Water Flow In Unsaturated Soils Research Articles

A New Reduced-Dimension Iteration Two-Grid Crank–Nicolson Finite-Element Method for Unsaturated Soil Water Flow Problem

The main objective of this paper is to reduce the dimensionality of unknown coefficient vectors of finite-element (FE) solutions in two-grid (CN) FE (TGCNFE) format for the nonlinear unsaturated soil water flow problem by using a proper orthogonal decomposition (POD) and to design a new reduced-dimension iteration TGCNFE (RDITGCNFE). For this objective, a new time semi-discrete CN (TSDCN) scheme for the nonlinear unsaturated soil water flow problem is first designed and the existence, stability, and error estimates of TSDCN solutions are demonstrated. Subsequently, a new TGCNFE format for the nonlinear unsaturated soil water flow problem is designed and the existence, unconditional stability, and error estimates of TGCNFE solutions are demonstrated. Next, a new RDITGCNFE format with the same FE basis functions as the TGCNFE format is built by the POD method and the existence, unconditional stability, and error estimates of RDITGCNFE solutions are discussed. Ultimately, the rightness of theory results and the superiority of the RDITGCNFE format are verified by two sets of numerical tests. It is worth noting that the RDITGCNFE format differs completely from all previous reduced-dimension methods, including the authors’ previous works. Therefore, the study of this paper can not only provide a new theoretical method for the dimensionality reduction of numerical models for nonlinear problems but also provide an algorithm implementation technology for the numerical simulation of practical engineering problems.

High-Resolution Estimation of Soil Saturated Hydraulic Conductivity via Upscaling and Karhunen–Loève Expansion within DREAM(ZS)

Quantification of the soil hydraulic conductivity is key to the study of water flow and solute transport in unsaturated soils. Rapid advances in measurement technology have provided a large number of observations at different scales, offering unprecedented opportunities and challenges for the estimation of hydraulic parameters. This paper proposes an inverse estimation method for downscaling of observations on coarse scales to estimate hydraulic parameters on high-resolution scales. Due to the significant spatial heterogeneity, the inversion faces the problems of dynamics-based integration of data at different scales, model uncertainty due to hundreds and thousands of parameters, and computational consumption due to the large number of forward simulations. To overcome these problems, this paper uses an efficient Bayesian optimization DREAM(ZS) as an inverse framework, and incorporates an analytical upscaling method and Karhunen–Loève (KL) expansion to infer finer-scale saturated hydraulic conductivity distribution conditioned on coarse-scale measurements. The efficient upscaling method is used to link measurements and hydraulic parameters at different scales, and Karhunen–Loève (KL) expansion is incorporated to greatly reduce the dimension of the parameter to be estimated. To further improve the efficiency of the inversion, a locally one-dimensional (LOD) algorithm is used to solve the multidimensional water flow model at coarse scales. The proposed inverse model is applied in a series of numerical experiments to demonstrate its applicability and effectiveness under different flow boundary conditions, different levels of ratio between coarse- and fine-scale grids, different densities of observation points, and different degrees of statistic heterogeneity of soil mediums.

Research on hydro-thermal coupling model of canal foundation soil based on particle grading curve predicting soil-water characteristic curve

Hydro-thermal coupling is the essence of the freeze-thaw process, and theoretical studies of this coupled process have been hot topics in the field of frozen soil. Darcy's law of unsaturated soil water flow, heat conduction theory, and relative saturation and solid-liquid ratio are based on this paper. According to the principle that the cumulative curve of particle gradation of canal foundation soil is similar to soil-water properties. A soil-water characteristic curve is derived using the cumulative particle gradation curve. VG model is then used to fit soil-water characteristic curves to obtain the canal foundation soil's hydraulic characteristic parameters, and the established hydro-thermal coupling model is modified to reflect canal foundation soil hydro-thermal evolution more objectively. A closed system one-way freezing test method is used to verify the feasibility of the proposed method in this part. The results show that the optimal parameters of the VG model of the subsoil are a = 0.06, n = 1.2, and m = 0.17, and the temperature and water fields obtained from the simulation are in good agreement with the measured data, showing the utility of the hydro-thermal coupling model in predicting hydraulic parameters. Analysis of the multi-field interaction mechanism and dynamic coupling process of the canal foundation soil during freezing and thawing. This has great importance for preventing freezing damage in canals and protecting agricultural safety.

Large-Scale Freezing and Thawing Model Experiment and Analysis of Water–Heat Coupling Processes in Agricultural Soils in Cold Regions

Heilongjiang Province, the largest commercial grain base in China, experiences significant challenges due to the environmental effects on its soil. The freezing and thawing cycle in this region leads to the transport of water and heat, as well as the exchange and transfer of energy. Consequently, this exacerbates the flooding disaster in spring and severely hampers farming activities such as plowing and sowing. To gain a better understanding of the freezing and thawing mechanisms of farmland soil in cold regions and prevent spring flooding disasters, this study focuses on Heilongjiang Province as a representative area in northeast China. The research specifically investigates the frozen and thawed soil of farmland, using a large-scale low-temperature laboratory to simulate both artificial and natural climate conditions in the cold zone. By employing the similarity principle of geotechnical model testing, the study aims to efficiently simulate the engineering prototypes and replicate the process of large-span and long-time low temperatures. The investigation primarily focuses on the evolution laws of key parameters, such as the temperature field and moisture field of farmland soil during the freeze–thaw cycle. The findings demonstrate that the cooling process of soil can be categorized into three stages: rapid cooling, slow cooling, and freezing stabilization. As the soil depth increases, the variability of the soil temperature gradually diminishes. During the melting stage, the soil’s water content exhibits a gradual increase as the temperature rises. The range of water content variation during thawing at depths of 30 cm, 40 cm, 50 cm, and 80 cm is 0.12% to 0.52%, 0.47% to 1.08%, 0.46% to 1.96%, and 0.8% to 3.23%, respectively. To analyze the hydrothermal coupling process of farmland soil during the freeze–thaw cycle, a theoretical model of hydrothermal coupling was developed based on principles of mass conservation, energy conservation, Darcy’s law of unsaturated soil water flow, and heat conduction theory. Mathematical transformations were applied after defining the relative degree of saturation and solid–liquid ratio as field functions with respect to the relative degree of saturation and temperature. The simulated temperature and moisture fields align well with the measured data, indicating that the water–heat coupling model established in this study holds significant theoretical and practical value for accurately predicting soil temperature and moisture content during the spring sowing period, as well as for efficiently and effectively utilizing frozen soil resources in cold regions.

Modeling water flow in unsaturated soils through physics-informed neural network with principled loss function

Modeling water flow in unsaturated soils is crucial in geotechnical practice. Nowadays, the physics informed neural network (PINN) is gaining popularity in solving the Richardson Richards equation (RRE) thanks to its mesh-free, physics-constrained, and data-driven properties. Despite several successful applications in modelling 1D infiltration problems, its capability and stability to deal with more complicated boundary conditions and multidimensional problems still need to be examined. This paper investigates the impacts of the loss weights and random state on the performance of the RRE-solving PINNs and possible solutions to mitigate such impacts. Two loss-balanced PINNs, GN-PINN and PLF-PINN, were compared to the baseline PINN in modelling three unsaturated groundwater flow problems. The results show that the performance of the baseline PINN severely depends on the loss weight configurations and random states. While GN-PINN’s tendency to ignore the train loss term makes it infeasible for solving RRE, PLF-PINN can strike a good balance between loss terms and hence enhance PINN’s robustness against loss weight initialization and random state greatly.

Modeling Water Flow and Solute Transport in Unsaturated Soils Using Physics‐Informed Neural Networks Trained With Geoelectrical Data

AbstractAccurate modeling of water flow and solute transport in unsaturated soils is of significant importance for precision agriculture, environmental protection and aquifer management. Traditional modeling approaches are considerably challenging since they require well‐defined boundaries and initial conditions. Physics‐informed neural networks (PINNs) have recently been developed to learn and solve forward and inverse problems also constrained to a set of partial differential equations and are more flexible than traditional modeling approaches. However, hydrological applications of PINNs used so far spatial measurements of hydraulic head, water content and/or solute concentrations, which were well distributed in the subsurface for training the system. Such measurements are hard to obtain in real‐world applications. Here, we propose to train PINNs with non‐invasive geoelectrical tools for simulating two‐dimensional water flow and solute transport during infiltration and redistribution processes with unknown initial conditions. Two‐dimensional flow and transport numerical simulations were used as benchmarks to examine the suitability of the described approach. Results have shown that the trained PINNs system was able to reproduce the spatiotemporal distribution of both water content and pore‐water salinity during both processes with high accuracy, using five time‐lapse geoelectrical measurements and matric head measurements at a single location. The trained PINNs system reconstructed the initial conditions of both state parameters at both processes. It was also able to separate the measured electrical signal into its two components, that is, water content and pore‐water salinity. The subsurface geoelectrical tomograms were significantly improved compared to those obtained from a classical inversion of the raw geoelectrical data.

ANALYTICAL SOLUTION OF UNSATURATED SOIL WATER FLOW FROM A POINT SOURCE

Water flow into unsaturated porous media is governed by the Richards’ partial differential equation expressing the mass conservation and Darcy’s laws. The Richards’ equation may be written in three forms,where the dependent variable is pressure head or moisture content, and the constitutive relationships between water content and pressure head allow for conversion of one form into the other. In the present paper, the “moisture-based" form of Richards’ equation is linearized by applying Kirchhoff’s transformation, whichcombines the soil water diffusivity and soil water content. Then the similarity method is used to obtain the analytical solution of wetting front position. This exact solution is obtained by means of Lie’s method of infinitesimal transformation groups. The predicted results of the analytical solution agreed well with available results of experiments and numerical solutions.

FluxPAW: A standalone software to calculate flux-based plant available water

<h2>Abstract</h2> Soil water contents at field capacity (FC), the limiting point (LP), and the wilting point (WP) delimit the fractions of soil water availability, the plant available water (PAW). FC is primarily a function of soil hydraulic properties, whereas LP and WP depend on soil, plant, and atmospheric conditions. The difference between FC and WP is called total available water, whereas FC minus LP is the readily available water. The flux-based method (FBM) predicts PAW from unsaturated soil water flow, rooting characteristics, and atmospheric water demand. This paper presents FluxPAW, a software to calculate PAW using the FBM algorithm.

Influence of Soil Heterogeneity on Water Flow and Solute Transport Characterized by Dye Tracer Experiments

Soil heterogeneity is a key factor affecting horizontal water flow and solute transport in unsaturated soil, in which particle distribution, pore structure, and pore connectivity in the soil all present fractal characteristics. The initial water content of soil has significant influence in the resulting heterogeneity of the soil structure, which can affect the water flow and solute transport. In this study, through a series of horizontal transport laboratory simulation experiments in soil columns, the initial water content dependency of transport was elucidated using the fractal Richards’ equation (FRE) model. We first observed that water flows faster in soil columns with higher initial water content, but the law is opposite in the initial process of water flow. By analyzing the variation of parameter values in the FRE model, we speculated that the abnormal phenomenon of early diffusion is caused by the heterogeneity of the soil structure. To elucidate the mechanisms of the water content’s dependency of the solute transport process and to identify the circumstance of altering of flow regime, a series of dye tracer experiments were conducted. Results show that in soil columns with low water content, the inflow of external fluid leads to the agglomeration of soil particles. Combined with matric suction, the heterogeneity of the soil structure is enhanced, which affects the formation of water flow and solute transport paths. It even leads to preferential flow in unsaturated soil columns with low water content.

UnSaLuDo: The development of an educational board game on unsaturated soil mechanics

Non-conventional tools of teaching are effective in delivering deep learning of scientific facts as well as skills associated with scientific disciplines. A well-known method to promote student engagement and learning in general is the use of games in classrooms. Games can provide a flexible option that could engage students, create active learning experiences, and enable them to experiment in safe playful environments. This paper presents UnSaLuDo, a board game developed by the GeoFUN group. The goal was to develop a tool that enables students to experience fun and integrational activities in a classroom setting while learning complex scientific targeted threshold concepts related to unsaturated soil mechanics. The topics covered by UnSaLuDo included fundamental concepts of cohesion and adhesion in liquids, surface tension, contact angle and capillary rise; water retention and water flow in unsaturated soils; volume change, deformation, and shear strength of unsaturated soils. This paper provides background information on the motivation for the development of the board game and an overview of the topics covered. The paper also presents all the components of the board game developed and suggests approaches for how it can be incorporated into geotechnics courses.

Numerical analysis of two-dimensional unsaturated soil water flow problems with two-grid finite element methods

&lt;p style='text-indent:20px;'&gt;In this paper, we investigate a two-grid finite element method (FEM) for two-dimensional nonlinear unsaturated soil water flow problems. The application of two-grid methods can transform the solution of nonlinear problems on the fine grid to the solution of nonlinear problems on the coarse grid and the solution of linear problems on the fine grid. Based on the error results of standard FEM, we derive the continuous and discrete time error estimates in the energy norm for the two-grid method. Numerical experiments are carried out with the linear element as an example to illustrate that the two-grid method can achieve the similar optimal accuracy as standard FEM with less time as long as the two mesh sizes satisfy &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ H = O(\sqrt{h}) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;.&lt;/p&gt;

The Large-Scale Hydraulic Conductivity for Gravitational Fingering Flow in Unsaturated Homogenous Porous Media: A Review and Further Discussion

Gravitational fingering often occurs for water flow in unsaturated porous media. This paper reviews a recent effort in developing a macroscopic theory to describe the gravitational fingering flow of water in homogeneous and unsaturated soils, based on the optimality principle that water flows in unsaturated soils in such a manner that the generated flow patterns correspond to the minimum global flow resistance. The key difference between the new theory and the conventional unsaturated flow theories is that the hydraulic conductivity in the new theory is not only related to water saturation or capillary pressure, but also proportional to a power function of water flux, because the water flux is closely related to the fingering flow patterns and the power function allows for large hydraulic conductivities at locations where water fluxes are large as well to minimize the global flow resistance. The resultant relationship for the fraction of fingering flow zone is compared with that obtained from a parallel effort based on the fractal nature of fingering flow patterns. The relationships from the two efforts are found to be essentially identical for gravity-dominated water flow in unsaturated soils and can both be expressed as a power function of the water saturation. This work also demonstrates that the theoretical values for the exponent of the power function vary in a relatively narrow range between 0.75 and 0.80 for most soils, which is supported by observations from previous field tests. This remarkable finding makes it easy to apply the new theory to field sites where experimental data are not readily available for estimating the exponent value. The potential limitations of the theory and the suggested future research topics in the area are also discussed.

Investigations of Unsaturated Slopes Subjected to Rainfall Infiltration Using Numerical Approaches—A Parametric Study and Comparative Review

In the past 30 years, research on rainfall-induced landslides has grown remarkably. The contribution of matric suction to soil strength and the physics of water flow in unsaturated soils are widely accepted phenomena among researchers. However, the adoption of unsaturated soil mechanics in geotechnical engineering practice has been relatively slow, in part due to the practicality of design solutions available to the engineer. This paper conducts a parametric study on unsaturated silty slopes under a vertical steady flow rate to identify the suitable slope and hydrologic conditions to incorporate unsaturated conditions for preliminary stability analysis. Notably, the contribution of suction is most significant for silt/clay slopes with a water table located below the mid-height of the slope. For slopes with slope height ≥20 m and a fairly high water table, the slope height is a primary controlling factor of slope stability. Two case studies based on distinct failure mechanisms are presented to review the application of common geotechnical software in rainfall seepage and stability analyses of unsaturated slopes. Focus is placed on the pre-failure and failure stages of each case study. The slip surface search method, failure mode, and coupling approach integrated into each computer program caused notable differences in output results.

Forward and inverse modeling of water flow in unsaturated soils with discontinuous hydraulic conductivities using physics-informed neural networks with domain decomposition

Abstract. Modeling water flow in unsaturated soils is vital for describing various hydrological and ecological phenomena. Soil water dynamics is described by well-established physical laws (Richardson–Richards equation – RRE). Solving the RRE is difficult due to the inherent nonlinearity of the processes, and various numerical methods have been proposed to solve the issue. However, applying the methods to practical situations is very challenging because they require well-defined initial and boundary conditions. Recent advances in machine learning and the growing availability of soil moisture data provide new opportunities for addressing the lingering challenges. Specifically, physics-informed machine learning allows both the known physics and data-driven modeling to be taken advantage of. Here, we present a physics-informed neural network (PINN) method that approximates the solution to the RRE using neural networks while concurrently matching available soil moisture data. Although the ability of PINNs to solve partial differential equations, including the RRE, has been demonstrated previously, its potential applications and limitations are not fully known. This study conducted a comprehensive analysis of PINNs and carefully tested the accuracy of the solutions by comparing them with analytical solutions and accepted traditional numerical solutions. We demonstrated that the solutions by PINNs with adaptive activation functions are comparable with those by traditional methods. Furthermore, while a single neural network (NN) is adequate to represent a homogeneous soil, we showed that soil moisture dynamics in layered soils with discontinuous hydraulic conductivities are correctly simulated by PINNs with domain decomposition (using separate NNs for each unique layer). A key advantage of PINNs is the absence of the strict requirement for precisely prescribed initial and boundary conditions. In addition, unlike traditional numerical methods, PINNs provide an inverse solution without repeatedly solving the forward problem. We demonstrated the application of these advantages by successfully simulating infiltration and redistribution constrained by sparse soil moisture measurements. As a free by-product, we gain knowledge of the water flux over the entire flow domain, including the unspecified upper and bottom boundary conditions. Nevertheless, there remain challenges that require further development. Chiefly, PINNs are sensitive to the initialization of NNs and are significantly slower than traditional numerical methods.

Comparison of Seven Weibull Distribution Models for Predicting Relative Hydraulic Conductivity

AbstractModeling water flow in unsaturated soils requires accurate characterization of relative hydraulic conductivity (RHC) and water retention curve (WRC). The overall objective of this study is to investigate the performances of seven Weibull distribution models for predicting RHC using the Assouline et al. (1998), https://doi.org/10.1029/97WR03039 WRC. Specifically, a new RHC model was proposed via the approach of Assouline (2001), https://doi.org/10.1029/2000WR900254 with the assumptions of Burdine (1953), https://doi.org/10.2118/225-G and Kosugi (1999), https://doi.org/10.2136/sssaj1999.03615995006300020003x. The proposed model, together with the other six models, was then compared with data from 40 soils to explore their predictive performances for RHC. The results showed that the proposed model provides the best agreement with measured data and yields a 16.1% improvement in RHC prediction compared to the widely used Assouline et al. (1998), https://doi.org/10.1029/97WR03039‐Mualem (1976), https://doi.org/10.1029/WR012i003p00513 model. The proposed model can be used as RHC parameterization for water flow modeling in the unsaturated zone.

Review of Rain or Shine: An Introduction to Soil Physical Properties and Processes. Tyson Ochsner. Published by Oklahoma State University Libraries under the Creative Commons Attribution 4.0 International License. Version 6. Last updated 19 Jan. 2022. DOI: https://doi.org/10.17605/OSF.IO/Z4RBT

Review of <i>Rain or Shine: An Introduction to Soil Physical Properties and Processes</i>. Tyson Ochsner. Published by Oklahoma State University Libraries under the Creative Commons Attribution 4.0 International License. Version 6. Last updated 19 Jan. 2022. DOI: https://doi.org/10.17605/OSF.IO/Z4RBT

Numerical Solution of Water Flow in Unsaturated Soil with Evapotraspiration

Flow movement in unsaturated soil can be expressed by Richards equation. This equation can be obtained by applying the mass conversation law and the Darcy law. In this work, we solve one-dimensional Kirchhoff transformed Richards equation with loss of water due to the evaporation of unsaturated porous media (soils) and transpiration of plants numerically using Crank-Nicolson method. The result has compared with evapotranspiration function and without it in the governing equation. It has found that an additional work in time and flow movement is needs to reach the given boundary condition for the model without evapotranspiration.

The potential impact of biochar: Soil hydraulics and responses of maize under soil drying cycles

Biochar (BC) has been shown to positively impact soil hydraulic properties. However, its effect on water flow in unsaturated soil and root access to water have been poorly understood. This study was conducted to investigate the potential impact of BC application on soil water accessibility and plant responses under soil drying cycles.A grapevine wood-based BC was mixed with a sandy loam soil at the rates of 0, 20, and 40 g kg−1 and preincubated in the PVC pots for two months. Then, maize plants were grown in preincubated pots and regularly irrigated during their vegetative phase (64 days) to maintain soil moisture at optimum condition (0.30 cm3 cm−3). Thereafter, plants were subjected to repeated soil drying (via plant transpiration) and wetting cycles (via irrigation from the top) with soil water content varying between 0.18 and 0.30 cm3 cm−3, respectively. Some selected plant growth parameters and plant physiological responses to soil drying cycles were determined. In parallel, soil water retention and hydraulic conductivity curves of different treatments were measured by a combination of the sandbox, pressure plate apparatus, and Hyprop.The results showed that plant dry shoot biomass was slightly increased by the BC application (9% on average). In contrast, the leaf area and root volume did not show any significant difference. As soil dried during a soil drying cycle, the transpiration rate decreased among all treatments. However, the drop in transpiration rate was smaller in the BC-amended soils than in the control. At soil water content of 018 cm3 cm−3, plants grown in BC-amended soils had on average 83% and 164% higher transpiration and photosynthesis rate than plants grown in control soil, respectively. These comparisons suggest that BC improved water accessibility to plants during soil drying, resulting in 76 and 58% increase in grain yield and grain water productivity. The application of BC had little effect on soil water retention in the wet range and had no impacts in the drier range, while the unsaturated hydraulic conductivity (Kh) and calculated matric flux potential (φ) were positively impacted in a wide range of soil water content.The findings of this study revealed that BC mixed with sandy loam soil could prevent a big drop in Kh as soil dries, thereby improving root access to water and transpiration rate under drought stress.

Transition Between Constitutive Equations and the Mechanics of Water Flow in Unsaturated Soil: Numerical Simulations

Richards’ partial differential equation normally governs water flows in soil. It can be utilized to investigate water infiltration in the soil. A case study of the Haverkamp’s water infiltration simulation into Yolo Light Clay was carried out. A common practice to change between the hydraulic functions (from Haverkamp to van Genuchten) could cause a discrepancy in the simulation results. Hence, the first objective is to modify van Genuchten’s equations to reproduce the water infiltration result that was obtained using the Haverkamp constitutive functions. The method used was able to recreate water infiltration by increasing the fitting parameters of the van Genuchten constitutive functions. The second objective is to identify the mechanism governing the flow of water. The soil water diffusivity and the hydraulic conductivity were responsible for the flux of water in the soil. The Richards’ equation was discretized using a finite difference method, and the algebraic solution was coded into Simply Fortran 2008.

Richards’ Equation: Transition Between Constitutive Equations and the Mechanics of Water Flow in Unsaturated Soil

Richards’ Equation: Transition Between Constitutive Equations and the Mechanics of Water Flow in Unsaturated Soil