In this work, we study the behavior of saturation fronts for two-phase flow through a long homogeneous porous column. In particular, the model includes hysteresis and dynamic effects in the capillary pressure and hysteresis in the permeabilities. The analysis uses traveling wave approximation. Entropy solutions are derived for Riemann problems that are arising in this context. These solutions belong to a much broader class compared to the standard Oleinik solutions, where hysteresis and dynamic effects are neglected. The relevant cases are examined and the corresponding solutions are categorized. They include nonmonotone profiles, multiple shocks, and self-developing stable saturation plateaus. Numerical results are presented that illustrate the mathematical analysis. Finally, we discuss the implication of our findings in the context of available experimental results.

中文翻译：

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两相多孔介质流动问题的前沿：滞后和动态毛细作用的影响

在这项工作中，我们研究了通过长均质多孔柱的两相流动的饱和前沿行为。特别是，该模型包括毛细管压力中的滞后和动态效应以及渗透率中的滞后。该分析使用行波近似。熵解是为在这种情况下出现的黎曼问题推导出来的。与忽略滞后和动态效应的标准 Oleinik 解决方案相比，这些解决方案属于更广泛的类别。对相关案例进行了审查，并对相应的解决方案进行了分类。它们包括非单调剖面、多次冲击和自我发展的稳定饱和平台。给出了说明数学分析的数值结果。最后，