ARTICLE IN PRESS

Chemical Engineering Science 65 (2010) 1212–1226

Contents lists available at ScienceDirect

Chemical Engineering Science

0009-25

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/ces

Computer simulation of gas generation and transport in landfills. IVModeling of liquid–gas flow

Raudel Sanchez, Theodore T. Tsotsis, Muhammad Sahimi �

Mork Family Department of Chemical Engineering and Materials Science, University of Southern California, Los Angeles, CA 90089-1211, USA

a r t i c l e i n f o

Article history:

Received 2 March 2009

Received in revised form

24 September 2009

Accepted 29 September 2009Available online 7 October 2009

Keywords:

Landfills

Porous media

Two-phase flow

Computer simulations

Biodegradation

Gas transport

09/$ - see front matter & 2009 Elsevier Ltd. A

016/j.ces.2009.09.076

esponding author.

ail address: moe@iran.usc.edu (M. Sahimi).

a b s t r a c t

In the first three parts of this series a three-dimensional (3D) model was developed for transport and

reaction of gaseous mixtures in a landfill. An optimization technique was also utilized in order to

determine a landfill’s spatial distributions of the permeability, porosity, the tortuosity factors, and the

total gas generation potential, given a limited amount of experimental data. In the present paper we

develop the model further by including the flow of both the leachate and the gases. A 3D dynamic

model is developed that accounts for the generation of the four main gases of a landfill, along with the

dissolved organic acids and the carbon in the presence of the leachate. The model is then utilized,

through extensive numerical simulations, to study the effect of the various factors on the concentrations

of the gases and the micro-organism. In particular, we demonstrate the strong effect of the

heterogeneities of a landfill, represented by the spatial distribution of the local porosities, as well as

an anisotropic distribution of the local permeabilities, on the behavior of a landfill, and in particular

pressure buildup in it.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Strict environmental regulations have motivated considerableresearch on understanding the dynamics of landfills, and developingmethods of minimizing the hazards associated with the gases thatare generated by them. At the same time, given the known energyresources of the world, landfill gas (LFG), and in particular methane,are also viewed as a promising renewable energy source, henceproviding additional incentive for studying the dynamic behavior oflandfills. An essential tool for gaining a better understanding of thedynamics of landfills is an accurate model that can not only predicttheir future behavior, but also provide reliable estimates for theamount of CH4 that one may expect to extract from them. In orderto develop such a model, one must have a comprehensive under-standing of the reaction, generation, and transport of gases inlandfills. The LFG typically includes CH4, CO2, O2, N2, and a traceamount of other organic compounds. Natural, as well as man-controlled, factors, such as temperature, refuse concentration,moisture, and the pH influence the waste decomposition process,and control the LFGs composition.

In three previous papers (Hashemi et al., 2002; Sanchez et al.,2006, 2007, hereafter referred to, respectively, as Parts I, II, and III)

ll rights reserved.

we presented a comprehensive three-dimensional (3D) modelwhich accounts for the generation and transport of the four majorgaseous components of the LFG. Given that a landfill is essentiallya large-scale porous medium, the model developed in Parts I–IIIallowed for arbitrary spatial distributions of the permeability,porosity, and tortuosity factor in a landfill and its surrounding soil(if the landfill does not have any liners), as well as an arbitrarynumber of wells for extraction/monitoring of the LFG.

Part I studied the behavior of landfills under quasi-steady-statecondition, which pertains to those that have been closed for a longtime, and investigated the effect of various important parameters.Part II investigated the dynamics of a landfill under variousconditions, such as, for example, when (a) some of the monitoringand/or extraction wells are shut down; (b) some new wells aredrilled in the landfill, after it has been closed for sometime, inorder to collect additional gases or meet the environmentalregulations, and (c) the landfill’s cover is, for some reason,damaged. Part III proposed a new approach to the developmentof an accurate model of a landfill by addressing a key question:Given a limited amount of data for one or a few properties of a

landfill, what are the optimal spatial distributions of its porosity,

permeability, tortuosity factors, and the wastes’ gas generation

potentials that not only honor (preserve) the existing data, but also

provide accurate predictions for its future behavior? The study inPart III formulated the problem as one of optimization, andutilized a technique based on the genetic algorithm—a powerfulmethod of finding the optimal solutions—to address the problem.

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R. Sanchez et al. / Chemical Engineering Science 65 (2010) 1212–1226 1213

In addition to various gases, landfills usually contain leachate.Water enters a landfill by different means, such as precipitationfrom rain, moisture that the wastes contain when they are storedin the landfill, and recirculation of leachate throughout thelandfill. When it flows inside a landfill, the water comes intocontact with the solid wastes and is contaminated by them, hencegenerating the leachate. The production of CO2 and CH4 from thebiodegradation of the wastes in a landfill is the result of a series ofcomplex processes involving the leachate. Fig. 1 shows theschematic of the various sections of a typical landfill thatcontains a leachate collection system.

In Parts I and II we ignored the presence of the leachate, andincluded in the model only the rates of reaction and gasgeneration. Thus, in the present paper we develop the modelfurther, to be utilized in a future study with the optimizationmethod developed in Part III, by focusing on its extension to thecase in which the flow of both the leachate and the gases isincluded in the model. We develop a 3D dynamic model thataccounts for the generation and consumption of the four maincomponents of the LFG, along with the dissolved organic acids andthe carbon, in the presence of the leachate. The model is thenutilized for studying of gas generation, as well as transport of theliquids and the LFG, and the effect of several importantparameters that the control such phenomena. To our knowledge,a 3D dynamic model of the type that we develop in the presentstudy has not been previously proposed and studied.

There are already several models in the literature for flow ofleachate in a landfill. Demetracopoulos et al. (1986) proposed a 1Dtransient model for leachate production in municipal landfills thataccounted for the effect of generation and transport of contaminants,

Waste

Sand

Sand

Clay

Clay

Sand

Top Soil

Firs

t Sub

prof

ileS

econ

d S

ubpr

ofile

Third

Sub

prof

ile

1

23

4

5

6

7

8

910

11

VerticalPercolation Layer

Lateral Drainage Layer

Geomembrane Liner

Barrier Soil Layer

Precipitation

VerticalPercolation

Layer

Lateral Drainage Layer

Lateral Drainage Net

Geomembrane Liner

Lateral Drainage Layer

reniLlioSreirraB

Veg

Fig. 1. Various sections of a typical landfill c

as well as the effect of the concentration of micro-organisms. It alsoincluded the effect of both mass transfer and kinetic contributions.The reactions were modeled using a modified Monod equation (seealso Parts I–III). Celia et al. (1989) and Kindred and Celia (1989)developed a 1D contaminant transport model with biodegradation,which was represented by both aerobic and anaerobic reactions,along with multisubstrate and multipopulation biodegradation.Schroeder et al. (1994) developed the so-called HELP (HydrologicEvaluation of Landfill Performance) model for predicting the move-ment of leachate entering and leaving a landfill. The HELP model is aquasi-2D model that computes the flow of water entering differentsections of a landfill, and provides predictions for the verticalpercolation and lateral drainage layers, geomembrane and barriersoil liners, and lateral drainage nets. It is widely used for predictingthe water movement for long periods of time, but has been shown tobe inaccurate for predicting the daily leachate flow inside a landfill.El-Fadel et al. (1996a, b, 1997) used bio-kinetics equations to describethe biodegradation of solid wastes. A multicomponent gaseousmixture was considered, as were aqueous acetate acid and variousmicro-organisms. Huang et al. (1998) developed a 1D transientmodel that accounted for both moisture and contaminant transportin fly-ash landfills. The model landfill consisted of several layers, soas to take into account the biodegradation of the various layers of therefuse. Kouzeli-Katsiri et al. (1999) used a two-equation model usingfirst-order kinetics, which modeled the landfill as a single fullymixed reactor, in order to predict the leachate quality. Schroederand Aziz (1999) developed a model for the contamination ofleachate in storage facilities, using the advection–dispersion equa-tion, in order to predict the amount of the contaminants. Maraqaet al. (1999) used a 1D transient model to predict the concentration

EvapotranspirationRunoff

Infiltration

Percolation

Lateral Drainage (From Cover)

Lateral Drainage (Leachate Collection)

Leakage

epiPeganiarD

Lateral Drainage (Leakage Detection)

etation

Cap

or C

over

Geo

mem

bran

eLi

ner S

yste

mC

ompo

site

Line

r Sys

tem

ontaining a leachate collection system.

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R. Sanchez et al. / Chemical Engineering Science 65 (2010) 1212–12261214

of the contaminants in groundwater flow, and studied the effect ofthe retardation coefficient.

Khanbilvardi et al. (1995) developed a 2D transient model forflow of moisture, in order to simulate leachate flow in a landfill.They included the effects of evapotranspiration, precipitation, andrunoff. Mohammed et al. (2000) developed a model to predictsolute transport due to evaporation. White et al. (2004) and Al-Thani et al. (2004) modeled the biochemical degradation of solidwastes in landfills, and included in the model both leachatetransport and gas generations. Yildiz et al. (2004) presented adetailed model for the leachate production by using biodegrada-tion of wastes in the acidogenesis and methanogenesis phases.Suk et al. (2000) and Lee et al. (2001) used biological reactivetransport to predict biodegradation of the contaminants inleachate flow. Their model was 1D transient, intended formodeling of flow of leachate in landfills.

The rest of this paper is organized as follows. In the nextsection we describe the essentials of the model. Section 3describes the numerical method used for solving the governingequations. The results are presented in Section 4, where theirimplications are also described and discussed. The paper issummarized in the last section, where we also discuss furtherdevelopment of the model.

2. The model

Modeling two-phase flow in landfills that are filled with alltypes of materials is a complex problem—a result of the physical,chemical, and biological decompositions that occur simulta-neously in a landfill, and produce various byproducts by thebiodegradation of the municipal wastes. The products includesolids, liquids, and gaseous compounds. The most importantdecomposition process is, however, biological, since it is the onlymechanism for producing CH4 (and CO2) throughout a landfill.

For biological decomposition to occur, a suitable environment forthe micro-organisms to cultivate must be present. In order for themicro-organisms to survive and biodegrade the solid wastes intogaseous products, moisture must be present, without which no CH4

is produced within a landfill. Since moisture in a landfill is importantto the survival of the micro-organisms, knowledge of the moisturecontent is crucial for predicting the amount of the gases produced.

Biological decomposition in landfills takes place in three stages.The first is aerobic decomposition that occurs mostly at the initialstages, when the refuse is placed in the landfill, but also occurs solong as O2 is present. Aerobic decomposition represents only a

Ana

erob

ic, N

on-M

etha

noge

nic

Aer

obic CO

2

CH4

N

H2

O2

Land

fill g

as c

ompo

sitio

n

10

40

30

20

0

50

60

70

80

90

100

T

Fig. 2. Typical landfill gas composition for the aerobic, nonmethan

small portion of the overall biological decomposition. Aerobicmicro-organisms degrade the organic materials into CO2, water,and partially degraded organics, but no CH4, with the process beingexothermic. The second stage is the acid-phase anaerobic ornonmethanogenic decomposition, which occurs after the O2 withinthe landfill is depleted and the acid-producing micro-organismstarts to degrade the solid wastes. Organic acids, ammonia,hydrogen, and CO2 are produced by nonmethanogenic biodegrada-tion. The final stage of the biological decomposition occurs whenmethanogenic micro-organisms biodegrade the organic acidsproduced in the nonmethanogenic phase, in order to generateCH4 and CO2. Fig. 2 presents the typical landfill gas composition forthe various stages, namely, the aerobic, nonmethanogenic,transient anaerobic, and steady-state anaerobic.

Biodegradation of the wastes has been described in severalpapers, such as those of El-Fadel et al. (1996a, b, 1997), El-Fadel andAbou Najm (2002), Suk et al. (2000), Lee et al. (2001), McBean et al.(1995), Garcia de Cortazar et al. (2002), and Mora-Naranjo et al.(2004). The model that we use for describing biodegradation andleachate flow in landfills is based on the work of Suk et al. (2000)and Lee et al. (2001) and others, which we extend to 3D landfills andmodel them as porous media in which the permeability, porosity,and the tortuosity factor vary spatially. However, the methodologythat we present is completely general and may be used with anyother model of biodegradation. The Richardson equation is used,

f@Sw

@h

� �@h

@t¼

@

@zKzzkrw

@h

@z� 1

� �� �þ@

@yKyykrw

@h

@y

� �þ@

@xKxxkrw

@h

@x

� �;

ð1Þ

that relates the hydraulic head h to the water saturation Sw. All othernotations are given at the end of the paper. Several equations havebeen proposed in the past for relating Sw to h, and the relativepermeability krw to Sw, which can be utilized in the above equation.In the present paper we use the following equations that are due toVan Genuchten (1980) (clearly, any other sets of equations may alsobe used):

Sw ¼ Srþð1� SrÞ1

1þðahÞn

� �ð1�1=nÞ

; ð2Þ

Krw ¼ S1=2e f1� ½1� Sn=ðn�1Þ

e �ðn�1Þ=ng2; ð3Þ

Se ¼Sw � Sr

1� Sr: ð4Þ

In all the simulations described below we used, Sr ¼ 0:01, for theirreducible water saturation, and, a¼ 1:0 and n¼ 2, as the porous

Ana

erob

ic, M

etha

noge

nic

-Ste

ady

Ana

erob

ic, M

etha

noge

nic

-Uns

tead

y

2

55%

40%

5%

ime

ogenic, transient anaerobic, and steady-state anaerobic stages.

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R. Sanchez et al. / Chemical Engineering Science 65 (2010) 1212–1226 1215

media parameters. The saturated hydraulic conductivity was takento be, K ¼ 0:0183 m=day.

The governing equations for the concentrations of thedissolved organic carbon ðCCÞ, O2 ðCOÞ, and organic acids ðCacÞ inthe landfill are given by

eSwR@CC

@t¼= � ðeSwD=CCÞ � v �=CC � Cc1bc1eSw

CC

Kacc1þCC

CO

Kdoc1þCO

�CðnmÞac baceSw

CC

KacacþCC1�

CO

KdoacþCO

� �þk0eSw

S

SoðCST � CCÞ;

ð5Þ

eSw@CO

@t¼= � ðeSwD=COÞ � v �=CO � Cc1bc1Fc1eSw

CC

Kacc1þCC

CO

Kdoc1þCO

�Cc2bc2Fc2eSwCac

Kacc2þCac

CO

Kdoc2þCO� qðC�O � COÞ; ð6Þ

eSw@Cac

@t

¼= � ðeSwD=CacÞ � v �=CacþCðnmÞac bacPaceSw

CC

KacacþCC1�

CO

KdoacþCO

� �

�Cc2bc2eSwCac

Kacc2þCac

CO

Kdoc2þCO

�CmcbmceSwCac

KacmcþCac1�

CO

KdomcþCO

� �: ð7Þ

The last term on the right side of Eq. (5) represents the rate of masstransfer from the solid to the liquid phase. The first reaction kineticsterm on the right side of Eq. (5) expresses the consumption of theorganic carbon due to anaerobic biodegradation. As the O2 isdepleted, the consumption of the organic carbon from the aerobicmicro-organism loses its influence on the change of the concentra-tion of the dissolved organic carbon. The second reaction kineticsterm on the right side of Eq. (5) represents the consumption of thedissolved organic carbon from the acid-producing micro-organism.

The O2 concentration, consumed only in the aerobic phase, isgoverned by Eq. (6) that accounts for the O2 depletion in the aerobicphase, when the micro-organism biodegrades the dissolved organiccarbon, and is represented by the first reaction kinetics term on theright side. The second reaction kinetics term on the right side of Eq.(6) is the aerobic biodegradation of the dissolved organic acids. In Eq.(7), which governs the concentration of the organic acid, the firstreaction kinetics term on the right side expresses the production ofthe dissolved organic acid. In this process the acid-producing micro-organism biodegrades the dissolved organic carbon to produce theorganic acid in the nonmethanogenic phase. The effect of the aerobicdecomposition on the dissolved organic acid is expressed by thesecond reaction kinetics term on the right side of Eq. (7), the effect ofwhich decreases as the concentration of the dissolved O2 is depletedin the landfill. The final reaction kinetics term on the right side of Eq.(7) represents the effect of the methanogenic phase which is thesource of CH4 and CO2.

The rate equations for micro-organisms are given by

@

@tðfSwCc1Þ ¼ Cc1bc1Yc1fSw

CC

Kacc1þCC

CO

Kdoc1þCO� lc1Cc1fSw; ð8Þ

@

@t½fSwCðnmÞ

ac � ¼ CðnmÞac bacYacfSw

CC

KacacþCC1�

CO

KdoacþCO

� �� lacCðnmÞ

ac fSw;

ð9Þ

@

@tðfSwCc2Þ ¼ Cc2bc2Yc2fSw

Cac

Kacc2þCac

CO

Kdoc2þCO� lc2Cc2fSw; ð10Þ

@

@tðfSwCmcÞ ¼ CmcbmcYmcfSw

Cac

KacmcþCac1�

CO

KdomcþCO

� �� lmcCmcfSw:

ð11Þ

We assumed that the refuse is composed primarily of three types ofwastes, namely, the readily, moderately, and the least biodegrad-able materials. Then, the biodegradation process is modeledaccording to El-Fadel et al. (1996a, b)

@S

@t¼ � k0fSw

S

S0ðCST � CCÞ: ð12Þ

The equation that governs the concentration of the gas i is thestandard convection–diffusion–reaction (CDR) equation:

f@Ci

@t¼= � ðDi=CiÞ � v �=CiþRi; ð13Þ

where

R1 ¼ CmcbmcfSwCac

KacmcþCac1�

CO

KdomcþCO

� �gCH4

; ð14Þ

is the rate of the production of CH4, attributed only to thebiodegradation of the dissolved organic acid by the methanogenicmicro-organism. The CO2 production rate, given by

R2 ¼ Cc1bc1fSwCw

Kacc1þCW

CO

Kdoc1þCOgc1

þCc2bc2fSwCac

Kacc2þCac

CO

Kdoc2þCOgc2

þCðnmÞac bacfSw

Cw

KacacþCw1�

CO

KdoacþCO

� �gac

þCmcbmcfSwCac

KacmcþCac1�

CO

KdomcþCO

� �gmc; ð15Þ

represents the contributions by several sources, such as the aerobic(the first term on the right side), nonmethanogenic, and themethanogenic phases. The second term of Eq. (15) represents theproduction of CO2 by biodegradation of the dissolved organic acidby the aerobic micro-organism. When the concentration of thedissolved O2 decreases, the contributions of the first two termsbecome negligible. The third term on the right side of Eq. (15) is thecontribution to the concentration of CO2 by biodegradation of thedissolved organic carbon by the nonmethanogenic micro-organism,while the last term represents the contribution to the production ofCO2 arising from biodegradation of the dissolved organic acid bythe methanogenic micro-organism.

Let us emphasize that, instead of Eqs. (5)–(7), (8)–(11), and (14)and (15), any other set of equations that express the kinetics of thebiodegradation may be used. We utilize these equations because,(i) they seem to be most realistic and have been widely used in thepast, and (ii) one may make a direct comparison with thesimulations results presented in this paper that are for a 3Dmodel of landfills, and the previous ones that use the same set ofkinetic expressions but greatly simplified the model to 1D or 2D.In particular, one may also compare the results, described below,for a heterogeneous landfill with those obtained previously for ahomogeneous landfill (see below).

The boundary conditions for the gas phase in the landfill at theside walls and the bottom of the landfill are the no flux condition,

�Dkm@r@nþvnr¼ 0; ð16Þ

where n is the flow direction, and vn is the flow velocity in thatdirection. One must also invoke the continuity of the fluxes at theboundary between the landfill and its top cover,

�Dkm@r@nþvnr

� �z�¼ �Dkm

@r@nþvnr

� �zþ; ð17Þ

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Input Variables

Provide Initial Guess

Calculate Krw

Solve Richardson’s Equation

Convergence Criteria Met?

NO

Calculate Micro-OrganismConcentrations

Calculate Aqueous Phase Concentrations

Convergence Criteria Met?

NO

YES

Calculate Gas Phase Concentrations

Convergence Criteria Met?

Steady State Achieved?

NO

YES

YES

YES

NOStopPrint Solution

Fig. 3. The flow chart for the two-phase flow computations.

R. Sanchez et al. / Chemical Engineering Science 65 (2010) 1212–12261216

where z� and zþ denote two points on the two sides of theboundary between the landfill and its top cover.

3. Numerical simulation

Fig. 3 present the flow chart for the entire computations. Thegoverning equations were first made dimensionless (see theAppendix), and were then discretized. The derivatives withrespect to time were discretized using the Crank–Nicholsonmethod. Consider, first, the liquid phase. The boundaryconditions at the top of the landfill were set to be,CC ¼ CO ¼ Cac ¼ 0 mg=l. A constant infiltration rate of water, Q,was also assumed at the top of the refuse, whereas a free drainageboundary condition was applied at the bottom. Two different setsof initial conditions were utilized, with the main difference being

the initial concentration of the dissolved acid (nonmethanogenicmicro-organism) CðnmÞ

ac . One case corresponded to no initialdissolved acid in the landfill. The porosity of the landfill for thiscase was set to be 0.35. The second case simulated was one inwhich there was initially a large amount of the dissolved acid inthe landfill. In this case the porosity was set to be 0.5. We alsostudied the case in which the porosity was spatially distributed;see below. The two sets of initial conditions are listed in Tables 1and 2, respectively. In addition, the corresponding biodegradationparameters used in the simulations, which represent their typicalvalues, are listed in Tables 3 and 4.

At any given time (set by the time step Dt) we first solved forthe hydraulic head h using Eqs. (1)–(3). An initial saturationdistribution was assumed, in order to compute the relativepermeability and the saturation Sw in each block of the computa-tional grid. Newton’s method was then used and the set of the

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Table 1The first set of the initial conditions.

S 45,000

Cc1 1.0

Cc2 1.0

Cac 0.0

Mac 2.0

Cmc 0.1

CC 0.0

CO 8.0

CCH40.0

f 0.35

All the quantities (except f) are in mg/l.

Table 2The second set of the initial conditions.

S 80,000

Cc1 2.0

Cc2 2.0

Cac 4050.0

Mac 2.0

Cmc 2.0

CC 0.0

CO 8.0

CCH40.0

f 0.5

All the quantities (except f) are in mg/l.

Table 3The biodegradation parameters used in the simulations with the first set of the

initial conditions (Table 1).

Quantity xc1 xac xc2 xmc

b ðday�1Þ 2.036 0.906 1.40 0.40

Kac (mg/l) 5000 5000 0.1 0.01

Kdo (mg/l) 0.3 0.3 0.3 0.1

Y 0.04 0.1 0.3 0.08

lðday�1Þ 0.005 0.009 0.005 0.006

gCO210.0 0.002 3.3 8.0

F 3.0 – 3.0 –

P – 0.001 – –

gCH4– – – 4.0

Table 4The biodegradation parameters used in the simulations with the second set of the

initial conditions (Table 2).

Quantity xc1 xac xc2 xmc

b ðday�1Þ 2.33 0.806 2.33 0.60

Kac (mg/l) 20,000 1000 21,375 10

Kdo (mg/l) 0.3 0.3 0.3 0.1

Y 0.322 0.22 0.322 0.26

l ðday�1Þ 0.0025 0.001 0.0025 0.005

gCO210.0 0.2 10.0 0.008

F 3.0 – 3.0 –

P – 0.5 – –

gCH4– – – 0.009

Table 5Values of the parameters, used as the base case in the simulations.

Sector f=t Kx ¼ Ky (md) Kz (md) Thickness (m)

Landfill 0.125 3 1 28

Cover 0.025 1 1 2

t is the tortuosity in the gas phase.

R. Sanchez et al. / Chemical Engineering Science 65 (2010) 1212–1226 1217

linearized equations, which results from the discretization ofEq. (1), was solved by the biconjugate-gradient (BCG) method. Ifthe solution was convergent according to a convergence criterion,we would move on to the next step of the computations (seebelow). Otherwise, the relative permeabilities of the blocks and thespecific moisture saturation Sw were updated, and the BCG methodwas used again to recompute the hydraulic heads in the blocks. Theprocedure was repeated a few times until there was no significant

changes in the hydraulic heads between two consecutive iterations.Then, for that particular time step, the biomass reaction equationswere solved at every grid block using the fourth-order Runge–Kutta–Gill method.

With the hydraulic heads in every grid block calculated at thecurrent time step, the velocity and the dispersion coefficients forthe liquid phase were also computed. The solutions were thenused in the governing equations for the concentrations CC , CO, andCac , Eqs. (5)–(7), which, although nonlinear (due to the reactionterms), are also of the CDR-type equations. A novel numericalmethod for solving the nonlinear CDR equation was proposed inPart II, which was also utilized here. Thus, we first solved Eq. (5),at the current time step, for CC, the concentration of the dissolvedorganic carbon. Initial guesses for CC, CO, and Cac were assumed,and Newton’s together with the BCG methods were used to solvethe large system of linearized equations, resulting from thediscretization. Once the concentration profile for CC was com-puted after the first iteration in Newton’s method, it was used tocalculate the reaction term of the governing equation for theconcentration of the dissolved O2, Eq. (6), in order to compute itsconcentration profile by the same method that we used for CC. Theupdated solution for CC was then utilized to compute the CO

profile. The updated solutions for CC and CO were then used in (thediscretized) Eq. (7), in order to compute the concentration profileCac of the organic acid. The convergence criterion was thenchecked to see whether the numerical solutions obtained after thefirst Newton’s iteration satisfy the criterion. If not, the aboveprocedure was repeated as many times as necessary untilconvergence was achieved, after which the computations for thecurrent time step and the liquid phase were complete.

After the liquid concentrations for the current time step werecalculated, the reaction rates of CH4 and CO2 in each block of thecomputational grid were computed, and the CDR equations thatgovern the concentrations of the gases were solved by the sametechnique described above. The entire procedure was repeated forthe next time step. Because the simulations required a largeamount of CPU time, parallel computations with the domaindecomposition method was used, in order to reduce the requiredsimulation time (see Part III). Eight processors were utilized in theparallel computations.

The results described below were obtained for a landfill of size30 m� 30 m� 30 m, with a computational grid of size Lx � Ly � Lz

blocks and, Lx ¼ Ly ¼ 57 and Lz ¼ 39. The cover’s thickness was2 m, represented by the top 57� 57� 10 part of the grid. Table 5lists the typical values of the various parameters. The time step Dt

was dependent on the system’s configuration. In many cases, weused Dt¼ 1 day. The number of iterations for solving theRichardson equation was 20–40, and for solving the governingequations for CO, CC , and Cac was 15–30. Convergence to thenumerical solution of the governing equations for the gasesrequired 30–50 iterations.

4. Results and discussions

We carried out four sets of simulations. For the first two series,we assumed that the landfill is homogeneous but anisotropic,

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R. Sanchez et al. / Chemical Engineering Science 65 (2010) 1212–12261218

with the vertical permeability being 13 of the horizontal ones (the

permeabilities Kx and Ky in the horizontal planes were assumed tobe equal). The difference between the two sets of simulations wasmostly due to the different sets of the initial conditions (seeTables 1 and 2) that were utilized. For the third set of simulationswe distributed the permeabilities and porosities throughout thelandfill (see below), so that a completely heterogeneous landfillwas simulated. In the fourth series of the simulations wecompared the behavior of a highly heterogeneous landfill withand without the leachate. In what follows we describe the resultsfor each set, and discuss their similarities and differences, as wellas their implications.

4.1. Homogeneous landfills

Fig. 4 displays the production rates of CO2 for the first set ofparameters given in Tables 1 and 3. According to these results,over the first few days, most of the contributions to the totalproduction of CO2 are due to the aerobic micro-organisms xc1 andxc2, with the early peak in the CO2 production being an indicationof the strength of such contributions. But as oxygen is depleted bythe aerobic micro-organisms, their corresponding contributionsalso diminish quickly, and become negligible after a few weeks. Ifoxygen is introduced into the landfill, more CO2 would beproduced by the aerobic decomposition of both the dissolved

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

42000Tim

CH

4 P

rodu

ctio

n R

ate

(g/m

3 da

y)

Fig. 5. Same as in Fig. 4, but for t

0

2

4

6

8

10

12

0Time (days)

CO

2 P

rodu

ctio

n R

ate

(g/m

3 da

y)

CmcCacCc2Cc1Total

nm

100 200 300 400 500 600 700 800

Fig. 4. The production rate of CO2 for the first set of parameters (Tables 1 and 3)

used in the simulations.

organic carbon and organic acid. After about 100 days thecontributions due to xac (the dissolved organic acid, or thenonmethanogenic micro-organism) and xmc (the methane-producing micro-organism) to the production of CO2 becomeapparent. The production of CO2 due to xac appears after thedepletion of O2 and the aerobic phase. Most of the produced CO2

arises from the biodegradation of the dissolved organic acid by themethanogenic bacteria. The production of CO2 reaches a newmaximum before 400 days. The maximum contributions due toboth xac and xmc appear approximately at the same time.

Fig. 5 presents the time-dependence of the rate of productionof CH4. Unlike CO2, only micro-organism xmc contributessignificantly to the production of CH4. Almost no CH4 isproduced for the first 50 days, since the nonmethanogenicmicro-organism xmc has not yet decomposed into significantenough amounts to produce CH4. But, once the decomposition hasbecome significant enough, which happens after about 100 days,the CH4 production rate rises sharply, attaining its maximum overa relatively short time after about 250 days. Beyond themaximum, the production rate drops off just as quickly,reaching a quasi-steady-state stage after about 400 days. Thesetrends are clearly consistent with those shown in Fig. 4, whichshow the various contributions to the rate of production of CO2.

Fig. 6 displays the concentration profiles for the dissolvedorganic carbon and the organic acid, as well as the mole fractionprofiles of CH4 and CO2 (or, equivalently, their partial pressures),all computed at the center of the landfill. They demonstrate someinteresting features. For example, the concentration of thedissolved organic carbon reaches its maximum after about 200days, which is close to when the peak production of CO2 from xmc ,the methanogenic micro-organism, is attained; see Fig. 4. Overmuch of the same period, the concentration of the dissolvedorganic acid is very small, after which it increases sharply over ashort period of time, but also falls off just as quickly, so that itsproduction rate essentially vanishes after 250 days, which is whenthe nonmethanogenic micro-organism xac achieves its maximumconcentration (see Fig. 4). Over roughly the same initial period oftime, the amounts of the produced CH4 and CO2 increase sharply,and then reach an essentially steady state after about 300 days,which is the region of the methanogenic phase.

Fig. 7 shows the concentration profiles for the four micro-organism components that were included in the model, computedwith the first set of parameters (Tables 1 and 3). Theconcentrations of xc1 and xc2, the aerobic micro-organisms, con-tinuously decrease because the aerobic phase of biodegradation ofthe solid wastes occurs first in the landfill and over the first fewweeks. Depletion of oxygen inside the landfill causes the

80060000e (days)

he rate of production of CH4.

ARTICLE IN PRESS

0 200 400 600 8500

1000

2000

3000

4000

5000

6000

Time (days)

Org

anic

Car

bon

(mg/

L)

0 200 400 600 8500

1

2

3

4

Time (days)

Org

anic

Aci

d (m

g/L)

0 200 400 600 8500

0.2

0.4

0.6

Time (days)

CH

4 m

ole

fract

ion

0 200 400 600 8500

0.2

0.4

0.6

Time (days)C

O2

mol

e fra

ctio

n

Fig. 6. The concentration CC of the dissolved organic carbon, dissolved organic acid Cac , and the mole fractions of CH4 and CO2 at the center of the landfill.

0 200 400 600 8500

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (days)

Cc1

con

cent

ratio

n (m

g/L)

0 200 400 600 8500

0.5

1

1.5

Time (days)

Cc2

con

cent

ratio

n (m

g/L)

0 200 400 600 8500

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (days)

Cm

c co

ncen

tratio

n (m

g/L)

0 200 400 600 8500

200

400

600

800

1000

1200

1400

Time (days)

Cac

con

cent

ratio

n (m

g/L)

Fig. 7. The concentration profiles for the four micro-organisms species, computed using the first set of parameters (Tables 1 and 3) in the simulations.

R. Sanchez et al. / Chemical Engineering Science 65 (2010) 1212–1226 1219

continuous reduction of the concentration of the aerobic micro-organism. On the other hand, the concentration of thenonmethanogenic micro-organism xac increases first, sincebiodegradation of the dissolved organic carbon is continuouslygoing on by the nonmethanogenic bacteria in the moisture-richwastes. After the aerobic phase ends, the nonmethanogenic phasebegins, during which the concentrations of the nonmethanogenicmicro-organisms become dominant. As a result, the concentration

profiles of xmc and xac both reach a maximum after about 300days, which is also when the concentration of the dissolvedorganic acid (Fig. 6) attains its maximum value.

The concentration profiles of the dissolved organic carbon andthe dissolved organic acid, along with the mole fraction profiles ofCH4 and CO2, are shown in Fig. 8, all computed at the center of thelandfill using the second set of the initial conditions (Tables 2 and4). The maximum concentration of the dissolved organic carbon is

ARTICLE IN PRESS

0 200 400 600 8500

2000

4000

6000

8000

10000

12000

14000

Time (days)

Org

anic

Car

bon

(mg/

L)

0 200 400 600 8500

1000

2000

3000

4000

5000

Time (days)

Org

anic

Aci

d (m

g/L)

0 200 400 600 8500

0.01

0.02

0.03

0.04

0.05

0.06

Time (days)

CH

4 m

ole

fract

ion

0 200 400 600 8500

0.2

0.4

0.6

0.8

1

Time (days)C

O2

mol

e fra

ctio

n

Fig. 8. Same as in Fig. 6, but computed using the second set of the parameters (Tables 2 and 4) in the simulations.

0 200 400 600 8500

1

2

3

4

Time (days)

Cc1

con

cent

ratio

n (m

g/L)

0 200 400 600 8500

0.5

1

1.5

2

2.5

3

3.5

Time (days)

Cc2

con

cent

ratio

n (m

g/L)

0 200 400 600 8500

1000

2000

3000

4000

5000

Time (days)

Cm

c co

ncen

tratio

n (m

g/L)

0 200 400 600 8500

1000

2000

3000

4000

5000

Time (days)

Cac

con

cent

ratio

n (m

g/L)

Fig. 9. Same as in Fig. 7, but computed using the second set of the parameters (Tables 2 and 4) in the simulations.

R. Sanchez et al. / Chemical Engineering Science 65 (2010) 1212–12261220

reached after about 80 days, beyond which it rapidly decreasesover just a few days. It then reaches a quasi-steady-state. But,because the initial concentration of the dissolved organic acid isnot zero now, the maximum CO2 concentration occurs at anearlier time, since the nonmethanogenic anaerobic phase hasalready taken place. It reaches a steady state after about 100 days.A similar trend occurs for CH4. On the other hand, the organic acidis depleted continuously and rapidly. All the such phenomena areaided by the fact that the recirculating leachate stabilizes the

landfill, as a result of which the concentration profiles reachessentially steady states.

Fig. 9 presents biomass concentration profiles for the fourmicro-organism substrates, computed with the second set of theparameters. The results are for the landfill’s center. After attainingits maximum value very quickly, the concentration of thesubstrate xc1 continuously decreases, because the depletion ofoxygen takes place quickly, and the aerobic phase of thebiodegradation also occurs within only a few weeks. The same

ARTICLE IN PRESS

0 200 400 600 8500

1000

2000

3000

4000

5000

6000

Time (days)

Cc

conc

entra

tion

(mg/

L)

0 200 400 600 8500

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

Time (days)C

a co

ncen

tratio

n (m

g/L)

Fig. 10. The concentrations CC of the dissolved organic carbon, dissolved organic acid Cac , at the center of the landfill. The results are for an older landfill that contains only a

top cover with no side or bottom walls.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0Time (yrs)

Mol

e Fr

actio

n, C

H4

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0Time (yrs)

Mol

e Fr

actio

n, C

O2

200 400 600 800 200 400 600 800

Fig. 11. The time-dependence of the mole fractions of CO2 and CH4 that correspond to the concentration profiles of Fig. 10.

R. Sanchez et al. / Chemical Engineering Science 65 (2010) 1212–1226 1221

thing happens to the concentration of xc2. On the other hand, afterthe aerobic phase ends within the first few weeks, theconcentrations of the nonmethanogenic microorganism andmethanogenic organism increase, since the nonmethanogenicphase increases the production of the organic acid.

We also modeled an old landfill that has only the top layer, butno side or bottom liners (walls). As a result, the landfill actuallycommunicates with its surrounding soil. We assumed the soilbeing isotropic with a typical permeability, K ¼ 0:33 md, and,f=ts ¼ 0:05, where ts is the tortuosity factor of the soil. Shown inFig. 10 are the time-dependence of the concentrations of thedissolved organic carbon and the organic acid at the center ofthe landfill, computed using the first set of the parameters(Tables 1 and 3). The trends are qualitatively similar to what wediscussed above, namely, the maximum concentration ofthe dissolved organic carbon is reached after about 200 days.Then, the nonmethanogenic phase begins, which is the reason

why the maximum concentration of the dissolved organic acidsappears at a later time than that of the carbon, after about 250days.

The dependence of the CO2 and CH4 mole fractions on thetime, for the same ‘‘old’’ landfill and at its center, are shown inFig. 11. Initially, both mole fractions increase slowly. This isconsistent with the rates of production of both gases shown inFigs. 4 and 5. The two mole fraction then increase sharply, becauseas Figs. 4 and 5 indicate, there is a sharp maximum in their ratesof production. A steady state is then reached whereby the molefractions do not change any more. The patterns shown in Fig. 11are also consistent with the trends in the time-dependence ofthe concentrations of the dissolved organic carbon and theorganic acids.

Fig. 12 presents the time-dependence of the same quantities,but in the soil below the bottom of the landfill. The mostimportant conclusion drawn from the results shown in this figure

ARTICLE IN PRESS

0 200 400 600 8500

500

1000

1500

2000

2500

3000

Time (days)

Cc

conc

entra

tion

(mg/

L)

0 200 400 600 8500

1

2

3

4

5

6

7

8x 10−3

Time (days)

Cac

con

cent

ratio

n (m

g/L)

Fig. 12. Same as in Fig. 10, but computed at a location below the landfill in the soil.

R. Sanchez et al. / Chemical Engineering Science 65 (2010) 1212–12261222

is that, if groundwater flows under the landfill, it willbe contaminated by significant concentrations of the organicacid and the dissolved organic carbon. The correspondingconcentrations or mole fractions of CO2 and CH4 are flat andnearly zero and, hence, are not shown.

4.2. Heterogeneous landfills

An important factor that affects the behavior of a landfill, andin particular the flow and transport of CO2 and CH4, is itsheterogeneities. Due to the nature of the materials that are storedin a landfill, and the way they are stored, a landfill is, in fact, ahighly heterogeneous porous medium in which the local perme-abilities vary broadly. Therefore, in this section we study the effectof the spatial distributions of the local permeabilities Kx, Ky; andKz, and the porosity. The spatial distribution of the permeabilitiesof a real landfill has two important characteristics:

(1)

Due to compaction, the vertical permeability Kz is smallerthan the horizontal permeabilities, Kx and Ky.

(2)

Due to gradual filling and compacting of a landfill, thepermeabilities are depth-dependent, such that those sectorsthat are closer to the surface have higher permeabilities.

In the absence of any reliable data for the permeabilities for anactual landfill, we proceeded as follows. We generated what weconsider to be a realistic spatial distribution of the permeabilitiesby using the techniques that have been traditionally utilized inmodeling of oil reservoirs, which represent large-scale and veryheterogeneous porous media. To do so, one must have some data,even if limited, for the permeabilities. Therefore, we assumed thatthe horizontal permeabilities at five points of the landfill,assumed to be at the bottom near the five wells, are ‘‘known’’(clearly, the five points can be anywhere else in the landfill). The

idea is that in a typical landfill one may use the existing wells tomeasure the gases’ fluxes and pressures at the several points and,thus, obtain estimates of the local permeabilities there. Thehorizontal permeabilities were allowed to vary between 0.1 and100 md. The five ‘‘known’’ permeabilities were 0.5, 2.3, 4.5, 7.8,and 10.0 md. We then used the following procedure to generatethe spatial distribution of the permeabilities throughout themodel.

(i)

Using the five permeabilities at the bottom of the landfillmodel, we used the sequential Gaussian simulation (SGS)technique, which is used for generating the spatial distribu-tions of the permeability and porosity of highly heteroge-neous porous media (Sahimi, 1995; Deutsch and Journel,1998; Jensen et al., 2000), to develop the spatial distributionof the permeabilities Kx and Ky of the grid blocks thatrepresent the landfill’s bottom layer. As mentioned earlier, weassumed that Kx and Ky vary over three orders of magnitude,from 0.1 to 100 md. Given the range of the permeabilityvariations and their values at the five points as the input data,the SGS method generates the spatial distribution of thepermeabilities in the rest of the bottom layer (or even theentire grid, if need be), based on a combination of determi-nistic and stochastic techniques. It also honours the ‘‘mea-sured’’ data at the five points, i.e., it preserves their values.

(ii)

We assumed that the vertical permeability Kz of every gridblock is 1

3 of its horizontal permeabilities.

(iii) To construct the spatial distribution of the permeabilities in

the rest of the grid, we assumed that the permeabilities at thetop are 5 times larger than those at the bottom, and that theincrease in the permeabilities as one moves from the bottomto the top is linear. In this manner, we constructed the spatialdistribution of the permeabilities. Hereafter, we refer to themodel so generated as the heterogeneous landfill.

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R. Sanchez et al. / Chemical Engineering Science 65 (2010) 1212–1226 1223

Fig. 13 compares the pressure buildup at the center of thelandfill in the homogeneous and heterogeneous landfill,computed using the parameters listed in Tables 1 and 3. Theresults represent the average values for five independentrealizations of the system. Although the trends in both cases arequalitatively similar, the pressure in the heterogeneous landfill isalways larger than that in the homogeneous landfill. In particular,the maximum pressure in the heterogeneous landfill is about 50%larger than that in the homogeneous one. It would be most

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

0Time (days)

Gau

ge P

ress

ure

(kPa

)

HomogenousHetereogenous

100 200 300 400 500 600 700 800 900 1000

Fig. 13. Comparison of the pressures at the centers of a homogeneous and a

heterogeneous landfill in which the permeabilities have been distributed over

three orders of magnitude. The results were computed using the parameters listed

in Tables 1 and 3.

0123456789

10

0Time (days)

Gau

ge P

ress

ure

(kPa

) HetereogenousHomogeneous

0123456789

10

0Time (days)

Gau

ge P

ress

ure

(kPa

) HetereogenousHomogeneous

200 400 600 800

200 400 600 800

Fig. 14. Comparison of the pressures at four locations of a homogeneous and a heterogen

magnitude. The results were computed using the parameters listed in Tables 2 and 4. Th

(d) right top corner of landfill.

interesting to utilize a permeability distribution much broaderthan what we utilized in the simulations, which is what isexpected in practice, given the nature of the various wastes thatare stored in a typical large landfill.

To understand how the pressure varies at other locations in thelandfill, as well as the effect of the parameters that we haveutilized in the simulations (Tables 1–4), we present in Fig. 14 thepressure profiles at four points of the landfill (computationalgrid). The results were computed using the second sets of theparameters presented in Tables 2 and 4. The permeabilities weredistributed as in Fig. 13. The results shown are for a location at (a)the right bottom corner; (b) the center; (c) the left top corner, and(d) the right top corner of the landfill. Qualitatively, the results inFig. 14 are similar to those in Fig. 13, except that the pressuredecays more slowly after it reaches its maximum, which is due tothe different sets of the parameters used.

We also considered the case in which the porosity is spatiallydistributed throughout the landfill. The same procedure that wasused for generating the synthetic permeability distribution wasutilized, in order to generate the spatial distribution of theporosity. The permeabilities were held fixed (see above), in orderto distinguish their effect from that of a spatial distribution of theporosity. The porosity was varied between 0.01 and 0.7. The fiveassumed porosities that were used as the initial point in the SGSmethod were 0.15, 0.25, 0.40, 0.50 and 0.65. The results wereaveraged over five realizations.

Fig. 15 presents the results. The difference between the twopressures in the two model landfills is again large. But, whereasthe pressure is always larger than that in a homogeneous landfillwhen the permeabilities are distributed spatially, the opposite istrue when the porosity is distributed spatially. In particular,the maximum pressure in the homogeneous landfill is largerthan that in the heterogeneous one by a factor of slightly larger

0123456789

10

0Time (days)

Gau

ge P

ress

ure

(kPa

) HetereogenousHomogeneous

0123456789

10

0Time (days)

Gau

ge P

ress

ure

(kPa

) HetereogenousHomogeneous

200 400 600 800

200 400 600 800

eous landfill in which the permeabilities have been distributed over three orders of

e results are for (a) the right bottom corner; (b) the center; (c) left top corner, and

ARTICLE IN PRESS

R. Sanchez et al. / Chemical Engineering Science 65 (2010) 1212–12261224

than 2. Since one prime concern with the operation of a landfillis its pressure buildup, and the safety hazards that it posesto community surrounding the landfill, Figs. 13–15 demonstrateclearly the significance of including realistic spatial distribu-tions of the local permeabilities and porosities in models oflandfills.

The most realistic case is one in which both the porosity andpermeability are spatially distributed. Therefore, we generated amodel landfill in which the permeabilities were distributed overthree orders of magnitude, while as the porosity variations werethe same as above. The model was then utilized to compute thepressure at the center of the landfill for two cases. In one casethe leachate was ignored (as in Parts I–III), whereas the effect ofthe leachate was included in the second case (as in the casespresented above). The results were averaged over five realizations.

Fig. 16 presents and compares the results. Though the patternsof the time-dependence of the pressure for both cases are similar,the pressure in the two-phase case is always larger than the case

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0Time (days)

Gau

ge P

ress

ure

(kPa

)

HomogeneousHetereogeneous

100 200 300 400 500 600 700 800 900 1000

Fig. 15. Same as in Fig. 13, but in which the porosities have been distributed.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

0Time

Gau

ge P

ress

ure

(kPa

)

100 200 300 400 5

Fig. 16. Comparison of the pressures at the center of a heterogeneous land

in which the liquid phase is ignored. This is clearly due to thehigher rates of generation of the gases in the presence of the liquidphase. Most importantly, the peak pressure in the two-phasemodel is about 50% larger than that in the single-phase model,hence demonstrating the significance of two-phase modeling foraccurate forecasting, as well as for the safety of the areas thatsurround a landfill.

5. Summary

This paper presents a comprehensive model of two-phase flowin three-dimensional heterogeneous landfills. The biodegradationprocess was modeled by including both anaerobic and aerobicprocesses, from to the solid wastes to the dissolved organic acids.The results demonstrate, (i) the significance of including the liquidphase to any realistic modeling of biodegradation of solid wastesand generation of gases (and, in particular, CH4 and CO2), and (ii)the strong effect of the spatial distributions of the permeabilityand porosity throughout the landfill on the results, and inparticular on the gas pressure in the landfill.

The next step is to combine the present model with theoptimization technique developed in Part III, in order to develop arealistic model of a large landfill, based on actual data. Work inthis direction is in progress.

Notation

(days)00

fill in whi

bac

maximum organic acids utilization rate per unit mass of xac

bc1

maximum organic acids utilization rate per unit mass ofxc1

bc2

maximum organic acids utilization rate per unit mass ofxc2

bmc

maximum organic acids utilization rate per unit mass ofxc2

Cac

concentration of the dissolved organic acid

CðnmÞac

concentration of acid-producing nonmethanogenicmicro-organism xac

Cc1

total concentration of xc1

Two-PhaseSingle-Phase

600 700 800 900 1000

ch the permeabilities and porosities are distributed spatially.

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R. Sanchez et al. / Chemical Engineering Science 65 (2010) 1212–1226 1225

Cc2

total concentration of xc2

CC

concentration of dissolved organic carbon CCH4

methane concentration

Cmc

total concentration of xmc

CO

concentration of dissolved oxygen C�O equilibrium concentration of dissolved oxygen

CST

fluid phase saturation constant Di dispersion coefficient Fc1 ratio of oxygen to organic carbon consumed by xc1

Fc2

ratio of oxygen to organic acids consumed by xc2

gc1

ratio of CO2 produced to organic acids consumed by xc1

gc2

ratio of CO2 produced to organic acids consumed by xc2

gCH4

ratio of CH4 produced to organic acids consumed by xmc

gmc

ratio of CO2 produced to organic acids consumed by xmc

h

hydraulic head

k0

the dissolution rate from the solid to liquid (a rateconstant)

Kacac

organic acid half-saturation constant by xac (M L�3) Kacc1 organic acid half-saturation constant for xc1

Kacc2

organic acid half-saturation constant for xc2

Kacmc

organic acid half-saturation constant for xmc

Kdoac

oxygen half-saturation constant for xac (M L�3) Kdoac oxygen half-saturation constant for xac

Kdoc1

oxygen half-saturation constant for xc1

Kdoc2

oxygen half-saturation constant for xc2

Kdomc

oxygen half-saturation constant for xmc

Krw

relative permeability Kxx saturated hydraulic conductivity in the x direction

n

porous media parameter in the Van Genuchten equation Pac ratio of organic acids produced to organic carbon

consumed by xac

q

flow rate of oxygen injected or withdrawn per unitvolume

Q

infiltration rate of water R retardation factor S local mass per volume of the refuse available for transfer Sr irreducible water saturation Sw volumetric water content t time Vz Darcy velocity in the z direction

xac

nonmethanogenic micro-organism xc1 aerobic micro-organism xc2 aerobic micro-organism xmc methane-producing micro-organism Yac yield coefficient of xac

Yc1

yield coefficient (ratio of micro-organisms produced toorganic carbon biodegraded) of xc1

Yc2

yield coefficient (ratio of micro-organisms produced toorganic acid biodegraded) of xc2

Ymc

yield coefficient of xmc

Greek letters

a

porous media parameter in the Van Genuchten equation a1 production rate of CH4

a2

production of CO2

lac

endogenous decay rate of xac

lc1

endogenous decay rate of xc1

lc2

endogenous decay rate of xc2

lmc

endogenous decay rate of xmc

f

porosity

Acknowledgments

The authors are grateful to the California Energy Commissionand the Chevron-Texaco for partial support of this work. The

computations were carried out using the large computer clusterand parallel machine of the University of Southern California.

Appendix A

The dimensionless variables used in the simulator are definedas follows:

x¼ X=Lx; y¼ Y=Ly; z¼ Z=Lz; t¼ mrL2z

KrP0t; ðA:1Þ

dx ¼ Lz=Lx; dy ¼ Lz=Ly; ðA:2Þ

where Lx, Ly, and Lz are the linear dimensions of the computationalgrid. Moreover,

Al ¼Drmr

KrP0

el

tl; Ac ¼

Drmr

KrP0

ec

tc; ðA:3Þ

where Dr is a reference diffusivity which we take it to be thebinary diffusivity of CH4 in CO2, mr is a reference viscosity, takento be the viscosity of the air, P0 is the ambient pressure, e‘ and ec

are, respectively, the local landfill’s and the cover’s porosities, Kr isa reference permeability, and t‘ and tc are, respectively, the locallandfill’s and the cover’s tortuosity factors. In addition, p¼ P=P0,and

lk ¼MkTr

MrT; lm ¼

X4

k ¼ 1

lkyk ¼MmTr

MrT; ðA:4Þ

where Mk is the molecular weight of component k, and Mm is themolecular weight of the gas mixture. In addition, we define

wk ¼rkRTr

P0Mr;

X4

k ¼ 1

wk ¼X4

k ¼ 1

PkMk

RT

RTr

P0Mr

� �

¼X4

k ¼ 1

ykPMk

RT

RTr

P0Mr

� �¼X4

k ¼ 1

ðyklkpÞ ¼ lmp; ðA:5Þ

cxl¼

Kx

Kr

mr

mm

; cyl¼

Ky

Kr

mr

mm

; czl¼

Kz

Kr

mr

mm

; ðA:6Þ

cxc¼cyc

¼czc¼

Kc

Kr

mr

mm

: ðA:7Þ

Similar expressions can be written down for a correspondingquantity cs for the surrounding soil, if the landfill has permeablewalls:

Dk ¼Dekm

Dr; Dc ¼

Dec

Dr; B¼

RTrQ0mr

KrP20MrLz

; zkðzÞ ¼akðzÞL

3z

Q0; ðA:8Þ

where Dec is the effective gas diffusivity in the landfill’s cover, andQ0 is the total amount of the gases generated in the landfill duringthe time that it took to fill up the landfill, which is given by

Q0 ¼1

t1

X4

k ¼ 1

CTkVLF

X3

m ¼ 1

Amð1� e�lmt1 Þ: ðA:9Þ

We used, A1 ¼ 0:15, A2 ¼ 0:55, and A3 ¼ 0:3.

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