Pore-scale investigation of wettability effects on two-phase flow in porous media

Transcription

1 Pore-scale investigation of wettability effects on two-phase flow in porous media A thesis submitted to the University of Manchester for the degree of Doctor of philosophy in the Faculty of Science and Engineering 2017 Harris Sajjad Rabbani School of Chemical Engineering and Analytical Science

2 Contents List of Figures... 4 List of Tables... 9 Abstract Declaration Copyright Statement Dedication Acknowledgements Chapter 1: Introduction Wettability Aims and objectives Simplifications in the numerical model Spurious oscillations Outline of thesis Chapter 2: Suppressing viscous fingering in structured porous media Abstract Introduction Results Experiments Numerical simulations Theoretical analysis Discussion Conclusion Methods and materials References Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores Abstract Introduction Numerical model Governing equations Modelling specifications Initial and boundary conditions Spatial and temporal discretization Calculation of interface capillary pressure Results and discussions

3 3.3.1 Entry capillary pressure Comparison with analytical solutions Summary and conclusion Appendix References Chapter 4: Pore geometry control of apparent fractional wetting of a porous medium Abstract Introduction Fluid-fluid displacement in single capillaries Result and Discussion Curvature reversal of the interface in converging-diverging capillary Co-existence of concave and convex interfaces in 2D micromodels Apparent wettability number Summary and conclusion References Chapter 5: New insights on the complex dynamics of two-phase flow in porous media under intermediate-wet conditions Abstract Introduction Direct Numerical Simulation Result and Discussion Intermediate-wet porous media and interface dynamics Non-monotonic recovery of defending fluid as a function of wettability Summary and conclusions Materials and methods Simulation domain Validation of the numerical simulation References Chapter 6: Conclusions and Future perspectives Main results Implications of scientific findings in industrial applications Future perspectives 146 3

4 List of Figures Figure 1.1. Displacement patterns as a function of Ca and M observed by Lenormand et al. [5] and Lenormand [6] (after Sinha and Wang [7])...16 Figure 1.2. Shows distribution of fluids in (a) imbibition and (b) drainage displacement. Blue, red and black color represents invading, defending fluid and grains respectively.18 Figure 1.3. Sessile drop method showing different states of wettability...19 Figure 1.4. Scanning electron microscopy (SEM) images taken on a Philips XL30 ESEM in wet mode, using a Peltier cooling stage. The red circle shows oil-wet regions, while green circle indicates water-wet region (courtesy of Dr. Jim Buckman from Heriot-Watt University). 23 Figure 1.1. Showing variation in the (a) average velocity of interface directly computed from the numerical results v (-), (b) actual velocity of interface v* (-) measured as the ratio of distance travelled by the tip of fluid-fluid interface over time and (c) the error in mass balance m* as the function of Ca. v and v* are normalized with respect to injection velocity. (d) Indicating the boundedness of volume indicator function f along the length of capillary Figure 1.6. Displaying behaviour of p c against G at Ca = 10-7 and θ = 45 o. p c represents the difference in the entry capillary pressure computed from numerical simulations and Mayer-Stowe- Princen (MS-P) technique. G is the shape factor of capillary, the above results are shown for G = (square), G = (equilateral triangle) and G = (irregular triangle)..27 Figure 2.1. (a) Design of the microfluidic device. The length of the ordered medium l and the pore radius r are indicated. (b) Experimental results for Ca = and for a uniform porous medium with λ = 0 and a non-uniform medium with λ = The invading fluid is water with red dye and the displaced fluid is transparent mineral oil; the flow is from the bottom to the top...40 Figure 2.2. Simulation results demonstrating displacement fluid front morphologies for different values of capillary number Ca and the pore size gradients λ at the instance where the invading fluid reaches the outlet (the direction of displacement is from bottom to top). The colour bar represents the length travelled by the invading fluid. The viscosity of invading fluid µ 1 and defending fluid µ 2 were kept constant at 10-3 Pa s and 10-1 Pa s, respectively. The viscosity ratio of defending fluid over invading fluid Μ is 100. The results illustrate that fingering is suppressed as λ become more negative. Moreover, for λ = , and the critical capillary number Ca c at which 4

5 crossover from viscous fingering to compact displacement takes place is , and , respectively. 42 Figure 2.3. (a, b) Effects of the sign of λ on the morphology of displacement patterns in porous media. Negative (a) and positive (b) value of λ correspond to the cases when the large and small pores are placed at the injection point, respectively. In both cases, the capillary number was kept constant at Ca = and the viscosity of the invading (white) and defending (orange) fluids were constant at 10-3 Pa s and 10-1 Pa s, respectively. The direction of displacement is from bottom to top. The observed contrast in the displacement patterns indicates the substantial impact of sign of λ on the interface morphology and dynamics and the resulting macroscopic response. (c) Fractal dimension D f, (d) interface length L f normalized with respect to r i, (e) displacement efficiency E f (%) defined as the ratio of the recovered volume of the defending fluid to its initial volume and (f) normalized fingertip velocity V 44 Figure 2.4. Logarithmic plot showing relationship between generalized capillary number Ca and λ. The solid line indicates the analytically predicted Ca c, which is a solution of the stability criterion derived from linear stability analysis. The symbols are the results obtained by the direct numerical simulation where filled, half-filled and open points represent the stable front, Ca c and unstable front, respectively. Yellow and red regions mark analytically predicted stable and unstable regions separated by the solid line predicted by Eq. (2.4). The insets illustrate the displacement front morphology at Ca = computed by the direct numerical simulation Figure 2.5. Pore-scale mechanisms controlling the stability of a front at the macroscopic level. Schematic illustration of the pore-scale (a) overlap and (b) intermittent burst mechanisms influencing the dynamics of displacement. In (a) and (b), red, blue, black and green represents invading fluid, defending fluid, grains and the positions of interfaces before the respective pore-scale displacement mechanism. The direction of displacement is from bottom to top and movement of each individual interface is shown by the arrows. During the overlap mechanism (a), two neighbouring interfaces coalesce to form a stable interface (the white dashed line). In the intermittent burst mechanism, one interface advances towards downstream pores (the white dashed line), while the other interface recedes towards the upstream pore (the black dashed line). The 2D colour map shown in (c) and (d) indicates the frequency distribution of overlaps f O (%) and intermittent bursts f b (%) versus Ca and λ. (e) Evolution of the phase pressure difference δp (m -1 ) defined as p i p o r i p o at Ca = , where p i and p o correspond to the inlet and outlet (open to atmosphere) pressure, respectively 53 5

6 Figure 2S1. Schematic of the model used for the analytical analysis. White represents grains and red shows the interface between two immiscible fluids 1 and 2, where fluid 1 represents the invading fluid and fluid 2 represents the defending fluid..55 Figure 3.1. (a c) 3D views of the pores with different cross sections. (d) A side view of the simulation domain model. Invading phase (fluid 2) is displacing the receding phase (fluid 1) at a capillary number of Figure 3.2. Post-processing of simulations to calculate the capillary pressure. (a) Phase distribution with red and blue representing receding phase and invading phase respectively. All other colors indicate the transition zone. (b) The interface extracted from the transition zone corresponding to γ = 0.5 (c) Curvatures along the interface with the values represented by the color map...73 Figure 3.3. Configuration of receding phase (red) and invading phase (blue) in pore models with different wetting conditions. Each image is labeled with receding phase saturation at the junction inside the pore. G indicates the shape factor. The front view of capillary is shown with cross-section of reservoir (bottom) and pore (top). Invading phase is displacing receding phase at C a of Figure 3.4. Evolution of entry capillary pressure at meniscus entering pores with different cross sections and contact angles. X-axis represents the cross sectional saturation of the invading fluid at the pore entrance..75 Figure 3.5. Variation in interface velocity as the interface enters the pore with θ = 10 o All velocity values are scaled with respect to 1.0 x 10-4 m/s.77 Figure 3.6. Vector plot showing flow direction of fluids as advancing phase (blue) penetrates and displaces receding phase (red) in the equilateral triangle pore. The black arrow represents flow direction for fluid residing the corner 76 Figure 3.7. Schematic of water flooding to mobilize the trapped oil phase. The flow direction is from left to right. The driving force acting on imbibition interface marked blue tends to increase the pressure gradient between oil and water phase at the drainage interface. To remobilize the oil, it is vital for the driving force to overcome the capillary forces that are resisting mobilization of the trapped phase (figure adapted and modified from [33]) 80 Figure 3.8. Comparison between theoretical and simulation results for (a) 10 o, (b) 45 o and (c) 60 o contact angles. (d) Shows variation in interface shape as it advances towards the apex at different contact angle and corner angle...83 Figure 3A1. Showing Grid independence test results 85 6

7 Figure 3A2. Demonstrating variation in morphology of interface as interface moves from the reservoir to pore under different contact angles 87 Figure 3A3. (a) Representing configuration of forces under hydrostatic equilibrium (b) the viscous forces at one of the corners of angular pore can be resolved in form trapezium...88 Figure D representation of capillaries used in the presented numerical simulation Figure 4.2. (a) Movement of fluid-fluid interface in uniform and converging-diverging capillaries at different contact angles θ. Colour lines denote the position of interface at the vertical plane of symmetry of the 3D capillaries (See Fig. 3.1). In the case of θ = 45 o and 135 o, the interface shape is insensitive to the geometry of the capillary. However, at θ = 90 o the morphology of interface in the uniform capillary is significantly different from that shown in the converging-diverging capillary. In converging-diverging capillary, there is a transition in the interface curvature from concave to convex as interface moves from converging to the diverging section, whereas in the uniform capillary the interface remains flat. (b) Evolution of capillary pressure p c (Pa) as the interface moves from the converging to the diverging section of the capillary at different θ values. The non-monotonic relationship between p c and capillary radius r for various θ values illustrates the contribution of β in regulating p c trend. (c) Phase-diagram that distinguishes the (ii) pore geometry controlled apparent wetting regime (shaded by green) from (i,iii) pore geometry independent apparent wetting regime (shaded by yellow and brown). Note the circle and triangle symbols correspond to the pair of θ and β values where the interface is flat. Circle and triangle represent converging (θ = π β) and the 2 diverging (θ = π + β) section of the capillary, respectively Figure 4.3. (a) The distribution of phases influenced by the contact angle θ in the 2D micromodel. For θ = 45 o, interfaces are convex; as a result the invading fluid (non-wetting fluid) bypasses the pore throats and drains the defending fluid (wetting fluid) from relatively larger pore bodies. When θ = 135 o there is the dominance of concave interfaces (imbibition), which further illustrates that invading fluid is acting as a wetting fluid and displacement is governed by the cooperative displacement mechanism [29]. For θ = 90 o, the co-existence of concave and convex interfaces (indicated by arrows) can be observed, which has a significant impact on the recovery efficiency of defending fluid as shown in Rabbani et al. [32]. This type of condition is similar to a fractional wetting scenario. Besides, entrapment of defending fluid in pore throats shown by the red dotted circle which is typical in drainage displacement, confirms that the micromodel at θ = 90 o manifests apparently fractional wet conditions and thus capillary forces cannot be neglected in such a porous medium. Blue, yellow and black indicates defending fluid, invading fluid and grains, respectively. The displacement occurs from 7

8 bottom to top. (b) Evolution of recovery efficiency of defending fluid versus time t under different contact angle θ indicated in the legend 108 Figure 5I. Showing snap-off occurring in pore throats when wetting fluid (blue) invades the porous media and displaces the defending fluid (red) at contact angle θ = 5 o.118 Figure 5.1. (a) The main interfacial features observed during immiscible two-phase flow in intermediate-wet porous media (θ = 60 ) at 2.8 s. (b) Curvature distribution of interfaces shown in Fig. 5.1(a). (c) Dynamics of concave (labelled as 1 ) and convex (labelled as 2 ) interfaces during displacement in the porous medium with θ = 60. Pinning of convex interface and reverse displacement mechanism as a result of co-existence of concave and convex interface is observed. (d) Interface instability in a single pore. In the phase distribution shown in Fig. 5.1(a,c-d), red, blue and green represents defending fluid, invading fluid and the fluid-fluid interface, respectively. The pressure field shown in Fig. 5.1(c d) indicates the pressure values normalized with respect to the outlet pressure. The direction of injection in all images is from bottom to top Figure 5.2. (a) Fluid phase and pressure distribution under different wetting conditions at the end of simulation. White colour represents pathway of invading phase. Pressure is normalized with respect to the outlet pressure and it indicates the pressure in the defending phase. (b) Distribution of blobs size of defending fluid under different wettability scenarios. The inset illustrates the maximum blob size as a function of the contact angle. (c) The non-monotonic dependency of the defending phase recovery on the wettability of porous media Figure 5.3.Comparison between the blob-size distributions computed numerically and the ones measured by the microfluidic experiments for the fluids PMX 200 Silicone Fluid and water with water injection rate of 1.0 ml/hr Figure 5S1. The porous medium (numerical domain) used for simulation of immiscible displacement under different wettability conditions. Black and white colours represent void and solid phase, respectively. The porous medium is based on the pore-scale image obtained by X-ray tomography of a sand pack [32]

9 List of Tables Table 3.1. Fluid properties in the simulations 71 Table 3A1. Number of grid blocks per simulation. The grid block size is scaled with respect to inscribed radius of pore shape 86 Table 5S2. Physical description of porous media as given in Norouzi Rad et al.[32].136 9

10 Abstract of thesis submitted by Harris Sajjad Rabbani for the Degree of Doctor of Philosophy and entitled Pore-scale investigation of wettability effects on two-phase flow in porous media on 1st December 2017 Physics of immiscible two-phase flow in porous media is relevant for various industrial and environmental applications. Wettability defined as the relative affinity of fluids with the solid surface has a significant impact on the dynamics of immiscible displacement. Although wettability effects on the macroscopic fluid flow behaviour are well known, there is a lack of pore-scale understanding. Considering the crucial role of wettability in a diverse range of applications; this research aims to provide a pore-scale picture of interface configuration induced by variations in the wetting characteristics of porous media. Besides, this study also relates the pore-scale interfacial phenomena with the macroscopic response of fluids. Highresolution direct numerical simulations (DNS) at multiscale (single capillary and a highly heterogeneous porous media) were performed using computational fluid dynamics (CFD) approach in which the Navier-Stokes equation coupled with the volume of fluid method is solved to represent immiscible displacement. Numerical results demonstrate that at pore scale as the wettability of porous media changes from strong to intermediate wet the effects of pore geometry (that includes corner angle and orientation angle) on the interfacial dynamics also enhances. This was demonstrated by the non-monotonic behaviour of entry capillary pressure at the junction of pore, curvature reversal in the converging-diverging capillary and the coexistence of concave and convex interfaces in heterogeneous porous media with uniform contact angle distribution. In addition to simulations, theoretical argument is also presented that rationalize the underlying physics of complex, yet intriguing interfacial phenomena shown by DNS. Overall this research extends the fundamental understanding of multiphase flow in porous media and paves the way for future studies on porous media. 10

11 Declaration I declare that no portion of the work referred to in the thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. Harris Sajjad Rabbani 11

12 Copyright Statement i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the Copyright ) and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trademarks and other intellectual property (the Intellectual Property ) and any reproductions of copyright works in the thesis, for example graphs and tables ( Reproductions ), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. iv. Further information on the conditions under which disclosure, publication and commercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see 0), in any relevant Thesis restriction declarations deposited in the University Library, The University Library s regulations (see and in The University s policy on Presentation of Theses. 1 This excludes the chapters which are already published in academic journals for which the publisher owns the copyright 12

13 Dedication To the angels of my life my parents, Sajjad and Shahina, who have sacrificed a lot for me to reach this position in life. They believed and taught me to follow my dreams. And, my brother, Farhan, the best friend, who has been always supporting me. This thesis is also dedicated to my late grand father Gullam Rabbani, grand father Shaukat Malik, and grandmother Shamim Rabbani and Fahmidha Bano. 13

14 Acknowledgements Thanks to Allah for his blessing and mercy, which is new every morning, and great is his faithfulness. I would like to express my deepest gratitude and respect to my supervisor, Dr. Nima Shokri. I am grateful for his patience, enthusiasm, immense knowledge and selfless dedication which helped both my personal and academic development. I have been extremely blessed to work with such a serious yet very friendly supervisor and cannot think of a better one to have. I would also like to thank Dr Vahid J Niasar my co-supervisor for his incredible support and commitment to my work. I am grateful for his knowledge and insight. Both Nima and Vahid are legends and I sincerely appreciate their efforts and support. I am also thankful to my research group members Dr. Kofi Osei-Bonsu, Salomé Shokri- Kuehni, Abdulkadir Osman, Mina Bergstad, Michael Lacey, Mohammad Javad Shojaei, Amirhossein Hassani and Delphi Everest Dana Shokri-Kuehni. I enjoyed working with them; they have been great support for me during my Ph.D. I would also like to thank IMPRES members Dr. Nikolaos K. Karadimitriou, Rimsha Aziz and Sharul Nizam Hasan. I would like to thank my entire family. Special thanks to my aunt Neelam Malik, uncle Tariq Malik and dear friend Faisal Ishaq for taking care of me and providing home cooked food while my mother was away. I would like to thank the UK Engineering and Physical Sciences Research Council (EPSRC) for providing the funding for my PhD and University of Manchester IT Services for their assistance and the use of the high performance computing (Computational Shared Facility). I would also like to thank Dr. Jim Buckman from Heriot-Watt University for providing the SEM image. 14

15 Chapter 1 Introduction 1.1 Multiphase flow in porous media Multiphase flow in porous media is the branch of fluid dynamics that deals with the simultaneous flow of two or more fluids in pore spaces. Physics of multiphase flow through porous media is essential in several applications including fuel cells, printing technology, drug delivery, spacecraft propulsion system, oil recovery, carbon dioxide sequestration, aquifer remediation, salinization of soils and infiltration of rain into the subsurface. Depending upon the chemistry of fluids, the displacement process can either be miscible or immiscible. In comparison to miscible displacement, there exists an interface (in some texts known as meniscus) which separates fluids from one another during immiscible displacement. The influence of geometrical and chemical properties of porous media on the dynamics of interface displacement is the focus of this thesis. Porous media topology and geometry that includes pore size distribution, aspect ratio (ratio of pore body to pore throat size), pore shape and coordination number (among other factors) plays a crucial role in governing the dynamics of immiscible displacement. Al-Shalabi and Ghosh [1] performed experiments under water-wet conditions in 2-D micromodel and 3-D cores showing that decreasing average pore size increases the efficiency of oil recovery during water injection. Chatzis et al. [2] showed that in regular pore geometry the impact of pore size on the efficiency of immiscible displacement is negligible. They further concluded that the aspect ratio of porous media has a strong influence on the saturation profile of fluids such that increasing the aspect ratio intensifies the entrapment of defending fluid. This is later 15

16 Chapter 1: Introduction confirmed by pore network modelling studies of Joekar-Niasar et al. [3] where aspect ratio was kept very small and thus no entrapment of defending fluid was observed. Mason and Morrow [4] theoretically investigated the impact of pore angularity on retention of fluid in the corners. Their studies revealed that increasing angularity of pores enhances the retention of fluid and therefore can reduce the efficiency of immiscible displacement. In addition to the geometrical properties of porous media, properties of fluids such as the viscosity of invading fluid μin, the viscosity of defending fluid μdf, injection velocity U and interfacial tension σif that is defined as energy per unit area required to form an interface between two immiscible fluids are also critical parameters controlling the immiscible displacement process. Dimensionless capillary number Ca and mobility ratio M are commonly used in literature to quantify effects of the above-mentioned parameters on the immiscible displacement in porous media. The capillary number refers to the ratio of viscous to capillary forces acting at the interface and can be written as Ca = Uμ in σ if, whereas mobility ratio is defined as M = μ in μ df. Lenormand et al. [5] and Lenormand [6] used the combination of Ca and M as the diagnostic tools to distinguish among various flow regimes that may occur during displacement depending on the boundary conditions (See. Fig. 1.1). 16

17 Chapter 1: Introduction Figure 1.1 Displacement patterns as a function of Ca and M observed by Lenormand et al. [5] and Lenormand [6] (after Sinha and Wang [7]). They observed three distinctive regimes known as capillary fingering, viscous fingering, and stable regime. At very small Ca, the capillary forces (forces acting along the interface) dominate viscous forces, and the associated displacement pattern is known as capillary fingering. At high Ca, if M > 1 (viscosity of invading fluid is greater than defending fluid) then stable displacement prevails. However, if M is < 1 then finger-like protrusions form along fluid-fluid displacement front and therefore corresponding displacement regime is known as viscous fingering. Rodríguez de Castro et al. [8] investigated displacement efficiency at a wide range of capillary numbers and mobility ratios. For M = 1, they showed that displacement efficiency continuously increases with Ca. However, when M < 1 the trend was non-monotonic with highest displacement efficiency at Ca=

18 Chapter 1: Introduction The early stage of this research is focused on the fundamental understanding of the mechanisms controlling viscous fingering as a function of geometrical properties of porous media as well as the fluid properties. The key objective was to develop novel approaches to prevent or trigger viscous fingering depending on the application under consideration. One of the key parameters influencing the dynamics of fluid distribution and patterns in porous media is wettability which was extensively investigated in this research. 1.2 Wettability Wettability is another important feature of porous media that influences the distribution and flow of fluids in porous media [9-11]. It is defined as the relative affinity of fluid with the solid surface in the presence of another immiscible fluid. At the molecular scale, wettability is a result of competition between fluid cohesive and surface adhesive forces. During immiscible displacement, the fluid that wets the porous media is known as wetting phase, while the other fluid is regarded as the non-wetting phase. If invading fluid is acting as wetting phase then capillary forces assist the movement of interface and displacement process is known as imbibition. However, in cases where invading fluid is behaving as non-wetting phase, then capillary forces resist the interface movement, and the corresponding displacement process is called drainage. In porous media, wetting phase always resides in crevices, pore throats and depending upon disjoining pressure can also be located as a thin film between the surface and non-wetting fluid. In contrast, non-wetting phase tends to occupy relatively larger pore bodies. Fig. 1.2 shows a typical illustration of fluid distribution during imbibition and drainage phenomena. 18

19 Chapter 1: Introduction Figure 1.2. Shows distribution of fluids in (a) imbibition and (b) drainage displacement. Blue, red and black color represents invading, defending fluid and grains respectively. Wettability of porous media can be categorized into i) strong-wet, ii) weak-wet, iii) intermediate-wet or neutral-wet, iv) fractional-wet, v) mixed-wet. Under strong and weak wet conditions surface adheres to one of the immiscible fluids, while under intermediate-wet (neutral-wet) conditions this preferential adherence quality of the surface is absent, and therefore both fluids have almost equal affinity with the solid surface. Fractional wetting states occur when there is spatial variation in the wetting properties of porous media. Mixed wettability is a special case of fractional wettability where pore body and pore throats are wetted by different fluids [10]. Most commonly employed technique to delineate wettability of porous media is sessile drop method (shown in Fig. 1.3). 19

20 Chapter 1: Introduction Figure 1.3. Sessile drop method showing different states of wettability. In this method, a drop that is surrounded by another immiscible fluid is placed on a horizontal surface. The surface used in this technique has chemical properties similar to that of the porous media. The contact angle θ that droplet makes with the flat surface defines the wetting characteristics of porous media. The relationship between contact angle θ and surface forces can be described by Young s law [12]. For water and oil case θ is measured through the water as its density is greater than that of oil. Young's law for water-oil system can be written as; σ cos(θ) = σ of σ wf (1.1) where σ, σ of and σ wf are the interfacial tension between fluids, the surface tension between oil and solid, and surface tension between water and surface, respectively. According to Eq. (1.1), the surface exhibits strong or weak wet conditions θ < 90 o (σ of > σ wf ) or θ > 90 o (σ of < σ wf ). In the former case, the water shows affinity with the surface, while in the latter case oil preferentially adheres to the surface. Young's law further indicates that surface displays intermediate-wet (neutral-wet) conditions when θ = 90 o (σ of = σ wf ). In addition to 20

21 Chapter 1: Introduction sessile drop method, some other techniques that can be used to determine the wettability of porous media are Amott test [13] and USBM(U.S. Bureau of Mines) index. The Amott test involves spontaneous imbibition and forced displacement of oil and water. Initially, porous media is at residual oil saturation, and then oil is allowed to invade and displace water by spontaneous imbibition and forced displacement. This gives rise to I o that can be written as; I o = water volume displaced by spontaneous oil imbibition Total amount of water displaced by spontaneous oil imbibition and forced oil invasion (1.2) Similar procedure can be performed for oil displacement by water that results in I w. I w = oil volume displaced by spontaneous water imbibition Total amount of oil displaced by spontaneous water imbibition and forced waterinvasion (1.3) The Amott wettability index I is computed as I w I o, it approaches 1 for strongly water-wet porous media and -1 for strongly oil-wetting conditions. In cases where I is close to 0 the wettability conditions can either be neutral-wet, fractional-wet, or mixed-wet. Under mechanical equilibrium, there is a pressure difference between non-wetting and wetting phase. Commonly, the pressure of non-wetting phase is higher than wetting phase pressure. This pressure difference across interface is known as capillary pressure p c and can be represented by Young-Laplace equation as; p c = σk (1.4) where k is the curvature of the interface. For pores with cylindrical geometry Eq. (1.4) can be specified as; p c = 2 σ cos(θ) r (1.5) 21

22 Chapter 1: Introduction where r is the radius of capillary. As one can inspect from Eq. (1.5) that capillary pressure is an inverse function of pore size, decreasing pore size increases the capillary pressure and vice versa. Furthermore, Eq. (1.5) indicates that contact angle θ also has a significant impact on the magnitude and direction of capillary pressure. As the wettability of porous media changes from strong to intermediate-wet the strength of capillary forces (capillary pressure) decreases. The USBM wettability index is based upon area under the capillary pressure-saturation curve. The area during oil invasion, and then during water invasion is calculated, greater the area is more non-wetting the invading fluid is. 1.3 Aims and objectives In many studies the effect of wettability on the macroscopic behavior of fluids has been shown [14-20]. However, there is ambiguity in the literature, as some of these studies reveal that the recovery efficiency of defending fluid is highest under weak-wet conditions [14-15,18], while other investigations show optimum recovery efficiency when the solid surface has an equal affinity with both fluids (intermediate-wet states) [16-17]. Moreover, despite the availability of powerful imaging tools, there is a lack of pore-scale understanding of wettability impacts on dynamics of immiscible displacement. This is mainly because, experimentally it is quite difficult to control and vary the wetting properties of porous media. There are many investigations where pore network modeling approach has been employed to elucidate the control of wettability on various principles governing multiphase flow in porous media [16-19]. However, these studies are not conclusive as the pore network modeling approach involves a set of local filling rules and simplification of geometry that limits the physical understanding of processes [21]. Therefore, many open questions need to be addressed regarding the fundamental role of wettability on the dynamics of immiscible displacement at the pore scale. 22

23 Chapter 1: Introduction Motivated by the importance of wettability on defining the dynamics of multiphase flow in porous media, the specific objectives of this thesis are to delineate the pore-scale behaviour of immiscible displacement as the wetting properties of porous media varies, with a particular focus on the intermediate-wet conditions which has not received enough attention in the literature. To do so, direct numerical simulations using computational fluid dynamics (CFD) approach were performed in single capillaries as well as in natural porous media. In comparison to pore network modeling approach, direct numerical simulation can provide more detailed understanding of interfacial phenomena under complex boundary conditions [22-25]. The simulations were conducted using OpenFOAM (version 2.30 and 3.01). The findings of this research enhance the understanding of wetting dynamics in porous media, which is essential to optimize various applications occurring in geological and engineered porous media. The knowledge gained from this research can assist in building a more robust and reliable continuum-scale models. 1.4 Simplifications in the numerical model The objective of this research is primarily to enhance the knowledge of fundamental principles governing the two-phase flow in porous media. Therefore, to highlight the silent features of interface dynamics reported in this research simplifications were introduced in the simulated CFD pore-scale models. The simplifications reduce the complexity of problem while preserving the adequate physics and the quality information produced by CFD simulations. One of the key simplifications introduced in the boundary condition was assigning constant θ value to capillaries and 2D micromodels. However, in real applications occurrence of uniform wetting conditions is quite rare. Figure 1.4 clearly points out that oil-wet and waterwet surfaces can co-exist within the single pore in geological reservoirs. Nevertheless, in 23

24 Chapter 1: Introduction some industrial applications involving paper drying and fuel cells, the wetting conditions are kept uniform. Figure 1.4. Scanning electron microscopy (SEM) images taken on a Philips XL30 ESEM in wet mode, using a Peltier cooling stage. The red circle shows oil-wet regions, while green circle indicates water-wet region (courtesy of Dr. Jim Buckman from Heriot-Watt University). Contact angle hysteresis is a well-known phenomenon that occurs during multiphase flow in porous media, and has been extensively studied before [26]. The simulations reported in this study ignores the influence of contact angle hysteresis on the immiscible displacement, which was made necessary in order to highlight the synergistic effect of pore angularity and contact angle on interface dynamics explained in chapter 3,4 and 5. The capillary models shown in chapter 3 represents idealized pore spaces. However, the real pores spaces have quite complex geometry; exactly modeling them is a challenging task. According to Mason and Morrow [4] pores with same shape factor G defined as A where A P2 is the cross-sectional area, and P is the perimeter demonstrate the approximately similarly 24

25 Chapter 1: Introduction capillary behavior. As a result, one can idealize the pore space with a polygonal shape having G similar to that of real pore geometry. Although, the flow domains shown in chapter 3 are an idealized representation of natural pores, they are found in microfluidics devices and have been investigated in many previous studies [27-28]. Chapter 4 discusses the interface dynamics in a converging-diverging capillary, almost similar pore configuration can be found in natural porous media as shown in Fig Spurious oscillations The interface tracking approach employed in this study is the volume of fluid (VOF) [29], and capillary forces are computed with continuum surface (CSF) method [30]. It is well known that at a very small Ca value (capillary dominated flow regime) CFS method causes generation of artificial velocities at the interface which are known as spurious or parasitic currents. These spurious currents manifest in the form of nonphysical oscillations at the interface. In order to overcome this problem many different variants of CSF methods have been proposed which include semi-sharp surface model (SSF) and filtered surface force FSF [31]. However, in this research CSF is used because it is easily available and despite the presence of spurious currents can capture the distribution of phases accurately (which is the main focus of this thesis). Ferrari and Lunati [27] used CSF method to investigate inertia effects in a single capillary channel. Later Ferrari et al. [32] implemented CSF approach to model immiscible displacement in highly heterogeneous and uniform porous media and found good agreement between simulation and experiment results. In this section spurious currents are analysed, it will be shown that spurious currents are only limited at the interface and therefore do not affect the global dynamics of problem. 2D numerical simulations are performed in square capillary under drainage conditions. The invading fluid (viscosity μ in = 10-3 Pa.s, density ρ in = 1000 kg/m 3 ) is allowed to displace 25

26 Chapter 1: Introduction defending fluid (viscosity μ df = 10-3 Pa.s, density ρ df = 1000 kg/m 3 and interfacial tension σ = 0.07 N/m) at constant flow rate Q. The fluid-fluid interface makes an angle of contact θ with solid surface equal to 45 o. The numerical simulations are performed at three different Ca values 10-4, 10-5 and 10-6, the results are shown in Figure 1.5. Figure 1.5. Showing variation in the (a) average velocity of interface directly computed from the numerical results v (-), (b) actual velocity of interface v* (-) measured as the ratio of distance travelled by the tip of fluid-fluid interface over time and (c) the error in mass balance m* as the function of Ca. v and v* are normalized with respect to injection velocity. (d) Indicating the boundedness of volume indicator function f along the length of capillary. Fig. 1.5(a) indicates variations in the average velocity of the interface directly computed from numerical results v (-) (normalized with respect to injection velocity) against Ca. It can be 26

27 Chapter 1: Introduction seen form Fig. 1.5 (a) that v (-) increases as Ca decreases which is the result of spurious currents. Nevertheless one can depict from Fig. 1.5 (a) that spurious currents intensify only at Ca smaller than 10-5 as v (-) remains almost constant from Ca= 10-4 to 10-5, and then rapidly shoots up. Alternatively, the velocity of the interface can be calculated by dividing the distance travelled by the interface with time it takes to cover the distance, which is referred to as actual velocity v* (-) (normalized with respect to injection velocity). Fig. 1.5(b) shows that v* (-) remains almost equal to 1 irrespective of Ca which is consistent with conservation of mass principle, clarifying that the artificial velocities (spurious currents) present at the interface do not impact the actual velocity of interface v* (-). This method has been used to compute velocity in Fig. 2.3(f) of chapter 2. Furthermore, error in mass balance m* is computed which represents the difference between the mass of invading fluid entering the capillary and mass of defending fluid withdrawn from the capillary. One can clearly see that regardless of spurious currents numerical simulations do not violate the principle of conservation of mass as the order of magnitude of error is significantly smaller than the tolerance value of The VOF interface capturing approach utilizes volume indicator function f to distinguish between invading fluid, defending fluid and interface. In this case invading fluid is represented by f = 0, defending fluid f = 1 and interface is characterised by intermediate values (0<f>1). The key requirement to get accurate phase distribution and interface shape is the boundedness of f in between 0 and 1. Fig. 1.5(d) portrays boundedness of f at all simulated Ca, advocating the inability of spurious currents to deform the fluid-fluid interface shape and therefore impacting the capillary behaviour in porous media. To further confirm that presence of artificial oscillations at the interface (spurious currents) does not play a role in controlling the interface dynamics Fig. 1.6 is presented. 27

28 Chapter 1: Introduction Figure 1.6. Displaying behaviour of p c against G at Ca = 10-7 and θ = 45 o. p c represents the difference in the entry capillary pressure computed from numerical simulations and Mayer-Stowe- Princen (MS-P) technique. G is the shape factor of capillary, the above results are shown for G = (square), G = (equilateral triangle) and G = (irregular triangle). Fig. 1.6 delineates the variation in p c which signifies the difference in the entry capillary pressure computed from numerical simulations and Mayer-Stowe-Princen (MS-P) [33-34] technique against G (shape factor of capillary) at Ca = 10-7 (spurious currents are present at the interface). The details on the MS-P equations employed for estimation of entry capillary pressure are given in chapter 3. The entry capillary pressure from numerical results was obtained by extracting the contour map (interface shape) of f = 0.5 and then implementing ParaView (software use to visualize the numerical results) filter to calculate the interface curvature k which was then used as an input parameter in Eq. (1.4) to get the entry capillary pressure. Fig. 1.6 represents non-monotonic behaviour of p c with respect to G. Close 28

29 Chapter 1: Introduction inspection of Fig. 1.6 specifies that p c is well below the error generated by spurious currents (which is at an order of 90%) (Fig. 1.5 (a)). Consequently, it can be said that difference between capillary pressure obtained from numerical simulations and MS-P approach is due to inherent physics which MS-P is not able to capture. For example, simulations were performed under dynamic conditions where viscous effects are important while MS-P approach is limited to static conditions. In some of the simulations shown in this thesis (excluding the ones performed at Ca 10-5 ) spurious currents are present. However as Figs. 1.5 and 1.6 confirm that spurious currents do not influence the interface dynamics, and therefore findings reported in this thesis reveals the true interfacial physics. 1.6 Outline of thesis The rest of thesis is laid out as follows: In chapter 2 comprehensive analyses were conducted to extend the fundamental understanding of the mechanisms affecting the dynamics of twophase flow in porous media. In particular, the focus was on viscous fingering and how to prevent or trigger it during immiscible flow in porous media. This chapter highlights the importance of pore geometry in controlling the displacement patterns. The research presented in subsequent chapters will demonstrate the synergistic effect of pore geometry and wettability on governing the interfacial phenomena in porous media. Chapter 3 delineates the effect of wettability on entry capillary pressure in angular pores. In chapter 4 the role of pore geometry in controlling the apparent wettability of porous media is illustrated. Detail analysis of interface configuration in intermediate-wet porous media and their impact on the macroscopic behavior of fluids is discussed in chapter 5. The final chapter, chapter 6, is a conclusion of the thesis. This thesis has been prepared in alternative format and contains 4 29

30 Chapter 1: Introduction journal papers (chapters 3 and 5 have been already published). Chapters 2 is in review and chapter 4 will be submitted for publication. 30

31 Chapter 1: Introduction References [1] Al-Shalabi, E.W. and Ghosh, B., 2016.Effect of pore-scale heterogeneity and capillaryviscous fingering on commingled waterflood oil recovery in stratified porous media. Journal of Petroleum Engineering, [2]Chatzis, I., Morrow, N.R. and Lim, H.T., Magnitude and detailed structure of residual oil saturation. Society of Petroleum Engineers Journal, 23(02), pp [3] JoekarNiasar, V., Hassanizadeh, S.M., Pyrak Nolte, L.J. and Berentsen, C., 2009.Simulating drainage and imbibition experiments in a high porosity micromodel using an unstructured pore network model. Water resources research, 45(2). [4] Mason, G. and Morrow, N.R., Capillary behavior of a perfectly wetting liquid in irregular triangular tubes. Journal of Colloid and Interface Science, 141(1), pp [5] Lenormand, R., Touboul, E. and Zarcone, C., Numerical models and experiments on immiscible displacements in porous media. Journal of fluid mechanics, 189, pp [6] Lenormand, R., Liquids in porous media. Journal of Physics: Condensed Matter, 2(S), p.sa79. [7] Sinha, P.K. and Wang, C.Y., Pore-network modeling of liquid water transport in gas diffusion layer of a polymer electrolyte fuel cell. ElectrochimicaActa, 52(28), pp [8] Rodríguez de Castro, A., Shokri, N., Karadimitriou, N., Oostrom, M. and Joekar Niasar, V., 2015.Experimental study on nonmonotonicity of Capillary Desaturation Curves in a 2 D pore network. Water Resources Research, 51(10), pp

32 Chapter 1: Introduction [9] Anderson, W.G., Wettability literature survey-part 6: the effects of wettability on waterflooding. Journal of Petroleum Technology, 39(12), pp [10] Donaldson, E.C. and Alam, W., Wettability.Elsevier. [11] Al-Dhahli, A.R., Geiger, S. and van Dijke, M.I., Three-phase pore-network modeling for reservoirs with arbitrary wettability. SPE journal, 18(02), pp [12] Young, T., An essay on the cohesion of fluids. Philosophical Transactions of the Royal Society of London, 95, pp [13] Amott, E., Observations relating to the wettability of porous rock. [14] Jadhunandan, P.P. and Morrow, N.R., Effect of wettability on waterflood recovery for crude-oil/brine/rock systems. SPE reservoir engineering, 10(01), pp [15] Zhao, B., MacMinn, C.W. and Juanes, R., Wettability control on multiphase flow in patterned microfluidics. Proceedings of the National Academy of Sciences, 113(37), pp [16] Blunt, M.J., Physically-based network modeling of multiphase flow in intermediate-wet porous media. Journal of Petroleum Science and Engineering, 20(3), pp [17] Zhao, X., Blunt, M.J. and Yao, J., Pore-scale modeling: Effects of wettability on waterflood oil recovery. Journal of Petroleum Science and Engineering, 71(3), pp [18] Ryazanov, A.V., Sorbie, K.S. and Van Dijke, M.I.J., Structure of residual oil as a function of wettability using pore-network modelling. Advances in Water Resources, 63, pp

33 Chapter 1: Introduction [19] Suicmez, V.S., Piri, M. and Blunt, M.J., Effects of wettability and pore-level displacement on hydrocarbon trapping. Advances in water resources, 31(3), pp [20] Morrow, N.R., Wettability and its effect on oil recovery. Journal of Petroleum Technology, 42(12), pp [21] Joekar-Niasar, V. and Hassanizadeh, S.M., Analysis of fundamentals of two-phase flow in porous media using dynamic pore-network models: A review. Critical reviews in environmental science and technology, 42(18), pp [22] Zaretskiy, Y., Geiger, S. and Sorbie, K., Direct numerical simulation of pore-scale reactive transport: applications to wettability alteration during two-phase flow. International Journal of Oil, Gas and Coal Technology, 5(2-3), pp [23] Ferrari, A., Jimenez Martinez, J., Borgne, T.L., Méheust, Y. and Lunati, I., Challenges in modeling unstable two phase flow experiments in porous micromodels. Water Resources Research, 51(3), pp [24] Rabbani, H.S., Joekar-Niasar, V. and Shokri, N., Effects of intermediate wettability on entry capillary pressure in angular pores. Journal of colloid and interface science, 473, pp [25] Rabbani, H.S., Joekar-Niasar, V., Pak, T. and Shokri, N., New insights on the complex dynamics of two-phase flow in porous media under intermediate-wet conditions. Scientific Reports, 7(1), p [26] Ma, S., Mason, G. and Morrow, N.R., Effect of contact angle on drainage and imbibition in regular polygonal tubes. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 117(3), pp

34 Chapter 1: Introduction [27] Ferrari, A. and Lunati, I., Inertial effects during irreversible meniscus reconfiguration in angular pores. Advances in water resources, 74, pp [28] Martic, G., Gentner, F., Seveno, D., Coulon, D., De Coninck, J. and Blake, T.D., A molecular dynamics simulation of capillary imbibition. Langmuir, 18(21), pp [29] Hirt, C.W. and Nichols, B.D., Volume of fluid (VOF) method for the dynamics of free boundaries. Journal of computational physics, 39(1), pp [30] Brackbill, J.U., Kothe, D.B. and Zemach, C., A continuum method for modeling surface tension. Journal of computational physics, 100(2), pp [31] Raeini, A.Q., Blunt, M.J. and Bijeljic, B., Modelling two-phase flow in porous media at the pore scale using the volume-of-fluid method. Journal of Computational Physics, 231(17), pp [32] Ferrari, A., Jimenez Martinez, J., Borgne, T.L., Méheust, Y. and Lunati, I., Challenges in modeling unstable two phase flow experiments in porous micromodels. Water Resources Research, 51(3), pp [33] Mayer, R.P. and Stowe, R.A., Mercury porosimetry breakthrough pressure for penetration between packed spheres. Journal of colloid Science, 20(8), pp [34] Princen, H.M., Capillary phenomena in assemblies of parallel cylinders: II. Capillary rise in systems with more than two cylinders. Journal of Colloid and Interface Science, 30(3), pp

35 Chapter 2 Suppressing viscous fingering in structured porous media This chapter has been submitted and is currently under review. This research presents a novel strategy to control viscous fingering in engineered porous media. The theory was first validated by experiments, and later simulations were performed to elaborate the theory further. High resolution, 2D direct numerical simulations, and experiments were performed on ordered porous media which refers to porous media with continuous variation in pore size. The simulations were conducted using OpenFOAM (version (3.01)) at various pore size gradient, λ, and capillary number, Ca. During experiments and simulations, the viscosity of defending fluids was kept two orders of magnitude greater than the viscosity of invading fluids, therefore representing viscous fingering regime. In addition, theoretical analysis has also been performed to derive a generalized capillary number Ca and stability criterion capable of classifying displacement patterns. The results presented in this chapter suggest that continuous decrease in pore size along the flow direction can alter the invasion behaviour from unstable (viscous fingering) to stable (compact displacement). In order to stabilize the displacement pattern at higher flow rates, pore size gradient has to be increased. Theoretical analysis provides the physical explanation for such a modification in the morphology of displacement front, indicating that reducing pore size intensifies viscous dissipation which promotes stability in the displacement patterns. In the opposite case, where pore size is increasing along the flow direction, the viscous fingering prevails. 35

36 Chapter 2: Suppressing viscous fingering in structured porous media Suppressing viscous fingering in structured porous media Harris Sajjad Rabbani 1, Dani Or 2, Y. Liu 3, C.-Y. Lai 3, N. Lu 4, S.S. Datta 4, H.A.Stone 3 and Nima Shokri 1 (2017) (in review) 1 School of Chemical Engineering and Analytical Science, The University of Manchester, Manchester, M13 9PL, United Kingdom 2 Soil and Terrestrial Environmental Physics, Department of Environmental Sciences, ETH Zurich, Zurich, Switzerland 3 Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA 4 Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ 08544, USA 36

37 Chapter 2: Suppressing viscous fingering in structured porous media Abstract Finger-like protrusions that form along fluid-fluid displacement fronts in porous media are often excited by hydrodynamic instability when low viscosity fluids displace high viscosity resident fluids. Such interfacial instabilities are undesirable in many natural and engineered displacement processes. We report a phenomenon whereby gradual variation of pores sizes along the front path suppresses viscous fingering, that stands in contrast with conventional expectation of enhanced instability with pore variability. In addition to direct measurements and pore-scale numerical simulations, we developed an analytical model for displacement front morphology as a function of pore size gradient. Our results suggest that a smooth reduction of pore sizes restrains viscous fingering as anticipated also by gradient percolation theory. The resulting insights permit a new degree of control over interfacial instability in porous media based on the target application, and may offer new ways for controlled immiscible fluid displacement in engineered porous media such as fabric, paper and membranes. 37

38 Chapter 2: Suppressing viscous fingering in structured porous media 2.1 Introduction The unstable growth of fluid-fluid interfacial perturbations has been a subject of considerable interest. From fuel cells [1], chromatographic separation of solvents [2], to glass ceramics [3], the infiltration of water into soils [4], oil recovery from underground reservoirs [5-6], and carbon dioxide sequestration [7-8] to list a few. Hill [9] and Saffman and Taylor [10] were the first to quantify the highly ramified morphology of an interface resulting from displacement of viscous fluid by a fluid of lower viscosity and so document the emergence of finger-like invasion patterns (viscous fingering). Hill 9 investigated the process using a packed bed, while Saffman and Taylor [10] employed fluid filled Hele-Shaw cells to study viscous fingering. An excellent review on viscous fingering is provided by Homsy [11]. Although the fundamental principles governing interfacial instability are relatively well understood, their manifestation in porous media with rich morphologies of displacement fronts remain an active field of research. Fluid viscous fingering in porous media is of concern for various applications. In oil recovery from geologic reservoirs, viscous fingering can result in early breakthrough of the invading fluid (often water or brine) thus diminishing the efficiency of oil recovery and at times rendering it uneconomical [5-6]. In environmental applications, viscous fingering can potentially result in the early arrival of pollutants to underlying groundwater resources. The technological challenges presented by viscous fingering have prompted numerous theoretical and experimental studies [12-20]. Some of the studies have shown that the use of non- Newtonian fluids [13] or nonlinear control of injection rate [17] stabilize the fluid-fluid interface. Other studies[14-16] suggest that alteration of wetting properties of the porous medium offer a potential remedy for eliminating viscous fingering. However, for many 38

39 Chapter 2: Suppressing viscous fingering in structured porous media applications the alternation of wetting characteristics of the porous medium is not trivial; hence other solutions must be developed to control viscous fingering. The theory of percolation under a gradient introduced by Wilkinson [21] has been used to describe displacement patterns in porous media [22-24]. The classical work of Xu et al. [23] combines gradient percolation with conventional invasion percolation to derive the known classic phase diagram of fluid front stability in random porous media of Lenormand et al. [25]. Yortsos et al. [24] extended the approach to model flow profiles in spatially correlated pore networks. In the studies above, percolation gradient was introduced in the form of externally applied pressure drop using a set of local pore filling rules. Inspired by the concept of gradient percolation, we demonstrate the influence of regular pore size variations in the porous medium as a new means for suppressing the growth of viscous fingers. Such a statement may appear counter intuitive at a first glance, because the literature considers pore size variations as a factor that enhances fluid front tip-splitting frequency and thus intensifies the fingering phenomenon [15]. We report a novel paradigm of percolation gradient in the form of ordered porous media in which pore sizes vary monotonically along the direction of flow. From the physical point of view, such ordered porous media is intriguing as it allows simultaneous control over viscous and the capillary forces in the same direction, which is rare in random porous media and has not been studied before. Yortsos et al. [24] conjectured that gradient of pore network correlation could act to trigger or suppress viscous fingering, yet to date, no experimental or theoretical evidence supports this conjecture. Although recent studies [18-19] have indicated that gradual variation in the thickness of Hele-Shaw cell can significantly restrain viscous fingering, no study have shown how pore size gradient would affect viscous fingers in porous media (which is a far more complicated system than a Hele- Shaw cell). 39

40 Chapter 2: Suppressing viscous fingering in structured porous media In this study we combined experiments, numerical simulations and theoretical analysis to demonstrate how a porous medium with ordered pore sizes controls (triggers or suppresses) viscous fingering during immiscible flow in porous media. We begin with porous media (uniform and ordered) saturated with high viscosity defending fluid where low viscosity invading fluid displaces the defending fluid at various flow rates. Our results show that fluid fronts traversing a porous medium where the pore size is gradually reduced along the flow direction results in a velocity-dependent morphological transformation of the front from unstable to stable. Moreover, we show that stabilization of the invasion front at high injection rates requires an increase in the pore size gradient along the flow path. These results provide a means for inhibiting or triggering viscous fingering and interfacial instability in engineered porous materials. The insights gained from this study pave the way to new design of chromatographic columns, membranes, sensors and other porous media such that displacement front morphology is unconditionally stable (under prescribed operations conditions), and improve fundamental understanding of viscous fingering in porous media. 2.2 Results Experiments We conductedfluid-displacement experiments in micro-models of ordered porous media (see Fig. 2.1 (a)). The microfluidic device consists of pillar arrays made of PDMS with height H = 160 μm and variable pillar diameter spanning the width of the ordered region W= 30 mm. The pillar diameters and pores were ordered along the direction of the flow with a pore size gradient λ = r o r i, where 2r l o = 520µm is the pore width at the outlet, 2r i = 880 µm is the pore width at the inlet and l = 2.95 mm is the length of ordered region. We denote by λ < 0 a reduction in pore size along the flow path and vice versa for λ > 0. We first filled the device with mineral oil (viscosity µ 2 = 200 mpa s), and then injected water (coloured with red dye, 40

41 Chapter 2: Suppressing viscous fingering in structured porous media viscosity µ 1 = 1 mpa s) at a constant flow rate Q to displace the mineral oil. The capillary number is defined as Ca = µ 1U σ where U = Q/(HW) is the area-averaged (Darcy) velocity and σ = 28.2 mn/m is the interfacial tension between the two fluids. The experimental results for two different capillary numbers Ca = and are shown in Fig. 2.1(b). The displacement is unstable at both Ca for a uniform porous medium (λ = 0), as expected according to the traditional viscous fingering criterion. However, for a non-uniform porous medium (λ = ), stable displacement is achieved at the lower Ca, and becomes unstable at higher Ca. These results suggest that there is a capillary number range where the reduction in pore size along the flow direction can stabilize the displacement of a fluid with higher viscosity during injection of a fluid with lower viscosity. We next turn to numerical simulations to develop more understanding of the macroscopic and microscopic features of these observations. Figure 2.1. (a) Design of the microfluidic device. The length of the ordered medium l and the pore radius r are indicated. (b) Experimental results for Ca = and for a uniform porous 41

42 Chapter 2: Suppressing viscous fingering in structured porous media medium with λ = 0 and a non-uniform medium with λ = The invading fluid is water with red dye and the displaced fluid is transparent mineral oil; the flow is from the bottom to the top Numerical simulations Direct numerical simulation (DNS) has emerged as a powerful tool for diagnosing pore-scale multiphase flow problems with complex boundary conditions [26-29] enabling parameterization of macroscopic quantities [30]. In the present study, we utilized DNS within a computational fluid dynamics (CFD) framework to investigate how the proposed pore size arrangement influences the general dynamics of two phase-flow in porous media and stability of the front. Additional details regarding the numerical algorithm employed in this study is provided in Rabbani et al. [29] and thus not repeated here. For the two-dimensional simulations, we assumed an invading fluid of viscosity µ 1 = 10-3 Pa s, displacing an immiscible fluid (defending fluid) of viscosity µ 2 = 10-1 Pa s. The resulting viscosity ratio of defending fluid with respect to invading fluid was Μ = 100. The contact angle θ between interface and the solid surface measured along the defending fluid was kept uniform at 30 o (i.e., the defending fluid acts as the wetting phase). The values of l and r i were kept constant at 8 mm and 0.17 mm, respectively while r o was allowed to vary based on simulated λ value. The simulations were performed at several capillary numbers Ca ranging from to and the pore size gradients λ ranging from to , respectively. Fig. 2.2 shows the morphology of displacement patterns obtained from the numerical simulations at different values of Ca and λ. 42

43 Chapter 2: Suppressing viscous fingering in structured porous media Figure 2.2. Simulation results demonstrating displacement fluid front morphologies for different values of capillary number Ca and the pore size gradients λ at the instance where the invading fluid reaches the outlet (the direction of displacement is from bottom to top). The colour bar represents the length travelled by the invading fluid. The viscosity of invading fluid µ 1 and defending fluid µ 2 were kept constant at 10-3 Pa s and 10-1 Pa s, respectively. The viscosity ratio of defending fluid over invading fluid Μ is 100. The results illustrate that fingering is suppressed as λ become more negative. Moreover, for λ = , and the critical capillary number Ca c at which crossover from viscous fingering to compact displacement takes place is , and , respectively. The morphology of the invading fluid-fluid interface reflects the combined effects of Ca and λ, as shown in Fig Results show that at high displacement rates with Ca = no stabilizing effect of negative λ on viscous fingering (for the range of λvalues considered) 43

44 Chapter 2: Suppressing viscous fingering in structured porous media resulting in almost similar displacement patterns for all. However, at lower capillary numbers, there exists a critical capillary number (Ca c ) that indicates a transition in front propagation from viscous fingering (VF) to compact displacement (CO) for a prescribed value of the pore size gradient λ. The simulation results in Fig. 2.2 suggest that λ is related to Ca c approximately as Ca c ~λ 2 (we will provide more analysis on this next). The experiments and simulations shown in Figs.2.1 and 2.2, respectively, highlight the interplay of capillary and viscous forces on multiphase flow displacement in porous media and the role of pore size gradient that may affect both forces simultaneously and the resulting front morphology. To provide additional insights, we performed simulations where all properties are kept constant except changing the sign of λ (reversing the direction of fluid injection). A typical example in Fig. 2.3(a) and Fig. 2.3(b) depicts displacement front patterns in porous media with λ = and λ = , respectively. Based on its definition, positive and negative values of λ correspond to cases where either smaller or larger pores are present at the injection plane, respectively. Inspection of the patterns in Figs. 2.3(a) and 2.3(b) illustrates the dramatic effect of displacement front flow direction with respect to the pore size gradient. Although the pore size distribution, porosity, wetting, and fluid properties were identical in both cases, Figs. 2.3(a) and 2.3(b) show that the gradient in pore size relative to front flow direction resulted in vastly different displacement patterns. For the scenario where λ= with continuously increasing pore sizes along the direction of flow, viscous fingering is accentuated as the lower viscosity fluid preferentially flows through the least resistant pathway. In contrast, when flow direction is reversed and λ = the fluid-fluid interface becomes more stable (compact) and front spans the entire width of the domain. For certain range of negative λ, we observe a highly ramified front with 44

45 Chapter 2: Suppressing viscous fingering in structured porous media local short-fingers of the order of pore sizes (an example is presented in Fig. 2.3(a)) which is referred to as micro-fingering in this study. Figure 2.3. (a, b) Effects of the sign of λ on the morphology of displacement patterns in porous media. Negative (a) and positive (b) value of λ correspond to the cases when the large and small pores are placed at the injection point, respectively. In both cases, the capillary number was kept constant at Ca = and the viscosity of the invading (white) and defending (orange) fluids were constant at 10-3 Pa s and 10-1 Pa s, respectively. The direction of displacement is from bottom to top. The observed contrast in the displacement patterns indicates the substantial impact of sign of λ on the interface morphology and dynamics and the resulting macroscopic response. (c) Fractal dimension D f, (d) interface length L f normalized with respect to r i, (e) displacement efficiency E f (%) defined as the ratio of the recovered volume of the defending fluid to its initial volume and (f) normalized fingertip velocity V. 45

46 Chapter 2: Suppressing viscous fingering in structured porous media To systematically quantify front behaviours observed in Fig. 2.2, we computed four metrics aimed to characterise front displacement patterns as functions of prescribed Ca and λ: I) front fractal dimension Df, which measures the interface roughness, II) the fluid-fluid interface length Lf scaling the length of the interface between invading and defending fluids normalized with respect to r i, III) displacement efficiency Ef (%) defined as the ratio of the recovered volume of the defending fluid relative to its initial volume and IV) normalized finger-tip velocity V. The results are presented in Fig. 2.3 (c-f). The fractal dimension was computed using the box-counting method following Shokri et al. [31]. To calculate Lf and Ef, each image shown in Fig. 2.2 was segmented in ImageJ [32] and used to calculate these parameters. The finger-tip velocity was directly measured as the ratio of the maximum distance travelled by the front over time, and then normalized with respect to the injection velocity. As shown qualitatively in Fig. 2.3, all metrics corresponding to Ca = remained insensitive to λ for λ < 0, while for other values of Ca, the pore size gradient λ < 0 exerted a significant impact on each of the metrics presented in Fig. 2.3(c-f). A closer inspection of the results displayed in Figs. 2.2 and Fig.2.3 reveals that the maximum value of Df as a function of λ corresponds to the case when a transition from a stable to an unstable displacement pattern is observed. This maximum value is used to classify the observed patterns as stable or unstable with the corresponding critical capillary number denoted Ca c Theoretical analysis Our experimental and numerical results confirm that a prescribed gradient in pore sizes (λ < 0 ) along the fluid displacement flow direction can significantly affect the onset of viscous fingering. In this section we develop an analytical generalized capillary number Ca 46

47 Chapter 2: Suppressing viscous fingering in structured porous media that incorporates the contribution of the pore size gradient λ, and quantifies the relative importance of viscous and capillary forces. Furthermore, using linear stability analysis, we derived an analytical solution for the conditions that separate stable and unstable displacement patterns. Such stability criterion can aid in designing of structured porous media to achieve desired result (fingering or stability depending on the application) under given boundary conditions. Schematic of the porous medium used to formulate theoretical framework is presented in the supplementary information (Fig. 2S1). Note that while inertia was considered in the conducted direct numerical simulation using CFD, following Saffman and Taylor [10] and Al-Housseiny et al.[18], in the analytical analysis presented below, effects of inertia on two-phase displacement were neglected (justified by the relatively slow flows in porous materials). It should be noted that the gravitational force is included in the theoretical analysis. The generalized capillary number Ca was derived by applying the force balance as the interface moves from position a to position b (Fig. 2S1). The force balance is expressed as τ va + + τ vb + τ g = τ ca + τ cb (the subscription a and b corresponds to the position a and b) where τ v is viscous forces per unit interfacial length, τ g is the gravitational force that drives the interface forward and τ c is the local resistive capillary forces per unit interfacial length. Viscous forces over a characteristic length l x can be evaluated as τ v μvl x r(x) 2 while capillary forces are described by Young-Laplace equation as τ c 2σ cos(θ) r(x) and gravitational forces τ g (ρ 1 ρ 2 )(l l a )g. Expressing v = U ω (with ω being the porosity), ρ 1 ρ 2 = ρ and implementing the boundary conditions; l a = 0, l b = l, r 1 = r i and r 2 = r o enabled us to develop an equation for a generalized capillary number Ca that quantifies the relative importance of each force. 47

48 Chapter 2: Suppressing viscous fingering in structured porous media The new generalized capillary number Ca is defined as Ca = ( µ 1 Ul + 2σr 2 o cos(θ)ω ρgl ) [ r ir o ]. Substituting µ 1U and ρgr ir o by the capillary number 2σ cos(θ) r i + r o σ 2σcos (θ) Ca and bond number Bo (ratio of gravitational to capillary forces) in the derived expression for Ca and using r o = r i + λl gives rise to: Ca = Ca r i l cos(θ)ω (4r i 2 + 6r i λl + 2λ 2 l 2 ) + Bol 2r i + λl (2.1) Eq. (2.1) represents the generalized capillary number Ca in the form of conventional capillary number Ca and bond number Bo. Furthermore, using linear stability analysis, we further derived an analytical solution capable of distinguishing between the stable and unstable displacement patterns that takes into account the gradient of pore size λ along the flow direction (among other parameters). The analytical solution predicts the critical value of the generalized capillary number denoted Ca, above which the displacement pattern becomes unstable. The approach we adopted to c derive the stability criterion is similar to that described by Saffman and Taylor [10]. However, we have modified the dynamic boundary conditions to include the effect of λ on viscous fingering. The governing equations in our case are Darcy s law u i = k i μ i (p i + ρ i gx) for i = 1, 2 and continuity equation. u i = 0 for an incompressible flow, where u i, k i (m 2 ), μ i and p i (pa) are velocity, permeability, viscosity and average pressure of fluid i. The subscripts 1 and 2 refer to the invading and defending fluid, respectively. Darcy law can further be written in terms of flow potential φ i = k i μ i (p i + ρ i gx) and substituted into the continuity equation to form Laplace equation for each fluid flow potential 2 φ i = 0. Saffman and Taylor 10 obtained φ 1 = Uh e αt+iγy + αh eγx e αt+iγy and φ γ 2 = Uh e αt+iγy 48

49 Chapter 2: Suppressing viscous fingering in structured porous media αh e γx e αt+iγy as the solution to the fluid flow potential equation. Unlike the case γ investigated by Saffman and Taylor 10 where p 1 = p 2, here we state that the pressure difference across the interface (the dynamic boundary condition) is governed by the Young- Laplace equation which introduces the capillary pressure across the interface. Thus: [Uh e αt+iγy αh γ eαt+iγy ] µ 2 k 2 [Uh e αt+iγy + αh γ eαt+iγy ] µ 1 k 1 ρgh e αt+iγy = 2σcos (θ) r(x) (2.2) where α is the growth rate, γ is the wavenumber, h e αt+iγy is the position of perturbed displacement front relative to the base state and x = x Ut. In contrast to Hele-Shaw cell, k i in porous media is a function of both pore geometry and fluids saturation. Therefore, k i can be computed as k i = dk i dr (dr) dx, expressing dk i dx dr = 2r(x), dr dx = ωλ n and dx = h e αt+iγy results in k i = 2ωλr(x)h e αt+iγy n, where 2ωλr(x) and h e αt+iγy n quantifies the effects of porous media geometry and the fluid saturation (which varies with position of the front) on k i, respectively. Furthermore, the stability analysis has been performed around the region where k 1 = k 2, this results in: α γ [Μ + 1] = ω4σ cos(θ)λ nuµ 1 [1 Μ] + 2ωλ ρr(x)gh e αt+iγy (2.3) nuµ 1 Eq. (2.3) indicates that for stable regime ω4σ cos(θ)λ nuµ 1 [1 Μ] + 2ωλ ρr(x)gh e αt+iγy < 0, nuµ 1 while fingering persists if ω4σ cos(θ)λ nuµ 1 [1 Μ] + 2ωλ ρr(x)gh e αt+iγy > 0. Note that α > 0 nuµ 1 corresponds to unstable patterns and α < 0 indicates the stable displacement. Therefore, to determine the stability criterion, one can solve Eq. (2.3) for α = 0 and combine the result with the definition of Ca. This result in an analytical expression for the critical value of the generalized capillary number denoted as Ca c which can be used as a diagnostic tool to classify displacement patterns in porous media: 49

50 Chapter 2: Suppressing viscous fingering in structured porous media Ca c = 2λr i l n(2r i 2 + 3r i λl+ λ 2 l 2 )[1 Μ G] + Bol 2r i + λl (2.4) where G = 2ωλ ρr(x)gh e αt+iγy nuµ 1 represents the ratio of gravitational to viscous forces, Μ = µ 2 µ 1 indicating viscosity ratio of defending fluid over invading fluid and n is the number of pores perpendicular to the displacement direction (transverse direction). Eq. (2.4) is an analytical tool that enables us to predict the critical value of the generalized capillary number Ca c which distinguishes stable and unstable displacement patterns. The theoretical results illustrating how Ca varies as a function of λ (both positive and negative) under different capillary numbers along with the stability criterionca are presented in Fig. 2.4 c Figure 2.4. Logarithmic plot showing relationship between generalized capillary number Ca and λ. The solid line indicates the analytically predicted Ca c, which is a solution of the stability criterion 50

51 Chapter 2: Suppressing viscous fingering in structured porous media derived from linear stability analysis. The symbols are the results obtained by the direct numerical simulation where filled, half-filled and open points represent the stable front, Ca c and unstable front, respectively. Yellow and red regions mark analytically predicted stable and unstable regions separated by the solid line predicted by Eq. (2.4). The insets illustrate the displacement front morphology at Ca = computed by the direct numerical simulation. Fig. 2.4 illustrates that the analytically predicted Ca agrees well withthe numerical resultsin c distinguishing between stable and unstable displacement fronts for all combinations of capillary number and pore size gradientλ. The slight discrepancy in the classification of some points is attributed to the simplifying assumptions made for the derivation of the analytical stability criterion (e.g. ignoring thin wetting films [18] and trapped fluids behind the displacement front). Examination of Eqs. (2.1) and (2.4) suggests that during imbibition (the displacement of a non-wetting phase by a wetting phase), a positive λ would delay the onset of viscous fingering, whereas unstable fronts would always persists for negative λ. Although imbibition processes were not investigated in this study, such a conclusion deduced from Eqs. (2.1) and (2.4) is experimentally supported by the results of Al-Housseiny et al. [18]. An important result of our simulations is that under a same capillary number, when λ > 0 the competition between capillary and viscous forces remains the same (as reflected on the constant value of Ca for λ > 0 in Fig. 2.4), therefore viscous fingering continues. However, for λ < 0 and under the same capillary number, Ca increases that results in stable displacement front. The result presented in Fig. 2.4 further show that there is an analogy between viscosity of the invading fluid µ 1 and negative λ. According to the phase-diagram introduced by Lenormand et al. [25] increasing µ 1 transforms the invasion behaviour from viscous fingering to a stable regime. Similarly, our results indicate that increasing the gradient of pore size λ (more negative values) stabilizes the displacement front. Therefore, 51

52 Chapter 2: Suppressing viscous fingering in structured porous media overall the trend observed in Fig.2.4 suggests that it is the viscous dissipation rather than capillary forces that govern stability of the displacement front (due to increase in Ca as the absolute value of λ increases), in agreement with the previous investigations [14-16, 20]. Negative λ enhances the dissipation of viscous forces which further promotes lateral growth in disturbances and augments the relaxation time of interfaces, thus not allowing the system to reach capillary equilibrium [33-34]. At the macro-scale, these non-equilibrium effects instigate decline in velocity of displacement front V (Fig. 2.3(f)) and development of a smooth invasion pattern. 2.3 Discussion Our results demonstrate the impact of λ on the nature of immiscible displacement in porous media. We show that the viscous fingering, which is traditionally considered as a function of flow rate, viscosity ratio and wetting properties of porous media, is controlled by λ as well. Depending upon the wettability of porous media, for a given Ca of the invasion process, both positive and negative λ can inhibit or trigger the growth of viscous fingers. Our numerical and experimental analyses at the pore-scale enabled us to identify two pore-scale invasion mechanisms responsible for suppressing viscous fingering: I) overlap [35-36] (Fig. 2.5(a)) and II) intermittent burst (Fig. 2.5(b)). Although fundamentally different, these pore-scale displacement mechanisms are non-local and exhibit cooperative behaviour. During overlap two neighbouring interfaces merge to form a new stable interface (the white dash line in Fig. 2.5(a)). The key role of overlap in the formation of a uniform displacement front has been observed in many investigation [14-16]. However, neighbouring interfaces show the opposite effect during the intermittent burst mechanism (Fig. 2.5(b)) as one interface recedes (the black dash line) assisting the adjacent interface to move forward (the white dash line) [33-34,37-38]. In comparison to the viscous instability that occurs during viscous fingering, 52

53 Chapter 2: Suppressing viscous fingering in structured porous media waiting (pinning) time [37] for the interface to drain a pore is significantly longer during intermittent burst as indicated in Fig. 2.3(f). It should be noted that the micro-fingering phenomenon illustrated earlier is a macroscopic result of the intermittent burst mechanism. Moreover, the frequency of overlap f O (%) and intermittent burst f b (%) mechanism has been computed for each simulation case (Fig. 2.5(c-d)). It can be seen from Fig. 2.5(c-d) that increasing flow rate weakens the impact of overlap and intermittent burst mechanism on the displacement dynamics, causing instabilities to govern the interfacial displacement and thus resulting in viscous fingering. Furthermore, Fig. 2.5(c-d) demonstrates the change in the dominant pore-scale displacement mechanism from the overlap to intermittent burst mechanism as the magnitude of λ increases. The identified pore-scale mechanisms control not only the morphology of displacement front but also the evolution of phase pressure difference δp (defined as p i p o r i p o where p i and p o correspond to the inlet and outlet pressure (atmospheric pressure), respectively) as illustrated in Fig. 2.5(e). Although, increasing δp is one of the characteristics of the stable displacement regime [39], how it really increases depends strongly on the processes occurring at the pore-scale. Our results show that when the overlap is the dominant mechanism, δp gradually increases, however in ordered media where the intermittent burst is the dominant displacement mechanismδp increases exponentially (Fig. 2.5(e)). 53

54 Chapter 2: Suppressing viscous fingering in structured porous media Figure 2.5. Pore-scale mechanisms controlling the stability of a front at the macroscopic level. Schematic illustration of the pore-scale (a) overlap and (b) intermittent burst mechanisms influencing the dynamics of displacement. In (a) and (b), red, blue, black and green represents invading fluid, defending fluid, grains and the positions of interfaces before the respective pore-scale displacement mechanism. The direction of displacement is from bottom to top and movement of each individual interface is shown by the arrows. During the overlap mechanism (a), two neighbouring interfaces coalesce to form a stable interface (the white dashed line). In the intermittent burst mechanism, one interface advances towards downstream pores (the white dashed line), while the other interface recedes towards the upstream pore (the black dashed line). The 2D colour map shown in (c) and (d) indicates the frequency distribution of overlaps f O (%) and intermittent bursts f b (%) versus Ca and λ. (e) Evolution of the phase pressure difference δp (m -1 ) defined as p i p o r i p o and p o correspond to the inlet and outlet (open to atmosphere) pressure, respectively. at Ca = , where p i 2.4 Conclusion In this research we have employed a novel design of porous media in the form of ordered media to supress viscous fingering. This study has implications in a number of industrial applications from the design of stable exchange porous columns for analyses and separation 54

55 Chapter 2: Suppressing viscous fingering in structured porous media science to designing new membranes and porous products for suppression of spurious viscous fingering. We envision potential applications related to optimization of reactant transport and phase distribution in fuel cells, sensors and control of fluid flow in spacecraft under microgravity [40] and more. 2.5 Methods and materials The polydimethylsiloxane (PDMS) microfluidic device was made by photolithography. The positive photoresist and plasma etching were used to make the silicone mold for the PDMS in order to obtain uniform height of the channels. The ratio between the cross-linker and the elastomomeric base was chosen as 1.5:10 to enhance the stiffness of the channels. The finished channel was hydrophobic and oleophilic. The triangular area at the inlet was designed for stabilizing the interface before it reached the porous medium. The displaced fluid was phenylmethylsiloxane oligomer (PDM-7050) purchased from Gelest Inc. The invading fluid was de-ionized water mixed with 0.1 wt. % food dye for visualization. Considering the small weight ratio of the dye, its effects on the water viscosity and the wateroil interfacial tension was negligible. We started the experiment at a low capillary number Ca = until a stable interface reached the first row of the pillars. Then, the flow rate was increased to a specified value and the time evolution of the displacment process was recorded by a Nikon camera. Supplementary material Theoretical analysis is performed on a 2D porous medium with the length and width in x and y direction denoted as l and w, respectively (Fig. 2S1). The pore size distribution r(x) is characterised as r(x) = r i + λx, where r i is the pore size at the inlet and λ is the pore size gradient. There is no variation of pore size in the y direction. The pore size at the outlet r o is 55

56 Chapter 2: Suppressing viscous fingering in structured porous media dictated by λ defined as λ = r o r i, where λ < 0 indicates r l o < r i and λ > 0 indicates r o > r i. The number of pores n along the y direction is uniform and the porosity of the model is given as ω. The invading fluid, of density ρ 1 and viscosity µ 1, displaces the defending fluid, of density ρ 2 and viscosity µ 2, at a constant flow rate Q (Fig. 2S1). The front advances in the x direction at a constant injection velocity of U and pore velocity of v. The fluids are immiscible with interfacial tension given as σ and the contact angle between interface and grain surface along the defending fluid is denoted as θ. The viscosity ratio of defending fluid and invading fluid is represented by Μ = µ 2 µ 1. Figure 2S1. Schematic of the model used for the analytical analysis. White represents grains and red shows the interface between two immiscible fluids 1 and 2, where fluid 1 represents the invading fluid and fluid 2 represents the defending fluid. 56

57 Chapter 2: Suppressing viscous fingering in structured porous media References [1] Medici, E. F., & Allen, J. S. (2009). Existence of the phase drainage diagram in proton exchange membrane fuel cell fibrous diffusion media. Journal of Power Sources, 191(2), [2] Shalliker, R.A., Catchpoole, H.J., Dennis, G.R. and Guiochon, G., Visualising viscous fingering in chromatography columns: high viscosity solute plug. Journal of Chromatography A, 1142(1), pp [3] Haas, S., Hoell, A., Wurth, R., Rüssel, C., Boesecke, P., & Vainio, U. (2010). Analysis of nanostructure and nanochemistry by ASAXS: Accessing phase composition of oxyfluoride glass ceramics doped wither3+/yb3+. Physical Review B, 81(18). [4] Cueto-Felgueroso, L. and Juanes, R., Nonlocal interface dynamics and pattern formation in gravity-driven unsaturated flow through porous media. Physical Review Letters, 101(24), p [5] Lake, L.W., Enhanced oil recovery. [6] Sahimi, M., Flow and transport in porous media and fractured rock: from classical methods to modern approaches. John Wiley & Sons. [7] Wang, Y., Bryan, C., Dewers, T., Heath, J.E. and Jove-Colon, C., Ganglion dynamics and its implications to geologic carbon dioxide storage. Environmental science & technology, 47(1), pp [8] Islam, A., Chevalier, S., & Sassi, M. (2013). Experimental and numerical studies of CO2 injection into water-saturated porous medium: capillary to viscous to fracture fingering phenomenon. Energy Procedia, 37,

58 Chapter 2: Suppressing viscous fingering in structured porous media [9] Hill, S., Channeling in packed columns. Chemical Engineering Science, 1(6), pp [10] Saffman, P.G. and Taylor, G., 1958, June. The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences(Vol. 245, No. 1242, pp ). The Royal Society. [11] Homsy, G.M., Viscous fingering in porous media. Annual review of fluid mechanics, 19(1), pp [12] Løvoll, G., Méheust, Y., Toussaint, R., Schmittbuhl, J. and Måløy, K.J., Growth activity during fingering in a porous Hele-Shaw cell. Physical Review E, 70(2), p [13] Lindner, A., Bonn, D. and Meunier, J., Viscous fingering in complex fluids. Journal of Physics: Condensed Matter, 12(8A), p.a477. [14] Holtzman, R. and Segre, E., Wettability stabilizes fluid invasion into porous media via nonlocal, cooperative pore filling. Physical review letters, 115(16), p [15] Holtzman, R. Effects of pore-scale disorder on fluid displacement in partially-wettable porous media. Scientific Reports 6, (2016). [16] Zhao, B., MacMinn, C.W. and Juanes, R., Wettability control on multiphase flow in patterned microfluidics. Proceedings of the National Academy of Sciences, 113(37), pp [17] Dias, E.O., Parisio, F. and Miranda, J.A., Suppression of viscous fluid fingering: A piecewise-constant injection process. Physical Review E, 82(6), p

59 Chapter 2: Suppressing viscous fingering in structured porous media [18] Al-Housseiny, T.T., Tsai, P.A. and Stone, H.A., Control of interfacial instabilities using flow geometry. Nature Physics, 8(10). [19] Jackson, S.J., Power, H., Giddings, D. and Stevens, D., The stability of immiscible viscous fingering in Hele-Shaw cells with spatially varying permeability. Computer Methods in Applied Mechanics and Engineering, 320, pp [20] Pihler-Puzović, D., Illien, P., Heil, M. and Juel, A., Suppression of complex fingerlike patterns at the interface between air and a viscous fluid by elastic membranes. Physical review letters, 108(7), p [21] Wilkinson, D., Percolation effects in immiscible displacement. Physical Review A, 34(2), p [22] Chaouche, M., Rakotomalala, N., Salin, D., Xu, B. and Yortsos, Y.C., Invasion percolation in a hydrostatic or permeability gradient: experiments and simulations. Physical Review E, 49(5), p [23] Xu, B., Yortsos, Y.C. and Salin, D., Invasion percolation with viscous forces. Physical Review E, 57(1), p.739. [24] Yortsos, Y.C., Xu, B. and Salin, D., Delineation of microscale regimes of fullydeveloped drainage and implications for continuum models. Computational Geosciences, 5(3), pp [25] Lenormand, R., Touboul, E. and Zarcone, C., Numerical models and experiments on immiscible displacements in porous media. Journal of fluid mechanics, 189, pp

60 Chapter 2: Suppressing viscous fingering in structured porous media [26] Zaretskiy, Y., Geiger, S. and Sorbie, K., Direct numerical simulation of pore-scale reactive transport: applications to wettability alteration during two-phase flow. International Journal of Oil, Gas and Coal Technology, 5(2-3), pp [27] Ferrari, A., Jimenez Martinez, J., Borgne, T.L., Méheust, Y. and Lunati, I., Challenges in modeling unstable two phase flow experiments in porous micromodels. Water Resources Research, 51(3), pp [28] Rabbani, H.S., Joekar-Niasar, V., Pak, T. and Shokri, N., New insights on the complex dynamics of two-phase flow in porous media under intermediate-wet conditions. Scientific Reports, 7(1), p [29] Rabbani, H.S., Joekar-Niasar, V. and Shokri, N., Effects of intermediate wettability on entry capillary pressure in angular pores. Journal of colloid and interface science, 473, pp [30] Kang, Q., Lichtner, P.C. and Zhang, D., An improved lattice Boltzmann model for multicomponent reactive transport in porous media at the pore scale. Water Resources Research, 43(12). [31] Shokri, N., Sahimi, M. and Or, D., Morphology, propagation dynamics and scaling characteristics of drying fronts in porous media. Geophysical Research Letters, 39(9). [32] Schneider, C.A., Rasband, W.S. and Eliceiri, K.W., NIH Image to ImageJ: 25 years of image analysis. Nature methods, 9(7), pp [33] Furuberg, L., Måløy, K.J. and Feder, J., Intermittent behavior in slow drainage. Physical Review E, 53(1), p

61 Chapter 2: Suppressing viscous fingering in structured porous media [34] Armstrong, R.T. and Berg, S., Interfacial velocities and capillary pressure gradients during Haines jumps. Physical Review E, 88(4), p [35] Martys, N., Cieplak, M. and Robbins, M.O., Critical phenomena in fluid invasion of porous media. Physical review letters, 66(8), p [36] Cieplak, M. and Robbins, M.O., Influence of contact angle on quasistatic fluid invasion of porous media. Physical Review B, 41(16), p [37] Moebius, F. and Or, D., Interfacial jumps and pressure bursts during fluid displacement in interacting irregular capillaries. Journal of colloid and interface science, 377(1), pp [38] Berg, S., Ott, H., Klapp, S.A., Schwing, A., Neiteler, R., Brussee, N., Makurat, A., Leu, L., Enzmann, F., Schwarz, J.O. and Kersten, M., Real-time 3D imaging of Haines jumps in porous media flow. Proceedings of the National Academy of Sciences, 110(10), pp [39] Osei-Bonsu, K., Grassia, P. and Shokri, N., Investigation of foam flow in a 3D printed porous medium in the presence of oil. Journal of colloid and interface science, 490, pp [40] Weislogel, M.M., Jenson, R., Chen, Y., Collicott, S.H., Klatte, J. and Dreyer, M., The capillary flow experiments aboard the International Space Station: Status. Acta Astronautica, 65(5), pp

62 Chapter 3 Effects of intermediate wettability on entry capillary pressure in angular pores This chapter has been published in Journal of Colloid and Interface Science, volume 473 in The objective of this work was to investigate the effect of wettability on the dynamic entry capillary pressure profile in angular pores. High-resolution numerical simulations were performed using OpenFOAM (version 2.3) at various static contact angles 10 o, 45 o and 60 o. The simulation domain consisted of a reservoir which was connected to an angular pore of three different cross-sections square, equilateral triangle, and irregular triangle. During simulations, the invading fluid was allowed to displace the defending fluid at capillary number = The results presented in this chapter suggest that entry capillary profile is strongly governed by the interplay of contact angle and pore angularity. When the contact angle = 10 o or 45 o the behaviour of entry capillary pressure profile is similar to that reported in the literature i.e. increasing the angularity of pore (from square to triangle) enhances the pressure required for the interface to enter the pore. However, under intermediate-wet conditions (contact angle = 60 o ) there is a decline in the entry capillary pressure just at the entrance of the pore, which contrary to strong and weak wet conditions results in the easier advancement of an interface in irregular pores compare to pores with smooth corners. This study has implications in the fields of ganglion mobilization, for example, oil recovery and aquifer remediation. In addition to numerical simulation, a generalized analytical equation is proposed that can predict entry capillary pressure under equilibrium conditions for given pore geometry and 62

63 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores contact angle. The analytical equation along with numerical results were validated with other theoretical equation presented in the literature (verified by experiments). 63

64 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores Effects of intermediate wettability on entry capillary pressure in angular pores *This chapter has been published in Journal of Colloid and Interface Science Rabbani, H., Joekar-Niasar, V. and Shokri, N. (2016). Effects of intermediate wettability on entry capillary pressure in angular pores. Journal of Colloid and Interface Science, 473, pp

65 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores Abstract Entry capillary pressure is one of the most important factors controlling drainage and remobilization of the capillary-trapped phases as it is the limiting factor against the two-phase displacement. It is known that the entry capillary pressure is rate dependent such that the inertia forces would enhance entry of the non-wetting phase into the pores. More importantly the entry capillary pressure is wettability dependent. However, while the movement of a meniscus into a strongly water-wet pore is well-defined, the invasion of a meniscus into a weak or intermediate water-wet pore especially in the case of angular pores is ambiguous. In this study using OpenFOAM software, high-resolution direct two-phase flow simulations of movement of a meniscus in a single capillary channel are performed. Interface dynamics in angular pores under drainage conditions have been simulated under constant flow rate boundary condition at different wettability conditions. Our results shows that the relation between the half corner angle of pores and contact angle controls the temporal evolution of capillary pressure during the invasion of a pore. By deviating from pure water-wet conditions, a dip in the temporal evolution of capillary pressure can be observed which will be pronounced in irregular angular cross sections. That enhances the pore invasion with a smaller differential pressure. The interplay between the contact angle and pore geometry can have significant implications for enhanced remobilization of ganglia in intermediate contact angles in real porous media morphologies, where pores are very heterogeneous with small shape factors. 65

66 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores 3.1 Introduction Understanding of the pore scale physics of immiscible displacement is imperative for maximizing efficiency of many industrial applications, most notably in petroleum industry for enhanced oil recovery [1 3] as well as soil remediation practices [4]. The capillary forces acting on the interfaces separating two immiscible fluids play the primary role in subsurface capillary trapping of hydrocarbons and low recovery factors. To mobilize the capillary trapped ganglia, the mobilization force should overcome entry capillary pressure, which is controlled by wettability conditions and pore size [1]. From computational point-of view, entry capillary pressure is an essential entity required for simulation of two phase flow in pore network models [5]. Delineating the impact of different variables on entry capillary pressure in angular pore geometries is not as straightforward as in cylindrical pore shapes [6]. Due to the presence of angular corners, interface dynamics and its mathematical formulation become complicated [7]. To calculate the entry capillary pressure, Mayer and Stowe [8] devised an ingenious technique that relied on force balance in non-circular pore geometries. The technique was later further developed by Princen [9] and is referred to as Mayer- Stowe-Princen (MS-P) approach. The principle of MS-P approach has been applied in many analytical [5,10,11] and semi-analytical approaches [12-13] to compute entry capillary pressure in different polygonal cross sections. However, despite its significant value and its applicability in pore-network modelling, MS-P can only be employed under equilibrium conditions. Direct two phase flow simulation provides a unique opportunity to investigate interface dynamics under different wettability conditions at different pore geometries. Unlike traditional pore-network modelling approach where phase distribution is controlled by local 66

67 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores filling rules [14], the direct two-phase flow simulation relies on energy balance to simulate displacement process. Ferrari and Lunati [15] performed direct two phase flow simulation to study the impact of capillary number (C a ) and viscosity ratio ( µ inv µ dis ) on entry capillary pressure in strongly water-wet porous system, where µ dis is the viscosity of displacing fluid and µ inv is the viscosity of the invading phase. Capillary number is defined as the ratio of viscous to capillary forces: Ca = μ invq inv, where μ σ inv is viscosity of invading fluid, q inv is velocity of invading fluid and σ is the interfacial tension. The numerical simulation results demonstrated insensitivity of entry capillary pressure to viscosity ratio and capillary number. This is due to the two-dimensionality of the domain, which is not a representative of real porous medium. Further, Raeini et al. [16] formulated an algorithm to simulate two-phase flow in star-shaped pore channels [17], showing increasing entry capillary pressure as the pore body to pore throat contraction ratio increases. Similar to [15], the pore scale study conducted by Raeini et al. [17] was restricted to perfectly water-wet conditions. 3.2 Numerical model Governing equations We use OpenFOAM (Open Field Operation and Manipulation) to simulate dynamics of twophase flow in porous media. The code has been developed in C++ programming language and has been successfully implemented in several porous-media related research studies [15,17-18]. The equations governing incompressible and immiscible displacement in a pore include mass and momentum balances, as follows: Continuity equation reads. u = 0 (3.1) 67

68 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores Momentum balance equation reads ρu t + (ρuu) = p + (μ( u + ut ) + f sa (3.2) where u is the velocity vector (m s -1 ), ρ is the density (kg m -3 ), p is the pressure (kg m -1 s -2 ), μ is the viscosity (kg m -1 s -1 ) and f sa is the body force due to the capillary forces acting at the interface (kg m -2 s -2 ). In the above system of equations the influence of gravity on flow domain is eliminated and fluid properties (density and viscosity) will vary depending upon the phase present in the computational cell [15]. For interface tracking, the Volume of Fluid (VOF) method introduced by [19] has been used. VOF is relatively straightforward to implement and can accurately model complex interfacial geometries [20]. It involves computation of governing equations using volume indicator function (γ), which represents fraction of phases in each grid block as; (0,1) Interface; γ { [1] Fluid 1; [0] Fluid 2. At the interface the volume weighted fluid properties that includes density and viscosity are coupled to VOF algorithm as; ρ = γρ 1 + (1 γ)ρ 2 (3.3) μ = γμ 1 + (1 γ)μ 2 (3.4) The volume indicator function follows the advection transport as shown in Eq. (3.5): γ t + (γu) + (γ(1 γ)u r) = 0 (3.5) 68

69 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores where ρ 1 is the density of fluid 1 (kg m -3 ), ρ 2 is the density of fluid 2 (kg m -3 ), μ 1 is the viscosity of fluid 1 (kg m -1 s -1 ), μ 2 is the viscosity of fluid 2 (kg m -1 s -1 ) and u r is the relative velocity between two fluids (m s -1 ). One of the drawbacks of VOF approach is the numerical diffusion of interface, which causes the interface to spread over many cells. This problem is referred to as smearing [21]. Eq. (3.5) is the modified version of conventional volume of fluid transport equation as it includes additional convection term, known as artificial compression [21] (the last term in Eq. (3.5). The motive of introducing this term is to minimize the smearing problem at the fluid-fluid interface which can otherwise be quite cumbersome to resolve with conventional volume of fluid method [22-21]. In addition to interface capturing algorithm, the surface force f sa acting as a source term in momentum equation was described by Continuum Surface Force (CSF) method [23] that can be represented as; f sa = σ 12 κ γ (3.6) Where σ 12 is the interfacial tension between fluid 1 and fluid 2 (kg s -2 ) and κ is the curvature of the interface (m -1 ) computed as; κ = ( γ γ ) (3.7) Eqs. ( ) are discretized using finite volume approach and solved numerically at each time step to obtain primary unknowns p, u and γ [17]. Furthermore, the coupling between pressure and velocity equation was based on PISO (Pressure Implicit with Splitting of Operator) algorithm developed by [24]. 69

70 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores Modelling specifications The numerical domain represents a pore connected to a large reservoir. Different cross sections including square, equilateral triangle and irregular triangle have been considered. The 3D views of pores are shown in Fig. 3.1(a c) and the side view of the pore and its connecting reservoir has been shown in Fig. 3.1(d). Each pore can be described by its shape factor (G) expressed as the ratio of cross-sectional area to the square of perimeter. Furthermore, in all simulations pores are kept much longer than the reservoir. Fluid properties are shown in Table

71 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores Figure 3.1. (a c) 3D views of the pores with different cross sections. (d) A side view of the simulation domain model. Invading phase (fluid 2) is displacing the receding phase (fluid 1) at a capillary number of

72 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores Table 3.3. Fluid properties in the simulations Parameter Symbol Value Unit Interfacial tension σ Kg s -2 Receding phase viscosity μ Kg m -1 s -1 Invading phase viscosity μ Kg m -1 s -1 Receding phase density ρ Kg m -3 Invading phase density ρ Kg m Initial and boundary conditions The boundary conditions implemented in simulations can be seen in Fig. 3.1(d). Initially the interface is located 0.1 mm away from the junction inside the reservoir. As simulation starts the invading phase (fluid 2) is injected in the z direction at a constant flowrate and displaces the receding phase (fluid 1), while the outlet was kept at atmospheric pressure. The equilibrium contact angle, θ, is defined through receding phase imposed to the solid boundaries. In addition, to eliminate the hysteresis effect the advancing and receding contact angles were kept equal and no-slip boundary condition was imposed on the walls [25]. Contact angles of θ = 10 o, 45 o and 60 o are simulated. At θ = 45 o the corner interface would be flat in square pore geometry and this would happen in equilateral triangle pore with θ = 60 o. 72

73 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores Spatial and temporal discretization In order to avoid dependency of the interface thickness on the grid size, numerical simulations at different grid resolutions have been carried out. The results of grid independence tests are presented in Fig. 3A1, showing the variation of γ versus length of the simulation domain. The green curves show the grid size where the interface thickness does not change with grid resolution. Number of grid blocks per simulation are shown in Table 3A1. For a typical run the numerical time step was made adjustable according to the courant number ( u t ), where t is the time step and x is the grid block size. The maximum courant x number was kept at 0.15 for square geometry and 0.4 in the case of regular and irregular triangle geometry. The simulations were performed using 12-node Intel CPU cluster (Xeon X GHz) with 4GB of memory. Only the drainage process was simulated, where invading fluid was acting as a non-wetting phase and receding fluid as a wetting phase unless stated otherwise Calculation of interface capillary pressure The post-processing of numerical results was conducted with Paraview; an open-source visualization application [26]. Capillary pressure at each time step can be obtained by calculating the curvature of the interface; p c = σ 12 κ (3.8) where p c is the capillary pressure (kg m -1 s -2 ). Following [27], the iso-surface of γ=0.5 is considered as the fluid-fluid interface. Then the built-in Paraview plugin was used to evaluate the distribution of mean curvature over the 73

74 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores entire interface. Finally, the median value of curvature distribution was used to calculate the capillary pressure using Eq The complete post-processing route is shown in Fig Figure 3.2. Post-processing of simulations to calculate the capillary pressure. (a) Phase distribution with red and blue representing receding phase and invading phase respectively. All other colors indicate the transition zone. (b) The interface extracted from the transition zone corresponding to γ=0.5 (c) Curvatures along the interface with the values represented by the color map. 3.3 Results and discussions Entry capillary pressure During the drainage while the non-wetting phase enters the pore, wetting phase will move out from the pore. The interface between the fluids which moves normal to the flow direction is referred to as main terminal meniscus (MTM) [28]. Also some parts of the wetting phase remains in the corners. The fluid-fluid interface in the corners is referred to as arc meniscus (AM) as shown in Fig. 3.3 [29]. In cases where θ is equal or greater than critical contact angle (the contact angle at which AM becomes flat) the AM disappears from the corners, resulting in entrapment of receding phase at the junction in form of pendular rings (see Fig. 3.3) [6]. The critical contact angle for square geometry is 45 o and in the case of equilateral triangle it is 60 o. When the contact angle is equal to half corner angle, the interface would be flat, indicating almost zero capillary pressure. 74

75 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores Figure 3.3. Configuration of receding phase (red) and invading phase (blue) in pore models with different wetting conditions. Each image is labeled with receding phase saturation at the junction inside the pore. G indicates the shape factor. The front view of capillary is shown with cross-section of reservoir (bottom) and pore (top). Invading phase is displacing receding phase at C a of Variations in entry capillary pressure profile as MTM passes the junction are shown in Fig

76 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores Figure 3.4. Evolution of entry capillary pressure at meniscus entering pores with different cross sections and contact angles. X-axis represents the cross sectional saturation of the invading fluid at the pore entrance. 76

77 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores In a water-wet pore (contact angle of 10 o ) the entry capillary pressure in square cross section increases monotonically until reaching the maximum value as it moves into the pore. However in the case of other two geometries, there is a small dip before the entry capillary pressure increases. This localized reduction in entry capillary pressure may correspond to enhanced local and temporal unstable fluid meniscus. Instability in the meniscus triggered due to rapid change in the size of confined region, makes it less resistant against the flow and elevates the local velocities as seen in Fig It is evident from Fig. 3.5 that in square geometry the rise in velocity occurs quite late and is not steep compared to both triangular geometries; as a result the impact of instability on the dynamics of interface is more significant in triangular pore geometries than in square pore. To further illustrate the phenomenon, Fig. 3.6 has been presented which shows vector plot of flow domain as invading fluid enter the pore. The motion of interface can be described in four key steps. First, there is a coflow of immiscible fluids (both invading and receding fluid from the corners of junction flow in same direction), that allows the meniscus to relax against capillary forces [30] as shown in Fig. 3.6(a). Second, as meniscus makes contact with the pore wall and inflates, the intensity of coflow process decreases due to change in trajectory of corner flow shown in Fig. 3.6(b). During this phase the entry capillary pressure increases linearly with decrease in receding fluid saturation. Third, flow of fluid in the corners is directed downward from tip to neck of MTM [31], at this point the entry capillary pressure is maximum as shown in Fig. 3.6(c). The numerical results presented in this investigation indicates that for a pore of given inscribed radius, the maximum entry capillary pressure is inversely related to the shape factor and contact angle which is in agreement with previous studies [6,10,32]. Fourth, imbibition initiates at the neck of MTM inaugurating Plateau- 77

78 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores Rayleigh instability (Fig. 3.6(d)); this period is clearly visible in Fig. 3.4(c) after reaching maximum value, the entry capillary pressure in irregular triangle geometry decreases. Evolution of the interface in different pore cross sections as it advances from reservoir into the pore under different wetting conditions are presented in Fig. 3A2. Figure 3.5. Variation in interface velocity as the interface enters the pore with θ = 10 o All velocity values are scaled with respect to 1.0 x 10-4 m/s. 78

79 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores Figure 3.6. Vector plot showing flow direction of fluids as advancing phase (blue) penetrates and displaces receding phase (red) in the equilateral triangle pore. The black arrow represents flow direction for fluid residing the corner. 79

80 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores The oscillations exhibited in the entry capillary pressure profile near the junction as wettability changes from perfect to partial wet (θ = 45 o and 60 o ) become more pronounced with decrease of the shape factor in angular pores. Inspection of Fig. 3.3 indicates that as the interface progress into the partially-wet pore, thermodynamically driven rupturing of receding phase film occurs at the junction, this along with coflow process declines the capillary pressure trend of all three pore models as shown in Fig. 3.4(b) and (c). At θ = 60 o the capillary behaviour of interface as a consequence of film rupturing changes from drainage to imbibition in both triangle pore models, whereas in square pore model the change is not observed. This phenomenon can be rationalized by examining the variations in morphology of the interface shown in Fig. 3A2. In equilateral and irregular triangle pore cross-section, the capillary forces imbibe the corners of interface while the MTM is just at the junction. During this phase MTM is almost flat causing the overall curvature of the interface to be dictated by the AM (change in curvature of AM as seen in triangle pore models; see Fig. 3.3). However, in square pores the rupturing of film is initiated while the MTM is above the junction, causing the entry capillary pressure to remain positive. Results reported in Figs. 3.3 and 3.4 clearly manifest that in pore network models similar to the pore body and pore throat, it is important to model junction as a separate entity. Ignoring it can lead to unrealistic flow scenarios. Practical implication of our numerical results in terms of remobilization of trapped phase is demonstrated schematically in Fig

81 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores Figure 3.7. Schematic of water flooding to mobilize the trapped oil phase. The flow direction is from left to right. The driving force acting on imbibition interface marked blue tends to increase the pressure gradient between oil and water phase at the drainage interface. To remobilize the oil, it is vital for the driving force to overcome the capillary forces that are resisting mobilization of the trapped phase (figure adapted and modified from [33]). During water flooding, in order to invade the pore throat, the local pressure gradient across the interface should overcome the entry capillary pressure. More importantly under same wettability conditions the entry capillary pressure across different pore geometries varies significantly. For example according to Fig. 3.4 (c), at θ = 60 o in both triangular pore geometries the entry capillary pressure at the moment of entree can drop sharply, while this behaviour is less visible in square pore geometry. This drop of entry capillary pressure may indicate an easier remobilization of the trapped nonwetting phase for intermediate saturations when the shape factor is smaller. It is worthwhile to mention that in real porous media and rock formation the shape factor can vary mostly in the range of triangular pores [32,34]. 81

82 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores Comparison with analytical solutions In this section the maximum entry capillary pressure estimated from simulation results will be compared with the one derived from MS-P method and equation proposed in the present study. The entry capillary pressure for rectangular cross sections can be calculated using the following analytical solution [2]: 1 p c = σ 12 ( (a+b)cos (θ)+ (a+b)2 cos 2 θ+4ab( π 4 θ 2 cos(π 4 +θ)cosθ) ) 4( π 4 θ 2 cos(π 4 +θ)cosθ) (3.9) where p c is the entry capillary pressure (pascal), a is the length (m) and b is the width (m) of the capillary cross-section. For equilateral triangular geometries, the following equation can be used as reported in [6]; p c = σ 12 (cosθ + 1 π/6+θ (sin2θ + π(1 ))) (3.10) 2 3 π/2 For irregular geometries, the entry capillary pressure can be calculated using Eq derived by [35] as; p c = σ 12(1+2 πg)cosθ ( 1+ 4GD 1+ cos 2 θ r i 1+2 πg ) (3.11) where r i is the radius of inscribed circle (m) and D = π (1 θ π/3 ) + 3sinθcosθ cos2 θ 4G. Using simple trigonometry relationships, we proposed the following equation to estimate the maximum entry capillary pressure in angular pore shapes. 82

83 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores p c = 2πr i σ 12 cosθ (1 β π ) sin (π 2β) G P 2 sin (β) (3.12) where β is the half corner angle (degrees) and P is the perimeter of the cross section (m). The comparison between entry capillary pressures predicted by theoretical equations against simulation results are presented in Fig Although the simulation results are generally in agreement with both theoretical predictions, there is slight discrepancy which could be due to the following two reasons: Firstly, due to the 2D nature of the analytical solutions, the longitudinal curvature of the interface (in flow direction) has been assumed equal to zero [17]. Secondly, our simulation results indicate that even at Ca = 10-7, the dynamic effects are present. Not only the contact angle will be different from the equilibrium one (see Fig. 3.8(d)) but also the interface will not have equal mean curvature over the whole interface as shown in Fig. 3A2. Near three phase contact line, the viscous forces commensurate with capillary forces that results in contact angle to be different from its actual value [36-37]. Moreover, as the corner of pore gets smaller the dissipation of viscous stress at three-phase contact line escalates; this enhances the deformation of interface caused by the dynamic effects [36]. 83

84 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores Figure 3.8. Comparison between theoretical and simulation results for (a) 10 o, (b) 45 o and (c) 60 o contact angles. (d) Shows variation in interface shape as it advances towards the apex at different contact angle and corner angle. 84

85 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores 3.4 Summary and conclusion In this paper, direct two phase flow simulation was performed to investigate variations in the interface behaviour in angular pores under a wide range of wetting conditions. Our results offer new and fundamental insights regarding entry capillary pressure into pores; essential information for accurate description of flow in porous media. Under partially-wet conditions interface behaviour at the junction is highly unstable, inducing non-monotonicity in the entry capillary pressure trend, which was found to increase with pore angularity. The numerical results show that at θ = 60 o angular pores with smaller shape factors induce enhancement of the meniscus movement that leads to smaller entry capillary pressure. This is an important phenomenon as it suggests that in natural porous media composed of pores of different shape factors the impact of wettability on entry capillary pressure will be highly non-uniform. One may speculate that the remobilization of trapped oil phase will be easier in triangle or silt type pore geometries rather than circular or square pore throats. Taking into account the non-linear temporal evolution of the entry capillary pressure in the dynamic pore network modelling approach can allow manifestation of multi-phase flow scenarios pertinent to real oil reservoir rocks. 85

86 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores Appendix Appendix A1. Grid independence Figure 3A4. Showing Grid independence test results 86

87 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores Table 3A1. Number of grid blocks per simulation. The grid block size is scaled with respect to inscribed radius of pore shape. Pore network Scaled grid block size Grid blocks Square Equilateral triangle Irregular triangle

88 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores Appendix 3A2. Iso-Surface map Figure 3A5. Demonstrating variation in morphology of interface as interface moves from the reservoir to pore under different contact angles. 88

89 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores Appendix 3A3. Derivation of proposed equation Figure 3A6. (a) Representing configuration of forces under hydrostatic equilibrium (b) the viscous forces at one of the corners of angular pore can be resolved in form trapezium. Using sine rule one can show that τ f = sin (180 2β) τ sin (β) f At three phase line of contact viscous forces and capillary forces are comparable (37) so; τ f = sin (180 2β) sin (β) σ 12 cosθ(2πr i )( 2β 360 ) The pressure gradient at the interface (P nw pw) that is the total driving force (D) as soon as interface makes contact with solid wall can be written as D = sin (180 2β) sin (β) σ 12 cosθ(2πr i ) 89

90 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores sin (180 2β) sin (β) takes into account the angularity of pore shape. From force balance equation, one can conclude that the total driving force equals to the summation of capillary force and viscous force. Thus capillary force can be written as Capillary force = sin (180 2β) sin (β) σ 12 cosθ(2πr i ) sin (180 2β) sin (β) σ 12 cosθ(2πr i )( 2β 360 ) = sin (180 2β) sin (β) σ 12 cosθ(2πr i ) (1 2B 360 ) Entry capillary pressure = sin (180 2β) σ sin (β) 12 cosθ(2πr i )(1 2B 360 ) G P 2 It is important to note that for irregular pore shapes, smallest corner angle should be used.. 90

91 Chapter 3: Effects of intermediate wettability on entry capillary pressure in angular pores References [1] Dullien, F.A., Porous media: fluid transport and pore structure. Academic press. [2] de Castro, A.R., Oostrom, M. and Shokri, N., Effects of shear-thinning fluids on residual oil formation in microfluidic pore networks. Journal of colloid and interface science, 472, pp [3] Rodríguez de Castro, A., Shokri, N., Karadimitriou, N., Oostrom, M. and Joekar Niasar, V., Experimental study on nonmonotonicity of Capillary Desaturation Curves in a 2 D pore network. Water Resources Research, 51(10), pp [4] Abdel-Moghny, T., Mohamed, R.S., El-Sayed, E., Mohammed Aly, S. and Snousy, M.G., Effect of soil texture on remediation of hydrocarbons-contaminated soil at El-Minia district, Upper Egypt. ISRN Chemical Engineering, [5] Joekar Niasar, V., Hassanizadeh, S.M., Pyrak Nolte, L.J. and Berentsen, C., Simulating drainage and imbibition experiments in a high porosity micromodel using an unstructured pore network model. Water resources research, 45(2). [6] Ma, S., Mason, G. and Morrow, N.R., Effect of contact angle on drainage and imbibition in regular polygonal tubes. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 117(3), pp [7] Orr, F.M., Scriven, L.E. and Rivas, A.P., Menisci in arrays of cylinders: numerical simulation by finite elements. Journal of Colloid and Interface Science, 52(3), pp [8] Mayer, R.P. and Stowe, R.A., Mercury porosimetry breakthrough pressure for penetration between packed spheres. Journal of colloid Science, 20(8), pp

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96 Chapter 4 Pore geometry control of apparent fractional wetting of a porous medium This chapter will be submitted for publication. The focus of this investigation is to delineate the impact of pore geometry on the apparent wettability of porous media. This study provides fundamental understanding on the local wetting characteristics of porous media (apparent wettability), and highlights the relation between apparent wettability and the macroscopic properties of fluids. In this research, computational fluid dynamic simulations were performed using OpenFOAM (version 3.01) in uniform capillary, non-uniform capillary and 2D micromodel at three different static contact angles 45 o, 90 o and 135 o. The non-uniform capillary converges and diverges at angle of 21.8 o. One of the main outcomes from this research is that pore geometry effect on the apparent wetting characteristics of porous media is limited to intermediate contact angles. When the static contact angle is close to 0 o or 180 o (which are 45 o and 135 o in this research), the pore orientation angle has negligible influence on the intrinsic wettability of porous media. Moreover, this research reveals that characterization of wettability of porous media using surface wettability should be restricted to strong or weak wet conditions. When the static contact angle is close to 90 o, then curvature of interface (intrinsic measure of porous media wettability) becomes highly susceptible to the pore geometry and therefore the local wettability of porous media differs from the surface wetting properties. The crucial role of pore geometry on apparent wettability was indicated by the curvature reversal of interface in converging-diverging capillary and the co-existence of concave and convex interfaces in micromodel. In addition, derivation of apparent wettability number W is also presented that 96

97 Chapter 4: Pore geometry control of apparent fractional wetting of a porous medium demarcates pore geometry controlled apparent wetting regime from pore geometry independent apparent wetting conditions. 97

98 Chapter 4: Pore geometry control of apparent fractional wetting of a porous medium Pore geometry control of apparent fractional wetting of a porous medium Harris Sajjad Rabbani 1, Benzhong Zhao 2, Ruben Juanes 2 and Nima Shokri 1 (2017) (to be submitted) 1 School of Chemical Engineering and Analytical Science, The University of Manchester, Manchester, M13 9PL, United Kingdom 3 The Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA 98

99 Chapter 4: Pore geometry control of apparent fractional wetting of a porous medium Abstract This research demonstrates the contribution of pore geometry in controlling the apparent wetting characteristics. We show that the reversal of interface curvature in the single converging-diverging capillary and the co-existence of concave and convex interfaces in porous media are the manifestation of the non-trivial effect of pore geometry on the apparent wettability. Our results illustrate that the influence of pore geometry on apparent wetting characteristics is limited to intermediate contact angles θ. Furthermore, we also present a detailed theoretical argument that underpins and defines the limiting conditions for physically induced apparent wettability. Our findings advance the fundamental understanding of wettability in confined geometries like porous media, and can be applied in wide range of applications from fuel cells to enhanced oil recovery mechanisms. 99

100 Chapter 4: Pore geometry control of apparent fractional wetting of a porous medium 4.1 Introduction Wettability is the preferential affinity of a fluid with the solid surface in the presence of another immiscible fluid [1-2]. In porous media, wettability plays a crucial role in controlling fluid flow and phase distribution [3-5].Understanding of wetting dynamics in porous media is essential for many applications including, hydrocarbon recovery [6], carbon dioxide sequestration [7], fuel cells [8], fluid flow in microgravity [9], photosynthesis in plants [10], infiltration of rainwater into soils [11], salt crystallization [12], preservation of historical monuments [13], and crack formation [14]. Traditionally, wettability of surfaces is characterized using static contact angle θ(dictated by the chemical properties of surface) between the smooth horizontal solid surface and fluidfluid interface (ideal system). Surface wetting properties are classified as strong-wet, weakwet and intermediate-wet. Strong and weak wet surfaces, adheres to one of the immiscible fluids, whereas intermediate-wet surfaces have an equal affinity with both fluids. Based upon fundamental assumption of local phase equilibrium, the relationship between θ and surface forces is described by Young s law as [1]; σ cos θ = σ if σ df (4.1) where σ, σ if and σ df are the interfacial energies of the fluid-fluid, invading fluid-solid, and, defending fluid-solid interfaces, respectively. In the present study θ was measured through defending fluid. According to Young s law, the surface exhibits strong or weak wet conditions when θ < 90 o (σ if > σ df ) or θ > 90 o (σ if < σ df ). In the former case, the defending fluid shows affinity with the surface, while in the latter case the invading fluid preferentially adheres to the surface [2]. Young's law further indicates that surface can exhibit intermediate-wet conditions when θ = 90 o (σ if = σ df ). 100

101 Chapter 4: Pore geometry control of apparent fractional wetting of a porous medium Many previous studies have demonstrated the impact of surface wettability (characterized by θ) on the different aspects of multiphase flow in porous media [3-5]. However, some studies reveal that contact angle hysteresis [15-19], can extensively deviate intrinsic contact angles of porous media from θ. Moreover, Dullien [20] indicates that considering only the intrinsic contact angle as a measure of porous media wettability is not complete and that in addition to the intrinsic contact angle, the angle of orientation of pore also influences the apparent wettability or in other words the intrinsic wetting behaviour of porous media. Therefore, one can conclude that surface wettability is not a true indicator of the wettability of porous media, as it can be significantly different from the apparent wetting characteristics of porous media. Although, there are many studies where the wettability of porous media is directly measured [21-23], according to our knowledge almost no study has been able to compute the local wettability of porous media while the interface is moving in angular pores. As a result, there are many open questions concerning the impact of apparent wettability on interfacial phenomena still remains. In this research, we performed Computational fluid dynamics (CFD) simulations in a single capillary channel and in a well-characterised porous medium at three different θ values of 45 o, 90 o and 135 o, to delineate the impact of apparent wettability on the pore scale immiscible displacement. The specific objectives of this research are divided into two folds. Firstly, we assess the validity of Dullien's [20] theory that whether in the absence of contact angle hysteresis such that intrinsic contact angle = θ, does the pore geomatery impact the apparent wettability. Secondly, how does apparent wettability influences the interfacial dynamics and the macroscopic behaviour of fluid flow in porous media with uniform θ distribution. This investigation extends the fundamental understanding wetting dynamics in porous media and opens a gateway for future porous media studies. 101

102 Chapter 4: Pore geometry control of apparent fractional wetting of a porous medium 4.2 Fluid-fluid displacement in single capillaries We performed computational fluid dynamics (CFD) simulations, where we solved Navier- Stokes equations coupled with the volume of fluid method using a finite volume approach, details of the numerical formulation are provided in Rabbani et al. [24], Deshpande et al. [25] and Jasak [26]. We simulated fluid-fluid displacement in two capillaries with square crosssection were; one with uniform and the other one with non-uniform (converging-diverging) geometry as shown in Fig The length of capillaries was kept constant equal to 2 mm. Figure D representation of capillaries used in the presented numerical simulation. The non-uniform capillary converges and diverges at an angle of orientation β = 21.8 o. The capillaries were first filled with defending fluid (viscosity µ d = Pa.s and densityρ d = 102

103 Chapter 4: Pore geometry control of apparent fractional wetting of a porous medium 1000 kg.m -3 ), which is displaced by the invading fluid (viscosity µ i = Pa.s, density ρ i = 1000 kg.m -3 and interfacial tension σ =70 mn.m -1 ) at constant flow rate of Q = 7.0 x m 3.s -1. The interfacial tension between fluids is σ = N.m -1. For all simulation scenarios, the outlet was kept at atmospheric conditions. The static contact angle θ, which corresponds to the angle between defending fluid and solid surface, was defined as an input parameter in the solver. The numerical simulations were performed at three different θ values: 45 o (drainage), 90 o (intermediate-wet) and 135 o (imbibition). During two phase flow the contact angle hysteresis and influence of dynamic conditions on the contact angle was neglected. The relative importance of viscous to capillary forces was characterised using the capillary number Ca = µ i v i σ, where v iis the injection velocity of the invading fluid [27]. All simulations were performed at Ca = 10-7 where capillary forces dominate viscous forces. Because fluid densities are identical gravity forces do not play a role in our 3D simulations. 4.3 Result and Discussion Curvature reversal of the interface in converging-diverging capillary The movement of the interface in the uniform and converging-diverging capillary at different values of contact angle θ is shown in Fig. 4.2(a). To provide a quantitative analysis of interface morphology exhibited by converging-diverging capillary, variations in capillary pressure p c against capillary radius r is presented in Fig. 4.2(b). The capillary pressure p c represents the pressure difference across the interface and is estimated by the Laplace equation as; p c = σk (4.2) 103

104 Chapter 4: Pore geometry control of apparent fractional wetting of a porous medium where k is the curvature of the interface computed following the methodology used in Rabbani et al. [24].The overlay (e.g., the converging section of θ = 45 o and the diverging section of θ = 90 o ) and the non-monotonic trend displayed by the p c curve in Fig. 4.2(b) demonstrates that at micro-scale in addition to θ, r and the corner angle [24,28] β also has a significant impact on the p c variation. Although the influence of β on k has been theoretically shown by Dullien [20], and Cieplak and Robbins [29], in the present study we visualize and provide numerical evidence of such behaviour. Figure 4.2. (a) Movement of fluid-fluid interface in uniform and converging-diverging capillaries at different contact angles θ. Colour lines denote the position of interface at the vertical plane of symmetry of the 3D capillaries (See Fig. 4.1). In the case of θ = 45 o and 135 o, the interface shape is insensitive to the geometry of the capillary. However, at θ = 90 o the morphology of interface in the uniform capillary is significantly different from that shown in the converging-diverging capillary. In converging-diverging capillary, there is a transition in the interface curvature from concave to convex 104

105 Chapter 4: Pore geometry control of apparent fractional wetting of a porous medium as interface moves from converging to the diverging section, whereas in the uniform capillary the interface remains flat. (b) Evolution of capillary pressure p c (Pa) as the interface moves from the converging to the diverging section of the capillary at different θ values. The non-monotonic relationship between p c and capillary radius r for various θ values illustrates the contribution of β in regulating p c trend. (c) Phase-diagram that distinguishes the (ii) pore geometry controlled apparent wetting regime (shaded by green) from (i,iii) pore geometry independent apparent wetting regime (shaded by yellow and brown). Note the circle and triangle symbols correspond to the pair of θ and β values where the interface is flat. Circle and triangle represent converging (θ = π β) and the 2 diverging (θ = π + β) section of the capillary, respectively. 2 Fig. 4.2 (a) and (b) show that in the case of θ = 135 o and 45 o the shape of interface and sign of p c is independent of capillary configuration. When θ = 45 o, invading fluid is the nonwetting fluid (convex shape of interface and positive p c ) and displacement process is regarded as drainage. However at θ = 135 o the morphology of interface is concave, and p c is negative indicating wetting of the surface by the invading fluid (imbibition). This change in the wetting behaviour of fluids due to the variation in θ values is a well-known phenomenon. Contrary to previous scenarios, when θ = 90 o there is a difference in the morphology of interface in uniform and converging-diverging capillary. The interface is flat and the resulting p c is negligible (k~0) in the uniform capillary, demonstrating intermediate-wet conditions. However, interface undergoes a morphological transformation from concave to convex in the converging-diverging capillary (Fig. 4.2(a)), which is also reflected in Fig. 4.2(b). Since the absolute value of p c > 0, it implies that at θ = 90 o capillary forces are still present. Furthermore, the change in the direction of p c (Fig. 4.2(b)) elucidates that there is a change in the direction of capillary forces as interfaces moves from converging to diverging section. 105

106 Chapter 4: Pore geometry control of apparent fractional wetting of a porous medium Understanding the sign of k as an indicator of apparent wettability [20] can allow us to establish a correlation between surface wettability and apparent wettability. Fig. 4.2 (a-b) suggests when θ is 45 o or 135 o surface wettability corresponds to apparent wettability as the sign of k is similar in uniform and converging-diverging capillary. Therefore, for uniform capillaries at all θ, and irregular capillaries when the θ is close to 0 o or 180 o, Young s law or mere measurement of θ on a flat surface is sufficient to delineate apparent wetting properties. However, in the case of intermediate θ values, β values have strong control over apparent wettability [20]. As a result, Young s law or surface wettability cannot describe the apparent wetting characteristics of irregular capillary channels under such conditions. Fig. 4.2(a-b) clearly shows that when θ = 90 o, although uniform capillary is manifesting intermediate-wet conditions, the apparent wetting state of non-uniform capillary is fractional-wet. The invading fluid imbibes in converging section and drains defending fluid from diverging section. To further investigate the physics of curvature reversal in the converging-diverging capillary at θ = 90 o,we have obtained a generic analytical model that relates all essential capillary parameters with p c. The generalized analytical equation is an integration of cos(θ ± β) [20,26] into the equation proposed by Rabbani et al. [24] p c = 2πrσcos(θ)(1 h π ) sin(π 2h) GP 2 sin(h). The resulting analytical equation to describe the capillary pressure in the converging-diverging capillary can be written as; p c = 2πrσcos(θ±β)(1 h π ) sin(π 2h) GP 2 sin(h) (4.3) where h, Pand G represents the half corner angle of capillary cross-section, perimeter of capillary cross-section and shape factor of the capillary, respectively. In Eq. (4.3), cos(θ + 106

107 Chapter 4: Pore geometry control of apparent fractional wetting of a porous medium β) represents p c in converging section and cos(θ β) represents p c in diverging section. The comparison between simulation results and the above analytical solutions is shown in Fig. 4.2(b).It is evident from the match between the numerical results and analytical equations that in contrast to the conventional analysis, the algebraic sum of both θ and β controls the sign of interface curvature k [20,29]. In the cases when θ = 45 o and 135 o, θ has a stronger control over the sign of k, thus the apparent wettability is consistent with surface wettability. However, for intermediate θ values (which is 90 o in our case), β has a substantial influence on the morphology of the interface, principally due to cos(θ) ~ 0. Moreover, Eq. (4.3) indicate that due to the difference in the sign of β, the converging and diverging section of capillary has an opposite effect on controlling the interface shape. Therefore, due to the pivotal role of β as well as the difference in the effect of converging and diverging section on interface dynamics, not only the apparent wetting state of converging-diverging capillary is different from that of uniform capillary, but also there is a contrast in the apparent wettability of converging and diverging section (fractional-wet) as shown in Fig. 4.2 (a) and (b). Later, it will be illustrated that this complex yet intriguing behaviour displayed by the interface at θ = 90 o in converging-diverging capillary has a significant impact on the fluid flow in porous media at macro-scale. The analytical model (Eq. (4.3)) further enables us to develop phase diagram presented in Fig. 4.1 (c).three interfacial regimes are shown by the phase diagram: i) where the shape of interface is controlled by θ and both converging and diverging sections of capillary show convex interface, ii) Curvature of interface reverses as it moves from converging to diverging section, in this regime β governs the apparent wetting characteristics and iii) both converging and diverging sections of the capillary display concave interface and the impact of β on the apparent wettability is negligible. The boundaries corresponding to the green shaded region 107

108 Chapter 4: Pore geometry control of apparent fractional wetting of a porous medium (ii) are the solutions of θ = π 2 β for converging (circles) and θ = π 2 + β for diverging (triangles) section. Note that for all irregular capillaries (β > 0) with θ = 90 o surface wettability does not relate to apparent wettability as portrayed by Figs. 4.1 (a) and (b). Moreover, for given β value there is range of θ where the interface is flat and apparent wettability resembles intermediate-wet conditions [20] Co-existence of concave and convex interfaces in 2D micromodels In order to investigate the macroscopic response of geometrically induced appparent wetting conditions (as illustrated in Fig. 4.2), immiscible two-phase flow was simulated in a 2D porous medium (micromodel). The design of 2D micromodel is based on the cross-sectional image of a real sand pack obtained by 3D X-ray micro-tomography [30]. The pore size distribution of the micromodel ranges from to mm with the average pore size r pof mm. The model has absolute permeability of 12mD. The pore size distribution was obtained by first implementing MATLAB function bwdist on the binary image that calculated the distance between all pixels and the boundary [31]. Later, Matlab function bwmorph was used to keep only the middle pixel of pore, which allowed us to determine the distance between boundary and middle pixel only [31].As for the simulation, the fluid properties and boundry conditions were similar to the ones used for the numerical simulation of the single capillary. However, the nature of simulation in this case was 2D and Ca was changed to It is important to note that the contactangle θ was kept uniform in the micromodel. Fig. 4.3 (a) illustrates typical examples of phase distributions in micromodel at different contact angles θ. 108

109 Chapter 4: Pore geometry control of apparent fractional wetting of a porous medium Figure 4.3. (a) The distribution of phases influenced by the contact angle θ in the 2D micromodel. For θ = 45 o, interfaces are convex; as a result the invading fluid (non-wetting fluid) bypasses the pore throats and drains the defending fluid (wetting fluid) from relatively larger pore bodies. When θ = 135 o there is the dominance of concave interfaces (imbibition), which further illustrates that invading fluid is acting as a wetting fluid and displacement is governed by the cooperative displacement 109

110 Chapter 4: Pore geometry control of apparent fractional wetting of a porous medium mechanism [29]. For θ = 90 o, the co-existence of concave and convex interfaces (indicated by arrows) can be observed, which has a significant impact on the recovery efficiency of defending fluid as shown in Rabbani et al. [32]. This type of condition is similar to a fractional wetting scenario. Besides, entrapment of defending fluid in pore throats shown by the red dotted circle which is typical in drainage displacement, confirms that the micromodel at θ = 90 o manifests apparently fractional wet conditions and thus capillary forces cannot be neglected in such a porous medium. Blue, yellow and black indicates defending fluid, invading fluid and grains, respectively. The displacement occurs from bottom to top. (b) Evolution of recovery efficiency of defending fluid versus time t under different contact angle θ indicated in the legend. Closer inspection of the phase distributions presented in Fig. 4.3 (a) and the interface dynamics presented earlier (Fig. 4.2) demonstrates a clear relationship between the results obtained from simulation in a single capillary and the ones obtained from flow simulation in micromodel. When θ = 45 o, the interfaces are convex (defending fluid is the wetting fluid); as a result, the capillary forces hinder the movement of interfaces, and displacement process (drainage) is impeded by narrow pore throats, giving rise to a ramified fluid pattern associated with invasion-percolation process. In the case of θ = 135 o the displacment process is imbibition and interfaces are concave (invading fluid is the wetting fluid), thus cooperative pore filling mechanism [29] promotes smooth propagation of the front. Interestingly, one can notice despite the uniform distribution of θ; there is co-existence of concave (imbibition) and convex (drainage) interfaces in micromodel at θ = 90 o. The co-existence of the concave and convex interfaces is an attribute of reversal of interface curvature from concave to convex in the converging-diverging capillary (Fig. 4.2) and has been observed before in the numerical studies performed by Rabbani et al. [32]. This further suggests that at θ = 90 o, the micromodel is behaving as a fractional-wet system as the local wettability of micromodel will vary depending upon the geometry of pores. Also, bypassing of the defending fluid [33] (red 110

111 Chapter 4: Pore geometry control of apparent fractional wetting of a porous medium dotted circle in Fig. 4.3(a)) in pore throats which is a prominent feature of drainage displacement confirms that at θ = 90 o capillary force has a significant impact on the interfacial dynamics. It is due to fractional wetting cherecteristics of micromodel that recovery efficiency of defending fluid E d declines from θ = 135 o to θ = 90 o, as illustrated in Fig. 4.2 (b). The trend shown in Fig. 4.3 (b) is consistent with previous studies performed by Cieplak et al. [29], Trojer et al. [5], Zhao et al. [3], Rabbani et al. [32] and Jadhunandan et al. [34] and Ryazanov et al. [35] Apparent wettability number One can deduce from the micromodel results (Fig. 4.2) that converging-diverging channels can better replicate natural porous systems. Modeling a complex pore network as an array of the regular capillaries can obscure true nature of interfacial dynamics espacially under intermediate contact angles conditions. While in real porous systems irregularity associated with pore size distribution makes the determination of β and therefore identification of conditions where pore geometry controls the apparent wetting dynamics (the green shaded region in Fig. 4.2(c)) a formidable task. Here, we have proposed a simple relationship between pore geometrical and surface chemical properties that can be used as a powerful diagnostic tool to describe and predict the interface configuration and wetting properties in complex porous media. This relationship is referred to as apparent wettability number W. Using trigonometric identity cos(θ + β) = cos θ cos β sin θ sin β, Eq. (4.3) can be transformed into; p c GP 2 sin(h) 2πrσ(1 h π ) sin(π 2h) + 1 = cos(θ)r b sin(θ) (r b r t) (4.4) where, r b and r t are the average pore body size and average pore throat size, respectively. Moreover, substituting p c GP 2 sin(h) 2πrσ(1 h π ) sin(π 2h) + 1 with W, and r t with relationship r t = 111

112 Chapter 4: Pore geometry control of apparent fractional wetting of a porous medium φο π n pr b 2 n t, and furthermore assuming close packing of equal spheres such that number of pore throats n t = N, where N is the number of solid particles (grains) and number of pore bodies n p = N, the following generalized relationship can be deduced; 4 W = r pcot (θ) r p φο πn r p 2 4 (4.5) where r p, φ and ο are the average pore size, the porosity, and the total area of the porous medium (excluding depth), respectively. In our simulation, the values of these parameters were mm, 0.51, 128 mm 2 and 439, respectively. Similar derivation can be performed for cos(θ β) = cos θ cos β + sin θ sin β, however in this case W will be representing p c GP 2 sin(h) 2πrσ(1 h π ) sin(π 2h) 1. Using the limiting conditions (p c = 0 ) of the green shaded region (shown in Fig. 4.2(c)), one can deduce range of W where the pore geometry effects dominates the surface chemistry effects in governing the apparent wettability, which is 1 W 1. As a result, co-existence of concave and convex interfaces (fractional wettability) will be observed in such porous media regardless of the uniform θ distribution. For the micromodel pattern used in our simulation, Eq. (4.5) indicates that pore geometry effects on wettability are noticeable when θ ranges between 54.7 o and o, with the maximum recovery of defending fluid expected at θ = o (due to cooperative pore filling mechanism). Our proposed relationship (Eq. (4.5)), agrees with Fig. 4.3(b), as well with the results from Rabbani et al. [32] where similar flow domain was employed for simulations and the maximum recovery of defending fluid was observed at θ = 135 o. 112

113 Chapter 4: Pore geometry control of apparent fractional wetting of a porous medium 4.4 Summary and conclusion Our results illustrate that treating pore geomatery and wettability of porous media (apparent wettability) as separate parameters should be limited to porous media with contact angles θ close to 0 o or 180 o. However, for porous media with intermediate contact angles (θ = 90 o in this research), apparent wettability is a strong function of the angle of orientation of pore geometry β. Consequently, determining the wettability of such porous media by simply measuring θ on a horizontal surface (surface wettability) is inadequate. Incorporating this classical approach in modelling practices may oversimplify this less-understood problem and potentially obscure the true physics governing immiscible two-phase displacment in complex porous media. As a consequence of geometrical effects on apparent wettability, reversal of curvature from concave to convex was observed in our simulation of interface displacement in a single converging-diverging capillary at θ = 90 o. Macroscpic response of such transition in interface curvature is the co-existance of concave and convex interfaces in the porous medium with unifrom θ distribution, which indicates a fractional wettability condition. Our results further demonstrated the maximum recovery efficeny of defending fluid at θ = 135 o rather than θ = 90 o. This is counter-intuivative as conventionally θ = 90 o relates to conditions where the capillary effects on interfacial dynamics are negligble and thus recovery of defending fluid is expected to be the highest. Our results sugget that due to the pore geomatery induced fractional wetting scenerio, there is a decrease in the recovery efficeny of defending fluid as θ varies from 135 o to 90 o. In addition to the numerical simulations, a detailed theoretical explanation along with demarcation of pore geomatery controlled wetting condition in the form of wettability number W, is also presented. 113

114 Chapter 4: Pore geometry control of apparent fractional wetting of a porous medium References [1] De Gennes, P.G., Wetting: statics and dynamics. Reviews of modern physics, 57(3), p.827. [2] Bonn, D., Eggers, J., Indekeu, J., Meunier, J. and Rolley, E., Wetting and spreading. Reviews of modern physics, 81(2), p.739. [3] Zhao, B., MacMinn, C.W. and Juanes, R., Wettability control on multiphase flow in patterned microfluidics. Proceedings of the National Academy of Sciences, 113(37), pp [4] Holtzman, R. and Segre, E., Wettability stabilizes fluid invasion into porous media via nonlocal, cooperative pore filling. Physical review letters, 115(16), p [5] Trojer, M., Szulczewski, M.L. and Juanes, R., Stabilizing fluid-fluid displacements in porous media through wettability alteration. Physical Review Applied, 3(5 ), p [6] Morrow, N.R., Wettability and its effect on oil recovery. Journal of Petroleum Technology, 42(12), pp [7] Saraji, S., Goual, L., Piri, M. and Plancher, H., Wettability of supercritical carbon dioxide/water/quartz systems: simultaneous measurement of contact angle and interfacial tension at reservoir conditions. Langmuir, 29(23), pp [8] Meng, C., Wang, B., Gao, Z., Liu, Z., Zhang, Q. and Zhai, J., Insight into the Role of Surface Wettability in Electrocatalytic Hydrogen Evolution Reactions Using Light- Sensitive Nanotubular TiO2 Supported Pt Electrodes. Scientific Reports,

115 Chapter 4: Pore geometry control of apparent fractional wetting of a porous medium [9] Weislogel, M.M., Jenson, R., Chen, Y., Collicott, S.H., Klatte, J. and Dreyer, M., The capillary flow experiments aboard the International Space Station: Status. Acta Astronautica, 65(5), pp [10] Hanba, Y.T., Moriya, A. and Kimura, K., Effect of leaf surface wetness and wettability on photosynthesis in bean and pea. Plant, Cell & Environment, 27(4), pp [11] Cueto-Felgueroso, L. and Juanes, R., Nonlocal interface dynamics and pattern formation in gravity-driven unsaturated flow through porous media. Physical Review Letters, 101(24), p [12] Bergstad, M. and Shokri, N., Evaporation of NaCl solution from porous media with mixed wettability. Geophysical Research Letters, 43(9), pp [13] Walker, R.A., Wilson, K., Lee, A.F., Woodford, J., Grassian, V.H., Baltrusaitis, J., Rubasinghege, G., Cibin, G. and Dent, A., Preservation of York Minster historic limestone by hydrophobic surface coatings. Scientific reports, 2, p.880. [14] Ghosh, U.U., Chakraborty, M., Bhandari, A.B., Chakraborty, S. and DasGupta, S., Effect of surface wettability on crack dynamics and morphology of colloidal films. Langmuir, 31(22), pp [15] Martic, G., Gentner, F., Seveno, D., Coulon, D., De Coninck, J. and Blake, T.D., A molecular dynamics simulation of capillary capillary imbibition. Langmuir, 18(21), pp [16] Zhou, M.Y. and Sheng, P., Dynamics of immiscible-fluid displacement in a capillary tube. Physical Review Letters, 64(8), p

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117 Chapter 4: Pore geometry control of apparent fractional wetting of a porous medium [26] Jasak, H., Error analysis and estimation for finite volume method with applications to fluid flow. [27] Lenormand, R., Touboul, E. and Zarcone, C., Numerical models and experiments on immiscible displacements in porous media. Journal of fluid mechanics, 189, pp [28] Ma, S., Mason, G. and Morrow, N.R., Effect of contact angle on drainage and imbibition in regular polygonal tubes. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 117(3), pp [29] Cieplak, M. and Robbins, M.O., Influence of contact angle on quasistatic fluid invasion of porous media. Physical Review B, 41(16), p [30] Rad, M.N., Shokri, N., Keshmiri, A. and Withers, P.J., Effects of Grain and Pore Size on Salt Precipitation During Evaporation from Porous Media. Transport in Porous Media, 110(2), pp [31] MATLAB and Image Processing Toolbox Release 2016B The MathWorks, Inc., Natick, Massachusetts, United States. [32] Rabbani, H.S., Joekar-Niasar, V., Pak, T. and Shokri, N., New insights on the complex dynamics of two-phase flow in porous media under intermediate-wet conditions. Scientific Reports, 7(1), p [33] Datta, S.S., Ramakrishnan, T.S. and Weitz, D.A., Mobilization of a trapped nonwetting fluid from a three-dimensional porous medium. Physics of Fluids, 26(2), p [34] Jadhunandan, P.P. and Morrow, N.R., Effect of wettability on waterflood recovery for crude-oil/brine/rock systems. SPE reservoir engineering, 10(01), pp

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119 Chapter 5 New insights on the complex dynamics of twophase flow in porous media under intermediatewet conditions This chapter has been published in Nature Scientific Reports, volume 7 in This chapter provides detail analysis on the configuration of interfaces in porous media with intermediate contact angles. Digital image of real sand pack obtained from X-ray micro-tomography was used to construct the simulation domain. Later, flow simulations were performed using OpenFOAM (version 2.30) at capillary number Ca = 10-5 under wide range of contact angles. Results from this study demonstrate that the complex interaction of pore geometry and contact angle induces the co-existence of concave and convex interfaces in intermediate-wet porous media. That manifests different interfacial dynamics including i) pinning of convex interfaces, ii) reverse displacement, and iii) interface instability. Furthermore, a recovery efficiency curve of displacement is also presented which indicates maximum recovery of defending fluid when invading fluid is weakly wetting the porous media. Figure 5I. Showing snap-off occurring in pore throats when wetting fluid (blue) invades the porous media and displaces the defending fluid (red) at contact angle θ = 5 o. 119

120 Chapter 5: New insights on the complex dynamics of two-phase flow in porous media under intermediate-wet conditions New insights on the complex dynamics of two-phase flow in porous media under intermediate-wet conditions *Note this chapter has been published in Nature Scientific Reports Rabbani, H.S., Joekar-Niasar, V., Pak, T. and Shokri, N., New insights on the complex dynamics of two-phase flow in porous media under intermediatewet conditions. Scientific Reports, 7(1), p

121 Chapter 5: New insights on the complex dynamics of two-phase flow in porous media under intermediate-wet conditions Abstract Multiphase flow in porous media is important in a number of environmental and industrial applications such as soil remediation, CO2 sequestration, and enhanced oil recovery. Wetting properties control flow of immiscible fluids in porous media and fluids distribution in the pore space. In contrast to the strong and weak wet conditions, pore-scale physics of immiscible displacement under intermediate wet conditions is less understood. This study reports the results of a series of two-dimensional high resolution direct numerical simulations with the aim of understanding the pore-scale dynamics of two phase immiscible fluid flow under intermediate-wet conditions. Our results show that for intermediate wet porous media, pore geometry has a strong influence on interface dynamics, leading to co-existence of concave and convex interfaces. Intermediate wettability leads to various interfacial movements which are not identified under imbibition or drainage conditions. These porescale events significantly influence macro-scale flow behaviour causing the counter-intuitive decline in recovery of the defending fluid from weak imbibition to intermediate-wet conditions. 121

122 Chapter 5: New insights on the complex dynamics of two-phase flow in porous media under intermediate-wet conditions 5.1 Introduction The physics of immiscible two-phase flow in porous media is a subject of intense study in a number of applications including enhanced oil recovery [1], CO 2 sequestration [2], remediation of contaminated aquifers [3], drying of porous media [4] and drug delivery [5]. Wettability, defined as the tendency of a fluid to spread over a solid surface in the presence of another fluid [6], has a significant impact on the dynamics of immiscible displacement [7 10]. The wetting conditions of a solid surface as a result of the relative importance of the adhesive and cohesive forces - can be classified into strong-wet, intermediate-wet, and weakwet. In strong-wet and weak-wet cases, one of the fluids has substantial preferential affinity to a solid surface. Alternatively, if both fluids have similar affinity to the solid surface, the surface is referred to as intermediate-wet. Under strong and weak wet conditions, fluid displacement processes in porous media are referred as drainage and imbibition. In drainage, the defending fluid is the wetting phase, while in imbibition the invading fluid wets the solid surface. The pore-scale displacement mechanisms that have been identified to occur in strong and weak wet porous media are snap-off [11-12], piston like displacement [13], corner flow [8], cooperative pore filling [7], Haines jump [14] and droplet fragmentation [15]. However, pore-scale displacement for intermediate-wet conditions remains less understood, despite the fact that intermediate-wet conditions occur in many natural porous media [9]. Here, we use Computational Fluid Dynamics (CFD) modelling to perform direct numerical simulation of two-phase immiscible fluids displacement in a porous medium, which is designed based on the pore-scale X-ray tomography image of a real sand pack. Performing direct numerical simulations on 2D [16-17] and 3D [18-19] images of real porous media is an advanced tool that allows capturing more detailed fluid dynamics information compared to 122

123 Chapter 5: New insights on the complex dynamics of two-phase flow in porous media under intermediate-wet conditions pore-network modelling approach [20 23], specifically for complex pore morphologies. We present results of direct 2D numerical simulations performed on a wide range of wettability conditions with a particular focus on intermediate-wet condition. Our results demonstrate the co-existence of concave and convex interfaces under intermediate-wet conditions emanated from the interplay between the wetting characteristics and pore geometry. Such a phenomenon promotes (i) pinning of convex interface, (ii) pore-level reverse displacement and (iii)interface instability. These complex yet intriguing pore-scale displacement events provide novel explanations to the classical non-monotonic behaviour of recovery of defending fluid as a function of porous media wettability. 5.2 Direct Numerical Simulation Immiscible two-phase flow is simulated through a heterogeneous 2D porous medium (patterns presented in Supplementary Information, Fig. 5S1). The Navier-Stokes equation coupled with Volume of Fluid algorithm (interface tracking approach) is numerically solved using OpenFoam (Open Field Operation and Manipulation). Complete information on the equations governing multiphase flow in porous media is provided in Supplementary section. The wettability of porous media is defined by the contact angle, θ, between the fluid-fluid interface (through the invading phase) and the grain surface, which is an input parameter to the solver. A series of numerical simulations are performed with different θ values ranging from 5 to 140. Contact angle ranging from 5 to 15 represents strong-imbibition, from 30 to 45 indicates weak-imbibition, from 60 to 100 shows intermediate-wet and from 120 to 140 represents drainage condition. In order to eliminate the effect of contact angle hysteresis, the advancing and receding θ are kept equal, resulting in uniform distribution of contact angle across the simulation domain. The invading fluid (with the viscosity of

124 Chapter 5: New insights on the complex dynamics of two-phase flow in porous media under intermediate-wet conditions Pa.s) is injected in the porous medium initially saturated with defending fluid (with the viscosity of Pa s) at a constant flow rate of 1.8ml/ hr for 6.5 s. 5.3 Result and Discussion Intermediate-wet porous media and interface dynamics The capillary forces in intermediate-wet porous media are weak. This leads to occurrence of various interfacial phenomena that are not present in strong and weak wet porous media. The key interfacial feature observed under uniformly distributed contact angle in the range of is the presence of both concave and convex interfaces. This is illustrated in Fig. 5.1(a). Figure 5.1. (a) The main interfacial features observed during immiscible two-phase flow in intermediate-wet porous media (θ = 60 ) at 2.8 s. (b) Curvature distribution of interfaces shown in 124

125 Chapter 5: New insights on the complex dynamics of two-phase flow in porous media under intermediate-wet conditions Fig. 5.1(a). (c) Dynamics of concave (labelled as 1 ) and convex (labelled as 2 ) interfaces during displacement in the porous medium with θ = 60. Pinning of convex interface and reverse displacement mechanism as a result of co-existence of concave and convex interface is observed. (d) Interface instability in a single pore. In the phase distribution shown in Fig. 5.1(a,c-d), red, blue and green represents defending fluid, invading fluid and the fluid-fluid interface, respectively. The pressure field shown in Fig. 5.1(c d) indicates the pressure values normalized with respect to the outlet pressure. The direction of injection in all images is from bottom to top. The co-existence of both concave and convex interfaces stems from the increasing dependence of the interface morphology on the angularity of pores (the angle at which a pore converges or diverges) as the wettability changes from strong-wet to intermediate-wet conditions. In other words, the complex interplay between contact angle and pore angularity influence the direction of capillary forces leading to the variations of the interface curvature [24-26]. Statistical analysis of interface curvature presented in Fig. 5.1(b) illustrates the comparison between the positive (convex) and negative (concave) curvature under intermediate-wet conditions. The co-existence of concave and convex interfaces influences the displacement dynamics and overall flow pattern at micro and macro-scale. The micro-scale interface topology has been illustrated in Fig. 5.1(c-d). Figure 5.1(c) shows that while the concave interface (interface 1) is displaced upwards, the convex interface (interface 2) is temporarily pinned at the junction of pore body. The behaviour of the interfaces 1 and 2 can be explained using the computed pressure fields presented in Fig. 5.1(c). Due to the contrast in the morphology of the interfaces 1 and 2, the pressure gradient developed within the invading phase causes the preferential flow of invading phase towards interface 1. This restricts the displacement of interface 2 as shown in Fig. 5.1(c). Over time, pressure gradient across interface 2 declines; 125

126 Chapter 5: New insights on the complex dynamics of two-phase flow in porous media under intermediate-wet conditions as a consequence, the interface is forced back into the pore throat against the direction of the main stream flow. Such a mechanism has been observed in Berg et al. [14] and Joekar- Niasar et al. [27]. We refer to this phenomenon as the reverse displacement. It is important to note that as interface 2 enters the pore throat, its curvature changes from convex to concave. This analysis demonstrates the impact of pore angularity in dictating the curvature of the interface for the contact angle θ of 60. The obtained high resolution numerical results allow us to investigate another complex interfacial process occurring in intermediate-wet porous media that is related to the instability of interface in a single pore (Fig. 5.1(d)). As explained before, in the presence of intermediate-wet condition, the curvature of an interface can change from convex to concave or vice versa. Figure 5.1(d) illustrates that such morphological transformation of interface is not spontaneous, but occurs through an intermediate stage where the interface is instable. The morphology of the instable interface is significantly different from its stable counter parts that are concave and convex. Figure 5.1(d) shows that across one single interface, the sign of capillary pressure (defined by the difference between the pressures across the interface) changes. At macroscopic-scale, this will lead to non-uniform distribution of the capillary pressure. The instable interface depicted in Fig. 5.1(d) manifests that near the pore wall 1, the interface is convex, while at the pore wall 2 the interface is concave. The sharp variation in the curvature of interface induces pressure gradient within invading phase similar to what has been discussed before. However, unlike the case illustrated in Fig. 5.1(c), both concave and convex sides of the interface are attached and facilitates the movement of each other exhibiting cooperative behaviour. As a result of the pressure gradient, the invading phase 126

127 Chapter 5: New insights on the complex dynamics of two-phase flow in porous media under intermediate-wet conditions tends to flow from high pressure region (convex) to low pressure region (concave) indicated with black arrows in Fig. 5.1(d). This ceases the advancement of convex interface momentarily, but provides impetus for the concave interface to move forward Non-monotonic recovery of defending fluid as a function of wettability Under intermediate-wet conditions, interaction of interface with pore surface leads to the coexistence of concave and convex interface (Fig. 5.1(a)) which has been observed in different pores (Fig. 5.1(c)) and even within a single irregular pore (Fig. 5.1(d)). To investigate the influence of these displacement events on the macroscopic flow behaviour, we quantified the recovery efficiency of the defending fluid as a function of the wettability of porous media with the results being presented in Fig

128 Chapter 5: New insights on the complex dynamics of two-phase flow in porous media under intermediate-wet conditions Figure 5.2. (a) Fluid phase and pressure distribution under different wetting conditions at the end of simulation. White colour represents pathway of invading phase. Pressure is normalized with respect to the outlet pressure and it indicates the pressure in the defending phase. (b) Distribution of blobs size of defending fluid under different wettability scenarios. The inset illustrates the maximum blob size as a function of the contact angle. (c) The non-monotonic dependency of the defending phase recovery on the wettability of porous media. 128

129 Chapter 5: New insights on the complex dynamics of two-phase flow in porous media under intermediate-wet conditions Figure 5.2(a) shows the distribution of phases under different wetting conditions. Visual inspection of this figure along with Fig. 5.2(b) shows that under intermediate-wet conditions the blobs of defending fluid are more widespread compared to other wetting conditions which might be attributed to the interface coalescence. Furthermore, the recovery efficiency of the defending fluid (the area represented by white in Fig. 5.2(a)) as a function of the wettability of porous media is quantified and shown in Fig. 5.2(c). Traditionally, the contact angle measured on flat surface is known to be a major indicator of change in wettability of porous media [8], which can be mathematically defined according to Young-Dupre law, i.e. σ 1 = σ o cos(θ) + σ 2 where σ 1 is the surface tension between defending fluid and solid surface, σ o is the interfacial tension between invading and defending fluid and σ 2 is the surface tension between invading fluid and solid surface. Since on the basis of Young-Dupre law, the capillary forces are the weakest under intermediate-wet conditions (or to be more specific at contact angle θ of 90 ), one may expect the highest recovery efficiency under intermediate-wet condition. However, our results do not support this conclusion. As indicated in Fig. 5.2(c), the recovery efficiency is a non-monotonic function of wettability of porous media, but the highest recovery efficiency is found to be under weak imbibition conditions. Figure 5.2(c) shows that the recovery of defending fluid reduces when the contact angle θ increases from 45 to 100 which is counter intuitive. Although, the trend indicated in Fig. 5.2(c) has been observed previously by Ryazanov et al. [28] and Zhao et al. [8], the underlying physical processes were remained elusive which are discussed next. Our numerical results delineate the underlying mechanisms of the counter-intuitive decline of defending-fluid recovery from the weak imbibition to the intermediate-wet condition. We found that this non-monotonic behaviour is governed by a critical contact angle θ c. The 129

130 Chapter 5: New insights on the complex dynamics of two-phase flow in porous media under intermediate-wet conditions critical contact angle when the arc interface (i.e. the interface residing in corners of pores) is flat is a function of the corner angle [29]. The relationship between corner angle of pore and critical contact angle θ c can be mathematically defined as θ c = π β, where β is the corner 2 angle. For a typical micro-model (which is the simulation domain of present investigation), β = 90 which results in θ c = 45. Detailed analysis of the role of corner angle on capillary pressure and interface dynamics under various wetting conditions has been presented in Ma et al. [29] and Rabbani et al. [30] thus not repeated here. According to Fig. 5.2(c), the maximum recovery in our system occurs at weak imbibition condition (at the contact angle close to 45 ) which is indeed in agreement with the microfluidic experimental results reported in Zhao et al. [8] and 3D investigation performed by Singh et al. [31]. Furthermore, Fig. 5.1 suggests that under intermediate-wet conditions (θ = ), pore angularity (i.e. converging-diverging angle) plays a crucial role in dictating the curvature of the interface. Different direction of capillary forces acting along the interfaces that are residing in different pores induces dramatic decline in the mobility of convex interface which eventually reduces the recovery efficiency (Fig. 5.1(c)). Although, interface instability shown in Fig. 5.1(d) can be regarded as a phenomenon that inhibits the entrapment of defending phase (due to cooperative behaviour of concave and convex interface), its influence is localized within single pores. In contrary to the interface instability, the effects of pinning of convex interfaces and reverse displacement phenomena (as a consequence of pinned convex interface) shown in Fig. 5.1(c) dominate the dynamics of displacement in intermediate-wet condition. Since the conventional Young-Dupre law does not accommodate the role of pore geometry (corner angle and converging-diverging angle), the characterization of recovery efficiency curves by mere definition of wettability based on the flat surfaces can be misleading and can 130

131 Chapter 5: New insights on the complex dynamics of two-phase flow in porous media under intermediate-wet conditions obscure the true physics controlling the recovery curve (as illustrated in our results obtained by the direct numerical simulation). 5.4 Summary and conclusions Wetting characteristics of porous media significantly influence multiphase flow and transport processes. In the present study, we conducted a comprehensive series of investigation by means of direct numerical simulation to delineate the pore-scale mechanisms controlling immiscible two-phase flow in porous media under different wettability scenarios with a particular focus on intermediate-wet conditions which has been rarely discussed in literature. The present pore-scale analysis helps to rationalize the physics governing some of the unexplained previous observations [8,28]. With the current experimental tools available, it is not feasible to experimentally observe some of the effects induced by the wettability condition which ultimately determine the dynamics of displacement in porous media (such as the pressure field developed at pore-scale influencing the interface dynamics as illustrated in Fig. 5.1(c-d)). Inspection and visualization of our numerical results enabled us to gain insights on the complex pore level dynamics controlling the displacement mechanisms as a function of wetting properties of porous media and the resulting macroscopic displacement patterns that emerge. Our numerical results revealed a non-monotonic dependence of defending fluid recovery on the wetting characteristics of porous media with the recovery efficiency being the highest under the weak imbibition condition. At pore-scale, our results confirms the presence of both concave and convex interfaces under intermediate-wet conditions. We show that for a uniform contact angle, both concave and convex interface exists in heterogeneous porous media. This co-existence of concave and convex interface leads to several interfacial 131

132 Chapter 5: New insights on the complex dynamics of two-phase flow in porous media under intermediate-wet conditions processes influencing the dynamics of multiphase flow. The illustrated processes including pinning of convex interface and reverse displacement causes decline in the recovery efficiency of defending fluid. Furthermore, we illustrate that linking the contact angle measured on flat surfaces to the recovery efficiency of defending fluid is not sufficient to describe the governing mechanisms and that the geometry of pore is another important parameter which must be taken into consideration that controls the recovery efficiency. 5.5 Materials and methods Simulation domain We have used pore-scale images obtained by 3D X-ray micro-tomography of a real sand pack [32] as the simulation domain. The 2D image that was used for simulation in the present study is shown in Fig. 5S1 of the Supplementary information which illustrates the grain arrangement at the central cross section of the sand pack. More information about the pore and grain size is given in Table 5S1 of the Supplementary information. Rhinoceros (CAD software) was used to extract the pore network skeleton from digital images of the porous medium which was imported into the simulator as an STL (STereoLithography) file. The numerical domain was first converted into triangulated surface geometry, which was later discretised into small elements by means of the mesh generator in OpenFoam [16]. The final arrangement of these elements was almost unstructured, near the grain surface it was split-hexahedrals and hexahedrals elsewhere [16]. The meshing algorithm employed in this research has been successfully used by Ferrari et al.[33]. According to the grid independence analysis performed in Rabbani et al. [30], the optimum size of the spatial element chosen for the computational domains scaled with respect to the average pore size was

133 Chapter 5: New insights on the complex dynamics of two-phase flow in porous media under intermediate-wet conditions Validation of the numerical simulation In addition to the numerical simulations, microfluidics experiments were conducted to evaluate the performance of the numerical model. A micromodel was fabricated using the same pore-scale 2D image obtained by 3D X-Ray micro-tomography of a sand pack. The micromodel was fabricated in a silicon wafer using standard photolithography and inductively coupled plasma-deep reactive ion etching (ICP-DRIE) methods. Further detail about the fabrication procedure can be found in Willingham et al. [34]. The contact angle of the micromodel was The micromodel was saturated with PMX 200 Silicone Fluid having viscosity of Pa s (provided by Dow Corning) at flow rate of 100 ml/hr and then displaced by de-ionized water at 1.0 ml/hr. Dynamics of the displacement was recorded using an optical microscope (Leica M205C, 20.5:1 zoom, µm resolution, equipped with a Leica DFC 3000G high resolution digital camera). More detail about the experimental procedure can be found in Rodríguez de Castro et al. [35-36]. We have quantified the distribution of the trapped blobs of the defending fluid obtained by the simulation and experiment (results presented in Fig. 5.3). 133

134 Chapter 5: New insights on the complex dynamics of two-phase flow in porous media under intermediate-wet conditions Figure 5.3.Comparison between the blob-size distributions computed numerically and the ones measured by the microfluidic experiments for the fluids PMX 200 Silicone Fluid and water with water injection rate of 1.0 ml/hr. The comparison shows that the numerical prediction slightly underestimates the experimental results. A possible explanation for this discrepancy could be related to the edges of the grains. In the micromodel, the grains could have some roughness, which is not presented in the domain used for the numerical simulation. Roughness of the grains enhances the entrapment of smaller blobs. Other possible reasons of this discrepancy could be related to the measured contact angle of the micromodel as well as the experimental values obtained based on the segmented images. These could be possible sources of the difference observed between numerically determined residual saturation (30.4%) and the experimentally measured value (32%). 134

135 Chapter 5: New insights on the complex dynamics of two-phase flow in porous media under intermediate-wet conditions Supplementary information The porous medium used for numerical modelling in the present study is shown in Fig. 5S1. Figure 5S1. The porous medium (numerical domain) used for simulation of immiscible displacement under different wettability conditions. Black and white colours represent void and solid phase, respectively. The porous medium is based on the pore-scale image obtained by X-ray tomography of a sand pack [32]. In this research direct numerical simulation was performed using C++ library called OpenFoam (Open Field Operation and Manipulation).The pore scale dynamics of immiscible displacement is governed by mass and momentum equation, as follows 30 ; Mass balance. u = 0 (5.1) 135