We present a multiphase level set method with local volume conservation for capillary-controlled displacement in porous structures. Standard numerical formulations of the level set method for capillary-controlled (or, curvature-driven) motions assume phase pressures and interface properties are spatially uniform and disregard the fact that separate phase ganglia typically have distinct pressures. This is a major problem for the suitability of such methods to simulate capillary trapping in porous rocks as it will lead to severe mass loss. The method presented here preserves volumes of individual phase ganglia, while it predicts capillary pressures between ganglia and surrounding phases. A conservative volume redistribution algorithm handles ganglia breakup and coalescence. The method distinguishes between three-phase systems, where separate level set functions describe the different phases, and two-phase systems, where one level set function represents interfaces. We present sequential and parallel algorithms for the new method and emphasize important aspects specific to the patch-based parallel implementation.

We validate the method numerically by applying local volume conservation to simulations of two and three phase systems in both two and three spatial dimensions. The model is tested for both saturation and pressure controlled systems and handles both merging and splitting of phase ganglia.

我们提出了一种局部体积守恒的多相能级设置方法，用于多孔结构中毛细管控制的位移。用于毛细管控制（或曲率驱动）运动的水平集方法的标准数值公式假定相压力和界面特性在空间上是均匀的，并且忽略了独立相神经节通常具有不同压力的事实。对于这种方法在多孔岩石中模拟毛细管捕集的适用性而言，这是一个主要问题，因为这将导致严重的质量损失。这里介绍的方法保留了单个相神经节的体积，同时预测了神经节和周围相之间的毛细压力。保守的卷重新分配算法可处理神经节破裂和合并。该方法区分三相系统，其中，单独的级别集功能描述了不同的阶段；对于两阶段系统，其中一个级别集功能表示了接口。我们介绍了该新方法的顺序算法和并行算法，并着重说明了基于修补程序的并行实现的重要方面。

我们通过在两个和三个空间维度上将局部体积守恒应用于两相和三相系统的仿真，以数值方式验证了该方法。该模型针对饱和系统和压力控制系统进行了测试，并且可以处理相神经节的合并和分裂。