In this work, we consider a mathematical model for describing flow in an unsaturated porous medium containing a fracture. Both the flow in the fracture as well as in the matrix blocks are governed by Richards' equation coupled by natural transmission conditions. Using formal asymptotics, we derive upscaled models as the limit of vanishing ε, the ratio of the width and length of the fracture. Our results show that the ratio of porosities and permeabilities in the fracture to matrix determine, to the leading order of approximation, the appropriate effective model. In these models the fracture is a lower dimensional object for which different transversally averaged models are derived depending on the ratio of the porosities and permeabilities of the fracture and respective matrix blocks. We obtain a catalogue of effective models which are validated by numerical computations.

在这项工作中，我们考虑一个数学模型来描述包含裂缝的非饱和多孔介质中的流动。裂缝以及基质块中的流动都受自然传播条件耦合的理查兹方程控制。使用形式渐近线，我们推导了升阶模型作为ε消失的极限，裂缝的宽度与长度之比。我们的结果表明，裂缝与基质的孔隙率和渗透率之比决定了合适的有效模型，逼近的主导顺序为。在这些模型中，裂缝是低维物体，根据裂缝和相应基质块的孔隙率和渗透率的比率，可以得出不同的横向平均模型。我们获得了有效模型的目录，该目录已通过数值计算验证。