# Numerical Simulation of Unstable Preferential Flow during Water Infiltration into Heterogeneous Dry Soil

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Mathematical Model of Unsaturated Flow in Porous Media

#### 2.2. Finite Element Implementation

#### 2.3. Capillary Pressure and Relative Permeability Functions

#### 2.4. Problem Setup

#### 2.5. Random Heterogeneity Fields

## 3. Results

#### 3.1. Impact of Correlation Length on the Patterns of Unstable Infiltration: Isotropic Permeability Fields

#### 3.2. Impact of Correlation 0 on the Patterns of Unstable Infiltration: Anisotropic Permeability Fields

#### 3.3. Infiltration into Soils with Large Contrast between Spatial Correlation Lengths

#### 3.4. Impact of Infiltration Rate on the Patterns of Infiltration into Heterogeneous Dry Soil

## 4. Discussion and Conclusions

- Soil heterogeneity did not suppress fingering instabilities, but it may actually enhance their effects of preferential flow and channeling during water infiltration. In particular, soil heterogeneity enhanced fingering at low infiltration rates for all explored spatial structures of intrinsic permeability. At large infiltration rates, the wet area tended to cover the entire soil, so that the finger width was comparable with the system size.
- Fingering patterns strongly depended on soil structure, in particular the correlation length and anisotropy of the permeability field. While the finger size and flow dynamics were only slightly controlled by the correlation length in isotropic fields, layering led to significant finger meandering and bulging, changing arrival times and wetting efficiencies.
- Fingering and soil heterogeneity needed to be considered when upscaling the constitutive relationship of multiphase porous media from the finger to field and basin scales. While relative permeabilities remained unchanged upon upscaling for stable displacements, the inefficient wetting due to fingering led to relative permeabilities at the field scale that were significantly different from those at the Darcy scale. These effective relative permeability functions also depended, although less strongly, on heterogeneity and soil structure.
- Novel experimental observations of unsaturated flow and fingering in heterogeneous soils, such as those in [33,34,46], will help guide the refinement of mathematical models such as the one used in the present study, in particular regarding the calibration of the strength of the second-gradient extension of Richards’ equation used to reproduce fingering instabilities.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Rodriguez-Iturbe, I. Ecohydrology: A hydrologic perspective of climate-soil-vegetation dynamics. Water Resour. Res.
**2000**, 36, 3–9. [Google Scholar] [CrossRef][Green Version] - Richards, L.A. Capillary conduction of liquids through porous mediums. Physics
**1931**, 1, 318–333. [Google Scholar] [CrossRef] - Horton, R.E. The role of infiltration in the hydrologic cycle. Trans. Am. Geophys. Union
**1933**, 14, 446–460. [Google Scholar] [CrossRef] - Philip, J.R. The theory of infiltration 1. The infiltration equation and its solution. Soil Sci.
**1957**, 83, 345–357. [Google Scholar] [CrossRef] - Philip, J.R. Theory of infiltration. In Advances in Hydroscience; Chow, V.T., Ed.; Academic Press: New York, NY, USA, 1969; pp. 215–296. [Google Scholar]
- DiCarlo, D.A. Stability of gravity-driven multiphase flow in porous media: 40 years of advancements. Water Resour. Res.
**2013**, 49, 4531–4544. [Google Scholar] [CrossRef] - Xiong, Y. Flow of water in porous media with saturation overshoot: A review. J. Hydrol.
**2014**, 510, 353–362. [Google Scholar] [CrossRef] - Hill, D.E.; Parlange, J.Y. Wetting front instability in layered soils. Soil Sci. Soc. Am. J.
**1972**, 36, 697–702. [Google Scholar] [CrossRef] - Glass, R.J.; Parlange, J.Y.; Steenhuis, T.S. Wetting front instability, 2. Experimental determination of relationships between system parameters and two-dimensional unstable flow field behaviour in initially dry porous media. Water Resour. Res.
**1989**, 25, 1195–1207. [Google Scholar] [CrossRef] - Ritsema, C.J.; Dekker, L.W.; Nieber, J.L.; Steenhuis, T.S. Modeling and field evidence of finger formation and finger recurrence in a water repellent sandy soil. Water Resour. Res.
**1998**, 34, 555–567. [Google Scholar] [CrossRef][Green Version] - Méheust, Y.; Løvoll, G.; Måløy, K.J.; Schmittbuhl, J. Interface scaling in a two-dimensional porous medium under combined viscous, gravity, and capillary effects. Phys. Rev. E
**2002**, 66, 051603. [Google Scholar] [CrossRef][Green Version] - Leroux, N.R.; Pomeroy, J.W. Modelling capillary hysteresis effects on preferential flow through melting and cold layered snowpacks. Adv. Water Resour.
**2017**, 107, 250–264. [Google Scholar] [CrossRef] - Aubrecht, R.; Lánczos, T.; Schlögl, J.; Audy, M. Small-scale modelling of cementation by descending silica-bearing fluids: Explanation of the origin of arenitic caves in South American tepuis. Geomorphology
**2017**, 298, 107–117. [Google Scholar] [CrossRef] - Diment, G.A.; Watson, K.K. Stability analysis of water movement in unsaturated porous materials 3. Experimental studies. Water Resour. Res.
**1985**, 21, 979–984. [Google Scholar] [CrossRef] - Glass, R.J.; Cann, S.; King, J.; Baily, N.; Parlange, J.Y.; Steenhuis, T.S. Wetting front instability in unsaturated porous media: A three-dimensional study in initially dry sand. Transp. Porous Media
**1990**, 5, 247–268. [Google Scholar] [CrossRef] - Selker, J.S.; Leclerq, P.; Parlange, J.Y.; Steenhuis, T. Fingered flow in two dimensions, 1. Measurement of matric potential. Water Resour. Res.
**1992**, 28, 2513–2521. [Google Scholar] [CrossRef][Green Version] - Selker, J.S.; Parlange, J.Y.; Steenhuis, T. Fingered flow in two dimensions, 2. Predicting finger moisture profile. Water Resour. Res.
**1992**, 28, 2523–2528. [Google Scholar] [CrossRef][Green Version] - Lu, T.X.; Biggar, J.W.; Nielsen, D.R. Water movement in glass bead porous media, 2. Experiments of infiltration and finger flow. Water Resour. Res.
**1994**, 30, 3283–3290. [Google Scholar] [CrossRef] - Bauters, T.W.J.; DiCarlo, D.A.; Steenhuis, T.S.; Parlange, J.Y. Preferential flow in water-repellent sands. Soil Sci. Soc. Am. J.
**1998**, 62, 1185–1190. [Google Scholar] [CrossRef] - Bauters, T.W.J.; DiCarlo, D.A.; Steenhuis, T.S.; Parlange, J.Y. Soil water content dependent wetting front characteristics in sands. J. Hydrol.
**2000**, 231, 244–254. [Google Scholar] [CrossRef] - Yao, T.; Hendrickx, J.M.H. Stability analysis of the unsaturated water flow equation 2. Experimental verification. Water Resour. Res.
**2001**, 37, 1875–1881. [Google Scholar] [CrossRef] - Flekkøy, E.G.; Schmittbuhl, J.; Løvholt, F.; Oxaal, U.; Måløy, K.J.; Aagaard, P. Flow paths in wetting unsaturated flow: Experiments and simulations. Phys. Rev. E
**2002**, 65, 036312. [Google Scholar] [CrossRef] [PubMed][Green Version] - Wang, Z.; Jury, W.A.; Tuli, A.; Kim, D.J. Unstable flow during redistribution: Controlling factors and practical implications. Vadose Zone J.
**2004**, 3, 549–559. [Google Scholar] [CrossRef] - Wei, Y.; Cejas, C.M.; Barrois, R.; Dreyfus, R.; Durian, D.J. Morphology of rain water channeling in systematically varied model sandy soils. Phys. Rev. Appl.
**2014**, 2, 044004. [Google Scholar] [CrossRef][Green Version] - Cueto-Felgueroso, L.; Juanes, R. Nonlocal interface dynamics and pattern formation in gravity-driven unsaturated flow through porous media. Phys. Rev. Lett.
**2008**, 101, 244504. [Google Scholar] [CrossRef] [PubMed][Green Version] - Cueto-Felgueroso, L.; Juanes, R. A phase-field model of unsaturated flow. Water Resour. Res.
**2009**, 45, W10409. [Google Scholar] [CrossRef] - Gomez, H.; Cueto-Felgueroso, L.; Juanes, R. Three-dimensional simulation of unstable gravity-driven infiltration of water into a porous medium. J. Comput. Phys.
**2013**, 238, 217–239. [Google Scholar] [CrossRef][Green Version] - Rätz, A.; Schweizer, B. Hysteresis models and gravity fingering in porous media. Z. Angew. Math. Mech.
**2014**, 94, 645–654. [Google Scholar] [CrossRef][Green Version] - Brindt, N.; Wallach, R. The moving-boundary approach for modeling gravity-driven stable and unstable flow in soils. Water Resour. Res.
**2017**, 53, 344–360. [Google Scholar] [CrossRef] - Beljadid, A.; Cueto-Felgueroso, L.; Juanes, R. A continuum model of unstable infiltration in porous media endowed with an entropy function. Adv. Water Resour.
**2019**. under review. [Google Scholar] - Ritsema, C.J.; Dekker, L.W. Soil moisture and dry bulk density patterns in bare dune sands. J. Hydrol.
**1994**, 154, 107–131. [Google Scholar] [CrossRef] - Hendrickx, J.M.H.; Flury, M. Uniform and preferential flow mechanisms in the vadose zone. In Conceptual Models of Flow and Transport in the Fractured Vadose Zone; The National Academies Press: Washington, DC, USA, 2001; pp. 149–187. [Google Scholar]
- Cremer, C.J.M.; Schuetz, C.; Neuweiler, I.; Lehmann, P.; Lehmann, E.H. Unstable Infiltration Experiments in Dry Porous Media. Vadose Zone J.
**2019**, 16, 13. [Google Scholar] [CrossRef] - Sililo, O.T.N.; Tellam, J.H. Fingering in unsaturated zone flow: A qualitative review with laboratory experiments in heterogeneous systems. Ground Water
**2000**, 38, 864–871. [Google Scholar] [CrossRef] - Glass, R.J.; Steenhuis, T.S.; Parlange, 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] - Ommen, H.C.V.; Dijksma, R.; Hendrickx, J.M.H.; Dekker, L.W.; Hulshof, J.; Heuve, M.V.D. Experimental assessment of preferential flow paths in a field soil. J. Hydrol.
**1989**, 105, 253–262. [Google Scholar] [CrossRef] - Liu, Y.; Steenhuis, T.S.; Parlange, J.Y. Formation and persistence of fingered flow fields in coarse grained soils under different moisture contents. J. Hydrol.
**1994**, 159, 187–195. [Google Scholar] [CrossRef] - Ritsema, C.J.; Dekker, L.W. How water moves in a water repellent sandy soil 2. Dynamics of fingered flow. Water Resour. Res.
**1994**, 9, 2519–2531. [Google Scholar] [CrossRef] - Ritsema, C.J.; Steenhuis, T.S.; Parlange, J.Y.; Dekker, L.W. Predicted and observed finger diameters in field soils. Geoderma
**1996**, 70, 185–196. [Google Scholar] [CrossRef] - Ritsema, C.J.; Dekker, L.W. Wetting patterns and moisture variability in water repellent Dutch soils. J. Hydrol.
**2000**, 231, 148–164. [Google Scholar] - Ritsema, C.J.; Dekker, L.W. Preferential flow in water repellent sandy soils: Principles and modeling implications. J. Hydrol.
**2000**, 231, 308–319. [Google Scholar] [CrossRef][Green Version] - Wang, Z.; Wu, Q.J.; Wu, L.; Ritsema, C.J.; Dekker, L.W.; Feyen, J. Effects of soil water repellency on infiltration rate and flow instability. J. Hydrol.
**2000**, 231, 265–276. [Google Scholar] [CrossRef] - Wallach, R.; Jortzick, C. Unstable finger-like flow in water-repellent soils during wetting and redistribution—The case of a point water source. J. Hydrol.
**2008**, 351, 26–41. [Google Scholar] [CrossRef] - Wallach, R.; Margolis, M.; Graber, E.R. The role of contact angle on unstable flow formation during infiltration and drainage in wettable porous media. Water Resour. Res.
**2013**, 49, 6508–6521. [Google Scholar] [CrossRef] - Glass, R.J.; Oosting, G.H.; Steenhuis, T.S. Preferential solute transport in layered homogeneous sands as a consequence of wetting front instability. J. Hydrol.
**1989**, 110, 87–105. [Google Scholar] [CrossRef] - Wildenschild, D.; Jensen, K.H. Laboratory investigations of effective flow behavior in unsaturated heterogeneous sands. Water Resour. Res.
**1999**, 35, 17–27. [Google Scholar] [CrossRef] - Wildenschild, D.; Jensen, K.H. Numerical modeling of observed effective flow behavior in unsaturated heterogeneous sands. Water Resour. Res.
**1999**, 35, 29–42. [Google Scholar] [CrossRef] - Jiménez-Martínez, J.; Skaggs, T.; van Genuchten, M.; Candela, L. A root zone modelling approach to estimating groundwater recharge from irrigated areas. J. Hydrol.
**2009**, 367, 138–149. [Google Scholar] [CrossRef] - Jiménez-Martínez, J.; Candela, L.; Molinero, J.; Tamoh, K. Groundwater recharge in irrigated semi-arid areas: Quantitative hydrological modelling and sensitivity analysis. Hydrogeol. J.
**2010**, 18, 1811–1824. [Google Scholar] [CrossRef] - Crosbie, R.S.; Scanlon, B.R.; Mpelasoka, F.S.; Reedy, R.C.; Gates, J.B.; Zhang, L. Potential climate change effects on groundwater recharge in the High Plains Aquifer, USA. Water Resour. Res.
**2013**, 49, 3936–3951. [Google Scholar] [CrossRef][Green Version] - Berg, A.; Sheffield, J. Climate Change and Drought: The Soil Moisture Perspective. Curr. Clim. Chang. Rep.
**2018**, 4, 180–191. [Google Scholar] [CrossRef] - Samaniego, L.; Thober, S.; Kumar, R.; Wanders, N.; Rakovec, O.; Pan, M.; Zink, M.; Sheffield, J.; Wood, E.F.; Marx, A. Anthropogenic warming exacerbates European soil moisture droughts. Nat. Clim. Chang.
**2018**, 8, 421–426. [Google Scholar] [CrossRef] - Muskat, M.; Meres, M.W. The flow of heterogeneous fluids through porous media. Physics
**1936**, 7, 346–363. [Google Scholar] [CrossRef] - Muskat, M. Physical Principles of Oil Production; McGraw-Hill: New York, NY, USA, 1949. [Google Scholar]
- Bear, J. Dynamics of Fluids in Porous Media; Elsevier: New York, NY, USA, 1972. [Google Scholar]
- Sciarra, G. Phase field modeling of partially saturated deformable porous media. J. Mech. Phys. Solids
**2016**, 94, 230–256. [Google Scholar] [CrossRef][Green Version] - Leverett, M.C. Capillary behavior of porous solids. Trans. AIME
**1941**, 142, 152–169. [Google Scholar] [CrossRef] - COMSOL. COMSOL Multiphysics Structural Mechanics Module User’s Guide v5.2a; COMSOL: Stockholm, Sweden, 2016. [Google Scholar]
- Brooks, R.H.; Corey, A.T. Properties of porous media affecting fluid flow. J. Irrig. Drain. Div. Proc. Am. Soc. Civ. Eng.
**1966**, IR2, 61–88. [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] - Gelhar, L.W.; Axness, C.L. Three-Dimensional Stochastic Analysis of Macrodispersion in Aquifers. Water Resour. Res.
**1983**, 19, 161–180. [Google Scholar] [CrossRef]

**Figure 1.**Problem setup. We simulate constant-rate infiltration into heterogeneous soil. (

**a**) Sample realization of a random, spatially-correlated permeability field. (

**b**) Snapshot of the water saturation field from a simulation with the permeability field shown in Panel (

**a**).

**Figure 2.**Statistics of the random permeability fields. (

**a**–

**d**) Maps of isotropic intrinsic permeability fields, which are random and characterized by the minimum and maximum permeability, ${10}^{-13}$ m${}^{2}$ and 2 × ${10}^{-10}$ m${}^{2}$, receptively, and correlation lengths, ${\lambda}_{x}$ and ${\lambda}_{y}$. Sample permeability fields with (

**a**) ${\lambda}_{x}$=${\lambda}_{y}$=$0.8$ cm, (

**b**) ${\lambda}_{x}$=${\lambda}_{y}$=$2.4$ cm, (

**c**) ${\lambda}_{x}$=${\lambda}_{y}$=$4.8$ cm, and (

**d**) ${\lambda}_{x}$=${\lambda}_{y}$=$7.2$ cm. (

**e**) Sample probability density functions of permeability for the fields shown in Panels (

**a**–

**d**).

**Figure 3.**Statistics of the random permeability fields. (

**a**–

**d**) Anisotropic intrinsic permeability fields, which are random and characterized by the minimum and maximum permeability, ${10}^{-13}$ m${}^{2}$ and 2 × ${10}^{-10}$ m${}^{2}$, receptively, and correlation lengths, ${\lambda}_{x}$ and ${\lambda}_{y}$. Sample permeability fields with (

**a**) ${\lambda}_{x}$=$0.8$ cm and ${\lambda}_{y}$=$3.2$ cm, (

**b**) ${\lambda}_{x}$=$0.8$ cm and ${\lambda}_{y}$=$6.4$ cm, (

**c**) ${\lambda}_{x}$=$3.2$ cm and ${\lambda}_{y}$=$0.8$ cm, and (

**d**) ${\lambda}_{x}$=$6.4$ cm and ${\lambda}_{y}$=$0.8$ cm. (

**e**) Sample probability density functions of permeability for the fields of Panels (

**a**–

**d**).

**Figure 4.**Impact of correlation length on unstable infiltration: isotropic permeability fields. Snapshots of the saturation field, at times $t=2000,5000,8000$, and 15,000 s, for a flux ratio, ${R}_{s}$=$0.01$. The correlation lengths in permeability are ${\lambda}_{x}$=${\lambda}_{y}$=$0.8$ cm (

**a**–

**d**) and ${\lambda}_{x}$=${\lambda}_{y}$=$2.4$ cm (

**e**–

**h**). Lines in grey scale are isocontours of permeability, corresponding to the field in Panels (

**a**,

**b**) in Figure 2.

**Figure 5.**Impact of correlation length on unstable infiltration: isotropic permeability fields. Snapshots of the saturation field, at times $t=2000,5000,8000$ and 15,000 s, for a flux ratio, ${R}_{s}$=$0.01$. The correlation lengths in permeability are ${\lambda}_{x}$=${\lambda}_{y}$=$4.8$ cm (

**a**–

**d**) and ${\lambda}_{x}$=${\lambda}_{y}$=$7.2$ cm (

**e**–

**h**). Lines in grey scale are isocontours of permeability, corresponding to the fields in panels (

**c**,

**d**) in Figure 2.

**Figure 6.**(

**a**) Evolution of the domain-averaged saturation for various correlation lengths. (

**b**) Evolution of the volumetric flux at the outlet (bottom boundary).

**Figure 7.**(

**a**–

**d**) Maps of water saturation at time t= 15,000 s, for four different pairs of correlation lengths: (

**a**) ${\lambda}_{x}$=$0.8$ cm and ${\lambda}_{y}$=$3.2$ cm, (

**b**) ${\lambda}_{x}$=$0.8$ cm and ${\lambda}_{y}$=$6.4$ cm, (

**c**) ${\lambda}_{x}$=$3.2$ cm and ${\lambda}_{y}$=$0.8$ cm, and (

**d**) ${\lambda}_{x}$=$6.4$ cm and ${\lambda}_{y}$=$0.8$ cm. (

**e**–

**h**) Permeability fields for the simulations in Panels (

**a**–

**d**).

**Figure 8.**Patterns of unstable infiltration: anisotropic permeability fields. (

**a**–

**d**) Maps of water saturation at times t=$2000,5000,8000$, and 15,000 s, for two different pairs of correlation lengths: (

**a**–

**d**) ${\lambda}_{x}$=$6.4$ cm and ${\lambda}_{y}$=$0.8$ cm and (

**e**–

**h**) ${\lambda}_{x}$=$0.8$ cm and ${\lambda}_{y}$=$6.4$ cm. Contours in grey scale represent isocontours of permeability, corresponding to the permeability field in Panels (

**f**,

**h**) in Figure 7.

**Figure 9.**(

**a**) Evolution of domain-averaged water saturation for the various anisotropic permeability fields. (

**b**) Evolution of the volumetric flux at the outlet (bottom boundary).

**Figure 10.**Patterns of infiltration into nearly-layered media. (

**a**–

**d**) Maps of water saturation at times $t=2000,5000,8000$ and 15,000 s, for correlation lengths ${\lambda}_{x}$=$25.6$ cm and ${\lambda}_{y}$=$3.2$ cm and minimum and maximum permeability ${10}^{-12}$ m${}^{2}$ and 2 × ${10}^{-10}$ m${}^{2}$, respectively. (

**e**–

**h**) Maps of water saturation at times $t=2000,5000,8000$ and 15,000 s, for correlation lengths ${\lambda}_{x}$=$25.6$ cm and ${\lambda}_{y}$= 4 cm and minimum and maximum permeability ${10}^{-11}$ m${}^{2}$ and 2 × ${10}^{-10}$ m${}^{2}$, respectively.

**Figure 11.**Impact of infiltration rate on infiltration into heterogeneous dry soil. We show maps of water saturation at t= 15,000 s. Along the rows, saturation fields for flux ratios ${R}_{s}=0.01,0.05,0.1$, and $0.2$ and permeability fields with correlation lengths (

**a**) ${\lambda}_{x}$=${\lambda}_{y}$=$0.8$ cm, (

**b**) ${\lambda}_{x}$=${\lambda}_{y}$=$4.8$ cm, and (

**c**) ${\lambda}_{x}$=$0.8$ cm and ${\lambda}_{y}$=$6.4$ cm.

**Figure 12.**(

**a**) Evolution of the domain-averaged water saturation for the several flux ratios and correlation lengths ${\lambda}_{x}$=${\lambda}_{y}$=$0.8$ cm. (

**b**) Evolution of the volumetric flux at the outlet (bottom boundary).

**Figure 13.**(

**a**) Evolution of the domain-averaged water saturation for the several flux ratios and correlation lengths ${\lambda}_{x}$=${\lambda}_{y}$=$4.8$ cm. (

**b**) Evolution of the volumetric flux at the outlet (bottom boundary).

**Figure 14.**(

**a**) Evolution of the domain-averaged water saturation for the several flux ratios and correlation lengths ${\lambda}_{x}$=$0.8$ cm and ${\lambda}_{y}$=$6.4$ cm. (

**b**) Evolution of the volumetric flux at the outlet (bottom boundary).

**Figure 15.**Impact of unstable infiltration and heterogeneity on the effective conductivity of dry soils. We plot the volumetric flux as a function of mean water saturation, for three different pairs of correlation lengths. These results correspond to the simulations with different flux ratios of Figure 11, Figure 12, Figure 13 and Figure 14. The dashed lines show popular choices of relative conductivity curves assuming compact infiltration fronts (power-law functions).

Name | Value | Unit | Description |
---|---|---|---|

$\varphi $ | 0.3 | – | Soil porosity |

$\rho $ | 1000 | kg/m${}^{3}$ | Water density |

$\mu $ | 0.001 | Pa·s | Water viscosity |

$\sigma $ | 0.016 | N/m | Air-water surface tension |

g | 9.81 | m/s${}^{2}$ | Acceleration of gravity |

${k}_{\mathrm{min}}$ | ${10}^{-13},{10}^{-12},{10}^{-11}$ | m${}^{2}$ | Minimum permeability |

${k}_{\mathrm{max}}$ | 2 × ${10}^{-10}$ | m${}^{2}$ | Maximum permeability |

${\lambda}_{x}$ | [0.8,25.6] | cm | Horizontal correlation length |

${\lambda}_{y}$ | [0.8,6.4] | cm | Vertical correlation length |

${h}_{\mathrm{cap}}$ | $\frac{0.016}{\rho g\sqrt{k(x,y)/\varphi}}$ | m | Capillary height |

${K}_{\mathrm{s}}$ | $\frac{\rho gk(x,y)}{\mu}$ | m/s | Saturated hydraulic conductivity |

${S}_{e}$ | $\frac{{S}_{w}-{S}_{rw}}{1-{S}_{rw}}$ | – | Effective water saturation |

${S}_{rw}$ | 0.1 | – | Irreducible water saturation |

${R}_{s}$ | ${q}_{w}/{K}_{\mathrm{s}}(x,0)$ | – | Flux ratio (top boundary) |

${k}_{r}$ | ${S}_{e}^{a}$ | – | Relative permeability function |

J | ${S}_{w}^{-1/\alpha}\left[1-exp\left(\beta \left({S}_{w}-{v}_{e}\right)\right)\right]\left(1+\beta \frac{\alpha}{\alpha -1}{S}_{w}\right)$ | – | Leverett J-function |

$\kappa $ | ${h}_{\mathrm{cap}}{\delta}^{2}\frac{\alpha}{\alpha -1}{S}_{w}^{\frac{\alpha -1}{\alpha}}\left[1-exp\left(\beta \left({S}_{w}-{v}_{e}\right)\right)\right]$ | m${}^{3}$ | Gradient energy multiplier |

$\delta $ | ${h}_{\mathrm{cap}}$ | m | Characteristic gradient energy length |

a | 7 | – | Exponent of relative permeability function |

$\alpha $ | 10 | – | Parameter of the Leverett J-function |

$\beta $ | 40 | – | Parameter of the Leverett J-function |

${v}_{e}$ | 1 | – | Parameter of the Leverett J-function |

© 2020 by the authors. 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Cueto-Felgueroso, L.; Suarez-Navarro, M.J.; Fu, X.; Juanes, R. Numerical Simulation of Unstable Preferential Flow during Water Infiltration into Heterogeneous Dry Soil. *Water* **2020**, *12*, 909.
https://doi.org/10.3390/w12030909

**AMA Style**

Cueto-Felgueroso L, Suarez-Navarro MJ, Fu X, Juanes R. Numerical Simulation of Unstable Preferential Flow during Water Infiltration into Heterogeneous Dry Soil. *Water*. 2020; 12(3):909.
https://doi.org/10.3390/w12030909

**Chicago/Turabian Style**

Cueto-Felgueroso, Luis, María José Suarez-Navarro, Xiaojing Fu, and Ruben Juanes. 2020. "Numerical Simulation of Unstable Preferential Flow during Water Infiltration into Heterogeneous Dry Soil" *Water* 12, no. 3: 909.
https://doi.org/10.3390/w12030909