Muzemder, ATMSH., M.S. December 2020 Geology PORE-SCALE GEOMETRY and INTRA-PORE TORTUOSITY CONTROLS on FLOW ENHANCEMENT

Muzemder, ATMSH., M.S. December 2020 Geology

PORE-SCALE GEOMETRY AND INTRA-PORE TORTUOSITY CONTROLS ON FLOW

ENHANCEMENT DUE TO BOUNDARY SLIP (72 pp.)

Thesis Advisor: Kuldeep Singh

ABSTRACT

At the solid-liquid interface in porous media, fluid velocity profiles can be significantly modified in the existence of a slip boundary condition. The boundary slip, in contrast to conventional no-slip boundary condition exists due to the hydrophobic nature of media grains and can significantly control the flow behavior in media with small pore-throat, e.g. siltstones and shale. The degree of slip is represented by slip-length which is the distance from the wall- fluid interface to where the linearly extrapolated fluid velocity profile reaches the velocity of wall or u = 0. Most pore-scale studies to date, presume a no-slip boundary condition, and a few who have investigated aspects of slip-boundary presume a simplified straight tube or a non- tortuous pore domain, such as in pore-network modeling. Therefore, how the degree of tortuosity and pore-geometry control flow enhancement in porous media during slip flows remains unexplored. In this computational study, we design two distinct types of pore geometries;

1) Diverging-converging pores (DCST) which consists of variable amount of tortuosity resulting from a staggered packing of sediment grains, 2) capillary tubes which take form of; a) sinusoidal and b) helical shapes with variable amount of tortuosity. To represent size variations in pore-throats (d), domains are isotropically scaled in log-space for all DCST and capillary pore channels with d between 0.1 µm to 100 µm, which are obtained from fine siltstones to sandstone

reservoirs. By using Comsol Multiphysics FEM simulator, Navier-Stokes (N-S) equation accounting for no-slip and slip boundary conditions are solved numerically for examining the modifications in flow behavior. To quantify flow behavior modifications and its causes, we examine variations in velocity, permeability, and energy dissipation form the of pore domains.

Results demonstrate that pore geometry has a great contribution to modifying the emergent

Darcy flow behavior and flow enhancement factor (E) with differences over several order of magnitude as systematically increase in boundary slip from no-slip condition. In DSCT pores, flow enhancement factor shows limited flow enhancement of ‘S’-shaped characteristics curve in log-log space with high or low boundary slips. In comparison, in capillary sine and helix pores, E increases linearly in an unlimited manner in log-log space with an increase of boundary slip.

Furthermore, pore-throat sizes have a great control on how a given slip length enhances effective flow. Larger pore-throats have a marginal impact on flow enhancement whereas smaller pore- throats (i.e., < 10 µm) have substantial influence on the flow enhancement. In DCST pores, the diverging-converging nature of pores offers an increase in energy dissipation, resulting resistance to flow and therefore, contribute to the asymptote in the flow enhancement factor in low to high boundary slips. The energy dissipation rate (ɛ) of capillary straight tube pores (both in distribution and magnitude) remains stable when no-slip or slip boundary conditions are employed. However, as tortuosity increases, i.e., uncompacted sine and helix pores, there is a marginal amount of flow resistance offers, which leads to an infinite rise in the flow enhancement. Outcomes of this study may relevant to the applications of contaminant transport in aquifers, storage of chemical wastes, hydrocarbon recovery and many other engineering applications.

PORE-SCALE GEOMETRY AND INTRA-PORE TORTUOSITY CONTROLS ON

FLOW ENHANCEMENT DUE TO BOUNDARY SLIP

A thesis submitted

to Kent State University in partial

fulfillment of the requirements for the

Degree of Master of Science

A T M Shahidul Huqe Muzemder

December 2020

Except for previously published materials

Thesis written by

A T M Shahidul Huqe Muzemder

B.S., Shahjalal University of Science and Technology, 2012

M.S., Kent State University, 2020

Approved by

Kuldeep Singh , Advisor

Daniel Holm , Chair, Department of Geology

Mandy Munro-Stasiuk , Interim Dean, College of Arts and Sciences

TABLE OF CONTENTS

TABLE OF CONTENTS…………………………………………………………………... v LIST OF FIGURES………………………………………………………………………... vii ACKNOWLEDGEMENTS………………………………………………………………... ix CHAPTER 1: INTRODUCTION………………………………………………………….. 01 1.1 MOTIVATION AND PROBLEM STATEMENT…………………………………. 01 1.2 THESIS GOAL AND RESEARCH OBJECTIVES……………………………...... 06 1.3 SIGNIFICANCE OF RESEARCH…………………………………………………. 08 1.4 THESIS OUTLINE…………………………………………………………………. 08 CHAPTER 2: METHODOLOGY…………………………………………………………. 09 2.1 PORE GEOMETRY DESIGN……………………………………………………… 09 2.1.1 DIVERGING-CONVERGING PORES…………………………………... 10 2.1.2 CAPILLARY TUBES…………………………………………………….. 11 2.1.2.1 SINUSOIDAL CAPILLARY……………………………… 11 2.1.2.2 HELICAL CAPILLARY………………………………...... 12 2.2 COMPUTATIONAL FLUID DYNAMICS (CFD)………………………………… 12 2.3 DARCY’S LAW……………………………………………………………………. 13 2.4 DIMENSIONLESS PARAMETERS……………………………………………….. 14 2.4.1 GEOMETRIC TORTUOSITY……………………………………………. 14 2.4.2 REYNOLDS NUMBER…………………………………………………… 15 2.4.3 HYDRAULIC SHAPE FACTOR…………………………………………. 15 2.4.4 FLOW ENHANCEMENT FACTOR…………………………………...... 16 2.4.5 ENERGY DISSIPATION FACTOR……………………………………… 16 CHAPTER 3: RESULTS AND DISCUSSIONS………………………………...... 17 3.1 RESULTS…………………………………………………………………………… 17 3.2 COMPUTED FLOW FIELDS……………………………………………………… 18 3.2.1 DCST……………………………………………………………………… 18 3.2.2 SINUSOIDAL AND HELICAL PORES…………………………………. 20 3.3 FLOW ENHANCEMENT………………………………………………………….. 23 3.3.1 DCST……………………………………………………………………… 23

3.3.2 SINUSOIDAL AND HELICAL PORES…………………………………. 27 3.4 THEORETICAL MODEL FOR FLOW ENHANCEMENT……………………….. 29 3.4.1 STRAIGHT CAPILLARY PORE………………………………………… 30 3.4.2 DCST……………………………………………………………………… 31 3.4.3 APPLICABILITY OF THE THEORETICAL MODELS……………...... 31 3.5 THE COEFFICIENT, C WITH INTRA-PORE TORTUOSITY AND PORE GOMETRY……………………………………………………………………………. 35 3.6 THE LIMITS ON FLOW ENHANCEMENT IN DCST PORES…………………... 37 3.7 PHYSICAL MECHANISM FOR VARIATIONS IN FLOW ENHANCEMENT .... 38 3.7.1 ENERGY DISSIPATION………………………………………………… 39 3.7.1. 1 SPATIAL DISTRIBUTION OF ENERGY DISSIPATION IN DCST PORES…………………………………………………………………………… 39 3.7.1. 2 SPATIAL DISTRIBUTION OF ENERGY DISSIPATION IN CAPILLARY PORES...... 40 3.8 ENERGY DISSIPATION FACTOR ……………………………………………….. 43 3.8.1 DCST PORES……………………………………………………………... 43 3.8.2 SINUSOIDAL AND HELICAL PORES…………………………………. 45 3.9 THE LIMIT OF ENERGY DISSIPATION VS. THE LIMIT OF FLOW ENHANCEMENT………………………………………………………………………. 47 3.10 DISCUSSIONS……………………………………………………………………. 49 CHAPTER 4: SUMMARY………………………………………………………………… 52 REFERENCES ……………………………………………………………………………. 55

LIST OF FIGURES

Figure 1: Representation of different pore geometries. (a) An axis symmetric diverging converging staggered tortuous pore and 2D section of an axis symmetric tortuous pore channel (b) Different tortuous diverging converging staggered tortuous pores, (c) Sinusoidal capillary tube pores, (d) Helical capillary tube pores.…………………………………………….. 10

(τG = 1.02) is minimum, i.e., when pore geometry is close to straight tube and slip length is maximum (b = 10-4 [m]) and with the increases of intra- pore tortuosity, velocity also decreases…………………………………….. 18

straight tube, i.e., (τG = 1) and (m-t) changes in spatial velocity distribution in 3D helix capillary pores. In five different tortuous geometries, velocity is maximum in straight tube (l) when maximum slip length is applied (b =10-4 [m]). As tortuosity increases in both the sine and helix tubes, the average velocity decreases as well..………………………………………… 20

Figure 4: (a-c) represents flow enhancement behavior with respect to different slip lengths (b) for different pore throats diameter ranges from 0.1µm to 100 µm in DCST pores, (e-g) in sine tube capillary pores and (i-k) in helix tube capillary pores. Legends for subfigures (a-k) are show in subfigures (i & g). (d, h, and l) show how all computed flow enhancement for various pore-throats collapse into a single curve as function of non-dimensional slip length (b/d)..……………...... 25

Figure 5: (a) shows how computed flow enhancement factor (E) for various pore- throats (d) (0.1µm to 100 µm) in different tortuous pores show good fit to the equation (19) for DCST pores,(b) for sine tube capillary pores, (c) for helix capillary pores as a function of dimensionless slip length (b/d)……… 33

vii

Figure 6: Relationship between geometric tortuosity (τG) with theoretical constant C (equation 19). a) in DCST pore, b) in sine pore, c) in helix pore…………... 35

Figure 7: Maximum change in in flow enhancement (∆E) and the coefficient C from the theoretical model (equation 19) show exponential relationship with the hydraulic shape factor (β) in different pore throats (d) (0.1µm to 100 µm) for various DCST pore geometries…………………………………………. 37

Figure 8: (a-l) variations in energy dissipation rate, ɛ [Wm-3] in 2D axis-symmetric DCST pore geometries as a result of no-slip (b = 0) to various slip boundary conditions. As intra-pore tortuosity decreases, energy dissipation rate decreases with the increase of slip length, which is opposite to velocity distributions………………………………………………………………… 39

Figure 9: (a-h) an illustration of changes in energy dissipation rate, ɛ [Wm-3] in 3D sine tube tortuous capillary pores; (i-l) variations in energy dissipation rate

in 3D straight tube, (τG = 1) and (m-t) changes in energy dissipation rate in 3D helix capillary pores in respect to no-slip (b = 0) to slip boundary conditions. In straight tube, energy dissipation rate is uniform with the increases of slip length whereas in tortuous capillary pores energy dissipation gradually increases with the increase of intra-pore tortuosity as well as slip length…………………………………………………………... 40

Figure 10: (a) energy dissipation factor (Ꞷ) for various pore throats (d) (0.1µm to 100 µm) in different DCST pores, (b) for sine tube capillary pores, (c) for helix capillary pores as a function of dimensionless slip length (b/d)……………. 43

Figure 11: (a) represents relationship between flow enhancement factor (E) and energy dissipation factor (Ꞷ) for three different DCST pore geometries for different pore throats diameter (d) (0.1µm to 100 µm), (b) for different tortuous sine tube capillary pores and (c) for different tortuous helix tube capillary pores………………………………………………………………. 45

Figure12: Relationship between maximum change in flow enhancement (∆E) and energy dissipation (∆Ꞷ) in different pore throats (d) (0.1µm to 100 µm) for various DCST pore geometries, which is 1:1………………………………. 47

viii

ACKNOWLEDGEMENTS

I would like to express my gratitude to my thesis advisor, Dr. Kuldeep Singh for his inspiring encouragement, constant guidance, constructive criticism, and valuable suggestions during my thesis work. His supreme knowledge of pore-scale simulations has motivated me to pursue this fundamental research. Without his help and support, I would not be able to overcome the difficulties I faced during my study.

I would like to thank the Department of Geology at Kent State University (KSU) for this opportunity and the funding to conduct this research.

I would also like to thank my committee members, Dr. Anne Jefferson, and Dr. Joseph

Ortiz for their helpful suggestions and discussions which were extremely valuable.

Grateful thanks are also extended to Hydro-lab group at KSU (Jacob Bradley and Erika

Hiwiller) for their encouragement and guidance during the research.

Finally, my sincere gratitude goes to my family members and to all my friends in

Bangladesh for their love, compassion and involvement in my past two years residing in the

United States.

Thanks to all of you. Thank you very much!!!

CHAPTER 1: INTRODUCTION

1.1 Motivation and problem statement

The investigation of the physics of fluid flow in porous media is of substantial interest for a wide variety of applications in areas of science and engineering, e.g., in the area of hydrology, infiltration through soils, contaminant transport in aquifers, storage of chemical wastes, and production of hydrocarbons (Dullien, 1979; Pisani, 2011; Vafai, 2012). However, how the physics of fluid-solid interactions in porous media manifest as the continuum flow behavior or the Darcy law is not clearly understood and well represented. The structure of porous media is known for inherent complexity due to diversity of pore geometries and tortuosity. The complexity offered by the pore topology is further magnified by lack of understanding about how fluids interact with the mineral surface, i.e., what is the appropriate boundary condition? We are motivated to address how these relevant complexities found at pore-scale manifest as the continuum flow behavior.

At the solid-liquid interface, the no-slip boundary condition is a well-accepted assumption, which states that the velocity of a fluid in contact with a solid boundary equals the velocity of the solid. If this assumption is invalid the wall molecules have non-zero momentum or the "slip" along the boundary, thereby modifying the profile of velocity and net flow rate.

Based on numerous experiments, a consensus have been developing that the no-slip boundary condition may no longer be a suitable approximation for pore-scale natural flow conditions

(Afsharpoor and Javadpour, 2016; Moghaddam and Jamiolahmady, 2016; Neto et al., 2005;

Priezjev et al., 2005; Rothstein, 2010; Sahraoui and Kaviany, 1992).

In (1823), Navier was first to introduce the slip boundary condition. He presented the prospect of fluid slip which assumes that the velocity, 푈푥 at a solid surface is proportional to the shear rate at the surface,

휕푈 푈 = 푏 푥 (1) 푥 휕푦 where b is the slip-length. Here, for b = 0 the no-slip boundary condition is retrieved. If b is finite, fluid slip happens at the wall, however its effect depends on the length scale of confined flow (Javadpour et al., 2015; Neto et al., 2005).

The slip-length (b) is calculated form laboratory experiments or molecular dynamics

(MD) simulations. Using these methods, slip lengths on the order of 8 nm to 10’s of µm have been reported, i.e. up to a several orders of magnitude greater than the length-scale of confinement (Li et al., 2018; Majumder et al., 2005; Neto et al., 2005; Priezjev et al., 2005;

Voronov et al., 2007; Zhou et al., 2020). Specially, surface wettability (Neto et al., 2005;

Rothstein, 2010; Voronov et al., 2007) and surface roughness (Kucala et al., 2017; Priezjev,

2007) are significant contributors to influencing the magnitude of slip length at the liquid-solid interface. For example, slip length (b) of 8-9 nm was reported on perfectly wetting mica and glass surface (Bonaccurso et al., 2002), in 10 µm deep hydrophilic glass microchannel slip length was noticed around 50 nm (Joseph and Tabeling, 2005), from rough hydrophilic glass surface b of 85 nm was recorded (Ahmad et al., 2017). Similarly, b of 0.1 µm to ~10 µm was found from hydrophobic glass surfaces (Joseph and Tabeling, 2005), and b of 20 µm to ~200 µm are reported from rough-patterned superhydrophobic surfaces (Choi and Kim, 2006).

Natural porous media exhibits a large variation in wettability and surface roughness

(Dullien, 1979; Geistlinger and Zulfiqar, 2020) and hence, resulting in a wide range of slip length in porous media. The influence of slip flow can be negligible if slip length is smaller than pore-throat sizes, but if slip length is greater than the size of pore-throats, slip flow may significantly modify the flow behavior (Singh, 2020). In most aquifers, reservoirs, and soils, the size of pore-throats is on the order of 0.1 µm to 100 µm (Nelson, 2009). However, at what pore- throat sizes, slip flows become significant and therefore, modify Darcy behavior has not been well studied.

The coupled effects of boundary slip and the length-scale of confinement are evaluated by calculating either the pressure loss or the flow enhancement factor which is the ratio of the computed or the measured volumetric flux with boundary slip to an expected volumetric flux with the no-slip boundary condition (Majumder et al., 2005; Mattia et al., 2015; Yu et al., 2015).

The majority of recent research on the effects of boundary slip is focused on fluid hydraulics in carbon nanotubes (CNTs), which show a linear dependence of flow enhancement to a boundary slip. Studies on the effects of boundary slip in porous are limited. A wide body of literature exists about gas slip flow in porous media; however, it focuses on cumulative production with emphasis to Knudsen diffusion; it does not explore whether or not pore geometry could contribute to the continuum scale flow behavior.

Generally, most theoretical studies of fluid dynamics in porous media require some simplification of the pore geometry. At first, the capillary bundles were used to define the theory of fluid flow in porous media (Scheidegger, 1974). More recently, representation of pore geometry using ‘curved’ capillaries with circular and non-circular sections has been consistently

used (Cai et al., 2014; Jackson, 2008; Wang et al., 2018). The usage of straight tube capillary to study the pore-scale effect does not capture the mechanics of fluid-structure interaction, for example, the role of energy dissipation and drag forces are not included in evaluation of continuum scale flow behavior. Therefore, investigations using capillary pores with circular and non-circular sections may only yield partial information about the continuum flow. A more complex representation of pore structure is achieved by combining a network of capillary tubes to pore bodies known as the ‘pore-network modeling’ (Balhoff and Wheeler, 2009; Bryant and

Blunt, 1992; Van Marcke et al., 2010), Pore-network modeling is based on solution scheme which solves Hagen–Poiseuille equation, and therefore, the model are limited in capturing the physics of fluid-pore structure interactions. These fluid-pore structure interactions are key for evaluating how pore geometry offer resistance to flow and manifest as the permeability. As a result, pore network modeling forecasts seldom match with the experimental data (Ghanbarian et al., 2013).

The fluid flow behavior within the porous media is directly dependent on the inherent complex pore geometry (Vanson et al., 2017). The natural pore geometry comprises the interstitial spaces between a pack of sediment grains, which take form of a diverging-converging geometry between pore-throats and pore body and consist of an intra-pore tortuosity. The diverging-converging nature of pore geometry is known to transform the streamlines, which contributes to fluid velocity variability along the streamlines and decreases the permeability

(Dullien, 1979). This diverging-converging nature of pores offer energy dissipation and form drag which has been shown to limit the flow enhancement due to boundary slip (Nelson, 1968;

Pilotti et al., 2002; Popadić et al., 2014) . In comparison, the straight tube capillary models do not capture the physics of fluid-structure interactions and can erroneously predict large amounts

of flow enhancement due to the boundary slip (Sisan and Lichter, 2011; Walther et al., 2013;

Zhou et al., 2016). Studying the effects of natural pore geometry has important implications; it allows us to ascribe the relevant aspects of fluid physics and mechanisms of the pore-scale to representative upscaling.

To better understand the fluid flow phenomenon of porous media, at first, it is important to characterize the pore structures, i.e., pore and pore-throat size distribution and pore morphology of a porous medium, and then capillary pores within the pore structures (Bear and

Bachmat, 1990; Dullien, 1979; Vallabh et al., 2011). In porous media, for example, cap rocks, aquifers, and soils, fluid capillary pores are not linear, rather they are tortuous and take on meandering shape (Bear, 1972; Dullien, 1979; Scheidegger, 1974). The magnitude of tortuosity depends on the amount of mechanical compaction and other diagenetic processes. A large tortuosity may indicate that the fluid will likely take a longer and more complex path that creates a larger flow resistance. Thus, tortuosity has been known to have a significant impact on the transport properties of porous media, which directly has influence on infiltration, storage of wastes, and recovery of hydrocarbons (Bear and Bachmat, 1990; Ghanbarian et al., 2013;

Sobieski et al., 2012).

The concept of tortuosity was first introduced by Carman in the well-known Kozeny-

Carman equation (1937) to account for the tortuous character of flow through a granular bed. Since then, tortuosity is an inferred property of porous media as opposed to a direct measurement. While the concept of tortuosity seems straightforward in the Kozeny-Carman equation, it is one of the most overlooked parameters in rock physics due to a lack of understanding and the lack of proper approaches to quantify tortuosity. Tortuosity calculation involves visualization or accurate tracing of the flow paths in natural rocks without disturbing the samples, something that is difficult to

accomplish. The latter is now possible by using the processes of digital rock physics. The inferred tortuosity is based on empirical relationships utilizing the measured quantity of porosity. Today, we know of different forms of tortuosity, for example, geometric tortuosity, hydraulic tortuosity, electric tortuosity, or diffusive tortuosity (Matyka et al., 2008). Nonetheless, the concept of these forms of tortuosity is identical, and presented in a dimensionless form. Indirect method or an empirical equation is generally used to relate one of the forms of tortuosity based on the nature of physics to evaluate and represent the pore-scale phenomenon. Thus, the direct or the discreet control of tortuosity in modifying the pore-scale fluid flow are not well known.

1.2 Thesis Goal and Research objectives

In this study, we begin with fundamental investigations by studying the role of intra-pore tortuosity in modifying the flow behavior as a result of the boundary slip. To perform these investigations, we need to develop pore-scale simulation frameworks that incudes pore geometry design, pore-scale simulators for single-phase flows and high-performance computing infrastructure.

First, we want to design a set of diverging-converging pore geometries which form due to the staggered packing of sediment grains. The sediment grains are compressed to represent the mechanical compaction which introduces a variable amounts of intra-pore tortuosity. Results from these grain pack geometries are then contrasted to two different forms of capillary ‘tubular’ pores with a circular cross section. The capillary pores also consist of a variable amount of tortuosity, for example, a straight to a sinusoidal capillary pore and a straight to a helical capillary pore.

Second, we want to isotropically scale all the pore domains over three orders in magnitude to represent the pores found naturally from the mudrocks to sandstones or

unconsolidated soils and sands. These pore domains will be used to conduct computational fluid dynamic simulations which includes a sensitivity study about how the boundary slip modifies the flow fields. Such pores will be designed to have a wide variation in pore-throat sizes (i.e., 0.1

μm-100 μm) that are normally present in geologic porous media. A large range in the boundary slip or the slip-length from 10-9 m to 10-4 m, i.e., as reported in literature will be used for the sensitivity study.

Although, existence of slip flow in porous media is a well-known phenomenon

(Moghaddam and Jamiolahmady, 2016; Rothstein, 2010), the relationship among pore geometry, intra-pore tortuosity, and flow enhancement due to boundary slip is still not well understood. In this study we will investigate the research question, “How does tortuosity and pore-scale geometry control the flow enhancement due to various slip flow scenarios in porous media?”

The following hypotheses will be tested in this study:

1. Tortuous pore geometries will limit the flow enhancement with increasing slip-length or

slip flows due to an increase the intra-pore tortuosity in both diverging-converging and

capillary pores.

2. Due to irregular shape of the pore-body, intra-pore tortuosity effects will be more

evident in diverging-converging staggered pores as opposed to capillary pores (i.e.,

sinusoidal, and helical pores).

3. In both diverging-converging and capillary pores, smaller pore throats diameter (i.e., d

<10 µm) will trigger the magnitude of flow enhancement as compared to larger pore

throats (i.e., d ≥ 10 µm).

4. Energy dissipation rate will control the magnitude of flow enhancement in both

diverging-converging and capillary pores.

The objectives of my research are to test the above hypotheses, quantify the impact of

tortuosity and pore-scale geometry for different slip-lengths on fluid flow and quantify how

they control the flow enhancement in porous media.

1.3 Significance of research

Estimation of permeability is challenging due to the small size of the flow channel in tight and shale gas reservoirs. Study of tortuosity with the effect of slip-length in micro and nanoscale porous system will provide an advantageous perspective for the estimation net flow rates in consolidated tight and shale reservoirs, reservoir exploration and development, soil and groundwater pollution control, biological systems, and in the area of micro and nano fluidics systems in civil and mechanical engineering (Kannam et al., 2017; Moghaddam and

Jamiolahmady, 2016; Nair and Sameen, 2015).

1.4 Thesis outline

The outline of the thesis is as follow: Chapter 1 presents introduction of this study includes the motivations, problem statement, research goal and the objectives. Chapter 2 describes how various pore geometries (i.e., diverging-converging, and capillary pores) are designed, the solution scheme of Navier-stokes equations and different dimensional parameters that used in this study. Chapter 3 presents different findings of this numerical study which explained how the magnitude of intra-pore tortuosity contributes to flow modification and flow enhancement due to the boundary slip. Then, detailed discussions of results are presented. Finally, in Chapter 4, main findings of this study are outlined in the summary section.

CHAPTER 2: METHODOLOGY

2.1 Pore geometry design

In sedimentary rocks, the characteristics of the pore geometry begin with the depositional fabric and develop through diagenetic modifications over the geological time-scale. During primary deposition, the sediment grains generally settle in a rhombohedral close packing or the staggered pattern (Figure 1). The resulting pore spaces take the form of a diverging-converging geometry between pore-throats and pore bodies and consist of an intra-pore tortuosity. Previous studies or methods based on simplified geometry were perhaps the need of the hour in absence of computational resources and may have worked well for studying certain aspects of pore-scale fluid physics. However, with recent advancement in computational resources, we take its advantage to investigate the role of rather complex interactions of fluid and the pore structure. In this study, we design two distinct types of pore geometries; 1) Diverging-converging pores which consists of variable amount of tortuosity resulting from a staggered packing of sediment grains, 2) capillary tubes which take form; a) sinusoidal and b) helical shapes with variable amount of tortuosity.

To represent size variations in pore-throats (d), domains are isotropically scaled in log- space for all diverging-converging staggered tortuous and capillary pore channels with d between

0.1 µm to 100 µm, the large variation in the size of pores is included in the pore geometry design to account for a large variation in the size of pores and pore-throats found from fine siltstone to sandstone reservoirs (Nelson, 2009).

2.1.1 Diverging-converging pores

Diverging-converging nature of pore geometry is one of the important characteristics of natural porous media that has significant influence on the transport properties of fluid flow at pore-scale. Thus, for this study, we designed a series of diverging-converging pores formed due to rhombohedral or staggered arrangement of sediment grains in two-dimensions (2D). The 2D grains are aligned along the line of axial symmetry, which allows us to study the three- dimensional (3D) effects of flow around sediment grains. Thus, the resulting pore space consists of pore-throats and pore-body with an intra-pore tortuosity (Figure 1). To attain pore shapes with variable amounts of intra-pore tortuosity, i.e., a set of pores which can represent change in pore space due to mechanical compaction related to the geologic burial processes, the 2D sediment grains are compressed sequentially in the vertical or the z-direction. This process is repeated to obtain 9 domains which consist of pores with tortuosity spanning between 1.02 and 1.75, i.e.,

representative of natural porous media (Ghanbarian et al., 2013). We refer to this set of diverging-converging ‘staggered’ tortuous pores as DCST pores in rest of the manuscript.

2.1.2 Capillary tube pores

Many pore scale studies use a simpler description of pore-scale topology, for example, some use circular vs noncircular section capillary tubes (Dong et al., 1995; Keller et al., 1997;

Ransohoff and Radke, 1988; Watanabe and Flury, 2008); Some use sinusoidal tubes (Nissan,

2016; Selvarajan et al., 1988; Tsangaris and Leiter, 1984); and pore network modeling (Balhoff and Wheeler, 2009; Blunt et al., 2002; Bryant and Blunt, 1992). In all these models, the diverging-converging nature of actual porous media and related effects are overlooked, it is generally used as a simple link between continuum scale properties and microstructure of porous media. In our study, we designed capillary tubes comprised of sinusoidal and helical shape with variable amount of tortuosity, to quantify the tortuosity effect on ‘capillary’ types of pores

(Figure 1), and compared them with the more realistic pore geometries, i.e., diverging- converging staggered tortuous (DCST) pores.

2.1.2.1 Sinusoidal capillary

The sinusoidal 3D tortuous capillary tube pore boundaries are described by,

2휋푟 휋 ℎ(푟) = ℎ̅ + ℎ̀ (푠𝑖푛 − ) (2) 퐿 2 where r is the radial coordinate, h is the average channel height, h ' is the amplitude of pore body, and L is the length of a single pore.

The aspect ratio (푒) of pore is defined as ratio of pore size to pore throat size (Bolster et al.,

2009; Dykaar and Kitanidis, 1996),

2ℎ̅ 푒 = (3) 퐿 and the fluctuation ratio (푎) which is the ratio between the amplitude of the pore fluctuation and mean half aperture of the pore is denoted by,

ℎ̀ 푎 = (4) 2ℎ̅

2.1.2.2 Helical Capillary

The helical pore channel, which is a curve in 3D space composed of x, y, and z components, described by following parametric equations:

푥 = 푟푐표푠(푡), 푦 = 푟푠𝑖푛(푡), 푧 = 푝푡 (5) where the radius of the curve is r, t is the winding angle, and p is the spacing between turns.

The radius of pore throat (d) and straight length (L) of a single pore are the same for all tortuous capillary tube domains where L and d are 1.0⨯10-6 m, 0.1⨯10-6 m, respectively.

2.2 Computational fluid dynamics (CFD)

The study is explained by conducting CFD simulations through numerically solving

Navier-Stokes and continuity equations using finite element method. Under the assumptions of steady state incompressible flow, for single phase fluid Navier-Stokes equation in pore channels is governed by,

∇p = µ ∇2 u – ρ (u. ∇) u, (6)

∇ . u = 0, (7)

where 휌 is fluid density, u is velocity vector in x and r directions, μ is dynamic viscosity and p is total pressure. Standard properties of water, ρ =1000 kg·m-3, and μ = 0.001 Pa·s is used. In the periodic inlet – outlet boundaries, pressure gradient, ∇푝 = 1 (Pa m-1) is applied. The laminar flow

(low number of Reynolds, i.e., Re <1) interface was used in this research to compute the velocity field for the flow of a single-phase fluid passing through the above-mentioned tortuous pore domains.

Initially, boundaries are considered with a no-slip boundary condition and for quantifying the effect of slip flows, a range of slip-length between 10-9 m – 10-4 m are applied at the boundary for different tortuosity. The inlet and outlet boundaries are the periodic boundaries that help to achieve a fully developed flow and a flow field representative from a single pore of an infinite series of pores. A Boundary Layers mesh was adopted on the pore domain walls to accurately capture boundary slip-induced effects of the flow field and accurately determine the physics of the laminar boundary layer. Finite element mesh was used to discretize the pore domain with a wide variance in b and d which are integral to FEM meshing methods. The tortuous pore domain was discretized with a fine mesh size (1.24 × 10-9 m ~ 2.41× 10-8 m) at the middle of the domain and an extremely fine mesh size (0.62 × 10-9 m ~ 1.205× 10-8 m) near the boundaries. Finally, numerical solutions of the incompressible Navier-stokes equation were obtained by using COMSOL Multiphysics software.

2.3 Darcy’s law

Darcy’s law is derived theoretically by using a description of pore spaces at the pore- scale (Berg, 2014; Dullien, 1979; Hubbert, 1957) and is considered valid for the laminar flow with a low Reynolds number, Re <10 (Bear, 2018; Fand et al., 1987). Darcy’s law defines a

linear relationship between fluid pressure gradient and volumetric flux with a mobility term, which consists of permeability and the fluid viscosity. Darcy’s law is given as;

푘 푞 = 훻푝 (8) µ where 푞 is the Darcy velocity, 푘 the permeability, µ is the viscosity, and 훻푝 is the pressure gradient. In this study, the pore domains have similar topology in many ways to the pore spaces used for the theoretical derivation of Darcy’s law. The simulated steady flow fields are on the order of Re <1. We thus, use the linear relationship between the imposed pressure gradient and computed volumetric fluid flux to calculate the permeability following the equation 8.

2.4 Dimensionless Parameters

2.4.1 Geometric tortuosity

In this study, we used Geometric tortuosity (τG) to quantify the relationship between intra-pore tortuosity and flow enhancement, i.e., how intra-pore tortuosity modifies pore-scale fluid flow. Geometric tortuosity (τG) is a dimensionless parameter, which is the ratio of the actual length of the geometric capillary pores across the medium to the straight-line length through the porous medium in the direction of flow (Bear, 1972; Dullien, 1979; Ghanbarian et al., 2013). Geometric tortuosity (τG) calculated as:

퐿 휏 = 푒 (9) 퐺 퐿

where 퐿푒 and L are the actual length of the flow path and the straight length of the flow path.

Geometric tortuosity is always greater than one (τG > 1) and τG =1, which means capillary pores is similar to capillary straight tube pore or the pipe geometry.

2.4.2 Reynolds number

To determine the relative significance of inertial forces to viscous forces and to contrast flow fields between a variety of pore geometries, we used Reynolds number (Re) which is calculated as:

휌 U 푑 푅푒 = a (10) µ

where, Ua is average velocity and d is the pore-throat diameter.

2.4.3 Hydraulic Shape Factor

We used a hydraulic shape factor, β which is a dimensionless metric of pore geometry, based on the hydraulic radius theory (Chaudhary et al., 2013). Hydraulic Shape Factor (β) calculated as:

푆 ⨯퐿 훽 = 퐴 (11) 푉

where 푆퐴 is the surface area of pores, 푉 is the pore volume, and 퐿 is the length of domain. We used 훽 as a metric to quantify the sensitivity of hydraulic properties to pore geometry. Hydraulic shape factor, 훽 is analogous to the specific surface and hydraulic radius concepts (Bear, 1972;

Saeger et al., 1991).

2.4.4 Flow enhancement factor

To investigate how net variations in average velocity within a microscopic pore contribute to effective hydraulic behavior of the continuum-scale as a function of pore geometry, intra-pore tortuosity, a wide variation in the size of pore-throats, and a large variation in the amount of boundary slip, flow enhancement factor, E is used in this study. Flow enhancement factor, E is defined as the ratio of the observed flow rate (in the presence slip) from expected flow rate (with no-slip BC) for a steady state flow of an incompressible fluid, which can be expressed as in terms of permeability (Mattia et al., 2015; Kannam et al., 2017),

푘 퐸 = 푠푙푖푝 (12) 푘푛표−푠푙푖푝 where 푘푠푙𝑖푝 is the average permeability during slip flow, 푘푛표−푠푙𝑖푝 is the average permeability with the no-slip boundary. The flow enhancement factor allows both to assess the permeability magnitude change with an increase in slip flows and analyze the modified permeability adjustments related to pore-throat size variations.

2.4.5 Energy dissipation factor

To quantify how energy dissipation rate (ɛ) may control the flow enhancement factor, E, we define a dimensionless energy dissipation factor (Ꞷ), which is the ratio of energy dissipation rate, (Ꜫ) slip boundary condition to no-slip boundary condition.

Ꜫ Ꞷ = 푠푙푖푝 (13) Ꜫ푛표−푠푙푖푝 where Ꜫ푠푙𝑖푝 is the average energy dissipation rate during slip flow, Ꜫ푛표−푠푙𝑖푝 is the average energy dissipation rate with the no-slip boundary

CHAPTER 3: RESULTS AND DISCUSSIONS

3.1 Results

The results of how the magnitude of intra-pore tortuosity contribute to flow modification and flow enhancement due to the boundary slip are presented. Investigations of the effects of intra-pore tortuosity are presented from two distinct types of pore geometries, i.e., a set of diverging-converging staggered tortuous (DCST) pores and a set of capillary tube pores with sinusoidal and helical shapes. Large variations in the size of pore-throats and the length of boundary slip are considered for its application to diverse porous media. Results from the computational fluid dynamics (CFD) simulations are presented as an investigation of

Low-Reynolds number hydrodynamics interactions of a single-phase fluid with the tortuous structure of the media. The computed velocity fields at the pore-scale allow investigations of how pore geometry and intra-pore tortuosity contribute to microscopic fluid flow behavior, which gets manifested as the flow phenomenon of the continuum-scale.

In the following sections, how velocity varies within pores as a function of pore geometry, geometric tortuosity, and boundary slip are presented. The magnitude of effects is quantified by using the permeability and the flow enhancement factor. A set of theoretical models are presented which can predict flow enhancement for various pore geometries.

Parameters that quantify pore geometry, i.e., tortuosity and hydraulic shape factor are used to determine new empirical equations which can be used to upscale the microscopic flow behavior to a macroscopic scale. Finally, if and how energy dissipation can regulate and clarify

inconsistencies in flow enhancement factor linked to velocity variations inside pores are addressed and discussed. Results from DCST are presented first followed by capillary pores.

3.2 Computed flow fields

3.2.1 DCST

The computed steady-state flow fields from divergent-convergent staggered

tortuous (DCST) pores (d = 0.1µm) demonstrate notable differences in how the velocity

varies in different regions of pores, and due to the transition from a homogeneous no-slip

to a homogeneous slip boundary condition (i.e., the same boundary condition

everywhere; Figures 2a-l ).

Figure 2: (a-l) representation of changes in velocity field in 2D axis-symmetric DCST pore geometries as a result of no-slip (b = 0) to slip boundary conditions. (l) Maximum velocity obtained when intra-pore tortuosity (τG = 1.02) is minimum, i.e., when pore geometry is close to straight tube and slip length is maximum (b =10-4 [m]) and with the increases of intra-pore tortuosity, velocity also decreases.

In uncompacted pores (i.e., τG = 1.54), the velocity varies from negligible flow around the far-end parts of the pore-body (i.e., dead-end pore throats) to the maximum velocity near the flow-through pore-throats. These large variations in velocity within a pore are found to get diminished in pore domains with a decrease in their intra-pore tortuosity, which is meant to represent change in intra-pore tortuosity due to pore deformation related to mechanical compaction. As the boundary slip is increased, higher velocity which is focused near the regions of pore-throats, is found to get the most enhancement relative to velocity in far reaches of the pore-body (Figures 2a-l).

At no-slip or partial slip boundary conditions (i.e., b ≤ 10-8 m), the DCST pores with a decrease in their intra-pore tortuosity are found to have a reduced average velocity (figures 2a,

2b, 2e, 2f, 2i & 2j). For example, for pores with the no-slip boundary condition, i.e., b = 0 and d

= 0.1µm, velocity decreases by a factor of ∼ 1.55 as τG decreases from 1.54 to 1.02. In contrast, with an increase in the boundary slip, i.e., when b > 10-7 m, the average velocity is found to increase with a decrease in intra-pore tortuosity of DCST pores. For example, the 0.1µm DCST pores with a fully-slip boundary condition (i.e., b = 10-4 m), shows an average velocity increase by a factor of 1.77, respectively as τG decreases from 1.54 to 1.02 (Note; the size of pore-throats remains same when considering change in intra-pore tortuosity).

In summary, the intra-pore tortuosity of DCST pores, despite the identical size of pore- throat, is found to play a vital role in modifying the microscopic flow behavior. The decrease in the intra-pore tortuosity reduces the variations in velocity and its magnitude, i.e., when considering the no-slip boundary condition. However, with an increase in boundary slip, the

compacted DCST pores lead to a larger increase in flow; as such, the pore with the least tortuosity is found to have the maximum increase in velocity (Figure 2l).

3.2.2 Sinusoidal and Helical pores

The computed steady-state flow fields from sinusoidal and helical capillary tube pores (d

=0.1µm) show that the velocity variations are limited to the theoretical ‘parabolic profile’. The

‘profile’ velocity is found to increase uniformly in pore domains with the decrease in tortuosity and with an increase in boundary slip (Figures 3a-3t).

Figure 3: Flow-fields showing spatial velocity distribution from no-slip (b = 0) to various slip boundary conditions. (a-h) spatial velocity distribution in 3D sine tube tortuous capillary pores; (i-l) changes in velocity field in 3D straight tube, i.e., (τG = 1) and (m-t) changes in spatial velocity distribution in 3D helix capillary pores. In five different tortuous geometries, velocity is maximum in straight tube (l) when maximum slip length is applied (b=10-4 [m]). As tortuosity increases in both the sine and helix tubes, the average velocity decreases as well.

The decrease in tortuosity of sine and helix pores (i.e., representative of mechanical compaction), contribute to a net increase in average velocity at all boundary conditions, albeit small differences between the two capillary pore shapes. Among the two shapes, sine pores with a lower tortuosity (i.e., τG < 1.1) have relatively higher velocity (Figures 3e & 3m), but at a larger tortuosity (i.e., τG ≥1.1) helix pores have relatively higher velocity (Figures 3d & 3t). With the no-slip boundary condition (i.e., b = 0), the pore with the least tortuosity, i.e., straight tube capillary pore (τG = 1) is found to consist of the largest average velocity relative to sinusoidal and helical pores. For example, in straight capillary tube pore with b = 0 and d = 0.1µm, velocity increase by a factor of ∼ 1.53 relative to the sine pore of τG = 1.52 and by a factor of ∼1.29 as compared to helix pore of τG = 1.51 (figure 3i, 3a & 3q).

With an increase in boundary slip, velocity increases uniformly proportional to the boundary slip in both sinusoidal and helical capillary pores. However, the tortuosity tends to subdue the net increase in velocity with noticeable differences between sinusoidal and helical capillary pores. For example, in 0.1µm size pores with an intermediate boundary slip, i.e.,

-7 b = 10 m, the sinusoidal pores with τG = 1 (i.e., straight capillary tube), τG = 1.05, and τG = 1.54, has velocity increase relative to the no-slip BC by a factor of ~ 8.98, ~ 8.96, and ~ 8.68, respectively. In comparison, helical pores with τG = 1.05, τG = 1.54, show velocity increase by a factor of ~ 8.90 and ~ 8.52, respectively. However, at a full-slip boundary condition, i.e., b = 10-4 m, 0.1µm size sine pores with τG = 1, τG = 1.05, and τG = 1.54, show velocity increase by a factor of ~ 7955, ~ 7065, and ~ 1190, respectively. In comparison, helix pores with τG = 1.05 and

τG = 1.54, show velocity increase by a factor of ~ 5650 and ~ 3430, respectively.

Therefore, with a partial boundary slip, the tortuosity of helical pores tends to subdue the velocity increase more than the sinusoidal pores, albeit these differences are small. However, with a large boundary slip (a.k.a a full-slip), helical pores with a smaller tortuosity (i.e., τG < 1.1 subdue the velocity increase more than the sinusoidal pores. But when the tortuosity is large (i.e.,

τG ≥ 1.1), the sinusoidal pores tend to significantly limit the velocity increase relative to the helical pores. In contrast, the straight tube capillary pores always display the highest velocity at all boundary conditions. The magnitude of difference in velocity increase due to boundary slip between the straight tube capillary pore and helical and sinusoidal pores are, thus, found to increases both with larger boundary slip and tortuosity.

To summarize, in both sine and helix shape capillary tube pores, even though uniform cross section everywhere, intra-pore tortuosity and slip flows play an important role in modifying the microscopic flow behavior. The decrease in intra-pore tortuosity reduces the pressure and friction loss and thus, increase the magnitude of velocity at any boundary slip or no-slip conditions. However, with the increase of boundary slips, velocity increases largely in compacted sine and helix pores. Therefore, the pore with the least tortuosity, i.e., straight tube capillary pore is found to have the maximum velocity at full-slip boundary condition (Figure 3l).

Due to diverging-converging nature of pore channel, intra-pore tortuosity effects are more noticeable in DCST pores as compared to sine and helix pores in no-slip to various boundary slip conditions. At no-slip or partial slip conditions (i.e., b ≤ 10-8 m) in DCST pores, velocity decreases with the decrease of tortuosity and velocity increases at higher slip flows (i.e., b ≥ 10-5 m) with the decrease of tortuosity, whereas in all capillary tube pores velocity increases with the decrease of tortuosity at any slip or no-slip boundary conditions. In addition, velocity

enhancements are uniform everywhere in capillary tube pores, nonetheless, in DCST pores, maximum velocity is found near the pore throats and negligible flow around the far-end parts of pore-body. Moreover, in both DCST and capillary tube pores, least tortuous pores are found to have maximum velocity in fully-boundary slip condition.

3.3 Flow enhancement

At micro- and nanoscale, an increase in average velocity or permeability due the liquid slip at the solid surface is well documented (Holt et al., 2006; Majumder et al., 2005). The ratio of net change in average velocity or permeability from slip to no-slip flows is known as the flow enhancement factor (E; equation-12) (Kannam et al., 2017; Mattia et al., 2015). Thus, the flow enhancement factor (E) is used in this study to investigate how net variations in velocity within a microscopic pore lead to effective hydraulic behavior of the continuum-scale as a function of pore geometry, intra-pore tortuosity, a wide variation in the size of pore-throats, and a large variation in the amount of boundary slip (Figures 4a-l). The flow enhancement factor, E allows us both to quantify the magnitude of changes due to boundary slip (Figures 4a-c, 4e-g & 4i-k) and investigate a normalized variation in E related to size variation of pore throats (figures 4d, 4h

& 4l), and thus, allow a discrete upscaling of the microscopic flow behavior to a macroscopic or

Darcy-scale.

3.3.1 DCST

The calculated flow enhancement factor, E from DCST pores shows that there are insignificant changes for smaller boundary slip relative to the size of pore-throat, i.e., when b << d. However, when b < d to b ~ d, the flow enhancement factor tends to a linear relationship with the boundary slip. And, for a larger boundary slip, this linear relationship is found to get

‘limited’ marked by an asymptote when b > d. This ‘limited’ character of the flow enhancement factor takes on a sigmoidal or ‘S’ -shaped characteristics curve in the log-log space (Figures 4a- c). The ‘S’-shaped characteristics defines an asymptote for both the high or low boundary slip conditions, which indicate negligible flow enhancement both at high and low boundary slips relative to the size of pore-throats, i.e., b >> d and b << d, respectively. Therefore, the size of pore-throats has a significant control in how the microscopic effects of a given boundary slip gets manifested as the continuum flow behavior.

Figure 4: (a-c) represents flow enhancement behavior with respect to different slip lengths (b) for different pore throats diameter ranges from 0.1µm to 100 µm in DCST pores, (e-g) in sine capillary pores and (i-k) in helix capillary pores. Legends for subfigures (a-k) are show in subfigures (i & g). (d, h, and l) show how all computed flow enhancement for various pore- throats collapse into a single curve as function of non-dimensional slip length (b/d).

The magnitude of changes in the flow enhancement factor due to the boundary slip vs. the size of pore-throats can be accounted for by introducing a dimensionless slip length parameter defined as b/d. The usage of b/d allows us to collapse the variations in E due to b and d on to a single curve. Figure 4d shows an example of such ‘collapsed’ curves (i.e., collapsed from the curves shown in Figure 4a-c) from three DCST pores with different intra-pore tortuosity which together display the relationship between the flow enhancement factor, E as a function of b/d in a dimensionless log-log space. This usage of dimensionless space allows to evaluate the sensitivity of cumulative effects associated with modification in microscopic flow to relative differences in the boundary slip and the size of pore-throats. Here, E, which clearly demonstrates the sigmoidal or ‘S’ -shaped characteristics, is found to be trivial for a system when b/d < 10−1. It, however, shows a linear relationship for 10−1 < b/d < 101 which tends to asymptote for b/d ~1 and remains unchanged for b/d > 10 (Figure 4d). The onset of a linear trend, the slope of the linear relationship, and the magnitude of an asymptote in E are all found to be directly dependent on the intra-pore tortuosity.

The intra-pore tortuosity of the DCST pore is found to magnify the response of the flow enhancement to changes in boundary slip and size of pore-throats. With the decrease in intra- pore tortuosity, which is meant to represent microscopic changes in pore topology related to mechanical compaction, the flow enhancement factor, E shows an amplified response to changes in b/d. For a pore system with b/d < 10−1, the trivial (i.e., < 10%) changes in E becomes increasingly non-trivial (i.e., ∼ 30-70%) for compacted pores with a smaller tortuosity. Likewise, the slope of a linear relationship for 10−1 < b/d < 101 increases for the lower tortuosity pores.

Furthermore, the maximum increase in flow enhancement, i.e., the difference in E from the

‘trivial’ to an asymptote value (Figure 4d), is found to increase non-linearly, i.e., in a amplified manner, between uncompacted pores of τG 1.54 to the compacted pores of τG 1.05 (Figure 4d).

In summary, the intra-pore tortuosity is found to play a vital role in how the microscopic effects of boundary slip gets manifested as the continuum flow behavior or the flow enhancement factor. The sigmoidal or ‘S’ -shaped characteristics of the flow enhancement factor found from the DCST pores can be better evaluated for the relative controls of the boundary slip vs. the size of pore-throats by using a dimensionless slip length, i.e., b/d. The usage of b/d provides thresholds which can be used to determine the relative significance of a probable boundary slip for a known porous media of some average pore-throat size. Lastly, the intra-pore tortuosity of the DCST pore is found to magnify the nature of flow enhancement related to boundary slip. With the decrease in intra-pore tortuosity or an increase in the mechanical compaction, the flow enhancement factor is found to display an amplified response to changes in dimensionless slip length, i.e., b/d.

3.3.2 Sinusoidal and Helical pores

With a decrease in intra-pore tortuosity (i.e., representative of mechanical compaction) from 1.52 to 1, there is no change observed in the magnitude of E in both sine and helix pores

(Figures 4f, 4g, 4j & 4k). However, at higher slip flows (i.e., b >10-6 m), with the decrease of intra-pore tortuosity, E increases significantly in both sine and helix pores. For example, in a sine pore with the full-slip boundary conditions, i.e., b =10-4 m and d = 1µm, E increases by a factor

-4 of 1.62 as τG decreases from 1.52 to 1; in helix pores, at b =10 m and d = 1µm, E increases by a factor of 1.14 as τG decreases from 1.51 to 1.05.

The change in flow enhancement factor in different capillary tube pores is further assessed by using a dimensionless slip length, b/d which allows us both to collapse the variations in E due to b and d on to a single curve and to evaluate the sensitivity of cumulative effects associated with modification in microscopic flow to relative differences in the boundary slip and the size of pore-throats. Figures 4h and 4l represent the relationship between the flow enhancement factor, E with dimensionless slip length (b/d) for different tortuous sine and helix pore geometries, where the concept of asymptote is nearly vanished in tortuous sine and helix pores. Furthermore, E is inconsequential (i.e., < 10%) for b/d < 10−1 and becomes significant

(i.e., ∼ 30-70%) and linearly increases for b/d > 10-1. Therefore, the concept of maximum change in E is no longer applicable in sine and helix pores. However, comparing sine and helix pores, at higher tortuosity, i.e., τG > 1.3, the magnitude of flow enhancement is always found to be larger in helix pores as opposed to sine pores for all b/d > 10-1 conditions.

To sum up, the depiction of porous media by capillary sine, and helix tube pore geometries as compared to a more realistic DCST pore geometry to analyze slip flow characteristics or the upscaling of relevant hydraulic parameters can lead to several order of magnitude over predicting flow enhancement, flow rate and pressure loss. Furthermore, intra- pore tortuosity effects are more noticeable in DCST pores due to diverging-converging nature of flow path as opposed to different capillary tube pores and larger slip length always increases the magnitude of flow enhancement in all types of pore geometries. Moreover, smaller pore-throats

(i.e., d < 10 µm), in both DCST and capillary tube pores trigger the magnitude of flow enhancement as compared to larger pore throats (i.e., d ≥ 10 µm).

3.4 Theoretical model for flow enhancement

For understanding the dynamics of fluid flow at micro-and nanoscale, i.e., if and how different variables modify the continuum flow behavior, it is important to use a theoretical model. In addition, theoretical models not only capture the physics of fluid flow, they allows to predict the flow enhancement and provide researchers an opportunity to consider their shortcomings.

In this study, the proposed theoretical model for flow enhancement is based on the

Haagen–Poiseuille equation for steady state, laminar flow (i.e., Re<<1) through a diverging- converging pore and straight capillary tube pore with a uniform cross section considering boundary slip. Finally, it allows us to validate our numerical results and compare with previous outcomes. Furthermore, theoretical model for flow enhancement is an idealization of a real porous media at microscale, which has ramifications for improving the predictive nature of macro-scale models and therefore, it can be used in representative upscaling of pore-scale phenomenon. Moreover, the predictive ability of the proposed flow enhancement model can be useful for contaminant transport in aquifers, storage of chemical wastes, hydrocarbon recovery and many other engineering applications.

3.4.1 Straight capillary pore

Considering no-slip boundary condition, fluid flow in a capillary pore is given by the

Hagen-Poiseuille (HP) equation:

휋푟4 ∆푃 푄 = (14) 퐻푃 8휇 퐿

where 푄퐻푃 is volumetric flux with a no-slip boundary condition, 푟 is the radius of capillary pore, and 퐿 is the length of the pore. For a slip boundary condition with a prescribed slip length (b), fluid flow in a capillary pore is given by a modification to the Hagen-Poiseuille (HP) equation

(Panigrahi and Asfer, 2008):

푏 푄 = 푄 (1 + 4 ) (15) 푠푙𝑖푝 퐻푃 푟

where 푄푠푙𝑖푝 is volumetric flux with a slip boundary condition of the slip length, b. Using the equation (14) and (15) in combination to the definition of equations (12), respectively, flow enhancement factors E for the capillary pore or the pipe geometry, i.e., 퐸푝 can be calculated as;

푏 퐸 = 1 + 4 (16) 푐 푟

3.4.2 DCST

Fluid flow in a diverging-converging tortuous pore, i.e., flow through a system of pore- throat and pore body with a no-slip boundary condition is given after (Roscoe, 1949; Weissberg,

1962) as;

8휇퐿 퐶휇 푄 = ∆푃( + )−1 (17) 푛표−푠푙𝑖푝 휋푟4 푟3

where 푄푛표−푠푙𝑖푝 is volumetric flux at the outlet boundary of a diverging-converging tortuous pore and 퐶 is a constant with a value of 3 for small length pore throats (Popadić et al., 2014;

Weissberg, 1962). Fluid flow with boundary slip from diverging- converging tortuous pore, i.e., a geometry that has a thin pore- throat which diverges in a larger pore body can be calculated as

(Sisan and Lichter, 2011),

8휇퐿 퐶휇 푄 = ∆푃( + )−1 (18) 푠푙𝑖푝 4휋푟3푏+휋푟4 푟3

where 푄푠푙𝑖푝 is volumetric flux of a diverging-converging tortuous pore with a slip boundary condition of slip length, b, and 퐶 once again is the same constant as in equation (17). Using the equations (17) and (18) in combination to the definition of equation (12), respectively, flow enhancement factors and for diverging-converging tortuous pore, i.e., can be calculated as;

1 퐶휋 −1 퐸퐷퐶 = ( 푏 + 퐿 ) (19) 1+4 8 푟 푟

Here, if the coefficient C is 0, equations 19 transforms into equation 16, i.e., theoretical flow enhancement equation for straight capillary tube pore and the coefficient C is a function of pore geometry, and different values of C are a manifestation of differences in magnitude of the bending of streamlines in different tortuous pores.

3.4.3 Applicability of the theoretical models

The applicability of the theoretical models to predict the flow enhancement is validated from a variety of DCST, sine, and helix pores. The results of if and how the theoretical models fit to computed data are presented as a function of dimensionless slip length, i.e., b/d which include both a large variation in size of pore-throats, d and a large variation in boundary slip, b.

We find that the theoretical equation (19) can be used to predict the computed flow enhancement from DCST pores. Figure 5a shows the fit of equation (19) to a variety of DCST pores which consist of different amounts of intra-pore tortuosity. Different values of the coefficient C (listed in Figure 5a) are used in equation (19) to fit to the computed flow enhancement data from different tortuosity DCST pores. The coefficient C is known to be a function of pore geometry, and different values of C are a manifestation of differences in magnitude of the ‘bending of streamlines’ which are a result of energy dissipation when flow transits from a pore-throat to a pore-body (Dagan et al., 1982 ; Gravelle et al., 2014). Thus, the values of coefficient C are likely a function of intra-pore tortuosity – an aspect which is further explored in the section 3.4.

As defined by theory, the theoretical equation (16) for the straight tube capillary pore is found to fit well to the computed flow enhancement data (Figures 5 b & c: τG =1). The equation

(16) also fits well to the computed E data from both sine and helix pores, but only when the dimensionless slip length, i.e., b/d < 101. The fit tends to diverge from the computed E data when b/d > 101. We find that this deviation is both dependent on the shape of tortuous capillary pore, i.e., sine vs helix pores, and its magnitude is found to be dependent on the amount of tortuosity.

The magnitude of deviation, i.e., the difference between the model fit from the computed data is found to increase with an increase in tortuosity, and among the two shapes, sine pores show the most deviation. This deviation in sine pores with a larger tortuosity and when b/d > 101 exhibits the limited nature of flow enhancement, which tend to be similar in behavior to the DCST pore, albeit, there are over a few order in magnitude differences E.

The limited nature of flow enhancement from sine and helix pores is likely due the

‘bending of stream-lines’ and a result of related energy dissipation due to tortuosity. Therefore, we use equation (19) to fit the E data from sine and helix pores. Here again, we find a good fit

with small values of coefficient C (listed in Figures 5 b & c). Note, that C = 0, reduces the equation (19) to the equation (16). Thus, the coefficient C is a good indicator of resistance to flow offered by pore geometry and intra-pore tortuosity, exclusively. Between the two shapes of capillary pores, C is found to be relatively smaller for sine pores when τG < 1.1 and which becomes significantly larger (i.e., ~ 10 times greater) for sine pores with τG > 1.1.

Figure 5: (a) shows how computed flow enhancement factor (E) for various pore-throats (d) (0.1µm to 100 µm) in different tortuous pores show good fit to the equation (19) for DCST pores, (b) for sine tube capillary pores, (c) for helix tube capillary pores as a function of dimensionless slip length (b/d). All the solid lines are from equation 19 with different C values.

3.5 The coefficient, C with intra-pore tortuosity and pore geometry

The coefficient C has been known to have a value of 3 from a circular orifice (Roscoe,

1949) and from small length pore-throats that transit to a significantly large pore-body which also studied as the ‘end effects’ of carbon nano tubes (CNTs) (Popadić et al., 2014; Weissberg,

1962). Likewise, values of C > 3 or C < 3 are also known which are found to be dependent on the pore geometry (Dagan et al., 1982 ; Gravelle et al., 2014). Therefore, we led the investigation to test if there is relationship between coefficient C and a metric of pore geometry, for example, the intra-pore tortuosity, τG.

We find that there exists a direct relationship between C and τG for all pore systems considered. From the DCST pore, the values of C are found to be exponentially dependent on the

τG (Figure 6a). In contrast, the two capillary pores, i.e., sine and helix, exhibit uniquely different relationship relative to each other. In sine pores, the values of C are found to be linearly dependent on the τG (Figure 6b), whereas, in helix pores, C is found to be exponentially dependent on the τG (Figure 6c).

For uncompacted DCST pores, C is ~ 4, which decreases to ~ 1.5 with the decrease in intra-pore tortuosity or an increase in ‘mechanical compaction’. On the other hand, the values of coefficient C are significantly smaller (i.e., 102 to 103 smaller relative of DCST pores) from the two capillary pores. And, among the two shapes of capillary pores considered, C from sine pore is ~ 10 times larger than the helix pores. These large differences in the values of C, which are a function of pore geometry, are a direct indicator of the amount of resistance offered by each type of pore geometry, and thus, provide a basis to determine how much impact does the choice of pore geometry may likely have on upscaling the microscopic flow to a continuum.

Figure 6: Relationship between geometric tortuosity (τG) with theoretical constant C (equation 19). a) in DCST pore, b) in sine pore, c) in helix pore.

3.6 The limits on flow enhancement in DCST pores

We examine the maximum possible flow enhancement as a limit specific to a type of pore geometry. Due to asymptotic nature of the flow enhancement factor, E found from DCST pores, the maximum change in E i.e., ∆E is calculated for different tortuosity pores. ∆E is the difference between flow enhancement factor from 0.1 µm size pores with the full-slip length

(i.e., b=10-4 m) and 100 µm size pores with the minimum slip length (i.e., b=10-9 m). This allows us to evaluate ∆E in a dimensionless space marked on Figure (5a). In contrast, the straight capillary pore lead to an ‘unlimited’ nature of flow enhancement, and both the tortuous capillary pores, i.e., sine and helix, also exhibit a continuous change in flow enhancement factor, E.

Therefore, the maximum possible flow enhancement cannot be evaluated from capillary pores and is presented only from DCST pores.

We find that the maximum change in magnitude of flow enhancement, i.e., ∆E is exponentially dependent on the hydraulic shape factor, β (figure 7). β is a yet another dimensionless metric of pore geometry which based on the hydraulic radius theory (equation 11).

∆E is found to increase exponentially in DCST pores with a larger β or the more ‘compacted’ pores. Likewise, we find that the coefficient C of equation (19) is also exponentially dependent on the hydraulic shape factor, β, but inversely relative to the ∆E. The empirical ‘curve-fit’

훽 훽 − equation for ∆E is found as ∆E ∝ 푒 5 and for the coefficient C, the equation is C ∝ 푒 5 , both with a R2 = 0.99 (Figure 7). With these equations, it will be possible to predict the relative impact of a pore geometry on flow enhancement or pressure drop because of a potential boundary slip.

3.7 Physical mechanism for variations in flow enhancement

In low-Reynolds number flows, (i.e., Re <1), additional source of inertial forces might evolve from the frequent changes of flow direction following a tortuous flow path across a pore space, however, even such inertial forces are usually negligible in comparison with the viscous forces at Re <1 (Pia and Sanna, 2014; Wei et al., 2018). Therefore, the flow resistance can be viewed as entirely due to viscous forces, and thus, proportional to the flow rate (Heller, 1972;

Larson, 1981). Viscous losses are known to increase when the fluid deviates from a straight flow path, e.g. when it flows around a structure or through a tortuous channel such as the DCST pore.

These viscous losses contribute to the dissipation of energy which is manifested as permeability of porous media, and hence, integral to evaluation of the flow enhancement factor, E (Nelson,

1968; Pilotti et al., 2002).

3.7.1 Energy dissipation

To investigate if and how differences in energy dissipation related to pore geometry can explain for the observed differences in the flow enhancement factor of all pore systems considered, energy dissipation rate (ɛ) is computed from pore domains at the no-slip and slip boundary condition as;

휀 = 흉: 훻풖 (20) where, τ is stress tensor and 훻푢 is strain rate tensor.

3.7.1. 1 Spatial distribution of Energy dissipation in DCST pores

The computed energy dissipation rate (ɛ) for different DCST pores of d = 0.1 µm with no-slip to different slip boundary conditions are shown in figure (8a-l). At the no-slip boundary condition (i.e., b = 0), energy dissipation largely occurs around pore-throats (i.e., constricted sections) and minor amounts of energy dissipation is found from the pore-body (i.e., wider section away from pore throats). However, with an increase of boundary slip (i.e., b > 0), energy dissipation rate (ɛ) increases in the region of pore-body. In addition, at partial boundary slip condition (i.e., b << d), energy dissipation rate remains similar to the case of the no-slip in all

DCST pores. However, at intermediate to large-boundary slip condition (i.e., when b ∼ d to b > d), energy dissipation rate increases with an increase in boundary slips. For example, in 0.1µm size pores with τG = 1.14, energy dissipation rate increases by a factor of ∼20, respectively, as boundary slip increases from no-slip to full-slip (Figures 8e & 8h).

Figure 8: (a-l) variations in energy dissipation rate, ɛ [Wm-3] in 2D axis-symmetric DCST pore geometries as a result of no-slip (b=0) to various slip boundary conditions. As intra-pore tortuosity decreases, energy dissipation rate decreases with the increase of slip length, which is opposite to velocity distributions.

With the decrease in intra-pore tortuosity (i.e., pores undergo mechanical compaction) of

DCST pores, energy dissipation rate (ɛ) is found to decreases at all boundary conditions (i.e., no- slip to various boundary slip conditions), which is opposite to the velocity distribution. For example, in 0.1µm size pores with no-slip BC (i.e., b = 0), energy dissipation rate decrease by a factor of ∼8.02 (Figures 8a & 8i), at intermediate boundary slip, i.e., b = 1.8⨯10-7 m, by a factor of ∼9.64 (Figures 8c & 8k), and at a full-slip boundary condition, i.e., b = 10-4 m, by a factor of

∼2.93 (Figures 8d & 8l), respectively, as τG decreases 1.54 to 1.02.

3.7.1. 2 Spatial distribution of Energy dissipation in capillary pores

The spatial distribution of energy dissipation rate (ɛ) found from capillary pores is presented in Figures (9i-l). In capillary pores with the no-slip boundary condition (i.e., b = 0), the parabolic velocity profile consist of the largest gradient near the boundary which results in the largest viscous dissipation found near the edge of boundaries. In straight capillary tube pore,

with an increase in boundary slip, the straight capillary pore shows no changes in distribution of the energy dissipation (figures 9i-l).

Figure 9: (a-h) an illustration of changes in energy dissipation rate, ɛ [Wm-3] in 3D sine tube tortuous capillary pores; (i-l) variations in energy dissipation rate in 3D straight tube, (τG = 1) and (m-t) changes in energy dissipation rate in 3D helix capillary pores in respect to no-slip (b = 0) to slip boundary conditions. In straight tube, energy dissipation rate is uniform with the increases of slip length whereas in tortuous capillary pores energy dissipation gradually increases with the increase of intra-pore tortuosity as well as slip length.

However, in tortuous sine pores, both with an increase in boundary slip and tortuosity (i.e., b = 0

& τG >1), the magnitude of energy dissipation increases whereas the largest amount of dissipation gets focused towards crest and trough. In addition, with an increase in boundary slip the flow velocity increases, however, due to change in the direction of flow path, i.e., due to tortuosity, additional source of inertia emerges which increases the viscous dissipation.

Therefore, the regions with the most change in flow path or the largest curvature, i.e., crest and

trough tends to offer the most amount of energy dissipation which is found get further amplified with an increase in boundary slip (Figures 9b-d & 9f-h). For example, at τG =1.05 with d =

0.1µm, the calculated energy dissipation rate is found to increase by a factor of ∼1858, respectively as boundary conditions change from no-slip to full-slip (i.e., b = 0 to b = 10-4 m) .

In comparison, in helix pores, similar to sine pores, the magnitude of energy dissipation also increases with the boundary slip, albeit it is relatively smaller than the sine pore. However, both with an increase in boundary slip and intra-pore, tortuosity the largest amount of dissipation gets focused towards the ‘outer edges’ of the pore (Figures 9n-p & 9r-t). Here, as the increased velocity due to boundary slip undergoes a change in the direction of flow path due to helical tortuosity, the direction of new emergent inertia points obliquely ‘outwards’ just like the

‘centrifugal force’ would in a rotating helical path. The emergent inertia leads to the loss of energy which gets magnified both with an increase in the boundary slip and an increase in the tortuosity, even though, the net amount of energy loss remains smaller in helical pores relative to sine pores. For example, at τG =1.05 with d = 0.1µm, the calculated energy dissipation rate is found to increase by a factor of ∼1504, respectively as boundary conditions change from no-slip to full-slip.

With the increase in intra-pore tortuosity of sine pores, i.e., in uncompacted sine pores, the distribution of energy dissipation tends to increase the amount of larger dissipation towards the region of larger curvature, i.e., crest and trough. In comparison, with an increase in intra-pore tortuosity of the helix pores, there are no significant changes in spatial curvature, which therefore, show insignificant changes in the distribution of energy dissipation rate (ɛ). In addition, intra-pore tortuosity effects are more noticeable at intermediate to full-slip conditions as opposed to no-slip to partial boundary slip conditions in both sine and helix pores. For

example, pores with 0.1µm size with full slip, the energy dissipation rate is decreases by a factor of ∼2.69 and ∼1. 26 in sine and helix pore, respectively, as τG increases from 1.05 to ∼1.5, respectively.

3.8 Energy dissipation factor (Ꞷ)

To determine how energy dissipation rate (ɛ) may control the flow enhancement factor,

E, we define a dimensionless energy dissipation factor (Ꞷ) , which is the ratio of energy dissipation rate with slip to no-slip boundary conditions (equation 13). Figure (10a) shows the calculated Ꞷ as a function of dimensionless slip length, (b/d). The relation of Ꞷ is explored with b/d for the same reason as E with b/d which described in subsection 3.2.1.

3.8.1 DCST pores

In DCST pores, energy dissipation factor (Ꞷ) follows a sigmoidal or ‘S’ -shaped characteristics curve in the log- log space (Figure 10a) which is similar to the behavior of flow enhancement factor E (Figure 5a). The ‘S’-shaped characteristics curve exhibits an asymptote for slip flows with high or low boundary slips. The energy dissipation factor remains constant for b/d ≤ 10−1 and shows a linear relationship for 10−1 < b/d < 101 which tends to asymptote for b/d

~1 and after that, i.e., b/d > 101, remains unchanged.

For comparison, figure (11a) shows relation between flow enhancement factor (E) and energy dissipation factor (Ꞷ) for three distinct tortuosity DCST pores. This figure enables examination of if and how Ꞷ controls the E with respect to any change in b/d. Here, both the magnitude of energy dissipation factor, Ꞷ and flow enhancement factor, E remain constant up to b/d ∼5⨯10-2. However, after that, E continues to increase linearly, whereas Ꞷ, remains constant

until for b/d ∼3⨯10-1. As a result of this significant lag between E and Ꞷ, the enhanced flow observed in different DCST pores. Nonetheless, for b/d ≳ 3⨯10-1, the energy dissipation factor also begins increasing dramatically and combine with flow enhancement factor for b/d ∼101, thus, disappeared the lag between Ꞷ and E. Therefore, flow enhancement gets limited and asymptote to negligible change noticed for b/d > 101.

Due to diverging-converging nature of pore channel, energy dissipation rate increases with the increase of intra-pore tortuosity and boundary. Therefore, pores with higher slips and larger tortuosity offer more viscous losses or energy dissipation, which ultimately, reduces the average velocity and limits the flow enhancement.

3.8.2 Sinusoidal and helical pores

The change in energy dissipation factor, Ꞷ from capillary tube pores is examined in the dimensionless space, i.e., with b/d. Here, the variations in Ꞷ due to b and d are collapsed on to a single curve. In both tortuous sine and helix capillary pores, energy dissipation factor remains unchanged for b/d ≤ 10, beyond which an increase in b/d leads to an increase in the energy dissipation factor (Ꞷ) in an ‘unlimited’ manner (Figures10b &10c). The b/d threshold beyond which Ꞷ tends to an ‘unlimited’ type of increase is found to depend on the tortuosity. In tortuous sine pores, energy dissipation factor, remains constant up to b/d ∼1 and after that, begin to increase linearly for lower tortuous pores (i.e., τG ≲ 1.5). However, at τG ≳ 1.5, the magnitude of

Ꞷ tends to decrease, i.e., shifted from unlimited to limited manner of increase with an increase in b/d after b/d ≅102 (figure 10b). In contrast, in helix pores, Ꞷ remains unchanged until b/d ∼10, which is ∼ 10 times larger than the sine pores and after that start increasing in an ‘unlimited’

manner for τG up to ∼ 2. However, at τG ≳ 2, the magnitude of Ꞷ begins to “limited manner” of increase with an increase in b/d after b/d ≅ 7⨯102 (Figure 10c).

Figure11: (a) represents relationship between flow enhancement factor (E) and energy dissipation factor (Ꞷ) for three different DCST pore geometries for different pore throats diameter (d) (0.1µm to 100 µm), (b) for different tortuous sine tube capillary pores and (c) for different tortuous helix tube capillary pores.

For comparison, figures (11b-c) show relation between flow enhancement factor (E) and energy dissipation factor (Ꞷ) related to dimensionless slip length (b/d) for three distinct geometric tortuosities in sine (Figure 11b) and helix pores (Figure 11c). In both sine and helix pores, Ꞷ remains constant for b/d ≤ 10 and after that, with an increase in b/d, Ꞷ begins to increase in an unlimited manner. However, the magnitude of flow enhancement factor is found to remains unchanged for b/d ≤ 10-1 and shows a linear increase for b/d > 10-1. Therefore, within

10-1 ≤ b/d ≤10, due to the significant amount of lag between Ꞷ and E in sine and helix pores compare to DCST pores, flow enhancement increases linearly in an ‘unlimited’ manner.

In summary, due to the lack of diverging-converging nature in sine and helix pores, the intra-pore tortuosity effects are found comparatively insignificant in sine and helix pores as opposed to DCST pores. The energy dissipation factor remains constant for b/d ≤ 10 in sine and helix pores, whereas in DCST pores energy dissipation factor starts to increase for b/d ∼ 10-1. In addition, the lag between Ꞷ and E is more significant in sine and helix pores as opposed to

DCST pores. Furthermore, in all tortuosity DCST pores, for b/d ≥ 10, Ꞷ starts to merge with E and thus, flow enhancement gets limited and asymptote to negligible change noticed for b/d >

10. However, in sine pores, at higher tortuosity, i.e., τG ≳ 1.5 and in helix pores, at τG ≳ 2, the flow enhancement factor, E , the energy dissipation factor, Ꞷ tends to decrease, i.e., shifted from

‘unlimited’ to ‘limited’ manner of increase for b/d ∼103.

3.9 The limit of energy dissipation vs. the limit of flow enhancement

As an extension of our investigations about the maximum limit of flow enhancement presented in section 3.5, here we determine the maximum limit, if any, in energy dissipation rate,

and examine if and how this limit may entirely or partially explain for the nature of ‘limit’ found in flow enhancement, i.e., ∆E. The maximum change (i.e., limit) in the energy dissipation rate, denoted here by ∆Ꞷ is calculated similar to ∆E, i.e., it is the difference between Ꞷ of 0.1 µm size pores with the full-slip length (i.e., b=10-4 m) and Ꞷ of 100 µm size pores with the minimum slip length (i.e., b=10-9 m) (Figure 12). Due to the nature of asymptote found only from DCST pores, this analysis is only possible from DCST pores.

Figure 12: relationship between maximum change in flow enhancement (∆E) and energy dissipation (∆Ꞷ) in different pore throats (d) (0.1µm to 100 µm) for various DCST pore geometries, which is 1:1

Here, we find that there exist a direct 1:1 relationship between ∆Ꞷ and ∆E, which indicate that the energy dissipation rate can completely account for the ‘limit’ found in the flow enhancement factor. The uncompacted pores (i.e., with larger tortuosity) which can lead to the

‘smallest’ amount of ∆E also contribute to the ‘smallest’ and identical amount of ∆Ꞷ (i.e., maximum loss of energy). The same holds for the compacted pores. For example, the DCST pore with an intra-pore tortuosity, τG =1.05, can lead to the ‘largest’ amount of ∆E which also

contribute to the ‘largest’ and identical amount of ∆Ꞷ (i.e., maximum loss of energy). Thus, while the limit of maximum possible dissipation in energy is dependent on the pore geometry and exponentially related to intra-pore tortuosity, it is still bounded by the maximum possible flow, yet of Re <1, in a domain.

3.10 Discussions

This study defines how intra-pore tortuosity and pore geometry play an important role in modifying flow behavior as a result of boundary slip. For the periodic DCST pore geometries, the numerical results have indicated that the flow enhancement factor at pore-scale strongly depends on the intra-pore tortuosity and boundary conditions in porous media. In addition, due to diverging-converging nature of pore channel, the effects of intra-pore tortuosity are more evident in DCST pores as opposed to capillary tube pores. Furthermore, it is noticed that when slip flows increases from no-slip to large slip lengths, E exhibits an ‘asymptote’ or a ‘limited’ increase in

DCST pore geometries, whereas in sine and helix capillary tube pore geometries contribute to an

‘unlimited’ liner increase in flow enhancement factor. However, their magnitude varies depending on the intra-pore tortuosity. Likewise, asymptote in the flow enhancement factor or pressure loss has also been mentioned from computational studies from carbon nanotubes (CNT)

(Popadić et al., 2014) and studying end effects of nanochannels (Calabrò et al., 2013; Sisan and

Lichter, 2011)

To explore the physical mechanism behind why DCST pore geometries, contribute to an asymptote in the flow enhancement factor and why capillary tube pores, lead to a linear increase in the flow enhancement factor, unified energy dissipation rate for all DCST and capillary tube

pore domains is examined. The computed energy dissipation rate from the DCST pore geometries indicates that energy dissipation rate increases with an increase in intra-pore tortuosity and boundary slip. In addition, at lower tortuosity i.e., compacted DCST pores, with an increase of slip length energy dissipation occurs all over the pore domain. However, as tortuosity increases, i.e., uncompacted pores, flow enhancement mostly occurs in the middle of the pore while negligible change in energy dissipation in the dead-end part of the pore body. In contrast, the computed energy dissipation rate from the straight capillary tube pores indicates that there is no fluid deformation in the capillary pores channel as such the energy dissipation is mainly attributable to the viscosity and remains constant for flows with an increase in boundary slip. Similarity, the energy dissipation rate is almost insignificant in different tortuous sine and helix capillary tube pores at partial slip conditions. However, at fully developed slip flows, energy dissipation slightly decreases in sine and helix pores with the decrease of intra-pore tortuosity and hence, flow enhancement factor gets limited by ∼20-30%, respectively.

This research illustrates a basic physical mechanism inherent in fluid mechanics considering boundary slip situations in various tortuous pore geometries and how magnitude of slip flows and intra-pore tortuosity uniquely modify Darcy flow behavior when considering more realistic DCST pore geometries as opposed to sine and helix capillary pore geometries. The representation of porous media by capillary sine and helix tube pore geometries as opposed to a more realistic DCST pore geometries to examine slip flow characteristics or the upscaling of related hydraulic parameters can lead to several order of magnitude over predicting flow enhancement, flow rate and pressure loss. The diverging-converging nature of pores offer an increase in energy dissipation, resulting resistance to flow and therefore contribute to the asymptote in the flow enhancement factor. Previously, most of the studies, considered straight

capillary tube geometries to represent porous media and over predicted the magnitude of flow enhancement factor (E). Therefore, flow enhancement factor (E) are clearly influenced by the pore geometry and intra-pore tortuosity with different pore throats, and the findings are consistent with the enhanced water flow in hydrophobic CNTs (Secchi et al., 2016; Sisan and

Lichter, 2011). We studied the influence of the pore geometry and intra-pore tortuosity on fluid flow in porous media considering boundary slip and validated with the theoretical model. These investigations will shed light on the complex interaction of pore geometry and tortuosity in a homogeneous ideal porous medium and facilitate a better understanding of fluid flow in complex pore structures. Therefore, outcomes of this study may be relevant to the applications of contaminant transport in aquifers, storage of chemical wastes, hydrocarbon recovery and many other engineering applications.

CHAPTER 4: SUMMARY

4. Summary

a) In this study, pore-scale simulation frameworks were developed for numerical simulations of microscopic fluid flow in porous media. The frameworks include straight sine tube capillary pores, tortuous sine & helix pores, and more realistic divergent-converging staggered tortuous (DCST) pores, simulators for single-phase flow and a variety of boundary slip lengths (i.e., 10-9 m- 10-4 m) for efficient pore-scale simulations. Such pores are designed to have a wide variation in pore-throat sizes (i.e., 0.1 μm-100 μm) that are normally present in geologic porous media.

b) The single-phase fluid movement in porous media is regulated by the Navier-Stokes equation in different tortuous pore geometry, and the permeability on the microscopic scale reflects the hydraulic resistance imposed by the solids on the liquid. The findings of the flow and transport simulation results were validated against theoretical model and previous studies.

Finally, energy dissipation rate is investigated to clarify the physical mechanisms related to the observed differences in the evolving hydraulic behavior and flow enhancement factor from the two distinct pore geometries.

c) Results demonstrate that pore geometry has a great contribution to modify the emergent Darcy flow behavior and flow enhancement factor with differences over several order of magnitude as systematically increase in boundary slip from no-slip condition. In DSCT pores, flow enhancement factor shows limited flow enhancement of ‘S’-shaped characteristics curve in

log-log space with high or low boundary slips. In comparison, capillary sine and helix pores, E increases linearly and unlimited manner in log-log space with an increase of boundary slip.

c) The intra-pore tortuosity is found to play a vital role in how the microscopic effects of boundary slip gets manifested as the continuum flow behavior or the flow enhancement factor.

The characteristics of the flow enhancement factor found from the DCST and capillary tube pores can be better evaluated for the relative controls of the boundary slip vs. the size of pore- throats by using a dimensionless slip length, i.e., b/d. The usage of b/d provides thresholds which can be used to determine the relative significance of a probable boundary slip for a known porous media of some average pore-throat size. In DCST pore, with the decrease in intra-pore tortuosity or an increase in the mechanical compaction, the flow enhancement factor is found to display an amplified response to changes in dimensionless slip length, i.e., b/d. In contrast, in capillary pores, with the decrease in intra-pore tortuosity, there is negligible change observed in flow enhancement to changes in dimensionless slip length.

e) The computed flow enhancement factor, E shows a good fit with the theoretical model

(equation 19) for various tortuous pores. The coefficient C is a function of pore geometry, and different values of C are a manifestation of differences in magnitude of the bending of streamlines which are a result of energy dissipation when flow transits from a pore-throat to a pore-body. In sine and helix tortuous pores, C values are lesser than one, while in DCST pores,

C values are within ∼1.5-4, depending on the magnitude of the bending of streamlines. The

magnitude of maximum change of flow enhancement (∆E) and the theoretical model coefficient

훽 훽 − relations are ∆E ∝ 푒 5 and C ∝ 푒 5 with a R2 = 0.99.

f) In DCST pores, the diverging-converging nature of pores offers an increase in energy dissipation, resulting resistance to flow and therefore, contribute to the asymptote in the flow enhancement factor in low to high boundary slips. The energy dissipation rate (ɛ) of capillary straight tube pores (both in distribution and magnitude) remains stable when no-slip or slip boundary conditions are employed. However, as tortuosity increases, i.e., uncompacted sine and helix pores, there are marginal amounts of flow resistance offers, which leads to an infinite rise in the flow enhancement.

f) The relation between maximum change in flow enhancement factor (∆E) and energy dissipation factor (∆Ꞷ) for various DCST pores is 1:1. However, the premise of maximal change in flow enhancement is no longer applicable in sine and helix pores, as a consequence of limitless forms of increasing flow enhancement and energy dissipation factor.

g) This study analyzes the influence of the pore geometry and intra-pore tortuosity on fluid flow in porous media considering boundary slip and validated with the theoretical model.

These investigations will shed light on the complex interaction of pore geometry and intra-pore tortuosity in a homogeneous ideal porous medium and facilitate a better understanding of fluid flow in complex pore structures. Therefore, outcomes of this study may relevant to the applications of contaminant transport in aquifers, storage of chemical wastes, hydrocarbon recovery and many other engineering applications.

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