The Impact of Wettability and Surface Roughness on Fluid Displacement and Capillary Trapping in 2‐D and 3‐D Porous Media: 2. Combined Effect of Wettability, Surface Roughness, and Pore Space Structure on Trapping Efficiency in Sand Packs and Micromodels

A comprehensive understanding of the combined effects of surface roughness and wettability on the dynamics of the trapping process is lacking. This can be primarily attributed to the contradictory experimental and numerical results regarding the impact of wettability on the capillary trapping efficiency. The discrepancy is presumably caused by the surface roughness of the inner pore‐solid interface. Herein, we present a comparative μ‐CT study of the static fluid‐fluid pattern in porous media with smooth (glass beads) and rough surfaces (natural sands). For the first time, a global optimization method was applied to map the characteristic geometrical and morphological properties of natural sands to 2‐D micromodels that exhibit different degrees of surface roughness. A realistic wetting model that describes the apparent contact angle of the rough surface as a function surface morphology and the intrinsic contact angle was also proposed. The dynamics of the trapping processes were studied via visualization micromodel experiments. Our results revealed that sand and glass beads displayed opposite trends in terms of the contact angle dependence between 5° and 115°. Sand depicted a nonmonotonous functional contact angle dependency, that is, a transition from maximal trapping to no trapping, followed by an increase to medium trapping. In contrast, glass beads showed a sharp transition from no trapping to maximal trapping. Since both porous media exhibit similar morphological properties (similar Minkowski functions: porosity, surface density, mean curvature density, Euler number density), we deduce that this difference in behavior is caused by the difference in surface roughness that allows complete wetting and hence precursor thick‐film flow for natural sands. Experimental results on micromodels verified this hypothesis.


Introduction
The morphology and connectivity of pore space and the wettability and roughness of the pore-solid interface determine (i) the fluid pattern formation during multiphase flow, (ii) the geometry of the displacement front, and (iii) the capillary trapping efficiency of the defending fluid by the invading fluid.
Recent research has focused on capillary trapping and the impact of wettability on fluid pattern formation. Wang et al. (2019) studied the combined effect of viscous and capillary forces, local disorder, and wettability using Lattice-Boltzmann simulations. The 2-D triangular stochastic lattice consisted of circular grains with a uniformly distributed grain diameter. The authors found that the trapping efficiency increased with the local disorder, whereas a reduced trapping of the defending fluid occurred when then capillary number was increased. Moreover, the authors found an interesting wetting transition from no trapping to trapping with increasing contact angle for low disorder and small capillary numbers (≈10 −5 ). This dynamical phase transition associated with changes in local growth modes at the displacement front was first discussed by Cieplak and Robbins (1988). For small contact angles (<θ c ≈ 90°, i.e., water-wet rocks) cooperative filling results in a strong smoothing effect of the displacement front. For large θ (i.e., oil-wet rocks), the displacement front becomes fractal. Intuitively, one would expect the invasion front to become increasingly compact with decreasing θ. However, depending on the morphology and connectivity of the pore space, a second wetting transition from compact duct flow to fractal corner flow is possible. Hu et al. (2017Hu et al. ( , 2018silica micromodels) and Zhao et al. (2016;polymer-based micromodels) observed this second wetting transition in 2-D micromodels at a critical θ c of 45°. Both micromodels exhibited a smooth pore-solid interface and rectangular pore walls, that is, the grains are cylinders. For such a rectangular pore wall geometry, corner flow occurs at a θ of 45°. A question remains regarding whether this transition can be observed in 3-D porous media with smooth surfaces (e.g., glass beads) (see Part 1 of the study; Geistlinger & Zulfiqar, 2020), or in 2-D micromodels with different pore space structure and connectivity. AlRatrout et al. (2018) proved that surface roughness and wettability are correlated at least at the pore scale using μ-CT. Surface roughness leads to a wide distribution of the apparent contact angle for the mixed-wet state. Phenomenologically, this can be described by Wenzel's model (also denoted as Wenzel's argument;Blunt, 2017;Wenzel, 1936), where the apparent contact angle, θ * ¼ arccos(r × cos(θ 0 )), is a function of surface roughness (r) (i.e., the ratio of the surface area to the flat surface area) and the intrinsic contact angle (θ 0 ) (i.e., the contact angle on a flat surface). A rough surface with varying degrees of r causes a distribution of apparent contact angles and magnifies the underlying wetting properties. For the mixed-wet state (θ 0 ≈ 90°), the apparent contact angle deviates in both directions and thus causes a wide θ * distribution.
This simple model was criticized by de Gennes (1985) because small values of r (≈1.1) caused complete wetting (θ * ¼ 0) on rough glass surfaces (θ 0 ≈ 20°). However, this was not observed experimentally (Bico et al., 2001;Shibuichi et al., 1996). A more realistic wetting model has to include the morphology of the rough surface, for example, that the penetrating wetting film only wets the "valleys" of the rough surface. In the present paper, a realistic wetting model is proposed that describes the apparent contact angle as a function of the dry and wet parts of the rough surface as well as the intrinsic contact angle. Even though complete wetting is supposed to occur on rough silica surfaces (AlRatrout et al., 2018;Blunt, 2017;Wang et al., 2020), experimental proof is still lacking.
It is questionable if μ-CT with a resolution of 2 μm (AlRatrout et al., 2018) can resolve the surface roughness of natural sand of a few micrometers in diameter (Kibbey, 2013). Furthermore, the μ-CT study only analyzed the static fluid-fluid distribution. Therefore, an understanding of the combined effect of surface roughness and wettability on the dynamics of the trapping process is currently lacking.
Representative 2-D micromodels with varying degrees of surface roughness could be an alternative for studying the process dynamics. These micromodels must represent the characteristic geometrical and morphological properties of the 3-D porous medium.
The only recent publication that reported on a study of the impact of surface roughness on the fluid-fluid displacement process (i.e., air displaces the wetting fluid oil; θ 0 ≅ 20°and μ oil ¼ 0.15 Pa s) within 2-D-glass micromodels is that of Mehmani et al. (2019). The pore network in this study consisted of a regular quadratic lattice of circular grains (grain diameter ¼ 270 μm, grain height ¼ pore depth ¼ 10-20 μm, and porosity ≅ 0.5). We note that a regular (not stochastic) lattice was used to represent a 3-D porous medium (sandstone). The roughness was introduced by placing a cream etchant on the glass substrate prior to thermal bonding. Roughness was measured by the averaged hillock-height-to-pore-depth ratio (Ω). The drainage experiments were conducted for different degrees of surface roughness (Ω (%) ¼ 0, 0.2, 6.0, 13.0) at unrealistically small capillary numbers (≈10 −8 ). The authors observed an increased sweep efficiency of 14% when roughness was increased from 6% to 13%. However, the authors could not quantitatively differentiate the finger morphology and topology on the basis of surface roughness. Perimeter-to-area ratio values and Euler numbers showed negligible differences among the finger paths within the different micromodels. Accordingly, we hypothesize that a higher sweep efficiency is caused by the denser pore network because the addition of hillocks between the regular grains creates a stochastic lattice of hillocks and cylindrical grains. The classical phase diagram of Lenormand et al. (1988) gives at Log (μ air /μ oil ) ≈ −4 and Log (Ca) ≈ −8 viscous fingering (see Figure 4a in Lenormand et al., 1988).
The micromodels used in previous studies were thus neither representative of the pore space morphology, grain geometry, or surface roughness of natural sands (Kibbey, 2013). In current literature, there is limited information on 2-D sand analogs (i.e., representative micromodels) with different degrees of surface roughness.
As an instructive example we discuss the dynamics of the trapping process in a water-wet system (siliceous porous media with a rough surface, such as natural sand or glass ceramic micromodels; intrinsic contact angle θ 0 ≈ 20°). The different capillary flows are governed by the order of driving capillary pressures. First, a precursor thick-film flow (TFF) will "spontaneously" invade the porous media assuming that surface roughness leads to complete wetting of the rough surface. This causes a continuous TFF throughout the entire porous media and a redistribution of the water phase. Figure 1 shows a spontaneous invasion experiment. In this range of capillary numbers, both the invading and the defending fluids can be continuous in 2-D porous media (micromodels); that is, TFF allows bicontinuity of both phases, which is only possible for duct flow (DF) in 3-D porous media (Adler & Brenner, 1988).
When the capillary number is increased up to 10 −6 , CF occurs and capillary trapping is controlled by the interplay of TFF and CF in front of the main DF-displacement front (blue clusters in Figure 1). This interplay leads to very efficient trapping mechanisms in rough, water-wet porous media. The continuous TFF (complete wetting) can reach any region of the pore space and "feed" CF. This swelling of the corner water phase can lead to unstable fluid-fluid interfaces depending on the pore-channel geometry. The unstable interface will "jump" across the whole duct cross section and form a more stable interface, that is, a full duct meniscus. This is the basic mechanism of snap-off trapping.
We emphasize that this efficient snap-off trapping mechanism needs two necessary requirements: (i) spontaneous continuous TFF (complete wetting on rough surfaces) and (ii) CF that can lead to unstable fluidfluid-interface configurations (an irregular geometrical shape of the throats or grains). In short, we denote the first requirement as the complete wetting condition, and the second requirement as the geometrical instability or snap-off condition.
In Geistlinger and Zulfiqar (2020), we studied the impact of wettability on fluid pattern formation and capillary trapping in 3-D glass bead packs during fluid invasion at a capillary number of 10 −7 using μ-CT. The invading fluid was water and the defending fluid was air. The contact angle of the glass beads was altered between 5°and 115°using Piranha cleaning and salinization. We found a sharp transition in the trapping efficiency at a contact angle of ≈90°. The key question is whether this interesting dynamical phase transition will occur in (i) sediments with rough surfaces (e.g., sand or sandstones), and (ii) sediments with more irregularly shaped grains compared to spherical glass beads. Therefore, part two of our current study focuses on natural sands and micromodels with a rough pore-solid interface. The first objective is to investigate the trapping efficiency in natural sands by changing the contact angle between 5°and 115°using μ-CT and to then analyze the static fluid-fluid pattern. The second objective is to visualize the trapping dynamics using representative 2-D micromodels (2-D sand analogs) with different pore space structures and different surface roughness values to test the complete wetting and the geometrical instability condition using high-resolution optics and fluorescence microscopy.

Sands and Fluids
Natural sand (NS, filtered and fine quartz sand, mgsdekorations, Germany) with a sieve fraction of 0.75-1.25 mm and an average diameter of 1.04 ± 0.07 mm was used for this study. Close random packing (Dullien, 1992) was achieved for all experiments with a packing density of 1,572 kg/m 3 and a porosity of 0.39. Water ( ρ w ¼ 998 kg/m 3 , μ w ¼ 1.002 × 10 −3 Pa s, σ gw ¼ 72.86 × 10 −3 N/m) and air ( ρ g ¼ 1.184 kg/m 3 , μ g ¼ 1.84 × 10 −5 Pa s) were used as the invading and defending fluids, respectively.

Experiments
The experimental procedure for wettability alteration, experimental setup, and methodology follows the same protocols described in Part 1 (Geistlinger & Zulfiqar, 2020). We studied NS with different wettability: Figure 1. Schematic: Spontaneous precursor TFF (green area) and water redistribution over the rough pore-solid interface of the entire pore network (glass ceramic micromodel) at time t > t 0 . Water was injected into the micromodel on the left side and injection was stopped at t 0 . Water is drawn from the complete saturated region (red dashed line). The interplay of precursor TFF and CF leads to snap-off trapping of the defending fluid (blue clusters) in front of the water-saturated region. NS1: strongly water wet (θ ¼ 5°), NS2: weakly water wet (θ ¼ 33°), NS3-5: intermediate wet (θ ¼ 81°, 88°, 99°), and NS6: weakly air wet (θ ¼ 115°).
For the fluid invasion experiment, a Plexiglas column of 200 mm length and 19.1 mm inner diameter was packed with the NS. For all experiments, close random packing with approximately identical pore size distribution (PSD, 0.290 ± 0.12 mm) was achieved. The PSD was derived by the first Minkowski function of the pore space (Geistlinger et al., 2015). To avoid grain rearrangement during fluid invasion, lead spheres (diameter ¼ 3.1 mm and density ¼ 11.35 × 10 3 kg/m 3 ) were placed for a height of 30 mm on top of the NS. The six different experiments are denoted according to their increasing contact angle (NS1 to NS6).
A series of 24 fluid invasion experiments were conducted (four replicates for each specific NS-type, Table 1). Air-saturated water was injected with a volumetric flow rate q w ¼ 0.82 mm 3 /s with a front velocity of 0.007 mm/s, corresponding to a capillary number of 10 −7 . The air-saturated water was injected using a high-precision bidirectional syringe pump with a step resolution of 0.046 μm (Fusion 200, Chemyx, Stanford, USA).

Computed Microtomography and Image Processing
The fluid-fluid pattern and the pore space structure were recorded after each injection experiment, and analyzed at three different sections of the column using computed X-ray microtomography. An Industrial X-ray scanner with a focal spot of 3 μm was used (X-Tek XTH 225, Nikon Metrology, Belgium). The projections were recorded on a Digital Perkin Elmer 1620 Flat Panel Detector with a resolution of 2,000 × 2,000 pixels. To maneuver full range of gray values, optimum energy settings (155 μA, 120 kV, 18.6 W) were used and beam-hardening artifacts were reduced by using a 0.1 mm copper filter.
The spatial resolution of the CT image (0.013 mm) is important, as it determines the threshold at which a fluid meniscus in a pore throat can be detected. For close random packing, the mean pore-throat diameter can be estimated by d k,min ¼ 0.31 × 0.92 mm ¼ 0.29 mm (¼minimal pore diameter, Geistlinger et al., 2006). Hence, the fluid meniscus can be reasonably resolved by 22 voxels along the diameter. Image processing was carried out according to the protocol described in Part 1 of the study (Geistlinger & Zulfiqar, 2020).

Micromodel Experiments 2.4.1. Microstructures
Experiments were conducted using two different stochastic pore space structures, which exhibit different morphology and connectivity of the pore space and different geometrically shaped grains and pore channels.  (Brücher & Bottlinger, 2006). Strong connected pore systems possess a high fraction of long chords, whereas dense pore systems possess a high fraction of short chords. A high peak in the grain-CLD shows that the particle size distribution is narrow. A second weak peak in the grain-CLD indicates contact points between neighboring grains. Nonspherical grains show an asymmetry between horizontal and vertical chords. The CLDs of a 1 mm sand (details in Geistlinger et al., 2015) were used for the 3-D-2-D mapping, so that the irregular MM4 microstructure has similar morphological, topological, and geometrical properties. The global optimization method Simulated Annealing (Schlueter et al., 2010) starts with a randomly generated pixel distribution that has the average porosity of the 3-D structure (¼0.39) and changes the pixel configuration until the 2-D structure has the specified properties of the 3-D structure, that is, the same CLDs.
Regular pore structure/micromodel MM5 ( Figure 2b): The square lattice microstructure contains 80 × 80 equidistant sites connected by lognormal distributed throats (Matlab-generator with a throat width

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d throat ¼ 400 ± 62 μm). We note that this microstructure is similar to that used by Lenormand et al. (1988) for deriving their famous phase diagram on fluid pattern formation. Their chemical etched square-lattice microstructure consists of rectangular throats (ducts), which have a log-normal distributed width and a constant etching depth of 1 mm. A discrete frequency distribution was used with seven size classes from 100-600 μm. The mean throat width of 350 μm corresponds to the throat width of 400 μm of the regular MM5 microstructure.

Production of Silicon Micromodels
A microfabrication method involving photolithography, inductively coupled plasma-deep reactive ion etching (ICP-DRIE) and anodic bonding was used to fabricate the micromodel in a silicon wafer. An interval-based ICP-DRIE technology (Kuechler et al., 2003;Zue et al., 2013) was applied for anisotropic etching. This resulted in high edge steepness and a true mapping of the lattice structure with depth ( Figure 2b), that is, only minimal underetching is produced (1.2°deviation from the vertical line, and maximal deviation from the horizontal bottom line of 10 μm). This 1.2°deviation was achieved for a ninefold deeper microstructure compared to the microstructure produced by Zue et al. (2013). The production process is described in Geistlinger et al. (2019) to which the reader is referred for additional details.

Production of Glass Ceramic Micromodels
The 2-D micromodel was constructed by using a photolithographic etching process (Invenios Europe GmbH). As solid material, a photosensitive glass ceramic FOTURAN (Schott-GmbH) was used. The chemical etched microstructure was covered with a plain glass plate by thermal-diffusion bonding (500°C), and hydraulic tube connections were installed at the inlet and outlet triangle. After complete construction, the etched structures were cleaned by isopropanol flushing, subsequent water flushing and final drying at 200°C.
The SEM images of the MM5 micromodel ( Figure 3) show the microstructure at different magnifications indicating the high accuracy of the anisotropic photolithographic etching process compared to isotropic chemical etching (underetching) of glass micromodels. Considering the sharpness/slope of the grain wall, this accuracy is in the order of 10 μm. The main feature of the microstructure is the surface roughness, which is caused by the glass ceramic. The mean roughness was determined by the grain size distribution of the sintered glass powder which is 1-3 μm as provided by the producer (Invenios Europe GmbH) (see Figure 3b). The estimated surface roughness parameters (ISO-standard, 1997; Voburger & Raja, 1990) are

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Experimental Setup and Visualization
To perform imbibition experiments at a constant flow rate (Ca ¼ 10 −7 to 5 × 10 −5 ), a high-precision bidirectional syringe pump with a step resolution of 0.046 microns (Fusion 200, Chemyx, Stanford, USA) was used.
As the wetting fluid, water was used, and air was used as the nonwetting fluid.
For visualization, we used an SLR-camera Canon EOS7D with a 100-mm Macro lens (Canon EF 100mm F2.8 USM Macro lens) both in the single-shot mode (spatial resolution: 5,184 × 3,456 pixel) and in the video mode with 24 frames/s (spatial resolution: 1,920 × 1,080 pixel). Furthermore, a fluorescence microscope in combination with the SLR-camera was used (Leica Leitz DMRB microscope; LM digital SLR wide-field adapter, micro-tech-lab Ltd.). Uranin was used as a fluorescent tracer. Figure 4 compares the contact angle dependency of the trapped gas saturation for sands and glass beads. When the contact angle is increased sand and glass beads displayed opposite trends in terms of the contact angle dependence: Sand shows a nonmonotonous functional contact angle dependency, that is, a transition from maximal trapping (residual gas saturation S g,res ≅ 13%) to no trapping and then an increase to medium trapping (S g,res ≅ 7%). Glass beads show a monotonous behavior, that is, a sharp transition from no trapping to maximal trapping (S g,res ≅ 7%).

Experimental Results: Trapped Gas Saturation and Cluster Shape
Since both porous media exhibit similar morphological properties (similar Minkowski functions: porosity, surface density, mean curvature density, Euler number density), we deduce that this difference in behavior is caused by the difference in surface roughness that allows complete wetting and hence precursor thick-film flow. This causes a highly efficient snap-off trapping mechanism, which yields a maximal residual gas saturation of approximately 13% (experiment NS1, Table 1). Increasing the contact angle up to 33°the residual gas saturation is still high and about 11% (Table 1).

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We think that the reason for this small decrease is the heterogeneous wettability of NS caused by its nonhomogeneous surface roughness, that is, some parts of the surface are smooth while other parts are rough (see Figure 1 in Kibbey, 2013). Therefore, the degree of surface roughness r is smaller compared to that of micromodels produced by glass ceramic and, consequently, Wenzel's argument will lead to a smaller range of contact angles, where complete wetting is possible.
The shape of the trapped gas clusters and their spatial distribution are shown in Figure 5x. Because the clusters are trapped within the macropores, they are extended over many pores and their shape is ganglia-like. The cluster size distribution of 1 mm sand packs (Figure 5x) shows universal scaling behavior. This supports the dynamical trapping model for low flow rates (Lenormand & Zarcone, 1984; in the following denoted as L&Z84), namely that the trapping mechanisms can be described by Ordinary Percolation with trapping (Wilkinson, 1984). The cluster shape and size analysis of the trapped gas clusters within glass ceramic micromodels and packs of natural sand will be published elsewhere.
Based on Wenzel's argument one expects a transition in trapping efficiency with an increase in contact angle to some critical value around 90°. Indeed at θ ¼ 81°(experiment NS3) no gas is trapped. This is a strong indicator that the complete wetting and hence precursor TFF is not possible. Therefore, the highly efficient snap-off trapping mechanism is suppressed for contact angles larger than 81°. In the range of neutral contact angles (≈90°; NS3 and NS4) capillary forces are nonexistent, which leads to flat menisci and to a flat front advance. Therefore, capillary bypass trapping is not possible.
Increasing the contact angle up to 99°(NS5) and 115°(NS6), the sand becomes hydrophobic and the invading fluid water is now the nonwetting fluid. The character of the displacement process changes from imbibition to drainage. Therefore, the displacement process for NS5 and NS6 can be understood as a drainage process with bypass trapping. This leads to a residual gas saturation of 6.6%, that is, the trapping efficiency for high contact angles is approximately half of the trapping efficiency for small contact angles. Inspecting the shape of the trapped gas clusters and their spatial distribution shown in Figure 5y, one recognizes that gas as the wetting fluid is now trapped within the small pores (see the occupational frequency of small pores, Figure 9 in Geistlinger & Zulfiqar, 2020).
It is instructive to compare the normalized trapped gas phase for 2-D sand analogs (micromodels MM4 glass ceramic, MM4-Si) and 3-D porous media (sand packs and GBS packs) shown in Figure 4. S MM4 − glass g; max (¼73%) is used as a normalization factor for the micromodels (red symbols) and S sand g; max ð≅13%) is used for 3-D porous media (sand: blue rhombs, GBS: gray circles).  The main conclusion from Figure 6 is that the transition from snap-off trapping to bypass trapping causes a reduction of trapping efficiency of about 50% in all cases. Interesting is that the drainage process at high contact angles (θ ¼ 115°) leads to bypass trapping with nearly the same trapping efficiency S g,res ≅ 7% (the symbols for sand and glass beads-blue rhomb and gray circlecover each other).

Discussion: Physical Mechanisms and Micromodel Experiments
In this section we will discuss physical mechanisms and constraints, which control the trapping dynamics using visualization micromodel experiments. We will test these trapping constraints (conditions) using micromodels with different pore space structure and surface roughness. We will show that the interplay of geometrical and surface properties of the pore space lead to different capillary trapping mechanisms. We emphasize that one has to distinguish CF bypass trapping from standard DF bypass trapping. The discussion in the previous section was based on the hypothesis that complete wetting on rough silica surfaces can occurs for an intrinsic contact angle of ≈20°. For the first time we will provide experimental evidence for the complete wetting hypothesis.

Physical Mechanisms 4.1.1. Complete Wetting Condition: Impact of Precursor Thick-Film Flow on Capillary Trapping
The wetting behavior of an ideal flat solid is fixed by its chemical composition and local or intrinsic contact angle θ 0 (Young's relation). But real solids are rough, which affects their wettability (Bico et al., 2001;Cazabat & Stuart, 1986;Johnson & Dettre, 1964;Netz & Andelman, 1997). Figure 7 shows a rough surface in contact with a water reservoir and a water film propagating along the rough surface. The Young relation is locally valid and for partial wetting the water flow stops and a certain section of the rough surface remains dry. At this position the Free Energy F of the three-phase system "water-air-solid" has reached its minimum (thermodynamical equilibrium: dF ¼ 0).
A spontaneous TFF occurs (blue arrow in Figure 7) if its Free Energy can be lowered by moving forward, that is, if the corresponding increment dF is negative for a given θ 0 . In this case the TFF will further advance and cover the entire rough surface, that is, complete wet the rough surface.
Let us considers a water film of length L LV (the length of the flat gas-water contact line; Figure 7) that covers a rough siliceous surface with a contact line L SL , then the change in Free Energy in the x direction is given by where γ denotes the free energy per area of the solid/vapor (S/V), solid/liquid (S/L), and liquid/vapor (L/V) interface, also known as the surface tension, ϕ S denotes the solid fraction that remains dry (i.e., Figure 7. Thermodynamical complete wetting condition for spontaneous thick-film flow (blue arrow) on a rough surface (length L SL ): A small film advance from x ¼ L LV + dx leads to a negative increment of the Free Energy (dF < 0). The thick film penetrates the rough surface and wets the "valleys" completely and leaves the "hills" dry. The dry part of the rough surface is denoted by ϕ s (Bico et al., 2001;de Gennes, 1985). In the case of oxidic solid surfaces, the Free Energy (γ SL + γ LV ) is always smaller than γ SV (de Gennes, 1985). Therefore, the system lowers its Free Energy if wetting proceeds (dx > 0; first term in Equation 1b ¼ flat surface term). The wetting behavior is enhanced on rough surfaces because the roughness term (second term in Equation 1b) leads also to a decrease in Free Energy if wetting proceeds. Hence, roughness magnifies hydrophilic wetting behavior (Wenzel's argument). Searching for the minimum of F and using Young's equation (γ LV cos(θ) ¼ γ SV − γ SL ), the critical contact angle (dF ¼ 0) follows from Equation 1a and the complete wetting condition is given by ( 2) where θ 0 is the equilibrium contact angle of the liquid on an ideal flat surface of the same chemical composition (¼ intrinsic contact angle) and 1 − e ϕ s · L LV is the surface area of the flat water film (Bico et al., 2001).
The physical meaning of Equation 2 is that for all θ 0 < θ c , the increment dF of the Free Energy is negative (shown in Figure 7). This forces the TFF to advance as long as the entire surface is covered by a water film. The range of the critical contact angle is determined by the degree of surface roughness r. For a flat surface (r ¼ 1), complete wetting and hence spontaneous TFF is only possible for θ 0 ¼ 0.
For large values of r ≫ 1 and this is the typical case, the critical contact angle is π/2. Consequently, complete wetting of rough surfaces is possible for a broad range of intrinsic contact angles: θ 0 < π/2. Equation 2 yields for the rough surface of the glass ceramic micromodels (roughness parameter: δ ¼ 1 μm, λ ¼ 2 μm) a critical contact angle of 57°, where a hemisphere geometry was assumed, that is, an array of sintered spherical powder grains. The radius of a single hemisphere was estimated by λ and the thickness of the penetrating water film by δ/2. Consequently, one expects complete wetting for the glass ceramic micromodels (θ 0 ≈ 20°). Figure 8 shows spontaneous TFF advance at three subsequent times and at very low capillary numbers (<10 −6 ) after water injection was stopped. The TFF redistributes the water of the saturated region (region left from the red dashed line shown in Figure 1) over the whole rough surface of the micromodel. This interesting wetting behavior of continuous capillary flow at no water injection, that is, complete wetting of a rough silica surface, is shown in a video provided in the supporting information (Movie S1-MM4-glass ceramic-TFF). The experimental observation of spontaneous capillary flow at no water injection verifies the complete wetting hypothesis.

Geometrical Instability Condition
Besides the wetting properties, capillary trapping is also controlled by the pore space geometry. In order to understand the geometrical constraints one has to consider the capillary pressures for certain meniscus configurations. Following L&Z84 the capillary pressure for piston-like displacement (duct flow) in a quadratic duct with width a (Figure 9a; curvature of the whole meniscus) is given by The snap-off capillary pressure is

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The corresponding water phase pressures (air gauge: p air ¼ 0) are and For the special case θ ¼ 0 the three different temporal states during the swelling process of the corner flow are shown in Figure 9a) and indicated by 1, 2, 3. State 2 shows the unstable gas-water interface (red dashed line). Its curvature is 2/a and the snap-off capillary pressure is given by Equation 4b. The temporal course of the capillary pressure (red curve) and the water phase pressure (blue curve) is shown in Figure 9b). Physically, the air-water configuration is unstable, because any small perturbation will force the thermodynamical system to jump into the next stable State 3. Hence, the water meniscus will snap over the whole cross section and divide or split the defending gas phase.
The obvious problem is that such Haines jumps are not possible in a straight quadratic capillary, because water phase pressure for snap-off is always larger than for duct flow (Equation 4aand 4b); thus, duct flow will always occur before snap-off. This statement is also true for the general case of a rectangular duct (L&Z84). Therefore, snap-off trapping is unlikely to occur in pore networks with rectangular ducts.
For concave-shaped pore channels shown in Figure 10 the corresponding water phase pressure at the meniscus location L 0 and at the bottleneck of the pore channel L 2 are given by (L&Z84) and Figure 9. (a) Gas-water interface configurations at three subsequent times during swelling corner flow. Interface 1 is a stable interface, whereas Interface 2 (red dashed line) is an unstable interface. The interface jumps across the whole cross section to a stable meniscus across the duct, that is, to its end point 3. (b) Corresponding capillary-and water phase pressures. The water phase pressure (air gauge: p air ¼ 0) is at its maximum at Interface 2 and thermodynamical system will jump into the stable State 3 with lower water phase pressure. This Haines jump or snap-off is the basic mechanism of snap-off trapping. Fluid-fluid interface configurations (red lines) across the pore channel at different locations L. The interface at L 2 is unstable, because a further CF swelling leads to a smaller curvature and higher capillary pressure (lower water phase pressure, compare with Figure 9) and the interface will jump across the pore channel (Haines jump; snap-off).

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where d is depth of the pore channel. Hence, the geometrical snap-off condition for corner flow is given by For our irregular micromodels MM4 L 0 is about 1 mm and the etching depth d ¼ 0.3 mm. Snap-off will occur at all bottlenecks of the micromodel, where L 2 is smaller than 0.2 mm assuming a contact angle zero. Figure 11 shows this complex trapping process within the 2-D sand analogs (glass ceramic micromodels) at six subsequent times at a capillary number 10 −6 . At this capillary number CF occurs and the interplay of both flow types leads to a very efficient snap-off trapping mechanism. First, TFF covers the whole surface and provides enough water for water bridges at geometrical bottlenecks indicated by white dashed rectangles.
If the geometrical snap-off condition is satisfied snap-offs (Haines jumps) occur at these bridge locations and CF develops around closely adjacent grains (black dashed lines). Note that this CF has no connected path to the water saturated region; it is fed by TFF alone. Further snap-off leads to trapped, ganglia-like gas clusters indicated (Figure 11f). The video of the snap-off trapping process is provided in the supporting information (Movie S2-MM4-Glass-TFF-CF-SO-trapping). Figure 11. Snap-off trapping in MM4 glass ceramic micromodels at six subsequent times. (a) Precursor TFF leads to unstable bottleneck and water bridges occur (white dashed rectangles) and CF arises around closely adjacent grains (black dashed lines). This precursor CF has no connected path to the water saturated region shown in Figure 1. In the following time events it can be seen that snap-off occurs at the bridge locations, CF surrounds closely adjacent grains and trapped gas clusters occur (Image f) (Uranin tracer, fluorescence microscopy).

Figure 12.
Trapped gas clusters within a representative window (34.5 mm × 23.5 mm) of the MM4 glass ceramic micromodel shown in Figure 3a). Seventy-three percent of the pore space is occupied with gas clusters. Grains and the water phase are not shown. Figure 12 shows the spatial distribution of trapped gas clusters of different sizes at breakthrough of DF at Ca ¼ 10 −6 . The mean residual gas saturation of the whole micromodel is 73%. We note that the cluster size distribution shows universal scaling.

Capillary Trapping in 2-D Micromodels With Smooth Surface
In this section we study the impact of pore space structure on the geometry of the displacement front and on trapping efficiency. We focus on the geometrical snap-off condition and suppress complete wetting. Therefore, we consider only micromodels with smooth surfaces, that is, we compare the trapping behavior of MM4-Si (irregular structure, 2-D sand analog, Figure 2a) with that of MM5-Si (stochastic lattice

10.1029/2020WR027965
Water Resources Research structure, Figure 2b). Note that the averaged contact angle of Si micromodels is 42°. Based on this comparison we will discuss some experimental results in recent literature, which seem to contradict our main conclusions about the contact angle dependency of the trapping behavior.
MM4 Si micromodels, irregular pore space structure. The mean trapped gas saturation within the MM4-Si micromodel is 32%, that is, compared to the glass ceramic micromodel it is reduced by a factor 0.44. This strong decrease in trapping efficiency indicates a trapping transition, that is, the trapping process must have changed, qualitatively. The trapping process is now bypass trapping due to CF, because complete wetting and precursor TFF is not possible in MM4-Si micromodels (see Figure 13). Now the CF path is connected to the water-saturated region of DF. If the geometric snap-off condition is fulfilled then a water bridge at a bottleneck occurs (topmost image of Figure 13) with subsequent CF around the adjacent grain. The fractal CF front leads to CF-bypass trapping with a reduced trapping efficiency of about 50% compared to the TFF-snap-off trapping in the MM4-glass ceramic micromodel. We emphasize that both micromodels exhibit the same irregular pore space structure (2-D sand analogs). The qualitative difference between the two trapping mechanisms is only determined by the surface roughness. For the MM4-glass ceramic micromodel both trapping conditions are fulfilled, whereas for the MM4-Si micromodel the complete wetting condition is violated. This leads to a transition in trapping efficiency from 73% to 32% gas trapping. The video of corner flow bypass trapping process is shown in a video provided in the supporting information (video 3-MM4-Si-CF-BP-trapping). To the best of our knowledge such an experimental study of CF-bypass trapping mechanisms is lacking in current literature. We note that this trapping mechanism has to be distinguished from the well-known DF-bypass trapping.
MM5-Si micromodels, regular pore space structure. Figure 14 compares the piston flow capillary pressure (black curve, Equation 3a) with the snap-off capillary pressure (red curve, Equation 3b) of a square capillary for the contact angle of silicon. For a straight capillary with a width of 400 μm (mean channel width of the micromodel MM4, see Figures 15b and 15d) the piston flow pressure is always larger (Factor 20). For a pore space structure (topology), which possess narrow throats with a width of about 0.02 mm the snap-off pressure equals the piston flow pressure and snap-off becomes possible. Inspecting the pore size distribution of MM5-Si (Figure 15d), such small pores are not present. Therefore, snap-off is not possible, and the geometrical trapping condition is violated. This leads to flat front advance and no trapping for both MM5-micromodels (silicon and glass ceramic; MM5-Si: Figures 15b and 15f); MM5-glass ceramic is not shown). The flat front advance of DF is caused by interface smoothing and cooperative filling and is similar to that within GBS-monolayer discussed in part one of this study. The video is provided in the supporting information (Movie S4-MM5-Si-flat-front).
In contrast the irregular pore space structure of the micromodel MM4-Si (2-D sand analog) exhibits a lot of small throats as its pore size distribution shows (Figure 15c). Therefore, the geometrical trapping condition is satisfied and precursor CF leads to a fractal displacement front ( Figure 15e) and to bypass trapping of about 30% of the pore space (Movie S3-MM4-Si-CF-BP-trapping).
Experimental and theoretical results for 2-D micromodels with smooth surface and ideal circular grains show critical contact angles θ c in the range of 60-80°for low capillary numbers Ca ≈ 10 −6 : Cieplak and Robbins (1988) The key question is whether a compact displacement front and no trapping is observed, because θ Si ¼ 42°< θ c ≈ 60°. As shown above the answer depends on the pore space structure of the porous media. The MM5-Si micromodel shows a compact displacement front with no trapping, whereas the MM4-Si micromodel shows fractal CF front with trapping.
As an instructive example we analyzed the micromodel experiments conducted by Jung et al. (2016). They studied the displacement process in high-porosity (0.85) 2-D micromodels with ideal circular grains and smooth surface for Ca between 3 × 10 −9 and 10 −4 . To cover a contact angle range from 46°to 180°they used various combinations of fluids and wall materials. They found a transition between stable compact displacements with no trapping for θ < 80°to capillary fingering with trapping for θ > 120°. Figure 1c in Jung Our counter example MM4-Si shows that depending on the pore space structure a fractal displacement front is possible. Hence, this smoothing effect that is intensively discussed and studied experimentally and numerically in literature (see references in Geistlinger & Zulfiqar, 2020) is connected to artificial porous media, which exhibit a smooth surface, a high porosity (0.6-0.85), and ideal circular grains.

Conclusions
We studied the impact of pore space structure (morphology and connectivity of the pore space) and the physicochemical surface properties (wettability and roughness of the pore-solid interface) on capillary trapping in 2-D and 3-D porous media. We used sand packs as the 3-D porous media and micromodels with different pore space structures and surface roughness as the 2-D porous media. We developed a new 3-D-2-D mapping algorithm to represent important morphological properties of the 1 mm sand pack. The Simulating-Annealing algorithm maps the porosity and the chord length distributions of the sand pack to the 2-D microstructure (MM4, 2-D sand analogs). We also used micromodels with a stochastic lattice structure to study the impact of pore space structure on displacement and trapping. To represent the rough surface of natural sands the micromodels were produced from a glass ceramic. All microstructures were produced by glass ceramic and silicon (smooth surface), in order to study the impact of surface roughness. The glass ceramic micromodels were constructed using an anisotropic photolithographic etching process based on the photosensitive glass ceramic FOTURAN. For the production of the Si-micromodels an interval-based ICP-DRIE-technology was applied for anisotropic etching.
Our main conclusion is that capillary trapping in porous media depends on two necessary conditions, the complete wetting condition and the geometrical snap-off condition: i. If both conditions are satisfied the highly efficient snap-off trapping mechanism is possible, which is caused by precursor thick-film flow (experimental proof: trapping in natural sands and micromodels MM4-glass ceramic with rough surfaces). ii. If the complete wetting conditions is not satisfied the trapping behavior changes qualitatively from snap-off trapping to bypass trapping. The trapping efficiency is reduced by approximately 50% (experimental proof: increasing the contact angle in sand packs from 5°to 115°; comparing micromodels with irregular pore space structure and different degree of surface roughness). iii. If the geometrical snap-off condition is not satisfied trapping is not possible. This shows the strong impact of pore space structure (frequency of small throats/bottlenecks) on trapping (experimental proof: comparing silicon micromodels (smooth surfaces) with different pore space structure (irregular and lattice structure, which have approximately the same porosity and the same mean pore diameter).
Sand and glass beads show a completely different and opposite functional contact angle dependency. Glass bead packs show a crossover from no-trapping to maximal trapping with an increase in the contact angle. Conversely, sand packs show inverse nonmonotonous behavior; that is, at low contact angles (5-33°) maximal trapping occurs, followed by a crossover to no trapping (81-88°) and then medium trapping (99-115°). Since both porous media exhibit similar morphological properties (similar Minkowski functions: porosity, surface density, mean curvature density, Euler number density, Geistlinger et al., 2015), we deduce that this difference in behavior is caused by the difference in surface roughness that allows complete wetting and hence precursor TFF for natural sand. Consequently, the smoothing effect that leads to this sharp transition in glass bead packs (as has been intensively studied in recent literature) is only relevant in artificial porous media (i.e., those which have a smooth surface, high-porosity, and circular grains).

Data Availability Statement
The data can be accessed at this site (https://www.ufz.de/index.php?en¼39791).