可压缩两相流固耦合模型的间断Galerkin有限元方法

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

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δs和δf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;δs和δf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.

Understanding the coupled multiphase flow and solid deformation processes in porous media is a significant issue in the area of developing and utilizing underground resources. This study first established the coupled modeling of compressible two-phase flow and deformation of porous media, which considers capillarity and gravity. Meanwhile, the strong form and the corresponding weak form of coupled multiphase flow and solid deformation model were presented.Then, the capacity of the proposed DG method for the coupled hydromechanical model was verified by comparison with analytical and numerical results of the one-dimensional Terzaghi consolidation problem. Subsequently, the two-and three-dimensional cases were performed to study the flow behaviors and deformation characteristics, respectively. In addition, the effects of the penalty factors δ s and δ f on the stability of the numerical results were analyzed. The simulation results show that gas saturation and pore pressure continually increase with the injection of gas. The increment of pore pressure reduces the effective stress, which results in deformation and expansion of the porous medium. The gas floats up and gathers at the top boundary due to gravity. The decrease of the penalty factors δ s and δ f trends to cause the fluctuation of saturation, pressure, effective stress, and displacement. The increases in penalty factors are beneficial to suppress the discontinuity of the finite element function crossing the elements.