您的位置：山东大学 -> 科技期刊社 -> 《山东大学学报（理学版）》

山东大学学报（理学版） ›› 2017, Vol. 52 ›› Issue (8): 25-34.doi: 10.6040/j.issn.1671-9352.0.2017.040

• • 上一篇    下一篇

可压多孔介质流的扩展混合元解法

于金彪1,任永强2,3,曹伟东1,鲁统超2,程爱杰2,戴涛1   

  1. 1.中国石化股份胜利油田分公司勘探开发研究院, 山东 东营 257015;2.山东大学数学学院, 山东 济南 250100;3. 齐鲁工业大学理学院, 山东 济南 250353
  • 收稿日期:2017-02-10 出版日期:2017-08-20 发布日期:2017-08-03
  • 作者简介:于金彪(1970—),男,高级工程师,研究方向为油藏数值模拟. E-mail:yjbxx@slof.com
  • 基金资助:
    国家科技重大专项资助基金(2011ZX05011-004,2011ZX05052)

Expanded mixed finite element method for compressible miscible displacement in heterogeneous porous media

YU Jin-biao1, REN Yong-qiang2,3, CAO Wei-dong1, LU Tong-chao2, CHENG Ai-jie2, DAI tao1   

  1. 1. Research Institute of Petroleum Exploration &
    Development, Shengli Oilfield Company Ltd., Dongying 257015, Shandong, China;
    2. School of Mathematics, Shandong University, Jinan 250100, Shandong, China;
    3. School of Mathematial and Physical Sciences, Qilu University of Technology, Jinan 250353, Shandong, China
  • Received:2017-02-10 Online:2017-08-20 Published:2017-08-03

PDF (PC)

661

摘要： 讨论多孔介质中两种可压缩流体混溶驱动问题数值方法,假定介质是各向异性的,渗透率系数为张量形式。压力方程采用扩展混合元方法求解压力变量、梯度变量,以及速度变量;浓度方程采用标准有限元方法求解,这一方法对各向异性渗透率多孔介质流可以获得更可靠的数值解。构造了半离散数值格式,通过理论分析得到了压力、速度以及浓度等变量的最优L2模误差估计,对浓度变量获得了H1模最优误差估计。

关键词: 扩展混合元, 渗透率张量, 混溶驱动问题, 多孔介质, 可压缩, 误差估计

Abstract: We consider the numerical methods for the mathematical model describing the transport and diffusion process of porous media flow. There are two kinds of fluids, one is displaced by another, which are compressible and miscible. The media is supposed to be heterogeneous, so the permeability is of full tensor form. For the pressure equation, an expanded mixed finite element method is introduced to solve the variables of pressure, gradient, and velocity. For the concentration equation, a Galerkin finite element formulation is constructed to solve the variable of concentration. This approach aims at obtaining more reliable numerical solutions for porous media flow with heterogenous permeability. By means of theoretical analysis, optimal error estimates in L2-norm for pressure and in H1-norm for concentration are derived.

Key words: porous media, compressibility, error analysis, tensor permeability, miscible displacement, expanded mixed finite element method

中图分类号: 

  • O241.82
[1] JR DOUGLAS J, ROBERTS J E. Numerical methods for a model for compressible miscible displacement in porous media[J]. Mathematics of Computation,1983, 41:441-459.
[2] YNGVE Aasum. Effective properties of reservoir simulator grid blocks[D]. Oklahoma: University of Tulsa, 1992.
[3] LEE Jaedong. Analytical upscaling of permeabilities for reservoir simulation grid blocks[D]. Oklahoma: University of Tulsa, 1996.
[4] 张建松,羊丹平. 多孔介质中可压缩驱动问题的全离散分裂正定混合元方法[J]. 山东大学学报(理学版),2006,41(1):1-10. ZHANG Jiansong, YANG Danping. A fully-discrete splitting positiVe definite mixedelement scheme finite for compressible miscible displacement in porous media[J]. Journal of Shandong University(Natural Science), 2006, 41(1):1-10.
[5] CHEN Chunguang. Mixed method for compressible miscible displacement with dispersion in porous media[J]. Numerical Mathematics, 2007, 16(1):74-82.
[6] WANG K. An optimal-order estimate for MMOC-MFEM approximations to porous medium flow[J]. Numerical Methods for Partial Differential Equations, 2009, 25:1283-1302.
[7] WHEELER M F, YOTOV I. Mixed finite element methods for modeling flow and transport in porous media, Mathematical Modeling of Flow through Porous Media[C] // Bourgeat A, Carasso C, Luckhaus S, et al. London:World Scientific, 1995, 337-358.
[8] ARBOGAST T, WHEELER M F, YOTOV I. Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite difference[J]. SIAM Journal of Numerical Analysis, 1997, 41:828-852.
[9] CHEN Z. Expanded mixed finite element methods for linear second order elliptic problems 1[J]. RAIRO Model Meth Anal Numer, 1998, 32:478-499.
[10] CHEN Z. Expanded mixed finite element methods for quasilinear second order elliptic problems 2[J]. RAIRO Model Meth Anal Numer, 1998, 32:500-520.
[11] RUI Hongxing, LU Tongchao. An expanded mixed covolume method for elliptic problems[J]. Numerical Methods for Partial Differential Equations, 2005, 21:8-23.
[12] WOODWARD C S, DAWSON N D. Analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media[J]. SIAM Journal of Numerical Analysis, 2000, 37:701-724.
[13] CHEN Huanzhen, WANG Hong. An optimal-order error estimate on an H1-Galerkin mixed method for a nonlinear parabolic equation in porous medium flow[J]. Numerical Methods for Partial Differential Equations, 2010, 26:188-205.
[14] ZHOU Zhaojie, WANG Weiwei, CHEN Huanzhen. An H1-Galerkin expanded mixed finite element approximation of second-order nonlinear hyperbolic equations[J]. Abstract and Applied Analysis, 2013, 4:1-12.
[15] SONG Huailing, JIANG Lijian, CHEN Gaojie. Convergence analysis of hybrid expanded mixed finite element method for elliptic equations[J]. Computers & Mathematics with Application, 2014, 68:1205-1219.
[16] SONG Huailing, YUAN Yirang. The expanded upwind-mixed multi-step method for the miscible displacement problem in three dimensions[J]. Applied Mathematics and Computation, 2008, 195:100-109.
[17] SONG Huailing, YUAN Yirang, LIU Gongjie. The expanded upwind-mixed method on changing meshes for positive semi-definite problem of two-phase miscible flow[J]. International Journal of Computer Mathematics, 2008,85:1113-1125.
[18] CHEN Yanping, CHEN Luoping, ZHANG Xiaochun. Two-Grid method for nonlinear parabolic equations by expanded mixed finite element methods[J]. Numerical Methods for Partial Differential Equations, 2013, 29:1238-1256.
[19] CHEN Yanping, LIU Huanwen, LIU Shang. Analysis of two-grid methods for reaction-diffusion equations by expanded mixed finite element methods[J]. International Journal for numerical methods in Engineering, 2007, 69:408-422.
[20] RUSSELL T F, WHEELER M F. Finite element and finite difference methods for continuous flows in porous media[M]. Philadephia: Society for Industrial and Applied Mathematics, 1983: 35-106.
[1] 丁凤霞,程浩. 椭圆方程柯西问题磨光正则化参数的后验选取[J]. 山东大学学报（理学版）, 2018, 53(2): 18-24.
[2] 史艳华1,石东洋2*. 伪双曲方程类Wilson非协调元逼近[J]. J4, 2013, 48(4): 77-84.
[3] 邓利华,尚海锋. 具强非线性源的多孔介质方程的Cauchy 问题[J]. J4, 2012, 47(8): 50-54.
[4] 吴志勤1,石东伟2. 带弱奇异核非线性积分微分方程的收敛性分析[J]. J4, 2012, 47(8): 60-63.
[5] 李娜,高夫征*,张天德. Sobolev方程的扩展混合体积元方法[J]. J4, 2012, 47(6): 34-39.
[6] 马效培，孙同军. 多孔介质中不可压缩混溶驱动问题的一类特征积分平均非重叠型区域分解方法[J]. J4, 2012, 47(6): 49-56.
[7] 于金彪1,2,席开华3*,戴涛2,鲁统超3,杨耀忠2,任永强3,程爱杰3. 两相多组分流有限元方法的收敛性[J]. J4, 2012, 47(2): 19-25.
[8] 宋丹丹,原保全*. 可压缩磁流体方程组的显式爆破解[J]. J4, 2012, 47(2): 26-30.
[9] 周家全,孙应德,张永胜. Burgers方程的非协调特征有限元方法[J]. J4, 2012, 47(12): 103-108.
[10] 乔保民,梁洪亮. 一类非线性广义神经传播方程Adini元的超收敛分析[J]. J4, 2011, 46(8): 42-46.
[11] 边东芬，原保全*. 可压缩磁流体方程组整体弱解的不存在性[J]. J4, 2011, 46(4): 42-48.
[12] 郑瑞瑞,孙同军. 一类线性Sobolev方程的迎风混合元方法[J]. J4, 2011, 46(4): 23-28.
[13] 马宁. 含弥散项渗流问题的正交配置法[J]. J4, 2011, 46(2): 78-81.
[14] 周建伟1,羊丹平2*. 二维区域Legendre-Galerkin谱方法的后验误差估计[J]. J4, 2011, 46(11): 122-126.
[15] 陆瑶1,李德生2,杨洋1. 非线性SchrÖdinger方程的Fourier谱逼近[J]. J4, 2011, 46(1): 119-126.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
版权所有 © 2018 《山东大学学报(理学版)》编辑部　
地址: 山东省济南市山大南路27号山东大学中心校区（250100）　电话: +86-0531-88366917　E-mail: xblxb@sdu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发