SLIPPAGE SOLUTION OF GAS PRESSURE DISTRIBUTION IN PROCESS OF LANDFILL GAS SEEPAGE

Applied Mathematics and MechanicsPublished by Shanghai UniversityEnglish Edition) Vol 26 No 12 Dec 2005Shanghai, ChinaCEditorial Committee of Appl. Math. Mech., ISSN 0253-4827 Article ID: 0253-4827(2005 )12-1623-11SLIPPAGE SOLUTION OF GAS PRESSURE DISTRIBUTIONIN PROCESS OF LANDFILL GAS SEEPAGEXUE Qiang(薛强), FENG Xia-ting(冯夏庭), LIANG Bing(梁冰)2(1. Institute of Rock and Soil Mechanics, Chinese Academy of ScienceWuhan 430071, P R China2 Department of Mechanics and Engineering Sciences, Liaoning Technical UniversityFuxin, Liaoning 123000, P. R. ChinCommunicated by LI Jia-chunAbstract: A mathematical model of landfill gas migration was established underpresumption of the effect of gas slippage. The slippage solutions to the nonlinearmathematical model were accomplished by the perturbation and integral transformationmethod. The distribution law of gas pressure in landfill site was presented under theconditions of considering and neglecting slippage effect, Sensitivity of the model inputparameters was analyzed. The model solutions were compared to observation valuesResults show that gas slippage effect has a large impact on gas pressure distributionLandfill gas pressure and pressure gradient considering slippage effect is lower than thatneglecting slippage effect, with reasonable agreement between model solution andmeasured data. It makes clear that the difference between considering and neglectingslippage effect is obvious and the effects of coupling cannot be ignored. The theoreticalbasis is provided for engineering design of security control and decision making of gasexploitation in landfill site. The solutions give scientific foundation to analyzing welltest data in the process of low-permeability oil gas reservoir exploitationKey words: landfill gas; gas slippage effect; mathematical model; perturbationmethod; slippage solutionChinese Library Classification: 0357. 3; TE312Document code: A2000 Mathematics Subject Classification: 76S05; 93 A30; 65M60: 65L15IntroductionWith the development of economics and growth of population, municipal solid wasteyield increase 8%-10% every year, Large sanitary landfill is constructed in some municipaReceived Jul 06, 2004; Revised May 23, 2005ject supported by the National Natural Science中国煤化工500053eMajor State Basic Research Development Prog2002CB412708); the Sunshine Young Project in WCNMH Gram)(NoCIty on nina INO. 2005500305933); the Open Fund of State Professonal Lab of Geological Hazard Prevention and Engineering Geological Environment Protection( No. GZ22004-011)Coresponding author XUE Qiang, Associate Professor, Doctor, E-mail: qiang@ whrsm ac cn16231624XUE Qiang, FENG Xia-ting and LIANG BingLandfill gas consists of four primary components, which are methane, carbon dioxideoxygen and nitrogen, and a trace amount of other organic compounds i. The landfill gaswas given off during the landfilling of the garbage and other refuse and after the landfillclosure. The landfill gas can flow through the soil or sandy soil around landfill site,andmethane can be measured in the buildings beyond 100 m-200 m. If the landfill site does notuse liner and gathering system, it can migrate to the further surrounding areas of the landfillsite along smaller routes, then releases to atmosphere, which causes air pollution andaccidents. Methane in volumetric concentration of 5%-15% is explosive. In order tocontrol the air pollution and hazard from the gases produced from the solid waste in thelandfills, gas collection systems are installed. In any exploitation within 250 m near landfillsite, the migration laws of landfill gas must be made through filed measurement, and itspotential risk should be fully estimated. So it has important meanings to investigatelandfill gas migration in landfill site for security control engineering designsExperiments indicate that gas migration represents the slippage phenomenon that itspeed is not zero along solid wall 3-5. Gas seepage law does not coincide with linearDarcys law, but appears to slippage effect, thereby result in the change of gaspermeability. The slippage effect is more obvious for low permeability porous media.Thereis not a fully successful method for measuring gas flux in soil. The information of methaneconcentration and gas pressure gradient is usually obtained by filed measurement. Thereforeit is necessary to study the pressure distribution law of gas in landfill siteSeveral models have been developed for describing landfill gas migration, productionand extraction through landfill sites. For example, Youngo developed a more realisticmodel of LFG generation and transport in a landfill cell of rectangular shape, which couldaccount for flow of a single gas inside a non- isotropic porous medium,within thepresumption of small variation of gas pressure and density. Rathfelder and Mackay 71developed a numerical model for the distribution law of volatile organic gas under soil vaporextraction.Douglas(8)et al. calculated the change of concentration distribution of landfillgas in saturated soil by adopting numerical method. Lu and Kuntzl9) developed a ID radialflow model which uses measurements of lfG pressures and the pressure changes resultingfrom the withdrawal of gas to calculate the landfills rate of methane production. Chen etal. o proposed a numerical model of landfill gas migration under the condition of neglectingmedium deformation and single flow in unsaturated soil. Xue et al. in developed anunsteady coupling seepage flow model of landfill gas in soil. However, because of thecomplexity of the seepage characteristics of landfill gas in porous medium, mostly adoptednumerical solution in simulating the dynamic behavior landfill gas transport and the mostimportant assumption of those models were that gas migration coincides with Darcys lawA mathematical model of landfill gas migration hasresumption ofthe effect of gas slippage in this paper. The analytic中国煤化tical model areaccomplished by the perturbation and integral transfoCNMHGa v, introducing adimensionless transformation factor The distribution law of gas pressure has beinvestigated quantitatively, and sensitivities of the model parameters and model dependabilityare analyzedpage Solution of Landfill Gas Seepage1625I Slippage Mechanism of Landfill Gas MigrationLandfll gas is derived from the decomposition of the moist organic fraction of thewaste. The production and accumulation of landfill gas within the landfill raises the gaspressure in the landfill above atmospheric pressure. The resulting pressure gradient acts as adriving force causing the gas to diffuse out of the landfill, into the surrounding soil strata ornto the airThere are mainly two ways of landfill gas migration concentration gradient resulting ingas diffusion and pressure gradient causing gas convection. The concentration of gaseousmixture, which is produced by anaerobic decomposition and contains methane and carbondioxide, is greater than air. Methane and carbon dioxide will diffuse from landfill site to airCompared with convection the diffusion effect is secondary in gas migration. There arenumerous factors involved in landfill gas migration, including: cover layer and beddingcourse materials, geological conditions, hydrographic conditions and atmospheric pressure.Considering the secondary pollution of landfill site, low permeability covering layer isusually applied to permit gas diffusion to air after closure of landfill. So the landfill gastransport is mainly determined by transverse migration in low permeability porous mediaIn the gas permeability test, Klinkenberg presented the slippage effect when flowingthrough capillary tubes. The molecular acting forces between gas and solid are much lowerthan the ones between liquid and solid. Some of the gas molecules on pipe wall are also inthe motion state but not adhered to pipe wall. On the other hand, because of the momentuminterchange the gas molecules in the adjacent layers can migrate directionally along the pipewall with the gas molecules on the pipe wall, i. e, slippage effect. Much less is the averagepressure, more serious is the slippage effect.The expression of permeability isk g krsh(1)where k, is the gas permeability at mean pressure, P, and k is relative permeability of gasFgw is the intrinsic permeability of gas; c is a proportionality factor; a is the mean free path ofthe gas and r is the average radius of the capillariesAccording to Klinkenberg experiments, the mean free path of gas is inverselyproportional to the mean pressure Pm, and Eq.(1)is reproduced as follows(2)where P is the mean pressure which is defined by P=(P+P)/2, P and P are inlet andoutlet pressures of the pores, respectively; b is the gas slip factor, which is defined asbAPSampath and Keighin(1982)studied the corthe intrinsic gas permeability of gas based on a largeYH中国煤化工 slip factor andCNMHG. and obtainedthe empirical equation as followsb=0.0955-8(4)1626XUE Qiang, FENG Xia-ting and LIANG Bingwhere is the porosity of the core sample.2 Model Formulation2.1 Mathematical model developmentThe governing equation is based on the following assumptions1)The landfill gas is considered as ideal gas. The effect of media deformation iseligible.2)The medium is isotropic, that is to say, the porosity and the absolute intrinsicpermeability are constant3)Gas seepage is constant temperature and single-phase flow. The affects of gravityand capillary force are negligible4)Kaluarachchi's assumption should be satisfealu), that is to say, the mean pressurePm should be the gas pressure P at any point in porous mediaEquation for specificke aP_ krg6where q: is Darcys velocity, P is the gas pressure, P is the coefficient of viscosity of gasGas state equation(6where R is the gas constant, T is the absolute temperature, Mg is the mole mass of gas (7)Mass conservation equationa(p2)+(p2g)=0.(8)Introducing(5)-(7)into Eq.(8),we may obtain the governing equation oflandfill gas seepap a(1+p2It is assumed that a =AEOkconsidered. Under the condition of Ea. In this paper, only transverse migration of gas isone dimension, gas pressure distribution Eq(9)may bewritten as(10)2.2 Initial and boundary conditionsThe initial and boundary conditions of landfill gas seepage problems are describedfollows中国煤化工Initial conditionCNMHGP(x,)boundary conditionP(x, t)Isso= Pl, P(a, t)Slippage Solution of Landfill Gas Seepage16273 Slippage Solutions to Mathematical ModelCombining Bq. (10), the initial condition(11)and the boundary condition(12),wecan obtain a mathematical model for landfill gas migration under the condition of effect ofgas slippage. It is a second order nonlinear partial differential equations set, and the exactsolution is unavailablel15). Therefore, the analytic solutions to this mathematical model areaccomplished by the perturbation and integral transformation method and by introducing theimensionless transformation factor a. The progress to obtain the solution is as followsThe regular perturbation problem is obtained by combining Eqs. (10),(11)and(12)Using dimensionless transformation method to solve Eq (10)AssumingP2-P2P2-P2Introducing Eq (13)into Eq(10), we obtainPP)2F(14)whereT=The initial and boundary conditions Eqs.(11)and(12)becomeP0,P(15)It is known from the T Theorem for dimensional analysis that, the definite-solutionproblem, which is composed of Eqs. (14 )and( 15), has a similarity solution P(5)Assuming ar?' we may rewrite Eq (14)and Eq (15)β)d 2pd(16)P(自)l=1,P()lea0=0(17)Assuming(18)PwP)-a2(F。+op…)Introducing Eqs. (18 )and(19)into Eq (16), so we中国煤化工d 2p(1+B)+o(1+d PiCNMHGdd22d2d p+a(1+dg-(2则)1628XUE Qiang, FENG Xia-ting and LIANG BingdPds dE(20)The boundary conditions becomeP0(∞)+oP1(PP0(0)+uP1(0)+a2P2(0)0.22Combining the same power parts in eqs.(20),(21)and(22), we obtaind 2 pdpβ(23)P0(∞)=1,P0(0)=0;d p2dpd21+Bd·2(1+β)42(24)P1(∞)=0,P1(0)=d'_ dP2- Pod P./P, Pald po1 +B dsd2d(25)P2(∞)=0,P2(0)=0.Using integral-transform method, we obtain the solution to Eq.(23)erf(√y26),where y = 2/(1+B)Using the method of constant coefficients variation to solve Eq( 24), by means ofintegral transformation we obtain1ds+SEedsxle"*" le"w"dsdt +y()emends(27)Using the condition of P, (oo)=1, to simplify Eg(27), we can obtainF,=m3÷(YI,e-m/(28)Similarly, the solution to Eq(25), may be obtained by introducing Eqs.(26)and( 28)Into it=21叫(如-“)4(叫引Y/Eel6√√26(5-y2)e16m2Introducing Eqs.(26),(28)and(29 )into中国煤化工 che second-ordippage solution to the mathematic modelCNMHGP=P+oP, +o2P叫43)-24(2ke+)-x+pSlippage Solution of Landfill Gas Seepage162916m32l6√(5-y5) e yosefx)ef2)+*32m(30)4 Theoretical Analysis and Verification of Slippage Solution1)When gas slippage effect is negligible, namely 6=0, we can obtain theapproximate analytic solution( Darcy solution), under the satisfaction of Darcy law. It canP()Lerf(5)(2+{ef()(1+√2me-e"2))、1-er(]+ael6m(5-y5)eef(4)]2+4(2-5)eaef(9)32163)-erf()(31)2)For the gas seepage problem in oil-gas reservoir engineering is difficult to solve, wemay simplify the mathematical model (10)by Robinson linearization method( 16 In thisase,the governing equation should beap aP(32)where m a/Pe, P is the intrinsic pressure in the original formationUnder the condition of using the same initial and boundary conditions(11)and 12)we obtain its analytic solution( Robinson solution )asP(x, t)=erf(s) where s(33)In order to simulate the dynamic distribution characteristics of gas pressure in landfillgas migration under the condition of gas slippage effect, we compare the slippage solution inthis paper with inner gas pressure distribution characteristics of a landfill in Shenzhen underthe condition of single well extraction. The parameters of the mathematic model are listed inTable 1 and solution results illustrated in Figs. 1-6( Notation: DS- Darcy Solution; SSSlippage Solution; RS-Robinson Solution; MC-Measured Curve: FC-Fitted CurveThe distribution of gas pressure with the temporal and spatial variation in landfill sitewere analyzed in Fig. 1 and Fig. 2 under considering中国煤化工 Page effect,respectively. It shows that the gas pressure considerinCNMHGwer than thatneglected, and the differences are greater with the time of air extraction and the distancefrom the well increasing. So, the slippage effect should be considered in the gas migration inlandfill site1630XUE Qiang, FENG Xia-ting and LIANG BingTable 1 Model parameters valueEffective Initial Boundary Effective Temperature Coefficient Density Length of Height ofParameters drainage pressure pressure gasof viscosityoL Po/MPa P /MPa k/'m T/K P,(Pas)p/(ke/'m)L/mH/mValue0.350.100.0951.2×10301.7×1051.2110051.0- x 10 m, SSx= 10 m DS0.81=20 days, SSx=20 m. sS4=40 days, Ds0.6t=40 days, ss30m,S1-60 days, DS06x=30 m, DS1=60 days, SsA040.202dFig. 1 Landfill gas pressure along levelFig 2 Landfill gas pressure along time atdistance at different time underdifferent location under consideringconsidering and neglecting slippageand neglecting slippage effecteffectIn Fig 3, the curves of gas slip factor b affecting the pressure distribution of landfill gasshow that slippage phenomenon is more obvious with the slip factor increases and the gaspressure decreases. It is consistent with the testing results by Wang m7, and the difference isgreater as time increases. The main reason of gas pressure decrease is that under thecondition of gas extracted, the pore radius decreases, the solid media will be compressedand the gas permeability decreases, then the gas pressure decreasesThe effect of the perturbation parameter transformation factor o variation on gas pressuredistribution is discussed in Fig. 4. The calculation result shows that the variation has moreb=00050.50=0.10060620404中国煤化工CNMHGFig3 Gas slip factor b effect on landfillFig 4 Perturbation parameter w effect ongas pressure distributionlandfill gas pressure distributionSlippage Solution of Landfill Gas Seepage1631effect on the gas pressure distribution when the value perturbation parameter is in the rangeof0. 01 w< 1; the effect becomes lower as the perturbation parameter e decreases. Onlywhen o <0. 01, the parameter has no effect on the gas pressure distribution. So, as we usethe perturbation method to solve the nonlinear problem, the condition1 must besatisfied, otherwise the precision of model solutions will be reduced. It provides a referenceto determine the perturbation parameter.The relationship between landfill gas output and time wased in Fig. 5. Itobviously shows that gas output is the least considering the slippage effect; the one satisfyingthe Darcys law is secondary and the one calculated by robinson linearization method is themost. The decrease of gas output under the effect of slippage results from the lower of gasconductivity and the slower change of the pressure gradient in the landfill site. It is consistentwith the change of the pressure gradient in Fig. 1. Namely, the gas pressure gradientconsidering the effect of slippage is smaller than the one neglecting slippage effect.Ifcalculating the gas resources exploitation by the Darcy's law in low permeability landfillate, the gas output will be greater. Then the exploitation index will be estimated excessivelyhigh in decision makingFigure 6 shows that the comparison and verification is obtained through gas pressuredata of ten measured points in a landfill site of Shenzhen. Compared results indicate that theslip solution is closer to the curves fitted using the measured data than the darcy's solutionand the robinson solution. The reason of the difference near the extraction well is that thegas pressure reducing suddenly from Po to P,14×104日1.0×0Hs DSDS60×103MC虽00972.0×10300965060009560Fig. 5 The relation of landfill gas outputFig 6 Comparison curve among differentalong timesolutions for landfill gas pressuredistribution5 ConclusionsThe following conclusions can be obtained) A mathematical model of landfill gas中国煤化工 ablished undconsidering the effect of gas slippage. The analytiTHCN Gr mathematicalmodel are accomplished by the perturbation and integral transformation method2)The dynamic distribution characteristics of landfill gas pressure shows that theslippage has an important effect on gas pressure distribution, and the difference will be more1632XUE Qiang, FENG Xia-ting and LIANG Bibvious as the increasing time and distance. So the slippage effect can't be neglected forinvestigating gas migration in landfill site3)By analyzing the sensitivities of model parameter b and w, it shows that the changeof their values will affect directly the gas pressure distribution in the landfill site. Thecondition of w<1 must be satisfied if you want to apply the perturbation method, otherwiseit will affect the precision of the solution. The gas slip factor b should be determined bycolumn permeability experiments for ensuring the precision of parameter values4)Contrasted with the measured data, the slip solutions obtained by the model in thispaper are more precise and coincide with the practical gas migration law. It shows that it isimportant to consider the slippage effect in the landfill gas migration. The perturbationtheory method provides a way for solving the nonlinear partial differential equations.5) The slippage solutions give specific reference to investigating the dynamicdistribution characteristics of gas pressure and analyzing well tests data in the process of low-permeability oil gas reservoir exploitationAcknowledgements The authors would like to thank Professor Jagath J. Kaluarachchiand Liu Ci-qun for their enthusiastic guidance and helpReferences[1] Chen Y C, Chen K S, Wu C H Numerical simulation of gas flow around a passive vent in asanitary landfll[J]. 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