Similarity theory for the physical simulation of natural gas hydrate reservoir development

Availableonlineatwww.sciencedirect.comMININGScienceDirectSCIENCE ANDTECHNOLOGYELSEVIERMining Science and Technology 20(2010)0782-0788sevier. com/locate/jcumtSimilarity theory for the physical simulation ofnatural gas hydrate reservoir developmentLIU Yaping.2., CHEN Yueming, BAI Yuhu,LI ShuxiaSchool of Petroleum Engineering, China University of Petroleum, Dongying 257006, Chine2Globe Research Center, Exploration and Production Research Institute of SINOPEC, Beijing 100083, ChinaTechnology Research Department, CNOOC Research Center, Beijing 100027, ChinaAbstract: In order to apply physical simulation results to natural gas hydrate reservoir parameters to provide a theoretical frame-work for the design of a development plan, an analytical equation method was used to obtain the similarity criteria of natural gashydrate reservoir development by physical simulation, based on a mathematical model of natural gas hydrate development. Giventhe approach of numerical simulation, a sensitivity analysis for all parameters was carried out, which specifically demonstrated thatnitial temperature is the most important parameter. Parameters of thermal conductivity coefficients are not necessary for conduct-ing the NGH dissociation process, which will fundamentally simplify the design and establishment of the model. The analysisvides a sound theoretical basis and design principles for particular similarityKeywords: mathematical model; equation analysis; similarity criteria; nature gas hydrate; sensitivity analysis1 Introductionexperiments involving the same parameters withidentical measurements may still provide very differ-At present, Natural Gas Hydrate (NGH), a high- ent results. Developing such a physical simulationquality, clean and new solid energy source, repre- device does not simply require scaling down thesenting high-density enrichment, has a wide applica- NGH field conditions. It requires an understandingion and development potential to help meet and control of a large number of multi-scale physicallong-term energy demands. Based on a scientific and chemical phenomena, spanning in scale fromestimate the methane resources in NGH account for decimeters to hundreds of meters. Consequently, italmost twice the total methane resources in all fossil becomes necessary to apply the principles of similfuels(coal, oil and natural gas)worldwide. Along ity theory as a guide in physical simulation experiwith in-depth theoretical studies, laboratory research ments. Based on similarity theory, a similarity physi-and field practices for NGH, commercial exploitation cal model of NGH was constructed by scaling downof NGh has gradually become of primary interest". the prototype parameters. Subsequently, the physicalThe hydrate dissociation process involves many processes of the prototype would be reproduced accomplex mechanisms, which cannot be explained cording to specific similarity relations where, throughsatisfactorily by means of existing theories, field short-term, small-scale simulation experiments, thepractices or physical simulation(5-6) Compared with seepage process of hydrate reservoirs would be ob-field practices, physical simulation has many advan-served rapidly and the required parameters deter-tages, such as low-cost, short time prerequisites and minedeasy operation. A NGH physical simulation devicetypically integrates many fluid components and con- 2 Basic principles of the similarity theorytains functions simulating actual NGh reservoir con-ditions. Although a large number of hydrate dissocia-Similarity theory is a theory expressing similar re-tion experiments have been carried out, a unified un-tionships between the same phenomena, which canderstanding of NGH has not been formulated so far be described by the same formal equations, requiringand without the guidance of similarity theory, varioussImI中国煤化工 kind of physicalReceived 11 December 2009: accepted 22 February 2010*Corresponding author. Tel: 86 10 823381 14CNMHGonstructs of simiwcorem. the similardoi:10.1016/S16745264(09)60281-7and its inverseLIU Yaping et alSimilarity theory for the physical simulation ofSimilarity I theoremover the pressure-temperature equilibrium condition兀1=∫(丌2,丌3y…,mn)the hydrate in the neiwell dissociatest is a set of dimensionless groups. In this set, theThe NGH development process is a very complexvarious groups are independent from each other. Each process of seepage, which involves many intricatedimensionless group can be expressed by the multi- factors, such as the hydrate phase change(solid-liq-plication of the power of the other group. According uid-gas), hydrate dissociation dynamics, the changeto eq (1), all the relevant physical parameters of the of parameters, such as porosity, permeability andtwo systems can be exchanged by a constant ratio. As saturation, the flow dynamics between water and gaswell, if the physical process involves m basic dimen- and heat transfer and latent heat of phase change. Insions, the physical process can be expressed by n-m the present study, the following assumptions aredimensional groups, made up of n variablesmade: i)the aqueous phase contains only water,The similarity theorem: for all similar phenomena, methane gas dissolved in water is not considered; ii)at a corresponding moment, the same numerical crite- the synthesis and decomposition of NGH is unrelatedria have the same numerical value on the corre- to the salt concentration in the reservoir; iii)thesponding points.seepage of water and gas is consistent with DarcysThe inverse similarity theorem: for the same kind law and iv) the quality of transmission does not conof phenomena, if the single value conditions are same sider molecular and hydrodynamic diffusion.and under the condition of a single value, the corre- 3.2 Model formulationsponding criteria of stereotypes are the same as well,then these phenomena are called similarity phenom- 3.2.1 Infiltration equationprocess of dimensionless mathematical equations, or tinuity equations for different species are given by 9 aenaThe governing equations used to solve for thThe derivation process of the similarity criteria for multi-phase flow conditions during the hydrate dissephysical phenomenon is similar to the derivation ciation process are outlined in this section. The cothe derivation process of dimensional analysis forvarious variables, based on descriptive phenomena P. PK sV(Ps+Pg82)+m +e a(pg5g)(2)from similarity theory. There are two methods to de-gduce similarity criteria: equation analysis and dimen-sions of lier Atia ing month which desnrbedia v(=v(p, +p. 2)physical process, similarity criteria obtained bymeans of equation analysis have the advantage of aclear physical interpretation. Therefore, equationa(5atanalysis would be the preferred method to deduce thesimilarity criteria if the physical process can be iden- where m is the rate of generation/dissociation pertified by definite mathematical equations, otherwise unit area, p the porosity of the core, s saturation,pthe dimensional analysis constitutes the only avail- pressure and k permeability. p is the phase density, Qable method. In our study, equation analysis is used the density rate of output, u phase viscosity and g theto deduce the similarity criteria of NGH reservoir gravitational acceleration constant. The subscripts h,development in physical simulation, since the w and g stand for hydrate, water and gas phases, remathematical description of the NGH development spectively. The density rate of methane gas production per unit length Qg, and the density rate of wateroduction per unit length @w can be given as:3 Similarity criteria in NGH physical si-丌nP2k,(pp-p)mulationQ,“2m()-4(x-x)0-y)(53.1 Hydrate dissociation modelQ=29.6x-x1)6(-y)We assumed a methane hydrate reservoir withressure P and temperature T, containing stable solid丌nDk(pv-p)hydrate and free methane gas. When wells are drilledD(x-x)6(-y)(6)into the reservoir, the pressure in the drilled area2uw In(o/ro)drops to the pressure Pk of the production wells, orwhere qr is the volume of the water injected into athe temperature in the drilled area decreases to that of layer中国煤化工time: Rell thethe injection wells, i. e, Tw. Given sufficiently low ductipressure Ps below the pressure-temperature equilibwellCN MH Gtion well and pro-rium condition, or sufficiently high temperature Tw duction well; ro the well radius; pi pi; x the coorMining Science and Technology20No.5dinates in the x-direction; and y the coordinates in the=0,l=P。-P4y-direction. Subscripts 8, W, P, and I indicate gas,(15)water, production well, injection well, respectively.3. 2.2 Energy balance equationsdp0,l=g,w,i=1,2,3The heat energy conservation equation for an ef-fective medium may be written as:(closed boundary conditions)where Pep and Tgp are bottomre and tempera-CT)=V[P, C-V(P,+P,)ture of production well, respectively3.2.5 Boundary conditionsThe boundary conditions are given+P.W. v(P: +p, 82)71Pleo=Pi, s-o=Shi,+V(K,VT)-r,AHST+CT oy d(x-x,8(y-y))(7) where Pi, Shi, Swis and Ti are initial pressure, hydratein whichsaturation, water saturation and temperature, respecC,=pP, Cw+P, Cv +P,,C)tively.3.2.6 Supplemental formula+PC,(1-p)(8)Because of the special nature of hydrates, a supplemental formula needs to be included:K,=os,Kw+p ke +os,Kh+(1-p)K,(9) 1)Hydrate decomposition kineticsAs the hydrate dissociates into gas and water, awhere T is the temperature, Cvg and Cpg are the con- dissociation front forms that divides the core into twostant volume and pressure of specific heat,C is the regions: one containing solid hydrate and the otherspecific heat, K the coefficient of thermal conductiv- dissociated gas and water. The local mass rate of gasity, AH the enthalpy change in hydrate dissociationand K, the effective thermal conductivity. Subscripts rm, generated by hydrate dissociation may be con-nd h indicate rock and hydrate, respectively.trolled by the Kim-Bishnoi model3.2.3 Auxiliary equationm,=kgA, (ea-f)(16)For saturations of various phases, the following where A, is the specific surface area of the porousuality holds:(10) librium fugacity. Fugacity differences can be replacedwhere sp, Sw and sh are saturation of gas, water, and by pressure differences. ka is the dissociation constanthydrate phases, respectively. The effective porosity of The quantities of r, i, and r, in the equa-the medium pe is giventions below denote the corresponding local mass ratesp=(-)(D) of hydrate, gas and water produced per unit volume,Water pressure pw, gas pressures Pg and capillary respectively. The relationship betweenpressure Pe are related according to the capillary forceand m can be shown as followsPe=Pw-PsAssuming that the reservoir rock and fluid aremw+me =mhslightly compressible, water and its state equationsN,M +Mare respectively given byri N,MP=Po[l+c(P,-Pwo)where Mw, M, is the molecular weights of water and中=41+cP,+P, Pwo + Pgo(14)2where Cw and ce are the compressibility of water and cules in the hydrate. Then mm can be identified byporous media, respectively and pwo and Pro the refer- the following formulaence pressures of water and gaNM+M3.2. 4 Initial conditionsA(DThe initial conditions are given as中国煤化工MP,=Pp, T=TRsurreyCNMHGequilibrium pres-P, are used toLIU Yaping et alSimilarity theory for the physical simulation of785approximate fea and f. The surface area of hy- the effective water saturation s" is defined as fol-drate particles is expressed as a function of the num- lows:ber of methane moles in the hydrate(N,). This func-/(1-Sh)tion satisfies several conditions such as the invariantcomposition of the hydrate the constant number of Similarly, the effective gas saturation sg can bemethane moles per unit volume of hydrate, a uniform expressed as:decomposition rate, a constant number of particlesduring decomposition, etc./(1-Sh)(23)For methane hydrates, if N,=6, M =18Power Model was used to estimate the local abso-M=16, thermn=7.75 a A, (Pea-P)k=Q-4)(24)((-)m,=6.75kA,(Pq-P)where k is the local absolute permeability, Po the2)Equilibrium formulawhole porosity, ko the reference absolute permt the dissociation front are functions of time. ability, relative to the point of porosity po,9. theThe phase equilibrium relation between equilibrium effective porosity which takes into account the hy-temperature T( K)and equilibrium pressure Pdrate saturation and ice saturation and B is the in-(Pa)at the dissociation front is given as%1dex which determines absolute permeability changesalong with the changes in porosityIg Po=a(T-To)+b(T-To)(19)The state equation of gas iswhere t. is 273. 15 K and a, b and c are em-mpirical constants, which depend on the hydrate com-position obtained from the equilibrium pressurewhere p is the absolute pressure, v the gas voltemperature data of methane hydrate3)Endothermic process during NGh phase transi-ume, m the mass of gas, M is the molecularM=0016The process of hydrate dissociation at the dissociakg/mol; z is the gas compressibility factor, R thetion front is an endothermic phase-change process. universal gas constant(J/(mol/K))and T the abso-The latent heat per kilogram of hydrate in J/kg is lute temperature. Therefore,AH=AT+BPkZRTwhere T is the dissociation temperature and A and Bare constants given asd,M,pM,1、dzA=-1050J/kg,B=3527000J/(kgK)dp ZRT'Rr丶z24)Special treatment of physical NGH parametersDuring the process of hydrate decomposition, theRT Zspace volume shared by the gas phase is constantlychanging, i.e, the mobile space of the fluid is con- Then the natural gas compressibility c, can bestantly changing. Therefore the parameter A, alsoexpressed aschanges with time in the Hydrate Decomposition Ki-netic equation as follows"-3.Cr14p1(26)A,=(p/2k)3(21)dpThis equation shows the changes of A, during NGH development has been established. No analyti-At this point, a complete mathematic model ofhydrate dissociation process. o(t) means moment t cal solution is available for this governing equationduring which the total volume is occupied by water Therefore, a numerical solution needs to be consid-and gas per unit pore volume, i. e, its effective mobile ered. We used the difference method of IMPES to中国煤化工 s above and com-In the relative permeability model, water saturation pilel simulation softand gas saturation need to be re-defined, using the wareCNMHGaturation of the effective mobile space. At this pointMining Science and TechnologVoL 20 No53.3 Similarity criteria of physical simulationgcos eUsing the analytical equation method, the four ba-L'Psk Cv Pvariables, i. e, length, quality, time and temperature were restated in non-dimensional form. These.0q丌y= HWLk, Ps V9Proglfour non-dimensional variables are presented as folHingthe number of similarity criteria for physical simula-to SnOoP ShiPP, y2kmation in a NGH reservoir development is 43. Accord-ing to the Buckingham-theorem, after removing thewhere subscript d denotes the dimensionless vari- four basic variables, i.e., length, quality, time andables xp, yD and zp as dimensionless coordi-temperature, the number of independent parameters isnates and x, y and z are the coordinates; tp is 39, which is as same as the number of similarity cri-teria derived earlier. No matter which method wedimensionless time; L is the characteristic length, used, the number of similarity criteria is the same, butw the characteristic width and H the characteristic different in the formnessPo is the maximal porosity and kthe maximal permeability, when the hydrate satura. 4 Criteria sensitivity analysis for nGh detion is zero; Pen is the production pressure differpressurization developmentence;to is the characteristic time, i.e., the time re- Given the limitations imposed by the condition ofquired for the complete decomposition of a unit vol- the laboratory, it is difficult to achieve an experimen-ume of NGH, given the condition of a bottom-hole tal model which is similar to the prototype. With thepressure-drivennumerical approach of a sensitivity analysis to quanGiven the similarity theory and our stated mathe- tify the effect of physical parameters on hydrate dis-matical model, the similarity criteria group for NGH sociation, according to different physical simulationphysical simulation for reservoir development, purposes, the dominant parameters have been deter兀1~兀39, has been derived as follows:mined and their degree of relative importance evaluated therefore, secondary parameters can be relaxedLBased on the numerical approach to sensitivity analysis, the sensitivity factor S, can be defined asH△F/FSPPF,=SRO.V2KKKKAF=1R,V,…V,…V,n)CAHPR.(. VCC TV,…,V,n)drPwhere F is a target function concerned in the depres-兀n=Pn,surization system and is the function of all the parameters. R(Vi, V2,., Vi,..., Vn, n) is the dissociation72sPxpc, 26= PgCo, n=Sn, 28=rg,ratio and T is the dissociation time span. The subscript p denotes the dissociation system and the subscript d means deviation from the parameters studied./ u中国煤化工 of the sensitivityfaof the target func-tIonCnmhgParameteRs.com-u,k, Vopared with the values of thequantitative effect of the physical parameters on theeory for the physical simulation ofhydrate dissociation can be determined. Thus, the coefficient w of each parameter to be 1% of its trueimportant factors could be conveniently singled out in coefficient and keep the others fixed, the sensitivitythe system devolved by us. If we let the deviation can be derived as shown in tableTable 1 Sensitivity factors of all parametersariable Sensitivity Variable Sensitivity Variable Sensitivity Variable Sensitivity Variable Sensitivity Variable Sensitivity393×01△H735×10cTcc353×103|K,15×10KnP877×103gP,3.03x15.65x108.71x105.10x10Table 1 shows that the values of the sensitivity can affect the gas saturation distribution. Given thisIn our system, T is the most important parameter the sensitivity of all physical parameter ameters andfactors cover a wide domain, ranging from 10 to 10. analysis, we can clarify the governing parawith a sensitivity factor over 10, while Peg,Po, cOnclusionsPi, P, and Sm are relatively important parame-Using similarity criteria to guide our experimentaParameters such as the coefficients of thermal con- design and model operation, we studied hydrate res-ervoir fluid flows within various scaled ratios andductivity, Kw, K,, Kg, are not necessary for con- realized the theoretical convbetween physicalducting the dissociation process, which will funda- simulation results and NGH reservoiparametersmentally simplify the design and establishment of the which will provide strong technical support for themodeldesign and optimization of NGH reservoir develop-The initial temperature T is the most important ment projectsparameter. This can be explained as follows. HydrateWe suggest furthermore, a numerical approach ofis stable only if the initial pressure is equal to orsensitivity analysis in order to quantify the degree ofabove the phase equilibrium pressure. According to results show that dissociation is sensitive to the initialthe equilibrium equation, a small increase of initialthe initial and equilibrium pressures. Under this con- pressure, the porosity, the initial hydatequilibriumtemperature reduces the pressure difference between temperature and pressure, the phasedition, with the same production pressure, hydrateand hydrate densityresults providedissociates over a wide region in a short time. In con-theoretical basis andprinciples fortrast, a small reduction of the initial temperature willsimilaritycause an increase in the pressure difference betweenthe initial and equilibrium pressures, which results inAcknowledgementsa narrow dissociation region. The equilibrium pres-sure directly affects the driving force Pr -p ofThis work was financially supported by the chinaPetroleum and Chemical Corporation(No. P06070hydrate dissociation, thus playing an important role in and the National Natural Science Foundation ofdissociation. The effect of porosity on the dissocia- China (No. 50404003)sessedhand, porosity is related to a specific surface area,thus affecting the dissociation rate directly; on theReferencesother hand, according to the Kozeny-Carmantion, we have k-P, so that as o changes,[1] Zhang B Y, Wu Q Sun D L. Effect of surfactant Tweenon induction time of gas hydrate formation. Journal ofpermeability must be adjusted to be consistent. TheChina University of Mining Technology, 2008, 18(1)variation in permeability affects the flow of fluid and18-21therefore the pressure distribution, thus affecting the [2] Ray B. Resource potential of methane hydrate comingdriving force of hydrate dissociation. The effect ofinto focus. 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