# 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

## References

- Flury, M.; Flühler, H.; Jury, W.A.; Leuenberger, J. Susceptibility of soils to preferential flow of water: A field study. Water Resour. Res.
**1994**, 30, 1945–1954. [Google Scholar] [CrossRef] - Šimůnek, J.; Jarvis, N.J.; van Genuchten, M.T.; Gardenas, A. Review and comparison of models for describing non-equilibrium and preferential flow and transport in the vadose zone. J. Contam. Hydrol.
**2003**, 272, 14–35. [Google Scholar] [CrossRef] - Beven, K.; Germann, P. Macropores and water flow in soils. Water Resour. Res.
**1982**, 18, 1311–1325. [Google Scholar] [CrossRef][Green Version] - Gerke, H.H.; van Genuchten, M.T. A dual-porosity model for simulating the preferential movement of water and solutes in structured porous media. Water Resour. Res.
**1993**, 29, 305–319. [Google Scholar] [CrossRef] - Warren, J.E.; Root, P.J. The behavior of naturally fractured reservoirs. SPEJ
**1963**, 3, 245–255. [Google Scholar] [CrossRef][Green Version] - Liu, H.H.; Doughty, C.; Bodvarsson, G.S. An active fracture model for unsaturated flow and transport in fractured rocks. Water Resour. Res.
**1998**, 34, 2633–2646. [Google Scholar] [CrossRef] - Glass, R.J.; Steenhuis, T.S.; Parlarge, J.Y. Wetting front instability as a rapid and far-reaching hydrologic process in the vadose zone. J. Contam. Hydrol.
**1988**, 3, 207–226. [Google Scholar] [CrossRef] - Wang, Z.; Feyen, J.; Elrick, D.E. Prediction of fingering in porous media. Water Resour. Res.
**1998**, 34, 2183–2190. [Google Scholar] [CrossRef] - de Rooij, G.H. Modeling fingered flow of water in soils owing to wetting front instability: A review. J. Hydrol.
**2000**, 231–232, 277–294. [Google Scholar] [CrossRef] - Liu, H.H. Fluid Flow in the Subsurface: History, Generalization and Applications of Physical Laws; Springer: Cham, Switzerland, 2017. [Google Scholar]
- Beven, K. A century of denial: Preferential and nonequilibrium water flow in soils, 1864–1984. Vadose Zone J.
**2018**, 17, 180153. [Google Scholar] [CrossRef] - Brooks, R.H.; Corey, A.T. Hydraulic Properties of Porous Media; Hydrology Paper no. 3; Civil Engineering Department, Colorado State University: Fort Collins, CO, USA, 1964. [Google Scholar]
- van Genuchten, M.T. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J.
**1980**, 44, 892–898. [Google Scholar] [CrossRef][Green Version] - Liu, H.H. A conductivity relationship for steady-state unsaturated flow processes under optimal flow conditions. Vadose Zone J.
**2011**, 10, 736–740. [Google Scholar] [CrossRef][Green Version] - Eagleson, P.S. Ecohydrology: Darwinian Expression of Vegetation Form and Function; Cambridge University Press: New York, NY, USA, 2002. [Google Scholar]
- Bejan, A. Shape and Structure, from Engineering to Nature; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Leopold, L.B.; Langbein, W.B. The Concept of Entropy in Landscape Evolution; US Government Printing Office: Washington, DC, USA, 1962.
- Howard, A.D. Theoretical model of optimal drainage networks. Water Resour. Res.
**1990**, 26, 2107–2117. [Google Scholar] [CrossRef] - Rodriguez-Iturbe, I.; Rinaldo, A.; Rigon, A.; Bras, R.L.; Marani, A.; Hijas-Vasquez, E. Energy dissipation, runoff production and the three-dimensional structure of river basins. Water Resour. Res.
**1992**, 28, 1095–1103. [Google Scholar] [CrossRef] - Rinaldo, A.; Rodriguez-Iturbe, I.; Rigon, A.; Bras, R.L.; Ijjasz-vazquez, E.; Marani, A. Minimum energy and fractal structures of drainage networks. Water Resour. Res.
**1992**, 28, 2183–2191. [Google Scholar] [CrossRef] - Liu, H.H.; Zhang, G.; Bodvarsson, G.S. The active fracture model: Its relation to fractal flow patterns and an evaluation using field observations. Vadose Zone J.
**2003**, 2, 259–269. [Google Scholar] [CrossRef] - Gelhar, L. Stochastic Subsurface Hydrology; Prentice Hall: Hoboken, NJ, USA, 1993. [Google Scholar]
- Weinstock, R. Calculus of Variations with Applications to Physics and Engineering; Dover Publications, Inc.: New York, NY, USA, 1974. [Google Scholar]
- Flury, M.; Flühler, H. Modeling solute leaching in soils by diffusion-limited aggregation: Basic concepts and applications to conservative solutes. Water Resour. Res.
**1995**, 31, 2443–2452. [Google Scholar] [CrossRef] - Persson, M.; Yasuda, H.; Albergel, J.; Berndtsson, R.; Zante, P.; Nasri, S.; Ohrstrom, P. Modeling plot scale dye penetration by a diffusion limited aggregation (DLA) model. J. Hydrol.
**2001**, 250, 98–105. [Google Scholar] [CrossRef] - Feder, J. Fractals; Plenum Press: New York, NY, USA, 1988. [Google Scholar]
- Smith, J.E.; Zhang, Z.F. Determining effective interfacial tension and predicting finger spacing for DNAPL penetration into water-saturated porous media. J. Contam. Hydrol.
**2001**, 48, 167–183. [Google Scholar] [CrossRef] - Mualem, Y. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res.
**1976**, 12, 513–522. [Google Scholar] [CrossRef] - Sheng, F.; Wang, K.; Zhang, R.D.; Liu, H.H. Characterizing soil preferential flow using iodine–starch staining experiments and the active region model. J. Hydrol.
**2009**, 367, 115–124. [Google Scholar] [CrossRef][Green Version] - Liu, H.H.; Zhang, R.; Bodvarsson, G.S. An active region model for capturing fractal flow patterns in unsaturated soils: Model development. J. Contam. Hydrol.
**2005**, 80, 18–30. [Google Scholar] [CrossRef] [PubMed][Green Version] - Öhrström, P.; Persson, M.; Albergel, J.; Zante, P.; Nasri, S.; Berndtsson, R.; Olsson, J. Field-scale variation of preferential flow as indicated from dye coverage. J. Hydrol.
**2002**, 257, 164–173. [Google Scholar] [CrossRef] - Engstrom, E.; Liu, H.H. Modeling bacterial attenuation in on-site wastewater treatment systems using the active region model and column-scale data. Environ. Earth Sci.
**2015**, 74, 4827–4837. [Google Scholar] [CrossRef] - Beljadid, A.; Gueto-Felgueroso, L.; Juanes, R. A continuum model of unstable infiltration in porous media endowed with an entropy function. Adv. Water Resour.
**2020**, 114, 103684. [Google Scholar] [CrossRef] - Eliassi, M.; Glass, R.J. On the porous-continuum modeling of gravity-driven fingers in unsaturated materials: Extension of standard theory with a hold-back-pile-up effect. Water Resour. Res.
**2002**, 38, 1234. [Google Scholar] [CrossRef][Green Version] - Kmec, J.; Fürst, T.; Vodák, R.; Šír, M. A two dimensional semi-continuum model to explain wetting front instability in porous media. Sci. Rep.
**2021**, 11, 3223. [Google Scholar] [CrossRef]

**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.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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