An Abstract of the Thesis Of

AN ABSTRACT OF THE THESIS OF

Diba Behnoudfar for the degree of Master of Science in Chemical Engineering presented on April 27, 2020.

Title: Exploring the Wettability States and Contact Angle Variation in Porous Media Multi-Phase Flow Systems.

Abstract approved: ______Dorthe Wildenschild

The wetting of a surface by a liquid is a crucial part of many natural and industrial processes. Despite numerous existing studies, some elements of wetting-dewetting such as contact angle variation are still poorly understood. Knowledge of contact angle behavior during the flow is necessary for modeling fluid displacements in capillary-dominated flows. In the context of multi-phase flow in porous medium, the lack of direct contact angle measurements inside pores (in-situ), adds to the ambiguity. This work consists of performing in-situ contact angle measurements on X-ray micro-computed tomography images of multi-phase fluid systems during quasi-static flow and investigating the effects of pore geometry and interfacial forces on contact angle. An algorithm enabled automated analysis of contact angle throughout the three-dimensional images. Observations revealed two unique contact angle variation patterns for oil (dodecane)-water and air-water systems with larger hysteresis for the oil-water. Introduction of a third phase (air) and altering surface chemistry (wettability) of a portion of solid phase, influenced oil-water contact angle near unaltered surfaces, but the overall trend remained the same. Results of this investigation show that contact angle correlates with the variations of fluid-fluid interfacial curvature and the pore radius at the wetting front.

Exploring the Wettability States and Contact Angle Variation in Porous Media Multi- Phase Flow Systems

by Diba Behnoudfar

A THESIS

submitted to

Oregon State University

in partial fulfillment of the requirements for the degree of

Master of Science

Presented April 27, 2020 Commencement June 2021

Master of Science thesis of Diba Behnoudfar presented on April 27, 2020

APPROVED:

Major Professor, representing Chemical Engineering

Head of the School of Chemical, Biological, and Environmental Engineering

Dean of the Graduate School

I understand that my thesis will become part of the permanent collection of Oregon State University libraries. My signature below authorizes release of my thesis to any reader upon request.

Diba Behnoudfar, Author

ACKNOWLEDGEMENTS

The author expresses sincere appreciation to Dorthe Wildenschild for her expert advice and insightful comments, Douglas Meisenheimer for his technical assistance and my family for their support.

TABLE OF CONTENTS Page

1. Introduction ...... 1

2. Background ...... 3

2.1 Flow processes in porous media: drainage and imbibition ...... 3

2.1.1 Capillary pressure ...... 5

2.2 Governing equations ...... 6

2.2.1 Small capillary numbers and quasi-static regime ...... 8

2.3 Contact angle ...... 9

2.3.1 Contact angle hysteresis ...... 10

2.3.2 Contact line pinning ...... 13

2.3.3 Origins of contact angle hysteresis ...... 15

2.3.4 Contact angle measurement ...... 17

2.4 X-ray micro-computed tomography ...... 19

3. Methods ...... 21

3.1 Fluid flow experiments ...... 21

3.2 Algorithm for contact angle measurement ...... 23

3.2.1 Identifying the contact points ...... 23

3.2.2 Creating contact line ...... 23

3.2.3 Generating normal slice ...... 25

3.2.4 Calculating contact angles ...... 25

3.2.5 Testing of the approach ...... 27

3.2.6 Limitations of the method ...... 30

3.3 Image analysis ...... 30

3.3.1 Object separation ...... 30

TABLE OF CONTENTS (Continued)

Page

3.3.2 Measuring curvature ...... 33 3.3.3 Estimating wetted pore radius ...... 34 4. Results ...... 36 4.1. Contact angle-saturation and distribution of contact angles ...... 36 4.1.1. Air-water fluid pair ...... 36 4.1.2 Oil-water fluid pair in water-wet system ...... 36 4.1.3 Oil-water fluid pair in mixed-wet system ...... 40 4.1.4 Overview of contact angle variation near water-wet beads ...... 42 4.2 Contact angle spatial distribution ...... 43 5. Discussion ...... 45 5.1 Conical pore model ...... 45 5.1.1 Wetted pore radius variation ...... 46 5.1.2 Curvature and capillary pressure variation ...... 49 5.1.3 Predicting the contact angle variation ...... 51 5.2 Effect of wettability ...... 53 6. Conclusion ...... 56 References ...... 58

LIST OF FIGURES

Figure Page

1.1 Contact angle hysteresis ...... 2

2.1 2D cross-sections of a column of glass beads ...... 4

2.2 The stages of coalescence of pendular rings associated with a pore throat ...... 4

2.3 Capillary action and water rise in a glass capillary tube ...... 5

2.4 Capillary pressure versus saturation relationship for water-oil in a pack of glass beads ...... 6

2.5 Interface between two immiscible fluids ...... 7

2.6 Meniscus in a cylindrical capillary ...... 8

2.7 Effect of surface roughness on apparent contact angle ...... 10

2.8 Schematic of static and dynamic contact angle hysteresis ...... 11

2.9 Schematic of (a) liquid column trapped in a capillary tube and (b) a raindrop ...... 11

2.10 Critical capillary number vs viscosity ratio (M) and static contact angle 12

2.11 Contact line pinning and deformation of interface at a fixed contact point as time develops ...... 13

2.12 Photographs of a liquid bridge formed from glycerol during (a) separation and (b) approach ...... 14

2.13 (a) pendular liquid bridge (b) a schematic of the relationship between the wetting and the force hysteresis ...... 14

2.14 Equilibrium states in a biconical pore segment ...... 16

2.15 (a) Three-phase contact line highlighted on the smoothed mesh (b) Normal vectors defined on both the oil/brine and brine/rock interfaces 18 for the contact line set ......

2.16 Illustration of X-ray micro-CT technique with a cone-beam configuration ...... 20

LIST OF FIGURES (Continued)

Figure Page

3.1 (a) Schematic of sample holder for 3-phase flow system; in 2-phase experiments, the air ports on the sides of the column are absent (b) Schematic of entire experimental set-up ...... 22

3.2 Depiction of contact lines in the region of interest ...... 24

3.3 Visualization of a 2D image obtained as a slice of a segmented image taken along the plane locally normal to the three-phase contact line 25

Representation of the 2D slice of 3D segmented images where fitted 3.4 circles and contact angle is computed. Oil is green, water blue and rock grey ...... 26

3.5 Spatial distribution of contact points in the imaged volume ...... 27

3.6 Illustration of manual contact angle measurement using angle tool in ImageJ ...... 28

3.7 Histogram of measured contact angles using automated (blue bars) and manual (brown bars) methods ...... 29

3.8 Trapped oil ganglia SSb isolated from the Ketton dataset ...... 29

3.9 The Watershed principle ...... 31

3.10 Depiction of (a) overlapping objects in a binary image (b) distance map (c) and separated objects ...... 32

3.11 Snapshots of separated beads before (a) using BorderKill module and (b) after using the module ...... 32

3.12 Size distribution of the beads in mixed-wet system ...... 33

3.13 Depiction of (a) wetted pore radius and (b) perimeter of fluid interface 34

3.14 Example of fluid-fluid interfaces in an air-water system ...... 35

4.1 The (a) average contact angle for air-water system and the distributions for: (b) drainage, (c) imbibition ...... 37

4.2 The (a) average contact angle for two-phase oil-water system and the distributions for oil-dodecane system: (b) drainage, (c) imbibition 38

LIST OF FIGURES (Continued)

Figure Page

4.3 The (a) average OW contact angle for oil-water-air system and its distributions: (b) drainage, (c) imbibition ...... 39

The (a) average OW contact angle for the mixed-wet air-oil-water 4.4 system and its distributions: (b) drainage for water-wet beads, (c) imbibition for water-wet beads ...... 41

4.5 The distributions of OW contact angle for oil-wet beads in the mixed- wet air-oil-water system: (a) drainage, (b) imbibition ...... 42

4.6 Summary of contact angles near water-wet beads ...... 43

4.7 Spatial distribution of contact angle during imbibition for the oil-water system: (a) low saturation (0.13), (b) medium saturation (0.45) ...... 44

5.1 Menisci in a conical capillary ...... 46

(a)-(b) Transition of wetting phase from a web of pendular rings to the 5.2 funicular stage, (c) wetted pore radius variation for air-water (AW) and 2-phase oil-water (OW) systems ...... 47

5.3 Mean curvature variation in air-water (AW) and 2-phase oil-water (OW) systems ...... 48

5.4 Capillary pressure variation in air-water (AW) and 2-phase oil-water (OW) systems ...... 48

5.5 Capillary pressure – saturation curve for an oil-water system ...... 50

5.6 Pinning of fluid-fluid interface (red surface) as the flow develops during imbibition (a-c) and their respective cross-sections (d-f) ...... 51

5.7 Predicted contact angles (offset by φ) for air-water system ...... 52

5.8 Predicted contact angles (offset by φ) for oil-water system ...... 52

5.9 Mixed-wet capillary tube ...... 53

5.10 Pore types in the mixed-wet system formed by water-wet beads (orange) and oil-wet beads (green) and configuration of interfaces ...... 55

LIST OF FIGURES (Continued)

Figure Page

6.1 Demonstration of effect of curvature (a) and pore radius (b) variations on contact angle ...... 56

LIST OF TABLES

Table Page

3.1 Dataset characteristics ...... 21

3.2 Comparison of manually and automatically measured contact angles ... 28

3.3 Comparison of contact angle measurement results for dataset SSb using the two methods ...... 30

3.4 Mixed wet system ...... 31

1 1. Introduction

Flow of immiscible fluids through a porous matrix is of great importance in many natural and technological processes, such as secondary oil recovery from reservoir rocks, the storage of CO2 in deep saline aquifers or depleted oil and gas fields, water retention in soils, water purification, metal infiltration, subsurface fluid contaminant transport, ink spreading on paper, fluid transport in membrane fuel cells, and drying processes in the paper and textile industry to name a few [1].

When two or more immiscible fluids flow through a porous medium, they readily form separate phases. The result is a physically discernible interface between the two fluids, and a contact line which is the meeting place of the fluid-fluid interface and the solid surface. One of the conditions that distinguishes this flow, is the significance of a pore-scale effect called capillarity, active on the contact line of fluids and porous solid. In slow flow regime, capillary forces dominate other forces such as gravity, as the pore system becomes progressively smaller, at a scale less than roughly two millimeters [2]. This effect can be the main driving force in a variety of small-scale processes arising in biology and environmental science such as water uptake in plant roots.

Capillary forces are strongly affected by wettability of the pore walls, one of the most important properties of a porous medium that controls the dynamics of slow fluid displacement and determines the microscopic distribution of fluid phases in the pore space. The wettability expresses the relative tendency of fluids to adhere to a solid and is conventionally quantified by the angle the fluid interface forms with the solid surface in mechanical equilibrium. Fluids with a contact angle smaller than 90◦ are called wetting. If the contact angle of the interface exceeds 90◦, the fluid is called nonwetting [1].

The contact angle for the situation of stable fluid configurations is termed static. For a smooth and homogeneous surface, static contact angle is traditionally assumed to be an invariant property of a system (that is if the flow direction remains unchanged). This means that it is not influenced by the geometry of the system or non-capillary forces (such as fluid pressure induced by external forces) active on the interfaces. This assumption partly stems from the work of Thomas Young [3] in which he asserted that “the contact angle depends only on the materials, and in no other way on the conditions of the problem”. However, advances in pore-scale imaging technologies have allowed the determination of contact angle in situ from imaged porous

2 medium, and recent studies recognize that contact angle value depends on the kinetics of flow and thus there is a distribution of contact angles at static condition within the single-wet porous material [4-7]. For example, there is still no fully agreed explanation for the phenomena such as contact line pinning and contact angle hysteresis which is defined as the difference between the advancing contact angle and the receding contact angle for a contact line moving in opposite direction at the same velocity [8]; this difference usually persists after the cessation of the flow (Figure 1.1).

Figure 1.1 Contact angle hysteresis

It is natural to take contact angle as the boundary condition for determining the free fluid interfaces in modelling and numerical simulation of multiphase flow, on the basis of Young’s assertion [9]. Although the treatment of interfaces in such works is nowadays well established, the description of contact lines and corresponding contact angles is still under development [10]. Most previous models suffer uncertainty originating from the lack of a direct contact angle measurement inside pores. The contact angle is generally treated as an input parameter to the models and to capture the effect of contact angle hysteresis, different constant values are applied for each flow process. A robust pore-scale numerical model requires knowledge of in-situ contact angle values for different modes during the flow. To obtain a better understanding of how the contact angle varies during multi-phase flow, this work presents experimentally measured distributions of contact angles inside a porous matrix at different flow stages. This is done using an algorithm capable of extracting a large number of measurements from X-ray micro computed tomography three-dimensional images of two and three-phase fluid systems inside a glass bead packs. In addition, we investigate the dependence of contact angle on pore geometry and pressure at the interface.

2. Background

2.1 Flow processes in porous media: drainage and imbibition

If a wetting fluid is displaced by an immiscible and nonwetting fluid from a porous medium (e.g., air invading an initially water-filled glass matrix), the situation is called drainage. The nonwetting fluid (air) avoids regions with high specific surface area to minimize contact with the pore walls and selectively advances through large pore bodies and pore throats [1]. Pore bodies are the wide openings of the void space of the medium, while pore throats are the narrow regions connecting two neighboring pores [1]. Imbibition, is the reverse situation to drainage when a nonwetting fluid is displaced by a wetting fluid. Due to tendency for the solid matrix, the wetting fluid that displaces a nonwetting fluid is preferentially in contact with the pore walls, causing different fluid patterns [1]. Regions of the pore space with a high specific surface area, such as narrow pore bodies and throats, are filled first by the wetting fluid before it enters the wide regions of the void space [1]. These flow processes are depicted in Figure 2.1. The fluid interfaces during imbibition are typically less curved than those during drainage (Figure 2.1). These processes lead to the trapping of the invading (wetting) fluid by the non-wetting fluid, typically in small pores and throats as well as in pore corners and crevices during drainage and of the non-wetting fluid during imbibition.

If the pores of a column of beads that have been initially filled with water, are allowed to drain and then imbibe (secondary imbibition), the water is initially found in discrete fluid bodies and each mass is a ring of liquid wrapped around the contact point of a pair of neighboring beads, as shown in Figure 2.2. Liquid must accumulate in these positions rather than a uniform layer over the bead surfaces, otherwise the Gibbs free energy of the retained liquid system would not be a minimum [11]. A very thin wetting layer, however, exists over the bead surfaces that connects the discrete masses and forms a web of rings. The single liquid rings are called “pendular” rings [11]. As the imbibition progresses in the column, the rings increase in size until they begin to coalesce into more complicated masses. This stage is called “funicular” or “transition”. Finally, a stage is reached where the more complicated masses of the funicular stage merge and form a continuum extending completely across the pore space excluding those portions where there is trapped residual nonwetting phase present.

a) b) c) 0 0 0

Sw = 0.7 Sw = 0.3 Sw = 0.1

d) e) f) 0 0 0

Sw = 0.1 Sw = 0.3 Sw = 0.7

Figure 2.1 2D cross-sections of a column of glass beads (dark blue) with water as wetting fluid (light blue) and oil as non-wetting fluid (red) at different water saturations (Sw), (a)-(c) drainage, (d)-(f) imbibition [31]

Figure 2.2 The stages of coalescence of pendular rings associated with a pore throat (transition from pendular to funicular stages)

2.1.1 Capillary pressure

Capillary effect is reflected in a parameter called “capillary pressure” which is commonly used to quantify the flow behavior of porous media containing two or more immiscible fluid phases. Capillary pressure is defined as the pressure difference across the interface or between the wetting and non-wetting fluid phases. To illustrate this, rise of water in an air-filled glass capillary tube can be used as a simple example (see Figure 2.3). The only driving force in this process is the adhesion between the glass surface and water. At equilibrium, the capillary force acting on the base of the meniscus equals the weight of the water column; therefore, capillary pressure is equal to height of the water column. This relationship permits the expression of capillary pressure in terms of head units, where the head value is equivalent to the height of the water in the capillary tube.

Figure 2.3 Capillary action and water rise in a glass capillary tube; Pc is the capillary pressure.

For a porous medium, it is customary to describe the multiphase flow by the capillary pressure- saturation relationship where the saturation is defined as:

푉표푙푢푚푒 표푓 푓푙푢𝑖푑 푝ℎ푎푠푒 𝑖 푆 = 푖 푇표푡푎푙 푎푐푐푒푠𝑖푏푙푒 푝표푟푒 푣표푙푢푚푒

An example of the capillary pressure versus saturation relationship for water-oil in a pack of glass beads is shown in Figure 2.4. Evidently the relationship between the capillary pressure and

800

700 Drainage

Imbibition 600

500

400

300

200 Capillary Capillary pressure [Pa]

100

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -100 Water saturation

Figure 2.4 Capillary pressure versus saturation relationship for water-oil in a pack of glass beads [31]

saturation is not unique but depends on the flow direction and saturation history. The separation between the capillary pressure-saturation curves is known as capillary pressure hysteresis. It is generally considered to have two causes: hysteresis of contact angle and pore structure effects [11]. Also, as mentioned in section 2.1, wetting and non-wetting fluids, each, selectively flow through a group of pores which leads to different displacement patterns for drainage and imbibition.

2.2 Governing equations

There are two basic models of multiphase flow in pore networks: dynamic and quasi-static. In dynamic models, capillary, gravity, and viscous forces in the fluids are accounted for simultaneously [12]. The laminar motion of an incompressible Newtonian fluid of uniform density ρ and dynamic viscosity μ is traditionally described by the Navier-Stokes equations, which represent a continuum statement of Newton’s laws [2]. This mathematical description can be applied to the fluids inside a pore:

Figure 2.5 Interface between two immiscible fluids; r1 and r2 are the principle radii of curvature of the fluid interface; n is the unit normal to the surface

휕풖 휌 ( + 풖 ∙ 훁풖) = 훁 ∙ 푷 + 푭 (2.1) 휕푡

푷 = −푝푰 + 휇(훁풖 + (훁풖)푇) (2.2) where u, P and p are the velocity, the stress tensor and the pressure in the bulk; F is the external body force density; I is the unit tensor. Surface tension acts only at the interface; consequently, it does not appear in the Navier-Stokes equations, but rather enters through the boundary conditions. The normal stress at an interface must be balanced by the curvature pressure associated with the surface tension [2]:

풏 . 푷ퟏ . 풏 − 풏 . 푷ퟐ . 풏 = 훾(훁 ∙ 풏) (2.3)

P1 and P2 are stress tensors of fluid 1 and 2 at the interface; n is the unit normal to the surface; 훾 is the uniform surface tension at the interface. The surface tension is the stress that causes the stretching or tension of the fluid interface and has the dimension of force per unit length. As shown in Figure 2.5, the interface is subjected to a surface tension around the base of the cap and two stresses exerted by the fluids at each point on the surface.

In a non-dimensional form, Navier-Stokes equation may be rewritten as follows [13]:

휕풖 푊푒 ( + 풖 ∙ 훁풖) = −훁푝 + 퐶푎 ∆풖 + 퐵표 푭 (2.4) 휕푡 where 퐶푎 = 휇푈/훾 (the capillary number), 푊푒(= 푅푒퐶푎) = 휌퐿푈2/훾 (the Weber number) and 2 퐵표 = 퐹0퐿 /훾 (the Bond number). L, U and F0 are characteristic variables for scaling length, velocity and external force density, respectively. The capillary number indicates the relative magnitudes of viscous and curvature forces within a fluid, and the Weber number those of inertial and capillary forces.

Figure 2.6. Meniscus in a cylindrical capillary

2.2.1 Small capillary numbers and quasi-static regime

As the flow rate of the system becomes smaller, capillary forces dominate and the relative importance of surface tension over viscous forces increases. At the limit of zero capillary number, the flow reaches “quasi-static” regime. In quasi-static models, the microscopic fluid distributions are frozen at each level of the capillary pressure and interface shape can be described using the equations for the static situation [12]. Invasion by either of the fluids does not occur until this capillary pressure exceeds a specific threshold capillary pressure that is different for the different displacement mechanisms and pore geometries [12]. In this study we adopt the quasi-static approach and ignore the effects of viscous forces.

At mechanical equilibrium, Equation 2.3 reduces to:

1 1 푝1 − 푝2 = 훾(훁 ∙ 풏) = 훾 ( + ) (2.5) 푟1 푟2

which is generally known as the Young-Laplace equation and gives the interfacial pressure difference or “capillary pressure”. Principle radii of curvature of the fluid interface are denoted by r1 and r2 (Figure 2.5). It is customary to introduce the mean radius of curvature defined by:

1 1 1 1 = ( + ) (2.6) 푟푚 2 푟1 푟2

from which the Young-Laplace’s equation becomes:

2훾 푝1 − 푝2 = (2.7) 푟푚

For a single cylindrical capillary (pore) with a smooth surface (Figure 2.6), the above equation can be written in terms of contact angle (휃):

2훾 푝 − 푝 = cos 휃 (2.8) 1 2 푅 where R is the radius of the capillary.

2.3 Contact angle

For ideal solids whose surfaces are homogeneous, isotropic, smooth, and rigid, and when surrounding fluids are inert to such a solid (no chemical reaction or specific adsorption, dissolution, swelling, or rearrangement of phases, molecules, and functional groups), thermodynamic analyses predict that each materials system should exhibit a unique contact angle given by Young’s equation. This equation relates the contact angle to the three interfacial energies and is defined as [3]:

훾 − 훾 cos 휃푌 = 1푠 2푠 (2.9) 훾12

Here 훾1푠, 훾2푠 and 훾12 are, respectively, the solid–fluid1, solid–fluid2 and fluid1–fluid2 interfacial force per unit length at the contact line, i.e. surface tension, and 휃푌 is Young’s contact angle.

The “apparent” contact angle should be distinguished from the microscopic contact angle, i.e. Young's angle. If the surface of the solid is rough, the observed angle (which is conventionally measured at macro scale) between the fluid interface and the apparent surface of the solid will then vary with the position on the line of contact as schematically shown in Figure 2.7. It is believed that Young’s equation is applicable to each flat element on the rough surface [11].

Figure 2.7 Effect of surface roughness on apparent contact angle (휃푎). 휃푇 is the contact angle measured on a smooth, flat surface [11].

2.3.1 Contact angle hysteresis

Contrary to Young’s prediction, contact angles on most ambient surfaces are hysteretic, depending on the history of the contact line motion [14]. The hysteresis is characterized by two angles: the advancing contact angle measured after increasing the solid-liquid contact area and the receding angle measured after decreasing the contact area (Figure 1.1). As sketched in Figure 2.8, one can observe that this difference in fact consists of two parts. First, there is a discontinuity in the contact angle at zero velocity known as static hysteresis [8]. The static hysteresis is extremely stable, i.e., a droplet will remain in a state where it maintains a contact angle between the two limiting angles, without relaxation or motion (assuming no evaporation occurs) [8]. However, time dependence of contact angle hysteresis should be considered due to deformations of the substrates caused by the unbalanced surface normal component of the surface tension [8].

Figure 2.8 Schematic of static and dynamic contact angle hysteresis [15]; Ca and U stand for capillary number and velocity, respectively.

Figure 2.9 Schematic of (a) liquid column trapped in a capillary tube and (b) a raindrop [2]; g is the gravity acceleration.

Two well-known phenomena that are attributed to contact angle hysteresis, are a liquid column trapped in a capillary tube and a raindrop on a window pane (Figure 2.9). The difference in top and bottom contact angles in these two situations is conventionally referred to as contact angle hysteresis. In both cases, an equilibrium is only possible if the two angles are different, thus, the weight of the slug or drop may be supported by capillary force associated with the contact angle difference.

The dynamic component of the hysteresis is caused by the interplay of the liquid motion with the solid surface, and specially the liquid’s inability to simply flow over it [8]. For a slow-moving drop on a rough surface (that is, all solids but single crystal structures like mica/graphene) the static hysteresis will dominate, but for high velocities or low static-contact-angle-hysteresis surfaces (specially on liquids or liquid-soaked solids) the dynamic hysteresis becomes extremely important [8]. A critical capillary number can be introduced to judge whether the dynamic contact model should be incorporated into calculations. Numerical and experimental studies have been performed to investigate the effect of different factors on dynamic contact angle. In a recent study, Zhu et. al [17] simulated the flow of an oil slug in a capillary tube and obtained a three- dimensional universal chart of critical capillary number, as a function of static contact angle and viscosity ratio (Figure 2.10). The obtained critical capillary number is based on a predefined relative error limit, associated with the meniscus displacement. However, more investigation is needed to verify whether the meniscus displacement alone is a good estimator for the extent of contact angle variation during the flow.

Figure 2.10. Critical capillary number vs viscosity ratio (M) and static contact angle [17].

2.3.2 Contact line pinning

A characteristic of above-mentioned hysteretic systems is that, when a force acts on the fluid to change the flow direction, the three-phase contact line will remain stationary while the contact angle changes between two limits: receding and advancing contact angles; this is termed pinning [17]. When either of these limits is reached, the three-phase contact line will slip. In this case, there is a range of capillary pressure where the invading interface remains pinned. For the case of a porous medium, as the capillary pressure drops, the interface remains fixed in place and the contact angle adjusts to a new value, 휃ℎ (Figure 2.11). The hinging angle, 휃ℎ , can acquire any value between receding and advancing angles [17].

Figure 2.11. Contact line pinning and deformation of interface at a fixed contact point as time develops; 휃푟 , 휃ℎ , 휃푎 are receding, hinging and advancing contact angles

Existence of contact angle hysteresis and pinning has been shown experimentally using a number of methods including tilted plane, sessile drop and Wilhelmy methods [8]. Willett et. al. [18] investigated the influence of wetting hysteresis on the behavior of pendular rings both theoretically and experimentally for liquid-air system. Photographs of a liquid bridge formed using glycerol on smooth sapphire spheres are shown in Figure 2.12. Separating and approaching of the two spheres, causes the receding and advancing of the contact line.

The relationship between the capillary force and wetting hysteresis found in above study, is shown schematically in Figure 2.13. If the contact angle is greater than the receding value, an increase in the separation distance will result in an increase in the force due to pinning (a→b)

Figure 2.12. Photographs of a liquid bridge formed from glycerol during (a) separation and (b) approach [18].

Figure 2.13. (a) pendular liquid bridge (b) a schematic of the relationship between the wetting and the force * * hysteresis [18]. F is the dimensionless capillary force and S is the dimensionless half-separation distance. 휑푎 and 휑푟 are advancing and receding contact angles; 훽 is the filling angle.

[18]. This will be accompanied by a reduction in the contact angle until the receding value is achieved when the force decreases (b→c). If the separation distance is then decreased, the contact angle will increase until the advancing value is achieved (c→d). That is, pinning will again occur which will correspond to a reduction in the force. A further reduction in the separation distance to the initial value position will occur by slipping at the three-phase contact line with the contact angle fixed at the advancing value (d→a) [18].

The energies determining the contact angle can be divided into two competing parts: the “elastic energy”, due to the fluid-fluid surface tension and gravity, and the “pinning energy”, due to thepreference of the solid surface for either of the fluid phases [14]. For stable contact line configurations, the pinning force is related to the tension of the contact line which equals the elastic restoring force [14]. Boruvka et al. [19] modified the Young’s equation (Equation 2.9) to take into account the three-phase contact line tension:

휏퐾 cos 휃 = cos 휃푌 − (2.10) 훾12 where 휏 is the contact line tension and K is geodesic curvature of the contact line. Note that θ is generally not constant, because a nonzero 휏 exerts a force along the contact line thereby changing Young’s classical force balance [20].

Another form of pinning can happen when one fluid passes from a narrow opening in the pore throat into a wider pore body displacing the other fluid. This often occurs during the flow in the porous medium and is usually followed by another phenomenon known as “Haines jump” [21]. Haines jump is the result of instabilities that cause the fluid interface burst into the next pore in an accelerated manner [22]. Hilpert et al. [20] modelled the two-phase flow through a pore throat using a biconical pore segment (diverging and converging cones) and showed that the contact line is pinned in the center of the throat while it slides elsewhere. Accounting for the contact line tension using Equation 2.8, authors provided analytical solutions that described contact angle hysteresis in the pore throat (Figure 2.14).

2.3.3 Origins of contact angle hysteresis

Despite advances in both theoretical analysis and experimental methods, contact angle hysteresis remains poorly understood and mostly unpredictable for a variety of liquids on a variety of solids[23]. As of today, contact angle hysteresis is mainly attributed to surface chemical heterogeneity and roughness acting as sources of contact line pinning [23]. It can also be caused by the hysteresis in the wetting process itself. Molecular dynamics (MD) simulation and experimental investigation by atomic force microscope (AFM) have shown that adhesion and

Figure 2.14. Equilibrium states in a biconical pore segment [20]. (a) five selected menisci (b) curvature (J) versus contact line position z. (c) contact angle θ versus z. The circles in plots (b) and (c) represent the contact line positions of the menisci in plot (a).

separation of two surfaces are inherently irreversible processes, even when they proceed infinitely slowly [24]. Some potential energy between atoms or molecules is dissipated as

vibration (heat) to the environment as bonds are formed or snapped [24]. An additional cause for contact angle hysteresis is intrinsic mechanical instability of a deformable solid [23]. Polymers and other organic substrates demonstrate surface chemical irreversibility during the time of contact angle measurements that is associated with reorganization, realignment, and diffusion of polar functionalities, which contribute to hysteresis and its time dependency as well [23].

2.3.4 Contact angle measurement

Conventionally, contact angles have been measured indirectly using methods such as capillary rise or pressure measurements using a capillary tube or thin plate filled with powder of a certain material based on Lucas–Washburn and Young–Laplace equations, capillary pressure curves, USBM (United States Bureau of Mines), or Amott methods [25]. These indirect methods only yielded a statistical sense of microscopic behavior of the medium. Values of in-situ contact angles and local capillary pressure are not obtainable using these methods. In a different approach, static and dynamic contact angles have been vastly measured on a flat surface representing a specific mineral surface (i.e., silica, mica, or natural rock) using various methods such as sessile drop, captive bubble, Wilhelmy plate, and dual-drop dual-crystal (DDDC) methods [25]. High discrepancies in contact angles measured in the literature have been observed. This is caused by the use of different measurement methods, substrate imperfections, and surface cleaning methods [25]. Whether the contact angle on a flat surface can properly represent wettability inside of a porous medium pore has been a great concern. Unlike smooth flat surface used for lab measurements of contact angle, real pore walls are microscopically rough and chemically heterogeneous, making the contact angle measured on a smooth flat surface an unreliable one for pores inside porous media [25].

The advances in micron-resolution imaging tools have made it possible to study pore-scale configurations of fluid phases. In particular, X-ray methods are now widely used as they allow non-destructive imaging of the medium and fluids in the pore space under varying pressure, temperature, salinity and wetting conditions [6]. X-ray imaging has been used to measure contact angles in capillary tubes, however to date, few studies have measured contact angle in real

porous medium. Andrew et al. [26] measured contact angle in situ in a sample of Ketton limestone for a water-wet CO2 /brine system where the contact angles were measured manually on a raw image of the plane perpendicular to the three-phase contact line. The problem with this approach is that it is time-consuming and not automated. Scanziani et al. [7] extended this approach by fitting a circle on the fluid/fluid interface and a line to the fluid/rock interface on two-dimensional slices perpendicular to the three-phase contact line. Klise et al. [27] developed another automated contact angle measurement that was applied to fluids in a bead pack with varying wettability characteristics. In this approach planes were fitted to the fluid/fluid and rock/fluid interfaces and the angle between them computed. The last two methods were developed based on fitting planes/lines to voxelized images. More recently, AlRatrout et al. [6] presented another approach which involved meshing and smoothing the fluid/fluid and fluid/solid interfaces. The two vectors that have a direction perpendicular to both extracted surfaces were then computed on the three-phase contact line. Contact angle was found from the dot product of these vectors where they meet at the contact line. A schematic of this method is shown in Figure 2.15.

Figure 2.15. (A) Three-phase contact line highlighted on the smoothed mesh (B) Normal vectors defined on both the oil/brine and brine/rock interfaces for the contact line set. The cosine of the contact angle is calculated from the dot product of these two normals [6]

While these methods have allowed the estimation of contact angle directly from images of the fluids within the pore space, one should not ignore the errors associated with these techniques.

Some of the possible error sources include: segmentation artifacts, resolution of images, and error originated from fitting and surface extraction.

2.4 X-ray micro-computed tomography

Laboratory X-ray micro–computed tomography (micro-CT) is a fast-growing method in scientific research applications that allows for non-destructive imaging of morphological structures [28]. Computed or computerized axial tomography involves the recording of two- dimensional (2D) X-ray images from various angles around an object, followed by a digital three-dimensional (3D) reconstruction. The resulting 3D-rendered volume allows for the multi- directional examination of an area of interest and dimensional, volumetric, or other more advanced measurements to be made [28].

The principle of micro-CT is similar to other X-ray based analysis, whereas it records the differences in X-ray attenuation by the object. Attenuation is described as the proportion of the X-ray that interacts with the material and represented by the gray intensities in the reconstructed slice images. The interaction between the material and the X-ray beam decreases the intensity of the X-ray as it passes through the volume [29]. This decrease of intensity is described by the Beer–Lambert Law:

퐼(푥) = 퐼0푒−휇푥 where I(x) is the intensity measured at the detector, I0 is the intensity of original incident beam from the X-ray source, x is the length of the X-ray path within the material, and µ is the attenuation coefficient of the material, which depends on the material atomic number and density. Water and oil that are used in multi-phase flow experiments have very similar attenuation coefficients, however, a heavier contrast agent can be added to alter their attenuation properties by making use of photoelectric absorption edges [30].

The fundamental components of any micro-CT instrument are (i) penetrating ionizing radiation, (ii) a rotation stage, and (iii) a detector [28]. The X-rays are directed through a sample before being collected on a 2D X-ray detector (Figure 2.16). The detector records the attenuated

radiation, which passes through the sample along a straight line, into the 2D digital images. In this way, many hundreds or thousands of 2D projection images are recorded during the scan process. After scanning, these images are used to reconstruct a 3D data set by making use of filtered back-projection algorithms [28]. Effectively, every volumetric pixel (or voxel) is imaged from many angles, and the sum of its view from every angle produces a representation of the actual X-ray density and hence brightness of that voxel.

Figure 2.16. Illustration of X-ray micro-CT technique with a cone-beam configuration [32]

3. Methods

This study uses existing datasets based on the works of Paustian [30], Meisenheimer et. al. [33] and Schlüter et. al. [31]. The reconstructed 3D images acquired from X-ray micro-tomography are processed according to the procedures described in the above-mentioned sources. The processing consists of steps such as contrast enhancement, filtering and segmentation, i.e., the partitioning of an image into the different phases like the porous medium and the involved fluids. The segmentation steps were reported in the previous publications.

3.1 Fluid flow experiments

The quasi-static flow experiments are conducted with glass bead packing. The glass beads are naturally water-wet, however for part of the study, the wettability of the beads is altered, so some experiments are conducted using a water-wet system, and some with a mixed-wet system. Water is used as the wetting fluid and oil and/or air as the nonwetting fluids. The bottom of the sample is connected to a syringe pump filled with water. The top of the sample is connected to an oil reservoir at atmospheric pressure. In the case of three-phase system, two air vents located on the sides of the column allow air to enter or leave when open. Experiments are run with a full cycle of primary drainage (PD), main imbibition (MI), and main drainage (MD) at very slow flow rates (). The following table summarized the experimental characteristics for each dataset. Figure 3.1 is a representative schematic of the experimental set-up and sample holders used.

Table 3.1 Dataset characteristics

Source Fluid system Oil type System wettability Paustian [30] Water-Oil-Air Decane Water-wet Paustian [30] Water-Oil-Air Decane Mixed-wet Paustian [30] Water-Oil Decane Water-wet Schlüter et. al. [31] Water-Oil Dodecane Water-wet Miesenheimer et. al. [33] Water-Air - Water-wet

Figure 3.1 (a) Schematic of sample holder for 3-phase flow system; in 2-phase experiments, the air ports on the sides of the column are absent (b) Schematic of entire experimental set-up [30]

3.2 Algorithm for contact angle measurement

The local contact angles are measured using an algorithm based on the work of Scanziani et al. [7] which was further developed by Meisenheimer [35]. The outline of the algorithm is depicted below:

1. The contact points, defined as the meeting point of the three (fluid–fluid–solid) phases, are identified within the segmented three-dimensional images. 2. A contact line is created by putting together all of the points identified in step 1. 3. For each contact point, a 2D slice is generated which is normal to the contact line at each point of contact. 4. The contact angle is then automatically measured on this slice.

3.2.1 Identifying the contact points

The contact points are identified by first dilating each phase in the segmented 3D image by one voxel. An arithmetic operation is then performed to add the three dilated phases by summing the corresponding voxel values. The overlaying voxels with a value of three in the resulting image, are the points of contact between three phases. All the operations on images are completed using Avizo®, a 3D visualization and analysis software [34].

3.2.2 Creating contact line

In the 2nd step, the identified contact points are ordered to create a contact line and calculate the normal unit vector components of each point. A curve with a complex spatial pattern connects all of the contact points as illustrated in Figure 3.2. The resulting three-phase contact curve is then smoothed through a moving average filter which is a technique commonly used for noisy signal smoothing [7]. The moving average filter operates by averaging a number of points from the input signal to produce each point in the output signal. In equation form, this is written as:

푁−1 1 푦 = ∑ 푥 (3.1) 푖 푁 푖−푗 푗=0 where xi is the input signal, yi is the output signal, and N is the number of points used to compute the moving average. N is set to 4 for all the measurements in this work, based on an optimization study by Scanziani et al. [7]. After smoothing, the three normal unit vectors of each point in the direction from the selected point to the next point is calculated as components of vector 푣⃗⃗⃗푛⃗⃗⃗,⃗푘 [7]:

푣⃗⃗⃗푛⃗⃗⃗,⃗푘 = (푥푘+1 − 푥푘)푖 + (푦푘+1 − 푦푘)푗 + (푧푘+1 − 푧푘)푘⃗ (3.2)

In Eq. (4) xy , yk and zk are the three Cartesian coordinates of the kth contact point, and 푣⃗⃗⃗푛⃗⃗⃗,⃗푘 is a three-component vector with the desired direction, i.e., aligned with the three-phase contact curve. The normal unit vector 푛⃗⃗⃗⃗푘 is then found dividing 푣⃗⃗⃗푛⃗⃗⃗,⃗푘 by its modulus |푣⃗⃗⃗푛⃗⃗⃗,⃗푘 |:

푣⃗⃗⃗푛⃗⃗⃗,⃗푘 푛⃗⃗⃗⃗푘 = (3.3) |푣⃗⃗⃗푛⃗⃗⃗,⃗푘 |

The calculations for this step are done using Matlab®.

Figure 3.2 Depiction of contact lines in the region of interest

3.2.3 Generating normal slice

At each contact point, the segmented 3D image is cut using a plane normal to three-phase contact line at that point. The contact angle has to be estimated on each slice perpendicular to the contact curve. Figure 3.3 depicts an example of one of these extracted 2D slices. The normal direction of in Avizo®, using the normal vectors computed in previous step and is repeated for every 10 points on each contact line.

Figure 3.3 Visualization of a 2D image obtained as a slice of a segmented image taken along the plane locally normal to the three-phase contact line [7]

3.2.4 Calculating contact angles

The local contact angles on each slice, can now be estimated according to the following protocol performed in Matlab®:

1) Locating components: the algorithm loops through the slices and identifies each point in the labeled image that belongs to the wetting or non-wetting phases, solid-fluid interface and fluid- fluid interface. The triple-point is then found by determining the point where the smallest distance between the solid points and fluid points is.

2) Fitting: estimating the contact angle in this study is based on the assumption that the curvature of the fluid-fluid interfaces is constant near the solid surface. It is also assumed that the solid-

fluid and fluid-fluid interfaces are smooth surfaces, ignoring the atomic scale surface roughness; therefore the method yields an “effective” angle. To that aim, two circles are fit to the solid-fluid and fluid-fluid interfaces according to the direct least-squares fitting method by Vaughan Pratt [36]. This fitting is done in the close vicinity of triple point at a distance of 5 pixels for the solid- fluid interface and 3 pixels for fluid-fluid interface, in a ~3 µm resolution image. The distance limits are chosen based on the visual inspection of sample fitted lines.

휃

Figure 3.4 Representation of the 2D slice of 3D segmented images where fitted circles and contact angle is computed. Oil is blue, water green and solid red

3) Computing the angle: at the point of contact, the angle between the following two straight lines is the sought contact angle: the tangent to the circle representing the solid-fluid interface and the tangent to circle representing the fluid-fluid interface (red dashed lines in Figure 3.4). In this study the contact angle is defined as measured through the wetting phase, therefore, one needs to choose between the two complementary angles and select the one that surrounds the wetting phase. To implement this, a polar coordinate system is used so that each point is determined by a distance from the contact point and an angle from a fixed reference direction. Two sample points are then chosen on each interface, at an appropriately close distance from the

contact point. If the angle between the two points with reference to the contact point is smaller than 90°, the acute angle is selected, otherwise, the obtuse angle is to be selected as the contact angle on the normal slice.

The contact angles are measured in the whole volume of the imaged section in the column (indicated by the yellow area in Figure 3.1). A visualization of spatial distribution of contact points in the imaged volume is shown in Figure 3.5.

Figure 3.5 Spatial distribution of contact points in the imaged volume; grey denotes wetting phase and colored balls are the contact points; color bar represents the contact angle.

3.2.5 Testing of the approach

To test the accuracy of the estimation procedure mentioned in section 3.2.4, we apply the algorithm on two sets of data. The first dataset is the mixed-wet glass bead sample with the three-phase fluid system. We make a comparison with the manually-measured contact angles for

a total of 203 contact points in the same sample. The manual contact angle measurements were obtained using the standard angle measurement tool within ImageJ [37] (Figure 3.6). The distributions of calculated contact angles for both approaches are presented in Figure 3.7. The mean contact angles calculated using the automated and manual methods are 97 °and 92 °, respectively (Table 3.2). The large spread of obtained contact angles is typical of such porous systems and is due to variations in the geometry, curvature and wettability, as will be discussed in next chapters. These measurements are also sensitive to segmentation quality and resolution of images.

Figure 3.6 Illustration of manual contact angle measurement using angle tool in ImageJ

Table 3.2 Comparison of manually and automatically measured contact angles

Case Manual Automated Contact points measured 203 203 Mean contact angle (degrees) 97.3 92.6 Standard deviation 25.3 24.2

Frequency

Contact angle (degrees)

Figure 3.7 Histogram of measured contact angles using automated (blue bars) and manual (brown bars) methods

Figure 3.8 Trapped oil ganglia SSb isolated from the Ketton dataset [7]

In addition, we study a sub-volume of a water-wet Ketton dataset (SSb) used in the work of Scanziani et. al. [7] to compare their method with our implementation of the algorithm. The method used here is different, in that a circular fit was applied for the solid-fluid interface, since for the medium with a small average pore size, fitting a linear approximation to the surface is not

30 always optimal [35]. One ganglion of oil is selected from the mentioned dataset (Figure 3.8) and the algorithm described in Section 3.2 is applied to segmented images to obtain contact angle estimations in a total of 758 of its three-phase contact points. A comparison is rendered between contact angle values through both methods for the dataset SSb (Table 3.3).

3.2.6 Limitations of the method

There are two main sources of error in every image-based method of contact angle measurement which could affect the accuracy of results. The resolution of the 3D images is the primary source of inaccuracy. As the resolution of the image decreases, the boundaries in the image become more pixelated and are less able to capture the shape of real interfaces. As discussed in section 2.2.4, this method estimates an effective contact angle by fitting a circle to interfaces. The quality of segmentation also greatly impacts these measurements since it defines the shape of the fluid-fluid and solid-fluid boundaries and the uncertainties are reported to be greatest at these locations [4].

Table 3.3 Comparison of contact angle measurement results for dataset SSb using the two methods

Case Scanziani et. al. method This work Number of contact points 4025 758 Mean contact angle (degrees) 42.2 38.0 Standard deviation (degrees) 14.6 25.4

3.3 Image analysis

3.3.1 Object separation

The mixed-wet system used in the experiments is a mixture of differently sized glass beads as described in Table 3.4. In this system, the largest sized beads, 1000 μm-1400 μm, were oil-wet and the two smaller sizes were water-wet, resulting in 65% of the total surface area being oil-wet [30].

Table 3.4 Mixed wet system

Bead Diameter (µm) Weight percent 650 26% 850 26% 1000 - 1400 48%

To be able to separately measure the contact angle on each group of beads, we need to separate them using image analysis. This is done with a series of modules in Avizo® as described below:

A Separate Object module is first applied to the thresholded binary image of beads. This module is a combination of watershed, distance transform and numerical reconstruction algorithms. The watershed is a classical algorithm used for segmentation, that is, for separating different objects in an image. Starting from user-defined markers, the watershed algorithm treats pixels values as a local topography (elevation). The algorithm floods basins from the markers until basins attributed to different markers meet on watershed lines (see Figure 3.9). In many cases, markers are chosen as local minima of the image, from which basins are flooded.

The local topography is computed using a Chamfer distance map function which calculates the distance of each voxel from the boundary as the sum of successive erosions in diagonal direction. In the example shown in Figure 3.10, two overlapping circles are to be separated. To do so, one computes a distance map an image that is the distance to the background. The maxima of this distance are chosen as markers and the flooding of basins from such markers separates the two circles along a watershed line.

Figure 3.9 The Watershed principle

Figure 3.10 Depiction of (a) overlapping objects in a binary image (b) distance map (c) and separated objects

The objects touching the boundaries of the sample are not completely located inside the selection volume and should be removed from analysis. This is done by BorderKill module which identifies the intersection of image with the frame and reconstructs the boundary objects starting from the indentified voxels; logical difference between the original image and the recunstructed image gives the desired filtered image. The results of object separation and removing boundary objects are shown in Figure 3.11.

Figure 3.11 Snapshots of separated beads before (a) using BorderKill module and (b) after using the module

Separation of oil-wet and water-wet beads is done based on size difference of these two types. A Label Analysis module computes the equivalent diameter of each separated component which is equal to diameter of a spherical particle of the same volume. The result of the size analysis is presented in Figure 3.12.

3.3.2 Measuring curvature

The curvature information used in this study is based on the measurements performed by Paustian [30] and Schlüter et. al. [31]. Briefly, the Curvature module in Avizo®, computes the mean curvature for a discrete triangular surface generated using the surface generation function in Avizo®. The algorithm works by approximating the surface locally by a quadric form. The eigenvalues and eigenvectors of the quadric form correspond to the principal curvature values and to the directions of principal curvature.

10 9 8 7 6 5

counts 4 3 2 1 0 500 600 700 800 900 1000 1100 1200 1300 1400 1500 Bead Diameter (microns)

Figure 3.12 Size distribution of the beads in mixed-wet system

Due to the segmentation, the curvature values that are close to the solid bead surface are not as accurate as the curvatures far away from the beads [30]. To account for this, Li [38] developed a method that weighted the average curvatures based on their distance to a solid surface. To reduce

34 the effect of the edges on the curvature value even further, the surfaces are also clipped. The distance of each triangle to the edge of the surface is measured, and those triangles that fall within 20% of the maximum distance from the edge are removed from the distance weighted curvature average. Li [38] and Paustian [30] validated the use of curvature-based capillary pressure measurements by comparing the measurements from the transducer to the pressure values obtained via the Young-Laplace equation.

3.3.3 Estimating wetted pore radius

A pore is defined as a portion of pore space bounded by solid surfaces (pore body) and by planes where the cross-sectional area of the pore space exhibits minima (pore throats). It is customary to use hydraulic radius as a measure of pore size, defined as [11]:

푎푟푒푎 표푓 푐푟표푠푠 푠푒푐푡𝑖표푛 푅 = (3.4) 퐻 푙푒푛푔푡ℎ 표푓 푝푒푟𝑖푚푒푡푒푟 표푓 푐푟표푠푠 푠푒푐푡𝑖표푛

For irregular capillaries, the minimum value of the ratio given by the right-hand side of Equation 3.4 must be found by varying the orientation of the sectioning plane about the same fixed point inside the capillary. We propose “wetted pore radius”, defined as the hydraulic radius of the pore at the wetting front position. Assuming the curvature of fluid-fluid interface is small, it is possible to approximate the area of cross section of the pore at the wetting front, with the fluid- fluid interfacial area (Figure 3.13).

Figure 3.13 Depiction of (a) wetted pore radius (R) and (b) perimeter of fluid interface (white line). Solid phase is blue and fluid-fluid interface is red.

Figure 3.14 Example of fluid-fluid interfaces in an air-water system; the mean wetted pore radius is obtained by dividing this surface by the length of the white curve enclosing the surface

The “mean wetted pore radius” can be estimated by image analysis and using the following relationship:

푡표푡푎푙 𝑖푛푡푒푟푓푎푐𝑖푎푙 푎푟푒푎 푅 = 2 × (3.5) 푡표푡푎푙 푙푒푛푔푡ℎ 표푓 푝푒푟𝑖푚푒푡푒푟 표푓 𝑖푛푡푒푟푓푎푐푒

Interfacial surfaces can be generated at the intersection between two different phase classes via the marching cube algorithm, which locates the interface and creates a triangular mesh approximating a smooth surface. The surface generation function on Avizo® allows for different smoothing settings, type and extent. We use constrained smoothing, which does not allow any labels to be altered. The smoothing value is chosen based on an analysis by Paustian [30] for each dataset. The interfacial area is determined by summing the area of each triangle of the mesh on the generated interfacial surface.

4. Results

In this chapter, the average static contact angles throughout the flow experiments and their distributions are presented. In chapter 5, we discuss the observed trends and explain the relationship between contact angle, capillary pressure and morphology.

4.1. Contact angle-saturation and distribution of contact angles

The algorithm for contact angle is applied to measure several hundred in situ contact angles in the entire image for each water saturation state (Sw). These measurements are done for air-water pair in air-water system and oil-water fluid pair in two and three-phase systems with different wettability.

4.1.1. Air-water fluid pair

We see a rather wide distribution of contact angles for the air-water pair in contact with the glass bead pack (Figure 4.1). The obtained average value is within the range of 27°-39°, while for the flat glass slide, the manually measured value is ~8°. As the water saturation in the system decreases, the distribution becomes slightly wider with the average contact angle shifting towards higher values. In both drainage and imbibition, the average contact angle decreases as the saturation increases, however the curves separate gradually, with a steeper slope for imbibition (Figure 4.1-a).

4.1.2 Oil-water fluid pair in water-wet system

The oil-water (OW) average contact angle exhibits similar variation trends in both the two-phase and three-phase systems (Figures 4.2-3); during the drainage, it drops sharply at the beginning and becomes smoother as the saturation decreases; in the imbibition, it gradually increases and separates from the drainage curve; both curves appear to be meeting at a saturation point of approximately 0.7, however for the three-phase system, this happens at a higher contact angle.

37 50 a) Drainage 45 Imbibition 40

25 Contact Contact angle (degrees)

20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Water saturation

30 b) D1 (saturation = 0.62) 25 D3 (saturation = 0.47) D4 (saturation = 0.29) D6 (saturation = 0.14) 20

10 Frequency Frequency (%)

0 0 20 40 60 80 100 120 140 160 180 Contact angle (degrees)

40 c) I1 (saturation = 0.13) 35 I2 (saturation = 0.18) 30 I3 (saturation = 0.31) I5 (saturation = 0.47) 25 I6 (saturation = 0.70) 20

15 Frequency Frequency (%) 10

0 0 20 40 60 80 100 120 140 160 180 Contact angle (degrees)

Figure 4.1. The (a) average contact angle for air-water system and the distributions for: (b) drainage, (c) imbibition based on data from [33]

38 90 80 70 60 50 40 30 Drainage (oil: decane) [30] 20 Imbibition (oil: decane) [30]

Contact Contact angle (degrees) Imbibition (oil:dodecane) [31] 10 Drainage (oil: dodecane) [31] 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Water saturation

30 b) D1 (Sw = 0.70) 25 D2 (Sw = 0.69) D3 (Sw = 0.67) 20 D6 (Sw = 0.28) D7 (Sw = 0.11) 15

10 Frequency Frequency (%)

0 0 20 40 60 80 100 120 140 160 180 Water saturation

25 c) I1 (Sw = 0.13) 20 I2 (Sw = 0.32) I3 (Sw = 0.44) I5 (Sw = 0.65) 15 I7 (Sw = 0.71)

10 Frequency Frequency (%) 5

0 0 20 40 60 80 100 120 140 160 180 Water saturation

Figure 4.2. The (a) average contact angle for two-phase oil-water system and the distributions for oil-dodecane system: (b) drainage, (c) imbibition

39 100 a) 90

50 Drainage

Contact Contact angle (degrees) 40 Imbibition 30 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Water saturation

45 b) D1 (saturation = 0.72) 40 D2 (saturation = 0.58) 35 D3 (saturation = 0.42) D4 (saturation = 0.20) 30 D6 (saturation = 0.08) 25 20

15 Frequency Frequency (%) 10 5 0 0 20 40 60 80 100 120 140 160 180 Contact angle (degrees)

30 c) I1 (saturation = 0.09) I2 (saturation = 0.10) 25 I3 (saturation = 0.15) 20 I4 (saturation = 0.56)

10 Frequency Frequency (%)

0 0 20 40 60 80 100 120 140 160 180 Contact angle (degrees)

Figure 4.3. The (a) average OW contact angle for oil-water-air system and its distributions: (b) drainage, (c) imbibition (based on data from [30])

The contact angle distribution for the three-phase system, shifts significantly toward smaller values at the beginning of drainage and then becomes more even, in accordance with the changes of average contact angle; during the imbibition, the distribution shifts gradually to larger values.

4.1.3 Oil-water fluid pair in mixed-wet system

The contact angles for the mixed-wet system are measured separately for the chemically treated (oil-wet) and non-chemically treated (water-wet) glass bead pack. As the flow develops, the distribution switches between two peaks for water-wet beads, with one averaging below 90° and the other over 90° (Figure 4.4, b-c); there exists an inflection point for imbibition (Sw = 0.17) at which the first peak becomes less significant and the second peak starts to grow; the second peak averages about the same value as the oil-wet beads (~130°) (see Figure 4.5), which suggests that at low saturation states of the non-wetting phase (or end of imbibition), the water-wet beads have a wetting behavior similar to the oil-wet beads; this opens up the discussion that fluid-fluid interfaces at a point of contact are not influenced only by the surfaces close to the contact point but also by the adjacent surfaces. For the oil-wet beads (Figure 4.5), the distribution has a larger spread at low saturation states of the wetting phase (beginning of imbibition and end of drainage) partly covering contact angles below 90°; at higher saturations the distribution shifts toward larger values over 90°.

The change of average oil-water contact angle near the water-wet beads, shows a similar trend as the water-wet system in previous section: a sequence of dropping and becoming even during drainage and gradual rise in imbibition (Figure 4.4-a); the overall change of contact angle is approximately 60° at its highest. Conversely, near the oil-wet beads, the contact angle rises approximately 20° as the saturation increases at early stages and becomes almost flat afterwards for both drainage and imbibition; this behavior is completely different from the water-wet beads for which the drainage and imbibition curves separate dramatically at low saturations.

41 140 a)

120

100

60 Drainage (water-wet) Imbibition (water-wet)

Contact Contact angle (degrees) Drainage (oil-wet) 40 Imbibition (oil-wet) Imbibition (combined) Drainage (combined) 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Water saturation

27 b) D1 (saturation = 0.73) D3 (saturation = 0.35) 22 D4 (saturation = 0.18) D5 (saturation = 0.07)

Frequency Frequency (%) 7

0 20 40 60 80 100 120 140 160 180 -3 Contact angle (degrees)

27 c) I1 (saturation = 0.04) 22 I2 (saturation = 0.08) I3 (saturation = 0.17) 17 I4 (saturation = 0.93)

Frequnecy Frequnecy (%) 7

0 20 40 60 80 100 120 140 160 180 -3 Contact angle (degrees)

Figure 4.4. The (a) average OW contact angle for the mixed-wet Air-Oil-Water system and its distributions: (b) drainage for water-wet beads, (c) imbibition for water-wet beads (based on data from [30])

42 40 a) D1 (saturation = 0.73) 35 D2 (saturation = 0.18) D3 (saturation = 0.07) 30 D4 (saturation = 0.04) 25

15 Frequency Frequency (%) 10

0 0 20 40 60 80 100 120 140 160 180 Contact angle (degrees)

35 b) 30 I1 (saturation = 0.04) 25 I2 (saturation = 0.08) I3 (saturation = 0.17) 20 I4 (saturation = 0.93)

Frequency Frequency (%) 10

0 0 20 40 60 80 100 120 140 160 180 Contact angle (degrees) Figure 4.5. The distributions of OW contact angle for oil-wet beads in the mixed-wet air-oil-water system: (a) drainage, (b) imbibition

4.1.4 Overview of contact angle variation near water-wet beads

Figure 4.6 summarizes all the observed behaviors of oil-water interfaces near the water-wet beads. A hysteresis pattern is clearly recognizable for the oil-water system. The contact angles associated with imbibition significantly separate from the drainage-associated contact angles and have greater values for all three systems. The opposite is true for the air-water system which shows small hysteresis and a flipping of drainage and imbibition contact angle curves. These behaviors will be further discussed in next chapter.

140 OW 2-phase (imbibition) (oil: dodecane) [31] 120 OW 2-phase (drainage) (oil: dodecane) [31] OW 3-phase water-wet 100 (drainage) [30] OW 3-phase water-wet 80 (imbibition) [30] OW 3-phase mixed-wet (drainage) [30] 60 OW 3-phase mixed-wet (imbibition) [30] AW 2-phase (drainage) 40 [33] Contact Contact angle (degrees) AW 2-phase (imbibition) [33] 20 OW 2-phase (imbibition) (oil: decane) [30] 0 OW 2-phase (drainage) 0 0.2 0.4 0.6 0.8 1 (oil: decane) [30] Water saturation

Figure 4.6. Summary of contact angles near water-wet beads

4.2 Contact angle spatial distribution

The spatial distributions of contact angles for the two-phase oil-water system is illustrated in Figure 4.7 for two saturation states. It can be seen that at both flow stages, the contact angles are distributed rather evenly in the imaged volume and no strong correlation is observed between the location of contact points and the contact angle.

Figure 4.7. Spatial distribution of contact angle during imbibition for the oil-water system: (a) low saturation (0.13), (b) medium saturation (0.45); the grey patches represent the wetting phase, and colored balls the contact angle

5. Discussion

In order to explain the characteristic patterns of contact angle variation observed during the drainage and imbibition experiments, we rely on the relationship between the capillary pressure, pore structure and the contact angle. The force balance on the interface of two immiscible fluids at mechanical equilibrium, indicates whether the above variables are inter-related. To describe such relationship, we approximate the pores with an idealized geometry similar to the approach in pore-network modelling. Pore networks are often used to simulate multiphase flow in porous media, because flow processes are easier to describe in geometrically simple pore networks than in a real geometry. Geometrically well-defined pore networks consist of pore bodies, such as spheres or right hexahedra, that are typically connected by cylindrical, triangular, or rectangular tubes, which may be either straight or of converging/diverging type.

The interplay of contact angle, pore geometry and capillary pressure is, of course, more complicated in real porous medium; however, the simplified models used in pore networks have proven to be powerful tools for extrapolating limited measured data and for developing valuable insight into complex multiphase flow phenomena such as capillary pressure and relative permeability hysteresis, the effect of wettability, and three-phase flow [17].

5.1 Conical pore model

A conical capillary with variation in cross section and cone angle, is used here to model the pores. We chose this model because it is able to represent both the pore body and pore neck by assigning different values to cone angle. The contact angle is related to the pore radius (R), capillary pressure (∆푃), surface tension (훾), curvature of the interface (r) and geometry of the pores (휑), as expressed in the following equations [11]:

푅 휃 = cos−1 − 휑 (5.1) 푟 where 2훾 푟 = (5.2) ∆푃

which follows directly from the geometry of the menisci and Young-Laplace’s equation as shown in Figure 5.1. The expression R/cos(휃 + 휑), is the mean radius of curvature of the meniscus. The effect of orientation of solid surfaces, expressed by the angle 휑, on the radius of curvature may be visualized by imagining that the solid boundary lines are tilted over a wide range of values of 휑 [11].

Figure 5.1. Menisci in a conical capillary

In the following sections, we separately examine the variations of the two factors, namely, wetted pore radius and fluid-fluid curvature, and then put them together using the above- mentioned model to reproduce the average contact angle.

5.1.1 Wetted pore radius variation

Figures (5.2, a-b) demonstrate different configurations of wetting phase during imbibition, as it transitions from a web of pendular rings to the funicular stage. As the flow develops, the fluid- fluid interfaces move from corner spots to larger pore spaces. Consequently, the area of pore cross-section at wetting front, or wetted pore radius, increases. The difference in the pore radius variation curves for drainage and imbibition implies that the flow paths for these events are not identical. This is partly the cause for hysteretic behavior observed in contact angle curves. Looking at wetted pore radius variation, it is also evident that these values vary more for the air- water system compared to the oil-water system and there is greater separation between drainage

and imbibition. This reveals the difference in displacement patterns for the two systems which could be due to different selectivities of wetting fluid for certain pore spaces in each system.

a) b)

)

c) c) d )

0.18 Imbibition (AW) c) 0.16 Drainage (AW) 0.14 Drainage (OW) Imbibition (OW) ) 0.12 0.1 0.08 0.06

Wettedpore radius [mm] 0.04 0.02 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Water saturation

Figure 5.2. (a)-(b) Transition of wetting phase from a web of pendular rings to the funicular stage, (c) wetted pore radius variation for air-water (AW) and 2-phase oil-water (OW) systems

7 Drainage (AW) Imbibition (AW) 6 Drainage (OW) Imbibition (OW) 5

3 Curvature Curvature [1/mm] 2

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Water saturation

Figure 5.3. Mean curvature variation in air-water (AW) and 2-phase oil-water (OW) systems

900

800

700

600

500 Imbibition (AW) Drainage (AW) 400 Drainage (OW) 300 Imbibition (OW)

Capillary Capillary pressure [Pa] 200

100

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -100 Water saturation

Figure 5.4. Capillary pressure (experimentally measured) variation in air-water (AW) and 2-phase oil-water (OW) systems

5.1.2 Curvature and capillary pressure variation The capillary pressure in a porous medium is an increasing function of the non-wetting phase or, alternately, a decreasing function of the wetting phase saturation [11]. Radius of curvature of the fluid-fluid interface is related to capillary pressure through Equation 2. The image analysis on air-water and oil-water systems show that the curvature follows the same trend as the capillary pressure, which is in agreement with Equation 5.2, assuming a constant surface tension (Figures 5.3-4). The curvature hysteresis, or the separation of curvature values between drainage and imbibition, is significantly greater for the oil-water system. As will be shown further on, this is one of the main causes for the different behaviors of the air-water and oil-water systems.

5.1.2.1 Effect of capillary pressure hysteresis and pinning

For a system with uniform wettability, one would expect that the capillary pressure required for flow, is independent of flow direction based on Young- Laplace equation and follow similar trends for drainage and imbibition. Some hysteresis is, of course, expected due to differences in flow path as shown in section 5.2. However, looking at Figure 5.5, it is seen that there is significant capillary pressures hysteresis, particularly for the oil-water system. It is also apparent that increasing the capillary pressure during drainage, at first results in only a very small change in saturation causing a sudden pressure rise. This effect is followed by the “Haines jumps”, events showing a sudden rise or drop in saturation when one phase passes from a pore neck into a wider pore body displacing the other phase, which were first recognized more than 80 years ago [39]. Conversely, during imbibition, the capillary pressure sharply drops at the beginning of the process. An analogous phenomenon occurs when the wetting phase front advances during wetting or imbibition. When the wetting layers in a narrow opening between two pores touch and coalesce, an instability called snap-off occurs, leading to disconnection and trapping of the nonwetting phase [39]. These phenomena ultimately lead to differences in fluid occupation of pores and thus to hysteretic behavior in water retention curves.

Figure 5.5. Capillary pressure – saturation curve for an oil-water system [31]

We have evidence to believe that these events are closely linked with the pinning of fluid-fluid interfaces, as confirmed by other researchers ([14];[17]-[20]). Image analysis on the oil-water interfaces during early stages of imbibition, reveals that the interface remains approximately fixed in place while the curvature decreases and the contact angle adjusts to a new value. Figures 5.6 illustrates one example of such events at a location near the pore neck. As the capillary pressure decreases, the oil-water interface (red surface) becomes less bent and appears to be swelling up. The contact angle at the point denoted by the red circle in the image cross-section, increases from 39.8° to 82.0°.

As discussed in section 1.3, the equilibrium states in a pore can be categorized into sequences of sliding in pore bodies and pinning in the center of pore throats. The classical form of the Young- Laplace equation is not able to explain the pinning effect, since it does not account for the interfacial and lineal tensions that cause nonconstant contact angles [20].

Figure 5.6. Pinning of fluid-fluid interface (red surface) as the flow develops during imbibition (a-c) and their respective cross-sections (d-f)

5.1.3 Predicting the contact angle variation

We are now able to estimate the average contact angle, knowing the pore radius and the curvature of each interface at the point of contact, based on Equation 5.1. Figures 5.7-8 presents the calculated contact angles (offset by the cone angle, 휑) for air-water and oil-water-air systems. Interestingly, the obtained trend agrees with that of measured contact angles using image analysis, confirming that the two factors of interfacial curvature and pore radius, dictate the contact angle. In other words, Equation 5.1 can approximately describe the contact angle variation of the system.

This correlation (Equation 5.1) can also explain the observed difference in the behaviors of air- water and oil-water systems. The smaller separation between the imbibition and drainage contact

80 80

70 70

60 60

50 50 (model)

° 40 40 φ

θ+ θ+ 30 30 º (directly º(directly measured)

Imbibition (model) θ 20 20 Drainage (model) 10 Drainage (directly measured) 10 Imbibition (directly measured) 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Water saturation

Figure 5.7 Predicted contact angles (offset by 휑) for air-water system

100 100

90 90

80 80

70 70

(model) 60 60

° φ

+ + 50 50 θ

Drainage (model) (directly measured)

40 40 °

Imbibition (model) θ 30 Drainage (directly measured) 30 Imbibition (directly measured) 20 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Water saturation

Figure 5.8. Predicted contact angles (offset by 휑) for oil-water system

angle curves for air-water system and its decreasing trend, is the result of smaller curvature hysteresis for this system which is adjusted by the rate of variation in pore radius. Also, the flipping of drainage and imbibition contact angle curves for air-water system originates from the

smaller imbibition-associated wetted pore radius compared to the drainage-associated values of this quantity in the flow path.

5.2 Effect of wettability

For a mixed-wet pore, it is no longer valid to assume that the interfacial tension is uniform across the interface. To explain the formation of a multi-curvature interface, we derive a simple model based on a cylindrical capillary tube with different surface properties at each half-wall (Figure 5.9) which relates the pressure drop across the interface to its curvature:

Figure 5.9 Mixed-wet capillary tube

The element dL on the fluid-fluid-solid contact line is subjected to a force 훾푑퐿 whose projection along the normal PN is [11]:

푅 훾푑퐿 sin 휑 = 훾휑푑퐿 = 훾 ( ) 푑퐿 (5.3) 푟

푅 since the proportion of and hence 휑, is supposed to be very small [11]. For this particular 푟푖 case, 푟푖 and 훾푖 are defined as functions of L but have constant values along each half of the boundary associated with the upper or lower surface of the tube. Integrating over the boundary line gives the total force induced by surface tension:

훾 훾 휋푅2 ( 1 + 2) (5.4) 푟1 푟2

For mechanical equilibrium of the surface, this force is to be balanced by the pressure forces and therefore we must have:

훾 훾 휋푅2∆푃 = 휋푅2 ( 1 + 2) (5.5) 푟1 푟2 which reduces to:

훾 훾 ∆푃 = 1 + 2 (5.6) 푟1 푟2

Writing the above equation in terms of contact angles we get:

1 ∆푃 = (훾 cos 휃 + 훾 cos 휃 ) (5.7) 푅 1 1 2 2

This example shows that for a pore with different wettability, the individual contact angles on one contact line are not independent from each other. We can also see that capillary pressure is equal to the linear contributions from each surface element.

The effect of peak transitioning in the contact angle distribution of mixed-wet system (section 4.1.3), can be explained using this relationship. There are three possible pore types in a mixed- wet system, in terms of wettability; the pores that are formed by water-wet beads (type 1), those that are formed by a mixture of water-wet and oil-wet beads (type 2) and finally the pores that are formed by only oil-wet beads (type 3).

We hypothesize that at lower saturation states of the wetting phase, the majority of contact points are located inside pore type 1 (Figure 5.10a) since the water phase tends to accumulate in the

corners of water-wet beads to minimize the surface free energy. This corresponds to the peak with the contact angle averaging below 90° in the distribution plot. As the flow develops, the interfaces move to configurations b and c in Figure 5.10 contributing to the contact angles in the transitioning region at ~90° in the distribution. At high saturations of wetting phase or low saturations of non-wetting phase, the nonwetting phase tend to escape to the regions close to oil- wet beads shifting the average contact angle towards values over 90° as demonstrated by Figure 5.10d.

Figure 5.10. Pore types in the mixed-wet system formed by water-wet beads (orange) and oil-wet beads (green) and configuration of interfaces; blue regions represent water and grey region, the oil

6. Conclusion

This study explored the variations of contact angle for different fluid systems in a porous medium and investigated the correlation of contact angle and parameters related to flow physics and system geometry. Unlike the universally accepted belief that contact angle is a metric for wettability, we showed that it is essentially the result of force balance on the interface of the two immiscible fluids in contact with the solid phase and is controlled not only by wettability but also pressure deformation of the interface and geometry.

Figure 6.1 Demonstration of effect of curvature (a) and pore radius (b) variations on contact angle

Our proposed approach also suggest that the problem of contact angle variation can be viewed from a geometry perspective; the fluid-fluid contact angle is just another angle formed by the interface of fluids and the solid surface; this view transforms the problem into a geometry problem. The curvature of interface and wetted pore radius, the two of the factors identified to control the contact angle, are each responsible for variations of one side of this angle. This is

illustrated in Figure 6.1 in which the two variables are each changed separately at a time while the other one is kept constant. Figure 6.1a shows the effect of curvature on the resulting contact angle, whereas Figure 6.1b shows how changes in pore radii for a fixed curvature control contact angle. Evidently, increasing the pore radius causes a decrease in contact angle while decreasing the curvature increases the contact angle. The combination of these effects plus surface forces determine the final value for contact angle.

Although the correlation discussed here can help developing insights into contact angle hysteresis and identifying variables to predict contact angle variation, however, more work needs to be done to find the underlying causes for changes of these variables during fluid displacement events. As mentioned earlier, wetting/de-wetting and pinning phenomena remain poorly understood as of today. These effects directly influence the curvature of fluid-fluid interfaces and cannot be explained solely by the Young-Laplace equation. The current form of the above correlation does not capture the variations of the solid surface orientation which needs to be incorporated into the model for more accurate results.

References

[1] Singh, Kamaljit, Michael Jung, Martin Brinkmann, and Ralf Seemann. "Capillary-dominated fluid displacement in porous media." Annual Review of Fluid Mechanics (2019). [2] Bush, John. “18.357 Interfacial Phenomena”. Fall 2010. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu/. [3] Young, Thomas. "III. An essay on the cohesion of fluids." Philosophical transactions of the royal society of London 95 (1805): 65-87. [4] Blunt, Martin J., Qingyang Lin, Takashi Akai, and Branko Bijeljic. "A thermodynamically consistent characterization of wettability in porous media using high-resolution imaging." Journal of colloid and interface science 552 (2019): 59-65. [5] AlRatrout, Ahmed, Martin J. Blunt, and Branko Bijeljic. "Spatial correlation of contact angle and curvature in pore‐space images." Water Resources Research 54, no. 9 (2018): 6133-6152. [6] AlRatrout, Ahmed, Ali Q. Raeini, Branko Bijeljic, and Martin J. Blunt. "Automatic measurement of contact angle in pore-space images." Advances in Water Resources 109 (2017): 158-169. [7] Scanziani, Alessio, Kamaljit Singh, Martin J. Blunt, and Alberto Guadagnini. "Automatic method for estimation of in situ effective contact angle from X-ray micro tomography images of two-phase flow in porous media." Journal of colloid and interface science 496 (2017): 51-59. [8] Eral, H. B., and J. M. Oh. "Contact angle hysteresis: a review of fundamentals and applications." Colloid and polymer science 291, no. 2 (2013): 247-260. [9] [10] Huber, Manuel, Franz Keller, Winfried Säckel, Manuel Hirschler, Philip Kunz, S. Majid Hassanizadeh, and Ulrich Nieken. "On the physically based modeling of surface tension and moving contact lines with dynamic contact angles on the continuum scale." Journal of Computational Physics 310 (2016): 459-477. [11] Dullien, Francis AL. Porous media: fluid transport and pore structure. Academic press, 2012. [12] Al‐Futaisi, Ahmed, and Tad W. Patzek. "Impact of wettability alteration on two‐phase flow characteristics of sandstones: A quasi‐static description." Water Resources Research 39, no. 2 (2003). [13] Shikhmurzaev, Yulii D. "Spreading of drops on solid surfaces in a quasi-static regime." Physics of Fluids 9, no. 2 (1997): 266-275. [14] Nadkarni, G. D., and S. Garoff. "An investigation of microscopic aspects of contact angle hysteresis: Pinning of the contact line on a single defect." EPL (Europhysics Letters) 20, no. 6 (1992): 523.

[15] Blake, Terence D. "The physics of moving wetting lines." Journal of colloid and interface science 299, no. 1 (2006): 1-13. [16] Zhu, Guangpu, Jun Yao, Lei Zhang, Hai Sun, Aifen Li, and Bilal Shams. "Investigation of the dynamic contact angle using a direct numerical simulation method." Langmuir 32, no. 45 (2016): 11736-11744. [17] Oren, P-E., Stig Bakke, and Ole Jakob Arntzen. "Extending predictive capabilities to network models." SPE journal 3, no. 04 (1998): 324-336. [18] Willett, C. D., M. J. Adams, S. A. Johnson, and J. P. K. Seville. "Effects of wetting hysteresis on pendular liquid bridges between rigid spheres." Powder Technology 130, no. 1-3 (2003): 63-69. [19] Boruvka, L., and A. W. Neumann. "Generalization of the classical theory of capillarity." The Journal of Chemical Physics 66, no. 12 (1977): 5464-5476. [20] Hilpert, Markus, Cass T. Miller, and William G. Gray. "Stability of a fluid–fluid interface in a biconical pore segment." Journal of colloid and interface science 267, no. 2 (2003): 397-407. [21] Haines, William B. "Studies in the physical properties of soil. V. The hysteresis effect in capillary properties, and the modes of moisture distribution associated therewith." The Journal of Agricultural Science 20, no. 1 (1930): 97-116. [22] Singh, Kamaljit, Hagen Scholl, Martin Brinkmann, Marco Di Michiel, Mario Scheel, Stephan Herminghaus, and Ralf Seemann. "The role of local instabilities in fluid invasion into permeable media." Scientific reports 7, no. 1 (2017): 1-11. [23] Drelich, Jaroslaw W. "Contact angles: From past mistakes to new developments through liquid-solid adhesion measurements." Advances in colloid and interface science (2019). [24] Yang, X. F. "Equilibrium contact angle and intrinsic wetting hysteresis." Applied physics letters 67, no. 15 (1995): 2249-2251. [25] Jafari, Mohammad, and Jongwon Jung. "Direct measurement of static and dynamic contact angles using a random micromodel considering geological CO2 sequestration." Sustainability 9, no. 12 (2017): 2352. [26] Andrew, Matthew, Branko Bijeljic, and Martin J. Blunt. "Pore-scale contact angle measurements at reservoir conditions using X-ray microtomography." Advances in Water Resources 68 (2014): 24-31. [27] Klise, Katherine A., Dylan Moriarty, Hongkyu Yoon, and Zuleima Karpyn. "Automated contact angle estimation for three-dimensional X-ray microtomography data." Advances in water resources 95 (2016): 152-160. [28] Du Plessis, Anton, Chris Broeckhoven, Anina Guelpa, and Stephan Gerhard Le Roux. "Laboratory x-ray micro-computed tomography: a user guideline for biological samples." GigaScience 6, no. 6 (2017): gix027.

[29] Guntoro, Pratama Istiadi, Yousef Ghorbani, Pierre-Henri Koch, and Jan Rosenkranz. "X-ray Microcomputed Tomography (µCT) for Mineral Characterization: A Review of Data Analysis Methods." Minerals 9, no. 3 (2019): 183. [30] Paustian, Rebecca L. "Analyzing the Relationship between Capillary Pressure, Saturation, and Interfacial Area for a Three-Phase Flow System in Porous Media." (2018): Oregon State University. [31] Schlüter, S., S. Berg, M. Rücker, R. T. Armstrong, H‐J. Vogel, R. Hilfer, and D. Wildenschild. "Pore‐scale displacement mechanisms as a source of hysteresis for two‐phase flow in porous media." Water Resources Research 52, no. 3 (2016): 2194-2205. [32] Wildenschild, Dorthe, and Adrian P. Sheppard. "X-ray imaging and analysis techniques for quantifying pore-scale structure and processes in subsurface porous medium systems." Advances in Water Resources 51 (2013): 217-246. [33] Meisenheimer, Douglas E., James E. McClure, Mark L. Rivers, and Dorthe Wildenschild. "Exploring the effect of flow condition on the constitutive relationships for two-phase flow." Advances in Water Resources (2020): 103506. [34] Thermo Fisher Scientific: Amira-Avizo 3D Software, http://www.fei.com/software/avizo3d/. [35] Sun, C., McClure, J.E., Mostaghimi, P., Herring, A.L., Meisenheimer, D.E., Wildenschild, D., Berg, S. and Armstrong, R.T. (Expected 2020). Unraveling the robust and efficient nature of topological principle for characterizing wetting behavior. In Preparation for Journal of Colloid and Interface Science. [36] Pratt, Vaughan. "Direct least-squares fitting of algebraic surfaces." ACM SIGGRAPH computer graphics 21, no. 4 (1987): 145-152. [37] Abràmoff, Michael D., Paulo J. Magalhães, and Sunanda J. Ram. "Image processing with ImageJ." Biophotonics international 11, no. 7 (2004): 36-42. [38] Li, Tianyi, Steffen Schlüter, Maria Ines Dragila, and Dorthe Wildenschild. "An improved method for estimating capillary pressure from 3D microtomography images and its application to the study of disconnected nonwetting phase." Advances in Water Resources 114 (2018): 249- 260. [39] Berg, Steffen, Holger Ott, Stephan A. Klapp, Alex Schwing, Rob Neiteler, Niels Brussee, Axel Makurat et al. "Real-time 3D imaging of Haines jumps in porous media flow." Proceedings of the National Academy of Sciences 110, no. 10 (2013): 3755-3759.