低渗透多孔介质渗流动边界模型的解析与数值解

考虑启动压力梯度的低渗透多孔介质非达西渗流模型属于强非线性动边界问题, 分别利用相似变量变换方法和基于空间坐标变换的有限差分方法, 对内边界变压力情况下、考虑启动压力梯度的一维低渗透多孔介质非达西渗流动边界模型进行了精确解析与数值求解研究. 研究结果表明:该动边界模型存在唯一的精确解析解, 且所求得的精确解析解可严格验证数值解的正确性;且当启动压力梯度值趋于零时, 非达西渗流动边界模型的精确解析解将退化为达西渗流情况下的精确解析解. 由求解结果作出的非零无因次启动压力梯度下的地层压力分布曲线表现出紧支性特点, 其与达西渗流模型的有显著不同. 因此, 研究低渗透多孔介质中非稳态渗流问题时, 应该考虑动边界的影响. 研究内容完善了低渗透多孔介质的非达西渗流力学理论, 为低渗透油气藏开发的试井解释与油藏数值模拟技术提供了理论基础.

RESEARCH ON ANALYTICAL AND NUMERICAL SOLUTIONS OF A MOVING BOUNDARY MODEL OF SEEPAGE FLOW IN LOW-PERMEABLE POROUS MEDIA

The modes of non-Darcy seepage flow in low- permeable porous media with threshold pressure gradient belong to the moving boundary problems with strong nonlinearity. In this paper, a moving boundary model of one- dimensional non-Darcy seepage flow in low- permeable porous media with threshold pressure gradient is studied for the case of a variable pressure at the inner boundary. A similarity transformation method and a spatial coordinate transformation based finite difference method are applied to obtain the exact analytical solution and numerical solution of the moving boundary model, respectively. Research results show that the moving boundary model has a unique exact analytical solution, which also strictly verifies the accuracy of the numerical solution; and when the value of threshold pressure gradient tends to zero, the exact analytical solution of the moving boundary model of non-Darcy seepage flow can reduce to the one corresponding to Darcy's seepage flow. The formation pressure distribution curves with non- zero dimensionless threshold pressure gradient from these solutions exhibit the characteristics of compact support, which is obviously different from those corresponding to Darcy's flow model. Therefore, the study on the unsteady seepage flow problem in low- permeable porous media should take into account the effect of moving boundary condition. The presented research improves the theory of non-Darcy seepage flow in low- permeable porous media, and also builds theoretical foundation for the technologies of well testing interpretation and reservoir numerical simulation involved in the development of low- permeable oil and gas reservoirs.