Numerical Study on Dissociation of Gas Hydrate and Its Sensitivity to Physical Parameters

China Ocean Engineering, Vol. 2I, No 4, Pp. 625-636C 200 China Ocean Press, ISSN 0890-$487Numerical Study on Dissociation of Gas Hydrate and ItsSensitivity to Physical ParametersBAI Yu-hu(白玉溯), LI Qing-pi吗g(李清平),YU Xi-chong(喻西崇) and FENG Guo-zhi(冯国智)Technology Research Dept, China National Ofshore Oil Corp, Research CenterBeying 100027, China(Received 28 Auguest 2006; received revised form 24 July 2007; accepted 5 September 20m7)ABSTRACTThe natural gas hydrate resource is tremendous. How to utilize the gas from hydrates safely is researchers concemIn this paper, a one-dimensional model is developed to simulate the hydrate dissociation by depressurization in hydrate-bearing porous medium. This model can be used to explain the effects of the flow of multiphase fluids, the endothermicprocess of hydrate dissociation, the variation of permeability, the convection and conduction on the hydrate dissociationNumerical results show that the hydrate dissociation can be divided into three stages: a rapid dissociation stage mainlygoverned by hydrate dissociation kinetics after an initially slow dissociation stage govemed mainly by now, and finally aslow dissociation stage. Moreover, a numerical approach of sensitivity analysis of physical parameters is proposed, withwhich the quantitative effect of all the parameters on hydrate dissociation can be evaluated convenientlyKey words: gas hydrate; depressurization; mathematical modeling; sensitivity analysis; hydrate dissociation1. IntroductionAccording to the researchers' estimates, 90% ocean bottom are suitable for hydrate formation, asare regions in and beneath the permafrost, which covers more than 23% of the earth s surface( Burhears et al., 1986). Therefore the gas hydrate reserve predicted is tremendous. Even the most con-servative estimates of the total quantity of gas in hydrate is two times of the energy content of the totalfuel fossil reserves recoverable by conventional methods( Sloan, 1998). Each volume of methane-hydrate can contain as many as 170 volumes of gas at standard condition. So gas hydrate is regarded as akind of future altemative resourcesHow to release gas from hydrate is the key problem. Some researchers(Ma1997; Ji etal., 2003; Ahmadi et al. 2004; Tsypkin, 2001)viewed hydrate dissociation as a moving boundaryablation process as solid melting, and adopted the classical Stefans equation to describe the processIn this model, a dissociation front is assumed to exist to separate the hydrate reservoir into a gas and ahydrate zone. Yousif et al.(1991)developed a three-phase 1-D model to simulate the process ofproduction from Berea sandstone samples containing methane hydrate by depressurization. For thetime,their model coupled the Kim- Bishnoi(Kim et al.,中国煤化工However, theThis work is financially supported by the National High TechnologCNMHGram of China(863CorrespondingauthorEmailbyh_2002@163.comBAI Yu-hu et al. /China Ocean Engineering, 21 (4), 2007, 625-636hydrate dissociation process was assumed to be isothermal. In some circumstances, hydrate reservoirsitha free gas reing the fronatural gas reservoir, the equilibrium condition of hydrate is broken and hydrate dissociates. Holderand Anger( 1982)firstlydepressurization process of this combination reservoir in whichthe heat of dissociation came from the sensible heat of reservoir itself. In his model hydrate dissociationwas assumed to occur only at the interface between hydrate"cap"and free gas strata. In energy equation, only heat conduction was considered. Results indicate that hydrate could contribute about 20%30% of the total gas production. Moridis(2002)designed EOSHYDR2 by adding a module for hydrate dissociation into the toUGH2 general-purpose reservoir simulator. It can model the flow of gasand water, the non-isothermal gas release and phase behavior. Sun et al.(2005)developed a thermal, three-phase, one-dimensional model to simulate two regimes of gas production by depressurize-tion: the dissociation-controlled regime and the flow-controlled regime. a dimensionless parameter Rnamely dissociation-flow time-scale ratio was defined and employed to identify the two regimes. Atlarge values of Rr, flow is a faster process and the system is increasingly controlled by dissociation;otherwise, the system is controlled by flow. Li et al.(2005)analyzed some parameters effect on gashydrate production with isothemal model. Tang et al.(2006)conducted sensitivity analysis of somearameters on hydrate dissociation behavior with the ToUGH-Fx hydratePreviously extensive research focused on the models of depressurization and employment on thegas production from hydrate reservoir. Few papers reported the model simultaneously involving thesemechanisms such as the flow of water and gas, the variation of the permeability, the dissociation kinetics, convective and conductive heat transfer, etc. Mehran(2004)recommended that the early modelsshould include all these mechanisms until more is leamed about the effect of the individual mechanismson the over all process, which then may lead to reasonable simplifying assumptions. The objective ofour work is to develop a 1-D model associated with all these mechanisms for quick interpretation of thedissociation process of depressurization. Based on this simulation, a numerical approach of sensitivityanalysis is proposed to identify the important physical parameters for the laboratory experiment and fieldproduction2. Mathematical ModelWith consideration of one-dimensional hydrate-bearing porous medium of length L, the produwell is at x=0, where the bottom pressure is below the equilibrium pressure. Then the hydrate disso.ciation occurs in the porous medium and the pressure and saturation develop throughout the systemWith these concemed, the transport equations goveming hydrate dissociation and the flow of gas andwater through this porous medium area ekg a)+mg中国煤化工1)a(ky e)+m= atCNMHG(2)BA/ Yu hu et al. / China Ocen Engineering, 21(4), 2007, 625-636where, p, P, k, s, and A are the density, pressure, permeability, saturation and viscosity, respectively; y is the porosity; the subscripts g, w, and h denote the gas, water and hydrate, respectively. The quantities of mh, mg and mu in equations denote the corresponding local mass rate of hydrategas and water produced per unit volume, respectively. These are related byin which Mw and Mk represent the molecular weight of water and gasin this paper, methane),re-spectively, and Nh is hydrate number. The local mass rate of gas generated by hydrate dissociation maybe controlled by the Kim- Bishnoi model( Kim et al., 1987)mg= kdA, e-f)(5)where, A, is the specific surface area of porous medium; f is the local gas fugacity; and fe is the gasequilibrium fugacity. The local pressure gas Pg and the methane-water-hydrate equilibrium pressure pegare used to approximate and fe, respectively, as in the work by Yousif et al.(1991)and Sun etal.(2005).kd is the dissociation constant.ThereforekdA, (pea-P)N,M+ Me)For methane hydrate, N=6, M,=18, M, 16The relationship of saturation satisfi1The water and gas pressures are related according to the capillary force equationPe P- PIn our model, the porous medium and water are regarded slightly compressive, and their state e-quations are respectively given byPw =Prol 1 +c(Pu-Puo)Iφ=s1+q4(Pk-patp)],(10)where,cw and c are the compressibility of water and porous medium, respectively; Puo and Pgo arewater anThe state equation ofg=mArTwhere,Z is the gas deviation factor; m, the mass of the gas; V, the gas volume; and R, the univer-sal gas constant. Then we obtain the gas compressibility c, as follows1dk_(1中国煤化工CNMHG(12)PAThe phase equilibrium pressure peo is evaluated by( Makogon, 1997)BA/ Yuhu e al. China Oean engineering, 21 (4), 2007, 625-636lgpeg a(T-To)+b(T-To)2where pea is in Pa and T in Kelvin; To is 273. 15 K; a, 6, and c are empirical constants; and a0.034K-1,b=0.0005K-2,c=6.4804The energy balance equation can be written in terms of temperature as follow:(Cr)Pukg apsm△H(14)xin which∮(psC+ pskOv+PMs(C)+PC(1-∮)=C,K+内K++(1-∮K=Kwhere,Cw,Ch, and Cr are the specific heat of water, hydrate and rock, respectively: Cvs and Cpgare the constant volume and pressure specific heat Ky, Ke, K, and Kh are the coefficients of thermal conductivity of water, gas, porous medium and hydrAH is the enthalpy changein hydrate dissociation. The process of hydrate dissociation ishermit phase change processThe latent heat for per kilogram of hydrate in J/kg is given by( Kamath, 1983)AH AT +B15where A and B are constants given byA=-1050J/(kg·K),B=3527000J/kgThe boundary conditions are imposed as follows:x=0,p=m,r=nx=,=0.2=0and the initial conditions are specified asp(x,0)=p1,54(x,0)=5,5(x,0)=5,7(x,0)=THere, Pep and Ta are bottom pressure and temperature of production well, respectively; Pi, Shi, swiand T; are initial pressure, hydrate saturation, water saturation and temperature, respectivelyDuring the dissociation process, the specific gurface area A, in Eq. (5)may change. We estimate A, with the method by Amys et al. (1960)A中3/(2k)(16)where k is the absolute permeability. y(t)is the porosity and is evaluated byWith the dissociation of the hydrate, the absolute permeability will change too. Because of the signifcant discrepancy among different measurements for the hydrate-bearing formation, no permeability mod-el has yet emerged(Mehran, 2004). We use the empirical relationship(Yousif et al. 1991)between e(t)and k from field data for Berea sandstone r中国煤化工When dealing with the capillary force and the relatiCNMHGases,rding to the followinBAI Yuhu e al./China Ocean Engineering, 21(4), 2007, 625-636Fig. 1. The relationship between porosity and per- Emeabilitysabit(mayNote that, sy and sx are based on the total pore volume occupied by fuid phases and hydrates, ands as and sk denote the effective pore volume-based saturation, respectively. The data of the relativepermeability and capillary force are the same as Yousif et aL. (1991)3. Numerical Solutionno analytical solution is available for the above goveming equations. Therefore, a numerical solution is considered. The goverming equations are discretized with finite difference and the difference equations are solved in a sequential manner. The hydrate saturation is solved first explicitly; then thepressure is solved implicitly, followed by explicit solution of the water saturation. Finally, the temperature is implicitly solvedThe hydrate saturation can be solved in an explicit scheme(φon)Discretizing the water and gas mass balance equations, the following difference equations can be yieldgi-1+ cegpgi+ bep中2cx+ Pgisgipoc;)中)(s41-5)(20)aspri - j+ CaPi+buPai+l =-(Psw ne Co wisuPocy)Pu(中)(21)"V山中国煤化工o)An];沙CNMHG630BAI Yu-hu d al. /China Ocean Engineering, 21 (4), 2007, 625-636Adding Eg. (20)to Eq.(21)multiplied by the coefficient Pai/ poi and combining the relationships ofsaturation and capillary force, we obtain pressure difference equation of gas phase as follows:qPs+如g+1=Sm+sm+sk6= bg+AbnAcAt: ASm=-kdA,Peg-6.65Ak Peg; Spc A(auPe-I+CaPa buPc+i);(4P埏64 g2a(Psip.cp中o4)△Aoish-shThe implicit difference equation for energy balance can be derived asch+1)T+ kcr+myAH(23)Pvi+!二PCkg、,Px2k1(C)2a、p4+(22);Axmain physical parameters in the model are listed in Table ITable 1Values of main physical parametersrametervalueParameteValue( MPa)C (J/K kg)840Pgp(MPa)2.5Ke( W/mk)T(K)288K,(W/mk)T(K)K (W/m K)0.257K (W/m K)(kg/mr)1000k,(kg/m.Pas)4.4E-16P(kg/m)C(J/K·kg)2206e, ( kg/m)C (J/Kkg)(%)Ch(J/K kg)1800Dynamic Process of Hyd中国煤化工From the physical view, the dissociation processCNMHGe-fluid flow anddissociated fluids, conductive and convective heat flow, and kinetic dissociation at the hydrate inter-BAl Yu-hu e a. /China Ocean Engineering, 21 (4), 2007, 625-636631face. Gas production from the hydrate results in pressure reduction. When pressure falls below the e-quilibrium pressure of hydrate, the hydrates will dissociate till their vapor pressure is re-attained underthis pressure. As hydrates dissociate, they adsorb heat from surroundings and thus result in the de-crease of the temperature. The generated temperature difference between hydrates and surroundings actsas the driving force to initiate heat flowing from the surroundings and fluids to the cooled area. The hydrates will continue to dissociate until they generate enough gas to raise the pressure up to their vaporpressure at new temperature which is lower than the original. By keeping the pressure reduction, thehydrates continue to dissociate, thus staying cold. Therefore, the temperature gradient and heat flux tothe hydrates can also be maintainedFig. 2 shows evolutions of gas pressure distribution for t=2000, 10000, and 4000000 secondsIt is clear that the typical pressure distribution for (=10000 seconds can usually be divided into threedistinct zones: the sharp front zone, the gradual variation zone and the uniform zone. The sharp frontzone is adjacent to the production well behind the gradual variation zone. Whereas, the uniform zonefar away from the production well is ahead of the sharp front zone. In the initial stage of production,alarge pressure gradient exists close to the production well and the gradual variation zone is absentwhich means the dissociation occurs in a narrow zone and can be regarded as a dissociation front. Theuniform zone is"untouched"zone lying adjacent to the no-flow boundary. In this region, the pressureis kept above the equilibrium pressure of hydrate and maintains the initial pressure distribution, thusno dissociation takes place. With the continuous production, the gradual variation zone extends and thepressure gradient decreases, manifesting the dissociation of most hydrate in this region. The uniformzone reduces and the sharp zone becomes gradual transition with the time lapsing and the dissociationzone is not a narrow one any more but an extended zoneE+076E+6Dimensionless distanceDimensionless distanceFig. 2. The evolutions of gas pressure in hydrateFig. 3. The evolutions of temperature in hydrabearing formationbearing formation中国煤化工Fig. 3 shows the evolutions of temperature in hydrate-C Gee that with thecontinuous production of gas form hydrate the temperature decrease in the areas due to the endothermicBA/ Yu-hu d al./ China aen Engineering, 21(4), 2007, 625-636dissociation process. Thus heat flows from the surroundings to the hydrate dissociation zone under thedriving force of temperature difference. We can reasonably deduce that under certain conditions thetemperature may fall below the ice point of water and hence ice appears, which complicates the flow ofmultiphase fluid and the hydrate dissociationFig. 4 shows the evolutions of hydrate saturation. Obviously, a sharp front exists in the evolutionprofile of the initial dissociation stage. In this stage, the dissociation occurs in a narrow zone with lesshydrate dissociation. Therefore, the surrounding medium can supply enough heat for hydrate dissociation.As the dissociation continues, hydrate dissociates in a wide region and a larger amount of heat isin need, thus, the heat cannot be supplied immediately. Consequently, hydrate cannot dissociate in ashort time and a wide dissociation zone forms025Dimantlonkass dhtanc4. Hydrate saturation evolutions in hydrateFig. 5. Gas saturation evolutions in hydratebearing formationFig. 5 shows the evolutions of gas saturation. The plot shows that gas saturation rapidly decreasein the region adjacent to production well in the initial stage of production on account of the existence ofthe free gas, the reduction of well bore pressure causes the free gas flow to the well bottom quickly. Inaddition,in this stage, hydrate dissociation just commences and can not generate enough gas to makeup the loss of free gas. The profile of t= 10000 seconds shows a relative high gas saturation in the region adjacent to production well, meaning widely hydrate dissociation and a large amount of gas genera-tion. The gas saturation in regions far from the production well still keeps the initial distribution, whichmeans that pressure front does not transport to this region yetFig. 6 shows the evolutions of water saturation. As hydrate dissociation continues, water satura-tion about 55% appears in production well This is in accordance with the large water production(upto 84%)indicated by the simulation result by Yousif et al. (1991). This effect may indicate a significant production hindrance during the recovery of natural中国煤化工Fig 7 shows relationship between the dissociationCNMHGdissociation ratiodefined as the ratio of dissociated hydrate to total hydrate reserves, thus it can be used to evaluateBAl Yu-hu e al./China Ocean Engineering, 21(4), 2007, 625-636the hydrate recovery. Obviously, the curve of the dissociation ratio can be divided into three stagerapid increase stage after slow increase stage and followed by slow increase stage. Slow increase stageof the dissociation ratio means the less hydrate dissociation, and rapid increase stage means the largerhydrate dissociation. In the first stage, the gas production from well is mainly govemed by flows andhydrate dissociation occurs in a narrow region. In the second stage, the dissociation ratio increasesrapidly, which means that the dissociation occurs in a wide region and the gas productionby dissociated gas. In the third stage, the increase tendency of dissociation ratio becomescan be explained as that most of hydrates have dissociated in the second stage and large quantity of en-ergy has been consumed. The reduction of energy greatly restricts the hydrate dissociation0.1Dimenalonkes dstanceTime (secondFig. 6. Water saturation evolutions in hydrateFig. 7. The relationship between dissociation5. Numerical Approach of Sensitivity AnalysisIn the following section, we propose a numerical approach of sensitivity analysis to quantify theeffect of physical parameters on the hydrate dissociation. On this basis, the goverming parameters aredetermined and their relative importance degree is evaluatedFirstly, a sensitivity factor Si comesponding to the parameter Vi is defined asa(F/F)where F is a target function concemed in the depressurization system. F is the function of all therameters and can be expressed as F= F(VI, V2, ",Vi,,V,). The subscript p denotes the disso-ciation system. Therefore, the physical meaning of the sensitivity factor is the relative variation ratio ofthe target function with respect to that of the physical parameter. Comparing with the values of the sen-sitivity factors, the quantitative effect of the physical para中国煤化工 ation can be determined. Thus, the important factors can be singled outCNMHGinvolved in thispaper,the function related to dissociation ratio is selected as the target function to denote the hydrateBAl Yu-hu et al. China Ocean Engineering, 21(4), 2007, 625-636dissociation, The target function can beR(V1,V2,…,V,…,Vn,t)dt(26)where R(V1, V2,", Vi,", V, t) is the dissociation ratio as shown in Fig. 7, and T' is the dissociation time spanIn numerical scheme, the sensitivity factor can be written as:AF:/FRn(V1,V2,…,V,t)dt△F=J。R(v,H,…阿,…,V,)-凡(V,V,…V,)dSet the deviation coefficient wi of each parameter to be 1% respectively and keep the others fixedThen, we can readily derive the sensitivity factors as listed in Table 2Table 2Sensitivity factors of all the parametersK,kκS|401E2「353E342E26.33E19.8E2735E12.61E71.57E52.86E4K wKrg654E6565E+159023.93E11.46E1E31.46E1871E31.86E3Ps3:6E+0449521.49E+02.52E+01.9+030E1105E+0510E1Table 2 shows that the values of sensitivity factors cover a wide domain ranging from 10-to 10our system, T; is the most important parameter with the sensitivity factor over 10, Peg, so,PiPh, and Sh are the relatively more important parameters with the sensitivity factors between 10 and10, and Cr, AH, Kd, Krg, Ag, Pw, and sui are the moderate important parameters with the sensitivity factorsbetween 10- and 10The initial temperature Ti is the most important parameter. This can be explained as follows. Hy-drates are stable existence only if the initial pressure is equal to or above the phase equilibrium preg-sure.According to the equilibrium equation, the small increase of initial temperature will cause relatively large increase of the equilibrium pressure with this initial temperature, which reduces pressuredifference between the initial and equilibrium pressure. Under this condition, with the same produc-tion pressure, hydrate dissociates in a wide region in a short time. On the contrary, the small reduction of the initial temperature will cause the increase of the中国煤化工 the initial andequilibrium pressures, which results in a namow dissociateCNMH Oressure directlyaffects the driving force Peg-p of hydrate dissociation, thus playing an important role on the dissocia-BAI Yu-hu d al. /China Oean Engineering, 21(4), 2007, 625-636635tion. The effect of porosity on the dissociation can be assessed in two aspects. On the one hand, theporosity is related with the specific surface area, thus affecting the dissociation rate directly; on theother hand, according to the Kozeny-Carman equation, we have k-p, thus as f changes, the permeability must be adjusted for consistency. The variation of the pemeability affects the flow of fluidsand then the pressure distribution, thus affecting the driving force of hydrate dissociation. The effect ofinitial pressure is analogous to that of the initial temperature. The initial hydrate saturation shi determines the lotal hydrate reserves, hence directly affecting the dissociation. Similarly, the hydrate density Ph included in the mass balance equation of hydrate cathe gas saturation distribution. Thelatent heat of hydrate dissociation is crucial for heat amount adsorbed from the surrounding environ-ment.With high latent heat, the unit mass of hydrate dissociation requires more adsorbed energy andthis will cause more temperature reduction, resulting in lower equilibrium pressure. Consequently,thedriving force of hydrate dissociation reduces, leading to a slow hydrate dissociation rate. In the presentsystem,we find that the dissociation constant is a moderate important parameter. However, we shouldpay more attention to it because we set the dissociation constant kd as small as 4.4 x 10-16 kg/(m.Pas)though Kim et al.(1987)considered it as about 10"for pure hydration. We can presumablydeduce that its sensitivity maybe increase with the dissociation constant. The relative permeability kras and gas viscosity p,the hydrate dissociation by their influence on thand thus on the pressure distribution in the hydrate-bearing formation. Now, from the above analysiswe can clarify the goverming parameters and the sensitivity of all physical parameters6. ConclusionsA one-dimensional model for gas hydrate dissociation by depressurization is developed, in whichthe effects of flow of multiphase fluids, the endothermic process of hydrate dissociation, the variation ofpermeability, and the convection and conduction are considered. The evolutions of the pressure,satu-rations of water, gas and hydrate and temperature distributions are plotted and described. We find outthat hydrate dissociation by depressurization can be divided into three stages: a rapid dissociation stagemainly govemed by hydrate dissociation kinetics after an initially slow dissociation stage governed main-ly by nows, and then followed by a slow dissociation stageFurthermore, a numerical approach of sensitivity analysis is suggested to quantify the importancedegree of all the physical parameters. Results show that the dissociation is more sensitive to the initialtemperature and pressure, the phase equilibrium pressure, the porosity, the initial hydrate saturationand hydrate density, etc. These results can be used as an initial support for laboratory experiments andhelp to the deep mathematical study of gas hydrate productionAcknowledgements-The authors wish to thank Prof. LI Jia-chun, academician of Chinese Academy of Sciences, for his中国煤化工ReferencesCNMHGAmyx,H. C, Bass, D. M. and Whiting, R. L, 1960. Petrodewn reservoi engineering physical properties, McGraw-636BAI YI-hu e al./ China Ocean Engineering, 21(4), 2007, 625-636Hill Book Co.. New YorkAhmadi, G, Ji, C. and Smith, D. H, 2004. Numerical solution for natural gas production from methane hydrate dissociation, Journal of Petroleum Science and engineering, 41, 169-385Burshears, M., O Brien, T. J. and Malone, R. D,, 1986. 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