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

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## Abstract

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## 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

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**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 | ${S}_{w}^{-1/\alpha}\left(\right)open="["\; close="]">1-exp\left(\right)open="("\; close=")">\beta \left(\right)open="("\; close=")">{S}_{w}-{v}_{e}$ | – | Leverett J-function |

$\kappa $ | ${h}_{\mathrm{cap}}{\delta}^{2}\frac{\alpha}{\alpha -1}{S}_{w}^{\frac{\alpha -1}{\alpha}}\left(\right)open="["\; close="]">1-exp\left(\right)open="("\; close=")">\beta \left(\right)open="("\; close=")">{S}_{w}-{v}_{e}$ | 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 |

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**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