Describe Water Flow Research Articles

Auto-Bäcklund transformation and exact solutions for a new integrable (2+1)-dimensional shallow water wave equation

Shallow water waves (SWWs) are often used to describe water flow and wave movement in shallow water areas. The article introduces a novel (2 + 1)-dimensional SWW equation. We prove that the equation is integrable and obtain an auto-Bäcklund transformation by truncating Painlevé expansion. Using the bilinear form of the equation, a new auto-Bäcklund transformation and some exact solutions are obtained. Besides, a convergent power series solution is derived using Lie symmetry method. These exact solutions can enrich mathematical modeling and help us understand nonlinear wave phenomena. Finally, conserved vectors are derived.

Dynamic and empirical methods for field capacity estimation in fine textured soils with a coarse interlayer

Field capacity (FC) is an important soil hydraulic concept in soil science and irrigation management. It is generally determined from soil water content in a soil layer when soil profile reaches a steady pressure head or negligible drainage flux from an initially saturated soil. However, the proposed criteria are mainly tested for uniform soils and vary with soil textures. To quantify FC in layered soils, a Richards equation-based model was used to describe water flow in fine-textured soils with a coarse interlayer. With calibrated soil hydraulic parameters for loam and sand from infiltration measurements, drainage from saturation was simulated in the loam with a sand interlayer. A relative drainage rate (δ) was defined as a function of water storage and drainage flux to analyze soil water status at FC. Soil water content in the upper loam layer of layered profiles was improved compared with that in the uniform loam, which was negatively correlated with buried depth but positively correlated with thickness of the sand layer for a specified δ. Under different buried depths and thicknesses, soil water content decreased with the decline of δ and decreased rapidly as δ reduced to 1 % d−1. The drainage flux at δ = 1 % d−1 changed within a range of 0.056–0.26 cm d−1, and soil water content reached to 0.278–0.346 cm3 cm−3, which accounted for 70–87 % of the saturated water content of loam. Although the FC in the upper fine-textured soil layer varied for different buried depths and thicknesses of coarse interlayer, the proposed dynamic method is reliable and universal to estimate the FC in the above layered soils at δ = 1 % d−1. An empirical equation was also developed to calculate the FC in fine-textured soils with different buried depths and thicknesses of a coarse interlayer based on the critical δ value.

Evaluation of Soil Hydraulic Properties in Northern and Central Tunisian Soils for Improvement of Hydrological Modelling

The hydrological cycle is strongly affected by climate changes causing extreme weather events with long drought periods and heavy rainfall events. To predict the hydrological functioning of Tunisian catchments, modelling is an essential tool to estimate the consequences on water resources and to test the sustainability of the different land uses. Soil physical properties describing water flow are essential to feed the models and must therefore be determined all over the watershed. A simple but robust ring infiltration method combined with particle size distribution (PSD) analysis (BEST method) was used to evaluate and derive the retention properties and the hydraulic conductivities. Physically based and statistical pedotransfer functions based on PSD were compared to test their potential use for different types of Tunisian soils. The functional sensitivity of these parameters was assessed by employing the Hydrus-1D software (PC Progress, Prague, Czech Republic) for water balance computations. This evaluation process involved testing the responsiveness and accuracy of the parameters in simulating various water balance components within the model. The evaluation of soil hydraulic parameters across the three used models highlighted significant variations, demonstrating distinct characteristics in each model. While notable differences were evident overall, intriguing similarities emerged, particularly regarding saturated hydraulic conductivity between BEST and Rosetta, and the shape parameter (n) between Arya–Paris and Rosetta. These parallels indicate shared hydraulic properties among the models, underscoring areas of agreement amid their diverse results. Significant differences were shown for scale parameter α for the various methods employed. Marginal differences in evaporation and drainage were observed between the BEST and Arya–Paris methods, with Rosetta distinctly highlighting a disparity between physically based models and statistical models.

Combining root and soil hydraulics in macroscopic representations of root water uptake

AbstractPlant water uptake and plant and soil water status are important for the soil water balance and plant growth. They depend on atmospheric water demand and the accessibility of soil water to plant roots, which is in turn related to the hydraulic properties of the root system and the soil around root segments. We present a simulation model that describes water flow in the soil–plant system mechanistically considering both root and soil hydraulic properties. We developed an approach to upscale three‐dimensional (3D) flow in the soil toward root segments of a 3D root architecture to a model that considers one‐dimensional flow between horizontal soil layers and radial flow to root segments in that layer. The upscaled model couples upscaled linear flow equations in the root system with an analytical solution of the nonlinear radial flow equation between the soil and roots. The upscaled model avoids simplifying assumptions about root hydraulic properties and water potential drops near roots made in, respectively, soil‐ and root‐centered models. Xylem water potentials and soil–root interface potentials are explicitly simulated and show, respectively, large variations with depth and large deviations from bulk soil water potentials under dry soil conditions. Accounting for hydraulic gradients in the soil around root segments led to an earlier but slower reduction of transpiration during a drought period and a better plant water status with higher nighttime plant water potentials.

Can Basic Soil Quality Indicators and Topography Explain the Spatial Variability in Agricultural Fields Observed from Drone Orthomosaics?

Crop growth is often uneven within an agricultural parcel, even if it has been managed evenly. Aerial images are often used to determine the presence of vegetation and its spatial variability in field parcels. However, the reasons for this uneven growth have been less studied, and they might be connected to variations in topography, as well as soil properties and quality. In this study, we evaluated the relationship between drone image data and field and soil quality indicators. In total, 27 multispectral and RGB drone image datasets were collected from four real farm fields in 2016–2020. We analyzed 13 basic soil quality indicators, including penetrometer resistance in top- and subsoil, soil texture (clay, silt, fine sand, and sand content), soil organic carbon (SOC) content, clay/SOC ratio, and soil quality assessment parameters (topsoil biological indicators, subsoil macroporosity, compacted layers in the soil profile, topsoil structure, and subsoil structure). Furthermore, a topography variable describing water flow was used as an indicator. Firstly, we evaluated single pixel-wise linear correlations between the drone datasets and soil/field-related parameters. Correlations varied between datasets and, in the best case, were 0.8. Next, we trained and tested multiparameter non-linear models (random forest algorithm) using all 14 soil-related parameters as features to explain the multispectral (NIR band) and RGB (green band) reflectance values of each drone dataset. The results showed that the soil/field indicators could effectively explain the spatial variability in the drone images in most cases (R2 > 0.5), especially for annual crops, and in the best case, the R2 value was 0.95. The most important field/soil features for explaining the variability in drone images varied between fields and imaging times. However, it was found that basic soil quality indicators and topography variables could explain the variability observed in the drone orthomosaics in certain conditions. This knowledge about soil quality indicators causing within-field variation could be utilized when planning cultivation operations or evaluating the value of a field parcel.

Investigating Soil-Root Interactions with the Numerical Model R-SWMS.

In this chapter, we present the Root and Soil Water Movement and Solute transport model R-SWMS, which can be used to simulate flow and transport in the soil-plant system. The equations describing water flow in soil-root systems are presented and numerical solutions are provided. An application of R-SWMS is then briefly discussed, in which we combine in vivo and in silico experiments in order to decrypt water flow in the soil-root domain. More precisely, light transmission imaging experiments were conducted to generate data that can serve as input for the R-SWMS model. These data include the root system architecture, the soil hydraulic properties and the environmental conditions (initial soil water content and boundary conditions, BC). Root hydraulic properties were not acquired experimentally, but set to theoretical values found in the literature. In order to validate the results obtained by the model, the simulated and experimental water content distributions were compared. The model was then used to estimate variables that were not experimentally accessible, such as the actual root water uptake distribution and xylem water potential.

A new integral model for predicting the hydraulic conductivity of saturated frozen soil

Hydraulic conductivity of frozen soil is indispensable for describing water flow process in freezing soils. However, this conductivity is difficult to measure in the laboratory. So, the objective of this study is to propose a simple and smooth hydraulic conductivity model for saturated frozen soil based on the soil freezing characteristic curve (SFCC). Considering that the volume of pore water approximately represents the pore volume, the distribution of effective pore radii changed as a function of temperature was determined by combining the SFCC and the Gibbs–Thomson equation. Based on the distribution of effective pore radii, Hagen-Poiseuille equation, Darcy’s law, relative hydraulic conductivity and tortuosity, a simple and smooth hydraulic conductivity model for saturated frozen soil based on the SFCC was proposed. To illustrate the model performance, eight existing experimental cases were analyzed. The results showed that predictions considering the three different tortuosity formulations were in good agreement with the tested data; the results were worse when not considering the tortuosity and the results were best when considering the tortuosity τF=B/rb2. Moreover, the predictive modeling power was related to the tortuosity function. Furthermore, the model was further compared with the previous models, the results of which showed the better performance of the proposed model. The model is quite simple and timesaving to use when the SFCC and the saturated hydraulic conductivity are available. This result provides a new idea for developing the hydraulic conductivity model of saturated frozen soil.

A Comparative Study of Conceptual Model Complexity to Describe Water Flow and Nitrate Transport in Deep Unsaturated Loess

AbstractUnderstanding nitrate migration through the deep vadose zone is essential for aquifer vulnerability assessments. The effect of variability of physical properties of the deep vadose zone on nitrate transport has been scarcely explored. Recently, deep nitrate storage profiles were determined in the vadose zone of the Loess Plateau of China. Using these observations along with measured soil properties, this study investigates the effect of loess vertical heterogeneity on water movement and nitrate transport through the deep vadose zone. Models of different complexity were established and calibrated. First, a simple piston flow and nitrate mass balance approach was calibrated to the observed nitrate storage. The results indicate that the total nitrate storage is estimated well, while the estimation of the distribution of nitrate is relatively poor. Subsequently, Richards' equation and the Advection‐Dispersion equation were evaluated. Three different conceptualizations of the numerical models were calibrated against deep vadose zone nitrate and water content observations: (1) one‐layer model assuming homogenous loess vadose zone; (2) a model that considers a hydraulic conductivity (Ks) decay function and (3) a model where the Miller‐Miller scaling factors are prescribed to account for changes of the hydraulic functions with depth. Accounting for the vertical Ks decay in the numerical models improved water flow performances. The study reveals the adequacy of implementing water flow and nitrate transport numerical models together with a simple representation of the vertical loess variability, for simulating nitrate migration in loess deep vadose zone environments.

Extending the Classical Continuum Theory to Describe Water Flow through Two-Dimensional Nanopores.

Water flow through two-dimensional nanopores has attracted significant attention owing to the promising water purification technology based on atomically thick membranes. However, the theoretical description of water flow in nanopores based on the classical continuum theory is very challenging owing to the pronounced entrance/exit effects. Here, we extend the classical Hagen-Poiseuille equation for describing the relationship between flow rate and pressure loss in laminar tube flow to two-dimensional nanopores. A totally theoretical model is established by appropriately considering the velocity slip on pore surfaces both in the friction pressure loss and entrance/exit pressure loss. Based on molecular dynamics simulations of water flow through graphene nanopores, it is shown that the model can not only well predict the overall flow rate but also give a good estimation of the velocity profiles. As the pore radius and length increase, the model can reduce to the equations applicable to the fluid flow in infinitely/finitely long nanotubes, thin orifices, and macroscale tubes, showing an accurate prediction of the existing experimental and simulation data of the water flow through nanotubes and nanopores in the literature. Namely, the presented model is a unified model that can uniformly describe the fluid flow from nanoscales to macroscales by modifying the classical continuum theory.

Application of the Darcy and Richards equations for modelling of water capillary rise in building materials

This paper presents the application of the Darcy and Richards equations, describing water flow and dynamic changes of the water content in porous medium, for numerical modeling of water capillary rise in the selected building materials. The numerical simulations of unsaturated and variably saturated water flow in the studied materials were performed in FEFLOW commercial computational software, by DHI-Wasy, Germany. The developed model represented a wall section constructed of aerated concrete blocks, covered by mortar prepared using blended cement with biomass ash addition. The physical and water retention characteristics of the modeled materials were based on the previously performed laboratory measurements and literature research. The assumed initial conditions reflected air dry wall with limited moisture, while the boundary conditions allowed modeling of capillary rise due to direct contact of the studied materials with water with variable pressure head. The results of numerical calculations allowed observation of the time-dependent changes in the moisture content of building materials in relation to their physical properties and variable boundary conditions.

A Stable Radial Basis Function Partition of Unity Method with d-Rectangular Patches for Modelling Water Flow in Porous Media

Richards’ equation is used to describe water flow in porous media. It is strongly nonlinear, which makes the approximation of its solution a challenging task. In this study, we aim to linearize Richards’ equation using the exponential Gardner models of hydraulic conductivity and moisture content variables. Then, we propose a new formulation of radial basis function partition of unity method (RBFPUM) based on d-rectangular patches as local subdomains in order to approximate the solution of the linearized Richards’ equation. In particular, the multivariate weights are expressed as a product of one dimensional Wendland compactly supported radial basis functions. With small values of the shape parameter contained in the Gaussian radial basis function, safe computations of the solution will be achieved by adopting RBF-QR algorithm (Forenberg et al. in SIAM J Sci Comput 33(2):869–892, 2011). Hence, an efficient numerical solution can be obtained not only in terms of safe computations but also in terms of accuracy, as we will see in the numerical examples. In order to deal with layered soils, another method is proposed for the 2D and the 3D cases. It is based on the domain decomposition principle coupled with RBFPUM by using d-rectangular patches and the RBF-QR algorithm. The matching conditions, i.e. the continuity of mass fluxes and pressure head, are imposed at the interface between zones via the Steklov–Poincare equation. The efficiency of the proposed method will be examined via some examples.

OpenMP Performance in Numerical Simulation of Dambreak Problem Using Shallow Water Equations

Numerical simulation of water surface waves is widely used to describe water flow and its impact for human life. For instance, numerical simulation of waves is elaborated to simulate Tsunami as an early warning system. Using numerical approach, the study of water flow will reduce cost and save time compared with the conventional approach (in laboratory). Shallow water equations (SWE) is one of the mathematical models which can be used to describe water flow. In numerical simulation of SWE, finite volume method is a robust method to approximate SWE. The result using numerical approach depends on the number of grids. High number of grids then smooth solution can be obtained. However, increasing number of grids leads to the increasing of computational cost. In this paper, parallel computing using OpenMP platform is given in order to reduce computational cost of numerical simulation. In parallel computing performances, Speedup andEfficiency of numerical simulation using 6400 grids points are obtained 4 times and 51% respectively.Moreover, by several numbers of cores from 2 to 8, CPU time of parallel computing is shown decreasing along the increasing number of computer cores.

Mean Velocity and Shear Stress Distribution in Floating Treatment Wetlands: An Analytical Study

AbstractFloating treatment wetlands (FTWs) are efficient at wastewater treatment; however, data and physical models describing water flow through them remain limited. A two‐domain model is proposed dividing the flow region into an upper part characterizing the flow through suspended vegetation and an inner part describing the vegetation‐free zone. The suspended vegetation domain is represented as a porous medium characterized by constant permeability thereby allowing Biot's Law to be used to describe the mean velocity and stress profiles. The flow in the inner part is bounded by asymmetric stresses arising from interactions with the suspended vegetated (porous) base and solid channel bed. An asymmetric eddy viscosity model is employed to derive an integral expression for the shear stress and the mean velocity profiles in this inner layer. The solution features an asymmetric shear stress index that reflects two different roughness conditions over the vegetation‐induced auxiliary bed and the physical channel bed. A phenomenological model is then presented to explain this index. An expression for the penetration depth into the porous medium defined by 10% of the maximum shear stress is also derived. The predicted shear stress profile, local mean velocity profile, and bulk velocity agree with the limited experiments published in the literature.

A note on separation of variables solutions of generalized nonlinear diffusion equations

In this letter we show how more cases of the generalized nonlinear diffusion equation that contain separable solutions to those found by Zhang et al. (2002, 2003) via the generalized conditional symmetries method, can be found. Importantly we demonstrate with an example that can be used to describe water flow in unsaturated soil, and which can provide new insights when plant-root uptake is affected by water speed.

Analysis of a hybrid numerical method – decomposing leaf hydraulic conductance

AbstractWe describe a hybrid numerical method to solve a boundary value problem where an unknown parameter of the model is chosen to satisfy an additional boundary condition. After the solution of the differential equation is approximated using a one-step method, a secant method is used to update the value of the unknown parameter. The model is a generalization of a model first used to describe water flow through roots, which was later used to describe water flow through the tank bromeliad Guzmania lingulata. In both cases, identification of the unknown parameter represents the decomposition of overall plant conductance into components in the radial and axial directions. We describe convergence of the one-step and secant portions of the method in a base case corresponding to previous applications of the model and in an intermediate case corresponding to a first approximation of the geometry of the leaf. We demonstrate that in the more general case, which better represents the geometry of G. lingulata, the...

Simulations of freshwater lens recharge and salt/freshwater interfaces using the HYDRUS and SWI2 packages for MODFLOW

Abstract The paper presents an evaluation of the combined use of the HYDRUS and SWI2 packages for MODFLOW as a potential tool for modeling recharge in coastal aquifers subject to saltwater intrusion. The HYDRUS package for MODFLOW solves numerically the one-dimensional form of the Richards equation describing water flow in variablysaturated media. The code computes groundwater recharge to or capillary rise from the groundwater table while considering weather, vegetation, and soil hydraulic property data. The SWI2 package represents in a simplified way variable-density flow associated with saltwater intrusion in coastal aquifers. Combining these two packages within the MODFLOW framework provides a more accurate description of vadose zone processes in subsurface systems with shallow aquifers, which strongly depend upon infiltration. The two packages were applied to a two-dimensional problem of recharge of a freshwater lens in a sandy peninsula, which is a typical geomorphologic form along the Baltic and the North Sea coasts, among other places. Results highlighted the sensitivity of calculated recharge rates to the temporal resolution of weather data. Using daily values of precipitation and potential evapotranspiration produced average recharge rates more than 20% larger than those obtained with weekly or monthly averaged weather data, leading to different trends in the evolution of freshwater-saltwater interfaces. Root water uptake significantly influenced both the recharge rate and the position of the freshwater-saltwater interface. The results were less sensitive to changes in soil hydraulic parameters, which in our study were found to affect average yearly recharge rates by up to 13%.

Analysis of a Hybrid Numerical Method – Decomposing Leaf Hydraulic Conductance

We describe a hybrid numerical method to solve a boundary value problem where an unknown parameter of the model is chosen to satisfy an additional boundary condition. After the solution of the differential equation is approximated using a one-step method, a secant method is used to update the value of the unknown parameter. The model is a generalization of a model first used to describe water flow through roots, which was later used to describe water flow through the tank bromeliad Guzmania lingulata. In both cases, identification of the unknown parameter represents the decomposition of overall plant conductance into components in the radial and axial directions. We describe convergence of the one-step and secant portions of the method in a base case corresponding to previous applications of the model and in an intermediate case corresponding to a first approximation of the geometry of the leaf. We demonstrate that in the more general case, which better represents the geometry of G. lingulata, the one-step method also converges as expected. Finally, we discuss the implications of including a better description of the geometry of the leaf in context of radial conductance and show that our modeling of the leaf geometry increases the component of the overall leaf conductance in the radial direction by as much as 25%.

Determination of water flow through clayey slurries using computed micro-tomography

The purpose of this study was to develop a clear understanding of water flow through clayey slurries. Computed micro-tomography was used to quantify slurry constituents and validate index properties and hydraulic conductivity. An image analysis method was developed to capture the mechanics of flow through clayey slurries that are difficult to quantify through conventional measurement techniques. A solids ratio was defined to identify actual flow paths within a mixture of the solids, water and hydrated grains . Likewise, hydraulic radius was re-defined to describe water flow paths (pore throat) thereby eliminating the measurement of shape factor and tortuosity. More importantly, hydraulic conductivity calculated through the new method (and based on Poiseuille's law) corroborated well with the empirical method of Pane & Schiffman (1997) for flocculated slurries.

Modeling of water flow in multi-channel river system in the Narew National Park

Abstract Modeling of water flow in multi-channel river system in the Narew National Park. Anastomosing rivers constitute a rare example of multi-channel systems, which used to be very common before the agricultural and industrial development. Presently few of them remain worldwide and the only example in Poland is the Upper River Narew within Narew National Park. Although hydraulic modeling using one-dimensional models is commonly used to describe water flow in rivers, for multi-channel rivers problem is more complicated. For this type of rivers it is expected that the feedback between process of plants growth (expressed by Manning’s coefficient) and distribution of flow in anabranches is high. However, assignment procedure on roughness coefficients in splitting and rejoining channels is laborious and difficult. Therefore, for efficient water flow modeling in multi-channel systems a stand-alone hydraulic model equipped with automatic optimization procedure was developed. Optimization and validation stages, based on field measurements data of discharge and water levels, indicated that the model accurately simulates water flow in multi-channel system.

Applying Process-Based Models for Subsurface Flow Treatment Wetlands: Recent Developments and Challenges

To date, only few process-based models for subsurface flow treatment wetlands have been developed. For modelling a treatment wetland, these models have to comprise a number of sub-models to describe water flow, pollutant transport, pollutant transformation and degradation, effects of wetland plants, and transport and deposition of suspended particulate matter. The two most advanced models are the HYDRUS Wetland Module and BIO-PORE. These two models are briefly described. This paper shows typical simulation results for vertical flow wetlands and discusses experiences and challenges using process-based wetland models in relation to the sub-models describing the most important wetland processes. It can be demonstrated that existing simulation tools can be applied for simulating processes in treatment wetlands. Most important for achieving a good match between measured and simulated pollutant concentrations is a good calibration of the water flow and transport models. Only after these calibrations have been made and the effect of the influent fractionation on simulation results has been considered, should changing the parameters of the biokinetic models be taken into account. Modelling the effects of wetland plants is possible and has to be considered when important. Up to now, models describing clogging are the least established models among the sub-models required for a complete wetland model and thus further development and research is required.