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## Numerical simulation of fractured-vuggy porous media based on gamblets

Zhang Qing-Fu, Huang Zhao-Qin, Yao Jun, Li Yang, Yan Xia
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• #### 摘要

缝洞型介质通常具有非均质性强、结构多尺度的特征. 传统数值方法在解决此类多尺度流动问题时, 难以兼顾计算精度与计算效率, 无法实际应用. 对此, 本文提出了多孔介质流体流动的多尺度分解法, 并应用于缝洞介质流动模拟, 能够大幅减小计算的复杂度, 同时可以通过控制均化程度控制计算精度. 该方法将求解空间分为若干个子空间的正交直和, 从而获得一个近线性的计算复杂度; 以分层计算的方式实现了快速计算, 另外这种方法是一种无网格方法, 具有较好的地层适应性. 同时, 采用离散缝洞模型简化缝洞结构, 进一步提高了计算效率. 详细阐述了基于多尺度分解法的多孔介质流体流动数值计算格式的建立, 重点介绍了如何在不同的层次上计算基函数. 数值结果表明, 本文提出的计算方法不仅能够准确捕捉多孔介质中的精细流动特征, 而且具有很高的计算效率, 是一种有效的流动模拟方法.

#### Abstract

Numerical simulation of a fractured-vuggy porous medium is a challenging problem. One reason is the coexistence of matrix, fractures and vugs on multiple scales that need to be coupled, and the other reason is that the high-resolution fractured-vuggy model may contain up to several millions of gridcells in applications, which brings severe computational challenges into the numerical methods. Therefore, the requirement for accurate and efficient technique is widely increasing. Fractured-vuggy porous medium is generally represented by triple-continuum model in which the matrix system, fracture system and vug system each are treated as a parallel continuous system. Although triple-continuum model is widely used because of its easy-implementation and high efficiency, it fails to capture the detailed flow patterns of reservoir with disconnected long fractures. Discrete fracture-vug network (DFVN) model can precisely model the fluid flow in fractures and vugs. However, the simulation of this model is deemed intractable even with the advent of supercomputers because of the large amount of calculation. In view of the fact that the multigrid method is now well known as one of the fastest method of solving elliptic problems, in this paper we introduce a nearly linear complexity multiresolution decomposition method for fluid flow in a fractured-vuggy reservoir. The detailed flow patterns are described by combing the advantages of continuum model and discrete model. That is, the homogenization theory is used to construct an equivalent permeability in each coarse grid block in which the vugs and small-scale fractures are represented by discrete fracture-vug network model. We decompose the solution space into several subspaces and then we compute the corresponding solutions of heterogeneous discrete fracture network model in each subspace. Gamblets are constructed and they are elementary solutions of hierarchical information games associated with the process of computing with partial information and limited resources. These gamblets have a natural Bayesian interpretation under the mixed strategy emerging from the game theoretic formulation. This method could realize its fast simulation by decomposing the solution space into a direct sum of linear subspaces that are orthogonal to each other. Finally, the pressure difference distribution of fractured-vuggy porous medium is obtained by combing the DFVN solutions of all subspaces. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed multiresolution decomposition method. The results show that this method is a promising method of numerically simulating the fractured-vuggy porous medium.

#### 作者及机构信息

###### 通信作者: 黄朝琴, emcgroup@163.com
• 基金项目: 国家科技重大专项(批准号: 2016ZX05060-010)、中央高校基本科研业务费(批准号: 17CX06007)和国家自然科学基金(批准号: 51404292)资助的课题.

#### Authors and contacts

###### Corresponding author: Huang Zhao-Qin, emcgroup@163.com
• Funds: Project supported by the National Science and Technology Major Project of the Ministry of Science and Technology of China (Grant No. 2016ZX05060-010), the Fundamental Research Fund for the Central Universities, China (Grant No. 17CX06007), and the National Natural Science Foundation of China (Grant No. 51404292).

#### 施引文献

• 图 1  缝洞型介质示意图

Fig. 1.  Schematic of fractured-vuggy porous medium.

图 2  区域$\varOmega$的网格剖分示意图

Fig. 2.  Schematic of grid partition of solution space.

图 3  分层基函数示意图

Fig. 3.  Illustration of gamblets $\psi _i^{\left( k \right)}$.

图 4  基函数$\chi _i^{\left( k \right)}$示意图

Fig. 4.  Illustration of basis functions $\chi _i^{\left( k \right)}$.

图 5  区域$\varOmega$的多尺度解示意图

Fig. 5.  Illustration of solution of the multiple spaces.

图 6  (a)小尺度缝洞型油藏几何模型; (b)等效渗透率场图; (c) 多层网格系统

Fig. 6.  (a) Geometrical model of a small-scale-fractured vuggy porous oil reservoir; (b) equivalent permeability of fractured-vuggy medium; (c) nested grid system.

图 7  对于小尺度缝洞模型, 参考解和多重网格解对比　(a) 参考解; (b) k = 3时的多重网格解

Fig. 7.  Comparison of reference solution and gamblets solution for a small-scale-fractured vuggy porous medium: (a) Reference solution; (b) gamblets solution with k = 3.

图 8  长裂缝介质模型

Fig. 8.  Geometrical model of a fractured-vuggy porous medium with a long fracture.

图 9  对于长裂缝模型, 参考解和多重网格解对比　(a)参考解; (b) k = 3时的多重网络解; (c) k = 2时的多重网格解

Fig. 9.  Comparison of reference solution and gamblets solution for a fractured-vuggy porous medium with a long fracture: (a) Reference solution; (b) gamblets solution with k = 3; (c) gamblets solution with k = 2.

图 10  大尺度缝洞介质几何模型

Fig. 10.  Geometrical model of a large-scale-fractured vuggy porous medium.

图 11  大尺度缝洞介质模型在各层上的解

Fig. 11.  Solutions in different levels for a large-scale-fractured vuggy porous medium.

图 12  对于大尺度缝洞介质模型, 参考解和多重网格解对比　(a) 参考解; (b) k = 3时的多重网格解; (c) k = 2时的多重网格解; (d) k = 1时的多重网格解

Fig. 12.  Comparison of reference solution and gamblets solution for a large-scale-fractured vuggy porous medium: (a) Reference solution; (b) gamblets solution with k = 3; (c) gamblets solution with k = 2; (d) gamblets solution with k = 1.

图 13  沿x = 5 m的参考解和不同均化程度下多重网格解对比曲线图

Fig. 13.  Comparison of reference solution and gamblets solution with different k when x = 5 m.

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##### 出版历程
• 收稿日期:  2018-08-31
• 修回日期:  2019-01-21
• 上网日期:  2019-03-01
• 刊出日期:  2019-03-20

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