中文摘要：

潜水含水层中的饱和地下水流动可以用两种方式描述。一种是严格的自由潜水面模型,用线性的三维连续性方程作为控制方程,在潜水面处用非线性自由潜水面边界条件。第二种是基于Dupuit假设和非线性Boussinesq方程的垂向平均模型。其中,第一种模型中的潜水自由面边界条件可以用两种方式描述,第一种只含有水头的偏导数,第二种则同时含有水头和潜水面标高的偏导数。前人分别从不同出发点推导了这两种边界条件,但是并未说明他们的等价性。用严格的数学推导证明了这两种边界条件的等价性,进一步完善了前人的工作。另一方面,已有人从第一种边界条件出发,对三维连续性水流方程求积分推导出了Boussinesq方程,但这种方法比较抽象,物理意义不太明确。从第二种边界条件出发直接推导Boussinesq方程,所用方法简单、物理意义明确,易于理解。在自由潜水面边界条件下,Boussinesq方程可以直接通过对三维连续性方程在整个饱和含水层厚度上求积分而得到。此外,潜水含水层的隔水底板可以推广为一个曲面。阐明了两种边界条件、三维连续性水流方程、Dupuit假设、Boussinesq方程之间的内在关系。

英文摘要：

The saturated groundwater flow in an unconfined aquifer can be described in two different approaches. The first is the strict free surface model including the linear,three-dimensional,flow continuity differential equation for the groundwater head derived from the mass balance and Darcy ＇s law and the nonlinear free surface boundary condition at the location of the water table. The second is the verticallyaveraged model based on the well-known nonlinear Boussinesq ＇s equation,which is derived from the mass balance,the Darcy＇s law and the Dupuit assumptions. Although both approaches have been widely used,to the knowledge of the authors,the direct link between the three-dimensional and two-dimensional models is not explicitly and directly stated in the literature. Results in this paper show that by means of the free surface boundary condition,the Boussinesq＇s equation can be directly obtained by integrating the three-dimensionalflow continuity differential equation of head over the whole saturated thickness. Additionally,the impermeable bottom of the unconfined aquifer can be generalized into a curved surface.