# Interplay between Fingering Instabilities and Initial Soil Moisture in Solute Transport through the Vadose Zone

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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. Transport of a Passive Solute

#### 2.3. Finite Element Implementation

## 3. Results: One-Dimensional Infiltration Fronts

#### 3.1. Problem Set-Up

#### 3.2. Impact of the Initial Saturation on Solute Transport

## 4. Results: Two-Dimensional Infiltration Patterns

#### 4.1. Problem Set-Up

#### 4.2. Statistics of the Initial Water Saturation Fields

#### 4.3. Sample Two-Dimensional Simulations

#### 4.4. Analysis: Impact of the Spatial Structure of the iNitial Saturation Field on Water and Solute Transport

#### 4.5. Analysis: Impact of the Solute Diffusivity/Dispersivity on the Effective Solute Dilution

#### 4.6. Analysis: Solute Fluxes and Overall Statistics of the Concentration and Saturation Fields

#### 4.7. Solute Transport in Stable and Unstable Infiltration Fronts

## 5. Conclusions

- To the best of our knowledge, these are the first numerical simulations of solute transport during unstable infiltration into soil. We show that the patterns of water flow and solute transport arising from unstable infiltration are very different from those obtained using Richards’ equation, which predicts compact, stable front of water saturation and solute concentration.
- The initial water content and its spatial distribution play a key role in the patterns of water infiltration and solute transport in unsaturated coarse soil. The structure of initial freshwater saturation controls solute transport through two mechanisms: (1) the interplay between the fingering instability and the increased conductivity at the wet patches; (2) the lateral dilution of solute into the ambient water. The extent to which the former mechanism controls transport is related to the anisotropy of the initial water saturation field and to the maximum saturations at the water pockets. Isotropic distributions of low initial water saturation lead to the classical straight, downward-moving fingers, with finger sizes and spacing controlled by the internal hydrodynamic length scales. Isotropic fields with large saturation lead to complex infiltration patterns, with finger meandering and efficient solute dilution into the ambient water. Anisotropic fields with preferentially vertical correlation, combined with the fingering instability, promote focused transport and reduce mixing to the extent of lateral dilution due to solute diffusion/dispersion. Water pockets of lenticular shape thicken the finger size and induce lateral mixing of the solute. Lateral mixing becomes very efficient as the horizontal correlation length increases.
- The effective diffusivity/dispersivity of solute is particularly important for solute dilution in dry soils, or in those where the initial water saturation field induces flow focusing (vertical correlation). When the initial saturation field leads to complex infiltration patterns (the presence of water lenses or isotropic fields with large saturation), the flow field is complex and enhances mixing, leading to more efficient dilution.
- The integrated solute fluxes at the outlet (bottom of the domain) are consistent with the one-dimensional prediction of a delayed arrival of the solute when the soil is initially wet.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic of 1D simulations of coupled unsaturated flow and transport in a soil column. (

**a**) We simulate constant-rate infiltration into soil with initial freshwater saturation. The infiltrating water has a constant solute concentration. (

**b**) The capillary pressure–saturation curve (Leverett’s J function) and relative permeability curves used in this study.

**Figure 2.**Solute and water front speeds in constant-rate infiltration into soil with initial fresh water saturation (Equations (29) and (30)). The water front speed is given by the slope of the secant line joining the relative mobilities of the initial and inflow saturations (blue and red dots, respectively), while the solute front speed is given by the slope of the secant line from zero to the inflow relative mobility.

**Figure 3.**One-dimensional profiles of solute transport during infiltration. For the same constant flux ratio ${R}_{s}$ = $0.5$, we show profiles for three initial saturations, ${S}_{0}$, at time t = 30 s. (

**a**) For ${S}_{0}$ = $0.1$, (

**b**) ${S}_{0}$ = $0.25$, and (

**c**) ${S}_{0}$ = $0.5$. Black (resp. blue) vertical dotted lines correspond to the water (solute) front positions predicted by Equations (29) and (30). Red (resp. blue) dots indicate the left-state (resp. right-state) water saturations, ${S}^{-}$ (resp. ${S}^{+}$) used to illustrate the different front velocities in Figure 2.

**Figure 4.**One-dimensional profiles of solute transport during infiltration. For the same constant flux ratio ${R}_{s}$ = $0.02$, we show profiles for three initial saturations, ${S}_{0}$, at time t = 520 s. (

**a**) For ${S}_{0}$ = $0.01$, (

**b**) ${S}_{0}$ = $0.1$, and (

**c**) ${S}_{0}$ = $0.2$. Black (resp. blue) vertical dotted lines correspond to the water (solute) front positions predicted by Equations (29) and (30).

**Figure 5.**Two-dimensional problem set-up. (

**a**) We simulate coupled unsaturated flow and solute transport during infiltration into soil with an initial distribution of fresh water. We simulate a pulse infiltration of water with a given solute concentration, as constant water flux at the top boundary for a short time period of 180 s. (

**b**,

**c**) We study the impact of initial soil moisture distribution on the patterns of water flow and transport, and on the solute flux at the bottom of the domain. We couple unsaturated flow with he advection–dispersion equation, to compute the evolution of (

**b**) water saturation (the fraction of pore volume occupied by water) and (

**c**) solute concentration.

**Figure 6.**Characteristics of isotropic initial saturation fields. (

**a**–

**d**) Maps of initial saturation, which are random isotropic fields characterized by the maximum initial saturation, ${S}_{0,\mathrm{max}}$, and correlation lengths, ${\lambda}_{x}$ and ${\lambda}_{z}$. The minimum initial saturation is the same in all cases, ${S}_{0,\mathrm{min}}=0.01$. Here we show sample fields with ${\lambda}_{x}$ = ${\lambda}_{z}$ = $1.6$ cm, and (

**a**) ${S}_{0,\mathrm{max}}$ = $0.25$, (

**b**) ${S}_{0,\mathrm{max}}$ = $0.35$, (

**c**) ${S}_{0,\mathrm{max}}$ = $0.45$, (

**d**) ${S}_{0,\mathrm{max}}$ = $0.55$. (

**e**) Probability density functions of initial saturation for the saturation fields shown in panels (

**a**–

**d**).

**Figure 7.**Characteristics of anisotropic initial saturation fields. (

**a**–

**c**) Maps of initial saturation, which are random fields characterized by the maximum initial saturation, ${S}_{0,\mathrm{max}}$, and correlation lengths, ${\lambda}_{x}$ and ${\lambda}_{z}$. The minimum initial saturation is the same in all cases, ${S}_{0,\mathrm{min}}$ = $0.01$. Here we show sample fields with ${S}_{0,\mathrm{max}}$ = $0.25$, and correlation lengths (

**a**) ${\lambda}_{x}$ = ${\lambda}_{z}$ = $1.6$ cm, (

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

**c**) ${\lambda}_{x}$ = $6.4$ cm, ${\lambda}_{z}$ = $0.8$ cm. (

**d**) Probability density functions of initial saturation for the saturation fields shown in panels (

**a**–

**c**).

**Figure 8.**Sample 2D simulations of unstable infiltration and solute transport in soil with random initial saturation. We simulate the transport of a solute deposited at the soil surface by a short water pulse. We consider constant solute concentration at the top ($c(x,0,t)$ = 1 kg/m${}^{3}$). The infiltration pulse is modeled as a constant water flux along the top boundary during a period of 180 s. The initial saturation field is isotropic, with maximum saturation ${S}_{0,\mathrm{max}}$ = $0.25$, and correlation lengths ${\lambda}_{x}$ = ${\lambda}_{z}$ = $1.6$ cm. (

**a**–

**d**) Maps of water saturation, ${S}_{w}$, at times 100 s, 300 s, 1500 s and 15,000 s. (

**e**–

**h**) Maps of net solute concentration, $c{S}_{w}$, at the same time levels.

**Figure 9.**Sample 2D simulations of unstable infiltration and solute transport in soil with random initial saturation. We simulate the transport of a solute deposited at the soil surface by a short water pulse. We consider constant solute concentration at the top ($c(x,0,t)$ = 1 kg/m${}^{3}$). The infiltration pulse is modeled as a constant water flux along the top boundary during a period of 180 s. The initial saturation field is isotropic, with maximum saturation ${S}_{0,\mathrm{max}}$ = $0.55$, and correlation lengths ${\lambda}_{x}={\lambda}_{z}=1.6$ cm. (

**a**–

**d**) Maps of water saturation, ${S}_{w}$, at times 100 s, 300 s, 1500 s and 15,000 s. (

**e**–

**h**) Maps of net solute concentration, $c{S}_{w}$, at the same time levels.

**Figure 10.**Sample 2D simulations of unstable infiltration and solute transport in soil with random initial saturation. We simulate the transport of a solute deposited at the soil surface by a short water pulse. We consider constant solute concentration at the top ($c(x,0,t)$ = 1 kg/m${}^{3}$). The infiltration pulse is modeled as a constant water flux along the top boundary during a period of 180 s. The initial saturation field is anisotropic, with maximum saturation ${S}_{0,\mathrm{max}}$ = $0.25$, and correlation lengths ${\lambda}_{x}$ = $1.6$ cm, ${\lambda}_{z}=12.8$ cm. (

**a**–

**d**) Maps of water saturation, ${S}_{w}$, at times 100 s, 300 s, 1500 s and 15,000 s. (

**e**–

**h**) Maps of net solute concentration, $c{S}_{w}$, at the same time levels.

**Figure 11.**Sample 2D simulations of unstable infiltration and solute transport in soil with random initial saturation. We simulate the transport of a solute deposited at the soil surface by a short water pulse. We consider constant solute concentration at the top ($c(x,0,t)$ = 1 kg/m${}^{3}$). The infiltration pulse is modeled as a constant water flux along the top boundary during a period of 180 s. The initial saturation field is anisotropic, with maximum saturation ${S}_{0,\mathrm{max}}$ = $0.45$, and correlation lengths ${\lambda}_{x}$ = $1.6$ cm, ${\lambda}_{z}$ = $12.8$ cm. (

**a**–

**d**) Maps of water saturation, ${S}_{w}$, at times 100 s, 300 s, 1500 s and 15,000 s. (

**e**–

**h**) Maps of net solute concentration, $c{S}_{w}$, at the same time levels.

**Figure 12.**Sample 2D simulations of unstable infiltration and solute transport in soil with random initial saturation. We simulate the transport of a solute deposited at the soil surface by a short water pulse. We consider constant solute concentration at the top ($c(x,0,t)$ = 1 kg/m${}^{3}$). The infiltration pulse is modeled as a constant water flux along the top boundary during a period of 180 s. The initial saturation field is anisotropic, with maximum saturation ${S}_{0,\mathrm{max}}$ = $0.25$, and correlation lengths ${\lambda}_{x}$ = $6.4$ cm, ${\lambda}_{z}$ = $0.8$ cm. (

**a**–

**d**) Maps of water saturation, ${S}_{w}$, at times 100 s, 300 s, 1500 s and 15,000 s. (

**e**–

**h**) Maps of net solute concentration, $c{S}_{w}$, at the same time levels.

**Figure 13.**Sample 2D simulations of unstable infiltration and solute transport in soil with random initial saturation. We simulate the transport of a solute deposited at the soil surface by a short water pulse. We consider constant solute concentration at the top ($c(x,0,t)$ = 1 kg/m${}^{3}$). The infiltration pulse is modeled as a constant water flux along the top boundary during a period of 180 s. The initial saturation field is anisotropic, with maximum saturation ${S}_{0,\mathrm{max}}$ = $0.55$, and correlation lengths ${\lambda}_{x}$ = $6.4$ cm, ${\lambda}_{z}$ = $0.8$ cm. (

**a**–

**d**) Maps of water saturation, ${S}_{w}$, at times 100 s, 300 s, 1500 s and 15,000 s. (

**e**–

**h**) Maps of net solute concentration, $c{S}_{w}$, at the same time levels.

**Figure 14.**Influence of the spatial structure of the initial saturation field on solute transport in unstable infiltration. We simulate the transport of a solute deposited at the soil surface by a short water pulse. We consider constant solute concentration at the top ($c(x,0,t)$ = 1 kg/m${}^{3}$). The infiltration pulse is modeled as a constant water flux along the top boundary during a period of 180 s. The initial saturation field is anisotropic, with maximum saturation ${S}_{0,\mathrm{max}}$ = $0.45$. (

**a**–

**c**) Maps of water saturation, ${S}_{w}$, at time 1500 s for an initial saturation field with correlation lengths (

**a**) ${\lambda}_{x}$ = $1.6$ cm, ${\lambda}_{z}$ = $1.6$ cm, (

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

**c**) ${\lambda}_{x}$ = $0.8$ cm, ${\lambda}_{z}$ = $12.8$ cm. (

**d**–

**f**) Maps of net solute concentration, $c{S}_{w}$, at the same time.

**Figure 15.**Influence of the spatial structure of the initial saturation field on solute transport in unstable infiltration. We simulate the transport of a solute deposited at the soil surface by a short water pulse. We consider constant solute concentration at the top ($c(x,0,t)$ = 1 kg/m${}^{3}$). The infiltration pulse is modeled as a constant water flux along the top boundary during a period of 180 s. The initial saturation field is anisotropic, with correlation length ${\lambda}_{x}$ = $6.4$ cm, ${\lambda}_{z}$ = $0.8$, and several values of the maximum saturation. (

**a**–

**d**) Maps of water saturation, ${S}_{w}$, at time 1500 s for an initial saturation field with maximum saturations (

**a**) ${S}_{0,\mathrm{max}}$ = $0.15$, (

**b**) ${S}_{0,\mathrm{max}}$ = $0.35$, (

**c**) ${S}_{0,\mathrm{max}}$ = $0.45$, (

**d**) ${S}_{0,\mathrm{max}}$ = $0.55$. (

**e**–

**h**) Maps of net solute concentration, $c{S}_{w}$, at the same time.

**Figure 16.**Influence of solute diffusivity on the mixing and dilution of a solute transported by unstable water infiltration into soil. Here we show the maps of water saturation, ${S}_{w}$, and solute concentration in the pore water, c, at time 1500 s, for five different isotropic initial saturation fields, with maximum saturation ${S}_{0,\mathrm{max}}$ = $0.15$, $0.25$, $0.35$, $0.45$, $0.55$ (distributed along columns), and three different solute diffusivities/dispersivities, $\varphi D$ = $2\xb7{10}^{-8}$ m${}^{2}$/s, $\varphi D$ = ${10}^{-7}$ m${}^{2}$/s, and $\varphi D$ = ${10}^{-6}$ m${}^{2}$/s (along rows). Lower diffusivities lead to focusing of the solute transport along the preferential paths created by the infiltrating water fingers, with little dilution into the ambient freshwater.

**Figure 17.**Integrated solute flux at the bottom outlet section. (

**a**) Evolution of solute flux for isotropic initial saturation field, ${\lambda}_{x}$ = ${\lambda}_{z}$ = $1.6$ cm, diffusivity $\varphi D$ = 2·${10}^{-8}$ m${}^{2}$/s, and three different maximum saturations ${S}_{0,\mathrm{max}}$ = $0.15,0.25,0.45$. (

**b**) Evolution of solute flux for isotropic initial saturation field, ${\lambda}_{x}$ = ${\lambda}_{z}$ = $1.6$ cm, diffusivity $\varphi D$ = ${10}^{-7}$ m${}^{2}$/s, and three different maximum saturations ${S}_{0,\mathrm{max}}$ = $0.15,0.25,0.45$. (

**c**) Evolution of solute flux for isotropic initial saturation field, ${\lambda}_{x}$ = ${\lambda}_{z}$ = $1.6$ cm, diffusivity $\varphi D$ = ${10}^{-6}$ m${}^{2}$/s, and three different maximum saturations ${S}_{0,\mathrm{max}}$ = $0.15,0.25,0.45$.

**Figure 18.**Time evolution of the probability density fuction (pdf) of solute concentration, $c{S}_{w}$, for isotropic initial saturation fields, ${\lambda}_{x}$ = ${\lambda}_{z}$ = $1.6$ cm, for the same diffusivity, $\varphi D$ = ${10}^{-7}$ m${}^{2}$/s, and four maximum initial saturations, (

**a**) ${S}_{0,\mathrm{max}}$ = $0.15$, (

**b**) ${S}_{0,\mathrm{max}}$ = $0.25$, (

**c**) ${S}_{0,\mathrm{max}}$ = $0.35$, (

**d**) ${S}_{0,\mathrm{max}}$ = $0.45$.

**Figure 19.**Time evolution of the probability density fuction (pdf) of water saturation, ${S}_{w}$, for anisotropic initial saturation fields, ${\lambda}_{x}$ = $6.4$ cm, ${\lambda}_{z}$ = $0.8$ cm, for four maximum initial saturations, (

**a**) ${S}_{0,\mathrm{max}}$ = $0.15$, (

**b**) ${S}_{0,\mathrm{max}}$ = $0.25$, (

**c**) ${S}_{0,\mathrm{max}}$ = $0.35$, (

**d**) ${S}_{0,\mathrm{max}}$ = $0.45$.

**Figure 20.**Comparison between water and solute breakthrough curves for an infiltration pulse into soil with heterogeneous intial saturation. We plot the evolution of integrated fluxes (water, black solid line, and solute, blue solid line), and two snapshots of water saturation (with contours of solute concentration at the same time level), for isotropic random initial saturation fields (${\lambda}_{x}$ = ${\lambda}_{z}$ = $1.6$ cm), and (

**a**) small initial saturation, ${S}_{0,\mathrm{max}}$ = $0.25$; and (

**b**) large initial saturation, ${S}_{0,\mathrm{max}}$ = $0.55$.

**Figure 21.**Solute transport in stable and unstable infiltration fronts. We compare simulations using the Richards equation (panels (

**a**,

**c**,

**e**)), with the present simulations including fingering instability (panels (

**b**,

**d**,

**f**)). Here we show snapshots of the water saturation field and solute concentration contours, at times $t=100,300$, and 1500 s, for low intial saturation (${S}_{0,\mathrm{max}}$ = $0.15$), with equal spatial correlation lengths, ${\lambda}_{x}$ = ${\lambda}_{z}$ = $1.6$ cm.

**Figure 22.**Profiles of water saturation and solute concentration, averaged along the x-direction, $\langle {S}_{w}\rangle $ and $\langle c{S}_{w}\rangle $, corresponding to the snapshots shown in Figure 21: (

**a**) $t=100\phantom{\rule{3.33333pt}{0ex}}s$, for low intial saturation (${S}_{0,\mathrm{max}}=0.15$, with equal spatial correlation lengths, ${\lambda}_{x}$ = ${\lambda}_{z}$ = $1.6$ cm); (

**b**) $t=300\phantom{\rule{3.33333pt}{0ex}}s$, for low intial saturation (${S}_{0,\mathrm{max}}=0.15$, with equal spatial correlation, lengths, ${\lambda}_{x}$ = ${\lambda}_{z}$ = $1.6$ cm); (

**c**) $t=1500\phantom{\rule{3.33333pt}{0ex}}s$, for low intial saturation (${S}_{0,\mathrm{max}}=0.15$, with equal spatial correlation lengths, ${\lambda}_{x}$ = ${\lambda}_{z}$ = $1.6$ cm).

**Figure 23.**Solute transport in stable and unstable infiltration fronts. We compare simulations using the Richards equation (panels (

**a**,

**c**,

**e**)), with the present simulations including fingering instability (panels (

**b**,

**d**,

**f**)). Here we show snapshots of the water saturation field and solute concentration contours, at times $t=100,300$, and 1500 s, for low intial saturation (${S}_{0,\mathrm{max}}$ = $0.55$), with equal spatial correlation lengths, ${\lambda}_{x}$ = ${\lambda}_{z}$ = $1.6$ cm.

**Figure 24.**Profiles of water saturation and solute concentration, averaged along the x-direction, $\langle {S}_{w}\rangle $ and $\langle c{S}_{w}\rangle $, corresponding to the snapshots shown in Figure 23: (

**a**) $t=100\phantom{\rule{3.33333pt}{0ex}}s$, for low intial saturation (${S}_{0,\mathrm{max}}=0.55$, with equal spatial correlation lengths, ${\lambda}_{x}$ = ${\lambda}_{z}$ = $1.6$ cm); (

**b**) $t=300\phantom{\rule{3.33333pt}{0ex}}s$, for low intial saturation (${S}_{0,\mathrm{max}}=0.55$, with equal spatial correlation, lengths, ${\lambda}_{x}$ = ${\lambda}_{z}$ = $1.6$ cm); (

**c**) $t=1500\phantom{\rule{3.33333pt}{0ex}}s$, for low intial saturation (${S}_{0,\mathrm{max}}=0.55$, with equal spatial correlation lengths, ${\lambda}_{x}$ = ${\lambda}_{z}$ = $1.6$ cm).

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

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

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

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

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

${S}_{0,min}$ | 0.01 | – | Minimum initial water saturation |

${S}_{0,max}$ | $[0.15,0.55]$ | – | Maximum initial water saturation |

${\lambda}_{x}$ | [0.8,6.4] | cm | Horizontal correlation length |

${\lambda}_{z}$ | [0.8,12.8] | cm | Vertical correlation length |

${h}_{\mathrm{cap}}$ | $0.02$ | m | Capillary height |

${K}_{\mathrm{s}}$ | 40 | cm/min | 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 |

$\varphi D$ | 2·${10}^{-8},{10}^{-7},{10}^{-6}$ | m${}^{2}$/s | Effective solute diffusivity/dispersivity |

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## Share and Cite

**MDPI and ACS Style**

Cueto-Felgueroso, L.; Suarez-Navarro, M.J.; Fu, X.; Juanes, R.
Interplay between Fingering Instabilities and Initial Soil Moisture in Solute Transport through the Vadose Zone. *Water* **2020**, *12*, 917.
https://doi.org/10.3390/w12030917

**AMA Style**

Cueto-Felgueroso L, Suarez-Navarro MJ, Fu X, Juanes R.
Interplay between Fingering Instabilities and Initial Soil Moisture in Solute Transport through the Vadose Zone. *Water*. 2020; 12(3):917.
https://doi.org/10.3390/w12030917

**Chicago/Turabian Style**

Cueto-Felgueroso, Luis, María José Suarez-Navarro, Xiaojing Fu, and Ruben Juanes.
2020. "Interplay between Fingering Instabilities and Initial Soil Moisture in Solute Transport through the Vadose Zone" *Water* 12, no. 3: 917.
https://doi.org/10.3390/w12030917