Liu (2011) proposed a theory to describe fingering flow in homogeneous porous media at the large scale. It was argued based on an optimality principle that water flows in unsaturated porous media in such a manner that the generated flow patterns correspond to the minimum global flow resistance. The key difference between the proposed theory and other conventional unsaturated flow approaches is that the hydraulic conductivity in the former is not only related to water saturation or capillary pressure, but also proportional to a power function of water flux that is closely related to the fingering flow patterns. The validity of the theory was demonstrated by the consistence between the related theoretical results and observations (Liu, 2011). However, the mathematical derivation of the theoretical relationship for hydraulic conductivity, based on the calculus of variations, was brief in Liu (2011), and the discussion of the application of the optimality principle was not complete either. The objective of this addendum is to provide a more comprehensive description of the optimality principle used by Liu (2011) and to clarify several key steps in the mathematical derivation of the hydraulic conductivity relationship.

In this addendum, all the equation numbers and definitions of variables refer to the original article.

In fact, when the optimality principle is applied to minimize the global flow resistance, the total energy expenditure rate defined in Equation 6 could return either the minimum or the maximum, depending on the given boundary conditions. For fixed E values at boundaries, the small flow resistance allows for large water flux, and thus the absolute value of the energy expenditure rate reaches its maximum. On the other hand, for fixed flux values at boundaries, the small flow resistance allows for small absolute value of the energy gradient along a flow path, and accordingly, the absolute value of the energy expenditure rate reaches its minimum. These are the direct results of the definition of the energy expenditure rate that is the product of water flux and energy gradient. This important relationship between the boundary conditions and the extrema of the total energy expenditure rate was not discussed by Liu (2011). In the original derivation, the boundary condition with fixed E values was only addressed. Consequently, the integral given in Equation 6 should correspond to the maximum in the original publication.

Liu（2011）提出了一种理论来大规模描述均质多孔介质中的指流。基于最佳原理，有人认为水在不饱和多孔介质中的流动方式应使生成的流动模式对应于最小的整体流动阻力。所提出的理论与其他常规非饱和流方法之间的主要区别在于，前者的水力传导率不仅与水饱和度或毛细压力有关，而且与与指流密切相关的水通量的幂函数成比例。模式。相关理论结果与观测值之间的一致性证明了该理论的有效性（Liu，2011）。然而，在Liu（2011）中，基于变化的演算，对水力传导率的理论关系的数学推导是简短的，并且关于最优性原理的应用的讨论也不完整。本附录的目的是对Liu（2011）所使用的最优性原理进行更全面的描述，并阐明水力传导率关系的数学推导中的几个关键步骤。

在本附录中，所有方程式编号和变量定义均参考原始文章。

实际上，当应用最优性原理来最小化整体流动阻力时，公式6中定义的总能量消耗率可能会返回给定的边界条件，为最小值或最大值。对于固定的E如果在边界处具有最大的数值，则较小的流阻允许较大的水通量，因此能量消耗率的绝对值达到最大值。另一方面，对于边界处的固定通量值，较小的流动阻力允许沿流动路径的能量梯度的绝对值较小，因此，能量消耗率的绝对值达到其最小值。这些是能量消耗率定义的直接结果，能量消耗率是水通量和能量梯度的乘积。Liu（2011）没有讨论边界条件和总能源消耗率极值之间的重要关系。在原始推导中，具有固定E的边界条件价值观只是解决。因此，公式6中给出的积分应对应于原始出版物中的最大值。