solid waste landfills

Moisture Distribution Through Solid Waste LandfillsB S Thandaveswara, FellowD Sivakumar, Non-member

The solid waste generated in urban areas is increasing day by day. Many Municipalities and Corporations are facing the problemof managing large volumes of solid waste generated in urban areas. Leachate from sanitary landfill is recognised as one of theimportant ground water pollutants. The contaminants are released from the solid waste to the water through physical, chemical,and microbial process and percolate through the unsaturated soil, polluting the ground water with organic and inorganic matters.Several models have been formulated to predict the leachate movement through solid waste landfills. All these models use hydrologicwater balance techniques and are used to predict the leachate volume at the landfill bottom. Therefore, the movement of leachateand the estimation of leachate rate in a landfill site are important steps in designing the collection systems or identifying the treatmentalternatives to reduce the migration of pollutant from the leachate to both the surface water and ground water. This paper mainlydealt with the movement of leachate (moisture content) with respect of depth and time variation using two dimensional numericalmodel techniques with central difference explicit finite difference scheme. The results obtained from the above model are used toprovide general guidelines for the design of new waste landfills with proper collection and treatment systems.

Keywords : Landfill; Solid waste; Leachate; Mathematical model and analysis

NOTATIOND : diffusivity coefficient [L2 ⁄ T]K : hydraulic conductivity [L ⁄ T]Ks : saturation hydraulic conductivity [L ⁄ T]P : precipitation [L]q : moisture flux [L3 ⁄ T]te : effective rainfall duration [T]

tp : ponding time [T]

θ : moisture content [L3 ⁄ L3]θs : saturation moisture content [L3 ⁄ L3]

Ψ : suction pressure [L]

Ψs : saturation suction pressure [L]

INTRODUCTIONLarge quantity of solid wastes are being disposed off on landis a common practice (Tchobanoglous12, Misra and Mani9).Generally, when the water that flows through the solid waste,the more pollutants are leached. The rate of generation ofcontaminated water (leachate) and the time taken by leachateto reach the surface and ground water depends on the move-ment of leachate through the solid waste. The estimate ofleachate rate in a landfill site is of considerable importance inthe design of an appropriate collection system or the treat-

ment alternatives to reduce the offsite migration that mightpollute surface water and ground water resources (Karfiatisand Demetracopoulos6). The movement of water through thesolid waste takes place in the vertical downward direction onlywhen the precipitation is usually distributed evenly over thelandfills. In this condition, the movement of water is governedby the one dimensional flow equation. As discussed else-where, several models have been formulated to predictleachate volumes discharged at the bottom of the solid wastelandfills and is based on water balance technique (Fenn5,Dass2, Perrier and Gibson10, and Khanbilvardi8). Demetra-copoulos4, Korfiatis and Demetracopoulos7, and Demetra-copoulos3 described the formulation and solution techniquesto compute the leachate mound head in the saturated zone ofa landfill.Various pollutants down through the unsaturated zone ofsolid waste and clay liner to the water table and its complexitieshave not been yet fully understood. In this paper, the govern-ing equations, boundary conditions and method to find thesolution of flow through solid waste are presented. Themovement of moisture content with respect to depth and timevariation using two-dimensional numerical model techniqueswith central difference explicit finite difference scheme isdiscussed11. Finally, a detailed parametric analysis has beenperformed to investigate the effects of various parametersassociated with the hydraulic properties of porous medium.

NUMERICAL MODELS

Governing Equations and Boundary Conditions

A solid waste column Yn, mm of height is divided into ‘n’equal increments of depth. Each soil increment is therefore,

B S Thandaveswara is with the Department of Civil Engineering,Indian Institute of Technology Madras, Chennai 600 036 and DSivakumar is with the Department of Civil Engineering, Anjali AmmalMahalingam Engineering College, Tiruvarur 614 403.This paper was received on August 19, 2002. The written comments on thepaper will be received till June 30, 2004.

Vol 84, March 2004 63

Yn/n mm thick. Assuming that the initial moisture distribu-tion is known and that values of the moisture content arespecified at upper and lower boundaries at all time t, thevertical transport of an incompressible fluid through andunsatured flow medium may be written as

∂θ∂t

= ∂K∂y

+ ∂∂x

D ∂θ∂x + ∂

∂y D ∂θ

∂y

(1)

K = Ks (θ ⁄ θs)B (2)

in which θ and θs are the moisture content and Saturationmoisture content of solid waste respectively, Ks is Saturationhydraulic conductivity, D is the Diffusivity coefficient and Bis a constant exponent is equal to 2b + 3 in which b is also aconstant and it is estimated from the relationψ = ψs (θ ⁄ θs)

− b, where, ψs is the Saturation suction head orsaturation suction pressure and b is found to be 2.1511. TheDiffusivity coefficient (Khanbilvardi8) is given by

D = K (∂ψ ⁄ ∂θ); ∂ψ ⁄ ∂θ = (ψs ⁄ θs) b (θs

⁄ θ)b + 1 (3)

in which ψ is the suction head. The solution of equation (1)requires initial and boundary conditions. Two cases of initialboundary conditions are considered. (i) For surface moisturebelow saturation, at

y = 0, P = K (θ) − D (θ) ∂ θ ⁄ ∂ y; 0 ≤ t ≤ tp (4)

in which P is the net precipitation intensity, (LT − 1) and tp isthe time required for surface to become saturated and wouldbe equal to ponding time. tp depends on the solid wastecharacteristics and ratio of P ⁄ Ks. (ii) For P smaller than Ks, thesaturation time (tp) is infinity. After the surface becomessaturated, and rainfall continues, the surface condition can beexpressed as follows. At

y = 0, P ≥ Ks, θ = θs and ∂θ ⁄ ∂y = 0; tp ≤ t ≤ te (5)

in which te is the time of effective precipitation and is equalto 2 h. However, due to the highly porous nature of solidwaste, "the moisture gradient being zero" never occurs. Theboundary condition presented herein physically implies thatsaturation has been attained at surface when the ponding timeis very small.

NUMERICAL SOLUTIONS OF MOISTUREMOVEMENT USING FINITE DIFFERENCEAPPROXIMATIONThe partial differential equations, in general, must be solvedusing numerical methods. Finite difference equations areformulated from the original partial differential equations forcontinuity and momentum. In numerical methods for solvingpartial differential equations the calculations are performed

on a grid placed over the y-t plane. The y-t grid is a networkof points defined by taking depth increments of length ∆ yand time increments of duration ∆ t. Numerical schemestransform the governing partial differential equations into aset of algebraic finite difference equations. The finite differ-ence equations represent the spatial and temporal derivativesin terms of the unknown variables on both the current timeline, i + 1 and the preceding time line, i, where all the valuesare known from the previous computation.A finite difference method may employ either an explicitscheme or an implicit scheme for solution. The main differ-ence between the two is that in the explicit method, theunknown values are solved sequentially along a time line fromone depth point to the next, while in the implicit method theunknown values on a given time line are all determinedsimultaneously. The explicit method is simpler but may beunstable, which means that small values of ∆ y and ∆ t are tobe selected for convergence of the numerical procedure of thenumerical procedure. The explicit method is as the resultsavailable given at the grid points. The implicit method ismathematically more complicated. The method is stable forlarge computation steps with little loss of accuracy and henceworks much faster than the explicit method. The finite differ-ence method is represented by the mesh points on the depth- time and mesh points on distance-depth-time plane. Assum-ing that at time t, the moisture content of solid waste is known.The simplest scheme determines the partial derivatives atpoint (i + 1, j + 1) in terms of the quantities at adjacent points(i, j + 1) (i, j ) and (i + 1, j ) using

θ i + 1 j + 1 = θ i

j + 1 +

D i

j + D i + 1 j

θ i

j − θ i + 1 j

∆ y2

+ D i

j + D i + 1 j

θ i

j − θ i + 1 j

∆ x2 +

K i

j + K i + 1j

∆ y

∆ t

2 (6)

In the present study, a forward difference scheme is used forthe time derivative and a central difference scheme is used forthe spatial derivative. It may be noted that the spatial deriva-tive is written using known terms on time line i.

PARAMETRIC ANALYSISEquation (1) indicates that the influence of the hydraulicconductivity and the diffusivity coefficient. The parametricanalysis of moisture movement through the solid waste of aperungudi solid waste dumping ground has been performedfor the present study. It is located 20 kms from Chennai anddetails of the perungudi dumping ground are given Table 1.The analysis was performed for an unsaturated condition.

64 IE(I) Journal-EN

About 700 tonnes of wastes are transferred daily throughlight mechanical vehicles 3 tonnes capacity and 1 tonne ca-pacity of Auto track trailers. This dumping ground is a marshywetland. The solid wastes come from (1) residential area -60%, (2) commercial area - 14%, (3) restaurants, hotels,marriage halls, schools, institution - 5 -12% and (4) industry -2%. The solid waste is generated from the total area of174 km2, and an estimated total waste per capita per day is500 g average (Ref. Chennai Corporation, Chennai).

Effect of Grid Size on the Moisture Profiles forUnsaturated Conditions

The parametric analysis has been performed for six conditionas indicated in Table 2. For all conditions, the moisturecondition of solid waste is assumed to be below the fieldcapacity of 46.4%. Therefore, the experiments were con-ducted using Perungudi Solid Waste with the initial moisturecontent of 24.3% and the effective porosity was found to be63.3%. These conditions are assumed for the numerical analy-sis to conduct the parametric analysis. Further, it is alsoassumed that the intensity of rainfall is less than the hydraulicconductivity (P/Ks is less than one). The parameter used inthe simulation are (i) rainfall intensity 45 mm/h, (ii) total depthof solid waste in Perungudi dumping yard is 1100 mm and(iii) the total length of area is 1875.0 m. The ponding time isdetermined by using Green - Ampt model technique (Appen-dix A) and is 0.96 h corresponding to a saturated hydraulicconductivity of 145 mm/h. The moisture profiles are com-

puted for different values of time, varying between zeros totime to ponding and are shown in Figures 1 to 6.When the results obtained from the NR 2 compared withNR 1, it may be observed that the results are affected bychanging the depth increment (ie, the movement of moisturecontent will be increasing to 400 mm from 300 m at the endof 0.48 h and increase to 450 mm from 425 mm at the end of0.96 h). Similarly the results obtained from NR 4 comparedwith NR 3, it may be observed that by changing the depthincrement the results are affected as in case of NR 1 andNR 2. And the similar observation has been found for NR 5and NR 6. From the observation made in the parametricanalysis, it may observed that the movement of moisture

Table 2 The summery of twelve numerical runs (NR) (effect of grid size) for unsaturated conditions

Run Nos ∆x, m ∆y , mm ∆t, h

NR 1 187.5 110 0.48

NR 2 187.5 220 0.48

NR 3 187.5 110 0.24

NR 4 187.5 220 0.24

NR 5 187.5 110 0.16

NR 6 187.5 220 0.16

Table 1 Details of dumping ground

Particulars Availability

Total area of solid waste dumpingground

323.76 ha

Total model area of solid wastedumping ground

122.10 ha (1875 m × 651.2 m)

Solid waste thickness 1100 mm

Soil cover (Clay loam) 300 mm

Hydraulic gradient 1.8 m from the top of soil cover

Slope of land 5% towards south

x = 187.5 m

y = 110 mm

t = 0.48 h

0.00

-200.00

-400.00

-600.00

-800.00

-1000.00

-1200.00

Dep

tho

fS

oli

dW

aste

inm

m

Time t = 0.0 h

Time t = 0.48 h

Time t = 0.96 h

0.20 0.25 0.30 0.35 0.40 0.45 0.50

Moisture Content, mm /mm3 3

Figure 1 Effect of grid size on the moisture profiles for unsaturated condition (NR 1)

0.00

-200.00

-400.00

-600.00

-800.00

-1000.00

-1200.00

Dep

tho

fS

oli

dW

aste

inm

m

0.20 0.25 0.30 0.35 0.40 0.45 0.50


x = 187.5 m

y = 220 mm

t = 0.48 h

Time t = 0.0 h

Time t = 0.48 h

Time t = 0.96 h


Vol 84, March 2004 65

content at the end of 0.48 h has reached to a depth is less thanthe movement of moisture content at end of 0.96 h. It wasdemonstrated that the changes in distance step do not affectthe moisture profile. It may also observed that for the timestep of 2 equal parts and for the time step of 6 equal parts gavethe same results of depth of penetration of moisture throughthe solid waste.

Effect of Ponding time on the Moisture Profile forUnsaturated Conditions

The parametric analysis has been performed for four differentcases on moisture profile as indicated in Table 3. In all thesefour cases, it is assumed that the moisture condition of solid

waste is below the field capacity of 46.4%. The other assump-tion regarding initial moisture content, effective porosity,rainfall intensity, size of the field and ponding time is same asin the earlier case. In all these four cases, the total length of

0.00

-200.00

-400.00

-600.00

-800.00

-1000.00

-1200.00

0.20 0.25 0.30 0.35 0.40 0.45 0.50

Dep

thof

Soli

dW

aste

inm

m

Time t = 0.0 h

Time t = 0.24 h

Time t = 0.48 h

Time t = 0.72 h

Time t = 0.96 h

x = 187.5 m

y = 110 mm

t = 0.24 h



0.00

-200.00

-400.00

-600.00

-800.00

-1000.00

-1200.00

Dep

tho

fS

oli

dW

aste

inm

m

0.20 0.25 0.30 0.35 0.40 0.45 0.50


Time t = 0.0 h

Time t = 0.24 h

Time t = 0.48 h

Time t = 0.72 h

Time t = 0.96 h

x = 187.5 m

y = 220 mm

t = 0.24 h




Table 3 The summery of five cases (effect of ponding time) for unsaturated conditions

Cases ∆x, m ∆y , mm ∆t, h

1 187.5 110 0.48

2 187.5 110 0.24

3 187.5 110 0.16

4 187.5 110 0.12

66 IE(I) Journal-EN

landfill is divided into 10 discrete parts such that the ∆ x isequal to 187.5 m and the depth of solid waste in the landfillhas divided into 10 equal parts (ie, ∆ y = 110 mm) but theponding time of 0.96 h only varying for these four cases. Theeffect of ponding time on the moisture profile for unsaturatedconditions is shown in Figures 7 to 10. It may be observedfrom the Figures 7 to 10, that the amount of moisture contentincreased as the ponding time increased for a a given depth.It was observed that at the end of ponding time of 4.96 h,5.96 h, 6.96 h and 7.96 h, the moisture content at the top ofthe solid waste are 46.55%, 49.91%, 53.08% and 55.18%respectively for the Cases 1, 2, 3 and 4 respectively. In additionto the above, the depth of movement of water through the

solid waste and the moisture content at the top of the solidwaste increases as the ponding time increases for a givenmoisture content.

Effect of Rainfall Intensity on Moisture Content ProfileAs in the previous case the total depth of solid waste inPerungudi Dumping Ground is 1100 mm and the total lengthof area is 1875.0 m. The parameters used in the simulation arerainfall intensity was shown in Figure 11(a). The total lengthof the landfill has been divided into 10 discrete parts such thatthe ∆ x is equal to 187.5 m and the depth of solid waste in thelandfill is divided into 10 equal parts (ie, ∆ y = 110 mm) butthe ponding time has been divided into only 2 equal parts isequal to 0.48 h (ponding time t = 0.96 h). Different moisture

0.20 0.30 0.40 0.50 0.60

-1000.00

-500.00

0.00

Dep

tho

fS

oli

dW

aste

inm

m

x = 187.5 m

y = 110 mm

t = 0.48 h

Ponding Time = 0.96 h





Moisture Content in mm / mm33

Figure 7 Effect of ponding time on the moisture profile for unsaturated condition (Case 1)







x = 187.5 m

y = 110 mm

t = 0.24 h

0.00

-250.00

-500.00

-750.00

-1000.00

-1250.00

Dep

tho

fS

oli

dW

aste

inm

m

0.20 0.30 0.40 0.50


0.60


Figure 9 Effect of ponding time on the moisture profile for unsaturatedconditions (Case 4)

0.00

-125.00

-250.00

-375.00

-500.00

-625.00

-750.00

-875.00

-1000.00

-1125.00

-1250.00

0.20 0.30 0.40 0.50 0.60

Moisture Content in mm / mm3 3

Dep

thof

Soli

dW

aste

inm

m

x = 187.5 m

y = 110 mm

t = 0.12 m










Vol 84, March 2004 67

profiles at different elevation are presented in Figure 11. Itmay be observed that at the top of solid waste (0.0 mm), asthe time increases the moisture penetration into the solidwaste increases and also as the rainfall increases, the amountof movement of moisture through the solid waste increasestill the ponding time reaches and vice versa. It has been observedthat the pattern of movement of water through the solid wasteis same as the depth increases from the ground level. Finallythe water profiles reaches the horizontal at the depth of 550mm from the ground level beyond which there is no penetra-tion of moisture (ie, all the solid waste below 550 mm havingthe initial moisture content of 24.3%). This above observationwas made for Effect of Moisture Content with respect toDepth and Varying Rainfall Intensity.

CONCLUSION

The movement of moisture content with respect to depth andtime variation with central difference explicit finite differencescheme has been analysed. A detailed parametric analysis wereperformed to investigate the effect of various parametersassociated with the hydraulic properties of the porous me-dium such that (i) Effect of grid size on the moisture profilesfor unsaturated conditions, (ii) Effect of ponding time on themoisture profile for unsaturated conditions and (iii) Effect ofrainfall intensity on moisture content profile. This model isalso used to predict the leachate quantities generated from the

existing solid waste landfills and to design the collectionsystems or indentifying the treatment alternatives to reducethe migration of various pollutants from the leachate to theground water. The spatial and temporal variation of moisturedistribution with respect to effect of grid size, ponding timeand rainfall intensity can be used to predict the direction ofmovement of mass transport (contaminant transport) withrespect to depth. Thus, the results obtained from the para-metric analysis are providing general guidelines for the appli-cation of the model in practical cases, the design of new solidwaste disposal facilities and evaluation of existing landfills.

REFERENCES1. V T Chow, D R Maidment and L W Mays. ‘Applied Hydrology.’McGraw-Hill-Book Company, New York, 1988.2. P Dass, G R Tamke and C M Stoffel. ‘Leachate Production at SanitaryLandfill Sites.’ Journal of Environmental Engineering, Preceedings of the ASCE,vol 103, no EE6, 1977, pp 981-988.3. A C Demetracopoulos, G P Korfiatis, E L Bourodimos and E G Nawy.‘Unsaturated Flow Through Solid Waste Landfills : Model and SensitivityAnalysis.’ Water Reso Bull, Am Water Reso Assocn, vol 22, no 4, 1986,pp 601-609.4. A C Demetracopoulos. ‘Overview of Landfill Bottom Liner Hydraulics.’Water Reso Bull, Am Water Reso Assoc, vol 24, no 1, 1988, pp 49-56.5.D G Fenn, K J Henley and T V Degree. ‘Use of Water Balance forPredicting Leachate Generation from Solid Waste Disposal Sites.’ Rep.S W - 168. U S Environmental Protection Agency, Wasignton, D C, vol 8-11, 1975.6. G P Karfiatis and A C Demetracopoulos. ‘Moisture Transport in a SolidWaste Column.’ Journal of Environmental Engineering, Proc. ASCE, vol 110,no 4, 1984, pp 780-796.7. G P Karfiatis and A C Demetracopoulos. ‘Flow Charecteristes of LandfillLeachate Collection Systems and Liners.’ Journal of Environmental Engineering,Preceedings of the ASCE, vol 112, no 3, 1986, pp 538-550.8. R M Khanbilvardi, S Ahmed and P J Gleason. ‘Flow Investigation forLandfill Leachate (FILL).’Journal of Environmental Engineering, Preceedings of theASCE, vol 121, no 1, 1996, pp 45-57.9. S H Misra and D Mani. ‘Pollution Through Solid Waste.’ Ashish PublishingHouse, New Delhi, 1993.10. E R Perrier and A C Gibson. ‘Hydrologic Simulation on Solid WasteDisposal Sites.’ EPA-SW-868, Cincinnati, Ohio, U S EPA, 1980.11. D Sivakumar. ‘Solid Waste Leachate Quantity and Quality Estimation.’M Tech Project Thesis. Department of Civil Engineering. Indian Institute of TechnologyMadras, 1999.12. G Tchobanoglous, H Theisen and R Eliassen. ‘Solid Wastes EngineeringPrinciples and Management Issues.’ McGraw-Hill Book Company, New York,1977.

APPENDIX ‘A’Green - Ampt Model Calculation for Ponding Time

(Chow1)

Based on the experiments carried out for the solid waste in the laboratoryfor Perungudi Dumping Yard, initial moisture content (θ), field capacity,Permanent wilting point (θr), total porosity η of solid waste were found tobe 24.3%, 46.4%, 11.8% and 63.3%, respectively. Similarly, the hydraulicconductivity of solid waste is found to be 145.0 mm/h. The duration (t)and intensity (i) of rainfall is assumed to be one hour and 45 mm/hrespectively. For the above condition, the following parameters were

0.00 20.00 40.00 60.00 80.00 100.00 120.00

0.20

0.25

0.30

0.35

0.40

0.45

0.50at 0.0 mm from G.L.

at 110.0 mm from G.L.





Time in min

Mo

istu

reC

on

ten

t,m

m/m

m

8.00

6.00

4.00

2.00

0.00

0.00 25.00 50.00 75.00 100.00 125.00

Time in min

Rai

nfa

llIn

ten

sity

,cm

/h

(a) Hyeto Graph

Figure 11 Moisture content profile with varying rainfall intensity

68 IE(I) Journal-EN

calculated to determine the ponding time using Green-Ampt Infiltrationmodel.

(1) The effective saturation Se is 0.243, using the equation

Se = θ − θrη − θr

(1)

(2) The effective porosity θe is 0.515; it is the difference betweenporosity (η) and the residual moisture content (θr).

(3) From the calibration curve (Figure A), the suction pressure Ψ is0.53 m of water, corresponding to the initial moisture content of24.3%.

(4) The pore size distribution index (λ) is 0.8 (Figure B); it is the slopeof log-log best-fit curve drawn between effective saturation (Se)and the suction pressure (Ψ).

(5) The bubbling pressure Ψb is 90.4 mm of water using the equation.

Se =

ΨbΨ

λ

(2)

(6) The wetting front capillary pressure Ψf is 57.8 mm of water, usingthe equation

Ψf = 2 × λ + 32 × λ + 2

(3)

(7) The cumulative infiltration and infiltration rate after one hour ofrainfall intensity of 45 mm/h on a clay loam soil are calculated asfollows

(i) Ψf ∆ θ = 57.8 × 0.248 = 29.767 mm

(ii) The ponding time (tp), for i = 45 mm/h,

tp = k × Ψf × ∆ θ

i (i − k)(4)

= 145 × 29.76745 (145 − 45)

= 0.96 h

(8) Thus, the ponding time for the solid waste is 0.96 h and it is usedfor parameteric analysis.

4.00

2.00

0.00

0.00 40.00 80.00

Moisture content, Percentage

Su

ctio

nP

ress

ure

,m

of

wat

er

Figure A Calibration curve

1.00

0.10

0.01 0.10 1.00

Suct

ion

Pre

ssure

,m

of

wat

er

Saturation Moisture content(Volume of water per volume of water)

Figure B Pore size distribution curve

Vol 84, March 2004 69

solid waste landfills

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