Research on the static-dynamic boundary switch and its rationality verification method
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摘要:
繁琐复杂的静动力边界转换处理是进行静动力耦合模拟的关键步骤,为了研究黏弹性边界条件下传统静动力边界转换的合理性及其验证方法的适用性,本文首先根据弹性叠加原理提出了一种验证静动力耦合模拟分析中静动力边界转换合理性的方法(参考解法);然后基于有限元软件ABAQUS,并结合自行研发的适用于成层介质的黏弹性边界施加和等效节点力计算程序VBEA2.0,对多土层自由场静动力耦合模型和土-地下结构静动力耦合模型进行了地震反应分析,对参考解法的适用性进行了探究;最后使用本文提出的参考解法,对一种典型的未进行合理静动力边界转换的自由场模型进行了分析,探讨了静动力边界转换产生振荡的原因和机理。研究结果表明:用于验证静动力边界转换合理性的参考解法,既适用于多土层自由场静动力耦合计算模型,又适用于土-结构相互作用的静动力耦合模型;静动力边界转换后产生的振荡是静动力耦合计算模型静力不平衡所导致,一般是因为未合理施加边界反力或单元应力。
Abstract:In the dynamic analysis of underground structure subjected to seismic loading, dynamic boundary (e.g., viscous-spring boundary condition, transient wave transmission boundary, etc.) is essential to be adopted to absorb the reflected wave. Generally, the dynamic boundary is not applicable to the static analysis in the simulation of seismic response of underground structure. Taking the viscous-spring boundary condition as an example, if this dynamic boundary condition is applied during the static analysis phase the subsequent dynamic response will be significantly affected, since this operation results in incorrect input of ground motion in the numerical model. To solve this problem, the traditional approach involves employing a fixed boundary during static analysis, which is then replaced by a dynamic boundary during dynamic analysis. Specifically, the dynamic calculation is performed using a viscous-spring boundary condition based on the results obtained in the static analysis with static boundary condition. The complex process of switching between static and dynamic boundary is a significant step in the static-dynamic coupling simulation. However, oscillations occur at the onset of dynamic response when switching from a static to a dynamic boundary, affecting dynamic response of acceleration, velocity, displacement, etc. To investigate the validity of the traditional scheme for switching static-dynamic boundary condition types (from static boundary to viscous-spring boundary) and the applicability of the verification method the following works were conducted: based on the superposition principle, a method for verifying the rationality of static-dynamic boundary switch is given (reference method). Using the finite element method software ABAQUS and the self-developed VBEA2.0 program, which can automatically set viscous-spring boundary and input seismic wave, the seismic response of layered free field and soil-underground structure models (Dakai metro station) considering integral static-dynamic analysis are analyzed. Based on these results, the applicability of the reference method is discussed. Adopting this reference method, an analysis is conducted on a free field model employing a typical, unreasonable static-dynamic switching method with the aim of shedding lights on the oscillations during the switching between the static and dynamic boundary condition. Finally, a potential mechanism for the observed oscillation in the process of switching boundary condition is proposed. The simulation results indicate that the introduced method of switching boundary conditions for the viscous-spring boundary condition in seismic analysis performs well, effectively avoiding oscillations at the beginning of dynamic analysis. The procedure should be as follows: First, perform the static analysis to balance the in-situ stress; then, carry out the switching between static and dynamic boundaries; and finally, calculate the dynamic analysis. In particular, the rational application of nodal reaction force, gravity, and element stress is a key factor influencing the rationality of static-dynamic boundary switching.In addition to that, the proposed reference method for verifying the rationality of static-dynamic boundary switching is applicable to both the layered free field model and the soil-underground structure model based on the seismic response of the free field and the field embedding an underground structure. In the reference method, dynamic results of the elastic and non-damped model, without considering the static-dynamic coupling effects (reference model), are used as the benchmark solution for checking the dynamic response analysis. Dynamic response of the numerical model with the correct static and dynamic boundary switching should align with the corresponding dynamic response recorded in the reference model, in terms of acceleration, displacement time history. The second-order Euclidean norm is recommended for calculating the relative errors between the benchmark model and the numerical model which needs to check the validity of the traditional scheme for switching static-dynamic boundary. This can intuitively indicate the presence of oscillations in the early stage of dynamic calculation and assess the degree of their impact on the later stage, offering an effective tool to examine the rationality of static and dynamic boundary switching for a reasonable static-dynamic coupling analysis of soil-underground structure. Lastly, to reveal the mechanism of oscillation in the switching process of static and dynamic boundary, a model with only nodal reaction forces on the truncated boundary extracted and applied for the dynamic phase, which is typically incorrect to switch the static boundary condition to dynamic boundary condition, is built and analyzed. This incorrect way generates substantial forces in the spring elements leading the inaccurate seismic wave input employing equivalent nodal forces in the dynamic phase, according to the magnitude of elastic fore for spring elements located at the viscous-spring boundary. This means the numerical model may underestimate the dynamic response as portion of the equivalent nodal forces are counteracted by these forces from the springs. Therefore, the oscillated response in this switching procedure of static-dynamic boundary is typically caused by the non-static equilibrium in the static-dynamic coupling calculation model, possibly resulting from incorrect static-dynamic boundary switching.
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水平井是石油钻井技术进步的产物,因其水平段在深度方向起伏很小,可以简化为地下几百米至几千米深度处的一个完全水平的空心管线,管线壁称为套管;在套管壁上射出若干簇孔,称为射孔;射孔使管线内外的流体可以进行交换,深度通常大于1 m;射孔指向方向垂直于套管壁及水平段走向, 人工压裂裂缝沿着最大水平主应力方向扩展,这是由人工裂缝方向判断最大水平主应力方向的理论依据,但其初裂缝方向沿着射孔指向方向(图 1).作用在射孔截面上的二向主应力是井周周向地应力集中及平行套管走向的地应力(刘建中等,1993).
除应力外,构造、原生裂缝和介质间断面等因素也是影响人工裂缝方向的因素,进而影响根据人工裂缝方向判断最大水平主应力方向的可信度(赵国石等,2012).剔除上述因素的影响,获得可靠的应力方向,并把结果应用于地学其它领域,一直是很多地壳动力学研究人员孜孜以求的目标.直井压裂获得的数据量少,可用于比较的参数很少,因此很难剔除上述3个因素的影响;而水平井分段压裂微地震监测获得的资料多,且监测区集中,可以对比不同压裂段的监测结果,更利于剔除构造、裂缝和介质不均匀等因素的影响,从而获得比较可靠的应力方向.
油田钻井必然受到三向原地应力的作用,其中一个原地应力为垂向主应力,其余两者为水平主应力.钻井的产状不同,其井周的应力状态不同.本文研究的是水平井,其钻孔壁和井壁射孔周边的应力状态可表示为
(1) (2) 式中:W为地层介质比重;H为水平段深度;P0为地层液柱压力;σV,σH1,σ1,σ2分别为垂向有效主应力,垂直于钻孔水平段走向的水平原地有效应力及最大、最小有效水平主应力;α为水平段走向与最小水平主应力方向的夹角.
岩石井壁的有效周向应力集中σθ1为
(3) 将矢径R写为r,则σθ1可用下述公式表示:
(4) 式中, σθ1为钻孔岩石壁上的周向应力集中,θ1是以钻孔轴为圆心、竖直方向为零度的圆周角,R为以井轴为原点的矢径,r为井孔半径(刘建中等,1993).
在射孔周向截面上存在二次应力集中,其二向主应力为钻孔岩石壁的周向有效应力集中和平行于钻孔的水平原地有效应力.通常射孔深度较大,有时可达1 m以上.距离井壁越远,岩石壁的周向应力集中越小,沿着整个射孔段的钻井周向应力集中的平均值为岩石壁周向应力集中的0.57倍(刘建中等,2002),则射孔段的平均周向应力集中σθ2可表示为
(5) 式中,β为以射孔轴为中心、平行钻孔方向为零度的圆周角.油田压裂中,初始破裂总是出现在射孔周向应力集中σθ2最小的位置.
平行于钻孔水平段走向的水平原地有效应力σH2为
(6) 图 1给出了各相关参量的相对方位示意图.可以看出,从水平段走向向最小水平主应力方向旋转与从人工裂缝方向向最大水平主应力方向旋转的方向及角度均相同.
根据式(5)绘制出在不同的α,θ1时,σθ2随β的变化曲线. α取值有多种,鉴于油田水平井在设计时,α通常取值较小,因此本文仅以α=30°为例绘出其对应关系(图 2). θ1可取值0°,60°,120°,180°,240°,300°, 从应力研究角度,0°和180°结果相同,60°,120°,240°,300°结果相同,因此仅需研究θ1为0°和60°时的应力状态.此次压裂,水平段深度为1500 m,根据大庆油田实测应力值, 设有效应力σV为17 MPa, σ1为20 MPa, σ2为14 MPa,若地层压力正常,则上述3项有效应力的原地应力值分别为32,35,29 MPa.从图中可以看出:当θ1=0°和180°时,存在最小周向应力集中,套管顶、底部的射孔首先发生破裂;当β接近0°时,初始破裂沿套管切线方向,形成垂直于水平段走向的直立裂缝.经计算,套管顶、底部的射孔首先发生破裂,且初始裂缝仅有两种走向:当α<55°时,β接近0°,初始破裂沿水平段套管切线方向传播,与水平段走向垂直;当α>55°时,β接近于90°,初始破裂走向平行于水平段走向.由于在油田水平井设计中,通常采用较小的α,因此本文认为,初始人工裂缝方向与水平段走向垂直.
人工裂缝形态用监测到的微震点分布进行描述,微震点的二元线性回归方向为该监测段的人工裂缝方向,实际人工裂缝方向受到初始裂缝和延伸裂缝方向的共同影响,后者总是趋于最大水平主应力方向(刘建中等,1994).沿着射孔,垂直于水平段走向的初始人工裂缝,在延伸过程中转向最大水平主应力方向;水平井各段的应力条件接近,各段初始裂缝方向、延伸趋势也趋于一致.如果α=0°,初始人工裂缝与最大水平主应力方向一致,裂缝始终沿着确定方向延伸,则各压裂段的人工裂缝均是直立、平直、沿最大水平主应力方向的裂缝,彼此间平行度好;如果α很小,人工裂缝方向的转向角度也小,则各压裂段的人工裂缝近于直立、平直、大体沿最大水平主应力方向,彼此间平行度较好;如果α很大,人工裂缝方向的转向角度也大,则各压裂段的人工裂缝近于直立,但不平直,初始人工裂缝方向影响人工裂缝的最终方向,与最大水平主应力方向有较大偏差,平行度很差.综上所述,α严重影响各压裂段人工裂缝方向的一致性, 以各压裂段人工裂缝方向与所有压裂段的平均方向的最大偏差角度作为判据,可以判断出以水平井平均人工裂缝方向确定最大水平主应力方向的可靠性.
油田不同井区的地层因构造、沉积、裂缝等因素影响,应力方向有所差别.若以水平井压裂裂缝方向判断应力方向,那么各压裂段人工裂缝走向的平行度可以作为可靠性的判据,以此得到不同井区不同于区域构造应力的应力方向.对大庆油田的3口水平井(PP5井、ZP6井和AP5井)进行微地震监测,用上述理论对人工裂缝方向资料进行分析,以期得到可靠的井区应力方向.
图 3为ZP6井压裂微地震人工裂缝监测结果俯视图,该图给出了各压裂段监测获得的微震分布及人工裂缝走向,自下而上为1—19段压裂段.从图中可以看出,该井整体水平段走向为NE177.9°, 其它两井的俯视图与此类似. 3口井的人工裂缝方位见表 1.
表 1 AP5,ZP6,PP5井监测层段方位统计Table 1. Statistics of monitoring interval orientations of wells AP5, ZP6 and PP5监测层段 人工裂缝方向 AP5井 ZP6井 PP5井 第一段 NE84.4° NE75.6° NE84.7° 第二段 NE86.4° NE84.8° NW89.2° 第三段 NE87.3° NE84.4° NE86.0° 第四段 NE79.7° NE84.7° NW86.2° 第五段 NE78.6° NE81.8° NE84.2° 第六段 NE82.3° NE76.9° NE86.2° 第七段 NE78.7° NE80.1° NE87.9° 第八段 NE88.6° NE86.9° NE84.6° 第九段 NE84.6° NE87.8° NE86.2° 第十段 NE82.9° NE84.2° NE87.0° 第十一段 NE84.7° NE82.4° NE80.9° 第十二段 NE84.9° NE82.5° NE84.4° 第十三段 NE85.4° NE84.9° 第十四段 NE79.2° 第十五段 NE78.2° 第十六段 NE84.8° 第十七段 NE88.4° 第十八段 NE84.3° 第十九段 NE81.7° 表 2列出了各水平井的平均人工裂缝方向及最大偏差.可以看出,3口井的人工裂缝方向差别不大,且各井的最大偏差均较小,最大偏差是指由微震点二元回归方向给出的各压裂段人工裂縫方向与整井平均人工裂缝方向的最大差值.水平段走向向左旋转,偏差趋小,可以判断最小水平主应力方向在水平段走向的左旋方向.依据图 1,人工裂缝方向向最大水平主应力方向与水平段方向向最小水平主应力方向的旋转方向相同.实际最大水平主应力方向在平均人工裂缝方向的左旋方向.因此,测试最大水平主应力方向减掉最大偏差,测试结果可能会更接近实际最大水平主应力方向.经偏差校正处理后的最大水平主应力方向差别趋小,AP5井、ZP6井和PP5井的最大水平主应力方向分别为NE78.5°, NE76.9°和NE78.8°,与油田动态及静态资料给出的应力方向一致.
表 2 各水平井平均人工裂缝方向及最大偏差Table 2. Average fracturing orientation and maximum gap of each horizontal well井名称 平均人工裂缝方向 最大偏差/° 水平段走向 压裂段数 AP5 NE83.6° 5.1 NE175.7° 12 ZP6 NE82.9° 6.0 NE177.9° 19 PP5 NE86.3° 7.5 NE1.23° 13 本文提出了一种通过分析水平井微地震监测资料提取更可靠的应力场方向的理论与方法,并根据大庆油田3口水平井的微地震监测数据得出监测井区的应力场方向.这3口井相距较近,原始资料给出的各井的应力方向最大相差3.4°;经偏差校正处理后,各井的应力方向最大相差仅为1.9°,且与油田生产的动、静态资料结果大体一致,也与原地应力场在很大范围内基本稳定的判断一致,因此根据平行度判断水平井人工裂缝方向与最大水平主应力方向的吻合程度是一个现实可行的方法.
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图 4 自由场地震反应计算结果
(a) 100 m×50 m模型峰值加速度;(b) 100 m×50 m模型最大相对位移;(c) 200 m×100 m模型峰值加速度;(d) 200 m×100 m模型最大相对位移;(e) 随深度变化的加速度、位移时程误差
Figure 4. Seismic dynamic response of free field
(a) Maximum acceleration for 100 m×50 m model;(b) Maximum relative displacement for 100 m×50 m model;(c) Maximum acceleration for 200 m×100 m model;(d) Maximum relative displacement for 200 m×100 m model;(e) Acceleration and displacement time-history errors with varying depths
图 7 带地下结构的成层场地地震动力反应计算结果
(a) 场地土PGA;(b) 场地土最大相对位移;(c) 车站中柱最大相对位移;(d) 车站结构重要部位最大主应力
Figure 7. Earthquake dynamic response of underground structure in the layered site
(a) Maximum acceleration of the site;(b) Maximum relative displacement of the site;(c) Maximum relative displacement of the central column;(d) Maximum principal stress at points in the station
图 8 带地下结构的成层场地左边界及中部节点水平向的加速度时程(左)和位移时程(右)
(a) 左边界顶部节点;(b) 地表中部节点;(c) 左边界底部节点;(d) 底边界中部节点
Figure 8. Horizontal acceleration (left) and displacement (right) time history of nodes at the left boundary and the vertical middle of underground structure in the layered site
(a) Node at the top of the left boundary;(b) Node at the centre of the surface;(c) Node at the bottom of the left boundary;(d) Node at the centre of the base
表 1 大开地铁车站数值模型场地条件参数
Table 1 Mechanical parameters of the soil layers for the Dakai metro station
土质 模型各层厚度/m 密度/(kg·m−3) 较软地层模型vS
/(m·s−1)真实地层模型vS
/(m·s−1)较硬地层模型vS
/(m·s−1)泊松比 人工填土 1 1 900 80 140 290 0.333 全新世砂土 4 1 900 80 140 290 0.320 全新世砂土 3 1 900 110 170 320 0.320 更新世黏土 3 1 900 130 190 340 0.400 更新世黏土 6 1 900 180 240 390 0.300 更新世砂土 5 2 000 270 330 480 0.260 基岩 20 2 000 1 000 1 000 1 000 0.260 -
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