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Adapting the Green and Ampt model to account for air compression and counterflow
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by Darwiche Sabeh.
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[Tampa, Fla.] :
University of South Florida,
2004.
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Thesis (M.S.C.E.)University of South Florida, 2004.
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ABSTRACT: One of the earliest functions to express infiltration as a function of time was introduced by Green and Ampt. In this study their formula was modified to account for air compression and counterflow. Physically,infiltration, air compression, and counterflow occur simultaneously, while in this model they are decoupled within a time step. Counterflow is calculated as a mass flux and pressure is found using the perfect gas law. First, a comparison of three infiltration methods, the original Green and Ampt formulation, a modified version incorporating air compression only, and the third version including air compression and counterflow, was conducted. Then sensitivity of the model accounting for both air compression and counterflow was explored. Results showed that accounting for both air compression and counterflow improves the predicted infiltration rate.Air effect on infiltration can be significant even for environments with an impervious layer as deep as 10m; while for very deep water table environments (100m) the three models give similar results. In shallow water table environments (0.5m), air effect on infiltration rate, cumulative infiltration, ponding time, and saturation time is substantial. The model accounting for air compression and counterflow was then tested for different parameters. It provided reasonable results compared to the Green and Ampt model and the modified version accounting for air compression only. The advantages of this model are that no additional data is required other than what's needed for the original Green and Ampt formulation, and it can be applied for any environment.The assumption of uniform soil moisture content is a limitation for the model, especially for shallow water table environments where the variations in the soil moisture profile within the wetting front depth is substantial.
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Adviser: Nachabe, Mahmood.
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rainfall.
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water table depth.
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Adapting the Green and Ampt Model to A ccount for Air Compression and Counterflow by Darwiche Sabeh A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Civil Engineering Department of Civil and Environmental Engineering College of Engineering University of South Florida Major Professor: Mahmood Nachabe, Ph.D. Mark Ross, Ph.D. Paul Zandbergen, Ph.D. Date of Approval: October 28, 2004 Keywords: rainfall, infiltration, shar p wetting front, water table depth Copyright 2004, Darwiche Sabeh
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DEDICATION I dedicate this work to my parents, my siblings, Nabil and Amale Sabeh, and to my brothers and sisters. Thank you for al l the support you have given me throughout my life, your care, and for the invaluable educat ion I have acquired from you. I could not do it without you. Choukran.
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ACKNOWLEGEMENTS I would like to thank the CMHAS pe rsonnel for making my stay at USF memorable. I extend special gratitude to Je ffrey Vomacka for his helpful assistance in this research. Thank you to Dr. Mahmood Nachabe for his constant guidance and commitment to quality research. Thank you also to Dr. Mark Ross and Dr. Paul Zandbergen for meeting with me and providi ng helpful insights. I would also like to recognize my cousins for their helpfu l support to my visit to the USA.
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i TABLE OF CONTENTS LIST OF TABLES.............................................................................................................iv LIST OF FIGURES............................................................................................................v ABSTRACT.....................................................................................................................vi ii CHAPTER 1: INTRODUCTION.......................................................................................1 1.1 Background...............................................................................................................1 1.2 Objectives and Scope................................................................................................2 1.3 Bridging Two Runoff Mechanisms..........................................................................4 1.3.1 Traditional Separation of the Runoff Mechanisms............................................4 1.3.2 Air Phase Effect on Infiltration..........................................................................6 1.4 Literature Review.....................................................................................................9 1.4.1 Runoff Mechanisms...........................................................................................9 1.4.2 Green and Ampt Equation...............................................................................11 1.4.3 Air Phase Effect on Infiltration........................................................................13 1.5 Contribution of This Study.....................................................................................21 CHAPTER 2: METHODOLOGY....................................................................................23 2.1 Empirical vs. Theoretical Approach.......................................................................23 2.2 Methodology Briefing.............................................................................................24 2.3 Air Compression using BoyleÂ’s Law......................................................................24
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ii 2.4 Modified Green and Ampt Approach (MODGA)..................................................25 2.5 Finite Difference Approach....................................................................................27 2.6 Air Pressure Quantification....................................................................................27 2.7 Air Mass Flux.........................................................................................................28 2.8 Algorithm for the Modified Green and Ampt Model (MODGA)..........................30 2.8.1 MODGAÂ’s Assumptions..................................................................................31 2.8.2 MODGAÂ’s Description....................................................................................31 2.8.2.1 Ponding Time Calculation........................................................................32 2.8.2.2 Infiltration Capacity Calculation...............................................................33 2.8.3 Flow Chart.......................................................................................................34 2.9 MODGAÂ’s Sensitivity to Time Step.......................................................................35 CHAPTER 3: RESULTS AND DISCUSSION................................................................40 3.1 Introduction.............................................................................................................40 3.2 Comparison of Three Infilt ration Modeling Approaches.......................................40 3.2.1 Shallow Water Table Environment (SWT)......................................................41 3.2.1.1 Modeling Air Pressure Ahead of the Wetting Front.................................41 3.2.1.2 Modeling Wetting Front Depth a nd Cumulative Infiltration....................43 3.2.1.3 Modeling the Infiltration Rate..................................................................45 3.2.2 Deep Water Table Environment (DWT)..........................................................47 3.3 Impact of DepthtoWater Table.............................................................................50 3.4 Impact of Initial Soil Moisture Content..................................................................53 3.5 Impact of Rainfall Intensity....................................................................................57 3.6 Impact of Saturated Hydraulic Conductivity..........................................................60
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iii 3.7 Impact of Soil Type................................................................................................63 CHAPTER 4: CONCLUSION.........................................................................................67 4.1 Comparison of Three Infiltration Approaches........................................................67 4.2 MODGA Sensitivity to Different Parameters.........................................................68 REFERENCES.................................................................................................................70 APPENDICES..................................................................................................................73 Appendix A. MODGA Programme d with Visual Basic............................................74 Appendix B. Air and Water Physical Properties at 20 C.............................................84
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iv LIST OF TABLES Table 1 Â– Infiltration Rates (cm/h) Imp acted by Air (Constantz et al., 1988)..................18 Table 2 Â– Comparison of Three Infilt ration Models (Wilson et al., 1982).......................20 Table 3 Â– Parameters Used for the Reference Simulation................................................35 Table 4 Ponding Time Sensitiv ity to Time Step (MODGA)..........................................39 Table 5 Â– Air Effect on Ponding and Saturation Times (SWT)........................................45 Table 6 Â– Impact of Depthto Water Table on Ponding Time..........................................53 Table 7 Â– Impact of Initial Water Content on Ponding Time...........................................56 Table 8 Â– Impact of Rainfall Intensity on Ponding Time.................................................59 Table 9 Â– Impact of Saturated Hydr aulic Conductivity on Ponding Time.......................62 Table 10 Â– Parameters Used for the Impact of Soil Type Simulation..............................63 Table 11 Â– Impact of Soil Type on Ponding Time............................................................66 Table 12 Â– Air and Water P hysical Properties at 20 C.....................................................84
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v LIST OF FIGURES Figure 1 Â– Runoff Mechanisms (Freeze, 1980)..................................................................5 Figure 2 Â– Air Compression Effect on Water Table Elevation on 2/22/02........................8 Figure 3 Â– Rain Event on 2/22/02.......................................................................................8 Figure 4 Â– Factors Influencing Runoff Mechanisms (Dunne, 1983)................................10 Figure 5 Â– Green and Ampt Infiltr ation Model (Chow et al., 1988).................................12 Figure 6 Â– Air Effect on Water Cont ent Profiles (Vachaud et al., 1974).........................14 Figure 7 Â– Air Effect on Cumulative Infiltration (Vachaud et al., 1974).........................15 Figure 8 Â– Air Effect on Infiltrati on Rate (Vachaud et al. 1974)......................................15 Figure 9 Â– Modified Green and Am pt Model (MODGA) Algorithm...............................34 Figure 10 Â– Air Pressure Se nsitivity to Time Step...........................................................35 Figure 11 Â– Air Pressure Sensitiv ity to Time Step (Zoom)..............................................36 Figure 12 Â– Wetting Front Depth Sensitivity to Time Step..............................................36 Figure 13 Â– Infiltration Sens itivity to Time Step..............................................................37 Figure 14 Â– Infiltration Rate Se nsitivity to Time Step......................................................37 Figure 15 Â– Infiltration Rate Sensit ivity to Time Step (Zoom)........................................38 Figure 16 Â– Air Pressure Ahead of the Wetting Front (SWT)..........................................41 Figure 17 Â– Air Pressure Ahead of the Wetting Front (SWT) (Zoom).............................42 Figure 18 Â– Wetting Front Depth (SWT)..........................................................................43
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vi Figure 19 Â– Cumulative In filtration (SWT)......................................................................44 Figure 20 Â– Infiltration Ra te Modeling (SWT).................................................................45 Figure 21 Â– Infiltration Cap acity at Saturation.................................................................46 Figure 22 Â– Air Pressure at Saturation..............................................................................47 Figure 23 Â– Air Pressure Ahead of the Wetting Front (DWT).........................................48 Figure 24 Â– Wetting Front Depth (DWT).........................................................................48 Figure 25 Â– Cumulative In filtration (SWT)......................................................................49 Figure 26 Â– Infiltration Ra te Modeling (DWT)................................................................49 Figure 27 Â– Impact of DepthtoW ater Table on Air Pressure.........................................51 Figure 28 Â– Impact of DepthtoW ater Table on Wetting Front.......................................51 Figure 29 Â– Impact of DepthtoWater Table on Cumulative Infiltration........................52 Figure 30 Â– Impact of DepthtoWater Table on Cumulative Infiltration Rate................52 Figure 31 Â– Impact of Initial Soil Moisture Content on Air Pressure...............................54 Figure 32 Â– Impact of Initial Wa ter Content on Wetting Front........................................54 Figure 33 Â– Impact of Initial Water Content on Cumulative Infiltration..........................55 Figure 34 Â– Impact of Initial Wate r Content on Infiltration Rate.....................................55 Figure 35 Â– Infiltration when Soil is Near Saturation.......................................................56 Figure 36 Â– Impact of Rainfall Intensity on Air Pressure.................................................57 Figure 37 Â– Impact of Rainfall Intensity on Wetting Front..............................................58 Figure 38 Â– Impact of Rainfall Intens ity on Cumulative Infiltration................................58 Figure 39 Â– Impact Rainfall Intensity on Infiltration Rate...............................................59 Figure 40 Â– Impact of Saturated Hydr aulic Conductivity on Air Pressure.......................60 Figure 41 Â– Impact of Saturated Hydr aulic Conductivity on Wetting Front....................61
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vii Figure 42 Â– Impact of Saturated Hydraulic Conductivity on Cumulative Infiltration......61 Figure 43 Â– Impact of Saturated Hydrau lic Conductivity on Infiltration Rate.................62 Figure 44 Â– Impact of Soil Type on Air Pressure.............................................................64 Figure 45 Â– Impact of Soil Type on Wetting Front..........................................................64 Figure 46 Â– Impact of Soil Type on Cumulative Infiltration............................................65 Figure 47 Â– Impact of Soil T ype on Infiltration Rate.......................................................65 Figure 48 Â– MODGAÂ’s Graphical Interface.....................................................................74
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viii ADAPTING THE GREEN AND AMPT MODEL TO ACCOUNT FOR AIR COMPRESSION AND COUNTERFLOW Darwiche Sabeh ABSTRACT One of the earliest functions to expre ss infiltration as a function of time was introduced by Green and Ampt. In this study their formula was modified to account for air compression and counterflow. Phys ically, infiltration, air compression, and counterflow occur simultaneously, while in th is model they are dec oupled within a time step. Counterflow is calculate d as a mass flux and pressure is found using the perfect gas law. First, a comparison of three infiltra tion methods, the original Green and Ampt formulation, a modified version incorporat ing air compression only, and the third version including air compression and counterflow, wa s conducted. Then sensitivity of the model accounting for both air compression a nd counterflow was explored. Results showed that accounting for both air compression and counterflow improves the predicted infiltration rate. Air e ffect on infiltration can be significant even for environments with an impervious layer as deep as 10m; while for very deep water table environments (100m) the three models give similar results. In shallow water table environments (0.5m), air effect on infiltra tion rate, cumulative infiltration, ponding time, and saturation time is substantial. The model accounting for air compression and counterflow was then tested for different parameters. It provided reasonable results
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ix compared to the Green and Ampt model a nd the modified versi on accounting for air compression only. The advantages of this model are that no additional data is required other than whatÂ’s needed fo r the original Green and Ampt formulation, and it can be applied for any environment. The assumpti on of uniform soil moisture content is a limitation for the model, especially for sh allow water table environments where the variations in the soil moisture profile with in the wetting front depth is substantial.
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1 CHAPTER 1: INTRODUCTION 1.1 Background Numerous formulations have been proposed to express infiltrati on as a function of time or of the total quantity of water infilt rated into the soil. One of the earliest was introduced by Green & Ampt in 1911 (Hillel 1998), whose theory has been found to apply particularly to infiltration into unifo rm, initially dry and coarsetextured soils, which exhibit a sharp wetting front (Hillel and Gardner, 1970). The formula is best applied for infiltration excess runoff (Hortoni an mechanism), where runoff occurs after rainfall intensity exceeds the in filtration capacity of the soil. In contrast, in shallow water table environments and soils with high hydraulic conductivity, it is believed that the dominating runoff mechanism is the satura tion excess runoff (Dunne mechanism) where the soil storage capacity between a shallo w water table and the ground surface is filled, and the remaining rainfall goes to runoff. However, due to the effect of the air phase on the infiltration process, the classification of runoff into one of these two mechanisms is questionable. In fact, research has shown that air entrapment, co mpression, and counterflow in this kind of environment greatly reduce the soil storage cap acity, as well as the infiltration rates (Vachaud et al., 1974; MorelSeytoux and Kh anji, 1975; Touma et al., 1984; Wang et al., 1997) Â– parameters that control the runoff pr ocess. Therefore, in environments where
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2 saturation excess is likely to occur, i.e., Central and South Florida, the air phase movement and its effect on infiltration can re duce the infiltration capaci ty of the soil to a point where Hortonian runoff might occur. This reduction in the infiltration capacity of the soil can reach a value as low as zero and all the rain will go into runoff, which is the same as saturation excess runoff, but with a nonfully saturated soil. Thus, it is suggested that saturation excess runoff should be defi ned as the runoff that occurs when the infiltration rate reaches zero and not when the soil storage is filled. Observations suggest that encapsulated air below water table will always prevent complete saturation. Thus, in shallow water table environments and soils with high hydrau lic conductivity, the traditional concept of saturation excess runoff may significantly underestimate the instantaneous and total volume of runoff. It is necessary to bridge between infiltration and saturation excess runoff based on the infi ltration capacity of the soil as impacted by the air phase. 1.2 Objectives and Scope The purpose of this research is to quan tify the infiltration/runoff phenomenon to account for air encapsulation, air compression, a nd counterflow. In particular, the simple and widely used formula of Green and Ampt will be adjusted to account for the air phase effect on infiltration rate. Unlike the original Green and Ampt concept where the air is considered to be at atmospheric pressure duri ng the infiltration and the flow of water is decoupled from the air flow, the infiltration pr ocess is approached as a twophase flow (waterair). A model is formulated that pr ovides coupling between air and water during the infiltration process by accounting for ai r pressure in the porous medium. The new
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3 formula accounts for the possibility of a Hort onian type runoff occurring due to the air phase effect in areas where a saturation exce ss runoff is anticipated; thus, accounting for a runoff before the soil storage is completely filled. Results of this research will offer a modi fied form of the Green and Ampt formula to account for air phase impact on the infiltrati on rate. Also, it will be show that it is essential to account for both air compression and counterflow to accurately quantify the infiltration. Also, the Green and Ampt formul a overestimates the infiltration capacity of the soil, and accounting for air compression onl y by using the BoyleÂ’s law underestimates it. Instead, coupling the two phase flow by appl ying the perfect gas law to the air phase in the porous medium after changing the volume occupied and mass every time step is a preferred approach. In addition, the impact of depth to water ta ble, rainfall rate, initial soil moisture content, and soil properties on infiltration are explor ed and discussed. The application of this formula gives a better estimation of in stantaneous and total runoff during a rainfall event. In f act, unlike the traditional con cept of saturation excess runoff where all the rain infiltrates until the soil storage is completely filled when all additional rain becomes overland flow, this approach ac counts for the possibility of having some runoff before saturation and reaching an infiltra tion rate of zero before the soil storage is completely filled. This work is divided into four chapters. Chapter 1 is the introduction and contains general background on infiltration and saturati on runoff mechanisms, the objective of this study, the need to bridge between the two mechanisms, and a literature review of previous work on twophase flow. Chapter 2 is a description of the methodology used in this research. Chapter 3 represents the resu lts of the modified Green and Ampt model.
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4 Finally, Chapter 4 discusses the results of the proposed model and summarizes the findings of the study. 1.3 Bridging Two Runoff Mechanisms This section contains a review of the tr aditional concept of infiltration excess and saturation excess runoff mechanisms and the eff ect of the air phase on total infiltration as well as infiltration rate. 1.3.1 Traditional Separation of the Runoff Mechanisms Runoff generation has been considered to be either from infiltration excess (Hortonian) or saturati on excess (Dunne). Hortonian runoff occurs when the rainfall rate exceeds the infiltration capacity of the soil and usually is observed in deep water table environments. Saturation excess runoff occurs wh en the soil is fully saturated, i.e., when the total infiltration depth ex ceeds the soil storage capacity. This mechanism occurs in highly conductive soils with shallow water tabl e where the infiltration depth fills the soil storage before any overland flow starts to occur.
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5 Figure 1 Â– Runoff Mechanisms (Freeze, 1980) Figure 1 represents a comparison between these two mechanisms as described by Freeze (1980). For infiltration excess runoff, Figure 1 a), all the rainfall infiltrates into the soil increasing the soil moisture content before ponding time, tp. After that time, infiltration proceeds at infiltration capacity of the soil and the excess rain goes into runoff. In contrast, for saturation excess runoff, Figure 1 b), all rain infiltrates into the soil until saturation is reached. After the soil storage is completely saturated, rainfall becomes runoff. A detailed literature review on this subject is pres ented in section 1.4.
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6 It is suggested that presence of air aff ects runoff generation substantially, and a Hortonian type runoff might occur in sha llow water table environments with highly conductive soils prior to sa turation excess runoff. 1.3.2 Air Phase Effect on Infiltration The vadose zone is a multiphase porous medium where the movement of one phase in the pore space is associ ated with a movement of the ot her. In particular, there are several mechanisms by which air affects the infiltration process: viscous resistance, compression effect, buoyancy effect, counterflow effect, hysteresis effect and several others (MorelSeytoux and Khanji, 1975). Because of air entrapment (or encapsu lation in bubble or bypassed pores), the soilwater content does not attain total satu ration but some maximal value lower than saturation, which has been calle d satiation (Hillel, 1998). This issue can be taken into account by considering that the maximum wate r content in a soil only reaches a value smaller than porosity known as natural sa turation or effective porosity (Charbeneau, 2000). During infiltration, air can be compressed in the vadose zone, especially if the water table is shallow and air can not escape to deeper layers. Research has shown that air compression can affect the infiltration pr ocess significantly. For instance, Culligan et al. (2000) found that even for a small increase in pressure relative to the case where the air was free to escape, e.g., < 1cm of water, th ere was a small but measurable reduction in infiltration.
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7 Compressed air can still make its way out to the atmosphere through the wetting front. This process is counterflow of air. According to MorelSe ytoux and Khanji (1975), because air flows upward, the wa ter content in the counterflow zone must decrease. They also stated that counterflow and hysteretic effects must be associated. However, they suggest that as long as the wa ter content is high in the zone affected by counterflow and the gradient of water content low, this hys teretic effect will not be pronounced (MorelSeytoux and Khanji, 1975). Impact on the water table level is another interest for quantifyi ng the air pressure. In fact, if water table level and soil moisture content are mo nitored at the same location, it is observed that rise in water table occurs prior to the wetting fr ont propagation to the capillary zone (Charbeneau, 2000) Figure 2 is a representation of this effect from data collected by Vomacka et al. (2002) in Lithia, Florida. Figure 2 shows water table depth in a monitoring well (dashed lines) at several periods during a rainfall event, as well as soil water content (continuous lines) obtained from soil moisture probes. The change in water content has not reached the saturated zone; th erefore, a change in water table elevation would not be expected. However, the monitori ng well at that location shows a water table rise. Therefore, the rise in the monitoring well is only an Â“apparent ri seÂ” of water table as a result of pressure increase in the vados e zone. The water table well is open to the atmosphere. Simply, due to air compression a pressure gradient between the compressed air in the vadose zone and the air in the monitoring well cause s a water level rise in the monitoring well. This water level rise in the we ll is a measurement of gage air pressure in the vadose zone. Figure 3 represents cumula tive rainfall for this event and continuous
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8 change in the depthtowater ta ble during this event period. A de tailed literature review of air phase effect on infiltration wi ll be presented in section 1.4.3. Soil Moisture and Water Table Depth0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0510152025303540 Water Content (%)Depth (ft) 2/22/02 10:45 2/22/02 16:45 2/22/02 22:45 2/23/02 4:45 2/23/02 10:45 2/22/02 10:45 2/22/02 16:45 2/22/02 22:45 2/23/02 4:45 2/23/02 10:45 Figure 2 Â– Air Compression Effect on Water Table Elevation on 2/22/02 Cumulative Rainfall and Water Table Depth 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.610:45 12:45 14:45 16:45 18:45 20:45 22:45 0:45 2:45 4:45 6:45 8:45 10:45TimeRainfall (in3.8 3.9 4.0 4.1 4.2 4.3 4.4DepthtoWater Table (ft) Cumulative Rainfall DepthtoWater Table Figure 3 Â– Rain Event on 2/22/02
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9 Later in this study, air phase effect on infiltration rate will be quantified. This quantification will provide a way to account fo r possibility of occurrence of a Hortoniantype runoff in environments where a satura tion excess runoff is considered the dominant runoff mechanism. Also, saturation excess r unoff will be defined herein as a runoff observed after infiltration capacity of the so il reaches zero, rather than the traditional concept of full saturation. To make this argument there is a need to review traditional runoff generation mechanisms in literature and to adapt an infiltration model, i.e., Green and Ampt, to bridge between the two runoff mechanisms. 1.4 Literature Review This section contains a review of runo ff mechanisms, rainfallrunoff studies in Florida, the Green and Ampt equation, and previous studies on air phase effect on infiltration. 1.4.1 Runoff Mechanisms Effective rainfall or runoff is traditionally defined as net liquid water supplied to channels at time scales comparable to durat ion of storm after evaporation, interception, surface retention, infiltration, and percolation to underlying aquifers (Bras, 1990). But quantifying runoff depends on the process that generated it and the modeling approach that is used. Soils, topography, climate and vegetation are factors influencing runoff mechanisms. Figure 4 (Dunne, 1983) shows these factorsÂ’ effects on the runoff processes. In general, an overland flow is considered to be either an infiltration excess runoff or a
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10 saturation excess runoff. This study will account for the necess ity to bridge between these two types due to the impact of air phase on infiltration rate. Arid to Subhumid Sparse Vegetation Humid Dense Vegetation Climate and Vegetation Soils and Topography Thin Soils High to Low Permeability Flatter Slopes Deep Soils High Permeability Steep Slopes Hortonian Overland Flow Dominant Subsurface Stormflow Dominant Saturation Overland Flow Dominant Figure 4 Â– Factors Influencing Runoff Mechanisms (Dunne, 1983) Horton (1939) found that infi ltration capacity of soil decr eases with rainfall time to reach a constant minimum value. He fitted the following exponential model to infiltration capacity with time, t K c cfe f f f f 0, Equation 1 where f is the infiltrationcapacity at time t; f0 is the initial infiltrationcapacity at t = 0; fc is the minimum constant infiltrationcapaci ty known also as permeability at natural saturation, Kns; and Kf is constant for a given curve. This type of runoff occurs when
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11 rainfall intensity exceeds infiltration capacity of soil causing ponding at surface. If soils are highly conductive, rainfall intensities mi ght not exceed the infiltration capacity and no runoff will be observed until soil storage is filled, which is known by saturation excess runoff. However, presence of air in the soil matrix may reduce infiltration rate substantially and to a point where a Hortonian runoff mi ght occur when a saturation excess runoff is expected. Touma et al. (1984) reported that the primary effect of air pressure in a confined column is to reduce in filtration rate to about onethird of its value in comparison with cases where air is free to escape. On a catchment scale, due to spatial variability in rainfall and catchment charac teristics, it is unlikely for runoff to be generated by one mechanism. Even at one location, simulations have shown that runoff generation can switch from infiltration to saturation excess depending on initial conditions and rainfall events (Loague and Abrams, 2001). A deeper insight of air phase effect is presented later in section 1.4.4. 1.4.2 Green and Ampt Equation In 1911, Green and Ampt suggested a theoretical approach for modeling infiltration in their paper Â“The Flow of Ai r and Water through SoilsÂ”. The authors main assumptions are that there exists a distinct and precisely definabl e wetting front during infiltration, and that although this wetting fr ont moves progressively downward as the process proceeds, it is characterized by a c onstant matric suction, regardless of time and position. Furthermore, this approach assumes that in the transmission zone behind the wetting front the soil is uniformly wet and of constant conductivity. The wetting front is
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12 thus viewed as a plane separating a uniforml y wetted infiltrated zone and asyet totally uninfiltrated zone (Hillel, 1998). Green and Ampt proposed the simplified picture of infiltration below (Chow et al., 1988). Figure 5 Â– Green and Ampt Infiltr ation Model (Chow et al., 1988) (Scanned and digitized copy using AutoCAD) Taking these assumptions into account, the Green and Ampt theory can be formulated as follow, F H K L L H H K dt dF fi s c ns c ns 10. Equation 2 By integrating between ponding time tp and time t, we get H0 L s Porosity i r
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13 F S F S SLn F T t K Fp p p ns, Equation 3 where f is infiltration rate [L/T]; Kns is hydraulic conductivity at natural saturation [L/T]; H0 is ponding depth at the surface [L] (a ssuming that ponded water becomes runoff, H0=0); L is depth to wetting Front [L]; s is saturated water content [L3/L3]; i is initial water content [L3/L3]; Hc is wetting front suction head [L]; F is cumulative infiltration [L] (F = L( s Â– i)); Fp is cumulative infiltration at p onding time [L]; t is time [T]; Tp is ponding time [T]; and S = Hc( s Â– i). The air phase being neglected in this a pproach, it is necessary to adapt this equation to account for air compression and the f act that we have a twophase flow in soil and that air in soil does not rema in at atmospheric pressure. 1.4.3 Air Phase Effect on Infiltration Under most applications, neglecting resistan ce to water flow caused by flow of air is not a problem. However, various exceptions arise, including that of infiltration under ponded conditions in shallow water table c onditions where this resistance cannot be ignored (Charbeneau, 2000). In addition to th eoretical and analytical studies, both field and laboratory experiments have been done to account for air phase effect on infiltration rate, infiltration depth, and water table fluctuation. Using a single vertical column of fine sa nd packed into an acrylic plastic cylinder 56 cm long and 5 cm inside diameter, Vacha ud et al. (1974) studied the effects of air movement and compression during ponded infilt ration. They showed that if air cannot escape freely, there is a cons iderable reduction in infiltration rate, the shape of water
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14 profiles is significantly differe nt, and air pressure gradient s are not negligible. Figure 6 and Figure 7 represent their documentation of air compression effect on water content profile and cumulative infiltration. Figure 6 Â– Air Effect on Water Cont ent Profiles (Vachaud et al., 1974) Water Content (cm3/cm3) Lateral Air Flow Air Compression
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15 Figure 7 Â– Air Effect on Cumulative Infiltration (Vachaud et al., 1974) Figure 8 Â– Air Effect on Infiltra tion Rate (Vachaud et al. 1974) (Scanned and digitized copy using AutoCAD)
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16 Figure 8 is a plot of the infiltration rate th ey obtained during their experiment. It shows that due to air compression, infiltration capaci ty of soil reaches zero (curve 2), while it reaches some constant value when air is allo wed to escape (curve 1). This confirms the hypothesis of defining the starting point of sa turation excess runoff when the infiltration capacity of soil reaches zero a nd not when storage is full. Also note that at any time, t, infiltration capacity of soil is less if the air is compre ssed. Thus, for highly conductive soils where saturation excess runoff is antici pated, we might observe a Hortonian runoff due to air compression effect. These findings support the purpose of this research to bridge between two runoffgeneration mechan isms by modifying the simple Green and Ampt model to accoun t for air phase. Air entrapment during groundwat er recharge can cause an anomalously large rise of water levels in observation wells in shallow unconfined aquifers during heavy rainstorms (Freeze and Cherry, 1979). Fayer and Hillel (1986b) concluded from a field experiment that encapsulated air is an im portant component of shallow water table fluctuations. In fact, they found that dependi ng on initial depth of water table and soil moisture characteristics, water table rises we re two to five times those when air was not encapsulated: the shallower the water table, the higher the rise. Fayer and Hillel (1986a) reported volumetric air encapsulation as f unction of depth in their paper Â“Air Encapsulation: I. Measurement in a Field Soil .Â” They measured volume of encapsulated air for their field experiment at 15 cm dept h intervals after water table rose to land surface from a depth of 1.5m. Air entrapped was assumed to be the difference between soil porosity and moisture content measured just after water tabl e had reached surface. They studied also the effect of rain intensit y on air encapsulation by sprinkling the site at
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17 different rates. Later, Constantz et al. ( 1988) conducted field and laboratory experiments to measure effects of air encapsulation on in filtration and reported a value of 19% of entrapped air for the Olympic Sand. Wang et al. (1998) reported that residual encapsulated air in an airconfining condition increased 7% on average in comparison to an airdraining condition. Wangemann et al. (2000) studi ed effects of antecedent soil water content and air entrapment on infiltration. The authors found that wetter initial su rface water content resulted in lower infiltration rates and attri buted this effect to more rapid aggregate breakdown and surface seal development under we tter initial conditions as compared to drier. In contrast, due to air entrapment, they found that wetter init ial conditions resulted in higher percolation rates: dr yer soil would have more air to block conducting pores. In shallow water table environments, as the wetting front moves downward through soil, air gets trapped between wetti ng front and water table, which impedes further infiltration and causes a reducti on in infiltration rate Wang et al. (1998) compared air effect on infiltr ation rate in a laboratory ex periment using some 45cm long columns packed with ovendried sand ( 4.5% clay, 11.3% silt, and 84.2% sand). Their results showed that infiltration rates for airdraining condition were 310 times larger than those obtained under airconfining conditi on. Under nonponding condition, infiltration rate, iw, decreased on average from 55% of the saturate d conductivity, Ks, for air draining condition to 18% of Ks for airconfining conditio n. Under ponded condition it was reduced by an average factor of 6. Wilson et al., (1982) reported a substantial difference between saturated hydraulic c onductivity values obtained in the lab and field conductivity rates obtained in the field using a modified Purduetype infiltrometer. They measured the
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18 field conductivity for Â“dryÂ” and Â“wetÂ” runs. Dr y runs refer to experiments that were performed on soils at initial moisture conten t occurring naturally in the field; wet runs were run 24 hours after dry runs to obtain highe r initial soil moisture content. They noted a difference between wet and dry runs, but they couldnÂ’t explain this in their study. Constantz et al. (1988) studied the effect of air on infilt ration rate also and reported substantial effect of air phase in both field and laboratory experiments. They compared infiltration rates of four soil types for two gases in the soil matrix: air and CO2, which was injected in soil to replace air and minimi ze air effect on infiltration. Their values are documented in Table 1, where Â“ControlÂ” refers to the experiments without CO2 pretreatment (infiltration controlled by air). Table 1 Â– Infiltration Rates (cm/h) Im pacted by Air (Constantz et al., 1988) Faybishenko (1995) introduced the term Â“quasisaturated soilsÂ” to define soils beneath water table which contain entrapped air. DarcyÂ’s coefficient accounting to air entrapment is called Â“quasisaturated hydrauli c conductivityÂ”. This terminology is used to distinguish between the terms unsaturated hyd raulic conductivity (used for unsaturated soils in the vadose zone) and saturated hydrau lic conductivity (used fo r saturated soils in the aquifer). The author distingu ished three stages in temporal behavior of quasisaturated Field Experiments Laboratory Experiments Los Gatos Gravelly Loam Diablo Sandy Loam Olympic Sand Aiken Loam Control 25.2 5.4 15.0 1.2 CO2 264 25.2 73.8 6.0
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19 hydraulic conductivity since it vari ed significantly with time. The first stage represents decreases in the quasisaturat ed hydraulic conductivity by as much as 58 times. He attributed this effect to en trapped air blocking large pores Â– this period lasted 0.5 to 2 days. During the second stage, entrapped air moves as free gas and as dissolved state in the water phase. As mobile air discharges progressively from soil, quasisaturated hydraulic conductivity incr eases slowly. When the remaini ng immobile air is discharged as a dissolved phase, quasisat urated hydraulic conductivity is increased by 2 orders of magnitude reaching the value of saturated c onductivity. During the third stage, decrease in hydraulic conductivity is attributed to surf ace sealing and microbiol ogical activities. In this study, work is in the first stage, sin ce modeling the infiltration process is during rainfall events which last for only a few hours. Several works have been done in modeling the infiltration process. Wilson et al. (1982) compared the results from three models: The GAMLUNMOD model developed by Mein and Larson (1973) using the infilt ration equation proposed by Green and Ampt (1911), the GAMLETA model to account for th e air entrapment effect, which is a modified version of the GAMLUNMOD, a nd GAMLART, which accounts for both air entrapment and resistance effect. They reported that GAMLUNMOD failed to predict ponding time due to use of unmodified saturate d conductivity values: in all but three of the simulations, ponding time was which means that no ponding is reached. GAMLETA did a better job in pred icting ponding time on dry soil s, yet it overpredicted infiltration on wetter soils. GAMLART did the best overall job in predicting infiltration but it also overpredicted infiltration on we t soils. Table 2 contains an average value comparison of these models.
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20 Table 2 Â– Comparison of Three Infi ltration Models (Wilson et al., 1982) MorelSeytoux and Khanji (1974) adjusted the Green and Ampt formula to account for viscous resistance due to air movement. Thei r modified infiltrati on rate equation is L L H H K fc ns 0, Equation 4 where the dimensionless total viscous resistance factor (like the effectiv e capillary drive Hc) is solely a function of soil and fl uid characteristics. For most soils is greater than 1, which may explain why the Green and Ampt e quation overpredicts infi ltration. In a later paper (1975) they modified the equation above to account for air compression and counterflow effects. The terms for air compression and counterflow were derived separately even though all eff ects occur simultaneously. For air compression effect the authors added a term to the numerator of Equation 4 using BoyleÂ’ s law to quantify air pressure in the vadose zone for deep wa ter tables. Their adjusted formula is, L D L H L H H K fatm c ns 0, Equation 5 where Hatm refers to atmospheric pressure, a nd D is depthtowater table. Unlike Equations 4 and 5, the formula MorelSeytoux and Khanji (1975) derived to account for Predicted Value/ Observed Value, Ponding Time: tp(PRE)/tp(OBS) Predicted Value/ Observed Value, Infiltration Rate: I(PRE)/I(OBS) UNMOD GAMLETAGAMLARTGAMLETA GAMLART Dry Soils 3.64 1.52 1.26 0.99 Wet Soils 4.63 2.03 2.95 2.12
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21 counterflow does not have th e Green and Ampt functional form. In fact, a term was added to the denominator of Equation 5 to acco unt for counterflow effect which tends to reduce the total viscous resistance. Equation 6 below is the model they presented to account for viscous resistance, ai r compression, and counterflow, i s ns atm c nst K L D L H L H H K f 0, Equation 6 where, dimensionless quantity, is a counterflow correction factor. 1.5 Contribution of This Study Even though effect of air on infiltration has been widely studied, in this paper mass flux of air will be quantified as well as air pressure in the porous medium. Researches done to date have decoupled ai r flux and compression. In fact, so far air pressure in the soil matrix has been calculated using BoyleÂ’s law, which assumes the air mass to remain constant during infiltration. So me formulas accounted for that by adding a term to the equation to account for counter flow. In addition, while using BoyleÂ’s law, water table was assumed to be deep; an assu mption that restrains these equations from being applied to shallow water table envir onments. However, in this research, the pressure of air in the porous medium is found by application of the perfect gas law to the remaining mass of air and the volume occupied at the beginning of each time step. The model provided can be used for both deep and shallow water table environments. Coupling between air compression and counterfl ow for one and with water infiltration on
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22 the other side is a new method of appr oaching the twophase flow. A complete description of methodology is pr ovided later in chapter 2.
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23 CHAPTER 2: METHODOLOGY 2.1 Empirical vs. Theoretical Approach First, this study was supposed to deri ve an empirical model from field observations of soil moisture and water table le vel. As seen in Figure 2 above, water table rise can be observed in the monitoring well before change in soil moisture reaches the capillary fringe. Thus, this rise can not be attributed to rechar ge. Instead, the only physical explanation is to attribute it to an air pressure difference between air in the porous medium (compressed air) and air in the monitoring well open to atmosphere (atmospheric pressure). Attempts to model th is rise empirically and use it as the air pressure term in the Green and Ampt model we re made. Rise of water table represents air pressure to include in the Green and Ampt equation. Yet, this phenomenon could not be modeled empirically due to th e large number of paramete rs and factors included, the complexity of the coupling between air compre ssion and counterflow, and the infiltration of water. To overcome this issue, a con ceptual model was built where the parameters variation is limited and the water and air pha se were decoupled. A description of this model can be seen below.
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24 2.2 Methodology Briefing This chapter describes the modified Gr een and Ampt model proposed to account for air phase impact on infiltration. The mode l is similar to the Green and Ampt except for an additional term accounting for air pressu re. This suggested model is similar to the one presented by MorelSeytoux in Equations 5 and 6 above, but instead of accounting for air pressure in the nu merator using BoyleÂ’s law (Hatm*L/D) and counterflow in the denominator, air pressure in the porous medi um (due to compression and counterflow) is introduced in the numerator, but will not be found using BoyleÂ’s law. Instead, at each time step air pressure is calculated by estim ating air flux out of the soil and applying the perfect gas law for the remaining mass and volum e of air ahead of the wetting front. This way air compression and counterflow are coup led, which is not the case when using BoyleÂ’s law. 2.3 Air Compression using BoyleÂ’s Law As seen already in the literature review air pressure ahead of the wetting front was found using BoyleÂ’s law. Since a different approach is proposed here, the reasons for which BoyleÂ’s law will not be used needs to be highlighted. BoyleÂ’s law states that under isothermal conditions, and for a perfect gas (l ike air), the pressure of gas is inversely proportional to the volume it occupies, i.e., P*V = constant. Applying this formula for air mass ahead of the wetting front, replacing pressure by pressure head and volume by dept htowater table, yields to:
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25 Hatm*D = Hp*(D Â– L) = Constant. If Ha is the gage air pressure head in the porous medium, Ha = Hp Â– Hatm, then L D L H H L D D H Hatm atm atm a *. For deep water tables, i.e., D>>>L, this equation can be written as, D L H Hatm a* Equation 7 As explicitly seen in the derivation of Equation 7 above, this approach is only valid for deep water tables where wetting fron t depth can be neglecte d in front of depthtowater table. Also, implicitly included in the application of BoyleÂ’s law, is the fact that mass of air in the porous medium remains th e same during infiltration, since counterflow is neglected, and air pressure continues to increase. Some research previously done accounted for counterflow by adding two terms to the Green and Ampt approach to account for air compression and counterflo w. In this paper air compression and counterflow effect will be lumped into one term, which simplifies the equation and allows accounting for mutual effect of air compression and counterflow on one another. 2.4 Modified Green and Ampt Approach (MODGA) The pressurized air in the porous medium will reduce the soilÂ’s ability to absorb water, i.e., infiltration capacity. To account for this effect air pressure in the porous medium will be plugged in the numerator of Equation 2 above. The proposed model is represented by the following equations, f > i for t < Tp, and for t > Tp, L H L H H K fa c ns 0, Equation 8
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26 where, Ha is the gage air pressure head in the porous medium. Below is a the description of the air pressure quantification. For simplification purposes, consider that thereÂ’s no ponding at the surface, i.e., H0 = 0. Knowing that F = L*( s Â– i), Equation 7 can be written as: F S S K dt dF fa ns1, where Sa = Ha*( s Â– i). Separation of variables leads to dt K S S F dF Fns a with S being constant for a soil type and Sa assumed constant during a time step. This equation, integrated for each time step as follows, t t t ns F F adt K S S F dF Fe b* where, Fb and Fe are the cumulative infiltration at the beginning and end of a time step, yields to the following equation, t K S S F S S F Ln S S F Fns a b a e a b e * ) (. Equation 9 This is the same form as the regular Green and Ampt equation, with the exception that the integration is ca rried by time step instead of carrying it between ponding time and any time, t, because the term Sa varies each time step. To solve this equation, i.e., find the cumulative infiltration, some iteration must be carried as an explicit form for Fe cannot be reached. Equations 8 and 9 apply after ponding becau se before ponding all rain infiltrates. Ponding occurs when f = i, where i is the ra infall rate. Incorporating f = i, and F = i*Tp into Equation 7 above yields to the following formula for ponding time,
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27 ns a ns pK i i S S K T *. Equation 10 The suction head, Hc, and therefore S, is a f unction of soil properties. 2.5 Finite Difference Approach The infiltration process is c ontinuous with time. Thus, infiltration capacity of soil, air pressure and mass in porous medium, and counterflow of air vary continuously with time. Due to the large amount of variables a ffecting the process, an approach through finite difference, i.e., each of these quantitie s will be considered constant within a time step, will be used to model infiltration. The model will be tested for convergence with time steps for results accuracy. 2.6 Air Pressure Quantification Instead of using BoyleÂ’s law to calculate pressure in porous medium as described in Equation 7 above, air pressure ahead of the wetting front will be calculated using the perfect gas law, P = *R*T, Equation 11 where P is the pressure in Pascal [Pa]; is the air density [Kg/m3]; R is the perfect gas constant; R = 286.9 [J/Kg.K]; a nd T is the temperature of the air in Kelvin [K]. Each time step air density changes because both the space volume available in soil and air mass change due to infiltration of water and counterflow of air. Volume of space available is the pore space between water table and wetting front, i.e., V = (D Â– L)*A*(n Â– i), where n is porosity of the soil, and A is area. Mass of air
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28 remaining in soil at the end of a time step is mass at the beginning of this time step minus counterflow. Thus, if assuming a constant mass flux of air during a time step, mass of air remaining in the soil is, me = mi Â– mf dt, where me and mi represents mass of air at the beginning and end of a time step [Kg]; mf mass flux of air [Kg/s]; and dt time step [s]. The following section represents the formula used to calculate air mass flux from soil during infiltration. 2.7 Air Mass Flux A description of the formula used to acc ount for counterflow is presented hereby. Flow of air will be approached as a mass flux. This approach is described by Charbeneau (2000) and will be reviewed here. The gene ral DarcyÂ’s equation for both compressible and incompressible fluids is k g P k q Equation 12 where k is the intrinsic permeability of the soil [m2]; is the dynamic viscosity of fluid [Ns/m2]; is the fluidÂ’s density [Kg/m3]; and g is the gravitational acceleration [m/s2]. If the fluid in consideration is air, gravita tional effect can be neglected and Equation 11 above yields to P k q Equation 13 For cross section A, and for onedim ensional flow, fluid mass flux is dL dP k A qA mf Equation 14
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29 For isothermal conditions the pressure is given by 0 0 P P where p0 and 0 are reference values for the fluid pressure and density at standard atmospheric condition. Thus, Equation 14 can be written as dL dP P k PA mo f 0 Variable separation yields to ) (0PdP P k A dL mo f Equation 15 For a wet soil, multiply the intrinsic pe rmeability of soil by relative permeability of air, kra, to account for pores filled with wate r. Since this model follows the same assumption of an advancing sharp wetting fr ont presented by Green and Ampt, air mass flux will cross a column of soil at natural saturation of a height equal to the wetting front depth. Integrating Equation 14 between we tting front and land surface results in L P P P k k A mp ra f2 .2 0 2 0 0 Equation 16 where Pp is pressure in porous medium ahead of wetting front. Relative permeability of air is function of soil water content and can be calculated using the following formula by Charbeneau (2000): ) 1 ( ) 1 () / 2 1 ( 2 rak Equation 17 where is the pore size dist ribution index; and r rn is the normalized water content in the Brooks and Corey Model (BC). The value used of water content at natural saturation is critical for this model because of its impact on counterflow: Equation 17 is a non linear equation. Some simulations were do ne to choose representative values of s:
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30 the formula / 12 1 ) ( r r sn derived using the BC model and BouwerÂ’s suggestion 2 K Ks (1966), didnÂ’t yield to physically a cceptable results. Thus, in this research a value for saturated water content close to porosity is assumed. The wetting front suction head can be calculated usi ng BC parameters and the equation below (Nachabe and Illangasekare, 1994), b ch H 3 1 3 2 Equation 18 where / 2 3 and hb is Brooks and CoreyÂ’s bubbling pressure. These equations have been programmed using Visual Basic to model the infiltration process. In addition to modeli ng infiltration to account for air compression and counterflow, two other models we re programmed: one accounting for air compression but neglecting counterflow, and a s econd where air phase in the soil matrix is neglected (original Green and Ampt appr oach). The following section represents the algorithm of this program. 2.8 Algorithm for the Modified Green and Ampt Model (MODGA) A description of the algorithm and use of the above formulas is presented below. In addition, the assumptions included in th e formulas derivation will be explicitly shown before the presentation of the programÂ’s flow chart.
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31 2.8.1 MODGAÂ’s Assumptions The derivation of formulas was based on the assumption of considering the variable parameters of the infiltration process as quasi steady. This means that different parameters which continuously vary with time will be considered as constant within a time step. Thus, MODGA is sensitive to the tim e step used and this sensitivity will be tested in the following section. The basic assumptions and limitations for this model are the following: i. Sharp wetting front is maintained during infiltration; ii. Constant initial soil moisture content; iii. Constant rainfall rate; iv. Uniform air pressure in porous medium; v. Air pressure ahead of the wetting front is constant during a time step, i.e., mass flux is constant during a time step, as well (when counterflow is n0t neglected); and vi. Constant infiltration capac ity during a time step. 2.8.2 MODGAÂ’s Description A literal description of MODGA and the AlgorithmÂ’s flow chart, drawn using Microsoft Visio , are represented hereby. Tw o main sections are distinguished in the program: before and after ponding time.
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32 2.8.2.1 Ponding Time Calculation Before surface ponding, all rainfall infiltrates into the soil and infiltration capacity of the soil is higher than rainfall intensity. Thus first step in the program is to find time at which surface ponding is reached, known as ponding time, Tp, using Equation 10. i. At time, t, Ha = constant until t + t ii. Sa = Ha*( s Â– i) iii. s a s pK i i S S K T iv. If Tp > t+ t, ponding does not occur in this time step and all rain infiltrates. v. Calculate L P P P k k A mp ra f2 .2 0 2 0 0 M = M Â– Mf t vi. Fe = Fb + i* t i s eF L V = (n Â– i)*(D Â– L) RT V M Pp If Pp < Patm, it is a mathematical cons equence that is physically incorrect of this finite difference approach, (Mf = constant). To correct it, set Pp = Patm and recalculate the mass of air in porous medium. vii. atm water p aH P H Fb = Fe viii. Repeat these steps until Tp < t+ t, which corresponds to ponding time. N.B.: If counterflow is neglected, BoyleÂ’s law is used to calculate Ha and Mf = 0. If air phase is neglected completely (Green and Ampt Approach), Ha and Mf are null. Clearly Equation 10 shows that air phase reduces ponding time, as will be seen in the following chapter.
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33 2.8.2.2 Infiltration Capacity Calculation After ponding, infiltration proceeds at soil Â’s capacity. To find the infiltration capacity, iterate using Equation 9. The steps are as follow: i. At time, t, Ha = constant until t + t ii. Sa = Ha*( s Â– i) iii. Calculate L P P P k k A mp ra f2 .2 0 2 0 0 M = M Â– Mf t iv. t K S S F S S F Ln S S F Fs a b a e a b e * ) ( To find total infiltration at the end of a tim e step, iterations are n eeded to solve this equation. Incremental values are functions of infiltration capacity at the previous time step (or saturated hyd raulic conductivity), and time step. v. t b e cF F f (if fc < Ks/100 fc = 0) vi. i s eF L V = (n Â– i)*(D Â– L) RT V M Pp (same condition as before) vii. atm water p aH P H Fb = Fe viii. Repeat these steps until end of the storm is reached. The steps presented above are a major description of the proposed model. A detailed description is provi ded in the flow chart below (soil saturation, storm ending without ponding the soilÂ…). Th e complete Visual Basic model is documented in Appendix A.
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34 2.8.3 Flow Chart Figure 9 Â– Modified Green and Ampt Model (MODGA) Algorithm
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35 2.9 MODGAÂ’s Sensitivity to Time Step For a reference soil type and properties, water table depth, and rainfall rate, MODGA was run for different time steps until convergence was reached. Parameters related to the reference simulation are documen ted in Appendix B. Time steps that were used are: 5 min., 3 min., 1 min., 30 sec., 15 sec., and 9 sec. The figures below show sensitivity to time step of air pressure, de pth to wetting front, total infiltration, and infiltration rate. Table 3 Â– Parameters Used for the Reference Simulation Soil Type: Sandy Loam Water Table and Rainfall n s r i Ks (cm/hr)Hb(m)Hc(m)kra D(m) i(cm/hr) 0.41 0.39 0.065 0.2071.0 0.13 0.1650.017 0.89 0.5 3.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 020406080100120Time (min)Gage Pressure Head (m) 5 min. 3 min. 1 min. 30 sec. 15 sec. 9 sec. 6 sec. Figure 10 Â– Air Pressure Sensitivity to Time Step ZOOM
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36 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.00.51.01.52.02.53.03.54.0Time (min)GageAir Pressure (m) 30 sec. 15 sec. 9 sec. 6 sec. Figure 11 Â– Air Pressure Sensitivity to Time Step (Zoom) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 020406080100120Time (min)Wetting Front Depth (m) 5 min. 3 min. 1 min. 30 sec. 15 sec. 9 sec. 6 sec. Figure 12 Â– Wetting Front Depth Sensitivity to Time Step
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37 0.00 0.50 1.00 1.50 2.00 2.50 3.00 020406080100120Time (min)Cumulative Infiltration (cm 5 min. 3 min. 1 min. 30 sec. 15 sec. 9 sec. 6 sec. Figure 13 Â– Infiltration Se nsitivity to Time Step 0.00 0.50 1.00 1.50 2.00 2.50 3.00 020406080100120Time (min)Infiltration Rate (cm/hr ) 5 min. 3 min. 1 min. 30 sec. 15 sec. 9 sec. 6 sec. Figure 14 Â– Infiltration Rate Sensitivity to Time Step ZOOM
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38 2.50 2.60 2.70 2.80 2.90 3.00 12.012.212.412.612.813.013.213.413.613.814.0Time (min)Infiltration Rate (cm/hr ) 1 min. 30 sec. 15 sec. 9 sec. 6 sec. Figure 15 Â– Infiltration Rate Sens itivity to Time Step (Zoom) Figures 12 and 13 above show that for dt<=1min, wetting front depth, i.e., cumulative infiltration, is not affected by time step. Figures 14 and 15 above, in addition to Table 4 below, reflect sensitivity of ponding time and infiltration rate to time step: for dt<=1 min., convergence is reached and curves are smooth. In contrast, Figures 10 and 11 above show that air pressure and infiltration rate are more sensitive to time step than other parameters. Therefore, the smaller the time step, the shorter the oscillation period and amplitude. These oscillations are in part due to the finite difference approach and in part can be considered as physically justifiabl e. In fact, air mass flux, which is inversely proportional to wetting front depth, is larg e at the beginning of the event when wetting front is not deep enough to reduce air flux.
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39 Table 4 Ponding Time Sensitiv ity to Time Step (MODGA) To minimize numerical erro rs finer time steps are needed. Oscillations occur before ponding, thus variable time steps can be used. Before ponding the model will run for a time step equal to 1/60 of the time step af ter it. For the rest of the simulations in this study, the model is run for a time step of 15 sec., the calculations before ponding are done at a time step of 0.25 sec consequently. Note: this dual time step is used to model infiltration while accounting for air phase; for original Green and Ampt approach, only one time step will be used for the entire simulation. Time Step 5 min. 3 min. 1 min. 30 sec. 15 sec. 9 sec. 6 sec. Ponding Time (min) 5 3 12.06 12.06 12.06 12.07 12.07
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40 CHAPTER 3: RESULTS AND DISCUSSION 3.1 Introduction This chapter, containing the results of different simulations run using MODGA, is divided into six sections in addition to the introduction. First section contains a comparison between the results of three infilt ration approaches: original Green and Ampt, a modified approach accounting for air co mpression but only, and MODGA. The five other sections underline MODGAÂ’s sensitivity to (1) depth to water tabl e, (2) initial water content, (3) rainfall intens ity, (4) saturated hyd raulic conductivity, and (5) soil type. 3.2 Comparison of Three Infilt ration Modeling Approaches Three approaches are compared: original Green and Ampt model that neglects air effect on infiltration, a model accounti ng for air compression only, and the MODGA approach described in Chapter 2. Comparis on including a descripti on of air pressure trends, propagation of the wetting fronts (cumulative infiltration), effect on ponding times, and infiltration rates obtained from the three models. This comparison is conducted for shallow and deep water table environments (SWT/DWT).
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41 3.2.1 Shallow Water Table Environment (SWT) The comparison of three inf iltration models will be conducted in a shallow water table environment (D = 0.5m). Other parameters are those used for reference simulation. 3.2.1.1 Modeling Air Pressure Ahead of the Wetting Front For the original Green and Ampt approach, air in the porous medium is assumed to remain at atmospheric pressure. The fi gures below represent a comparison of the pressure build up in the porous medium in case counterflow is either neglected or accounted for. (For the referen ce simulation described above). 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 020406080100120Time (min)Gage Pressure Head (m) Air Compression and Counterflow Air Compression Only Figure 16 Â– Air Pressure Ahead of the Wetting Front (SWT) ZOOM
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42 0.0000 0.0005 0.0010 0.0015 0.0020 0.000.020.040.060.080.100.120.140.160.180.20Time (min)Gage Pressure Head (m) Air Compression and Counterflow Figure 17 Â– Air Pressure Ahead of the Wetting Front (SWT) (Zoom) The red curve in Figure 16 shows a rapi d increase of pressu re head in case counterflow of air is neglect ed, which results in an early and complete blockage of infiltration when soil is not yet saturated. In contrast, when counter flow is accounted for, pressure increases with a lower gradient a llowing for soil to reach natural saturation before infiltration shutsoff completely (this will be discussed in details later in this chapter). The blue curve in Figure 16 can cl early be broken into two lines based on the slope: a high slope before ponding time (dashe d blue line) and a milder slope after ponding. The high slope is a result of high infiltration rate before ponding: the more infiltration, the higher the air space decrease, thus the higher the pressure increase. Even though pressure builds up at a lower gradient when counterflow is accounted for, it can reach a value higher than that reached when ai r counterflow is neglec ted: the blue curve is higher than the red by end of storm event. Therefore, even though pressure is higher,
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43 infiltration wonÂ’t stop because the wetting front is deeper and thus requires a higher pressure to block its further downward moveme nt. As discussed in Chapter 2, Figure 17 shows oscillations of pressure at the beginning of infiltration: for the first three seconds counterflow and air compression effects are of similar magnit ude in that air pressure keeps going back to atmospheric pressure. Once the wetting front is deep enough to reduce the counterflow, a linear trend starts to be seen. 3.2.1.2 Modeling Wetting Front Depth and Cumulative Infiltration The figures below represent wetting front depth and cumulative infiltration for the same physical parameters and storm event us ing three different modeling approaches. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 020406080100120Time (min)Wetting Front Depth (m) Air Compression and Counterflow Air Compression Only Green & Ampt Water Table Figure 18 Â– Wetting Front Depth (SWT)
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44 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 020406080100120Time (min)Cumulative Infiltration (cm Air Compression and Counterflow Air Compression Only Green & Ampt Figure 19 Â– Cumulative Infiltration (SWT) As discussed previously, high pressure built up ahead of the wetting front if counterflow of air is neglected will shut off infiltration and results in an underestimation of amount of water infiltrated. On the othe r hand, using the original Green and Ampt approach results in an underestimation of runoff by overestim ating infiltration. Accounting for both air compressi on and counterflow yields to different results of the infiltration process. Note that for a st orm long enough soil will never reach natural saturation if air compression onl y is accounted for, while wetting front reaches the water table if air effect was neglected, or if both air compression and counterflow were accounted for. Saturation time comparison of the three models for the reference simulation is represented in Tabl e 5 of the following section.
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45 3.2.1.3 Modeling the Infiltration Rate In this section the impact of air pressu re on ponding time and infiltration rate will be discussed. As seen in Equation 10, the higher the air pressure the faster the ponding. For the reference simulation, ponding time is documented in Table 5 and Figure 20 below. Table 5 Â– Air Effect on Ponding and Saturation Times (SWT) Air Compression & Counterflow Air Compression Only Green and Ampt Ponding Time (min) 12.03 2.64 30.27 Saturation Time (min) 647 Infiltration stops after 4.75 min. without saturating the soil. 309 0.00 0.50 1.00 1.50 2.00 2.50 3.00 020406080100120Time (min)Infiltration Rate (cm/hr ) Air Compression and Counterflow Air Compression Only Green & Ampt Saturated Conductivity Figure 20 Â– Infiltration Rate Modeling (SWT)
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46 After ponding, infiltration proceeds at capacity and infiltration capacity of soil at any point in time obtained from the MODGA approach is lo wer than that obtained from the Green and Ampt approach. Even though trends of infiltration rates of the two models are similar in the beginning, infiltrati on capacity drops below saturated hydraulic conductivity in the MODGA approach, while it a pproaches it asymptotically in the Green and Ampt. Figure 21 below represents infilt ration rate directly before saturation: infiltration capacity of soil dropped below saturated hydraulic conductivity due to air and oscillations at the end were due to high pressure built up then the air release as seen in Figure 22. 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 642643644645646647648Time (min)Infiltration Rate (cm/hr ) Air Compression and Counterflow Saturated Conductivity Figure 21 Â– Infiltration Ca pacity at Saturation
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47 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 642643644645646647648Time (min)Gage Pressure Head (m) Air Compression and Counterflow Figure 22 Â– Air Pressu re at Saturation 3.2.2 Deep Water Table Environment (DWT) Simulations above were repeated for e nvironments with deep water table, (D=10m, and D=100m). Conclusions are the sa me regarding the general comparison of the three models. Oscillations of pressure associated with MODGA do not exist in these environments since pressure increases are slower due to large pore space available. Results for a deep water table environment ar e shown in Figures 23 to 26 below. It is clear that the deeper the wate r table, the more similar are the results which is physically correct since for deeper water table pore space is available: For a 100m deep water table, the three models give approximately the same results, substantial differences are not seen in the figures below, yet for a 10m deep wate r table, it is important to account for air phase in the infiltration model.
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48 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 020406080100120Time (min)Gage Pressure Head (m) Air Compression and Counterflow (10m) Air Compression Only Air Compression and Counterflow (100m) Air Compression Only Figure 23 Â– Air Pressure Ahead of the Wetting Front (DWT) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 020406080100120Time (min)Wetting Front Depth (m) Air Compression and Counterflow (10m) Air Compression Only Air Compression and Counterflow (100m) Air Compression Only Green & Ampt Figure 24 Â– Wetting Front Depth (DWT)
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49 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 020406080100120Time (min)Cumulative Infiltration (cm Air Compression and Counterflow (10m) Air Compression Only Air Compression and Counterflow (100m) Air Compression Only Green & Ampt Figure 25 Â– Cumulative Infiltration (SWT) 0.00 0.50 1.00 1.50 2.00 2.50 3.00 020406080100120Time (min)Infiltration Rate (cm/hr ) Air Compression and Counterflow (10m) Air Compression Only Green & Ampt Saturated Conductivity Air Compression and Counterflow (100m) Air Compression Only Figure 26 Â– Infiltration Rate Modeling (DWT)
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50 In conclusion, accounting for both air compression and counterflow improves infiltration modeling: even for a water table as deep as 10m, the differences between the three models were substantial (for a relativel y high initial uniform soil moisture content, i = 33KPa). High sensitivity of the model to time step, as well as oscillations of air pressure at the beginning of the rainfall even t make it essential to minimize time steps for decoupling both effects. The remaining part of this chapter includes impacts of depth to water table, initial soil moisture content, rainfall intensity, and soil type on MODGA. 3.3 Impact of DepthtoWater Table In addition to the reference simulation, MODGA was run for a depthtowater table of 1, 3, 10 and 100m to assess the impact of water table depth on infiltration. Since all parameters of the reference simulation are kept the same, differences in results reflect the impact water table depth. Figures 27 to 30 and Table 6 show that the deeper the water table the less the air effect on infiltration. Yet, as seen in section 3.2.2 above, air affects the infiltration for a water table as deep as 10 meters (under same soil moisture conditions).
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51 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 020406080100120Time (min)Gage Pressure Head (m) 0.5m 1m 3m 10m 100m Figure 27 Â– Impact of Depthto Water Table on Air Pressure 0.00 0.05 0.10 0.15 0.20 0.25 020406080100120Time (min)Wetting Front Depth (m) 0.5m 1m 3m 10m 100m Figure 28 Â– Impact of Depthto Water Table on Wetting Front
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52 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 020406080100120Time (min)Cumulative Infiltration (cm 0.5m 1m 3m 10m 100m Figure 29 Â– Impact of DepthtoWat er Table on Cumulative Infiltration 0.00 0.50 1.00 1.50 2.00 2.50 3.00 020406080100120Time (min)Infiltration Rate (cm/hr ) 0.5m 1m 3m 10m 100m Saturated Conductivity Figure 30 Â– Impact of DepthtoWater Table on Cumulative Infiltration Rate
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53 Table 6 Â– Impact of Depthto Water Table on Ponding Time Results above show sensitivity of infiltr ation to depthtowater table in the MODGA model. Clearly, in sh allower water table ponding occu rs earlier and more runoff is generated: infiltration capacity is reduced and the air pressure gr adient is higher. For deeper water table, more water infiltrates into the soil. For very deep water tables (D=100m), air pressure is of no great influe nce and the regular Green and Ampt approach can be used as described in the previous s ection (air pressure le ss than 3cm after 2 hours of a 3cm/hr storm). 3.4 Impact of Initial Soil Moisture Content A major factor known to aff ect infiltration is anteceden t soil moisture content. A wet soil absorbs less water than a dry one. The effect of initial soil moisture content on infiltration using MODGA is represented below: results include a comparison between different soil moisture contents ( i = r = 0.065 (dry), i = 0.15, i =0.30 in addition to the reference simulation). Also, a simulation is run showing the case where soil is almost saturated before a storm. D (m) 0.5 1 3 10 100 Tp (min) 12.03 13.03 16.14 21.62 28.83
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54 0.00 0.05 0.10 0.15 0.20 0.25 020406080100120Time (min)Gage Pressure Head (m) i=0.207 i= r=0.065 i=0.15 i=0.30 Figure 31 Â– Impact of Initial Soil Moisture Content on Air Pressure 0.00 0.05 0.10 0.15 0.20 0.25 0.30 020406080100120Time (min)Wetting Front Depth (m) i=0.207 i= r=0.065 i=0.15 i=0.30 Figure 32 Â– Impact of Initial Water Content on Wetting Front
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55 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 020406080100120Time (min)Cumulative Infiltration (cm i=0.207 i= r=0.065 i=0.15 i=0.30 Figure 33 Â– Impact of Initial Wate r Content on Cumulative Infiltration 0.00 0.50 1.00 1.50 2.00 2.50 3.00 020406080100120Time (min)Infiltration Rate (cm/hr ) i=0.207 i= r=0.065 i=0.15 i=0.30 Saturated Conductivity Figure 34 Â– Impact of Initial Water Content on Infiltration Rate
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56 Table 7 Â– Impact of Initial Water Content on Ponding Time 0 5 10 15 20 25 30 35 40 0102030405060Time (min)Gage Pressure Head (m ) 0.0 0.5 1.0 1.5 2.0 2.5 3.0Infiltration Rate (cm/hr) Gage Air Pressure Infiltration Rate Figure 35 Â– Infiltration when Soil is Near Saturation Figure 31 above shows that the dryer the so il, the lower the air pressure. In fact, for a dry soil, pore space is larger and as pre ssure is inversely proportional to volume the result is physically correct. For wet soils, gradie nt of air pressure is higher since the same amount of infiltration results in a deeper wett ing front depth, i.e., higher reduction in air volume (Figure 32). Figure 33 shows that a dr yer soil absorbs more water than a wet one for a same storm event (~2.2cm vs. ~3.7cm). Ponding time and infiltration capacity sensitivity to initial soil mois ture content are represented in Figure 34 and Table 7 above: infiltration rate is higher for a dry soil and it takes more time to reach ponding. The case i 0.065 0.15 0.207 0.30 Tp (min) 21.85 15.97 12.03 5.63
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57 where soil is nearly saturated ( i=0.38~ s) is represented in Figur e 35 above: oscillations in air pressure are due the small air volume available, which causes abrupt increases in pressure and thus higher counterflow. 3.5 Impact of Rainfall Intensity As the source of infiltration is rainfall, checking the rainfall intensity effect on infiltration and air pressure is intuitive: Figures 36 to 39 below show this effect, and Table 8 shows variation of ponding time with rainfall intensity. 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 020406080100120Time (min)Gage Pressure Head (m) 3cm/hr 1.5cm/hr 5cm/hr 10cm/hr Figure 36 Â– Impact of Rainfall Intensity on Air Pressure
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58 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 020406080100120Time (min)Wetting Front Depth (m) 3cm/hr 1.5cm/hr 5cm/hr 10cm/hr Figure 37 Â– Impact of Rainfall Intensity on Wetting Front 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 020406080100120Time (min)Cumulative Infiltration (cm 3cm/hr 1.5cm/hr 5cm/hr 10cm/hr Figure 38 Â– Impact of Rainfall Inte nsity on Cumulative Infiltration
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59 0.00 2.00 4.00 6.00 8.00 10.00 0.0020.0040.0060.0080.00100.00120.00Time (min)Infiltration Rate (cm/hr ) 3cm/hr 1.5cm/hr 5cm/hr 10cm/hr Saturated Conductivity Figure 39 Â– Impact Rainfall Intensity on Infiltration Rate Table 8 Â– Impact of Rainfall Intensity on Ponding Time Figure 36 shows that air pressure is hi ghly dependant on rainfall rate before ponding. This is attributed to the fact that before ponding all the rain infiltrates and thus air compression is a function of rainfa ll intensity, while af ter ponding infiltration proceeds at soilÂ’s capacity and a higher rainfa ll intensity yields to a higher runoff but infiltration is not affected. This explains why after ponding air pressure curves are close regardless of rainfall intensity The greatest effect of rain fall intensity is on ponding time as it can be seen in the results above. After ponding infiltration capacity curves are also I (cm/hr) 1.5 3 5 10 Tp (min) 56.51 12.03 4.23 1.1
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60 similar. Of course, for low rainfall intensity (1.5cm/hr) the propagation of wetting front is slow, which allows for a larger counterfl ow and thus air pressure reduction. 3.6 Impact of Saturated Hydraulic Conductivity Soil conductivity is of criti cal importance as it directly affects infiltration and counterflow. Thus, the necessity to study the modelÂ’s sensitivity to this parameter, especially since the values us ed for conductivities can vary by more than an order of magnitude. Figures 40 to 43 and Table 9 belo w are the results obtai ned after running the model for Ks = 0.5, 0.8, 1.2, 1.5, and 2.0cm/hr. 0.00 0.05 0.10 0.15 0.20 0.25 020406080100120Time (min)Gage Pressure Head (m) Ks=1.0cm/hr Ks=0.5cm/hr Ks=0.8cm/hr Ks=1.2cm/hr Ks=1.5cm/hr Ks=2.0cm/hr Figure 40 Â– Impact of Saturated Hydr aulic Conductivity on Air Pressure
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61 0.00 0.05 0.10 0.15 0.20 0.25 020406080100120Time (min)Wetting Front Depth (m) Ks=1.0cm/hr Ks=0.5cm/hr Ks=0.8cm/hr Ks=1.2cm/hr Ks=1.5cm/hr Ks=2.0cm/hr Figure 41 Â– Impact of Saturated Hydr aulic Conductivity on Wetting Front 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 020406080100120Time (min)Cumulative Infiltration (cm Ks=1.0cm/hr Ks=0.5cm/hr Ks=0.8cm/hr Ks=1.2cm/hr Ks=1.5cm/hr Ks=2.0cm/hr Figure 42 Â– Impact of Saturated Hydraulic Conductivity on Cumulative Infiltration
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62 0.00 0.50 1.00 1.50 2.00 2.50 3.00 020406080100120Time (min)Infiltration Rate (cm/hr ) Ks=1.0cm/hr Ks=0.5cm/hr Ks=0.8cm/hr Ks=1.2cm/hr Ks=1.5cm/hr Ks=2.0cm/hr Figure 43 Â– Impact of Saturated Hydrau lic Conductivity on Infiltration Rate Table 9 Â– Impact of Saturated H ydraulic Conductivity on Ponding Time Figure 40 shows the effect of saturate d hydraulic conductivity, i.e., intrinsic permeability, on air compression. Air pressure behavior can be divided into two categories: before and afte r ponding time. Before ponding, at any point of time, the higher the hydraulic conductivity the lower the air pressure. In fact, for any time we have similar amount of infiltration thus same air volume change. But a higher intrinsic permeability (proportional to hydraulic conduc tivity) yields to a higher counterflow, which explains the lower air pressure fo r a higher conductivity before ponding. After ponding, conductivity has an opposite effect: th e higher the saturate d conductivity, the Ks (cm/hr) 0.5 0.8 1.0 1.2 1.5 2.0 Tp (min) 5.89 9.45 12.03 14.79 19.35 28.26
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63 higher the air pressure. In fact, a higher c onductivity means more in filtration is allowed into the soil. Of course, counterflow is sti ll higher also, but since counterflow proceeds at a low relative permeability, the effect of infiltration is of greater importance. This effect can also be seen in Figures 41 to 43, whic h show that a higher infiltration (cumulative infiltration and infiltration capacity) is associated w ith a higher conductivity. 3.7 Impact of Soil Type A rainfall rate of 3cm/hr and a 0.5m de pthtowater table are maintained during the simulations while all soil properties are ch anged (Table 10). All soils are considered dry for this set of simulations. Three soil types will be compared: Sandy Loam, Loam, and Clay Loam. Table 10 below includes the so ilsÂ’ physical properties. Results are shown in Figures 44 to 47 and Table 11 below. Table 10 Â– Parameters Used for the Impact of Soil Type Simulation Soil type n s r i Ks (cm/hr)Hb(m)Hc(m) kra Sandy Loam 0.41 0.39 0.0650.0652.18 0.13 0.165 0.0170.89 Loam 0.43 0.40 0.0780.0781.32 0.28 0.384 0.0280.56 Clay Loam 0.41 0.39 0.0950.0950.20 0.53 0.805 0.0120.31
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64 0.00 0.10 0.20 0.30 0.40 0.50 0.60 020406080100120Time (min)Gage Pressure Head (m) Sandy Loam Loam Clay Loam Figure 44 Â– Impact of So il Type on Air Pressure 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 020406080100120Time (min)Wetting Front Depth (m) Sandy Loam Loam Clay Loam Figure 45 Â– Impact of Soil Type on Wetting Front
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65 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 020406080100120Time (min)Cumulative Infiltration (cm Sandy Loam Loam Clay Loam Figure 46 Â– Impact of Soil T ype on Cumulative Infiltration 0.00 0.50 1.00 1.50 2.00 2.50 3.00 020406080100120Time (min)Infiltration Rate (cm/hr ) Sandy Loam Loam Clay Loam Figure 47 Â– Impact of Soil Type on Infiltration Rate
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66 Table 11 Â– Impact of Soil Type on Ponding Time Results obtained in this section are in fact a combination of several physical parameters at a time. First, results here unde rline the combined eff ect of conductivity and relative permeability: in fact, Figure 44 shows th at air pressure for Sandy Loam (S.L) and Loam (L.) are comparable even though we have large differences in hydraulic conductivity; while air pressure for Clay Loam (C.L.) is higher. This can be interpreted by looking at the combined value of conduc tivity and relative permeability of air: (Ks.kra)(S.L.) (Ks.kra)(L.) 0.037 > (Ks.kra)(C.L.) = 0.0024 Â– the higher the ratio, the higher the counterflow, and the lower the air pressu re. Of course, the amount of available pore space is an influencing factor too, but in th is case we almost have similar initial air volumes. Even though saturated hydrau lic conductivity of Sandy Loam is almost twice that of Loam, total infiltration into Loam is the highest. This can be explained by ponding occurring later for Loam resul ting in a lower infiltration in to Sandy Loam between for the duration between the two ponding times. As for the ponding time difference, itÂ’s a combination of Ks, Hc and Ha that resulted in having an earlier ponding for Sandy Loam: air pressure is almost the same for both soils, and the higher hydraulic conductivity results in a later ponding but a lower sucti on head results in an earlier ponding. Soil Type Sandy Loam Loam Clay Loam Tp (min) 58.65 88.29 18.68
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67 CHAPTER 4: CONCLUSION 4.1 Comparison of Three Infiltration Approaches An objective of this study is to underline the effect of air on infiltration using a simple model. For this, the original Green and Ampt model that does not account for air phase was compared to two other models: on e accounting for air compression only, and one accounting for air compression and counterflow (MODGA). To account for air effect, only one term was added to the Green a nd Ampt model: air pre ssure in the porous medium. For the air compression only model, air pressure was calcu lated using BoyleÂ’s law, considering that air mass remains the same during the infiltration process (counterflow = 0), whereas for MODGA counterflow was accounted for using Equation16 to calculate the mass flux of air a nd the perfect gas law to find air pressure. First, the models were compared for sha llow water table environments (0.5m). If counterflow is neglected, ai r pressure builds up quickly in soil to a point where infiltration is shutoff without saturating th e soil. As for the Green and Ampt model, ponding occurs later and infiltration rate decr eases with time reaching saturated hydraulic conductivity asymptotically. Whereas for th e MODGA approach, ponding occurs earlier than Green and Ampt ponding time yet not as dramatically as for air compression only, and infiltration rate drops below saturated hyd raulic conductivity. Li ke in the original Green and Ampt model, the soil reaches natura l saturation, but it takes a longer time for
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68 that since the infiltration rate is overestimat ed in the regular Green and Ampt model. For deep water table environments, two cases we re explored: D = 10m, and 100m. Results showed that the air effect on infiltration is substantial for water tables as deep as 10m. While for D = 100m, we can simply use the Gr een and Ampt approach and the minor air effect on infiltration can be neglected. Theref ore, for areas where the water table is very deep, using a model that accounts for the ai r phase is contraindicated, while it is essential in environments with shallow impervious layers. 4.2 MODGA Sensitivity to Different Parameters Remainder of the results derived in this research represents sensitivity of MODGA to different parameters. First, for deeper water tables, infiltration, i.e., infiltration capacity, increases because of re duction in air pressure. Initial soil moisture content is of significance on in filtration as well: a dryer soil absorbs more water and the dryer the soil the larger the air volume a nd the less the air compression. For general applications, a more practical step is to lump these two parameters into a single dimensionless parameter, space volume over water table depth, especially if the model is modified to account for variable initial soil moisture with depth. This can be the subject for future research. On the other hand, the rain fall rateÂ’s effect is almost restricted to ponding time only. While soil conductivity has a major effect, since it influences both infiltration and counterflow: a higher hydr aulic conductivity results in a higher counterflow, which explains the slower pressure build up before ponding. After ponding the soil conductivity limits infiltration and the higher the conductivity, the more infiltration and air compression. Also, a dimensi onless parameter to lump rainfall rate and
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69 soilÂ’s conductivity can be the subject of further research. Dimensionless analysis, i.e., lumping different parameters together, might result in the possibility of defining a sharp threshold on whether to account to air or not. Finally, three simulations with three different soils were run to see a combined effect of all soilÂ’s parameters: air bubbling pressure also has a major role on infiltration, as well as the relative permeability of the air pore size index.
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70 REFERENCES Bouwer, H., (1966). Â“Rapid Field Measurem ents of Air Entry Value and Hydraulic Conductivity of Soil as Significant Parame ters in Flow System Analysis.Â” Water Resources Research, 2, 729738. Bras, R., (1990). Â“Flow in Unsaturate d Porous Media and Infiltration.Â” Hydrology: An Introduction to Hydrologic Cycle, AddisonWesley Publishing Company, 349393. Charbeneau, R., (2000). Groundwater Hydraulics and Pollutant Transport, PrenticeHall inc., 1890. Chow, V., Maidment, D., Mays, L. (1988). Â“Subsurface Water.Â” Applied Hydrology, McGrawHill inc., 99126. Constantz, J., Herkelrath, W., and Mur phy, F. (1988). Â“Air Encapsulation During Infiltration.Â” Soil Science Society of America Journal, 52(1), 1016. Culligan, P., Barry, D., Parlange, J., Steenhuis, T., and Haverkamp, R. (2000). Â“Infiltration with Controlled Air Escape.Â” Water Resources Research, 36(3), 781785. Dunne, T. (1983). Â“Relation of Field Studies and Modeling in the Prediction of Storm Runoff.Â” Journal of Hydrology, 65(13), 2548. Faybishenko, B. (1995). Â“Hydrau lic Behavior of QuasiSatu rated Soils in Presence of Entrapped Air: Laboratory Experiments.Â” Water Resources Research, 31(10), 24211435. Fayer, M., and Hillel, D. (1986a). Â“Air Encapsulation: I. Measurement in a Field Soil.Â” Soil Science Society of America Journal, 50, 568572. Fayer, M., and Hillel, D. (1986b). Â“Air Encapsulation: II. Profile Water Storage and Shallow Water Table Fluctuations.Â” Soil Science Society of America Journal, 50, 572577. Freeze, R., and Cherry, J. (2000). Â“Gr oundwater and the Hydrologic Cycle.Â” Groundwater, PrenticeHall inc., 192236.
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71 Freeze, R., (1980). Â“A Stochastic Conceptual Analysis of RainfallRunoff Process on a Hillslope.Â” Water Resources Research, 16(2), 395. Hillel, D., (1998). Â“Entry of water into soil.Â” Environmental Soil Physics, Academic Press, 385426. Hillel, D., and Gardner, W. (1970). Â“Transient Infiltration into CrustTopped Profiles.Â” Soil Science, 109(2), 6976. Horton, R. (1939). Â“Analysis of Runoff Plot Experiments with Varying Infiltration Capacity.Â” Transactions, American Geophysical Union, Part IV, 693711. Loague, K., and Abrams, R. (2001). Â“Stochasti cConceptual Analysis of NearSurface Hydrological Response.Â” Hydrological Processes, 15(14), 27152728. MorelSeytoux, H., and Khanji, J. (1974). Â“D erivation of an Equa tion of Infiltration.Â” Water Resources Research, 10(4), 795800. MorelSeytoux, H., and Khanji, J. (1975). Â“E quation of Infiltration with Compression and Counterflow Effect.Â” Hydrological Science Bulletin, 20(4), 505517. Nachabe, M., and Illangasekare, T. (1994). Â“Use of Tension Inf iltrometer Data with Unsaturated Hydraulic Conductivity Models.Â” Ground Water, 32(6), 10171021. Touma, J., Vachaud, G., and Parlange, J.Y. (1984). Â“Air and Water Flow in a Sealed, Ponded Vertical Soil Column: Experiment and Model.Â” Soil Science, 137(3), 181187. Vachaud, G., Gaudet, J.P., and Kuraz, V. (1974). Â“Air and Water Flow During Ponded Infiltration in a Vertical Bounded Column of Soil.Â” Journal of Hydrology, 22, 89108. Vomacka, J., Thompson, D., Ross, M., Nach abe, M., and Tara, P. (2002). Â“Measurement of Surficial Aquifer Recharge, ET, Rainfall, Runoff and StreamAquifer Interaction Characteristics of the Centra l and Southern Region of the Southwest Florida Water Management District.Â” A Field Project Providing Hydrologic Data Collection and Analysis. Center for Modeling Hydrologic and Aquatic Systems. Department of Civil and Environmental Engi neering. University of South Florida. Wang, Z., Feyen, J., Nielsen, D., and va n Genuchten, M. (1997). Â“Twophase Flow Infiltration equations accounting for air entrapment effects.Â” Water Resources Research, 33(12), 27592767. Wang, Z., Feyen, J., van Genuchten, M., and Ni elson, D. (1998). Â“Air Entrapment Effects on Infiltration Rate and Flow Instability.Â” Water Resources Research, 34(2), 213222.
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72 Wangemann, S., Kohl, R., and Molumeli, P. (2000). Â“Infiltration and Percolation Influenced by Antecedent Soil Water Content and Air Entrapment.Â” Transactions of the American Society of Agricultural Engineers, 43(6), 15171523. Wilson, B., Slack, D., and Young, R. (1982) Â“A Comparison of Three Infiltration Models.Â” Transactions of the American Socie ty of Agricultural Engineers, 25(2), 349356.
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73 APPENDICES
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74 Appendix A. MODGA Program med with Visual Basic Figure 48 Â– MODGAÂ’s Graphical Interface Option Explicit 'Variables Defined in the Graphical Interface: 'Txti: Rainfall intensity (cm/hr) 'TxtTetai: Initial Water Content (%) 'Txtn: Porosity (%) 'txtTetar: Residual Water Content (%) 'txtTetas: Saturated Water Content (%) 'Txtdt: Time Step (min) 'TxtD: Depth to Water Table (m) 'TxtKs: Saturated Hydraulic Conductivity (cm/hr) 'TxtHb: Bubling Pressure (m) 'TxtSt: Storm Duration (hr) 'TxtSoilType: Soil Type 'TxtLambda: Brooks and Corey Pore Size Distribution Index 'OpCompCount: Option Button for Compression and Counterflow Calculations 'OpComp: Option Butt on for Compression while Neglecting Counterflow Calculations 'OpNoAir: Option Button for the Option Neglecting the Air Phase in the Calculations 'Perfect Gas Constant for air: Const R = 286.9 'm2/(s2.K) (or J/Kg.K). (P=Rho*R*T) 'Standard Atmospheric Pressure: Const Patm = 101000# 'N/m2 (Pa) 'Gravitationnal acceleration: Const g = 9.807 'm/s2 'Temperature in Kelvine (20C) Const Tk = 293# 'K 'Physical properties of air at st andard atmospheric pressure and temperature = 20C Const GammaAir = 11.81 'N/m3 Const RhoAir = 1.204 'Kg/m3 Const MuAir = 1.82 10 ^ (5) 'Ns/m2 Const NuAir = 1.51 10 ^ (5) 'm2/s 'Physical properties of water at temperature = 20C Const GammaWater = 9789# 'N/m3 Const RhoWater = 998.2 'Kg/m3 Const MuWater = 1.002 10 ^ (3) 'Ns/m2 Const NuWater = 1.004 10 ^ (6) 'm2/s 'SubtimeStep Const c = 60# Private Sub CmdExit_Click() End End Sub Private Sub CmdRun_Click() 'Definition of the different Variables Dim t As Single 'time in minutes Dim Tp As Single 'Ponding time in minutes Dim CapTeta As Double 'Dimensionless Water Content = (TetaTetar)/(PorosityTetar) Dim Kra As Double 'Relative Permeability of air Dim k As Double 'Intrinsic permeability(cm2)= Saturated Hydraulic Conductivity x MuWater/GammaWater Dim Fb, Fe As Double 'Total infiltration at Beginning and End of a time step (cm) Dim Re As Double 'Excess Rainfall (cm/hr) Dim Pp, Hp As Double 'Pressure (pa) and Pressure Head (m of water) of the air phase in porous medium Dim Pptmp As Double 'Temporary variable use in the pressure iterations
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75 Appendix A. (Continued)Dim Hatm As Double 'Standa rd Atmospheric Pressure Head (m) Dim H As Double 'Pressu re Head difference between air inside and outside porous medium (m of water) Dim Hc As Double 'Suction Head (m) = (2+3Lambda)*Hb/(1+3Lambda) Dim L As Double 'Wetting Front Depth (m) Dim V As Double 'Volume of air in porous medium for a unit area (m3) Dim m As Double 'Mass of air in porous medium (Kg) Dim Mf As Double 'Mass Flux of air from porous medium (Kg/s) Dim fc As Double 'Infiltration Capacity of the soil (cm/hr) Dim S, Sa As Double 'S=Hc(TxtTetastetai); Sa=H(TxtTetastetai) Dim count As Integer 'Counter Dim Msg As String 'Message Box 'Creating an Excel File Dim oXL As Object 'Excel application Dim oBook As Object 'Excel workbook Dim oSheet As Object 'Excel Worksheet Dim Table() 'Table of values to be stored Dim iRow As Long 'Index variable for the current Row Dim nRow As Long 'Number of Rows in the table Dim nCol As Integer 'Number of Columns in the table 'Start Excel and create a new workbook Set oXL = CreateObject("Excel.application") Set oBook = oXL.Workbooks.Add Set oSheet = oBook.Worksheets.Item(1) 'Define the table iRow = 1 nRow = 65536 nCol = 13 ReDim Table(1 To nRow, 1 To nCol) If OpCompCount Then 'Infiltra tion While Accounting for Air Compression and CounterFlow Table(iRow, 1) = "Infiltr ation While Accounting for Air Compression and CounterFlow" iRow = iRow + 1 Table(iRow, 1) = "Time (min)" Table(iRow, 2) = "Rainfall (cm/hr)" Table(iRow, 3) = "Air Volume (m)" Table(iRow, 4) = "Air Mass (Kg)" Table(iRow, 5) = "Air Mass Flux (Kg/hr)" Table(iRow, 6) = "Absolute Air Pressure Head (m)" Table(iRow, 7) = "Gage Air Pressure Head (m)" Table(iRow, 8) = "Weeting Front Depth (m)" Table(iRow, 9) = "Total Infiltration (cm)" Table(iRow, 10) = "Incremental Infiltration (cm)" Table(iRow, 11) = "Inf iltration Capacity (cm/hr)" Table(iRow, 12) = "Excess Rainfall (cm/hr)" Table(iRow, 13) = "Comments" iRow = iRow + 1 'Initial Values t = 0 count = 0 Table(iRow, 1) = t / 60 Table(iRow, 2) = "" Table(iRow, 5) = "" Table(iRow, 10) = "" Table(iRow, 11) = "" Table(iRow, 12) = "" CapTeta = (TxtTetas TxtTetar) / (Txtn TxtTetar) Kra = (1 CapTeta ^ 2) (1 CapTeta ^ (1 + 2 TxtLambda)) k = TxtKs MuWater / (GammaWater 100 3600 10 ^ (4)) Hc = (2 + 3 TxtLambda) TxtHb / (1 + 3 TxtLambda) Fb = 0 Table(iRow, 9) = Fb Fe = 0 Re = 0 Pp = Patm Hp = Pp / GammaWater Table(iRow, 6) = Hp Hatm = Patm / GammaWater H = Hp Hatm Table(iRow, 7) = H L = 0.000001 Table(iRow, 8) = L V = ((Txtn TxtTetai) / 100) TxtD Table(iRow, 3) = V m = Pp V / (R Tk) Table(iRow, 4) = m S = Hc (TxtTetas TxtTetai) / 100 Sa = H (TxtTetas TxtTetai) / 100 iRow = iRow + 1 'Estimation of ponding time Tp = ((S Sa) 100 TxtKs / (Txti (Txti TxtKs))) 3600 While (Tp >= t + Txtdt / c And t < TxtSt 3600 And L < TxtD) count = count + 1 If count >= c Then count = 0 End If t = t + Txtdt / c Table(iRow, 2) = Txti Table(iRow, 1) = t / 60 Mf = k 10 ^ (4) Kra RhoAir (Pp ^ 2 Patm ^ 2) / (MuAir Patm 2 L) Table(iRow, 5) = Mf m = m Mf Txtdt / c Fe = Fb + Txti Txtdt / (3600 c) Table(iRow, 10) = Fe Fb L = (Fe / 100) / ((TxtTetas TxtTetai) / 100) Table(iRow, 8) = L V = ((Txtn TxtTetai) / 100) (TxtD L) Table(iRow, 3) = V Pp = m R Tk / V If (Pp < Patm) Then Pp = Patm m = Pp V / (R Tk) End If Table(iRow, 4) = m Hp = Pp / GammaWater Table(iRow, 6) = Hp H = Hp Hatm Table(iRow, 7) = H Sa = H (TxtTetas TxtTetai) / 100 Tp = ((S Sa) 100 TxtKs / (Txti (Txti TxtKs))) 3600 If (Tp < t) Then Tp = t End If
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76 Appendix A. (Continued) Fb = Fe Table(iRow, 9) = Fb Table(iRow, 11) = "" Table(iRow, 12) = 0 iRow = iRow + 1 Wend If (L < TxtD) Then If (t < TxtSt 3600) Then 'Infiltration up to ponding time If (Tp = t) Then 'Step to prevent having the ponding time appear twice in the table Table(iRow 1, 11) = Txti Table(iRow 1, 13) = "Ponding Time" Else Table(iRow, 1) = Tp / 60 Table(iRow, 2) = Txti Mf = k 10 ^ (4) Kra RhoAir (Pp ^ 2 Patm ^ 2) / (MuAir Patm 2 L) Table(iRow, 5) = Mf m = m Mf (Tp t) Fe = Fb + Txti (Tp t) / 3600 Table(iRow, 10) = Fe Fb L = (Fe / 100) / ((TxtTetas TxtTetai) / 100) Table(iRow, 8) = L V = ((Txtn TxtTetai) / 100) (TxtD L) Table(iRow, 3) = V Pp = m R Tk / V If (Pp < Patm) Then Pp = Patm m = Pp V / (R Tk) End If Table(iRow, 4) = m Hp = Pp / GammaWater Table(iRow, 6) = Hp H = Hp Hatm Table(iRow, 7) = H Sa = H (TxtTetas TxtTetai) / 100 Fb = Fe Table(iRow, 9) = Fb Table(iRow, 11) = Txti Table(iRow, 12) = 0 Table(iRow, 13) = "Ponding Time" iRow = iRow + 1 End If 'Calculations for the rest of the time step corresponding to ponding time 'FeFb(SSa)Ln[(Fe+SSa)/(Fb+SSa)]=Ks*dt If (L < TxtD) Then t = t + Txtdt (c count) / c Table(iRow, 1) = t / 60 Table(iRow, 2) = Txti Mf = k 10 ^ (4) Kra RhoAir (Pp ^ 2 Patm ^ 2) / (MuAir Patm 2 L) Table(iRow, 5) = Mf m = m Mf (t Tp) While (Fe Fb (S Sa) 100 Log((Fe + (S Sa) 100) / (Fb + (S Sa) 100)) < TxtKs (t Tp) / 3600 And (Fe Fb) < Txti (t Tp) / 3600) Fe = Fe + TxtKs (t Tp) / (3600# 1000#) Wend If ((Fe Fb) > Txti (t Tp) / 3600) Then Fe = Fb + Txti (t Tp) / 3600 End If If ((Fe Fb) <= TxtKs (t Tp) / (3600# 100#)) Then Fe = Fb End If Table(iRow, 10) = Fe Fb fc = ((Fe Fb) / (t Tp)) 3600 Table(iRow, 11) = fc Re = Txti fc Table(iRow, 12) = Re L = (Fe / 100) / ((TxtTetas TxtTetai) / 100) Table(iRow, 8) = L V = ((Txtn TxtTetai) / 100) (TxtD L) Table(iRow, 3) = V Pp = m R Tk / V If (Pp < Patm) Then Pp = Patm m = Pp V / (R Tk) End If Table(iRow, 4) = m Hp = Pp / GammaWater Table(iRow, 6) = Hp H = Hp Hatm Table(iRow, 7) = H Sa = H (TxtTetas TxtTetai) / 100 Fb = Fe Table(iRow, 9) = Fb t = t + Txtdt iRow = iRow + 1 Else Table(iRow 1, 13) = "Ponding Time and Soil Fully Saturated" iRow = iRow + 1 Table(iRow + 1, 1) = "Soil Type: Table(iRow + 1, 6) = TxtSoilType Table(iRow + 2, 1) = "Porosity: Table(iRow + 2, 6) = Txtn Table(iRow + 3, 1) = "Saturated Water Content: Table(iRow + 3, 6) = TxtTetas Table(iRow + 4, 1) = "Residual Water Content: Table(iRow + 4, 6) = TxtTetar Table(iRow + 5, 1) = "Initial Water Content: Table(iRow + 5, 6) = TxtTetai Table(iRow + 6, 1) = "Saturated Hydraulic Conductivity (cm/hr): Table(iRow + 6, 6) = TxtKs Table(iRow + 7, 1) = "Bubbling Pressure (m): Table(iRow + 7, 6) = TxtHb Table(iRow + 8, 1) = "Suction Head (m): Table(iRow + 8, 6) = Hc Table(iRow + 9, 1) = "Pore Size Distribution Index: Table(iRow + 9, 6) = TxtLambda Table(iRow + 10, 1) = "Depth to Water Table (m): Table(iRow + 10, 6) = TxtD Table(iRow + 11, 1) = "Relative Permeability of Air: Table(iRow + 11, 6) = Kra 'Export Results to Excel oSheet.Range("A1").Resize(nRow, nCol).Value = Table() 'Make Excel Visible oXL.Visible = True oXL.UserControl = True Exit Sub End If Else
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77 Appendix A. (Continued) Table(iRow 1, 13) = "Storm ends without ponding the soil" iRow = iRow + 1 Table(iRow + 1, 1) = "Soil Type: Table(iRow + 1, 6) = TxtSoilType Table(iRow + 2, 1) = "Porosity: Table(iRow + 2, 6) = Txtn Table(iRow + 3, 1) = "Saturated Water Content: Table(iRow + 3, 6) = TxtTetas Table(iRow + 4, 1) = "Residual Water Content: Table(iRow + 4, 6) = TxtTetar Table(iRow + 5, 1) = "Initial Water Content: Table(iRow + 5, 6) = TxtTetai Table(iRow + 6, 1) = "Saturated Hydraulic Conductivity (cm/hr): Table(iRow + 6, 6) = TxtKs Table(iRow + 7, 1) = "Bubbling Pressure (m): Table(iRow + 7, 6) = TxtHb Table(iRow + 8, 1) = "Suction Head (m): Table(iRow + 8, 6) = Hc Table(iRow + 9, 1) = "Pore Size Distribution Index: Table(iRow + 9, 6) = TxtLambda Table(iRow + 10, 1) = "Depth to Water Table (m): Table(iRow + 10, 6) = TxtD Table(iRow + 11, 1) = "Relative Permeability of Air: Table(iRow + 11, 6) = Kra 'Export Results to Excel oSheet.Range("A1").Resize(nRow, nCol).Value = Table() 'Make Excel Visible oXL.Visible = True oXL.UserControl = True Exit Sub End If Else Table(iRow 1, 13) = "Soil Fully Saturated" iRow = iRow + 1 Table(iRow + 1, 1) = "Soil Type: Table(iRow + 1, 6) = TxtSoilType Table(iRow + 2, 1) = "Porosity: Table(iRow + 2, 6) = Txtn Table(iRow + 3, 1) = "Saturated Water Content: Table(iRow + 3, 6) = TxtTetas Table(iRow + 4, 1) = "Residual Water Content: Table(iRow + 4, 6) = TxtTetar Table(iRow + 5, 1) = "Initial Water Content: Table(iRow + 5, 6) = TxtTetai Table(iRow + 6, 1) = "Saturated Hydraulic Conductivity (cm/hr): Table(iRow + 6, 6) = TxtKs Table(iRow + 7, 1) = "Bubbling Pressure (m): Table(iRow + 7, 6) = TxtHb Table(iRow + 8, 1) = "Suction Head (m): Table(iRow + 8, 6) = Hc Table(iRow + 9, 1) = "Pore Size Distribution Index: Table(iRow + 9, 6) = TxtLambda Table(iRow + 10, 1) = "Depth to Water Table (m): Table(iRow + 10, 6) = TxtD Table(iRow + 11, 1) = "Relative Permeability of Air: Table(iRow + 11, 6) = Kra 'Export Results to Excel oSheet.Range("A1").Re size(nRow, nCol).Value = Table() 'Make Excel Visible oXL.Visible = True oXL.UserControl = True Exit Sub End If 'Calculations of in filtration at soil capacity While (t <= TxtSt 3600) If (L < TxtD) Then Table(iRow, 1) = t / 60 Table(iRow, 2) = Txti Mf = k 10 ^ (4) Kra RhoAir (Pp ^ 2 Patm ^ 2) / (MuAir Patm 2 L) Table(iRow, 5) = Mf m = m Mf Txtdt If (Sa 100 < Fb + S 100) Then While (Fe Fb (S Sa) 100 Log((Fe + (S Sa) 100) / (Fb + (S Sa) 100)) < TxtKs Txtdt / 3600 And (Fe Fb) < Txti Txtdt / 3600) If (fc = 0) Then Fe = Fe + TxtKs Txtdt / (3600# 1000#) Else Fe = Fe + fc Txtdt / (3600# 1000#) End If Wend If ((Fe Fb) > Txti Txtdt / 3600) Then Fe = Fb + Txti Txtdt / 3600 End If If ((Fe Fb) <= TxtKs Txtdt / (3600# 100#)) Then Fe = Fb End If Else Fe = Fb End If Table(iRow, 10) = Fe Fb fc = ((Fe Fb) / Txtdt) 3600 Table(iRow, 11) = fc Re = Txti fc Table(iRow, 12) = Re L = (Fe / 100) / ((TxtTetas TxtTetai) / 100) Table(iRow, 8) = L V = ((Txtn TxtTetai) / 100) (TxtD L) Table(iRow, 3) = V Pp = m R Tk / V If (Pp < Patm) Then Pp = Patm m = Pp V / (R Tk) End If Table(iRow, 4) = m Hp = Pp / GammaWater Table(iRow, 6) = Hp H = Hp Hatm Table(iRow, 7) = H Sa = H (TxtTetas TxtTetai) / 100 Fb = Fe Table(iRow, 9) = Fb t = t + Txtdt iRow = iRow + 1 Else Table(iRow 1, 13) = "Soil is fully saturated" iRow = iRow + 1 Table(iRow + 1, 1) = "Soil Type: Table(iRow + 1, 6) = TxtSoilType Table(iRow + 2, 1) = "Porosity: Table(iRow + 2, 6) = Txtn Table(iRow + 3, 1) = "Saturated Water Content: Table(iRow + 3, 6) = TxtTetas Table(iRow + 4, 1) = "Residual Water Content: Table(iRow + 4, 6) = TxtTetar
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78 Appendix A. (Continued) Table(iRow + 5, 1) = "Initial Water Content: Table(iRow + 5, 6) = TxtTetai Table(iRow + 6, 1) = "Saturated Hydraulic Conductivity (cm/hr): Table(iRow + 6, 6) = TxtKs Table(iRow + 7, 1) = "Bubbling Pressure (m): Table(iRow + 7, 6) = TxtHb Table(iRow + 8, 1) = "Suction Head (m): Table(iRow + 8, 6) = Hc Table(iRow + 9, 1) = "Pore Size Distribution Index: Table(iRow + 9, 6) = TxtLambda Table(iRow + 10, 1) = "Depth to Water Table (m): Table(iRow + 10, 6) = TxtD Table(iRow + 11, 1) = "Relative Permeability of Air: Table(iRow + 11, 6) = Kra 'Export Results to Excel oSheet.Range("A1").Resize(nRow, nCol).Value = Table() 'Make Excel Visible oXL.Visible = True oXL.UserControl = True Exit Sub End If Wend iRow = iRow + 1 Table(iRow + 1, 1) = "Soil Type: Table(iRow + 1, 6) = TxtSoilType Table(iRow + 2, 1) = "Porosity: Table(iRow + 2, 6) = Txtn Table(iRow + 3, 1) = "Saturated Water Content: Table(iRow + 3, 6) = TxtTetas Table(iRow + 4, 1) = "Residual Water Content: Table(iRow + 4, 6) = TxtTetar Table(iRow + 5, 1) = "Initial Water Content: Table(iRow + 5, 6) = TxtTetai Table(iRow + 6, 1) = "Saturated Hydraulic Conductivity (cm/hr): Table(iRow + 6, 6) = TxtKs Table(iRow + 7, 1) = "Bubbling Pressure (m): Table(iRow + 7, 6) = TxtHb Table(iRow + 8, 1) = "Suction Head (m): Table(iRow + 8, 6) = Hc Table(iRow + 9, 1) = "Pore Size Distribution Index: Table(iRow + 9, 6) = TxtLambda Table(iRow + 10, 1) = "Depth to Water Table (m): Table(iRow + 10, 6) = TxtD Table(iRow + 11, 1) = "Relative Permeability of Air: Table(iRow + 11, 6) = Kra 'Export Results to Excel oSheet.Range("A1").Resize (nRow, nCol).Value = Table() 'Make Excel Visible oXL.Visible = True oXL.UserControl = True ElseIf OpComp Then 'Infiltra tion While Accounting for Air Compression but Neglecting the CounterFlow Table(iRow, 1) = "Infiltr ation While Accounting for Air Compression Only" iRow = iRow + 1 Table(iRow, 1) = "Time (min)" Table(iRow, 2) = "Rainfall (cm/hr)" Table(iRow, 3) = "Air Volume (m)" Table(iRow, 4) = "Air Mass (Kg)" Table(iRow, 5) = "Air Mass Flux (Kg/hr)" Table(iRow, 6) = "Absolute Air Pressure Head (m)" Table(iRow, 7) = "Gage Air Pressure Head (m)" Table(iRow, 8) = "Weeting Front Depth (m)" Table(iRow, 9) = "Total Infiltration (cm)" Table(iRow, 10) = "Incremental Infiltration (cm)" Table(iRow, 11) = "Inf iltration Capacity (cm/hr)" Table(iRow, 12) = "Excess Rainfall (cm/hr)" Table(iRow, 13) = "Comments" iRow = iRow + 1 'Initial Values t = 0 count = 0 Table(iRow, 1) = t / 60 Table(iRow, 2) = "" Table(iRow, 10) = "" Table(iRow, 11) = "" Table(iRow, 12) = "" CapTeta = (TxtTetas TxtTetar) / (Txtn TxtTetar) Kra = (1 CapTeta ^ 2) (1 CapTeta ^ (1 + 2 TxtLambda)) k = TxtKs MuWater / (GammaWater 100 3600 10 ^ (4)) Hc = (2 + 3 TxtLambda) TxtHb / (1 + 3 TxtLambda) Fb = 0 Table(iRow, 9) = Fb Fe = 0 Re = 0 Pp = Patm Hp = Pp / GammaWater Table(iRow, 6) = Hp Hatm = Patm / GammaWater H = Hp Hatm Table(iRow, 7) = H L = 0.00001 Table(iRow, 8) = 0 V = ((Txtn TxtTetai) / 100) TxtD Table(iRow, 3) = V m = Pp V / (R Tk) Table(iRow, 4) = m S = Hc (TxtTetas TxtTetai) / 100 Sa = H (TxtTetas TxtTetai) / 100 iRow = iRow + 1 'Estimation of ponding time Tp = ((S Sa) 100 TxtKs / (Txti (Txti TxtKs))) 3600 While (Tp >= t + Txtdt / c And t < TxtSt 3600 And L < TxtD) count = count + 1 If count >= c Then count = 0 End If t = t + Txtdt / c Table(iRow, 2) = Txti Table(iRow, 1) = t / 60 Fe = Fb + Txti Txtdt / (3600 c) Table(iRow, 10) = Fe Fb L = (Fe / 100) / ((TxtTetas TxtTetai) / 100) Table(iRow, 8) = L V = ((Txtn TxtTetai) / 100) (TxtD L) Table(iRow, 3) = V Pp = m R Tk / V Hp = Pp / GammaWater Table(iRow, 6) = Hp H = Hp Hatm Table(iRow, 7) = H
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79 Appendix A. (Continued) Sa = H (TxtTetas TxtTetai) / 100 Tp = ((S Sa) 100 TxtKs / (Txti (Txti TxtKs))) 3600 If (Tp < t) Then Tp = t End If Fb = Fe Table(iRow, 9) = Fb Table(iRow, 11) = "" Table(iRow, 12) = 0 iRow = iRow + 1 Wend If (L < TxtD) Then If (t < TxtSt 3600) Then 'Infiltration up to ponding time If (Tp = t) Then 'Step to prevent having the ponding time appear twice in the table Table(iRow 1, 11) = Txti Table(iRow 1, 13) = "Ponding Time" Else Table(iRow, 1) = Tp / 60 Table(iRow, 2) = Txti Fe = Fb + Txti (Tp t) / 3600 Table(iRow, 10) = Fe Fb L = (Fe / 100) / ((TxtTetas TxtTetai) / 100) Table(iRow, 8) = L V = ((Txtn TxtTetai) / 100) (TxtD L) Table(iRow, 3) = V Pp = m R Tk / V Hp = Pp / GammaWater Table(iRow, 6) = Hp H = Hp Hatm Table(iRow, 7) = H Sa = H (TxtTetas TxtTetai) / 100 Fb = Fe Table(iRow, 9) = Fb Table(iRow, 11) = Txti Table(iRow, 12) = 0 Table(iRow, 13) = "Ponding Time" iRow = iRow + 1 End If 'Calculations for the rest of the time step corresponding to ponding time 'FeFb(SSa)Ln[(Fe+SSa)/(Fb+SSa)]=Ks*dt If (L < TxtD) Then t = t + Txtdt (c count) / c Table(iRow, 1) = t / 60 Table(iRow, 2) = Txti While (Fe Fb (S Sa) 100 Log((Fe + (S Sa) 100) / (Fb + (S Sa) 100)) < TxtKs (t Tp) / 3600 And (Fe Fb) < Txti (t Tp) / 3600) Fe = Fe + TxtKs (t Tp) / (3600# 1000#) Wend If Fe Fb <= TxtKs (t Tp) / (3600# 100#) Then Fe = Fb End If Table(iRow, 10) = Fe Fb fc = ((Fe Fb) / (t Tp)) 3600 Table(iRow, 11) = fc Re = Txti fc Table(iRow, 12) = Re L = (Fe / 100) / ((TxtTetas TxtTetai) / 100) Table(iRow, 8) = L V = ((Txtn TxtTetai) / 100) (TxtD L) Table(iRow, 3) = V Pp = m R Tk / V Hp = Pp / GammaWater Table(iRow, 6) = Hp H = Hp Hatm Table(iRow, 7) = H Sa = H (TxtTetas TxtTetai) / 100 Fb = Fe Table(iRow, 9) = Fb t = t + Txtdt iRow = iRow + 1 Else Table(iRow 1, 13) = "Ponding Time and Soil Fully Saturated" iRow = iRow + 1 Table(iRow + 1, 1) = "Soil Type: Table(iRow + 1, 6) = TxtSoilType Table(iRow + 2, 1) = "Porosity: Table(iRow + 2, 6) = Txtn Table(iRow + 3, 1) = "Saturated Water Content: Table(iRow + 3, 6) = TxtTetas Table(iRow + 4, 1) = "Residual Water Content: Table(iRow + 4, 6) = TxtTetar Table(iRow + 5, 1) = "Initial Water Content: Table(iRow + 5, 6) = TxtTetai Table(iRow + 6, 1) = "Saturated Hydraulic Conductivity (cm/hr): Table(iRow + 6, 6) = TxtKs Table(iRow + 7, 1) = "Bubbling Pressure (m): Table(iRow + 7, 6) = TxtHb Table(iRow + 8, 1) = "Suction Head (m): Table(iRow + 8, 6) = Hc Table(iRow + 9, 1) = "Pore Size Distribution Index: Table(iRow + 9, 6) = TxtLambda Table(iRow + 10, 1) = "Depth to Water Table (m): Table(iRow + 10, 6) = TxtD Table(iRow + 11, 1) = "Relative Permeability of Air: Table(iRow + 11, 6) = Kra 'Export Results to Excel oSheet.Range("A1").Resize(nRow, nCol).Value = Table() 'Make Excel Visible oXL.Visible = True oXL.UserControl = True Exit Sub End If Else Table(iRow 1, 13) = "Storm ends without ponding the soil" iRow = iRow + 1 Table(iRow + 1, 1) = "Soil Type: Table(iRow + 1, 6) = TxtSoilType Table(iRow + 2, 1) = "Porosity: Table(iRow + 2, 6) = Txtn Table(iRow + 3, 1) = "Saturated Water Content: Table(iRow + 3, 6) = TxtTetas Table(iRow + 4, 1) = "Residual Water Content: Table(iRow + 4, 6) = TxtTetar Table(iRow + 5, 1) = "Initial Water Content: Table(iRow + 5, 6) = TxtTetai Table(iRow + 6, 1) = "Saturated Hydraulic Conductivity (cm/hr): Table(iRow + 6, 6) = TxtKs Table(iRow + 7, 1) = "Bubbling Pressure (m): Table(iRow + 7, 6) = TxtHb
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80 Appendix A. (Continued) Table(iRow + 8, 1) = "Suction Head (m): Table(iRow + 8, 6) = Hc Table(iRow + 9, 1) = "Pore Size Distribution Index: Table(iRow + 9, 6) = TxtLambda Table(iRow + 10, 1) = "Depth to Water Table (m): Table(iRow + 10, 6) = TxtD Table(iRow + 11, 1) = "Relative Permeability of Air: Table(iRow + 11, 6) = Kra 'Export Results to Excel oSheet.Range("A1").Resize(nRow, nCol).Value = Table() 'Make Excel Visible oXL.Visible = True oXL.UserControl = True Exit Sub End If Else Table(iRow 1, 13) = "Soil Fully Saturated" iRow = iRow + 1 Table(iRow + 1, 1) = "Soil Type: Table(iRow + 1, 6) = TxtSoilType Table(iRow + 2, 1) = "Porosity: Table(iRow + 2, 6) = Txtn Table(iRow + 3, 1) = "Saturated Water Content: Table(iRow + 3, 6) = TxtTetas Table(iRow + 4, 1) = "Residual Water Content: Table(iRow + 4, 6) = TxtTetar Table(iRow + 5, 1) = "Initial Water Content: Table(iRow + 5, 6) = TxtTetai Table(iRow + 6, 1) = "Saturated Hydraulic Conductivity (cm/hr): Table(iRow + 6, 6) = TxtKs Table(iRow + 7, 1) = "Bubbling Pressure (m): Table(iRow + 7, 6) = TxtHb Table(iRow + 8, 1) = "Suction Head (m): Table(iRow + 8, 6) = Hc Table(iRow + 9, 1) = "Pore Size Distribution Index: Table(iRow + 9, 6) = TxtLambda Table(iRow + 10, 1) = "Depth to Water Table (m): Table(iRow + 10, 6) = TxtD Table(iRow + 11, 1) = "Relative Permeability of Air: Table(iRow + 11, 6) = Kra 'Export Results to Excel oSheet.Range("A1").Re size(nRow, nCol).Value = Table() 'Make Excel Visible oXL.Visible = True oXL.UserControl = True Exit Sub End If 'Calculations of in filtration at soil capacity While (t <= TxtSt 3600) If (L < TxtD) Then Table(iRow, 1) = t / 60 Table(iRow, 2) = Txti If (Sa 100 < Fb + S 100) Then While (Fe Fb (S Sa) 100 Log((Fe + (S Sa) 100) / (Fb + (S Sa) 100)) < TxtKs Txtdt / 3600 And (Fe Fb) < Txti Txtdt / 3600) If (fc = 0) Then Fe = Fe + TxtKs Txtdt / (3600# 1000#) Else Fe = Fe + fc Txtdt / (3600# 1000#) End If Wend If Fe Fb <= TxtKs Txtdt / (3600# 100#) Then Fe = Fb End If Else Fe = Fb End If Table(iRow, 10) = Fe Fb fc = ((Fe Fb) / Txtdt) 3600 Table(iRow, 11) = fc Re = Txti fc Table(iRow, 12) = Re L = (Fe / 100) / ((TxtTetas TxtTetai) / 100) Table(iRow, 8) = L V = ((Txtn TxtTetai) / 100) (TxtD L) Table(iRow, 3) = V Pp = m R Tk / V Hp = Pp / GammaWater Table(iRow, 6) = Hp H = Hp Hatm Table(iRow, 7) = H Sa = H (TxtTetas TxtTetai) / 100 Fb = Fe Table(iRow, 9) = Fb t = t + Txtdt iRow = iRow + 1 Else Table(iRow 1, 13) = "Soil is fully saturated" iRow = iRow + 1 Table(iRow + 1, 1) = "Soil Type: Table(iRow + 1, 6) = TxtSoilType Table(iRow + 2, 1) = "Porosity: Table(iRow + 2, 6) = Txtn Table(iRow + 3, 1) = "Saturated Water Content: Table(iRow + 3, 6) = TxtTetas Table(iRow + 4, 1) = "Residual Water Content: Table(iRow + 4, 6) = TxtTetar Table(iRow + 5, 1) = "Initial Water Content: Table(iRow + 5, 6) = TxtTetai Table(iRow + 6, 1) = "Saturated Hydraulic Conductivity (cm/hr): Table(iRow + 6, 6) = TxtKs Table(iRow + 7, 1) = "Bubbling Pressure (m): Table(iRow + 7, 6) = TxtHb Table(iRow + 8, 1) = "Suction Head (m): Table(iRow + 8, 6) = Hc Table(iRow + 9, 1) = "Pore Size Distribution Index: Table(iRow + 9, 6) = TxtLambda Table(iRow + 10, 1) = "Depth to Water Table (m): Table(iRow + 10, 6) = TxtD Table(iRow + 11, 1) = "Relative Permeability of Air: Table(iRow + 11, 6) = Kra 'Export Results to Excel oSheet.Range("A1").Resize(nRow, nCol).Value = Table() 'Make Excel Visible oXL.Visible = True oXL.UserControl = True Exit Sub End If Wend iRow = iRow + 1 Table(iRow + 1, 1) = "Soil Type: Table(iRow + 1, 6) = TxtSoilType Table(iRow + 2, 1) = "Porosity: Table(iRow + 2, 6) = Txtn Table(iRow + 3, 1) = "Saturated Water Content:
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81 Appendix A. (Continued) Table(iRow + 3, 6) = TxtTetas Table(iRow + 4, 1) = "Residual Water Content: Table(iRow + 4, 6) = TxtTetar Table(iRow + 5, 1) = "Initial Water Content: Table(iRow + 5, 6) = TxtTetai Table(iRow + 6, 1) = "Saturated Hydraulic Conductivity (cm/hr): Table(iRow + 6, 6) = TxtKs Table(iRow + 7, 1) = "Bubbling Pressure (m): Table(iRow + 7, 6) = TxtHb Table(iRow + 8, 1) = "Suction Head (m): Table(iRow + 8, 6) = Hc Table(iRow + 9, 1) = "Pore Size Distribution Index: Table(iRow + 9, 6) = TxtLambda Table(iRow + 10, 1) = "Depth to Water Table (m): Table(iRow + 10, 6) = TxtD Table(iRow + 11, 1) = "Relative Permeability of Air: Table(iRow + 11, 6) = Kra 'Export Results to Excel oSheet.Range("A1").Resize (nRow, nCol).Value = Table() 'Make Excel Visible oXL.Visible = True oXL.UserControl = True ElseIf OpNoAir Then 'Infiltra tion While Neglecting the Air Phase in the Porous Media Table(iRow, 1) = "Infiltration While Neglecting the Air Phase" iRow = iRow + 1 Table(iRow, 1) = "Time (min)" Table(iRow, 2) = "Rainfall (cm/hr)" Table(iRow, 3) = "Air Volume (m)" Table(iRow, 4) = "Air Mass (Kg)" Table(iRow, 5) = "Air Mass Flux (Kg/hr)" Table(iRow, 6) = "Absolute Air Pressure Head (m)" Table(iRow, 7) = "Gage Air Pressure Head (m)" Table(iRow, 8) = "Weeting Front Depth (m)" Table(iRow, 9) = "Total Infiltration (cm)" Table(iRow, 10) = "Incremental Infiltration (cm)" Table(iRow, 11) = "Inf iltration Capacity (cm/hr)" Table(iRow, 12) = "Excess Rainfall (cm/hr)" Table(iRow, 13) = "Comments" iRow = iRow + 1 'Initial Values t = 0 Table(iRow, 1) = t / 60 Table(iRow, 2) = "" Table(iRow, 10) = "" Table(iRow, 11) = "" Table(iRow, 12) = "" CapTeta = (TxtTetas TxtTetar) / (Txtn TxtTetar) Kra = (1 CapTeta ^ 2) (1 CapTeta ^ (1 + 2 TxtLambda)) k = TxtKs MuWater / (GammaWater 100 3600 10 ^ (4)) Hc = (2 + 3 TxtLambda) TxtHb / (1 + 3 TxtLambda) Fb = 0 Table(iRow, 9) = Fb Fe = 0 Re = 0 Pp = Patm Hp = Pp / GammaWater Table(iRow, 6) = Hp Hatm = Patm / GammaWater H = Hp Hatm Table(iRow, 7) = H L = 0.00001 Table(iRow, 8) = 0 V = ((Txtn TxtTetai) / 100) TxtD Table(iRow, 3) = V Table(iRow, 4) = m S = Hc (TxtTetas TxtTetai) / 100 iRow = iRow + 1 'Estimation of ponding time Tp = (S 100 TxtKs / (Txti (Txti TxtKs))) 3600 While (Tp >= t + Txtdt And t < TxtSt 3600 And L < TxtD) t = t + Txtdt Table(iRow, 2) = Txti Table(iRow, 1) = t / 60 Fe = Fb + Txti Txtdt / 3600 Table(iRow, 10) = Fe Fb L = (Fe / 100) / ((TxtTetas TxtTetai) / 100) Table(iRow, 8) = L V = ((Txtn TxtTetai) / 100) (TxtD L) Table(iRow, 3) = V Tp = (S 100 TxtKs / (Txti (Txti TxtKs))) 3600 Fb = Fe Table(iRow, 9) = Fb Table(iRow, 11) = "" Table(iRow, 12) = 0 iRow = iRow + 1 Wend If (L < TxtD) Then If (t < TxtSt 3600) Then 'Infiltration up to ponding time If (Tp = t) Then 'Step to prevent having the ponding time appear twice in the table Table(iRow 1, 11) = Txti Table(iRow 1, 13) = "Ponding Time" Else Table(iRow, 1) = Tp / 60 Table(iRow, 2) = Txti Fe = Fb + Txti (Tp t) / 3600 Table(iRow, 10) = Fe Fb L = (Fe / 100) / ((TxtTetas TxtTetai) / 100) Table(iRow, 8) = L V = ((Txtn TxtTetai) / 100) (TxtD L) Table(iRow, 3) = V Fb = Fe Table(iRow, 9) = Fb Table(iRow, 11) = Txti Table(iRow, 12) = 0 Table(iRow, 13) = "Ponding Time" iRow = iRow + 1 End If 'Calculations for the rest of the time step corresponding to ponding time 'FeFb(SSa)Ln[(Fe+SSa)/(Fb+SSa)]=Ks*dt If (L < TxtD) Then t = t + Txtdt Table(iRow, 1) = t / 60 Table(iRow, 2) = Txti While (Fe Fb S 100 Log((Fe + S 100) / (Fb + S 100)) < TxtKs (t Tp) / 3600 And (Fe Fb) < Txti (t Tp) / 3600) Fe = Fe + TxtKs (t Tp) / (3600# 1000#) Wend
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82 Appendix A. (Continued) If Fe Fb <= TxtKs (t Tp) / (3600# 100#) Then Fe = Fb End If Table(iRow, 10) = Fe Fb fc = ((Fe Fb) / (t Tp)) 3600 Table(iRow, 11) = fc Re = Txti fc Table(iRow, 12) = Re L = (Fe / 100) / ((TxtTetas TxtTetai) / 100) Table(iRow, 8) = L V = ((Txtn TxtTetai) / 100) (TxtD L) Table(iRow, 3) = V Fb = Fe Table(iRow, 9) = Fb t = t + Txtdt iRow = iRow + 1 Else Table(iRow 1, 13) = "Ponding Time and Soil Fully Saturated" iRow = iRow + 1 Table(iRow + 1, 1) = "Soil Type: Table(iRow + 1, 6) = TxtSoilType Table(iRow + 2, 1) = "Porosity: Table(iRow + 2, 6) = Txtn Table(iRow + 3, 1) = "Saturated Water Content: Table(iRow + 3, 6) = TxtTetas Table(iRow + 4, 1) = "Residual Water Content: Table(iRow + 4, 6) = TxtTetar Table(iRow + 5, 1) = "Initial Water Content: Table(iRow + 5, 6) = TxtTetai Table(iRow + 6, 1) = "Saturated Hydraulic Conductivity (cm/hr): Table(iRow + 6, 6) = TxtKs Table(iRow + 7, 1) = "Bubbling Pressure (m): Table(iRow + 7, 6) = TxtHb Table(iRow + 8, 1) = "Suction Head (m): Table(iRow + 8, 6) = Hc Table(iRow + 9, 1) = "Pore Size Distribution Index: Table(iRow + 9, 6) = TxtLambda Table(iRow + 10, 1) = "Depth to Water Table (m): Table(iRow + 10, 6) = TxtD Table(iRow + 11, 1) = "Relative Permeability of Air: Table(iRow + 11, 6) = Kra 'Export Results to Excel oSheet.Range("A1").Resize(nRow, nCol).Value = Table() 'Make Excel Visible oXL.Visible = True oXL.UserControl = True Exit Sub End If Else Table(iRow 1, 13) = "Storm ends without ponding the soil" iRow = iRow + 1 Table(iRow + 1, 1) = "Soil Type: Table(iRow + 1, 6) = TxtSoilType Table(iRow + 2, 1) = "Porosity: Table(iRow + 2, 6) = Txtn Table(iRow + 3, 1) = "Saturated Water Content: Table(iRow + 3, 6) = TxtTetas Table(iRow + 4, 1) = "Residual Water Content: Table(iRow + 4, 6) = TxtTetar Table(iRow + 5, 1) = "Initial Water Content: Table(iRow + 5, 6) = TxtTetai Table(iRow + 6, 1) = "Saturated Hydraulic Conductivity (cm/hr): Table(iRow + 6, 6) = TxtKs Table(iRow + 7, 1) = "Bubbling Pressure (m): Table(iRow + 7, 6) = TxtHb Table(iRow + 8, 1) = "Suction Head (m): Table(iRow + 8, 6) = Hc Table(iRow + 9, 1) = "Pore Size Distribution Index: Table(iRow + 9, 6) = TxtLambda Table(iRow + 10, 1) = "Depth to Water Table (m): Table(iRow + 10, 6) = TxtD Table(iRow + 11, 1) = "Relative Permeability of Air: Table(iRow + 11, 6) = Kra 'Export Results to Excel oSheet.Range("A1").Resize(nRow, nCol).Value = Table() 'Make Excel Visible oXL.Visible = True oXL.UserControl = True Exit Sub End If Else Table(iRow 1, 13) = "Soil Fully Saturated" iRow = iRow + 1 Table(iRow + 1, 1) = "Soil Type: Table(iRow + 1, 6) = TxtSoilType Table(iRow + 2, 1) = "Porosity: Table(iRow + 2, 6) = Txtn Table(iRow + 3, 1) = "Saturated Water Content: Table(iRow + 3, 6) = TxtTetas Table(iRow + 4, 1) = "Residual Water Content: Table(iRow + 4, 6) = TxtTetar Table(iRow + 5, 1) = "Initial Water Content: Table(iRow + 5, 6) = TxtTetai Table(iRow + 6, 1) = "Saturated Hydraulic Conductivity (cm/hr): Table(iRow + 6, 6) = TxtKs Table(iRow + 7, 1) = "Bubbling Pressure (m): Table(iRow + 7, 6) = TxtHb Table(iRow + 8, 1) = "Suction Head (m): Table(iRow + 8, 6) = Hc Table(iRow + 9, 1) = "Pore Size Distribution Index: Table(iRow + 9, 6) = TxtLambda Table(iRow + 10, 1) = "Depth to Water Table (m): Table(iRow + 10, 6) = TxtD Table(iRow + 11, 1) = "Relative Permeability of Air: Table(iRow + 11, 6) = Kra 'Export Results to Excel oSheet.Range("A1").Re size(nRow, nCol).Value = Table() 'Make Excel Visible oXL.Visible = True oXL.UserControl = True Exit Sub End If 'Calculations of in filtration at soil capacity While (t <= TxtSt 3600) If (L < TxtD) Then If (Sa < Fb + S) Then Table(iRow, 1) = t / 60 Table(iRow, 2) = Txti While (Fe Fb S 100 Log((Fe + S 100) / (Fb + S 100)) < TxtKs Txtdt / 3600 And (Fe Fb) < Txti Txtdt / 3600) If (fc = 0) Then Fe = Fe + TxtKs Txtdt / (3600# 1000#)
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83 Appendix A. (Continued) Else Fe = Fe + fc Txtdt / (3600# 1000#) End If Wend If Fe Fb <= TxtKs Txtdt / (3600# 100#) Then Fe = Fb End If Table(iRow, 10) = Fe Fb fc = ((Fe Fb) / Txtdt) 3600 Table(iRow, 11) = fc Re = Txti fc Table(iRow, 12) = Re L = (Fe / 100) / ((TxtTetas TxtTetai) / 100) Table(iRow, 8) = L V = ((Txtn TxtTetai) / 100) (TxtD L) Table(iRow, 3) = V Fb = Fe Table(iRow, 9) = Fb Else Table(iRow, 1) = t / 60 Table(iRow, 2) = Txti Fe = Fb Table(iRow, 10) = Fe Fb fc = ((Fe Fb) / Txtdt) 3600 Table(iRow, 11) = fc Re = Txti fc Table(iRow, 12) = Re L = (Fe / 100) / ((TxtTetas TxtTetai) / 100) Table(iRow, 8) = L V = ((Txtn TxtTetai) / 100) (TxtD L) Table(iRow, 3) = V Fb = Fe End If t = t + Txtdt iRow = iRow + 1 Else Table(iRow 1, 13) = "Soil is fully saturated" iRow = iRow + 1 Table(iRow + 1, 1) = "Soil Type: Table(iRow + 1, 6) = TxtSoilType Table(iRow + 2, 1) = "Porosity: Table(iRow + 2, 6) = Txtn Table(iRow + 3, 1) = "Saturated Water Content: Table(iRow + 3, 6) = TxtTetas Table(iRow + 4, 1) = "Residual Water Content: Table(iRow + 4, 6) = TxtTetar Table(iRow + 5, 1) = "Initial Water Content: Table(iRow + 5, 6) = TxtTetai Table(iRow + 6, 1) = "Saturated Hydraulic Conductivity (cm/hr): Table(iRow + 6, 6) = TxtKs Table(iRow + 7, 1) = "Bubbling Pressure (m): Table(iRow + 7, 6) = TxtHb Table(iRow + 8, 1) = "Suction Head (m): Table(iRow + 8, 6) = Hc Table(iRow + 9, 1) = "Pore Size Distribution Index: Table(iRow + 9, 6) = TxtLambda Table(iRow + 10, 1) = "Depth to Water Table (m): Table(iRow + 10, 6) = TxtD Table(iRow + 11, 1) = "Relative Permeability of Air: Table(iRow + 11, 6) = Kra 'Export Results to Excel oSheet.Range("A1").Resize(nRow, nCol).Value = Table() 'Make Excel Visible oXL.Visible = True oXL.UserControl = True Exit Sub End If Wend iRow = iRow + 1 Table(iRow + 1, 1) = "Soil Type: Table(iRow + 1, 6) = TxtSoilType Table(iRow + 2, 1) = "Porosity: Table(iRow + 2, 6) = Txtn Table(iRow + 3, 1) = "Saturated Water Content: Table(iRow + 3, 6) = TxtTetas Table(iRow + 4, 1) = "Residual Water Content: Table(iRow + 4, 6) = TxtTetar Table(iRow + 5, 1) = "Initial Water Content: Table(iRow + 5, 6) = TxtTetai Table(iRow + 6, 1) = "Saturated Hydraulic Conductivity (cm/hr): Table(iRow + 6, 6) = TxtKs Table(iRow + 7, 1) = "Bubbling Pressure (m): Table(iRow + 7, 6) = TxtHb Table(iRow + 8, 1) = "Suction Head (m): Table(iRow + 8, 6) = Hc Table(iRow + 9, 1) = "Pore Size Distribution Index: Table(iRow + 9, 6) = TxtLambda Table(iRow + 10, 1) = "Depth to Water Table (m): Table(iRow + 10, 6) = TxtD Table(iRow + 11, 1) = "Relative Permeability of Air: Table(iRow + 11, 6) = Kra 'Export Results to Excel oSheet.Range("A1").Resize (nRow, nCol).Value = Table() 'Make Excel Visible oXL.Visible = True oXL.UserControl = True Else Msg = MsgBox("Choose a Model to Calculate Infiltration", vbExclamation) End If End Sub
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84 Appendix B. Air and Water Physical Properties at 20 C Table 12 Â– Air and Water P hysical Properties at 20 C Specific Weight (N/m3) Density (Kg/m3) Dynamic Viscosity (Ns/m2) Kinematic Viscosity (m2/s) Air 11.81 1.204 1.82.105 1.51. 105 Water 9789 998.2 1.002.103 1.004.106 i. Perfect Gas Constant for Air: R = 286.9 J/Kg.K ii. Standard Atmospheric Pressure: Patm = 101,000 N/m2 (Pa) iii. Gravitationnal Acceleration: g = 9.807 m/s2
