Mathematical Prediction Model of the Infiltration Deterioration Due to
Clogging in Pervious Pavement Based on Pore/Particle Size Distribution
By
Ahmed Sharaby
A Thesis Submitted in Partial Fulfillment of the
Requirements for the Degree of
MASTER OF APPLIED SCIENCE
in the Department of Civil Engineering
©Ahmed Sharaby, 2019
University of Victoria
All rights reserved. This thesis may not be reproduced in whole or in part,
by photocopy or other means, without the permission of the author.
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Mathematical Prediction Model of the Infiltration Deterioration Due to
Clogging in Pervious Pavement Based on Pore/Particle Size Distribution
By
Ahmed Sharaby
Supervisory Committee
Dr. Rishi Gupta, Supervisor
Department of Civil Engineering
Dr. Caterina Valeo, Co-supervisor
Department of Mechanical Engineering
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ABSTRACT
Permeable pavement structures (PPSs) are one of the significant LID systems that have potential positive effect on the ecosystem. Yet, the performance of permeable pavements is still questionable. Further studies on the hydrological performance of the system need to be addressed for better design criteria and maintenance during the operation. The infiltration through the pavement is a crucial parameter that projects the system performance. Several factors affect its deterioration. The entrapment of suspended materials associated with the infiltrated stormwater through the system is one of the major factors that affect its performance. Factors that promote the entrapment of particles were discussed thoroughly through the literature and are explained in this study. Many previous studies were focused on performing experimental work and developing empirical models to study the hydraulic performance of the system. Yet, prediction models on the infiltration deterioration need to be addressed and theoretical analysis needs to be performed in order to determine the empirical coefficients with defined parameters that were introduced in the previous literature. Furthermore, the sensitivity of the pore and particle size distribution and mass loading rate of the suspended materials on the infiltration rate need to be addressed. The study focuses on investigating performance of PPSs with examining the variation effect of pore and particle size distribution on it. A prediction model was made and simulated using Matlab software, in which pore and particle size means and standard deviations are taken as inputs. Further, the variation in these parameters on infiltration is examined. Critical levels, that infiltration decline would reach, were defined based on the introduced mechanisms from the previous literature. Based on the variation of pore and particle size means and standard deviations, these critical levels were studied through the analysis of the obtained results from the simulated model.
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Table of Contents
Supervisory committee ii
Abstract iii
List of figures vii
List of tables x Acknowledgement xi Nomenclature xii Chapter 1: Introduction 1 1.1. Research background 1 1.2. Thesis layout 3
Chapter 2: Literature review 5
2.1. Overview 5
2.2. Factors affect the hydraulic performance of PPS 5
2.2.1. Water Runoff Intensity 6
2.2.2. PPS Porosity 6
2.2.3. PPS Permeability 7
2.2.4. Variation of PPS domains 9
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2.2.6. Tortuosity 9
2.2.7. Water Runoff Sediment and Clogging Effect 10
2.2.7.1. TSS clogging effect 10
2.2.7.2. TSS removal percentage 12
2.2.7.3. Inflow TSS concentration 12
2.3. Clogging mechanisms of pervious pavements 14
2.4. Primary equations 15
2.5. Knowledge gaps 19
Chapter 3: Research Objectives & Contribution of Work 23 Chapter 4: Prediction Model of the Infiltration Reduction Variation with Pore/Particle
Size Distribution 27
4.1. Introduction 27
4.2. System Model 30
4.3. Particle/Pore Size Correlation Attempt 31
4.4. Classification Index/Removal Efficiency Correlation Attempt 33 4.4.1. Trapping of particles due to straining effect 33 4.4.2. Trapping of particles due to Bed Filtration and Brownian motion 39 4.5. Algorithm used for the derivation of Kss and KDBF 41
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Chapter 5: Model Simulation of the Infiltration Reduction Variation with Pore/Particle
Size Distribution 44
5.1. Introduction 44
5.2. Base Case Simulation Results and Observations 45
5.3. Simulation Model 47
5.4. Main Features of Simulation (Key Assumptions) 50
5.5. Results and Analysis 50
5.5.1. Dimensional Analysis 51
5.5.2. Variation Analysis
5.5.3. Variation of pore/particle size distribution effect on the straining and deep-bed filtration behavior
51 54 5.5.4. Variation of pore/particle size distribution effect on the infiltration
reduction behavior 60
Chapter 6: Conclusion & Recommendations 75
References 79
Appendix I 87
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LIST OF FIGURES
Figure (2-1) Average annual numbers of rain events according to the weather condition 2-4
Figure (2-2) Indication of tortuosity of pervious pavement 2-6
Figure (2-3) Schematic illustration of the flow through an elementary volume of the porous pavement
2-12
Figure (3-1) Infiltration deterioration behavior over the duration of operation 3-1
Figure (3-2) Schematic illustration of the three stages of the infiltration deterioration behavior due to surficial straining, deep-bed filtration, and Brownian motion 3-3
Figure (4-1) Schematic illustration of the retention of suspended particles with different
sizes over the porous pavement 4-2
Figure (4-2) Hierarchy of the Model System 4-4
Figure (4-3) Schematic illustration of the correlation of the two Particle/Pore probability
density functions 4-5
Figure (4-4) Schematic illustration of the fine pores clogging in the plane view of the porous media due to straining effect
4-7
Figure (4-5) Schematic illustration of the particles entrapment through the pores cross section due to Size Exclusion or Brownian motion based on the model presented in it
4-12
Figure (5-1) Observed Particle Size density function based on the experimental work
made in Sansalone et al. (2012) 5-3
Figure (5-2) Hierarchy of the simulation model 5-6
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LIST OF FIGURES (Cont.)
Figure (5-4) Illustration chart of the 1st phase variation analysis 5-10
Figure (5-5) Observed Results of the classification index CDF of (i) the surficial straining and (ii) deep-bed filtration critical levels at dp Mean = 500 µm and a) dp STD = 125 µm; b) dp STD = 250 µm; dp STD = 500 µm.
5-13
Figure (5-6) Observed Results of the classification index CDF of (i) the surficial straining and (ii) deep-bed filtration critical levels at dp Mean = 1000 µm and a) dp STD = 125 µm; b) dp STD = 250 µm; dp STD = 500 µm.
5-14
Figure (5-7) Observed Results of the classification index CDF of (i) the surficial straining and (ii) deep-bed filtration critical levels at dp Mean = 1500 µm and a) dp STD = 125 µm; b) dp STD = 250 µm; c) dp STD = 500 µm.
5-15
Figure (5-8) Observed results of the (i) PDF or (ii) CDF of the infiltration rate deterioration at dp Mean = 500 µm; dp STD = 250 µm; and (a) dn Mean = 150 µm & STD = 50 µm; (b) dn Mean = 150 µm & STD = 150 µm; (c) dn Mean = 350 µm & STD = 50 µm; (a) dn Mean = 350 µm & STD = 150 µm.
5-22
Figure (5-9) Observed results of the (i) PDF or (ii) CDF of the infiltration rate deterioration at dp Mean = 500 µm; dp STD = 500 µm; and (a) dn Mean = 150 µm & STD = 50 µm; (b) dn Mean = 150 µm & STD = 150 µm; (c) dn Mean = 350 µm & STD = 50 µm; (a) dn Mean = 350 µm & STD = 150 µm.
5-23
Figure (5-10) Observed results of the (i) PDF or (ii) CDF of the infiltration rate deterioration at dp Mean = 800 µm; dp STD = 250 µm; and (a) dn Mean = 150 µm & STD = 50 µm; (b) dn Mean = 150 µm & STD = 150 µm; (c) dn Mean = 350 µm & STD = 50 µm; (d) dn Mean = 350 µm & STD = 150 µm.
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LIST OF FIGURES (Cont.)
Figure (5-11) Observed results of the (i) PDF or (ii) CDF of the infiltration rate deterioration at dp Mean = 800 µm; dp STD = 500 µm; and (a) dn Mean = 150 µm & STD = 50 µm; (b) dn Mean = 150 µm & STD = 150 µm; (c) dn Mean = 350 µm & STD = 50 µm; (d) dn Mean = 350 µm & STD = 150 µm.
5-25
Figure (5-12) Observed results of the (i) PDF or (ii) CDF of the infiltration rate deterioration at dp Mean = 1200 µm; dp STD = 250 µm; and (a) dn Mean = 150 µm & STD = 50 µm; (b) dn Mean = 150 µm & STD = 150 µm; (c) dn Mean = 350 µm & STD = 50 µm; (d) dn Mean = 350 µm & STD = 150 µm.
5-26
Figure (5-13) Observed results of the (i) PDF or (ii) CDF of the infiltration rate
deterioration at dp Mean = 1200 µm; dp STD = 500 µm; and (a) dn Mean = 150 µm & STD = 50 µm; (b) dn Mean = 150 µm & STD = 150 µm; (c) dn Mean = 350 µm & STD = 50 µm; (d) dn Mean = 350 µm & STD = 150 µm.
5-27
Figure (5-14) Observed results of the (i) PDF or (ii) CDF of the infiltration rate
deterioration at dp Mean = 1500 µm; dp STD = 250 µm; and (a) dn Mean = 150 µm & STD = 50 µm; (b) dn Mean = 150 µm & STD = 150 µm; (c) dn Mean = 350 µm & STD = 50 µm; (d) dn Mean = 350 µm & STD = 150 µm.
5-28
Figure (5-15) Observed results of the (i) PDF or (ii) CDF of the infiltration rate deterioration
at dp Mean = 1500 µm; dp STD = 500 µm; and (a) dn Mean = 150 µm & STD = 50 µm; (b) dn Mean = 150 µm & STD = 150 µm; (c) dn Mean = 350 µm & STD = 50 µm; (d) dn Mean = 350 µm & STD = 150 µm.
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LIST OF TABLES
Table (2-1) Pervious Pavement Categories 2-9
Table (4-1) Derivation procedure of the mathematical formulas of Kss and KDBF 4-14
Table (5-1) Selected Conditions for the 2nd stage variation analysis based on the
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ACKNOWLEDGEMENT
I’d like to thank the HAL research group and department of civil and mechanical enginering for their support in this study and help with all the necessary data and previous literature made from their side in this field and that serves this point of study.
I’d like to gratefully thank my supervisors Dr. Caterina Valeo and Dr. Rishi Gupta for their continuous support that always keeping up my inspirations and performance in this study.
My special thanks to Dr. Valeo for giving me this opportunity to participate in the ongoing HAL project and aiding me to overcome the obstacles that I faced either at the beginning of the program or during the research timeline. I appreciate her continuous guidance, and support during my presence in the University of Victoria. I learned from her creativeness and patience to solve the difficult matters and the understanding she gave to me during the whole period of research time that would project significantly in my future life.
My special thanks to Dr. Gupta for accepting me in his research group and continuous support to his students through the regular meetings he made for them. He brought the spirit inside his research team that inspires everyone to help and support the others towards the obstacles they may face.
Last but not least, I wish to dedicate this dissertation to my parents for all their wholehearted devotions to carrying me to this stage.
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NOMENCLATURE V Volume of the fluid in the elementary porous sample
Qin The volumetric inflow rate
Qout The volumetric outflow rate
C Particles concentrationpassed through the pore throats Ci Initial particles concentrationpassed through the pore throats
Cm Particles concentration in the mobilized phase (fluid)
Cim Particles concentration in the immobilized phase (solid)
C* Concentration of particles on the boundaries that denotes the saturation level ὺ Average flow velocity
K Ko
Hydraulic infiltration Initial infiltration rate
∇H Gradient operator of the hydraulic head Pa The pressure measurement in the air phase Pw The pressure measurement in the water phase σaw The surface tension of the air-water interface
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v𝑐𝑐 The critical velocity
vm The linear velocity of the mobilized medium (fluid velocity)
𝑉𝑉𝑤𝑤 The linear velocity of the water
𝑉𝑉𝑝𝑝 The linear velocity of the particles
μ The sphericity of gravel, which ranges from 0 to 1 depending on the shape of the gravel
d The gravel diameter
g The gravitational acceleration
l The kinematic viscosity of infiltrated water Le/l The pore tortuosity
A* The specific surface area of elementary volume Ѳt the total porosity of elementary volume
S0
a
The shape of the pores, ranges from 2 to 3 depending on the assumption of either circular or rectangular pores
Porosity reduction coefficient
s
Ɣ
Power exponent
Volumetric reaction rate 𝑚𝑚 The hydraulic radius
xiv 𝜆𝜆 Kss KDBF Mt Nt Ѳss ѲDBF
The classification index, denoted as (dp/dm)
The infiltration rate after particle entrapment due to surficial straining The infiltration rate after particle entrapment due to deep-bed filtration Mass loading rate of particles
The number of pores per unit area of the media
The system porosity after particle entrapment due to surficial straining The system porosity after particle entrapment due to deep-bed filtration
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CHAPTER 1: INTRODUCTION
1.1. Research background
The adverse effects on the urban water environment resulting from the use of impervious areas for urbanization creates major problems on the infrastructure system that leads to high cost of maintenance and shortens the life span of their utilization. Infrastructure systems built for managing the mass water runoff resulted from rainfall requires high cost of implementation and frequent periodic maintenance. Furthermore, replacement and rehabilitation of these systems takes long time for execution especially in highly populated areas. This is accompanied by deterioration in the urban traffic systems, and other adverse consequences including water and air pollution, and the inability to cope with the mass flow of water runoff (Huong & Pathirana 2013, Leopold 1968, Semadeni-Davies et al. 2008, Zhou et al. 2012, Arora & Reddy 2013, Hatt et al. 2004, Wang et al. 2008).
The future approach is to highlight other important aspects in the design of the utility systems as elaborated by Chocat et al. (2007), Echols (2007), Stahre (2006), Ferguson (1991), and France (2002). This includes the quality of the drained stormwater with the partial removal of accompanied pollutant and sediments with the flow, the protection of the ecosystem, and building multifunctional utility systems such as the integrated hybrid system of pervious pavements with bioretention cells. Furthermore, there are major concerns about the limited capacity and workability of the conventional systems to cope with the weather change and the new urban expansions (Krebs & Larsen 1997, Arisz & Burrell 2006, Zhou 2014).
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Low Impact Development (LID) systems are an ecological alternative to the impervious urban infrastructure for long-term sustainability in the system design procedure (Larsen & Gujer 1997, United Nations. Agenda 21 1992, Ashely et al. 2007, Charlesworth 2010, Fryd et al. 2012). Permeable pavement system (PPS) is an effective alternative LID solution to the impervious urban one. PPS is used widely for enhancing water circulation system. This is achieved by the reduction of water runoff, peak discharge and delaying the outflow (Lin et al. 2014, Schluter & Jefferies 2001). Furthermore, PPSs are sustainable systems as they mimic the discharge of water runoff through natural systems with their open pore structure (Henderson & Tighe, 2011). This happens as the high porosity allows surface water to drain through the PPS structure and naturally infiltrate into the subgrade (Henderson & Tighe, 2011).
PPS has significance in the removal of over 90% of the pollutants and suspended solids associated with the water runoff (Huang, 2015). This gives mass accumulation of these suspended solids on the surface and through the pores of PPS, and consequently, this leads to the reduction of water infiltration through the system. Several articles discussed the clogging effect on infiltration of PPS including (Wo & Huang, 2000), (Yu et al., 2015), (Alawi et al., 2013). Most of these studies were made experimentally giving empirical equations for the deterioration of porosity or infiltration. However, limited articles that modelled the reduction rate of infiltration with the time of operation. This study aims accordingly to provide a mathematical model of the infiltration deterioration with the variation of pore and particle size distribution. The essence of studying the infiltration decline with the variation of pore and particle size distribution is for investigating the mechanisms of deterioration and determining the level that infiltration would reach as a percentage from its initial value after the clogging of pavement that varies with each mechanism.
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Thesis layout
Following this chapter, a general literature review is conducted in chapter 2 in which previous studies towards the infiltration deterioration through the porous pavements are highlighted. The hydraulic performance and clogging of the system are discussed including water runoff intensity, system porosity, permeability, connectivity, and tortuosity.
The factors that affect the systems clogging are highlighted including the effect of suspended solids mass loading rate. The variation of suspended solids concentration is hence discussed from the previous related literature including the statistical data based on the environment and climate change.
Furthermore, the mechanisms of porous pavement clogging, the primary equations, and experimental models used in the previous literature to determine the infiltration rate of the system and its deterioration are discussed.
In chapter 3, the research gaps from the previous literature presented in chapter 2 are elaborated to develop a mathematical model and examine the deterioration of the infiltration rate. Several points are highlighted that need to be studied in order to define the unknown parameters given in the primary equations that determines the reduction rate of infiltration. From the defined points, the objectives of the research are discussed. The objectives focus on studying the variation of pore and particle size gradation and their effect on the deterioration of infiltration rate.
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In chapter 4, a prediction model is conducted to determine the critical levels that the infiltration decline could reach based on the introduced mechanisms in McDowell-Boyler et al. (1986), Teng & Sansalone (2004). Two main critical levels are investigated through a mathematical analysis based on the defined behavior of the two dominant mechanisms, surficial straining and deep-bed filtration.
In chapter 5, a simulation is made to the predicted model using Matlab software and the output observed graphs are analyzed to determine the variation of the infiltration reduction with pore/particle size distribution. The analysis is made through a comprehensive variation analysis that aims to determine the effect of pore/particle size distribution on 1st the pore to particle size
ratio (classification index) and 2nd the deterioration of infiltration of the pervious pavement system
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CHAPTER 2: LITERATURE REVIEW
2.1. Overview
Permeable pavement system (PPS) is a green alternative solution to the storm water conventional collection systems. Despite the challenges that face PPS for implementation in large scales, they are a better choice for cost reduction from the implementation of the conventional collection systems through pipes and efficient for treating the runoff water from pollutants for direct reuse in landscaping (Scholz and Grabowiecki 2007). Accordingly, PPS is a hybrid system with three main functions as runoff reduction when the stormwater infiltrates through the porous system, partial treatment when the associated suspended particles with stormwater get entrained through the system, and reuse for vegetation when the system is integrated with bioretention cells through connected network pipelines.
Several studies took place in improving the engineering specifications and enhancing the hydrological performance of these systems including the infiltration of runoff water, PPS clogging reduction, and the removal efficiency of existed pollutants in the runoff water (Huang et al. 2016). However, few researchers investigated the decrease of infiltration and change in removal efficiency of pollutants of PPS with time, depth and space. This research aims to run a mathematical modeling to study the variation of pore and particle size distribution and their effect on the deterioration of infiltration rate.
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Since infiltration is one of the main parameters that measures the hydraulic performance of PPS, it is essential to formulate its performance according to several factors that reflect the aging of the pervious material. This formulation helps in the prediction of the PPS infiltration and clogging after certain duration in order to acquire the required maintenance for recovering its initial efficiency. According to several articles such as Tong (2011), Mishra et al. (2013), Sansalone et al. (2008), the main factors that lead to the increase of PPS clogging and reduction of infiltration with time are the structure porosity, water runoff intensity, pore connectivity and size distribution, tortuosity, pollutant concentration and particles size distribution.
2.2.1 Water Runoff Intensity
It is notable that the increase of rain intensity is proportional to the increase of the PPS infiltration and clogging rate. This is due to the increase of storage capacity of PPS with the high rain intensity while the outflow reaches its maximum discharge Lin, Ryu, and Cho (2014). The storage capacity of PPS is the maximum volume of water that can be stored in the pavement without the outflow of the water effluent from the system. This leads to an increase in the void to total volume ratio (porosity) of PPS. This was proven where the author remarked a fall in the infiltration rate of PPS with the decrease of rainfall intensity below certain limits.
2.2.2 PPS Porosity
Porosity is an essential factor to be considered in our calculations since it governs the infiltration by indicating the available void space inside the pervious pavement. Porosity values of the porous pavement merely vary according to the type of pavement, and the gradation of the aggregate material used for construction. Porous Asphalt (PA) and Pervious Concrete (PC) have similar porosity while the porosity of Pervious Interlock Concrete Paver (PICP) is lower. This is due to
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the filling material in the joints between the interlocking concrete blocks of PICP. Porosity of PA and PC is proportional to the aggregate size, while it is not desired to increase the void percentage more than 40% as this affect pavement strength to traffic loading.
The following statistical data give an overview about the initial conditions of the porosity of pervious pavements, and used as a reference in estimating the initial porosity in the model. According to Ferguson (2005), aggregates with diameter ranged between 0.45 and 0.75 mm have void percentage of 26% and 28%, respectively.
For pervious concrete, porosity was denoted in several articles that has an overall range between 0.15-0.35 (Mishra et al., 2013), (Huang et al., 2016), (Martin & Kaye, 2014), (Tennis et al., 2014). For porous asphalt, the porosity was denoted in several articles with an overall range between (0.15-0.25) (Chen & Wong, 2018), (Huang et al., 2016), (Martin & Kaye, 2014), and denoted not to be less than 18% in (ASTM 2008). For PICP, the porosity was denoted in (Huang et al., 2016) to be 10-12%.
The relation between infiltration and porosity is proportional, where increasing the void to total volume ratio will enhance the increase of water flow through the pore-throats which are the voids connections from the surface to the bottom of the porous pavement.
2.2.3 PPS permeability
The hydraulic permeability or conductivity of the porous pavement refers to its maximum infiltration rate under saturated conditions. The permeability differs according to the type of pavement and gradation of the aggregate.
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Several articles tested the permeability of Porous Asphalt (PA), Pervious Concrete (PC), and Pervious Interlock Concrete Paver (PICP), where there are wide range of its values. For PA, the average hydraulic conductivity was denoted equal to 340 in/hr in Martin & Kye (2014), 1723 in/hour in Huang et al. (2016), while it was equal to 240 in/hr for PA with aggregate size 0.4 inch, and 350 in/hour for PA with aggregate size 0.75 inch (Ferguson, 2005).
For PC, the average hydraulic conductivity was denoted equal to 340 in/hour in (Max & Kye, 2014), 444 in/hour in (Huang et al., 2016). For PICP, the average hydraulic conductivity was denoted equal to 297 in/hour (Huang et al., 2016), and ranged between 1.9-42.5 in/hour for tested PICP in Netherlands (Boogaard et al., 2014). Low values were denoted in Ferguson (2005) as in figure (2-1) that vary according to the gradation of the filling material in the joints of PICP.
Figure 2-1: Average hydraulic conductivity based on unbounded aggregate gradation Ferguson (2005)
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The relation between permeability and aggregate size gradation is directly proportional. This is due to the increase of the spacing between aggregates with the increase of their size which gives higher void content and porosity of the medium.
2.2.4 Variation of PPS domains
In the presence of two pervious pavements with different porosities, local nonequilibrium exists in the hydraulic performance due to the change in hydraulic parameters with the change of pore size and shape. The effect of this contrast could not be investigated through single porosity model, but through double porosity model (Lewandowska et al., 2004). The authors in (Lewandowska et al., 2004) developed a theoretical model and verified experimentally in (Lewandowska et al., 2005) to investigate the flow behavior of the fluid through the double-porosity media which projects the concept of two overlapping subdomains. The double-porosity concept with the two overlapping subdomains aims to create two distinct water pressures inside the system that enhances the hydraulic conductivity and reduces suspended particles retention inside the system, and hence, it delays the clogging of the system.
2.2.5 Pore Connectivity
Pore connectivity remarks the effective porosity of the pervious pavement. There are two types of porosity as open or effective and closed porosity. Open porosity are the only active void space in pervious pavement structure to permeate the flow through itSansalone, Kuang, and Ranieri (2008).
2.2.6 Tortuosity
Tortuosity (τ) is a crucial parameter that considers the pore orientation, connectivity, and size variation. This parameter could be simply illustrated as the ratio between the length of the effective
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flow path (le) divided by the length of the straight path (l), while Qin and Qout denotes the inflow
and outflow of the fluid as shown in Figure 2-2:
Figure 2-2: Indication of tortuosity of pervious pavement according to Mishra et al. (2013) It is notable that the infiltration rate is inversely proportional to the tortuosity, when the hydraulic path is longer, and due to the friction losses that happens and reduces the flow velocity.
[2-1] τ = 𝑙𝑙𝑒𝑒
𝑙𝑙
2.2.7 Water Runoff Sediment and Clogging Effect
Clogging is the main factor behind the reduction of infiltration with time from the initial one as the pore size decreases either due to suspended solids deposition or the growth of bacteria and living organisms. 2.2.7.1. TSS clogging effect Le L Qin Qout Flow direction
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There are several articles that discussed the clogging effects of pervious concrete pavement such as Tong (2011), Mata (2008), and Neithalath (2009). The clogging testing methods, sample preparations, sedimentation assumptions, conclusions and weakness are slightly different for each of these studies. However, most of these experiments were conducted based on the principle similar to falling head permeability cell, and the gradual reduction of permeability of specimens was measured. The difference between the clogging test and falling head permeability test is to infiltrate the fluids that includes the suspended clogging materials through pervious concrete specimens instead of a pure water.
The author in Mata (2008) discussed the clogging effect of sediments on pervious pavement layers. His model was to study the effect of several types of sediments which are sand (particle size ≤ 75 µm), clayey silt (particle size between 2000 - 75 µm), and clayey silty sand (composed of 40% sand and 60% clay) on the hydraulic performance of pervious pavements. It was found in this study, as described in Tong (2011), that sand has negligible effect on the clogging of PPS. Clayey silt has a notable effect on PPS due to the straining of its large grains on the surface of the medium. Clayey sand silt has a tremendous effect on effective infiltration since the clogging happens due to the straining of particles on the surface and retention of particles in the pore throats of the medium, where traditional maintenance is not preferred. It was also noted that sand improves the PPSs frost resistance.
Pore structure features of PPS were discussed in Neithalath (2009) and the effect of clogging on them. Fine and coarse sand with different aggregate sizes (#4, #8 and 3/8”) and described in Tong (2011). The study defined a new term called “clogging potential” which points to the easiness of certain PPS to get clogged. It was investigated in this study that fine deposits make severe clogging to the pore-throats that gives permeability reduction. It was noted that pore-throats with large pore
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sizes have less clogging effect than small pore sizes due to the higher pore size distribution and crushed aggregates have higher clogging effect than single sized aggregates.
Tong (2011) gave conclusion based on Tan et al. (2001) study on recommended PSS structures as following:
• PPS with larger voids are recommended for acceptable infiltration rate after clogging. • High ratios of the clogging material particle to the permeable size meets decrease in
infiltration of PPS under clogging effect.
• The gradation, particle shape and amount of deposits have to be considered in terms of clogging effect.
2.2.7.2. TSS removal percentage
Tennis et al. (2014) and EPA (2004) denoted that the removal percentage of the total suspended solids (TSS), due to the infiltration of the stormwater and retention of TSS through the system, as the main deposits is ranged between 82-95%. Other articles such as Huang et al. (2012) and Huang et al. (2016) have results that support this range 85-90%. This range of data would help in quantifying the retained particles over a period of time and determining the reduction rate of porosity.
2.2.7.3. Inflow TSS concentration
The concentration of TSS is an important parameter since other pollutants are attached to sediment particles that affect significantly the clogging rate. Several articles discussed the characteristics of the TSS in storm water and its concentration. In Huang et al. (2016), TSS concentration was taken in range between 491–539 mg/l in a study made according to the geographical and cold weather
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conditions of the city of Calgary. However, Rossi et al. (2005) performed a thorough study on the behavior of TSS concentration variation during the rainfall event. They explained the behavior as it takes a lognormal distribution during the rainfall event, where most of the sediment get infiltrated during the first-third of the rainfall event time. We can conclude that the TSS concentration reached its peak ~ 570 mg/l after approx. 45 minutes of the rainfall event which is slightly higher than the range given in Huang et al. (2016). The rainfall precipitation was followed by low TSS concentration ~ 170 mg/l during the last 2 hours of the rainfall event.
This indicates that taking the TSS concentration peak value and normalizing it throughout the whole period is not feasible to determine an approximate value of the true clogging rate of the pavement. Authors in the same article presented a derived equation for the TSS concentration that accompanies the water runoff, and proposed accordingly a default range of values between 12-372 mg/l with a median value of 68 mg/l.
The variation of sediments concentration practically depends on the environmental condition and traffic loading rate on the pavement. The traffic on roads differs according to their use. They are classified into five categorize (A to E) according to California Department of Transportation based on the traffic load as in table (2-1).
Table (2-1):Pervious Pavement Categories (Pervious Pavement Design Guidance, 2013)
Category Example Traffic loading
A Landscaped areas, sidewalks and bike paths (with no vehicular access), miscellaneous pavement to accept run-on from adjacent impervious areas (e.g. roofs)
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B Parking lots, park & ride areas, maintenance access roads, scenic overview areas, sidewalks and bike paths (with maintenance/vehicular access), maintenance vehicle pullout
Low
C Rest areas, maintenance stations Moderate
D Shoulders, some low volume roads, areas in front of noise
barriers (beyond the traveled way) Moderate
E Highways, weigh stations High
This categorization of road type help for proper estimation of the TSS concentration based on the level of traffic loading in a given range of TSS concentration values as presented in Rossi et al. (2005).
It is notable that concentration of salt and sediments infiltrated through asphalt or concrete pavements are relatively higher in the winter than the summer. This is due to the associated high amount of sediments and residues of vegetation that come with the water runoff in the Fall-Winter season.
2.3. Clogging mechanisms of pervious pavement
Apart from studying the characteristics and properties of the porous pavement, the mechanism of sediment retention with its transport through the porous pavement takes similar analogue to the Deep Bed Filtration (DBF) process. As Teng & Sansalone (2004) discussed the filtration mechanism based on the particle to pore size ratio considering the aggregate and pore diameters are equal. Three main mechanisms were highlighted that cause the retention of particles through the porous media which are the Surficial-Straining (dp/dn < 10), Deep-bed filtration (10 < dp/dn <
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behavior to the porous pavement. The surficial straining records the highest retention rate of particles since it depends on preventing the suspended particles with bigger size from infiltration through the media leading to either the accumulation of sediments on the media or displacement with the runoff. This was demonstrated experimentally in Sansalone et al. (2012) where particles with large size had the highest removal efficiency of the total retained suspended particles.
2.4. Primary equations
Since infiltration is one of the main parameters that measures the hydraulic performance of PPS, it is essential to formulate its performance according to several factors that reflect the aging of the pervious material. The governing equations, that will be discussed, reflect the calculation of fluid flow in an elementary volume of a surface porous medium. These equations are varied according to several parameters as following: the equations of fluid motion, material properties and geometry, and material clogging effect.
The conservation of mass is represented through the continuity equation which works on the principle that mass is neither created nor destroyed.
According to the continuity equation, the rate of change of the volume with time of an elementary volume of the fluid is equal to the change in the volumetric flow rate of its boundaries. Accordingly, the continuity equation will be expressed as following:
[2-2] 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = ∑iQin - ∑jQout
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Continuity equation could help in the calculation of the saturation levels of the fluid and the maximum outflow rate of the porous media. Exceeding the inflow rate that comes from the rainfall or seepage of the neighbor mediums will lead to the occurrence of runoff on the surface of the elementary sample (i.e: ∑iQin ≤ ∑jQout).
Figure 2-3: Schematic illustration of the flow through an elementary volume of the porous pavement
The second governing equation that analyzes the conservation of momentum of fluid inside the elementary porous medium is Darcy’s equation. The momentum balance equation aims to determine the microscopic flow quantities (average velocity and pressure gradient) and represented as following:
[2-3] ὺ = K × ∇H
Where, ὺ denotes the Darcy velocity, K denotes the hydraulic conductivity of the medium, ∇H denotes the hydraulic pressure head.
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Material geometry is another factor that controls the fluid flow inside the porous media. The shape and density of the solid part play the major role towards the water infiltration rate and the occurrence of the water runoff on its surface. Several factors control the hydraulic performance of the porous media are the shape porosity, pore connectivity, and tortuosity. These factors are calculated through different governing equations which get assessed according to the limitations and boundary conditions.
Porosity is an essential factor to be considered in our calculations since it governs the infiltration as the available void space inside the pervious pavement is calculated.
The relation between infiltration and porosity is proportional, where increasing the void to total volume ratio will enhance the increase of water flow through the pore-throats of the porous pavement. The relation could be represented through Carmen-Kozney equation as following: [2-4] 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 = 1801 (1−Ѳ)Ѳ3 2∙ (𝜑𝜑𝜑𝜑)2∙ 𝑔𝑔
𝜇𝜇𝜕𝜕
Pore connectivity remarks the effective porosity of the pervious pavement. Since there are two types of porosity as open or effective and closed porosity, open porosity is the only active void space in pervious pavement structure to permeate the flow through it.
Tortuosity is another crucial parameter that considers the pore orientation, connectivity, and size variation. This parameter could be simply illustrated as the ratio between the length of the effective flow path divided by the length of the straight path.
Tortuosity could be illustrated inside the empirical equation of infiltration as a function of porosity after combining Darcy’s law with pore geometrics as seen in equation [2-5] as following:
18 [2-5] 𝐾𝐾 = Ѳ𝑡𝑡3
𝑆𝑆𝑜𝑜{𝜕𝜕𝑒𝑒⁄ }𝑙𝑙 02(1−Ѳ)2𝐴𝐴∗2
𝛾𝛾 𝜇𝜇
Furthermore, clogging is the main factor of the infiltration decline with time due to the retention of accompanied suspended particles with stormwater on the surface of the porous system pores and through their pore-throats.
As the duration of time between each maintenance of the pervious material increases, the infiltration rate and total suspended solids (TSS) decreases. Prediction of the infiltration rate with time has no universal governing equation in which our model will work on deriving an empirical equation that considers the effect of clogging, rain intensity, porosity, and tortuosity as a function of time and distance.
The variation of rain intensity could be marked with the change in the flow velocity through the porous media. It was explained in Alawi et al. (2013) that at velocity (𝑣𝑣𝑐𝑐), the clogging particles
retention takes place on the surface of deposition. (𝑣𝑣𝑐𝑐) is termed as the critical velocity where the
clogging rate decreases after it due to the partial drew of the clogging particles with the flow of the fluid. The rate of particle deposition on the surface was expressed using modified 𝜕𝜕𝜕𝜕𝑖𝑖
𝜕𝜕𝑑𝑑
Gruesbeck and Collins’s model, where (𝑣𝑣 i) denotes the volume of deposited particles. This is to
determine the rate of change of the volume ratio of particles on the surface to the total volume of the pervious media with time as presented in Liu et al. (1989), Liu et al. (1993), Liu et al. (1994), as following:
[2-6] 𝜕𝜕𝜕𝜕𝑖𝑖
𝜕𝜕𝑑𝑑 = �
𝛼𝛼𝑑𝑑,𝑖𝑖 |𝑣𝑣𝜔𝜔| 𝐶𝐶𝑖𝑖 𝑤𝑤ℎ𝑒𝑒𝑒𝑒 𝑣𝑣𝜔𝜔 ≤ 𝑣𝑣𝑐𝑐
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Alternatively, the rate of change of the volume of entrapped particles in pore throats to the total volume of the pervious media 𝜕𝜕𝜕𝜕∗𝑖𝑖
𝜕𝜕𝑑𝑑 could be expressed as following:
[2-7] 𝜕𝜕𝜕𝜕𝑖𝑖∗
𝜕𝜕𝑑𝑑 = 𝛼𝛼𝑝𝑝𝑑𝑑,𝑖𝑖 |𝑣𝑣𝜔𝜔| 𝐶𝐶𝑖𝑖
2.5. Knowledge gaps
As several articles discussed the clogging effect and behavior on the deterioration of infiltration rate and porosity with time, few mathematical models were made to draw general formulas for this behavior. Previous literature such as in Huang et al. (2016) focused on modelling the infiltration reduction with empirical exponents and coefficients. These empirical parameters are based on their performed experimental work in which they vary from one experiment to another based on the selected fixtures and lab conditions. Hence, a theoretical model to simulate the behavior of infiltration reduction is still questionable, where few studies were made to derive universal formulas from the primary Karmen-Cozney equation to model the deterioration of infiltration in pervious pavements.
Huang et al. (2016) made a thorough experimental study on the degradation of porosity with time. They concluded according to their experimental work several empirical equations of porosity reduction. These empirical equations are able to determine the reduction of porosity in an exponential form where they vary according to the type of pavement and initial porosity of each type. The introduced empirical equations in Huang et al. (2016) take the same form given in equation (2-8) which have coefficient (A) and exponent (B) with undefined parameters that their values were r-anged in the article according to the type of pavement and initial porosity of each type.
20 [2-8] Ѳ(t) = A × Ѳo × exp(-B × t)
Where, (Ѳ) denotes the porosity after time (t); (Ѳo) denotes the initial porosity; (A) & (B) are the
empirical exponents that differ based on the type of pavement and initial porosity. Further studies are required to continue on the work of Huang et al. (2016) in order to draw a relation between porosity, infiltration reduction, and time with defined parameters.
Furthermore, the sensitivity of the introduced parameters that affect infiltration are not yet examined including the heterogeneity of the system, the number of rainfall events, the concentration and gradation of suspended particles, etc. Yet, previous studies such as Sharma & Yortsos (1987) focused on investigating the clogging mechanisms of the porous systems brought their attentions on the loading rate of the clogging materials, their size and morphology. Brakel (1975) focused also on modelling the fluid transport based on the structure of the media. General prediction or stochastic models of the porous system clogging were also introduced in Fallah et al. (2012), Santos & Bedrikovetsky (2006), and Gomes et al. (2017). However, these models were focusing on developing theoretical equations that predict the instant fluid motion and system clogging. They also differ based on the scale of the applied system, in which many studies focused on the micro or small scale models that simulate the clogging of membranes and filter systems. Hence, it is necessary to predict the macro-scale that represents the porous pavement systems from the introduced primary equations in section (2-4), and to derive the governing equations based on the introduced clogging mechanisms.
The analysis of the basic equations is accordingly required to draw a relation between the infiltration and porosity with time. Starting with the primary Karmen-Cozney equation as given in equation [2-5], a derived relation between infiltration and time was obtained through a series of
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theoretical equations as explained in appendix (I). This analysis aimed to perform a mathematical derivation of the governing equation for infiltration degradation with time. The deterioration behavior of infiltration rate was based on the change of porosity with time as presented in equations [2-9]. The porosity relation with time was taken in a power function as in equation [2-10], in which its change is dependent on coefficient (a), the reduction coefficient of porosity, which projects the empirical coefficients investigated from the experimental results in Huang et al. (2016).
[2-9]𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑑𝑑 𝑑𝑑Ѳ
.
𝑑𝑑Ѳ 𝑑𝑑𝑑𝑑 [2-10] 𝑑𝑑Ѳ 𝑑𝑑𝑑𝑑 = -a. Ѳ𝑠𝑠 Where, (𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑) denotes the change of infiltration rate with time; ( 𝑑𝑑𝑑𝑑
𝑑𝑑Ѳ
)
denotes the change of infiltration rate with porosity due to clogging of the pervious pavement and entrapment of suspended solids; (𝑑𝑑Ѳ𝑑𝑑𝑑𝑑) denotes the change of porosity with time; (Ѳ) is the porosity at time (t); (s) power exponent; (a) porosity reduction coefficient.
Part of the introduced factors in section (2-2) that affect the clogging of the pavement are sensitive to the variation of pore and particle size, particle concentration, the fluid intensity, and the heterogeneity of the system. The introduced equations in liu et al. (1993) as given in equations [2-6, 2-7] in section (2-4) provide a defined linear relation between the porosity deterioration, initial concentration and superficial fluid velocity. Yet, the given retention of particles coefficient (α) in the equation is not theoretically represented in a relation with defined parameters. Furthermore, it does not give a relation with the different mechanisms that are responsible for the infiltration and porosity deterioration as introduced in McDowell-Boyler et al. (1986), Teng & Sansalone (2004),
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and which justifies the temporal variation of infiltration deterioration and its significant decline during the first period of system operation. Accordingly, the critical levels that infiltration rate would decline to shall be defined based on the introduced mechanisms. A variation analysis is required to determine the effect of the change in pore and particle size distribution on the infiltration deterioration.
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CHAPTER 3: RESEARCH OBJECTIVES & CONTRIBUTION OF WORK
In the gaps of knowledge presented in section (2.5) of chapter 2, the essence of developing a theoretical model to determine the infiltration deterioration and hydraulic performance of pervious pavements was elaborated. The experimental work performed in previous articles investigated the decline in infiltration over the operational time with empirical equations. These formulas are made in a simpler linear form with empirical coefficients and exponents. These coefficients and exponents vary with each experiment based on the input parameters and the adopted fixed conditions. Hence, developing a mathematical model is necessary to address the deterioration behavior of infiltration prior to the design of pavement.
Based on the previous studies, the deterioration behavior takes the exponential form over the operational time. From the analysis of previous results as shown in figure (3-1), the major decline
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infiltration (~ 50%) is reached within the first 3 to 12 months. The decline continues to reach (~ 30%) of its initial value within the first 12 to 24 months. The last operational period that ranges between 24 to 60 months meets a slight linear decline in infiltration that would reach 20% of its initial value.
From these analysis, it is notable that the change in infiltration decline over time is based on the clogging mechanism involved in the deterioration behavior. Hence, in order to develop a mathematical model that reflects this behavior, the infiltration decline due to each mechanism should be modelled independently over the operational time.
Determining the boundary levels of the infiltration decline due to each clogging mechanism is essential prior to modelling the infiltration over the operational time based on the sediment mass loading rate. Determining the boundary levels of the infiltration decline due to each clogging mechanism should be addressed in the form of a prediction model that is based on the size distribution of the pores of the pavement and suspended particles infiltrated through the system. Hence, the study objective is to perform this prediction model to determine the boundary values of infiltration after complete clogging of pavement due to each clogging mechanism. Further, the characteristics of the suspended particles varies based on the type of pavement use and environmental conditions. Hence, the prediction model is followed by investigating the variation significance of the pore and particle size distribution on the infiltration deterioration.
According to the described clogging mechanisms in section (2.3) in chapter 2, these three mechanisms are surficial straining, deep-bed filtration, and clogging due to Brownian motion. The dominant mechanism is based on the ratio of pore to suspended particle size and the mass loading rate of the suspended particles. Based on (Teng & Sansalone, 2008), the surficial straining is the
25
first clogging mechanism that the system is subjected to it since it depends on the retention of suspended particles with large size on the surface of the medium. This is followed by the deep-bed filtration that depends on the entrapment of medium-size particle through the pore throats prior the retention of small particles on the boundaries of pore throats due to Brownian motion. Clogging due to Brownian motion takes longer for particles to retain on the throat boundaries depending on the concentration of suspended particles.
Hence, it is considered in our model that the major decline of infiltration in the first 3 to 12 months is due to the surficial straining, identified as stage I, in which infiltration would reach a certain value denoted as Kss. The same manner was followed for determining the decline of infiltration in
the first 12 to 24 months due to deep-bed filtration, identified as stage II, in which infiltration would reach a certain value denoted as KDBF. The last period which meets slight decrease in
infiltration is due to the retention of particles on the boundaries of pore throats due to Brownian motion, and identified as stage III as illustrated in figure (3-2).
Figure (3-2): Schematic illustration of the three stages of the infiltration deterioration behavior due to surficial straining, deep-bed filtration, and Brownian motion
26
Accordingly, the prediction model is presented in chapter 4 while the results of the variation analysis based on a set of different values of pore and particle size means and standard deviations are discussed in chapter 5.
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CHAPTER 4: PREDICTION MODEL OF THE INFILTRATION REDUCTION VARIATION WITH PORE/PARTICLE SIZE DISTRIBUTION
4.1. Introduction
As explained in chapter 3, several authors focused their research on the infiltration behavior with time under the clogging effect resulted from the suspended materials in the stormwater. Many of these studies were conducted through experimental work to examine the reduction of pavement infiltration under certain conditions. As the flow rate of stormwater and mass loading rate of the suspended sediments are the dominant parameters to the clogging behavior of porous pavement, few studies highlighted their objectives on the effect of pore/particle size distribution. The change of suspended particle to pavement pore size meets a change in the straining effect on porous pavements under fixed parameters of flow rate and sediment concentration.
Based on the explained clogging mechanisms in section (2.3) in chapter 2. Pore and particle size distributions play an important role on DBF process if the straining effect is operative. A common relation to understand the straining effect is through the ratio of pore to particle size. For instance, when the pore to particle size ratio dp/dn < 10, considering the pore and aggregate size are equal,
straining mechanism would be the dominant one and hence the removal efficiency of sediment material is 100%. The polarization phenomena would then take place on the surface of the pavement leading to rapid decline in water infiltration unless the runoff would flush the rested sediment. However, if dp/dn > 10, the retention mechanism differs from straining to deep-bed
filtration and physical chemical effects. The dominant mechanism would depend on how far the dp/dn ratio is from the boundary values.
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The particle sizes of sediments are not a fixed value but have a wide range of variation in the macro-scale. Similarly, the pore sizes in the porous media are not homogenous and their range of values follow the same manner of the particle size distribution in the macro-scale. Accordingly, it is essential to understand distribution system for both the sediment particles and pores of the system. This takes place by running a model system that mimics the particle migration through the pavement.
A mathematical model is proposed in this chapter for predicting the pore/particle distribution effect on the infiltration of water through the pavement system.
Q
in, C
inQ
out, C
outRetained particles due to Straining effect (dm > dp)
Retained particles due to Brownian motion & Diffusion effect (dm << dp)
Retained particles due to Interception effect (dm < dp)
Figure (4-1): Schematic illustration of the retention of suspended particles with different sizes over the porous pavement
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4.2. System Model
This work has a model system that consists of suspended particle that migrates through a porous media through pores with different sizes. The model focuses on examining the straining of suspended particles on the surface of the media based on the suspended particles size distribution (SSD) or the pore size distribution (PSD).
As Gaussian and log-normal are the most common systems in size distribution, the normal distribution probability function was used to represent SSD or PSD as in equations 1] and [4-2] along with other prospective derived equations for the model. The fraction of suspended particles that retained in the porous media due to straining effect are quantified.
[4-1] 𝑓𝑓(𝑟𝑟𝑚𝑚) = √2𝜋𝜋𝜎𝜎1 2 𝑒𝑒 −(𝑟𝑟𝑚𝑚−µ )2 2𝜎𝜎2 [4-2] 𝑔𝑔�𝑟𝑟𝑝𝑝� = √2𝜋𝜋𝜎𝜎`1 2 𝑒𝑒 (𝑟𝑟𝑝𝑝−µ` )2 2𝜎𝜎`2
Where 𝑓𝑓(𝑟𝑟𝑚𝑚) and 𝑔𝑔�𝑟𝑟𝑝𝑝� are the probability density functions (PDFs) of the suspended particles
and pore size distribution of the system, respectively. 𝑟𝑟𝑚𝑚 and 𝑟𝑟𝑝𝑝 are the radius of the suspended
particles and pores, respectively. 𝜎𝜎 and 𝜎𝜎` are the standard deviations of the suspended particles and pores, while µ and µ` are the means of the suspended particles and pores, respectively.
The straining of suspended particles over the surface of the pavement leads to an apparent reduction in the connectivity of the system and hence a major decline in water infiltration through the system. The model also regards the deep-bed filtration mechanism which is due to particle interception with pore boundaries and chemical-physical retention mechanism which is due to the Brownian motion of particles based on the classification index (𝜆𝜆), where 𝜆𝜆 = 𝑟𝑟𝑚𝑚 𝑟𝑟
𝑝𝑝 � . A
30
mathematical solution would be made to derive the governing differential equation of the model. This would be followed by a numerical solution in order to obtain the discrete equation of the PDF. The hierarchy of this model is represented in figure (4-2).
4.3. Particle/Pore Size Correlation Attempt
In this section, we proceed to define the governing equations for the probability function of the classification index rm/rp. Following Gomes et al. (2017), the concentration distribution of particles
(number of particles with radii between rm and rm + drm per unit volume of the fluid) is defined as
F(rm) drm. Similarly, the concentration distribution of pores (number of pores with radii between
rp and rp + drp per unit surface area of the porous media) is defined as G(rp) drp.
Hence the particle size probability function 𝑓𝑓(𝑟𝑟𝑚𝑚) presented in the previous section is expressed
as the ratio of concentration distribution between rm and rm + drm to the total concentration of
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suspended material as in equation [4-3]. Similarly, the pore size probability function 𝑔𝑔�𝑟𝑟𝑝𝑝� is the
ratio of concentration distribution between rp and rp + drp to the total concentration of suspended
material as in equation [4-4]. [4-3] 𝑓𝑓(𝑟𝑟𝑚𝑚) = ∫ F(𝑟𝑟𝑚𝑚) 𝑑𝑑𝑟𝑟𝑚𝑚 𝑟𝑟𝑚𝑚+𝑑𝑑𝑟𝑟𝑚𝑚 𝑟𝑟𝑚𝑚 ∫ F(𝑟𝑟0∞ 𝑚𝑚) 𝑑𝑑𝑟𝑟𝑚𝑚 [4-4] 𝑔𝑔�𝑟𝑟𝑝𝑝� = ∫𝑟𝑟𝑝𝑝𝑟𝑟𝑝𝑝+𝑑𝑑𝑟𝑟𝑝𝑝F(𝑟𝑟𝑝𝑝) 𝑑𝑑𝑟𝑟𝑝𝑝 ∫ F(𝑟𝑟0∞ 𝑝𝑝) 𝑑𝑑𝑟𝑟𝑝𝑝
Each probability density function of the particles or the pores is assumed to follow the normal distribution curve. [4-5] ℎ �𝑟𝑟𝑚𝑚 𝑟𝑟𝑝𝑝� = 𝑓𝑓(𝑟𝑟𝑚𝑚) . 𝑔𝑔�𝑟𝑟𝑝𝑝� (gdp, max; fdm, max) (gdp, max; fdm, min) (gdp, min; fdm, max)
Figure (4-3): Schematic illustration of the correlation of the two Particle/Pore probability density functions
32
The probability function of the classification index ℎ �𝑟𝑟𝑚𝑚
𝑟𝑟𝑝𝑝� or ℎ(λ) depends accordingly on the variation of 𝑓𝑓(𝑟𝑟𝑚𝑚) and 𝑔𝑔�𝑟𝑟𝑝𝑝�. Each discrete segment of the domain 𝑟𝑟𝑚𝑚 has its probabilistic function
and a corresponding probabilistic function of the other domain 𝑟𝑟𝑝𝑝 as shown in figure (4-3).
4.4. Classification Index/Removal Efficiency Correlation Attempt
Removal percentage of suspended materials is proportional to their size and concentration. Retained particles in the porous media increase accordingly with the same proportion, leading to an increase in clogging and reduction of porosity and infiltration. The size of pores in the media plays the inverse role with its proportionality to the size of suspended materials, the classification index, in which as long the pore size is bigger, the probability of porosity and infiltration reduction is low. This section aims to develop a prediction model to be able to quantify the removed amount of suspended materials in proportion with the classification index.
4.4.1. Trapping of particles due to straining effect
When the diameter of suspended materials is larger than the diameter of pores, the predominant mechanism is the surficial straining. Hence, a fraction of the total mass of suspended materials is excluded from their concentration in the influent. This excluded mass of particles retains on the surface of the porous media leading to apparent closure of pores, and hence, a rapid decline in its surface porosity and infiltration. The actual amount of particle concentration in the influent (Cin)
counts on the subtracted portion that has no surficial straining effect (i.e: dp/dn > 10). The
mathematical calculation of (Cin) would then be as following:
33
Where, Cin, total denotes the total mass of suspended materials associated with the fluid; Cin, actual
denotes the actual mass passed through the porous media after the deduction of the excluded fraction part of the mass retained due to surficial straining effect; P(dp/dn<10) denotes the
probability of suspended materials that has large size to be retained on the surface of the porous media due to surficial straining.
When we need to examine the removal efficiency of suspended materials (Ʒss) due to straining
effect, the removal efficiency equation is considered as denoted in [4-7]. it is observed that [4-7] Ʒss = 𝐶𝐶𝑖𝑖𝑖𝑖,𝑡𝑡𝑜𝑜𝑡𝑡𝑡𝑡𝑡𝑡𝐶𝐶 −𝐶𝐶𝑖𝑖𝑖𝑖,𝑡𝑡𝑐𝑐𝑡𝑡𝑎𝑎𝑡𝑡𝑡𝑡
𝑖𝑖𝑖𝑖,𝑡𝑡𝑜𝑜𝑡𝑡𝑡𝑡𝑡𝑡 ×100%
For calculating the reduction in surface porosity and infiltration due to surficial straining effect, a modelling attempt was made to determine the reduced values of these two parameters. The model is based on the probability distribution function of the pore size. The closure of fine pores that corresponds to the fraction of the classification index (dp/dn < 10) is assumed to occur due to the
Figure (4-4): Schematic illustration of the fine pores clogging in the plane view of the porous media represented in the hatched circles due to straining effect and accumulation of large particles on the surface of these pores.
Qin
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surficial straining. The proportion of pore size that meets this condition is excluded from the proportionality of the surface porosity as in figure (4-4), and hence, a fraction of the initial porosity is deducted to estimate the reduced one due to surficial straining. This is mathematically expressed by concerning the reduction in surface porosity is part of the total surface area of the existing pores of the surface layer. The mathematical equation of the infiltration reduction portion due to straining effect is subsequently derived and denoted over the following set of equations, [4-8] - [4-28]. [4-8] Ѳin = 𝑆𝑆𝐴𝐴𝑆𝑆𝐴𝐴𝑝𝑝 × 𝑑𝑑 𝑚𝑚 ×𝑑𝑑 [4-9] Ѳin = ∑ 𝜋𝜋𝑁𝑁𝑖𝑖𝑟𝑟𝑝𝑝,𝑖𝑖 2 𝑖𝑖 𝑖𝑖=1 𝑆𝑆𝐴𝐴𝑚𝑚 [4-10] Ni = Nt × 𝑓𝑓𝑖𝑖 [4-11] Ѳin = 𝜋𝜋𝑁𝑁𝑡𝑡 ∑ 𝑓𝑓𝑖𝑖𝑟𝑟𝑝𝑝,𝑖𝑖 2 𝑖𝑖 𝑖𝑖=1 𝑆𝑆𝐴𝐴𝑚𝑚
Where (Ѳin) denotes the initial porosity of the porous media; (𝑆𝑆𝑆𝑆𝑝𝑝) and (𝑆𝑆𝑆𝑆𝑚𝑚) denote the surface
area of the pores and whole media, respectively; (𝜑𝜑) denotes the thickness of the media; (Nt)
denotes the total number of pores presented on the surface layer of the media; (Ni) denotes the
number of pores of the ith term in the set of different pore sizes. Porosity is equal to the ratio between the volume of voids to the total volume of the media. Considering the voids are represented in the pores along the depth of the medium (d), hence its ratio is equal to the ratio of the surface area of these pores to the total surface area of the media as represented in equation [4-8]. The initial porosity could hence be a function of the particle size and number of pores based on the proportionality of each particle size of the ith term as part of the pores normal distribution curve as in [4-11]. The new calculated porosity (Ѳss) is estimated after the deduction of the fraction
35
part of pores number that has the predominance to get clogged due to the straining effect as in equations [4-12] and [4-13]. [4-12] Ѳss,i. = 𝜋𝜋𝑁𝑁𝑡𝑡 𝑓𝑓𝑖𝑖,𝑠𝑠𝑠𝑠𝑟𝑟𝑝𝑝,𝑖𝑖 2 𝑆𝑆𝑆𝑆𝐴𝐴𝑚𝑚 [4-13] Ѳss. = 𝜋𝜋𝑁𝑁𝑡𝑡 ∑ 𝑓𝑓𝑖𝑖,𝑠𝑠𝑠𝑠𝑟𝑟𝑝𝑝,𝑖𝑖 2 𝑖𝑖 𝑖𝑖=1 𝑆𝑆𝑆𝑆𝐴𝐴𝑚𝑚 [4-14] 𝑓𝑓𝑠𝑠𝑠𝑠,𝑖𝑖 = 𝑓𝑓𝑖𝑖 − 𝑓𝑓𝑖𝑖(dpdn< 10) [4-15] Ѳss,i. = 𝜋𝜋𝑁𝑁𝑡𝑡 (𝑓𝑓𝑖𝑖− 𝑓𝑓𝑖𝑖� dp dn<10�)𝑟𝑟𝑝𝑝,𝑖𝑖2 𝑆𝑆𝑆𝑆𝐴𝐴𝑚𝑚 [4-16] Ѳss. = 𝜋𝜋𝑁𝑁𝑡𝑡 ∑ (𝑓𝑓𝑖𝑖− 𝑓𝑓𝑖𝑖� dp dn<10�)𝑟𝑟𝑝𝑝,𝑖𝑖2 𝑖𝑖 𝑖𝑖=1 𝑆𝑆𝑆𝑆𝐴𝐴𝑚𝑚
Where, (𝑓𝑓𝑠𝑠𝑠𝑠,𝑖𝑖) denotes the remained probability fraction part of the normal size distribution of
surface pores where the inflow of stormwater is discharged to the porous system through it. (𝑓𝑓𝑖𝑖(dpdn< 10)) denotes the probability fraction of pores that is subjected to clogging due to straining
effect based on the classification index range of values. (Ѳss) could hence be a function of the
particle size and number of pores. This is based on the proportionality of each particle size of the ith term as part of the pores normal distribution curve after the deduction of the fraction part subjected to clogging due to straining effect as represented in [4-15] and [4-16].
Recalling the derived equation of infiltration rate by Karmen-Cozeny, we can have a direct relation between it and the porosity of the media as in equation [4-17].
[4-17] Kin = Ѳ𝑖𝑖𝑖𝑖 3
𝑆𝑆𝑖𝑖𝑖𝑖2(1−Ѳ𝑖𝑖𝑖𝑖)2
𝜆𝜆𝜌𝜌𝑤𝑤𝑔𝑔 𝜂𝜂
36
Where, (Kin) denotes the initial infiltration rate of the porous media; (
𝜌𝜌
𝑤𝑤) denotes the density ofthe fluid; (𝜆𝜆) denotes the shape factor of the media due to the tortuosity effect; (𝜂𝜂) denotes the dynamic viscosity of the fluid; (𝑆𝑆𝑖𝑖𝑖𝑖) denotes the initial specific surface area. (𝑆𝑆𝑖𝑖𝑖𝑖) is expressed as
a function of the initial porosity (Ѳ𝑖𝑖𝑖𝑖) and pore diameter (𝜑𝜑𝑝𝑝) as in equation [4-18]. (g) denotes the
gravitational acceleration force.
The particle diameter of the ith term is calculated based on its fraction portion in the probability size distribution of pores. Hence, the infiltration rate is calculated through a bar model in which (Kin,i) denotes the initial infiltration rate of the fluid discharged through the fractional part of pore
size with ith term into the system. (Kin,i) expression form is hence derived mathematically as a
function of (𝑟𝑟𝑝𝑝,𝑖𝑖) as in equation [4-20]. [4-18] Sin = 6(1−Ѳ𝑖𝑖𝑖𝑖) 𝑑𝑑𝑝𝑝 [4-19] 𝜑𝜑𝑝𝑝,𝑖𝑖
=
𝑓𝑓𝑖𝑖 × 2 𝑟𝑟𝑝𝑝,𝑖𝑖 [4-20] Kin,i = (𝜋𝜋𝑁𝑁𝑡𝑡 𝑓𝑓𝑖𝑖𝑟𝑟𝑝𝑝,𝑖𝑖𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚2 ) 3 𝑆𝑆2�1−𝜋𝜋𝑁𝑁𝑡𝑡 𝑓𝑓𝑖𝑖𝑟𝑟𝑝𝑝,𝑖𝑖2 𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚 � 2𝜆𝜆𝜌𝜌𝑔𝑔𝜂𝜂 [4-21] Kin = (𝜋𝜋𝑁𝑁𝑡𝑡 ∑𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚𝑒𝑒𝑖𝑖=1𝑓𝑓𝑖𝑖𝑟𝑟𝑝𝑝,𝑖𝑖2 ) 3 𝑆𝑆2�1−𝜋𝜋𝑁𝑁𝑡𝑡 ∑𝑒𝑒𝑖𝑖=1𝑓𝑓𝑖𝑖𝑟𝑟𝑝𝑝,𝑖𝑖2 𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚 � 2𝜆𝜆𝜌𝜌𝑔𝑔𝜂𝜂37 [4-22] Kin = (𝜋𝜋𝑁𝑁𝑡𝑡 ∑𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚𝑒𝑒𝑖𝑖=1𝑓𝑓𝑖𝑖𝑟𝑟𝑝𝑝,𝑖𝑖2) 3 ⎝ ⎜ ⎛6�1− 𝜋𝜋𝑁𝑁𝑡𝑡 ∑𝑒𝑒 𝑓𝑓𝑖𝑖𝑟𝑟𝑝𝑝,𝑖𝑖2 𝑖𝑖=1 𝑆𝑆𝑆𝑆𝑚𝑚 � ∑𝑒𝑒𝑖𝑖=1𝑓𝑓𝑖𝑖×2 𝑟𝑟𝑝𝑝,𝑖𝑖 ⎠ ⎟ ⎞ 2 �1−𝜋𝜋𝑁𝑁𝑡𝑡 ∑𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚𝑒𝑒𝑖𝑖=1𝑓𝑓𝑖𝑖𝑟𝑟𝑝𝑝,𝑖𝑖2� 2 𝜆𝜆𝜌𝜌𝑔𝑔 𝜂𝜂
The new infiltration rate (Kss) is expressed in equations [4-23] and [4-24] which denotes the
modified value after the clogging of fractional part of pore size distribution due to the straining effect. (Kss,i) expression form is hence derived mathematically as a function of (𝑟𝑟𝑝𝑝,𝑖𝑖) as in equation
[4-24]. This derived equation considers the change of porosity and pore size of each ith term due to the straining effect, where the total infiltration rate is the cumulative value of (Kss,i) as in
equation [4-27]. [4-23] Kss = Ѳ𝑠𝑠𝑠𝑠 3 𝑆𝑆2(1−Ѳ𝑠𝑠𝑠𝑠)2 𝜆𝜆𝜌𝜌𝑔𝑔 𝜂𝜂 [4-24] Kss = (𝜋𝜋𝑁𝑁𝑡𝑡 ∑ (𝑓𝑓𝑖𝑖− 𝑓𝑓𝑖𝑖� dp dn<10�)𝑟𝑟𝑝𝑝,𝑖𝑖2 𝑒𝑒 𝑖𝑖=1 𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚 ) 3 𝑆𝑆𝑠𝑠𝑠𝑠2�1− 𝜋𝜋𝑁𝑁𝑡𝑡 ∑ (𝑓𝑓𝑖𝑖− 𝑓𝑓𝑖𝑖� dp dn<10�)𝑟𝑟𝑝𝑝,𝑖𝑖2 𝑒𝑒 𝑖𝑖=1 𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚 � 2𝜆𝜆𝜌𝜌𝑔𝑔𝜂𝜂 [4-25]Sss,i = 6(1−Ѳ𝑠𝑠𝑠𝑠) 𝑑𝑑𝑝𝑝,𝑖𝑖,𝑠𝑠𝑠𝑠 [4-26]Sss,i = 6(1−Ѳ𝑠𝑠𝑠𝑠) 𝑓𝑓𝑖𝑖,𝑠𝑠𝑠𝑠×2 𝑟𝑟𝑝𝑝,𝑖𝑖
Where (Sss,i) denotes the new specific surface area of the porous media after the deduction of the
fraction part of clogged pores due to the straining effect. It is expressed as a function of the modified porosity (Ѳ𝑠𝑠𝑠𝑠) and pore diameter (𝑟𝑟𝑝𝑝,𝑖𝑖,𝑠𝑠𝑠𝑠) as in equations [4-35] and [4-36].
