An Abstract of the Thesis Of

AN ABSTRACT OF THE THESIS OF

Lisa Marie Windom for the degree of Master of Science in Soil Science and Water Resource Science presented on March 19, 2020.

Title: Calcium Ion Transport via Seeping Flow Events through Epikarst Microfissures.

Abstract approved: ______

Maria Dragila John Selker

Solute transport models in karst groundwater must consider variable and complex flow regimes. Within fissures less than 2 mm in aperture, during unsaturated flow events, seeping flow may flow as films or under capillary tension as a capillary rivulet. This project focuses on exploring the mass transport characteristics of seeping film by quantifying the transport effectiveness of capillary rivulets in comparison to a flat, thin film. A laboratory constructed fissure comprised of a limestone slab and a glass plate was used to quantify the mass of calcium ions transferred from the rock to the film during the passage of seeping water as either a film or capillary rivulet. While film flow extracted 170% more calcium ion mass than the capillary rivulets, when normalized by the wetted area, the capillary rivulet extracted 300% more calcium ions than the film flow. As mass flux, capillary rivulets exhibit greater potential for solute transfer across the rock-liquid interface than film flow. ©Copyright by Lisa Marie Windom March 19, 2020 All Rights Reserved Calcium Ion Transport via Seeping Flow Events through Epikarst Microfissures

by Lisa Marie Windom

A THESIS

submitted to

Oregon State University

in partial fulfillment of the requirements for the degree of

Master of Science

Presented March 19, 2020 Commencement June 2020 Master of Science thesis of Lisa Marie Windom presented on March 19, 2020

APPROVED:

Co-Major Professor, representing Soil Science

Co-Major Professor, representing Water Resource Science

Head of the Department of Crop and Soil Science

Director of the Water Resources Graduate Program

Dean of the Graduate School

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

Lisa Marie Windom, Author ACKNOWLEDGEMENTS

I wish to share sincere gratitude and appreciate to Dr. Maria Dragila for her steady vision, patience and kindness. In a line of work so often frenzied with chaos, Maria brings focused calm to her every encounter. Though my heart aches from my non- traditional path through graduate school, Maria has remained a loving ally. Her compassion, love and respect towards her students knows no bounds, and for that we are all changed for the better. The faculty and staff within the Crop and Soil Science Department and the Water Resource Science group have given me endless encouragement, support and wisdom throughout my time in graduate school and I cannot thank them enough. I send my gratitude to Noam Weisbrod for his creative and valuable input towards project design. And I wish to acknowledge the United States-Israel Binational Science Foundation for their financial support.

TABLE OF CONTENTS

Title Page

Chapter 1: Introduction ...... 1 Project Motivation ...... 1 Project Focus ...... 2 Scientific Background ...... 2 Hydrodynamics through Fractured Rock ...... 2 Hydrodynamics of Modes of Flow ...... 13 Hydrodynamic Exchange with Porous Media ...... 18 Calcium Mass Continuity ...... 20 Chapter 2: Experimental Methods, Results, and Discussion ...... 21 Materials and Methods ...... 21 Experimental Design ...... 21 Statistical Analysis ...... 28 Results ...... 30 Experimental Controls ...... 30 Experimental Results ...... 32 Data Analysis ...... 41 Discussion ...... 47 Discussing the Results ...... 47 Reflection on Competing Mechanisms ...... 48 Chapter 3: Conclusion...... 51 References ...... 54

LIST OF FIGURES

Figure Page

Figure 1.1 Defining the coordinate system...... 3 Figure 1.2 The Nusselt height signifies the representative height of wavy film flow...... 5 Figure 1.3 Top-down view of contact area calculation scheme...... 6 Figure 1.4 Cross-sectional view of film flow profile...... 7 Figure 1.5 Cross-section of capillary rivulet: Force diagram show net force vectors along meniscus surface...... 8 Figure 1.6 Cross-section of capillary rivulet showing meniscus corner geometry...... 9 Figure 1.7 Linear method for calculating the wetted area estimate for the capillary rivulet...... 10 Figure 1.8 Applying the conservation of energy, the effective angle can be derived ...... 11 Figure 1.9 Visualization of mult-segment method...... 12 Figure 1.10 Predominant capillary rivulet behavior within this experiment. With distance, the capillary rivulet either remains as a single rivulet, splits and recombines (Y), or splits into two more narrow rivulets called a double...... 13 Figure 1.11 Film Flow: Semi-parabolic velocity profile of film with distance from rock surface for a film height of 250 microns...... 15 Figure 1.12 Capillary rivulet: Parabolic velocity profile of rivulet with distance form rock for an apperture of 700 microns...... 16 Figure 1.13 Mass exchange between film and matrix pores...... 18 Figure 2.1 Edwards Formation limestone sample and set up...... 22 Figure 2.2 Capillary rivulet created between limestone and glass...... 23 Figure 2.3 Design of Mariotte bottle supplying the influent solvent...... 25 Figure 2.4 Subsection scheme for single, Y and double rivulet behavior...... 34 Figure 2.5 Time series of calcium ion concentration of effluent measured in each captured effluent sample over time by trial ...... 39 Figure 2.6 Statistical analysis of the data for the calcium ion concentration of the effluent...... 40 Figure 2.7 Statistical analysis (box and wisker plot) of effluent calcium ion concentration for the capillary rivulet, grouped by flow behavior (single, Y, and double)...... 41 Figure 2.8 Time series of calcium ion mass discharge for film flow and capillary rivulet flow. . 42 Figure 2.9 Time series of calcium ion flux for film and capillary rivulet...... 43 Figure 2.10 Statistical analysis (box and whisker plot) of calcium ion flux...... 44 Figure 2.11 Effluent calcium ion versus magnesium ion concentration ...... 45 Figure 2.12 Calcium vs magnesium flux per unit surface area...... 46 Figure 2.13 Cross-section of film and limestone normal to the limestone surface; water flows gravitationally to the right along the x-axis...... 48 Figure 2.14 The velocity profile of the film and capillary rivulet created in this experiment ...... 49 Figure 2.15 Representation of a meandering capillary rivulet flow with time and distance...... 50

LIST OF TABLES

Table Page

Table 2.1 Analytical and propagated error flow rate and contact area measurements ...... 26 Table 2.2 Minerology results of calcareous rock samples ...... 28 Table 2.3 Order of flow mode randomly generated for each trial ...... 28 Table 2.5 Methods of estimating surface area over time and space...... 29 Table 2.5 Propagated error based on analytical error for the film and capillary rivulet flow results...... 30 Table 2.6 Influent solvent chemistry ...... 31 Table 2.7 System constants for water are given for 20°C ...... 32 Table 2.8 Film flow geometry...... 32 Table 2.9 Film flow contact area...... 33 Table 2.10 Calculated parameters from system constants ...... 33 Table 2.11 M1: Linear method ...... 35 Table 2.12 M2: Meandering method...... 36 Table 2.13 Applying empirical observation to estimate capillary rivulet contact area...... 37 Table 2.14 Mass balance of carbon pools dissolved vs. released as gas ...... 38 Table 2.15 Summary table: dimensions, effluent calcium ion concentration, calcium ion mass discharge and calcium ion flux for film vs capillary rivulet flow ...... 47

DEDICATION

I dedicate this thesis to my family and loving partner. No mountain too high and challenge too grand for the stubbornness of perseverance.

Chapter 1: Introduction

Project Motivation

This study looks at solute transport through microfissues in Karst terrain with the goal of advancing our understanding of contaminant migration and of karst development. Globally, 20% of the world relies on a karst aquifer for their water; and in the United States, it is estimated that 40% of its groundwater is pumped from a karst aquifer (Ford Williams, 2007; U.S. Geologic Survey, 2012; White et al., 1997). These aquifers are a substantial source of water, but due to their high hydraulic conductivity and short residence time, they are also highly susceptible to contamination (Hillebrand et al., 2015; Katz et al., 2010; Morales et al., 2010). Data from tracer studies can be used to elucidate many of the features of karst transport, but the mechanisms involved are still being investigated (Hauns et al., 2001; Hillebrand et al., 2015). Fractured vadose zones exist in more than karst systems, Yucca Mountain, Nevada is comprised of networks of fractures within volcanic tuff. Researchers studying Yucca Mountain, Nevada, as a trial storage site for toxic waste, found elevated concentrations of bomb-pulse Chlorine 36 several hundred meters deep, deeper than expected from predictions of models of pore water seepage through the matrix. The observation, indicated migration of water from surface to these depths in a time period of several decades (Liu et al., 1995). The study identified contaminant transport by seepage through the fissure network as the primary mechanism for rapid transport rather than the pore matrix (Dragila and Wheatcraft, 2001). Investigations into how water moves through fissures began in earnest in the late 1990’s as part of the Yucca Mountain characterization study (Su et al., 1999; Tokunaga and Wan, 1997). Su et al describes two modes of flow, free-surface films where water is in contact with only one wall and seepage that bridges across the aperture with simultaneous contact onto both walls, here called capillary rivulets (Su et al., 1999). Through their exploration, Su et al described capillary rivulets as a pervasively unsteady and intermittent flow which bridged the fissure gap through capillary forces. While films may flow down fractures of almost any aperture (as long as it is wider than the film thickness), rivulets require microfissures to not exceed 2 mm in aperture, so that the bridged connection can be maintained through capillary forces. It is here posited that if contaminated waters were leaking into an epikarst network where the fissures were

predominantly less than 2 mm in aperture, accelerated transport via films and by capillary rivulets may become a significant mechanism of dissolved solute transport. Which brings into question, if this capillary connection influences mass transport across the matrix-liquid interface such that films and rivulets may exhibit different transport properties? Project Focus

This project explores solute flux through a microfissure in a porous matrix for the case in which the influent liquid chemically interacts with the matrix of the microfissure. The study used a fissure in a limestone rock with the influent solvent being deionized water saturated with carbon dioxide. Calcium ion concentration is measured at the outflow. Two flow modes are investigated, free-surface films and capillary rivulets. The objective is to explore solute transport through capillary rivulets, an under studied mode of non-saturated flow, in comparison to film flow to inform our understanding of mass transport through microfissures. Scientific Background

Hydrodynamics through Fractured Rock

Anatomy of Modes of Flow Because the walls of natural epikarst fissures are rough and riddled with asperities, and the fissure aperture varies, the configuration of the seeping water flow can take many forms. Under non-saturated conditions, water can move as films down the sidewalls, or if an asperity causes hydraulic connection with the opposing wall, a capillary rivulet may form. The capillary rivulet results from the initial and maintained bridging of the flowing liquid between the two walls by capillary forces. Depending on the fracture inclination, a flow of the same volumetric flow rate may differ in flow thickness or cross section. For a solute originating from the rock face, solute mass transport is limited by diffusive transport normal to the wall and advective transport along the wall. Quantification of mass flux rates requires that the wetted contact area and the internal velocity profile of the flow be quantified. Figure 1.1 defines the coordinate system used and the anatomy of the modes of flow.

Figure 1.1 Defining the coordinate system. The x-axis runs parallel to the flow down the limestone, the y-axis spreads the width of the flow, and the z-axis spans the gap between the fissure walls. (a) The Side View shows the coordinates from the side, perpendicular to the flow. (b) The Top-Down View shows the coordinates from the view straight at the dissolving wall. (c) The Flow Cross-section View shows coordinates as the flow comes directly into the page, through the fissure. The angle of inclination, θ, is measured from the horizontal to the plane of flow. The open circle indicates normal pointing into the page, and the circle with a central dot indicates normal pointing out of the page.

Anatomy of Film Flow As seeping water freely falls along fissure walls, it can create a flowing film (Phan and Narain, 2007; Vesipa et al., 2015; Yu and Cheng, 2014). In the case of laminar flow, with a Reynolds number, Re < 2100 (Equation 1), the balance between viscous shear stress and gravity dictates the thickness of the film (Dreybrodt and Kaufmann, 2007; Lin and Dreybrodt, 1997; Nusselt, 1916). At this thinness, the flow is relatively flat but wavy; sinusoidal instabilities will occur along the water/air interface and will increase in amplitude with flow distance. These long- wave instabilities for a film moving down an incline plane were first solved by Benjamin (1957), and defined to occur in film flow above a critical Reynolds number, Recrit (Equation 2) (Benjamin, 1957). Later Yih confirmed this solution by an expansion of the disturbance stream function and of the growth rate in powers of real wavenumbers, α (Eagles, 1990; Yih, 1955). By combining Equation 1 with 2, the critical film thickness at which instabilities begin to form can be estimated (Equation 3). Equation 3 demonstrates that experimental conditions can be managed to maintain a film thickness for which waves will not grow.

휌2푔푠𝑖푛휃ℎ3 (1) 푅푒 = 2 2휇 5 (2) 푅푒푐푟𝑖푡 = 푐표푡휃 4 1/3 5휇2 푐표푡휃 (3) ℎ = 푐푟𝑖푡 2휌2푔 푠𝑖푛휃

With this understanding that natural films can be wavy, Nusselt (1016) developed an equation to estimate the mean film thickness (Equation 4). Since these experiments were designed so as to suppress waves to attain as flat a film as possible, which was verified experimentally, the Nusselt calculations were deemed appropriate (Adachi, 2013; Nusselt, 1916; Phan and Narain, 2007). Derivation of the Nusselt formulation assumes a film of infinite width. The equation is here adapted by replacing the flow rate with the specific flow rate defined as the volumetric flow rate per unit film width (Equation 5). The formulations are still considered applicable because the width was much greater than the thickness. The Nusselt height represents the median height between the waves and troughs (Figure 1.2). The film height, hz, represented ’ by the Nusselt height, hN (Equation 4), is a function of the specific flow rate, Qin , gravitational acceleration, g, and fissure slope or inclination, θ, the fluid density, ρ, and viscosity, µ. 1 (4) ′ 3 3휇푄𝑖푛 ℎ푧 = ℎ푁 = 휌푔푠𝑖푛휃

′ 푄𝑖푛 (5) 푄𝑖푛 = 푤

Figure 1.2 The Nusselt height signifies the representative height of wavy film flow. Note that the amplitude has been magnified to facilitate definitions. With distance, a surface of film flow will develop waves of wavelength much longer than the film thickness. The Nusselt height bisects the extremes and provides a constant film height with distance.

Application of Nusselt Theory to Films of Varying Width The film width is not constant because of surface tension effects along the edges that tend to pull the film inward. For the same flow rate, a changing width will change the height, and thus the velocity of the film (Figure 1.3). Although an average width could be used to calculate rock- liquid area, it would not represent the variable change in velocity and thus weathering potential with distance. To improve the average area approximation, the flow is divided into polygons and a weighted average width for each polygon, wCXF, is calculated by averaging the top and bottom widths, wi, of each polygon and multiplying it by the proportion of the length the polygon, xi, to the total distance, L, of the film on the rock (Equation 6).

푤1 + 푤2 푥1 푤2 + 푤3 푥2 푤3 + 푤4 푥3 (6) 푤 퐶푋,퐹 = ∗ + ∗ + ∗ 2 퐿 2 퐿 2 퐿 Once an average width is calculated, Equation 4 and 5 are used to calculate a representative height for the film flow, then these are used to calculate the wetted area and film cross-sectional area. The contact area of a film, ACT, F, is calculated by summing the area of each subdivided polygon (Equation 7). To calculate the film flow cross-sectional area, ACX,F (Equation 8), the film is here treated like a simple rectangle (Figure 1.4), needing only the Nusselt height (from Equation 4) and an average width of flow (from Equation 6).

푤 + 푤 푤 + 푤 푤 + 푤 (7) 퐴 = 1 2 ∗ 푥 + 2 3 ∗ 푥 + 3 4 ∗ 푥 퐶푇,퐹 2 1 2 2 2 3

1 (8) ′ 3 3휇푄𝑖푛 퐴퐶푋,퐹 = 푤 퐶푋 ∗ 휌푔푠𝑖푛휃

Figure 1.3 Top-down view of contact area calculation scheme. Because film flow is not always rectangular, the contact area is calculated by combining multiple polygons that better represent the contact area geometry. Width, wi, and length, xi, are measure for each subsection; the total distance represented by the L.

Figure 1.4 Cross-sectional view of film flow profile. Width of wetted surface, wCX, is measured empirically and averaged and height is calculated as the Nusselt Height, hN. Anatomy of Capillary Rivulet Flow The second mode of flow investigated are capillary rivulets, which have a unique geometry and flow pattern that determines the hydraulic contact area and the velocity profile over time and space. These capillary rivulets form when water flowing as a film along one wall connects with the opposing wall and maintains this connected water bridge through capillarity. If the net flow volume is less than the saturated potential of the fracture, the fissure will remain air filled except for where the rivulet sustains local contact with both walls. For wettable rock surfaces, the two air-water interfaces will form two concave menisci; the capillary force vector points into the air phase towards the center of curvature of the menisci (Figure 1.5) (Dragila and Weisbrod, 2004). Because these force vectors point in towards the center of curvature and out from the capillary rivulet, the pressure within the capillary rivulet is less than the pressure of the gas phase. This pressure difference could drive advective fluid exchange between the pore- matrix of the mineral and the body of flow within the capillary rivulet.

Figure 1.5 Cross-section of capillary rivulet: Force diagram show net force vectors along meniscus surface. The net force along the water surface, is maintained by surface tension, points towards the center of curvature.

While the core of the rivulet dominates flow, with regards to solute exchange the edges of the menisci increase contact area with the rock and may be important to solute exchange with the rock. Effectively, the cross-sectional profile can be subdivided into two regions: the bulk flow and the menisci corners (of which there are four). As described in Figure 1.6, the radius of the menisci curvature can be calculated trigonometrically (Equation 9) based on the height of the right triangle, b, equal to half the aperture of the crack, and the corner angle, α, equal to the water-mineral contact angle (here shown as zero). The wetted width, ww, extends beyond the rectangular, bulk width, wb, and includes the two menisci corners, xo (Equation 10). The extension of the menisci corner shrinks with a larger contact angle (Equation 11). Since aperture and contact angle are constant, xo will be constant over the entire rivulet path; fluctuation in volumetric flow rate, Qb, or the average velocity, 푢̅x, will only change wb (Equation 12 and 13). (9)

푤푤 = 푤푏 + 2푥표 (10)

(11)

(12)

푄푏 (13) 푤푤 = + 2푥0 푢̅푟 (2푏)

Figure 1.6 Cross-section of capillary rivulet showing meniscus corner geometry. The capillary rivulet encompasses the aperture, to where half the aperture, b, also defines the center of the menisci curve. The radius, b, of the menisci curvature creates an isosceles, right-angle triangle with half the aperture. The distance between the bulk width and the corner extension is defined as corner width, xo and the contact angle in this schematic is zero.

As for film flow, the calcium concentration delivered by the capillary rivulet needs to be normalized by the rock-liquid contact area to calculate the transport rate across this interface. Capillary rivulets were found to meander rather than maintain the linear path that had been expected, thus three methods were developed to account for the changes in flow shape across both time and space. Three approaches to estimate the wetted area for the capillary rivulet are here discussed: linear method (M1), meandering method (M2), and multi-segment method (M3).

Linear Method, M1 The linear method, M1, assumes that the flow path of the capillary rivulet is straight, does not meander or split (Figure 1.7). For a straight path the average width, ww, across time and ’ space is given by Equation 15, and the contact area ACT,R calculated by Equation 14 (linear method, aka., M1). ′ 퐴퐶푇,푅 = 푤푤 ∗ 퐿 (14)

Figure 1.7 Linear method for calculating the wetted area estimate for the capillary rivulet. Method assumes capillary flow path is linear. Flow length represented by L.

Meandering Method, M2 The meandering method, M2, assumes that capillary rivulets follow a meandering flow path. Meandering reduces the effective gravitational gradient resulting in slower velocities requiring a wider rivulet (Figure 1.8b). The effect of the meanders can be incorporated into the equation by finding an effective gravitational angle ( ) that accounts for the varying gravitational influence. Conservation of energy is used to obtain the effective angle, , for a single meandering rivulet as presented in Figure 1.8c, the potential energy difference between the top and bottom of distance, L, along slope, , is given by H. This is the same potential energy difference between the top and bottom of the meandering distance, l, along effective slope,  (Equation 15). Algebraically, the sine of the effective angle can be solved (Equation 16) and substituted into the average velocity equation to obtain the effective velocity, 푢̅푒푓푓 for the

meandering rivulet (Equation 17). The decreased angle will decrease the average velocity and increase the average, effective width, weff (Equation 18). The rock-liquid contact area of a single ’’ meandering rivulet, ACT,R , is the product of the effective width and the traveled distance, l, using the meandering method equation (aka, M2, Equation 19). 푔퐻 = 푔푠𝑖푛휃퐿 = 푔푠𝑖푛훽푙 (15)

퐿 (16) 푠𝑖푛훽 = 푠𝑖푛휃 푙

휌푔푠𝑖푛훽 2 (17) 푢̅푒푓푓 = (2푏) 12휇 푄 (18) 푤 = 푡표푡 + 2푥 푒푓푓 푢̅ (2푏) 0 푒푓푓 퐴 ′′ = 푤 ∗ 푙 (19) 퐶푇,푅 푒푓푓

Figure 1.8 Applying the conservation of energy, the effective angle of a meandering rivulet can be derived. (a) The potential energy difference of a linear rivulet flowing the total distance, L, at fissure slope, θ is the same as (b/c) the potential energy difference of a meandering rivulet flowing the traveled distance, l, at a decreased slope angle, β (also known as the effective slope).

Multi-Segment Method, M3 Not only do capillary rivulets meander but the flow splits and recombines across time and space as well. In this third method, the contact area of the entire capillary rivulet is calculated by the sum of individual contact area segments (Figure 1.9). Snapshots of the splitting and meander capillary rivulet are divided into subsections of similar widths, wi, where the length of each subsection is measured as the length traveled, li (Equation 20). This approach is defined here as the multi-segment method (aka, M3). Values for the widths and lengths are obtained from measurements of photographic data.

퐴 ′′′ = 푤 ∗ 푙 + 푤 ∗ 푙 … + ⋯ 푤 ∗ 푙 (20) 퐶푇,푅 1 1 2 2 𝑖 𝑖

Figure 1.9 Visualization of multi-segment method. (a) Seeping flow as a capillary rivulet flowing with an irregular flow path. (b) Irregular wetted area captured by subsectioning flow of similar widths, multiplied by the length of the subsection.

Capillary Rivulet Flow Nomenclature The irregular flow pattern of the capillary rivulet was simplified to three categories of flow: the single rivulet, the “Y” rivulet, and the double rivulet (Figure 1.10). The single rivulet does not split flow paths but does meander with distance. The “Y” rivulet splits and recombines flow with distance and does not meander as broadly as the single rivulet. The double rivulet divides the discharge across two narrower rivulets which each meandering independently. The “Y” rivulet

captures the influence of splitting and recombing on solute extraction. The double rivulet captures the influence from the decreased bulk width, wb, when discharge is divided between two flow paths. Note for the double rivulet, the contact area for the capillary corners does not decrease but there are twice as many capillary corners (which will be discussed in Hydrodynamics of Capillary Flow, Equation 33).

Figure 1.10 Predominant capillary rivulet behavior within this experiment. With distance, the capillary rivulet either remains as a single rivulet, splits and recombines (Y), or splits into two more narrow rivulets called a double.

Hydrodynamics of Modes of Flow

The velocity profile and average velocity for these two flow regimes, films and capillary rivulets, are derived from the Navier-Stokes equation (Equation 21). The Navier-Stokes equation is

simplified to Equations 22 and 23 by assuming that the flow is incompressible, laminar, non- wavy and steady state. Both modes of flow move through the fissure in response to gravity, but the fluid dynamics are different in response to different boundary conditions. Free-surface film flow has been modeled by Stokes law which describes the balance between the gravitational pull and resisting shear force. While capillary rivulet flow behaves as flow between two parallel plates and is representative of Poiseuille flow which describes the flow of a liquid between two parallel of infinite width. The velocity profile in the direction normal to the plane of flow for either flow regime is obtained by integrating Equation 23 and using the appropriate boundary conditions. 휕푢 1 휇 1 (21) + 푢 + ∇ 푢 = − ∆푃 + ∇2푢 + 퐹 휕푡 휌 휌 휌

2 (22) 휇∇ 푢 = 퐹 푑2푢 퐹 (23) 2 = − 푑푧 휇 Hydrodynamics of Film Flow The velocity profile of a laminar film can be obtained assuming a film that is of a specific thickness but with infinite extension in the x-y direction, and by discretizing the film thickness into infinitesimal layers. By integrating Equation 23, with the boundary conditions that the velocity is equal to zero at the wall (ux = 0, z = 0), and that the maximum velocity is reached at the air-liquid interface (ux = umax, z = hN) and more specifically that the shear stress at the air- liquid interface is zero (du/dz = 0, z = hN), one attains the Stokes equation (Equation 24). Stokes law describes a semi-parabolic velocity profile increasing with distance from the rock surface to the air/water interface (Figure 1.11) (Sinkunas et al., 2005). The average velocity of such a flow is given by integrating the parabolic velocity profile across the film thickness (Equation 25). 휌푔푠𝑖푛휃 푧2 (24) 푢 푧 = ∗ ℎ 푧 − 휇 푁 2

휌푔푠𝑖푛휃 2 (25) 푢̅푥 = ( )ℎ푁 3휇

Figure 1.11 Film Flow: Semi-parabolic velocity profile of film with distance from rock surface for a film height of 250 microns.

Hydrodynamics of Capillary Rivulet Flow Because capillary rivulets span across both fracture walls simultaneously, rivulets can be locally modeled as Poiseuille flow between two plates. The equation is derived from Equation 23 by integrating and applying the two boundary conditions: no slip condition (zero velocity) at both rock surfaces and zero sheer stress at the midpoint between the two rock surfaces. The parameter b is half the aperture distance (Equation 26). It assumes laminar flow, and creates a fully parabolic velocity profile with maximum speed through the center plane of the fracture (Figure 1.12) (Eagles, 1990; Sinkunas et al., 2005). The average velocity of the rivulet is calculated by integrating across the full fracture aperture (Equation 27) (Adachi, 2013; Dragila and Weisbrod, 2004; Kim and Infante Ferreira, 2009). A significant assumption to this solution is that a narrow rivulet can be assumed to follow Poiseuille flow, however, this simplification was used by

Dragila and Weisbrod (2004). The velocity profile equation for a meandering rivulet can be easily adjusted by reducing the effective strength of the driving gravitational force which is accounted for by substituting beta for theta in Equations 26 and 27 (Figure 1.8b and Equation 16). 휌푔푠𝑖푛휃 (26) 푢 푧 = ∗ 푧 2푏 − 푧 휇

휌푔푠𝑖푛휃 2 (27) 푢̅푥 = ∗ 2푏 12휇

Figure 1.12 Capillary rivulet: Parabolic velocity profile of rivulet with distance form rock for an apperture of 700 microns.

The importance of menisci corners on flow dynamics depends on the relative width of the bulk flow region and the corners. To calculate flow rate by a rivulet it is convenient to separate the cross-section of the rivulet into the sum of the flow through central bulk cross-section

(rectangle), Qb, and the flow through the four corners, Qcorner (Equation 28). The flowrate

through the bulk cross-section of the capillary rivulet is defined by the product of the average velocity, 푢̅̅푟̅, and its cross-sectional area as defined by the sum of the bulk width, wb, and two times the fissure aperture, b (Equation 29, bulk width shown in Figure 1.6). The capillary rivulet corners have an air/water interface formed by curved menisci with a flow rate given by Equations 30 and 31 (variables defined in Figure 1.6) (Dragila and Weisbrod, 2004). The speed of flow in the corners is also semi-parabolic, resembling the internal flow of the main body of the rivulet, and is continuous with that main body, but the maximum speed is modified to account for the curving geometry of the interface. (28) (29)

휌푔푠𝑖푛휃 4 (30) 푄푐표푟푛푒푟 = 푏 퐹(훼) 3휇 (31)

Advectively, the volumetric flow through the corners, Qcorner, of the rivulet is often very small and velocity is near zero, relative to the bulk flow for the system used in this study. However, diffusion and dissolution still occur within these corners and they may provide an important contribution. When most models neglect the extension of the menisci corners, they can potentially underestimate the possible reactive contact area. Only when the volumetric flow rate decreases below a critical flow rate, Qcrit, (Equation 32) is the inclusion of the corner flow significant (Dragila and Weisbrod, 2004). At this critical flow rate, the low volumetric flow narrows the bulk width so that the sum of the flow through all four corners equals the bulk flow and the extension of the corners is of the same magnitude as the bulk width. Because quantifying the area of the rock-water interface is necessary to normalize the calcium ion flux, the contribution of the corners must be explored for accuracy.

푄푐푟𝑖푡 = 4푄푐표푟푛푒푟푠 (32) And more so, corner flow becomes more critical if the rivulet flow splits and recombines. Because of the irregular geometry of the capillary rivulet, with each split, more flow is diverted into four new corners, and adds to the collective proportion of corner flow (Equation 33). (33) 푄푡표푡 ,푠푝푙𝑖푡 = 푄푏,1 + 4푄푐표푟푛푒푟 + 푄푏,2 + 4푄푐표푟푛푒푟 +. . +(푄푏,𝑖 + 4푄푐표푟푛푒푟 )

Hydrodynamic Exchange with Porous Media

Though the majority of the water moves rapidly down the fissure, it can seep into and out of the limestone matrix losing mass or extracting pore water with high solute concentration (Figure 1.13). Because concentration is highest within the pores, advective flow exchange with the pore matrix will also increase the mass flux of dissolved ions into the film/rivulet. Advective movement into the limestone matrix corresponds to a net mass loss from the film. Advective movement out of the rock will increase the mass of the flow and concentration of calcium ions in the flow, corresponding to a net mass gain in the film. Pressure pulses created by the wavy film surface and the pressure gradient imposed by the capillary tension of the capillary rivulet may create advective pore-water exchange.

Figure 1.13 Mass exchange between film and matrix pores. Schematic shows advective incursion of film water into near surface pores clearing out near surface pores of solutes (wavy curve), or net water mass loss from film into rock matrix or net mass gain of solute laden water from rock matrix into film (both shown as wide arrows).

The pressure within the capillary rivulet, caused by the curvature of the capillary menisci can be calculated by the Young-Laplace Equation (Equation 34). The Young-Laplace Equation describes the capillary pressure of the menisci of water climbing a tube by adhesive and cohesive forces. The pressure is a function of the fluid surface tension, γ, and the aperture of the curvature (Hillel, 1998). The curvature is the sum of the inverse of the two principal radii of curvature

(C = 1/Ry+1/Rz) of the tube, which for a cylindrical tube is C = 2/R, where R is the tube radius, assuming a zero-contact angle. For a fissure there is only one menisci of curvature that is important since the other principal curvature is infinite. Thus, for a fissure where C = 1/R, Equation 34 applies. (34)

The pressure difference between the capillary rivulet and the matrix pore water creates a pressure gradient. Because capillary rivulets are unstable in that they meander, with each lateral sweep of the meandering capillary rivulet, the water filled pores are pulsed by the change in pressure. The pressure pulse will cause advective-driven pore-water movement out of the pore matrix. In addition to this pressure pulse there is also a net pressure gradient from matrix to rivulet or film by dynamic pressure, defined by the Bernoulli Equation (Equation 35). The dynamic pressure, pdyn, is a function of the water density and the fluid velocity. This would be true for the film as well as the capillary rivulet.

1 2 (35) 푝푑푦푛 = 휌푢 2 The net pressure difference for the film will be the sum of capillary pressure pulses caused by the film surface waves and the dynamic pressure (Equation 36, where “” is the half wavelength of the surface waves and is used as an approximation for the radius of curvature of the surface waves). For the rivulet, as it sweeps laterally past a region of rock, the pressure difference is caused by the capillarity of the menisci and the dynamic pressure (Equation 37). Water will move out of the limestone pores by Darcian flow as a result of these pressure differential. The advective movement represented by the Darcian flux is a function of a pressure gradient described here by the sum of capillary and dynamic pressures (Equations 36 and 37).

The path length is assigned as one pore length, . The resulting velocity, udyn, of the flow into and out of the pores is given by Equation 38. 2 푝푑푦푛 + 푝푐 1/2휌푢 + 훾/휆 (36) 푞 = −퐾 = −퐾 푝표푟푒−푓푖푙푚 훿 훿 2 푝푑푦푛 + 푝푐 1/2휌푢 + 훾/푏 (37) 푞 = −퐾 = −퐾 푝표푟푒−푟푖푣푢푙푒푡 훿 훿

푞푑푦푛 (38) 푢푑푦푛 = 휙 Where  is the porosity. Any increase in advective exchange will increase the translocation of pore water solutes. Explained further in the following section, the mass transport capacity of pore solutes can increase as a result of these advective, pressure-driven movements. Calcium Mass Continuity

Given that dissolution only happens on and within the limestone surface, solute continuity in the bulk fluid flow is governed only be transport. The mass transport of solutes depends on a compilation of the addition of mass from advection and diffusion: Addition of mass into Addition of mass in Rate of change of mass in = control volume due to + control volume due to control volume advection diffusion

Alterations to the system were made to eliminate the possibility of solute precipitation and control the dissolution rate to simplify the boundary conditions of the control volume. The alteration to the system, discussed in the Materials and Methods, made the rate of calcium carbonate dissolution not chemically limited and maintained undersaturation of calcium ions.

Chapter 2: Experimental Methods, Results, and Discussion

Materials and Methods

Within the Materials and Methods, both the experimental design and the methods used for data analyses are explained. The experimental design explains the choice of limestone, the solvent chemistry, the flow rate, the saturation of the limestone and the creation of both film and capillary rivulet flow. The data analysis reviews the methods of calculating water to limestone contact area (also referred to as the “surface area” of the mode of flow), and the process of normalizing the measured effluent calcium ion concentration to mass volume discharge and mass flux per unit wetted area. Experimental Design

The experimental design section explains the chosen parameters and the preparation method. The limestone, flow rate, and fissure configuration (aperture, slope, length) determine the physical parameters of the film and capillary rivulet. The preparation of the solvent and the limestone slab define factors that influence dissolution: the preparation of the solvent defines the influent chemistry and the saturation of the limestone with water prevents advection into the pore matrix. Only two flows are generated for this study: film and capillary rivulet. Choice of Limestone After informative, preliminary experiments the limestone source was selected. Considerations in selection of limestone sample included ease of access, mineral homogeneity within the sample, mineral reactivity to hydrochloric acid (indicating carbonate content), surface texture, and physical size of the sample. Ease of access tended to consist of whether the sample was available at the local quarry or provided by reasonable shipping. Most all mineral samples were homogeneous. The sample could not be polished with a glaze or coating but the surface texture itself needed to be smooth to reduce the creation of fluid turbulence by surface topography. The sample was coarsely sanded by the quarry to create a flat surface. The limestone sample was calcium carbonate (mineralogical analysis discussed later) obtained from the Edwards Balcones of Texas, USA, known for its karst hydrology (White, 1983). Lastly, sample dimension and

weight were limited based on logistics and safety to ensure the slab could be manipulated by one individual. Multiple 30 cm by 30 cm by 5 cm in dimension unpolished slabs were acquired. Fissure Configuration To mimic a fissure of constant aperture, a 15 cm by 30 cm piece of glass was placed upon four 0.07 cm thick metal plates (Figure 2.1). The glass allowed for the mode of flow to be observed as well as maintained a microfissure of constant aperture. The angle of inclination controls the Nusselt film height. It was found that an angle greater than ~15° caused waves to form in the film flow. Though wavy flow can be laminar, it increases advective motion from rock pores, and it was decided to keep the film as flat as possible. Therefore, to ensure confidence in the fluid dynamic exhibited across all trials, the angle slope was set at 15°.

(a) (b)

Figure 2.1 Edwards Formation limestone sample and set up. (a) Front view highlights the visibility through the glass and the path of flow. (b) Side view highlights the 15° angle of incline.

Controlling Flow Rate The influent was prepared in a 19 L Mariotte bottle positioned on a ledge above the experiment. The head was set to 106 cm between the air intake of the solvent Mariotte bottle and the outlet that drained into the experimental fissure. The flow rate was set at 0.68 mL/s and remained constant throughout each trial. The flow rate was checked constantly throughout and between experiments to verify its consistency. Creating the Modes of Flow Creation of either a film or a rivulet required careful control at the flow inlet. The influent, of constant flowrate, saturated a piece of felt at the top of the fissure. From this saturated piece of felt, water would gravitationally seep as a film or capillary rivulet. Before each trial, the felt was

saturated in deionized water. The flow rate was measured at the inlet between trials, and at the fissure effluent throughout the trials to ensure the volumetric discharge remained constant. To create film flow, the saturated felt was spread uniformly across the section of rock above the fissure and as influent flow poured onto the felt, the displaced water within the felt seeped out as a film. To create a capillary rivulet the felt was moved a millimeter closer to the glass plate forming the fissure. The water seeping from the felt bridged the gap between the glass and the rock and subsequently flowed as a rivulet down the rock, maintaining capillary connection (Figure 2.2). Variability in resulting rivulet structures were discussed in Figure 1.10.

(a) (b)

Figure 2.2 Capillary rivulet created between limestone and glass. (a) Stable, single capillary rivulet flowing through the 0.07 cm gap between the limestone and glass. The rivulet was then

intercepted by a mesh at the lower end of the fissure to guide the flow towards the sample vials. (b) The width of this single rivulet at a 0.68 mL/sec flow rate is measured to be 0.5 cm.

Preparation of Influent To prepare the influent, 0.3-0.8 lbs solid carbon dioxide (dry ice) was placed into 19 L deionized

(DI) water (Figure 2.3). Once the dry ice all converted to CO2 gas, and the gas dissolved into the

DI water and the air phase within the bottle was filled with CO2(g), the lid to the Mariotte bottle was secured and the water was left to equilibrate to laboratory temperature for 14 hours. Because each experiment consumed roughly 7 L, only two trials per day could be performed. The process of dissolving carbon dioxide was repeated at the end of each day in preparation for the next day, across four sequential days.

Figure 2.3 Design of Mariotte bottle supplying the influent solvent. The air inlet and water outlet are labeled to define the functionality of the bottle. The line defining atmospheric pressure (i.e., zero energy head) remained constant throughout the experiment. Dissolved CO2 content increased in solvent by adding dry ice and allowing CO2 gas to bubble through the solvent. Rock Preparation Details The limestone was saturated in deionized water to ensure pore saturation and to reduce seepage of flow water into rock during the experiment. To saturate, the limestone sample was placed horizontally in a bucket; water was filled around the limestone so that the top surface was still exposed to the atmosphere, allowing the release of displaced air. The limestone was saturated for 14 hours and remained in solution until moments before the start of the experiment. Moments before the start of the experiment, the limestone was removed from the water and flipped so the more saturated “under side” of the sample became the surface upon which the experiment occurred. Debris and small particles were flushed from the limestone surface with a squirt bottle of deionized water. Excess moisture on the rock face was removed with a squeegee. Once saturated with deionized water, cleaned of loose particles, secured at an angle, and the glass plate placed at the desired aperture atop the limestone, the experiment was ready to begin. Sample Collection and Analysis The bottom of the limestone slab created a capillary barrier that impeded direct drainage of the capillary rivulet or film. At this barrier water tended to pool. To overcome the capillary barrier and to direct effluent water to the collection vials, coarse mesh netting at the end of the limestone slab intercepted the flow and directed flow into the sample vials. This mesh was saturated with water with a hydraulic conductivity unique to the mesh. This specific mesh was chosen because its hydraulic conductivity matched the flowrate through the fissure; it was confirmed that the mesh did not decelerate the flow rate and that the discharge from the mesh matched the discharge into the fissure. If the mesh inhibited calcium ion transport, it was assumed it would inhibit transport uniformly across both film and capillary rivulet and not skew the comparison between the two modes of flow. Effluent that was not captured in sample vials dripped into a tub placed atop an analytical scale. In between sample measurements, the accumulation of volume over time provided a mid-experiment verification of flow rate (mL/s). Full tubs would be exchanged with empty tubs to reduce the weight on the analytical scale and prevent overflow.

The influent solvent was sampled at the beginning of the experiments for the day; recall that the solvent was prepared the night before the experiment by dissolving dry ice so that ample time to reach equilibrium was given. Each influent sample is representative of the influent chemistry of that experiment day. Trial 1 was performed with sample Influent 1, Trial 2-3 were performed with sample Influent 2, and Trial 4 performed with Influent 3. Each trial was video recorded to capture the behavior of the modes of flow and to track the discharge rate throughout the experiment. The video captured the mode of flow, the mass balance data, and the sample ID for each sample collected. For each trial the effluent was sampled every two-minutes across 60 minutes (20 mL vials), pausing at 10, 30, and 60 minutes to collect partial pressure of carbon dioxide (pCO2) samples (380 mL beer bottles). Collection method is described below. Samples were collected for three separate types of analysis: cations, pH and pCO2. The analytical error is shown in Table. 2.1.

Table 2.1 Analytical and propagated error flow rate and contact area measurements.

(+/-) # Flow Rate, Q (mL/sec) 1 Concentration, C (umol/mL) 0.001 Width (cm) 0.1 Length (cm) 0.1 Contact Area (cm2) 0.14 Flow Rate, Q (mL/s) 0.03 Calcium Ion Mass Discharge, C*Q (μmol/s) 1 Flux (μmol/s*cm2) 1

Cation samples were collected in 20 mL volumes, then filtered through Whitman #1 filter paper to removed large dislodged particles caused by physical weathering rather than chemical dissolution. After filtration, samples were analyzed for calcium ions (Ca) and magnesium ions (Mg) by inductively coupled plasma optical emission spectrometry (ICP-OES) in the Oregon State University Central Analytical Lab within the Dept. of Crop and Soil Science. The

analytical precision of the ICP-OES provided calcium ion concentrations measurements within ± 0.001 µmol/L. Effluent samples to be sent to the ICP-OES were first filtered as a precaution to protect the equipment from damage potentially caused by small particles; this also focused our work on the diffusion or advection mechanisms, rather than the physical removal of small particles. From the 380 mL beer bottles, the partial pressure of carbon dioxide, total carbon dioxide, and total alkalinity were measured and calculated using methods used to measure carbon in oceanic systems (Bandstra et al., 2006). The influent and effluent solvent were captured in beer bottles, preserved with 280 µL mercury chloride (added through three drops with an eye dropper) to prevent microbial activity and capped to seal with a beer bottle cap press. Within the Hales Laboratory of the Oregon State University College of Earth, Atmosphere and Ocean

Science (CEOAS), the pCO2 and total carbon dioxide (TCO2) were measured using non- dispersive infra-red (NDIR) gas analyzers coupled with the Lamont Pumping SeaSoar system.

From the pCO2 and TCO2 data, the Hales Lab calculated the speciation of carbonic acid (H2CO3), - 2 bicarbonate ions (HCO3 ), and carbonate ions (CO3 ) (Weiss, 1974). The TCO2 samples were diluted 100:1 in order to be within the measurement range of our equipment (Hales et al., 2004). Mineralogy A subsample of the limestone was prepared for mineral analysis by the SX 100 Electron Probe Microanalyzer. The preparation and analysis of the sample was performed by the Electron Microprobe Laboratory at CEOAS. The preparation process included taking a subsample of the limestone less than a square inch in size (and less than a quarter inch thick), placing in a mold to be imbedded in an epoxy one-inch disk, polished down to expose a uniform surface and coated with gold (Reed, 1993). Two limestone slabs were used, one for the flow experiments, and one to use for mineralogical analysis. Each slab was homogenous, null of surface texture or roughness and the sample taken for the mineral analysis was representative of the greater whole (Table 2.2). The calcareous rock sample used for these experiments is limestone made of calcium carbonate (99% pure, with 0.02% magnesium oxide impurities). The crystal structure was not measured nor the mineral definition defined, but the sample had an intercrystal pore type, was light in color, and porous (Maclay and Small, 1984).

Table 2.2 Minerology results of calcareous rock samples

Calcium Oxide Magnesium Oxide Carbon Dioxide

(CaO) (MgO) (CO2) Weight %/Formula Weight 1.02 0.01 1.00 Calculated Stochiometric 1 0.02% of impurity 1 Ratio

Compiled Mineral CaCO3 with (0.02% MgO impurity) Composition Statistical Analysis

Because the same slab was used for multiple experiments, the order of flat film and rivulet flows were randomized to avoid crossover effect (Table 2.3), meaning that the weathering from a film type structure would impact future weathering from a rivulet, and vice versa. Spatial replications were performed by dividing the limestone into three sections on both side: a total of six replicates were possible per piece of limestone. Four replicates were successfully performed. The time increment between each sample collected remained the same across all experiment replications, therefore, the mean calcium ion concentration at each sample grab-time could be averaged across each trial. Data considered to be impacted by experimental design rather than the targeted modes of flow were removed. The mean calcium ion concentration over time of each mode of flow was normalized to calcium ion flux. Because a comparison of means was applied, the standard error was used to quantity the precision of the mean estimate. The capillary rivulet exhibited irregular flow paths which were captured on video. The occurrence of flow behavior (single, Y, or double, see Figure 1.10) were noted for each trial.

Table 2.3 Order of flow mode randomly generated for each trial.

Trial Mode Created First Mode Created Second 1 Film Capillary Rivulet

2 Capillary Rivulet Film 3 Film Capillary Rivulet 4 Capillary Rivulet Film

Surface Area Calculations Each experiment was video recorded and photographed to accurately document flow parameters and from these, the film and (for methods M2 and M3) the capillary rivulet width and length were captured and analyzed. From the photographs, the physical boundaries were marked. A metric ruler lined the glass, providing an accurate measure of the length and width recorded in the photographs. Multiple methods were applied to estimate the water to limestone contact area, these methods are referenced in Table 2.4. Sub-polygons were defined based on relative likeness of flow width. The contact area of the film was divided into representative sub- polygons; the total contact area of each mode of flow included a summation of the sub-polygon areas. The film height was calculated by the estimated Nusselt height. For the capillary rivulet, three methods were used to obtain 3 estimates of the area.

Table 2.4 Methods of estimating surface area over time and space.

Mode of Method Brief Description Flow Geometric Assumes average film width and estimated Nusselt height (based on Film volumetric flow rate and angle of flow) provides a representative Flow cross-sectional area. Equation 9 Linear Assumes flow is linear and width is constant. Width can then be (M1) estimated by volumetric flow rate and aperture to estimate the cross- Capillary sectional area. Equation 16 Rivulet Meandering Assumes flow meanders at a constant path, reducing the effective Flow (M2) angle of flow, the instantaneous volumetric flow rate, and therefore, narrowing the width and the cross-sectional area. Equations 21

Multi- Assumes flow meanders at a dynamic and constantly changing path, segment creating a spectrum of effective angles of flow over time, and a (M3) subsequent range of instantaneous volumetric flow rates. Therefore, the width widens and narrows constantly, as does the cross-sectional area. Equation 22 Results

The results are presented in three subsections: Experimental Control; Experimental Results (film then rivulet); and Data Analysis. The experimental controls describe the solvent chemistry, flow structure and flow dynamics. The experiment results show the calcium ion concentration of the effluent either transported by film or capillary rivulet. Data is presented normalized to flow rate and to rock-liquid contact area. Methods used for normalizing to contact area are presented. Experimental Controls

Certain parameters were held constant to simplify normalization calculations and limit the influence of variables on solute extraction. These constants include the solvent chemistry, fissure structure (slope, aperture, length, and flowrate), and limestone minerology; data on the first wo is shown below, and limestone minerology was already discussed. Analytical and Propagated Error Each method contained some level of analytical error which propagated through each calculation. Table 2.5 defines the propagated error of the calculations provided within the results based on the analytical error inherent to the accuracy limitations of the tools applied to make these measurements.

Table 2.5 Propagated error based on analytical error for the film and capillary rivulet flow results.

Film Flow Propagated Error Based on Analytical Error Width (cm) 0.50 Length (cm) 2.34 Capillary Rivulet Propagated Error Based on Analytical Error Linear Method (M1) Width (cm) 0.06 Length (cm) 0.21

Meandering Method (M2) Width (cm) 0.09 Length (cm) 2.11 Multi-Segment Method (M3) Width (cm) 0.05 Length (cm) 0.19

Solvent Chemistry In addition to calcium ion concentration, pCO2 and TCO2 were measured and from them calculated speciation was provided by the Hales Lab (as shown in Table 2.6). The pH of the influent solution remained between pH 5-6. Within this pH range, total dissolved carbon dioxide

(DIC) predominantly exists in water as aqueous CO2, the remaining 10% of the dissolved inorganic carbon had converted to bicarbonate. Calcium dissolution will occur when exposed to this solvent chemistry.

Table 2.6 Influent solvent chemistry

- 2- Sample pCO2 pH DIC [H2CO3*] Alk [HCO3 ] [CO3 ] (atm) (µmol/kg) (µmol/kg) (µeq/kg) (µmol/kg) (µmol/kg) Influent 1 0.556 5.71 24,253 21,767 4,075 4,664 0.00 Influent 2 0.527 5.06 23,798 20,632 -276 987 0.00 Influent 3 0.396 5.26 18,871 15,516 235 1,177 0.00

Average 0.493 5.35 22,307 22,307 1,345 2,276 0.00 Std Dev 0.085 0.33 2,984 2,985 2,378 2,070 0.00 Std Error 0.049 0.19 1,723 1,723 1,373 1,195 0.00 ± % Error 10% 36% 8% 10% 102% 53% -

Fissure Structure The aperture, slope, and length of the fissure were held constant and controlled the flow shape and rock-liquid contact area of both the film and capillary rivulet flow. Values measured during the experiment are shown Table 2.7. The aperture defines the thickness of the capillary rivulet and the slope controls the capillary rivulet surface area. The length of the fissure impacts

both surface area calculations for the film and capillary rivulet flow. The flow rate was the same for both the film and capillary rivulet and was checked continuously throughout the experiment for accuracy.

Table 2.7 System constants for water are given for 20°C (average laboratory temperatures).

System Constants Gravitational Constant, g (cm/s2) 980 Water Density (g/cm3) 1 Water viscosity at 20°C, μ (g/cm*s) 0.0089 Slope Angle, θ (radians) 0.26 Aperture, 2b (cm) 0.07 Flow rate, Q (cm3/s) 0.68

Experimental Results

The Experimental Results include the raw measurements and data analysis. Raw measurements include flow geometry and calcium ion concentration. Data analysis includes calculations of surface area, calcium ion mass discharge and calcium ion flux across the rock- liquid interface. Film Geometry After the flow made its initial pass to down the limestone, it maintained a constant flow path, film thickness and contact area geometry. The width of the film flow was widest at the inflow and narrowed with distance down the fissure. The film geometry was measured from photographic data (Table 2.8).

Table 2.8 Film flow geometry. Observed widths and lengths of the film flow subsections are measured empirically.

2 Width, wi (cm) Length in Distance, xi (cm) Area (cm ) Geometric Top Bottom Avg. Top Bottom Absolute Area of Subsection i Width Width Width Distance Distance Distance subsection (cm) (cm) (cm) (cm) (cm) (cm)

Subsection 1 6 5.7 5.9 3.1 18.5 15 90.1 Subsection 2 5.7 2.2 4.0 18.5 23.4 4.9 19.4 Subsection 3 2.2 2.2 2.2 23.4 25.5 2.1 6.8

Weighted 5.0 Average Total Area, AF,CT 116.3 Film Contact Area For films, the contact area was photographed, and the total area calculated using the method described by Equation 7: summing the area of multiple subsections of similar width (analytical error +/- 0.2 cm for each measurement). The width of the film started at 6 cm width and narrowed with distance down the limestone fissure to 2.2 cm (details in Table 2.8). The average width along 23.4 cm length, L, was 4.6 cm. The total hydraulic contact area was 116.3 cm2. The film height (film thickness) was estimated from the average Nusselt height, which, based on a 0.68 mL/sec flow rate, is 240 µm (Table 2.9) (Equation 4).

Table 2.9 Film flow contact area. Observed widths and lengths of the film flow subsections are summed from Table 2.8 to estimate the total contact area.

Film Flow Geometric Subsection i Area (cm2) Subsection 1 90.1 Subsection 2 19.4 Subsection 3 4.6 Total Contact Area (cm2) 116.30

Table 2.10 Calculated parameters from system constants (as from Table 2.7). Table 2.7 values are applied to produce the specific discharge, Nusselt height for the film flow. Specific flow rate is defined as volumetric flow rate divided by average film width.

Calculated Parameters

Weighted Average Film Flow Width, 푤 퐶푋,퐹 (cm) 5.0 Specific Flow Rate, Q’film (cm/s) 0.136 Nusselt Height, hN (cm) 0.024

Capillary Rivulet Geometry Measurements for capillary rivulet thickness, length and width were as straight forward as for film flow. The flow thickness spanned the aperture gap, which was kept at 0.7 mm. The

width of the capillary rivulet would be constant if the rivulet stayed straight, because the width is controlled by the flow rate (as represented by M1). But as discussed, the rivulet meandered thus the velocity and width differed greatly across space and time. Figure 2.4 shows the subsections applied to calculate the width and length for each trial. Trial 1 exhibited a single rivulet as the dominate flow shape, trials 2 and 3 were predominantly Y rivulets, and trial 4 flowed predominantly as the double rivulet.

Figure 2.4 Subsection scheme for single, Y and double rivulet behavior. Applying this subsection scheme, the areas within each subsection were subdivided by similar width.

Capillary Rivulet Contact Area For the capillary rivulet, three methods were used as described below. Methods are summarized in Table 2.4. The estimated contact area for each method are summarize in Table 2.15. The follow three sections show the width and lengths collected for each method and how these contact areas were calculated.

M1: Linear Method The linear model assumes the bulk width of the capillary rivulet is uniform and the flow path linear (Equation 14). Recall that the capillary rivulet has four capillary menisci corners extending out along the fissure walls; their contact width (i) against the limestone is a function of

the interfacial energies, not the bulk width (wb). The bulk width could grow with increased discharge, but the menisci corners remain that same size. In this case the aperture and the discharge are constant, so are the widths of capillary menisci corners and the bulk width are constant. Bulk width is calculated from the aperture and flow rate (Table 2.11). The bulk width was used to calculate the estimated contact area. Substituting experiment parameters and variables from Table 2.7 the average velocity was calculated to be 14.2 cm/s. The volumetric flow passing through the menisci corners was insignificant compared to what was passing through the bulk width of the capillary rivulet (4%). But they were included consistently across all methods. The estimated contact area for the theoretical, linear method (M1) equaled 16.1 cm2.

Table 2.11 M1: Linear method. Equations 10-14, defined parameters and calculations.

Equation Parameter (units) Calculated

Average Velocity, 푢̅̅푥̅ (cm/s) 14.2

Bulk Width, wb (cm) 0.60

Total Width, ww (cm) 0.64

Total Length, L (cm) 24.9 (M1) Est. Contact Area (cm2) 16.1

M2: Theoretical, Meandering Method Because there was significant meandering by the rivulet, it was proposed that the bends in flow path were lowering the effective slope of the flow, the effective velocity and therefore increasing the average width of the rivulet. To accommodate for this variability in widths, M2 was used to better quantify contact area. The meandering method measured the path length of the capillary, from which an effective gravitational gradient was calculated, to estimate the effective increase in cross-sectional width and decrease in velocity. Flow parameters and variables shown in Table 2.7 and travel length were applied to Equation 15-19 to estimate an effective slope, effective

flow velocity, and an effective width (Table 2.12). The effective velocity differed for each trial, with a mean effective velocity of 14.4 cm/s. The average velocity of the meandering curves (M2) increased by 0.23 cm/s from the average velocity of a straight capillary rivulet (M1). The subsections of Trial 2-4 contained splitting and recombining flow which complicated calculations. The mean contact area across the trials equaled 16.9 cm2.

Table 2.12 M2: Meandering method. Table shows path length traveled, l, estimated effective velocity 푢̅푒푓푓, (Equation 17), split flow between multiple rivulets, and estimated effective width, 푤푒푓푓 (Equation 18). The rock length, L, is the absolute distance along the x-axis and does not track the bends of the meandering flow. The ratio of total length to effective length are applied to determine the effective angle of flow, sinβ, (Equation 15-16) which is then used to calculate the

effective velocity. All to calculate the effective contact area of each trial, ACT,R” (Equation 19).

) )

2 2

/s)

Area (cm Area (cm Area

L/l

Effective Sub Effective

[

Trial

Total Length Total

]

Flow(cm

(radians)

Traveled Length Traveled

Effective Angle Effective

Subsection

% of Total Flow Total of %

Effective (cm) Width Effective

Sub

L (cm) L

(cm)

Contact Contact

Effective Contact Total Effective

Sin

Effective (cm/s) Velocity Effective

1 A 25 24.1 0.96 0.250 13.72 100% 0.677 0.68 17.1

17.1 2 A 16 15.8 0.99 0.256 14.05 48% 0.328 0.35 5.5

B 15 15.8 1.05 0.273 14.99 52% 0.349 0.35 5.2

C 7.4 7.9 1.07 0.276 15.19 100% 0.677 0.62 4.6

15.3 3 A 15.5 16 1.03 0.267 14.69 49% 0.333 0.34 5.2

B 15 16 1.07 0.276 15.18 51% 0.344 0.34 5.1

C 8 7.8 0.98 0.252 13.87 100% 0.677 0.68 5.4

15.7

4 A 24 23.2 0.97 0.250 13.75 53% 0.359 0.38 9.2

B 27.1 23.2 0.86 0.222 12.18 47% 0.318 0.38 10.3 19.5 (M2) Estimated Contact Area 16.9 Method 3: Empirical Method Lastly, capillary rivulet widths and lengths for each subsection of similar width were measured from photographic samples, one photo was taken for each trial. As seen in Table 2.13, for each trial one photograph was taken and used. Photographic area for the capillary rivulet was broken down into smaller subsections. Each subsection was of similar width and the area within each subsection was summed to equal the estimated water to limestone contact area for that trial. The areas of each trial were then averaged to create the average contact area. From the contact area of each trial, the mean contact area was calculated as 19.4 cm2.

Table 2.13 M3: Applying observation to estimate capillary rivulet contact area.

Trial Subsection Width (cm) Length (cm) Area by Total Contact Area per (A, B or C) Subsection (cm2) Trial (cm2)

1 1-A 0.5 25 12.5 12.5

2 2-A 0.3 16 4.8 17.3 2-B 0.3 15 4.5 2-C-1 0.5 2.7 8 2-C-2 1 1.7 2-C-3 0.7 3 3 3-A 0.4 15.5 6.2 18.9 3-B 0.5 15 7.5 3-C-1 0.6 4 5.2 3-C-2 0.7 4 4 4-A 0.4 24 9.6 29.6 4-B-1 0.5 16 20.0 4-B-2 1.7 4.2 4-B-3 0.7 3.2 4-B-4 0.7 3.7 (M3) Estimated Contact Area 19.6 Carbon Dioxide Loss The difference between the dissolved inorganic carbon concentration of the influent versus the effluent were assumed to be due to carbon dioxide gas loss from solution to the atmosphere

(Table 2.14). The only way carbon can be lost from the water sampled is through off gassing to the air phase in the fracture or mass transport into the limestone media. Since it was calculated and verified through empirical measurements that no mass was lost to seepage into the limestone media, the only remaining vector of carbon loss is through carbon dioxide loss into the atmosphere. Between the influent and effluent of the fissure, the film lost 48% of CO2 from the solvent, and the capillary rivulet lost 27% of the solvent CO2 to the atmosphere.

Table 2.14 Mass balance of carbon pools dissolved vs. loss as gas

Film Capillary Rivulet Measured Pool Concentration Percent Concentration Percent (µmol/kg) of Initial (µmol/kg) of Initial Pre-exposure (Initial solvent) Dissolved Inorganic Carbon (DIC) 22,300 100% 22,300 100% - 2- as CO2+HCO3 + CO3 Aqueous Carbon Dioxide (CO2) 19,300 100% 19,300 100% Post-exposure (Effluent sample) Dissolved Inorganic Carbon (DIC) 10,700 48% 6,000 27% - 2- as CO2+HCO3 + CO3 Total Carbon lost as Carbon 9,000 53% 14,350 26% Dioxide to Atmosphere Remaining Aqueous Carbon 6,500 52% 12,450 73% Dioxide (CO2)

Calcium Ion Concentration The time series for calcium ion concentration in the fissure effluent is divided into two periods, the first is dominated by the experimental startup and the second shows distinct trends for each mode of flow. During the first 10 minutes there is a logarithmic-like increase which then levels out to a linear rate. This logarithmic increase, referred to as the experimental startup, is believed to be a result of dilution of the influent by the felt which had been saturated initially with DI water, thus it is due to the experimental design and not associated with the mode of flow. Following the experimental start up period, the data is controlled by the mass transport properties of the film or capillary rivulet. All comparisons between time series and statistics (box and whisker plots) exclude the data collected during the experimental startup period. Figure 2.5 captures the concentration of calcium ion within the effluent when exposed to either a film or capillary rivulet across all four trials. The film flow trials predominantly grouped

among each other except for trial 2. Trial 2 was performed on the second day of the experiment, where the film was the first mode of flow created. It is not clear why this trial had a lower mass transport than the other trials. Also performed that day with the same solvent were capillary rivulet trial 2, and both film and capillary rivulet flow for trial 3. Yet, these other trials did not group with the effluent chemistry of the trial 2 film flow. Trial 2 remains somewhat of an outlier, although it is included in all the statistics. Figure 2.6 presents the mean calcium ion concentration for each trial, excluding the experimental startup period. Based on the average calcium ion concentration (ignoring the first 10 minutes), the effluent flow from the film had, with statistical significance, a higher concentration of calcium ions, averaging three times more calcium ions than the capillary rivulet (Welch’s two-sample T-Test, n= 80, p-value << 0.01). Because both the flow rate was identical for all film and rivulet trials, comparing calcium ion concentrations required no normalization to flow rate.

Figure 2.5 Time series of calcium ion concentration of effluent measured in each captured effluent sample over time by trial for the film and the capillary rivulet flow by trial. The four trials of film flow are represented by black squares for trial 1, grey triangles for trial 2, black x’s

for trial 3, and grey-blue stars for trial 4. The four trials for capillary rivulet are represented by a solid burnt circle for trial 1, an orange diamond for trial 2, dark-rimmed, light peach filled circle for trial 3, and a brown dash for trial 4. The shaded region represents gaps in calcium ion data because carbon dioxide samples were being collected.

Figure 2.6 Statistical analysis of the data for the calcium ion concentration of the effluent. Box and whisker plot for film and capillary rivulet. The film flow (left) had a higher average dissolved calcium ion and a greater variability of measured concentrations than those observed from the capillary rivulet. The first 10 minutes of data (representing experimental startup) were removed.

The capillary rivulet trials group among each other, except for a few samples which exceeded 250 μmol/L collected during trial 3. Video recordings show the occurrence of capillary droplets 43 minutes into trial 3, which were not studied in this experiment, but have been found to induce solute uptake and internal circular mixing (Hay, 2008). When the effluent calcium ion concentration was grouped by the flow behavior (single, Y, or double), the Y rivulet behavior exhibited a statistically distinct average calcium ion concentrations than the single rivulet behavior (p-value << 0.001, Figure 2.7). The Y rivulet exhibited minimal meandering but contained mixing where the two flows combined. As a result of reduced lateral meandering, the variability for this behavior is less than those that meander. The single and double rivulets exhibited similar variability and were not statistically distinct from each other (p-value = 0.16)

and this is a result of similar lateral meandering behavior. The Y rivulet was also not statistically distinct from the double rivulet (p-value = 0.16), suggesting similarities as a result from dividing flow volume through multiple rivulets.

Figure 2.7 Statistical analysis (box and wisker plot) of effluent calcium ion concentration for the capillary rivulet, grouped by flow behavior (single, Y, and double). Flow geometry for single, Y and double are defined in Figure 2.4.

Data Analysis

Within the Data Analysis, the concentration of calcium ions within the effluent is normalized by volumetric flow rate as calcium ion mass discharge, and by rock-liquid contact area as flux. The effluent calcium ion concentration was multiplied by the volumetric flow rate 0.68 mL/s to create the calcium ion mass discharge. To normalize by contact area to obtain mass flux, the calcium ion mass discharge was divided by the calculated rock-liquid contact areas for the film and capillary rivulet flow shown within Table 2.15.

Calcium Ion Mass Discharge The concentration of calcium ions in the effluent was multiplied by the volumetric flow rate to obtain the Calcium Ion Mass Discharge. Because the same flow rate was used for all runs, this calculation does not change trends of either the film or capillary rivulets (Figure 2.8), it simply serves as a unit conversion. However, because a capillary rivulet would pull mass from both limestone walls of the fissure, if this experiment had not used glass for one of the walls the concentration would be doubled, thus the concentration of calcium ions for the capillary rivulet was multiplied by two. Though the gap between the two modes of flow decreased, they remained statistically distinct (Two Factor ANOVA, p-value << 0.001). All trends and relations identified in the Calcium Ion Concentration section are upheld.

Figure 2.8 Time series of calcium ion mass discharge for film flow and capillary rivulet flow. Capillary rivulet calcium ion concentration data was multiplied by two to represent the flux from two limestone fissure walls. Calcium ion mass discharge is calculated as calcium ion concentration times volumetric flow rate.

Calcium Ion Flux When normalized to the rock-liquid contact area, the data gives a value that is representative of the calcium ion flux being transported from the rock into the flowing liquid. The trend in the results shifted: capillary rivulets extracted more calcium ions per unit surface

area relative film (Figure 2.9). There is an apparent increase in the scatter of the rivulet data, but this is only because all 3 methods (M1, M2 and M3) for calculating the contact area are plotted on the same graph. The apparent scatter is just related to the differences between these methods. Even considering all rivulet methods, the difference between film and rivulet results are statistically significant (p-value << 0.001, Two Factor ANOVA). The increase in variability seen in the capillary rivulet data is a product of the increased uncertainty of the analytical methods applied. Proportionally the analytically error for measuring contact area (+/- 0.2 cm2) has a greater influence on the capillary rivulet than the film flow because of the difference in their widths.

Figure 2.9 Time series of calcium ion flux for film and capillary rivulet. The capillary rivulet is normalized by flow rate and multiplied by two to estimate the dissolution from two calcium carbonate walls.

It was observed that on average film flow had a normalized (per unit surface area) calcium flux of 2.9*10-3 µmol/s*cm2. Capillary rivulets are highly variable over time and space and attempts to capture the surface area proved highly variable as well (Figure 2.10). The three methods for capillary rivulet flow produced average, normalized calcium ion flux values: M1

estimated the average, normalized calcium ion flux as 7.3*10-3 µmol/s*cm2, M2 as 6.9*10-3 µmol/s*cm2, and M3 to be 6.4*10-3 µmol/s*cm2 .

Figure 2.10 Statistical analysis (box and whisker plot) of calcium ion flux. Capillary rivulet area was normalized by three methods: M1, M2, and M3. . Calcium vs. Magnesium Ion Mass Discharge Based on the limestone minerology there are two magnesium ions for every 100 calcium ions within the mineral structure. Although the dissolution kinetics for magnesium and calcium ions are different, it is expected that the ratio of the ions dissolved by the two flow modes would be similar for the film and capillary rivulet flow unless the physical processes were different for the two modes. As shown in Figure 2.11, the dissolved calcium ion to magnesium ion flux ratios for both the film and capillary rivulet flow track the mineral ratio with similar variability. The variability is greater for the magnesium ion values because the low concentration of magnesium results in a higher uncertainty in the measured values.

Figure 2.11 Effluent calcium ion versus magnesium ion concentration of the capillary rivulet (orange circles) and film flow (black squares). The dashed line signifies the mineral ratio of the limestone samples used for the experiment. Both the film and capillary rivulet ratios roughly track along this line.

When normalized to the rock-liquid contact area, however, the calcium to magnesium ion ratio of the film flow tracks the trend of the limestone mineralogy, and the results are tightly grouped. For the capillary rivulet flow, the ratio of dissolved-ion flux followed loosely the minerology ratio but exhibited high scatter with a trend towards much greater Mg/Ca ratio than the mineral fraction. When normalized to surface area (Figure 2.12), the multiple methods applied to normalize capillary rivulet flow given an apparent increase in scatter of the data. Each method (M1, M2, or M3) shifts the pattern slightly from the other methods.

Figure 2.12 Calcium vs magnesium ion flux per unit surface area. M1, M2, M3 compared for capillary rivulet flow. The mineral ratio of calcium to magnesium identified as the blue dotted line.

Discussion

The Discussion is organized by first discussing the experiment, second reflecting on the competing mechanisms originally identified, and third a discussion of other mechanisms or factors that could explain the differences in calcium ion mass discharge and calcium ion flux for film and capillary rivulet flow. Discussing the Results

For the same flow rate, the film flow transported three times more calcium ions by volume than the capillary rivulet, presumably due to the increase in the film’s contact area with the limestone surface (Table 2.15). But when normalized to the rock-liquid interfacial area the capillary rivulet transported twice the amount of calcium ions than the film flow. It is proposed that the increased mass flux by the capillary rivulet is the result of three mechanisms. First, pressure gradients across the rivulet and matrix pore interface caused the advective-driven transport of calcium ions into the capillary rivulet. Second, the meandering of the capillary rivulet exposed rivulet water to fresh limestone that contained calcium ion-rich pore water and may be picked up of colloids from the rock surface. And, lastly, meandering action drove advective circulation within the capillary rivulet that accelerated diffusion.

Table 2.15 Summary table: dimensions, effluent calcium ion concentration, calcium ion mass discharge and calcium ion flux for film vs capillary rivulet flow (M1, M2, M3)

Units Film Rivulet Rivulet Rivulet Flow Flow- Flow- Flow- M1 M2 M3

Average Effluent Calcium µmol/L 504.9 173.5 173.5 173.5 Ion Concentration Average Calcium Ion Mass µmol/s 0.34 0.12 0.12 0.12 Discharge Average Width cm 5.0 0.6 0.5 0.5 Total Length cm 23.4 24.9 23.7 Effective Length cm 38.3 38.3 Estimated Contact Area cm2 116.3 16.0 16.9 18.6 Average Calcium Ion Flux 10-3 µmol/s*cm2 2.9 7.3 6.9 6.4 Flux Ion Standard 4.34 29.36 27.87 24.07 Deviation Flux Ion Standard Error 0.05 0.37 0.35 0.30

Reflection on Competing Mechanisms

Two mechanisms were left uncontrolled, diffusion and advection, in order to explore their influence on the mass transport of calcium ions through a fissure. Diffusion drives calcium ion from the limestone surface out to the water column, from the region of high concentration to low. Advection by laminar flow moves water in layers of increasing velocity parallel to the limestone surface. The velocity increases perpendicular to the surface (Figure 2.13) and over the first few centimeters below the inlet it also increases in the direction of flow; the direction of flow is from a higher to lower gravitational energy. The overlap of these two drivers of mass transport create an interesting mechanism. It is suggested here that the relative combination of these two drivers could be the source of difference between the film and the capillary rivulet when normalized by water to limestone surface area.

Figure 2.13 Cross-section of film and limestone normal to the limestone surface; water flows gravitationally to the right along the x-axis. Laminar Flow Velocity Scheme shows the direction of flow, and the relative magnitude of the velocity within that layer of water. Diffusion moves perpendicular to the limestone surface, out into the water column above.

It was suspected that the velocity profiles between the modes of flow created in this experiment could be a source of discrepancy between the mechanisms of mass transport. But the velocity profiles for the film and the capillary rivulet in this experiment are very similar (Figure 2.14). Within the film profile, the velocity increases with distance from the limestone surface, and maximizes at 9 cm/s (Equation 25). For a meandering capillary rivulet, the average velocity approaches a similar maximum velocity at 9 cm/s (Equation 27). This model represented in Figure

2.14 eliminates the advective velocity profile as the cause for discrepancies between film and capillary rivulet mass transport mechanisms.

Figure 2.14 The velocity profile of the film and capillary rivulet created in this experiment. The velocity profiles for the film with the thickness of 243 nm, and a meandering rivulet with the velocity equal to the effective velocity calculated by Equation 19.

Flow Geometry The rock-liquid contact was estimated from a series of snapshots taken throughout the experiments. This method has a larger uncertainty for the capillary rivulet flow in comparison to film flow. Because the capillary rivulet was dynamic, it bent, meandered, split, recombined, and at times created capillary droplets. The width of the flow fluctuated as the capillary rivulet widened and narrowed with shifts in velocity. Different formations either sped or slowed passage through the fissure, altering the relative exposure to the limestone.

There are more precise methods of estimating instantaneous contact area, such as summing pixels of a photograph, that could be applied to reduce the contact area analytical error. In a recent study performed by Ruiz-Ripoll et al, pixel analysis could estimate area within 1 mm2 as compared to 20 mm2 capable within this experiment (Ruiz‐Ripoll et al., 2013). Even with this increased analytical accuracy, the variability of the calcium ion concentration as a result of the capillary rivulet will still be greater than the film flow if the capillary rivulet flow splits, meanders, and at times creates capillary droplets (Figure 2.15). Combing split flow, meandering and creating capillary droplets are each unique mechanisms that create mixing, and that facilitates the diffusion of solutes from the mineral surface up the flow profile. At the point where two paths of laminar flow merge, such as at the combination of two rivulets, there is also mixing (Enfield, 2003). When meandering laterally, the dynamic pressure imposed over the saturated limestone pores will induce advective motion and thus mixing into the capillary rivulet (as described by the Bernoulli Equation, Equation 35). And capillary droplets induce mixing as droplets break from the capillary rivulet flow path and rapidly pass through a new patch of limestone (Hay, 2008).

Figure 2.15 Representation of a meandering capillary rivulet flow with time and distance. Each shade represents an a theoretical snapshot in time based on the type of behavior observed.

Chapter 3: Conclusion

This project investigated the extraction of a solute from a porous matrix and transport by two flow modes, film flow and capillary rivulet. The solute measured at the outflow was calcium ions that had been generated by the interaction of the influent solution (CO2 enriched DI water) and the calcium carbonate composition of the limestone rock. For the same volumetric discharge, water passing through a fissure as a film will transport more dissolved solutes from the fissure surface than if that same volume of water passed as a capillary rivulet. Within this experiment, it was observed that films transported 170% more solute mass than capillary rivulet flow; this is predominantly due to greater liquid-rock contact area exhibited by the film flow in comparison to the capillary rivulet. When normalized to contact area, however, the capillary rivulet had 300% greater solute flux than the film flow. If both the film and the capillary rivulet flow were identical in all mechanisms of solute transport, normalizing to contact area should yield similar results instead, the results were statistically distinct from each other. The difference in transport effectiveness between the two flow modes indicates that there are different mechanisms acting at the rock-fluid interface for films as for rivulets. Both film and rivulet impose the same dynamic pressure gradient because data shows that they exhibit similar velocity profiles (Equation 25 and 27, Figure 2.14). However, for capillary rivulets, three additional mechanisms are proposed. First, the capillary menisci impose a pressure gradient across the capillary rivulet and matrix pore interface that can drive advective flow of pore water, and thus calcium ions into the capillary rivulet via Darcian flow (Equation 38). Second, the meandering of the capillary rivulet exposes this water to fresh limestone that contained solute rich pore water. And, third, the meandering action may drive advective circulation within the capillary rivulet that accelerates diffusion of the ions from the liquid-rock interface into the body of the rivulet. Determining the relative magnitude of this contribution is left for future research where tests can be made as to whether the system is diffusion or dissolution limited. The surface area within the pores susceptible to chemical dissolution is magnitudes greater than what was captured by the liquid-rock contact area calculations. It is also assumed in this study that mineral phase is dissolved homogenously across the limestone pore surface area.

But in fact, the heterogenous distribution of the dissolution reaction over the mineral surface and could limit access to certain regions of the pore surface (Saldi et al., 2017; Inks & Hahn, 1967). This study focused on the transport of a solute from the limestone matrix into the flow profile. But recent studies suggest that the physical removal of calcium carbonate colloids can be an equally important mechanism of mass loss at the limestone surface (Levenson and Emmanuel, 2017). Colloid removal from natural fractures has been observed in a number of studies (e.g., Weisbrod et al., 2002). The repulsive forces between calcite grains weaken the rock surface, facilitating the transport of calcite grains less than 1 μm in diameter, with a grain-grain contact area of 5 μm2 (Levenson and Emmanuel, 2017). This study filtered out any grains larger than 11μm in diameter, which would not exclude the transport of these colloids. Within the time between when the samples were collected and they were analyzed by the ICP-OES, it is suggested that these colloids dissolved and were included in the measured calcium ion concentration. This investigation has highlighted a number of questions that should be explored in future studies such as these three questions: how colloid transport might be facilitated by capillary rivulet flow, how do pressure pulses from meandering flow influence ionic interactions within the pores, and where in the tracer breakthrough curves could this flow contribute to mass transport. First, the transport of calcite colloids and the physical smoothing of the limestone surface on a microscale as a result of these mechanisms of transport should be explored with regards to chemical and physical erosion of the limestone surface. Because the physical transport of calcite colloids contributes to calcium carbonate erosion, mass transport facilitated by capillary rivulet could be a contributing mechanism. Second, the pressure pulses caused by capillary rivulet meandering could overcome locked ion interaction within the matrix, such as those leading to the heterogenous distribution of the dissolution reaction. And third, facilitated mixing between a matrix and capillary rivulet flow because of the lateral meandering is not limited to application in karst aquifers. The contribution of these seeping flows to mass transport events could be captured by tracer studies. The tracer breakthrough curve often exhibits skewness with a long tail (Hauns et al., 2001). The initial peak is described as the rapid mass flow through the large conduits and caverns, and the long tail as seeping flow draining the remnants of the flow event. Exploration into the seeping flow could include capillary rivulet

transport, deepening our understanding of the mechanisms that contribute to the shape of these breakthrough curves. This study adds to our understanding of unsaturated flow through microfractures, and more importantly it identifies additional mechanisms unique to the capillary rivulet that are not acting in film flow. There have been many studies of solute transport from porous media to films (e.g., Wasden and Dukler, 1992); however, this is the first study of solute transport into capillary rivulets. As discovered in this study, for the same flowrate, films are more effective than capillary rivulets because films contact more surface area. But capillary rivulets have a greater mass flux (solute and colloid) than film flow, per liquid-rock contact area. Having unique mass transport properties, capillary rivulets may have many applications as well as occurrences yet to be discovered.

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