A coupled model for prediction of settlement and gas flow in MSW landfills

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech. 2010; 34:1169–1190Published online 26 October 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nag.856

A coupled model for prediction of settlementand gas flow in MSW landfills

L. Yu∗,†, F. Batlle and A. Lloret

Department of Geotechnical Engineering and Geosciences, Technical University of Catalonia, Barcelona, Spain

SUMMARY

Prediction of long-term settlement and control of gas pollution to the environment are two principleconcerns during the management of municipal solid waste (MSW) landfills. The behavior of settlementand gas flow in MSW landfills is complicated due to the combined effect of mechanical deformation ofthe solid skeleton and continuous biodegradation of the waste. A one-dimensional settlement and gas flowmodel is presented in this paper, which is capable of predicting time evolution of settlement as well astemporal and spatial distribution of gas pressure within multi-layered landfills under a variety of operatingscenarios. The analytical solution to the novel model is evaluated with numerical simulation and fieldmeasurements. The resulting efficiency and accuracy highlight the capability of the proposed model toreproduce the settlement behavior and gas flow in MSW landfills. The influences of operating conditionsand waste properties on settlement and gas pressure are examined for typical MSW landfills. Copyrightq 2009 John Wiley & Sons, Ltd.

Received 16 September 2008; Revised 3 September 2009; Accepted 7 September 2009

KEY WORDS: municipal solid waste landfill; settlement; gas flow; Laplace transform; analyticalformulation

1. INTRODUCTION

Sanitary landfills are recognized to be an economic method for the disposal of municipal solidwaste (MSW). Prediction of long-term settlement and control of gas pollution to the environmentare two principle concerns during the management of MSW landfills.

MSW landfills suffer large long-term settlement that is associated with volume reduction causedby biodegradation of organic solids, and also by creep of the MSW skeleton [1, 2]. The impact ofexcess settlement on liner system and final cover, landfill gas (LFG)/leachate extraction system aswell as post-closure development of MSW landfill is significant. Therefore, accurate estimation of

∗Correspondence to: L. Yu, Department of Geotechnical Engineering and Geosciences, Technical University ofCatalonia, Barcelona, Spain.

†E-mail: [email protected]

Copyright q 2009 John Wiley & Sons, Ltd.

1170 L. YU, F. BATLLE AND A. LLORET

settlement becomes critical to the successful operation and further maintenance of the MSW land-fills. Early models used to predict long-term settlement for MSW landfills are either adjusted fromsoil mechanics [1] or empirical functions based on best-fit approximation of field measurements[3–5]. These models are used extensively in the practice for their simplicity. Abundant experienceshave been accumulated in the determination of the model parameters.

For the prediction of gas flow within MSW landfills, many models have been proposed byassuming that waste is porous medium and gas velocity is governed by Darcy’s law. In additionto the models that simulate gas flux around vertical gas extraction well [6–8], some other modelswere developed to predict gas flux in horizontal gas extraction systems. Townsend et al. [9] gave ananalytical solution for steady-state one-dimensional gas flow to assist in the assessment and designof horizontal LFG collection systems. Findikakis and Leckie [10] presented a one-dimensionalmulti-component gas flow model to simulate the vertical distribution of gas pressure and gascomposition. Young [11] modeled advective gas flow in MSW sites when a series of horizontalextraction wells are present.

The settlement and gas flow models mentioned above focus only on one phase neglecting thefact that the MSW landfill is an interacting multi-phase medium (solid, gas and liquid) with eachphase exhibiting significant temporal and spatial variations [12]. Recently some work was reportedon landfill settlement incorporating the fluid flow in MSW landfills. Durmusoglu et al. [13] gave aone-dimensional multi-phase numerical model considering the interactions among solid, gas andliquid phases. In this model, Maxwell viscoelastic model, which describes an infinite compressionof solid skeleton with time, was used as mechanical constitutive law. The three-phase settlementmodel by Hettiarachchi et al. [14, 15] used the laboratory compression curve as the constitutivelaw for stress-induced settlement. The coupling effect between mechanical compression and fluidflow was built with the assumption that the void volume formed during biodegradation transformstotally into the landfill settlement. Liu et al. [16] proposed a solid–gas coupled model based onthe linear unsaturated consolidation theory. In the model, mechanical compression of the wastewas assumed to vary linearly with gas pressure. Yu et al. [17] developed a model to predict two-dimensional radial transient gas flow to a vertical gas extraction well in deformable MSW landfillsusing K-H rheological model. The analytical solutions given in the Laplace transform domainfacilitate the optimum design of LFG systems. McDougall [18] gave a conceptual frameworkfor coupled analysis in relation to hydraulic, biodegradation and mechanical processes. In theframework, an unsaturated flow model, a two-stage anaerobic digestion model and a mechanicalmodel consisting of load, creep and biodegradation-induced settlement are incorporated. Benteet al. [19] are carrying out a numerical simulation considering mechanical deformation and reactivetransport processes within MSW landfills.

A typical landfill is deposited in layers and usually takes up to decades to be filled. It is verycommon that at the same spot, the new cell is constructed many years after the previous cell wasfinished. During the time span before construction of the new layer, the old waste has alreadyundergone some degree of degradation. Permeability, waste density and mechanical properties ofthe old waste are much different from those in the newly constructed cell. Therefore, it is muchmore reasonable to assume that the properties of waste over the life of the landfill vary from layerto layer. Quite few existing models simulate the settlement and gas pressure in layer-depositedMSW landfill. After back analysis of parameters using Gibson & Lo model in four sites, Edil et al.[4] pointed that old refuse has a lower secondary compressibility compared with refuse recentlysurcharged. Bleiker et al. [20] developed a settlement model to predict the compression of eachrefuse layer in response to the varying weight of overlying refuse in deep landfills.

Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2010; 34:1169–1190DOI: 10.1002/nag

PREDICTION OF SETTLEMENT AND GAS FLOW IN MSW LANDFILLS 1171

Most of the existing multi-phase models rely on numerical method (FEM/FDM). This featuremakes the reproduction and application of these models inconvenient especially for engineerswithout much experience of numerical techniques. This paper presents an analytical solution basedon multi-phase model to predict both settlement and gas pressure for single-layered as well asmulti-layered MSW landfill under various operating scenarios. The analytical solution considerstime-dependent compression of solid skeleton, as well as generation and dissipation of LFG.Analytical method serves as a valuable tool because it offers quick insight into the physics of thecoupled system and can provide preliminary and scoping calculations. Moreover, the analyticalsolution given in this paper offers a helpful benchmark for validation of the solutions employingnumerical techniques.

2. MODEL DEVELOPMENT

Some basic assumptions are made to develop the governing equation:

(1) Solid particles are assumed incompressible due to its much higher stiffness than solidskeleton for MSW;

(2) LFG is normally considered to behave as an ideal gas under the pressures and temperaturescommonly encountered in MSW landfill, and Darcy’s law can be applied to the gas flowprocess [9–11, 15];

(3) According to the study of Young [11], the temperature within the MSW landfill tends to befairly uniform (usually around 35.40◦C). Therefore, the effects of temperature variation isneglected and isothermal condition is assumed;

(4) Gas diffusion is disregarded based on the study of Townsend et al. [9] that the maximumratio of diffusive flux to advective flux was 1.1×10−3 for LFG.

2.1. Mass balance equation

One-dimensional gas flow in a single-layered MSW landfill, as illustrated in Figure 1, is considered.The single-layered MSW landfill was assumed to be constructed instantaneously and at the sametime. Mass balance equation for the gas phase is given by

�(�a�a)

�t=−�(Ja)

�z+� (1)

where �a is volumetric gas content expressed as �a=�Sg , � is the porosity and Sgis degree of gassaturation defined as the volumetric fraction of voids occupied by gas phase; �a is the density ofgas (kg/m3); Ja is gas flux in z direction (kg/m2/s); and � is source term of gas production ratedue to waste biodegradation (kg/m3/s).

One-dimensional gas flux in z direction is written as

Ja=−kag

(�ua�z

−�ag

)(2)

where ka is gas conductivity in z direction (m/s) and is assumed to be constant along the depthof the waste column; ua is absolute gas pressure (kPa) and expressed as ua=ua+uatm, uatm isatmospheric pressure (101.3 kPa), ua is excess gas pressure (kPa); g is gravitational acceleration(9.807m/s2).



Figure 1. One-dimensional gas flow in a single-layered landfill.

Biodegradation of solid waste and gas generation in MSW landfills are governed by a seriesof chemical and biological reactions through which solid organic particles are solubilized andconverted to methane and carbon dioxide. The most widely used approach for quantitativelydescribing gas production rate in MSW landfills is based on the first-order kinetic equation([6, 10, 13–16], etc.)

�=GT �e−�t (3)

where GT is gas production potential per unit volume of waste (kg/m3); � is reaction-rate constantof the waste (s−1); t is time elapsed since waste deposition (s).

LFG is considered to behave as an ideal gas in the environment of MSW landfills. Therefore

�a=uaM

RT(4)

where R is universal gas constant equal to 0.008314 kJ/mol/K; T is absolute temperature of gasand assumed to be 310K; M is average molecular weight of gas and taken to be 0.03 kg/mol.

Considering that porosity may change because of both compression of the solid matrix andenlargement of pore volume due to solid degradation, while using porosity for �a, variations inmass storage for a dry MSW landfill (Sg =1) are given by

�(�a�a)

�t=�a

��

�t+�

��a�t

=�a

(�εv

�t+ Y�

�s+ �

�a

��a�t

)(5)

where εv is volumetric strain of solid matrix; �s is the density of solid phase (kg/m3); Y is gasyield coefficient of refuse (mass of solid phase degraded/mass of gas phase generated and it isassumed to be 1). The term, Y�/�s , represents the rate of volume enlargement due to degradationof the organic matter.

By using Equations (2)–(5), and neglecting gas flux due to gravitational gradient, Equation (1)can be written as

�a�εv

�t+ ��a

ua

�ua�t

= kag

�2ua�z2

+GT �e−�t[1− Y�a

�s

](6)

Excess gas pressure in MSW landfill normally is a few kPa. Its variation can be neglectedcompared with uatm (101.3 kPa). Therefore there exist ua≈ ua0 and �a≈�a0, where



E

E0

'

Figure 2. Schematic representation of K-H viscoelastic model.

ua0=uatm+ua0, ua0 is initial excess gas pressure (kPa), �a0 is initial gas density (kg/m3).Considering that porosity of waste does not vary much under the combined effect of compressionand mass loss, we assume a first-order approximation of porosity, that is �≈�0, where �0 is theinitial porosity. Rearranging Equation (6) gives

�a0�εv

�t+ �0�a0

ua0

�ua�t

= kag

�2ua�z2

+GT �e−�t[1− Y�a0

�s

](7)

2.2. Mechanical deformation

Continuous degradation of solid matter weakens the micro-structure of waste through conversionof the solid mass into biogas. The durable settlement that happened in MSW landfills is the resultof collapse of the weakened micro-structure waste skeleton. There is no rigorous model describingthe relationship between mass loss and the corresponding settlement of the landfill. The response ofsettlement due to collapse of the weakened micro-structure waste skeleton may be locally erratic,however, a smoothed time-strain curve is capable of describing the overall behavior of settlementat macroscopic level. In this paper, the settlement process in MSW landfills is represented by aviscoelastic K-H rheological model (VOIGT form of the standard linear solid model as illustratedin Figure 2), which reflects both the stress-dependent and the time-dependent deformations, andrequires limited parameters easy to be determined. In K-H rheological model, the Hookean springis used to simulate the primary settlement and the Kelvin element is used to simulate the secondarysettlement.

According to unsaturated soil mechanics of Fredlund and Rahardjo [21], the compressibility ofsolid is controlled by two stress invariants: net normal stress, �−ua and matric suction, �, where� is the total stress, ua is excess gas pressure. When there is no external load applied on the landfill,the total stress within the refuse is gravitational stress caused by self-weight. For the typical rangeof moisture content in MSW landfills, matric suction is negligible compared with net normal stressbased on the retention curve measured by Kazimoglu et al. [22]. Thus, the volumetric strain thathappened in the waste at depth of z can be expressed as

εv =mak (�0gz−ua) (8)

where mak is compression coefficient of solid skeleton (kPa−1); �0 is bulk density of refuse

(kg/m3).



The expression for volumetric strain can be split between that ε1, corresponding to the Hookeanspring of stiffness E0, and ε2, corresponding to the Kelvin element. There satisfy

ε1 = �′/E0 (9a)

�′ = Eε2+ε2 (9b)

where �′ is effective stress equal to �0gz−ua (kPa); E0 is elastic modulus of waste for primarysettlement (kPa); E is residual elastic modulus of the solid matrix for secondary settlement (kPa); is viscosity of solid skeleton (kPas). The observed response of settlement in MSW landfilldisplays a very close relationship with the progress of biodegradation. Settlement is controlled bythe interaction of soil skeleton strength and stress state [23]. Here it is assumed that the stiffness ofthe solid matrix decreases at the same rate with the degradation of solid mass, there is E/(�)=1.Therefore, the parameter can be determined by the residual modulus and the degradation rateconstant of waste.

The explicit expression of mak can be expressed by employing the Laplace transform. Trans-

forming equations (9a) and (9b), and adding the corresponding transformed strain, lead to

εv = mak �

′ =−(

1

E0+ 1

E+s

)(�0gz

s− ua

)(10)

where εv =∫∞0 εv(t)e−st dt is the Laplace transform of the volumetric strain at depth z, s is the

Laplace operator; mak , �′ and ua are the Laplace transforms of corresponding variables as defined

above. Here the increase of pore volume is assumed to be positive; thus, the coefficient mak is

negative.Incorporating K-H rheological constitutive law of solid matrix (Equation (10)) into the gas mass

balance equation (Equation (7)) in the Laplace transform domain gives

�2ua�z2

− s

Cav

ua+ Czv

Cav

z+ Cd

Cav

1

s+�+Cu0

Cav

ua0=0 (11)

whereCa

v = −RTka/((mak ua0−�0)Mg);Cz

v =�0gmak ua0/(m

ak ua0−�0)

Cd = −(RT/M−Y ua0/�s)GT �/(mak ua0−�0);Cu0=−�0/(m

ak ua0−�0)

Equation (11) will be used to find the excess gas pressure within the MSW landfill. The term(Cz

v/Cav )z in Equation (11) is the coupling term caused by the compression of solid matrix under

gravitational stress. The term (Cd/Cav )/(s+�) describes the effect of biodegradation of solid mass

and the term Cu0ua0/Cav is generated due to the initial excess gas pressure. Equation (11) is able to

demonstrate intrinsic relationships among a great number of parameters involved in the problem,and makes the prediction of effects due to variation of some key parameters straightforwardly.

The total compression of landfill in the Laplace transform domain for one-dimensional case canbe expressed as

S=∫ H

0εv(z)dz (12)

where S is the total compression of a waste column with height of H ; εv is Laplace transform ofvolumetric strain at depth z and is determined by Equation (10).



3. SOLUTION FORMULATION

The proposed model developed above is firstly applied into a single-layered MSW landfill. Analy-tical solutions are obtained for three different operating scenarios. Furthermore, the analyticalformulation will be extended to a multi-layered landfill system.

3.1. Solutions for single-layered landfill

Solution for excess gas pressure within the MSW landfill requires specifying boundary conditions.The following three operating scenarios are commonly encountered in MSW landfills.

Case 1: Free-venting at top surface, and fixed gas flux is specified at the bottom. This is tosimulate an influx of gas from a possible saturated layer at the base [11]. This solution is usefulfor the dry part above leachate table in the landfill.

ua = 0 at z=0 (13a)

−kag

�ua�z

= Jb at z=H (13b)

where Jb is the Laplace transform of the gas influx rate, Jb (kg/m2/s), at the base.Case 2: Fixed gas pressure is specified at the top surface to simulate vacuum pressure applied

beneath the impermeable geomembrane cap, and fixed gas pressure at the bottom base to simulategas extraction through the leachate collection system (LCS) by applying the vacuum pressure tothe LCS [9].

ua = ut at z=0 (14a)

ua = ub at z=H (14b)

where ut and ub are the Laplace transforms of the vacuum pressure applied at the top and bottom,respectively.

Case 3: A thin layer with thickness dl (m) and gas permeability kl (m/s) is added at the topsurface to simulate the low-permeable cover. Gas flow is predominantly vertical [6, 11] when theheight of landfill H �dl and thus the boundary condition at landfill cover is

�ua�z

= kldlka

ua at z=0 (15a)

At the bottom of the landfill, it is assumed to be impermeable

−kag

�ua�z

=0 at z=H (15b)

The initial condition for excess gas pressure throughout the landfill for all three cases isdefined as

ua=ua0 at t=0, 0�z�H (16)

At the time of closure, the initial excess gas pressure, ua0, is assumed to be zero in the followinganalysis. The solutions of excessive gas pressure, total compression of the waste column and gas



Figure 3. Multi-layered landfill.

flux at the top for one-layered landfill are obtained in the Laplace transform domain and given inAppendix A. Crump’s method [24] is used in this paper to perform the numerical inversion of theLaplace transform in order to get the solutions in the real-time domain.

3.2. Solutions for multi-layered landfill

The multi-layered MSW landfill considered in this section is illustrated in Figure 3. It consists ofn homogeneous and perfectly bonded layers. Any layer j ( j=1,2, . . . ,n), occupying the regionZ j−1�z�Z j with finite thickness Hj = Z j −Z j−1, has its respective gas conductivity kaj, reaction-rate constant � j , bulk density �0 j , gas production potential GTj and mechanical parameters E0 j ,E j , j , etc.Equations (2) and (11) can be rewritten as a matrix ordinary differential equation

dX

�z= A ·X+B (17)

where

X={ua

Ja

}, A=

⎛⎜⎝

0 −g/ka

− s

Cav

kag

0

⎞⎟⎠ and B=

⎧⎪⎨⎪⎩

0

Czv

Cav

kagz+ Cd

Cav

kag

1

s+�+ ka

g

Cu0

Cav

ua0

⎫⎪⎬⎪⎭

The solutions of the matrix ordinary differential equation (17) in the Laplace transform domainat depth z within the j th layer (Z j−1<z<Z j ) can be expressed as [25, 26]

X(z)=T j (z−Z j−1) ·X(Z j−1)+S j (z−Z j−1) (18)



where X(Z j−1) is the vector of excess gas pressure and gas flux at the top surface of the j th layerin the Laplace transform domain; X(z) is the vector of excess gas pressure and gas flux at depthof z in the Laplace transform domain; T j and S j are transferring matrices of the j th layer (seeAppendix B).

The continuity conditions for any interface between the j th layer and ( j+1)th layer isexpressed as

X(Z−j )= X(Z+

j ) (19)

By applying the forward transfer matrix method to the sequence of finite layers from the topsurface of the multi-layered landfill (z=0) downwards and by using the solution of the vector X(0)given in Appendix B, the vector solution, X(z) at any depth z within the j th layer (Z j−1<z<Z j−1)

can be expressed as

X(z)=TT(z) ·X(0)+SS(z) (20)

where X(0) is the vector of excess gas pressure and gas flux at the top surface of the multi-layeredlandfill in the Laplace transform domain. TT and SS take the following forms:

TT(z) =Tj(z−Z j−1)Tj−1(Hj−1) . . .T2(H2)T1(H1) for (Z j−1<z<Z j−1) (21a)

SS(z) =j∑

i=1Tj(z−Z j−1)Tj−1(Hj−1) . . .Ti+1(Hi+1)Si (Hi ) for (Z j−1<z<Z j−1) (21b)

The expressions of transferring matrices Tj and Sj, and the solution of the vector X(0) underthree operating scenarios (Equations (13)–(16)) are presented in Appendix B.

4. MODEL TESTING

In this section, the correctness of the analytical solution is verified by comparison with resultsfrom the numerical method. The applicability of the solution for predicting long-term settlementand gas flux is verified through comparison with field measurements at two MSW landfills.

Methods used to determine material parameters that are needed in the proposed model can befound in the previous publications. Initial waste porosity �0 has its typical value varying within0.4–0.5 suggested by Oweis et al. [27], and a default value of 0.5 is assumed in the followinganalysis. Bulk density of waste �0 can be determined by large-scale borehole test or test pit. Gasproduction potential GT and reaction-rate constant � are evaluated by fitting the exponential modelto the gas recovery data at the MSW landfill [7]. Gas conductivity ka is estimated from pumpingtest or air-injection test. By curve-fitting the actual strain versus time in log scale, primary andresidual modulus of solid skeleton, E0 and E , are estimated [4].

4.1. Verification of the analytical solution

Solutions of settlement and excess gas pressure are compared with numerical results obtainedwith CODE BRIGHT. CODE BRIGHT is a well-established finite element code developed forcalculating displacements, liquid pressure, gas pressure, temperature and salt content for boundaryvalue problems in saturated or unsaturated soil [28]. Modifications have been implemented into the



Lower layer

H1 =20m

H2 =10m

Upper layer

Free venting surface ua=0.0

=8×10-10s-1 , E0=5000kPa, E=200kPa

0=760kg/m3, ka=1.2×10-6 m/s , GT=220kg/m3

=8×10-8s-1 ,E0 =1×1020kPa,E =1000kPa

0 =1000kg/m3 , ka=5×10-7m/s, GT =100kg/m3

Impermeable bottom

0.01

Time (days)

-50

-40

-30

-20

-10

0

Tota

l ave

rage

str

ain

(%)

0

5

10

15

20

25

30

Exc

ess

gas

pres

sure

at t

he b

otto

m (

kPa)

analytical results

numerical simulation

Total average strain

Excess gas pressure

0

Excess gas pressure (kPa)

30

20

10

0

Dep

th (

m)

1 day

100 days

250 days

1000 days

0.1 1 10 100 1000 10000 100000

5 10 15 20 25

(a)

(b)

(c)

Figure 4. (a) Illustration of a two-layered landfill; (b) comparison between numerical simulation andanalytical results for a two-layered landfill; and (c) temporal and spatial distributions of excess gas pressure

within a two-layered landfill.



Table I. Parameter values applied in the model.

Name of parameter Notation Value Unit

Height of landfill H 20 mInitial porosity of waste∗ �0 0.5Bulk density of waste† �0 760 kg/m3

Gas generation potential† GT 157 kg/m3

Gas conductivity‡ ka 8.6×10−7 m/sReaction-rate constant of the waste§ � 1.62×10−9 s−1

Density of solid phase �s 1500 kg/m3

Thickness of the cover¶ dl 1.0 mGas conductivity of the cover¶ kl 8.6×10−8 m/sResidual modulus of solid skeleton‖

E 350 kPaPrimary modulus of solid skeleton‖

E0 10 000 kPa

∗Value is within the range mentioned by Oweis et al. [27].†Value is suggested by Nastev et al. [7] at the CESM landfill.‡Based on the typical value of intrinsic permeability and gas viscosity mentioned by Townsendet al. [9].§ In the absence of site-specific data, the default value is suggested by Oweis [29] which isequivalent to half life t1/2=− ln(0.5)/�=13.6 years.¶The cover properties are based on the values used in Arigala et al. [6].‖Values are within the range summarized by Edil et al. [4].

code to incorporate gas generation and porosity change due to degradation of solids. By ‘turningoff’ three sub-modulus related to liquid pressure, temperature and salt concentration, two massbalance equations of solid phase and gas phase are solved.

The numerical simulation was conducted for a one-dimensional two-layered landfill with imper-meable bottom and free-venting surface (Figure 4(a)). The boundary and initial conditions arelisted in Equations (13a), (13b) and (16) with Jb=0. The lower layer contains a type of readilydegradable waste. Compared with the waste from the lower part, the waste in the upper layer isfresh and looser with relatively higher gas conductivity. The parameters for two layers are shownin Figure 4(a). In the remainder of the paper, unless noted specifically, the other default inputparameters are the same as those in Table I, which are within the typical ranges suggested fromthe previous publications.

Comparisons between analytical solution given in this paper and numerical simulation byCODE BRIGHT is shown in Figure 4(b) for both total average strain and excess gas pressure builtup at the bottom. Total average strain mentioned here refers to the ratio of the total compression ofthe landfill over its initial height. It is a dimensionless indicator of the settlement happened at thelandfill. As shown in Figure 4(b), the results given by two methods are very close to each other,which proves the correctness of the given solution.

Time evolution of total average strain (Figure 4(b)) shows clearly two different stages. Most ofthe settlement that happened in the first stage (t<2000 days) comes from the secondary compressionhappened in the lower layer, and settlement in the later stage (t>2000 days) is mainly due to thecompression of the fresh waste from the upper layer. The evolution of total average strain indicatesthat the proposed model is capable of considering different waste properties for each layer in amulti-layered landfill.



A-2B-2

C-2E-1

MSW

25002000

400

440

480

520

560

Ele

vatio

n (m

)

S

Original valley

X coordination (m)

Section S-S

S

10

Time (days)

-30

-25

-20

-15

-10

-5

0

Tota

l ave

rage

str

ain

(%)

Time (days)

-30

-25

-20

-15

-10

-5

0

Tota

l ave

rage

str

ain

(%)

Field measurements

Model simulation

Field measurements

Model simulation

B-2A-2

100 1000 10000 100000 10 100 1000 10000 100000

Time (days)

-30

-25

-20

-15

-10

-5

0

Tota

l ave

rage

str

ain

(%)

Time (days)

-30

-25

-20

-15

-10

-5

0

Tota

l ave

rage

str

ain

(%)

Field measurements

Model simulation

Field measurements

Model simulation

C-2 E-1

10 100 1000 10000 100000 10 100 1000 10000 100000

(a)

(b)

S

Leachate treatment reservoir

Settlement control points

Figure 5. (a) Plan view of Coll Cardus landfill and (b) comparison between field measurements and modelsimulation for Coll Cardus landfill.



Table II. Parameter values for Coll Cardus landfill.

Control points Layer no. Constructed time (year) Layer height (m) E0 (kPa) E (kPa)

Group 1 A-2 1 1997 22.1 10 000 550B-2 1 1997 18.6 3000 750

Group 2 C-2 1 2001 8.43 10 000 3002 1992 32.7 1×1020 700

E-1 1 2002 9.22 10 000 3502 1998 12.6 1×1020 10003 1992 32.2 1×1020 1000

The gradual dissipation of excess gas pressure within the landfill is shown in Figure 4(c). Gasflows upwards into the atmosphere and the maximum gas pressure built up at the bottom of thelandfill. The abrupt transition in the gas pressure curve at the intersection of the two layers iscaused by the distinct values of gas conductivity for two layers.

4.2. Comparison with field measurements

4.2.1. Coll Cardus landfill. Coll Cardus sanitary landfill is an active municipal and industrial solidwaste disposal center in the region of Catalonia, located in a natural valley in east-northern Spain.The landfill began in mid of 80s and the average depth of waste reaches about 60m (Figure 5(a)).Two groups of total four settlement control points are selected for the purpose of analysis. Group 1includes A-2 and B-2, which are situated on the front slope and both are single-layered. Group 2includes C-2 and E-1, which are located in the middle of the landfill and both have more thanone layer of refuse. The detailed construction information at each point is listed in Table II. Thelandfill was treated as impermeable at the base and free-venting at the top surface. Because therewere no field records for gas flux or gas pressure, default value of 1.62×10−9 s−1 (0.05year−1)

for reaction-rate constant suggested by Oweis et al. [27] was adopted in the analysis.Model simulations were performed using the analytical solutions given in the Appendix with

the parameters listed in Tables I and II. Mechanical properties were adjusted in a reasonable rangeto best fit the field records. The simulated results in Figure 5(b) show a good agreement with fieldmeasurements at four settlement control points. This proves the ability of the proposed model inpredicting settlement for both single-layered and multi-layered MSW landfills.

4.2.2. Mountain View Controlled landfill. Large-scale test was conducted under different opera-tional management practices at the Mountain View Controlled landfill in north-east of California,U.S.A. by El-Fadel et al. [30, 31]. The test included six cells and each cell was 30m in length,30m in width and 15m in depth. All the six cells were filled with municipal solid refuse, whichwas placed in the cell at the same time, and each cell was lined with a 1.5m thick compacted clayliner at the base as well as at the side wall. The bulk density recorded in the test is 704kg/m3.The gas mixture flowing out of each cell was collected and measured at the top and cell settlementwas monitored at the cell surface over a period of 4 years. Among all the six cells, cell F wasarranged to be the control cell. The averaged gas flux measured at the top surface of cell F isabout 1.23×10−5 kg/m2/s (Figure 6(a)) and the maximum vertical strain at the end of 1576 daysmeasured at cell F reaches 13% (Figure 6(b)).



10

Time (days)

-50

-40

-30

-20

-10

0

Tota

l ave

rage

str

ain

(%)

0

Time (days)

0

0.5

1

1.5

2

2.5

Field measurements

Model simulation

Averaged gas fluxmeasured at the top

Model simulation

400 800 1200 1600 2000 100 1000 10000 100000

(a) (b)

Gas

flux

at t

he to

p su

rfac

e (

10×

-5kg

/m2 /

s)

Figure 6. Comparison between field measurements and model simulation at Mountain View Controlledlandfill: (a) gas flux at top surface and (b) total average strain.

Table III. Parameter values for Mountain View Controlled landfill.

Parameter Value Unit

H 15 m�0 704 kgm−3

� 3×10−9 s−1

GT 330 kgm−3

E 150 kPaE0 20 000 kPa

Simulation of both settlement and gas flow rate was carried out for cell F using the proposedmodel. The cell was treated as impermeable at the base and free-venting at the top surface.Parameter values are listed in Table III with other default values in Table I. Reaction-rate constantand mechanical properties were adjusted within the reasonable range to best fit test measurements.

Simulated gas flux at the top surface decreases with time due to first-order kinetic gas generationlaw. The averaged value during the first 2000 days agrees with the field records (Figure 6(a)).Simulated total average strain fits well with field measurements as shown in Figure 6(b). The resultsof the comparison prove that the proposed model is capable of predicting both the settlement andgas production for MSW landfills.

5. EVALUATION OF PARAMETERS

Based on the solutions given in Appendix A, influences of various operating scenarios and materialproperties on landfill behavior are investigated in this section for a single-layered landfill to



Excess gas pressure (kPa)

-220

15

10

5

0

Dep

th (

m)

base case: free-venting top surface,impermeable bottomcase 1: free-venting top surface,J b=-1.32 10-5 kg/m2/s

case 2: Ut=Ub =-2kPa,gas is collected from top and bottom

case 3: with cover, impermeable bottom

-1 0 1 2 3 4

Figure 7. Distribution of excess gas pressure in a single-layered landfill under differentscenarios, 5 years after closure.

gain insight into the characteristics of settlement evolution and gas pressure distribution. Defaultparameter values are listed in Table I unless noted specifically.

5.1. Effects of different operating scenarios

Distribution of excess gas pressure under different operating scenarios is presented in Figure 7.Base case represents a landfill with free-venting top surface and impermeable bottom. For case 1,the specified flux rate at the bottom, Jb, is taken to be −1.32×10−5 kg/m2/s (negative meansinflux), which is equivalent to the gas emission rate from a landfill with height of 30m (based onthe recommended gas generation rate of 4.4×10−7 kg/m3/s mentioned by Townsend et al. [9]).In case 2, gas collection is implemented at both top surface and bottom with vacuum pressure of−2kPa. Case 3 is used for illustrating the effect of a low-permeable top cover on the distributionof the excess gas pressure.

Results show that in the based case, gas flows upward into the atmosphere and the highest gaspressure about 0.5 kPa was built up at the bottom. Much higher gas pressure built up in case 1indicates that gas pressure at the leachate level in a high landfill increases dramatically if there isno other gas extraction system available. While in case 2, the application of vacuum of −2kPa atboth ends results in negative gas pressure all over the landfill. For a covered landfill as in case 3,excess gas pressure builds up beneath the cover. The comparison between the base case and case3 shows that the cover retards the gas flux at the top surface; thus, the gas pressure within thelandfill is higher than the case without cover.

5.2. Evolution of bulk density

The temporal and spatial variation of bulk density in a single-layered, covered landfill (case 3) isexamined in this section. The initial bulk density and porosity are supposed to be homogeneous



10

Time (days)

600

700

800

900

1000

1100

1200

Bul

k de

nsity

(kg

/m3 )

At depth of 10m

at depth of 5m

At depth of 20m

100 1000 10000 100000

Figure 8. Evolution of bulk density with time at different depths of the landfill.

within the landfill. The vertical stress is equal to the gravitational stress for the closed-landfill.Time evolution of the bulk density at different depths in the landfill is shown in Figure 8. Densityexhibits different evolution trends for waste from different depths. The bulk density for the wastein the lower part increases continuously with time because the mechanical compression is largerthan void enlargement due to mass degradation. The decreased bulk density with time in the upperpart indicates that the newly formed void volume is greater than the void decreasing due to skeletoncompression owing to the less stress level in the upper part of the landfill.

5.3. Effect of gas conductivity

The influence of gas conductivity on the settlement and gas pressure is examined for case 3. Typicalvalues of gas conductivity for the waste are within the range from 1.16×10−7 to 1.74×10−6m/sbased on the report of Lang and Tchobanoglous [32]. Result in Figure 9 clearly shows that theexcess gas pressure within the landfill decreases with increasing value of ka. Settlement curvesdemonstrate that within the typical range of ka, the influence of gas conductivity on settlementbehavior can be ignored because excess gas pressure built up within the landfill is only a few kPa.

5.4. Effect of cover properties

The solution for case 3 (Appendix A) indicates that the effect of cover properties is determined byan integrated parameter, Lc, defined as kl/(dlka). Lower value of Lc can be obtained by increasingthe thickness of the cover or decreasing the gas permeability of the cover material. The dependenceof total average strain and excess gas pressure on Lc (case 3) is shown in Figure 10 at the time of5 years after closure. Variable values of Lc are obtained through setting kl/ka=0.1 and varyingdl with all the other input parameters remaining the same as those in Table I.

Smaller value of Lc (less permeable of the cover) causes larger gas pressure due to moretardiness for the gas flowing outwards through the top cove. Settlement rate of the landfill isdelayed correspondingly due to the correspondingly smaller effective stress. When Lc>0.1, whichis equivalent to a 1-meter-thick cover with gas conductivity of one order less than gas conductivityof the waste, the top cover will no longer affect the behavior of a 20-meter-high landfill after5 years of closure.



0

2

4

6

8

10

Exc

ess

gas

pres

sure

(kP

a)

-25

-20

-15

-10

-5

0

Tota

l ave

rage

str

ain

(%)

10

Time (days)

ka1

ka2

ka1=1.16 10-7m/s

ka2=1.74 10-6m/s

100 1000 10000 100000

×

×

Figure 9. The effect of gas conductivity, ka, on the time evolution of total average strain and excess gaspressure at the bottom of the landfill.

0.01

Cover property parameter Lc (m-1)

-6

-5

-4

-3

Tota

l ave

rage

str

ain

(%)

0

2

4

6

8

10

Exc

ess

gas

pres

sure

(kP

a)

Total average strain

Excess gas pressure

0.1 1

Figure 10. Impact of cover parameter, Lc, on the total average strain and excess gas pressure at the bottomof the landfill, 5 years after closure.

6. CONCLUSIONS

Analytical solutions to predict one-dimensional settlement and gas flow for both single-layeredand multi-layered MSW landfills have been presented. Mechanical compression of the solidskeleton is coupled with gas pressure using K-H viscoelastic model. The solutions of settlementsas well as excess gas pressure within the landfill have been obtained analytically in the Laplacetransform domain. The solutions have been verified by comparisons with numerical simulation.



Comparisons with field records at two landfill sites prove that the proposed model can wellreproduce the time evolution of settlement and predict gas flux in the horizontal LFG collectionsystems.

Analytical solution always serves as a useful and convenient tool for parametric studies. Theeffects of gas conductivity, ka, and cover property, Lc, on the rate of settlement and gas pressureare examined. The results show that the rate of settlement is not so much influenced by ka and Lc,although excess gas pressure is sensitive to these parameters. In the typical MSW landfill, excessgas pressure is of several orders less than gravitational stress and the coupling effect between gaspressure and mechanical compression is too little to be noticeable. The coupling effect may becomeapparent under the cases of low gas conductivity, deep landfills, bioreactor landfills or inefficientgas extraction system. Under these circumstances, the increased gas pressure may apparently delaythe settlement of the landfill.

The model presented in this paper focuses only on solid phase and gas phase. It is applicablefor closed-landfills or waste part above the leachate level. This model will be extended to includeliquid phase and consider the interaction among solid, gas and liquid phases in the future.

APPENDIX A

The solutions of excess gas pressure within the landfill at depth of z, ua(z), total compressionof the waste column, S, and gas flux at the top, Jt, of one-layered landfill for three differentboundary scenarios listed in Equations (13)–(16) are given in the Laplace transform domain asfollows:

1. The solutions for case 1 are

ua(z) =(

Cd

s(s+�)+Cu0ua0

s

)[1− exp(−(2H−z))+exp(−z)

1+exp(−2H)

]

−(Cz

v

s+ Jb

s

g

ka

)exp(−(H−z))−exp(−(z+H))

1+exp(−2H)+Cz

v

sz

S = mak

s

{�0gH

2

2+ua0H−

(Cd

s+�+Cu0ua0

)[H+ exp(−2H)−1

(1+exp(−2H))

]

+(Cz

v + gJbka

)[(1−exp(−H))2

2(1+exp(−2H))

]−Cz

v

H2

2

}

Jt = −kag

{(Cd

s+�+Cu0ua0

)

s

[1−exp(−2H)

1+exp(−2H)

]

−1

s

(Cz

v + gJbka

)[2exp(−H)

1+exp(−2H)

]+Cz

v

s

}

where =√−s/Cav .




ua(z) =(

Cd

s(s+�)+Cu0ua0

s

)

×[exp(−(H−z))−exp(−(2H−z))+exp(−z)−exp(−(H+z))

exp(−2H)−1+1

]

+ubs

exp(−(H+z))−exp(−(H−z))

exp(−2H)−1+ ut

s

exp(−(2H−z))−exp(−z)

exp(−2H)−1

+Czv

sH

[exp(−(H−z))−exp(−(H+z))

exp(−2H)−1+ z

H

]

S = mak

s

{�0gH

2

2+ua0H−

(Cd

s+�+Cu0ua0

)H−Cz

v

H2

2

−[2

(Cd

s+�+Cu0ua0

)−ub−ut+Cz

vH

](1−exp(−H))2

(exp(−2H)−1)

}

Jt = −kag

⎡⎢⎢⎢⎣

s

(− Cd

s+�−Cu0ua0

)(1−exp(−H))2

exp(−2H)−1− 2ub

s

exp(−H)

exp(−2H)−1

+uts

exp(−2H)+1

exp(−2H)−1+Cz

v

sH

(2exp(−H)

exp(−2H)−1+ 1

H

)⎤⎥⎥⎥⎦


ua(z) =(

Cd

s(s+�)+ Cu0ua0

s

)[1− Lc exp(−(2H−z))+Lc exp(−z)

(Lc−)exp(−2H)+Lc+

]

+Czv

s

⎡⎢⎢⎣exp(−(2H−z))− Lc+

exp(−(H−z))+exp(−z)+ Lc−

exp(−(z+H))

(Lc−)exp(−2H)+Lc++z

⎤⎥⎥⎦

S = mak

s

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

�0gH2

2+ua0H−

[H+ Lc exp(−2H)−Lc


](Cd

(s+�)+ Cu0ua0

)

+Czv

2

[−Lc(1−exp(−H))2

(Lc−)exp(−2H)+Lc++ H2

2

]⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

Jt = − kag

{

s

(Cd

s+�+Cu0ua0

)[ −Lc exp(−2H)+Lc


]

+Czv

s

[Lc(1−exp(−H))2


]}

where Lc=kl/dlka.



APPENDIX B

The transfer matrices at depth z (Z j−1<z<Z j−1) within j th layer are expressed as

T j (�)=

⎛⎜⎜⎜⎝

ch j� − g

kaj

sh j�

j

− s

V ajv

g

kaj

sh j�

jch j�

⎞⎟⎟⎟⎠ , S j (�)=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

(Cdj

Cav j

1−ch j�

s+� j+Cz

v j

Cav j

(�− sh j�

j

))1

2j(Cdj

Cav j

1

s+� j

sh j�

j−Cz

v j

Cav j

1−ch j�

2j

)kajg

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

where j =√

−s/Cav j and � is the depth from the top surface and equal to z−Z j−1.

The solutions for the vector of excess gas pressure and gas flux at the top surface of the landfillin the Laplace transform domain are

For scenario 1:

X(0)={ua(0)

J (0)

}=⎧⎨⎩

0

Jb/s−SS2TT22

⎫⎬⎭

For scenario 2:

X(0)={ua(0)

J (0)

}=

⎧⎪⎪⎨⎪⎪⎩

uts

−TT11

TT12

uts

+ 1

TT12

ubs

− SS1TT12

⎫⎪⎪⎬⎪⎪⎭

For scenario 3:

X(0)={ua(0)

J (0)

}=

⎧⎪⎪⎨⎪⎪⎩

− SS2TT22

ka1dlkl

−SS2TT22

⎫⎪⎪⎬⎪⎪⎭

where TTij and SSi are components of matrices TT(Zn) and SS(Zn):

TT(Zn) =[TT11 TT12

TT21 TT22

]=Tn(Hn) . . .T2(H2)T1(H1)

SS(Zn) =[SS1

SS2

]=

n∑i=1

Tn(Hn) . . .Ti+1(Hi+1)Si (Hi )

ACKNOWLEDGEMENTS

The first author is supported by the Catalonia Government, Spain with FI doctoral scholarship. Thecomputing resources (CODE BRIGHT and GID) are provided by the Technical University of Catalunya(UPC) and International Center for Numerical Methods in Engineering(CIMNE). The authors gratefully



acknowledge Group HERA for the supply of the field data at Coll Cardus landfill. The authors wouldalso wish to acknowledge the constructive suggestions from two anonymous reviewers.

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A coupled model for prediction of settlement and gas flow in MSW landfills