The model of spatial analogous hyperboloid for three-dimensional rock strata movement in mining engineering
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摘要: 基于大量现场调研及岩层移动内外“类双曲线”移动模型分析，本文对哈拉沟煤矿厚松散岩层煤层开挖展开了物理相似模型试验，验证了内外“类双曲线”整体移动模型的有效性；将其推广到三维空间，提出采动岩层全空间“类双曲面”立体移动模型。该模型包含“类单叶”、“类双叶”两类双曲面，能近似描述不同岩性条件下全空间采动岩层立体移动与地表沉陷特征。“类单叶双曲面”模型为空间岩层水平移动边界模型，隐含于厚松散层覆岩内部；“类双叶双曲面”模型为空间地表沉降与覆岩裂隙拱、冒落拱垂向移动边界模型，均在主关键层近似关于“原点”对称，具体表现为采掘扰动下地表沉陷及采场围岩拱形垮落等外部形态。通过理论分析、相似模型试验及3DEC数值模拟，本文建立并验证了厚松散层近水平煤层开采条件下空间采动岩层立体移动与地表沉陷“类双曲面”模型。同时，分析了共渐近锥面“类双曲面”模型的构成条件、影响因素及整体运移规律。研究结果表明：在薄基岩厚松散层近水平煤层开采条件下，“类双曲面”理论模型与3DEC模拟结果吻合较好，表明采动空间覆岩运移和地表沉降呈“类双曲面”特征。Abstract: Based on a large number of field investigations and the analysis of the "analogous hyperbolic" overall movement model proposed by the author, this paper further extends it to three-dimensional space, and proposes a full-space "analogous hyperboloid"three-dimensional movement model of the mining strata.There are two types of hyperboloids, "quasi-single-leaf"and "quasi-double-leaf", which can approximately describe the three-dimensional movement and surface subsidence characteristics of full-space mining strata under different lithological conditions.The "analogous single leaf hyperboloid" model is a horizontal moving boundary model of the spatial rock layer, which is implicit in the thick loose layer overlying rock; the "analogous double leaf hyperboloid" model is the vertical movement of the spatial surface subsidence and the overlying fissure arch and caving arch.The boundary models are approximately symmetrical with the "origin" in the main key layer, and are embodied in external forms such as surface subsidence under mining disturbance and arch caving of the surrounding rock in the stope. Through theoretical analysis, similar model test and 3DEC numerical simulation, this paper deduces and proves in detail the "analogous hyperboloid" model of spatially mined rock strata movement and surface subsidence under the condition of near-horizontal coal seam mining in thick loose layers.At the same time, this study analyzed the composition conditions, influencing factors and overall migration law of the co-asymptotic conical surface "analogous hyperboloid" model.Results show that under the condition of near-horizontal coal seam mining with thin bedrock and thick loose layer, the "quasi-hyperboloid" theoretical model is in good alignment with the 3DEC simulation results, indicating that the overlying rock migration and surface subsidence in space mining are "analogous hyperboloid" feature.
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图 4 哈拉沟煤矿相似模型试验
Figure 4. State diagram of similarity model experiment before and after mining
图 6 采动岩层空间“类单叶双曲面”模型
k—x方向主关键层破断长度的1/2；p—y方向主关键层破断长度的1/2；r—工作面煤层到地面高度的1/2；Di与Dj—分别为x方向和y方向的地表沉陷距离；x、y、z—分别为工作面推进方向、工作面布置方向以及埋深方向
Figure 6. Analogous single leaf hyperboloid model of mining strata space
图 7 采动岩层空间“类双叶双曲面”模型
Figure 7. Analogous double leaf hyperboloid model of mining strata space
图 8 不同工作面布置下俯视剖面变化
Figure 8. Variation of top view section under different working face arrangements
图 10 岩层移动与地表沉降“类双曲面”形成位移云图
Figure 10. The "hyperboloid" of rock strata movement and surface subsidence forms the displacement cloud map
序号 岩性 厚度Hr/m 抗压强度
σc/MPa抗拉强度
σt/MPa弹性模量
E/GPa泊松比μ 内摩
擦角φ/(°)黏聚力
c/MPa密度
ρ/(kg·m-3)1 风积沙 0~12 0 0 0.05 0.37 21.03 0 1 830 2 细砂岩 12~27 38.74 2.16 6.00 0.36 36.12 4.99 2 161 3 中砂岩2 27~35 31.69 1.96 5.96 0.34 38.11 5.33 2 296 4 泥岩2 35~44 46.08 5.63 6.06 0.32 41.43 2.08 2 316 5 中砂岩1 44~55 24.62 2.86 3.95 0.35 38.08 3.70 2 048 6 泥岩1 55~61 56.83 1.51 6.51 0.33 40.30 3.31 2 291 7 砂岩2 61~69 55.62 2.63 5.98 0.33 40.99 2.65 2 390 8 煤层 69~74 39.59 2.31 3.21 0.32 27.06 1.37 1 482 9 砂岩1 74~124 54.59 2.58 6.06 0.33 39.56 2.78 2 298 
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表 2 平行截割“类单叶双曲面”模型
Table 2. Parallel cutting of analogous single leaf hyperboloid model
平行截割 第1组 第2组 第3组 坐标平面 截面位置 z=0 x=0 y=0 方程 $\left\{\begin{array} { l } { \frac { x ^ { 2 } } { k ^ { 2 } } + \frac { y ^ { 2 } } { p ^ { 2 } } = 1 } \\ { z = 0 } \end{array}\right.$ $\left\{\begin{array} { l } { \frac { y ^ { 2 } } { p ^ { 2 } } - \frac { z ^ { 2 } } { r ^ { 2 } } = 1 } \\ { x = 0 } \end{array} \right.$ $ \left\{\begin{array}{l} \frac{x^2}{k^2}-\frac{z^2}{r^2}=1 \\ y=0 \end{array}\right.$ 任意平面 截面位置 z=h x=h y=h 方程 $\left\{\begin{array}{l}\frac{x^2}{k^2}+\frac{y^2}{p^2}=1-\frac{h^2}{r^2} \\ z=h\end{array}\right.$ $\left\{\begin{array}{l}\frac{y^2}{p^2}-\frac{z^2}{r^2}=1-\frac{x^2}{k^2} \\ x=h\end{array}\right.$ $\left\{\begin{array}{l}\frac{x^2}{k^2}-\frac{z^2}{r^2}=1-\frac{h^2}{p^2} \\ y=h\end{array}\right.$ 截面图 
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表 3 平行截割“类双叶双曲面”模型
Table 3. Parallel cutting of analogous double leaf hyperboloid model
平行截割 第1组 第2组 第3组 坐标平面 截面位置 z=0 x=0 y=0 方程 不相交 $\left\{\begin{array} { l } { \frac { z ^ { 2 } } { t ^ { 2 } } - \frac { y ^ { 2 } } { s ^ { 2 } } = 1 } \\ { x = 0 } \end{array}\right.$ $\left\{\begin{array}{l} \frac{z^2}{t^2}-\frac{x^2}{w^2}=1 \\ y=0 \end{array}\right.$ 任意平面 截面位置 z=h x=h y=h 方程 $\left\{\begin{array} { l } { \frac { x ^ { 2 } } { w ^ { 2 } } + \frac { y ^ { 2 } } { s ^ { 2 } } = - 1 + \frac { z ^ { 2 } } { t ^ { 2 } } } \\ { z = h } \end{array} \right.$ $\left\{\begin{array} { l } { \frac { y ^ { 2 } } { s ^ { 2 } } - \frac { z ^ { 2 } } { t ^ { 2 } } = - 1 - \frac { x ^ { 2 } } { w ^ { 2 } } } \\ { x = h } \end{array}\right.$ $\left\{\begin{array}{l} \frac{x^2}{w^2}-\frac{z^2}{t^2}=-1-\frac{y^2}{s^2} \\ y=h \end{array}\right.$ 截面图 
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表 4 不同开采条件下“类单叶双曲面”模型的拓扑演化规律
Table 4. Topological evolution law of analogous single leaf yperboloid model under different mining conditions
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