A Thesis entitled Modeling Diffusion Using an Agent-Based ...

A Thesis

entitled

Modeling Diffusion Using an Agent-Based Approach

by

Pratibha Sapkota

Submitted to the Graduate Faculty as partial fulfillment of the

requirements for Masters of Science degree in Civil Engineering

Dr. Defne Apul, Committee Chair

Dr. Daryl F. Dwyer, Committee member

Dr. Gursel Serpen, Committee member

Dr. Patricia R. Komuniecki, Dean

College of Graduate Studies

The University of Toledo

May 2010

iii

An Abstract of

Modeling Diffusion Using an Agent-Based Approach

by

Pratibha Sapkota

Submitted to the Graduate Faculty as partial fulfillment of the

requirements for Masters of Science degree in Civil Engineering

The University of Toledo

May 2010

Arsenic is one of the toxic substances introduced in groundwater by various

anthropogenic and natural sources. Understanding fate and transport of arsenic in

groundwater and wetlands is crucial for remediation. Previously fate and transport of

arsenic have been modeled using various equation based methods (EBM) such as

ordinary differential equations (ODE) and partial differential equations (PDE), which

encompass rigorous mathematics and assume that only one species of arsenic is present.

But in reality, various forms of arsenic are present in groundwater. Based on the

availability of oxygen, arsenic transforms from one form to another, thus creating a

heterogeneous mix. Therefore, the equations used to describe the relation among

parameters of interest become non-linear. The agent-based method (ABM) has emerged

as a potential tool to model multidisciplinary and highly complex environmental

problems. The goal of this research was to develop an ABM for the transport of arsenate

(H2ASO4-) in water and soil. First, the diffusion of arsenate from contaminated water into

the overlying uncontaminated water was modeled and second, the diffusion of arsenate

iv

from contaminated soil to the overlying uncontaminated water was modeled. Since this is

the first time the model was developed using ABM, the results obtained from both

models were compared with results from HYDRUS 1-D for verification. Although

HYDRUS can model diffusion process, it is unable to model processes such as reduction

and oxidation of arsenic, which is an important process for arsenic remediation.

Therefore, HYDRUS is used for initial comparison purpose. The results obtained from

ABM and HYDRUS-1D for diffusion in water showed good agreement with each other.

However, the results obtained for diffusion in soil using ABM and HYDRUS 1-D were

not in complete agreement with each other. The difference in the results obtained was due

to the relation on which each model focused upon. Specifically, HYDRUS results were

obtained by assigning diffusivity coefficient value and monitoring variability over time

by using partial differential equations. However, ABM results were obtained by allowing

each individual contaminant to move freely in the porous soil. Another reason for the

difference was due to tortuosity. In HYDRUS, tortuosity depends on porosity (i.e. τ = ε-

1/3), but ABM model does not have a specific relation between tortuosity and porosity,

therefore, the results obtained differed.

v

Acknowledgements

First of all, I would like to thank my advisor Dr. Defne Apul for her constant support and

guidance throughout my research. I am thankful for her encouragement during my time in

UT. I also thank her for giving me an opportunity to work under her guidance.

I would also like to thank Dr. Daryl Dwyer and Dr. Gursel Serpen, my committee

members for their presence in my master’s thesis. I would like to thank them for giving

valuable suggestions and feedbacks.

Special thanks to my friends Chirjiv Anand and Jill Shalabi for their valuable help and

suggestions throughout my thesis.

Lastly, I am very thankful to my family. They have always supported, encouraged, and

inspired me in every step of my life and work. Finally, and most of all, I want to thank

my husband, who makes everything I do worthwhile. His love, support, and patience

make me very happy. I am very thankful for every moment I spend with him and this

thesis is dedicated to him.

vi

Table of Contents

An Abstract of .................................................................................................................... iii

Acknowledgements ............................................................................................................. v

Table of Contents ............................................................................................................... vi

List of Tables ...................................................................................................................... ix

List of Figures ..................................................................................................................... x

1. Introduction ..................................................................................................................... 1

1.1 Arsenic contamination as a widespread problem ...................................................... 1

1.2 Remediation of arsenic contaminated waters ............................................................ 2

1.3 Fate and transport of arsenic in a wetland ................................................................. 2

2. Motivation ....................................................................................................................... 7

3. Background .................................................................................................................... 8

3.1 Contaminant transport processes ............................................................................... 8

3.1.1 Diffusion ............................................................................................................. 8

3.2. Agent-Based Modeling (ABM) .............................................................................. 11

3.2.1 Introduction to ABM......................................................................................... 11

3.2.2 What is an agent? .............................................................................................. 12

3.3 Equation Based Modeling (EBM) and Agent-Based Modeling (ABM) ................. 13

3.4 Comparison of Various ABM tools ......................................................................... 14

3.5 NetLogo: The ABM of preference .......................................................................... 15

3.5.1 Features and Structure of NetLogo ................................................................... 16

3.5.2 NetLogo in various disciplines ......................................................................... 16

vii

4. Overview of Approach and Objectives ......................................................................... 19

4.1 Objectives ................................................................................................................ 19

5. Methodology ................................................................................................................ 20

5.1 Description of NetLogo platform ............................................................................ 20

5.2 Agents in NetLogo .................................................................................................. 23

5.3 Input/ Output ........................................................................................................... 25

5.4 Problem Description ................................................................................................ 25

5.5 Problem 1: Diffusion in water ................................................................................. 26

5.5.1 Model setup in NetLogo ................................................................................... 26

5.5.2 Sensitivity Analysis .......................................................................................... 29

5.5.3 Modeling diffusion in water using HYDRUS .................................................. 30

5.6 Problem 2: Diffusion in saturated soil ..................................................................... 32


5.6.2 Model setup in HYDRUS ................................................................................. 33

5.7 Problem 3: Diffusion in saturated soil with varying porosity ................................. 35


5.7.2 Model setup in HYDRUS ................................................................................. 36

6. Results and Discussion .................................................................................................. 37

6.1 Results from problem 1: Diffusion in water ............................................................ 37

6.1.1 Results from NetLogo model ............................................................................ 37

6.1.2 Result from HYDRUS model ........................................................................... 42

6.1.3 Comparison of HYDRUS and NetLogo results ................................................ 43

6.2 Results from problem 2: Diffusion in soil ............................................................... 45

6.2.1 Results from HYDRUS and NetLogo............................................................... 45

6.3 Results from problem 3: Diffusion in soil with varying porosity............................ 46

6.3.1 Results from HYDRUS and NetLogo............................................................... 46

7. Conclusion and Future work ......................................................................................... 48

7.1 Conclusion ............................................................................................................... 48

7.2 Future Work ............................................................................................................. 49

References ......................................................................................................................... 50

viii

Appendices ........................................................................................................................ 59

Appendix A: Information Tab for Enzyme Kinetics and B-Z reaction ............................. 59

Appendix B: Information Tab for Diffusion in Water ...................................................... 61

Appendix C: Information Tab for Diffusion in Porous Media.......................................... 62

Appendix D: NetLogo Code for Diffusion in Water ........................................................ 63

Appendix E: NetLogo code for Diffusion in Soil ............................................................. 65

Appendix F: NetLogo code for Diffusion in Soil with Varying Porosity ......................... 67

ix

List of Tables

Table 1: Comparison between various agent-based platforms (taken from Railsback,

(2006)) ............................................................................................................................... 14

Table 2: Wetland processes and similar examples in NetLogo library ............................. 18

Table 3: Mean, Standard deviation, and Coefficient of Variation (CV) of cumulative flux

(mg/m2) for a turtle ............................................................................................................ 40

Table 4: Different number of runs with 1 turtle representing 1mg ................................... 42

x

List of Figures

Figure 1: Conceptual model [adapted from Zhang et al. 2008] .......................................... 3

Figure 2: Conceptual description of an agent (Macal and North, 2005) ........................... 12

Figure 3: Screen shot of interface tab. The black box is the view. ................................... 21

Figure 4: Screen shot of procedure tab.............................................................................. 22

Figure 5 NetLogo world [Adapted from Xin and Li, 2008]............................................. 24

Figure 6: Uniformly Contaminated water ........................................................................ 27

Figure 7: Screenshot of Coordinates ................................................................................. 28

Figure 8: Pink represents contaminated water layer and blue represents uncontaminated

water layer ......................................................................................................................... 31

Figure 9: Uniformly contaminated saturated soil .............................................................. 32

Figure 10: Pink represents the contaminated soil layer and blue represents

uncontaminated water in HYDRUS setup ........................................................................ 34

Figure 11: Uniformly contaminated saturated soil ............................................................ 35


uncontaminated water in HYDRUS setup ........................................................................ 36

Figure 13: 1 turtle equivalent to 100mg. The graph was obtained by running the same

simulation 50 times with 1 turtle equal to 100mg. The figure shows that the cumulative

flux varied between 2050 to 2250 mg after 40 days. ........................................................ 38

Figure 14: 1 turtle representing 10mg. The graph was obtained by running the same

simulation 50 times with 1 turtle equal to 10mg. .............................................................. 39

Figure 15: Plot obtained by assigning a turtle 1mg each. The plot is smooth although it

shows some variation among the 50 simulations. ............................................................. 39

xi

Figure 16 : The plot is obtained by assigning 0.1 mg to a turtle. The plot is very smooth

and the variation after 50 simulations is also very small. ................................................. 40

Figure 17: Cumulative flux after 40 days ......................................................................... 43

Figure 18: Cumulative flux after 40 days ......................................................................... 44

Figure 19: HYDRUS vs. NetLogo for diffusion in water ................................................. 44

Figure 20: Diffusion in soil ............................................................................................... 45

Figure 21: HYDRUS vs. NetLogo for diffusion in soil .................................................... 46

Figure 22: Cumulative flux after 40 days for diffusion in soil with varying porosity ...... 47

1

1. Introduction

1.1 Arsenic contamination as a widespread problem

Arsenic is a toxic metal that is introduced in the subsurface by mining activities,

irrigation practices, and disposal of industrial wastes. Arsenic contaminated groundwater

is a widespread problem and poses health problems such as black disease, diabetes,

kidney and lung disease, high blood pressure, and reproductive disorder (WHO, 2004).

Among the various risks, drinking arsenic contaminated groundwater poses the greatest

threat to human health. The United States Environmental Protection Agency (USEPA)

has categorized arsenic as carcinogenic and has lowered the contaminant level deemed

safe for drinking water from 50 ppb to10 ppb (USEPA, 2001). Arsenic concentrations

above the given levels have been found in groundwater (often the only source of drinking

water) in many countries including Bangladesh, India, Vietnam, China, and United

States; this contamination was attributed to anthropogenic sources (Nordstom,2002;

Smedley and Kinniburgh, 2002). Therefore, it is of utmost importance to find a way to

remove arsenic from the contaminated water.

2

1.2 Remediation of arsenic contaminated waters

Recently there has been a keen interest in using constructed wetlands for water

quality improvement (Green et.al., 1997; Goulet et. al., 2001; Kadlec and Reddy, 2001).

Constructed wetlands are recognized as energetically sustainable systems because they

use natural energy to reduce pollutants. Various efforts have been made in the past to

treat arsenic laden wastewater using wetlands and algae ponds (La Force et al., 2000;

Wilkin and Ford, 2006; Kalbitz and Wenrich, 1998; Buddhawong et al., 2005). These

studies have shown that the removal capacities are higher in soil based wetlands than in

algae ponds.

Mechanisms involved in arsenic removal from water in a wetland are very

complex, comprising a large array of physical, chemical, and microbiological reactions.

Numerous studies have given details on complex processes involved in arsenic removal

from water using wetlands (Smith et al., 1998; Mahimairaja et al., 2005). These studies

showed that wetlands with proper soil type, plants, and microorganisms are efficient for

reducing arsenic below influent concentrations.

1.3 Fate and transport of arsenic in a wetland

The biogeochemistry of arsenic in wetlands is a complex process which includes

chemical and microbiological reactions (Figure 1). Several processes such as adsorption,

reduction-oxidation, precipitation, plant uptake, and microbial activity contribute towards

3

removing arsenic from water. Adsorption on soil colloids (clay, oxides, Al, Fe,

Mn, CaCO3, organic matter) is one of the most important processes in arsenic removal

from soil solutions. Precipitation of solid phase is another mechanism of arsenic removal

from the soil solutions. Reduction-Oxidation (Redox) of arsenic compounds also plays a

vital role in removing arsenic by changing arsenic compound into different forms. Micro-

organisms such as bacteria, fungi also help in reducing arsenic concentration from the

soil solutions by changing arsenic compound into volatile form.

Figure 1: Conceptual model [adapted from Zhang et al. 2008]

The fundamental processes for moving and mixing contaminants in wetlands are

advection, diffusion, and dispersion. Advection is the transport of contaminant along the

Organic As As (V)

As (III)

As compounds

As(V) compounds

Sorbed As (V) compounds

Sorbed As (III) compounds

Input

Oxidation

Reduction

Adsorption

Desorption

Reduction

Microbial activity

Precipitation Dissolution

Demethylation

Oxidation Methylation

Adsorption

Desorption

Plant uptake Precipitation

Output

Advection Advection

Diffusion Diffusion

4

flow of water. Diffusion and dispersion signify the mixing of contaminants due to

concentration gradients. The most commonly used equation for the transport of

contaminants in porous media, the advection dispersion equation, is given by

��

��=

� ��

��−

��

��

Where

� = linear velocity [L/T]

D=Hydrodynamic dispersion [L2/T]

Hydrodynamic dispersion is the sum of dispersion and diffusion (D =De+ αL �) .

Where

De = effective diffusion coefficient [L2/T]

αL = dispersion[L]

Under flow conditions, diffusion is insignificant, therefore it is neglected. However, in

quiescent conditions ( � = 0), diffusion dominates because of random motion of

contaminant molecules. Fick’s law, which is used to describe diffusion, states that the

mass diffusing is proportional to the concentration gradient. In one dimension, the

transport process is given by

��

��= −�(

��

��)

5

Where, D= Diffusion coefficient [L2/T]

dC/dx=Concentration gradient[M/L3/L]

Numerical models have been widely used to gain insight into the fate of the

chemicals in wetlands. Finite difference method (FDM), finite element method (FEM),

integrated finite difference method (IFDM), boundary integral equation method, and

analytical elements are some of the methods used for fate and transport modeling

(Alhumaizi, 2004; Narasimhan and Witherspoon, 1978; Ligget and Liu, 1983; Strack,

1987). Among them, FDM and FEM are more commonly used to solve flow problems.

Studies have shown that FDM requires relatively large computational time and is often

unstable when dealing with steep concentration gradients (Alhumaizi, 2004; Botte, Ritter

and White, 2000). Piatkowski et al. (2003) showed that FEM is more useful than FDM

for irregular boundaries and is relatively stable than FDM when dealing with steep

concentration gradients. To summarize, FDM is easy to program, understand, and is a

good method when dealing with simpler boundaries. However, FEM is better at

approximating complex boundaries and is able to simulate point sources and sinks,

seepage faces, and moving water table.

The transport of arsenic in wetlands is determined by the presence of adsorption-

desorption sites, the effect of humic substance, and the presence of competing anions.

Adsorption-desorption behavior of arsenic is responsible for non equilibrium transport of

arsenic in soil, making it difficult to model using existing fate and transport models.

Because arsenic cycling in wetland is a complex process and adsorption in soils is

nonlinear (Zhang and Selim, 2006), it is necessary to understand the relationship among

6

the individual components in order to make future predictions accurately. Hellweger and

Kianirad (2007) have shown that for a system that is heterogeneous and non-linear,

existing modeling methods produce significant errors because variability within species is

not accounted for. Therefore, research is needed for characterizing and quantifying the

sources of variability so that accurate models can be developed.

The agent-based method (ABM) has emerged as a potential tool to deal with the

multidisciplinary and highly complex environmental problems. The growing interest in

this technique is due to its ability to incorporate more realistic assumptions than existing

fate and transport models; therefore, ABM is proving effective in simulating many real

world fate and transport problems. Salgado and Aranda (2007) developed and compared

the performance of ABM to equation based modeling (EBM) and found that for modeling

adsorption on solid surfaces, results obtained from ABM were more stable than the

results obtained from EBM. Gujer (2002) and Schuler (2005) showed that EBM

introduces error while simulating water quality models because it uses an averaging

assumption for nutrient uptake across the entire population. Hellweger (2007) has

suggested ABM as an alternative to EBM in such a situation as it does not make any

averaging assumptions.

7

2. Motivation

ABM is a simulation method that has proven its usefulness in handling

complexities in diverse fields such as ecology (Takasu, 2009; Uchmanski et al., 2008),

medicine (Gary and Wilensky, 2009; Eapen, 2009), and economics (Remenik, 2009;

Damaceanu, 2008), yet ABM has not been used as much in environmental modeling.

ABM allows the incorporation of heterogeneity and variability within the population and

does not make any averaging assumptions; thus it is a promising method for modeling

fate and transport compared to existing EBM methods. For example, arsenic cycling in

wetland is a very complex process which requires monitoring of various arsenic species

simultaneously. The above studies have proved the capabilities of ABM for handling

complexities. Therefore, the overarching goal of this study was to model wetland

processes using agent-based approach. However, since this was a daunting task, this

thesis addressed only the diffusion process in wetlands or any other porous media. To

my knowledge, this study is the first attempt to model fate and transport of contaminants

through porous media at the individual level using ABM.

8

3. Background

3.1 Contaminant transport processes

The basic transport processes governing the transport of chemicals are advection,

diffusion, and dispersion. All of these processes can occur simultaneously during the

transport of chemicals or the transport can be dominated by a single process. For

example, when flow velocity is zero, chemical movement takes place through diffusive

transport and when flow velocity is large, advection and dispersion effects dominate.

Computational approaches to problems involving advection-diffusion equations are

relatively well studied, and in depth discussions about transport phenomenon and

applications can be found in various textbooks (Bear, 1972; Freeze and Cherry, 1979;

Fetter, 1993).

3.1.1 Diffusion

Diffusion is one of the widely studied transport processes in fate and transport

problems (Selvadurai, 2004; Bulavin et al., 2008; Rebour et al., 1997). Fick introduced

the concept of diffusion and stated that diffusive flux is directly proportional to

9

concentration gradient (Fetter, 1993). Einstein studied the probabilistic nature of

diffusion process and explained that the Brownian motion of the particles in a fluid use

probability density functions (Weber, 1996). He showed that the movement of each

particle in the fluid is independent of the other particles, and that diffusion coefficient

increases as a function of movement of the particles.

Existing experimental methods only provide the effective diffusion coefficient i.e.

coefficient averaged over diffusion zone (Weber, 1996). Gathering information relating

to individual diffusion mechanisms, molecular level characteristics using the existing

methods is also very difficult, if not impossible. Individual level modeling appears to be

the viable method for gaining insights into the complexities of diffusion, and it provides a

fundamental framework for solving diffusion problems.

At the microscopic level, diffusion results from individual displacement of

diffusing particles, also referred as diffusive jumps. Diffusive jumps are usually single-

molecule jumps of fixed length. Diffusivity can be expressed in terms of physical

quantities that describe elementary jump processes such as jump rates and jump distance

of a molecule. The jump rate of a molecule depends on its individual energy, Boltzmann

constant, and the absolute temperature (Flynn, 1972).

Γ=νo

exp (-∆G/kBT)

Γ is the jump rate, ∆G is the Gibbs free energy, kB is the Boltzment constant, νo is the

Debye frequency, and T the absolute temperature. If the jump distance of the molecule is

given by λ [L]and the jump rate from one plane to the neighboring one is Γ [L/T], then

10

without a driving force, forward and backward jumps occur with the same jump rate.

The net flux J [amount L-2

T-1

]

from one point to another is given by

J= Γn1 – Γn2. …………………………………………… 1

where n1[L-2

] and n2[L-2

] are related to volume concentrations of diffusing molecules

C1=n1/ λ, C2=n2/ λ ………………………………………2

where C1 and C2 represent concentration [parts L-3

] at point 1 and 2.

Concentration C(x,t)changes slowly as a function of the distance variable x in terms of

distances between two jumps. First order Taylor expansion of C(x,t) results in

C1-C2 =- λ ∂C/∂x……………………………………………… 3

Inserting equation 2 and 3 in 1 results in

J= - λ2 Γ∂C/∂x………………………………………………….4

After comparison with Fick’s law the diffusion coefficient is determined to be

D= λ2 Γ………………………………………………………….5

This shows that the diffusion coefficient is related to the distance squared and jump rate.

Additional details on the equation and its derivation can be found elsewhere (Flynn,

1970; Franklin, 1975; Bennett, 1975)

11

3.2. Agent-Based Modeling (ABM)

3.2.1 Introduction to ABM

ABM is a form of computational simulation that is gaining popularity in various

disciplines (Bonabeau, 2002; Macal and North, 2005). ABM enables building models

where individual entities and their interactions are directly represented. In ABM, a

system is modeled as a collection of autonomous decision-making entities called agents.

These agents are adaptive to their surrounding environment and can react to the

environmental conditions. ABM allows visualizing the emergence of macro level

behavior from micro level. A simple ABM can exhibit complex behavior patterns and

provide valuable information about the system that it emulates. In addition, agents may

be capable of evolving, allowing unanticipated behaviors to emerge.

ABM is also known by several other names such as Agent-Based Systems (ABS)

and Individual Based Modeling (IBM). All of these terminologies are extensively used

(Macal and North, 2005), but ABM is used throughout this thesis. All the agents in the

system are modeled within a region called an environment. The environment is a virtual

world where the agents act.

Cellular Automata (CA) is yet another form of computational modeling that was

used to model complex environment but it is not as sophisticated as ABM. In CA the grid

is homogeneously populated with agents; whereas in ABM, the agents are heterogeneous

and therefore do not necessarily occupy all spaces within the grid. In CA, agents don’t

12

interact with other agents outside their immediate neighborhood, but in ABM agents are

more flexible in interaction with other agents.

3.2.2 What is an agent?

An agent is defined as an autonomous entity with its own set of characteristics

and that can act on its own. The important feature of an agent is shown in Figure 2. An

agent has a set of characteristics that allows it to act independently. An agent is able to

perceive its environment. Thus, agents can determine what agents or objects are located

near them. An agent is able to move within the space, can send or receive messages from

other agents, and can interact with the environment (perform) and it has a set of goals that

it pursues with its own initiative (making it proactive).

Figure 2: Conceptual description of an agent (Macal and North, 2005)

Agent

� Autonomous unit

� Social

� reactive

� Proactive

EN

VIR

ON

ME

NT

EN

VIR

ON

ME

NT

13

3.3 Equation Based Modeling (EBM) and Agent-Based Modeling (ABM)

The fundamental difference between ABM and EBM is the relationship on which

they focus. EBM begins with a set of equations that has a pre-determined relationship

among the parameters of interest. The equations are used to monitor parameter

variability over time by using ordinary differential equations (ODE) or over time and

space by using partial differential equations (PDE). The results obtained using these

equations are the outcomes of individual behaviors but those behaviors have no explicit

representation in EBM. For example, the advection dispersion equation presented in

section 1.3 models concentrations which is an aggregate parameter arising from a

collection of individual; individual behaviors are not explicitly represented in the

advection dispersion equation. In contrast, ABM begins by representing the behavior of

each individual, and then allows each individual component to interact, which produces

the ultimate outcome. Thus relationships among the parameters of interest are an output

of modeling process; they are not the input to the models (Xin and Li, 2008). In most

cases, EBM tends to aggregate the values when modeling the system. This aggregation of

the values is imprecise, and eventually leads to erroneous results (Hellweger, 2007).

To summarize, ABM has several advantages over EBM. ABM captures emergent

phenomena of global behavior from local interactions and provides a natural description

of the system that is close to reality (Bonabeau, 2002). Apart from that it is more flexible

than EBM. For example, more agents can be easily added during the simulation. The

major advantage of ABM lies in its ability to handle complexity with ease. It can also

14

encapsulate the randomness of complex systems. Moreover, it does not require as much

mathematical knowledge as is required for traditional methods [EBM] (Wishart et al.,

2004).

3.4 Comparison of Various ABM tools

ABM has been popular in various disciplines including economics, social science,

medicine, and ecology (Bonabeau, 2002), therefore, various agent-based models have

been developed. Table 1 shows the comparison of some of the popular ABM tools. ABM

platforms have been compared with respect to some of the most important features.

More information on these platforms can be found elsewhere (Berryman, 2008;

Railsback, 2006).

Table 1: Comparison between various agent-based platforms (Railsback, (2006))

ABM

platforms

User base Modeling

Language

Speed of

execution

Ease of

learning and

programmin

g

User materials

Ascape Diminishing Java Moderate Moderate Good

documentation

Mason Increasing Java Fastest Moderate Limited

documentation

Repast Large Java,

python

Fast Moderate Limited

documentation

.etLogo Large NetLogo Moderate Good Extensive

documentation

SWARM Diminishing Objective

C,Java

Moderate Poor Good

documentation

15

SWARM is one of the oldest and most stable ABM’s that is able to support

complex models, but has weak error-handling capability. Repast is one of the powerful

agent-based modeling platforms. However, it requires an extensive knowledge of Java

and is suitable for computationally intensive models. Mason is also a good simulation

tool and is a good choice for an experienced programmer but is computationally

intensive. NetLogo has a user friendly environment and has its own programming

language that is simpler than Java or Objective-C. It also has good documentation and

visualization abilities. From the above comparison, NetLogo was found to be a suitable

platform for modeling since it is able to handle a large number of agents. Time spent on

developing this model is less than time required for some other library-based platforms

such as Repast, SWARM. Also, studies in different fields have proven the efficacy of

NetLogo for handling complexity (Chitnis & Itoh, 2004; Damaceanu, 2008; Xin et al.,

2008)

3.5 .etLogo: The ABM of preference

NetLogo is a modeling environment based on agents designed for simulations of

natural phenomena. It was designed by Uri Wilensky in 1999 and it is in a process of

continuous development and modernization at the Center for Connected Learning and

Computer-Based Modeling – Northwestern University. NetLogo is written in Java

language and can be run on all major platforms (Mac, Windows, Linux). NetLogo is a

freeware and can be downloaded from the following web address:

http://ccl.northwestern.edu/netlogo/.

16

3.5.1 Features and Structure of .etLogo

NetLogo has three main entities: agents, landscape, and observer. All of these

entities can run code, and interact with each other. Variables can be defined globally and

all entities have access to them. This structure gives modelers a great deal of flexibility

while creating models. Furthermore, NetLogo is easy to use and has excellent

documentation.

3.5.2 .etLogo in various disciplines

NetLogo has emerged as a powerful and elegant language that can incorporate

complexity and heterogeneity in a few lines of code with relative ease. Although

NetLogo has been increasingly used in many disciplines, a few examples of its use are

discussed below.

Lockta Volterra model (predator prey model, Forrestor, 1971), one of the popular

models in ecology, was traditionally simulated using EBM. Lockta Volterra model is

used to make future predictions about population level, growing population, and the rate

of consumption of natural resources. EBM solves differential equations that calculate the

value of a variable at the next time step for the given population using the value at the

current time step, but each individual in the population is not uniquely represented. This

makes it hard to model heterogeneity among individuals. It is also hard to represent

individual behaviors that depend on its past experience. But NetLogo has proved its

capability in such a scenario. When the same model was simulated in NetLogo, each

agent’s past behavior along with heterogeneity was modeled with ease (Wilensky, 1998).

This feature is very helpful for the current research. Figure 1 shows that arsenic species

17

changes from arsenate to arsenite and vice versa, thus making it heterogeneous. The

above example shows that NetLogo allows modeling such scenarios which are

impossible using existing EBM as it can only model homogeneous individuals.

NetLogo is also popular in medical and biomedical disciplines because of its

ability to deal with finer details and track numerous parameters at a time. For example,

Eapen (2009) developed a NetLogo model for laser hair removal. Modeling laser hair

removal requires monitoring numerous parameters simultaneously which is difficult

using existing modeling methods, but NetLogo proved effective in tracking the numerous

parameters. Modeling wetland processes require monitoring various parameters at a time.

Factors such as pH, Fe and Mn oxides, sulfides, and organic matter should be monitored

simultaneously as they impact reduction and oxidation of arsenic. The above example

proves the capability of NetLogo for monitoring numerous parameters at a time which is

very important for modeling reduction and oxidation.

Similarly, NetLogo has also been used for modeling complex economic systems.

For example, Damaceanu (2008) modeled wealth distribution among upper, middle, and

lower class. Each patch had some amount of resource and the turtles collected some of

that resource in order to survive. The amount of resource that each turtle collected was its

wealth. The above example proves the capability of NetLogo to model processes such as

adsorption and desorption as arsenate adsorbs on soil matrix rich in Fe and Al. Such

scenarios are difficult to model using EBM as it requires rigorous mathematics and may

not be solvable.

18

Table 2 shows the wetland processes of interest and existing NetLogo models that

resemble those processes. Left most column shows the general equation used for the

wetland processes. Middle column gives an example of equation used. The last column

shows examples of similar processes modeled in NetLogo which is useful for modeling

the wetland processes. Further details about the NetLogo models can be found in the

appendix section.

Table 2: Wetland processes and similar examples in NetLogo library

Wetland processes Example Related .etLogo models

Adsorption/Desorption

A +B AB

Adsorption

AsO43-

+ FeOH+3H+=

FeH2ASO4+H20

Enzyme Kinetics(Appendix A)

Reduction/Oxidation

A B

2MnO2+H3ASO3=

2MnOOH+H3AsO4

2MnOOH+H3AsO3=

2MnO+H3AsO4+H2O

B-Z reaction(Appendix A )

Precipitation/Dissolution

A + B AB

Dissolution

FeAsO4.2H2O+H+=

H2AsO4-+Fe(OH)

2++H2O


Plant uptake and Microbial

uptake

A + E B C+E

��

��=

��

�� + ��∗ ��

�� = (�� + ��)/��

Where,KM is called the

Michaelis Constant


Oxidation

Reduction

K1 K2

Precipitation

Dissolution

Adsorption

Desorption

K-1

19

4. Overview of Approach and Objectives

The primary objective of this study was to model diffusion using an agent-based

approach. With an agent-based approach, a greater understanding of transport of

contaminants may be possible because the results obtained are the outcomes of individual

interactions. Also, an agent-based approach provides an opportunity to examine the

impact of assumptions made by equation based approach.

4.1 Objectives

The objectives of the current study was to

� Select an ABM platform for modeling diffusion.

� Develop an agent-based model for diffusion in water and porous media using an

agent-based approach.

� Perform sensitivity analysis to determine a suitable agent size.

� Perform sensitivity analysis to evaluate intrinsic randomness of the agent-

based model.

� Verify the results obtained from diffusion in water and porous media against a well

established FEM based fate and transport model, HYDRUS.

20

5. Methodology

5.1 Description of .etLogo platform

The NetLogo window has three tabs: interface tab, the information tab, and

procedures tab as shown in the top left of Figure 3. Only one tab is visible at a time but

one can switch between different tabs by clicking on the tabs at the top of the window.

The interface tab is used to visualize the output of the simulation and to control it. The

information tab provides text-based documentation about the simulation and expected

results. The procedure tab is the workspace where the code of the model is stored.

21

Figure 3: Screen shot of interface tab. The black box is the view.

In NetLogo, two-or three-dimensional models can be created. In the 2D version of

NetLogo, the interface tab includes a black square called view which is made up of

patches. A patch is a spatial environment on which the agents move. The simulation

program instructs the agents to move and act from patch to patch. The results can be seen

in the view (Figure 3). The interface tab is a visual editor in which one can edit graphical

elements such as buttons, sliders, switches, monitor, and output (Figure 3). Interface tab

also allows changing the world dimension, and the dimension of the overall setup.

The information tab provides an environment for the programmer to describe the

model. By default, the information tab includes the following sections: ‘What is it?’,

22

‘How it works’, ‘How to use it’, ‘Things to notice’, ‘Things to try’, ‘Extending the

model’, ‘NetLogo features’, ‘Related models’, and ‘Credit and References’. The

programmer can edit these sections and add equations and methods used. The

information tab for the current research can be found in Appendices B and C.

In the procedure tab commands are written in a specific format as shown in Figure

4. (The code used in this study is given in Appendices D, E and F) The program has three

parts: first, the global variables are defined; second, setup procedure is written to

initialize the simulation; and third, go procedure is written that is repeatedly executed by

the system. The go procedure tells each agent to carry out the given instruction

independently. Agents in NetLogo model are referred as turtles. Typically, a population

of turtles is initialized and procedures are written that control the behavior of the turtles.

The turtles represent physical entities whose behaviors result in movements around the

two dimensional world.

Figure 4: Screen shot of procedure tab

23

5.2 Agents in .etLogo

NetLogo has four different types of agents and each agent can follow instructions

and carry out its own activity. These agents are turtles, patches, links, and observer

(Figure 5). Turtles are the functional agents and the patch is a square ground over which

the turtles move. Initially, the world is empty and the turtles are created by the observer.

In addition, the patches can create turtles, too. Patches and turtles have their coordinates

determined by the variables xcor and ycor for turtles, and pxcor and pycor for patches.

The patch with the coordinate (0, 0) is the origin which can be placed anywhere in the

view box based on the model requirement. The total number of patches is determined by

the parameters min-pxcor, max-pxcor, min-pycor, and max-pycor.

By default, NetLogo has fixed patch coordinate values, when the model starts that

can be changed accordingly. For example, when NetLogo starts, min-pxcor, max-pxcor,

min-pycor, and max-pycor values are -16, 16, -16, and 16 respectively (Figure 5). This

means pxcor and pycor both range from -16 to 16, so there are 33 times 33, or 1089

patches. The patches coordinates are only integers whereas the turtles coordinates are

whole numbers and fractions. This means the turtles can move anywhere on the patch. A

link is an agent that connects two turtles if there is relation between them.

24

Figure 5 NetLogo world [Adapted from Xin and Li, 2008]

In NetLogo world, the patches can wrap around indicating that when a turtle

moves to the edge of the world it disappears and then reappears on the opposite end. If

this approach is used, a one dimensional simulation can be run using a two dimensional

view box. Time is discrete in NetLogo and turtles act on each tick. A tick represents one

step of the model, i.e., one cycle of the main loop. Ticks can be either seconds, hours,

days based on the simulation requirement. If the daily events are simulated, then the time

scale is one tick per day. If hourly events are simulated, then one tick is an hour. The real

time the model takes to run is different than the model time. For example, if time scale is

one tick per year, and the model only takes one second to run, then 100 years simulation

can be simulated in 100 seconds.

Patch

Link

Turtle

Observer

max pxcor, max pycor

(16, 16)

max pxcor, min pycor

(16,-16)

min pxcor ,min pycor

(-16,-16)

min pxcor, max pycor

(-16, 16)

25

5.3 Input/ Output

Once the code is finalized, input parameters are provided via various buttons in

the interface tab. Output from the simulation is viewed in the interface tab without

changing the window. Basic buttons required for NetLogo are setup and go buttons which

are created by the user. Other buttons such as sliders, switches, and monitors are created

by the user based on the requirement. Output plots can easily be exported to an excel

sheet for further analysis.

5.4 Problem Description

Two problems were simulated using NetLogo and HYDRUS. These two

problems were adopted from Thoma et al.’s work (1993). Thoma et.al (1993) studied the

transport of 2, 4, 6-trichlorophenol (TCP), a hydrophobic pollutant released from lake

bed sediments. The model consisted of a uniformly contaminated layer at the bottom of a

vertical column; sediment cap layer in the middle, and the water column at the top. The

sediment cap layer and the water column were initially contamination free. With time, the

contaminant in the contaminated sediment bed diffused into the cap layer and the water

column. Two diffusion scenarios were studied. First, the transport of contaminant in

sediment without capping was studied, and then the transport of contaminant with

sediment cap was studied. The model results were also compared with lab results, which

showed good agreement with each other, indicating that the transport mechanism is well

understood.

26

The setup for the current research is similar to that of Thoma et al (1993) but

without the sediment cap. The transport of H2AsO-4 was studied instead of TCP. The

molecular diffusion coefficient of the species is 7.82*10-5

m2/day (Li and Gregory, 1974).

The problem is analyzed in two parts; first, diffusion of contaminant in water was

modeled, second, diffusion of contaminant in soil was modeled. Thoma et al. (1993) was

chosen as one of the standards for verifying NetLogo model because of its focus on

diffusion process. Alshawabkeh et al. (2005) used the Thoma et al. (1993) model to

develop criteria for evaluation and design of contaminated sediment. Similarly Valsaraj et

al. (1997) used Thoma’s equation to study diffusion of organic compounds through

porous media. Other authors (Valsaraj et al. (1998); Eek et al. (2008); Chen et al. (2009)

etc.) have also used the concept of Thoma et al. because of the simplicity of problem

analyzed.

5.5 Problem 1: Diffusion in water

5.5.1 Model setup in .etLogo

A beaker of water uniformly contaminated to a certain depth was simulated

(Figure 6). Initially the two sections of the beaker are envisioned to be separated from

each other using a thin plate which is then instantaneously removed and with time, the

chemical diffuses throughout the entire beaker of water. The amount of chemical

reaching the top of the beaker as a function of time was measured as flux. This basic

numerical experiment was performed for the preliminary evaluation of NetLogo model.

The success of this model provides insights into the limitations and strengths of NetLogo

for modeling the diffusion process.

27

Figure 6: Uniformly Contaminated water

The simulation was performed with an initial contaminant (H2AsO-4)

concentration of 150mg/L. Therefore, the net concentration (per unit area) in the beaker

for the depth of 15mm of contaminated layer is 2,250mg/m2

(150mg/L *103/m

3*1L

*0.015)). The dimension in the model is entered as patches and the user decides the unit

dimension a patch represents. For example, a NetLogo dimension of 20mm can be

populated with 20 patches each 1 mm2, or 40 patches each 0.5 mm

2. For the current

simulation, a total of 120 patches were used which was equivalent to a vertical distance

of 22mm, therefore the size of each patch was 0.1833 mm. The min-pycor, max-pycor,

min-pxcor and max-pxcor for the current simulation are -82, 38, -60, and 60 (Figure 7).

Since the model wraps horizontally (turtles appear on left side of view when it disappears

on right side), it is considered a 1D model and the horizontal patches were not taken into

account.

22mm

15mm

28

Figure 7: Screenshot of Coordinates

According to Weber (1995), individual molecules inside the beaker have their

respective kinetic energies and undergo frequent collisions with each other. As a result of

these collisions, contaminants move in random directions. These resulting collisions carry

individual molecules from the region of higher concentration to lower or vice versa.

For the given simulation, the contaminants move randomly inside the beaker

based on a diffusivity value assigned to them. The diffusivity value is the number of

patches over which a turtle moves in a tick. Molecular Diffusion coefficient of H2AsO-4

is given by

0.0543mm2/min

= 0.78 mm2/14.4min

29

=0.78 mm2/tick where, 1 tick = 14.4 min

=4.3 * 0.1833 mm2/tick

=4.3 patches/tick

Where, 1 patch = 0.183mm

For the given simulation, the turtles move over 4.3 patches in a tick which is equivalent

to the diffusion coefficient of H2AsO-4. Equation 5 showed that diffusion coefficient is

related to the distance travelled by an individual molecule (D= λ2 Γ). Unlike Brownian

particles, the contaminants were not assigned individual energies since it added to the

complexity of the model. Keeping track of energies for each molecule would be

cumbersome, thus making it difficult to model.

5.5.2 Sensitivity Analysis

Since the contaminants were represented as agents in the model, the given

contaminant concentration (150mg/L) needed to be changed into agent form. Sensitivity

analysis was performed to obtain a proper balance of data points and runtime. The

simulation was run with a turtle representing various amounts (100mg, 10mg, 1mg, and

0.1 mg) of contaminants. For example, when each turtle represented 1mg, 2250 turtles

were used to represent 2250mg of contaminants. Similarly, when each turtle represented

10mg, 225 turtles were used to represent 2250mg. The simulation was run 50 times in

each case to see if the variation increased with run time.

NetLogo also has built-in randomness, therefore, the simulation results vary slightly each

time the model is run. After determining the suitable amount a turtle should represent, the

30

same simulation was run over and over to see if randomness increased. The same

simulation was run 100 times to monitor net randomness of NetLogo model.

5.5.3 Modeling diffusion in water using HYDRUS

The HYDRUS program is a finite element model that solves Fickian-based

advection-diffusion equation for solute transport. The standard diffusion equation in one

dimension is given by

��

��= ��

��

��

Where Dw [L2/T] is the diffusion coefficient in water and

�

�! is the change in

concentration with time. Molecular diffusion coefficient of contaminant in water was

7.82*10-5

m2/day (Li and Gregory, 1974). Contaminant concentration of 150mg/L was

entered in the contaminated layer via the graphical editor as shown in Figure 8. The

contaminated sediment depth was 15 mm and the overall depth of the beaker was 22mm

as in the NetLogo model.

31

Figure 8: Pink represents contaminated water layer and blue represents uncontaminated

water layer

The profile window allows the user to define the spatial distribution of

parameters. All the parameters in this window can be added or edited using the edit

window. The user selects the part of the domain to add a particular value for the selected

variable. The profile window is very user friendly; it is possible to select an individual

node, part of the domain, or the entire domain. For the given simulation, the

concentration of 150mg/L was assigned by selecting the rectangular domain represented

by pink as shown in Figure 8. Color-coded window on the left side of the window shows

the values represented by the domains. Blue represents a concentration of 0 and pink

represents a concentration of 150mg/L.

32

5.6 Problem 2: Diffusion in saturated soil


In problem 2, diffusion of a contaminant from saturated soil to overlying water

was simulated. It was assumed that the soil was uniformly contaminated throughout the

depth of 15mm (Figure 9). The setup is similar to Thoma et al (1993) without a cap

sediment layer. With time the contaminant diffused into the water. The amount of

chemical reaching the top of the beaker was quantified as flux. The basic buttons required

for simulation are ‘setup’ and ‘go’. Other buttons used were ‘porosity’, ‘count agents’,

and ‘count void spaces’.

Figure 9: Uniformly contaminated saturated soil

Brown patches represented soil and white particles represented contaminant. It

can be seen that the soil was uniformly contaminated to a depth of 15 mm and the overall

Uniformly

contaminated layer

(15mm)

Water

(7mm)

33

depth of the setup was 22mm. Initially, the water above 15mm was assumed to be

contaminant free. The porosity of soil was 0.49. The contaminants were uniformly

distributed throughout the depth. As discussed in section 5.5.1, the contaminants move

over 4.3 patches at each tick. When the contaminants hit a soil particle (brown patch), it

bounces off of it at an angle. According to particle-wall collision theory, when a particle

hits a smooth wall with an angle θ, it rebounds with a reflection angle of (180–θ).

However, in reality the reflection angle depends on factors such as frictional force, wall

roughness, and particle spin and is a complicated process (El Hor et al., 2008;

Sommerfield, 1999). The contaminants also follow particle-wall collision principle

(Weber, 1996). Therefore, in order to keep the model simple, smooth wall reflection was

assumed for the current simulation.

5.6.2 Model setup in HYDRUS

The standard diffusion equation as discussed in section 5.3.3 was used for

modeling diffusion in soil. The model setup was similar to diffusion in water. In soil,

diffusion cannot proceed as fast as it can in water because the contaminants must follow

longer pathways as they travel around soil particles. To account for this, an effective

diffusion coefficient is used instead of molecular diffusion coefficient in water.

The effective diffusion coefficient of a solute in a soil is estimated from the solute's

diffusion coefficient in water, soil porosity, and soil water content using MQ model

(Millington and Quirk, 1961). Since the contaminant follow tortuous path in the porous

soil, the effective diffusivity equation for the simulation is given by De = Dw (ε/τ)

(Millington and Quirk, 1961) and Tortuosity (τ) is given by

34

τ = ε-1/3

Therefore, the input parameters for diffusion in soil model are porosity and diffusivity.

The porosity value of 0.49 and the diffusivity value of 7.82*10-5

m2/day were used, and

then the simulation was run for 40 days. The contaminated sediment concentration was

entered through the graphical editor as shown in Figure 10. Pink represents uniformly

contaminated soil with depth of 15 mm and the blue domain represents water with depth

of 7mm.


uncontaminated water in HYDRUS setup

35

5.7 Problem 3: Diffusion in saturated soil with varying porosity


In problem 3, diffusion of contaminant from saturated soil to overlying water

layer was simulated. The setup was similar to problem 2, but with varying porosity.

Figure 11 shows the model setup in NetLogo. The basic buttons required were ‘setup’

and ‘go’ buttons. Other buttons used were ‘porosity’, ‘count agent’ and ‘count void

spaces’.

Figure 11: Uniformly contaminated saturated soil

Soil was represented by brown patches and contaminants were represented by

white particles. The soil was uniformly contaminated to the depth of 15mm. Initially the

water above 15 mm was assumed to be contaminant free. Lower half of the soil was 30

percent porous and the upper half of the soil was 49 percent porous. The contaminants

Water

(7mm)

Uniformly contaminated

layer (7.32 mm)

Porosity (0.49)

Uniformly contaminated

layer (7.68 mm)

Porosity (0.3)

36

move over 4.3 patches in a tick as discussed in section 5.6.1.The simulation was

performed with an initial contaminant concentration (H2AsO-4) of 150mg/L. The net

concentration in the beaker for the depth of 7.32 mm was 1098mg/m2

(150mg/L*1000m3*7.32/1000). Similarly, the net concentration in the beaker for the

depth of 7.68 mm was 1152 mg/m2 (150mg/L*1000m

3*7.68/1000).Therefore, the net

concentration in the beaker was 2250mg/m2.

5.7.2 Model setup in HYDRUS

The setup was similar to the setup in 5.6.2 and since the HYDRUS model was

not able to take varying porosities into account, the molecular diffusion coefficients from

both of the layers were averaged and entered as follows.

(3.02*10-5

m2/day + 1.57*10

-5m

2/day) /2 = 2.29*10

-5m

2/day.

The overall depth of the beaker was 22mm and the contaminated sediment depth was 15

mm as in the NetLogo model (Figure 12).


uncontaminated water in HYDRUS setup

37

6. Results and Discussion

6.1 Results from problem 1: Diffusion in water

6.1.1 Results from .etLogo model

The results obtained from sensitivity analysis for one turtle representing various

masses of contaminants are shown in Figures 13, 14, 15, and 16. Figure 13 shows that the

plot obtained by assigning 100mg to 1 turtle is zigzag in nature. It also shows that when

the same simulation was run multiple times, the resulting flux varies. For example,

cumulative flux after 10 days varies from 1950 mg to 2250 mg. The mean value of

cumulative flux was 2137.5mg in 10days and the standard deviation on 10th

day was

112.5(Table 3). Similarly, when the same simulation was run again with relatively small

amount representing a turtle, then the curve became smoother as shown in Figure 14.

Figure 13: 1 turtle equivalent to 100mg. The graph was obtained by running the same

simulation 50 times with 1 turtle equal to 100mg. The figure shows that the

cumulative flux varied between 2050 to 2250 mg afte

Figure14 shows that the curve obtained by assigning 1 turtle equal to 10mg is

much smoother than the one obtained by assigning 1 turtle equal to 100mg. Even

smoother curves are seen when a tu

16), and the variation after

amount a turtle represents the

continuous distribution of data and facilitates easy comparison of data. How

large number of turtles to represent the given concentration slows the simulation run

significantly. The simulation run time for a turtle representing 100mg was ½ minute,

whereas the simulation run time for a turtle representing 10 mg and 1

approximately 1 and 2 minutes.

representing 0.1mg, the simulation run time increased to 13 minutes, which is

slower compared to the earlier cases.

38

: 1 turtle equivalent to 100mg. The graph was obtained by running the same


cumulative flux varied between 2050 to 2250 mg after 40 days.

shows that the curve obtained by assigning 1 turtle equal to 10mg is


smoother curves are seen when a turtle was assigned 1mg (Figure 15) and 0.1 mg (Figure

), and the variation after multiple runs was also very small. This shows that smaller

amount a turtle represents the smoother the curve becomes. A smoother curve represents

continuous distribution of data and facilitates easy comparison of data. How


The simulation run time for a turtle representing 100mg was ½ minute,

whereas the simulation run time for a turtle representing 10 mg and 1 mg were

minutes. However, when the simulation was run with a turtle

representing 0.1mg, the simulation run time increased to 13 minutes, which is

compared to the earlier cases.

: 1 turtle equivalent to 100mg. The graph was obtained by running the same


shows that the curve obtained by assigning 1 turtle equal to 10mg is


) and 0.1 mg (Figure

multiple runs was also very small. This shows that smaller the

smoother the curve becomes. A smoother curve represents

continuous distribution of data and facilitates easy comparison of data. However, using a


The simulation run time for a turtle representing 100mg was ½ minute,

g were

However, when the simulation was run with a turtle

representing 0.1mg, the simulation run time increased to 13 minutes, which is much

Figure 14: 1 turtle representing 10mg. The graph was obtained by running the same

simulation 50 times with 1 turtle equal to 10mg.

Figure 15: Plot obtained by assigning a turtle 1mg each. The plot is smooth although it

shows some variation among the 50 simulations.

.

39

tle representing 10mg. The graph was obtained by running the same

simulation 50 times with 1 turtle equal to 10mg.

: Plot obtained by assigning a turtle 1mg each. The plot is smooth although it

among the 50 simulations.

tle representing 10mg. The graph was obtained by running the same

: Plot obtained by assigning a turtle 1mg each. The plot is smooth although it

Figure 16 : The plot is obtained by assigning 0.1 mg to a turtle. The plot is very smooth

and the variation after 50 simulations is also very small.

Table 3: Mean, Standard deviation, and Coefficient of Variation (CV) of cumulative flux

(mg/m2) for a turtle

Mass

equivalent

1 turtle=

100mg

mean

std.dev

CV

1 turtle=

10mg

mean

std.dev

CV

1 turtle=

1mg

mean

stdev

CV

1 turtle=

0.1mg

mean

std.dev

CV

40

: The plot is obtained by assigning 0.1 mg to a turtle. The plot is very smooth

and the variation after 50 simulations is also very small.

d deviation, and Coefficient of Variation (CV) of cumulative flux

Simulation duration (days)

1 10 20 30

375 2137.5 2212.5 2225

128.17 112.6 74.4 70.71

0.34 0.05 0.033 0.031

416.25 2097.5 2182.5 2202.5

64.79 24.34 23.75 19.82

0.155 0.011 0.01 0.008

436.5 2097 2180.3 2204

16.51 10.9 9.86 6.76

0.037 0.005 0.004 0.003

422.96 2094.32 2180.98 2203.74

6.02 4.08 1.42 1.81

0.014 0.001 0.0006 0.0008

: The plot is obtained by assigning 0.1 mg to a turtle. The plot is very smooth

d deviation, and Coefficient of Variation (CV) of cumulative flux

40

2225

70.71

0.031

2210

13.09

0.005

2210.2

5.86

0.002

2209.9

1.1

0.0004

41

Table 3 shows that deviation is very high after a day when turtles were assigned

100mg , but the deviation gradually decreased when turtles were assigned 10, 1, and, 0.1

mg. When small mass equivalents were assigned to each turtle, the deviation decreased

along with coefficient of variation. The small CV represents small variation among the

cumulative flux simulations and indicates a good fit for the current scenario. The table

obtained displays the expected result, which is also in agreement with Figures13-16 in

showing reduced variation in data for turtles with smaller concentrations. Based on the

results obtained it was decided to represent 1 turtle as 1mg for the current model.

Representing 1 turtle as 1mg yields similar result to the one representing 1 turtle as 0.1

mg but takes less time to run the simulations.

The results for 1 turtle representing 1mg obtained by running the same simulation

multiple times is shown in Table 4 along with mean, standard deviation, and CV values.

Even after running the simulation 100 times the CV was less than 0.1 and the deviation

was also small. This confirms that the variation among different simulation runs

remained negligible when 1 turtle represented 1mg.

42

Table 4: Different number of runs with 1 turtle representing 1mg

6.1.2 Result from HYDRUS model

Cumulative flux is quantified as the amount of contaminant reaching the top of

the beaker. Figure 17 shows a steep increase in cumulative flux until day 10 and a

stabilized flux after that. The contaminants had already reached the top of the beaker by

the10th

day, therefore the cumulative flux stabilized. The result obtained from HYDRUS

was deterministic; therefore, sensitivity and randomness analyses were not performed for

the given simulation.

.umber Simulation duration (days)

of runs

1 10 20 30 40

10-

runs mean 436.5 2097 2180.3 2204 2210.2

Std

dev 16.51 10.9 9.86 6.76 5.86

CV 0.037 0.005 0.003 0.003 0.002

50-

runs

mean 424.46 2098.1 2182.72 2204.76 2210.74

Std

dev. 18.51 12.59 8.53 6.91 6.3

CV 0.043 0.006 0.003 0.003 0.002

100-

runs

mean 428.46 2097.03 2180.71 2203.48 2209.93

Std

dev. 18.25 12.41 7.84 6.59 6.47

CV 0.042 0.005 0.003 0.002 0.002

43

Figure 17: Cumulative flux after 40 days

6.1.3 Comparison of HYDRUS and .etLogo results

When the HYDRUS and NetLogo results were plotted, a close resemblance was

found between HYDRUS and NetLogo results. Figure 18 shows that the cumulative flux

predicted by HYDRUS was higher than the cumulative flux predicted by NetLogo. More

contaminant particles stayed inside the beaker in the case of NetLogo than in HYDRUS.

Therefore, the amount of flux from NetLogo results was less than the amount from

HYDRUS results. For example, the flux after 20 days in NetLogo was 2175mg/m2,

whereas in HYDRUS it was 2230mg/m2.

-500.00

0.00

500.00

1000.00

1500.00

2000.00

2500.00

0 10 20 30 40 50

Cu

m f

lux

[mg

/m2

]

time[days]

44

Figure 18: Cumulative flux after 40 days

When HYDRUS and NetLogo results were plotted against each other (Figure 19),

a good fit was found between the two models. Again when the results were compared

with a 45 degree line, NetLogo results slightly underpredicted HYDRUS results.

Figure 19: HYDRUS vs. NetLogo for diffusion in water

-500

0

500

1000

1500

2000

2500

0 10 20 30 40 50

Cu

m f

lux

[m

g/m

2]

time[days]

HYDRUS

NetLogo

0

500

1000

1500

2000

2500

0 500 1000 1500 2000 2500

Ne

tLo

go

(mg

/m2

)

HYDRUS (mg/m2)

NetLogo vs. HYDRUS

45 degree line

45

6.2 Results from problem 2: Diffusion in soil

6.2.1 Results from HYDRUS and .etLogo

It can be seen from Figure 20 that both models produced similar results during the

first 10 days but after 10 days flux values from NetLogo were less than HYDRUS. The

difference in the result obtained was due to the relation on which each model focuses

upon. For example, HYDRUS results were obtained by assigning diffusivity coefficient

value and monitoring variability over time by using partial differential equations.

However, NetLogo results were obtained by allowing each individual contaminant to

move freely in the porous soil, which produced the given output. Another reason for the

difference is due to the tortuosity. Tortuosity depends on porosity (i.e. τ = ε-1/3

), but

NetLogo model does not have a specific relation between tortuosity and porosity,

therefore, the result obtained differ. Also, using the ideal scenario for contaminant

bouncing may have caused the results to differ.

Figure 20: Diffusion in soil

-500

0

500

1000

1500

2000

2500

0 10 20 30 40 50

Cu

m f

lux

[mg

/m2

]

time[days]

HYDRUS

NetLogo

46

When HYDRUS and NetLogo results were plotted against each other (Figure 21),

a good fit was found between the two models. However, when the results were compared

with a 45 degree line, the NetLogo results underpredicted HYDRUS results most of the

time. This could be because of contaminants getting trapped inside dead-end pores of soil

in the case of NetLogo.

Figure 21: HYDRUS vs. NetLogo for diffusion in soil

6.3 Results from problem 3: Diffusion in soil with varying porosity

6.3.1 Results from HYDRUS and .etLogo

Figure 22 shows that the cumulative flux predicted by NetLogo was higher than

the cumulative flux predicted by HYDRUS during the first 20 days. This could be

because in NetLogo model the upper part of the soil was highly porous. This means the

NetLogo setup has a higher molecular diffusion coefficient initially than that of

HYDRUS, which uses the average value. Therefore, NetLogo slightly overpredicted

0

500

1000

1500

2000

2500

0 500 1000 1500 2000 2500

Ne

tLo

go

HYDRUS

NetLogo vs. HYDRUS

45 degree line

47

HYDRUS results for the first 20 days. The flux values from NetLogo were less than the

flux values from HYDRUS after 20 days. This could be because of contaminants getting

trapped inside dead-end pores of soil in the case of NetLogo.

Figure 22: Cumulative flux after 40 days for diffusion in soil with varying porosity

-500

0

500

1000

1500

2000

2500

0 10 20 30 40 50

Cu

m f

lux

[mg

/m2

]

time[days]

HYDRUS

NetLogo

48

7. Conclusion and Future work

7.1 Conclusion

This is an exploratory research to model wetland processes using an agent-based

method. Since modeling wetland processes is a daunting task, only diffusion process was

modeled. The model results obtained for the diffusion have been verified against

HYDRUS-1D, a FEM based modeling platform. The results obtained from both models

for diffusion in water were in good agreement with each other. However, the HYDRUS

and NetLogo results for diffusion in soil differed. The difference is attributed to tortuosity

and assumption of simple particle-wall collision of contaminants.

This is the first time diffusion has been modeled using NetLogo; therefore, much

emphasis is placed on verifying results with HYDRUS, a popular fate and transport

modeling platform. While comparing NetLogo results with HYDRUS, it appears that

HYDRUS is used to determine a time frame that a tick should represent in NetLogo.

However, both the models were run independently, and comparison was done against

each other. For the current diffusion model, a tick can represent any period of time (e.g.

day, year), with the diffusion coefficient changed accordingly. Since HYDRUS was run

with a time increment of 14.4 minutes, a tick for the current NetLogo model represented

49

14.4 minutes. Both models were run upto a period of 40 days with a time increment of

0.01 days or 14.4minutes.

7.2 Future Work

Representing a soil matrix is a complex process in which organic matter, mineral

(sand, silt, and clay) and micro-pores and macro-pores are organized differently

according to their size. The soil model can be further developed by adding effects of

organic matter, and micro and macro-pores. Furthermore, more research is needed to add

processes such as adsorption-desorption, redox to the existing model simultaneously

along with diffusion.

50

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59

Appendices

Appendix A: Information Tab for Enzyme Kinetics and B-Z reaction

Enzyme Kinetics

This model demonstrates the kinetics of single-substrate enzyme-catalysis. The

interactions between enzymes and substrates are often difficult to understand and the

model allows users to visualize the complex reaction.

The standard equation for this reaction is shown below.

| Kc Kr

| E + S <=======> E-S ------> E + P

Kd

Here E represents Enzyme, S Substrate, E-S Enzyme-Substrate complex, and P product.

The rate constants are Kc for complex formation, Kd for complex dissociation, Kr for

catalysis. The first step in catalysis is the formation of the E-S complex. This can consist

of either covalent or non-covalent bonding. The rates of complex formation and

dissociation are very fast because they are determined by collision and separation of the

molecules. The next step is for the enzyme to catalyze the conversion of substrate to

product. This rate is much slower because the energy required for catalysis is much

higher than that required for collision or separation.

The model demonstrates several important properties of enzyme kinetics. Enzyme

catalysis is often assumed to be controlled by the rate of complex formation and

dissociation, because it occurs much faster than the rate of catalysis. Thus, the reaction

becomes dependent on the ratio of Kc / Kd. The efficiency of catalysis can be studied by

observing catalytic behavior at different substrate concentrations.

By measuring the rate of complex formation at different substrate concentrations, a

Michaelis-Menten Curve can be plotted. Analysis of the plot provides biochemists with

the maximum rate (Vmax) at which the reaction can proceed. As can be seen from the

model, this plot is linear at low levels of substrate, and non-linear at higher levels of

substrate. By examining the model, the reasons for this relationship can be seen easily.

60

Enzyme catalysis can also be controlled using inhibitors. Inhibitors are molecules that are

structurally similar to substrate molecules that can complex with the enzyme and interfere

with the E-S complex formation. Subsequently, the shape of the Michaelis-Menten

Curve will be altered. The model demonstrates the effects of inhibitors on catalysis.

More information on this model can be found at

http://ccl.northwestern.edu/netlogo/models/EnzymeKinetics. Center for Connected

Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

B-Z reaction

The Belousov-Zhabotinsky reaction (or B-Z reaction for short) is an unusual chemical

reaction. Instead of steadily moving towards a single equilibrium state, it oscillates back

and forth between two such states. Before this "chemical oscillator" was discovered, it

was thought that such a reaction could not exist.

If you do the reaction in a beaker, the whole beaker regularly changes color from yellow

to clear and back again, over and over. In this case, we say that the reaction is oscillating

in time. However, if you do the reaction in a thin layer of fluid trapped between two glass

plates, then a beautiful pattern emerges of concentric or spiral waves of color change

passing through the fluid. Here, the reaction is oscillating in both time and space.

This model is a cellular automaton (or CA) that produces spiral waves that resemble

those produced by the B-Z reaction. Similar spiral waves have also been observed in

biological systems, such as slime molds.

The B-Z reaction is a redox reaction that periodically moves between an oxidized and a

reduced state, and has been demonstrated for various chemicals. This model does not

attempt to replicate the actual mechanism of the chemical reaction, which is quite

complex (including 18 reactions and 21 species, according to the Fields-Koros-Noyes

model). The abstract features shared by the real reaction and this model include:

1. Two end states.

2. A positive feedback mechanism.

3. A negative feedback mechanism.

The positive feedback mechanism acts to push the system further in the direction that it is

already going, reinforcing and amplifying the initial change. (In the chemical reaction,

positive feedback comes from auto-catalysis.) The negative feedback mechanism pushes

the system back in the opposite direction once a threshold is reached, suppressing or

counteracting the effected change. More information on this model can be found at

http://ccl.northwestern.edu/netlogo/models/B-ZReaction. Center for Connected Learning

and Computer-Based Modeling, Northwestern University, Evanston, IL.

61

Appendix B: Information Tab for Diffusion in Water

WHAT IS IT?

The model shows diffusion in liquid. The beaker is uniformly contaminated to the depth

of 15mm and the overlying water layer is initially contamination free.

HOW IT WORKS

The turtles move on certain number of patches randomly at each tick. The turtles

reaching the top of the model vanish.

HOW TO USE IT

The buttons used for this model are SETUP, GO, DIFFUSE, and NUMBER

SETUP button initializes the model

GO button runs the simulation

DIFFUSE slider allows the user to assign the number of patches turtles move in a tick.

NUMBER slider allows the user to change the number of turtles during the simulation.

THI�GS TO TRY

Diffusive slider can be changed before or during the simulation run. Changing diffusive

slider affects the outcome of the simulation.

Number slider can be used to add more turtles before or during the simulation run. The

user can see the difference in the run time of the model.

EXTE�DI�G THE MODEL

The model can be further extended by adding temperature effect.

RELATED MODELS

Gaslab Adiabatic Piston

Gaslab Free Gas

62

Appendix C: Information Tab for Diffusion in Porous Media

WHAT IS IT?

This model shows diffusion in saturated soil. The soil is uniformly contaminated to the

depth of 15mm and the overlying water layer is initially contamination free.

HOW IT WORKS

The turtles move randomly inside the beaker and the turtles reaching the top of the beaker

vanish. The turtles move over certain number of patches at a tick. When the turtles hit

the brown patches (soil) they turn around 180 degrees and start moving elsewhere.

HOW TO USE IT

The buttons used for this model are SETUP, GO, DIFFUSE, and NUMBER

SETUP button initializes the model; GO button runs the simulation

DIFFUSE slider allows the user to assign the number of patches turtles move in a tick.

NUMBER slider allows the user to change the number of turtles before and during the

simulation run.

THI�GS TO �OTICE

This model shows how turtles look ahead before they move. By looking ahead, a turtle

determines what is in front of it and take a necessary action. Looking ahead is

appropriate in situations where the turtle is not supposed to go "on top of" certain agents.

The turtles look ahead of them and if they see brown patches they turn around 180

degrees and start moving randomly.

THI�GS TO TRY

Diffuse slider can be changed before or during the simulation run. Changing diffusive

slider affects the outcome of the simulation.

Number slider can be used to add more turtles before or during the simulation run. The

user can see the difference in the run time of the model.

Soil Porosity can be changed to see how the flux varies when one changes the porosity.

EXTE�DI�G THE MODEL

The model can be extended by separating micropores, macropores, and adding the effects

humic substances.

RELATED MODEL

Bounce example

63

Appendix D: .etLogo Code for Diffusion in Water

Comments are indicated by semi-colon (;;). Detail about NetLogo codes can be

downloaded from

http://cc.northwestern.edu/netlogo/docs.

to setup ;; begins defining a procedure named "setup".

ca ;; resets the world to an initial, empty state

setup-patches ;; moves to setup-patches procedure

setup-reaction ;; moves to setup- reaction procedure

do-plots ;; moves to plot

end ;; end the setup procedure

to setup-patches

ask patches [set pcolor white] ;; set up all the patches white

ask patches with [pycor = min-pycor] [set pcolor brown ]

end

to setup-reaction

crt numbers [set shape "circle" set color red set size 0.001 setxy random-xcor random

minpycor]

;; create given number of turtles (given via slider), with shape circle, size 0.183

;;µm, which are distributed uniformly throughout the depth of 15mm.

end

to go

if ticks >= 4000 [stop] ;;it the ticks is greater than 4000 the simulation stops.

ask turtles [

ifelse [pcolor] of patch-ahead dist = nobody [set heading (heading – random 180)]

[fd dist]] ;; ask turtles if they reach at the bottom of the beaker bounce back, else it is

safe to ;; move the given distance(dist =3.4,i.e., .0432mm2/min)

enterwatercolumn

tick ;; increases time count by 1

do-plots

end

64

to enterwatercolumn

ask turtles [ if ycor >= max-pycor ;;if turtles reach the top of the beaker ask them to

[ die ] ] ;;die

end

to do-plots

set-current-plot "cumulative-flux"

set-current-plot-pen "cumulative-flux"

plot number - count turtles ;; plots cumulative flux

set-current-plot "instantaneous-flux"

set-current-plot-pen"instantaneous-flux"

plot count turtles with [pycor = max-pycor ] ;; plots instantaneous flux

end

65

Appendix E: .etLogo code for Diffusion in Soil

Comments are indicated by semi-colon (;;). Detail about NetLogo codes can be

downloaded from

http://cc.northwestern.edu/netlogo/docs.

globals [flux]

patches-own [depth] ;; patch variable

to setup

ca

setup-patches

end

to setup-patches

ask patches with [pycor < 0] ;;the coordinates are customly arranged. Patches with

ycor < 0 ;; is modeled as uniformly contaminated

soil

[let n random-float 1.0

[

if n <= 0.51

[ set pcolor brown ] ;; ask 51% of the patches to set color brown; this means the

;; soil is 49% porous

]

ask patches with [ pycor = min-pycor ] [set pcolor brown ] ;;bottom of the beaker

ask patches with [pycor >= 0 and pycor <= max-pycor] ;;water layer

[set pcolor pink - 4 ] ;; different color set up for clear visualization

ask n-of 2250 (patches with [pcolor = black ]) ;;ask n number of black void space out

of ;;2250

66

[sprout 1 [set color white set size .001 ] ] ;; create 1 turtles on each black

;; patch and not on top of soil

end

to go

if ticks >= 4000 [stop] ;; stop if ticks is greater than 4000

ask turtles

[

ifelse (patch-ahead temp = nobody ) or ([pcolor] of patch-ahead temp = brown )

;; the world does not wrap vertically so when the turtles reach the bottom of beaker

;;or hit the soil it performs do-edge routine

;;else it start moving the given distance

[do-edge-routine]

[fd dist]

enterwater ]

do-plot

tick ;;increase counter by 1

end

to enterwater

if ycor >= max-pycor ;; if turtles move beyond max-pycor i.e. top of beaker,

[die]

end

to do-edge-routine

set heading (heading - random 180)

end

to - plot

set-current-plot "cumulative-flux"

set-current-plot-pen "cumulative-flux"

set flux count turtles

plot number - count turtles

end

67

Appendix F: .etLogo code for Diffusion in Soil with Varying Porosity

globals[flux]

to setup

ca

setup-colour

setup-patches

end

to setup-colour

ask patches with [pycor <= -40 ] ;; different color for clear visualization

[set pcolor gray ]

end

to setup-patches

ask patches with [pycor <= -40 ] ;;the coordinates are custom arranged. Patches with

;;ycor < 0 is modeled as uniformly contaminated soil

[

let n random-float 1.0

if n <= 0.7

[ set pcolor brown - 0.1] ;; ask 70% of the patches to set color brown; this means the

;; soil is 30% porous

]

ask patches with [pycor > -40 and pycor < 0]

[let n random-float 1.0

if n <= 0.52 ;; ask 52% of the patches to set color brown; this means the soil is

;;48% porous

[ set pcolor brown ] ]

ask patches with [pycor >= 0 and pycor <= max-pycor]

[set pcolor pink - 4 ] ;; one way to set up color for clear visualization

ask n-of 1098 (patches with [pcolor = black ]) ;; ask n number of black void space in soil

;;out of 1098;which is the concentration for the

given depth.

68

[ sprout 1 [set color white set size .001 ] ] ;; create 1 turtles on each black patch

and ;; not on top of soil

ask n-of 1152 (patches with [pcolor = gray ]) ;;ask n number of gray void space in soil

out ;;of 1152; which is the concentration for the given depth

[ sprout 1 [set color white set size .001 ] ]

end

to go

if ticks >= 4200 [stop]

ask turtles

[

ifelse (patch-ahead temp = nobody ) or ([pcolor] of patch-ahead temp = brown ) or

([pcolor] of patch-ahead temp = brown - 0.1 )

;; the world does not wrap vertically so when the turtles reach the bottom of beaker

or hit the soil it performs do-edge routine

;;else it start moving the given distance

[do-edge-routine]

[fd temp]

enterwater

]

do-plot

tick ;; increase time counter

end

to enterwater

if ycor >= max-pycor ;; if turtles move beyond max-pycor i.e. top of beaker, ask

them to die

[die]

end

to do-plot

set flux count turtles

plot 2250 - flux ;;869 is the total concentration in the beaker

end

to do-edge-routine

set heading (heading - random 180) ;; turn around 180 degree

end

A Thesis entitled Modeling Diffusion Using an Agent-Based ...

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