Finite analytic numerical method for three-dimensional fluid flow in heterogeneous porous media
Finite analytic numerical method for three-dimensional fluid flow in heterogeneous porous media
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DOI：
10.1016/j.jcp.2014.08.026
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Yan-Feng Wang;Zhifeng Liu;Xiao-Hong Wang
中科院分区：
文献类型：
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作者：
Yan-Feng Wang;Zhifeng Liu;Xiao-Hong Wang
了解流体在非均匀多孔介质中的流动是应用地球科学的基础。天然含水层或油藏连通性变化较大，使得等效渗透率具有较强的空间变异性。在对地下水流进行模拟时，不同网格单元之间的界面上的渗透率可能具有很强的不连续性。利用传统的数值格式模拟强非均质介质中的流动，需要大幅度提高网格单元的精确度才能得到准确的结果。最近，我们提出了一种求解非均质多孔介质中二维流体流动的有限解析数值格式。该方法只需2×2或3×3的剖分，即可提供较精确的解。本文发展了求解非均质多孔介质中三维流体流动的有限分析数值方法。对于矩形网格系统，一般认为，根据典型的幂函数解，在垂直于连接不同渗透率区域的边缘的平面内，压力梯度在逼近边缘时趋于无穷大，并且由于压力的连续性，沿边缘的压力的切向导数必然是有限的。因此，在每条边周围的邻域内，三维流动将缩减为二维流动。然后将这种准二维行为应用于构造有限解析数值格式。数值算例表明，该格式只需2×2×2或3×3×3的剖分即可得到较高精度的解，且收敛速度与渗透率非均质性无关。由于该格式具有较高的计算效率，被用来验证著名的LLM(Landau，Lifshitz和Matheron)猜想，该猜想为各向同性对数正态多孔介质提供了k eq/k G=EXP(1 6σlnk2)。数值结果不支持大σlnk的这一猜想，但强烈地表明线性关系k eq/k G=1+1 6σlnk2。
Understanding fluid flows in heterogeneous porous media is fundamental to applied geosciences. The wide connectivity variations in the natural aquifer or oil reservoirs make the equivalent permeability have strong spatial variations. When performing the simulations for subsurface flows, the permeabilities may have strong discontinuities across the interfaces between different grid cells. Utilizing the traditional numerical schemes to simulate flows in strong heterogeneous media, the refinement ratio for the grid cell needs to increase dramatically to get an accurate result. Recently, we proposed a finite analytic numerical scheme to solve the two-dimensional fluid flows in heterogeneous porous media. With only 2× 2 or 3× 3 subdivisions, this scheme can provide rather accurate solutions. In this paper, we develop the finite analytic numerical method for solving the three-dimensional fluid flows in heterogeneous porous media. For the rectangular grid system, it is generally proposed that the pressure gradient in a plane normal to the edge joining different permeability regions will tend to infinite as approaching the edge according to a typical power-law solution and the tangential derivate of the pressure along the edge must be of limited value due to the pressure continuity. Consequently, the three-dimensional flow will reduce to the two-dimensional one in the neighborhood around each edge. Such quasi-two-dimensional behavior is then applied to construct a finite analytic numerical scheme. Numerical examples show that the proposed scheme can provide rather accurate solutions with only 2× 2× 2 or 3× 3× 3 subdivisions and the convergent speed is independent of the permeability heterogeneity. Due to its high calculation efficiency, the proposed scheme is utilized to test the well known LLM (Landau, Lifshitz and Matheron) conjecture, which provides k eq/k G= exp(1 6 σ ln k 2) for the isotropic log-normal porous medium. The numerical results do not support this conjecture for large σ ln k, but strongly suggest the linear relation k eq/k G= 1+ 1 6 σ ln k 2.