The spatial and temporal evolution of thermal stress after granite emplacement and its influencing factors
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摘要: 花岗岩与岩浆热液型矿床、油气成藏等有密切的成因关系。高温岩浆侵位到较冷的围岩中会形成岩浆热场和热应力，但热应力的大小和其影响范围尚缺乏系统研究。随着岩浆热耗散、与周围地层达到热平衡后，热应力会逐渐消失，因而数值模拟是定量研究岩浆热应力的常见方法之一。以往模拟岩浆热应力时往往采用岩石在常温下的线性热膨胀系数，但这与高温下岩石线性热膨胀系数存在较大差距。文章利用FLAC3D软件模拟花岗质岩浆侵位至上地壳范围内引起的热应力。求解物理方程包括热传导方程与线性热弹性本构方程，其中热场可通过温度差和线性热膨胀系数改变应力场，但应力场的变化不影响热场（即热场与应力场的单向耦合）。通过一系列数值模拟实验考察围岩岩性（花岗岩或碳酸盐岩）、杨氏模量、热学参数和岩浆侵位深度如何影响岩浆在上覆围岩产生的热应力。数值实验结果表明：岩石热传导系数通过传热快慢影响热应力的变化；围岩的杨氏模量越大，热应力也越大；由于花岗岩的平均杨氏模量大于碳酸盐岩，所以围岩为花岗岩时产生的热应力要高于碳酸盐岩；围岩无论是花岗岩还是碳酸盐岩，其在高温条件下的线性热膨胀系数比常温时高约1个数量级，产生的热应力最高可达100 MPa。花岗岩浆侵位后，围岩温度逐渐升高，对应的热应力不断增大；随着与岩浆房距离的增大，热应力不断减小，影响范围为岩浆房上方2 km以内；侵位深度浅的岩浆房冷却较快，其产生的热应力更有利于上覆围岩裂隙的形成和扩展。综合数值模拟结果可知，岩浆侵位所产生的热应力可影响岩体2 km内的应力场，这一局部存在且短瞬的热应力促使围岩破裂，为热液流体成矿提供运移通道或容矿空间。Abstract:
Objective Granitic magmas are genetically associated with magmatic-hydrothermal deposits and oil and gas reservoirs. The emplacement of granitic magmas into cooler rocks produced thermal anomaly and thermal stress, yet systematic studies on the spatial and temporal evolution of thermal stress still need to be completed. Previous numerical modeling often used rocks' linear thermal expansion coefficient at room temperature, but this parameter is highly temperature-dependent and reaches much higher levels at high temperatures. Therefore, the magnitude of the thermal stress caused by magma emplacement needs to be re-examined. A series of numerical experiments were carried out to investigate how the surrounding rock's lithology (granite or carbonate rocks), Young's modulus, thermal parameters, and the depth of magma emplacement affect the thermal stress generated by the magma in the overlying surrounding rocks. Methods Because the magma cools and eventually reaches thermal equilibrium with the surrounding strata, numerical simulation is one of the common methods to examine the thermal stress after magma emplacement quantitatively. This article used FLAC3D software to simulate the thermal stress caused by the emplacement of granitic magma into the upper crust. The differential equations we solved include the thermal conduction and linear thermoelasticity equations. The models' thermal field influences the stress field through temperature difference and the linear thermal expansion coefficient. However, changes in the stress field do not affect the thermal field, i.e., one-way coupling between the thermal field and the stress field. Results (1) Heat transfer is quicker on wallrocks with high thermal conductivity, causing a faster change in thermal stress. Compared to the high-thermal-conductivity case, the same thermal stress can be produced on wallrocks with a lower thermal conductivity after a more extended period of magma cooling. (2) The thermal stress produced by the surrounding rock's Young's modulus of 80 GPa is higher than the surrounding rock's Young's modulus of 60 GPa and 40 GPa. (3)The thermal stress simulated in the article is an order of magnitude larger than those generated using the linear coefficient of thermal expansion at room temperature. The thermal stress induced by granite surrounding rocks is nearly 30 MPa higher than that induced by carbonate rocks. (4) The thermal stress decreases with increasing distance from magma, approaching the initial stresses at nearly 2 km. (5) When the emplacement depth is shallow, both initial temperature and initial stress are lower than those in deeper emplacements; The magma room cools faster at shallow depths. Because the initial temperature of magma is the same, shallow emplacements will produce higher thermal stresses on overlying surrounding rocks. Conclusion The modeling results indicate that the thermal conductivity of surrounding rocks influences the change rate of thermal stress through the heat transfer rate. The thermal stress increases with the surrounding rock's Young's modulus. Since the average Young's modulus of granites is greater than that of carbonate rocks, the thermal stress on granite is greater than that on carbonate rocks. Either granites or carbonate rocks at high temperatures have a thermal expansion coefficient about one order of magnitude greater than that at room temperature, resulting in thermal stress of up to 100 MPa. The temperature of the surrounding rock gradually increases after the granite magma emplacement, corresponding to the increasing thermal stress. The thermal stress decreases with increasing distance from magma, exerting no influence on the initial stress of host rocks above 2 km of the granitic magma. When the magma emplacement is shallow, the combination of high thermal stress and low initial stress is more conducive to the formation and expansion of fractures in overlying surrounding rock. Significance The results of numerical simulations reveal that the thermal stresses generated by magma emplacement can affect the stress field 2 km above the magma. These localized and short-lived thermal stresses may fracture the overlying rocks, providing transport channels or ore-bearing spaces for later hydrothermal fluids. -
图 1 岩浆侵位二维数值模拟模型
三角代表固定两端水平方向的位移；圆形代表竖直方向可以向上产生运动但不可向下运动，水平方向也可自由移动；L1表示每隔200m记录的X、Z方向上的剪应力及温度的变化；L2表示每隔200 m记录的X、Z方向上的正应力及温度的变化；P1、P2表示系列实验1的记录点；图中所给的边界条件及初始条件见1.5节
Figure 1. The two-dimensional numerical model of magma emplacement
The boundary and initial condition are shown in the model are described in Section 1.5. The triangles represent that the horizontal displacement is fixed, and the circles represent that the bottom cannot move downward but can move horizontally. L1 indicates that shear stress in the X, Z direction and temperature change are recorded every 200 m; L2 indicates that normal stress in the X, Z direction and temperature change are recorded every 200 m. P1 and P2 are the points used in the numerical experiment series 1.
图 2 不同岩石类型的线性热膨胀系数随温度的变化特征
a—花岗岩（花岗岩1—花岗岩3中斜长石含量高于石英）；b—碳酸盐岩
Figure 2. The trend of the linear thermal expansion coefficient with temperature for various rock kinds
(a) Granite (granite1–granite 3 with higher plagioclase content than quartz); (b) Carbonate rock
图 3 花岗质岩浆房中心温度变化曲线变化趋势
Figure 3. The trend of temperature variation for granitic magma centers
图 4 重力引起的初始应力场
a—Z方向初始正应力（最大为315 MPa）；b—X方向初始正应力（最大为105 MPa）
Figure 4. The initial stress field caused by gravity
(a) Initial normal stress in Z-direction (max. 315 MPa); (b) Initial normal stress in X-direction (max. 105 MPa)
图 5 岩浆侵位8 ka后岩浆房周围温度场及热应力场
a—温度场；b—Z方向正应力；c—XZ方向剪应力；d—X方向正应力
Figure 5. The distribution of temperature and thermal stress around the granitic magma after 8 ka
(a) Temperature field; (b) Normal stress in Z-direction; (c) Shear stress in XZ-direction; (d) Normal stress in X-direction
图 6 距岩浆房上方1 km处的温度和正应力随时间的变化曲线
k—热传导系数a—温度场；b—Z方向正应力；c—X方向正应力
Figure 6. Time changes of temperature and normal stress at 1 km in the upper part of the granitic magma
(a) Temperature field; (b) Normal stress in Z-direction; (c) Normal stress in X-directionk—thermal conductivity
图 7 距岩浆房左上角上方1 km处温度和剪应力随时间变化曲线
k—热传导系数a—温度场；b—XZ方向剪应力
Figure 7. Temperature and shear stress variation curves over time at 1 km in the upper left corner of the granitic magma center
(a) Temperature field; (b) Shear stress in XZ-directionk—thermal conductivity
图 8 侵位时间8 ka后岩浆房上方（图1中L2）的温度和正应力变化曲线
E—杨氏模量a—温度场；b—Z方向正应力；c—X方向正应力
Figure 8. Temperature and normal stress change in the upper part of the granitic magma center (see L2 in Fig. 1) after 8 ka
(a) Temperature field; (b) Normal stress in Z-direction; (c) Normal stress in X-directionE—Young's modulus
图 9 侵位时间8 ka后岩浆房左上角上方（图1中L1）的温度和剪应力变化曲线
E—杨氏模量a—温度场；b—XZ方向剪应力
Figure 9. Temperature and shear stress change in the upper left corner of the granitic magma center (see L1 in Fig. 1) after 8 ka
(a) Temperature field; (b) Shear stress in XZ-directionE—Young's modulus
图 10 岩浆侵位至具有不同线性热膨胀系数的围岩所产生的热应力变化曲线
$ {\alpha }_{{\rm{t}}} $—线性热膨胀系数a、b—Z方向正应力；c、d—X方向正应力；e、f—XZ方向剪应力
Figure 10. Thermal stress change from granitic magma emplacement to surrounding rock with varying linear thermal expansion coefficients
(a, b) Normal stress in Z-direction; (c, d) Normal stress in X-direction; (e, f) Shear stress in XZ-direction$ {\alpha }_{t} $—The linear thermal expansion coefficients
图 11 岩浆房上方（图1中L2）的温度和正应力变化曲线
a—温度场；b—Z方向正应力；c—X方向正应力
Figure 11. Spatial changes of temperature and normal stress in the upper part of the granitic magma center (see L2 in Fig. 1)
(a) Temperature field; (b) Normal stress in Z-direction; (c) Normal stress in X-direction
图 12 岩浆房左上角上方（图1中L1）温度和剪应力变化曲线
a—温度场；b—XZ方向剪应力
Figure 12. Spatial changes of temperature and shear stress in the upper left corner of the granitic magma center (see L1in Fig. 1)
(a) Temperature field; (b) Shear stress in XZ-direction
图 13 侵位深度为3 km侵位时间8 ka后的温度场和热应力场
a—温度场；b—Z方向正应力；c—XZ方向剪应力；d—X方向正应力
Figure 13. The distribution of temperature field and thermal stress around the granitic magma center at 3 km emplacement depth after 8 ka
(a) Temperature field; (b) Normal stress in Z-direction; (c) Shear stress in XZ-direction; (d) Normal stress in X-direction
图 14 实验5与实验4的温度和正应力结果对比曲线
a—温度场；b—Z方向正应力；c—X方向正应力
Figure 14. The trend of temperature and normal stress for Experiment 5 compared to Experiment 4
(a) Temperature field; (b) Normal stress in Z-direction; (c) Normal stress in X-direction
图 15 实验5与实验4的温度和剪应力结果对比曲线
a—温度场；b—XZ方向剪应力
Figure 15. The trends of temperature and shear stress for Experiment 5 compared to Experiment 4
(a) Temperature field; (b) Shear stress in XZ-direction
表 1 数值模拟实验中采用的岩石力学和热学参数
Table 1. Rock mechanics and thermal parameters used in numerical experiments
实验 序号 围岩 密度/
($ \mathrm{k}\mathrm{g}/{\mathrm{m}}^{3} $)杨氏模量/
$ \mathrm{G}\mathrm{P}\mathrm{a} $泊松比 抗拉强度/
$ \mathrm{M}\mathrm{P}\mathrm{a} $摩擦角/
（°）黏聚力/
$ \mathrm{M}\mathrm{P}\mathrm{a} $比热容/
$ \mathrm{J}/(\mathrm{k}\mathrm{g}\cdot \text{℃}) $热传导系数/
$ \mathrm{W}/(\mathrm{m}\cdot \text{℃}$)线性热膨胀系数/
(1/$ \text{℃} $)系列实验1 1-1 花岗岩 2700 60 0.25 10 50 23 800 2 由公式(7) 得出 1-2 2.5 1-3 3 系列实验2 2-1 花岗岩 2700 40 0.25 10 50 23 800 3 由公式(7) 得出 2-2 60 2-3 80 系列实验3 3-1 花岗岩 2700 60 0.25 10 50 23 800 3 由公式(7) 得出 3-2 花岗岩 $ 1.8\times {10}^{-6} $ 3-3 碳酸盐岩 45 由公式（8）得出 3-4 碳酸盐岩 $ 1.8\times {10}^{-6} $ 实验4 1-3、2-3、3-1 花岗岩 2700 60 0.25 10 50 23 800 3 由公式(7) 得出 实验5 — 花岗岩 2700 60 0.25 10 50 23 800 3 由公式(7) 得出 
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