# The Large-Scale Hydraulic Conductivity for Gravitational Fingering Flow in Unsaturated Homogenous Porous Media: A Review and Further Discussion

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. The Relationship Based on the Optimality Principle

_{x}, q

_{y}and q

_{z}are large-scale Darcy flux along x, y and z directions, respectively.

_{sat}is the saturated hydraulic conductivity, ${k}_{r}$ is the relative permeability, and the square of the energy gradient ${S}_{*}$ is defined by

_{w}with an unknown function w:

_{x}, w

_{y}, and w

_{z}are derivatives of w with respect to x, y, and z, respectively. When I

_{w}reaches its extrema, the unknown function w follows the Euler equation [23]:

_{h}is a function of h and a is a constant. The term ${K}_{sat}{F}_{h}\left(h\right)$ in Equation (22) is considered unsaturated hydraulic conductivity on a local scale (Figure 1) simply because it is a function of h (the capillary pressure head in fingering flow zone) only. The power function term

#### 2.2. The Relationship Based on the Fractal Flow Pattern

_{e}is effective water saturation defined by

_{1}. However, a nonequilibrium process may still exist on the scale l

_{1}. Thus, we continue the process for deriving Equation (32), on an average sense, for a smaller control volume with a characteristic length l

_{1}and a cutoff scale l

_{2}< l

_{1}at which we have:

_{w,1}is the average water volume for the small control volumes covering the fingering flow part and determined by ${V}_{w}/{N}^{*}\left({l}_{1}\right)$.

#### 2.3. The Consistency between the Relationships Derived from the Two Methods

_{f}is hydraulic conductivity within the fingering zone. Inserting Equation (38) into Equation (23) gives

## 3. Discussion

## 4. Concluding Remarks

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Demonstration of the “box” counting procedure for a two-dimensional fingering flow pattern. The shaded zones correspond to the boxes covering fingering.

**Figure 3.**Parameter $\gamma $ as a function of the index of pore size distribution $\lambda $ between 0.2 and 12.

**Figure 4.**The fraction of fingering flow zone f for ${S}_{e}$ = 0.5 and $\gamma $ between 0.75 and 0.95.

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**MDPI and ACS Style**

Liu, H.-H. The Large-Scale Hydraulic Conductivity for Gravitational Fingering Flow in Unsaturated Homogenous Porous Media: A Review and Further Discussion. *Water* **2022**, *14*, 3660.
https://doi.org/10.3390/w14223660

**AMA Style**

Liu H-H. The Large-Scale Hydraulic Conductivity for Gravitational Fingering Flow in Unsaturated Homogenous Porous Media: A Review and Further Discussion. *Water*. 2022; 14(22):3660.
https://doi.org/10.3390/w14223660

**Chicago/Turabian Style**

Liu, Hui-Hai. 2022. "The Large-Scale Hydraulic Conductivity for Gravitational Fingering Flow in Unsaturated Homogenous Porous Media: A Review and Further Discussion" *Water* 14, no. 22: 3660.
https://doi.org/10.3390/w14223660