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首页» 过刊浏览» 2025» Vol.10» lssue(2) 309-325     DOI : 10.3969/j.issn.2096-1693.2025.02.013
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无偏隐压显饱数值方法 —— 两相渗流模拟的新算法
陈黄鑫, 陈玉祥, 孙树瑜.
1 厦门大学数学科学学院,厦门 361005 2 同济大学数学科学学院,上海 200092
Unbiased implicit pressure explicit saturation schemes: Novel algorithms for the simulation of two-phase flow in porous media
CHEN Huangxin, CHEN Yuxiang, SUN Shuyu.
1 School of Mathematical Sciences, Xiamen University, Xiamen 361005, China 2 School of Mathematical Sciences, Tongji University, Shanghai 200092, China

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摘要  多孔介质中多相渗流问题是油气藏开发领域重要的研究内容,由于我国地质条件复杂,岩石性质如渗透率、孔隙度等分布不均匀,复杂多相渗流问题的数值求解需克服系统的多变量、强非线性、计算量大以及保持变量的物理属性等难点。对于传统的不可压不混溶两相渗流模型,隐压显饱(IMPES)的半隐格式是求解该类问题的一类广泛使用的重要算法,即隐式求解压力方程和显式更新饱和度,但传统的IMPES方法在更新饱和度时需计算饱和度梯度,因此在求解复杂非均匀介质中的两相流问题中并不适用,Hoteit 和Firoozabadi提出了改进的IMPES方法,使得改进后的方法可以预测非均匀介质中饱和度不连续的情况。由于前两种IMPES方法在更新饱和度时只选取其中一相流体的质量守恒方程进行计算,因此无法保证另一相流体亦满足局部质量守恒。这两种IMPES方法对压力方程的推导都是在偏微分方程连续层面加合各相的体积守恒方程而得,然后对压力方程和饱和度方程用不完全匹配的空间离散方法,所以无法同时保证两相流体逐相局部质量守恒。本文基于课题组近几年发表的求解两相渗流问题的几类新型IMPES半隐格式,提出了一种新型推导IMPES中压力方程的框架,即先对每相的体积守恒方程用局部守恒的空间离散方法做离散,然后加合每相离散的体积守恒方程,从而实现了压力方程和饱和度方程在空间离散上的完全匹配,从本质上克服了以往文献中的IMPES半隐方法无法同时保证两相流体均满足局部质量守恒的难点,使得新型IMPES方法保各相流体均满足局部质量守恒、饱和度保界,计算格式为无偏求解,且适用于求解非均匀介质中具有不同毛管力分布的两相渗流问题。本文提出的新型逐相守恒IMPES框架还有一个传统IMPES没有的优势,即新型逐相守恒IMPES框架中只需要定义体积守恒或质量守恒方程的空间离散方法,不需要单独定义压力方程的空间离散方法。课题组近几年发表的几类新型的IMPES半隐格式求解可以认为是本文提出的新型逐相守恒IMPES框架的特例,本文IMPES框架还可以应用于更复杂的多组分多相渗流,构造更多的新颖格式。本文同时通过非均质多孔介质数值算例,验证了新型IMPES方法在处理复杂地质条件下两相流问题的有效性和优越性,相比传统方法更具适应性,同时更稳定也更高效。
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关键词 : 两相渗流,隐压显饱方法,迎风型混合有限元方法,局部质量守恒,保界
Abstract

Multiphase flow in porous media is an important research topic in the field of oil and gas reservoir development. Due to the complex geological conditions in China, properties of rocks, such as permeability and porosity, are complex and heterogeneous. The numerical solution for the complex multiphase flow problems needs to overcome challenges such as the system’s multiple variables, strong nonlinearity, large computational cost, and the preservation of physical properties. For the traditional incompressible and immiscible two-phase flow model, the IMplicit Pressure Explicit Saturation (IMPES) semi-implicit scheme is a widely-used important algorithm for solving such problems, where the pressure equation is solved implicitly, and the saturation is updated explicitly. However, the traditional IMPES scheme requires the calculation of saturation gradients when updating the saturation. Therefore, it is not suitable for solving the two-phase flow problems in complex heterogeneous media. Hoteit and Firoozabadi proposed an improved IMPES method, allowing the method to reproduce discontinuous saturation in heterogeneous media. However, these two IMPES methods only update the saturation through the mass conservation equation of one phase of fluid, they cannot guarantee that the other phase of fluid also satisfies the local mass conservation property. The derivations of the pressure equations for these two IMPES methods are obtained by adding the volume conservation equations of each phase at the continuous level of partial differential equations, and then using incompletely matched spatial discretization methods for the pressure equation and the saturation equation. Therefore, it is impossible to simultaneously ensure the local mass conservation of each phase for the two-phase fluid. In this paper, based on several types of novel IMPES semi-implicit schemes for solving two-phase flow in porous media that we have published in recent years, we propose a new framework for deriving the pressure equation in IMPES. That is, we first discretize the volume conservation equation of each phase using a spatial discretization method with local conservation, and then add up the discretized volume conservation equations of each phase. In this way, a complete match in spatial discretization between the pressure equation and the saturation equation is achieved. Essentially, it overcomes the difficulty in previous literatures that the IMPES semi-implicit method cannot simultaneously ensure that both phases of the fluid satisfy local mass conservation. The novel IMPES method ensures that each phase of the fluid satisfies local mass conservation, the saturation is bounded, the computational scheme is an unbiased solution, and it is suitable for solving two-phase flow problem with different capillary pressure distributions in heterogeneous porous media. The novel phase-wise conservation IMPES framework proposed in this paper also has an advantage that the traditional IMPES does not have. That is, in the novel phase-by-phase conservation IMPES framework, it is only necessary to define the spatial discretization method of the volume conservation or mass conservation equation, and there is no need to separately define the spatial discretization method of the pressure equation. The solutions of several types of novel IMPES semi-implicit schemes that we have published in recent years can be regarded as special cases of the novel phase-by-phase conservation IMPES framework proposed in this paper. The IMPES framework in this paper can also be applied for more complex multi-component and multi-phase flow in porous media to construct more novel schemes. At the same time, through numerical examples of heterogeneous porous media, this paper verifies the effectiveness and superiority of the novel IMPES method in dealing with two-phase flow problems under complex geological conditions. Compared with the traditional method, it is more adaptable, more stable, and more efficient.


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收稿日期: 2025-04-30     
PACS:    
基金资助:国家自然科学基金面上项目(12471345) 资助
通讯作者: suns@tongji.edu.cn
引用本文:   
陈黄鑫, 陈玉祥, 孙树瑜. 无偏隐压显饱数值方法 —— 两相渗流模拟的新算法. 石油科学通报, 2025, 10(02): 309-325
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