ANALYTICAL SOLUTION OF LIMIT STORAGE PRESSURES FOR TUNNEL TYPE LINED GAS STORAGE CAVERNS
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摘要: 隧洞式内衬储气库是一种新型能源储存方法, 有助于平衡供需, 推动国家由化石能源向绿色能源的持续过渡, 有利于国家“碳中和、碳达峰”目标的实现. 本文采用极限平衡方法和弹塑性分析方法推导隧洞式内衬储气库极限储存压力的解析解. 在极限平衡方法中, 考虑上覆围岩自重、破裂面受力和极限储存压力, 选用刚性锥模型, 推导了上限压力表达式; 在弹塑性分析方法中, 根据围岩中应力分布规律和剪切、抗拉强度, 推导获得了弹塑性条件下上限与下限压力表达式. 最终综合考虑两方法求得的结果, 确定极限储存压力解析解. 结果表明: 极限平衡方法求得上限压力与埋深呈二次函数关系, 且随着侧压力系数的增大而增大; 弹塑性分析方法确定的上限压力和下限压力与埋深呈线性关系, 下限压力随着侧压力系数的增大而减小, 且Ⅰ级围岩条件下的内衬储气库不用考虑下限压力. 在侧压力系数λ = 1.2时上限压力最大, 因此应尽量在侧压力系数为1.2的围岩条件下修建隧洞式储气库. 最后根据典型工况下上限和下限压力曲线给出内衬洞室推荐压力范围.Abstract: Tunnel lined cavern gas storage is a new energy storage method, which helps balance supply and demand, promotes the continuous transition from fossil energy to green energy, and facilitates the realization of national goal of "carbon neutralization and carbon peak". In this paper, the ultimate equilibrium method and the elastoplastic analysis method are used to derive the analytical solution of the ultimate storage pressure of tunnel lined rock cavern gas storage. In the ultimate equilibrium method, the self-weight of the overlying surrounding rock, the force of the fracture surface and the ultimate storage pressure are considered, the rigid cone model is selected, and the upper limit pressure expression is derived. In the elastoplastic analysis method, according to the stress distribution law and shear and tensile strength in the surrounding rock, the upper and lower pressure expressions under elastoplastic conditions are derived. Finally, the analytical solution of the ultimate pressure is determined with considering the results obtained by the two methods. The results show that the relationship between the upper limit pressure and the buried depth is quadratic function, and increases with the increase of lateral pressure coefficient; The upper limit pressure and lower limit pressure determined by the elastoplastic analysis method are linear with the burial depth, and the lower limit pressure decreases with the increase of the lateral pressure coefficient, and the lower limit pressure is not considered for the lined gas storage under the condition of class I surrounding rock. When the lateral pressure coefficient is 1.2, the upper limit pressure is the largest, so the tunnel type gas storage should be built as far as possible under the surrounding rock condition with the lateral pressure coefficient of 1.2. Finally, the recommended pressure ranges of lined rock caverns are given according to the upper and lower limit pressure curves under typical working conditions.
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引 言
中国已将减碳列入国家重要发展目标, 中国的能源结构正逐渐向清洁低碳转型[1]. 地下储气库将与气电转换、储氢储氦、压气蓄能、大规模 CO2 埋存等新技术和应用深度融合发展. 已有多个国家应用内衬洞库储存氢气或天然气. 瑞典最早应用人工内衬岩洞来储存高压天然气[2]. 而后, 日本、韩国相继将该技术应用到压气储能岩洞中, 并尝试用高分子材料代替钢衬作为密封层[3-5]. 国内储库研究多为盐岩储库[6-9], 内衬岩洞的研究大多服务于压气储能项目[10-12]. 建设内衬储气库的目的是储存压缩气体, 因此储气库内可储存气体的上限压力和下限压力至关重要. 上限和下限压力不仅关系到储气库的安全稳定问题, 而且可以反映储气库对储存气体的利用程度, 这对评估建设内衬储气库的可行性和经济性有着重大意义.
内衬储气库设计的一个关键问题就是极限储存压力, 有些学者通过数值模拟和实验方法对其进行了评估[13-15], 但却很少给出确切解析解. 上限压力Pmax需要等于或小于上覆岩层压力以确保岩洞的稳定性[16], 目前常用的计算模型包括刚性锥模型, 对数螺旋模型和直线破坏模型[17-18]. Kim等[19]根据直线破坏模型提出了一种简单的分析方法, 针对内衬岩洞上覆岩层表面隆起对地下储库进行稳定性评估. 直线破坏模型忽略了破坏面的不确定性, 计算结果过于保守. 挪威准则为计算上限压力提供了一种方法[20], 上覆岩层重量是抵抗岩体抬升的唯一阻力, 气体的压力与上覆岩层的重力达到力学平衡. Vezole等[21]认为, 抗抬升力不仅包括上覆岩体的重量, 还包括由于上覆岩体抗拉强度而产生的抵抗力, 这两种抵抗力与气体压力力学平衡. 这两种假设都是基于刚性锥模型而提出的. Damasceno等[22]建立了有限元模型, 评估了两种计算方法的合理性, 结果表明这两种方法得到的结果明显小于数值模拟结果, 这是由于这两种方法未考虑上覆岩体破坏面上的剪切阻力. Kim等[23]考虑破裂面剪切阻力进行了极限平衡分析, 但是未给出确切计算公式, 且其选用的破坏模型为直线破坏模型.
除了使用极限平衡方法得到储气库上限压力, 还可以使用弹塑性分析得到储气库的上限压力和下限压力. 目前内衬储气库的弹塑性分析大多运用数值模拟进行, 王其宽等[24]使用数值模拟方法对内衬储气库进行弹塑性分析, 以位移和塑性区为指标研究了内径等因素对稳定性的影响规律. Chen等[25]使用数值模拟方法得到了内衬储气库位移和应力的变化规律, 给出了确定岩洞稳定性的埋深、压力等参数. 上述研究只是通过数值模拟得到位移、塑性区等因素的变化规律以确定岩洞埋深等参数, 并未提出解析解. 徐英俊等[26]基于极限分析上限定理推导得到了上限压力表达式, 未考虑储气库的下限压力. Zhang等[27]采用解析法计算了在给定盐岩储气库深度、尺寸、水平与垂直地质应力比和内部压力条件下, 椭球形岩洞顶部的环向和径向应力, 并结合剪切强度和抗拉强度得到椭球形储库的上限和下限压力解析解, 但是该解析解并不适用于隧洞式内衬洞室.
国内外学者对于储气库极限压力的研究更偏向于上限压力的研究, 对于下限压力的关注度不够. 相关论文列举了一些案例的最小运行压力[28], 但并未给出相应计算方法和确切的计算公式. Kim等[23]根据抗拉强度得到了椭球形储库的下限压力, 但是并不适用于隧洞式洞室. 目前缺少对内衬洞室下限压力的研究. 许多学者只是根据已有内衬储气库项目下限压力的数值进行后续研究[29-30], 而对如何得到下限压力未做详细说明.
本工作采用极限平衡方法和弹塑性分析方法推导隧洞式内衬岩洞极限储存压力的解析解, 结合岩洞埋深、岩洞尺寸、侧压力系数和Ⅰ ~ Ⅳ级围岩的物理力学参数, 分别得到极限平衡条件下的上限压力和弹塑性条件下的上限和下限压力随埋深的变化规律曲线. 对比极限平衡理论得到的上限压力和弹塑性分析方法得到的上限、下限压力解析解, 确定将由弹塑性分析方法确定的上限、下限压力解析解来确定极限储存压力, 以期确定隧洞式内衬岩洞可储存气体的压力范围.
1. 问题提出
常见的内衬岩洞形状为隧洞形, 隧洞式内衬岩洞的横断面是圆形, 半径为r, 如图1所示. 影响岩洞稳定性的关键参数包括岩洞的埋深、岩洞类型和规模(半径和高度)、岩洞内气体压力等. 建设隧洞式储气库之前, 岩洞所能承受的极限压力是需要提前考虑和确定的. 因此, 本文推导得到隧洞式储气库下限和上限压力的解析解. 首先选用刚性锥模型[31], 根据围岩的物理参数求得隧洞式内衬岩洞极限平衡条件下的上限压力解析解; 然后分析弹塑性条件下内衬洞室的上限和下限压力, 根据弹塑性理论得到围岩的应力场, 结合剪切和抗拉强度, 推导得到内衬岩洞弹塑性条件下上限和上限压力解析解.
2. 极限平衡条件下上限压力
计算隧洞式储气库上限压力时选用刚性锥模型, 如图2(a)所示, 上覆岩体CC'BB'上部长度为2L, 下部长度为2r, 高度为H, BC面和B'C'面为上覆围岩破裂面, 破裂面角度为
$ 45^\circ - \varphi /2 $ [32]. 上覆岩体受到自重W, BC面和B'C'面受到压力$ {F_{{\text{out}}}} $ 和剪切阻力T, CC`面为地面, 不受外力作用, BB'面受到气体压力P. 并做以下假设: 破裂面处剪切力满足Mohr-Coulomb强度准则; AB面受静止土压力和剪切力作用. 图2(b)为块体ABC的受力分析图, 块体受到自重W1; AC面为地平面, 不受力的作用; BC面受到压力$ {F_{{\text{out}}}} $ 和剪切阻力$ T $ ; AB面上受到压力F和剪切阻力$ T' $ . AB面上的静止土压力合力为![]()
图 2 极限平衡条件下上限压力分析模型Figure 2. Analysis model of upper limit pressure under ultimate equilibrium$$ F = \int_0^H {\lambda \gamma z{\rm{d}}z = } \frac{1}{2}\lambda \gamma {H^2} $$ (1) 式中,
$ \lambda $ 为侧压力系数,$ \gamma $ 为围岩重度,$ H $ 为洞室埋深,$ F $ 为AB面上静止土压力的合力.块体ABC在水平方向上的力学平衡为
$$ F = {F_{{\text{out}}}} \cos \left( {45^\circ - \frac{\varphi }{2}} \right) + T \sin \left( {45^\circ - \frac{\varphi }{2}} \right) $$ (2) 式中,
$ {F_{{\text{out}}}} $ 为BC面上总压力,$ \varphi $ 为围岩的内摩擦角,$ T $ 为BC面上总剪切力.BC面上的正应力和剪切力满足Mohr-Coulomb强度准则, 则BC面上总的剪切力T的表达式为
$$ T = {F_{{\text{out}}}} \tan \varphi + cH\sec \left( {45^\circ - \frac{\varphi }{2}} \right) $$ (3) 式中,
$ c $ 为围岩黏聚力.将式(1) ~ 式(3)联立得
$$ {F_{{\text{out}}}} = \dfrac{{\dfrac{1}{2}\lambda \gamma {H^2} - cH\tan \left( {45^\circ - \dfrac{\varphi }{2}} \right)}}{{\cos \left( {45^\circ - \dfrac{\varphi }{2}} \right) + \tan \varphi \sin \left( {45^\circ - \dfrac{\varphi }{2}} \right)}} $$ (4) 对整个上覆岩体CC`BB`竖直方向受力进行分析
$$ \begin{split} & W + 2T \cos \left( {45^\circ - \frac{\varphi }{2}} \right) =\\ &\qquad{P_{\max }} \cdot 2r \cdot {F_{{\text{safe}}}} + 2{F_{{\text{out}}}} \sin \left( {45^\circ - \frac{\varphi }{2}} \right)\end{split} $$ (5) 式中,
$ {P_{\max }} $ 为上限压力,$ r $ 为洞室半径,$ {F_{{\text{safe}}}} $ 为安全系数, 一般取1.3 ~ 1.5, W为上覆岩体重力. 上覆岩体的重力为$$ W = \gamma \left[ {2r + H\tan \left( {45^\circ - \frac{\varphi }{2}} \right)} \right]H $$ (6) 将式(3) ~ 式(6)联立
$$ {P_{\max }} = \frac{{\alpha {H^2} + \beta H}}{{{F_{{\text{safe}}}}}} $$ (7) 式中
$$ \alpha = \dfrac{{\gamma \tan \left( {45^\circ - \dfrac{\varphi }{2}} \right)}}{{2r}} + \dfrac{{\lambda \gamma \left[ {\tan \varphi - \tan \left( {45^\circ - \dfrac{\varphi }{2}} \right)} \right]}}{{2r\left[ {1 + \tan \varphi \tan \left( {45^\circ - \dfrac{\varphi }{2}} \right)} \right]}} $$ (8) $$ \beta = \frac{{c{{\sec }^2}\left( {45^\circ - \dfrac{\varphi }{2}} \right)}}{{r\left[ {1 + \tan \varphi \tan \left( {45^\circ - \dfrac{\varphi }{2}} \right)} \right]}} + \gamma $$ (9) 3. 弹塑性条件下上限和下限压力
3.1 隧洞式内衬储气库围岩应力场
以塑性区贯通作为洞室失稳的判据[33], 因此对于圆形洞室来说, 当围岩内壁全部进入塑性区时洞室失稳. 为了得到弹塑性条件下上限和下限压力解析解, 需要先得到隧洞式储气库围岩弹性应力场分布情况.
图3为隧洞式储气库受力图, 则可以把外边界条件看作无限边界, 由于内衬层厚度相比于岩洞半径来说是很小的, 且围岩假定为理想弹塑性材料, 服从Mohr-Coulomb强度准则, 因此不考虑衬砌, 假设气体压力直接作用在围岩内壁. 这里使用极坐标进行求解, 首先将直边变换为圆边. 为此, 以一无限大长度R0为半径, 以坐标原点为圆心, 作一个大圆. 根据圣维南原理, 在半径为R0的外圆圆周处, 应力情况与无孔时相同[34], 即
$$ {\sigma _x} = - \lambda {q_0},\;\;\;\;{\sigma _y} = - {q_0},\;\;\;\;{\tau _{xy}} = 0 $$ (10) 式中,
$ {q_0} $ 为上覆岩体施加的重力. 应力分量由直角坐标向极坐标的变换为$$ {\sigma _\rho }\left| {_{\rho = {R_0}}} \right. = - \frac{{\left( {\lambda + 1} \right)}}{2}{q_0} + \frac{{\left( {1 - \lambda } \right)}}{2}{q_0}\cos (2\phi) $$ (11) $$ {\sigma _\varphi }\left| {_{\rho = {R_0}}} \right. = - \frac{{\left( {\lambda + 1} \right)}}{2}{q_0} + \frac{{\left( {\lambda - 1} \right)}}{2}{q_0}\cos (2\phi) $$ (12) $$ {\tau _{\rho \varphi }}\left| {_{\rho = {R_0}}} \right. = \frac{{\left( {\lambda - 1} \right)}}{2}{q_0}\sin (2\phi) $$ (13) $$ {q_0} = \gamma H $$ (14) 在半径为r的储气库内边界上的应力边界条件为
$$ {\sigma _\rho }\left| {_{\rho = r}} \right. = - P $$ (15) $$ {\tau _{\rho \varphi }}\left| {_{\rho = r}} \right. = 0 $$ (16) 式中, P为岩洞内壁受到的压力.
根据基尔希公式[35], 得到应力分量表达式为
$$ \begin{split} &{\sigma _\rho } = \frac{{\left( {\lambda + 1} \right)}}{2}{q_0}\left( {\frac{{{r^2}}}{{{\rho ^2}}} - 1} \right) - P\frac{{{r^2}}}{{{\rho ^2}}} + \\ &\qquad\frac{{1 - \lambda }}{2}\left( {1 - 4\frac{{{r^2}}}{{{\rho ^2}}} + 3\frac{{{r^4}}}{{{\rho ^4}}}} \right){q_0}\cos (2\phi)\end{split} $$ (17) $$ \begin{split} &{\sigma _\phi } = P\frac{{{r^2}}}{{{\rho ^2}}} - \frac{{\left( {\lambda + 1} \right)}}{2}{q_0}\left( {1 + \frac{{{r^2}}}{{{\rho ^2}}}} \right)+\\ &\qquad \frac{{\lambda - 1}}{2}{q_0}\left( {1 + 3\frac{{{r^4}}}{{{\rho ^4}}}} \right)\cos (2\phi) \end{split}$$ (18) $$ {\tau _{\rho \phi }} = \frac{{\lambda - 1}}{2}\left( {{q_0} + 2{q_0}\frac{{{r^2}}}{{{\rho ^2}}} - 3{q_0}\frac{{{r^4}}}{{{\rho ^4}}}} \right)\sin (2\phi) $$ (19) 3.2 强度准则确定极限压力
根据弹性解析解得到洞壁处弹性应力表达式, 若以压应力为正, 则洞壁处径向、切向应力分别为
$$ \qquad\qquad\qquad{\sigma _\rho } = P $$ (20) $$ \qquad\qquad\qquad{\sigma _\phi } = Q - P $$ (21) $$ \qquad\qquad\qquad{\tau _{\rho \phi }} = 0 $$ (22) 式中
$$ Q = \left( {\lambda + 1} \right){q_0} - 2\left( {\lambda - 1} \right){q_0}\cos (2\phi) $$ (23) 设隧洞轴线方向为z方向, 则该方向上的应力为
$$ {\sigma _z} = \nu \left( {{\sigma _\rho } + {\sigma _\phi }} \right) $$ (24) 式中,
$ \nu $ 为围岩泊松比.将得到的最大、最小主应力的值和强度准则的主应力形式引入
$ {\sigma _1} - {\sigma _3} $ 空间, 如图4所示, 为了便于分析, 假设应力路径最大和最小主应力分别取径向和切向应力弹性解. 如果应力路径与剪切强度准则相交, 如图4(a)所示, 随着气体压力由零逐渐增大, 应力路径从A点开始斜向下变化, 其中AD段在剪切强度准则上方, 实际情况中围岩应力场不存在AD段, 因此该段路径使用虚线表示. 当内压较小时, 应力路径斜向下变化与强度准则相交, 则该交点处的压力值即为所需的下限压力. 当内压较大时, 应力路径变化方向相反, 会再次与剪切强度准则相交, 交点处的压力值即为上限压力; 若应力路径与剪切强度准则不相交, 则可以根据抗拉强度确定上限压力. 剪切强度准则选用Mohr-Coulomb强度准则,$ {\sigma _1} > {\sigma _3} $ , 且压应力为正, 这与弹性解析解相反. Mohr-Coulomb强度准则为一条直线. 剪切强度准则的主应力表达形式为$$ {\sigma _1} = \xi {\sigma _3} + {R_c} $$ (25) $$ \xi = \frac{{1 + \sin \varphi }}{{1 - \sin \varphi }},\;\;{R_c} = \frac{{2c\cos \varphi }}{{1 - \sin \varphi }} $$ (26) 由于洞壁处各点只有
$ {\sigma _\rho } $ ,$ {\sigma _\phi } $ 和$ {\sigma _z} $ 三个应力, 其他应力都为零, 则这三个应力为主应力. 随着气压的增大, 三个主应力的大小顺序依次存在4种情况: (1)$ {\sigma _\rho } < {\sigma _z} < {\sigma _\phi } $ , (2)$ {\sigma _z} < {\sigma _\rho } < {\sigma _\phi } $ , (3)$ {\sigma _z} < {\sigma _\phi } < {\sigma _\rho } $ , (4)$ {\sigma _\phi } < {\sigma _z} < {\sigma _\rho } $ .根据三个主应力的应力表达式及其大小关系, 分别得出4种情况所适用的压力范围
$$\qquad\qquad\qquad 0 \leqslant P < \nu Q $$ (27) $$ \qquad\qquad\qquad\nu Q \leqslant P < 0.5Q $$ (28) $$\qquad\qquad\qquad 0.5Q \leqslant P < \left( {1 - \nu } \right)Q $$ (29) $$ \qquad\qquad\qquad P > (1 - \nu )Q $$ (30) 根据上述压力范围和主应力表达式, 可以作出应力路径图, 如图4所示.
$ A( \to D) \to B $ 过程为情况(1),$ B \to C $ 为情况(2),$ C \to B $ 为情况(3),$B( \to D) \to A$ 为情况(4).当
$ Q \geqslant {R_c} $ 时, 如图4(a)所示, 随着内压P不断增大, 内衬储气库仅发生剪切破坏, 此时内衬储气库存在上限和下限压力, 其中AD段在剪切强度准则上方, 实际情况中围岩应力场不存在AD段, 因此该段路径使用虚线表示.当
$ Q < {R_c} $ 时, 如图4(b)所示, 随着内压P不断增大, 内衬储气库仅发生拉伸破坏, 此时内衬储气库仅存在上限压力.当围岩内壁全部进入塑性区时洞室失稳. 当内压较小时, 切向应力是洞壁处最大主应力, 径向应力是最小主应力, 最小主应力值是确定的, 则取最大主应力最小位置应力情况进行判断, 即
$$ {\sigma _3} = {\sigma _\rho } = P $$ (31) 当
$ \lambda > 1 $ 时, 则最大主应力取左右两侧的切向应力$$ {\sigma _1} = \left( {\lambda + 1} \right){q_0} + 2\left( {1 - \lambda } \right){q_0} - P $$ (32) 当
$ \lambda \leqslant 1 $ 时, 则最大主应力取顶部和底部的切向应力$$ {\sigma _1} = \left( {\lambda + 1} \right){q_0} - 2\left( {1 - \lambda } \right){q_0} - P $$ (33) 将式(25)、式(31)、式(32)和式(33)分别联立得到
$$ {P_{\min }} = \left\{ \begin{split} &{\dfrac{{\left( {\lambda + 1} \right){q_0} - 2\left( {1 - \lambda } \right){q_0} - {R_c}}}{{\xi + 1}},\;\lambda \leqslant 1} \\ &{\dfrac{{\left( {\lambda + 1} \right){q_0} + 2\left( {1 - \lambda } \right){q_0} - {R_c}}}{{\xi + 1}},\;\lambda > 1} \end{split}\right. $$ (34) 当内压较大时, 径向应力是洞壁处最大主应力, 切向应力是最小主应力, 则取切向应力最小位置处的应力进行判断, 因此得到上限压力
$$ {P_{\max }} = \left\{ \begin{split} &{\frac{{\xi \left[ {\left( {\lambda + 1} \right){q_0} + 2\left( {1 - \lambda } \right){q_0}} \right] + {R_c}}}{{1 + \xi }},\;\;\lambda \leqslant 1} \\ &{\frac{{\xi \left[ {\left( {\lambda + 1} \right){q_0} - 2\left( {1 - \lambda } \right){q_0}} \right] + {R_c}}}{{1 + \xi }},\;\;\lambda > 1} \end{split}\right. $$ (35) 若应力路径与破坏准则没有交点, 如图4(b)所示, 则不考虑下限压力, 而上限压力则需要根据抗拉强度确定, 由于围岩抗拉强度很小, 因此取抗拉强度为零. 因此, 洞室边界上的切应力不超过抗拉强度的条件是
$$ {\sigma _\phi } \geqslant 0 $$ (36) 得到上限压力为
$$ {P_{\max }} = \left\{ \begin{split} &{\left( {\lambda + 1} \right){q_0} + 2\left( {1 - \lambda } \right){q_0},\;\;\lambda \leqslant 1} \\ &{\left( {\lambda + 1} \right){q_0} - 2\left( {1 - \lambda } \right){q_0},\;\;\lambda > 1} \end{split} \right. $$ (37) 极限平衡条件只能得到上限压力, 弹塑性条件可以得到上限和下限压力. 因此取弹塑性条件下的下限压力表达式(34)的值为下限压力. 取极限平衡条件和弹塑性条件下的上限压力的最小值为最终的上限压力, 极限平衡条件下上限压力与埋深呈二次函数关系, 而弹塑性条件下上限压力与埋深呈线性关系. 因此, 当洞室埋深很小时, 极限平衡条件下的上限压力比较小; 当埋深较大时, 弹塑性条件下的上限压力比较小.
4. 典型围岩条件下极限压力
4.1 围岩参数选取
对于隧洞式内衬岩洞极限压力, 可以根据得到的极限压力解析解进行计算. 围岩条件分别取Ⅰ ~ Ⅳ级围岩, 围岩的重度
$ \gamma $ 、黏聚力$ c $ 和内摩擦角$ \varphi $ 按照《工程岩体分级标准》GB/T 50218–2014规定进行选取. 计算各级围岩条件下内衬储气岩洞的下限和上限压力时, 取岩洞半径$ R = 17.5 \;{\rm{m}} $ ; 取不同侧压力系数λ = 0.8, 1.0, 1.2. 岩洞推荐埋深为100 ~ 200 m. 最终得到各级围岩条件下极限压力与岩洞埋深的解析解和极限压力随埋深关系曲线.4.2 极限平衡条件下的上限压力
将表1中的数据代入式(7)中, 得到极限平衡条件下Ⅰ ~ Ⅳ级围岩中隧洞式内衬储气库上限压力与埋深的关系表达式, 埋深取为0 ~ 200 m, 得到上限压力与埋深关系曲线.
图5(a) ~ 图5(c)分别是侧压力系数为0.8, 1.0, 1.2时隧洞式储气库上限压力曲线, 安全系数取1.5. 隧洞式储气库上限压力曲线结果表明: 上限压力随岩洞埋深呈现二次函数关系, 随着埋深增大, 上限压力不断上升且增长率不断增大.
表 1 围岩参数Table 1. Parameters of surrounding rockGrade Ⅰ Ⅱ Ⅲ Ⅳ γ/(kN·m−3) 28 26.5 24.5 22.5 $ \varphi $/(°) 62 56 44 0.7 c/ MPa 2.4 1.5 0.7 0.2 ν 0.2 0.25 0.3 0.35 ![]()
图 5 上限压力与埋深关系曲线Figure 5. Curves of relationship between upper limit pressure and depth相同埋深时, Ⅰ ~ Ⅳ级围岩条件下隧洞式储气库上限压力随着侧压力系数的增大而增大, 因此应尽量在侧压力系数为1.2的围岩条件下修建隧洞式储气库. λ = 1.2时, 埋深范围为100 ~ 200 m时, Ⅰ ~ Ⅳ级围岩情况下内衬洞室上限压力分别为: 17 ~ 51 MPa, 12.5 ~ 38 MPa, 8 ~ 24 MPa, 5 ~ 16 MPa.
4.3 弹塑性条件下的下限压力
分别作出λ = 0.8, 1.0, 1.2时下限压力曲线(图6). 显然, 下限压力与埋深呈现正比例关系.
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图 6 下限压力与埋深关系曲线Figure 6. Curves of relationship between lower limit pressure and depth当λ = 0.8, 1.0和1.2时, 在Ⅰ级围岩中, 应力路径与剪切强度准则没有交点, 即对下限压力没有限制. 当围岩物理参数和埋深相同时, 三种侧压力条件下的下限压力λ = 1.2时最小, λ = 0.8时最大, 在Ⅲ和Ⅳ级围岩中建设储气洞库所需的垫气量远大于Ⅰ和Ⅱ级围岩, 侧压力系数0.8, 1.0和1.2条件下建设的储气库垫气量依次减少. 因此应尽量在侧压力系数为1.2的围岩条件下修建隧洞式储气库, λ = 1.2、埋深范围为100 ~ 200 m时, Ⅰ级围岩情况下不用考虑下限压力, Ⅱ ~ Ⅳ级围岩情况下内衬洞室下限压力分别为: 0 ~ 0.34 MPa, 0.47 ~ 1.44 MPa和1.04 ~ 2.25 MPa.
4.4 弹塑性条件下的上限压力
根据计算公式得到弹塑性条件下Ⅰ ~ Ⅳ级围岩中内衬储气库的上限压力与埋深的关系表达式, 并得到Ⅰ ~ Ⅳ级围岩中上限压力与埋深关系曲线.
图7是侧压力系数为λ = 0.8, 1.0, 1.2的围岩中隧洞式储气库弹塑性条件下的上限压力曲线. 结果表明: 围岩质量越好, 相同埋深下岩洞可承受的极限压力越大. 当埋深较小时, 岩洞内壁可能产生拉伸破坏, 当埋深增大到一定数值时, 岩洞内壁发生剪切破坏. 随着埋深的增大, 围岩破坏形式可能由拉伸破坏转变为剪切破坏, 围岩质量越好, 破坏形式变化时的埋深越大. Ⅰ级围岩条件下破坏形式一直都是拉伸破坏. 当λ = 1.0时上限压力最小, λ = 0.8和λ = 1.0条件下的上限压力基本相同, 且破坏形式变化时的埋深也差别不大. 上限压力在λ = 1.2时最大, 因此应尽量在侧压力系数为1.2的围岩条件下修建隧洞式储气库.
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图 7 上限压力与埋深关系曲线Figure 7. Curves of relationship between upper limit pressure and depth对比图5和图7, 相比于弹塑性条件, 极限平衡条件下求得的上限压力更大, 但是弹塑性条件下求得的上限压力更加保守、安全. 因此选用弹塑性条件下上限压力作为储气库上限压力. λ = 1.2时, 埋深范围为100 ~ 200 m时, Ⅰ ~ Ⅳ级围岩情况下内衬洞室上限压力分别为: 7.3 ~ 14.6 MPa, 6.9 ~ 13.4 MPa, 5.9 ~ 11.3 MPa, 4.8 ~ 9.5 MPa.
现有工程一般修建于Ⅰ ~ Ⅲ级围岩中, 目前, 韩国和瑞典等国家已经建立了试验岩洞以用于研究[3,36-38], 它们的上限压力如表2所示. 这些上限压力都在本文极限压力范围内.
表 2 经典案例参数Table 2. Parameters of classic casesCases Country Depth/m Pressure/MPa Jongson Korea 100 5 pilot plant Korea 100 5 pilot plant Sweden 450 8 Kamimasagawa Japan 450 8 5. 结 论
采用极限平衡方法和弹塑性分析方法推导得到隧洞式内衬岩洞极限压力解析式, 并比较分析了三种侧压力系数(0.8, 1.0, 1.2)情况下建设隧洞式内衬储气库的可行性和优劣性. 选用刚性锥模型, 考虑上覆围岩自重和破裂面处剪切力和岩洞内部压力, 采用极限平衡方法推导内衬岩洞的上限压力解析解. 考虑围岩中应力分布规律和剪切、抗拉强度, 采用弹塑性分析方法推导内衬岩洞的上限压力和下限压力解析解, 获得如下结论.
(1)极限平衡条件下的上限压力随岩洞埋深呈现二次函数关系, 随着埋深增大, 上限压力不断上升且增长率不断增大. 相同埋深时, Ⅰ ~ Ⅳ级围岩条件下隧洞式储气库上限压力随着侧压力系数的增大而增大.
(2)弹塑性条件下的下限压力与埋深呈线性关系, 且在Ⅰ级围岩情况下内衬洞室的下限压力为零. 上限压力与埋深呈线性关系. 对比极限平衡条件下上限压力曲线, 一般只有在埋深小于50 m时弹塑性条件下的上限压力才可能比极限平衡条件下的上限压力低. 由于内衬洞室埋深不会那么小, 因此选用弹塑性条件下得到的上限压力解析解来确定内衬洞室上限压力.
(3)综合论文中得到典型工况下的极限压力, 给出推荐方案: 侧压力系数λ取1.2, 埋深范围为100 ~ 200 m时, Ⅰ ~ Ⅳ级围岩情况下内衬洞室上限压力分别为: 7.3 ~ 14.6 MPa, 6.9 ~ 13.4 MPa, 5.9 ~ 11.3 MPa, 4.8 ~ 9.5 MPa; Ⅰ级围岩情况下不用考虑下限压力, Ⅱ ~ Ⅳ级围岩情况下内衬洞室下限压力分别为: 0 ~ 0.34 MPa, 0.47 ~ 1.44 MPa, 1.04 ~ 2.25 MPa.
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图 2 极限平衡条件下上限压力分析模型
Figure 2. Analysis model of upper limit pressure under ultimate equilibrium
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图 5 上限压力与埋深关系曲线
Figure 5. Curves of relationship between upper limit pressure and depth
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图 6 下限压力与埋深关系曲线
Figure 6. Curves of relationship between lower limit pressure and depth
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图 7 上限压力与埋深关系曲线
Figure 7. Curves of relationship between upper limit pressure and depth
表 1 围岩参数
Table 1 Parameters of surrounding rock
Grade Ⅰ Ⅱ Ⅲ Ⅳ γ/(kN·m−3) 28 26.5 24.5 22.5 $ \varphi $/(°) 62 56 44 0.7 c/ MPa 2.4 1.5 0.7 0.2 ν 0.2 0.25 0.3 0.35 
下载: 导出CSV
表 2 经典案例参数
Table 2 Parameters of classic cases
Cases Country Depth/m Pressure/MPa Jongson Korea 100 5 pilot plant Korea 100 5 pilot plant Sweden 450 8 Kamimasagawa Japan 450 8
下载: 导出CSV
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