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可压缩两相流固耦合模型的间断Galerkin有限元方法

马天然 沈伟军 刘卫群 XuHao

马天然, 沈伟军, 刘卫群, Xu Hao. 可压缩两相流固耦合模型的间断Galerkin有限元方法. 力学学报, 2021, 53(8): 2235-2245 doi: 10.6052/0459-1879-21-177
引用本文: 马天然, 沈伟军, 刘卫群, Xu Hao. 可压缩两相流固耦合模型的间断Galerkin有限元方法. 力学学报, 2021, 53(8): 2235-2245 doi: 10.6052/0459-1879-21-177
Ma Tianran, Shen Weijun, Liu Weiqun, Xu Hao. Discontinuous Galerkin FEM method for the coupling of compressible two-phase flow and poromechanics. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(8): 2235-2245 doi: 10.6052/0459-1879-21-177
Citation: Ma Tianran, Shen Weijun, Liu Weiqun, Xu Hao. Discontinuous Galerkin FEM method for the coupling of compressible two-phase flow and poromechanics. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(8): 2235-2245 doi: 10.6052/0459-1879-21-177

可压缩两相流固耦合模型的间断Galerkin有限元方法

doi: 10.6052/0459-1879-21-177
基金项目: 国家自然科学基金资助(U1762216, 11802312), 江苏省自然科学基金(BK20180636)资助项目
详细信息
    作者简介:

    沈伟军, 副研究员, 主要研究方向: 渗流流体力学和油气田开发工程. E-mail: wjshen763@imech.ac.cn

  • 中图分类号: O359+.1, TE312

DISCONTINUOUS GALERKIN FEM METHOD FOR THE COUPLING OF COMPRESSIBLE TWO-PHASE FLOW AND POROMECHANICS

  • 摘要: 认识多孔介质内多相流动和固体变形耦合特征是地下资源开发利用的关键问题. 针对这一问题, 首先建立了考虑毛细压力和重力作用的可压缩两相渗流与孔隙介质变形耦合方程. 在此基础上, 推导了流体方程的非对称内罚Galerkin弱形式和固体变形方程的非完全内罚Galerkin弱形式. 其次, 通过对比一维Terzaghi固结问题的理论解和数值解, 验证了间断Galerkin方法在计算流固耦合问题方面的可行性和准确性. 然后, 针对性开展了二维平面算例和考虑重力效应作用的三维算例, 分析了加罚因子$ {\delta }_{\rm{s}} $$ {\delta }_{\rm{f}} $对数值结果的影响. 结果表明, 随着气体的持续注入, 气体饱和度和孔压增加, 有效应力降低, 继而引发多孔介质膨胀变形; 气体在重力影响下上浮聚集于顶部边界; $ {\delta }_{\rm{s}} $$ {\delta }_{\rm{f}} $的降低会导致流体和力学信息在局部出现不同程度的波动, 提高加罚因子可以有效抑制有限元函数在跨越单元时的不连续性.

     

  • 图  1  几何区域、边界条件和网格示意图

    Figure  1.  Domain with boundary conditions and mesh skeleton

    图  2  Terzaghi固结问题几何示意图

    Figure  2.  Sketch of Terzaghi’s consolidation problem

    图  3  Terzaghi固结问题理论解和数值解的比较

    Figure  3.  Comparison of the analytical and numerical results for Terzaghi’s consolidation problem

    图  4  模拟几何尺度、初始和边界条件

    Figure  4.  Simulation geometrical configuration with boundary and initial conditions

    图  5  第1500 s气体压力$ {p}_{{\rm{n}}{\rm{w}}} $、水饱和度$ {S}_{{\rm{w}}} $、水平位移$ u $和水平有效应力$ {\sigma }_{x}^{{{'}}} $的分布云图

    Figure  5.  The distribution of gas pressure $ {p}_{{\rm{n}}{\rm{w}}} $, water saturation $ {S}_{{\rm{w}}} $, horizontal displacement $ u $ and horizontal effective stress $ {\sigma }_{x}^{{{'}}} $ at t = 1500 s

    图  6  第1500 s时气体压力$ {p}_{{\rm{n}}{\rm{w}}} $、水饱和度$ {S}_{{\rm{w}}} $、水平位移$ u $和水平有效应力$ {\sigma }_{x}^{{{'}}} $沿监测线的分布

    Figure  6.  The profiles of gas pressure $ {p}_{{\rm{n}}{\rm{w}}} $, water saturation $ {S}_{{\rm{w}}} $, horizontal displacement $ u $ and horizontal effective stress $ {\sigma }_{x}^{{{'}}} $ along the monitoring line at t = 1500 s

    图  7  第1500 s时不同加罚因子$ {\delta }_{\rm{s}} $条件下水平位移$ u $和水平有效应力$ {\sigma }_{x}^{{{'}}} $沿监测线的分布

    Figure  7.  The profiles of horizontal displacement $ u $ and horizontal effective stress $ {\sigma }_{x}^{{{'}}} $ along the monitoring line at t = 1500 s with different values of penalty parameter $ {\delta }_{\rm{s}} $

    图  8  第1500 s时$ {\delta }_{\rm{s}}=0.01 $水平位移$ u $和水平有效应力$ {\sigma }_{x}^{{{'}}} $分布云图

    Figure  8.  The distribution of horizontal displacement $ u $ and horizontal effective stress $ {\sigma }_{x}^{{{'}}} $ with $ {\delta }_{\rm{s}}=0.01 $ at t = 1500 s

    图  9  第1500 s时不同加罚因子$ {\delta }_{\rm{f}} $条件下气体压力$ {p}_{{\rm{n}}{\rm{w}}} $、水饱和度$ {S}_{{\rm{w}}} $沿监测线(0, 0.05)−(10, 9.95)的分布

    Figure  9.  The profiles of gas pressure $ {p}_{{\rm{n}}{\rm{w}}} $ and water saturation $ {S}_{{\rm{w}}} $ along monitoring line at t = 1500 s with different values of $ {\delta }_{\rm{f}} $

    图  10  不同加罚因子$ {\delta }_{\rm{s}} $条件下间断有限元和传统有限元方法计算水平位移和水平有效应力的比较

    Figure  10.  Comparison of the horizonal displacement and horizonal effective stress between DG and the CG results with different values of penalty parameter $ {\delta }_{\rm{s}} $

    图  11  三维模型几何尺度以及模拟初始和边界条件

    Figure  11.  Geometrical configuration with boundary and initial conditions of 3D case

    图  12  注气1 d后气体压力$ {p}_{{\rm{n}}{\rm{w}}} $、气体饱和度$ {S}_{{\rm{n}}{\rm{w}}} $、垂直位移$ w $和水平有效应力$ {\sigma }_{x}^{{{'}}} $的分布图

    Figure  12.  The distribution of gas pressure $ {p}_{{\rm{n}}{\rm{w}}} $, gas saturation $ {S}_{{\rm{n}}{\rm{w}}} $, vertical displacement $ w $ and horizontal effective stress $ {\sigma }_{x}^{{{'}}} $ after 1 day of injection

    图  13  考虑和不考虑重力效应算例中点a, bc处气饱和度$ {S}_{{\rm{n}}{\rm{w}}} $随时间变化图

    Figure  13.  The temporal evolution of gas saturation $ {S}_{{\rm{n}}{\rm{w}}} $ at points a, b and c in the cases with and without gravity

    表  1  两相流固数值模拟的属性参数

    Table  1.   Parameters for two-phase flow simulation

    ParameterValueUnit
    elastic modulus, $ E $$ 30.00 $$ {\rm{G}}{\rm{Pa}} $
    poisson's ratio, $ \nu $$ 0.25 $$ - $
    permeability, $ k $$ 1.00\times {10}^{-14} $$ \;{\rm{m}}^{2} $
    porosity, $ \phi $$ 0.20 $$ - $
    water density, $ \;{\rho }_{{\rm{w}}0} $$ 1000.00 $$ {\rm{kg}}/{\rm{m}}^{3} $
    water viscosity, $ \;{\mu }_{{\rm{w}}} $$ 1.00\times {10}^{-3} $$ {\rm{Pa}}\cdot {\rm{s}} $
    water compressibility[32], $ {c}_{{\rm{w}}} $$ 3.84\times {10}^{-10} $$ {{\rm{Pa}}}^{-1} $
    hydrogen density, $\; {\rho }_{{\rm{n}}{\rm{w}}0} $$ 3.18 $$ {\rm{kg}}/{\rm{m}}^{3} $
    hydrogen viscosity, $\; {\mu }_{{\rm{n}}{\rm{w}}} $$ 9.02\times {10}^{-6} $$ {\rm{Pa}}\cdot {\rm{s}} $
    hydrogen compressibility[32], $ {c}_{{\rm{n}}{\rm{w}}} $$ 7.71\times {10}^{-7} $$ {{\rm{Pa}}}^{-1} $
    entry pressure, $ {p}_{\rm{e}} $$ 1.00\times {10}^{4} $$ {\rm{Pa}} $
    distribution index, $ \lambda $$ 0.46 $$ - $
    relative permeability variable, $ \omega $$ 2.00 $$ - $
    variable, $ {\omega }_{\rm{f}} $1.00$ - $
    penalty factor[27], $ {\delta }_{\rm{f}} $1.00$ - $
    variable, $ {\omega }_{\rm{s}} $0.00$ - $
    penalty factor[27], $ {\delta }_{\rm{s}} $10.00$ - $
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出版历程
  • 收稿日期:  2021-03-23
  • 录用日期:  2021-07-30
  • 网络出版日期:  2021-07-31
  • 刊出日期:  2021-08-18

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