Numerical simulation of mud inrush of tunnels with coupled LBM-DEM
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摘要: 为了探究隧道突泥灾害的灾变机制,开发了耦合格子Boltzmann–离散元法（LBM-DEM）数值计算平台并对隧道突泥破坏的演化过程进行了模拟,分析了致灾介质颗粒黏结强度、水压力及突泥口尺寸等因素对隧道突泥破坏特征的影响。结果表明：基于LBM-DEM模拟能再现隧道突泥破坏“启动、加速、缓慢和稳定”等连续4个阶段的演化过程；无黏结的致灾介质突泥破坏形态近似为直线,有一定黏结强度的致灾介质突泥破坏形态总体呈圆弧或抛物线状,突泥破坏区扩展范围和稳定后的突泥量随着颗粒间黏结强度的增大而逐渐减小；水压力越大,突泥灾害发生后突泥量增长越快,最终的突泥量也越大,且颗粒间黏结强度较大时水压力的这种影响越显著；当致灾介质颗粒间无黏结时,不同突泥口尺寸的模型在稳定后突泥量和破坏区范围基本相同,而当颗粒间形成一定强度的黏结后,突泥口尺寸越大,突泥灾害发生后突泥量增长越快,稳定后的突泥量也越多；隧道突泥破坏是致灾岩土介质、水压和开挖三者综合作用的结果。
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关键词:
- 隧道工程 /
- 突泥 /
- 格子Boltzmann方法（LBM） /
- 离散元法（DEM） /
- 颗粒–流体耦合
Abstract: To better understand the catastrophic mechanism of mud inrush disasters of tunnels, a numerical computing platform based on the coupled lattice Boltzmann method-discrete element method (LBM-DEM) is developed and used to simulate the evolution process of mud inrush of tunnels. According to the simulated results, the effects of particle bonding strengths of disaster-causing media, groundwater pressures and sizes of mud inrush holes on the characteristics of mud inrush of tunnels are analyzed. The results show that coupled LBM-DEM simulation can well reproduce the evolution process of four successive stages of mud inrush of tunnels: "starting, accelerating, decelerating and stabilizing". The failure form of unbonded disaster-causing media after mud inrush is approximately straight, whereas the failure form of disaster-causing media with certain bond strength is generally arc or parabolic. The expanded range of failure zone and mud inrush mass both decrease with the increase of the inter-particle bond strength of disaster-causing media. The higher the water pressure is, the faster the mud inrush mass increases after the occurrence of mud inrush disasters, and the more the final mud inrush mass is, which is more remarkable when the inter-particle bond is much stronger. When there is no bond between particles of the disaster-causing media, the models with different sizes of mud inrush holes have basically the same failure zone and mud inrush mass after stabilization. However, when a certain strength of bond is formed between particles, the mud inrush mass increases faster and the final mud inrush mass is more with the increase of the sizes of mud inrush holes. The mud inrush disasters are the result of the combined action of disaster-causing geo-materials, groundwater pressure and tunnel excavation. -
0. 引言
钙质砂是由珊瑚骨骼、贝类、虫黄藻类等海洋生物残骸沉积而成，其主要组成成分是碳酸钙[1-3]，是我国南海岛礁吹填的主要材料。因其生成环境、成因以及物质组成等因素影响，钙质砂具有颗粒易破碎、形状极不规则、内孔隙发育、微观结构复杂等显著区别于陆源石英砂的特点[4-6]。随着“一带一路”国家战略和建设“海洋强国”政策方针的推进，研究钙质砂工程力学特性具有重要意义[7-9]。
钙质砂作为填方工程的天然材料，其应力状态复杂多变，土体受到各向异性应力状态影响而产生初始静剪应力，在建（构）筑物的自重和动荷载（波浪、地震和交通荷载等）作用下，易引起地基强度降低、变形过大以及液化失稳等灾害。实际上，动荷载作用下剪切应力做功将导致材料损伤效应的累积，不排水条件下表现为孔压增长。因此，可以将孔压的升高与土体颗粒在运动或重排过程中所耗损的能量建立关联。损耗能作为标量，相较于应力、应变等矢量，可直接数学叠加，大幅度降低分析难度。Nemat-Nasser等[10]首先提出了耗散能量的概念，建立其与残余孔压的关系，来有效地评估孔隙水压力的产生和发展过程。Kokusho[11]和Pan等[12]提出了土骨架破坏产生的单位体积耗散能与应变和残余孔压累积直接相关，为评价砂土在不规则循环应力条件下的抗液化能力提供了有效方法。总体而言，上述研究主要针对石英砂，能否适用于钙质砂仍需进一步探究。
本文以饱和钙质砂为研究对象，开展不排水条件下循环剪切三轴试验，探究相对密实度、初始静剪应力以及循环应力对其孔压发展的影响；同时，引入能量法，建立钙质砂孔压与损耗能之间联系，提出基于能量损耗的液化评价方法，为钙质砂地基稳定性分析提供理论依据和技术支撑。
1. 试验材料与方案
1.1 试验材料
本文试验材料为中国南海某岛礁的天然钙质砂，颗粒多呈灰白色，形状有片状、块状、棒状等，颗粒内孔隙多、微观结构复杂，如图 1所示。
经过现场取材、清水冲洗、烘干等过程后，对粒径大于5 mm的颗粒进行剔除，处理后颗粒分布级配曲线如图 2所示，主要基本物理性质参数见表 1。不难发现，试样基本不含0.1 mm以下的细颗粒，不均匀系数和曲率系数分别为6.84和0.78，属于不良级配砂土。
表 1 钙质砂物理性质指标Table 1. Physical properties of calcareous sand相对质量密度 d50/
mm不均匀系数 曲率系数 最大孔隙比 最小孔隙比 2.79 2.0 6.84 0.78 1.15 0.87 1.2 试验方案
结合实际工况，采用CKC三轴试验系统模拟复杂应力条件下饱和钙质砂循环剪切试验，先进行有效围压为100 kPa的等向固结后再根据试验设计的初始静偏应力状态进行非等向固结，具体方案如表 2所示。初始静剪应力比SSR和循环应力比CSR可通过式（1）和（2）计算。
表 2 不排水循环剪切试验方案Table 2. Summary of undrained cyclic triaxial tests试验系列 相对密实度
Drqs/
kPaqcyc/
kPaSSR CSR Nf Ⅰ 70%
（密砂）0 20 0 0.1 232 0 25 0 0.125 74 0 30 0 0.15 17 0 40 0 0.2 6 20 30 0.1 0.15 168 20 45 0.1 0.225 19 20 50 0.1 0.25 3 50 50 0.25 0.25 53 50 60 0.25 0.3 11 50 70 0.25 0.35 6 80 70 0.4 0.35 14 80 80 0.4 0.4 7 -10 25 -0.05 0.125 78 -10 30 -0.05 0.15 39 -10 35 -0.05 0.175 8 -20 20 -0.1 0.1 210 -20 25 -0.1 0.125 11 -20 30 -0.1 0.15 8 -40 20 -0.2 0.1 57 -40 25 -0.2 0.125 16 -40 30 -0.2 0.15 8 Ⅱ 30%
（松砂）0 15 0 0.075 943 0 20 0 0.1 120 0 25 0 0.125 37 0 30 0 0.15 18 24 30 0.12 0.15 61 24 35 0.12 0.175 16 24 40 0.12 0.2 5 40 15 0.2 0.075 175 40 20 0.2 0.1 9 50 12.5 0.25 0.0625 17 50 15 0.25 0.075 2 -10 12.5 -0.05 0.0625 382 -10 15 -0.05 0.075 180 -10 20 -0.05 0.1 11 -20 10 -0.1 0.05 246 -20 12.5 -0.1 0.0625 202 -20 15 -0.1 0.075 12 -40 5 -0.2 0.025 104 -40 7.5 -0.2 0.0375 13 -40 10 -0.2 0.05 2 SSR=qs2p′0， (1) CSR=qcyc2p′0。 (2) 式中：qs为初始静剪偏应力；qcyc为循环偏应力；p′0为平均有效正应力。
2. 试验结果与分析
2.1 孔压特性发展规律
图 3给出不同初始偏应力作用下饱和密砂的孔压发展规律曲线。孔隙水压力可分为两类：①随着循环荷载作用实时变化的孔压，即实线所示的瞬态孔压，这种孔压会随着循环荷载的卸载而快速消散；②每个循环加载结束，试样未及时恢复的孔压，即虚线所示的残余孔压。从图 3（a）中可以看出，对于等向固结的试样，残余孔压在前期随着荷载的施加而逐渐累积，而在后期快速增长，直至达到荷载施加前的有效围压，ulim=100 kPa。如图 3（b）所示，在压缩静偏应力作用下，孔压在加载初期迅速累积，随着循环荷载持续进行，残余孔压逐渐趋于稳定，ulim=64.6 kPa。在拉伸静偏应力作用下，孔压发展与压缩静偏应力时有类似的变化趋势，孔压在加载初期累积较快而后基本保持不变，ulim=34.68 kPa。
同时，通过式（3）和（4）定义固结应力比Kc和残余孔压比ur。
Kc=σv0σh0， (3) ur=uσh0。 (4) 式中：σv0和σh0分别为初始有效竖向应力和水平应力，u为残余孔压。
图 4给出了饱和密砂的极限残余孔压比和固结应力比的关系曲线。从图中可以看出，饱和密砂的极限残余孔压比随着固结应力比的增大呈先增大后减小的趋势，在Kc=1（等向固结）时，极限残余孔压比达到最大值ur, lim=1，且大致上呈线性分布，与循环应力幅值大小无明显关系。
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图 4 极限残余孔压比与固结应力比的关系Figure 4. Relationship between ultimate residual pore pressure ratio and consolidation stress ratio对于同一材料的砂土，其在循环荷载作用下有效应力路径将沿着平行于等向固结线（ICL）的方向逐渐靠近临界状态线（CSL），而与循环应力幅值无关，如图 5所示。因此，对于给定的初始应力状态（σh0，σv0），会与临界状态线相交于一点，且理论上初始应力点与最终应力点之间的水平距离Δu为试验中的极限残余孔压，如式（5）所示。根据ur和Kc定义，可得到两者关系如式（6），符合图 4所示的线性关系。
ulim=Δu=σh0−σv0−σh0m−1， (5) ur, lim=ulimσh0=1−Kc−1m−1。 (6) ![]()
图 5 初始静剪应力状态对残余孔压影响示意图Figure 5. Influences of initial static shear stress on residual pore pressure2.2 损耗能演化规律
动荷载作用下饱和砂土损耗的能量主要用于颗粒的相对运动和重新排列。因此，引入能量法，提出基于损耗能的砂土液化评价方法。循环加载过程中一个振次的损耗能W可用应力-应变滞回圈的面积表示，即：
W=n−1∑i=112(qi+1+qi)(εa, i+1−εa, i)。 (7) 式中：n为计算增量的总个数，qi和εa, i分别为第i个增量的偏应力和轴向应变。
图 6分别给出饱和钙质砂在不同初始静偏应力作用下残余孔压比与正交化损耗能的内在关系，正交化损耗能Wn为损耗能W与初始有效水平正应力σh0的比值。结果显示：饱和密砂的残余孔压初期增长缓慢，随着Wn的增大而较快增长，最后趋于稳定；在饱和松砂中也观察到类似的变化趋势。这说明残余孔压与损耗能的关系主要取决于初始应力条件。
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图 6 残余孔压比与损耗能的关系Figure 6. Relationship between residual pore pressure ratio and dissipated energy从图 7可以看出，饱和钙质砂在失稳破坏时所积累的损耗能随着初始静剪应力的增加而增加；对于同一初始应力状态，密砂所需能量始终大于松砂。研究表明[13-14]，饱和砂土在循环荷载作用下损耗能主要与初始应力和相对密实度有关，受循环荷载幅值影响极小，具体可用式（8）表示：
W′n=10a(Dr−0.78)10b(SSR−1.0)。 (8) ![]()
图 7 损耗能与初始静剪应力比的关系Figure 7. Relationship between dissipated energy and initial static shear stress ratio式中：a和b为经验参数，根据本次试验数据可分别取0.65，1.5。图 8对损耗能的试验实测值与通过式（8）所得的预测值进行对比，发现两者基本落在斜率为1的对角线两侧，表明能量模型可较好地预测不同试验条件下饱和钙质砂的损耗能。
3. 结论
（1）饱和钙质砂的极限残余孔压比随固结应力比呈先增大后减小的趋势，在Kc=1时存在最大值，临界状态理论可以解释此现象。
（2）不排水循环加载条件下饱和钙质砂的损耗能与试样的初始静剪应力比和相对密实度有关，受循环应力比影响极小，可通过构建的能量模型较好地预测不同试验条件下饱和钙质砂所累积的损耗能。
本文在LBM-DEM耦合计算程序的开发和测试过程中得到了美国Los Alamos国家实验室Wang Min博士的悉心指导,在此致以衷心的感谢！ -
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图 2 DKT算例不同时刻的颗粒位置与流场
Figure 2. Position of particle and contours of fluid velocity at different instants for DKT case
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图 3 DKT过程中两颗粒的位置变化
Figure 3. Variation of horizontal and vertical positions of particles during DKT process
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图 5 不同黏结强度下的突泥量时变曲线
Figure 5. Time-varying curves of mud inrush mass of models with different inter-particle bond strengths
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图 6 B0-P2-O80模型突泥不同时刻的颗粒分布与流场
Figure 6. Distribution of particles and contours of fluid velocity at different instants during mud inrush for B0-P2-O80 model
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图 7 B14-P2-O80模型突泥不同时刻的颗粒分布与流场
Figure 7. Distribution of particle and contours of fluid velocity at different instants during mud inrush for B14-P2-O80 model
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图 8 不同黏结强度下模型最终的破坏形态
Figure 8. Final failure modes of models with different inter-particle bond strengths
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图 9 不同入口水压力下的突泥量时变曲线
Figure 9. Time-varying curves of mud inrush mass of models with different inlet water pressures
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图 10 不同入口水压力下模型最终的破坏形态
Figure 10. Final failure modes of models with different inlet water pressures
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图 11 不同突泥口尺寸下的突泥量时变曲线
Figure 11. Time-varying curves of mud inrush mass of models with different sizes of mud inrush hole
表 1 隧道突泥的LBM-DEM模拟参数
Table 1 LBM-DEM simulation parameters for mud inrush of tunnel
固体参数 取值 流体参数 取值 颗粒密度/(kg·m-3) 2650 流体密度/(kg·m-3) 1000 法向刚度/(N·m-1) 1×107 运动黏度/(m2·s-1) 1×10-6 切向刚度/(N·m-1) 5×106 弛豫时间 0.5001 摩擦系数 0.5 格子步长/m 0.001 
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