\phi_l = \phi_{l3}$, we increase it to $\phi_{l4}$, subsequently covering the intermediate cases shown in red, i.e., $\phi_{l1}$ and $\phi_{l2}$, before returning to $\phi_{l3}$. The significant rise and fall of the red pressure curve, between $\Delta P_{r2}$ and $\Delta P_{r1}$, is due to the manipulation of valves that allow foam to be extracted for measuring bubble sizes. This can temporarily alter the feed conditions of the column where the pressure sensor is located. (b) Typical measurements for the raw pressure drop $\Delta P_r$ (filled squares) and corrected pressure drop $\Delta P$ (filled circles) as a function of the foam flow rate, $Q_f(P_0)$. $\Delta P$ is obtained by subtracting the so-called "dead pressures" (see inset) from the raw pressure drop $\Delta P_r$. Measurements are conducted for bubble sizes $R_b \approx 95$ µm at two different liquid fractions, $\phi_l$, as indicated, using grains of size $R_g = 1000$ µm and a column length $L = 4$ cm. (inset) Measurements of the "dead pressure", obtained without grains (see main text for more details), as function of the foam flow rate $Q_f(P_0)$. Solid lines indicate power-law fits to the "dead pressure" measurements.</figcaption></figure>

driving pressure $\Delta P_r$ is such that $\Delta P_r/P_0 < 0.3$. For optimal grain compaction, the column is filled incrementally, with one-third of the column filled at a time and compacted using a 7 mm diameter rod.

The initial step involved determining the pore volume occupied within the packing. We measured the volume fraction of the pores in our packings for all grain sizes. The measurement consists of determining the mass of the introduced grains, which is then converted into volume $V_{grains}$ using their density, and measuring the column volume $V_{column}$ delimited by the two grids confining the
<table><caption>TABLE I. Radii $R_g$ of the grains (glass beads) used to make the packings and the resulting pore volume fraction $\phi_p$ and intrinsic permeability $K$, calculated by Eq.3, at different packing heights $L$.</caption><thead><tr><th></th><th>2500</th><th>1500</th><th>1000</th><th>750</th><th>400</th></tr></thead><tbody><tr><th>R<sub>g</sub> (μm)</th><td>2500</td><td>1500</td><td>1000</td><td>750</td><td>400</td></tr><tr><th>L (cm)</th><td>8</td><td>4</td><td>4 &amp; 2</td><td>4 &amp; 2</td><td>2</td></tr><tr><th>φ<sub>p</sub></th><td>0.42</td><td>0.40</td><td>0.36</td><td>0.41</td><td>0.35</td></tr><tr><th>K (μm<sup>2</sup>)</th><td>30590</td><td>8890</td><td>2530</td><td>2475</td><td>360</td></tr></tbody></table>

packing. The pore volume fraction is then calculated by $\phi_p = 1 - V_{grains}/V_{column}$. The intrinsic permeability $K$ of the packing is then estimated using the Kozeny-Carman equation:

$$K = \frac{\phi_p^3 (2R_g)^2}{36c_K (1 - \phi_p)^2}, \qquad (3)$$

where $c_K$ is the Kozeny-Carman constant, the value of which is generally set to $c_K = 5$ for packs of spheres²¹. Values of $\phi_p$ and $K$ are presented in table I.

The top of the column is fitted with a head equipped with a pressure sensor featuring a 19 mm diameter movable membrane (MPM251 supplied by Microsensor), which is in direct contact with the flowing foam in the head chamber and separated from the packed grains by the upper grid. The pressure sensor, capable of measuring up to 0.7 bars, outputs a DC voltage signal read by a multimeter DVM1200 (where 5V correspond to 0.7 bars). The multimeter is connected to a computer and a Python script allows live monitoring of the pressure difference $\Delta P_r$ between the top of the column and atmospheric pressure. Both the head and tail of the cell are secured with long screws to ensure the cell's rigidity during flow and pressure buildup in the system.

### C. Experimental procedure

After preparing the cell, the foam is continuously produced at a liquid fraction $\phi_l$. Using valves, the foam is diverted from the measurement cell for sampling, in order to ensure that the targeted bubble size $R_b$ is achieved at a constant flow rate $Q_g(P_0)$. After ensuring that foam production is stable, the flow is then directed into the cell, where the pressure begins to increase. We determined that injecting a foam volume equivalent to twice the pore volume is sufficient to reach a stable pressure plateau $\Delta P_r$. This plateau remains stable with a maximum variation of 3% from the average value. At constant $Q_g(P_0)$, we gradually increase the foam's liquid fraction $\phi_l$ to target
<figure><img src="image_5.png" /><figcaption>FIG. 4. Bubble size $R_b$, measured at the inlet (open symbols) and the outlet (filled symbols) of the flow cell, versus foam flow rate $Q_f(P_0)$ obtained for grain size $R_g=1000~\mu\text{m}$ at both liquid fractions: (a) $\phi_l \approx 5\%$ and (b) $\phi_l \approx 20\%$. Red triangles: blocked-coarsening conditions using $R_b \approx 60~\mu\text{m}$; Blue squares: blocked-coarsening conditions using $R_b \approx 95~\mu\text{m}$; Gray circles: blocked-coarsening conditions using $R_b \approx 180~\mu\text{m}$; Blue diamonds: free-coarsening conditions using the same filters to produce initial bubble size $R_{b,0} \approx 95~\mu\text{m}$. The dashed lines represent the average bubble size $R_b$ for the blocked-coarsening conditions.</figcaption></figure>

values, recording the corresponding pressure plateaus $\Delta P_r$. Increasing $\phi_l$ leads to a decrease in pressure (Fig. 3a), which is consistent with the expected foam properties: the apparent viscosity of the foam decreases with increasing liquid fraction¹. In order to get our data, we repeat some measurement points, starting initially from a liquid fraction higher than 5% and then reducing it, which shows no significant difference with the previous measurement, within an error margin of approximately 2% (Fig. 3a). Once all targeted liquid fractions are recorded, the flow is stopped, the granular packing is replaced with dry grains, and the setup is prepared to repeat measurements at different $Q_g(P_0)$, following the same procedure.

Measurements of the bubble size were performed at the inlet and the outlet of the flow cell to detect any bubble size evolution. The role of the coarsening-blocking agent can be observed in Fig. 4, in comparison with nitrogen used alone for foaming (without passing through the $C_6F_{14}$ bottle, see Fig. 1), using the same configuration of filters to obtain a bubble radius equal to $\approx 95~\mu\text{m}$. A significant evolution in bubble size $R_b$ is observed between the inlet and the outlet of the flow cell under free-coarsening conditions. The measured increase in $R_b$ correlates with the decrease of the foam flow rate $Q_f(P_0)$ due to the associated increase in the foam's residence time
in the cell. The residence time ranges from 20 seconds at $Q_f(P_0) \approx 50$ mL/min to 360 seconds at $Q_f(P_0) \approx 2.5$ mL/min, resulting in approximately a twofold increase in $R_b$ between the inlet and the outlet. Moreover, the bubble size also increases significantly during foam transport from the bubble breaker to the flow cell inlet, a process lasting up to 30 seconds at $Q_f(P_0) \approx 2.5$ mL/min. Such a huge evolution shows that the bubble size is not a controllable parameter if the coarsening is not blocked. Note that at the lowest flow rate and liquid fraction, we observe a slight increase in bubble size despite the use of the coarsening-blocking agent, but it is negligible compared to that observed without the blocking agent (see Fig. 4a).

Note that we encountered some limitations regarding the bubble-to-grain size ratio. Specifically, for ratios approaching 0.5, only foam with liquid fractions below 8% could be studied, and for higher ratios, it was simply not possible to achieve flow without breaking the bubbles, regardless of the liquid fraction.

By shortening the length of the measurement column to reduce gas compressibility effects as much as possible, we increase the relative impact of head losses due to the flow through the non-granular parts of the system (including the upstream pressure chamber, the grids and the downstream pressure chamber). This requires us to determine this contribution as accurately as possible, by systematic measurements with the empty columns (no grains) and for various flow rates. These baseline pressures, termed "dead pressures", allow us to correct the raw data by subtraction (see Fig. 3b) and to isolate the effective pressure associated solely with foam flow through the granular packing. To facilitate this correction, a power-law is first fitted to the dead pressure data (inset of Fig. 3b). The corrected driving pressure is then called $\Delta P$. As a precautionary measure with regard to our way of correcting raw data, we chose to use measurements only when the dead pressure did not exceed the total pressure by more than 30%, noting however that the average value of the dead pressures removed from our raw data is close to 16%. If we neglect the small effects of compressibility in our experiments (to which we will pay particular attention in the section II D), each measured pressure $\Delta P$ can be considered as the pressure required to push foam, with a specific liquid fraction $\phi_l$ and bubble size $R_b$, at a constant flow rate $Q_f$ through a column of height $L$ filled with grains of size $R_g$. On this basis, we can now develop a calculation that links macroscopic measurements to pore-scale foam rheology.

We start with local Darcy's law for the foam in the porous medium: $Q_f(x)\mu_{\text{app}}(x)dx = -KSdP$, where $Q_f(x)$ and $\mu_{\text{app}}(x)$ are respectively the flow rate of foam and the apparent viscosity of the foam, given by Eq. 1, at position $x$ in the column. Note that if the yield stress contribution
dominates over the contribution of the plastic viscosity in Eq. 1, we can write that $\mu_{app} \approx \frac{\sigma_y}{\delta}$, where $\delta$ is the effective shear rate that can be estimated from the characteristics of the porous medium, i.e. $\delta = 4Q_f/S\phi_p r_p^{22}$, with the equivalent pore radius $r_p$ defined as$^{23}$: $r_p = \sqrt{8K/\phi_p}$. From these expressions, Darcy's law can be written: $\sigma_y(x)dx = -\frac{4K}{\phi_p r_p}dP$. By integrating, we obtain the average value of the yield stress $\langle\sigma_y\rangle = \frac{1}{L}\int_0^L \sigma_y(x)dx$:

$$ \langle\sigma_y\rangle = \frac{4K}{L\phi_p r_p} \Delta P, \qquad (4) $$

which is equal to $\sigma_y(\phi_l, R_b, R_g)$ at $P_0$, based on our assumption that compressibility effects are negligible — or, equivalently, that the liquid fraction and bubble size remain constant. Note that, under this assumption, $\Delta P$ is independent of the flow rate, which is consistent with our experimental observations (see Fig. 3b). However, the compressibility effect of the gas cannot be entirely neglected: the effective flow rate is not exactly equal to $Q_f(P_0)$, and consequently, the liquid fraction and bubble size are likely affected, which could impact the values of the rheological parameters deduced from the experiment. Let's consider these effects.

## D. Compressibility effects

To address the compressibility effects, let us determine the average values of liquid fraction and bubble size: $\langle\phi_l\rangle/\phi_l(P_0) = (\frac{1}{\Delta P}\int_{P_0}^{P_0+\Delta P}\phi_l(P)dP)/\phi_l(P_0) \approx 1 + \frac{1-\phi_l}{2}\frac{\Delta P}{P_0} - \frac{(1-\phi_l)\phi_l}{3}\left(\frac{\Delta P}{P_0}\right)^2$ and $\langle R_b\rangle/R_b(P_0) = (\frac{1}{\Delta P}\int_{P_0}^{P_0+\Delta P}R_b(P)dP)/R_b(P_0) \approx 1 - \frac{1}{6}\frac{\Delta P}{P_0} + \frac{2}{27}\left(\frac{\Delta P}{P_0}\right)^2$, where we considered that $\frac{\Delta P}{P_0} \ll 1$, and we used $\phi_l(P) = \frac{P}{P+P_0Q_g(P_0)/Q_l}$ and $R_b(P) = R_b(P_0)(\frac{P_0}{P})^{1/3}$. We can observe in Fig. 5 that, within our experimental conditions, compressibility-induced variations for $\phi_l$ and $R_b$ are very limited, i.e., the maximal values are 13% and 4% respectively (3% and 1% as average values respectively), which justifies our assumption that the variations induced by compressibility on these two parameters do not affect the foam's yield stress, which can therefore be considered constant along the column. Furthermore, the observed independence of $\Delta P$ on the flow rate (Fig. 3b) also validates our initial assumption that the contribution of the yield stress dominates over that of the plastic viscosity, for which $\Delta P$ would instead increase with flow rate.

From the average foam flow rate, $\langle Q_f\rangle/Q_f(P_0) = \phi_l(P_0)/\langle\phi_l\rangle$ and the average effective shear rate $\langle\dot{\delta}\rangle = 4\langle Q_f\rangle/S\phi_p r_p$, we can determine the effective value for the apparent viscosity of the foam, i.e., $\langle\mu_{app}\rangle = \langle\sigma_y\rangle/\langle\dot{\delta}\rangle$, which reflects the rheological behavior of the flowing foam. As already explained, within our flow conditions the resistance to flow primarily arises from the con-
<figure><img src="image_6.png" /><figcaption>FIG. 5. Effect of gas compressibility on: (a) liquid fraction $\phi_l$ and (b) bubble size $R_b$. Solid lines correspond to the expressions given in the main text and the symbols represent the measurement points. The shaded areas show the maximum range of variation for each parameter of the experiment: $\frac{\Delta P}{P_0} < 0.3$, which induces a maximum effect of 13% on $\phi_l$ and of 4% on $R_b$.</figcaption></figure>

tribution of the foam’s yield stress, with very little from the plastic viscosity term, which allows us to write the associated stress of the foam $\sigma \approx \langle \sigma_y \rangle$. These conditions are the most relevant for foam applications in porous media, such as soils, for example⁵. As a result, we will focus primarily on exploring the characteristics of the yield stress, which will be presented and discussed in the following section. Note that, in order to simplify the notations, we will represent in the following section: the average effective shear rate $\langle \dot{\delta} \rangle$ as $\dot{\delta}$, the average yield stress $\langle \sigma_y \rangle$ as $\sigma_y$, the average apparent viscosity $\langle \mu_{app} \rangle$ as $\mu_{app}$, with values of $\phi_l$ and $R_b$ referred to the reference pressure $P_0$.
<figure><img src="image_7.png" /><figcaption>FIG. 6. Average foam apparent viscosity $\mu_{\text{app}}$ divided by $\gamma/R_b$ as function of the average effective shear rate $\dot{\delta}$, for different liquid fractions $\phi_l$, and bubble size $R_b \approx 95$ µm, considered as constant (blocked-coarsening conditions). Grain sizes: (a) $R_g = 1000$ µm and (b) $R_g = 400$ µm. The dashed lines represent power-laws of the form $\mu_{\text{app}}/(\gamma/R_b) = \tilde{\sigma}_y/\dot{\delta}$, where $\tilde{\sigma}_y = \sigma_y/(\gamma/R_b)$ and $\sigma_y$ is the average yield stress of the flowing foam. The red-filled squares at low $\dot{\delta}$ refer to data points higher by more than 20% from their corresponding dashed lines, while at high $\dot{\delta}$, they indicate that the “dead pressure” exceeds 30% of the raw pressure (see main text for more details). These red-filled squares were consequently excluded from our analysis.</figcaption></figure>

### III. RESULTS AND DISCUSSION

We begin by presenting the effects of liquid fraction, bubble size, and grain size on the yield stress of foam flowing through the pore space. We then highlight the crucial role of coarsening on the foam's yield stress and the need to suppress it in order to determine the intrinsic rheological law of a given liquid foam.

#### A. Effect of confinement degree

We present in Fig. 6 typical examples for the average apparent foam viscosity, i.e., $\mu_{\text{app}} \approx \sigma_y/\dot{\delta}$, divided by $\gamma/R_b$, which is the pressure scale in the foam, as a function of the average shear rate $\dot{\delta}$. This normalization by the bubble size is relevant here because accurate measurements of $R_b$ were performed, ensuring that this crucial parameter remains constant during flow. Note that 80 curves like these were obtained, representing more than 400 measurement points (see the Supplementary
Material). The first notable observation is the shear-thinning behavior of the confined foam: foam viscosity decreases with increasing shear rate, regardless of the liquid fraction or grain size. This behavior is characteristic of bulk foam¹ and it is widely observed in experiments of this type with foam¹⁰,¹⁵.

We introduce the normalized yield stress $\tilde{\sigma}_y = \sigma_y/(\gamma/R_b)$ to express the normalized foam viscosity: $\mu_{app}/(\gamma/R_b) \approx \tilde{\sigma}_y/\dot{\delta}$. For constant bubble size and liquid fraction, $\tilde{\sigma}_y$ is expected to be constant and $\mu_{app}/(\gamma/R_b)$ to describe a power-law relationship with $\dot{\delta}$ characterized by a slope of -1 in log-log scales. This behavior is precisely what we observe in Fig. 6. Therefore, by averaging the values obtained for $\tilde{\sigma}_y$ for the different flow rates, we determine a representative value of the foam's yield stress as a function of both $R_b$ and $\phi_l$, as varied in our study. In Fig. 6, for fixed $\dot{\delta}$, the decrease in apparent viscosity as liquid volume fraction increases is expected to be explained by Eq. 2, which sets the behavior for the yield stress: Increasing the liquid fraction diminishes the yield stress, ultimately transitioning the foam out of the viscoelastic regime once the critical liquid fraction, $\phi_l^* \approx 0.36$, is reached.

We want to point out that, in most cases at low flow rates, the experimental points lie slightly above the power law curves, with this effect becoming more pronounced as the liquid fraction increases. This phenomenon is inherent to yield-stress fluids in porous media and is expected at low flow rates, as the driving pressure is insufficient to initiate flow in all the pores of the medium²⁴. Gradual mobilization of the pore network transporting liquid foam has been previously observed in porous media using X-ray microtomography²⁵. In terms of the pressure-flow rate relationship, the presence of unyielded areas reduces the effective permeability of the porous medium, consequently leading to an apparent increase in viscosity and yield stress, manifested as higher measured pressure. The amplification of this effect with increasing liquid fraction can be attributed to the availability of liquid locally promoting the heterogeneity associated with this phenomenon. Consequently, in terms of data analysis, data points that are higher by more than 20% from average are then excluded (see Fig. 6).

All our measured values for $\tilde{\sigma}_y$ are plotted in Fig. 7 as a function of $\phi_l$. It is obvious that the yield stress of the confined foam depends on grain size: $\tilde{\sigma}_y$ increases as the grain size decreases. Moreover, whatever the chosen $\alpha$ value in Eq. 2, no dataset corresponding to fixed $R_g$ can be accurately described. This is especially evident for values with a high liquid fraction, where the Eq. 2 significantly underestimates them. Moreover, while the normalized yield stress $\tilde{\sigma}_y$ is not expected to depend on bubble size, our data show that, at a fixed grain size, $\tilde{\sigma}_y$ increases with larger
<figure><img src="image_8.png" /><figcaption>FIG. 7. Normalized average yield stress $\tilde{\sigma}_y = \sigma_y/(\gamma/R_b)$ as function of foam liquid fraction $\phi_l$ for all bubble sizes $R_b$ and grain sizes $R_g$. The black curves correspond to Eq. 2 (divided by $\gamma/R_b$), i.e., $\tilde{\sigma}_y = \alpha (\phi_l^* - \phi_l)^2$, where $\alpha$ was chosen to be: 0.5 for the solid curve and 1 for the dashed curve.</figcaption></figure>

$R_b$, which even goes against the trend defined by Eq. 2. This highlights the role of confinement in influencing the rheological behavior, specifically in relation to the bubble-to-grain size ratio.

To explain our results, we introduce the size ratio, $r = R_b/R_g$, which represents the degree of confinement of the foam. As already mentioned, we were able to reach a maximum value of $r$ close to 0.50, a value above which we were unable to study foam flows without observing significant changes in the bubble size distribution. In Fig. 8a, we show the normalized average yield stress $\tilde{\sigma}_y(r)$ divided by its corresponding normalized yield stress of the bulk foam $\tilde{\sigma}_y(0)$, as a function of $r$. The data for all bubble sizes, grain sizes and liquid fractions highlight an increase in the yield stress along with the increase in the degree of confinement. The data do not undergo any abrupt change at the value $r = 0.154$, which corresponds to the constrictions formed by three contacting grains in the packing. This suggests that the main mechanism driving the observed evolution is not related to the direct geometric interaction between the bubbles and the pore space walls.

We aim now to explain the abnormally large yield stress values induced by confinement. At this stage we refer to the recent phenomenon highlighted for liquid foam filling the porosity of granular packing$^{17,26}$: at equilibrium, liquid distributes in the pore space to form liquid bridges at contact between grains. More precisely, thanks to the volume exclusion of the bubbles within grain contact areas, liquid is extracted from the foam to form these liquid bridges. As a result, the effective liquid volume fraction of the foam in the pores, $\phi_l^{\text{eff}}$, is smaller than $\phi_l$. Moreover, it has been shown that $\phi_l^{\text{eff}}(r)$ is a decreasing function of $r$. Thus, it is highly likely that the mechanism responsible for the
<figure><img src="image_9.png" /><figcaption>FIG. 8. (a) Same data as in Fig. 7 for the normalized average yield stress $\tilde{\sigma}_y$, divided by the normalized yield stress of the corresponding bulk foam $\tilde{\sigma}_y(0) = \alpha (\phi_l^* - \phi_l)^2$, as function of the degree of confinement $r = R_b/R_g$, for all foam liquid fractions $\phi_l$, all the bubble sizes $R_b$ and grain sizes $R_g$; We used $\alpha = 0.6$ and $\phi_l^* = 0.36$. Crosses represent data found in the literature$^{15}$ but adapted to account for the effect of coarsening, which has been overlooked in the corresponding study. The analysis presented in the main text shows that the relevant bubble size to be considered is the one corresponding to $r \approx 0.5$. Solid lines are the predictions of Eq. 6. The horizontal dashed line corresponds to: $\tilde{\sigma}_y(r)/\tilde{\sigma}_y(0) = 1$, i.e., the behavior of the bulk foam. The vertical dashed line corresponds to $r = 0.154$, for which case the bubble size is equal to the size of the constriction made by the contacting grains in the packing, as illustrated by the diagram. (b) $\tilde{\sigma}_y$ as function of the effective liquid fraction of the foam $\phi_l^{\text{eff}}$ given by Eq. 5. The black curve corresponds to $\tilde{\sigma}_y = \alpha (\phi_l^* - \phi_l^{\text{eff}})^2$, with $\alpha = 0.6$.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>FIG. 9. Average foam yield stress $\sigma_y$, calculated using Eq. 4, versus average effective shear rate $\dot{\delta}$ obtained for grain size $R_g=1000~\mu\text{m}$ and (a) $\phi_l \approx 5\%$; (b) $\phi_l \approx 20\%$. Red triangles: blocked-coarsening conditions using $R_b \approx 60~\mu\text{m}$; Blue squares: blocked-coarsening conditions using $R_b \approx 95~\mu\text{m}$; Gray circles: blocked-coarsening conditions using $R_b \approx 180~\mu\text{m}$; Blue diamonds: free-coarsening conditions with an initial bubble size $R_{b,0} \approx 95~\mu\text{m}$. Dashed lines represent the average yield stress of the blocked-coarsening conditions at: $R_b \approx 60~\mu\text{m}$ (red), $R_b \approx 95~\mu\text{m}$ (blue) and $R_b \approx 180~\mu\text{m}$ (gray).</figcaption></figure>

increase in yield stress with the degree of confinement is as follows: as $r$ increases, the effective liquid fraction decreases, and according to Eq. 2, the yield stress increases. Note that here we assume that, despite the complexity of the foam flow through the pore space, the liquid distribution is similar to what has been found within static conditions. We will now quantitatively examine this hypothesis: The effective liquid fraction $\phi_l^{\text{eff}}(r)$ of polydispersed foam has been measured at different degree of confinement $r$ for $\phi_l = 20\%^{26}$. According to the theoretical approach proposed in$^{17}$, this result can be generalized for any $\phi_l$ value. Therefore, the general empirical expression for $\phi_l^{\text{eff}}(r)$ is the following$^{26}$:

$$ \frac{\phi_l^{\text{eff}}(r)}{\phi_l} = -56.191r^4 + 59.161r^3 - 13.585r^2 - 2.7915r + 1, \quad (5) $$

As a first test of our hypothesis, in Fig. 8b, we plot now the foam's yield stress $\tilde{\sigma}_y$ as a function of the effective liquid fraction $\phi_l^{\text{eff}}$ given by Eq. 5. We observe a clustering of our data in the form of a master curve that globally follows the trend observed for bulk foam (Eq. 2). Most of the data points that were previously far from the theoretical curve plotted using $\phi_l$ as a parameter, are now much better described using instead $\phi_l^{\text{eff}}$. Note that the agreement of Eq. 2 with our data is obtained using $\alpha = 0.6$, which falls within the expected range$^{1,27}$. This tends to validate our
approach based on the idea that the flowing foam has an effective liquid fraction lower than the overall liquid fraction of the pore space. This observation encourages us to take our analysis a step further: the resulting increase in yield stress is therefore expected to be given by:

$$ \frac{\tilde{\sigma}_y(r)}{\tilde{\sigma}_y(0)} = \frac{(\phi_l^* - \phi_l^{\text{eff}}(r))^2}{(\phi_l^* - \phi_l)^2}, \qquad (6) $$

where $\tilde{\sigma}_y(0)$ refers to the bulk foam's yield stress. Those predictions are plotted in Fig. 8a as solid curves, and they are describing well the experimental data normalized by their corresponding bulk foam yield stress (Eq. 2), using $\alpha = 0.6$. Fig. 8a quantifies the effect of confinement: For $\phi_l = 32\%$, yield stress values under confinement can be up to 50 times higher than their bulk counterparts, and this is fully explained by Eq. 6. This effect is however less pronounced for smaller liquid fractions. It is noteworthy that the increase in the effective yield stress tends to saturate for $r$ approaching 0.5. Thus, the majority of this increase is concentrated between $r$ values ranging from 0.05 to 0.3. Furthermore, we have verified that all the obtained curves can be accurately fitted by a sigmoidal function, suggesting that these curves describe a transition between the yield stress values characteristic of bulk foams and those defining the yield stress under strong confinement. As can be seen, Eq. 5 and Eq. 6, however, describe our data less accurately for low liquid fractions and high confinement ratios. This could suggest that our approach is insufficient or, let us say, incomplete. For instance, the assumption of liquid redistribution, as observed in static systems, might become inaccurate under dynamic conditions for low liquid fractions, due to the foam's low permeability to the liquid that is expected to redistribute. Investigating these aspects could be the subject of a dedicated follow-up study.

## B. Effect of coarsening

We showed, in section II C, measurements of the bubble size $R_b$ for experiments conducted under free-coarsening conditions. These experiments are performed with a granular packing of grain size $R_g = 1000$ µm, at both liquid fractions $\phi_l \approx 5\%$ and $20\%$, with produced bubbles of size $R_{b,0} \approx 95$µm. The measured average yield stress, $\sigma_y$ (Eq. 4), is shown in Fig. 9 as a function of the average effective shear rate, $\dot{\delta}$. For comparison, we present the results obtained for blocked-coarsening condition with three bubble sizes $R_b \approx 60, 95$ and $180$ µm. Note that, in agreement with what we discussed previously for blocked-coarsening condition, $\sigma_y$ is constant and doesn't depend on the shear rate. The behavior of the foam flowing within free-coarsening conditions at low liquid
fraction $\phi_l \approx 5\%$ is completely different: at low $\delta$, the yield stress is significantly smaller than that of its corresponding blocked-coarsening foam for $R_b \approx 95$ µm (blue squares) and closer to the yield stress obtained for $R_b \approx 180$ µm (gray circles), with this difference diminishing as $\delta$ increases (see Fig. 9a). This is fully consistent with the measured evolution for the size of the bubbles entering the flow cell (see Fig 4a), where the bubble radius is close to 180 µm at low flow rate and gets closer to 95 µm at higher flow rate. In contrast, for higher liquid fraction, i.e., $\phi_l \approx 20\%$, such evolution as a function of shear rate is strongly reduced (Fig. 9b), which accounts for the reduced bubble size increase reported in Fig 4b. With such variations in bubble size upon entering the column and during its passage through it, it becomes very difficult to extract rheological quantities that can be associated with a relevant bubble size. The corresponding values will primarily reflect the properties of the coarsened foam rather than those of the initially produced foam. In fact, a direct analysis that completely ignores these size variations would result in a completely incorrect flow law. For example, fitting the Herschel-Bulkley law to the $\sigma = f(\delta)$ data corresponding to Fig. 9a yields $\sigma_y \approx 4$ Pa and the stress increase is attributed to the plastic viscosity, whereas this is not the case: this increase corresponds to the yield stress associated with the smallest bubble sizes as the foam flow rate increases. Let us now analyze results found in the literature for foam flow in granular packing within coarsening-free conditions. Starting from bubble size $R_b \approx 11$ µm and liquid fraction $\phi_l = 15\%$, Omirbekov et al.¹⁵ highlighted a decrease in yield stress with an increase in grain size ($R_g$ ranging between 500 and 4000 µm). They assumed that, during the experiment, the bubble size was roughly fixed and the decrease in yield stress could be explained by the increase in the number of bubbles per pore. In fact, it is more likely that the effects of coarsening cannot be neglected, as we have shown previously, especially for residence times as long as those in Omirbekov's experiment (between 1 hour and 15 hours). Since the bubble size is initially small compared to the pore size, the foam is expected to coarsen similarly to bulk foam: the initial bubble size is expected to increase tenfold in less than 10 minutes²⁶,²⁸, which, in turn, increases the degree of confinement of the foam. As a result, it is very natural to assume that during the experiment, the bubble size increases dramatically to reach a size ratio $r$ close to 0.5, very close to the maximum value we were able to reach in our experiments. With this assumption, we can determine the relevant bubble size for their experiment¹⁵ and compute the corresponding dimensionless yield stress, which we plot in Fig. 8a, where they appear to be in line with our results. In particular, we note that their data, all corresponding to a liquid fraction of 15%, are generally between our data corresponding to 12% and 20%. This suggests that our framework for
describing the rheology of confined foam can be easily applied to both past and future studies,
provided the bubble size is known.

IV. CONCLUSION

We studied the flow of liquid foam in granular packing under well-controlled parameters, i.e., bubble size $R_b$, grain size $R_g$ and foam liquid fraction $\phi_l$. Thanks to reduced compressibility effects, we used the pressure-flow relationship to characterize the rheological behavior of the confined foam, under conditions where most of the stress associated with the flowing foam is due to the yield stress. Thanks to the blocked-coarsening conditions we were able to impose, we obtained a -1 power-law relationship between the apparent viscosity of the foam and the equivalent shear rate applied by the grains. The study highlights the critical role of confinement in determining the rheological properties of liquid foam in granular media, expressed with the yield stress of the confined foam $\tilde{\sigma}_y = \sigma_y/(\gamma/R_b)$, which was found to be higher than that of the bulk, even at high liquid fractions, where it was expected to be negligible. The yield stress increases with the decrease of the grain size $R_g$ at constant bubble size $R_b$, regardless of the liquid fraction $\phi_l$. Even at constant $R_g$, we reported an increase in $\tilde{\sigma}_y$ with the increase of $R_b$. Such a behavior suggests that the appropriate parameter to describe our results is the degree of confinement $r$ represented by the bubble-to-grain size ratio $r = R_b/R_g$. As the degree of confinement increases, the normalized yield stress of the confined foam increases for all liquid fractions $\phi_l$. Moreover, our results show that this increase is consistent with a smooth transition-like description, between the yield stress values characteristic of bulk foams, observed at very low values of the confinement parameter, and those defining the yield stress under strong confinement, i.e. $r \approx 0.5$. This suggests that the geometric effect associated with the passage of bubbles through the constrictions of the porous network does not play a significant role in this problem.

We have shown that the increase in the yield stress can be explained by the particular liquid
distribution in the granular packing, a phenomenon that has been reported recently in literature$^{17}$:
The liquid imported by the foam is redistributed at the level of the grains to form liquid bridges
at their contacts, leaving the foam with an effective foam liquid volume fraction smaller than the
global liquid fraction contained in the pore space, i.e. $\phi_l$. This explanation was found to well
describe our experimental data as governed by the confinement parameter $r$, excepted for low
liquid fractions and high confinement ratios, suggesting that a dedicated follow-up study is needed
to understand this issue.

Finally, we showed that, in order to better understand the flow of foam in porous media, we need to arrest the coarsening mechanism naturally found in foam, which would lead to an increase in the average bubble size. Coarsening would add some complexity to the flow by reducing the yield stress and varying the flow characteristics; hence, making it difficult to predict these parameters. We believe that the present study helps to predict the foam rheological properties in confined space. This will challenge us to predict the rheology complicated by coarsening and gas compressibility found in the applications. Such effects remain to be understood in detail.

## SUPPLEMENTARY MATERIAL

We provide in the Supplementary Material a table that contains all the values of $\tilde{\sigma}_y$ shown in Figures 7 & 8, with their corresponding experimental parameters ($R_b, R_g, \phi_l$, and $L$). We also provide all the rheological curves, i.e., average foam apparent viscosity $\mu_{app}$ divided by $\gamma/R_b$ as function of the average effective shear rate $\dot{\delta}$.

## ACKNOWLEDGEMENTS

The authors gratefully acknowledge funding from the Labex MMCD provided by the national program Investments for the Future of the French National Research Agency (ANR-11-LABX-022), funding from the French National Research Agency (ANR-23-CE51-0030-01) and co-funding from the European Space Agency (reference I-2020-02017).

## AUTHOR DECLARATIONS

The authors have no conflicts to disclose.

## AUTHOR CONTRIBUTIONS STATEMENT

O.P., V.L. and A.S. contributed to the preparation of the experiment. R.C. and A.S. did the measurements. O.P., V.L. and A.S. contributed to the analysis of the results. All the authors participated in the discussions and writing of the manuscript.
DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

REFERENCES

¹I. Cantat, S. Cohen-Addad, F. Elias, F. Graner, R. Höhler, O. Pitois, F. Rouyer, and A. Saint-Jalmes, *Foams: Structure and Dynamics* (Oxford University Press, Oxford, 2013).

²P. Stevenson, *Foam Engineering: Fundamentals and Applications* (John Wiley & Sons, New York, 2012).

³J. M. Alvarez, H. J. Rivas, and W. R. Rossen, “Unified Model for Steady-State Foam Behavior at High and Low Foam Qualities,” in *SPE Annual Technical Conference and Exhibition* (1999) https://onepetro.org/SPEATCE/proceedings-pdf/99ATCE/99ATCE/SPE-56825-MS/3506853/spe-56825-ms.pdf.

⁴A. R. Kovscek and C. J. Radke, “Fundamentals of foam transport in porous media,” (American Chemical Society, 1994) Chap. 3, pp. 115–163, https://pubs.acs.org/doi/pdf/10.1021/ba-1994-0242.ch003.

⁵G. Hirasaki, C. Miller, R. Szafranski, J. Lawson, and N. Akiya, “Surfactant/Foam Process for Aquifer Remediation,” in *SPE International Conference on Oilfield Chemistry* (1997) pp. SPE-37257-MS, https://onepetro.org/SPEOCC/proceedings-pdf/97OCS/97OCS/SPE-37257-MS/3754671/spe-37257-ms.pdf.

⁶C. Portois, C. Boeije, H. Bertin, and et al., “Foam for environmental remediation: Generation and blocking effect,” *Transport in Porous Media* **124**, 787–801 (2018).

⁷S. A. Jones, N. Getrouw, and S. Vincent-Bonnieu, “Foam flow in a model porous medium: II. the effect of trapped gas,” *Soft Matter* **14**, 3497–3503 (2018).

⁸G. J. Hirasaki and J. B. Lawson, “Mechanisms of Foam Flow in Porous Media: Apparent Viscosity in Smooth Capillaries,” *Society of Petroleum Engineers Journal* **25**, 176–190 (1985), https://onepetro.org/spejournal/article-pdf/25/02/176/2644542/spe-12129-pa.pdf.

⁹Q. Xu and W. R. Rossen, “Effective viscosity of foam in periodically constricted tubes,” *Colloids and Surfaces A: Physicochemical and Engineering Aspects* **216**, 175–194 (2003).
<footnote n="10">¹⁰A. Mauray, M. Chabert, and H. Bodiguel, “Yield stress fluid behavior of foam in porous media,” Phys. Rev. Fluids 5, 094004 (2020).</footnote>

<footnote n="11">¹¹A. H. Falls, J. J. Musters, and J. Ratulowski, “The apparent viscosity of foams in homogeneous bead packs,” SPE (Society of Petroleum Engineers) Reserv. Eng.; (United States) 4:2 (1989), 10.2118/16048-PA.</footnote>

<footnote n="12">¹²L. Pedroni and L. Nabzar, “New insights on foam rheology in porous media,” in *Rio Oil and Gas Expo and Conference* (2016).</footnote>

<footnote n="13">¹³S. A. Jones, N. Getrouw, and S. Vincent-Bonnieu, “Foam flow in a model porous medium: I. the effect of foam coarsening,” *Soft Matter* **14**, 3490–3496 (2018).</footnote>

<footnote n="14">¹⁴S. Nonnekes, L.E. and Cox and W. Rossen, “Effect of gas diffusion on mobility of foam for enhanced oil recovery,” *Transport in Porous Media* **106**, 669–689 (2015).</footnote>

<footnote n="15">¹⁵S. Omirbekov, H. Davarzani, and A. Ahmadi-Senichault, “Experimental study of non-newtonian behavior of foam flow in highly permeable porous media,” *Industrial & Engineering Chemistry Research* **59**, 12568–12579 (2020), https://doi.org/10.1021/acs.iecr.0c00879.</footnote>

<footnote n="16">¹⁶S. Omirbekov, H. Davarzani, S. Colombano, and A. Ahmadi-Senichault, “Experimental and numerical upscaling of foam flow in highly permeable porous media,” *Advances in Water Resources* **146**, 103761 (2020).</footnote>

<footnote n="17">¹⁷O. Pitois, A. Salamé, Y. Khidas, M. Ceccaldi, V. Langlois, and S. Vincent-Bonnieu, “Daisy-shaped liquid bridges in foam-filled granular packings,” *Journal of Colloid and Interface Science* **638**, 652–660 (2023).</footnote>

<footnote n="18">¹⁸F. Rouyer, O. Pitois, E. Lorenceau, and N. Louvet, “Permeability of a bubble assembly: From the very dry to the wet limit,” *Physics of Fluids* **22**, 043302 (2010).</footnote>

<footnote n="19">¹⁹N. Galvani, M. Pasquet, A. Mukherjee, A. Requier, S. Cohen-Addad, O. Pitois, R. Hohler, E. Rio, A. Salonen, D. J. Durian, and D. Langevin, “Hierarchical bubble size distributions in coarsening wet liquid foams,” *Proc. Natl. Acad. Sci. U. S. A.* **120**, e2306551120 (2023).</footnote>

<footnote n="20">²⁰F. G. Gandolfo and H. L. Rosano, “Interbubble gas diffusion and the stability of foams,” *Journal of Colloid and Interface Science* **194**, 31–36 (1997).</footnote>

<footnote n="21">²¹J. Kozeny, “Über die kapillare leitung des wassers im boden. sitzungsberichte.” *Royal Academy of Science* **136**, 271–306 (1927).</footnote>

<footnote n="22">²²R. Darby, R. Darby, and R. Chhabra, *Chemical Engineering Fluid Mechanics* (CRC Press, 2001).</footnote>

<footnote n="23">²³M. Kaviany, *Principles of Heat Transfer in Porous Media* (Springer, New York, 1995).</footnote>
<footnote n="24">²⁴D. Bauer, L. Talon, Y. Peysson, H. B. Ly, G. Batôt, T. Chevalier, and M. Fleury, “Experimental and numerical determination of Darcy’s law for yield stress fluids in porous media,” Physical Review Fluids 4, 063301 (2019).</footnote>

<footnote n="25">²⁵R. Poryles, N. Gland, A. King, E. Rosenberg, L. Barré, and T. Chevalier, “Foam trapping in a 3d porous medium: in situ observations by ultra-fast x-ray microtomography,” Soft Matter 16, 6354–6361 (2020).</footnote>

<footnote n="26">²⁶A. Salame, V.-T. Nguyen, V. Langlois, A. Petit, B. Soltner, O. Pitois, and S. Vincent-Bonnieu, “Enhanced coarsening induced by pore confinement,” Phys. Rev. E 111, L013401 (2025).</footnote>

<footnote n="27">²⁷F. Gorlier, Y. Khidas, and O. Pitois, “Yielding of complex liquid foams,” Journal of Rheology 61, 919–930 (2017).</footnote>

<footnote n="28">²⁸M. Pasquet, N. Galvani, A. Requier, S. Cohen-Addad, R. Höhler, O. Pitois, E. Rio, A. Salonen, and D. Langevin, “Coarsening transitions of wet liquid foams under microgravity conditions,” Soft Matter 19, 6267 (2023).</footnote>