Computers & Mathematics with Applications ( IF 2.9 ) Pub Date : 2022-07-14 , DOI: 10.1016/j.camwa.2022.06.004 Mehdi Dehghan, Zeinab Gharibi
Wormhole propagation, arising in petroleum engineering, is used to describe the distribution of acid and the increase of porosity in carbonate reservoir under the dissolution of injected acid and plays a very important role in the product enhancement of oil and gas reservoirs. In this paper, a fully mixed virtual element method (VEM) is employed to discretize this problem, in which mixed VEM is used not only for the Darcy flow equations but also for approximation the concentration equation by introducing an auxiliary flux variable to guarantee full mass conservation. The stability, existence and uniqueness of solution of the associated mixed VEM are proved by fixed point theory. Also, we obtain unconditionally optimal error estimate for concentration and auxiliary flux variable of convection-diffusion equation, as well as for the velocity and pressure of Darcy equations in the norm. Finally, several numerical experiments are presented to support the theoretical analysis of convergence and to illustrate the applicability for solving actual problems.
中文翻译:
石油工程虫洞传播全混合虚元法的统一分析
虫洞传播是石油工程中出现的一种,用来描述酸的分布和碳酸盐岩储层在注入酸的溶解作用下孔隙度的增加,对油气藏的增产具有非常重要的作用。在本文中,采用完全混合虚拟元法(VEM)来离散这个问题,其中混合VEM不仅用于达西流方程,而且通过引入辅助通量变量来近似浓度方程以保证全质量保护。用不动点理论证明了关联混合VEM解的稳定性、存在性和唯一性。此外,我们获得了对流扩散方程的浓度和辅助通量变量的无条件最优误差估计,规范。最后,提出了几个数值实验来支持收敛的理论分析,并说明解决实际问题的适用性。