煤层气在非饱和水流阶段的非定常渗流摄动解.pdf

第19卷 第4期应用力学学报Vol. 19 No. 4 2002年12月CHINESE JOURNAL OF APPLIED MECHANICSDec. 2002 文章编号10002493920020420101204 煤层气在非饱和水流阶段的非定常渗流摄动解 Ξ 孙可明 梁 冰 薛 强 辽宁工程技术大学 阜新 123000 摘要煤层甲烷由煤层的割理裂隙系统流入生产井一般经历单相水流、 非饱和流和气、 水两相饱和 流三个阶段,在非饱和流阶段,储层压力降至临界解吸压力之后,储存在煤基质中的吸附气体少量 被解吸出来形成互不连续的气泡并阻止水的流动,含气量尚未达到饱和程度。同时煤层甲烷运移 包含渗流场、 变形场和应力场的动态耦合过程。本文考虑渗流过程中水-气两相不溶混流体与固 体耦合作用,建立了非饱和水流阶段非定常渗流问题的流固耦合数学模型,对该强非线性一维数学 模型采用摄动法和积分变换法进行解析求解,并讨论了其压力动态特征,分析了压力随饱和度S及 时间t变化的规律和气相及耦合作用的影响,这些研究对煤层气、 石油和天然气的开采等地下工程 领域具有一定的指导意义。 关键词非饱和水流阶段;流固耦合;非定常渗流;摄动解 中图分类号O344. 6 ;TD313 文献标识码 A 1 引 言 煤层气不仅是宝贵的矿藏资源,而且它的开采 可从根本上消除瓦斯爆炸、 突出等灾害和减少对大 气的污染。煤层气一般与水共存,它由煤层的割理 裂隙系统流入生产井一般经历单相水流、 非饱和 流和气、 水两相饱和流三个阶段。在非饱和流阶段, 储层压力达到临界解吸压力之后,储存在煤基质中 的吸附气体少量被解吸出来形成互不连续的气泡并 阻止水的流动[1～3],使水的相对渗透率下降,水中 的含气量尚未达到饱和程度。煤层是固相、 气相和 液相并存的可变形多孔介质,煤层甲烷非饱和流阶 段的流动过程实质是流固耦合相互作用和水气两相 互不容混流体相互驱替的过程。对非饱和流阶段流 动问题的研究,在以往的分析和计算中一般没有考 虑流固耦合作用,特别是气相的作用,对其强非线性 数学模型的解析分析也未见报道。因此,本文考虑 水-气两相不溶混流体与固体耦合作用,建立了非 饱和流阶段渗流问题的流固耦合数学模型,对该强 非线性数学模型采用摄动法及积分变换法进行解析 求解,并对压力动态特征进行分析,分析了压力随饱 和度S及时间t变化的情况和气相以及耦合作用的 影响。这些研究对煤层气、 石油和天然气的开采等 地下工程领域具有一定的指导意义。 2 非饱和流阶段流固耦合数学模 型的建立 211 基本假设 1流体渗流为等温渗流且符合达西渗流定律; 2流体微可压缩,固体骨架和孔隙可压缩;3多孔 介质为均匀各向同性;4忽略毛管压力和重力影 响;5固体颗粒的密度为常数,气体对液体的相对 速度为零,即Vw Vg。 212 煤岩体变形场控制方程 将煤岩视为弹性介质,根据弹性多孔介质平衡 方程、 几何方程和物理方程得到煤岩体变形场控制 Ξ 基金项目高等学校骨干教师资助计划 来稿日期2001208227 修回日期2002202229 第一作者简介孙可明,男,1968年生,辽宁工程技术大学力学系讲师,博士生;研究方向地下流体力学 1 1995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved. 方程 G 2 ui G λ 5e 5xi α 5p 5xi 1 式中α为Biot系数,λ为拉梅常数。 将煤岩骨架 的体积应变e u代入1并对其两边取散度 div得变形场控制方程的另一形式 λ2 G 2 e α 2 p2 213 煤层甲烷和水两相流渗流场控制方程 设Vα是流体α相的真实速度, Vs是煤岩固体质 点的真实速度, Vrα是流体α相对煤岩固体基质质点 的流动速度,气、 水两相渗流达西定律为 φSα V α-Vs - Kkra μa p3 式中K为绝对渗透率,md; Kra为流体α相的相对渗 透率;μα为流体α相的粘度,Pa. s; p为流体的压力, Pa;φ为孔隙度;下标α分别代表气 g和水 w ; Sα 为α相饱和度;下标s代表固相,质量守衡方程为 液相质量守衡方程 φ ρwSVw 5φ ρwSw 5t 04 气相质量守衡方程 φ ρg1-S Vg 5φ ρg1- S w 5t 05 固相质量守衡方程 φ ρs1-φ V s 5φ ρs1-φ 5t 06 由于流体密度仅是压力p的函数,对于微可压 流体,定义压缩系数为 β α 1 ρa dp dρa ,gradραραβαgradp , 5ρ α 5t ραβα 5p 5t 7 对式3～7联立推导并忽略二阶项可得水流的 运动方程 Kkrw μw 2 p divVsφ β5 p 5t 8 其中,ββw 1-Sw / p , krw sw- swir 1- swir n 9 swir为残余水饱和度; n为时间指数 由于divVs 5e 5t 10 所以式8变为 Kkrw μw 2 p 5e 5t φ β5 p 5t 11 对式2两边积分后得 λ2G e αp f x , y , z , t12 当应力边界条件和孔隙压力边界值不随时间变化时 f x , y , z , t为常数 将式12代入式11得 Kkrw μw 2 p θφ β 5p 5t 13 其中θα / λ2 G 3 一维数学模型的解析求解 311 定解条件 考虑如下非饱和流阶段流体流动问题渗流场的 初始条件和边界条件。 ①p x ,0 p014 ②p h ,0 0,5 p 5x |x 0015 312 数学模型的解析求解 基本方程13和14 - 15构成的数学模型 属于二阶非线性偏微分方程组,本文采用摄动法进 行求解,对式13作如下变换, α ′ θφ β, β ′ φ1-Sw , 则式13变为 Kkrw μw 2 p α ′ β ′ p 5p 5t 16 其中β ′/ p项表示气相作用。 设εφ1- Sw , 因此 εφ1-Sw是小参数,所以方程16转化为 Kkrw μw 2 p α ′ ε p 5p 5t 17 由此可得由式17、14和15组成正则摄动问 题。 当小参数ε0时可以退化为一维抛物线方程 的初 2边值问题可以唯一求解[4],对p可作如下展 开式 p p1εp2ε 2 p3⋯18 将式18代入式17和式14、15 , 合并ε的同 次幂项可得 ε 0 Kkrw μw 2 p1 α ′ 5p1 5t p1 x , 0 p0, p1 h , t 0,5 p1 5x |x 00 19 ε 1 Kkrw μw 2 p2 α ′ 5p2 5t 1 p1 5p1 5t p2 x , 0 0, p2 h , t 0,5 p2 5x |x 00 20 ε 2 Kkrw μw 2 p3 α ′ 5p3 5t 1 p1 5p2 5t - p2 p21 5p1 5t p3 x , 0 0, p3 h , t 0,5 p3 5x |x 00 21 201应用力学学报第19卷 1995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved. 对于20式采用分离变量法进行求解可得如下解答 p1x ,t p06 ∞ m 1 - 1 m- 1 2 βmh cosβmx exp - Kkrwβ 2 m μw α ′ t 22 其中βm 2m - 1 2h π,对于式21、22均为非齐次 定解问题,它们的定解条件相同,只是非齐次项有差 别,其类型完全相同,可以统一写为 2 pi μw Kkrw α ′ 5pi 5t μw Kkrw f x , t pi x , 0 0, pi h , t 0, 5 pi 5x |x 00 23 对于式23采用积分变换法进行求解,经推导简 化推导过程可得式23的一阶渐近解 p x , t p06 ∞ m 1 - 1 m -1 2 βmhcos βmx exp - Kkrwβ 2 m μw α ′ t Kkrw μw 2 2φ1-Sw α ′ 2 6 ∞ m 1cos βmx exp - Kkrmβ 2 m μw α ′ t ∫ t 0∫ h 0cos βmx 6 ∞ p 1 - 1 p-1 - βpcosβpxexp - Kkrwβ 2 p μw α ′ t 6 ∞ q 1 - 1 q-11 βq cosβqxexp - Kkrwβ 2 q μw α ′ t exp Kkrwβ 2 m μw α ′ t dxdt24 对24解析式,用matlab计算, m取20时即可收敛。 4 算例及分析 考虑一个1m厚水平方向无限长含气煤层,弹 性模量5MPa ,泊松比0. 25 ,孔隙率5 ,流体的压缩 系数10- 3,Biot系数α0.26,水的饱和度变化值 100 -5 ,残余水饱和度为0,时间指数为3,水的 黏度为1.3110- 3Pa. s。初始条件 p x ,0 3Mpa,边界条件 p 0, t 0,计算结果如图1～4 所示。 图1表明耦合与非耦合的孔隙压力变化差别 较明显,考虑耦合时孔隙压力消散的程度较慢。 图3 和图4表明,不同水饱和度S在流动过程中对压力 曲线有影响,不管耦合和非耦合情况,随着水饱和度 的降低,即含气量增加,孔隙压力下降得越慢,当储 图1 s 0.95不同时间耦合与非耦合孔隙压力的水平变化 图2 t 8饱和度不同时压力随水平距离变化曲线 图3 s 1时间不同无气相时非耦合条件隙压力水平距离变化 图4 s 0.85时间不同非耦合条件孙隙压力随水平距离变化曲线 301第4期 孙可明,等煤层气在非饱和水流阶段的非定常渗流摄动解 1995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved. 层中仅有水相无气相时 s 1和含有少量气相如 s 0.85压力曲线变化相比,前者的孔隙压力较后 者孔隙压力下降得快,其原因是气体解吸出来而形 成互不连续的气泡存于水中,增加了水的流动阻力 从而降低了水的渗透率;这说明气相的存在对水运 移的阻滞作用。 图2和其它图比较表明时间不同时 的孔隙压力变化趋势较饱和度不同时孔隙压力变化 趋势快。 即渗流的非线性程度越高,孔隙压力消散的 程度越慢。 5 结 论 1 考虑流体的压缩性和固体变形及气相作用的 前提条件下,建立了较为真实地描述瓦斯与煤层的 耦合作用非饱和流阶段非稳态渗流问题数学模型, 并采用了摄动法对该强非线性数学模型进行求解。 摄动法是获得强非线性数学模型解析解非常有效的 方法。 2 耦合与非耦合的孔隙压力变化差别较明显,考 虑耦合时孔隙压力消散的程度较慢。 不管耦合和非 耦合情况,随着水饱和度的降低,即含气量增加,孔 隙压力下降得越慢,表明解吸出来而形成互不连续 的气泡存于水中,气体被压缩,孔隙压力增大,增加 了水的流动阻力从而降低了水的渗透率;这说明气 相的存在对水运移的阻滞作用。 3 时间不同时的孔隙压力变化趋势较饱和度不 同时孔隙压力变化趋势快。 即渗流的非线性程度越 高,孔隙压力消散的程度越慢。 参考文献 1 Massmann J W. , Applying groundwater flow models in vapor ex2 traction system design [J ] , Journal of Environmental Engineering. 1989. 1151 129～149 2 Peek A J. Moisture profile development and air compression during water uptake by bounded porous bodies [J ] ,Soil Sci ,1965. 100 333 ～340 3 钱凯,赵庆波等,煤层甲烷勘探开发理论与实验测试技术[M] ,北 京石油工业出版社,1996 66～70 4 林宗池,周明儒,应用数学中的摄动方法[ M] ,南京江苏教育出 版社,1995 50～55 401应用力学学报第19卷 1995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved. and dimension precision of the final parts. Based on rational assumptions , appropriate material model and engi2 neering analysis , the equations for governing the packing process are obtained using the basic equations of Non2 Newton , compressible , viscous fluid. A hybrid finite2element/finite2difference was used to model this processing , and a practical example are used to illustrate how to resolve some engineering problems , such as packing pressure , gate frozen time and clamp force. Keywords Injection2molding , packing stage , numerical simulation. The Analysis of Dynamic Interaction on Bulk Solids And Silo Structure and Its Foundation Yin Guansheng Huang Yi Xi ’an Architecture Science and Technical University , Science college , Xi’an 710 054 Abstract Considering the interaction of bulk solids and silo structure and its foundation , a systematic dynamic analysis of many kinds of cases to the bulk solid’s different calculating model of silo structure , which is put on the elastic foundation , was carried out in the paper. The conclusion is made after a comparing with the testing results made by authors To the horizontal impulse of foundation , the dynamic response of silo on elastic founda2 tion is stronger than that of silo on rigid foundation ; the relative motion between the silo wall and bulk solids has the effect of lightening the vibration of silo structure. Keywords Silo, Interaction , Rigid foundation , Elastic foundation , Bulk solid , Calculating model. Dynamic Character Analysis Including Material Macrostructure Effect Cao Zhiyuan Fu Zhiping Dept. of Engineering Mechanics and Technology , Key Laboratory of Solid Mechanics of MOE , Tongji University , Shanghai 200092 Abstract The microelement , which establishes the immediate relation between the microstructure of material and dynamic character of structural member in composite material , is discussed. The , firstly , divides the composite structural member into a set of macroelements , then analyzes the microstructure , and translates the variablesof the microstructure into variablesof nodesfor calculation. In this , any arbitrary pattern of microstructure and any tiny change will be reflected in macrocharacter without increasing the degrees of freedom. Keywords Microelement , composite material , dynamic character. The Perturbation Solution of Unsteady Seepage in Coalbed Methane Unsaturated Water Flow Phase Sun Keming Liang Bing Xue Qiang Dept. of mechanics , Liaoning Technical University , Fuxin 123000 Abstract Coalbeds are characterized by their dual porosity , which contains both matrix and cleats porosity sys2 tems. The matrix porosity system contains the vast majority of the gas volume , while the cleats porosity system ⅨNo. 4 CHINESE JOURNAL OF APPLIED MECHANICS 1995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved. provides the conduit for mass transfer to the wells. Coalbeds methane flow through cleats to production wells in2 cludes three phases single2phase water flow and both unsaturated water flow and saturated water flow. In the unsaturated water flow phase , when the cleats system pressure drops , gas molecules desorb from the matrix porosity interface and are released into cleats porosity system. As a result , desorbed gas s discontinue gas bubbles , which hold back water flow. At the same time , the coalbeds methane transport includes the coupling process of seepage field , deation field and stress field. In the paper the fluid2solid interaction and desorbed gas affect water transport are analyzed. A coupled mathematical model of unsteady seepage in coal methane un2 saturated flow phase is established based on the theory of fluid2solid interaction.The nonlinear mathematical model is investigated by using the perturbation and integral trans. Pore pressure along level distance at dif2 ferent saturation and time , and gas phase effect on seepage flow under coupling and without coupling conditions are discussed. Some valuable conclusions were got. Keywords Unsaturated water f low phase , f luid2solid interaction , unsteady seepage. A Research of of Dynamic Stiffness Test and Data Processing for Single Pile Zhang Gaihui Hu Shiyue Dept. of Eng. Mechanics , Xi’an Jiaotong University ,710049 Abstract In this paper , a synthetic test is presented based on the dynamic characteristic test criterion of pile. This combines sine excitation with shock excitation. Using this , the dynamic characteris2 tics of test piles in thermal power plant are tested. The model analytical technique is used to identify the model parameters of single pile. The vertical dynamic stiffness of the pile can be computed by these model parameters. Keywords Pile , dynamic stif f ness , synthetic test , data processing. Analysis of Laminated Shell Structure Using Piece2wise Ritz Based on Parametric Surface Description Xu Yanhai Jia Liping Zhang Jianw u Shanghai Jiaotong University , Shanghai , China 200030 Abstract A piece2wise Ritz to obtain a precise and valid solution used to analyze laminated shell struc2 ture is developed. In this , the laminated shell structure is divided into patches and represented by para2 metric surfaces that are connected in boundary according to C0 continuity. The trial functions of Ritz procedure with parametric surface characteristics are constructed to get functional and governing equations. Finally , an il2 lustrative example is given and the results show the correction and validity of present piece2wise Ritz . Keywords L aminated shell , parametric surf ace , piece2wise Ritz . Ⅹ CHINESE JOURNAL OF APPLIED MECHANICS Vol. 19 1995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.