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Performance prediction, state estimation and production optimization of a landfill
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Performance prediction, state estimation and production optimization of a landfill
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PERFORMANCE PREDICTION, STATE ESTIMATION AND PRODUCTION OPTIMIZATION OF A LANDFILL by Hu Li A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (CHEMICAL ENGINEERING) December 2012 Copyright 2012 Hu Li ii Dedication To my wife Yun Fang and my family. iii Acknowledgments First and foremost, I sincerely thank my advisors, Prof. Joe Qin, Prof. Muhammad Sahimi and Prof. Theodore T. Tsotsis. Without their guidance, encouragement, inspiration and support, this work could not be done. They brought me to the area of practical applications of advanced control theories to a landfill, an exciting and challenging research field. During my Ph.D. journey, they helped me to develop analytical and technical skills, which are desirable skills for my Ph.D. study and also my future career. Furthermore, I would also like to thank to Prof. Michael Safonov, another member of my dissertation committee, for his valuable advice and guidance. During my stay at USC, many other people also helped me to solve technical issues and inspired me to develop new ideas. I want to thank Dr. Raudel Sanchez, for his unconditional assistance to get me familiar with landfills. When I came to USC 4 years ago, I knew nothing about landfills and it is Dr. Sanchez who taught me the knowledge of landfills and details of the landfill simulator he developed before. The collaboration with Dr. Sanchez was really enjoyable and we finished one journal paper about ap- plication of Artificial Neural Network to landfill’s performance predictions together. I am also grateful to Zaoshi, Kyle and Dr. Rui Wang, who are ex- iv perts on Linux and HPCC system. Whenever I had problems in this aspect, they were always willing to help me out. I would also like to thank every- one in Dr. Qin’s research group, including Carlos, Zhijie, Yingying, Jingran, Yining, Johnny, Tao and Yu, for the valuable friendship and support. Finally, I want to express my deepest appreciation to my parents, Xinji Li and Lanling Song, and my wife Yun Fang, for their continuous un- derstanding and encouragement to my life and study. I am most grateful to my wife, who is always supporting and inspiring me in the past three years. v Table of Contents Dedication ii Acknowledgments iii List of Figures viii Abstract xii Chapter 1. Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Utilization of LFG . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Modeling of Landfill Gas Systems . . . . . . . . . . . . . . . . . 7 1.4 Landfill as a Heterogeneous System . . . . . . . . . . . . . . . . 14 1.5 Optimization of LFG Production . . . . . . . . . . . . . . . . . . 18 1.6 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Chapter 2. Use of Artificial Neural Network and the Genetic Algo- rithm for Short- and Long-Term Forecasting and Planning 26 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2 Model of The Artificial Neural Network . . . . . . . . . . . . . 32 2.3 Short-Term Forecasting Using The ANN . . . . . . . . . . . . . 39 2.3.1 The Landfill Data . . . . . . . . . . . . . . . . . . . . . . . 40 2.3.2 Results and Discussions . . . . . . . . . . . . . . . . . . . 42 2.3.2.1 Forecasting Methane Production . . . . . . . . . 42 2.3.2.2 Forecasting Carbon Dioxide Production . . . . . 44 2.3.2.3 Forecasting Oxygen Profile . . . . . . . . . . . . 45 2.3.2.4 Forecasting Temperature Profile . . . . . . . . . 46 2.4 Long-Term Forecasting Using The ANN and The Genetic Al- gorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.4.1 Model of The Genetic Algorithm . . . . . . . . . . . . . . 50 2.4.2 Generation of Synthetic Data . . . . . . . . . . . . . . . . 52 vi 2.4.3 Training ANN . . . . . . . . . . . . . . . . . . . . . . . . 56 2.4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . 56 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Chapter 3. Dynamic Updating of The Landfill Model Using The En- semble Kalman Filter 76 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.2 Generation of Synthetic Ensemble of Data . . . . . . . . . . . . 83 3.3 Initial Permeability Estimates . . . . . . . . . . . . . . . . . . . 86 3.4 Dynamic Updating: Ensemble Kalman Filter . . . . . . . . . . 88 3.4.1 Algorithm Description . . . . . . . . . . . . . . . . . . . . 88 3.4.2 Implementation of The EnKF . . . . . . . . . . . . . . . . 91 3.4.2.1 Low Rank Representation of Covariance Matrix 91 3.4.2.2 Unphysical Permeabilities . . . . . . . . . . . . . 93 3.4.2.3 Number of Ensembles . . . . . . . . . . . . . . . 93 3.4.2.4 Parallel Computing . . . . . . . . . . . . . . . . . 94 3.5 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . 95 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Chapter 4. Model-Based Production Optimization of A Landfill Gas System 113 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.2 Optimization Problem Formulation . . . . . . . . . . . . . . . . 120 4.3 Production Optimization Without Constraints . . . . . . . . . . 123 4.3.1 Ensemble Based Optimization Methods . . . . . . . . . . 123 4.3.2 A Simplified EnOpt . . . . . . . . . . . . . . . . . . . . . 125 4.3.3 Step Size Selection . . . . . . . . . . . . . . . . . . . . . . 126 4.3.4 Parallel Computing . . . . . . . . . . . . . . . . . . . . . 128 4.3.5 Implementation of The EnOpt . . . . . . . . . . . . . . . 129 4.3.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . 130 4.4 Production Optimization with Constraints . . . . . . . . . . . . 132 4.4.1 Techniques to Solve Constrained Optimization . . . . . 132 4.4.2 Parameterless GA . . . . . . . . . . . . . . . . . . . . . . 134 4.4.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . 135 vii 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Chapter 5. Conclusions 149 Bibliography 152 Appendix A. 162 viii List of Figures Figure 1.1 MSW generation rate from 1960 to 2010. (Adapted from [3]) 24 Figure 1.2 Comparison of time-dependence of the total pressures at two depths. The results shown are for the heterogeneous model with (HwD) and without (HwoD) mechanical dis- persion; the semi-heterogeneous model with (SHwD) and without (SHwoD) mechanical dispersion, and the homo- geneous model with (HowD) and without (HowoD) me- chanical dispersion. The results are for a landfill with one well. (Adapted from [72]) . . . . . . . . . . . . . . . . . . . . 25 Figure 2.1 The structure of the ANN used in simulations. . . . . . . . 61 Figure 2.2 The ANN iterative procedure. . . . . . . . . . . . . . . . . . 62 Figure 2.3 Variations of the CH 4 performance function during the iteration for its global minimization. . . . . . . . . . . . . . . 63 Figure 2.4 Comparison of the ANN predictions for the volume frac- tion of CH 4 with the data for the, (left) trained; (center) validated, and (right) tested parts of the data set. . . . . . . 64 Figure 2.5 Same as in Fig. 2.3, but for CO 2 . . . . . . . . . . . . . . . . . 65 Figure 2.6 Same as in Fig. 2.4, but for CO 2 . . . . . . . . . . . . . . . . . 66 Figure 2.7 Same as in Fig. 2.3, but for O 2 . . . . . . . . . . . . . . . . . . 67 Figure 2.8 Same as in Fig. 2.4, but for O 2 . . . . . . . . . . . . . . . . . . 68 Figure 2.9 Same as in Fig. 2.3, but for temperature. . . . . . . . . . . . 69 ix Figure 2.10 Same as in Fig. 2.4, but for temperature. . . . . . . . . . . . 70 Figure 2.11 Spatial distribution of the original reference permeability in a horizontal layer of depth 14 m. . . . . . . . . . . . . . . 71 Figure 2.12 Top view of the grid structure used in the computations and the locations of the wells. . . . . . . . . . . . . . . . . . 72 Figure 2.13 Comparison of the ANN predictions with the actual (syn- thetic) data for the time period (a) 0.5 - 0.7 year and (b) 0.7 - 1.0 year. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Figure 2.14 Comparison of the predicted CH 4 concentrations with the actual (synthetic data) determined after the permeability distribution was computed (a) by the GA alone, and (b) by the combination of GA and ANN methods. The results are for the period 0.7 - 1.0 year. . . . . . . . . . . . . . . . . . . . 74 Figure 2.15 Same as in Fig. 2.14, but year 2. . . . . . . . . . . . . . . . . 75 Figure 3.1 3D view of the grid structure, used in the computations. The locations of the four extraction wells are shown by the circles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Figure 3.2 3D view of the grid structure and the reference permeabil- ity distribution. The permeabilities are in md. The location of the extraction wells are shown by the circles. . . . . . . . 100 Figure 3.3 The reference permeability distribution in a horizontal layer at a depth of 15 m. The permeabilities are in md. . . . 101 Figure 3.4 Four realizations of the reference permeability distribu- tion in the same horizontal layer as in Fig. 3.3, generated from the same initial data. The permeabilities are in md. . . 102 Figure 3.5 Evolution of objective function with the iterations. . . . . . 103 x Figure 3.6 Spatial distribution of the permeability in a horizontal lay- er at the depth of 15 m, obtained by the genetic algorithm. The permeabilities are in md. . . . . . . . . . . . . . . . . . . 104 Figure 3.7 The flow chart of the entire computations. . . . . . . . . . . 105 Figure 3.8 The distributionN k of the original permeabilities (left) and their transformed distribution (right) after normal-score transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Figure 3.9 Evolution of the estimation error e for various number of ensembles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Figure 3.10 Evolution of, the error between each ensemble and the actual distribution (Eq. 3.14), with time for 80 ensembles. . 108 Figure 3.11 Evolution of , the standard deviation of all the ensem- bles, with the time for 80 ensembles . . . . . . . . . . . . . . 109 Figure 3.12 The spatial distribution of estimated permeability in a hor- izontal layer at a depth of 15 m. The permeabilities are in md. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Figure 3.13 Comparison of the estimated pressure (a) and CH 4 con- centration (b) with the data for a well near the center, and their evolution with the time. . . . . . . . . . . . . . . . . . . 111 Figure 3.14 Same as in Fig. 3.13, except that two of the wells were shut down after some time, as indicated by the jumps in the pressures. . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Figure 4.1 3D view of the grid structure used in the computations. The locations and numbering of the nine extraction wells are shown by the circles. . . . . . . . . . . . . . . . . . . . . . 137 xi Figure 4.2 3D view of the grid structure and the reference permeabil- ity distribution. The permeabilities are in mD. The location of nine extraction wells are shown by the circles. . . . . . . 138 Figure 4.3 The predicted natural gas price for the next 2 years. . . . . 139 Figure 4.4 Function of operation costh i (x i ) with respect to an extrac- tion well’s vacuum pressure x i . . . . . . . . . . . . . . . . . . 140 Figure 4.5 Diagram to show golden section search technique. (Adapted from [86]) . . . . . . . . . . . . . . . . . . . . . . . 141 Figure 4.6 Evolution of NPV with the iteration steps for conjugate gradient method (solid line) and steepest ascent method (dotted line). . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Figure 4.7 The optimal well pressure for 9 extraction wells obtained from the CGEnOpt. . . . . . . . . . . . . . . . . . . . . . . . 143 Figure 4.8 Evolution of NPV with the iteration steps for the GA method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Figure 4.9 Evolution of NPV with iterations using parameterless GA. 145 Figure 4.10 Evolution of number of species violating constraints with generations in parameterless GA. . . . . . . . . . . . . . . . 146 Figure 4.11 The optimal well pressure for 9 extraction wells obtained from parameterless GA. . . . . . . . . . . . . . . . . . . . . . 147 Figure 4.12 The CH 4 flow rate of all extraction wells with the same vacuum pressure of 0.5 atm: (solid lines) flow rate of well 3 and well 9; (dotted lines) flow rate of other extraction wells.148 xii Abstract In the United States, landfilling remains the primary choice for the disposal of municipal solid waste (MSW). The MSW in landfills undergoes anaerobic decomposition by various micro-organisms and produces large amounts of landfill gas (LFG). LFG, a gaseous mixture consisting mainly of CH 4 and CO 2 , can be utilized as a renewable source of green energy to produce electricity, heat and fuels. Even though there is a huge potential of converting LFG into energy, LFG has so far found fairly limited utility, because the economics of landfill energy projects are still not quite favor- able due to high operation costs. Among the key obstacle is that landfill engineers are not able to accurately predict the amount and quality of LFG emitted from a certain landfill and thus cannot determine optimal strategies to operate a landfill economically. In this dissertation, we are going to in- vestigate three essential aspects of efficient landfill management: short- and long-term performance prediction, real-time state estimation and model- based production optimization. Considering the importance of the short-term prediction, we first de- velop an artificial neural network (ANN) approach in order to make accu- rate short-term forecasting for several important quantities in a large land- xiii fill in Southern California, including the temperature, and the CH 4 , CO 2 , and O 2 concentration profiles. The ANN, a data-driven method, does not require the access to technical details about a landfill and thus is computa- tionally efficient. It is shown that the ANN can be successfully trained by the experimental data and can provide accurate predictions for the behavior of landfill at least in the short-term. For the long-term prediction, however, a model-based prediction method is commonly used, for which an accurate theoretical landfill model is needed. Hashemi et al. [37] and Sanchez et al. [72] presented a comprehensive 3D model to describe the LFG generation and transport, which has been validated by real landfill data. We further extend this landfill model, in conjunction with the ANN method and the genetic algorithm (GA) method, for the long-term prediction. The effec- tiveness of this novel approach is demonstrated by a synthetic landfill gas model. Another obstacle to accurately forecasting the amount and content of LFG is the lack of a realistic description of landfills, due to the fact that a landfill is a highly heterogeneous system, whereby the physical properties, such as permeability, porosity and tortuosity, are anisotropic and vary spa- tially. These properties are critical to determine generation and transport of LFG in a landfill. However, such information is difficult to obtain, be- cause only limited measurement data, such as the gas flow from a landfill xiv and its composition, are available. Therefore, it is important to characterize and estimate these factors that influence the accuracy of a landfill model, by assimilating the measurement data as soon as they become available. A real-time updating approach, based on a combination of the GA and the en- semble Kalman filter (EnKF), is proposed to solve this estimation problem. We then proceed to the problem of model-based production opti- mization for a landfill system, using the heterogeneous landfill model ob- tained from the real-time state estimation. The complexity of a landfill sys- tem will cause heavy computation burden to the optimization, where con- ventional optimization methods might fail. An effective ensemble based method is developed and successfully implemented to maximize the net present value (NPV) of a synthetic landfill model without constraints. The optimization with nonlinear flow rate constraints is handled by a GA based method, called parameterless GA, through modifying the objective function of the GA. 1 Chapter 1 Introduction 1.1 Background Municipal solid water (MSW), commonly consisting of everyday items that are discarded by the public, such as food, kitchen waste, waste plastics, glass, clothes, electrical appliances, etc. Generally, the MSW management industry utilizes three approaches to deal with the MSW: recycling, combustion with energy recovery and landfilling. As reported by United States Environmental Protection Agency (EPA) [3], in 2010, Amer- icans generated about 250 million tons of MSW and the waste generation per individual is 4.43 pounds per day (See Figure 1.1). Out of all the MSW generated in 2010, 34.1% was recycled, 11.7% was combusted for energy generation, and 54.2% was discarded in landfills. Therefore, landfilling still remains the primary choice for the disposal of MSW. The MSW in landfills undergoes anaerobic decomposition by various micro-organisms, and produces large amounts of landfill gas (LFG). LFG is a gaseous mixture, which consists of four major components, namely, CH 4 , CO 2 , O 2 , N 2 and a trace amount of other volatile compounds (VOC), including halogenated and organosulfur-type compounds. Typically, LFG production starts 2 immediately after the initial deposition of MSW, and reaches its peak production rate in about 10 years. Moreover, the production can continue for up to 40 years after the initial deposition [66]. One million tons of MSW are estimated to be able to produce approximately 432,000 standard cubic feet per day (scfd) of LFG. A typical landfill could produce approximately 4 - 5 million scfd of LFG, with larger landfills producing almost 50 million scfd. Such large amounts of LFG, which contains a significant amount of CH 4 , can be utilized as a renewable source of energy. The old-type landfills, where MSW was placed in an open dump, continually released toxic chemicals into the underground water and nox- ious gases into the air. After realizing these consequences, landfill engineers began to design landfills that can better protect the environment. In 1975, GSF Energy built the first full-scale sanitary landfill at Palos Verdes, Califor- nia. In this kind of modern landfills, trash is isolated from the surroundings by a bottom liner, side walls and also a daily covering of soil. A monitoring system is also required to test the quality of the underground water to de- termine if there is any contamination. Today, there are approximately 1150 modern landfills worldwide and this number is increasing every year [88]. Therefore, there is a huge potential of converting LFG to energy through various LFG utilization techniques. 3 1.2 Utilization of LFG LFG utilization is a process of collecting, treating and converting LFG to produce electricity, heat and fuels. The number of such landfill utiliza- tion projects increased from 399 in 2005 to 519 in 2009, according to the EPA. These technologies have gained popularity because the cost of LFG is rela- tively lower than that of other raw sources. Moreover, the efficient use of LFG is a promising approach, both in terms of conserving energy and re- ducing air pollution, due to the required treatments of LFG (e.g. removal of toxic compounds) before entering the energy producing equipment. The simplest and most direct utilization of LFG is by using it as a fuel for boilers or various other industrial processes (e.g., drying operations, kiln operations and cement production), where natural gas is commonly used. In these industries, LFG, as an alternative of natural gas, is transferred directly to the plant from the landfill. Compared to natural gas, LFG is cheaper and easier to obtain. Another advantage of using LFG is that only minimal condensate removal and filtration treatment are required, and thus only minor modification is needed for existing equipment. For typical boiler use, every 1 million metric tons of waste-in-place at a landfill can produce approximate 8,000 to 10,000 pounds of steam per hour [2]. However, since no economical way to store LFG exists, currently all recovered LFG must be used immediately. This requires the consumers to be located close to the 4 landfill site, usually less than 2 mils away [78]. As for the aspect of electricity generation, LFG can be used in inter- nal combustion engines and gas turbines to generate electricity. The internal combustion engine is the most common technology in LFG applications, be- ing used in more than 70% of LFG electricity generation projects. Internal combustion engines can achieve efficiency of 25% to 35% using LFG and this relatively high efficiency determines its popularity. However, the inter- nal combustion engine is only suitable for plants with gas capacity capable of generating 800 kW to 3 MW electricity. For larger LFG projects, a gas turbine is chosen, that can produce more than 5 MW electricity. Moreover, a turbine has relatively low maintenance cost as compared with a internal combustion engine with the same size of capacity. As an example, at the Ar- lington Landfill in Arlington, Texas, LFG is piped 4 miles to the Arlington Wastewater Treatment Plant and used to fuel two 5.2 MW gas turbines. Yet another important application of LFG is converting it to hy- drogen. Hydrogen is currently used in a variety of industries, including petroleum refining, chemical production and food industry. The demand for hydrogen in United States is around 9 million tons per year ([62]). The traditional way to produce hydrogen is through gasification of coal. As a result of economic development and population growth in the past few decades, the shortage of coal has brought a severe problem for these 5 traditional industries. Natural gas has the potential to be converted to hydrogen. The reaction of producing hydrogen from methane is through steam methane reforming (SMR): CH 4 + 2H 2 O!CO 2 + 4H 2 (1.1) Nowadays, 95% percent of hydrogen in the U.S. and about half of worldwide hydrogen, is produced from natural gas via SMR ([62]). Such SMR systems and equipment already run on natural gas can also be used with LFG, after removing the corrosive compounds existing in LFG. As for the aspect of green energy source, hydrogen can be used to fuel electric ve- hicles, which has the advantage of reducing or eliminating pollutant emis- sions. Together with oxygen, hydrogen can be used to produce electricity in a fuel cell. Because of the high efficiency, lower maintenance cost and zero emission, fuel cells are currently considered to be the best option for electricity generation. As an example, BMW is currently using hydrogen fuel cell to power nearly 100 vehicles at its new 1.1 million square meter assembly facility in South Carolina. Since 2003, LFG has been collected, cleaned and compressed from a local sanitary landfill and used to provide more than 50% of the plant’s total energy demand. The company claimed that it had reduced about 92,000 tons of CO 2 emission per year and saved about 5 million dollars in energy costs. 6 Moreover, LFG can also be used to produce high-Btu gas [63], such as pipeline-quality gas, compressed natural gas (CNG) and liquefied natu- ral gas (LNG). Pipeline-quality gas can be sold as natural gas to industries. CNG and LNG are used to power various facilities, ground transportation vehicles and fuel refuse-hauling trucks. The CNG facility at Puente Hills Landfill in Los Angeles converts LFG to CNG, which is then utilized to pow- er vehicles. It is able to convert an flow of 250 scfm with 55% of methane into 100 scfm of CNG with 96% of methane. This facility has been operated for more than 10 years. Besides producing CNG, this landfill also generates 63 MW electricity that is sold to Southern California Edison. On the other hand, LFG can be a significant source of greenhouse gases, mostly because of the CH 4 it contains. Compared with CO 2 , CH 4 is twenty times more hazardous in its greenhouse effect on a 100-year time horizon. According to the EPA, landfills released 25 to 40 million metric ton- s of methane in 1990. As a result, CH 4 emissions from landfills account for approximate 15% of current greenhouse gas emissions [13], and this fraction is increasing every year. Moreover, CH 4 is an explosive gas and thus may also create a risk of underground fires, if the ratio of CH 4 concentration and O 2 concentration reaches the flammability limit. In the last decade, there have been several fire reports due to the methane explosion in landfills [78]. Furthermore, LFG can be a significant source of pollution. For example, it 7 can migrate underground and then contaminate the underground water re- sources. As fugitive emissions, LFG can potentially create a variety of prob- lems, including odors, destruction of vegetation and occasional fires [66]. Therefore, considering these potential problems, developing LFG recovery projects is an effective way to reduce greenhouse gas emissions, improve air quality and eliminate odors. 1.3 Modeling of Landll Gas Systems Considering the urgency of utilizing LFG as an energy resource and preventing potential hazards, it is important to utilize LFG in power or ener- gy generation efficiently and economically. However, LFG has so far found fairly limited use, because landfill utilization technologies require landfills with high generation rate of LFG and sale price to justify the cost of land- fill operations. The benefits from these projects in some parts of the U.S. are not high enough to justify the cost of gas recovery. The EPA Landfill Methane Outreach Program (LMOP) reported 76 landfills with active LFG recovery projects in California alone. Besides, they listed another 274 land- fills as potential sites for energy recovery. This means a huge opportunity of recovering green energy throughout California and for the country as a whole, if the LFG utilization technologies can be applied to all the landfill sites. Among the key obstacle of utilizing LFG is that landfill engineers 8 are not able to predict the amount and quality of LFG that a given landfill may generate. This is one of the important reasons that landfill utilization technologies have not been used widely, since the benefits of landfill energy projects depend highly on the generation rate and quality of LFG. Current- ly, there are no effective models that have been validated by field data and also could give accurate predictions of the generation and migration of LFG. This makes it difficult to design a good landfill collection system, with most of the current designs being totally empirical. A good design of a collection system should be able to handle the maximum expected gas flow rate, col- lect gas at a sufficient extraction rate, reduce surface emissions and prevent pollution to the underground water. Typically, LFG with the best quality is generated in the deepest, most anaerobic regions of the landfill. If such LFG is not collected properly, it will rise to the surface, where it undergoes aerobic biodegradation and then CH 4 is converted into CO 2 . This will sig- nificantly impact the quality of LFG collected and will make it not suitable for energy production. Therefore, the ability of predicting where LFG with good quality is located is critical to the design of landfill gas collection sys- tems. For a good design of such system, an accurate model is needed, which can give an accurate prediction of the behavior of a landfill. In order to obtain an accurate landfill model, an effort which has gained intensive interest in recent years, it is critical to investigate the gas 9 reaction, generation and transport in landfills. Gas migration in porous me- dia has been studied for a long time, particularly by petroleum engineers. For the area of landfills, Mohsen [61] was the first to study the gas migration in landfills. The author assumed that LFG was in homogeneous condition and volume-averaged gas velocities were utilized in the model. The gas mi- gration out of the landfills was studied by Findikakis et al. [26] and Lee et al. [51]. They assumed the convective flow and diffusive flow to be equal. For describing production, migration and extraction of LFG through gas well- s, several theoretical models were developed, mostly one-dimensional (1D) or two-dimensional (2D) models. For example, Lu and Kuntz [57] devel- oped a 1D radial flow model, using measurements of LFG pressure. The pressure changes were caused by the withdrawal of gas to calculate the landfill’s methane production rate and gas-flow permeability. Young [89] proposed a more realistic 2D landfill model for a single gas to describe the LFG production, transport and extraction. In this model, each of the gases in the multi-component system was considered as a separate entity. Only the total rate of gas evolution was calculated from internal pressure changes and vertical permeability. Arigala et al. [8] developed a three-dimensional (3D) model for a landfill, which incorporated a more realistic description of biodegradation of MSW. However, Arigala et al. [8] also assumed LFG as a single gas, which meant that the model ignored the difference in the 10 transport properties of constituent gases of LFG and only used the pressure distribution in terms of the overall gas. Hashemi et al. [37] and Sanchez et al. [72] presented a comprehen- sive 3D model to describe the LFG generation and transport, which has been validated by real landfill data and proven to be a relatively accurate model. Moreover, this model works for an arbitrary number of extraction and monitoring wells, as well as arbitrary spatial distributions of perme- ability, porosity and tortuosity. In this model, the municipal solid waste (MSW) deposited in the landfill is divided into three classes, namely, readi- ly biodegradable wastes, moderately biodegradable materials, and the least biodegradable materials. The model is based on the convection-diffusion- reaction (CDR) equation, given by ` @ k @t +r ( k V) = k (Z) +r (D e km r k ) ; (1.2) where k is the mass concentration of componentk of the LFG, k (t) the gas generation rate of componentk, ` the local porosity of the landfill, V the gas mixture’s convective velocity, andD e km is the effective diffusion coefficient of gask in the mixture in the landfill. We assume that flow of the gases is slow enough that the Darcy’s law, V =(k=)rP , is applicable, wherek is the permeability, the viscosity, andP is the pressure. 11 The gas generation rate k (t) is given by k (t) = 3 X i=1 C T i A i i e i t ; (1.3) withi = 1; 2; and 3 corresponding to the readily, moderately, and slowly biodegradable fractions, respectively. k (t), the generation rate of gask, is in kg/m 3 of MSW per year,C T i is the total gas generation potential of waste of typei in kg/m 3 ,A i is the fraction of componenti of MSW, i is the gas generation constant ofi in yr 1 , andt is the time spent by the waste in the landfill. Note that a real landfill is loaded on a layer by layer basis, not filled all at once. So when the cover is placed on the top, the MSW at the bottom of the landfill already undergoes significant biodegradation. To take this into account, the timet in Eq. 1.2 is defined by t =t 0 +t f H L h : (1.4) Here,t 0 is the time since the cover was placed,t f the time to fill the landfill,H the depth of a specific layer, andL h the landfill’s total depth. To solve the governing equation, the dimensionless CDR equation and its non-reactive version for the cover are introduced and then solved numerically using the finite-volume (FV) method. A four-component gaseous mixture consisting of CH 4 , CO 2 , N 2 , and O 2 is considered. Not- 12 ing that, P 4 k=1 w k;i;j;l = m i;j;l p i;j;k (see the Appendix for the definitions of the various quantities), the discretized equations to be solved were given by, (n+1) m i;j;l p (n+1) i;j;l (n) m i;j;l p (n) i;j;l 4 + A ` 4 X k=1 8 > > > > > > > > > < > > > > > > > > > : d 2 x " D k;i+ (w k;i+1;j;l w k;i;j;l ) 4x i+ x i+1=2 x i1=2 + D k;i (w k;i1;j;l w k;i;j;l ) 4x i x i+1=2 x i1=2 # + d 2 y " D k;j+ w k;i;j+1;l w k;i;j;l 4y j+ y j+1=2 y j1=2 + D k;l w k;i;j1;l w k;i;j;l 4y j+ y j+1=2 y j1=2 # + " D k;l+ w k;i;j;l+1 w k;i;j;l 4z l+ z l+1=2 z l1=2 + D k;l w k;i;j;l1 w k;i;j;l 4z l z l+1=2 z l1=2 # 9 > > > > > > > > > = > > > > > > > > > ; +B 4 X k=1 k (z) + 8 > > > > > > > > > < > > > > > > > > > : d 2 x 2 " i+ m i+ p 2 i+1;j;l p 2 i;j;l 4x i+ x i+1=2 x i1=2 + i m i p 2 i1;j;l p 2 i;j;l 4x i x i+1=2 x i1=2 # + d 2 y 2 " j+ m ;j+ p 2 i;j+1;l p 2 i;j;l 4y j+ y j+1=2 y j1=2 + j m j p 2 i;j1;l p 2 i;j;l 4y j y j+1=2 y j1=2 # + 1 2 " l+ m ;l+ p 2 i;j;l+1 p 2 i;j;l 4z l+ z l+1=2 z l1=2 + l m l p 2 i;j;l1 p 2 i;j;l 4z l z l+1=2 z l1=2 # 9 > > > > > > > > > = > > > > > > > > > ; = 0; (1.5) with, i+ m i+ = 1 2 i+1;j;l m i+1;j;l + i;j;l m i;j;l , i m i = 1 2 i1;j;l m i1;j;l + i;j;l m i;j;l , D k i+ = 1 2 D k i+1;j;l +D k i;j;l ,D k i = 1 2 D k i1;j;l +D k i;j;l , x i+ = 1 2 (x i+1 x i ), and x i = 1 2 (x i x i1 ). Here, is the time step,n indicates the time step number at which the equations are solved ( = n) and, except for (n+1) m i;j;l p (n+1) i;j;l , all terms of Eq. 1.5 are evaluated at time step numbern. To find the solution of this 13 non-linear discretized equation, Hashemi et al. [37] and Sanchez et al. [72] used a biconjugate-gradient (BCG) method and solved it iteratively. The model developed by Hashemi et al. [37] and Sanchez et al. [72] is suitable for long-term planning for the operation of a landfill. However, it is too costly, in terms of the necessary computations, for short-term predic- tions. For example, a typical landfill operator would like to be able to pre- dict how the short-term trends in the spatial distribution of the temperature or gas composition of a landfill develop. Such short-term predictions can help operators take suitable actions, if any hazard is predicted to happen. The theoretical model described above needs to solve the CDR equation to obtain predictions, which is too costly for large-scale landfills, thus not suitable for short-term forecasting. Therefore, a faster prediction method is needed in this situation. The method we developed in this research is based on the use of artificial neural networks (ANNs). Generally speaking, an ANN is a computational model that is inspired by the way the human brain performs operations and computations, which consists of an inter- connected group of artificial neurons. In most cases, an ANN is an adaptive network system, aiming at modeling complex relationship between inputs and outputs. For the past two decades, the ANNs have been recognized as a powerful predictor and widely used for forecasting and planning in engi- neering. However, very few studies of implementing ANNs in the landfill 14 industry have been carried out. In this dissertation, we are going to use ANNs to perform short-term prediction for temperature and LFG composi- tion for a major landfill in Southern California. For the long-term prediction, considering the shortage of measurement data, the short-term predictions of the ANN are used in conjunction with the genetic algorithm (GA), so that an optimal model of a landfill could be developed, in terms of the spatial distributions of its permeability [73]. This novel combination is demonstrat- ed to be a powerful approach to significantly improve the accuracy of the long-term predictions, whereas the accuracy of the ANN predictions alone for the same time period deteriorates. 1.4 Landll as a Heterogeneous System As mentioned in the last section, different landfill models have been proposed to describe the gas generation and migration in landfills. How- ever, all these theoretical models assumed that physical properties, such as horizontal and vertical permeabilities, porosity and tortuosity, were con- stant throughout landfills. This assumption implies that a landfill is a ho- mogeneous porous medium. However, in reality, a landfill is a highly het- erogeneous porous medium. The great variety of wastes deposited in a landfill and the way they are compacted lead to a large-scale complex sys- tem, where the permeability is anisotropic and varies spatially. The same phenomena applies to porosity and tortuosity, which determine the effec- 15 tive diffusivity of LFG. Using a synthetic model, Sanchez et al. [72] demon- strated the strong influence of these physical parameters on behaviors of a landfill. Especially, they compared the effect of permeability distribution on the gas pressure measured at two different depths of a landfill with per- meable walls and 5 wells. Three different landfill models were considered: the first one was called homogeneous model, where the horizontal perme- abilities were the same throughout the landfill; in the second model, named as heterogeneous model, the horizontal permeabilities were spatially dif- ferent within each layer and decreased linearly from top to bottom; the last one was called semi-heterogeneous model, where only an effective horizon- tal permeability was used for each horizontal layer. For all three models, the vertical permeability was 1/3 of horizontal permeability. As shown in Fig. 1.2, the gas pressures for these three models were significantly different from each other. Therefore, an accurate landfill model requires a realistic estimate of such distributions of physical properties. However, it is difficult to obtain such information for a real landfill. The key obstacle is that only limited experimental data are available from landfill companies. Such limited da- ta are usually in terms of gas flow, composition and pressures measured at certain locations, usually in extraction/observation wells. There are very few measurement data of the various properties mentioned above. Even if 16 such data are available, landfill companies are not willing to disclose such information to the public. To solve this problem, intensive research has been focused on developing optimal geological model of large-scale porous me- dia, such as reservoirs and landfills, while honoring the production data. Such a process is commonly called history matching. In principle, history matching is formulated as an optimization problem, aiming to minimize the difference between measured and simulated data. Manual history matching was first proposed to adjust model parameters based on good engineering experience. However, it involves expensive trial and error procedure and might be biased by experience of the operator. Thus it is desirable to devel- op automatic history matching, free of bias and be able to provide accurate estimates of the parameter. Sanchez et al. [73] developed an automatic history matching approach to obtain optimal distributions of porosity, permeability and tortuosity, given limited experimental data from a property of the landfill. The optimization technique they are using is called genetic algorithm (GA). Compared with other optimization methods, such as gradient-based methods and simulated annealing (SA) methods, GA has two advantages: the first one is that it is not sensitive to the initial guess, so even an arbitrary initial guess can achieve the optimal solution; the second one is that GA is amenable for parallel computing, which helps decrease the 17 computation burden for a large-scale landfill system. The results indicated that the optimization techniques could not only accurately reproduce these distributions, but also provide accurate predictions for the future behavior of the landfill’s properties. However, history matching approach based on the GA does not allow for continuous model updating. When- ever new data are available, the whole updating procedure has to be repeated to assimilate new measurements. In recent years, the increased deployment of sensors for monitoring landfill behavior has brought an appealing incentive for engineers to study the continuous updating for landfill models. Considering the high frequency of data collecting from sensors, it is inefficient and computationally expensive to repeat the whole history matching procedure. In addition, it is critical to incorporate new data in the model as soon as they are available. The method we chose to deal with this problem is the ensemble Kalan filter (EnKF), which is actually a Monte-Carlo method, where the model states and uncertainty are represented by an ensemble of realizations of the system. Since first being proposed by Evensen in 1992 [22], the EnKF has gained increasing attention in continuous history matching problems, due to its conceptually simple formulation and relatively low computation. The EnKF has been proven to be very efficient and robust for real-time model updating for highly non-linear and complex systems [40, 38, 41]. Thus, for the landfill 18 system, we are proposing a novel approach using EnKF for estimating the permeability distribution, based on limited measurement data. 1.5 Optimization of LFG Production Another challenge of transforming LFG into a source of green ener- gy is the difficulty of producing LFG efficiently and economically and doing this safely, by smartly adjusting landfill settings. The ultimate goal for land- fill operators is to find an optimal strategy to build, operate and manage the landfill site, maximizing their benefits and at the same time reducing poten- tial hazards. This procedure is called production optimization in reservoir engineering, which uses optimal control techniques to increase the oil recov- ery. Specifically, production optimization for reservoirs can provide a way to maximize the cumulative oil production through manipulating critical variables, such as placement of wells, waterflood operations and injection rates. Inspired by potential benefits from production optimization, exten- sive efforts have been focused in this area and many efficient approach- es have been proposed. Similar to a reservoir system, a landfill is also a large-scale reactive porous medium. Therefore, we believe that we can use similar techniques as those being used in reservoir engineering, which, to our knowledge, have never been implemented in landfill’s production op- timization. One of the key hurdles of production optimization in landfills is that 19 a landfill is a complex biological and chemical reactor that responds sensi- tively to variations in weather conditions, pressure, temperature and oth- er landfill conditions. As illustrated in the previous section, a landfill is a highly uncertain system, with thousands of unknown variables, and thus continuous model updating is needed. Fortunately, as noted previously, we have developed an efficient approach, based on EnKF, to handle land- fill model uncertainties and their time evolution. The best estimates of the landfill’s physical properties can serve as initial conditions for production optimization. Another obstacle is that the traditional optimization meth- ods, such as gradient-based methods and stochastic methods, might not be suitable for production optimization in a landfill, which is a large-scale and highly nonlinear system. Moreover, for a typical landfill, where the LFG generation usually reaches the peak around 10 years, it is more desirable to maximize long-term benefits, for example, for up to 20 years. In order to obtain accurate performance predictions, a realistic and accurate theoretical landfill model is needed, which serves as the landfill forward model in the optimization. Because of its complexity, a landfill model needs thousands of simulation grids and thus even a single evaluation of long-term land- fill performance might take several hours. Traditional optimization meth- ods require running the landfill simulation hundreds of times, which will cause severe computation problems. Therefore, efficiency is essential for 20 our problem. The main objective of our work is to develop new algorithms that only need a minimal simulation model forward, but still can achieve a fairly good optimization performance. Furthermore, LFG production is often constrained by landfill conditions, handling capacity of surface facil- ities and also safety and environmental considerations. At the same time, LFG production must meet safety and environmental requirements from the EPA, in order to avoid potential hazards and protect the environment. Therefore, these constraints should also be considered in the optimization, which make the optimization problem much more complicated. For the past two decades, reservoir engineers have tried different ap- proaches to solve the problem of production optimization [75, 85, 84, 9]. An efficient approach, designed especially for large-scale system, was proposed by Sarma et al. [76], which used an adjoint model for calculation of gradi- ents. The authors demonstrated the effectiveness of their method for pro- duction optimization in both synthetic and real reservoirs. However, this adjoint-based method requires detailed knowledge of process model and access to the reservoir simulator. For a complex landfill model, it is difficult to obtain such information. To handle this issue, ensemble based optimiza- tion (EnOpt) was proposed, which regards the simulator as a black box, so access to the simulator was never required [16]. Since proposed, EnOpt has been used and proved to be very efficient in production optimization for 21 large scale reservoir system [14, 83, 52, 44]. In this dissertation, we are go- ing to implement EnOpt for unconstrained production optimization for a landfill gas system. For constrained production optimization, where EnOpt might fail due to heavy computation burden, we will propose another novel approach based on the GA, called parameterless GA. 1.6 Outline The dissertation is organized as follows. Chapter 2 presents an ANN approach, given a limited amount of ex- perimental data, aiming to making accurate short-term predictions for the behavior of a given landfill. We start with the description and generaliza- tion of ANN techniques used for the landfill problem. The proposed ANN approach is employed to analyze and forecast the gas composition and tem- perature profile for a major landfill in Southern California. Moreover, we show that a novel combination of the landfill theoretical model described in section 1.3, the optimization approach developed by Sanchez et al. [73], and the ANN approach is a powerful approach for developing an accurate model of a landfill for long-term predictions and planning. A simulation ex- ample is performed to demonstrate the effectiveness of this novel approach for long-term forecasting. Chapter 3 describes an automatic history matching approach, based on a combination of the GA and the EnKF, for the problem of generation 22 and dynamic updating of a landfill model. First, the GA is used to generate the initial spatial distribution of the permeability in a landfill by assimilat- ing the available measured data. Then, Sequential Gaussian Simulation (S- GS) is applied to generate multiple equiprobable permeability distributions, which serve as initial ensembles for the EnKF. Lastly, the EnKF is employed to continuously update the model using the data measured in real time. The effectiveness of this approach is demonstrated with the simulation of a model landfill and gas generation and transport therein, using synthetic data and a parallel computational strategy. Chapter 4 presents model based optimization techniques to solve the problem of production optimization of a landfill gas system. Conventional optimization methods are first introduced, such as gradient-based methods and stochastic methods. A simplified ensemble based optimization method, based on the EnOpt developed by Chen et al. [16], is developed to solve the unconstrained production optimization problem for a landfill gas system. A synthetic landfill gas system is used to demonstrated the efficiency of the EnOpt, using the heterogeneous landfill model obtained from Chapter 3. However, the operation of a real landfill involves all kinds of constraints, where the EnOpt might fail due to heavy computation burden. To handle this, a parameterless GA is proposed and successfully implemented on the same synthetic landfill model with constraints on the maximum flow rate 23 allowed in the extraction wells. Finally, Chapter 5 gives the conclusions for this dissertation. 24 Figure 1.1: MSW generation rate from 1960 to 2010. (Adapted from [3]) 25 8 9 10 11 12 0 0.2 0.4 0.6 0.8 1 1.2 Time (yrs) Gauge Pressure (kPa) 8 9 10 11 12 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time (yrs) Gauge Pressure (kPa) HwD HwoD SHwD SHwoD HOwD HOwoD HwD HwoD SHwD SHwoD HOwD HOwoD Z = 5m Z = 5m Z = 30m Figure 1.2: Comparison of time-dependence of the total pressures at two depths. The results shown are for the heterogeneous model with (HwD) and without (HwoD) mechanical dispersion; the semi-heterogeneous mod- el with (SHwD) and without (SHwoD) mechanical dispersion, and the ho- mogeneous model with (HowD) and without (HowoD) mechanical disper- sion. The results are for a landfill with one well. (Adapted from [72]) 26 Chapter 2 Use of Articial Neural Network and the Genetic Algorithm for Short- and Long-Term Forecasting and Planning 2.1 Introduction As mentioned in Chapter 1, landfills are complex systems to model, as they contain materials in three distinct phases, namely, solid, liquid, and gas. The solid wastes are biodegraded by various micro-organisms, and eventually produce significant amounts of CO 2 and CH 4 . The biodegra- dation process is usually exothermic, resulting in the generation of a sig- nificant amount of heat that is trapped in the landfill. The heat increases the landfill’s temperature, hence creating hazardous conditions and posing danger to the community adjacent to the landfill. A vital issue in landfill- s operation is the existence of hot spots, or enhanced oxidation zones (EOZ), where the temperature of the landfill can exceed 130 F. The EOZ condi- tions occur when significant amount of oxygen infiltrate landfills from the perimeter wells. The infiltration of air to sections of landfills transitions waste from an anaerobic phase to an aerobic phase. Aerobic biodegradation reactions have a higher heat of reaction than the anaerobic ones. If the heat 27 generated from aerobic biodegradation is trapped in the low permeability waste, it will gradually increase the temperature in the trapping section of the landfill. Some of the problems that arise from the elevated temperature and the development of the EOZ in a landfill include the loss of, or at least damage to, the extraction wells, underground fires, and rapid settlement of the wastes. The landfill gas (LFG) may also create the risk of underground fire, if the ratio of CH 4 and O 2 approaches its flammability limit. At the same time, large landfills produce very significant amounts of CH 4 that can be used as a relatively clean source of energy. They also produce large amounts of CO 2 that, if released into the atmosphere, will contribute to the Greenhouse phenomenon. Thus, due to the possibility of the underground fires and the danger to the surrounding community on the one hand, and the need for proper planning for the use of the produced CH 4 on the other hand, development of a computational model for predicting the temperature and the amounts of CH 4 and other important gases, such as O 2 and CO 2 , is vital to the safe operation of a landfill. Such a tool will enable the operators to identify the problematic areas in a landfill, and to take the proper precautions, in order to ensure that the emerging problems are addressed in a timely fashion. Conventional methods for predicting the temperature distribution and the concentrations of the various species throughout a landfill are 28 based on the numerical simulation of the governing equations for energy and mass transfer. In Chapter 1.3, we presented a comprehensive three- dimensional (3D) model developed by Hashemi et al. [37] and Sanchez et al. [72] that accounts for the generation and transport of the four major gaseous components of the LFG, namely, CH 4 , CO 2 , O 2 , N 2 , assuming that isothermal conditions prevail. In Chapter 1.3, we discussed the importance of short-term predictions and the computation cost involved in long-term forecasting using the theoretical landfill model. One purpose of this chap- ter is to describe one efficient approach, and use it for making short-term predictions for a large landfill in Southern California. The method that we develop is based on the use of artificial neural networks (ANNs). Generally, the ANNs consist of a class of computational tools that attempt to mimic the nature of human beings. They have been developed based on the way that the human brain performs operations and compu- tations, using millions of individual neurons that are highly interconnected with one another. Information, in the form of electrical pulses, from the output of other neurons is received by the cell at the connections that are known as synapses. The synapses connect to the cell inputs - the dendrites - with a single output of the neuron appearing at the axon. An electrical pulse is then sent down the axon when the total input stimuli from all of the den- drites exceed a certain threshold. An ANN is composed and operates in a 29 similar manner. It consists of individual models of the network, often in the form of distinct connection strengths associated with the synapses that it contains. The connections in an ANN are called the weights. The neurons are referred to as the nodes, when associated with an ANN. We will come back to the elements of the ANNs shortly. One uses a set of data to train an ANN and familiarize it with the trends in the data for the quantities of interest. The ANN is then used to perform a particular function, namely, computing by adjusting the weights between its elements. Typical ANNs are trained such that a particular input will produce a specific target output. To be effective, the ANNs must be accurate function approximators and excellent at recognizing patterns in the data. Even simple ANNs might be able to fit any practical function, if they are properly trained. But, as we describe in this chapter, use of an accurate ANN offers another advantage to modeling of landfills, namely, that it can broaden the range of the data that we need for determining the optimal spatial distributions of the important parameters of a landfill. Over the past two decades the ANNs have been recognized as uni- versal approximators, and have been widely used for forecasting analysis in engineering [25, 31, 30]. Applications of the ANNs to the problems in the landfill industry, as wells as identifying problems in groundwater flow that might somehow be linked to a landfill, have been made only very recent- 30 ly. Coppola et al. [17] successfully used an ANN to predict the hydraulic head using synthetic data and numerical simulations. They utilized the da- ta for a period of 5 years in order to train the ANN and predict the liquid head in a limestone aquifer. The inputs to the ANN included pumping rates and the climate conditions. Hamed et al. [32] used an ANN to predict the biochemical oxygen demand and the concentration of suspended solids in the effluent of a plant for wastewater treatment. Shamim et al. [65] used an ANN to predict groundwater contamination in a region of Pakistan, by predicting the concentrations of iron, copper, and lead in the groundwa- ter. Almasri and Kaluarachchi [5] utilized an ANN to forecast the nitrate concentration in the Sumas-Blaine aquifer in Washington state. The results were in agreement with the actual field data. Karaca and ¨ Ozkaya [47] utilized an ANN to study the effect of the meteorological and leachate characteristics on the predictions of the dai- ly amount of leachate at a major landfill in Turkey. The input parameters to their ANN were the leachate’s pH, temperature, and conductivity, the air temperature, cloudiness, pressure, relative humidity, and precipitation. Their study indicated that the ANN was able to accurately predict the daily quantity of the leachate in the landfill. ¨ Ozkaya et al. [47] used an ANN to study another landfill in Turkey. The input data included the pH, alkalinity, the concentration of the sulfate, the conductivity, the wastes’ temperature, 31 and the refuse age. Data collected over 34 months were used to validate the ability of the ANN for predicting some of the properties of interest in the landfill. Scozzari [77] used an ANN to determine the surface biogas flux in the municipal solid wastes landfills, with the input parameters being the meteorological parameters. The predictions provided by the ANN for the biogas flux were accurate. Singh and Datta [79] used an ANN to identify the unknown sources of groundwater pollution in terms of the magnitude, location, and the time of leakage of the pollutants. The ANNs have also been used for predicting and addressing many issues concerning modeling of oil and gas reservoirs, based on some limited data [4, 59, 60]. The work presented in this chapter differs from the previous stud- ies in that, we develop an ANN in order to predict the temperature and the amount of oxygen in a large landfill in Southern California. The goal is to identify the EOZ in the landfill that may lead to underground fires that would not only damage and even destroy the wells in the landfill - the typical cost of which is $50,000-100,000, but also pose a danger to the com- munity around the landfill. In addition, because the landfill is in the middle of a large residential community in southern California, the amount of CH 4 produced by the landfill must be predicted accurately, so as to prevent its migration toward the community. Moreover, we propose a novel combina- tion of an ANN and the GA, and show that it leads to more accurate models 32 of landfills for long-term predictions and planning. The plan of this chapter is as follows. In the next section we describe the ANN model that we utilize in the computations. Section 2.3 describes the landfill and the data that are used in the ANN computations and also the results of short-term predictions. In Section 2.4, we propose a combina- tion of an ANN with the GA, in order to develop more accurate models of landfills, to be used for accurate long-term predictions and planning. The last section presents a summary of the results. 2.2 Model of The Articial Neural Network In an ANN the input vector of all the data and the corresponding target output for a particular variable, say the temperature T , are used to train the network until it accurately minimizes a function - called the perfor- mance function - associated with the input vectors and the target output. The performance function is defined as the sum of the squares of the differences between the targets’ values and estimates. The targets are the measured data, and the performance function minimizes the difference between them and the estimates. We use an ANN that utilizes the back-propagation (BP) algorithm with a number of hidden layers. The term BP algorithm refers to the man- ner by which the gradient of the performance function is computed for the nonlinear multilayer networks of the type that we use in this work [69]; see 33 below. The hidden layers constitute the “interior” of the ANN that is not “seen” by the user, and represent neither the input nor the output layer. Their task is to obtain the optimal estimates of the weights that attribute the proper significance (weight) to each piece of the information used for training the ANN. The estimates are, of course, not totally unbiased and, hence, one also adjusts the biases for optimal performance. To move from one layer to the next one must specify the transfer functionF T that maps a layer’s output onto the next layer. The ANN that we utilize contains sig- moid hidden layers and a linear output layer. A sigmoid hidden layer is one for which the transfer function is given by, F T (x) = 1 1 + exp(x) : (2.1) For the output layer, on the other hand, we use,F T (x) =x. The two choices were made based on a series of preliminary simulations, in order to assess the performance of the various choices forF T (x). If an ANN with a BP algorithm is properly trained, it can produce reasonable output (predictions) if presented with input data that it has nev- er seen, provided that the patterns in the data that have never been provid- ed to the ANN are not too different from those that were utilized to train it. The way an ANN is set up is as follows. 1. One first assembles the data for the ANN training. 34 2. One then creates the ANN, i.e., specifies the number of the hidden layers, their type, etc. 3. The ANN is then trained using the existing data. 4. The trained ANN is further tested by a set of computations called the validation, in order to ensure that the optimal estimates of the weights and biases have been obtained. 5. The ANN is then utilized for predicting the quantities of interest. As an example, suppose that we wish to estimate the temperature T by minimizing its corresponding performance function. To do so, we assume thatT depends on many variables through the following functional form, T =F (vacuum pressures, wells’ locations, previously measuredT ,v i , W, B): (2.2) Here, the vacuum pressure refers to the static pressure of the wells, v i the measured volume fractions of gas i in the gaseous mixtures at the wells, and W and B represent the matrices of all the weights and biases used in the computations. The functionF is unknown, and a task of an ANN is to discover or recognize a pattern in the data, with the help of which it con- structs an empirical functional form forF , in order to estimateT . The mea- sured temperature log of the landfill at the wells represents the target, and 35 an important task of an ANN is to minimize the difference between the es- timated and target (measured) temperatures. The weights are the relations (connections) between the static pressures of the wells, the gases’ concen- tration, and the wells’ locations. Each input variable is related to all the other input data, but it must be given the proper weight so as not to over- or underestimate its significance. This is highly important for the ANN, as it attempts to discover a pattern in the data. Therefore, in essence the task of an ANN is solving a pattern recognition problem. The weights and biases are not constant, but vary during the itera- tion process for the minimization of the performance function. Their values are updated using the BP algorithm (see below), which is done during the training step in which the ANN minimizes the performance function. The relation between the old and new values of the weights is given by W n =W o rF p ; (2.3) whereW o andW n refer to the old and the newly updated values. A similar equation is used for updating the biases. , called a training parameter, is used to control the estimates of the weights and biases during the iteration process [18]. The total number of the weights isn!, wheren is the number of wells used in the study. Hence, the total number of all the weights is very large, as the data from a large number of wells are used in the present study. The more hidden layers are used in the ANN, the more accurate the 36 estimates for the weights and biases and, hence, the predictions will be, because the hidden layers improve the estimates by calculating the inter- mediate values that are used in the output layer. A larger number of layers also requires more computations, however. In this part of the present work we used 20 hidden layers, after carrying out preliminary simulations with 5, 10, 15, 20, and 25 hidden layers, and finding that 20 hidden layers yield results that are as accurate as those obtained with 25 layers. Once the proper weights and biases are calculated, one computes the neurons that are scalar functions used in the transfer function for calculating the output temper- ature and other quantities that we wish to estimate. Fig. 2.1 presents the schematic of the ANN used in our work. Figure 2.2 shows the iteration process for an ANN. The target vector is compared to the output generated by the ANN simulation. The perfor- mance function is evaluated after a number of iterations - also called the epochs. The weights and biases are adjusted until the convergence criterion is satisfied. We assumed that convergence was achieved when the value of the performance functions did not decrease any further after 20 extra iter- ations. The value of a neuron i in the layer m, n m i , is computed using the following equation, n m i = N m1 X i=1 W m ij a m1 j +B m i ; (2.4) whereW m ij is the weight between nodesi andj in layerm,a m1 j the input 37 from the layerm 1, andB m i the bias of nodei in the layerm. After com- puting the neuron, the output is computed by inserting the neuron into the particular transfer function that is being used, i.e. a m i =F m T (n m i ): (2.5) Here,F m T is the transfer function of layerm. The output from layerm is a vector a m , which serves either as the input to layerm+1 or, if it is the output from the last layer, represents the predictions. As described above, selecting the proper values of the weights and biases is crucial to the accuracy of an ANN. Their optimal values are ob- tained using the training data that consist of the input vectors and the tar- get vector T. In the present work we wish to make predictions, not only for the temperature, but also for the mole fraction profiles of several gases (see below). For example, in order to make predictions for the temperature, the following performance functionF p was used, F p = 1 N N X i=1 (T M T C ) 2 ; (2.6) whereN is the number of temperature readings used to minimize the per- formance function, T M the measured temperature, and T C the computed (predicted) value by the ANN. The goal of the training is to reduce the av- erage error over all the pairs of the input and target vectors. AdjustingW m ij and B m i reduces the value of the function F p and moves it toward its true 38 global minimum. The updated weights are computed by the following e- quation [see Eq. 2.3], W m ij j n =W m ij j o @F p @W m ij j o ; (2.7) where W m ij j n and W m ij j o represent, respectively, the new (updated) and the old weights. The updated values of the biases are computed similarly, B m i j n =B m i j o @F p @B m i j o : (2.8) Very small values of slows down the convergence of the algorith- m and increases the computational cost of the training and minimization of the performance functionF p . Large values of, on the other hand, may overemphasize adjustment with the data pair, as a result of which the global minimization of the performance function may become problematic. Large values of may also cause instability, hence leading to nonconvergence. An acceptable estimate of is often determined by performing a few prelimi- nary simulations and studying the results and their trends. Based on such preliminary simulations we chose, = 10 2 , as the optimal estimate of the training parameter. The BP algorithm uses a steepest-descent method in order to mini- mize the performance functionF p . We then compute a quantitys m i by s m i = @F p @n m i : (2.9) 39 in terms of which we obtain @F p @W m ij = @F p @n m i @n m i @W m ij =s m i a m1 j (2.10) and @F p @B m i = @F p @n m i @n m i @B m i =s m i (2.11) The relation betweens m i , the target output, and the performance function is given by, s m i = @F p @n m i = N X j=1 @n m+1 j @n m i @F p @n m+1 j = N X j+1 @n m+1 j @n m i s m+1 j ; (2.12) which enables us to expresss m i in terms of the transfer function, s m i =2(t i a m i ) @F T (n m i ) @n m i ; (2.13) where t i is the target value at i. Thus, we obtain the following equations that relate the updated weights and biases to the target variables and the transfer function, W m ij j n =W m ij j o + 2a m1 j (t i a m i ) @F T (n m i ) @n m i (2.14) and B m i j n =B m i j o + 2(t i a m i ) @F T (n m i ) @n m i : (2.15) 2.3 Short-Term Forecasting Using The ANN 1 1 This part of work was done in collaboration with Dr. Raudel Sanchez, Project Naviga- tor, Ltd., Brea, CA 40 2.3.1 The Landll Data The ANN that was described in the last section was used to ana- lyze and forecast the CH 4 , CO 2 , O 2 , and temperature profiles at a major landfill in Southern California. The landfill operators typically collect field data from the extraction wells using a gas analyzer and extraction monitor (GEM) unit. The GEM units are used to measure in-situ the pressure, tem- perature and gas concentrations. The amount of CH 4 is important to fore- cast because it is a crucial field parameter for being in compliance with the regulations imposed by the U.S. Environmental Protection Agency (EPA) for the perimeter wells adjacent to the community. Another reason for pre- dicting the amount of produced CH 4 is to accurately learn its quality ex- tracted from the wells. At the site that we study large quantities of produced CH 4 are utilized to generate electricity from five on-site turbines. The CO 2 distribution is also an important variable to gain knowl- edge about, as it indicates whether the landfill is in an anaerobic or aerobic phase in the section of the Site that produces it. If the CO 2 concentration is greater than 60% of the total gas produced, it would be indicate that the aerobic phase exists, and that the air might be infiltrating that section of the landfill. The CO 2 concentration is also an indicator that the EOZ is starting to develop at that section of the Site. The O 2 concentration is another important landfill parameter, as it 41 is a key contributor to the EOZ at the site. If the O 2 concentration is high and temperature is increasing, this would be indicative of an EOZ area, and the pressure of the extraction wells must be decreased in such a way that it would be in compliance with the Federal regulations. Finally, temperature is important to forecast, because it indicates that an EOZ condition exists and, thus, the proper measures need to be taken to control the EOZ. The site is a large landfill that operated from the 1940s until it was closed in the 1980s. The data utilized had been collected from 2004 - 2005, over a period of 18 months at 30 wells in the northeastern section of the landfill. At this particular landfill the wells are sampled once or twice per month, according to the Site’s Sampling and Analysis Plan (SAP) approved by the EPA. The wells are sampled twice a month if they are located in an EOZ area. Thus, we have nearly 1100 data points for each quantity that we wish to forecast. The data for the first 14 months were utilized for the ANN training, and the next 2 months for the validation, while those collected over the last 2 months were used to test the accuracy of the predictions. The data included such information as the temperatures measured at the 30 wells, the volume fractions of CH 4 , O 2 , and CO 2 in the gaseous mixtures, and the static (vacuum) pressures at the same wells, as well as the wells’ locations. It should be pointed out that each case that we studied took only a few CPU minutes on a standard desktop computer. The computer programs 42 may be executed on practically any desktop or laptop computer. As such, the ANN can be easily utilized by any landfill operator. 2.3.2 Results and Discussions In what follows we will describe the results and discuss their impli- cations. 2.3.2.1 Forecasting Methane Production Knowledge of CH 4 profile in any landfill is pivotal to avoiding po- tential fires caused by high concentrations of the gas, as well as ensuring compliance with the regulations of the EPA for protecting the communities adjacent to a landfill. At the same time, because the rate of production of CH 4 in a large landfill can be very significant, accurate predictions of the amount of CH 4 produced enable practical planning for its use as a relative- ly clean source of energy. Predicting the CH 4 distribution at interior wells helps the landfill operators to decide and implement the vacuum pressure of the interior extraction wells. If the CH 4 concentration is low at certain interior extraction wells, the landfill operators will decrease the vacuum at that particular well, whereas if the percentage of CH 4 in the gaseous mix- ture is approximately 50%, then the operators will increase the vacuum at the interior well. Data for the CH 4 profiles, measured at 30 extractions wells in the 43 landfill, were used to train the ANN. After carrying out the validation step, the trained ANN was utilized to forecast the volume fraction of the pro- duced CH 4 over a two month period at all the wells that were utilized in the ANN computations. Figure 2.3 presents the variations of the performance function for CH 4 during its iterative minimization, showing the results for the training, validation, and forecasting parts of the computations. As Fig. 2.3 indicates, after about 80 iterations the performance function converged completely to a constant value and attained its global minimum, although the convergence had already been achieved after 60 iterations. It is over the first 10 iterations, however, that the performance function decreases signif- icantly, after which it begins to converge to its global minimum. The values of the performance function for the training and validation steps decrease in similar manners, and are very close to each other, hence demonstrating that the ANN was properly trained and validated. Figure 2.4 compares the actual (measured) values of the produced CH 4 with those predicted by the ANN. If the data and the ANN predictions match exactly, they would all fall exactly on the 45 line. Approximately 60% of the data was used to determine the optimal values of the weight and biases. The validation step, using the next 20% of the data, was taken to ensure that there were no significant errors in the optimal values of the weights and biases. This helps the ANN to improve the accuracy of its 44 computations and predictions. As Fig. 2.4 indicates, most of the data are very close to the 45 line, hence demonstrating that the ANN was properly trained and, thus, was able to accurately predict the CH 4 volume fractions in the site’s northeastern section. The most important part of the computations is, however, testing the ability of the ANN for predicting that part of the data that was not used in the training. As Fig. 2.4 indicates, similar to the training and validation parts, the ANN is able to accurately predict the remaining 20% of the data for the CH 4 mole fraction profile at the site. Thus, the overall performance of the ANN is very good. 2.3.2.2 Forecasting Carbon Dioxide Production As already described, CO 2 is an indicator of an EOZ area in the par- ticular sections of a landfill where its concentration is significant. Therefore, predicting how much CO 2 may be produced by a landfill is important and, hence, we used the ANN to predict the amount of the CO 2 produced, after training it with 60% of the data. The input data in this case were the temper- ature, vacuum pressure, and the wells’ locations for about 80% of the data, 20% of which was used for the validation step. Figure 2.5 shows how the performance function decreases during its iterative global minimization. In this case about 100 iterations were used to ensure convergence to the global minimum of the performance function 45 F p , althoughF p did not change significantly after about 60 iterations. The trends are, however, completely similar to those for CH 4 . Figure 2.6 com- pares the training, validation, and testing steps of the ANN for the amount of CO 2 produced. As the validation part of Fig. 2.6 indicates, the ANN ap- pears to have been extremely well trained, which explains why it provides very accurate predictions for that part of the data that were not used in the training and validation steps. 2.3.2.3 Forecasting Oxygen Prole Knowledge of the amount of oxygen in a landfill is important for a variety of reasons. The anaerobic reactions that produce CO 2 occur in the p- resence of O 2 . The produced CO 2 is an indicator of a transition between the aerobic and anaerobic phases and a possible early stage of development of the EOZ. Knowledge of the amount of oxygen is also important in the sense that, once the ratio of oxygen to methane reaches a specific value, there would be a distinct possibility of an underground fire. Moreover, more oxy- gen contributes to enhanced oxidation processes at elevated temperatures in a landfill. In addition, if the oxygen concentration at an extraction well increases dramatically, it may be an indication of the existence of a tear in the landfill’s cover, in which case the static (vacuum) pressure of the well should be lowered in order to prevent, or at least limit, the EOZ. Thus, i- dentifying the problematic areas in which the oxygen concentration is high 46 is of utmost importance to its safe operation, and to avoiding potential un- derground fires at the Site. Figure 2.7 demonstrates how the performance function reduces as it- s minimization proceeds. In this case, the performance function required about 40 iterations to reach its global minimum, but we continued the com- putations for another 30 iterations in order to ensure that the function had attained its true minimum. Figure 2.8 shows the comparison of the mea- sured oxygen profile at the Site with the ANN predictions. As in the case of CH 4 and CO 2 , good agreement is obtained between the ANN-predicted oxygen profile and the data at the Site. Although the plots appear to be more scattered than those for CH 4 and CO 2 , the correlation coefficientR 2 is, in fact, higher than those for the previous two cases. 2.3.2.4 Forecasting Temperature Prole A landfill operator must be in compliance with the EPA regulation- s for the landfill’s perimeter. The main concern on the perimeter of the landfill is possible migration of CH 4 toward the community adjacent to the landfill. Because of this reason, the extraction wells near the perimeter are often operated at high vacuum, in order to guarantee that no CH 4 escapes the landfill. But, the high vacuum has the disadvantage that it helps the wells to pull in oxygen from the surrounding areas, which increases grad- ually the temperature in the landfill. This happens, in particular, around 47 the wells, because the heat of reaction of the exothermic aerobic reactions is larger than that of the endothermic anaerobic reactions. The heat is trapped in the low-permeability waste areas and gradually increases the tempera- ture in that specific area. The higher temperatures damage, and sometimes even destroy, the wells that are usually made of polymeric (plastic) mate- rials. Thus, forecasting the temperature distribution inside a landfill is im- portant. In addition, forecasting the temperature is important to prevent- ing underground fires that could threaten the community around a landfill. Moreover, burning of the wastes also poses a danger to the stability of the site, and may also contribute to enhanced settlement of the wastes. Rapid settlement of the wastes damages the top cover. Therefore, we utilized the ANN to predict the temperature in north- eastern part of the landfill. The input variables to the ANN in this case were the vacuum (static) pressures, the wells’ location, and the volume fractions of CH 4 , CO 2 , and O 2 at the extraction wells. Figure 2.9 shows how the per- formance function decreases during its global minimization. After only 30 iterations the performance function had, for all practical purposes, reached its minimum, but the computations were continued for another 60 iterations to ensure that the minimum reached was the true global one. Figure 2.10 compares the data with the ANN predictions. The a- greement is excellent. Such predictions may be used to identify the areas in 48 which enhanced oxidation may be occurring, which would indicate that the temperature in those areas is above 130 F. In that case, the static (vacuum) pressure is adjusted to lower the temperature at such problematic areas. 2.4 Long-Term Forecasting Using The ANN and The Ge- netic Algorithm As the results presented in the last section demonstrated, the ANN is an effective tool for providing accurate short-term predictions for the prop- erties of a landfill. On the other hand, a suitable optimization method, such as the Genetic Algorithm (GA), can generate an accurate model of a land- fill for long-term forecasting and planning, given enough information and data [73]. A major problem with modeling of landfills is, however, lack of experimental data as operators are often not willing to divulge their infor- mation and data that they have for the landfills that they run. The question is: how can one expand the data set that is needed as the input for an opti- mization method such as the GA, in order to generate an accurate model of a landfill for long-term forecasting and planning? Here, we propose a novel solution to the problem by combining the ANN and GA algorithms that, to our knowledge, has not been proposed in the past. To demonstrate the new approach, we use synthetic data. That is, we first generate a model of a landfill in which the permeability is distributed spatially, and compute its important properties, such as the spatial distri- 49 butions of a landfill’s main four gases, namely, CH 4 , CO 2 , O 2 , and N 2 . We then “pretend” that we do not have any data on the spatial distribution of the permeability of the landfill, but use a portion of the computed concen- tration profile of CH 4 in the landfill as the input data for training an ANN and making short-term predictions, and for use with the GA algorithm in order to determine the optimal spatial distribution of the landfill’s perme- ability and computing the CH 4 concentration profile. We then combine the same input data with accurate short-term predictions of the ANN in order to generate a broader set of input data for the GA, and study whether the accuracy of the results obtained by the GA-determined optimal distribution of the permeability increases. The proposed scheme has two advantages: (1) It provides more data samples for the GA in the case of limited training samples. (2) By presenting to GA the actual data and those predicted by the ANN, the algorithm helps the GA determines a solution to the optimization problem in a smaller part of the domain of all the plausible solutions to the optimization problem [73]. Thus, the GA results from such a scheme are smoother than those obtained by using that obtained by the original data alone, which could be more noisy in practice. In addition, the amount of input data is doubled, half of which are the original training data, while the other half is the data predicted by the ANN. 50 2.4.1 Model of The Genetic Algorithm The genetic algorithm is based on the concept from genetics to de- scribe the individual and Darwinian evolution theory to yield better solu- tions of the problem. In general, use of the GA involves three steps [58]: (1) The selection step for choosing the “species” (estimates of the pa- rameters - the permeability in our study) that generate the offsprings (new estimates), according to the objective function. (2) The crossover and mutation steps that generate new species that improve the optimal estimates. (3) The eliticism step to select species that lead eventually to the global minimum of the objective function. In Darwinian evolution theory, selection is the step to decide what species are chosen to produce off-springs, according to their abilities of adapting to the environment. From a biological point of view, all organisms consist of cells and each cell contains the same set of chromosomes, which are organized structures of DNA. A chromosome can be divided into genes, each of which encodes a particular kind of protein. Similarly, each species in GA can be represented by a set of parameters, which can be regarded as the genes of a chromosome described above. To facilitate the computation, these parameters are usually structured by a string of values in binary form. A objective function (OF) is used to evaluate the degree of fitness of a species to our objective. In our work, we want to study what is the optimal 51 distribution of a given landfill’s physical properties (permeability in our work), based on the available production data. In this case, each species can be represented by a spatial distribution of permeability and the OF will be the prediction error, defined by OF = (S d O d ) 2 ; (2.16) where S d and O d are simulated and measured production data. The species with a smaller OF has a greater possibility to be selected to produce off-springs for the next generation. In each generation, the set of all the possible species is called a population. In the selection step, a binary tournament method is used to choose a set of mating pairs, whose size is half of that of the population. Each mating pair can produce two off-springs in the crossover step, where a certain degree of mutation could happen. To speed up the evolution process, we also enforce the eliticism strategy, where the species with the smallest OF is inserted into the next generation at each generation. This can ensure that a global optimal solution is eventually achieved. After we finish all three steps, a new generation of species is pro- duced and then we have to repeat this entire procedure until the best species is obtained, when the OF of the best species converges. Since the GA only relies on the calculations of the OF and these cal- 52 culations are independent for each species, this determines that GA is a- menable for parallel computing, which is a very important advantage of the GA. Hence, we used a master-slave parallel computational strategy [73] for implementing the GA, with the slave processors solving the governing equation for gas generation and transport in the landfill given each species for the OF, and the master processor collecting all the evaluations of the OF to determine whether the best species has been generated. If not, the master node will repeat all the GA steps (e.g. selection, crossover, mutation and eliticism), during which it will pass the information to slave nodes to perform evaluations of the OF. 2.4.2 Generation of Synthetic Data As illustrated by the landfill model in Chapter 1.3, the landfill is rep- resented by a three-dimensional computational grid of cubic grid blocks. In our work, we assumed that the size of the computational grid is 202020, with the linear dimensions of the gridblocks being 1.5 m, except near the wells where the grid blocks are smaller (see below). We assume that the effective permeabilities of the grid blocks are spatially distributed. The per- meabilities are dependent upon the depth, as the sectors that are close to a landfill’s cover have higher permeabilities whereas, due to compaction and settling, those that are near or at the bottom of the landfill have the smallest permeabilities. Thus, in addition to be spatially distributed, the permeabili- 53 ty field is also anisotropic with the vertical permeability being smaller than those in the horizontal planes (parallel to the ground surface). To develop a realistic spatial distribution of the permeabilities, we must have some data, even if very limited. Thus, we assume that the per- meabilities of ten grid points (centers of the grid blocks) at the bottom layer of the landfill are known, having being estimated by, for example, collect- ing data for the fluxes and pressure of the gases through the monitoring wells in the landfill. Based on the data and the fact that the vertical perme- abilities are smaller than the horizontal ones, we generate the spatial dis- tribution of the permeabilities by using the Sequential Gaussian Simulation (SGS) method [45, 70]. The SGS method, used extensively in the petroleum industry, uses a set of permeability data for a porous formation, in order to generate the permeabilities in the rest of a model of the formation. The procedure that we use is as follows. (i) Using the known permeabilities of the grid blocks in the bottom layer, we use the SGS to generate the permeabilities for the rest of the grid blocks in the same layer. We assume that the permeabilities of the 10 grid blocks are in the range 1 - 8 md. (ii) We assume that the vertical permeability of any grid block is 1/3 of the horizontal permeabilities of the same block. (iii) To compute the permeabilites in the rest of the grid blocks, we assume that the permeabilities at the top layer are 5 times larger than those 54 at the bottom layer, and that the increase of the permeabilities from the bottom to top layer is linear. Thus, after generating the permeabilities for all the grid blocks in the bottom layer, the permeabilities of the remaining grid blocks are computed straightforwardly. Admittedly, the model is relatively simple. But, the purpose of the computations is demonstrating the potential of a combination of the ANN and GA algorithms for making accurate short- and long-term predictions for the landfill. Any other spatial distribution of the permeabilities may be used. The thickness of the landfill’s cover is assumed to be 2 m, with it- s permeability being isotropic and constant with a value of 0.1 md. The permeability of the surrounding soil is assumed to be 0.3 md. Figure 2.11 presents the synthetic permeability distribution of a horizontal layer at the depth of 14 m from the top. We assume that the landfill has 4 extraction and 6 monitoring wells; see Fig. 2.12 that shows the top view of the computa- tional grid and the wells’ positions. As described in Eq. 1.5, the dimensionless convective-diffusion re- action (CDR) equation and its non-reactive version for the cover are solved numerically using the finite-volume (FV) method, because the grid blocks’ sizes are not the same everywhere. The blocks around the wells are smaller than those in the rest of the computational grid (see Fig. 2.12). A four- component gaseous mixture consisting of CH 4 , CO 2 , N 2 , and O 2 is consid- 55 ered. In our simulation, the time step is 0.05 year, and we simulate gas generation and transport for a period of three years. The synthetic data that we use with the GA and for training the ANN are the computed CH 4 concentration profiles that are computed along the 10 wells in 10 vertical layers. After the CH 4 concentration profile throughout the landfill is com- puted, we use a portion of the results as the input data to train the ANN, and employ the rest of the results to test the accuracy of the trained ANN for making predictions. Likewise, a portion of the data is used in the GA to determine the spatial optimal distribution of the permeability and deter- mining the concentration profiles of the gases, and the rest is used to test the accuracy of the computed results. Finally, the ANN and GA are combined, as described below. Considering that a landfill is a complex system, computations with the GA are highly intensive, to the extent that it is not practical to carry them out on a single processor. Hence, we utilized parallel computing using func- tion decomposition since the GA lends itself to parallelism, as one is able to carry out in parallel the function evaluations needed for each generation of the GA iterative process. We used 800 processors, and each processor is assigned a specific number of function evaluations for each generation. 56 2.4.3 Training ANN The input data for training the ANN are the locations of the wells and the computed concentrations of CH 4 along the 10 wells, calculated for the first 0.5 year. The output of the ANN is the CH 4 concentrations for the same time period and longer. To keep the computations as simple as possible, we used an ANN with 4 hidden layers. Figure 2.13 compares the actual (synthetically-generated) CH 4 con- centrations with those generated by the ANN for two time periods. One, shown in Fig. 2.13(a), is for 0.5 - 0.7 year, whereas the second case, shown in Fig. 2.13(b), makes the same comparison for the time period 0.7 - 1.0 year. As Fig. 2.13(a) indicates, in the former case almost all the results are on the 45 line, implying that the (synthetic) data and the ANN predictions almost perfectly match, and that the ANN can indeed provide accurate short-term predictions. But, the accuracy of the ANN predictions over the longer time period of 0.7 - 1.0 year begins to deteriorate. In fact, the ANN predictions (not shown) beyond the first year are completely wrong. 2.4.4 Simulation Results Given a population of initial guesses for the optimal spatial distribu- tion of the permeability, the entire optimization process based on the GA consists of, (i) solving the CDR equation for the gases and computing the CH 4 57 concentration profiles, to be compared with (synthetic) data along the wells, and (ii) checking whether the convergence criterion for attaining the global minimum of the objective function (OF) of the GA has been satisfied. If so, the GA-based optimization process is completed. The convergence criterion is that the numerical value of the OF should not change within a set tolerance for a number of consecutive generations. The objective function that we use is the sum of the squares of the differences between the data and the computed CH 4 concentrations. The number of GA generations that we use is 1600. If convergence has not been achieved, then, (iii) the GA produces a new generation of possible solutions, and steps (i) and (ii) are repeated until the convergence criterion is satisfied. Using this procedure, two cases were simulated: (i) We use the GA to estimate the optimal permeability distribution using as the input the (synthetic) data over the first 0.5 year. We then com- pute the CH 4 concentrations for 0.5 - 3 year period (the concentrations of the remaining three landfill gases are also computed, but not used). (ii) Since the ANN provides very accurate predictions for the period 0.5 - 0.7 year [see Fig. 2.13(a)], we use these predictions together with the (synthetic) data for the CH 4 concentrations in the first 0.5 year as the input to the GA to estimate the optimal permeability distribution. We then compute the CH 4 concentrations for the 0.7 - 3 year period. 58 Figure 2.14 compares the (synthetic) data with the predicted CH 4 concentrations for the first 0.7 - 1.0 year. Clearly, the inclusion of the ANN short-term predictions in the GA input data has improved the accuracy of the predictions. The same is true for longer periods of time. For example, Fig. 2.15 compares the data with the predicted CH 4 concentrations com- puted with the GA-determined optimal permeability distribution, with and without inclusion of the ANN short-term predictions as the input. Once a- gain, inclusion of the short-term predictions of the ANN has improved the accuracy of the GA-based predictions. Why is the combined algorithm more accurate than either of the GA or ANN alone? The ANN does not just pick up the trends in the input da- ta. It also constructs a quantitative relationship, albeit empirically, between such input data as the wells’ positions, temperature and pressure, and the output - the concentration of the gases. In addition, The ANN enhances the quality of the data that are supplied to the GA by narrowing its focus on the underlying pattern in the data. It other words, it narrows the domain of the plausible solutions to the optimization problem, and hence increases its accuracy. 2.5 Summary In this chapter, an artificial neural network was developed to pro- vide accurate short-term (up to several weeks) predictions for several im- 59 portant quantities at a large landfill in Southern California. An ANN with a back-propagation algorithm was used The results indicate that the ANN can successfully be trained by a portion of the data, and then be utilized for providing accurate predictions for the CH 4 , CO 2 , O 2 , and temperature profiles in the sector of the landfill in which the data had been collected. The data for the site that we studied are collected once or twice a month from a total of 450 wells. Thus, a trained ANN may be used by the landfill’s operator for short-term prediction and planning. In addition, the ANN may be used to identify the enhance oxygen zones in landfills, detect damage to their cover, and search for those areas in which underground fires might occur, which is where the concentrations of CO 2 and O 2 are large. As more data become available, the training and validation steps of the ANN can also be updated, in order to improve the values of the weights and biases that the ANN uses. The updating will then improve the ability of the ANN for providing short-term predictions. We also demonstrated that if short-term predictions by the ANN are used in conjunction with the GA, the accuracy of the longer-term predic- tions improve significantly, whereas the accuracy of the ANN predictions alone for the same time period deteriorates. This is particularly important in the view of the fact that the amount of the publicly-available data that may be used for the optimization process is typically limited. Use of a trained 60 ANN, however, enables one to expand the range of the accurate data that can be utilized in the optimization process. Thus, a combination of the model and the optimization technique de- scribed in the previous parts of this series, and the ANN model presented in the present chapter represents a powerful approach to developing real- istic models of landfills, to be used for making both short- and long-term predictions and planning. Considering the estimation of permeability distribution using the GA is not real-time, the next step is to develop an efficient approach, which could continuously update the permeability distribution. Therefore, we could obtain more accurate long-term predictions. This will be studied in the next chapter. 61 Σ Σ Σ F1 F1 F1 Σ Σ Σ F2 F2 F2 B11 B12 B21 B22 B2n CH4, CO2, O2,P, Well 1 CH4, CO2, O2,P, Well n CH4, CO2, O2,P, Well 4 T @ well 1 T @ well n T @ well 2 Input Layer Output Layer Hidden Layer B1n W11 W21 W13 W12 W21 W22 W33 W23 W31 W22 W32 W12 W22 W33 W22 W32 W31 W21 W23 Figure 2.1: The structure of the ANN used in simulations. 62 Input Target Neural Network with Biases & Weights Converge Print NO YES Figure 2.2: The ANN iterative procedure. 63 0 10 20 30 40 50 60 70 80 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Iteration Performance function for CH4 Training Validation Forecast Figure 2.3: Variations of the CH 4 performance function during the iteration for its global minimization. 64 0 50 100 0 10 20 30 40 50 60 70 80 90 100 Data (%Volume) Predicted (%Volume) 0 50 100 0 10 20 30 40 50 60 70 80 90 100 Data (%Volume) Predicted (%Volume) 0 50 100 0 10 20 30 40 50 60 70 80 90 100 Data (%Volume) Predicted (%Volume) R 2 =0.9357 R 2 =0.9241 R 2 =0.9409 Figure 2.4: Comparison of the ANN predictions for the volume fraction of CH 4 with the data for the, (left) trained; (center) validated, and (right) tested parts of the data set. 65 0 10 20 30 40 50 60 70 80 90 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Iteration Performance function for CO2 Training Validation Forecast Figure 2.5: Same as in Fig. 2.3, but for CO 2 . 66 0 50 0 5 10 15 20 25 30 35 40 45 50 Data (%Volume) Predicted (%Volume) 0 50 0 5 10 15 20 25 30 35 40 45 50 Data (%Volume) Predicted (%Volume) 0 50 0 5 10 15 20 25 30 35 40 45 50 Data (%Volume) Predicted (%Volume) R 2 =0.9393 R 2 =0.9346 R 2 =0.9472 Figure 2.6: Same as in Fig. 2.4, but for CO 2 . 67 0 10 20 30 40 50 60 70 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Performance function for O2 Iteration Training Validation Forecast Figure 2.7: Same as in Fig. 2.3, but for O 2 . 68 0 10 20 0 5 10 15 20 25 Data (%Volume) Predicted (%Volume) 0 10 20 0 5 10 15 20 25 Data (%Volume) Predicted (%Volume) 0 10 20 0 5 10 15 20 25 Data (%Volume) Predicted (%Volume) R 2 =0.9612 R 2 =0.9579 R 2 =0.9561 Figure 2.8: Same as in Fig. 2.4, but for O 2 . 69 0 10 20 30 40 50 60 70 80 90 100 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Performance function for T Iteration Training Validation Forecast Figure 2.9: Same as in Fig. 2.3, but for temperature. 70 25 100 150 25 60 80 100 120 150 Data Predicted 25 100 150 25 60 80 100 120 150 Data Predicted 25 100 150 50 60 80 100 120 150 Data Predicted R 2 =0.912 R 2 =0.889 R 2 =0.907 Figure 2.10: Same as in Fig. 2.4, but for temperature. 71 5 10 15 20 25 Figure 2.11: Spatial distribution of the original reference permeability in a horizontal layer of depth 14 m. 72 Figure 2.12: Top view of the grid structure used in the computations and the locations of the wells. 73 0.18 0.19 0.2 0.21 0.22 0.23 0.18 0.19 0.2 0.21 0.22 0.23 Actual CH4 Concentration ANN CH4 Concentration R 2 =0.99019 (a) 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.2 0.21 0.22 0.23 0.24 0.25 0.26 Actual CH4 Concentration ANN CH4 Concentration R 2 =0.76386 (b) Figure 2.13: Comparison of the ANN predictions with the actual (synthetic) data for the time period (a) 0.5 - 0.7 year and (b) 0.7 - 1.0 year. 74 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.2 0.21 0.22 0.23 0.24 0.25 0.26 Actual CH4 Concentration GA CH4 Concentration R 2 =0.90677 (a) 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.2 0.21 0.22 0.23 0.24 0.25 0.26 Actual CH4 Concentration GA CH4 Concentration R 2 =0.96172 (b) Figure 2.14: Comparison of the predicted CH 4 concentrations with the actu- al (synthetic data) determined after the permeability distribution was com- puted (a) by the GA alone, and (b) by the combination of GA and ANN methods. The results are for the period 0.7 - 1.0 year. 75 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 Actual CH4 Concentration GA CH4 Concentration R 2 =0.84596 (a) 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 Actual CH4 Concentration GA CH4 Concentration R 2 =0.90658 (b) Figure 2.15: Same as in Fig. 2.14, but year 2. 76 Chapter 3 Dynamic Updating of The Landll Model Using The Ensemble Kalman Filter 3.1 Introduction Modeling of gas generation and migration in landfills is important for accurate forecasting of the potential hazards that are associated with the landfill gas (LFG) generation. Therefore, gaining a better understanding of gas generation and transport in landfills has witnessed significant research activities in recent years. The composition and transport rate of the LFG depend on many factors, including temperature, pressure and moisture, as well as the spatial distributions of the permeability, porosity, tortuosity of a landfill, and the rates of the reactions that occur there. A landfill is essen- tially a large three-dimensional (3D) highly heterogeneous porous medium (see Chapter 1.4). Thus, similar to other type of porous media, in order to es- timate the gas composition accurately and, thus, avoid potential hazards, it is critical to characterize and estimate the factors that influence the accuracy of a landfill model by incorporating process data as they become available. As described in Chapter 1.3, a comprehensive 3D landfill model has been developed [37, 72, 73, 74, 55], which accounts for the physical structure 77 of a landfill, such as the spatial distributions of its permeability and porosity, and generation and transport of the LFG in it. In this chapter, we focus on determining those important physical factors that cannot be measured directly and can be only inferred indirectly, while honoring the production data, as well as updating the model as more data become available in real time. Such a process is commonly called history matching. Sanchez et al. [73] and Li et al. [55] already reported the preliminary results of the study of this aspect of the problem. In the present chapter, we expand on the previous work, and propose an efficient method that, to our knowledge, has never been used in landfill modeling. In principle, history matching is formulated as an optimal control problem, aiming to minimize the difference between the observed (mea- sured data) and simulated values. Manual history matching was first pro- posed to adjust model variables using good engineering judgement and a workflow developed based on years of experience [87]. However, manual history matching involves extensive trial and error iterations and might be biased by experience of the user. For this reason, it is desirable to develop an automatic, free of bias, algorithm for history matching that is applicable to complex systems, but does not require a large amount of computing, in order to provide reasonable estimates of the parameters of interest. Thus, many automatic history matching methods, such as the evo- 78 lutionary algorithms [82, 68, 87, 73, 55], the so-called gradual deformation method [67, 42] and gradient-based methods (GBM) [7, 10, 53] have been developed. Other powerful techniques, such as simulated annealing [49], have also been developed and used by Hamzehpour and Sahimi [33, 35, 34] and others (for comprehensive reviews see Sahimi and Hamzehpour [71]; Sahimi [70]) for optimization of models of large-scale porous media based on limited data. Over the past several decades automatic history matching approach- es, especially the GBM, have been used widely in various industries. The GBM require the computation of a sensitivity coefficient matrix in order to further calculate the gradient of the objective function, i.e. the function whose minimization leads to the optimal estimates of the parameters (see below). Two approaches, namely, the sensitivity-equation approach [7] and the adjoint-system method [43], may be employed to compute the sensitiv- ity coefficient matrix. Both methods depend on the functional relationship between the model variables and the data, which is unique for a specific model and the associated numerical simulator. But, a specific method must be designed for each numerical simulator in order to calculate the sensitiv- ity coefficient matrix that is used in such methods (see below). Moreover, the calculation of the gradient of the objective function is computational- ly extensive, especially for highly nonlinear and large-scale systems, such 79 as models of landfills that we study. In addition, the GB history matching does not allow for continuous model updating, which requires that all the measured data to be used in order to carry out the history matching simulta- neously. When new data are available, the entire updating procedure must be repeated to assimilate all the measured data. In recent years, the increased deployment of sensors for monitor- ing temperature, pressure and gas composition in landfills has provided an appealing incentive for studying the application of continuous updating methods to modeling of landfills. On the one hand, considering the fre- quency of data output in industrial systems, it is inefficient to repeatedly use all the past data for estimating the unknown parameters. On the other hand, it is also critical to incorporate new data in the modeling as soon as they become available, so that the model and its parameters always honor the existing data. Both the heavy computational burden and high frequen- cy of data updating require a new history-matching approach for real time and continuous re-evaluation of the model. The Kalman filter (KF), devised by Kalman [46], has historically been used to continuously update the parameters of models of dynamical systems for assimilating production data. The KF is optimal only under the assumption that the system is linear and the measurement and process noise follow Gaussian distributions. For highly nonlinear systems, howev- 80 er, the KF fails. To address the failure, the extended KF (EKF) was proposed, whereby a statistical approximation of the nonlinear equation was used. More specifically, the nonlinear characteristics of the system’s dynamics were approximated by a version of the system that was linearized around the last state estimate. The EKF has gained popularity due to its ability for handling nonlinear systems and non-Gaussian noise. Evensen [22] studied the formulation of the EKF for a multilayer quasi-geostrophic ocean model, and found a closure problem associated with the EKF in the evolution equation for the error covariance. The linearization used in the EKF discards higher-order moments in the equation that governs the evolution of the error covariance. But, this kind of closure technique results in an unbounded error growth, which is more pronounced for large-scale, highly nonlinear systems. Therefore, the ensemble KF (EnKF) [23] was introduced to alleviate the closure problem. The EnKF is, in fact, a Monte Carlo method whereby the model states and uncertainty are represented by an ensemble of realizations of the system. Due to its conceptually-simple formulation and relatively low computation, the EnKF has gained increasing attention in history matching problems and continuous updating of models, as new data become available. Its power is due to the fact that, instead of computing the state covariance using a recursive method, the EnKF estimates the covariance matrix from a number 81 of realizations. By using the Monte Carlo method in the forecasting step, the EnKF updates a model’s variables as soon as new data are available and, hences, keeps the model current. The EnKF has been shown to be very efficient and robust for real-time updating in various fields, such as weather forecasting [41], oceanography [38] and meteorology [41]. The success indicates that the EnKF is capable of handling highly nonlinear and complex systems. Echevin et al. [21] applied the EnKF to the coastal version of the Princeton Ocean Model [1], a computer simulator for modeling of oceans, and studied the horizontal and vertical structure of the covariance functions. The EnKF was shown to be more consistent with the dynamic coastal system than the traditional statistical method. As far as application to problems involving large-scale porous media are concerned, Nævdal et al. [27] implemented the EnKF in monitoring of the near-well zones in an oil reservoir, in order to estimate the reservoir’s permeability distribution. Furthermore, they were able to obtain accurate predictions for the reservoir’s oil production. Gu and Oliver [29] examined the application of the EnKF to an oil reservoir model, and obtained accurate estimation of quantities of interest, using a fairly small ensemble and low computational cost. Skjervheim et al. [80] used the EnKF to incorporate 4D seismic data in the simulation of models of oil reservoirs. It was shown that the EnKF can handle a large amount of data resulting from 4D seismic 82 measurements. A landfill is a large-scale porous medium, and the phenomena that occur there are highly nonlinear. To model gas generation and transport in such a large-scale system, a large number of grid blocks must be used in the computational grid that represents a landfill. Each grid block is associated with several variables, such as the horizontal and vertical permeabilities, porosity, tortuosity and the reactivity. Owing to its advantages, the EnK- F is ideally suited for providing accurate estimates for the parameters of such a complex problem. A key obstacle to using the EnKF to continuously update a landfill model is that, only limited experimental data are avail- able for the physical properties. The limited data are usually available in terms of CH 4 /CO 2 concentrations and the pressures measured at certain extraction/observation wells. All history matching methods, including the EnKF, also need some accurate initial guess for the physical properties to be estimated. Therefore, our first objective in this chapter is to obtain ac- curate initial guess for the parameters, estimated from the initial available data. Multiple ensembles are then generated by a Monte Carlo approach, known as the sequential Gaussian simulation (SGS). The EnKF is then im- plemented to update the parameters - the permeabilities in the our work - by minimizing the covariance of the forecast error. Thus, in this chapter we propose a novel approach for estimating the 83 permeability distribution in a landfill, based on limited experimental data. As described below, the method is based on a combination of the GA and EnKF. Other parameters, such as the porosity and gas tortuosity factors, may also be estimated and updated, albeit at a higher computational cost. The rest of this chapter is organized as follows. In the next section, we describe the SGS and how it is used in the model. How the initial esti- mate of the spatial distribution of the permeabilities is obtained is described in Sec. 3.3. We then describe the use of the EnKF for estimation and updat- ing of the permeabilities in Sec. 3.4. The results are presented and discussed in Sec. 3.5, while our work is summarized in the last section. 3.2 Generation of Synthetic Ensemble of Data The model of gas generation and transport described in Chapter 1.3 is used to generate synthetic data based on a reference spatial distribution of the permeabilities. To obtain a realistic permeability distribution, we must have some information about the permeabilities, even if very limited. Here, we use the same approach as that in Chapter 2.4.2 to generate the reference permeability distribution. Sequential Gaussian simulation is an efficient method for generating multiple equiprobable realizations of the important parameters of models of large-scale porous media [20, 19, 70] that also honor the existing data, i.e. the known permeabilities at the ten grid blocks. It is quite straightforward 84 to use the SGS for generating the distribution of the permeabilities, condi- tioned on the given data. It uses the kriging algorithm [70] to set the mean and variance of the distribution. The kriging algorithm provides estimates with a minimum error variance of the unsampled values, can do so only for the data that follow a Gaussian distribution. The assumption of normality of the data is not, however, restrictive for real world data, because they can be transformed to another set of data that follow the Gaussian distribution. The steps in using the SGS involve [20, 19, 70], (1) setting up a computational grid and coordinates system; (2) performing normal-score transformation on the known data to transform them to a Gaussian distribution; (3) defining a random path that visits each grid point (block) once. The random path guarantees that multiple realizations generated by the SGS are equiprobable; (4) using kringing to construct the conditional probability distribu- tion function (CPDF). In the present work we include the permeabilities of the previously estimated grid blocks as data in order to preserve the proper covariance characteristics between the estimated values; (5) generating a synthetic permeability from the CPDF for each node and proceeding to the next node, and (6) back transforming the generated normally-distributed permeabil- ities. 85 Thus, we first construct a reference permeability distribution. Then, by incorporating the reference permeability into the landfill simulator, syn- thetic data for gas production, such as the compositions and pressures mea- sured at the extraction and observation wells, are generated. Based on the known permeabilities and the aforementioned two important attributes of the spatial distribution of the permeability, we generate a realistic reference spatial distribution by the following the same procedure as in Chapter 2.4.2, except that the permeabilities at the top layer below the cover are assumed to be ten times larger than those at the bottom. The physical size L x L y L z of the model landfill is assumed to be 30 m 30 m 30 m, represented by a computational grid of 57 57 37 blocks, where z denotes the vertical direction. The cover’s thickness is assumed to be 2 m with an isotropic permeability of 0.1 md. The grid blocks around the extraction wells are more resolved than the rest of the blocks. The landfill walls are assumed impermeable. Figure 3.1 depicts a 3D view of the structure of the computational grid that represents the landfill. Figure 3.2 displays the reference permeability distribution. Figure 3.3 presents the permeability distribution of a horizontal layer of the refer- ence system at a depth of 15 m from the top. The landfill model is assumed to contain four asymmetrically located extraction wells (see Figs. 3.1 and 3.2), where the vacuum pressure is assumed to be constant. To show the 86 permeability variations among the realizations, we show in Fig. 3.4 four re- alizations of the horizontal permeability distribution in one plane. Through this step together with the GA, multiple permeability ensembles are creat- ed, which serve as the initial ensembles for the EnKF updating (see below). 3.3 Initial Permeability Estimates To use the EnKF to update the permeability distribution, some accu- rate initial guess for the spatial distribution of the permeability must first be obtained, in order to generate the initial ensembles. Given a limited amount of data on the concentration and pressure of a gas, such as CH 4 , we must estimate the initial optimal distribution of the permeability. Several opti- mization methods are available (see Sahimi and Hamzehpour [71], for a comprehensive review), such as the gradient-based optimizers (GBO), sim- ulated annealing (SA) [49], and the Genetic Algorithm (GA). The GBO and SA have been used widely in many areas, including development of opti- mal models of large-scale porous media, such as oil reservoirs [33, 35, 34] , and have been shown to be effective in solving many optimization prob- lems [19]. One of their shortcomings is that they involve extensive com- putation. Moreover, they are not convenient for parallel computation. An accurate model of a landfill is characterized by several parameters for each grid block. A typical landfill model may contain tens of thousands of grid blocks, if not larger. Therefore, the problem is a large-scale optimization 87 one that involves heavy computations. In contrast, the computation time of the GA is small relative to the calculation of the objective functions. In addition, the calculations with the GA are amenable to parallel computing (see below). Thus, to obtain an initial estimate of the spatial distribution of the permeability, we use the GA for use in landfill modeling. But, since the problem to solve here is a large-scale one and our goal in using the GA is to obtain a rough initial guess of the permeability distribution, we do not need to estimate the permeabilities for every grid block. We, therefore, estimate them at some more important grid blocks, such as those that around the wells, and estimate the permeability of the remaining blocks by interpola- tion using the SGS, which speeds up the computations with the GA. In the present problem the CH 4 concentrations at four extraction wells, computed by solving the CDR equation in the computational grid and the reference permeability distribution over a time period of one year with the time step of 0.1 year, are used as (synthetic) the data to begin the optimization by the GA. The first generation of the permeabilities, to be used in the GA, is selected randomly within a specified range. For the subsequent generations, the selection is based on the minimization of the value of objective functionF , defined by F = 1 N N X i=1 (s i d i ) 2 ; (3.1) 88 wheres i andd i are simulated and measured data, andN is the number of data points. Figure 3.5 shows the typical evolution of the objective function over 400 generations. After 400 generations, the computations with the GA are terminated, and the resulting distribution of the permeability at the grid blocks are used in the subsequent computations. Figure 3.6 shows the per- meability distribution at a horizontal layer with the depth of 15 m, obtained by the method. 3.4 Dynamic Updating: Ensemble Kalman Filter 3.4.1 Algorithm Description Use of the EnKF consists of three steps: (1) a forecasting step that uses the landfill simulator to generate syn- thetic data by solving for the concentrations of the LFG; (2) an analysis step that computes the so-called Kalman gain (see below) based on the available measured data and the simulation results, and (3) updating of all the parameters - the permeabilities - using the EnKF updating scheme. One defines a state vector that includes two sets of parameters: those to be estimated - the permeabilities - and the measured variables that in- clude CH 4 partial pressures or concentrations computed (measured) at 100 observation wells and the four extraction wells. The state vector for theith 89 ensemble (realizations of the permeability distribution) is defined by i = k i d i ; (3.2) where k i and d i are, respectively, vectors of the estimated permeabilities and measured data in theith ensemble. The choice of the numberN e of the ensembles will be discussed shortly. The dimensionN p of the permeability vector is 57 57 29, including those of the cover, and the numberN d of the measurement (data) at the sensors is 312, with three sensors along each well. Thus, we write the state vector in the following way, Y = ( 1 ; ; Ne )2 R nNe ; n =N p +N d : (3.3) The EnKF uses the covariance of N e ensembles to approximate the true covariance matrix C y : C y = 1 N e 1 Y Y Y Y T ; (3.4) where Y, a vector of sizen, is the mean of the state vector, and T denotes the transpose operation. The ensembles of the observations are defined by, d i = d +" i ; i = 1; 2;:::;N e : (3.5) where " i are perturbations or noise added to the data. They are selected from a Gaussian distribution with zero average and a standard deviation that is a small percentage (5 percent in the current work) of the average 90 of d i . Here, D = (d 1 ;d 2 ;:::;d N ) T is related to the state vectors by, D = HY, with H being the matrix that selects the measured data from the state vector. Therefore, H has the form, H = [0jI], where 0 is a zero matrix with dimension ofN d (N y N d ), withN y being the dimension of Y. Note that random perturbations " i are added to the measured da- ta in order to create an ensemble of observations. As shown by Burgers et al. [12], if there is no random noise, the ensemble’s variance will be under- estimated and the EnKF updating will fail. The covariance matrix for the measurement error is given by C d = 1 N e 1 EE T ; (3.6) where E = (" 1 ;:::;" N )2 R N d Ne , and EE T is simply the co-variance of " i . The analysis step for the EnKF calculates the Kalman gain K as follows, K = C y H T (HC y H T + C d ) 1 = C y H T P 1 ; (3.7) where P is simply the sum of the two co-variance matrices. The state vari- ables are then updated using the Kalman updating scheme, a i = b i + K(d i H b i ); (3.8) with b i and a i being the state vectors before and after the updating, re- spectively. One iteration of the EnKF is completed when its three steps are 91 carried out. One then proceeds to the next time step at which new mea- sured data are available, using the updated permeability distribution. The flow chart of the entire procedure is illustrated in Fig. 3.7. 3.4.2 Implementation of The EnKF To produce a reliable and robust EnKF updating for landfill model- ing, we must also address the following important issues. 3.4.2.1 Low Rank Representation of Covariance Matrix In the expression for the Kalman gain, the covariance matrix is de- fined by P = HC y H T + C d = HA 0 (A 0 ) T H T + EE T ; (3.9) where A 0 = AAI Ne and I Ne is aN d N d matrix whose entries are all equal to 1=N e . But, because in the present problem, N d > N e , P is numerically singular and, thus, it is impossible to calculate its inverse directly. There- fore, we must use a pseudo-inverse method to calculate the Kalman gain. Evensen [24] proposed a stable pseudo-inverse method that computes the inverse in the (N e 1)-dimensional ensemble space, rather than in theN d - dimensional measurement space, which is as follows. Let S = HA 0 and, thus, the covariance, P = SS T +EE T . The singular- value decomposition of S, which represents its factorization, is given by S = U 0 0 V T 0 ; (3.10) 92 with U 0 2 R N d N d , 0 2 R N d Ne , and V T 0 2 R NeNe . Here, 0 = diag( 1 ;:::; Ne1 ; 0) is a diagonal matrix that contains all the singular values of S, 1 ; ; Ne1 , ordered from the largest to the smallest. U 0 is a left-multiplier unitary matrix that contains all the singular vectors of S, as is V 0 that is the same as U 0 , except that it is a right multiplier. S, a N d N e matrix withN d >N e . The sum of all of its rows is zero, indicating that they are not all independent. Thus, the rank of S is at most N e 1. In practice, given S and the fact that both U 0 and V 0 are unitary matrices (recall that for such matrices, U 0 U T 0 = I) suffice to determine 0 (determine 1 ; ; Ne1 ). Then, the inverse of S is, S 1 = V 0 1 0 U 0 . As last diagonal element of 1 0 is infinite, the direct inverse of S is singular. On the other hand, the original linear space of S contains all the sin- gular vectors corresponding to the singular values, including the smallest one, zero. To get around the singularity of S 1 we introduce a subspace , so as to eliminate the effect of the zero singular value. The subspace is defined by theN e 1 nonzero singular vectors of S. This then makes it possible to compute a stable pseudo-inverse of S, given by S 1 = V 0 1 0 U 0 ; 1 0 = diag( 1 1 ;:::; 1 Ne1 ; 0): (3.11) Therefore, the expression for P is given by P = SS T + EE T = SS T + (SS 1 )(EE T )(SS 1 ) T = SS T + EE T : (3.12) 93 By using E = (SS 1 )E, instead of E, we project E onto the subspace, which helps us avoid the loss of rank and, thus, compute a stable pseudo-inverse [48]. 3.4.2.2 Unphysical Permeabilities When implementing the EnKF, it is possible to obtain negative per- meabilities at some grid points, which is unphysical. One method to ad- dress the problem is to apply a normal-score transformation of all the per- meabilities, so that the transformed values follow a Gaussian distribution, and then utilize them in the computations, instead of the original perme- abilities. An example is shown in Fig. 3.8. After implementing the EnKF and obtaining new estimates of the permeabilities, we back transform all the computed permeabilities to obtain estimates in the original range. 3.4.2.3 Number of Ensembles Implementing the EnKF requires an adequate number of ensembles. A small number of ensembles might not guarantee an accurate approxima- tion of the true covariance matrix, hence underestimating it that will lead to the divergence of the EnKF. On the other hand, a large number of ensembles requires extensive computation. To select the appropriate number of en- sembles, we performed four series of simulations, using 40, 80, 150, and 200 ensembles. All the ensembles were generated by the SGS method described 94 earlier, using the same initial guesses and data for the permeability distri- bution that was computed by the GA. The average root mean-square error for the four cases are compared in Fig. 3.9, which indicates that N e = 80 ensembles yield accurate results. Hence, all the subsequent computations with the EnKF were carried out with 80 ensembles. 3.4.2.4 Parallel Computing As already mentioned, the landfill is represented by a large compu- tational grid that requires extensive computations. To implement the EnKF, we must solve the governing equations of gas generation and transport a large number of times, in order to generate the data for each ensemble. The advantage of the EnKF is that since each ensemble requires only the solu- tion of the governing equation for that ensemble, the calculations for each ensemble is independent of those for all other ensembles, implying that im- plementation of the EnKF is amenable to parallel computing. Hence, we used a master-slave parallel computational strategy [73] for implementing the EnKF, with the slave processors solving the governing equation for gas generation and transport in the landfill given each ensem- bles, and the master processor collecting all the results to perform real-time Kalman updating of the permeability distribution. The message-passing interface was the platform used for the parallel computing. The number of the slave processors used in the EnKF was 80, the same as number of 95 ensembles. 3.5 Results and Discussions To solve the governing equations we set the time step in the simu- lation to be 0.01 year, and assumed that the measured data, i.e. the con- centrations and pressures of the LFG, computed with the latest updated permeabilities, are available every two time steps over a period of 1.5 years. This means that we use the EnKF to update the permeabilities every two time steps, so that the total number of updating steps is 75. The quality of the estimates is assessed in terms of the root mean-square errors and the ensemble spread, which are computed as follows, i = v u u t 1 N g Ng X l=1 (k j l k j r ) 2 (3.13) i = v u u t 1 N g Ng X l=1 (k j l k j ) 2 ; (3.14) whereN g is the number of grid points, which is the same as the number of permeabilities to be estimated (excluding the permeability of the cover),k j l the estimated permeability at thejth grid block of thelth ensemble,k j r the reference permeability at thejth grid block of theith ensemble, andk j the average permeability of all the ensembles at thejth grid block. i measures the error between each ensemble i and the actual distribution, which, in the present problem, is known. In practical applications, however, one may 96 not know the real distribution of the physical properties to be estimated, in which case i is used to evaluate the performance of the EnKF, which actually denotes the standard deviation of all the ensembles. Figures 3.10 and 3.11 present, respectively, the evolution of and with the updating steps. They indicate the improvement of the estimates with time. Over the first few time steps, both and decrease sharply. This is because the measured (synthetic) data earlier times carry a signif- icant amount of new information about the system. After some time, the incoming data possess less new information and, thus, and decrease more slowly. Figure 3.12 illustrates the estimated horizontal permeability distribu- tion at the depth of 15 m after 75 updatings. We observe that the estimated permeabilities around the extraction wells are relatively smaller than the other grid blocks, which agrees with the same feature of the reference per- meability distribution. However, the estimated permeability distribution is not exactly the same as the reference one. This is caused by the error in the application of the GA, for which a coarser grid was used to speed up the computations. If the initial ensembles contain some errors and do not re- produce the reference permeability, there will also be some estimation error in the EnKF updating. In order to evaluate the performance of the EnKF updating, we car- ried out simulation of gas generation and transport in the landfill model for 97 a four year period, using the average of the permeabilities among all the estimated ensembles. Figure 3.13 compares the reference and predicted re- sults of the pressure and CH 4 concentrations, computed in an observation well near the center. The concentrations are in excellent agreement, while the pressures differ only slightly. Figure 3.14 presents the predicted concentrations and pressures un- der a dynamic conditions in which two wells were shut down after 0.5 and 1.5 years. Once again, there is excellent agreement between the measured and simulated CH 4 concentrations, but there is a small difference between the pressures. The relative error of pressure predictions under the steady- state condition is only 0.5 percent, while under the dynamic condition is about 2 percent. These results demonstrate the effectiveness and robust- ness of the EnKF estimation and updating of the parameters of a realistic 3D model of landfills. 3.6 Summary We presented a new approach for estimating and updating the spa- tial distribution of the permeability in a 3D model of landfills by assimi- lating limited available data. A combination of the genetic algorithm and the sequential Gaussian simulation was used to generate multiple initial en- sembles of the permeability distribution. The ensemble Kalman filter was successfully utilized in order to carry out the process of estimating and up- 98 dating the permeability distribution, by incorporating real-time (synthetic) data. Several issues that are relevant to the implementation of the EnKF, such as the low-rank representation of the covariance matrix, unphysical values of estimates, and the necessity of parallel computing, were also dis- cussed. The results demonstrated the effectiveness and applicability of the EnKF updating. After obtaining an accurate landfill model using the EnKF, the next step is to investigate how to operate an landfill efficiently and economically, by adjusting landfill settings. This procedure is commonly called produc- tion optimization and will be studied in the next chapter. 99 Figure 3.1: 3D view of the grid structure, used in the computations. The locations of the four extraction wells are shown by the circles. 100 Figure 3.2: 3D view of the grid structure and the reference permeability distribution. The permeabilities are in md. The location of the extraction wells are shown by the circles. 101 10 15 20 25 30 35 40 Extraction Wells Figure 3.3: The reference permeability distribution in a horizontal layer at a depth of 15 m. The permeabilities are in md. 102 10 20 30 40 50 60 70 (a) 10 20 30 40 50 60 70 80 90 (b) 10 20 30 40 50 60 70 80 (c) 10 20 30 40 50 60 70 80 (d) Figure 3.4: Four realizations of the reference permeability distribution in the same horizontal layer as in Fig. 3.3, generated from the same initial data. The permeabilities are in md. 103 0 50 100 150 200 250 300 350 400 0.005 0.01 0.015 0.02 0.025 Generation Number Objective Function Figure 3.5: Evolution of objective function with the iterations. 104 2 3 4 5 6 7 8 9 10 Extraction Wells Figure 3.6: Spatial distribution of the permeability in a horizontal layer at the depth of 15 m, obtained by the genetic algorithm. The permeabilities are in md. 105 Figure 3.7: The flow chart of the entire computations. 106 0 50 100 0 0.5 1 1.5 2 2.5 Original Permeability N k × 10 −4 −5 0 5 0 0.5 1 1.5 2 2.5 3 Transformed Permeability N k × 10 −4 Figure 3.8: The distributionN k of the original permeabilities (left) and their transformed distribution (right) after normal-score transformation. 107 0 5 10 15 20 15.5 16 16.5 17 17.5 18 18.5 19 Number of time steps δ e 40 ensembles 80 ensembles 150 ensembles 200 ensembles Figure 3.9: Evolution of the estimation error e for various number of en- sembles. 108 0 50 100 150 12 14 16 18 20 22 24 Number of time steps ξ 80 ensembles Figure 3.10: Evolution of, the error between each ensemble and the actual distribution (Eq. 3.14), with time for 80 ensembles. 109 0 50 100 150 0 2 4 6 8 10 12 14 16 18 Number of time steps χ 80 ensembles Figure 3.11: Evolution of , the standard deviation of all the ensembles, with the time for 80 ensembles . 110 15 20 25 30 35 Extraction Wells Figure 3.12: The spatial distribution of estimated permeability in a horizon- tal layer at a depth of 15 m. The permeabilities are in md. 111 0 50 100 150 200 250 300 350 400 1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04 1.045 1.05 Number of time steps Pressure (atm) (a) Data Simulated 0 50 100 150 200 250 300 350 400 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Number of time steps CH4 Concentration (b) Data Simulated Figure 3.13: Comparison of the estimated pressure (a) and CH 4 concentra- tion (b) with the data for a well near the center, and their evolution with the time. 112 0 50 100 150 200 250 300 350 400 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 Number of time steps Pressure (atm) (a) Data Simulated 0 50 100 150 200 250 300 350 400 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Number of time steps CH4 Concentration (b) Data Simulated Figure 3.14: Same as in Fig. 3.13, except that two of the wells were shut down after some time, as indicated by the jumps in the pressures. 113 Chapter 4 Model-Based Production Optimization of A Land- ll Gas System 4.1 Introduction As a result of economic and population growth in the past few decades, the current oil consumption is using up the world’s oil resources. It is reported that the world oil production is reaching a peak plateau and the number of significant discoveries is decreasing very year. At the same time, people are seeking for new energy resources to meet the high demand of energy. The biodegradation in a modern landfill can generate a large amount of landfill gas (LFG). LFG, which consists mainly of CH 4 and CO 2 , has become a promising source of green energy in recent years. In modern landfills, it is challenging and important to produce natural energy resources efficiently and economically by smartly adjusting landfill settings. This procedure is called production optimization in the reservoir engineering, which uses advanced control technologies to increase the oil recovery. Specifically, production optimization can provide a way to maximize the cumulative oil production through controlling many critical factors, such as the placement of wells, waterflood operations 114 and injection rates. Similar to a reservoir system, a landfill is also a large three-dimensional (3D) reactive porous medium. Therefore, we can apply techniques in reservoir engineering to increase the recovery of LFG, which, to our knowledge, have never been used in the production optimization of landfills. The problem of production optimization can be formulated as max- imizing an economic objective functionf(x) (e.g. net present value (NPV)) in a specific system by manipulating x. Here, x can be regarded as a set of control settings, such as landfill’s well placement, design of the landfill cover, and the extraction wells’ vacuum pressure. A variety of optimization techniques have been proposed and widely used in industries. The existing optimization methods can be classified into two major categories: gradient- based methods and gradient-free methods. The gradient-based methods, including steepest ascent method, Quasi-Newton method [28] and sequential quadratic programming (SQP) [11], need the gradient of objective function with respect to control vari- ables. The adjoint-based method, sensitivity method and finite difference method are proposed to compute the gradient. In recent decades, these gradient-based methods have been widely used in petroleum engineering and proved to be very efficient with fast convergence. The adjoint-based method obtains the gradient by solving an adjoint model using the Jacobian 115 matrix extracted from the simulator. The adjoint-based method was first proposed by Chen et al. [15] to compute gradients for history matching and authors thoroughly tested its practical use in a single-phase reservoir model. The method was shown to be effective and computationally effi- cient, compared to the standard constant-zone methods. For multi-phase systems, Wu et al. [75] developed a discrete adjoint method for computing sensitivity coefficients in a two-phase flow system. Their method could generate the sensitivity of production data to model parameters, which can facilitate the application of the fast convergent Gaussian-Newton algorithm. Li et al. [56] presented a general formulation of calculating sensitivity matrix for a fully implicit three-phase flow system. They also found that the coefficient matrix was simply the transpose of the Jacobian matrix in the Newton-Raphson solution of the implicit simulator. With the known Jacobian matrix, the process of deriving adjoint equations can be avoided. By following this idea, Sarma et al. [76] used an adjoint model for efficient calculation of gradients of NPV with respect to well control settings. These gradients were then used by SQP method to optimize the production. The authors demonstrated the success of their method for production optimization and closed-loop reservoir management. However, the adjoint-based method requires detailed knowledge of the process model and access to the process simulator, which are difficult to 116 obtain for complex and large-scale systems. By contrast, sensitivity and finite-difference methods treat the process simulator as a black box and only need the output of the simulator to evaluate the objective function. Jacquard and Jain (1965) introduced a method for calculating sensitivity coefficients of a linear system. They applied the method, together with the steepest ascent method, for history matching in a two-dimensional single phase reservoir by assimilating pressure measurements. Wang et al. [85] used SQP with numerical gradients to solve the production optimization problem in a conventional black-oil reservoir simulator. In their work, the gradient information was computed efficiently by finite difference methods estimated by Sparse Nonlinear Optimizer (SNOPT), a software package for solving large-scale optimization problems. The second category of optimization algorithms, gradient-free meth- ods, are actually stochastic methods, including the simulated annealing (SA) [50] and genetic algorithm (GA) [39]. The SA method has been used for developing optimal operating rules of reservoir systems, such as well placement, operation scheduling, and injection/extraction rates. Teegavarapu et al. [84] investigated and evaluated the effectiveness of SA for operation optimization to multi-reservoir systems. They showed that SA could be used for obtaining near-optimal solutions for multi- period reservoir operation problems that are computationally intractable. 117 Bangerth et al. [9] specifically examined the optimal positions of new injection/production wells to maximize the economic benefits of a given reservoir. They compared the performance of SA and finite difference gradient methods, and demonstrated that SA is very efficient in finding near optimal solutions of the optimization problem. The main disad- vantage of the SA is its incapability of being used in massively parallel computations. Therefore, for large-scale porous media systems, such as reservoirs and landfills, SA will fail due to the severe computation burden. By contrast, GA is amenable for parallel computing [54]. Therefore, GA is most commonly used in industries and has been applied for optimizing the well placement problem in petroleum engineering, where the objective is to optimize well type, location and scheduling. Harding et al. [36] applied GA to maximize NPV using problem-specific crossover operators. Comparing with the results of SA and SQP , they proved that GA has apparently significant performance improvements. Almedia et al. [6] used GA for the management of petroleum intelligent fields by selecting optimal valves control configurations. They even considered the existence of uncertainties in valve operations, such as the risk of valve failure. Their results showed the intelligent valve control using GA can reveal significant gains. Howev- er, gradient-free methods require tens of thousands of evaluations of the objective functions, which might still be time consuming for large-scale and 118 complex systems, such as reservoirs and landfills. To handle the problem of production optimization for reservoir systems, Chen et al. [16] proposed ensemble-based optimization (EnOpt), which can be regarded as a combination of gradient-based methods and stochastic methods. Similar to the Ensemble Kalman Filter (EnKF), where the covariance matrix of the state vector is approximated by an ensemble of model realizations, the gradient information of EnOpt is also acquired from an ensemble of control systems. The advantage of EnOpt is that it treats the simulator as a black box, so the access to the simulator code is not required. After the gradient information was obtained, Chen et al. [16] used the steepest ascent method to iteratively update the control variables. The applicability of the EnOpt was assessed through the use of a synthetic reservoir example, where a significant increase of NPV was observed. To improve the convergence speed, Chaudhri et al. [14] developed conjugate gradient EnOpt (CGEnOpt), where the search direction was determined by a conjugate gradient. This method was applied to a synthetic 3-D reservoir model and the effectiveness of the CGEnOpt to improve the convergence of optimization was demonstrated. Su et al. [83] showed the application of EnOpt on a large-scale reservoir model with 200,000 cells. Even for such complex system, the cumulative water production was reduced by 50% only after a few iterations. Leeuwenburgh et al. [52] discussed 119 several aspects of EnOpt, in terms of the ensemble size, perturbation, regularization and smoothing, by comparing it with the adjoint approach. They also investigated the effect of control time scale on the optimization performance for the well placement problem. EnOpt was also employed on the capacitance resistive model (CRM) to efficiently optimize large waterflooding operations (see Jafroodia and Zhang [44]). Their simulation result showed that the optimal injection rates could achieve a significantly higher oil production and a improved reservoir sweep efficiency. All these efforts show that EnOpt can work efficiently in the optimization of large-scale reservoir systems. Considering the similarity of reservoirs and landfills, we choose EnOpt for the production optimization of a landfill, which is a large-scale porous medium and the phenomena that occur there are highly nonlinear. The rest of this chapter is organized as follows. In the next section, we first briefly describe the formulation of the production optimization problem in the landfill system. In Section 4.3.1, we give a review of EnOpt and CGEnOpt methods proposed by Chen et al. [16] and Chaudhri et al. [14], respectively. Next, we show our interpretation of EnOpt from another aspect, which can further simply the algorithm. Some issues concerning with the application of EnOpt, such as step size selection, parallel comput- ing and implementation procedure, are also discussed. Then GA, EnOpt 120 and CGEnOpt methods are all applied to the production optimization of the same landfill problem without constraints. The comparison of the optimization performance for these algorithms are given in Section 4.3.6. We demonstrate that all the methods achieve a significant increase of NPV and CGEnOpt can get best convergence, which was also validated by Chaudhri et al. [14]. For constrained production optimization, we first look into the common techniques for solving it and then propose a novel parameterless GA algorithm in Sec. 4.4.2. Then, the applicability of the newly proposed method, parameterless GA, is tested and evaluated by the constrained optimization of the same landfill system in Section 4.4.3. The last section presents a summary of the results. 4.2 Optimization Problem Formulation For model-based optimization, the first step is to define a simulation model to describe the problem. Here, we use the same comprehensive 3D model of gas generation and transport as we used in the Chapter 3. The physical sizeL x L y L z of the model landfill is assumed to be 30 m30 m30 m, represented by a computational grid of 575737 block- s, wherez denotes the vertical direction. Figure 4.1 depicts a 3D view of the structure of the computational grid that represents the landfill. The landfil- l model is assumed to contain nine symmetrically located extraction wells (see Fig. 4.1). Due to the limited access to the real landfill, the landfill model 121 is subject to high geological uncertainty, which is permeability distribution in our case. Fortunately, we are able to retrieve permeability information from the measurement data by performing real-time history matching us- ing EnKF (see Chapter 3 for detail). Therefore, we can use the updated permeability distribution from EnKF to eliminate the geological uncertain- ty mentioned above. Figure 4.2 displays the final permeability distribution obtained from Chapter 3. We define x as the vector of control variables, which are vacuum pressures in the extraction wells, given by x = (x 1 ;x 2 ;:::;x Nx ); (4.1) where N x , the total number of control variables and also the number of extraction wells, is 9 in our current landfill model. The NPV over a period of 2 years, which is the objective function to maximize, is defined by g(x) = Nx X i=1 " Nt X j=1 f j i (x) j h i (x i ) # ; (4.2) whereN t is the total number of time steps,N x is the number of extraction wells, f j i (x) is the cumulative production of CH 4 for welli over time step 4t j , j is the price of natural gas over time step4t j , andh i (x i ) is the oper- ational cost associated with the vacuum pressure in theith extraction well. 122 Figure 4.3 presented the predicted gas price for the next 2 years, based on the historical gas price for the past 30 years. Figure 4.4 presents a simple profile ofh i (x i ) with respect to vacuum pressure in an extraction well. Moreover, the operation of the real landfill always has different kinds of constraints, such as constraints on production, budget and safety. For the landfill optimization problem, we consider a simple nonlinear constraint, the maximum flow rate for extraction wells. This is a common and reason- able constraint, due to the limitation on the ability of transportation and treatment. If excessive natural gas escapes to the environment, it would lead to emissions to the atmosphere which harms the environment. For the constrained optimization with the maximum flow rate con- straint, the production optimization can be formulated as max x g(x) subject to flow i (x)<;i = 1; 2;:::;N x (4.3) whereg(x) is the NPV defined in Eq. 4.2,flow i (x) is the flow rate for theith extraction well and is the maximum flow rate, which is assumed to be the same for all the extraction wells. The ensemble based algorithms we mentioned before, EnOpt and CGEnOpt, are designed for unconstrained optimization. The introduction of new flow rate constraints will lead to additional computation cost to the applicability of these methods. Therefore, we will discuss unconstrained 123 and constrained optimization in Sec. 4.3 and Sec. 4.4, respectively, using different approaches. 4.3 Production Optimization Without Constraints 4.3.1 Ensemble Based Optimization Methods In this section, we start with reviewing the general procedures of EnOpt and CGEnOpt. In EnOpt, Chen et al [16] utilized the steepest ascent method to update the control variables x, x l+1 = 1 l C x G T l + x l ; (4.4) wherel denotes the iteration index, l is a tuning parameter that determines the step size in the steepest ascent direction and is calculated by line search method, G l is the sensitivity of g(x) to the control variables x at the lth iteration step and C x is the smoothing matrix. Chen et al. [16] proved that C x G T l can be approximated by C x;g(x) , which is defined by C x;g(x) = 1 N e 1 Ne X j=1 (x l;j < x l >)(g(x l;j )<g(x l )>)); (4.5) where < x l >= 1 N e Ne X j=1 x l;j ; (4.6) and <g(x l )>= 1 N e Ne X j=1 g(x l;j ); (4.7) 124 whereN e denotes the number of ensembles, x l;j is thejth ensemble of con- trol variables generated by perturbingx l with zero mean Gaussian random noise, andg(x l;j ) is the corresponding NPV for thejth ensemble. Since the Gaussian noise is with a zero mean, we have< x l > x l and< g(x l ) > g(x l ). Therefore, the product of C x G T l can be approximated by C x;g(x) C x G T l ; (4.8) Through this approximation, the gradient can be estimated from the cross covariance of the ensemble. So the steepest ascent method in EnOpt becomes x l+1 = 1 l C x C x;g(x) + x l ; (4.9) To improve the convergence, Chaudhri et al. [14] developed the CGEnOpt, where the search direction was determined by a conjugate direction. By using the linear conjugate method, the steepest ascent method becomes x l+1 = 1 l C x G T l + l C x G T l1 + x l ; (4.10) where is determined by Fletcher-Reeves formula = (C x G T l )(C x G l ) (C x G T l1 )(C x G l1 ) (4.11) 125 4.3.2 A Simplied EnOpt However, the approximation by Eq. 4.8 will only be satisfied under the condition of zero mean Gaussian perturbance. For small number of ensembles, this is not usually guaranteed. Therefore, we interpret EnOpt from another aspect: multiple variable Taylor expansion. Firstly, we can define the vector ofith ensemble x l;i as x l;i = (x 1 l;i ;x 2 l;i ;:::;x Nx l;i ) (4.12) Here, x j l;i stands for thejth extraction well’s vacuum pressure ofith ensem- ble at iteration stepl. The current state vector of control variables x l is defined as x l = (x 1 l ;x 2 l ;:::;x Nx l ) (4.13) and the gradient G l is G l = (G 1 l ;G 2 l ;:::;G Nx l ) (4.14) From Taylor expansion, we know that the NPV of ith ensemble (g(x l;i )) can be linearized atith ensemblex l as g(x l;i ) g(x l ) +G 1 l (x 1 l;i x 1 l ) +G 2 l (x 2 l;i x 2 l ) +::: +G Nx l (x Nx l;i x Nx l ) = g(x l ) +G 1 l x 1 i +G 2 l x 2 i +::: +G Nx l x Nx i (4.15) 126 We can write it in the matrix form g e = 2 6 6 6 4 g(x l;1 ) g(x l;2 ) . . . g(x l;Ne ) 3 7 7 7 5 = 2 6 6 6 4 g(x l ) g(x l ) . . . g(x l ) 3 7 7 7 5 + G 1 l G 2 l G Nx l 2 6 6 6 4 x 1 1 x 1 2 x 1 Ne x 2 1 x 2 2 x 2 Ne . . . x Nx 1 x Nx 2 x Nx Ne 3 7 7 7 5 = g 0 + G l T A (4.16) We can use the least square method to solve the above equation G l = (AA T ) 1 A(g e g 0 ) T (4.17) From the derivations above, we can see that the simplified EnOp- t method can be applied towards non-Gaussian noise. Besides, it also has have the same advantage as the original EnOpt, which is suitable for paral- lel computing. After obtaining the gradient information, we can combined it with the steepest ascent or conjugate gradient method to perform the op- timization. 4.3.3 Step Size Selection To produce a reliable and robust production optimization using EnOpt, we must also choose an appropriate line search method to deter- mine the step size l . In the above sections, we assume that an appropriate step size is selected at each iteration of the algorithm. An optimal step size ^ l needs to satisfy the following rule ^ l = min () = min g( 1 G l + x l ) (4.18) 127 However, it is not easy to find the optimal step size that exactly sat- isfies Eq. 4.18. In general, we need an iterative method to find the near optimal step size. In literature, there are many methods for solving such a problem, such as binary search method, Cauchy search method and golden section search method. In our work, we use golden section search method. The golden section search method is a technique for finding the ex- tremum of a unimodal function by iteratively narrowing the range of val- ues inside which the minimum/maximum is known to exist. Figure 4.5 illustrates a single step of the technique to find the minimum of a function f(x). The horizontal direction is the x parameter and vertical direction is the value off(x). Suppose that we already evaluate the function values at x 1 , x 2 and x 3 , which are denoted as f 1 , f 2 and f 3 , respectively. For a uni- modal function f(x), if f 2 is smaller than f 1 and f 3 , it can be proved that the minimum lies between x 1 and x 3 . Now, we have 3 points x 1 , x 2 and x 3 , which form a triplet of points. For the next step in the golden section search method, we have to evaluate thef(x) at a new pointx 4 . An efficient selection ofx 4 , the interval c in this example, can be derived mathematically [64] b a = bc c (4.19) where the ratio of b a is the golden ratio 1.618033988. 128 If the evaluation off(x) at the new pointx 4 yieldsf 4a , the new triplet of points isx 1 ,x 2 andx 4 . Similarly, the new triplet of points isx 2 ,x 4 andx 3 for the case off 4b . 4.3.4 Parallel Computing As already mentioned, the landfill is represented by a large computa- tional grid that requires extensive computations. To implement the EnOpt, we must solve the governing equations of gas generation and transport a large number of times, in order to evaluate the objective function for each ensemble. The advantage of the EnOpt is that since each ensemble requires only the solution of the governing equation, the calculation of NPV for each ensemble is independent of those for all other ensembles, implying that im- plementation of the EnOpt is amenable to parallel computing. Similar to the implementations of the GA and the EnKF, we used a master-slave parallel computational strategy [73] for implementing the EnOpt, with the slave processors evaluating the NPV for each ensemble, and the master processor collecting all the NPV evaluations to perform gradient-based optimization. The message-passing interface was the plat- form used for the parallel computing. The number of the slave processors used in the EnOpt was 64, the same as the number of ensembles. 129 4.3.5 Implementation of The EnOpt The procedure of ensemble-based optimization is summarized as fol- lowings: 1. At the first time step l = 1, generate the initial state vector of control variables x l , which is sampled from a uniform distribution with lower and limits as 0.2 atm and 1 atm, respectively. At other time steps l6= 1, x l is set equal to the best guess at the previous step l 1. Then an ensemble of control variables x l;i ;i = 1; 2;:::;N e is generated by perturbing x l with Gaussian noise with zero average and a standard deviation that is a small percentage (10 percent in the current work) of the well pressure x l . 2. Run the landfill simulator for the given period to evaluate the NPV (g(x l;i )) for each ensemble using parallel computing described in Section 4.3.4. 3. Compute the gradient vector G l using the least-square method described in Section 4.3.2. 4. Determine the optimal step size l using the golden section search method. 5. Update the state vector x l+1 using either steepest ascent method or conjugate gradient method. At the same time, evaluate the NPVg(x l+1 ) for the updated state vector x l+1 . 6. The stopping criteria are g(x l+1 )g(x l ) g(x l ) < and jjx l+1 x l jj jjx l jj < , where and are tolerances forg(x l ) and x l , respectively. If the stopping criteria 130 are satisfied, terminate the loop of optimization, otherwise setl =l + 1 and go back to step 1. 4.3.6 Simulation Results We evaluate the performance of ensemble based optimization meth- ods for the landfill system described in Section 4.2. The main objective of this optimization problem is to find the optimal pressure settings of extrac- tion wells to maximum the NPV for a period of 2 years. Fig. 4.6 shows the evolutions of NPV obtained from steepest ascent method and conjugate gradient method. Both methods can achieve the significant increase of NPV within only 9 iteration steps. Since each evaluation of NPV takes almost 1 hour, both approaches can finish the optimization within approximate 10 hours, by using parallel computing. For a typical landfill, it is under dy- namic changes in terms of system properties, such as permeability, porosity and tortuosity. Other changes that can happen to landfills are weather con- ditions and LFG extractions. These dynamics will affect the landfill model and thus the optimal strategy need to be updated frequently for the new landfill system. Therefore, a daily optimization need to be performed. Since ensemble based optimization methods take less than one day to compute, they are implementable for landfill’s production optimization. For the optimization performance, the conjugate gradient method converges at the 5th iteration step and its maximum NPV is higher than 131 that of the steepest ascent method. Therefore, for this synthetic landfill op- timization problem, conjugate gradient method is relatively more efficient, which has also been validated by Chaudhri et al. [14] for a reservoir sys- tem. Fig. 4.7 presents the optimal well pressure for each extraction well gained from the conjugate gradient method. The optimal well pressures for all the extraction wells are within 0.4 - 0.7 atm. The reason is that both the flow rate and the operation cost will increase when the vacuum pressure decrease. Therefore, the optimal well pressure will be in the middle of the range, such as 0.4 - 0.7 atm in our result. Moreover, since the landfill sys- tem is a highly heterogeneous porus media, where physical properties are spatially distributed, the optimal vacuum pressures are different for each extraction well. Considering that the landfill system is a large-scale system, we can also use the parallelled GA to perform the optimization. Without consid- ering computation cost, GA is believed to be very efficient in finding the optimal solution within a specified search region. Similar to CGEnOpt, GA only needs the evaluation of the objective function, which is independent for each species. This determines that GA is amenable for parallel comput- ing. To be comparable to CGEnOpt, we set the number of species in GA to be 64, the same as the number of ensembles used in CGEnOpt. There- fore, for each iteration, both GA and CGEnOpt need to evaluate NPV for 64 132 ensembles. Moreover, CGEnOpt need additional evaluations of NPV , nor- mally less than 10 evaluations, to determine the optimal step size. Figure 4.8 compares the optimization results of GA and CGEnOpt. As indicated by the result, GA can achieve the same maximum NPV value as CGEnOpt does, but needs 25 iterations to converge. Therefore, even taking the com- putation of deciding optimal step size into account, we can still conclude that CGEnOpt is computationally more efficient than the GA. 4.4 Production Optimization with Constraints 4.4.1 Techniques to Solve Constrained Optimization The existing ensemble-based methods, EnOpt and CGEnOpt, cannot handle nonlinear constraints. Usually, constrained optimization can be converted to unconstrained optimization through several techniques, such as penalty function method, augmented Lagrange multiplier method, quadratic programming and gradient projection method. Here we are going to introduce two popular techniques to solve the constrained optimization. The penalty function method as described in Snyman [81] is as fol- lows max x " g(x) Nx X i=1 i (flow i (x)) 2 # (4.20) 133 The penalty parameter i is given by i = ( 0 ifflow i (x)< 0 ifflow i (x) (4.21) We need to find an appropriate value of i , because a large value of i can cause instability when deriving a solution with high accuracy. The se- quential unconstrained minimization technique (SUMT), which incremen- tally increase the penalty parameter, can be used to handle this. The augmented Lagrange method is proposed to solve for inequality constraints based on traditional Lagrange method. One possible augment- ed Lagrangian function is given by [81] max x fg(x) Nx X i=1 max( 1 2 i +(flow i (x)); 0) 2 g (4.22) where i is the Lagrange multiplier and is an adjustable penalty parame- ter. Similar to the penalty function, these parameters need to be determined by a iterative approximation. For both penalty function method and augmented Lagrange method, we have to perform the EnOpt with the modified objective functions (see E- qs. 4.20 and 4.22) iteratively, in order to obtain proper parameter values. This is really time consuming for large-scale landfill system, where each evaluation of NPV needs almost 1 hour. Due to the computation cost of es- timating optimal parameters, these ensemble based optimization methods will fail. 134 4.4.2 Parameterless GA To deal with the potential computation cost, as mentioned above, we propose a GA based method, called parameterless GA, which is actually a penalty method. However, it does not involve the estimation of parameters. As mentioned in Chapter 2.4.1, traditional GA involves three steps: selec- tion, crossover and mutation, and eliticism. In the selection step, the species with a larger objective function value (NPV in our work) has a higher pos- sibility to be selected to produce offsprings in the crossover step. For the selection of species, there are three common techniques used in GA: pro- portionate selection, ranking selection and tournament selection. The one we utilize in this work is called binary tournament, which is given as fol- lows [73]: we first set a prior probabilityp s of selection. Then we pick two random species from the current generation. A random numberr, uniform- ly distributed in (0; 1), is generated. Ifr<p s , we select species with a larger NPV . Otherwise, a species with a smaller NPV is selected. After repeating the same procedure multiple steps, we are able to choose enough species for the crossover step. In our work, we set p s = 0:85, large enough to en- sure an optimal solution. For parameterless GA, we modify the selection method by enforcing the following criteria: any species satisfying the con- straints is preferred over any species that violates any constraint. This can 135 be accomplished by modifying the objective function of GA as following max x g(x) = ( P Nx i=1 h P Nt j=1 f j i (x) j h i (x) i ifflow i (x)< 0 ifflow i (x) (4.23) From the modified objective function, we can see that if a species violates any constraint, it has a small possibility to be chosen, due to a zero objective function. 4.4.3 Simulation Results The effectiveness of this method is demonstrated by applying it to the same landfill system as that in Section 4.3. As illustrated by Figure 4.9, the objective function is able to converge to the maximum NPV in 120 it- erations. Figure 4.10 shows the evolution of number of species that violate constraints at each generation. As iterations proceed, the number of such species will decrease significantly. At the end of optimization, there are still several species, less than 5, violating constraints, which is acceptable and reasonable in GA. The reason is that the crossover and mutation step might generate species that violate constraints, although the selection step tends to choose a species satisfying constraints. In spite of this phenomena, the optimal species will definitely follow constraints, because its objection func- tion is not zero. Fig. 4.11 shows the optimal well pressure distribution from the parameterless GA. Compared to Fig. 4.7, well 3 and 9 have higher well pressure for constrained optimization. The reason is that the gas flow rate 136 of well 3 and 9 are larger than those of other wells with the same vacuum pressure (see Fig. 4.12). Typically, a smaller vacuum pressure in a extraction well will yield a higher flow rate. Therefore, with the same flow rate con- straint, the optimal well pressures for these two wells will be higher. This phenomena could be explained by the heterogeneous permeability distri- bution (see Fig. 3.12). Since permeabilities around well 3 and well 9 are relatively higher than those of other wells, the gas flow rate of these two wells tends to be higher. 4.5 Summary In this chapter, we presented a simplified ensemble-based optimiza- tion method. Other issues concerning with the application of EnOpt, such as step size selection, parallel computing technique and implementation procedure, were also studied. A synthetic landfill optimization problem without constraints was used to test the effectiveness and efficiency of the proposed ensemble based optimization. We also compared the performance of different optimization methods, such as steepest ascent method, conju- gate gradient method and the GA. Conjugate gradient method was proved to be able to gain a higher NPV with less computation. To handle the nonlin- ear constraints, a parameterless GA method was proposed and successfully applied to the same landfill model with constraints on the maximum flow rates. 137 9 8 7 5 6 4 3 2 1 Figure 4.1: 3D view of the grid structure used in the computations. The lo- cations and numbering of the nine extraction wells are shown by the circles. 138 Figure 4.2: 3D view of the grid structure and the reference permeability distribution. The permeabilities are in mD. The location of nine extraction wells are shown by the circles. 139 Jan Mar May Jul Sep Nov Jan Mar May Jul Sep Nov 0 1 2 3 4 5 6 Time (month) Natural Gas Price (dollars per thousand cubic feet) Figure 4.3: The predicted natural gas price for the next 2 years. 140 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 6 Well Pressure (atm) Operation Cost Figure 4.4: Function of operation cost h i (x i ) with respect to an extraction well’s vacuum pressure x i . 141 Figure 4.5: Diagram to show golden section search technique. (Adapted from [86]) 142 1 2 3 4 5 6 7 8 9 2.876 2.878 2.88 2.882 2.884 2.886 2.888 2.89 2.892 2.894 x 10 6 Iteration Number NPV Conjugate Gradient Method Steepest Ascent Method Figure 4.6: Evolution of NPV with the iteration steps for conjugate gradient method (solid line) and steepest ascent method (dotted line). 143 1 2 3 4 5 6 7 8 9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Well Number Well Pressure (atm) Figure 4.7: The optimal well pressure for 9 extraction wells obtained from the CGEnOpt. 144 5 10 15 20 25 2.865 2.87 2.875 2.88 2.885 2.89 2.895 x 10 6 Iteration Number NPV Figure 4.8: Evolution of NPV with the iteration steps for the GA method. 145 20 40 60 80 100 120 2.83 2.84 2.85 2.86 2.87 2.88 2.89 x 10 6 Generation Number Objective Function Figure 4.9: Evolution of NPV with iterations using parameterless GA. 146 20 40 60 80 100 120 0 10 20 30 40 50 60 Generation Number Number of Species Violating Constraints Figure 4.10: Evolution of number of species violating constraints with gen- erations in parameterless GA. 147 1 2 3 4 5 6 7 8 9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Well Number Well Pressure (atm) Figure 4.11: The optimal well pressure for 9 extraction wells obtained from parameterless GA. 148 0.25 0.5 0.75 1 1.25 1.5 1.75 2 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 Time (year) CH 4 Flow Rate (m 3 /s) Figure 4.12: The CH 4 flow rate of all extraction wells with the same vacuum pressure of 0.5 atm: (solid lines) flow rate of well 3 and well 9; (dotted lines) flow rate of other extraction wells. 149 Chapter 5 Conclusions In this dissertation, we present novel methods focusing on differen- t aspects of efficient landfill management, including short- and long-term performance predictions, real-time state estimation and model-based pro- duction optimization. An artificial neural network (ANN) approach is first developed to provide an accurate short-term predictions for important quantities at a large landfill in Southern California. The results show that the ANN can successfully be trained by a portion of the landfill data, then then be able to provide predictions of the temperature, CH 4 , CO 2 and O 2 profiles for a short time period. Thus, a trained ANN may be used by landfill operators for short-term forecasting and planning. Further, we demonstrate that if the short-term predictions by the ANN are used together with the genetic algorithm (GA) and the theoretical 3D landfill model, the accuracy of the long-term predictions could significantly be improved. The reason is that model-based long-term predictions rely on the amount of available production data, and use of a trained ANN could expand the range of accurate data that can be used in the GA. Therefore, the ANN model and 150 this novel combination represent powerful tools for developing realistic model of landfills, in order to make both short- and long-term predictions. To solve the problem of high uncertainty in the landfill model, a new approach for estimating and dynamic updating the spatial distribution of the permeability in a 3D model of landfills is presented. The GA and the sequential gaussian simulation are used to generate the initial ensembles of the permeability distribution. The ensemble Kalman filter is utilized to esti- mate and update the permeability distribution continuously, which is able to assimilate new data as soon as they become available. Several issues con- cerning with the implementation of this approach are also discussed, such as low-rank representation of the covariance matrix, unphysical permeabil- ity values and also parallel computing. The simulation results demonstrate the effectiveness and efficiency of this dynamic updating approach. After obtaining a landfill model with accurate estimates of physical properties, we develope an ensemble based optimization method to solve the problem of production optimization of a landfill gas system. Since a landfill is a large scale and complex system, conventional optimization methods, such as gradient based methods and evolutionary methods, are computationally costly. By contrast, the ensemble based optimization method, where the gradient information could be obtained from an ensem- ble of landfill models, is suitable for landfill’s production optimization. The 151 simulation results show that the proposed approach could increase the net present value (NPV) significantly, if no constraint is taken into account. For production optimization with nonlinear constraints, a GA based method, called parameterless GA, is presented and successfully implemented in the optimization of the same synthetic landfill. 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Moreover, A l = D r r K r P 0 l l ; A c = D r r K r P 0 c c ; (A.3) whereD r is a reference diffusivity which we take it to be the binary diffu- sivity of CH 4 in CO 2 , r is a reference viscosity, taken to be the viscosity of the air,P 0 is the ambient pressure, ` and c are, respectively, the local land- fill’s and the cover’s porosities, K r is a reference permeability, and ` and c are, respectively, the local landfill’s and the cover’s tortuosity factors. In addition,p =P=P 0 , and k = M k T r M r T ; m = 4 X k=1 k y k = M m T r M r T ; (A.4) whereM k is the molecular weight of componentk, andM m is the molecular weight of the gas mixture. In addition, we define w k = k RT r P 0 M r ; 4 X k=1 w k = 4 X k=1 P k M k RT RT r P 0 M r = 4 X k=1 y k PM k RT RT r P 0 M r = 4 X k=1 (y k k p) = m p; (A.5) 163 x l = K x K r r m ; y l = K y K r r m ; z l = K z K r r m ; (A.6) xc = yc = zc = K c K r r m : (A.7) Similar expressions can be written down for a corresponding quantity s for the surrounding soil, if the landfill has permeable walls. D k = D e km D r ; D c = D ec D r ; B = RT r Q 0 r K r P 2 0 M r L z ; k (z) = k (z)L 3 z Q 0 ; (A.8) whereD ec is the effective gas diffusivity in the landfill’s cover, andQ 0 is the total amount of the gases generated in the landfill during the time that it took to fill up the landfill, which is given by Q 0 = 1 t 1 4 X k=1 C T k V LF 3 X m=1 A m 1e mt 1 : (A.9)
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Asset Metadata
Creator
Li, Hu
(author)
Core Title
Performance prediction, state estimation and production optimization of a landfill
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Chemical Engineering
Publication Date
11/01/2012
Defense Date
10/02/2012
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
landfill,OAI-PMH Harvest,performance prediction,production optimization,state estimation
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Qin, S. Joe (
committee chair
), Safonov, Michael G. (
committee member
), Sahimi, Muhammad (
committee member
), Tsotsis, Theodore T. (
committee member
)
Creator Email
hli04tiger@gmail.com,tiger2004011471@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c3-106137
Unique identifier
UC11289955
Identifier
usctheses-c3-106137 (legacy record id)
Legacy Identifier
etd-LiHu-1264.pdf
Dmrecord
106137
Document Type
Dissertation
Rights
Li, Hu
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
landfill
performance prediction
production optimization
state estimation