Simulation experiment and mathematical model analysis for shale gas diffusion in nano-scale pores
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摘要: 深层页岩气在纳米孔隙中的扩散行为分为体相扩散（Fick和Knudsen扩散）和表面扩散。为了定量评价温度、压力等对扩散系数的影响，揭示深层页岩气的保存机理，以南方鄂西秭归茅坪地区寒武系牛蹄塘组页岩为实验对象，在不同温压条件下，通过等压扩散实验对纳米孔隙甲烷扩散进行实验模拟。结果表明：（1）扩散系数DF随压力增大而减小（当压力大于30 MPa时，DF趋于平稳），随温度升高而增大；（2）在高温高压环境下，DF受压力影响更大，总体趋于减小。随后，定量考虑了温度、压力、孔隙及岩性特征对各种扩散行为的影响，建立了数学模型。该模型与模拟实验结果相似，可以相互验证：（1）温度升高促使分子动能增大，导致体相和表面扩散系数都增大，而压力增大虽然会使Fick扩散和表面扩散作用稍微加强，但会显著限制Knudsen扩散并最终导致总扩散作用降低；（2）孔径增大加强了体相扩散作用，削弱了表面扩散作用。最后，结合具体研究区块，认为深层高压环境有利于页岩纳米孔隙气藏的保存，而地层抬升释放压力的过程是页岩气散失的主要阶段。Abstract: Gas diffusion in nano-scale porous media of deeply burried shale includes bulk diffusion (Fick and Knudsen diffusions) and surface diffusion. To reveal the migration mechanism of this process, the influence of temperature and pressure on diffusion coefficient needs to be quantitatively evaluated. A case study was made with the Cambrian Niutitang Formation in the Maoping area, Zigui, western Hubei, South China. Gas diffusion was simulated by isobaric diffusion experiments under different temperature and pressure conditions. The results indicated that: (1) The diffusion coefficient DF decreases with increasing of pressure (when the pressure is higher than 30 MPa, DF tends to be constant), and increases with increasing of temperature; (2) In the high temperature-pressure setting, DF is affected significantly by pressure and generally tends to decrease. Moreover, the impacts of temperature, pressure, porosity and lithology were quantitatively calculated, and a mathematical model of gas diffusion was established, which had comparable results with simulation experiment. The following conclusions were thus drawn: (1)Higher temperature will cause stronger molecular kinetic energy, resulting in increasing bulk and surface diffusion coefficients, while higher pressure will slightly strengthen the Fick and surface diffusions, but significantly limit the Knudsen diffusion, and cause lower total diffusion coefficient; (2) Larger pore size leads to stronger bulk diffusion, but weaker surface diffusion. Finally, according to the studies of a specific research block, high pressure setting is conducive to the preservation of nano-scale porous gas reservoir in shale, while the uplift accompanied by pressure release is the main stage of shale gas loss.
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Key words:
- methane /
- diffusion coefficient /
- simulation experiment /
- mathematical model
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图 1 鄂西秭归茅坪地区寒武系牛蹄塘组页岩孔隙直径大小及分布
Figure 1. Diameter size and distribution characteristics of shale pores in Cambrian Niutitang Formation, Maoping area, Zigui, western Hubei
图 2 鄂西秭归茅坪地区寒武系牛蹄塘组页岩SEM特征
Figure 2. SEM characteristics of shale in Cambrian Niutitang Formation, Maoping area, Zigui, western Hubei
图 3 扩散系数模拟实验装置简易图
Figure 3. Simplified diagram of simulator for diffusion coefficient analysis
图 4 扩散系数随单一变量(压力或温度)增加的变化趋势
Figure 4. Variation trend of diffusion coefficient with increased single variable (pressure or temperature)
图 5 温压耦合模拟实验中扩散系数变化趋势
Figure 5. Variation trend of diffusion coefficient in simulation experiment of temperature-pressure coupling
图 6 页岩纳米孔内甲烷扩散过程及机理示意
Figure 6. Schematic diagram of methane diffusion process and mechanism in shale nano-pores
图 7 扩散模拟实验与数学模型的相互验证
Figure 7. Mutual verification of simulation and mathematical model
图 8 纳米孔隙中不同条件下的扩散数学模型
Pconfine-P = 5 MPa
Figure 8. Mathematical model of nano-pores under different conditions
图 9 扩散数学模型中温度—压力共同影响下的扩散系数特征
Figure 9. Characteristics of diffusion coefficient affected by temperature and pressure in mathematical model
表 1 不同温压条件下模拟测试的甲烷扩散系数
Table 1. Methane diffusion coefficient tested by simulation under different temperature and pressure conditions
实验计划 实验条件 扩散系数/(10-8 cm2·s-1) 温度/℃ 环压/MPa 气体压力/MPa 温度保持不变，压力增高 30 8 2 2.528 30 15 10 1.433 30 25 20 0.539 30 35 30 0.231 30 50 45 0.500 30 60 55 0.404 压力保持不变，温度升高 30 2.528 50 2.758 80 8 2 3.013 110 3.239 温度、压力同时增高 30 8 2 2.528 50 16.5 11 2.684 90 27 20 0.876 110 35 30 1.815 110 40 35 0.943 110 45 40 0.561 
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表 2 扩散模拟实验和数学模型中的参数取值
Table 2. Parameter values in experimental simulation and mathematical model of this study
参数 参数意义 取值 参数 参数意义 取值 T/K 绝对温度 300~400 κb/(J·K-1) Boltzmann常数 1.308×10-23 P/Pa 气体压力 (2~55)×106 δ/dM /m 气体分子(碰撞)直径 3.8×10-10 Pconfine/Pa 施加压力/围压 (7~60)×106 M/(kg·mol-1) 气体摩尔质量 0.016 PL[28]/Pa Langmuir压力 4.48×106 μg/(Pa·s) 气体黏度 0.000 014 ΔH [29]/(J·mol-1) 等温吸附热 14 000 r0/m 初始孔隙半径 1.29×10-9 R/(J·mol-1·K-1) 通用气体常数 8.314 r/m 施压后孔隙半径 公式(7) κ 分子阻塞系数 0.5 α 比奥Biot系数 0.8 H(1-κ) Heaviside函数 1 φ 平均孔隙度 0.018 τ 孔隙迂曲度 10 εL 朗缪尔体积应变常数 0.05 K/Pa 页岩样品体积模量 8×108 ζreal-b 实际总扩散校正系数 0.01 ζreal-a 实际表面扩散校正系数 0.001 DF/(cm2·s-1) 实验测试扩散系数 表 1 Dtotal/(cm2·s-1) 理论总扩散系数 公式(16)
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表 3 单一变量下的扩散系数特征
Table 3. Parameter values in experimental simulation and mathematical model of this study
单一变量 Dknudsen/(10-8 cm2·s-1) Dfick/(10-8 cm2·s-1) Dsurface/(10-8 cm2·s-1) Dtotal/(10-8 cm2·s-1) 压力变化，其他条件不变
(T = 300 K；r0 = 1 nm；ζreal-b = 0.01)P=10 MPa 0.57 0.35 0.19 1.11 P=20 MPa 0.33 0.40 0.25 0.98 温度变化，其他条件不变
(P = 10 MPa；r0 = 1 nm；ζreal-b = 0.01)T=300 K 0.57 0.35 0.19 1.11 T=400 K 0.81 0.43 0.27 1.52 孔径变化，其他条件不变
(T = 300 K；P = 10 MPa；ζreal-b = 0.01)r0 = 1 nm 0.57 0.35 0.19 1.11 r0 = 5 nm 1.81 1.10 0.05 2.96 孔隙连通性变化，其他条件不变
(T = 300 K；P = 10 MPa；r0 = 1 nm)ζreal-b = 0.01 0.57 0.35 0.19 1.11 ζreal-b = 0.02 1.14 0.70 0.38 2.23
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