描述渗流流动的经典达西定律有适用范围，要求渗流速度、渗透性不能太强也不能太低，它所描述的压力与速度梯度的关系是线性关系。超出适用范围的渗流为非达西渗流，如低渗油气藏、致密气、页岩气以及高渗透油气藏，脑组织、肾组织中血液渗流等，流动规律复杂，其流速与压力梯度间的关系是非线性的。关系式中出现不光滑的绝对值函数；低渗透时还出现拟启动压力梯度，以及压力梯度不足时的不流动区域，导致数学描述和数值计算困难。对这类渗流，由于这种非线性关系，其单组分、单相两组分、两相流动问题的数学模型为强耦合非线性偏微分方程组。本课题将结合实际背景研究非达西渗流的数学模型和实用高效的数值方法，包括结构化网格和非结构网格的局部守恒有限差分法、有限体积法、多点通量方法，剖分单元网格线上速度通量的表示，算法的理论分析和高性能实现。选择低渗透致密气藏作为应用实例，全面研究气水两相渗流数值计算、应用中出现的各类计算问题。

The classic Darcy's law to describe porous media flow is applicable for flow with velocity not too fast nor too slow, and with permeability not so strong nor too low. In this case the relationship between the pressure and velocity is linear. Porous media flow which can not be describled by Darcy's law is non-Darcy flow, for example, low permeability petroleum reservoirs, low permeability gas reservoirs, tight gas reservoirs as well as some of the high permeability oil and gas reservoirs, the fluid within brain tissue, kidney tissue, etc. In this case the relationship between the flow rate and the pressure gradient is a nonlinear relationship with the no-smooth absolute value function. Forthmore in the case of low permeability flow also appears the pseudo threshold pressure gradient, and there exists no flow area where the pressure gradient is below the pseudo threshold pressure gradient. All this phenomena result in the difficulties in mathematical modelling and numerical calculations. Since the nonlinear relationship, the mathematical models for single-component flow, single-phase two-component flow, two-phase flow are strongly coupled non-linear partial differential equations. For this kind of flows, combining with the actual background, this project will study the mathematical modelling, the practical and efficient numerical methods including local-conservational finite difference method, finite volume method, multipoint-flux method, based on structured and unstructured grid. We will establish the theoretical analysis and high-performance implementation. Choosing low-permeability tight gas reservoir as an application example, we give a comprehensive study of the numerical solution of the two-phase gas-water seepage problem including other computational problems appear in the model.