MODELLING OF CALCIUM CARBONATE PRECIPITATION IN NATURAL

KARST ENVIRONMENTS UNDER HYDRODYNAMIC AND CHEMICAL

KINETIC CONTROL

A Thesis

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulllment

of the Requirements for the Degree

Master of Science

Brad L Justice

May, 2006 MODELLING OF CALCIUM CARBONATE PRECIPITATION IN NATURAL

KARST ENVIRONMENTS UNDER HYDRODYNAMIC AND CHEMICAL

KINETIC CONTROL

Brad L Justice

Thesis

Approved: Accepted:

Advisor Dean of the College Dr. Curtis Clemons Dr. Ronald F. Levant

Co-Advisor Dean of the Graduate School Dr. Eric Wright Dr. George R. Newkome

Faculty Reader Date Dr. Gerald Young

Department Chair Dr. Kevin Kreider

ii ABSTRACT

Rimstone dams are barriers composed mainly of calcium carbonate deposited from solution in ground and surface waters. These structures form a subclass of travertine formations which include owstone and stalactites, and often appear in close proxim- ity to these features. The initial formation of rimstone dams requires some degree of cave slope, a semi-continuous ow of water, and the preexistence of irregularities in the cave oor. These dams develop at heights from a few millimeters to several me- ters within free-surface streams, and create a self-propagating dam and pool structure which grows upward. The genesis and evolution of rimstone dams is theorized to be the result of hydrodynamic and chemical-kinetic control. The purpose of this paper is to develop a model, the scope of which encompasses both hydrodynamics and the reactive transport, which is qualitatively consistent with observed and experimentally derived results, and method for analyzing the mechanism governing the formation of these unique rimstone features.

iii TABLE OF CONTENTS

Page

LIST OF TABLES ...... vi

LIST OF FIGURES ...... vii

CHAPTER

I. INTRODUCTION ...... 1

II. CALCIUM CARBONATE PRECIPITATION MODEL ...... 7

2.1 Hydrodynamic Model ...... 7

2.2 Reactive Transport Model ...... 14

III. SOLUTION PROCEDURE ...... 22

3.1 Hydrodynamics Solution Procedure ...... 22

3.2 Reactive Transport Solution Procedure ...... 27

3.3 Free Boundaries Solution Procedure ...... 41

IV. RESULTS AND CONCLUSIONS ...... 44

4.1 Analysis of Reactants Driving Precipitation and Dissolution . . . . 44

4.2 Interpretation of Hydro-Chemical Dynamics ...... 45

4.3 Conclusions ...... 49

BIBLIOGRAPHY ...... 51

iv APPENDICES ...... 53

APPENDIX A. HYDRODYNAMIC CALCULATIONS ...... 54

APPENDIX B. REACTIVE TRANSPORT CALCULATIONS ...... 56

v LIST OF TABLES

Table Page

3.1 Non dimensional variables ...... 23

3.2 Summary of Geochemical Parameters of Groundwater in Tributaries of the Mystic River, Scott Hollow Cave [1]...... 31

3.3 Chemical kinetics constants and their values. M = mmol cm1. . . . . 33

3.4 Values of Parameters used in the Hydrodynamics and Reactive Transport Models ...... 38 3.5 Mathematically derived values for the backward reaction rate coef- cients at the water- mineral surface: M = mmol cm3 ...... 41

vi LIST OF FIGURES

Figure Page

1.1 Photograph montage of rimstone dams from Yellowstone National Park and two caves. Courtesy NPS...... 6 2.1 Schematic of Dam Initiation and Formation ...... 8

4.1 The concentration eld for a lm thickness of 0.002cm...... 46

4.2 Wave number k v. Growth rate for depth d = 0.002cm...... 47

4.3 Wave number k v. Growth rate for depth d = 0.00225cm...... 47

4.4 Wave number k v. Growth rate for depth d = 0.0025cm...... 48

4.5 Wave number k v. Growth rate for cave oor slope of 20 ...... 48

vii CHAPTER I

INTRODUCTION

Rimstone dams are barriers composed mainly of calcium carbonate deposited from solution in ground and surface waters. These structures form a subclass of travertine formations which include owstone and stalactites, and often appear in close proxim- ity to these features. Figure 1.1 shows the variety of environments, structures, and topological features which are inherent in rimstone dams. The initial formation of rimstone dams requires some degree of cave slope, a semi-continuous ow of water, and possibly the preexistence of irregularities in the cave oor. These dams develop at heights from a few millimeters to several meters within free-surface streams, and create a self-propagating dam and pool structure which grows upward. The genesis and evolution of rimstone dams is theorized to be the result of hydrodynamic and chemical-kinetic control.

The chemically controlled processes of calcium carbonate precipitation and dissolution have been rigorously studied. These chemical processes have been care- fully determined via extensive experimentalization. All such eorts are characterized by dissolved carbon dioxide from the atmosphere, transport of reactants from the solution to the mineral surface of the cave oor, and the eects of pH determining dissolution or precipitation.

1 Plummer et al. [2] pioneered investigations of this area by performing rigorous experiments on calcium carbonate dissolution and precipitation processes. The model employed was based on the surface reactions described by the experimentally derived rate law dubbed the PWP equation. This equation, assuming similar hydrodynamic conditions in the natural environment, relates chemical species, reaction mechanisms, and surface controlled kinetics to the dissolution and precipitation rates of calcium carbonate in travertine formations.

Dreybrodt et al. [3] [4] [5] utilized Plummer’s studies as a catalyst for further investigation of these chemical processes. Experiments were carried out on both dissolution and precipitation of calcium carbonate. Dreybrodt constructed a model which encompassed surface controlled deposition/precipitation, diusion of reactants in the bulk solution, and the slow conversion of CO2 to H2CO3. The PWP equation was utilized to describe the surface reactions. Thin lm water ow over rock surfaces was approximated by plug ow with constant velocity on a at plane of calcium carbonate for both laminar and turbulent ow characteristics. The reactants in the bulk solution were assumed to be fast in order to use mass balance laws to describe

+ the mutual dependence of CO2, OH , H , and HCO3 . These studies revealed the importance of lm thickness in explaining the dierent morphologies in the deposition of calcium carbonate. It was discovered that the model predicted a hydraulic jump from laminar to turbulent ow for sucient water lm thickness. It was theorized that this transition in ow characteristics contributed to the rate of dissolution and precipitation of calcium carbonate.

2 Chou et al. [6] was able to demonstrate that the dissolution and precipi- tation of many other carbonate minerals could be described by the PWP equation of Plummer et al. Later, Arakaki [7] extended the work of Plummer, Chou, and

Dreybrodt to include other surface speciation at the calcium carbonate surface. This study predicted the dominant role of pH on the dissolution and precipitation.

The investigation of hydrodynamic processes has been limited to observations at various sites of dam growth, and carefully derived experimental results. These eorts demonstrate observed relationships between ow characteristics, cave oor slope, and water lm thickness, and their inuences on rimstone dam initiation, structure, and evolution. The initiation and evolution of rimstone dams has been observed to exhibit complex dynamics since the processes of dam growth change the

ow characteristics.

Veni [8] observed abrupt hydrodynamic changes due to the geometry of the rimstone dam structures. The ow characteristics changed from laminar to turbulent

ows once the water passed over the lips of the dam and pool structures. He also developed a model assuming uniform water chemistry, uniform water depth, and dynamics following initial calcium carbonate precipitation. This model theorized the hydraulic jump from laminar to turbulent ow and demonstrated a relationship between cave oor slope and the height and frequency of rimstone dam growth. A mathematical relationship between slope and the velocity, gravitational acceleration, and the lm thickness of the water ow was utilized to assess this relationship.

3 It was found that rimstone dam height and frequency increased as the slope of the cave oor increased.

Xing et al. [9] investigated hydrodynamic conditions on various rimstone dam structures. This eort is based on thin lm water ows following the initial formation of rimstone dams and the eects of non-constant and semi- continuous

ows characteristic of seasonal changes and the eects of rimstone dam growth on the hydrodynamics. It was observed that dierent ow characteristics at dierent locations in caves aected the geometry of rimstone dams which is indicative of the dierent type of structures observed in caves. It was theorized that initial formation of rimstone dams required thin lm water ows and non-uniformities in the cave oor.

The cave evolution and its eects on the dynamics of the water ow and subsequent changes of the structure of rimstone dams was also discussed. It was found that after initial formation of the rimstone dams, the subsequent evolution in the shape of deposits was inuenced by the surface waves of the water ow which was theorized to be similar to the mechanism driving the deposition of beach sand by ocean waves.

It was also found that as the cave evolved, the supply of water to the rimstone dams became constant due to the formation of the dam pool structure inherent in these deposits.

Baker et al. [10] conducted a comparative study of the work of Plummer and Dreybrodt and the eects of thin lm water ow. This eort theorized that the inconsistencies between theoretical growth rates and experimental growth rates of calcium carbonate deposits resulted from the non-uniformities in the cave. It was

4 also observed that growth rates were sensitive to both Ca2+ and water lm thickness.

The purpose of this paper is to develop a model, the scope of which encom- passes both hydrodynamics and the reactive transport, which is qualitatively con- sistent with observed and experimentally derived results. This will be accomplished by utilizing an approach of a thin-lm uid ow approximation to the standard eld equations of uid mechanics, reactive transport of chemical species within the bulk

uid described by elementary chemical kinetics, and surface controlled kinetics driven by the reactive transport to the surface via surface reactions producing the precip- itant. The hydrodynamics and the reactive transport models are coupled through the free boundaries at the air-water interface and the water-mineral interface. This paper develops a method for analyzing the mechanism governing the formation of these unique rimstone features.

This eort also provides a catalyst for continuing study of geological, chem- ical, and physical processes which are similar. Mathematically, investigation in free boundary problems for precipitative pattern formation will be of further interest.

Details of this model will contribute to an understanding of the problem of scale deposition in industrial and water distribution systems which will subsequently allow for improvement in design, maintenance, and remediation of such systems.

5 Figure 1.1: Photograph montage of rimstone dams from Yellowstone National Park and two caves. Courtesy NPS.

6 CHAPTER II

CALCIUM CARBONATE PRECIPITATION MODEL

This chapter is devoted to the development and interpretation of the calcium car- bonate precipitation model. The following discussion develops the hydrodynamic and reactive transport systems separately. The hydrodynamics will be described by a thin-lm ow approximation for the equations of uid mechanics while reactive transport will be described by a system of advective-diusive equations. The model is fully realized by a systematic determination of the temporal and spatial proles of the air-water and water-mineral surfaces by coupling the shapes of these free surfaces with reactive transport and hydrodynamic mechanisms. The result is a robust model characterized by encompassing both these mechanisms.

2.1 Hydrodynamic Model

Figure 2.1 is a schematic of the genesis and evolution of the calcium carbonate pre- cipitant on the cave oor and subsequent rimstone dam formation. Let the x-axis coincide with the plane bed inclined at an angle with respect to the horizon, and the z-axis normal to the plane bed as depicted in Figure 2.1 Inset. The surface at the air-water interface is denoted by z = h(x, t), and the water-mineral interface is given by z = s(x, t).

7 Inset Water λ Flowstone calcite Source laminated secondary z calcite deposit

Water Film Flowing g x Water Bedrock of cave floor z=h(x,t) (Limestone) Calcite Bedrock of cave floor z=s(x,t) A (Limestone) θ

Water Water Dams 3 ) Source Source (CaCO Intial dams Pool

Pool

Pool Bedrock of cave floor (Limestone) Bedrock of cave floor (Limestone) B C

Figure 2.1: Schematic of Dam Initiation and Formation

8 In the literature reviewed there is no evidence for the growth of rimstone dams beneath deep water [3]. Additionally, the lack of documented observations and an available hydro-chemical model supporting features in such an environment has yet to be established; hence, this model considers a layer of thin lm ow for the initialization of rimstone dam formation. The ow must be relatively constant during periods of growth, since less ow would mean drying up, and more would coincide with increasing the water depth which is not amenable to dam growth.

Due to the nature of the cave environment, temperatures remain constant.

The density and viscosity are constant; thus, buoyancy driven forces due to tem- perature and concentration gradients are neglected. Although vorticity eects due to the ow over the dams occur, this model considers the initial formation of these features. In addition, characteristic velocities measured at various sites[3] [10] [11] support laminar ow. Lastly, the development of features transverse to the ow direc- tion will not be considered. Therefore, the ow is characterized as two-dimensional, incompressible, viscous, and irrotational.

The dynamics of the ow is described by a mass and momentum conservation law. The appropriate equation for the conservation of mass is given by the continuity equation

∂u ∂w + = 0, (2.1) ∂x ∂z

and the Navier-Stokes equations describing the conservation of linear momentum are 9 ∂u ∂u ∂u ∂p ∂2u ∂2u + u + w = g sin + + , (2.2) ∂t ∂x ∂z ∂x ∂x2 ∂z2 µ ¶ µ ¶

∂w ∂w ∂w ∂p ∂2w ∂2w + u + w = g cos + + , (2.3) ∂t ∂x ∂z ∂z ∂x2 ∂z2 µ ¶ µ ¶

where is the density of water, u and w are velocity components in the x and z- directions respectively, g is gravity, and is the viscosity of water.

The bottom boundary is a rigid surface; hence, a no-slip condition is applied.

Hence, at z = s(x, t)

u = v = 0. (2.4)

The top surface of the uid layer, h(x, t) is a free boundary; therefore, its position is unknown until the hydrodynamic and reactive transport systems are solved. It is necessary to specify a kinematic condition concerning its motion, and two dynamic conditions concerning the stresses acting on it. This is a material surface; hence, it must be composed of the same uid particles. The velocity at the surface is tangential to the moving surface at all times. Let the surface be described by

F (x, z, t) = z h(x, t). (2.5)

10 If a uid particle is located at ~x =< x, z > at time t, then in a small time interval d~x dt it moves to ~x + dt or ~x + V~ dt at time t + dt, where V~ =< u, w >. Hence, the dt new top surface is given by

F (~x + V~ dt, t + dt) = 0. (2.6)

Performing a Taylor expansion for small dt

∂F (~x, t) F (~x, t) + + V~ ∇F (~x, t) dt + O(dt2) = 0. (2.7) ∂t µ ¶

From (2.5) the rst term vanishes; hence,

∂F (~x, t) + V~ ∇F (~x, t) = 0 (2.8) ∂t

in the limit dt → 0. The following kinematic condition is now arrived at by (2.5)

∂h ∂h w = + u . (2.9) ∂t ∂x

11 Two dynamic conditions are necessary to describe the inuence of stresses on the shape of the surface. The stress-strain relation [12] is employed which is repre- sentative of this action,

P = 2S˙ pI, (2.10)

where P is the stress tensor, S˙ is the strain rate dyadic, and I is the identity.

In the cave environment there is no wind which generates stress in the tan- gential direction. This leads the to the expression

(Pijnj)i = 0, (2.11)

for the stress in the tangential direction, where Pij is an element of the stress tensor

P with the subscripts i and j representing the direction of the stress and the surface where the stress is applied respectively. The vectors nˆ and ˆ are the normal and tangential unit vectors to the surface and are given by

∂h ∂h , 1 1, ∂x ∂x nˆ = ¿ À and ˆ = ¿ À . (2.12) 2 2 ∂h ∂h 1 + 1 + ∂x ∂x s µ ¶ s µ ¶

The elements of the stress tensor P are dened by

12 ∂u P11 = p + , ∂x ∂u ∂w P12 = P21 = + , (2.13) ∂z ∂x µ ¶ ∂w P22 = p + . ∂z

Substituting these into (2.11), one nds the following stress condition

2 ∂h ∂u ∂w ∂h ∂u ∂w 2 + + 1 + = 0. (2.14) ∂x ∂x ∂z ∂x ∂z ∂x µ ¶ " µ ¶ # µ ¶

The normal component of the stress balances with the surface tension and the cur- vature of the surface which is represented by the expression

(Pijnj)ni = (2.15)

where is the surface tension and

∂2h 2 = ∂x (2.16) 2 3/2 ∂h 1 + ∂x à µ ¶ ! is the curvature of F (~x, t). This leads to the following stress condition in the normal direction

13 2 ∂u ∂h ∂h ∂u ∂w ∂w + + ∂2h ∂x ∂x ∂x ∂z ∂x ∂z " µ ¶ µ ¶ # ∂x2 p + 2 = . (2.17) 2 3/2 ∂h ∂h 1 + 1 + ∂x ∂x µ ¶ Ã µ ¶ !

An additional condition representative of the evolving boundary at the water-mineral interface, z = s(x, t), is needed. However, the dynamics of this surface are chemically driven; thus, details of this condition are delayed until the discussion of the reactive transport model is developed.

2.2 Reactive Transport Model

The uid is open to the atmosphere and is assumed to contain the basic constituents of the calcium carbonate reaction system. The reaction system is based on dissolved carbon dioxide and consists of six chemical species that participate in ve simultane- ous chemical reactions,

14 k1 CO2 + H2O ­ H2CO3

k1

k2 + CO2 + H2O ­ H + HCO3

k2

k3 CO2 + OH ­ HCO3 (2.18)

k3

k4 + H2CO3 ­ H + HCO3

k4

k5 + 2 HCO3 ­ H + CO3

k5

where ki and ki are the forward and backward reaction rate coecients respectfully for the ith reaction. This is a common reaction system in natural waters, and serves as a natural pH buer [13].

The following variables are dened with molar concentrations with units

3 2 + mmol cm : c1 = [CO2], c2 = [H2CO3], c3 = [HCO3 ], c4 = [CO3 ], c5 = [H ],

and c6 = [OH ]. Assuming elementary reaction dynamics one obtains the following rate laws for each reaction,

15 I1 = k1[CO2] + k1[H2CO3] = k1c1 + k1c2

+ I2 = k2[CO2] + k2[H ][HCO3 ] = k2c1 + k2c3c5

I3 = k3[CO2][OH ] + k3[HCO3 ] = k3c1c6 + k3c3 (2.19)

+ I4 = k4[H2CO3] + k4[H ][HCO3 ] = k4c2 + k4c3c5

+ 2 I5 = k5[HCO3 ] + k5[H ][CO3 ] = k5c3 + k5c4c5.

Hence, the net rate laws for each species are given by

P1 = I1 + I2 + I3

P2 = I1 + I4

P3 = I2 I3 I4 + I5 (2.20)

P4 = I5

P5 = I2 I4 I5

P6 = I3.

Explicitly rewriting I1 through I5 in terms of the concentration variables, the model equations for the carbonate system become:

16 ∂c1 2 ∂c1 ∂c1 = D1∇ c1 u w + (k1c1 + k1c2) ∂t ∂x ∂z

+ (k2c1 + k2c3c5) + (k3c1c6 + k3c3)

∂c2 2 ∂c2 ∂c2 = D2∇ c2 u w (k1c1 + k1c2) ∂t ∂x ∂z

+ (k4c2 + k4c3c5)

∂c3 2 ∂c3 ∂c3 = D3∇ c3 u w (k2c1 + k2c3c5) ∂t ∂x ∂z (2.21) (k3c1c6 + k3c3) (k4c2 + k4c3c5) + (k5c3 + k5c4c5)

∂c4 2 ∂c4 ∂c4 = D4∇ c4 u w (k5c3 + k5c4c5) ∂t ∂x ∂z

∂c5 2 ∂c5 ∂c5 = D5∇ c5 u w (k2c1 + k2c3c5) ∂t ∂x ∂z

(k4c2 + k4c3c5) (k5c3 + k5c4c5)

∂c6 2 ∂c6 ∂c6 = D6∇ c6 u w (k3c1c6 + k3c3). ∂t ∂x ∂z

Here, Di is the mass diusivity of the ith chemical species. The only other species which is being traced is Ca2+. This ion does not participate in any of the bulk reac- tions, but appears only in the surface reactions at the cave oor which are imposed as boundary conditions. Although calcium complexes do form in the bulk uid, this approach is sucient as a point of initiation. Hence, in the bulk uid, Ca2+ is a

2+ nonreactive tracer. Let c7 = [Ca ], then

∂c7 2 ∂c7 ∂c7 = D7∇ c7 u w . (2.22) ∂t ∂x ∂z

17 Each transport equation for the reactants (2.21) requires boundary conditions at each interface. At the air-uid interface, z = h(x, t), two types of conditions are necessary.

First, CO2 is a volatile gas, thus it may undergo degassing at this interface. So for c1 the boundary condition is

c1 = kH PCO2 , (2.23)

where kH is the Henry’s constant and PCO2 is the partial pressure of carbon dioxide as per Henry’s Law. All other species are non-volatile and satisfy no ux conditions

∂h , 1 ∂x (∇ci) nˆ = 0 where nˆ = µ ¶ . (2.24) 2 ∂h 1 + ∂x s µ ¶

At the uid-mineral interface, z = s(x, t), reactants may leave the uid via surface reactions; in particular, reactants in the bulk and the calcium ions form the calcium carbonate precipitant. Three reactions govern the precipitation and dissolu- tion of calcium carbonate at this interface [13]:

18 1 + 2+ CaCO3(s) + H ­ Ca + HCO3

1

2 2+ CaCO3(s) + H2CO3 ­ Ca + 2HCO3 (2.25)

2

3 2+ CaCO3(s) + H2O ­ Ca + HCO3 + OH

3

where i and i are the forward and backward reaction rate coecients respectfully for the ith surface reaction. The simplest method to model the rates of these reac- tions is to treat each as elementary. Therefore, the rate laws are given by:

+ 2+ J1 = 1[H ] + 1[HCO3 ][Ca ] = 1c5 + 1c3c7

2 2+ 2 J2 = 2[H2CO3] + 2[HCO3 ] [Ca ] = 2c2 + 2c3c7 (2.26)

2+ J3 = 3[H2O] + 3[HCO3 ][OH ][Ca ] = 3 + 3c3c6c7.

Precipitation of calcium carbonate occurs for high pH which corresponds to an in- crease in the concentration of OH and a decrease in the concentration of H+ which forces the rate laws J1, J2, and J3 positive. In the same way, these rate laws change sign, representative of calcium carbonate dissolution. This occurs when the concen- tration of OH decreases and the concentration of H+ increases which is characteristic

19 of low pH waters. The eects of pH on the surface kinetics will be discussed in more detail later.

The rate laws in (2.26) enable one to write the net reaction rate laws for the species c2, c3, c5, c6, and c7 at the water-mineral interface. All other rates are taken to be zero,

R1 = 0

R2 = J2

R3 = J1 + J2 + J3

R4 = 0 (2.27)

R5 = J1

R6 = J3

R7 = J1 + J2 + J3.

Applying these rates as forcing for ux conditions, one nds, the boundary conditions of the seven species at z = s(x, t) are given by

Dj∇cj nˆ = Rj. (2.28)

It is now possible to determine the boundary condition at the water-mineral inter- face for the free surface at z = s(x, t). Taking the normal velocity of the calcium carbonate front as driven by the surface reactions and subsequent deposition results in

20 ∂s c ∂t = McR7, (2.29) 2 ∂s 1 + ∂x s µ ¶

where Mc and c are the molar mass and density of calcium carbonate respectively.

The equation is a mass balance which states that the rate of increase in calcium carbonate mass is equal to the rate of loss of the chemical species due to the surface reactions. Note that R7 is a function of c2, c3, c5, c6, and c7 which in turn are functions of h(x, t) and s(x, t) through the boundary conditions. Hence, this equation is coupled with the previously determined kinematic condition (2.9) for the evolution of the top surface h(x, t).

The previous mathematical discussion for the hydrodynamic and reaction transport of chemical species provides a theoretical representation of the physical system. This precipitation model embodies the hydrodynamics, chemical control, and the eect of surface reactions producing the calcium carbonate precipitant. This includes changes to the morphology of the surface, which in turn, aects the dynamics of the uid ow.

21 CHAPTER III

SOLUTION PROCEDURE

This chapter consists of developing the solution procedure to the model equations from Chapter 2. A solution will be formulated rst for the dynamics of the uid ow and then the reactive transport model solution will be developed. Once these systems are known, the kinematic condition (2.9) and the equation characterizing the velocity of the calcium carbonate front (2.29) can be fully determined.

3.1 Hydrodynamics Solution Procedure

The uid equations are solved via asymptotic limits using the standard long-wavelength and lubrication theory approximations [14] [15] [16]. In this approach equations (2.1),

(2.2), (2.3), (2.4), (2.14), and (2.17) are used to dene solutions for the pressure p and velocities u and w. Due to the boundary conditions, the velocities will be functions of the unknown free boundaries h and s.

The underlying assumption is that the wavelength of the surface waves are larger in magnitude to the depth of the lm of water. Let d be the characteristic depth of the water lm and be the characteristic wavelength of the surface waves.

2d The scaling parameter k, called the wavenumber, is dened by k = . With these parameters a non-dimensionalization scheme can be developed. Table 3.1 summarizes

22 Table 3.1: Non dimensional variables Dimensional Variable Scaling Non-Dimensional Variable d x Longitudinal Dimension: x k d/k z Transverse Dimension: z d d gd2 u Longitudinal Velocity: u U = U w Transverse Velocity: w kU kU U p Pressure: p P = kd P d t Time: t T = c McR7 T the non-dimensional variables employed. Here U, P , and T are the characteristic ve- locity, pressure, and time respectively. Time T is scaled to the density of calcium carbonate c, the molar mass of calcium carbonate Mc, and the net rate law R7 as dened by (2.27). In this way, T >> 1, so the characteristic time is on a geological scale.

Applying this non-dimensionalization to the uid dynamics governing equations one

nds,

∂u ∂w + = 0 (3.1) ∂x ∂z

for the mass conservation equation (2.1), and

23 d ∂u ∂u ∂u ∂p ∂2u ∂2u kR + u + w = sin() + k2 + (3.2) e T Uk ∂t ∂x ∂z ∂x ∂x2 ∂z2 · ¸

and

d ∂w ∂w ∂w ∂p ∂2w ∂2u k3R + u + w = k cos() + k4 + k2 (3.3) e T Uk ∂t ∂x ∂z ∂z ∂x2 ∂z2 · ¸

for Navier Stokes equations (2.2) and (2.3) in the x and z-directions respectively,

Ud d where Re = is the Reynold’s number. The non-dimensional group T Uk << 1, since T >> 1 and k << 1 as per the long wavelength assumption.

At the water-mineral interface, z = s(x, t), the boundary conditions (2.4) become

u = w = 0. (3.4)

At the air-water interface, z = h(x, t), the tangential component of the stress (2.14) is given by

2 ∂h ∂u ∂w ∂h ∂u ∂w 2k2 + + 1 + k2 + k2 = 0, (3.5) ∂x ∂x ∂z ∂x ∂z ∂x µ ¶ " µ ¶ # µ ¶

and the non-dimensionalized normal component of the stress (2.17) is

24 ∂u ∂h ∂u ∂w ∂w 2 k2 k2 + k2 + ∂ h ∂x ∂x ∂z ∂x ∂z k ∂x2 p + · µ 2 ¶ ¸ = . (3.6) B 2 3/2 2 ∂h ∂h 1 + k 1 + k2 ∂x ∂x µ ¶ Ã µ ¶ !

gd2 Here B = k2B for thin lms where B = is the Bond number. Finally, the non-dimensionalized kinematic condition (2.9) is given by

d ∂h ∂h = w + u (3.7) T Uk ∂t ∂x

d Once again, the non-dimensional group T Uk << 1 which is representative of quasi- steady state dynamics. The hydrodynamics amenable for precipitation as outlined in this section suggests characteristic velocities between 0.3 and 0.5 cm s1 which in turn leads to Reynold’s numbers in a range of 0.05 and 0.1 which is indicative of laminar lm ow. This concludes the dimensional analysis of the hydrodynamics.

Assume the following asymptotic expansions for small k

u = u0 + ku1 +

w = w0 + kw1 + (3.8)

p = p0 + kp1 +

Substituting the expansions in for the system (3.1), (3.2), and (3.3) one nds the leading order problem is given by

25 ∂u0 ∂w0 + = 0 ∂x ∂z

2 ∂ u0 ∂p0 = sin() (3.9) ∂z2 ∂x

∂p0 = 0, ∂z subject to the conditions

u0 = w0 = 0 on z = s(x, t), (3.10) and

∂u0 = 0, p0 = 0 on z = h(x, t). (3.11) ∂z

The solution to this system is given in Appendix A. The system for the rst correc- tion is given by

∂u1 ∂w1 + = 0 ∂x ∂z

2 ∂ u1 d ∂u0 ∂u0 ∂u0 ∂p1 = R + u0 + w0 + (3.12) ∂z2 e T Uk ∂t ∂x ∂z ∂x µ ¶

∂p1 = cos(), ∂z subject to the conditions

26 u1 = w1 = 0, on z = s(x, t), (3.13)

and

2 ∂u1 1 ∂ h = 0, p1 = on z = h(x, t). (3.14) ∂z B ∂x2

The solution to this system is found in Appendix A. Substituting the velocities u and w and the pressure p into the non-dimensionalized kinematic equation (3.7), the evolving boundary can be determined. This nonlinear partial dierential equation has the form ∂h ∂h ∂2h ∂3h ∂4h = F1 x, t, s, h, , , , , (3.15) ∂t ∂x ∂x2 ∂x3 ∂x4 µ ¶

and is found in Appendix A. Hence, the shape of the air-water interface will couple to the shape of the water-mineral interface. The equation for the shape of the latter will be determined in the solution procedure for the reactive transport system.

3.2 Reactive Transport Solution Procedure

Since s(x, t) is chemically driven, the partial distribution of each concentration must be determined. This is accomplished by solving the reactive transport system. The system is a set of nonlinear and coupled partial dierential equations making it quite

27 dicult to solve; thus, a number of approximation schemes must be performed to make a solution tractable.

Based on the data collected at the sites of interest, the cave environment ensures relatively constant temperature and pH levels. Additionally, each chemical species has a mass diusivity on the order of 105cm2/s [3]. A characteristic mass diusivity is dened for the system as D 105cm2/s, and is taken to be the mass diusivity of each of the chemical species. This allows one to dene the following operator acting on each of the concentrations ci

∂ ∂ ∂ L = D∇2 + u + w . (3.16) ∂t ∂x ∂z

In this way system (2.21) may be written as L(ci) = Pi, where Pi was given in (2.20).

Rewriting Pi in terms of the rate laws Ii one arrives at the following matrix repre- sentation:

c1 1 1 1 0 0     I1    c2   1 0 0 1 0           I2         c3   0 1 1 1 1    L   =     . (3.17)      I3         c4   0 0 0 0 1               I4         c5   0 1 0 1 1               I5         c6   0 0 1 0 0           

28 It is well known that the rate laws for I4 and I5 reach equilibrium much faster than the others. If I4 and I5 are taken to be instantaneous compared to the other rate laws, then this system can be transformed by rotating the matrix using I4 and I5 as pivot points. One possible scheme is

c1 1 1 1 0 0     I1    c2   1 0 0 1 0           I2           0 0 0 0 0    L   =     , (3.18)      I3         c4   0 0 0 0 1               I4           1 1 0 0 0               I5         c6   0 0 1 0 0            where = c3 + 2c4 c5 + c6 and = c2 c4 + c5. Replacing I4 and I5 by their equilibrium conditions one nds

I4 = k4c2 + k4c3c5 = 0 (3.19)

I5 = k5c3 + k5c4c5 = 0.

Solving for c2 and c4 respectfully

c3c5 c2 = K4 (3.20) K4c3 c4 = , c5

29 where

k4 k5 K4 = and K5 = . (3.21) k4 k5

Using the algebraic expressions for c2 and c4, one nds the system is reduced to

k1 L(c1) = k1c1 + c3c5 k2c1 + k2c3c5 k3c1c6 + k3c3 K4

L(c6) = k3c1c6 + k3c3

L() = 0 (3.22) L() = k5c3 k5K5c3

c3c5 c2 = K4 K5c3 c4 = . c5

The system has been reduced to four coupled nonlinear partial dierential equations.

Further simplications can be made by examining the order of magnitude of each term in the system. Table 3.2 lists typical concentrations, with units mmol/cm3, of each species from two sites where there is active rimstone dam growth. The geochemical values shown in Table 3.2 are -log the activity unless indicated. Simplication to

(3.22) can be made based on order of magnitude. First note that c3 is the dominant term in the equation of ; hence, L() L(c3) = 0. The solution c3 identically a constant satises the equation. Additionally, c7 identically a constant satises

L(c7) = 0.

30 Table 3.2: Summary of Geochemical Parameters of Groundwater in Tributaries of the Mystic River, Scott Hollow Cave [1].

Location

Craig’s Creek John’s Flowstone

Sample Data 2/21/1998 5/17/1998 2/21/1998 5/17/1998

Temp ( C) 10.7 12.3 11.1 11.4

pH 7.78 7.41 7.8 7.18

TDS (mg/L) 196.9 191.1 191.1 204.6

Ca2+ 2.990 3.005 2.997 2.994

2 CO3 5.315 5.558 5.323 5.790

HCO3 2.616 2.509 2.648 2.500

H2CO3 3.940 3.477 3.996 3.230

H+ 7.78 7.41 7.8 7.18

OH 6.727 7.036 6.691 7.300

SICalcite 0.099 -0.144 0.096 -0.318

31 If one returns to the unrotated system it now has been reduced to three coupled nonlinear dierential equations. Notice that the terms I4 and I5 in the c5 equation vanish when replaced by their equilibrium conditions.

k1 L(c1) = k1c1 + c3c5 k2c1 + k2c3c5 k3c1c6 + k3c3 K4

L(c5) = k2c1 k2c3c5

L(c6) = k3c1c6 + k3c3

c3c5 c2 = (3.23) K4 K5c3 c4 = c5

c3 constant

c7 constant.

Table 3.3 lists the known coecients for the chemical transport system and the surface kinetics. The backward reaction rate coecients 1, 2, and 3 will be determined later and are presented in Table 3.5. K4 and K5 are found by

c3c5 K4 = c2 (3.24) c4c5 K5 = , c3

using Table 3.2 for the typical concentrations c2, c3, c4, and c5.

The expressions describing the concentrations c3 and c7 can be found through the boundary conditions (2.28) at the water-mineral interface z = s(x, t). Since the concentrations c3 and c7 are constants, the boundary conditions (2.28) are reduced

32 Table 3.3: Chemical kinetics constants and their values. M = mmol cm1.

Table of Constants Source

2 1 kH 5.3 10 M atm [3]

4 PCO2 3.4 10 atm [3]

2 1 k1 3.8 10 s [7]

1 1 k1 2.69 10 s [7]

3 1 k2 1.0 10 s [7]

4 1 1 k2 1.7 10 s M [7]

3 1 1 k3 8.5 10 s M [7]

4 1 k3 2.0 10 s [7]

2 1 1 4.30 10 cm s [2]

5 1 2 1.47 10 cm s [2]

7 1 3 1.05 10 cm s [2]

33 to an algebraic equation for both c3 and c7:

2 2 1c5 + 1c3c7 c3c5 2c3c7 3 + 3c3c6c7 = 0. (3.25) K4

The concentrations c5 and c6 once determined are taken to be their respective values at the water-mineral interface z = s(x, t). In this way c3 and c7 can be found:

2 2 2 1c7 + c5 3c6c7 + 1c7 + c5 3c6c7 + 423c7 K4 s K4 c3 = µ ¶ 22c7 2 1c5 + c3c5 + 3 K4 c7 = 2 . 2c3 + 3c3c6 1c3 (3.26)

The important role of the species c5 and c6 and their connection to the dynamics of the mineral surface will be discussed later in addition to the unknown reaction rate coecients 1, 2, and 3.

What remains to be solved is the coupled system of partial dierential equa- tions for c1, c5, and c6. The same order of magnitude assumptions employed for deriving (3.23) can be used to simplify the expressions for the net rate laws (2.20)

7 for the concentrations c1, c5, and c6. Neglecting terms O(10 ) and smaller results in the decoupling of the partial dierential equation for c6, and leaves two coupled advective-diusive equations describing c1 and c5, and a single partial dierential equation for c6:

34 ∂c1 2 ∂c1 ∂c1 k1 D∇ c1 + u + w = k1c1 + c3c5 k2c1 + k2c3c5 + k3c3 ∂t ∂x ∂z K4

∂c5 2 ∂c5 ∂c5 D∇ c5 + u + w = k2c1 k2c3c5 ∂t ∂x ∂z

∂c6 2 ∂c6 ∂c6 D∇ c6 + u + w = k3c3. ∂t ∂x ∂z (3.27)

Returning to the non-dimensionalization scheme Table 3.1 used in the hydro- dynamics solution procedure and applying it to this system one nds

d2 ∂c ∂2c ∂2c ∂c ∂c d2 i k2 i + i + kP u i + w i = P , (3.28) T D ∂t ∂x2 ∂z2 e ∂x ∂z D i µ ¶ µ ¶

Ud where Pe = D is the Peclet number. Analysis of the characteristic values and their order of magnitude demonstrates that the dominating method of transport of the re- actants at leading order is diusion of the chemical species from the air-water interface to the water-mineral interface. Therefore, the rate of change of the concentrations and advective transport can be neglected by the additional use of the geological time scale, yielding the following system of second order ordinary dierential equations in dimensional form:

35 2 ∂ c1 k1 k1c3 k2 k2c3 k3c3 2 = c1 c5 + c1 c5 ∂z D DK4 D D D

2 ∂ c5 k2 k2c3 = c1 + c5 (3.29) ∂z2 D D

2 ∂ c6 k3c3 = . ∂z2 D

Non-dimensionalization of the boundary conditions at the air-water interface, z = h(x, t), (2.23) and (2.24), are given by:

c1 = kH PCO2

2 ∂h ∂ci ∂ci k + (3.30) ∂x ∂x ∂z = 0 2 ∂h 1 + k ∂x s µ ¶

for the concentration c5 and c6 satisfying a no-ux condition at z = h(x, t). Hence, at leading order the concentrations of these chemical species at the air-water surface, in dimensional form, are given by:

c1 = kH PCO2

∂c5 = 0 (3.31) ∂z

∂c6 = 0. ∂z

For the water-mineral interface z = s(x, t), applying the non-dimensional scheme

36 leads to the respective boundary conditions (2.28)

∂s ∂c ∂c k2 i + i ∂x ∂x ∂z d = Ri, (3.32) 2 ∂s D 1 + k ∂x s µ ¶

for the concentrations ci of the species i = 1, 5, and 6. If the dominant terms in order of magnitude are taken to be the method of transport to the surface z = s(x, t), the following conditions, in dimensional form, can be found:

∂c1 D = 0 ∂z

∂c5 D = 1c5 1c3c7 (3.33) ∂z

∂c6 D = 3 + 3c3c6c7. ∂z

Finally, applying this scheme to (2.29) the evolution of the calcium carbonate front is given by

∂s ∂t T Mc = R7. (3.34) 2 ∂s cd 1 + k2 ∂x s µ ¶

At leading order, the evolution of the calcium carbonate front, in dimensional form, is given by:

∂s Mc = R7. (3.35) ∂t c

37 Table 3.4: Values of Parameters used in the Hydrodynamics and Reactive Transport

Models

Parameter Value Source

1 Molar Mass of Calcium Carbonate: Mc 0.100892g mmol [17]

3 Density of Calcium Carbonate: c 2.71g cm [17]

Density of Water: 0.99970g cm3 [12]

Surface Tension of Water: 74.2dyne cm1 [12]

Viscosity of Water: 0.013609g cm1 s1 [12]

Table 3.4 lists the values of the parameters used in the hydrodynamic and the reactive transport problems.

The proceeding simplications to the chemical transport of the model now allow for a more tractable method of solving the model equations. The solution of the concentration c6 has the form

2 c6 = a1z + a2z + a3, (3.36)

where a1, a2, and a3 are functions of x and t through the boundary conditions z = h(x, t) and z = s(x, t) via Appendix B. Next the system of coupled second order dierential equations describing the transport of the concentrations c1 and c5 can be

38 solved. The solutions have the form

1z 1z 2z 2z c1 = b11e + b21e + b32e + b42e + b5 (3.37)

1z 1z 2z 2z c5 = b1e + b2e + b3e + b4e + b5 and are given in Appendix B. The concentration eld for all seven chemical species is now fully determined:

1z 1z 2z 2z c1 = b11e + b21e + b32e + b42e + b5

c3 c2 = c5 K4

2 2 2 1c7 + c5 3c6c7 + 1c7 + c5 3c6c7 + 423c7 K4 s K4 c3 = µ ¶ 22c7

K5c3 c4 = c5

1z 1z 2z 2z c5 = b1e + b2e + b3e + b4e + b5

2 c6 = a1z + a2z + a3

2 1c5 + c3c5 + 3 K4 c7 = 2 . 2c3 + 3c3c6 1c3 (3.38)

39 Now that the velocity eld of the uid dynamics and the concentration eld for the chemical transport are known, it is now possible to determine the unknown boundaries h(x, t) and s(x, t). First it is necessary to nd the unknown reaction rate coecients 1, 2, and 3 in (2.25). These are found by using the known values from Table 3.3 and the net rate laws for the surface reactions (2.26). The concentra- tions c3 and c7, found in (3.26), contain these reaction rate coecients. The boundary condition for c5 and c6 contain the reaction rate coecients 1 and 3 respectively through the boundary conditions (2.28). These provide a system of three nonlinear algebraic equations for the three unknown reaction rate coecients:

c3 2 1c5(s) + 1c3c7 2 c5(s) + 2c3c7 3 + 3c6(s)c3c7 = 0, K4

∂c5 D 1c5(s) + 1c3c7 = 0, (3.39) ∂z ¯z=s ¯ ¯ ∂c6 ¯ D + 3 3c6(s)c3c7 = 0. ∂z ¯z=s ¯ ¯ ¯ Here the concentrations c3 and c7 are taken from Table 3.2. The values of 1,

2, and 3 listed in Table 3.5 are found in Appendix B, where all the values of the concentrations are taken at the water-mineral boundary z = h(x, t). These reaction rate coecients are pH dependent; hence, they are implicitly dependent on the water lm thickness. The values in Table 3.5 are calculated at a local depth of d = 0.002cm. The partial dierential equation describing the evolving calcium carbonate front (2.29) is now fully determined. The growth time for this front can

40 Table 3.5: Mathematically derived values for the backward reaction rate coecients at the water- mineral surface: M = mmol cm3

Reaction Rate Coecient Derived Value

4 1 1 1 3.3778 10 cm M s

5 2 1 2 2.3713 10 cm M s

2 2 1 3 8.0681 10 cm M s

be estimated by analysis of Equation (3.35). Using the values in Table 3.4 for the density of calcium carbonate c, the molar mass Mc, and the magnitude of R7 , the growth rate of the calcium carbonate front is 109cm s1 or approximately 1cm of deposition every 30 years which is consistent with the literature [10].

3.3 Free Boundaries Solution Procedure

Now that the velocity and concentration elds are fully determined it is possible to determine the shape of the free surfaces h(x, t) and s(x, t). These surfaces are dened by a coupled set of partial dierential equations. Although an analytic solution is quite intractable, it is possible to acquire qualitative information about these surfaces.

This will be accomplished by linearizing about a at surface. Stability analysis of the linearized problem will yield information on growth rate, wavelength, and the eects of cave slope on rimstone dam features.

41 First it is necessary to redimensionalize the kinematic condition (3.7) at the air-water boundary z = h(x, t). The result, in dimensional variables, is a coupled set of nonlinear partial dierential equations of the form

∂h ∂h ∂2h ∂3h ∂4h = F1 x, t, h, s, , , , ∂t ∂x ∂x2 ∂x3 ∂x4 µ ¶ (3.40) ∂s ∂h ∂s = F2 x, t, h, s, , . ∂t ∂x ∂x µ ¶

To obtain qualitative information on the solution of this system, the kinematic condi- tion z = h(x, t) is linearized about the local characteristic depth d, and the evolving calcium carbonate front at z = s(x, t) is linearized about a at interface. The follow- ing are dened for this purpose:

h(x, t) = d + H(x, t) (3.41) s(x, t) = 0 + S(x, t), where the terms H(x, t) and S(x, t) are small perturbations from the at surface.

Substituting Equations (3.41) into the boundary conditions, dierentiating with re- spect to the small parameter , and setting = 0, one nds the linearized system given in Appendix A and Appendix B. Assume the solutions have the form:

t ikx H(x, t) = 1e e (3.42) t ikx S(x, t) = 2e e ,

42 where is the growth rate with units s1 and k is the wave number with units cm1.

Substituting (3.42) into the linearized equations one nds, a system of two equations for the two unknowns 1 and 2. For nontrivial solutions, the determinant of this system, which is quadratic in , must vanish. For stability and physical considera- tions, the solution

(d, k, ) (3.43)

is the growth rate of the calcium carbonate front which is a function of the charac- teristic depth d, the wavenumber k, and the slope of the cave oor . Analysis of the growth rate function will determine characteristics such as water lm thickness, wavelength of the surface waves, and the slope of the cave oor which are amenable to rimstone dam growth.

43 CHAPTER IV

RESULTS AND CONCLUSIONS

This chapter interprets the mathematical model and solution procedure and resolves details in the geophysics and geochemistry of rimstone dam growth. Analysis of the dynamics of the evolving surfaces at the air-water interface and the water-mineral interface will provide information regarding features such as cave oor slope, wave- length of surface waves, and rate of deposition and their eects on morphological changes in the cave oor. Additionally, examination of the chemical kinetics at the mineral surface will provide insight into the inuence of reactants driving the precip- itation and dissolution of calcium carbonate. Finally, limitations and discussion on further investigation into this geological phenomenon are detailed.

4.1 Analysis of Reactants Driving Precipitation and Dissolution

The transport of chemical species is described by the concentration eld (3.38). Figure

4.1 is a concentration prole at a characteristic depth of 0.002cm. This prole illus- trates that the concentration of OH is greater than the concentration of H+ which is indicative of deposition of CaCO3 [2]. At the water-mineral boundary, z = s(x, t), the

+ concentrations of H2CO3 and H increase, and the concentration of OH decreases as a result of the production of the CaCO3 precipitant via the surface reactions (2.25).

44 Additionally, the concentration of CO2 rapidly decreases as the prole approaches the air-water boundary; this is evident of the degassing occurring at the interface z = h(x, t). The pH of the solution increases through the bulk uid which is an indication of the dependence of pH on the water lm thickness.

Utilizing the concentration eld (3.38) and analysis of the boundary data at the water-mineral interface (2.28), one nds the dominant reactants governing precipitation and dissolution can be found by examining the rate laws (2.26) for the surface reactions. Appealing to the values in Table 3.3 and the concentrations c3 and c7 given in Table 3.2, one nds that the dominant reactions are those dened by the rate laws J1 and J3 given in (2.26). In turn these rate laws are driven by the concentrations of H+ and OH. Therefore, pH appears to be the dominating chemical inuence in determining precipitation of calcium carbonate at the water- mineral interface. This analysis is consistent with both theoretical and experimental results in the literature [2] [7] [11].

4.2 Interpretation of Hydro-Chemical Dynamics

Morphological changes in the mineral surface of the cave oor is described by the rate of growth of dam formation, the wavelength of the surface waves, and the in-

uence of cave oor slope. The initiation of rimstone dam growth can be deter- mined qualitatively by analysis of the growth rate as dened in (3.43). Figures

4.2, 4.3, 4.4, and 4.5 demonstrate growth rate proles for the deposition of calcium carbonate. These proles show a range of the critical wavenumber which is neces- 45 0.100000000

CO2 H2CO3 0.010000000 -2 CO3 H+ OH- 0.001000000 ) -3 0.000100000

0.000010000

0.000001000 Concentration (mmol*cm

0.000000100

0.000000010

0.000000001 0 0.0005 0.001 0.0015 0.002 Film Depth (cm)

Figure 4.1: The concentration eld for a lm thickness of 0.002cm. sary for the deposition process. From this, one can nd that a wavelength in the range of 25 to 42cm is necessary for active deposition in a water depth of 0.002cm.

Similarly, 14cm < < 21cm and 11cm < < 16cm are minimum ranges of wave- lengths amenable for rimstone dam growth for water lm thicknesses of 0.00225cm and 0.0025cm respectively. These graphs indicate the dependence on the slope of the cave oor and depth of the water lm ow on the growth rate of rimstone dam features. First as the depth of the water lm increases, the growth rate increases and the wavelengths necessary for dam growth decrease. The growth rate also increases when the slope of the cave oor increases; this suggests that faster growth rates occur in areas where dam spacing is shorter.

46 0.000006 θ=10 Degrees θ=20 Degress θ=40 Degrees 0.000005 θ=60 Degrees

0.000004 ) -1 0.000003 (s σ

0.000002

0.000001

0.000000 0 0.05 0.1 0.15 0.2 0.25 k (cm-1)

Figure 4.2: Wave number k v. Growth rate for depth d = 0.002cm.

0.000025 θ=10 Degrees θ=20 Degrees θ=40 Degrees θ=60 Degrees 0.000020

0.000015 ) -1 (s σ

0.000010

0.000005

0.000000 0 0.1 0.2 0.3 0.4 0.5 k (cm-1)

Figure 4.3: Wave number k v. Growth rate for depth d = 0.00225cm.

47 0.000060 θ=10 Degrees θ=20 Degrees θ=40 Degrees 0.000050 θ=60 Degrees

0.000040 ) -1 0.000030 (s σ

0.000020

0.000010

0.000000 0 0.1 0.2 0.3 0.4 0.5 0.6 k (cm-1)

Figure 4.4: Wave number k v. Growth rate for depth d = 0.0025cm.

0.000070 d=0.0021 cm d=0.0022 cm 0.000060 d=0.0023 cm d=0.0024 cm

0.000050

0.000040 ) -1 (s σ 0.000030

0.000020

0.000010

0.000000 0 0.1 0.2 0.3 0.4 0.5 k (cm-1)

Figure 4.5: Wave number k v. Growth rate for cave oor slope of 20

.

48 These ndings are in agreement with observations and experiments previously under- taken [8] [9] [10] [11] and serve to validate qualitatively the precipitation model.

4.3 Conclusions

The proposed precipitation model encompasses both hydrodynamics and reactive transport, and has been shown to be qualitatively consistent with observed and ex- perimentally derived results. This was accomplished by utilizing an approach of a thin-lm uid ow approximation to the standard eld equations of uid mechanics, reactive transport of chemical species within the bulk uid described by elementary chemical kinetics, and surface controlled kinetics driven by the reactive transport to the surface via surface reactions producing the precipitant. The hydrodynamics and the reactive transport models inuence one another via the free surfaces at the air-water interface and the water-mineral interface. The mechanisms governing the formation of these rimstone features is thus described by the model.

The limitations of this model are its dependence of site specic data. Further application of the model at other sites of active rimstone growth is needed to arrive at more denitive conclusions of the utility of this model. Experimental study of the backward reaction rate coecients in (2.25) are needed to verify those derived through the solution procedure. Additionally, continued study of geological, chemi- cal, and physical similar processes may be performed with this proposed model; in particular, applications to other single component carbonate minerals that undergo natural growth through similar deposition mechanisms are of further interest. Solu- 49 tion procedures using more robust mathematical analysis of free boundary problems in concert with detailed numerical simulations may reveal other features absent from the procedure used in this paper. Details of this model may also contribute to an understanding of the problem of scale deposition in industrial and water distribution systems which will subsequently allow for improvement in design, maintenance, and remediation of such systems.

The development and subsequent analysis of the this model reveals key char- acteristics of the precipitation of CaCO3. The model uncovers the mutual dependence of hydrodynamic and reactive transport control on the deposition process. The af- fect of the cave oor slope demonstrated that as the slope increases, the wavelength decreases and the growth rate of the front increases. The role of water lm thickness was investigated and showed that increasing the thickness of the lm results in a decrease in wavelength and an increase in the growth rate of the CaCO3 precipitant.

Finally, the solution of the concentration eld in the model reected analysis of the chemical kinetics known to be conducive to the precipitation of calcium carbonate.

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52 APPENDICES

53 APPENDIX A

HYDRODYNAMIC CALCULATIONS

This appendix contains the Maple code used to simplify many of the calculations performed in the solution procedure Chapter 3. This appendix contains the Maple worksheet used for the uid dynamics calculations. This worksheet has been veried on Maple versions 8.0 and 9.5 both student and standard editions. > restart; > with(PDEtools): > with(linalg): > # > declare(h(x,t),s(x,t),_h(x,t),_s(x,t)): > # > # The leading order solutions > p[0]:=(x,z,t)->0: > u[0]:=(x,z,t)->-sin(theta)/2*z**2+h(x,t)*sin(theta)*z+ s(x,t)**2*sin(theta)/2-s(x,t)*h(x,t)*sin(theta): > w[0]:=(x,z,t)->-diff(h(x,t),x)*sin(theta)*z**2/2+ (diff(s(x,t),x)*h(x,t)*sin(theta)+s(x,t)*diff(h(x,t),x)*sin(theta)- s(x,t)*diff(s(x,t),x)*sin(theta))*z+diff(h(x,t),x)*sin(theta)* s(x,t)**2/2-(diff(s(x,t),x)*h(x,t)*sin(theta)+s(x,t)*diff(h(x,t),x)* sin(theta)-s(x,t)*diff(s(x,t),x)*sin(theta))*s(x,t): > # > # The problem for the first correction > p[1]:=(x,z,t)->-cos(theta)*z+h(x,t)*cos(theta)-diff(h(x,t),x$2)/B: > u_1:=int(int(R[e]*((d/(T*U*k))*diff(u[0](x,z,t),t)+ u[0](x,z,t)*diff(u[0](x,z,t),x)+w[0](x,z,t)*diff(u[0](x,z,t),z))+ diff(p[1](x,z,t),x),z),z): > f[1](x,t):=-eval(diff(u_1,z),z=h(x,t)): > f[2](x,t):=-eval(u_1+f[1](x,t)*z,z=s(x,t)): > u[1]:=(x,z,t)->u_1+f[1](x,t)*z+f[2](x,t): > w_1:=int(-diff(u[1](x,z,t),x),z): > f[3](x,t):=-eval(w_1,z=s(x,t)):

54 > w[1]:=(x,z,t)->w_1+f[3](x,t): > # > # Substituting the velocity field into the kinematic conditions at > # z=h(x,t) > surface1:=(d/(T*U*k))*diff(h(x,t),t)=eval(w[0](x,z,t)+ k*w[1](x,z,t),z=h(x,t))-eval(u[0](x,z,t)+k*u[1](x,z,t), z=h(x,t))*diff(h(x,t),x): > # > # The following commands redimensionalize the kinematic condition > algsubs(diff(h(x,t),x$4)=diff(_h(x,t),x$4)/d/(k/d)**4,surface1): > algsubs(diff(h(x,t),x$3)=diff(_h(x,t),x$3)/d/(k/d)**3,%): > algsubs(diff(h(x,t),x$2)=diff(_h(x,t),x$2)/d/(k/d)**2,%): > algsubs(diff(diff(h(x,t),x),t)=diff(diff(_h(x,t),x),t)/d/ (k/d)*T,%): > algsubs(diff(h(x,t),x)=diff(_h(x,t),x)/d/(k/d),%): > algsubs(diff(h(x,t),t)=diff(_h(x,t),t)/d*T,%): > algsubs(h(x,t)=_h(x,t)/d,%): > algsubs(diff(s(x,t),x$2)=diff(_s(x,t),x$2)/d/(k/d)**2,%): > algsubs(diff(diff(s(x,t),x),t)=diff(diff(_s(x,t),x),t)/d/ (k/d)*T,%): > algsubs(diff(s(x,t),x)=diff(_s(x,t),x)/d/(k/d),%): > algsubs(diff(s(x,t),t)=diff(_s(x,t),t)/d*T,%): > dim_surface1:=algsubs(s(x,t)=_s(x,t)/d,%): > # > # Linearizing the dimensionalized kinematic condition at z=h(x,t) > linear_H:=expand(eval(diff(eval(dim_surface1,_s(x,t)=epsilon* S(x,t),_h(x,t)=d+epsilon*H(x,t)),epsilon),epsilon=0)): >

55 APPENDIX B

REACTIVE TRANSPORT CALCULATIONS

This appendix contains the Maple code used to simplify many of the calculations performed in the solution procedure Chapter 3. This appendix contains the Maple worksheet used for the chemical transport equations calculations and is used to gen- erate the grpahs in Chapter 4. This worksheet has been veried on Maple versions

8.0 and 9.5 both student and standard editions. > restart; > with(linalg): > with(PDEtools): > with(plots): Warning, the name changecoords has been redefined

> Digits:=20: > declare(S(x,t),H(x,t)): > # Constants used in the solution procedure > k[1]:=3.8*10**(-2): > k[-1]:=26.9: > k[2]:=10**(-3): > k[-2]:=1.7*10**4: > k[3]:=8.5*10**3: > k[-3]:=2.0*10**(-4): > k[H]:=10**(108.3865-6919.53/283.7+0.01985076*283.7-40.45154* log10(283.7)+669365/(283.7**2)): > P[CO[2]]:=3.4*10**(-4): > kappa[1]:=10**(0.198-444./283.7): > kappa[2]:=10**(2.84-2177./283.7): > kappa[3]:=10**(-5.86-317./287.7): > c[3]:=10**(-2.616): > c[7]:=10**(-2.990): > K[4]:=10**(-7.78)*c[3]/10**(-3.940): 56 > Delta:=10**(-5): > M[c]:=0.100892: > rho[c]:=2.71: > > # The coupled system for c_1 and c_6 > SYS1:=Delta*diff(c[1](z),z$2)=k[1]*c[1](z)-k[-1]/K[4]*c[3]* c[5](z)+k[2]*c[1](z)-k[-2]*c[3]*c[5](z)-k[-3]*c[3], Delta*diff(c[5](z),z$2)=-k[2]*c[1](z)+k[-2]*c[3]*c[5](z): > BC1:=c[1](h)=k[H]*P[CO[2]],diff(c[5](h),h)=0,Delta* diff(c[1](s),s)=0,Delta*diff(c[5](s),s)=kappa[1]*c[5](s)-kappa[-1]* c[3]*c[7]: > solution1:=evalf(combine(dsolve(SYS1 union BC1, [c[1](z),c[5](z)]))): > # > # The system for c_6 > SYS2:=Delta*diff(c[6](z),z$2)=-k[-3]*c[3]: > BC2:=diff(c[6](h),h)=0,Delta*diff(c[6](s),s)=-kappa[3]+kappa[-3]* c[3]*c[7]*c[6](s): > solution2:=evalf(dsolve(SYS2,BC2,c[6](z))): > # Using a flat interface to find the unknown reation rate coefficients > unassign(’h’): > unassign(’s’): > #d:=0.0024: > h:=d: > s:=0.0: > lhs(solution1[1]): > # Solve the system of algebraic equations for the unknown coefficients > unassign(’kappa[-1]’): > kappa[-1]:=eval(Re(solve(eval(rhs(solution1[2]),z=s)=10**(-7.8), kappa[-1])),d=0.00225): > unassign(’kappa[-3]’): > kappa[-3]:=eval(solve(eval(rhs(solution2),z=s)=10**(-6.727), kappa[-3]),d=0.00225): > unassign(’kappa[-2]’): > kappa[-2]:=eval(Re(solve(Delta*eval(-kappa[1]*rhs(solution1[2])+ kappa[-1]*c[3]*c[7]-kappa[2]/K[4]*c[3]*rhs(solution1[2])+kappa[-2]* c[3]**2*c[7]-kappa[3]+kappa[-3]*c[3]*c[7]*rhs(solution2),z=s)=0, kappa[-2])),d=0.00225): > # Now linearize the ds/dt equation about a flat surface > unassign(’h’): > unassign(’s’):

57 > lhs(solution1[1]): > h:=d+epsilon*H(x,t): > s:=epsilon*S(x,t): > linearized_S:=eval(diff((M[c]/rho[c])*diff(s,t)=eval(-kappa[1]* rhs(solution1[2])+kappa[-1]*c[3]*c[7]-kappa[2]/K[4]*c[3]* rhs(solution1[2])+kappa[-2]*c[3]**2*c[7]-kappa[3]+kappa[-3]*c[3]* c[7]*rhs(solution2),z=s),epsilon),epsilon=0): > # > # Values of constants in hydrodynamics > U:=rho*g*d**2/mu: > R[e]:=rho*U*d/mu: > B:=rho*g*d**2/tau/k**2: > rho:=.99970: > g:=9.8*10**2: > mu:=0.013069: > tau:=74.2: > unassign(’theta’): > theta:=evalf(2*Pi/18): > # The Linearized kinematic condition at z=h(x,t) > linearized_H:=1/U/k*diff(H(x,t),t) = -1/3*1/U/k*d*R[e]*sin(theta)* diff(S(x,t),t,x)-3/40*1/k*d*R[e]*sin(theta)^2*diff(S(x,t),‘$‘(x,2))+ 3/40*1/k*d*R[e]*sin(theta)^2*diff(H(x,t),‘$‘(x,2))+1/3*cos(theta)* diff(H(x,t),‘$‘(x,2))*d/k+5/24*1/U/k*d*R[e]*sin(theta)* diff(H(x,t),t,x)-1/3*diff(H(x,t),‘$‘(x,4))/B/k^3*d^3-diff(H(x,t),x)/ k*sin(theta)+sin(theta)/k*diff(S(x,t),x): > # Assuming the form of the solution > H(x,t):=alpha[1]*exp(sigma*t)*exp(I*k*x): > S(x,t):=alpha[2]*exp(sigma*t)*exp(I*k*x): > # > # Substitution of the solution into the linearized equations > s[1]:=collect(expand(linearized_H*U*k*B/exp(sigma*t)/exp(I*k*x)), alpha[1],alpha[2],distributed): > s[2]:=collect(expand(linearized_S/exp(sigma*t)/exp(I*k*x)), alpha[1],alpha[2],distributed): > # > A:=matrix([[coeff(rhs(s[1]),alpha[1])-coeff(lhs(s[1]),alpha[1]), coeff(rhs(s[1]),alpha[2])],[coeff(rhs(s[2]),alpha[1]), coeff(rhs(s[2]),alpha[2])-coeff(lhs(s[2]),alpha[2])]]): > # > # For nontrivial solutions, the determinant must vanish > characteristic:=sort(collect(det(A)=0,sigma,distributed),sigma): > ch[1]:=eval(characteristic,d=0.0021): > ch[2]:=eval(characteristic,d=0.0022):

58 > ch[3]:=eval(characteristic,d=0.0023): > ch[4]:=eval(characteristic,d=0.0024): > s1:=solve(ch[1],sigma) assuming k>=0,sigma>=0: > s2:=solve(ch[2],sigma) assuming k>=0,sigma>=0: > s3:=solve(ch[3],sigma) assuming k>=0,sigma>=0: > s4:=solve(ch[4],sigma) assuming k>=0,sigma>=0: > > # > # The following rates are generated by examining fixed values of > # the cave floor slope and varying depth(slope) > # > rate1:=Re(evalc(eval(s1[1],theta=evalf(Pi/9)))) assuming k>=0: > rate2:=Re(evalc(eval(s2[1],theta=evalf(Pi/9)))) assuming k>=0: > rate3:=Re(evalc(eval(s3[1],theta=evalf(Pi/9)))) assuming k>=0: > rate4:=Re(evalc(eval(s4[1],theta=evalf(Pi/9)))) assuming k>=0: > > # > # This is a plot of the growth rate versus the wave number > plot([rate1,rate2,rate3,rate4],k=0...0.8,color=black,linestyle= [DOT,DASH,DASHDOT,SOLID],resolution=600,title="Cave Floor Slope 20 Degrees",labels=["Wavenumber","Growth Rate"], legend=["d=0.0021","d=0.0022","d=0.0023","d=0.0024"]): > f1:=unapply(rate1,k): > f2:=unapply(rate2,k): > f3:=unapply(rate3,k): > f4:=unapply(rate4,k): > # Write data points to external file for plotting > fd1:=fopen("angle20depth21.data",WRITE): > fd2:=fopen("angle20depth22.data",WRITE): > fd3:=fopen("angle20depth23.data",WRITE): > fd4:=fopen("angle20depth24.data",WRITE): > array1:=array(1...50): > array2:=array(1...50): > array3:=array(1...50): > array4:=array(1...50): > array5:=array(1...50): > for j from 1 to 50 do array1[j]:=f1((j-1)/100) end do: > for j from 1 to 50 do array2[j]:=f2((j-1)/100) end do: > for j from 1 to 50 do array3[j]:=f3((j-1)/100) end do: > for j from 1 to 50 do array4[j]:=f4((j-1)/100) end do: > for j from 1 to 50 do array5[j]:=(j-1)/100 end do: > for j from 1 to 50 do fprintf(fd1,"%e\t%e\n",array5[j],array1[j]) end do:

59 > for j from 1 to 50 do fprintf(fd2,"%e\t%e\n",array5[j],array2[j]) end do: > for j from 1 to 50 do fprintf(fd3,"%e\t%e\n",array5[j],array3[j]) end do: > for j from 1 to 50 do fprintf(fd4,"%e\t%e\n",array5[j],array4[j]) end do: > fclose(fd1); > fclose(fd2); > fclose(fd3); > fclose(fd4); >

60