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MODIFICATION OF THE KOZENY CARMAN EQUATION TO QUANTIFY FORMATION DAMAGE BY FINES IN CLEAN UNCONSOLIDATED POROUS MEDIA by Eva D. Krauss B.A. Biochemi stry, University of Colorado Boulder, 2009 B.A. Molecular, Cellular, and Developmental B iology, Univers ity of Colorado Boulder, 2009 A thesis submitted to the Faculty of the Graduate School of the University of Colorado Denver in partial fulfillment of the requirements for the degree of Master of Science Civil Engineering, Hydrology and Hydraulics 2012
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ii This thesis by Eva D. Krauss has been approved for the degree of Master of Science Civil Engineering, Hydrology and Hydraulics by Dr. David C. Mays Dr. Harihar Rajaram Dr. James Guo November 15, 2012
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iii Krauss, Eva D. ( M.S., Civil Engineering, Hyd rology and Hydraulics ) Modification of the Kozeny Carman equation to quantify formation damage by fines in clean unconsolidated porous media. Thesis directed by Assistant Professor David C. Mays ABSTRACT Accurately estimating formation permeability as a f unction of porosity, grain size, and the quantity and structure of fines is important for increasing hydrocarbon extraction from rock formations. Accurate formation permeability estimation can provide insight into the factors that lead to formation damage due to fines deposition. The Kozeny Carman equation can be used to estimate the permeability of unconsolidated media as a function of porosity and grain size, but does not account for the struc ture of fines. In this study, I show how incorporating a dimens ionless bulk factor into the Kozeny Carman equation can be used to quantify the permeability reduction resulting from deposition of fines in the pore space. I consider several experimental studies from the literature that use a variety of porosities, flui ds, fines (differentiated by diameter and type), porous media, and flow velocities. Results indicate that for all of the experiments analyzed, when other variables are held constant, experiments conducted at higher flow ve locity result in less plugging. Th e bulk value was adjusted in the Kozeny Carman equation to obtain a line of best fit. The root mean square m ethod was used to obtain the best bulk factor value that calculated a curve which had the best match for the experimental data. Fitted values of the bulk factor were then correlated with the Peclet number to investigate how the structure of fines in the pore space, quantified by the bulk factor depends on the
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iv characteristics of the porous media, the depositing colloids, and the flow v elocity. F or a p articular experimental set, at a lower Peclet Number, higher bulk factor s are observed, when diffusi ve transport dominates, which I speculate results from the presence of more dendritic deposits. At higher Peclet Numbers, lower bulk factor s are observed, w hen advecti ve transport dominates, which I speculate results from deposits that are m ore compact. By understanding the flow velocity dependence of permeability reduction in a formation, flow rates can be optimized in order to enable a more complete hydroca rbon recovery. The primary application of this work is to optimize well flow rates to prevent or manage formation damage (i.e., plugging) resulting from deposition of fines in initially clean unconsolidated sand. The form and content of this abstract are approved. I recommend its publication. Approved: David C. Mays
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v TABLE OF CONTENTS CHAPTER I. INTRODUCTION ................................ ................................ ................................ ....... 1 1.1 Background ................................ ................................ ................................ ....... 1 1.2 Formation Damage ................................ ................................ ............................ 3 1.3 Kozeny Carman Equation ................................ ................................ ................. 4 1.4 Fines Deposition Properties ................................ ................................ .............. 5 1.4.1 Dispersion and Diffusion. ................................ ................................ ........ 5 1.4.2 Advection. ................................ ................................ ................................ 6 II. MODELING ................................ ................................ ................................ ................ 7 ................................ ................................ .............................. 7 2.1.1 Specific Discharge. ................................ ................................ .................. 9 2.1.2 Average Linear Veloci ty. ................................ ................................ ......... 9 2.1.3 Intrinsic Permeability. ................................ ................................ .............. 9 2.2 Kozeny Carman Equation ................................ ................................ ............... 10 2.3 Specific Deposit ................................ ................................ .............................. 12 2.4 Modified Kozeny Carman Equati on ................................ ............................... 14 2.5 The Bulk Factor ................................ ................................ .............................. 15 III. ANALYSIS ................................ ................................ ................................ ................ 19 3.1 Data Sets Used ................................ ................................ ................................ 19 3.2 Fitting the Model to the Expe rimental Data ................................ ................... 21 3.3 Bulk Factor vs. Velocity Relationship ................................ ............................ 22 IV. DISCUSSION ................................ ................................ ................................ ............ 26 4.1 Analysis of the Bulk Factor vs. Velocity Correlation ................................ ..... 26 4.2 Peclet Number Analysis ................................ ................................ .................. 29
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vi 4.3 Slope Analysis ................................ ................................ ................................ 30 4.4 Multiphase Flow ................................ ................................ ............................. 33 V. CONCLUSION ................................ ................................ ................................ .......... 34 REFERENCES ................................ ................................ ................................ ................. 36 APPENDIX ................................ ................................ ................................ ....................... 39 A. Boller and Kavanaugh (1995) ................................ ................................ .................... 40 B. Chang (1985) ................................ ................................ ................................ ............. 43 C. Mays and Hunt (2007) Calcium Montmorillonite ................................ ..................... 47 D. Mays and Hunt (2007) Sodium Montmorillonite ................................ ...................... 51 E. Narayan et al (1997) ................................ ................................ ................................ .. 56 F. Perera (1982) ................................ ................................ ................................ ............. 61 G. Tobiason and Vigneswaran (1994) ................................ ................................ ............ 65 H. Veerapaneni and Wiesner (1997) d p = 0.044 m ................................ ...................... 69 I. Veerapaneni and Wiesner (1997) d p = 0.069 m ................................ ...................... 72 J. Veerapaneni and Wiesner (1997) d p = 0.090 m ................................ ...................... 89 K. Vi gneswaran and Chang (1989) ................................ ................................ ................ 93
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vii LIST OF TABLES TABLE III.1 Data Sets Used for Bulk Factor Analysis. ................................ ............................... 20 IV.1 Linear Regression Analysis of ................................ ........................ 32
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viii LIST OF FIGURES FIGURE II.1 Steady Flow through a Sand Column. ................................ ................................ .......... 8 II.2 Artistic Rendering of Bulk Sp ecific Deposit. ................................ ............................. 13 II.3 Artistic Rendering of Sand Grains. ................................ ................................ ............ 17 III.1 Schematic of the Experimental System Used to Collect Data. ................................ 23 III.2 Comparison of Model with the Bulk Factor and Without. ................................ ....... 24 III.3 Velocity Dependence of the B ulk Factor. ................................ ................................ 25 IV.1 Data Grouping of Bulk Factor vs. Velocity. ................................ ............................. 28 IV.2 Peclet Number vs. the Bulk Factor for Varied Conditions. ................................ ...... 31
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ix NOMENCLATURE average grain diameter gravitational acceleration permeability permeability without fines Peclet number water density specific discharge or Darcy velocity approach velocity volume of deposit total volume of granular media Greek Letters specific deposit dynamic viscosity porosity
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1 CHAPTER I. INTRODUCTION 1.1 Background Estimating the hydrocarbon formation permeability as a function of porosity, grain size, and the quantity and structure of fines in the pore space can be a challenge. The p ermeability of a hydrocarbon reservoir often needs to be estimated from data collected during a combined analysis of well logging and detailed core analysis. Well log ging can provide information about the porosity of a particular formation through measurements such as density poros ity logs, sonic porosity logs an d n eutron porosity logs. One of the drawbacks to counting on permeability and porosity values from well log data is that this combined method is only applicable in formations that have reached irreducible water saturations (Asquith, 2004, Chapter 7). Irreducible water saturation occurs when the lowest water saturation is reached through the displacement of form ation water by oil or gas (S a tter, Chapter 2 ). Since well log data is collected during the drilling process or soon after, it is difficult to determine whether a formation is at the irreducible water saturation. Incorporat ing core analysis to estimate the permeability of a reservoir formation also helps to obtain a more accura te measurement of the reservoir formation A drawback to this combined analysis is that there is a level of uncertainty during log measurements and it i s often difficult to maintain i n situ conditions during advanced core analysis. A linear relationship between log k and exists in sandstones, sand packs consolidated clays, c onsolidated sands and
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2 carbonate rocks and in unconsolidated sands ; where is permeability and is porosity (Nelson, 1994). In most cases, w ith an increase in the porosity of a formation, the permeabil ity also increases ; fine sands and clay s do not always display an increase in permeability with an increase in porosity possibly d ue to the reduction in pore thro at size (Nelson, 1994). Determining permeability from grain size can also pose a chal lenge due to the large variability present in each individual formation sample, which include s grain sorting, clay content, and the presence of vuggs which are large pores within a rock that are filled with mineral precipitates such as quartz and calcite A study by Beard and Weyl (1973) shows that there ex ists a correlation betw een grain size and permeability. T heir data shows that the permeability is proportional to the square of the grain size of the unconsolidated artificial sand packs in the range of coarse sands and very fine sands (Beard and Weyl, 197 3). A weakness also exists in this assumption because according to the Kozeny Carman equation, other parameters such as surface area and specific deposit needs to be incorporated to obtain an accurate permeability estimation, an additional error can occur in the permeability estimation using the methods of Beard and Weyl (1973) when the porosity data is i n accurate because the log k curves have a large positive slope (Vernik, 2000). The quantity and structure of fines such as clays, quartz, amorphous mat erials and other minerals i n the pore space also influence the ability to predict formation permeability (Muecke, 1978). Other models have attempted to account for an increase in clay content in a porous medium, for example t he W. Sal
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3 permeability model sh ows the permeability variability as a function of porosity and clay content, with an increase in clay content, the permeability decreases. The Kozeny Carman equation can be used to estimate the permeability of unconsolidated media as a function of porosit y and grain size, but the equation does not account for the structure of fines imbedded in the pores. Determining the structure of fines in porous media once they have been embedded has posed a big challenge for researchers. Deciphering the morphology of f ines in porous media during fluid flow i s difficult because there does no t exist a method for determining these structures at this time. Research is currently being carried out in this area (Armstrong and Ajo Franklin 2011, Mays et al. 2011). 1.2 Formation Damage The research discussed above can be applied to f ormation damage which occurs when there is a reduction in permeability of a hydrocarbon formation. The damage results in an ad verse effect on production of the fluids from a hydrocarbon formation (Civ an, 2000). Formation damage can stem from numerous different operations including drilling, cementing, acidizing, and production (Krueger, 1986). Damage can result from either mechanical or chemical mechanism which include fines migrations and rock fluid interactions, respectively (Civan, 2000). My s tudy focuses on how fines can im pa ir permeability in various different scenarios including but not limited to formation fines migration during production and fines invasion from drilling fluid Formation fines, as defined above, are present in naturally occurring formation rocks that differ from the rock matrix
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4 these fines can become mobile when the right conditions exist (Panda and Lake, 1995) Mobilization of these fines can occur during, changes in fluid flo w velocity, a change in fluid composition, or pressure changes occurring during well drawdown (Muecke, 1978). 1.3 Kozeny Carman Equation The Kozeny Carman equation is a widely used equation in the area s of groundwater, petroleum, and chemical engineering. This equation relates the porosity of a granular system with the permeability. Estimating the permeability in a porous bed filled with spherical grains, the Kozeny Carman equation is: (I.1) w here is the permeability, is the mean diameter of the spherical grains, and is the porosity (Xu, 2007). As a porous media accumulates fine s the porosity of the system decreases O ne way to account for this decrease in porosity is by including a specific deposit term defined in Chapter II in the Kozeny Carman equation to reduce porosity as shown here: (I.2) In this study, I show how incorporating a dimensionless bulk factor into the Kozeny Carman equation can be used to quantify the permeability reduction resulting from deposition of fines in the pore space. The bulk fac tor was first introduced by Ojha and Graham while characterizing flow thr ough deep bed fi lters for water treatment T he bulk factor is a term which converts the mass of
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5 fines to the volume occupied by the fine deposits in the porous media (Ojha and Graham 1992). Th e primary goal of th is work is to understand h ow formation permeability in clean unconsolidated sand differs at different flow velocities and different combinations of porous media and fine s I consider s everal experimental studies from the literature that use a variety of porosities, fluids, fines, porous media and flow velocities. The behavior of fines in pore network s is dependent on several different factors which include but are not limited to aqueous chemistry, mechanical stresses, swelling, and sorption (Quirk, 1994). The chemical conditions in each of these studies have been chosen to promote deposition. 1.4 Fines Deposition Properties 1.4.1 Dispersion and Diffusion The d ispersion of fines occur when the individual fines are more stable individually tha n they would be alongside other fines due to either collis ions or chemical interactions. Fines that remain separate will remain suspended in solution for a longer period and this dispersion characteristic will enable fines to remain mobile and it may hinder adsorption to the grain surfaces (Mays, 2007). Dispersio n is very similar to diffusion in that with a change in time a single fine, which remains suspended, and will move down the concentration gradient, on average Fines undergoing diffusion undergo Brownian motion when the flow velocity is low which states that a particle tends to move away from its original reference point over time due to collisions between the fines and the fluid molecules (Elimelech et al 1995). In this study,
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6 the fluid flow velocity through the porous media was la rge enough to mainly consider hydrodynamic dispersion. 1.4.2 Advection Advection in porous media containing fines refers to the fines moving passively with the fluid and not moving outside of the flow boundary. The assumption with advection is that the f ines move at the same velocity as the flowing fluid. Fluid velocity is seen to play a role in the degree of permeability impairment understanding the dependence of fluid velocity on how significantly a formation is damaged could possibly lead to optimizin g flow rates to minimize damage. To explain the relationship between fines accumulation, flow velocity, and the value of the bulk factor in the Kozeny Carman equation, I will first start by explaining how the modeling was carried out in Chapter II. The mo deling aspect study and following that by the defining the components of the the Kozeny Carman model and how it relates the porosity, specific deposit over time, the bulk fact or, and the reduction in permeability. Chapter III goes into how the analysis was carried out to determine the bulk factor and flow velocity relationship for eleven different series of experiments taken form the literature This is followed by Chapter IV which goes into a discussion of the significance of the data analysis and also relates the relationship between advection and diffusion using a dimensionless Peclet number
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7 CHAPTER II. MODELING The model presented here applies to systems that have fluid flowing throug h a homogeneous clean porous media wh ich does not initially c ontain any fines such as clays and silt. When working with homogeneous, clean porous media, the equation derived by French engineer, Henry Darcy ( 1856 ) relates volumetric flow to the head loss over the length of porous med ia Figure II.1 illustrates this concept using the classic Darcy testing apparatus Darcy showed that volumetric flow is proportional to head loss, ,and inversely proportional to the distance over which head loss is measured in a specific Law, relates the volumetric flow to four variables, (II.1 ) W here is the hydraulic conductivity, is the hydraulic grad ient, and is the cross sectional area of flow. The original Darcy equation considers volumetric flow in terms of hydraulic conductivity because it is assuming that flow is dependent on water flowing through a geologic medium. Hydraulic conductivity is a measure of the ability of a medium to transmit water and therefore hydraulic conductivity is a function of the size and distribution of the pores (Fitts, 2002, Section 3.2).
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8 Figure 0 1 Steady Flow thr ough a Sand Column This column has two manometers that measure the head difference betwee n two points in the sand column, is the distance between the two manometers paralleling the column
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9 2.1.1 S pecific Discharge Another version of the Darcy equation can be derived in terms of the specific discharge, which is also known as Darcy velocity, (II.2 ) where (II.3 ) where is the term for specific discharge. 2.1.2 Average Linear Velocity The average linear velocity of the fluid flowing through the porous media is numerically larger than the specific discharge because the average linear velocity only accounts for the space available in a granular media. Therefore, the equation for the average linear velocity is as follows: (II.4) where is the average linear velocity and is the effective porosity, which is the pore space available for fluid flow. In a clean sand granular media, the effective porosity is the total porosity of the system because it is assumed that 100% of the porosity is available for fluid flow. 2.1.3 Intrinsic Permeability In petroleum reservoirs, since the fluid composition is not limited to only water, it is important to consider the intrinsic permeability of a medium because it is only dependent on the composition of the granular media. The equation for the intrinsic permeability is as follows: (II.5)
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10 w here is the dynamic viscosity of water, is the density of water, and is the gravitational acceleration (Fitts, 2002, Section 3.3) 2.2 Koze ny Carman Equation To calculate hydraulic conductivity or intrinsic permeability based on porosity and average grain diameter, an equation developed by Kozeny in 1927 and modified by Carman in 1937 and in 1956, correlated porosity, which is the percent of the granular media that is pore space, and average grain diameter, The Kozeny Carman equation for calculating hydraulic conductivity is: (II.6) wher e, is porosity, is the average grain diameter, is the water density, is the gravitational acceleration, and is the dynamic viscosity (Bear, 1972). In my study, I am looking at the intrinsic permeability of a fluid, based on poros ity of the granular media and the average grain diameter. To use the Kozeny Carman equation to solve for permeability, the hydraulic conductivity term, needs to be replaced by the term for intrinsic permeability, This is done by using the permeabil ity equation developed above: (II.7) with the terms defined above. After solving for hydraulic conductivity, equation II.7 can be substituted into equation II.6, to derive the followi ng equation: (II.8)
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11 This equation is illustrated in Figure II.2, Panel A, where permeability is dependent only on the porosity of a clean granular bed with no fines depositi on.
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12 2.3 Specific Deposit Specific deposit is the volume of deposited particles divided by total volume of porous media. The equation for specific deposit, is as follows: (II.9) where is the volume of deposit and is the total volume of porous media. In a petroleum formation application, specific deposit is the volume of newly introduced fines drilling mud filtrate into a control volume and the volume of specific deposit increases with time as solids accumulate in the granular media. In this study, I am assuming that the deposits have zero porosity and the volume taken up by the fines is only a function of the direct volume occupied by the deposit being studied, this te rm is also known as the absolute specific deposit. If I am assuming that the deposits have porosity, the absolute specific deposit would equal the following: (II.10) where is the porosity of the deposited fines, is the volume taken up b y the porous fines, and is the specific deposit value if only considering the direct volume occupi ed by the accumulating fines (Iwasaki, 1937).
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13 Figure 0 2 Artistic Rendering of Bulk Specific Deposit Panel A shows a volume where the specific deposit is equal to zero Panel B shows where the dep osit has zero poro sity, and Panel C shows where the deposits have a certain porosity a nd they oc cupy a greater bulk of volume. A B C
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14 2. 4 Modified Kozeny Carman Equation With the intro du ction of fines into a granular syste m the permeability will decrease as a result of the decrease in available pore space for fluid flow. Equation II.8 assumes a clean homogeneous granular media from which permeability can be calculated. With the addition of fines into a granular volume, the porosity will decrease proportionall y. When there is zero fines migration, the specific deposit will equal zero and the porosity remains unchanged. When there are fines being deposited into the granular volume, the porosity will decrease as shown: (II.11 ) where is the porosity after the addition of fines, is the initial porosity and is the specific deposit. Substituting equation II.10 into the Kozeny Carman equation derived in e quation II.8, a new form of the equation can be obtained The new equation is as follows: (II.12 ) with the same terms as defined above. Equation II.11 as sumes that the porosity is decreased evenly throughout the clean porous bed through fines deposition that has zero porosity. In an attempt to illustrate the concept of evenly distributed fines in a porous media, I have created an artistic rendering of the conceptual model in the Google Sketchup Version 8, the illustrations are shown in Figure II. 3 Panel
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15 B It has been shown that fines deposition does not occur evenly throughout a granular media (Chen et al. 2008, Chen et al. 2009 ) The Modified Kozeny Car ma n equation that considers the addition of fines as an additional volume added to the grain and therefore a reduction in porosity will be shown in Chapter 3 to underestimate the loss in permeability with the addition of fines in the form of specific depos it. By this token, a correction needs to be made to the Modified Kozeny Carman equation to cor rectly model experimental data that accounts for uneven or clumped deposit morphology. 2.5 The Bulk Factor An empirically determined dimensionless constant the b u lk factor was first introduced by Ojha and Graham ( 1992 ) They proposed that the bulk factor can be used with the Kozeny Carman equation to correct for nonuniform addition of volume to the sand grain surfaces. The bulk factor is incorporated into the Mod ified Kozeny Carman equation by multiplying the specific deposit by a dimensionless constant, the bulk factor, : (II.13) where is the volume occupied by deposited fines with zero porosity. The new equation with the addition of the bulk factor is as follows: (II.14) For a conceptual understanding of what the bulk factor means, please refer back to Figure II.2. For fines deposition with zero porosity, the bulk factor will be
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16 equal to one and this would mean that the deposits form a dense compact structure where porosity is reduced only by the volume occupied by the actual volume of the deposits, this is an illustrated in Figure II.2, Panel B. When the fines are deposited in a non compact manner, the bul k factor will be greater than one, due to the fact that the volume occupied by the same mass of deposits is larger, this is shown in Figure II.2, Panel C. In this study, we will be looking at how the addition of the bulk factor to the Modified Kozeny Carme n equation can be used to better fit to experimental data obtained from clean bed filtration experiments.
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17 Figure 0 3 Artistic Rendering of Sand Grains Panel A shows sand grains as they would appe ar in a clean bed as shown in equation 11.8 Panel B is an artistic depiction of how deposits can evenly coat a sand grain as fines deposit, this would be illustrating the equation II.14 Panel C illustrates how dendritic deposits can deposit in a clean be d media, and this is an illustration of equation II.15 A B C
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18 Using the modified Kozeny Ca rman equation which incorporat es the bulk factor, the normalized reduction in permeability can be determined by the input of the specific deposit and the increase i n head loss over the course of the experiment. A normalized version of this model in regards to decreasing permeability with the addition of vol ume in the form of fines can be derived as shown below: (II.14) a nd with simplification, the following version of the Kozeny Carman equ ation can be obtained: (II.15) from which the reduction in porosity can be graphed as a function of varying amounts of accumulating fines over the course of an experimental run. In the following chapter, I will explain how this equation was used to obtain a bulk factor of best fit for experimental data and I will follow that by going step by step through how the results were analyzed.
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19 CHAPTER III. ANALYSIS 3 .1 Data Sets Used In this study, I analyzed 11 experimental data sets from 8 different filtration studies published in literature detailed in Table III.1 Each of these studies conducted tests in which head loss and fines accumulation were measured. These experiments varied in the use of fines, the porosity of the filtration system, filtration system composition, fluid viscosity and the flow velocity. Figure III.1 provides a schematic of the experimental system F or each experiment, the in fluent concentration was known and the effluent concentration was measured, so following equation was used by Mays and Hunt (2005) to determine the mass of fines accumulation over the course of the experiment: (III.1) where is the mass accumulated over time, is the volumetric flow, is time, is the influent concentration and is the effluent concentration. To c alculate the amount of fines in terms of specific deposit values, the volume of deposits were calculated from the mass of fines and the density of the deposits, the following equation was used: (III.2) where is the volume of deposits and is the density of the accumulating fines.
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20 Table 0 1 Data Sets Used for Bulk Factor Analysis. Fines Fines Diameter (um) Porous Media Porous Media Diameter (um) Porosity (%) Fluid Boller and Kavanaugh 1995 iron hydroxide 0.10 sand 1400 40 water Chang ( 1985 ) kaolinite 1.00 sand 1200 44 water Mays and Hunt ( 2007 ) Calcium Montmorillonite 1.00 sand 163 35 water Mays and Hu nt ( 2007 ) Sodium Montmorillonite 1.00 sand 163 35 water Narayan et al. ( 1997 ) Carbon black 8.00 glass 930 37 kerosene Perera ( 1982 ) kaolinite 1.00 sand 710 46 water Tobiason and Vigneswaran (1994) pol y mer polystyrene 0.27 glass 400 40 water Veerapanen i and Wiesner (1997) latex 0.044 glass 360 40 water Veerapaneni and Wiesner (1997) latex 0.069 glass 360 40 water Veerapaneni and Wiesner (1997) latex 0.090 glass 360 40 water Vigneswaran and Chang ( 1989 ) kaolinite 1.00 glass 350 40 water
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21 3 .2 Fitting the Model to the Experimental Data The data sets used from literature supplied both the accumulation of fines and the increase in head loss in the porous media as a function of time. Since the following relationship is true for permeability and head loss: (III.3) where is the permeability measured after a time is the initial permeability in the porous media, is the original head loss with zero fines accumulation and is the head loss in the porous media after a certain time The relationship between permeability, head loss and the accumulation of fines as a function of time was then taken to create a plot. The plot for each of the experiments analyzed demonstrated a relationship of a decrease in permeability with an increase in specific deposit, when there are no deposits in the porous media; the permeability is graphed at 1 because the original permeability is the same as t he measured permeability Using the permeability vs. specific deposit relationship, equation II.15 and the Goal Seek function from Excel was used to obtain a bulk factor which calculates a model of best fit to the experimental data. Figure III.2 shows th e magnitude of error for a typical experiment analyzed associated with using equation II.2 versus equation II.14 which accounts for the bulk factor. Notice that the original model fails to account for the true loss of permeability as seen from experimental resul ts. This discrepancy in mode l versus experimental results can be accounted for by incorporation the bulk factor
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22 into the equation to help the model better fit the true permeability loss with increasing deposit. The bulk factor multiplies the specific deposit by a certain factor to a ccount for fin e s structure that takes up more area than the immediate volume of the deposit. The dependence of the magnitude of the bulk factor will be shown in relation to the fluid flow velocity. As it is shown here, once the bulk factor is d etermined, a line of best fit can be modeled. The trend of lower bulk factor with increased fluid velocity indicates that there is less for bulk with increasing velocity. One reason for smaller bulk factor with increased velocity can possibly be attributed an inc rease in shear forces in the pore throats. 3.3 Bulk Factor vs. Velocity Relationship In the study, a common trend wa s observed as velocity increases With increased velocity, the bulk factor decreased for all of the experiments studied. After plotting the results on Figure III.3 a common trend of a decrease in the bulk factor with an increase in velocity can be seen.
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23 Figure 0 1 Schematic of the Experimental System Used to Collect Data The cylinde r above is representative of the p orous media column. T he in fluent par ticle concentration of the fluid was known, and the effluent concentration after flowing through the column was measured to calculate the fines trapped within the porous media A constant velocity, was imposed in each experiment analyzed. The pressure drop was also measured before and after the fluid was flowing through the column.
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24 F igure 0 2 Comparison of Model with the Bulk F actor and Without The bulk factor improves the estimation of permeability loss The ori ginal model was plotted using equation II.12 and the model that fits the experimental data points was plotted using equation II.14, this example is from Perera (1982) for u = 0.14 cm/s and a bulk factor of ( b ) = 67.4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0E+00 5.0E04 1.0E03 1.5E03 2.0E03 2.5E03 k / k o Specific Deposit ( ) Data p82sh5 Model Original Model
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25 Figure 0 3 Velocity Dependence of the Bulk Factor A common trend can be se en for the experiments analyzed W ith an increase in velocity, the bulk factor decreases. 1 10 100 1000 0.001 0.01 0.1 1 Bulk Factor (b) Velocity (cm/s) Boller and Kavanaugh (1995) Chang (1985) Mays and Hunt (2007) CaMontmorillonite Mays and Hunt (2007) NaMontmorillonite Narayan et al. (1997) Perera (1982) Tobiason and Vigneswaran (1994) Veerapaneni and Wiesner (1997) dp=0.044um Veerapaneni and Wiesner (1997) dp=0.069um Veerapaneni and Wiesner (1997) dp=0.090um Vigneswaran and Chang (1989)
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26 CHAPTER IV. DISCUSSION 4.1 Analysis of the Bulk Factor vs. Velocity Correlatio n Based on the porosity of the collector system, there does not appear to be a direct correlation with porosity and the magnitude of the bulk factor in this sample set. The porous media diameter, does not present a clear correlation with the tendency of the fines to bulk with increasing velocity as shown in Figure IV.1 The lack of correlation for porosity and particle size implies that the plugging is dependent on other factors. To make the analysis consistent, the data have been split into two groups as shown on Figure IV.1, Group 1 data has a higher bulk factor for the same velocity whereas Group 2 has a lower bulk factor for a set range of velocities. The porous media composition, which is similar to the makeup of a reservoir formation, appear s to have a correlation between the magnitude of the bulk factor and the composition of the porous media The surface of the porous media grains appears to affect the magnitude the bulk factor that is determined to best fit the experimental data. Of the experiments discussed in Chapter 3, 4 studi es used sand grains as the collector particle: Boller and Kavanaugh (1995) Chang (1985), Mays an d Hunt (2007) and Perera (1982) This could be as a result of uneven sand grain surfaces. In contrast, the glass beads have a lower overall bulk tendency which could result from l ess surface friction between the particles and the filter collector. This is shown in the following three experiments by:
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27 Narayan (1997), Tobiason and Vigneswaran (1993), and Vigneswaran and Chang (1989). Further research is requir ed to verify these hypotheses.
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28 Figure 0 1 Data Grouping of Bulk Factor vs. Velocity A Group 1 data is shown to have a higher bulk factor for a set range of velocities and Group 2 data is shown to have a lower bulk factor for a set range of velocities. 1 10 100 1000 0.001 0.01 0.1 1 Bulk Factor (b) Velocity (cm/s) Boller and Kavanaugh (1995) Chang (1985) Mays and Hunt (2007) CaMontmorillonite Mays and Hunt (2007) NaMontmorillonite Narayan et al. (1997) Perera (1982) Tobiason and Vigneswaran (1994) Veerapaneni and Wiesner (1997) dp=0.044um Veerapaneni and Wiesner (1997) dp=0.069um Veerapaneni and Wiesner (1997) dp=0.090um Vigneswaran and Chang (1989) Group 1 Group 2
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29 Looking at th e study that sim ulated homogeneou s sand with deposition by montmorillonite, we can see that the same trend is observed : as the velocity increases the bulk factor decreases (Mays and Hunt, 2007) In this data set, we want to highlight the difference between calcium montmoril lonite and sodium montmorillonite. Sodiu m montmo rillonite shows a greater tendency to reduce the permeability of the system compared with calcium montmorillonite. This is consistent with the known swelling behavior of sodium montmorillonite (Marand i, 2011) Calcium montmorillonite plugging is approximately 20 times less in comparison, and this can be attributed to the multi plate quasi crystals formed by t he calcium montmorillonite ( Mays and Hunt 2007 ). 4.2 Peclet Number Analysis The Peclet number is a dimensionless parameter that gives a ra tio of advection and diffusion: (IV.1) where u is the approach velocity, d 50g is the porous media d i ameter, and D p is the particle di ffusivity: (IV.2) where T is absolute temperature, is dynamic viscosity, and d p is the particle diameter. The term D p is the coefficient of
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30 hy drodyn amic dispersion, which is the sum of the effective diffusion coefficient and the dispersion coefficient. With an increase in Peclet n umber we will get a decrease in the bulk factor this can be attributed to the differences in porosity, flow velocity, and the types of colloids used in the experiment. Results with a low Peclet Number will have diffusion as the main contributor. Advection and diffusion effects were considered when fitted values of the bulk factor were correlated with the Peclet number to inv estigate how the structure of fines in the pore space depend on the characteristics of the porous media, the deposited colloids, and the flow velocity. The Peclet number is a dimensionless parameter that gives a ratio of advection and diffusion. Low Pecle t numbers indicate a diffusion dominated system ; when the Peclet numbers were high, which correlates to higher flow velocities, advective transport dominates. 4.3 Slope Analysis To determine a line of best fit through the Peclet Number dependent on the Bul k factor the linear regression equation and the standard error of the slope were determined for each set of ex perime nts using Microsoft Excel ( Table IV.1 )
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31 Figure 0 2 Peclet Nu mber vs. the Bulk Factor for Varied Conditions It can be seen that there is a decrease in the bulk factor with an increase i n the Peclet number (Pe) 1 10 100 1000 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 Bulk Factor (b) Pe Boller and Kavanaugh (1995) Chang (1985) Mays and Hunt (2007) CaMontmorillonite Mays and Hunt (2007) NaMontmorillonite Narayan et al. (1997) Perera (1982) Tobiason and Vigneswaran (1994) Veerapaneni and Wiesner (1997) dp=0.044um Veerapaneni and Wiesner (1997) dp=0.069um Veerapaneni and Wiesner (1997) dp=0.090um Vigneswaran and Chang (1989)
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32 Table 0 1 Linear Regression A nalysis of R 2 Slope Standard Deviation Slope Significant B oller and Kavanaugh (1995) 0.93 0.603 0.004 yes Chang ( 1985 ) 0.89 0.788 0.009 yes Mays and Hunt (2007) Ca Montmorillonite 0.98 0.458 0.006 yes Mays and Hunt (2007) Na Montm orillonite 0.66 0.151 0.016 yes Narayan et al. ( 1997 ) 0.78 0.487 0.042 yes Perera ( 1982 ) 0.92 0.449 0.003 yes Tobiason and Vigneswar an (1994) 0.80 2.033 0.158 yes Veerapaneni and Wiesner (1997) dp = 0.044 m 1.00 0.318 0.000 yes Veerapaneni and W iesner ( 1997 ) dp = 0.069 m 0.71 0.384 0.246 no Veerapaneni and Wiesner ( 1997 ) dp = 0.0 9 0 m 0.85 0.601 0.070 yes Vigneswaran and Chang ( 1989 ) 0.35 0.223 0.028 yes
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33 4.4 Multiphase Flow Even though the data analyzed in this study came from clean bed s ingle phase flow, real petroleum oil and gas reservoirs very often have two or even three phase flow. During multiphase flow, a combination of water, oil, and gas coexist within a reservoir. It has been shown that the greatest plugging will occur during on e phase flow, which would be either water or oil rather than when there is a flow of a multiphase reservoir fluid (Mueck, 1978). In the field, it is seen that the most fines are produced when there is a change in phase of the fluid flowing out of the well; this fines production can be attributed to fines that are water wet, dissolving into the water flowing phase that have been sealed in a water envelope when there are hydrocarbons flowing (Mueck, 1978).
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34 CHAPTER V. CONCLUSION This study looks at the particle plugging of clean unconsolidated porous media using various types o f fines. Since it has been shown that on average, only eleven weight percent of fines found in a formation is clay, it is important to look at the behavior of fines that have other chemical and physical properties (Mueck, 1978). Understanding fines plugging of porous media with applications to petroleum reservoirs is important for determining the reduction in permeability over the life of the reservoir. The original Kozeny Carman equation in terms of specific deposit, porosity, and average grain size diameter, fails to accurately predict the reduction in permeability with the addition of fines in the porous media. Incorporation the bulk factor proposed by Ojha and Graham into the Kozeny Carma n equation and using the Root Mean Square Method to determine the bulk factor that best fits the system, generates a model that fits the experimental data. This was shown using 11 different experiments with varying flow velocities. The results indicate th at the bulk factor decreases with an increase in flow velocity of the fluid. This could possibly be a result of higher fines compaction at high flow velocities and a more dispersed fines system when the flow velocities were lower. Higher bulk factors were correlated low Peclet whereas low bulk factors were correlated with high Peclet numbers.
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35 Creating better models to understand the magnitude of permeability reduction in porous media is very important in the field of reservoir engineering. The loss in perm eability of a petroleum reservoir as a result of man made interactions is often called formation damage ; f ines plugging is producing reservoirs is one of the biggest formation damage obstacles in petroleum engineering according to the Society of Petroleum Engineers. The goal of this work was to present a modified way of obtaining a model of best fit for the magnitude of permeability reduction with the addition of fines in a porous bed.
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36 REFERENCES Ahmed, T. ( 2006 ) Reservoir Engineering Handbook Third E dition. Burlington, MA. Elsevier Inc. Armstrong, R., and J. Ajo Franklin (2011), Investigating biomineralization using synchrotron based X ray computed microtomography, Geophysical Research Letters, 38, L08406, doi:10.1029/2011GL046916. Asquith, G. and Kr ygowski, D. (2004). Basic Well Log Analysis. Second Edition. The American Association of Petroleum Geologists. Tulsa, Oklahoma. Beard, D.C. and Weyl, P.K. (1973).Influence of Texture on Porosity and Permeability of Unconsolidated Sand. American Association of Petroleum Geologists Bulletin. 57(2): 349 369. Boller, M.A. and Kavanaugh, M.C. ( 1995 ) Particle Characteristics and Head loss Increase in G ranular Media Filtration. Water Res earch 29(4): 1139 1149. Carman, P.C. (1937). Fluid Flow through Granular Beds : Trans. Inst. Chem. Eng. 15, 150. Carman, P.C. (1948 ). Some Physical Aspects of Water Flow in Porous Media: Discuss. Faraday Soc. 3, 78. Carman, P.C. (1956 ). Flow of Gases through Porous Media. Butterworth Scientific Publication. Chang, J.W. (1985). Mathe matical Modeling of Deep Bed Filtration: Microscopic Approach. M.S. Thesis EV 85 3, Asian Institute of Technology, Bangkok, Thailand. Chen, C., A. I. Packman, and J. F. Gaillard (2008), Pore scale analysis of permeability reduction resulting from colloid d eposition, Geophys. Res. Lett., 35, L07 404, doi:10.1029/2007GL033077. Chen, C., B. Lau, J. F. Gaillard, and A. I. Packman (2009), Temporal E volution of Pore Geometry, Fluid F low, and Solute Transport Resulting from Colloid Deposition. Water Resour ce Res ear ch. 45, W06416, doi:10.1029/2008WR007252. Civan, F. ( 2000 ) Reservoir Formation Damage Fundamentals, Modeling, Assessment, and Mitigation. Houston, Texas: Gulf Publishing Company. Elimelech, M., Gregory, J., Jia, X., and Williams, R.A (1995). Particle Depo sition and Aggregation. Measurement, Modeling and Simulation. Butterworth Heinemann. Woburn, MA.
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37 Fitts, C. R. (2002). Groundwater Science. San Diego, CA: Academic Press. Iwasaki, T. ( 1937 ). Some Notes on Sand Filtration Journal American Water Works Associ ation 29: 1591. Kozeny, J. (1927). ber die kapillare Leitung des Wassers im Boden (Aufstieg Versickerung und Anwendung auf die Dewsserung): Sitz. Ber. Akad. Wiss. Wien, Math. Nat. (Abt. IIa). 136(a): 271 306. Krueger, R. F. (1986). An Overview of Formatio n Damage and Well Productivity in Oilfield Operations. Journal of Petroleum Technology. February 1986: 131 152. Liu, X., and Civan, F. (1996). Formation Damage and Filter Cake Buildup in Laboratory Core Tests: Modeling and Model Assisted Analysis. SPE Fo rmation Evaluation. Marandi, G. B., Mahdavinia, G. R., and Ghafary, S. (2011). Collagen g poly(Sodium Acrylate co Acrylamide)/ Sodium Montmorillonite Superabsorbent Nanocomposites: Synthesis and Swelling Behavior. Journal of Polymer Research 18: 1487 1499. Mays, D.C. (2007).Using the Quirk Schofield Diagram to Explain Environmental Colloid Dispersion Phenomena. Journal of Natural Resources and Life Science Education. 36: 45 52. Mays, D.C. (2010).Contrasting Plugging in Granular Media Filters, Soils, and Dead End Membranes. Journal of Environmental Engineering. 136(5): 475 480. Mays, D.C., O.T. Cannon, A.W. Kanold, K.J. Harris, T.C. Lei, and B. Gilbert (2011), Static light scatter ing resolves colloid structure in index matched porous media, Journal of Colloid and Interface Science, 363, 418 424. Mays, D.C. and Hunt, J.R. (2005).Hydrodynamic Aspects of Particle Plugging in Porous Media. Environmental Science and Technology. 39(2): 5 77 584. Mays, D.C. and Hunt, J.R. (2007).Hydrodynamic and Chemical Factors in Plugging by Montmorillonite in Porous M edia Environmental Science and Technology 41(16) : 5666 5671. Nelson, P.H. (1994). Permeability Porosity Relationships in Sedimentary Rocks. The Log Analyst. 35(3): 36 62. Narayan, R. and Coury, J. R. and Masliyah, J. H. an d Gray, M.R. 1997. Particle Capture and Plugging in Packed B ed Reactors. Indian Engineering Chemistry Resources. 36:4620 4627.
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38 Nooruddin, H. A. and Hossain, M. E. 2012. Modified Kozeny Carmen correlation for enhanced hydraulic flow unit characterization. Journal of Petroleum Science and Engineering. 80 (2012) 107 115. Muecke, T.W.( 1978 ) Formation fines and factors controlling their movement in porous media, in Third SPE Symposium on Formation Damage Control, February 15 16, 1978, Lafayette, Louisiana, S ociety of Petroleum E ngineers SPE Paper No. 7007, 84 91. Ojha, C. S. P. and Graham, J. D. ( 1992 ) .Appropriate Use of Deep Bed Filtration Models. Journal of Environmental Engineering. 18(6): 964 980. Panda, M.N., and Lake, L.W. (1995). A Physical Model of Cementation and Its Effects on Single Phase Permeability. AAPGBulletin. 79(3): 431 443. Perera, Y.A.P. (1982). Comparison of Performance of Radial and UpflowFilters. M.S.Thesis EV 82 10, Asian Institute of Technology, Bangkok, Thailand. Quirk, J.P. (1994).Interparticle forces: A Basis for the Interpretation of Soil Physical Behavior. Advanced Agronomy. 53: 121 183. Satter A., Iqbal, G. H., and Buchwalter A. L. ( ) Practical Enhanced Reservoir Engineering: Assisted with Simulation Software Tobiason J. E., and Vigneswaran, B. (1994), Evaluation of a Modified Model for Deep Bed Filtration.WaterResources. 28(2): 335 342. Valdez, J.R. and Santamarina, J.C. (2006). Particle Plugging in Radial Flow: Microscale Mechanisms. SPE Journal.193 198. Veerapaneni, S. and Wiesner, M.R. (1997 ). Deposit Morphology and Head Loss Development in Porous Media. Environmental Science and Technology. 31(10): 2738 2744 Vernik, L. (2000). Permeability Prediction in Poorly Consolidated Siliciclastics Based on Porosi ty and Clay Volume Logs. Petrophysics. 41(2): 138 147. Vigneswaran, S. and Chang, J. S. ( 1989 ) .Experimental Testing of Mathematical Models Describing the Entire Cycle of F iltration. WaterRes earch 23(11): 1413 1421. Xu, P. and Yu, B. (2007).Developing a new fo rm of permeability and Kozeny Carman constant for homogeneous Porous Media by Means of Fractal Geometry. Advances in Water Resources. 31: 74 81.
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39 APPENDIX This appendix contains the details of each of the experimental sets analyzed for this study. Each of the following graphs contains the data points obtained for the normalized permeability with increasing deposit within the porous media. The following studies are organized in alphabetical order and flow vel ocities along with the bulk factor for the best model fit is pre sented in each caption.
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40 APPENDIX A. Boller and Kavanaugh (1995) Figure A 1 : Fitted plugging model from equation II.14 for Boller and Kavanaugh ( 1995 ) for u = 0.19 cm/s, bulk factor (b) = 79.9 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 1.E03 2.E03 3.E03 4.E03 5.E03 6.E03 7.E03 k / k O Specific Deposit ( ) Data bk95sh1 Model
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41 Figure A 2 : Fitted plugging model from equation II.14 for Boller and Kavanaugh ( 1995 ) for u = 0.43 cm/s, bulk factor (b) = 56.9 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 5.E04 1.E03 2.E03 2.E03 3.E03 3.E03 4.E03 4.E03 5.E03 5.E03 k / k O Specific Deposit ( ) Data bk95sh2 Model
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42 Figure A.3 : Fitted plugging model from equation II.14 for Boller and Kavanaugh ( 1995 ) for u = 0.62 cm/s, bulk factor (b) = 37.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0E+00 5.0E04 1.0E03 1.5E03 2.0E03 2.5E03 3.0E03 3.5E03 k / k O Specific Deposit ( ) Data bk95sh3 Model
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43 B. Chang (1985) Figure B.1 : Fitted plugging model from equation II.14 for Chang (1985) for u = 0.13 cm/s, bulk factor (b)= 167 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0E+00 5.0E05 1.0E04 1.5E04 2.0E04 2.5E04 3.0E04 3.5E04 4.0E04 4.5E04 k / k O Specific Deposit ( ) Data c85sh1 Model
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44 Figure B.2 : Fitted plugging model from equation II.14 for Chang (1985) for u = 0.195 cm/s, bulk factor (b) = 135 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 1.E04 2.E04 3.E04 4.E04 5.E04 6.E04 7.E04 k / k O Specific Deposit ( ) Data c85sh2 Model
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45 Figure B.3 : Fitted plugging model from equation II.14 for Chang (1985) for u = 0.259 cm/s, bulk factor (b ) = 125 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0E+00 5.0E05 1.0E04 1.5E04 2.0E04 2.5E04 3.0E04 3.5E04 4.0E04 4.5E04 k / k O Specific Deposit ( ) Data c85sh3 Model
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46 Figure B.4 : Fitted plugging model from equation II.14 for Chang (1985) for u = 0.389 cm/s, bulk factor (b) = 67.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 1.E04 2.E04 3.E04 4.E04 5.E04 6.E04 7.E04 k / k O Specific Deposit ( ) Data c85sh4 Model
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47 C. Mays and Hunt (2007) Calcium Montmorillonite Figure C.1: Fitted plugging model from equation II.14 for Mays and Hunt (2007 ) Calcium Montmorillonite for u = 0.021 cm/s, bulk factor (b) = 28.3 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 1.E03 2.E03 3.E03 4.E03 5.E03 6.E03 7.E03 8.E03 k / k O Specific Deposit ( ) Data d05sh161 Series2
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48 Figure C.2: Fitted plugging model from equation II.14 for Mays and Hunt (2007 ) Calcium Montmorillonite for u = 0.21 cm/s, bulk factor (b) = 10.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 5.E04 1.E03 2.E03 2.E03 3.E03 3.E03 4.E03 k / k O Specific Deposit ( ) Data d05sh211 Series2
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49 Figure C.3: F itted plugging model from equation II.14 for Mays and Hunt (200 7 ) Calcium Montmorillonite for u = 0.53 cm/s, bulk factor (b) = 7.32 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 5.E04 1.E03 2.E03 2.E03 3.E03 3.E03 k / k O Specific Deposit ( ) Data d05sh151 Series2
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50 Figure C.4: Fitted plugging model from equation II.14 for Mays and Hunt (200 7 ) Calcium Montmorillonite for u = 0.53 cm/s, bulk factor (b) = 5.67 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 5.E04 1.E03 2.E03 2.E03 3.E03 3.E03 4.E03 k / k O Specific Deposit ( ) Data d05sh221 Series2
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51 D. Mays and Hunt (2007) Sodium Montmorillonite Figure D .1: Fitted plugging model from equation II.14 for Mays and Hunt (200 7) Sodium Montmorillonite for u = 0.021 cm/s, bulk factor (b) = 124 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 5.E04 1.E03 2.E03 2.E03 3.E03 3.E03 4.E03 k / k O Specific Deposit ( ) Data m05sh111 Series2
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52 Figur e D .2: Fitted plugging model from equation II.14 for Mays and Hunt (200 7) Sodium Montmorillonite for u = 0.11 cm/s, bulk factor (b) = 72.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 2.E04 4.E04 6.E04 8.E04 1.E03 1.E03 1.E03 2.E03 2.E03 2.E03 k / k O Specific Deposit ( ) Data m05sh121 Series2
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53 Figure D .3: Fitted plugging model from equation II.14 for Mays and Hunt (200 7) Sodium Montmorillonite for u = 0.11 cm/s, bulk factor (b) = 102 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 1.E04 2.E04 3.E04 4.E04 5.E04 6.E04 7.E04 8.E04 9.E04 k / k O Specific Deposit ( ) Data m05sh141 Model
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54 Figure D .4: Fitted plugging model from equation II.14 for Mays and Hunt (200 7 ) Sodium Montmorillonite for u = 0.53 cm/s, bulk factor (b) = 68.3 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 5.E05 1.E04 2.E04 2.E04 3.E04 3.E04 4.E04 k / k O Specific Deposit ( ) Data m05sh91 Series2
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55 Figure D .5: Fitted plugging model from equati on II.14 for Mays and Hunt (200 7) Sodium Montmorillonite for u = 0.53 cm/s, bulk factor (b ) = 80.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 5.E05 1.E04 2.E04 2.E04 3.E04 3.E04 4.E04 4.E04 5.E04 k / k O Specific Deposit ( ) Data m05sh131 Series2
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56 E. Narayan et al (1997) Figure E.1: Fitted plugging model from equation II.14 for Narayan et al. (1997) for u = 0.030 cm/s, bulk factor (b) = 13.5 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 0.0E+00 5.0E04 1.0E03 1.5E03 2.0E03 2.5E03 3.0E03 k / k O Specific Deposit ( ) Data ncmg97sh1 Model
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57 Figure E.2: Fitted plugging model from equation II.14 for Narayan et al. (1997) for u = 0.070 cm/s, bulk factor (b) = 8.88 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 0.0E+00 5.0E04 1.0E03 1.5E03 2.0E03 2.5E03 3.0E03 3.5E03 k / k O Specific Deposit ( ) Data ncmg97sh2 Model
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58 Figure E.3: Fitted plugging model from equation II.14 for Narayan et al. (1997) for u = 0.15 cm/s, bulk factor (b) = 7.87 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 0.0E+00 5.0E04 1.0E03 1.5E03 2.0E03 2.5E03 k / k O Specific Deposit ( ) Data ncmg97sh3 Model
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59 Figure E.4: Fitted plugging model from equation II.14 for Narayan et al. (1997) for u = 0.22 cm/s, bulk factor (b) = 7.00 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 0.E+00 5.E04 1.E03 2.E03 2.E03 3.E03 3.E03 4.E03 k / k O Specific Deposit ( ) Data ncmg97sh4 Model
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60 Figure E.5: Fitted plugging model from equation II.14 for Narayan et al. (1997) for u = 0.30 cm/s, bulk factor (b) = 3.33 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 0.E+00 5.E04 1.E03 2.E03 2.E03 3.E03 3.E03 4.E03 k / k O Specific Deposit ( ) Data ncmg97sh5 Model
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61 F. Perera (1982) Figure F.1: Fitted plugging model from equation II.14 for Perera (1982) for u = 0.14 cm/s, bulk factor (b) = 67.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0E+00 5.0E04 1.0E03 1.5E03 2.0E03 2.5E03 k / k O Specific Deposit ( ) Data p82sh5 Model
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62 Figure F.2: Fitted plugging model from equation II.14 for Perer a (1982) for u = 0.28 cm/s, bulk factor (b) = 57.3 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0E+00 5.0E04 1.0E03 1.5E03 2.0E03 2.5E03 3.0E03 k / k O Specific Deposit ( ) Data p82sh6 Model
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63 Figure F.3: Fitted plugging model from equation II.14 for Perera (1982) for u = 0.42 cm/s, bulk factor (b) = 45.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0E+00 5.0E04 1.0E03 1.5E03 2.0E03 2.5E03 3.0E03 3.5E03 k / k O Specific Deposit ( ) Data p82sh7 Model
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64 Figure F.4: Fitted plugging model from equation II.14 for Perer a (1982) for u = 0.56 cm/s, bulk factor (b) = 35.6 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0E+00 5.0E04 1.0E03 1.5E03 2.0E03 2.5E03 k / k O Specific Deposit ( ) Data p82sh8 Model
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65 G. Tobiason and Vigneswaran (1994) Figure G.1 : Fitted plugging model from equation II.14 for Tobiason and Vigneswaran for u = 0.14 cm/s, bulk factor (b) = 71.1 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 5.E04 1.E03 2.E03 2.E03 3.E03 k / k O Specific Depsoit ( ) Data tv94sh1 Model
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66 Figure G.2 : Fi tted plugging model from equation II.14 for Tobiason and Vigneswaran for u = 0.14 cm/s, bulk factor (b) = 126 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 5.E04 1.E03 2.E03 2.E03 3.E03 k / k O Specific Deposit ( ) Data tv94sh5 Model
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67 Figure G.3 : Fitted plugging model from equation II.14 for Tobiason and Vigneswaran for u = 0.14 cm/s, bulk factor (b) = 35.9 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 0.E+00 5.E04 1.E03 2.E03 2.E03 3.E03 k / k O Specific Deposit ( ) Data tv94sh7 Model
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68 Figure G.4 : Fitted plugging model from equation II.14 for Tobiason and Vigneswaran for u = 0.14 cm/s, bulk factor (b) = 7.99 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 1.01 0.E+00 1.E04 2.E04 3.E04 4.E04 5.E04 6.E04 k / k O Specific Deposit ( ) Data tv94sh8 Model
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69 H. Veerapaneni and Wiesner (1997) d p = 0.044 m Figure H. 1 : Fitted plugging model from equation II.14 for Veerapaneni and Wiesner (1997) d p = 44 nm for u = 0.0067 cm/s, bulk factor (b) = 3 63 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 5.E04 1.E03 2.E03 2.E03 3.E03 k / k O Specific Deposit ( ) Data v96sh103 Model
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70 Figure H. 2 : Fitted plugging model from equation II.14 for Veerapaneni and Wiesner (1997) dp = 44 nm for u = 0.033 cm/s, bulk factor (b) = 106 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 5.E04 1.E03 2.E03 2.E03 3.E03 3.E03 4.E03 4.E03 k / k O Specific Deposit ( ) Data v96sh105 Series2
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71 Figure H. 3 : Fitted plugging model from equation II.14 for Veerapaneni and Wiesner (1997) dp = 44 nm for u = 0.32 cm/s, bulk factor (b ) = 212 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 1.E03 2.E03 3.E03 4.E03 5.E03 6.E03 k / k O Specific Deposit ( ) Data v96sh104 Model
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72 I. Veerapaneni and Wiesner (1997) d p = 0.0 69 m Figure H. 4 : Fitted plugging model from equation II.14 for Veerapaneni and Wiesner (1997) dp = 69 nm for u = 0.0073 cm/s, bulk factor (b) = 543 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 2.E04 4.E04 6.E04 8.E04 1.E03 1.E03 1.E03 2.E03 2.E03 k / k O Specific Deposit ( ) Data v96sh68 Series2
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73 Figure H. 5 : Fitted plugging model from equation II.14 for Veerapaneni and Wiesner (1997) dp = 69 nm for u = 0.017 cm/s, bulk factor (b) = 158 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 2.E04 4.E04 6.E04 8.E04 1.E03 1.E03 1.E03 2.E03 2.E03 2.E03 k / k O Specific Deposit ( ) Data v96sh70 Series2
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74 Figur e H. 6 : Fitted plugging model from equation II.14 for Veerapaneni and Wiesner (1997) dp = 69 nm for u = 0.0 35 cm/s, bulk factor (b) = 158 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.E+00 2.E04 4.E04 6.E04 8.E04 1.E03 1.E03 1.E03 2.E03 2.E03 k / k O Specific Deposit ( ) Data v96sh61 Model
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75 Figure H. 7 : Fitted plugging model from equation II.14 for Veerapaneni and Wiesner (1997) dp = 69 nm for u = 0.050 cm/s, bulk factor (b) = 163 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 5.E04 1.E03 2.E03 2.E03 3.E03 3.E03 4.E03 4.E03 5.E03 k / k O Specific Deposit ( ) Data v96sh98 Series2
PAGE 85
76 Figure H. 8 : Fitted plugging model from equation II.14 for Veerapaneni and Wiesner (1997) dp = 69 nm for u = 0.070 cm/s, bulk factor (b) = 76.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 5.E04 1.E03 2.E03 2.E03 3.E03 3.E03 4.E03 4.E03 k / k O Specific Deposit ( ) Data v96sh71 Series2
PAGE 86
77 Figure H. 9 : Fitted plugging model from equation I I.14 for Veerapaneni and Wiesner (1997) dp = 69 nm for u = 0.11 cm/s, bulk factor (b) = 111 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 1.E03 2.E03 3.E03 4.E03 5.E03 6.E03 k / k O Specific Deposit ( ) Data v96sh90 Model
PAGE 87
78 Figure H. 10 : Fitted plugging model from equation II.14 for Veerapaneni and Wiesner (1997) dp = 69 nm for u = 0.14 cm/s, bulk factor (b) = 105 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 5.E04 1.E03 2.E03 2.E03 3.E03 3.E03 4.E03 k / k O Specifi Deposit ( ) Data v96sh62 Model
PAGE 88
79 Figure H. 11 : Fitted plugging model from equation II.14 for Veerapaneni and Wiesner (1997) dp = 69 nm for u = 0.14 cm/s, bulk factor (b) = 82.3 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 1.E03 2.E03 3.E03 4.E03 5.E03 6.E03 k / k O Specific Deposit ( ) Data v96sh66 Series2
PAGE 89
80 Figure H. 12 : Fitted plugging model from equation II.14 for Veerapaneni and Wiesner (1997) dp = 69 n m for u = 0. 14 cm/s, bulk factor (b) = 89.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 1.E03 2.E03 3.E03 4.E03 5.E03 6.E03 7.E03 k / k O Specific Deposit ( ) Data v96sh64 Model
PAGE 90
81 Figure H. 13 : Fitted plugging model from equation II.14 for Veerapaneni and Wiesner (1997) dp = 69 nm for u = 0.17 cm/s, bulk factor (b) = 94.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 1.E03 2.E03 3.E03 4.E03 5.E03 6.E03 7.E03 8.E03 k / k O Specific Deposit ( ) Data v96sh75 Series2
PAGE 91
82 Figure H. 14 : Fitted plugging model from equati on II.14 for Veerapaneni and Wiesner (1997) dp = 69 nm for u = 0.20 cm/s, bulk factor (b) = 123 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 1.E03 2.E03 3.E03 4.E03 5.E03 6.E03 k / k O Specific Deposit ( ) Data v96sh72 Series2
PAGE 92
83 Figure H. 15 : Fitted plugging model from equation II.14 for Veerapaneni and Wiesner (1997) dp = 69 nm for u = 0.22 cm/s, bulk factor (b) = 112 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 1.E03 2.E03 3.E03 4.E03 5.E03 6.E03 7.E03 8.E03 k / k O Specific Deposit ( ) Data v96sh91 Series2
PAGE 93
84 Figure H. 16 : Fitted plugging model from equation II.14 for Veerapaneni and Wiesner (1997) dp = 69 nm for u = 0.23 cm/s, bulk factor (b) = 79.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 1.E03 2.E03 3.E03 4.E03 5.E03 6.E03 7.E03 8.E03 9.E03 k / k O Specific Deposit ( ) Data v96sh76 Model
PAGE 94
85 Figure H. 17 : Fitted plugging model from equation II.14 for Veerapaneni and Wiesner (1997) dp = 69 nm for u = 0.26 cm/s, bulk factor (b) = 67.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 1.E03 2.E03 3.E03 4.E03 5.E03 6.E03 7.E03 k / k O Specific Deposit ( ) Data v96sh74 Series2
PAGE 95
86 Figure H. 18 : Fitted plugging model from equation II.14 for Veerapaneni and Wiesner (1997) dp = 69 nm for u = 0. 31 cm/s, bulk factor (b) = 76.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 1.E03 2.E03 3.E03 4.E03 5.E03 6.E03 7.E03 k / k O Specific Deposit ( ) Data v96sh65 Model
PAGE 96
87 Figure H. 19 : Fitted plugging model from equation II.14 for Veerapaneni and Wiesner (1997) dp = 69 nm for u = 0.33 cm/s, bulk factor (b) = 67.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 1.E03 2.E03 3.E03 4.E03 5.E03 6.E03 7.E03 8.E03 k / k O Specific Deposit ( ) Data v96sh102 Model
PAGE 97
88 Figure H. 20 : Fitted plugging model from equation II.14 for Veerapaneni and Wiesner (1997) dp = 69 nm for u = 0. 51 cm/s, bulk factor (b) = 74.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 1.E03 2.E03 3.E03 4.E03 5.E03 6.E03 7.E03 8.E03 k / k O Specific Deposit ( ) Data v96sh69 Model
PAGE 98
89 J. Veerapaneni and Wiesner (1997) d p = 0.0 90 m Figure H. 21 : Fitted plugging model from equation II.14 for Veerapaneni and Wiesner (1997) dp = 90 nm for u = 0.0073 cm/s, bulk factor (b) = 464 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 2.E04 4.E04 6.E04 8.E04 1.E03 1.E03 1.E03 k / k O Specific Deposit ( ) Data v96sh78 Model
PAGE 99
90 Figure H. 22 : Fitted plugging model from equation II.14 for Veerapaneni and Wiesner (1997) dp = 90 nm for u = 0. 034 cm/s, bulk factor (b) = 112 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 5.E04 1.E03 2.E03 2.E03 3.E03 3.E03 4.E03 4.E03 k / k O Specific Deposit ( ) Data v96sh79 Series2
PAGE 100
91 Figure H. 23 : Fitted plugging model from equation II.14 for Veerapaneni and Wiesner (1997) dp = 90 nm for u = 0.10 cm/s, bulk factor (b) = 58.3 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.E+00 1.E03 2.E03 3.E03 4.E03 5.E03 6.E03 7.E03 k / k O Specific Deposit ( ) Data v96sh77 Series2
PAGE 101
92 Figure H. 24 : Fitted plugging model from equation II.14 for Veerapaneni and Wiesner (1997) dp = 90 nm for u = 0.17 cm/s, bulk factor (b) = 84.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 1.E03 2.E03 3.E03 4.E03 5.E03 6.E03 7.E03 8.E03 9.E03 k / k O Specific Deposit ( ) Data v96sh82 Model
PAGE 102
93 K. Vigneswaran and Chang (1989) Figure I.1 : Fitted pl ugging model from equation II.14 for Vi gneswaran and Chang (1989) for u = 0.1389 cm/s, bulk factor (b) = 4.81 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 5.E03 1.E02 2.E02 2.E02 3.E02 3.E02 4.E02 k / k O Specific Deposit ( ) Data vc89Esh2 Model
PAGE 103
94 Figure I.2 : Fitted plugging model from equation II.14 for Vi gneswaran and Chang (1989) for u = 0.1736 cm/s, bulk factor (b) = 4.28 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 5.E03 1.E02 2.E02 2.E02 3.E02 3.E02 k / k O Specific Deposit ( ) Data vc89Esh3 Model
PAGE 104
95 Figure I.3 : Fitted plugging model from equation II.14 for Vigneswaran and Chang (1989 ) for u = 0.2083 cm/s, bulk factor (b) = 5.02 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 5.E03 1.E02 2.E02 2.E02 3.E02 k / k O Specific Deposit ( ) Data vc89Esh4 Model
PAGE 105
96 Figure I.4: Fitted plugging model from equation II.14 for Vi gneswaran and Chang (1989) for u = 0 .2431 cm/s, bulk factor (b) = 5.67 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 5.E03 1.E02 2.E02 2.E02 3.E02 k / k O Specific Deposit ( ) Data vc89Esh5 Model
PAGE 106
97 Figure I.5: Fitted plugging model from equation II.14 for Vigne swaran and Chang (1989) for u = 0.2778 cm/s, bulk factor (b) = 5.74 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 2.E03 4.E03 6.E03 8.E03 1.E02 1.E02 1.E02 2.E02 2.E02 2.E02 k / k O Specific Deposit ( ) Data vc89Esh7 Model
PAGE 107
98 Figure I.6: Fitted plugging model from equation II.14 for Vi gneswa ran and Chang (1989) for u = 0.3125 cm/s, bulk factor (b) = 4.28 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 2.E03 4.E03 6.E03 8.E03 1.E02 1.E02 1.E02 2.E02 2.E02 2.E02 k / k O Specific Deposit ( ) Data vc89Esh8 Model
PAGE 108
99 Figure I.7: Fitted plugging model from equation II.14 for Vi gneswaran and Chang (1989) for u = 0.3472 cm/s, bulk factor (b) = 4.38 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 2.E03 4.E03 6.E03 8.E03 1.E02 1.E02 1.E02 2.E02 2.E02 2.E02 k / k O Specific Deposit ( ) Data vc89Esh9 Model
PAGE 109
100 Figure I.8: Fitted plugging model from equation II.14 for Vi gneswaran and Chang (1989) for u = 0.3819 cm/s, bulk factor (b) = 4.28 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 2.E03 4.E03 6.E03 8.E03 1.E02 1.E02 1.E02 2.E02 2.E02 k / k O Specific Deposit ( ) Data vc89Esh10 Model
PAGE 110
101 Figure I.9: Fitted plugging model from equation II.14 for Vi gneswaran and Chang (1989) for u = 0.4167 cm/s, bulk factor (b) = 3.81 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 2.E03 4.E03 6.E03 8.E03 1.E02 1.E02 1.E02 2.E02 2.E02 k / k O Specific Deposit ( ) Data vc89Esh11 Model
PAGE 111
102 Fi gure I.10: Fitted plugging model from equation II.14 for Vi gneswaran and Chang (1989) for u = 0.4444 cm/s, bulk factor (b) = 3.81 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.E+00 2.E03 4.E03 6.E03 8.E03 1.E02 1.E02 1.E02 k / k O Specific Deposit ( ) Data vc89Esh12 Model
PAGE 112
103 Figure I.11: Fitted plugging model from equation II.14 for Vi gneswaran and Chang (1989) for u = 0.4722 cm/s, bulk factor (b) = 3.72 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.E+00 2.E03 4.E03 6.E03 8.E03 1.E02 1.E02 1.E02 2.E02 k / k O Specific Deposit ( ) Data vc89Esh13 Model
