Recent studies have underscored the significance of the capillary fringe in hydrological and biochemical processes. Moreover, its role in shallow waters is expected to be considerable. Traditionally, the study of groundwater flow has centered on unsaturated-saturated zones, often overlooking the impact of the capillary fringe. In this study, we introduce a steady-state two-dimensional model that integrates the capillary fringe into a 2-D numerical solution. Our novel approach employs the potential form of the Richards equation, facilitating the determination of boundaries, pressures, and velocities across different ground surface zones. We utilized a two-dimensional _{0}, we have shown the correlation between water table elevation and the upper limit of the capillary fringe.

The capillary fringe represents a highly dynamic region situated at the interface between the unsaturated zone and the water table, where significant hydrological and geochemical gradients are anticipated [

The capillary fringe is increasingly recognized for its substantial role in the generation of stream flow, influencing both the volume and swiftness of water mobilization [

The influence of the capillary fringe on physical and biochemical activities is intricately tied to soil types and pore sizes, such as sand, gravel, or coarse textures. While some studies, like Ronen et al. [

Another significant aspect of analyzing the capillary fringe pertains to its relationship with CO_{2} storage, one of the greenhouse gases contributing to climate change. Notably, the capillary fringe’s height can vary from a few centimeters in coarse materials to several meters in very fine materials, underscoring its importance in this context. However, experiments aiming to understand soil contamination involving stronger capillary forces approximations are less suitable [_{6}H_{6}O_{3}S) and the investigation of oil contamination [

Investigating water movement within a porous environment, where saturation levels vary, poses a significant hurdle because of the intricate equations dictating exchanges between saturated and unsaturated zones. This intricacy stems from the need to address the interplay between vertical flow in the unsaturated zone and horizontal flow in the groundwater, compounded by the uncertainty surrounding the water table’s initial state. Furthermore, accurately delineating the upper boundary of the region between these domains, known as the capillary fringe, continues to be a persistent obstacle [

Neglecting the capillary fringe often leads to the consideration of primarily vertical flow above the water table. Comprehensive studies focusing on the capillary fringe are relatively scarce, primarily due to its inherent complexity and challenging accessibility, particularly in field conditions. Furthermore, the absence of detailed measurements significantly contributes to the misunderstanding surrounding this critical zone. To address this, the potential approach integrates the widely utilized Gardner model, as extensively documented in literature [_{G}, quantifies the significance of gravitational forces concerning capillary forces. At the field scale, this parameter is assumed to follow a log-normal distribution. Remarkably, beyond being a fitting parameter, its inverse,

In this study, we propose a numerical model that integrates the capillary fringe, aiming to simulate two-dimensional steady-state water flow in variably saturated porous media. Our assumption hinges upon the flow being both homogeneous and incompressible. While previous studies heavily relied on experiments and observations to delineate the limits of the capillary fringe

To achieve this goal, we employed the steady-state potential form of the Richards equation, treating both the unsaturated and saturated zones as a unified continuum. The numerical modeling in this study relies on the finite element code

The laboratory tests involved a soil slab with dimensions of 300 cm in length and 200 cm in depth. The source of flow was provided by a rainfall simulator confined within a steel rectangle, which targeted a specific infiltration band measuring 50 cm in width [_{G} = 0.018 and

To enforce a consistent hydraulic head at the outlet, a reservoir was connected to the slab. The soil is assumed to possess homogeneity, rigidity, and isotropy. We adopted a coordinate system _{0}. To induce a consistent infiltration flow _{0} across the ground surface, this condition was assumed. Additionally, we placed the reservoir between two parallel trenches of equal length to the slab, ensuring they maintained a uniform piezometric level at a fixed depth of _{0}. The utilization of a Marriotte bottle system effectively regulated and upheld a constant water table level at the outlet. For a visual representation, please refer to

- A portion of the upper horizontal surface (ground level) undergoes the application of a constant flux _{0} = 15 cm/h, mimicking an infiltration band resembling an irrigation canal or artificial catchment basin. The selected value of _{0} was intentionally set below the saturated hydraulic conductivity of this particular soil.

- We utilize a nozzle rainfall simulator confined within a steel rectangle measuring 5 cm in height. Additionally, we establish a water table level at the outlet equivalent to _{0}. To mitigate mechanical disturbance to the soil structure during the hydrology experiment, we dispersed small stones across the surface of the infiltration strip, ensuring that the flow of water proceeded without disruption.

The process applied to fine sand ensures consistency and uniform distribution of grain sizes. We used 4 PR2/6 soil moisture probe system vertically in positions

Addressing the coupling problem between the unsaturated and saturated zones poses certain complexities, particularly in defining the boundary between these distinct areas. The water table’s free surface is identified by a null pressure head (

The unsaturated and saturated zones are considered as a cohesive continuum, divided by a capillary-dominant layer. Above the water table, there exists a zone with low capillary tension, leading to soil pores being nearly saturated with water while retaining a minimal amount of air. This transitional region is recognized as the capillary fringe, situated between the saturated and vadose zones [

According to Youngs [

The delineation between the unsaturated zone and the capillary fringe exhibits non-linearity and lacks a perfectly straight boundary. On a closer inspection of the upper limit curve at a finer scale, distinct irregularities become apparent. _{e}, validating this widely adopted approximation. In the capillary fringe, the spaces between pores are completely saturated, reflecting the moisture level at saturation. The key difference between this zone and the saturated region lies in the negative pressure within the pores, which remains lower than atmospheric pressure.

In the subsequent section, our focus will be on modeling flux within the unsaturated zone using the potential form of the Richards equation. Leveraging Gardner’s exponential model aids in mitigating the equation’s nonlinearity. Zone boundaries are defined in terms of potentials, and upon resolving the potential equations, we apply Kirchhoff’s inverse transform to determine the distribution of pressure and hydraulic head.

Several empirical models are employed to describe the hydrodynamic properties of soil, establishing relationships among water content, pressure head, and hydraulic conductivity [_{res} is disregarded due to its minimal quantity of moisture content _{e}

_{G} measures the relative significance of gravity against capillary forces within porous media. The determination of _{e} involves calculating it through theoretical or experimental relations. Rucker et al. [

Neglecting the compressibility of water, the steady-state flow in the unsaturated soil is based on the 2D Richards equation:

To have a potential form the Richards equation, we use the Kirchhoff integral transformation [

With _{ref} the referential potential often taken null for calculation facilities:

We use the Leibniz’s rule for differentiation with one variable:

The combination of the Gardner model and the Kirchhoff potential formula is:

Combining

We will derive an equation governing the flow based on the discharge potential in both the capillary fringe and the saturated zone. The application of the same equation to these zones is justified by the saturated nature of the pores within the capillary fringe. As highlighted by Luthin et al. [

Darcy’s law expresses relate the volumetric flux

The potential must satisfy Laplace’s equation in the saturated zone.

As observed, the sole distinction between the Richards equation in the unsaturated zone and that in the capillary fringe and saturated zone lies in a specific term _{G}

In general, throughout the domain, the potential must be finite, continuous everywhere and vanish at infinity. The boundary conditions on the surfaces are as follows:

On both sides of the area, we have a null horizontal flow,

Along the bottom, the vertical flow is null, so:

For x=0 and outside the infiltration strip, the flow is null

The vertical flow imposed on the infiltration band:

Along the outlet, a constant hydraulic head was specified,

After solving the problem in terms of potential, our next step involves establishing the relationship between pressure head and potential. Upon substituting Kirchhoff potential into the expression for the hydraulic conductivity function (

The pressure head

At saturation and based on

The Darcy flow vector expressed in terms of the potential:

Our determination of the boundary between the unsaturated and saturated zones is reliant on their respective definitions. In describing the unsaturated zone, we utilize both the pressure head and the water content:

Based on Gardner model:

Because _{G}

Hence:

Transferring

The boundary between the capillary fringe and the saturated zone is clearly defined by the water table.

The principal aim of this study is to model the capillary fringe employing the potential method. To achieve this objective, we utilized a dedicated code designed to compute both flux and hydraulic heads based on the φ-based equation. This entailed using mathematical models to solve two-dimensional partial differential equations (PDEs) and their associated boundary conditions through numerical simulations. We utilized the Galerkin finite element method along with

The outline of the field of study is divided into six parts. The flow is zero everywhere except in the infiltration inlet at the top and the outlet of the slab (reservoir). We initially solve the equation in Kirchhoff potential, then we convert the potential into hydraulic pressure depending on whether the flow is governed by

The problem: Find u a real function defined in

We iteratively refined the mesh until observing no further variations in the solution. However, it is important to note that an excessively fine mesh may lead to convergence issues. Our initial spatial discretization of the flow domain utilized an unstructured triangular mesh, comprising 770 elements and 426 vertices (see

For visualization the results are exported to .eps format. For this study’s purpose, we modified and tailored the code to address water flow challenges in unsaturated-saturated porous media. We utilized a polynomial space

To validate the model, we compared the calculated groundwater water table level with experimental measurements, utilizing _{G} _{s}

Polubarinova-Kochina [

We evaluated our model’s performance using statistical indicators, as outlined by [

The ^{2} indicator reaching 0.99, as depicted in

The main parameters to test are the saturated hydraulic conductivity _{s} and _{G}. The evaluation of the variation effect of each parameter is important to determine which measurement must be made with great precaution.

The thickness of the capillary fringe is associated with the particle size distribution of soils and is influenced by flow dynamics.

The thickness of the capillary fringe varies across the schema, with its greatest thickness observed in the infiltration band and at the outlet (with a fixed charge _{0}), surpassing the middle section (zone 2). Changes in the mean thickness depend on the conditions affecting the fringe. This disparity among the three zones highlights the model’s effectiveness in depicting the genuine dynamics of the fringe, surpassing the limitations of a simplistic static model [

The employed method involves systematically adjusting each entry parameter of the model by both −10% and +10% around its initial value. This percentage range is deemed acceptable, considering the inherent inaccuracies associated with the model’s input parameters.

Subsequently, the percentage of variation and sensitivity index [

_{i} the inputs, _{avg} the average.

_{i} the outputs, _{avg} the average.

This index serves as a quantitative measure to express the sensitivity of the model outputs concerning changes in the input variables. Specifically, we opted to analyze the soil characteristics for sensitivity assessment. The outcomes of this sensitivity analysis for both the water table and the capillary fringe are detailed in

Parameters | ||||||||
---|---|---|---|---|---|---|---|---|

CF | WT | CF | WT | |||||

Percent | −10% | +10% | −10% | +10% | −10% | +10% | −10% | +10% |

Variation (%) | 5.14 | 3.10 | 2.10 | 3.99 | 1.32 | 6.02 | 8.14 | 2.51 |

−0.98 | −0.65 | −0.43 | −0.87 | −0.13 | −0.65 | −0.74 | −0.26 |

Modifying inputs by ±10% induces output variations of approximately 5% for the capillary fringe and between 2% to 4% for the water table. Across the domain’s x-axis, the variation is uniformly distributed. Notably, a ± 10% alteration in α_{G} exhibits a corresponding impact on the rate of change in capillary fringe height.

The negative Sensitivity Index (_{G} and the capillary fringe level. Specifically, an _{cf−10%} near –1 implies that a variation in _{G} results in a proportional variation in the output. Comparatively, the _{cf+10%} exhibits a lower variation rate than the _{cf−10%}.

Similarly, the negative _{G} and the water table level. With an _{wt−10%} near −0.43, the variation rate between _{G} and the water table level is roughly half of the output’s rate variation.

In our analysis, we conducted a sensitivity test on the saturated hydraulic conductivity _{s}, as illustrated in _{s} results in an output variation of approximately 1.3% for the capillary fringe, while a +10% modification causes a 6% variation. Similarly, altering _{s} by +10% or −10% leads to a water table level variation of 2.51% and 8.14%, respectively. Remarkably, the most significant variance is observed within “zone 1”, gradually tapering off towards the outlet. Conversely, in “zone 2”, the variance is evenly dispersed across the x-axis. This finding indicates a diminishing lateral impact of hydraulic conductivity with increasing distance from the infiltration band or water source.

Furthermore, a decrease of _{s} by −10% induces a more pronounced change in the capillary fringe height compared to an increase of +10%. In instances of low soil permeability, the capillary fringe might ascend closer to the surface, particularly during the rainy season. However, it is crucial to prevent scenarios where the capillary fringe reaches the root zone, as plant roots require both water and air for optimal growth.

The negative Sensitivity Index (_{s}) and the elevation level. Interestingly, it is noted that for an equivalent rate of change, the rise in the water table level surpasses its decline. This discrepancy is due to the imposed hydraulic head at the outlet, which restricts a proportional decrease. Notably, a −10% variation in _{s} yields a change in the capillary fringe elevation of a certain magnitude, half that observed with a +10% variation. The _{wt−10%} value stands at 0.7, indicating that the variation rate of _{s} leads to about two-thirds of the variation rate in the output. These findings from the sensitivity analysis underscore the pivotal role of hydraulic conductivity as a key parameter influencing capillary fringe dynamics.

The pore water content is defined either by the volumetric water content or by the effective saturation. It is defined to scale the range between 0 and 1 and determined by the classical hydraulic parameters _{r}, _{s}. Typically, _{e}

The function of effective saturation _{e} concerning depth is visualized in _{e} experiences a rapid decline below unity. Typically, the upper boundary of the capillary fringe occurs at _{e}

Effective saturation serves as a useful indicator to establish the boundary between the unsaturated zone and the capillary fringe. However, it does not provide indications about the limit between the saturated zone and the capillary fringe (refer to _{e}

Displayed in

The thickness of the capillary fringe is markedly greater at the vertical level aligned with the infiltration band compared to its midpoint. This variation arises from heightened capillary forces exerted in this area, driven by substantial water movement. Upon reaching a steady state, the water table stabilizes at its maximum position. At this stage, the domain achieves hydrostatic balance, resulting in an equilibrium where the influx through the infiltration band matches the outflow through the aquifer’s outlet. The findings from

The behavior of vectors within the capillary fringe can be delineated into three distinct zones:

The initial zone, spanning from 0 to 50 cm beneath the infiltration band, is characterized by predominant vertical flow.

The middle zone exhibits significantly greater horizontal flow than vertical flow.

The last zone, near the exit (285–300 cm), demonstrates entirely vertical flow above the water table (refer to

Notably, within the middle zone, the horizontal component of the velocity vectors increases notably from the upper limit of the capillary fringe towards the groundwater level, expanding approximately two to fourfold. Numerical findings emphasize the significance of horizontal velocity within the capillary fringe, comparable in importance to the saturated zone. Hence, studies concerning contaminant transport should encompass the consideration of horizontal movement within the capillary fringe, extending beyond conventional limitations confined to the saturated zone.

The Wieringermeer effect has been documented in a variety of settings, having been first noticed in the Wieringermeer polder in Holland. This phenomena is characterized by a sharp drop in the aquifer’s water level and quick variation in its level. On the other hand, when the capillary fringe becomes closer to the surface, the aquifer rises quickly, which is an inverted manifestation. Reference [

As well examined by [_{0} to _{1}) at the infiltration band, expanding by 3cm. The discrepancy in the length of the capillary fringe depicted in _{2}) and (b_{2}) is attributed to variations in the outlet elevation.

To simulate water flow within variably saturated zones, incorporating the capillary fringe, we implemented a mathematical formulation that unifies these three regions into a cohesive continuum. Hydrological investigations employing the potential method have garnered comparatively less attention when juxtaposed with head and pressure methods. However, as demonstrated in this study, focusing on the potential form of the Richards equation remains pertinent and worthwhile. The amalgamation of discharge potential and Kirchhoff potential proves effective in delineating groundwater dynamics.

The numerical solution, employing the finite element method, maintains stability throughout simulations. These simulations underwent validation using experimental data derived from soils falling within the Gardner class. Furthermore, the model offers qualitative predictions of steady-state water table levels and upper limits of the capillary fringe.

The primary achievement of this study lies in precisely delineating the contours of the capillary fringe and seamlessly integrating it into the unsaturated-saturated continuum. Overall, the model demonstrates its capability to accurately simulate hydraulic dynamics within groundwater systems.

The results emphasize the significance of horizontal velocities within the capillary fringe, a key determinant affecting particle migration and transport phenomena. Neglecting the impact of the capillary fringe on hydrology and solute transport could result in misconceptions due to its complex nature and gradient fluctuations. Developing an alternative solution method capable of analyzing the capillary fringe’s dynamics in transient and three-dimensional scenarios is essential, especially when considering various soil permeability models such as van Genuchten or Brooks-Corey. The challenge of non-invasive pressure data collection in field-scale capillary fringe investigations remains a significant hurdle that needs resolution. Further exploration is warranted, particularly in heterogeneous soil conditions and variable water table scenarios. Exciting advancements in hydrogeology await by leveraging novel approaches to model the capillary fringe and porous media, taking inspiration from innovative technologies like neural networks and the groundbreaking contributions of researchers such as [

The Gardner model parameter depends on the size and distribution of pores

Relative conductivity exponent

Volumetric water content scale parameter, which is called the volumetric water content at natural saturation

Residual water content

Air-entry pressure is defined as the suction pressure at which air begins to move water from the pores (appearance of first bubbles)

Hydraulic conductivity

Saturated hydraulic conductivity

Matric flux potential

Effective particle diameter

Porosity

We extend our sincere gratitude to Dr. Mustapha Hachimi for his invaluable assistance and unwavering support throughout this research endeavor.

The authors received no specific funding for this study.

The authors confirm their contribution to the paper as follows: study conception and design: R. K, A. M; data collection: R. K; analysis and interpretation of results: R. K, A. M; draft manuscript preparation: R. K, A. M. All authors reviewed the results and approved the final version of the manuscript.

All the data used in the study is included in the manuscript.

The authors declare that they have no conflicts of interest to report regarding the present study.

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