致密砂岩水平井多裂缝扩展及转向规律研究

夏彬伟; 刘浪; 彭子烨; 高玉刚

doi:10.11779/CJGE202008021

致密砂岩水平井多裂缝扩展及转向规律研究 English Version

重庆大学煤矿灾害动力学与控制国家重点实验室,重庆　400044

基金项目:

国家自然科学基金基项目 51974042

山西省科技重大专项揭榜项目 20191101015

国家重点研发计划项目 2018YFC0808401

详细信息

作者简介:
夏彬伟(1978—),男,工学博士,副教授,博士生导师,从事水力压裂治理岩层顶板方面研究。E-mail：xbwei33@cqu.edu.cn

中图分类号: TU452
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出版历程
- 收稿日期: 2019-10-30
- 网络出版日期: 2022-12-05
- 刊出日期: 2020-07-31

Multi-fracture propagation and deflection laws of horizontal wells in tight sandstone

State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chonqing University, Chongqing 400044, China

摘要

摘要: 为研究致密砂岩水平井割缝压裂裂缝扩展及转向规律,采用四维水射流割缝装置和大尺寸真三轴相似物理模拟试验系统,开展了不同缝间距、应力差、压裂排量对水平井多裂缝扩展规律的试验和数值模拟研究,建立了单张开裂缝和多裂缝扩展的应力理论模型和一套室内割缝压裂物理试验方法。通过剖样观察和压力曲线特征的类比分析得到以下结论：①多裂缝起裂后后续压力曲线的典型波动峰值,是致密砂岩多裂缝相互干扰的一个明显特征；小间距使得邻近裂缝处于高诱导应力区域,增加了应力干扰和裂缝偏转程度；②大排量使得裂缝内部净水压增大,多裂缝偏转角度和程度增大,更容易形成纵向缝；且处在中间裂缝受到抑制,大角度偏离最大主应力方向延伸并趋于两侧裂缝最终停止扩展,而两侧裂缝延伸的距离更长；③高应力差条件下,诱导应力场难以改变原始主应力的大小,降低裂缝转向的角度,起裂后后续压力曲线波动较平稳,裂缝更易形成平行于最大主应力方向的横切缝。研究成果可用于多段割缝压裂施工参数的优化设计,从而为不同地质条件的砂岩储层油气开采和煤矿中水力压裂坚硬顶板治理强矿压提供参考。
- 水平井 /
- 割缝压裂 /
- 缝间应力干扰 /
- 裂缝偏转
Abstract: In order to study the multi-fracture slotted propagation and deflection laws of horizontal wells in tight sandstone, the influences of crack spacing, main stress difference and discharge capacity on the propagation geometry of multi-fractures are studied by using physical experiments and numerical simulations with FLAC^3D based on four-dimensional water jet slitting device and large-scale true triaxial hydraulic fracturing simulation system. A stress filed theoretical model of opening single and multi-fracture with water pressure and a set of indoor slotting-fracturing physical test method are established. The analysis of the characteristics of the sample splitting and the pressure curve reveals: (1) The typical fluctuation peak of the subsequent pressure curve after the initiation cracking is an obvious feature of the fracture mutual stress interference. The short spacing makes the adjacent fracture in the high induced stress zone, leading to strengthening the stress mutual interference and the degree of fracture deflection. (2) The angle and extent of the multi-fracture deflection increase greatly due to the high-volume pump increasing the internal water pressure of the fracture and short spacing, which forms the longitudinal hydraulic fracture. The middle fracture restrained nearly propagates in the direction perpendicular to the maximum principal stress and tends to stop propagating, while the extending distance between the middle fracture at both sides is longer. (3) The deflection angle declines because the induced stress is too difficult to change the original the stress field under the high stress difference. The subsequent propagation fluctuation is relatively stable, and the fracture is more likely to form a transverse hydraulic fracture parallel to the direction of the maximum principal stress. The research results can be used to optimize the design parameters of slotting multi-fracture and provide technical reference for oil and gas exploitation of sandstone reservoirs under different geological conditions and hydraulic fracturing of hard roof in coal mines to control the strong mine pressure.
- horizontal well /
- slotted fracturing /
- fracture mutual stress interference /
- crack deflection

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0. 引言

瑞利波传播过程中主要引起水平方向振动，这种振动与建筑物的自然频率相近时，会引起共振，导致更大的破坏。为进一步明确瑞利波传播特性和衰减规律，大量学者对其进行理论分析^[1]、数值模拟^[2]和试验研究^[3]。瑞利波在介质中的传播受多种因素影响，主要包括介质的弹性参数、密度、重力以及分层结构等^[4]。

桩基在瑞利波作用下的动力响应的研究成为抗震过程中的重要组成部分。Yang等^[5]在考虑桩顶柔性约束的情况下，研究了非饱和土-桩体系在瑞利波作用下的动力响应。此外Cai等^[6]采用刚性排桩作为隔离瑞利波的方法，考虑了软土地质所产生的影响。

通过以上讨论发现：考虑竖向荷载影响的瑞利波作用下桩水平动力响应分析比较复杂。且对于软土地基中桩在瑞利波作用下的动力响应的研究也相对较少。因此，本文建立考虑竖向荷载对瑞利波作用下饱和软土地基中桩的水平动力响应影响的计算模型。研究结果为瑞利波作用下桩基动力分析和设计提供理论指导。

1. 计算模型

1.1 问题描述

在瑞利波的作用下饱和土-桩体系的数学模型如图 1所示，其中桩长为L，半径为r_a，截面积为A_p，密度为ρ_p，杨氏模量为E_p，剪切模量为G_p。桩体模型简化为桩端为固定边界条件的Timoshenko梁，桩顶视为质量为M₀的刚性块体。桩周饱和土视为线弹性材料，泊松比、阻尼比和剪切模量分别取为 ${\nu _0}$ ， ${\varepsilon _0}$ ，s和G_s。其中瑞利波的作用方式是以波的形式穿过土体传播到桩体上。

图 1 瑞利波作用下桩在饱和土中的计算模型

Figure 1. Computational model for pile in saturated soils under Rayleigh waves

下载: 全尺寸图片幻灯片

1.2 基本方程

根据Biot’s理论，运动方程可以表示为

${\sigma _{ij, j}} = {\rho _{\text{s}}}{\ddot u_i} + {\rho _{\text{f}}}{\ddot w_i} \text{, }\;$

(1)

$- {p_{{\text{f}}, i}} = {\rho _{\text{f}}}{\ddot u_i} + {m_1}{\ddot w_i} + {r_1}{\dot w_i} 。$

(2)

式中： ${u_i}$ 为土体位移； ${w_i}$ 为流体位移； $\lambda$ ，μ为拉梅常数；m₁= ${\rho _{\text{f}}}$ /n为孔隙中流体密度与孔隙率的比值；r₁= ${\rho _{\text{f}}}$ g/k_d，k_d为土壤达西渗透系数， ${p_{\text{f}}}$ 为孔隙中流体压力；α和M分别为两相材料的Biot’s参数。

基于Timoshenko理论，建立桩段控制微分方程^[7]：

$\begin{array}{l} {A_{\text{p}}}{G_{\text{p}}}{k_{\text{0}}}\left( {\frac{{{\partial ^2}{u_{\text{p}}}}}{{\partial {z^2}}} - \frac{{\partial {\theta _{\text{p}}}}}{{\partial z}}} \right) + {\rho _{\text{p}}}{A_{\text{p}}}\frac{{{\partial ^2}{u_{\text{p}}}}}{{\partial {t^2}}} +\\ \;\;\;\;\;\;\;\;\;\;\;\;({k_{\text{h}}} + {\text{i}}{c_{\text{h}}})({u_{\text{p}}} - {u_{\text{r}}}\cos \theta ) = 0 \text{, }\; \end{array}$

(3)

$\begin{array}{l} {A_{\text{p}}}{G_{\text{p}}}{k_{\text{0}}}\left( {\frac{{\partial {u_{\text{p}}}}}{{\partial z}} - {\theta _{\text{p}}}} \right) - {\rho _{\text{p}}}{I_{\text{p}}}\frac{{{\partial ^2}{u_{\text{p}}}}}{{\partial {t^2}}} - \\ \;\;\;\;\;\;\;\;\;\;\;\;{E_{\text{p}}}{I_{\text{p}}}\frac{{{\partial ^2}{\theta _{\text{p}}}}}{{\partial {{\text{z}}^2}}} + {A_{\text{p}}}P\frac{{\partial {u_{\text{p}}}}}{{\partial z}} = 0 。 \end{array}$

(4)

式中： ${u_{\text{p}}}(z, t) = {\bar u_{\text{p}}}(z){{\text{e}}^{{\text{i}}\omega t}}$ ， ${\theta _{\text{p}}}(z, t) = {\bar \theta _{\text{p}}}(z){{\text{e}}^{{\text{i}}\omega t}}$ 分别为桩的水平位移和转角；k₀为Timoshenko梁理论中的修正剪切因子；P为由上部结构质量引起的竖向荷载。

2. 方程的求解

由式（2）得：

${w_{\text{r}}} = \frac{{{\omega ^2}{\rho _{\text{f}}}{u_{\text{r}}}}}{\vartheta } - \frac{1}{\vartheta }\frac{{\partial {p_{\text{f}}}}}{{\partial r}} \text{, }\; {w_\theta } = \frac{{{\omega ^2}{\rho _{\text{f}}}{u_\theta }}}{\vartheta } - \frac{1}{\vartheta }\frac{{\partial {p_{\text{f}}}}}{{r\partial \theta }} 。$

(5)

式中： $\eta _{\text{a}}^2 = 2/(1 - \nu )$ ， $\vartheta = {\text{i}}\omega {\rho _{\text{f}}}g/{k_{\text{d}}} - {m_1}{\omega ^2}$ 。

将式（5）代入式（1），（2）结合本构方程得

$\begin{array}{c} \eta _{\text{b}}^{\text{2}}\frac{\partial }{{\partial r}}\left[ {\frac{1}{r}\left( {\frac{{\partial (r{u_r})}}{{\partial r}} + \frac{{\partial {u_\theta }}}{{\partial \theta }}} \right)} \right] - \frac{1}{{{r^2}}}\frac{\partial }{{\partial \theta }}\left[ {\frac{{\partial (r{u_\theta })}}{{\partial r}} - \frac{{\partial {u_r}}}{{\partial \theta }}} \right] +\\ \frac{{{\partial ^2}{u_r}}}{{\partial {z^2}}} + {c_{\text{1}}}\frac{{{\rho _{\text{s}}}{\omega ^2}}}{{{G_s}}}{u_r} - {c_{\text{2}}}\frac{{\partial {p_{\text{f}}}}}{{\partial r}} = 0 \text{, }\; \end{array}$

(6)

$\begin{array}{c} \frac{\partial }{{\partial r}}\left[ {\frac{1}{r}\left( {\frac{{\partial (r{u_\theta })}}{{\partial r}} - \frac{{\partial {u_{\text{r}}}}}{{\partial \theta }}} \right)} \right] + \eta _{\text{b}}^{\text{2}}\frac{1}{{{r^2}}}\frac{\partial }{{\partial \theta }}\left[ {\frac{{\partial (r{u_r})}}{{\partial r}} + \frac{{\partial {u_\theta }}}{{\partial \theta }}} \right] +\\ \frac{{{\partial ^2}{u_\theta }}}{{\partial {z^2}}} + {c_{\text{1}}}\frac{{{\rho _{\text{s}}}{\omega ^2}}}{{{G_{\text{s}}}}}{u_\theta } - {c_{\text{2}}}\frac{{\partial {p_{\text{f}}}}}{{r\partial \theta }} = 0 。 \end{array}$

(7)

式中： ${c_{\text{1}}} = 1 + \frac{{{\omega ^2}\rho _{\text{f}}^2}}{{{\rho _{\text{s}}}\vartheta }};{c_{\text{2}}} = \frac{\alpha }{{2\mu }}\frac{{1 - 2\nu }}{{1 - \nu }} + \frac{{{\omega ^2}{\rho _{\text{f}}}}}{{{\mu _{\text{s}}}\vartheta }};$ $\eta _{\text{b}}^{\text{2}} = \frac{{2 - \nu }}{{1 - \nu }}$

忽略流体的垂直位移：

$\begin{array}{l} \left( {\frac{{{\omega ^{\text{2}}}{\rho _{\text{f}}}}}{\vartheta } - \alpha \frac{{1 - 2{\nu _{\text{0}}}}}{{1 - {\nu _{\text{0}}}}}} \right)\left( {\frac{{\partial {u_r}}}{{\partial r}} + \frac{{{u_r}}}{r} + \frac{1}{r}\frac{{\partial {u_\theta }}}{{\partial \theta }}} \right) \\ \;\;\;\;\;\;\;\;\;\;\;\; = \frac{1}{\vartheta }{\nabla ^2}{p_{\text{f}}} + \left( {\frac{{{\alpha ^2}}}{\lambda }\frac{{{\nu _{\text{0}}}}}{{1 - {\nu _{\text{0}}}}} + \frac{1}{M}} \right){p_{\text{f}}} 。 \end{array}$

(8)

水平位移和切向位移用两个势函数Φ和Ψ表示为

${u_r} = \frac{{\partial \phi }}{{\partial r}} + \frac{1}{r}\frac{{\partial \psi }}{{\partial \theta }} \text{, }\; {u_\theta } = \frac{1}{r}\frac{{\partial \phi }}{{\partial \theta }} - \frac{{\partial \psi }}{{\partial {\text{r}}}} 。$

(9)

将式（9）代入式（6）~（8），联立得

$\eta _{\text{b}}^2{\nabla ^2}\phi + \frac{{{\partial ^2}\phi }}{{\partial {z^2}}} + {c_{\text{1}}}\frac{{{\rho _{\text{s}}}}}{{{G_{\text{s}}}}}{\omega ^2}\phi = {c_{\text{2}}}{p_{\text{f}}} \text{, }\;$

(10)

${\nabla ^2}\psi + \frac{{{\partial ^2}\psi }}{{\partial {z^2}}} + {c_{\text{1}}}\frac{{{\rho _{\text{s}}}}}{{{G_{\text{s}}}}}{\omega ^2}\psi = 0 \text{, }\;$

(11)

${\nabla }^{2}{p}_{\text{f}}+\left(\frac{\vartheta {\alpha }^{2}}{\lambda }\frac{{\nu }_{\text{0}}}{1-{\nu }_{\text{0}}}+\frac{\vartheta }{M}\right){p}_{\text{f}}=\left({\omega }^{\text{2}}{\rho }_{\text{f}}-\alpha \vartheta \frac{1-2{\nu }_{\text{0}}}{1-{\nu }_{\text{0}}}\right){\nabla }^{2}\phi 。$

(12)

采用分离变量法，设 $\phi (r, \theta , z) = {\phi _1}(r, \theta )Z(z),$ 假设 $\varphi (r, \theta , z) = {\varphi _1}(r, \theta )Z(z)$ ， $\frac{{{{\text{d}}^2}Z}}{{{\text{d}}{z^2}}} + {a^2}Z = 0$ ，将Z(z)表示为

$Z(z)=B_0 \cos (a z)+B_1 \sin (a z)。$

(13)

联立式（10）~（12）并将式（13）代入得

${\nabla ^2}{\psi _{\text{1}}} - \left[ {{a^2} - {c_{\text{1}}}\frac{{{\rho _{\text{s}}}}}{{{G_{\text{s}}}}}{\omega ^2}} \right]{\psi _{\text{1}}} = 0 。$

(14)

$({\nabla ^{\text{2}}} - \zeta _{{\text{11}}}^{\text{2}})({\nabla ^{\text{2}}} - \zeta _{{\text{12}}}^{\text{2}}){\phi _{\text{1}}} = {\text{0}}$

(15)

这里设： ${d_{11}} = \frac{{{c_1}}}{{\eta _{\text{b}}^{\text{2}}}}\frac{{{\rho _{\text{s}}}}}{{{G_{\text{s}}}}}{\omega ^2} + \frac{\vartheta }{M} - \frac{{{a^2}}}{{\eta _{\text{b}}^{\text{2}}}} + \frac{{{c_2}\alpha \vartheta }}{{\eta _{\text{b}}^{\text{2}}}}\frac{{1 - 2{\nu _{\text{0}}}}}{{1 - {\nu _{\text{0}}}}} - \frac{{{c_2}{\omega ^{\text{2}}}{\rho _{\text{f}}}}}{{\eta _{\text{b}}^{\text{2}}}}$ + $\frac{\vartheta {\alpha }^{2}}{\lambda }\frac{{\nu }_{\text{0}}}{1-{\nu }_{\text{0}}}$ ， ${d_{{\text{12}}}} = \frac{\vartheta }{{\eta _{\text{b}}^{\text{2}}}}\left[ {{c_1}\frac{{{\rho _{\text{s}}}}}{{{G_{\text{s}}}}}{\omega ^2}\left( {\frac{{{\alpha ^2}}}{\lambda }\frac{{{\nu _{\text{0}}}}}{{1 - {\nu _{\text{0}}}}} + \frac{{\text{1}}}{M}} \right) - } \right.$ $\left. {\frac{{{a^2}{\alpha ^2}}}{\lambda }\frac{{{\nu _{\text{0}}}}}{{1 - {\nu _{\text{0}}}}} - \frac{{{a^2}}}{M}} \right] \text{, }\; \zeta _{{\text{11}}}^{\text{2}} = \frac{{ - {d_{11}} + \sqrt {d_{11}^2 - 4{d_{12}}} }}{2}$ , $\zeta _{{\text{12}}}^{\text{2}} =\frac{{ - {d_{11}} - \sqrt {d_{11}^2 - 4{d_{12}}} }}{2}$ ， ${v_{\text{s}}} = \sqrt {\frac{{{G_{\text{s}}}}}{{{\rho _{\text{s}}}}}}$ ，从而得到

$({\nabla ^{\text{2}}} + k_{{\text{a1}}}^2)({\nabla ^{\text{2}}} + k_{{\text{a2}}}^2){\phi _{\text{1}}} = {\text{0}} \text{, }\;$

(16)

$({\nabla ^2} + k_{{\text{b1}}}^2){\psi _{\text{1}}} = 0 。$

(17)

通过算子分解理论和分离变量法得：

$\begin{array}{l} {\phi _{\text{1}}} = {A_{{\text{s1}}}}\exp ( - {s_1}r\sin \theta - {\text{i}}{k_{\text{R}}}r\cos \theta ) +\\ \;\;\;\;{A_{{\text{s2}}}}\exp ( - {s_2}r\sin \theta - {\text{i}}{k_{\text{R}}}r\cos \theta ) \text{, }\; \end{array}$

(18)

${\psi _{\text{1}}} = {B_{\text{s}}}\exp ( - \gamma r\sin \theta - {\text{i}}{k_{\text{R}}}r\cos \theta ) 。$

(19)

式中：A_s1，A_s2，B_s分别为与边界条件有关的待定常数；k_R，V_R=ω/k_R分别为瑞利波的复波速和相速度； ${s_i} =$ $\sqrt {k_{\text{R}}^2 - k_{{\text{a}}i}^2}$ (i=1, 2)和 $\gamma = \sqrt {k_{\text{R}}^2 - k_{{\text{b1}}}^2}$ ，分别为压缩波和剪切波对应的衰减指数； ${k}_{{a}_{1}}^{2}=-{\zeta }_{11}^{2}, {k}_{{a}_{2}}^{2}=-{\zeta }_{12}^{2}，$ $k_{{b_{\text{1}}}}^2 =$ $- {a^2} + {\kappa _{\text{1}}}\frac{{{\rho _{\text{s}}}}}{{{G_{\text{s}}}}}{\omega ^2}$ 。

因此由式（13），结合边界条件即 ${\left. {{\tau _{rz}}} \right|_{z = 0}} = 0$ ， ${\left. {{u_r}} \right|_{z = L}} = {\left. {{u_\theta }} \right|_{z = L}} = 0$ 可知

$\left. \begin{array}{l} {B}_{0}={B}_{1}=0且\mathrm{cos}(aL)=0\text{ }\text{, }\;\\ {a}_{m}=\frac{(2m-1){\rm{ \mathsf{ π}}}}{2L}, m=1, 2, 3\text{ }。 \end{array} \right\}$

(20)

$\begin{array}{l} \varphi = [{A_{{\text{s1}}}}\exp ( - {s_1}r\sin \theta - {\text{i}}{k_{\text{R}}}r\cos \theta ) +\\ \;\;{A_{{\text{s2}}}}\exp ( - {s_2}r\sin \theta - {\text{i}}{k_{\text{R}}}r\cos \theta )]\sin ({a_m}z) \text{, }\; \end{array}$

(21)

$\psi = {B_{\text{s}}}\exp ( - \gamma r\sin \theta - {\text{i}}{k_{\text{R}}}r\cos \theta )\sin ({a_m}z) 。$

(22)

将式（21）代入式（10）得

$\begin{array}{l} \;\;\;\;{p_{\text{f}}} = \left[ { - \frac{{a_m^2}}{{{c_2}}} + \frac{{{c_1}}}{{{c_2}}}{{\left( {\frac{\omega }{{{V_{\text{s}}}}}} \right)}^2}} \right]\left[ {{A_{{\text{s1}}}}\exp \left( \begin{array}{l} - {s_1}r\sin \theta \hfill \\ - {\text{i}}{k_R}r\cos \theta \hfill \end{array} \right) + } \right. \hfill \\ \;\;\;\;\left. {{A_{{\text{s2}}}}\exp \left( \begin{array}{l} - {s_2}r\sin \theta \hfill \\ - {\text{i}}{k_{\text{R}}}r\cos \theta \hfill \end{array} \right)} \right]\sin ({a_m}z) + \hfill \\ \frac{{\eta _{\text{b}}^{\text{2}}}}{{{c_2}}}(s_1^2 - k_{\text{R}}^{\text{2}}){A_{{\text{s1}}}}\exp ( - {s_1}r\sin \theta - {\text{i}}{k_{\text{R}}}r\cos \theta )\sin ({a_m}z) + \\ \frac{{\eta _{\text{b}}^{\text{2}}}}{{{c_2}}}(s_2^2 - k_{\text{R}}^{\text{2}}){A_{{\text{s2}}}}\exp ( - {s_2}r\sin \theta - {\text{i}}{k_{\text{R}}}r\cos \theta )\sin ({a_m}z) 。\end{array}$

(23)

通过式（9）结合本构方程及式（21），（22）并代入边界条件 ${\left. {{\tau _{rz}}} \right|_{z = 0}} = 0, {\left. {{u_\theta }} \right|_{z = L}} = 0$ 可知

${A_{{\text{s2}}}} = {d_1}{A_{{\text{s}}1}}, {B_{\text{s}}} = {d_2}{A_{{\text{s1}}}} 。$

(24)

联立式（9）即可求得位移转角u_r、u_θ表达式。

因此自由场饱和土在瑞利波作用下位移由体积平均原理得

${\bar u_r} = (1 - n){u_r} + n{w_r} \text{, }\;$

(25)

${\bar u_\theta } = (1 - n){u_\theta } + n{w_\theta } 。$

(26)

3. 瑞利波作用下桩的动力响应

3.1 水平阻力

桩在瑞利波作用下的动力响应由桩周土体决定，采用动力Winkler模型来描述饱和土桩体系的水平动力响应，进而计算单位桩长上的水平阻力^[8-9]。因此设单位桩长上的水平阻力为

${q_h} = ({k_h} + {i} {c_h}){u_{{\bar{\text p}}}} 。$

(27)

式中：k_h和c_h分别为Winkler模型中弹簧刚度和阻尼系数。

3.2 考虑竖向荷载影响的桩的动力响应

将式（25），（27）代入式（3），（4）并省略因子e^iωt得

$\frac{{{{\text{d}}^{\text{4}}}{{\bar u}_{\text{p}}}}}{{{\text{d}}{z^4}}} + W\frac{{{{\text{d}}^2}{{\bar u}_{\text{p}}}}}{{{\text{d}}{z^2}}} + J{\bar u_{\text{p}}} = {H_1}{A_{{\text{s1}}}}\sin ({a_m}z) \text{, }\;$

(28)

$\frac{{{{\text{d}}^4}{{\bar \theta }_{\text{p}}}}}{{{\text{d}}{z^4}}} + W\frac{{{{\text{d}}^2}{{\bar \theta }_{\text{p}}}}}{{{\text{d}}{z^2}}} + J{\bar \theta _{\text{p}}} = {H_2}{A_{{\text{s1}}}}\cos ({a_m}z) 。$

(29)

其中， $W = \frac{{{\rho _{\text{P}}}{\omega ^2}}}{{{E_{\text{p}}}}} + \frac{{{\rho _{\text{P}}}{\omega ^2}}}{{{k_{\text{0}}}{G_{\text{p}}}}} + \frac{{{A_{\text{p}}}P}}{{{E_{\text{p}}}{I_{\text{p}}}}} - \frac{{({k_{\text{h}}} + {\text{i}}{c_{\text{h}}})}}{{{k_{\text{0}}}{A_{\text{p}}}{G_{\text{p}}}}}$ ， $J =$ $\frac{{\rho _{\text{p}}^{\text{2}}{\omega ^4}}}{{{k_{\text{0}}}{G_{\text{p}}}{E_{\text{p}}}}} - \frac{{{\rho _{\text{P}}}{A_{\text{p}}}{\omega ^2}}}{{{E_{\text{p}}}{I_{\text{p}}}}} - \left( {\frac{{{\rho _{\text{P}}}{\omega ^2}}}{{{k_{\text{0}}}{A_{\text{p}}}{G_{\text{p}}}{E_{\text{p}}}}} - \frac{1}{{{E_{\text{p}}}{I_{\text{p}}}}}} \right)({k_{\text{h}}} + {\text{i}}{c_{\text{h}}})$

方程（28）、（29）对应的解设为

$\begin{array}{c} {\bar u_{\text{p}}} = {M_{\text{1}}}\cos ({\lambda _{\text{1}}}z) + {M_{\text{2}}}\sin ({\lambda _{\text{1}}}z) + {M_{\text{3}}}ch({\lambda _{\text{2}}}z) +\\ {M_{\text{4}}}sh({\lambda _{\text{2}}}z) + {b_{\text{1}}}{A_{{\text{s1}}}}\sin ({a_m}z) \text{, }\; \end{array}$

(30)

$\begin{array}{c} {\bar \theta _{\text{p}}} = {M_{\text{1}}}{\chi _{\text{1}}}\sin ({\lambda _{\text{1}}}z) + {M_{\text{2}}}{\chi _{\text{2}}}\cos ({\lambda _{\text{1}}}z) + {M_{\text{3}}}{\chi _{\text{3}}}sh({\lambda _{\text{2}}}z) +\\ {M_{\text{4}}}{\chi _{\text{4}}}ch({\lambda _{\text{2}}}z) + {b_{\text{2}}}{A_{{\text{s1}}}}\cos ({a_m}z) \text{, }\; \end{array}$

(31)

式中： ${\lambda }_{1}=\sqrt{\frac{W+\sqrt{{W}^{2}-4J}}{2}}, {\lambda }_{2}=\sqrt{\frac{-W+\sqrt{{W}^{2}-4J}}{2}}，$ M_i(1，2，3，4)为与桩的边界条件有关的待定系数。其中 ${\chi }_{\text{1}}=-{\chi }_{\text{2}}=-\frac{{k}_{0}{A}_{\text{p}}{G}_{\text{p}}{\lambda }_{\text{1}}+{A}_{\text{p}}P{\lambda }_{\text{1}}}{{k}_{0}{A}_{\text{p}}{G}_{\text{p}}+{\rho }_{\text{p}}{I}_{\text{p}}{\omega }^{2}+{E}_{\text{p}}{I}_{\text{p}}{\lambda }_{\text{1}}^{2}}，$ ${\chi _{\text{3}}} =$ ${\chi _{\text{4}}} = \frac{{{k_0}{A_{\text{p}}}{G_{\text{p}}}{\lambda _{\text{2}}} + {A_{\text{p}}}P{\lambda _{\text{2}}}}}{{{k_0}{A_{\text{p}}}{G_{\text{p}}} + {\rho _{\text{p}}}{I_{\text{p}}}{\omega ^2} - {E_p}{I_{\text{p}}}\lambda _{\text{1}}^2}},$ ${b_{\text{1}}} = \frac{{{H_{\text{1}}}}}{{a_m^4 - Wa_m^2 + J}}$ ， ${b_{\text{2}}} = \frac{{{H_2}}}{{a_m^4 - Wa_m^2 + J}}$ 。

沿桩身的弯矩和剪力可由弹性力学推导为

$\begin{array}{l} {\bar M_{\text{p}}} = {E_{\text{P}}}{I_P}\left[ {{M_1}{\chi _1}{\lambda _1}\cos ({\lambda _1}z) - {M_2}{\chi _2}{\lambda _1}\sin ({\lambda _1}z)} \right. +\\ {M}_{3}{\chi }_{3}{\lambda }_{2}ch({\lambda }_{2}z)+{M}_{4}{\chi }_{4}{\lambda }_{2}sh({\lambda }_{2}z)-{b}_{\text{2}}{a}_{m}{A}_{\text{s1}}\mathrm{sin}({a}_{m}z)]。 \end{array}$

(32)

$\begin{array}{l} {{\bar Q}_{\text{p}}} = {k_0}{A_{\text{P}}}{G_{\text{P}}}\left[ { - {M_1}({\lambda _1} + {\chi _1})\sin ({\lambda _1}z) + {M_2}({\lambda _1} - {\chi _2})} \right. \cdot \hfill \\ \cos ({\lambda _1}z) + {M_3}({\lambda _2} - {\chi _3})sh({\lambda _2}z) + {M_4}({\lambda _2} - {\chi _4})ch({\lambda _2}z) + \hfill \end{array} \left. \\ \;\;\;\;\;\;\;\;\;\;{({b_{\text{1}}}{a_m} - {b_{\text{2}}}){A_{s{\text{1}}}}\cos ({a_m}z)} \right] 。$

(33)

桩顶柔性状态下，桩端处于固定状态，桩的边界条件为：

$\left. \begin{array}{l} {\overline{M}}_{\text{p}}(z)={K}_{1}{\overline{\theta }}_{c}+{K}_{2}\left[{\overline{\theta }}_{\text{p}}(0)-{\overline{\theta }}_{c}\right]\text{, }\;z=0\text{ }\text{, }\;\\ {\overline{Q}}_{\text{p}}(z)-{A}_{\text{p}}P{\overline{\theta }}_{\text{p}}(z)+{\omega }^{2}{M}_{\text{0}}{\overline{u}}_{\text{p}}(z)=0, z=0\text{ }\text{, }\;\\ {\overline{u}}_{\text{p}}(z)=0\text{, }\;{\overline{\theta }}_{\text{p}}(z)=0, z=L\text{ }。 \end{array} \right\}$

(34)

将边界条件代入桩体水平位移、旋转角度沿桩身的弯矩和剪力表达式，由此可以导出所有未确定的待定参数M_i（i=1, 2, 3, 4），A_s1。

4. 数值分析及讨论

通过数值算例验证，分析竖向荷载及桩顶柔性约束下各参数对于桩身位移、转角和弯矩的影响。各相关参数属性： ${\rho _{\text{s}}}$ =2.7×10³ kg/m³， ${\rho _{\text{f}}}$ =2.2×10³ kg/m³，E_a=2×10⁹ Pa， ${\nu _{\text{0}}}$ =0.4，n=0.4，α=0.9，k_d=1×10^-8 m/s，K_s=3.6×10¹⁰ Pa，K_f=2×10⁹ Pa，G_s=2.5×10⁶ Pa，d=1 m，r_a=0.5 m，L=20 m， ${\rho _{\text{p}}}$ =2.5×10³ kg/m³，E_p= 2.5×10¹⁰ Pa，k₀=0.75，I_p=π/64，ν_p=0.2，A_p=0.25π，M₀=1×10⁵ kg，P=10×10⁶ Pa。引入无量纲频率a₀=ωL_a/v_s。

4.1 验证

为验证模型的准确性，将Makris^[10]解与模型进行比较。在相同条件下，对本文的计算模型进行验证。图 2将水平位移与Makris^[10]解的结果进行比较。可以看出两种解法有较高的一致性。

图 2 本文解与Makris解的比较

Figure 2. Comparison between authors' and Makris' solutions

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4.2 参数分析

图 3，4给出了瑞利波作用下，竖向荷载对单桩的水平动力响应的影响，其研究了桩长、竖向荷载大小和无量纲频率对桩基水平振动的影响。从图 3可以看出，随着竖向荷载的增加，位移、转角和弯矩都在增大，且增加的趋势也越来越大。随着深度的增加，竖向荷载改变所引起的变化不再明显，最终趋于稳定。

图 3 竖向荷载对单桩横向响应沿桩长变化的影响

Figure 3. Influences of vertical load on lateral response of a single pile along pile length

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图 4 竖向荷载对单桩水平响应随无因次频率变化的影响

Figure 4. Influences of vertical load on horizontal response of a single pile with dimensionless frequency

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图 4显示了垂直载荷对位移、转角和弯矩的影响。当频率等于固有频率时，发生共振。在共振区，当竖向荷载较小时，随竖向荷载的增大，桩顶位移和桩端弯矩整体呈增大趋势。随无量纲频率的增大，竖向荷载的改变引起水平位移和转角的变化将不再明显。

5. 结论

考虑竖向荷载对瑞利波作用下饱和软土中单桩结构水平动力响应的影响，建立瑞利波作用下饱和软土中桩顶柔性约束下单桩水平动力响应的计算模型。通过数值计算结果，得出以下3点结论。

（1）随深度的增加，位移和转角先减小后增大；弯矩整体呈减小趋势；最终接近桩端固结端时桩的水平动力响应趋于稳定。

（2）随无量纲频率的增加，位移、转角和弯矩在共振区发生共振后最终趋于稳定。

（3）随竖向荷载的增加，位移、转角和弯矩都在增大。但是这种改变所引起的变化随着深度的增加不再明显，最终趋于稳定。

图 1 单裂缝产生诱导应力场示意图

Figure 1. Induced stress field generated by single fracture with water pressure

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图 2 多裂缝扩展及其相互影响应力模型

Figure 2. Multi-fracture propagation and its mutual influence stress model

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图 3 四维水射流割缝

Figure 3. Four-dimensional water jet slitting

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图 4 大尺寸真三轴压裂模拟试验系统

Figure 4. Large-scale true triaxial fracturing simulation test system

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图 5 室内真三轴压裂试验技术路线

Figure 5. Technical route of indoor true triaxial fracturing test

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图 6 试样样品

Figure 6. Sample of sandstone

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图 7 不同割缝间距水力压裂–时间曲线

Figure 7. Hydraulic fracturing-time curves under different spacings

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图 8 段间距对裂缝扩展形态影响

Figure 8. Influences of different spacings on fracture propagation morphology

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图 9 大排量和高应力差的水力压裂–时间曲线

Figure 9. Hydraulic fracturing-time curves under high displacement and large stress difference

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图 10 大排量和高应力差对裂缝扩展形态影响

Figure 10. Influences of large displacement and high stress difference on fracture propagation morphology

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图 11 数值模拟平面模型

Figure 11. Plane model of numerical simulation

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图 12 不同间距裂缝扩展形态和应力云图

Figure 12. Fracture propagation morphology and stress cloud under different spacings

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图 13 大排量和高应力差裂缝扩展形态和应力云图

Figure 13. Fracture propagation morphology and stress cloud under large displacement and high stress difference

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表 1 不同割缝间距下钻孔深度

Table 1 Drilling depths under different slot spacings (cm)

裂缝间距 15 30 45

钻孔深度 172.5 195 217.5

下载: 导出CSV

表 2 水力压裂试验参数

Table 2 Parameters of hydraulic fracturing tests

组数 σ_V/MPa σ_H/MPa σ_h/MPa 缝间距/mm 缝深/ mm 压裂排量/(mL·min^-1)

S-1 10 10 8 15 10 60
S-2 10 10 8 30 10 60
S-3 10 10 8 45 10 60
S-4 10 10 8 15 10 120
S-5 10 10 4 30 10 60

下载: 导出CSV

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0. 引言
1. 计算模型
1.1 问题描述
1.2 基本方程
2. 方程的求解
3. 瑞利波作用下桩的动力响应
3.1 水平阻力
3.2 考虑竖向荷载影响的桩的动力响应
4. 数值分析及讨论
4.1 验证
4.2 参数分析
5. 结论

组数	σ_V/MPa	σ_H/MPa	σ_h/MPa	缝间距/mm	缝深/ mm	压裂排量/(mL·min^-1)
S-1	10	10	8	15	10	60
S-2	10	10	8	30	10	60
S-3	10	10	8	45	10	60
S-4	10	10	8	15	10	120
S-5	10	10	4	30	10	60

组数	σ_V/MPa	σ_H/MPa	σ_h/MPa	缝间距/mm	缝深/ mm	压裂排量/(mL·min^-1)
S-1	10	10	8	15	10	60
S-2	10	10	8	30	10	60
S-3	10	10	8	45	10	60
S-4	10	10	8	15	10	120
S-5	10	10	4	30	10	60

致密砂岩水平井多裂缝扩展及转向规律研究 English Version

作者简介: 夏彬伟(1978—),男,工学博士,副教授,博士生导师,从事水力压裂治理岩层顶板方面研究。E-mail：xbwei33@cqu.edu.cn

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