Steady-State Dynamics of Ganglia Populations During Immiscible Two-Phase Flows in Porous Micromodels: Effects of the Capillary Number and Flow Ratio on Effective Rheology and Size Distributions

Abstract

We study experimentally the flow of non-wetting ganglia during the co-injection of n-heptane and water in a predominantly 2D PMMA micromodel, which is constructed based on a stochastic digital algorithm. The dynamics of the phase distribution patterns are recorded optically and post-processed using cluster identification and motion tracking algorithms in order to study the characteristics and the interactions between the mobile and stranded ganglia populations. We focus primarily on the effects of the capillary number (Ca) and the ratio of the injection flow rates (Q) on the observed ganglia size distributions and the effective two-phase rheology. Our experimental setup allows for the study of ganglia fragmentation and coalescence dynamics over five orders of magnitude (in terms ganglia sizes), thus offering novel physical insight on the pore-scale characteristics of different ganglia populations and on how their interactions determine the relative permeability of the non-wetting phase. We demonstrate that the rates of ganglia fragmentation and coalescence intensify at higher Ca values, as viscous forces become dominant over capillary ones, leading to a log-normal size distribution that shifts toward smaller mean values. This effect is directly correlated with the emergence of new flow paths that develop progressively through narrower pores-throats, where the continuous wetting phase sweeps ganglia with sizes smaller than the mean pore-throat diameter. These flow paths further contribute to the Darcy scale velocity of the non-wetting phase, thus leading to a power-law Darcian regime at intermediate Ca values with a scaling exponent that is found to be a function of Q.

1 Introduction

Immiscible two-phase (2p) flow through macroporous media is a physical process encountered in various environmental and energy-related applications. In the limit of low non-wetting (nw) phase saturations, the process is dominated by the formation of disconnected nw-phase blobs (or ganglia) that are typically stranded in the pore space due to strong capillary forces, despite the viscous stresses and pressure gradients imposed upon them by the flowing continuous wetting (w) phase. This phenomenon is commonly observed in various industrial applications and underground flow processes, such as enhanced oil recovery (EOR) (Payatakes et al. 1980), CO$_{2}$ sequestration (Andrew et al. 2014; Riazi et al. 2011; Al-Menhali et al. 2016), groundwater remediation (Sahloul et al. 2002; Reddi 1998), and air-sparging in saturated groundwater aquifers (Nyer and Suthersan 1993; Yang et al. 2005), among others. The formation of stranded disconnected nw-phase blobs under specific flow conditions has long been identified in the literature as the source of the so-called residual saturation in geologic flows (Chatzis et al. 1983, 1988; Mayer and Miller 1993).

At higher values of the nw-phase saturation, 2p flows are characterized by the presence of both stranded and mobile (i.e., flowing) ganglia populations with sizes ranging from very small ones (e.g., smaller than the mean pore size) to large ones covering several adjacent pores. Under steady-state flow conditions, these populations have been shown to continuously exchange mass between them through ganglia fragmentation and coalescence, thus leading to a dynamic equilibrium process (Yiotis et al. 2013; Talon et al. 2023). Significant insight in the complex process of ganglia flow and pore-scale interactions has been provided through earlier contributions by Payatakes and co-workers using elaborate pore-scale micromodels (Vizika et al. 1994; Avraam and Payatakes 1995). This approach allowed for the direct optical observation of ganglia dynamics under different flow conditions in predominantly 2D porous media. Based on these experiments, the authors verified the co-existence of mobile and stranded ganglia, and proposed mechanistic rules for their mobility based on their size and the stochastic spatial distribution of capillary pressure thresholds that were inversely proportional to the mean pore-throat size (Dias and Payatakes 1986a, b; Amili and Yortsos 2006). According to these studies, all nw-phase ganglia are subjected to a pressure difference imposed by the flowing continuous w-phase between their leading (downstream) and trailing (upstream) edges. When the pressure difference is greater than the local capillary threshold, then the ganglion was considered to be mobile, while otherwise it would be stranded.

More recent studies have focused on the effects of the fragmentation/coalescence (that takes place, in principle, at the pore scale) on the field-scale flow dynamics of both phases. At this scale (also known as the REV-scale), 2p flows have been traditionally modeled based on the classical generalized Darcy expressions (see, e.g., Hilfer 1998). This model relies on the simplified intrinsic assumption that each phase (either the w- or the nw-phase) flows within a dedicated flow channel under a pressure gradient which is assumed to be different for each phase. Therefore, the generalized Darcy approach, in its simpler form, assumes local capillary equilibrium between the two phases and neglects momentum transfer among them (an assumption which is intuitively and physically erroneous) (see, e.g., Sinha and Hansen 2012; Yiotis et al. 2019).

Several studies have attempted to resolve this inconsistency at the REV-scale description of the process (where the porous medium is treated as an effective continuum) by introducing a further generalization of Darcy’s law with the permeability being a second-order tensor with nonzero off-diagonal terms (De Gennes 1983; Rose 1990; Kalaydjian 1990). The proposed off-diagonal terms of the permeability tensor would account for momentum transfer across the interface, thus also allowing for the modeling of the flow of ganglia under the sweeping effect of the continuous w-phase. Despite their physical consistency, these early contributions proposed, however, a more generic approach without any specific consideration for specific pore-scale effects during ganglia flow. Therefore, the integration of such effects in the context of such models would require the calculation of all the elements of the permeability tensor based on either experimental or pore-scale studies (eventually rendering the actual problem even more complex compared to the intrinsic simplicity of the original Darcy formulation).

More recent numerical and experimental studies have focused on the specific case of 2p flows where ganglia flow is a major contributor to the nw-phase flow rates by considering such pore-scale effects. Tallakstad et al. (2009) and Sinha and Hansen (2012); Sinha et al. (2017) have shown both numerically and experimentally that the effective rheology through a porous medium is a nonlinear (power-law) function of the total pressure difference for intermediate values of the capillary number (Ca) with a practically constant scaling exponent. This has been attributed to the changes on the apparent permeability of the porous medium with Ca. Yiotis et al. (2013, 2019) have also studied the effects of fragmentation/coalescence on the mean field rheology using a robust Lattice Boltzmann model to resolve the immiscible flow process at sub-pore scale resolution in stochastically reconstructed 2D porous domains. They have also verified the above scaling expression but reported a somewhat different value for the scaling exponent. Furthermore, McClure et al. (2022) have recently shown that the relative permeability required for REV models can be derived by averaging in both space and time in case that the total pressure difference is considered as the driving force for both phases. These studies have clearly demonstrated the effects of the capillary number (Ca) on the apparent permeability of the medium due to the shifting dynamics of the fragmentation/coalescence process and the relevant pore-scale mechanisms (Amili and Yortsos 2006; Datta et al. 2014; Singh et al. 2017). Such effects can be modeled using a mean-field approach where the relative permeability of each phase is expressed as a function of both Ca and the time-averaged saturation during 2p flows.

In this contribution, we study the effective rheology of the nw-phase through porous media during 2p flows in the limit where ganglia flow is dominant. For this purpose, we perform an extensive experimental study of the fragmentation/coalescence process in order to determine how the structural properties (i.e., size and length) and the mobility of the ganglia populations contribute to the mean-field flow dynamics. The experimental setup is based on a 2D porous micromodel, constructed using a stochastic reconstruction algorithm, that leads to the formation of ganglia spanning over five orders of magnitude in size. We focus primarily on the effects of the capillary number (Ca) (i.e., viscous over capillary forces) and the flow rate ratio between the two fluids (Q) (i.e., ganglia crowding effects due to nw-phase saturation) on the ganglia populations and the contribution of different size groups in the effective rheology of the micromodel.

2 Pore-Scale Experiments

2.1 Micromodel Design and Construction

The micromodel used in this study consists of two transparent PMMA plates that are glued together using epoxy glue. The plates have a length of $L=325\,\text {mm}$, a width of $W=120\,\text {mm}$ and a thickness of approximately 18 mm. The inner surface of the bottom plate is engraved at a depth of $e=300\,\upmu \text {m}$ using a standard CNC (computer numerical control) drilling procedure with a drill bit of 0.8mm in diameter, following a pattern produced using a stochastic reconstruction algorithm (Yiotis et al. 2013; Chevalier et al. 2015). This algorithm produces a disordered digital pattern that consists of randomly shaped obstacles (or pillars) corresponding to the unengraved region of the plate. The solid obstacles have a correlation length (by design) of $\lambda _\textrm{d}=900\,\upmu \text {m}$, while the top-view porosity of the entire pattern is $\phi _\textrm{d}= 0.82$, as shown in Fig. 1a. The size of the produced obstacles in the design follows a log-normal distribution, as shown in Fig. 1d. Given the above parameters, the mean obstacle size (i.e., surface area) of the design is found equal to $\bar{V}_\text {s}=1.0\,\text {mm}^2$, which corresponds to a characteristic diameter of $D_\text {s}=1.15\,\text {mm}$ (assuming a circular obstacle shape). After the drilling process, the actual mean obstacle size of the engraved surface was measured optically and was found equal to $\lambda =1.40\,\text {mm}$, while the resulting porosity was found equal to $\phi =0.80\pm 0.02$, quite close to the original design value.

A single inlet and three outlet holes were drilled through the top plate at the opposite sides of the micromodel, as also shown in Fig. 1a. These served for fixing the inlet/outlet tubings and valves on the micromodel. A triangular region was also engraved just below each of these ports that served for better distributing the fluid flow at the inlet/outlets.

Fig. 1

a Top view of the porous media micromodel design used in this study. The black regions correspond to solid obstacles (unengraved surfaces of the PMMA plate), while the white regions correspond to the pore space with a depth of $e=300\,\upmu \text {m}$. Also shown in red circles are the locations of the flow input and output ports, and the transparent red area that corresponds to the field of view for the optical measurements. b−c Close-ups of the porous domain during a typical immiscible flow experiment. The non-wetting ganglia (dyed water) appear in blue color, while the solid obstacles and the wetting phase (n-heptane) are transparent in the snapshots. d Size distribution of the solid obstacles in the design rescaled with the average obstacle size $\bar{V}_\text {s}$

The intrinsic permeability of the micromodel was measured after fully saturating it by injecting n-heptane at different flow rates ranging between $0.1\le q_\textrm{w} \le 126\,\text {ml/min}$. At the same time, the pressure difference between the inlet and the outlets was measured using a high-precision differential digital manometer (see next section for further details). The results are plotted in Fig. 2 as a function of the Reynolds number, $\text {Re}=\frac{\rho _\textrm{w} q_\textrm{w} e}{\mu _\textrm{w} W e}$, where $\rho _\textrm{w}$ and $\mu _\textrm{w}$ are the density and viscosity of n-heptane, respectively.

A clear linear dependence between the flow rate, $q_\textrm{w}$, and the pressure drop, $\Delta P$, is observed for $\text {Re}<5$ (inset of Fig. 2), corresponding to the Darcian flow regime. For values of $\text {Re}>5$, a divergence from linearity is observed, which is clearly reminiscent of the Forchheimer regime in porous media, where the onset of inertia is first demonstrated. In this regime, it is expected that the pressure drop scales as $\Delta P \propto \left( \frac{q_\textrm{w}}{We}\right) ^2$. Based on the low Re number measurements in the linear regime (i.e., $\textrm{Re}<5$), we calculated the permeability of the micromodel to be equal to $k=1.65 \times 10^{-9} m^2$ (by fitting Darcy's law on the data). This value can be directly compared to the theoretical permeability for flow through two parallel plates (in the absence of obstacles) with the same distance, e, i.e., $k_0=e^2/12=7.5 \times 10^{-9}\,\text {m}^2$. We thus observe that $k=0.22\times k_0$.

Fig. 2

Volumetric flow rate, $q_\textrm{w}$, expressed in terms of the corresponding Reynolds number, $\textrm{Re}=\frac{\rho _\textrm{w} q_\textrm{w} e}{\mu _\textrm{w} W e}$, vs the pressure difference, $\Delta P$, across the micromodel during a single phase flow experiment. The inset shows a logarithmic close-up on the low Re number regime where a clear linear scaling is observed (i.e., scaling exponent equal to one)

2.2 Experimental Setup

The transparent micromodel described in the previous section was positioned just above a flat light source (Metaphase LED Backlight by Edmund Optics) in order to enhance the contrast between the two phases during the experiments, while a full-frame DSLR camera (Nikon D800 with Nikon AF-S Nikkor 35 mm f/1.8 lenses) was placed directly above it, as shown in Fig. 3. The camera allowed for recording the dynamics of the ganglia flow during each experiment within a field of view of 180 mm $\times$ 120 mm (transparent red area in Fig. 1a) and a maximum resolution of 24.5 $\upmu \text {m}$/pixel. The optical data were collected using either sequential imaging (at a rate of 0.1–0.2 fps and maximum resolution) or video recording (at a rate of 30 fps and a resolution of 94 $\upmu \text {m}$/pixel).

The fluids were injected simultaneously using two gear pumps (Ismatec BVP-Z) with interchangeable suction-shoe heads (Micropump Z-181), capable of delivering 0.1 to 100 ml/min, each. Each pump was used for injecting only one of the two fluids, either the wetting (i.e., n-heptane) or the non-wetting one (i.e., water). The fluids were continuously re-circulated during the experiments through the micromodel and were mixed together just before the single inlet using a single T-junction fitting. The 3 outlets of the micromodel were connected to a single tank, where the fluids were separated due to buoyancy, before being re-injected to the micromodel. The pressure difference between the inlet and outlet ports was monitored and recorded using a differential digital manometer (Digitron PM-80, 0–130 mbar).

Fig. 3

Right: Schematic of the experimental setup for the immiscible flow experiments used in this study. Left: Snapshot of the micromodel during a typical immiscible flow experiment. The dashed red area shows the field of view recorded by the camera

2.3 Immiscible Flow Experiments

We performed a series of immiscible flow experiments by simultaneously injecting in the micromodel n-heptane ($\mu _\textrm{w}=0.4\times 10^{-3}\,\text {Pa}\,\text {s}$, $\rho _\textrm{w}=700\,\text {kg/m}^3$), that served as the w-phase fluid, and water ($\mu _\textrm{nw}=10^{-3}\,\text {Pa}\,\text {s}$, $\rho _\textrm{nw}=1000\,\text {kg/m}^3$), that served as the nw-phase. The aqueous nw-phase was dyed using methyl blue by Sigma-Aldrich in order to visualize and record the flow of the nw-phase ganglia through the micromodel. The interfacial tension between the two fluids was found from the literature to be equal to $\gamma =52 \times 10^{-3}\,\text {N/m}$.

For each experiment, we defined the flow ratio of the injected fluids as $Q=q_\textrm{w}/q_\textrm{nw}$, where $q_\textrm{w}$ and $q_\textrm{nw}$ are the injection flow rates of n-heptane and water, respectively, and an effective capillary number as $\text {Ca}_\textrm{e}=\frac{\mu _\textrm{e} u_\textrm{t}}{\gamma }$, where $\mu _\textrm{e}=\frac{q_\textrm{w} \mu _\textrm{w} + q_\textrm{nw} \mu _\textrm{nw}}{q_\textrm{nw}+q_\textrm{nw}}$ is the volume-averaged viscosity of the fluids and $u_\textrm{t}=\frac{q_\textrm{w}+q_\textrm{nw}}{e W}$ is the total Darcy velocity of the fluids through the micromodel. This particular definition is based on the simplified assumption that the ratio of the injected flow rates is representative of the phase saturations within the field of view of our micromodel. Therefore, the effective viscosity is expressed as a volume-averaged property based on the injection ratio of the phases, that actually corresponds to the flowing volume of each phase.

A different approach would be to define the capillary number based on the actual average phase saturation that accumulates within the micromodel at steady state flow conditions. For the disconnected nw-phase, the observed saturation consists of both the mobile phase volume as well as the stranded volume, which may constitute a significant fraction of the total. In order to better understand the impact of the selected definition for the capillary number in our results, we also define a separate value for each phase as; $\text {Ca}_\textrm{w}=\mu _\textrm{w} u_\textrm{w}/\gamma$ and $\text {Ca}_\textrm{nw}=\mu _\textrm{nw} u_\textrm{nw}/\gamma$, where $u_\textrm{w}=q_\textrm{w}/\text {(We)}$ and $u_\textrm{nw}=q_\textrm{nw}/\text {(We)}$ is the Darcy-scale velocity of the w- and nw-phases, respectively. Based on these definitions, it is straightforward to show that $\text {Ca}_\textrm{nw}=\left( \frac{\mu _\textrm{nw}}{\mu _\textrm{w}}\right) \text {Ca}_\textrm{w}/Q$.

Initially, the micromodel was fully saturated by the w-phase by flushing n-heptane for several minutes at the maximum flow rate of the pump (i.e., 100 ml/min) to remove any air bubbles. When the micromodel became fully saturated, we started co-injecting both phases at constant Q and $\text {Ca}_\textrm{e}$ values, according to the values reported in Table 1, while measuring the pressure difference between the inlet and outlet ports. After a sufficient amount of time (ranging from a few minutes to several hours, depending on the total flow rate, $q_\textrm{t}=q_\textrm{w}+q_\textrm{nw}$, as well as on Q), the pressure difference reached a steady-state value. At the same time, the phase distribution patterns appeared to evenly cover the micromodel pore-space (on a time-averaged basis). From this point onward, both the average pressure difference and the phase distribution patterns were recorded over predefined time intervals in the order of $\delta t=5{-}15\,\text {s}$ for a period ranging from only a few minutes for experiments at higher $\text {Ca}_\textrm{e}$ values to several hours for those at very low $\text {Ca}_\textrm{e}$ values. After the completion of the imaging sequence, a video of a duration of 1−2 min was also recorded for each experiment.

In order to minimize “history effects” between subsequent experiments, we followed a strict experimental protocol by progressing from experiments of lower $\text {Ca}_\textrm{e}$ values toward higher ones, and from higher Q values toward lower ones (Chevalier et al. 2015). After the completion of each set of experiments at constant Q, the micromodel was flushed once again with n-heptane, in order to efficiently recover aqueous phase residuals. This procedure has proven to have a good sweeping efficiency leading to a typical residual saturation of water of approximately $S_\textrm{nw}\simeq 0.06$ (Chevalier et al. 2015). After flushing the micromodel, a new set of experiments was performed at a lower Q, according to the experimental sequence shown in Table 1 of Appendix, which also shows the complete set of the performed experiments.

2.4 Image Processing Methodology

After each experiment, the recorded steady-state images and videos were post-processed in order to calculate the average saturation of each phase and the ganglia size distributions within the field of view. This achieved by splitting the videos into still images (30 images/s) using the open-source video editing software FFmpeg (Fig. 4a). Each image was then binarized by applying a threshold value to isolate and characterize the ganglia (shown with red color in Fig. 4b) using a clustering algorithm developed in Matlab.

The produced binarized images for each experiment were then time-averaged in order to produce a mean phase distribution field. This describes the probability of any specific position in space being occupied by the nw-phase, i.e., $p({\textbf {x}})=\left\langle S_\textrm{g}({\textbf {x}}) \right\rangle _\textrm{t}=\frac{\int _{t=0}^{t_\textrm{max}}S_\textrm{g} \text {d}t}{t_\textrm{max}}$, where ${\textbf {x}}$ is the position vector and $S_\textrm{g}$ is the saturation of the nw-phase, which is equal to unity when a ganglion is present, and equal to zero, otherwise, and $t_\textrm{max}$ is the duration of the video. Therefore, darker red regions in Fig. 4c, where $p({\textbf {x}})=\left\langle S_\textrm{g}({\textbf {x}}) \right\rangle _t \simeq 1$, correspond to longer occupancy and thus mostly stranded ganglia, while lighter red and blue regions, where $p({\textbf {x}}) =\left\langle S_\textrm{g}({\textbf {x}}) \right\rangle _\textrm{t} < 1$, correspond to shorter temporal occupancy, which is a characteristic of flow paths.

By applying a probability threshold on the above field, i.e., $p({\textbf {x}})\ge 0.98$, we identify the regions of the micromodel that contain practically stranded ganglia over the entire video duration, i.e., $S_\textrm{im}=\left\langle p({\textbf {x}})\ge 0.98 \right\rangle _{{\textbf {x}}}$. We are thus able to calculate the stranded ganglia saturation, $S_\textrm{im}$, as well as the total (time-averaged) nw-phase saturation (that included both mobile and stranded ganglia), $S_\textrm{tot}$ by averaging over the nw-phase saturation that appears in all the images, i.e., $S_\textrm{tot}=\left\langle \left\langle S_\textrm{g}({\textbf {x}})\right\rangle _{{\textbf {x}}} \right\rangle _\textrm{t}$. The time-averaged mobile nw-phase saturation is thus calculated as $S_\textrm{mob}=S_\textrm{tot}-S_\textrm{im}$. The clustering algorithm was then applied on both the binarized and time-averaged frames (after applying the residency threshold) in order to recover the size distribution of the entire ganglia population (mobile and stranded) and that of the stranded population, respectively, as a function of $\text {Ca}_\textrm{e}$ and Q.

Fig. 4

Schematic of the video and image analysis procedure. a Actual still images extracted from the experimental videos. b Binarized image sequence after applying a color threshold to identify nw-phase ganglia (shown here in red color). c Time-averaged binarized image sequence that demonstrates the probability, p, of locating the nw-phase in any location. Here, the dark red colors correspond to $p=1$, while blue colors correspond to $p=0$. Also shown in black color is the location of the solid obstacles (pillars) of the micromodel. d Intensity map of the flow patterns that demonstrates the dominant flow channels of the ganglia. Here red colors correspond to $p'=1$ (i.e., heavier ganglia flow), while darker blue colors correspond to $p'=0$ (i.e., no ganglia flow)

An alternative approach was used to identify the pore-scale flow paths from where the ganglia preferably flowed through the micromodel. This was achieved by plotting a probability field that shows the spatial intensity of changes in saturation, $\left| \Delta S({\textbf {x}})\right|$ over subsequent images in time. When a change in saturation occurs at position ${\textbf {x}}$ between t and $t+\delta t$ (i.e., over subsequent time frames of the video) then $\left| \Delta S({\textbf {x}}) \right|$ takes a value of unity, while otherwise, it becomes zero. Then the time-averaged field $p^{'}({\textbf {x}})=\left\langle \left| \Delta S({\textbf {x}}) \right| \right\rangle _t=\frac{\int _{t=0}^{t_\textrm{max}}\left| \Delta S{\textbf {(x)}}\right| \text {d}t}{t_\textrm{max}}$ shows the intensity of ganglia flow in space and thus highlights the major flow paths of the nw-phase, as shown in Fig. 4d. The major 2p flow paths appear in lighter yellow-green color, while the stagnant flow regions appear in dark blue. These dark blue regions also correspond to the immobile ganglia regions. Note that the maximum intensity value, corresponding to the red color, appears only in very limited areas of these plots. We should also note that this approach is valid, as long as the nw-phase flows in a disconnected manner, rather than in the form of ganglia spanning across the entire domain.

3 Results and Discussion

3.1 Pore-Scale Ganglia Dynamics

The immiscible flow experiments performed under the previously discussed conditions reveal the complex dynamics of the ganglia fragmentation/coalescence process that eventually determine the field-scale rheology of the system. The dynamics of the process can be initially reviewed in the typical recorded experimental videos (prior to thresholding) for $Q=25.2$ and $Q=12.6$, that are provided as supplemental material for Exp1 (2023) ($\text {Ca}_\textrm{e}=9.9\times 10^{-5}$) and Exp5 (Original recorded video of Exp 5 (2023) ($\text {Ca}_\textrm{e}=5.4\times 10^{-5}$), respectively. Here, the nw-phase appears in blue color (i.e., water dyed with methyl blue), while the w-phase (i.e., n-heptane) is transparent. The solid pillars are also transparent in these videos, although their external boundaries are slightly visible. The corresponding binarized videos for the same experiments are also provided for Exp1 (2023) and Exp5 (2023). In this case, the nw-phase ganglia are clearly marked in red color by applying the thresholding algorithm previously described, while the w-phase appears in light blue color. The location of the solid pillars is also shown in back color after overlaying a mask that was constructed based on a high-resolution snapshot of the produced engraved micromodel.

These videos reveal that under our experimental conditions, the nw-phase flows predominantly in the form of non-percolating clusters (i.e., ganglia) with a very wide range of sizes. Depending on the flow conditions, these ganglia may be smaller than the typical pore size or span over several adjacent pores. Furthermore, they appear to undergo a continuous life-cycle of coalescence (into larger ones) and fragmentation (into smaller ones), as also earlier observed in the literature (Yiotis et al. 2019). At the same time, we can clearly identify two discrete ganglia populations, a mobile and a stranded one, that continuously exchange mass between them through fragmentation and coalescence events that take place at the pore-scale. Typically stranded ganglia may become mobilized when a mobile ganglion collides and coalesces with them. In this case, the resulting larger ganglion is subjected to a larger pressure difference across its leading and trailing edge, thus overcoming the capillary pressure threshold of the specific pore-throat and leading to its mobilization (Ng and Payatakes 1980). The opposite process of fragmentation typically leads to the stranding of at least one of the newly formed ganglia in the downstream direction, as long as the resulting ganglia are larger than the typical pore size (Talon et al. 2023). In case the ganglia are smaller than the pore size, then they are no longer subjected to capillary forces and they flow around the solid obstacles freely. These very small ganglia, however, exhibit a very short lifetime, as they tend to coalesce with other ganglia to form larger ones. This phenomenon is reminiscent of the Ostwald ripening process, in a completely different process, however (i.e., in the absence of ganglia dissolution) (Santos et al. 2017).

3.2 Size Distributions of Stranded and Total Ganglia Populations

3.2.1 Effects of $\text {Ca}_\textrm{e}$

In our experiments, the mean ganglion size appears to be a strong function of both $\text {Ca}_\textrm{e}$ and Q. We expect therefore that both these parameters affect the field-scale rheology of the system. Typically, smaller sizes appear at higher $\text {Ca}_\textrm{e}$ values, where the fragmentation process intensifies as viscous and pore-scale inertia forces become dominant over capillary ones. This is demonstrated in the characteristic snapshots shown in Fig. 5a–c, where the $\text {Ca}_\textrm{e}$ increases by approximately half an order of magnitude (i.e., from $9.86\times 10^{-5}$ to $4.96\times 10^{-4}$) for a fixed value of the injection ratio, $Q=25.2$.

Fig. 5

Snapshots of the phase distribution patterns for a flow ratio of $Q=25.2$, and a $\text {Ca}_\textrm{e}=9.86\times 10^{-5}$ (Exp1), b $\text {Ca}_\textrm{e}=1.97\times 10^{-4}$ (Exp2), and c $\text {Ca}_\textrm{e}=4.96\times 10^{-4}$ (Exp4). The nw-phase ganglia are shown in red, the w-phase in blue, and the solid obstacles in black color

An insightful comparison between these three experiments under constant Q is also provided in video (2010), where the frame rate has been rescaled by a factor of $\frac{q_\textrm{w,ref}+q_\textrm{nw,ref}}{q_\textrm{w}+q_\textrm{nw}}$, in order to highlight the differences in the size distributions of the entire ganglia population at approximately the same normalized velocity in all three experiments. Here, $q_\textrm{w,ref}$ and $q_\textrm{nw,ref}$ correspond to the w- and nw-phase flow rates, respectively, of the reference experiment, selected here as ‘Expl’. This reveals that the ganglia flow at a pore-scale velocity that is practically proportional to $\text {Ca}_\textrm{e}$, while the mean ganglion size is clearly a decreasing function of $\text {Ca}_\textrm{e}$.

We quantify this result by plotting the ganglia size distributions for these three experiments in Fig. 6. These plots show that, interestingly enough, the probability density follows a log-normal distribution for both the total (transparent red bars) and stranded populations (transparent blue bars) with a mean value in the vicinity of the mean pillar size, $\bar{V}_\textrm{s}$ (shown here as a green vertical line). The mean of both the total and stranded populations is found to be a decreasing function of $\text {Ca}_\textrm{e}$, with the total population being more sensitive to this parameter. This should be clearly attributed to the strong dependence of the size of the mobile ganglia on $\text {Ca}_\textrm{e}$, as also shown in the recorded videos (2023). It is also interesting to note that for $Q=25.2$ and $\text {Ca}_\textrm{e}=3.9\times 10^{-10}$ the average immobile population is larger in size that the total population. This reveals that the mobile ganglia should be much smaller than the stranded ones and the typical pore size of the micromodel. This regime is reminiscent of the drop traffic flow identified by Avraam and Payatakes (1995).

Fig. 6

Distribution of the ganglia size, $V_\textrm{g}$, normalized by the mean obstacle size, $\bar{V}_\textrm{s}$, for a flow ratio of $Q=25.2$, and a $\text {Ca}_\textrm{e}=9.86\times 10^{-5}$ (Exp1), b $\text {Ca}_\textrm{e}=1.97\times 10^{-4}$ (Exp2), and c $\text {Ca}_\textrm{e}=4.96\times 10^{-4}$ (Exp4). The size distribution of the total population is shown in red color, while the immobile population is shown in blue color. Also shown a log-normal distribution curve fitted on the experimental data for each population

3.2.2 Effects of Q

By decreasing the flow rate ratio, Q, in favor of the ganglia phase, the mean saturation of the nw-phase in the micromodel increases, as expected, allowing for the formation of larger ganglia. This is demonstrated in Fig. 7 for $Q=12.6$ and Fig. 9 for $Q=6.3$, respectively. A closer inspection of the snapshots corresponding to the lower $\text {Ca}_\textrm{e}$ values ($\text {Ca}_\textrm{e}=5.38\times 10^{-5}$ for ‘Exp5’ and $\text {Ca}_\textrm{e}=3.13\times 10^{-5}$ for ‘Exp10’) reveals the formation of very long ganglia that are almost spanning the entire micromodel along the main flow direction. As Q decreases, this flow regime becomes dominant for the nw-phase at lower $\text {Ca}_\textrm{e}$, resulting practically in a dominant flow channel for the nw-phase. This regime is reminiscent of the connected pathway flow identified by Avraam and Payatakes (1995).

An increase, however, in the value of $\text {Ca}_\textrm{e}$ under constant Q eventually leads to the intensification of the dynamic fragmentation process, thus producing initially large ganglia, that do not, however, span the entire domain, as shown in Fig. 7b, c and Fig. 9b, c. This is also clearly observed in the video for $Q=12.6$ (Layout 2010) where the running time for ˜Exp6 and ˜Exp7 has been rescaled to match the superficial velocity of ˜Exp5 that appears in actual time.

Fig. 7

Snapshots of the phase distribution patterns for a flow ration of $Q=12.6$, and a $\text {Ca}_\textrm{e}=5.38\times 10^{-5}$ (Exp5), b $\text {Ca}_\textrm{e}=1.08\times 10^{-4}$ (Exp6), and c $\text {Ca}_\textrm{e}=2.15\times 10^{-4}$ (Exp8). The nw-phase ganglia are shown in red, the w-phase in blue and the solid obstacles in black color

As expected, the decrease of Q leads to a clear increase of the mean ganglion size of the entire population at lower $\text {Ca}_\textrm{e}$ values, as shown in the probability density functions of Fig. 8a for $Q=12.6$ and Fig. 10a for $Q=6.3$, compared to the case of $Q=25$ (i.e., Fig. 5a). At the same time, however, the stranded population distribution appears to be more insensitive to the increasing values of $\text {Ca}_\textrm{e}$, reaching for all cases ($Q=6.3,12.6,25.2$) to practically the same size distribution curve with the same mean and variance (see Figs. 5c, 7c, 9c).

Fig. 8

Distribution of the ganglia size, $V_\textrm{g}$, normalized by the mean obstacle size, $\bar{V}_\textrm{s}$, for a flow ratio of $Q=12.6$, and a $\text {Ca}_\textrm{e}=5.38\times 10^{-5}$ (Exp5), b $\text {Ca}_\textrm{e}=1.08\times 10^{-4}$ (Exp6), and c $\text {Ca}_\textrm{e}=2.15\times 10^{-4}$ (Exp8). The size distribution of the total population is shown in red color, while the immobile population is shown in blue color. Also shown is a log-normal distribution curve fitted on the experimental data for each population

Fig. 9

Snapshots of the phase distribution patterns for a flow ratio of $Q=6.3$, and a $\text {Ca}_\textrm{e}=3.13\times 10^{-5}$ (Exp10), b $\text {Ca}_\textrm{e}=9.40\times 10^{-5}$ (Exp12), and c $\text {Ca}_\textrm{e}=1.88\times 10^{-4}$ (Exp14). The nw-phase ganglia are shown in red, the w-phase in blue and the solid obstacles in black color

Fig. 10

Distribution of the ganglia size, $V_\textrm{g}$, normalized by the mean obstacle size, $\bar{V}_\textrm{s}$, for a flow ratio of $Q=6.3$, and a $\text {Ca}_\textrm{e}=3.13\times 10^{-5}$ (Exp10), b $\text {Ca}_\textrm{e}=9.40\times 10^{-5}$ (Exp12), and c $\text {Ca}_\textrm{e}=1.88\times 10^{-4}$ (Exp14). The size distribution of the total population is shown in red color, while the immobile population is shown in blue color. Also shown is a log-normal distribution curve fitted on the experimental data for each population

This result is more clearly shown in Fig. 11 (right) where the mean ganglion size rescaled by the mean obstacle size is plotted vs $\text {Ca}_\textrm{e}$ for different values of Q. It is evident that the mean size of the stranded population decreases rapidly with $\text {Ca}_\textrm{e}$ for lower $\text {Ca}_\textrm{e}$ values, while it reaches asymptotically to the same value (i.e., $\text {ln}\left( \frac{V_\text {g}}{\bar{V}_\text {s}}\right) \simeq -0.4$) for larger values of $\text {Ca}_\textrm{e}$. This result should be attributed to the residual nw-phase saturation of the micromodel that depends more strongly on the pore-scale topology, rather than the specific flow conditions.

The average size of the entire population, however, appears to be a monotonically decreasing function of $\text {Ca}_\textrm{e}$ in our experiments. Furthermore, the average size appears to be larger than that of the stranded ganglia at low $\text {Ca}_\textrm{e}$, while it shifts toward a much smaller size at high $\text {Ca}_\textrm{e}$ values. Given that the stranded population reaches a steady-state value in the latter limit, we conclude that it is the mobile population that decreases significantly in size, leading to the Drop Traffic Regime, as previously discussed.

The presence of a characteristic residual immobile population at high $\text {Ca}_\textrm{e}$ values is also evident in Fig. 11 (left) that shows the average steady-state total and stranded saturations as a function of $\text {Ca}_\textrm{e}$. Here, the total saturation, S, is found to be a decreasing function of Q, as expected, due to the increasing fraction of the nw-phase in the flow inlet. It is interesting, however, that the immobile saturation exhibits the opposite behavior being a slightly increasing function of Q. This should be attributed to the overall higher saturation value at low Q values that leads to the formation of larger mobile ganglia (see also Figs. 8, 10) that provide enhanced hydraulic connectivity and increased coalescence probability for the nw-phase over longer distances. Therefore, the stranded ganglia appear to remain trapped in the same locations in our micromodel (regardless of Q), but they contain more mass (i.e., being somewhat larger) on a time-averaged basis as Q increases. This eventually corresponds to a gradual transition from a ganglia-based flow toward a continuous flow regime, where there are practically no ganglia (either stranded or mobile), and the nw-phase flow takes place through a separate flow path that is percolating at all times. This can be also explained in the context of percolation theory, where the probability for the formation of disconnected clusters (i.e., ganglia) decreases drastically when a single percolating cluster is formed that spans the entire micromodel. This occurs at lower Q values in our case, when the saturation of the nw-phase is sufficiently high.

Fig. 11

(Left) Total average saturation (dashed curves) and saturation corresponding to the immobile population (dotted curves) as a function of $\text {Ca}_\textrm{e}$ for different values of the flow ratio, Q. (Right) Rescaled mean ganglion size for the total (dashed curves) and the stranded (dotted curves) populations as a function of $\text {Ca}_\textrm{e}$ for different values of the flow ratio, Q

Fig. 12

Probability density maps for ganglia elongation, defined as $\frac{L_\textrm{max}}{L_\textrm{min}}$, as a function of size (i.e., $p\left( \log (\frac{L_\textrm{max}}{L_\textrm{min}}); log(V_\textrm{g}/\overline{V}_\textrm{s}\right)$) for all experiments at a $Q=3.8$, b $Q=6.3$, c $Q=12.6$, and d $Q=25.2$

Fig. 13

Bivariate histogram of rescaled ganglia length, $L_\textrm{g}/\sqrt{\overline{V}_\textrm{s}}$, versus their corresponding rescaled size, $V_\textrm{g}/\overline{V}_\textrm{s}$, for a $Q=6.3$ and $\text {Ca}_\textrm{e}=3.8\times 10^{-4}$ (Exp16), b $Q=12.6$ and $\text {Ca}_\textrm{e}=3.2\times 10^{-4}$ (Exp9), and c $Q=25.2$ and $\text {Ca}_\textrm{e}=3.9\times 10^{-4}$ (Exp4)

3.2.3 Pore-Scale Effects on the Apparent 2-Phase Rheology

The complex pore-scale ganglia interactions observed in our study are expected to have a profound effect on the effective rheology at the field-scale (i.e., time- and spatially averaged flow). Besides the effects of Ca_e and Q on the mean ganglia sizes, discussed above, it is expected that the shape of the ganglia is affected by these parameters, and in turn affects the flow behavior. Figure 12 shows the probability density maps of the ganglia deformation, expressed as a function of their length, $L_{\max }$ (typically along the main flow direction), over their width, $L_{\min }$ (typically normal to the main flow direction). These density plot are presented as a function of the rescaled size $V_\textrm{g}/\overline{V}_\textrm{s}$ and are averaged over all experiments bearing the same value of Q. Here, we observe a correlation between the ganglia size and their aspect ratio. Smaller ganglia, in particular, exhibit aspect ratios close to unity (i.e., circular shape), while bigger ones exhibit greater elongation. A clear power-law trend is evident in the data that supports a self-affine relationship. As previously stated, when the flow rate ratio increases, the occurrence of larger ganglia decreases noticeably. This decrease has a major influence on the measurement accuracy for larger-sized blobs at greater values of Q. Despite this effect, it is noteworthy that the flow rate ratio has little effect on the probability distribution map.

Figure 13 also shows the scaling of the rescaled ganglia perimeter, $P=\frac{L_\textrm{g}}{\sqrt{\overline{V}_\textrm{g}}}$ with respect to their size for different values of Q but practically the same value of $\text {Ca}_\textrm{e}=3.5\times 10^{-4}$. Here, we observe that the perimeter scales as $P\propto V_\textrm{g}^\textrm{a}$, where $a\simeq 0.68$ regardless of Q. This result also highlights the self-affine structure of the ganglia across different scales of size.

These changes in ganglia shape and size are expected to profoundly affect the Darcian-scale behavior of the process. Figure 14 (left) shows how the $\text {Ca}_\textrm{nw}$ scales with the measured pressure drop across the micromodel at steady-state conditions for different values of Q. The respective scaling for $\text {Ca}_\textrm{w}$ is proportional to that of $\text {Ca}_\textrm{nw}$, as described above. It is interesting that for $Q=2,3.8,6.3$ the scaling exhibits two clearly identified regimes. At high $\text {Ca}_\textrm{nw}$ values, we observe a practically linear scaling, i.e., $\text {Ca}_\textrm{nw}\propto \Delta P$, that demonstrates that the effect of permeability at this limit is fixed and independent of $\text {Ca}_\textrm{nw}$. However, at lower $\text {Ca}_\textrm{nw}$ the dependence of $\text {Ca}_\textrm{nw}$ on the $\Delta P$ is more important with a scaling of $\text {Ca}_\textrm{nw}\propto \Delta P^\alpha$, where $\alpha \simeq 1.35 \pm 0.05$. This nonlinear behavior has been also observed in previous numerical and experimental studies (Tallakstad et al. 2009, 2009; Sinha and Hansen 2012; Sinha et al. 2017; Zhang et al. 2021) and has been attributed to the increase of the effective permeability of the medium at increasing flow rates. In this specific case, the presence of stranded ganglia (due to capillary forces) in the micromodel blocks some of the available flow paths for both phases resulting in reduced effective permeabilities at lower flow rates. As the flow rate increases, viscous forces become progressively dominant over capillary ones leading to both the mobilization and fragmentation of stranded ganglia.

However, in our study we observe that the scaling exponent is also a decreasing function of Q (i.e., $\alpha =\alpha (Q)$), ranging between $1\le \alpha \le 1.4$ in our experiments. Namely, as the nw-saturation decreases in favor of the wetting one, the effective permeability of the medium increases toward its high-Ca value, which corresponds to the intrinsic permeability of the medium.

It is very interesting to note that all experiments for $Q\le 6.3$ collapse into a single scaling curve when we plot $\text {Ca}_\textrm{e}$ with respect to $\Delta P$ (Fig. 14 (right)). Here, the low-$\text {Ca}_\textrm{e}$ scaling exponent by fitting over all the data is found equal to $\alpha =1.43$, while for the large $\text {Ca}_\textrm{e}$ values the exponent is exactly equal to unity. The transition between the two regimes occurs at approximately $\text {Ca}_\textrm{e} \sim 1.5\times 10^{-4}$. This value is consistent with the one of Fig. 11 (right) where the mean ganglion size of the stranded population reaches a steady-state value.

The effect of the available flow paths on the transition between the low-$\text {Ca}_\textrm{e}$ nonlinear and the high-$\text {Ca}_\textrm{e}$ linear regimes is also visualized in Fig. 15 for $Q=6.3$. In this case, we have already observed two different scaling exponents in Fig. 14 (left). Figure 15a–c corresponds to $\text {Ca}_\textrm{e}=3.1 \times 10^{-5}$ that is below the critical value of $\text {Ca}_\textrm{e} \sim 1.5\times 10^{-4}$, as shown in Fig. 14 (right). Figure 15a corresponds to the time-averaged saturation field derived as discussed in the Methodology Section. Here darker red color corresponds to locations occupied by the nw-phase over 90% of the video frames. These regions could be either completely stranded ganglia or continuous flow paths for the nw-phase (in the case of percolating ganglia) (Rücker et al. 2015). Figure 15b corresponds to the time-averaged field of interface activity, as also discussed previously. Here, greener colors highlight the actual flow paths where the 2-phase flow takes place (as evidenced by the continuous changes in the occupying phase). Dark blue regions denote the absence of interface movements, but not necessarily the absence of single phase flow, either of the nw-phase or the w-phase.

By applying a color threshold in either of these two plots, we derive a logical field of the active flow paths as shown in Fig. 15c. Here, the white color corresponds to regions that 2-phase flow takes place (regardless of the local flow rate), while the black color shows the regions where 2-phase flow is absent (i.e., blocked regions for 2-phase flow).

It is evident that the flow paths increase substantially as $\text {Ca}_\textrm{e}$ increases from $\text {Ca}_\textrm{e}=3.1\times 10^{-5}$ (Fig. 15c) to $\text {Ca}_\textrm{e}=1.3\times 10^{-4}$ (Fig. 15f). The latter value is very close to the critical one, where the transition to linear scaling takes place. Based on this comparison, it is straightforward to conclude that within the nonlinear regime, the effective 2-phase permeability of the micromodel is very sensitive to changes in the imposed $\text {Ca}_\textrm{e}$ value. As $\text {Ca}_\textrm{e}$ increases (at fixed Q), viscous forces become progressively more important over capillary ones, mobilizing and intensifying the fragmentation of previously stranded larger ganglia, that were blocking the flow paths.

By further increasing $\text {Ca}_\textrm{e}$ well above the critical value as shown in Fig. 15-(i), that corresponds to $\text {Ca}_\text {e}=3.8\times 10^{4}$, the space available for 2p flow increases slightly and reaches its maximum value. Any further increase of $\text {Ca}_\textrm{e}$ would produce a negligible effect on the available flow space and thus on the effective permeability of the micromodel. During this regime, we observe therefore a linear scaling between $\text {Ca}_\textrm{e}$ and $\Delta P$, as predicted by Darcy’s law for a fixed value of the effective permeability.

Fig. 14

Scaling of the capillary numbers as a function of the pressure difference across the micromodel for different values of Q. Also shown are the least-square power law fittings separately for the lower and higher Ca values. (Left) $\text {Ca}_\textrm{nw}$ vs $\Delta P$, and (right) $\text {Ca}_\textrm{e}$ vs $\Delta P$

Fig. 15

Ganglia flow paths obtained by temporal averaging a, d, g and by tracking interface movements b, e, h over the entire duration of the video recordings for $Q=6.3$ and $\text {Ca}_\textrm{e}=3.1\times 10^{-5}$ (a–c), $\text {Ca}_\textrm{e}=1.3\times 10^{-4}$ (d–f), and $\text {Ca}_\textrm{e}=3.8\times 10^{-4}$ (g–i). The darker red color in figures (a, d, g) side denotes stranded ganglia, while the lighter green colors in figures (b, e, h) denote flow paths with heavier ganglia traffic. Figures (c, f, i) show the corresponding active flow path in white color by applying a color threshold in figures (b, e, h)

3.2.4 Contribution of Stranded Ganglia to Effective Rheology

An interesting feature observed in our experiments is that regions containing stranded ganglia, although they block the flow of the w-phase, they may actually participate in the transfer of the nw-phase in a temporarily discontinuous manner. As discussed in the previous section, at lower $\text {Ca}_\textrm{e}$ values the stranded ganglia block more pore-throats (i.e., narrower spaces between neighboring solid obstacles). The mobile ganglia then flow through three different mechanisms;

• 1. In larger pore-throats, they deform significantly under the effect of the pressure gradient imposed by the continuous w-phase across their leading and trailing edges and manage to by-pass these geometric restrictions without fragmentation.

• 2. In smaller pore-throats, however, capillary forces are dominant leading to the stranding of ganglia. A mobile ganglion often coalesces with a stranded one in such a restriction from its upstream side. The newly formed ganglion remains only temporarily stranded, while it deforms significantly (due to its large size) as it is being now subjected to a higher pressure difference along the flow direction. This often causes a snap-off in the downstream direction (rather than a mobilization of the entire ganglion), that results in the generation of a new small mobile ganglion (on the downstream side) and a new stranded one that remains back in the restriction. In this way, stranded ganglia in very narrow pore-throats serve as flow bridges for the mobile ones.

• 3. At very high $\text {Ca}_\textrm{e}$ values (and regardless of the size of the pore-throat), the fragmentation process is very intense leading to ganglia smaller than the average pore size, that flow without being restrained by capillary forces.

The latter two ganglia flow mechanisms through restrictions are demonstrated in the sequences of snapshots of Figs. 16 and 17. In the first case, which is dominant at lower $\text {Ca}_\textrm{e}$ values, a narrow pore-throat is persistently blocked by a resident immobile ganglion (highlighted with a black rectangle in Fig. 16). This restriction would appear as a closed flow path in Fig. 15. However, this region contributes to ganglia flow due to the mechanism described above. The mobile ganglion that approaches and coalesces from the upstream direction is highlighted with a green rectangle in Fig. 16a. The snap-off of the newly formed ganglion is shown in Fig. 16d and the produced mobile ganglion in the downstream direction is highlighted with a red rectangle in Fig. 16c. At the same time, the pore-throat remains blocked by the nw-phase at all times with respect to the flow of the w-phase, thus leading to a partial reduction of the effective medium permeability.

At higher $\text {Ca}_\textrm{e}$ values, this effect becomes progressively less important, as viscous forces become dominant over capillary forces that keep the ganglia stranded in narrower pore throats, thus opening up more blocked flow paths normal to the principal flow direction. Furthermore, the intensification of the fragmentation process leads to the generation of ganglia with sizes smaller than the average pore size that can flow through pore-throats without being restricted by capillary forces. This so-called drop traffic flow regime leads to the generation of 2p flow paths, that appear with very intense colors in Fig. 15. Figure 17 shows such a scenario of very small ganglia (highlighted with black squares) that flow through a large pore-restriction during a high $\text {Ca}_\textrm{e}$ experiment. Note that the fast flowing ganglia are surrounded by a number of large stranded ones that remain in place during the entire image sequence.

Fig. 16

Snapshot sequence showing the contribution of stranded ganglia in 2-phase flow in the form of single phase flow bridges in the lower $\text {Ca}_\textrm{e}$ regime. a A ganglion stranded in a narrow pore (black rectangle) throat coalesces with a mobile one (green rectangle) from its upstream side. b, c, d The newly formed ganglion deforms through the restriction, and e a fragmentation occurs leaving behind a stranded ganglion and producing a new mobile one (red rectangle) on the downstream side (thus bypassing the narrow pore-throat). The dominant flow direction is from left to right

Fig. 17

Snapshot sequence of the drop traffic flow (DTF) regime that becomes dominant at higher $\text {Ca}_\textrm{e}$ values. A series of ganglia with sizes smaller than the typical pore throat enters the field of view from the top left side in the field of view (a–e) and flows through the pore. Note that the small mobile ganglia are highlighted using a bounding box, while at the same time, the surrounding larger ganglia remain stranded

4 Conclusions

In this contribution, we performed a series of immiscible flow experiments in a stochastically reconstructed porous micromodel in order to study the effects of the capillary number and the injection flow ratio on the dynamics of the nw-phase ganglia. Our results demonstrated that the rates of ganglia fragmentation and coalescence intensify at higher $\text {Ca}_\textrm{e}$ values, as viscous forces become dominant over capillary ones, leading to a log-normal size distribution for both the mobile and total ganglia populations. While the mean size of the total population continuously shifts toward smaller values over increasing $\text {Ca}_\textrm{e}$ values, the stranded population is found to reach asymptotically to practically the same probability distribution curve (similar mean value and variance) for all Q values. This distribution corresponds to the irreducible residual saturation of the micromodel which is determined by its pore-scale characteristics.

Our results also revealed a different Darcy-scale behavior in the lower and higher of $\text {Ca}_\textrm{e}$ regions as a function of the measured pressure difference across the domain. We observed that the total flow rate scales with $\text {Ca}_\textrm{e}$ following a power-law scaling below a critical value of $\text {Ca}_\textrm{e}$. The scaling exponent is found to be an increasing function of the nw-phase saturation (as controlled by the flow rate ration Q) and ranges between $1\le \alpha \le 1.4$. This effect is directly correlated with the emergence of new flow paths that develop progressively through narrower pores-throats, where ganglia with sizes smaller than the mean pore-throat diameter are swept by the continuous wetting phase. These flow paths further contribute to the Darcy scale velocity of the non-wetting phase, thus leading to a nonlinear Darcian regime at intermediate Ca values. Finally, we observed the emergence of flow paths, that, despite being blocked by stranded ganglia, serve a flow ’bridges’ for the nw-phase only in a temporarily discontinuous manner.

Appendix

Table 1 shows the entire set of the experiments performed in this study and the corresponding flow conditions, $q_\textrm{w}$ and $q_\textrm{nw}$. The table also shows the measured pressure gradient, $\Delta P/L$, across the micromodel and the corresponding effective viscosity, $\mu _e$, and capillary numbers, $\text {Ca}_\textrm{e}$. As also discussed in the manuscript, the table shows the exact sequence that these experiments were performed.

Table 1 List of immiscible flow experiments and the corresponding flow conditions of this study