3. Principal flow regimes

We begin our discussion by exploring the five principal regimes of fluid–fluid displacement in porous media: (i) viscous fingering; (ii) stable displacement; (iii) invasion percolation; (iv) cooperative pore filling; and (v) corner flow. We anchor our discussion of principal flow regimes around a few key metrics that help to characterize and distinguish the regimes.

1. (i) Fractal dimension $D_f$ is a measure of how a pattern fills the space in which it is embedded. For a 2-D pattern, $D_f$ varies between 1 (for a line) and 2 (for a compact object). We calculate $D_f$ with the box-counting method (Kenkel & Walker Reference Kenkel and Walker1996). Following this method, we tile our flow patterns with boxes of size $\epsilon$ and count the number of boxes $N$ of that size needed to cover the pattern. We repeat this process for a sequence of $\epsilon$ and take $D_f$ to be the slope of $N$ against $\epsilon$ on a log–log plot (see Primkulov et al. (Reference Primkulov, Talman, Khaleghi, Rangriz Shokri, Chalaturnyk, Zhao, MacMinn and Juanes2018) for more details).

2. (ii) Finger width $w/a$ is the ratio of mean finger width to mean pore size. We estimate $w/a$ following a scheme detailed in Primkulov et al. (Reference Primkulov, Talman, Khaleghi, Rangriz Shokri, Chalaturnyk, Zhao, MacMinn and Juanes2018), which is an adaptation of an approach by Cieplak & Robbins (Reference Cieplak and Robbins1988, Reference Cieplak and Robbins1990). Briefly, we divide our images into slices and record the mean size $w$ of one-dimensional clusters containing the pattern. We repeat the same process for an image where we treat the entire pore space as a pattern and record the mean pore throat size as $a$.

3. (iii) Modified capillary number (${Ca}^{*}$) measures the fraction of characteristic viscous to capillary pressures in our setup. We take

   (3.1)\begin{equation} {Ca}^{*} = \frac{\Delta p_{visc}}{\Delta p_{cap}} = \frac{\max({Ca},{Ca}/M)}{|\Delta p_{crit}|}\frac{\gamma R}{a h}, \end{equation}
   after expanding the characteristic pressure drop as $\Delta p_{visc} = \max (\mu _i,\mu _d) u R /ah$, where $R$ is the radius of the Hele-Shaw cell. The term $\max ({Ca},{Ca}/M)$ ensures that the greater viscous forces are taken into account, and the magnitude of critical Laplace pressure $|\Delta p_{crit}|$ is taken directly from simulations.

All of these metrics are time dependent. We evaluate $D_f$ and $w/a$ at the moment of breakthrough, when the invading fluid first reaches the outer boundary of the flow cell. The characteristic velocity $u$ used in calculating ${Ca}$ and ${Ca}^{*}$ is taken as $Q/2{\rm \pi} r_{min}$, where $r_{min}$ is the radial distance of the post closest to centre of the Hele-Shaw cell. Additionally, we define a directional flow rate as the mean flow rate along different directions of the radial flow cell. We do so by dividing the flow cell into $10^{\circ }$ sectors and calculating the mean flow rate for each sector as time progresses.

3.1. Stable displacement ($D_f=1.93$, $w/a=37$, ${Ca}^{*}>1$)

When a more viscous fluid displaces a less viscous fluid ($M>1$), the displacement front is hydrodynamically stable (Saffman & Taylor Reference Saffman and Taylor1958) because viscous forces smooth perturbations.

Simulations at $M=10^{3}$, ${Ca}=10^{-1}$, $\theta \in [46^{\circ },180^{\circ }]$ produce nearly perfectly circular patterns (figure 4). The injection pressure increases as the displacement progresses (figure 4a), with most of the pressure drop taking place in the invading fluid (figure 4d). The flow rate is radially symmetric, decreasing with radius (figure 4b,e) and pattern symmetry is maintained throughout (figure 4c).

Figure 4.

Stable displacement in the benchmark geometry for ${Ca}=10^{-1}$, $M=10^{3}$ and $\theta =170^{\circ }$: (a) the injection pressure increases monotonically ($t_b$ is the breakthrough time); (b) flow rates within the network show radial symmetry and radially decreasing intensity ($\max (\boldsymbol {q}$) is the largest local flow rate at given $t$); (c) pore-invasion times reflect the radial symmetry in pattern growth; (d) pore-pressure distribution, where pressure gradients are significant only in the invading fluid ($\max (\boldsymbol {p}$) is the largest local pressure at given $t$); and (e) the evolution of the directional flow rate is indicative of continuous compact flow, where apparent ridges are artefacts due to discrete pore throats with high flow rates near the cell centre.

3.2. Viscous fingering ($D_f=1.63$, $w/a=2.1$, ${Ca}^{*}>1$)

Stable displacement is often desirable, but not always attainable in industrial applications such as oil recovery (Chuoke et al. Reference Chuoke, van Meurs and van der Poel1959) and sugar processing (Hill Reference Hill1952). Viscous fingers develop under potential flow when a less-viscous fluid displaces a more viscous one ($M<1$).

In figure 5, we highlight the signatures of viscous fingering for the benchmark pore geometry. The simulation in figure 5 is conducted for ${Ca}=10^{-1}$, $M=10^{-3}$, $\theta =170^{\circ }$. As the displacement advances, the injection pressure decreases (figure 5a) because the majority of the pressure drop takes place in the defending fluid (figure 5d). Although the pressure appears to decrease smoothly in time, removing the global trend from the signal would expose fluctuations due to intermittent activity at the displacement front (Primkulov et al. Reference Primkulov, Pahlavan, Fu, Zhao, MacMinn and Juanes2019). As the fingers develop and grow, they focus the flow along their main branches (figure 5b,e). The displacement pattern remains radially symmetric throughout (figure 5c). In fact, the diffusive signature of the pressure field in the defending fluid is what generates the striking similarity between viscous fingering and other patterns in nature, such as diffusion-limited aggregation (known as DLA) (Meakin et al. Reference Meakin, Tolman and Blumen1989), dielectric breakdown of materials (Niemeyer, Pietronero & Wiesmann Reference Niemeyer, Pietronero and Wiesmann1984) and spreading of fire fronts (Conti & Marconi Reference Conti and Marconi2010). The diffusive pressure field arises from Darcy flow and incompressibility, which lead to $\nabla ^{2}p=0$ in the defending fluid, which is identical to the diffusive solute concentration field in diffusion-limited aggregation (Paterson Reference Paterson1984).

Figure 5.

Viscous fingering in the benchmark geometry for ${Ca}=10^{-1}$, $M=10^{-3}$ and $\theta =170^{\circ }$: (a) the injection pressure decreases monotonically in time ($t_b$ is the breakthrough time); (b) flow rates within the network are pronounced along the main branches of the viscous fingers ($\max (\boldsymbol {q}$) is the largest local flow rate at given $t$); (c) pore-invasion times reflect the radial symmetry in pattern growth; (d) pore-pressure distribution, where most pressure changes occur within the defending fluid ($\max (\boldsymbol {p}$) is the largest local pressure at given $t$); and (e) the evolution of the directional flow rate shows persistent (rather than sporadic) growth of viscous fingers.

3.3. Invasion percolation ($D_f=1.8$, $w/a=3$, ${Ca}^{*}<1$)

When the invading fluid advances very slowly and viscous forces are negligible (${Ca}\rightarrow 0$), the flow is governed exclusively by capillary forces. In drainage ($\theta >90^{\circ }$), the invading fluid advances mainly through burst events and the flow is well captured by the invasion percolation model (Chandler et al. Reference Chandler, Koplik, Lerman and Willemsen1982; Wilkinson & Willemsen Reference Wilkinson and Willemsen1983; Lenormand & Zarcone Reference Lenormand and Zarcone1985).

We explore the characteristics of invasion percolation by simulating fluid–fluid displacement at ${Ca}=10^{-7}$, $M=1$, and $\theta =170^{\circ }$ on the benchmark pore geometry (figure 6). The pressure distribution in the invasion percolation regime is spatially uniform within each fluid (figure 6d), with the two fluid pressures differing by the Laplace pressure. As the displacement front advances, the pressure in the invading fluid is modulated by the sequence of lowest capillary entry pressures, and fluctuates sharply (figure 6a) (Måløy et al. Reference Måløy, Furuberg, Feder and Jøssang1992; Furuberg, Måløy & Feder Reference Furuberg, Måløy and Feder1996). This intermittency is also reflected in the flow field: only a small fraction of the pore space is active at any given time (figure 6b), and the flow direction changes frequently (figure 6e). As a result, the emerging flow pattern lacks radial symmetry throughout the displacement, with invasion-time patches reflecting invasion avalanches (figure 6c).

Figure 6.

Invasion percolation in the benchmark geometry for ${Ca}=10^{-7}$, $M=1$ and $\theta =170^{\circ }$: (a) the injection pressure fluctuates sharply due to pore-invasion events ($t_b$ is the breakthrough time); (b) flow rates within the network are very localized, only a small fraction of the pore space is hydrodynamically active at any given time ($\max (\boldsymbol {q}$) is the largest local flow rate at given $t$); (c) pore-invasion times show asymmetric pore invasion clusters; (d) the pore-pressure distribution is uniform within each fluid ($\max (\boldsymbol {p}$) is the largest local pressure at given $t$); and (e) the evolution of the directional flow rate shows intermittency in flow direction.

3.4. Cooperative pore filling ($D_f=1.93$, $w/a=15$, ${Ca}^{*}<1$)

Cooperative pore filling is a capillary-dominated regime that produces compact displacement patterns. Although cooperative pore filling can take place in viscous flow regimes, they are most prominent in weak imbibition and can dominate the displacement pattern when viscous forces are small. During cooperative pore filling, the displacement front advances mainly through overlap and touch events (see § 2), and the increased fraction of overlap events smoothens the displacement front (Cieplak & Robbins Reference Cieplak and Robbins1988, Reference Cieplak and Robbins1990; Holtzman & Segre Reference Holtzman and Segre2015; Primkulov et al. Reference Primkulov, Talman, Khaleghi, Rangriz Shokri, Chalaturnyk, Zhao, MacMinn and Juanes2018). As a result, the displacement front sweeps the defending fluid completely, producing compact displacement patterns (figure 7).

Figure 7.

Cooperative pore filling in the benchmark geometry for ${Ca}=10^{-7}$, $M=1$ and $\theta =46^{\circ }$: (a) the injection pressure is highly intermittent ($t_b$ is the breakthrough time); (b) flow rates within the network are localized, and only a small fraction of them have appreciable flow; (c) pore-invasion times reveal pore-invasion clusters ($\max (\boldsymbol {q}$) is the largest local flow rate at given $t$); (d) the pore-pressure distribution is uniform within each fluid phase ($\max (\boldsymbol {p}$) is the largest local pressure at given $t$); and (e) the evolution of the directional flow rate shows a high degree of intermittency in the flow direction.

Cooperative pore-filling simulations on the benchmark pore geometry at ${Ca}=10^{-7}$, $M=1$ and $\theta =46^{\circ }$ (figure 7) show many similarities to invasion percolation (§ 3.3). The pressure is uniform in each fluid phase (figure 7d), but exhibits sharp fluctuations in time (figure 7a). The flow field is highly intermittent (figure 7e), with only a small fraction of pores active at any given moment (figure 7b). This intermittency results in asymmetric and patch-like growth of the displacement pattern (figure 7c). Unlike invasion percolation, cooperative pore filling produces compact displacement patterns with no trapped patches of defending fluid. The difference stems from the nature of pore-scale invasion events: invasion percolation is dominated by burst events while cooperative pore filling is dominated by overlap and touch events (Cieplak & Robbins Reference Cieplak and Robbins1988, Reference Cieplak and Robbins1990; Holtzman & Segre Reference Holtzman and Segre2015; Primkulov et al. Reference Primkulov, Talman, Khaleghi, Rangriz Shokri, Chalaturnyk, Zhao, MacMinn and Juanes2018).

3.5. Corner flow ($D_f=1.54$, $w/a=0.8$, ${Ca}^{*}<1$)

In strong imbibition, the invading fluid no longer advances by filling the pores completely – instead, the invading fluid advances mainly through corner-flow events where it coats the corners at the intersection of posts with the top and bottom plates of the Hele-Shaw cell (figure 2c).

Figure 8 explores corner flow through simulations at ${Ca}=5 \times 10^{-7}$, $M=0.1$ and $\theta =4^{\circ }$ on the benchmark pore geometry. Corner flow shares many similarities with other capillary-dominated regimes. The spatial distribution of pressure is uniform within each fluid (figure 8d), while the injection pressure shows intermittency characteristic of capillary-dominated displacements (figure 8a). Only a small fraction of the pore space has appreciable flow (figure 8b), and flow changes direction frequently (figure 8e). The resulting pattern grows asymmetrically throughout the displacement (figure 8c).

Figure 8.

Corner flow in the benchmark geometry for ${Ca}=5\times 10^{-7}$, $M=0.1$ and $\theta =4^{\circ }$: (a) the injection pressure is highly intermittent ($t_b$ is the breakthrough time); (b) flow rates within the network are localized, and only a small fraction of them have appreciable flow ($\max (\boldsymbol {q}$) is the largest local flow rate at given $t$); (c) pore-invasion times show radial asymmetry; (d) the pore-pressure distribution is uniform within each fluid phase ($\max (\boldsymbol {p}$) is the largest local pressure at given $t$); and (e) the evolution of the directional flow rate shows a high degree of intermittency in the flow direction.

4. Crossover from viscous-dominated to capillary-dominated flow

We examine the difference in the invasion dynamics between high and low ${Ca}$ through the spatial and temporal distributions of pore-invasion events. In this section, we focus on unfavourable viscosity contrast displacement, $M=1/340$ (Zhao et al. Reference Zhao, MacMinn and Juanes2016, Reference Zhao2019). The effective ratio of viscous to capillary forces is therefore ${Ca}/M$, which we use in this section. Figure 9(a) shows histograms of the Euclidean distance $\Delta s$ between consecutive pore-invasion events. The distribution of $\Delta s$ indicates that consecutive pore-invasion events are significantly more likely to take place near each other for low ${Ca}/M$ than for high ${Ca}/M$. Furthermore, the time $\Delta t_{inv}$ between consecutive pore-invasion events at ${Ca}/M=10^{-7}$ shows that the median $\Delta t_{inv}$ increases as $\theta \rightarrow 46^{\circ }$ (figure 9b). As the wettability of the substrate changes from strong drainage to weak imbibition, the relative frequency of cooperative pore-filling events increases (Cieplak & Robbins Reference Cieplak and Robbins1990, Reference Cieplak and Robbins1988; Primkulov et al. Reference Primkulov, Talman, Khaleghi, Rangriz Shokri, Chalaturnyk, Zhao, MacMinn and Juanes2018). The increase in $\Delta t_{inv}$ is chiefly due to the increase in relative frequency of overlap events, which result in rapid invasion of several neighbouring pores. This in turn leads to significant retraction of the invading fluid from all of the throats at the displacement front. Thus, more time is needed to refill the pores at the displacement front, which results in the steady increase in $\Delta t_{inv}$ as $\theta$ decreases (figure 9b).

Figure 9.

(a) Histogram of the distance ($\Delta s$) between consecutive pore-invasion events. (b) Median time ($\Delta t_{inv}$) between consecutive pore-invasion events as a function of $\theta$. (c) Spatio-temporal autocorrelation of the normalized directional flow rate fields for $\theta =46^{\circ }$. (d–f) Temporal evolution of the normalized directional-flow-rate fields for $\theta =46^{\circ }$ and (d) ${Ca}/M = 10^{-3}$, (e) ${Ca}/M = 10^{-6}$, (f) ${Ca}/M = 10^{-7}$. The plots are complemented with the pore invasion time diagrams (insets).

The velocity distribution within the porous medium is also strikingly different at low and high ${Ca}/M$. We plot the temporal evolution of the directional flow rate for $\theta =46^{\circ }$ in figure 9(d–f). At ${Ca}/M=10^{-3}$, the invading fluid forms high velocity flow channels that persist until breakthrough (figure 9d). The pressure gradients in the defending fluid dominate the dynamics, and the invading fluid flows through growing viscous fingers. The displacement front advances with strong radial symmetry (figure 9d), as observed experimentally (Måløy et al. Reference Måløy, Feder and Jøssang1985; Løvoll et al. Reference Løvoll, Méheust, Toussaint, Schmittbuhl and Måløy2004; Holtzman, Szulczewski & Juanes Reference Holtzman, Szulczewski and Juanes2012). As ${Ca}/M$ decreases (figure 9e,f), the front velocity becomes increasingly intermittent. The pressure gradients within the fluids are negligible, and the pressure changes in the network are due almost exclusively to the Laplace pressure at the displacement front. Only portions of the displacement front are active at any given time (Ferer et al. Reference Ferer, Ji, Bromhal, Cook, Ahmadi and Smith2004; Holtzman et al. Reference Holtzman, Szulczewski and Juanes2012), and the front advances in asymmetric patches (figure 9e,f).

This transition from viscous-dominated to capillary-dominated flow can be quantified through the spatio-temporal autocorrelation of the normalized directional flow rate (figure 9c). The autocorrelation is calculated as $C(\alpha ,\tau )={\langle q(\alpha ,t)q(\alpha ,t+\tau ) \rangle }/\langle q(\alpha ,t) q(\alpha ,t) \rangle$, where $\langle \cdot \rangle$ indicates the ensemble average over time, $\alpha$ is the direction and $\tau$ is the time separation between the directional flow rate profiles. The average of $C(\alpha ,\tau )$ over all $\alpha$ is shown in figure 9(c) for $\theta =46^{\circ }$. The flow field becomes increasingly uncorrelated at low ${Ca}/M$, with a qualitative transition taking place below ${Ca}/M=10^{-5}$.

5. Extending Lenormand's phase diagram

We extend Lenormand's diagram by simulating fluid–fluid displacement over a wide range of $\theta$, ${Ca}$, $M$ on the benchmark pore geometry (7560 simulations in total). This thorough sweep of the parameter space is possible due to the relatively low computational cost of our model. For each simulation, we measure $D_f$, $w/a$ and ${Ca}^{*}$ at the moment of breakthrough. We use these variables to delineate regions corresponding to the different principal flow regimes.

First, we use the fractal dimension $D_f$ to separate compact patterns from non-compact patterns. Compact patterns include stable displacement and cooperative pore filling, both of which have $D_f>1.92$ (maroon isosurface in figure 10d). A threshold based on $w/a$ provides similar results (not shown).

Figure 10.

Evolution of $D_f$ in $M$–${Ca}$–$\theta$ space. Slices of the simulation data in (a) drainage, (b) weak imbibition and (c) strong imbibition. (d) The maroon isosurface corresponding to $D_f=1.92$ is used to draw the boundary between compact and non-compact displacement patterns. The black lines are the intersections of the isosurface with the cross-sections.

Next, we use ${Ca}^{*}$ to separate viscous-dominated flow regions (stable displacement and viscous fingering) from capillary-dominated flow regions (cooperative pore filling, invasion percolation and corner flow). The surface resulting from ${Ca}^{*} = 1$ in (3.1) is depicted in figure 11 in dark grey: the space above this surface is viscous-dominated, the space below it is capillary-dominated. The crease on the ${Ca}^{*} = 1$ surface originates from a vanishing out-of-plane contribution to Laplace pressure near $\theta =90^{\circ }$.

Figure 11.

Viscous-dominated and capillary-dominated regions of $M$–${Ca}$–$\theta$ space are separated by setting ${Ca}^{*} = 1$ in (3.1). This is depicted with a dark grey surface in this figure.

The combination of the maroon and grey isosurfaces from figures 10 and 11 is sufficient for delineating the principal flow regimes as follows.

1. (i) Invasion percolation is capillary dominated (${Ca}^{*}<1$) and non-compact ($D_f<1.92$).

2. (ii) Cooperative pore filling is capillary dominated (${Ca}^{*}<1$) and compact ($D_f>1.92$).

3. (iii) Corner flow is capillary dominated (${Ca}^{*}<1$) and non-compact ($D_f<1.92$).

4. (iv) Viscous fingering is viscous dominated (${Ca}^{*}>1$) and non-compact ($D_f<1.92$).

5. (v) Stable displacement is viscous dominated (${Ca}^{*}>1$) and compact ($D_f>1.92$).

Although we use sharp boundaries to outline regions that belong to different flow regimes, the transitions from one regime to another are smooth, as is evident from the cross-section images in figures 10 and 11.

Our extension of Lenormand's diagram with added wettability axis is presented in figure 12. Our model faithfully reproduces the original diagram in drainage (cross-section $\theta =180^{\circ }$ in figure 12), but reveals a more complete picture of the fluid–fluid displacement in porous media by augmenting the phase diagram with a wettability ($\theta$) axis.

Figure 12.

Extended Lenormand diagram constructed using ${Ca}^{*}$ and $D_f$ phase boundaries from figures 10 and 11 to separate the five principal flow regimes within the $M$–${Ca}$–$\theta$ parameter space: viscous fingering, stable displacement, invasion percolation and cooperative pore filling. Results from the ‘moving capacitor’ model are complemented with results from the quasi-static model that allows inferring the extent of cooperative pore filling in the limit ${Ca}\rightarrow 0$.

To assess the influence of pore-scale disorder on the displacement pattern, we run simulations on a pore geometry in which we can precisely define, and tune, the degree of geometric variability among realizations. To do so, we generate a regular triangular lattice with 2.8 mm spacing between vertices and place posts on its vertices. The radii of the posts are drawn from a uniform distribution $(r_0-\xi r_v,r_0+\xi r_v)$, where $r_0=1100\ \mathrm {\mu }\textrm {m}$ and $r_v=300\ \mathrm {\mu }\textrm {m}$ are selected to match the mean post size of the benchmark geometry and $\xi \in [0,1]$ is the index of disorder. When $\xi =0$, the medium is ordered and anisotropic; when $\xi =1$, the medium is disordered and isotropic. As demonstrated in Appendix B, the values of $D_f$ and $w/a$ do not change significantly with the degree of disorder $\xi$. Therefore, although the data in figure 12 were collected from simulations on a single benchmark pore geometry, the results apply generally to porous media with varying degree of disorder. The capillary-dominated region of the phase diagram (${Ca}^{*}<1$) is divided into invasion percolation, cooperative pore filling and corner flow. The boundary between compact and non-compact flow in the capillary-dominated region of figure 12 changes significantly with ${Ca}$: the upper and lower bounds (in $\theta$) of the cooperative pore filling region move apart as ${Ca}$ approaches the grey surface. When $M>1$, viscous forces stabilize the displacement front and aid cooperative pore filling events in making the patterns more compact (Hu et al. Reference Hu, Wan, Yang, Chen and Tokunaga2018).

The shape of the extended Lenormand diagram can be inferred outside the $M$–${Ca}$–$\theta$ parameter space probed with the ‘moving capacitor’ model in figure 12. In particular, the cooperative pore filling region extends farther into the $M<1$ region as ${Ca}$ decreases. This is evident from the quasi-static limit of the model, where cooperative pore filling boundaries are independent of $M$.

The extended Lenormand diagram in figure 12 is generated for a single pore geometry. While the overall shape of the diagram is expected to hold across different micromodels with a wide range of pore-scale disorder, spacing between the posts and gap thickness $h$, the boundaries between the principal flow regimes are likely to shift depending on the pore structure. For example, increasing the spacing between the post centres would bring the onset of cooperative pore filling to higher $\theta$ (Primkulov et al. Reference Primkulov, Talman, Khaleghi, Rangriz Shokri, Chalaturnyk, Zhao, MacMinn and Juanes2018). Larger spacing between the posts would also make corner flow less dominant in strong imbibition, as higher critical pressures would be needed to coat post corners. Therefore, compact displacement would occupy a greater proportion of the overall space in figure 12. The degree of disorder is also known to roughen the displacement front and shift the boundary between invasion percolation and viscous fingering (Holtzman & Juanes Reference Holtzman and Juanes2010; Holtzman Reference Holtzman2016; Hu et al. Reference Hu, Lan, Wei and Chen2019). Given that the pore geometry used in figure 12 is similar to one with $\xi =0.99$ in Appendix B, a pore space with smaller degree of disorder would make compact displacement more favourable, which in turn would enlarge the compact displacement region in figure 12 (stable displacement and cooperative pore filling).

One should not think of the boundaries between the principal flow regimes in figure 12 as sharp, because transitions from one regime to another are gradual. Regions of the $M$–$\theta$–${Ca}$ space near the maroon and grey boundaries correspond to crossover zones between principal flow regimes.

We summarize the findings from our comprehensive study with a schematic (figure 13). The classic phase diagram of Lenormand was developed for strong drainage (figure 13a), where displacement patterns advance through either viscous fingering, stable displacement or invasion percolation. This diagram undergoes a qualitative change when the system moves to weak imbibition (figure 13b), in which viscous fingers become significantly wider and invasion percolation is replaced by cooperative pore filling. Therefore, the majority of the $M$–${Ca}$ space leads to compact displacement patterns. Strong imbibition has only been sparsely studied (Zhao et al. Reference Zhao, MacMinn and Juanes2016; Odier et al. Reference Odier, Levaché, Santanach-Carreras and Bartolo2017; Primkulov et al. Reference Primkulov, Talman, Khaleghi, Rangriz Shokri, Chalaturnyk, Zhao, MacMinn and Juanes2018), but enough is known to outline the main modes of displacement (figure 13c). The displacement patterns advance through corner flow at low ${Ca}$, where the injected fluid occupies only a fraction of the pore space (denoted by darker shades in figure 13c). This mode of displacement has been explored experimentally by Zhao et al. (Reference Zhao, MacMinn and Juanes2016) and Odier et al. (Reference Odier, Levaché, Santanach-Carreras and Bartolo2017), and numerically in the quasi-static limit (Primkulov et al. Reference Primkulov, Talman, Khaleghi, Rangriz Shokri, Chalaturnyk, Zhao, MacMinn and Juanes2018). The invasion pattern advances through thin films on the solid surface for high ${Ca}$ and $M<1$ (Levaché & Bartolo Reference Levaché and Bartolo2014), while maintaining the viscous fingering morphology (Zhao et al. Reference Zhao, MacMinn and Juanes2016).

Figure 13.

A sketch of Lenormand's phase diagram in (a) strong drainage, (b) weak imbibition and (c) strong imbibition. The darker shades in strong imbibition represent partial pore-scale displacement. Art credit: Kamilla Omarova.

The simulations in figure 12 reproduce many experimental observations. First, as $\theta$ changes from $180^{\circ }$ to $46^{\circ }$, displacement patterns change from invasion percolation to cooperative pore filling (Trojer et al. Reference Trojer, Szulczewski and Juanes2015; Zhao et al. Reference Zhao, MacMinn and Juanes2016), and finger width increases in the viscous-fingering region of the diagram (Stokes et al. Reference Stokes, Weitz, Gollub, Dougherty, Robbins, Chaikin and Lindsay1986; Trojer et al. Reference Trojer, Szulczewski and Juanes2015; Zhao et al. Reference Zhao, MacMinn and Juanes2016). Second, the injection pressure fluctuates sharply in capillary-dominated regimes (Måløy et al. Reference Måløy, Furuberg, Feder and Jøssang1992; Furuberg et al. Reference Furuberg, Måløy and Feder1996), but instead varies monotonically with time in viscous-dominated regimes. Third, the model naturally reproduces the intermittent flow that is modulated by pore disorder in capillary-dominated flow. Finally, the model reproduces the interplay between imposed ordered post lattice and the flow morphology: snow flake patterns in viscous fingering (Chen & Wilkinson Reference Chen and Wilkinson1985; Chen Reference Chen1987), regular crystal-growth morphology in cooperative pore filling regime (Lenormand Reference Lenormand1990), perfect circles in stable displacement and disordered morphology in invasion percolation (Wilkinson & Willemsen Reference Wilkinson and Willemsen1983; Lenormand & Zarcone Reference Lenormand and Zarcone1985; Måløy et al. Reference Måløy, Furuberg, Feder and Jøssang1992).

While our ‘moving capacitor’ model is successful in reproducing the dynamics of the principal flow regimes (figure 13), it assumes piston-like displacement for burst, touch and overlap events. As a result, the model overestimates the invading fluid saturation at high ${Ca}$, as pointed out by Zhao et al. (Reference Zhao2019). Strong drainage and high ${Ca}$ features residual films of defending fluid (Bretherton Reference Bretherton1961; Zhao et al. Reference Zhao, MacMinn and Juanes2016). In strong imbibition, invading fluid films dominate the displacement patterns in viscous fingering and corner flow regimes (Levaché & Bartolo Reference Levaché and Bartolo2014; Zhao et al. Reference Zhao, MacMinn and Juanes2016; Odier et al. Reference Odier, Levaché, Santanach-Carreras and Bartolo2017). These regimes are captured more naturally through pore-scale 3-D continuum models (Zhao et al. Reference Zhao2019), which are unfortunately still prohibitively expensive for populating significant portions of the $M$–${Ca}$–$\theta$ parameter space in Lenormand's diagram (figure 12).