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This research focuses on elucidating a system of partial differential equations designed to model the process of water uptake by a single root within an unsaturated soil. This model extends a unidimensional root scenario by incorporating radial and axial hydraulic conductance in the root structure. The flow dynamics in the rhizosphere are represented by the Richards equation, while a diffusion equation is employed to characterize water movement within the root. The resulting system forms a strongly coupled, nonlinear parabolic-elliptic system. To establish the existence and uniqueness of a solution, we utilize the technique of semi-discretization and the Schauder-Tychonoff fixed point theorem. Additionally, we approximate the system using a mixed finite element method, and the subsequent numerical tests presented align well with the model predictions. This comprehensive approach contributes to our understanding of the intricate dynamics involved in root water uptake and provides a robust framework for further exploration in this field. [ABSTRACT FROM AUTHOR]

Modeling unsaturated flow through soils with water uptake by plant root has many applications in agriculture and water resources management. In this study, our aim is to develop efficient numerical techniques for solving the Richards equation with a sink term due to plant root water uptake. The Feddes model is used for water absorption by plant roots, and the van-Genuchten model is employed for capillary pressure. We introduce a numerical approach that combines the localized exponential radial basis function (EXP-RBF) method for space and the second-order backward differentiation formula (BDF2) for temporal discretization. The localized RBF methods eliminate the need for mesh generation and avoid ill-conditioning problems. This approach yields a sparse matrix for the global system, optimizing memory usage and computational time. The proposed implicit EXP-RBF techniques have advantages in terms of accuracy and computational efficiency thanks to the use of BDF2 and the localized RBF method. Modified Picards iteration method for the mixed form of the Richards equation is employed to linearize the system. Various numerical experiments are conducted to validate the proposed numerical model of infiltration with plant root water absorption. The obtained results conclusively demonstrate the effectiveness of the proposed numerical model in accurately predicting soil moisture dynamics under water uptake by plant roots. The proposed numerical techniques can be incorporated in the numerical models where unsaturated flows and water uptake by plant roots are involved such as in hydrology, agriculture, and water management. • We propose implicit EXP-RBF techniques for infiltration in soils with water uptake by plant roots. • Localized exponential radial basis function method is used in space. • BDF2 is used for temporal discretization. • The proposed techniques have advantages in terms of accuracy and computational efficiency. • The numerical model is accurate in predicting soil moisture under water uptake by plant roots. [ABSTRACT FROM AUTHOR]

The North China Plain is a crucial agricultural region in China, but irregular precipitation patterns have led to significant water shortages. To address this, analyzing the high-resolution dynamics of root-zone soil moisture transport is essential for optimizing irrigation strategies and improving water resource efficiency. The Richards equation is a robust model for describing soil moisture transport dynamics across multiple soil layers, yet its application at large spatial scales is hindered by its sensitivity to boundary conditions and model parameters. This study introduces a novel approach that, for the first time, employs a continuous time series of near-surface soil moisture as the upper boundary condition in the Richards equation to estimate high-resolution root-zone soil moisture in the North China Plain, thus enabling its large-scale application. Singular spectrum analysis (SSA) was first applied to reconstruct site-specific time series, filling in missing and singular values. Leveraging observational data from 617 monitoring sites across the North China Plain and multiple spatial covariates, we developed a machine learning model to estimate near-surface soil moisture at a 1 km resolution. This high-resolution, continuous near-surface soil moisture series then served as the upper boundary condition for the Richards equation, facilitating the estimation of root-zone soil moisture across the region. The results indicated that the machine learning model achieved a correlation coefficient (R) of 0.92 for estimating spatial near-surface soil moisture. Analysis of spatial covariates showed that atmospheric forcing factors, particularly temperature and evaporation, had the most substantial impact on model performance, followed by static factors such as latitude, longitude, and soil texture. With a continuous time series of near-surface soil moisture, the Richards equation method accurately predicted multi-layer soil moisture and demonstrated its applicability for large-scale spatial use. The model yielded R values of 0.97, 0.78, 0.618, and 0.43, with RMSEs of 0.024, 0.06, 0.08, and 0.11, respectively, for soil layers at depths of 10 cm, 20 cm, 40 cm, and 100 cm across the North China Plain. [ABSTRACT FROM AUTHOR]

We present a higher-order space-time adaptive method for the numerical solution of the Richards equation that describes a flow motion through variably saturated media. The discretization is based on the space-time discontinuous Galerkin method, which provides high stability and accuracy and can naturally handle varying meshes. We derive reliable and efficient a posteriori error estimates in the residual-based norm. The estimates use well-balanced spatial and temporal flux reconstructions which are constructed locally over space-time elements or space-time patches. The accuracy of the estimates is verified by numerical experiments. Moreover, we develop the hp-adaptive method and demonstrate its efficiency and usefulness on a practically relevant example. [ABSTRACT FROM AUTHOR]

In this study, we investigate numerical simulation models for water flow in variably saturated (unsaturated) soils. These models are crucial for addressing soil-related challenges and analyzing water-related risks, particularly in the context of water resource management, soil water-induced disasters, and the agricultural impacts of global environmental changes. The Richards equation is one of the most widely used models for simulating water flow in porous media, especially in unsaturated soils. However, as a highly nonlinear parabolic partial differential equation (PDE), it has limited analytic solutions, which often lack precision in practical scenarios. This necessitates the development of innovative and robust numerical methods for accurate simulations. We introduce a numerical procedure that linearizes the Richards equation using the Kirchhoff integral transformation, followed by discretization with various time-stepping schemes. This approach enables efficient and accurate modeling of water flow. To extend its application, we integrate the numerical solution with a hydrological infinite slope stability model to evaluate landslide hazards. Specifically, we calculate the factor of safety index based on an axisymmetric form of the Richards equation, which helps identify potential landslidenprone areas. Furthermore, our model provides a framework for predicting landslides by considering the interplay between water flow and the physical, geological, and topographical characteristics of a landscape. This integrated approach offers valuable insights for geohazard assessment and the mitigation of water-induced risks

Evapotranspiration (ET) is an important part of water cycle in nature, and the estimation of evapotranspiration on spatio-temporal scale has always been a hot issue. Remote sensing can estimate evapotranspiration on regional scale, but it is difficult to obtain evapotranspiration in continuous time series due to the limitation of satellite transit time. Soil moisture is an important controlling factor of evapotranspiration. Improving the remote sensing evapotranspiration model by combining soil moisture data is of great significance in improving the accuracy of remote sensing evapotranspiration estimation. However, most remote sensing methods give limited consideration to the characterization of soil moisture stress. This study used the evapotranspiration calculated by the vorticity correlation method as the actual evapotranspiration. combining with the single crop coefficient method recommended by FAO, the soil water content information was introduced into the Penman-Monteith formula to calculate the actual evapotranspiration. Based on Richards equation, the one-dimensional vertical soil water movement process under evaporation conditions was simulated to estimate the continuous daily evapotranspiration under soil water stress. Combining with remote sensing data, the regional scale evapotranspiration was estimated. The results show that the actual daily evapotranspiration calculated by the vorticity correlation method has a strong correlation with the potential daily evapotranspiration calculated by P-M formula, with the correlation coefficient of 0.918. With the introduction of soil water content information, the P-M formula improves the estimation accuracy of daily evapotranspiration significantly, and the RMSE reaches 0.133 mm/d. The estimated daily evapotranspiration under soil water stress based on Richards equation is close to the measured value, with the RMSE of 0.288 mm/d. The high value of daily evapotranspiration affected by the topography of the study area is concentrated in the water area and cultivated land area in the middle of the study area. The average daily evapotranspiration under different soil use types is water area > cultivated land > woodland > grassland > unused land, and the results on the regional scale show similar change with that measured in the the station in time series. This study provides basic information for understanding the influence mechanism of soil moisture on evapotranspiration and estimating regional evapotranspiration.

The purpose of this article is the mathematical analysis of a new class of models to describe the flow in shallow aquifers, as alternatives to the 3d-Richards model. This type of models was introduced in a previous work and consists of the coupling of an almost 1d vertical flow in the upper part of the aquifer with a 2d horizontal flow in the lower part. These two regions being separated by a time-dependent interface, an unknown of the problem. A result of existence of weak solutions is proved for a very general form of the hydraulic conductivity (anisotropic case). The strategy is based on the classical framework of parabolic equations in non-cylindrical domains. It also exploits the compressibility of the fluid to overcome the difficulty associated with the degeneracy in the time derivative term of Richards equation. [ABSTRACT FROM AUTHOR]

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Furrow infiltration interference results from the merging of the wetting plumes of neighboring furrows and reduces infiltration rates. Simulation models and computational procedures used to analyze furrow irrigation flows currently ignore this effect. An understanding of this process and its impact on infiltration is needed, and infiltration modeling approaches must be modified to account for interference effects, if those effects are substantial. A simulation study was conducted to characterize the transverse spread of furrow infiltration plumes subject to interference (i.e., constrained plumes) relative to unconstrained ones and its impact on infiltration rates. Constrained and unconstrained plumes were computed for the same furrow geometry, soil hydraulic properties, initial conditions, and boundary conditions and different combinations of these variables were tested. A relatively wide computational domain was used to compute the unconstrained plumes, whereas a relative narrow domain was used for the constrained plumes. Moment analysis was used to evaluate spread and fraction of infiltrated water contained within elliptical different regions of the plume. For any set of conditions, when a plume ceases to expand horizontally, its final horizontal standard deviation is approximately equal to the semiwidth of the constraining furrow divided by 1.7. This constant, and the resulting standard deviation, define an ellipse of an unconstrained infiltration plume that, at some point in time, matches the width of the constraining furrow. The onset of interference, and the resulting decrease in infiltration rate, can be predicted from the evolution of the standard deviation of the unfettered plume relative to the ultimate standard deviation of the constrained plume. These concepts were used to modify the lateral flow component of an existing semiphysical furrow infiltration model. The modified model predicts infiltration with reasonable accuracy in comparison with solutions computed with the two-dimensional Richards equation. Results suggest conditions under which interference is likely. [ABSTRACT FROM AUTHOR]

【Objective】The differences in grain filling and fruiting performance, quality traits and their correlation in different parts of adlay were explored, which provided reference for optimizing the yield performance of strong and weak adlay and breeding high-quality and high-yield cultivation varieties. 【Method】Using the single factor random block design and Richards equation, this study investigated the differences in correlation indicators such as grain dry weight, filling rate, grain plumpness and starch content in different parts (upper, middle, lower 1, lower 2, and lower 3) of different adlay varieties〔Guiyi No.1（GY.1) and Shizongheike (SZHK) 〕, as well as the correlation between filling and fruiting characteristics and filling parameters and quality traits.【Result】The grain filling equation was W＝78.76/(1+379259e0.27t) 1/7.91, the maximum grain filling rate was 1.82 mg/d, with an average of 1.12 mg/d. The active growth period was 73.61 days, with an average seed setting rate of 74.55% and an average filling degree of 76.84%. The 100-grain weight and the total grain weight per stem are 7.70 g and 13.49 g, respectively. The dry weight of grains gradually increased with the number of days after flowering, reaching its maximum value on the 49th day. The filling rate shows a trend of first increasing and then decreasing with the number of days after flowering, reaching its maximum value on the 31st day after flowering. As for the average grain filling rate, Lower 1 (1.26 mg/d) was significantly higher than other parts, and 21.10% higher than the Lower 3 (1.04 mg/d) ; The maximum grouting rate was also shown to be the highest in Lower 1 (2.03 mg/d), which was 18.70% higher than Lower 3 (1.71 mg/d) ; The overall performance of 100-grain weight and fullness in each part was as follows: Lower 1＞upper＞middle＞Lower 2＞Lower 3. Among the 100 grains weight, Lower 1 (8.37g) was significantly higher than other parts, with 12.95% and 14.81% higher than Lower 2 (7.41g) and Lower 3 (7.29g), respectively. In terms of grain plumpness, Lower 1 (80.23%) was 10.78% higher than Lower 3 (72.42%) ; The starch content of the first seed of adlay (38.10%) was 12.62% and 15.07% higher than that of Lower 2 (33.83%) and Lower 3 (33.11%), respectively. The correlation analysis showed that the 100-grain weight were highly significantly positively correlated with the average grain filling rate (0.78), maximum grain filling rate (0.89), starch content (0.95) . The starch accumulation rate (0.88) were highly significantly positively correlated with the plumpness of grains (0.87), and highly significantly negatively correlated with soluble sugar content (－0.93) .【Conclusion】The Lower 1 grain is superior to other parts in terms of grain weight, plumpness and starch content. It is preliminarily determined that the Lower 1 is a strong grain of adlay, while the Lower 2 and Lower 3 are weak grains. [ABSTRACT FROM AUTHOR]

The Richards equation is commonly used to model the flow of water and air through soil, and it serves as a gateway equation for multiphase flows through porous media. It is a nonlinear advection–reaction–diffusion equation that exhibits both parabolic–hyperbolic and parabolic–elliptic kind of degeneracies. In this study, we provide reliable, fully computable, and locally space–time efficient a posteriori error bounds for numerical approximations of the fully degenerate Richards equation. For showing global reliability, a nonlocal-in-time error estimate is derived individually for the time-integrated H^1(H^{-1}), L^2(L^2), and the L^2(H^1) errors. A maximum principle and a degeneracy estimator are employed for the last one. Global and local space–time efficiency error bounds are then obtained in a standard H^1(H^{-1})\cap L^2(H^1) norm. The reliability and efficiency norms employed coincide when there is no nonlinearity. Moreover, error contributors such as space discretization, time discretization, quadrature, linearization, and data oscillation are identified and separated. The estimates are also valid in a setting where iterative linearization with inexact solvers is considered. Numerical tests are conducted for nondegenerate and degenerate cases having exact solutions, as well as for a realistic case and a benchmark case. It is shown that the estimators correctly identify the errors up to a factor of the order of unity. [ABSTRACT FROM AUTHOR]

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.

We consider here a cell‐centered finite difference approximation of the Richards equation in three dimensions, averaging for interface values the hydraulic conductivity K=K(p)$$ K=K(p) $$, a highly nonlinear function, by arithmetic, upstream and harmonic means. The nonlinearities in the equation can lead to changes in soil conductivity over several orders of magnitude and discretizations with respect to space variables often produce stiff systems of differential equations. A fully implicit time discretization is provided by backward Euler one‐step formula; the resulting nonlinear algebraic system is solved by an inexact Newton Armijo–Goldstein algorithm, requiring the solution of a sequence of linear systems involving Jacobian matrices. We prove some new results concerning the distribution of the Jacobians eigenvalues and the explicit expression of their entries. Moreover, we explore some connections between the saturation of the soil and the ill conditioning of the Jacobians. The information on eigenvalues justifies the effectiveness of some preconditioner approaches which are widely used in the solution of Richards equation. We also propose a new software framework to experiment with scalable and robust preconditioners suitable for efficient parallel simulations at very large scales. Performance results on a literature test case show that our framework is very promising in the advance toward realistic simulations at extreme scale. [ABSTRACT FROM AUTHOR]

The paper studies the accuracy of modeling moisture transport under the conditions of sprinkler irrigation using evapotranspiration assessment methods that take into account the soil moisture conditions. Appropriate modifications of the Penman-Monteith and the Priestley-Taylor models are considered. Moisture transport modeling is performed using the Richards equation in its integer- and fractional-order forms. Parameters identification is performed by the particle swarm optimization algorithm based on the readings of suction pressure sensors. Results for the two periods of 11 and 50 days demonstrate the possibility of up to ~20% increase in the simulation accuracy by using a modified Priestley-Taylor model when the maintained range of moisture content in the root layer is 70%-100% of field capacity. When irrigation maintained the range of 80%-100% of field capacity, moisture content consideration within evapotranspiration assessment models did not enhance simulation accuracy. This confirms the independence of evapotranspiration from soil moisture content at its levels above 80% of field capacity as in this case actual evapotranspiration reaches a level close to the potential one. Scenario modeling of the entire growing season with the subsequent estimation of crop (maize) yield showed that irrigation regimes generated using evapotranspiration models, which take into account soil moisture data, potentially provide higher yields at lower water supply.

The predictor-corrector-type (P-C) numerical solution to the 1D Richards equation only requires one matrix inversion operation per time step, making it attractive in terms of computational cost. However, the mass conservation could be violated at the saturated-unsaturated interface. A new post-allocation procedure is designed for the P-C method, which redistributes moisture after the corrector step to achieve strict mass balance. A novel adaptive time-stepping strategy is proposed to further improve model efficiency and robustness. It adjusts time step size based on both moisture difference and the Courant number. The proposed solution method and time control strategies are tested and compared with an analytical solution, the previous P-C solution and other existing iterative solutions. The new method shows good conservation property and good agreements to the existing solutions. Compared to the iterative methods that occasionally experience convergence issues, the proposed P-C method is more robust. The new time-control strategy improves computational efficiency compared to the original P-C method, but it remains less efficient than iterative methods for most of the tested scenarios because of its explicit treatment of the corrector step.

The predictor-corrector-type (P-C) numerical solution to the 1D Richards equation only requires one matrix inversion operation per time step, making it attractive in terms of computational cost. However, the mass conservation could be violated at the saturated-unsaturated interface. A new post-allocation procedure is designed for the P-C method, which redistributes moisture after the corrector step to achieve strict mass balance. A novel adaptive time-stepping strategy is proposed to further improve model efficiency and robustness. It adjusts time step size based on both moisture difference and the Courant number. The proposed solution method and time control strategies are tested and compared with an analytical solution, the previous P-C solution and other existing iterative solutions. The new method shows good conservation property and good agreements to the existing solutions. Compared to the iterative methods that occasionally experience convergence issues, the proposed P-C method is more robust. The new time-control strategy improves computational efficiency compared to the original P-C method, but it remains less efficient than iterative methods for most of the tested scenarios because of its explicit treatment of the corrector step.

Flow in variably saturated porous media is typically modeled by the Richards equation, a nonlinear elliptic-parabolic equation which is notoriously challenging to solve numerically. In this paper, we propose a robust and fast iterative solver for Richards' equation. The solver relies on an adaptive switching algorithm, based on rigorously derived a posteriori indicators, between two linearization methods: L-scheme and Newton. Although a combined L-scheme/Newton strategy was introduced previously in [1] , here, for the first time we propose a reliable and robust criteria for switching between these schemes. The performance of the solver, which can be in principle applied to any spatial discretization and linearization methods, is illustrated through several numerical examples. [ABSTRACT FROM AUTHOR]

Spontaneous imbibition of water into ceramic tile is modeled using the well-known Richards equation (RE) employed for modeling unsaturated flow in the vadose zone. The Brooks and Corey model for relative permeability and Van Genuchten model for capillary pressure is employed to generate the final form of RE that predicts moisture migration in porous ceramic. COMSOL is employed to numerically solve RE in a cuboid, block like ceramic geometry. The simulation results, obtained using the mesh optimized through a grid independence test, are validated in three different ways. First, the simulation predictions are compared with the experimental results obtained from a neutron microscope imaging study of wetting of a ceramic block. Good qualitative and quantitative matches were obtained. Later, the simulation predictions for a 1-D moisture migration situation were compared with two different published analytical solutions, and once again, good agreements were observed. After the code validation, simulations for moisture migration through evolving saturation plots were carried for three different cases of the ceramic-block top getting wetted by falling drops: a single droplet, two droplets with a time gap, and continuous droplets. Evolution of saturation plots were studied where the third case was observed to cause the maximum water infiltration into ceramic. [ABSTRACT FROM AUTHOR]

In the present analysis, the natural transform decomposition and variational iteration transform methods have been employed to find an approximate analytical solution of the fractional order of Richards' equation. Some standard cases of Richards' equation have been discussed as an example to illustrate the high accuracy and reliability of proposed methods. The result obtained from the proposed method is very close to the exact solution of the problem. It is concluded that natural transform decomposition and variational iteration transform methods are better alternatives to some standard existing methods to solve some realistic problems arising in science and technology. [ABSTRACT FROM AUTHOR]

Earthen embankments are subjected to increasing threats because of climate change inducing sequences of severe drought periods followed by floods, possibly leading to overtopping of the structures. Consequently, the water saturation of the dike can vary significantly both in space and time, and the resulting groundwater flow can affect the free-surface flow in case of overtopping. Conversely, the free-surface flow can modify the pore water content, which controls erosion and slope instabilities. In this paper, a combined approach to such situations is presented, in which the degree of saturation and the flow through the embankment are simulated by solving the two-dimensional Richards equation on an unstructured mesh with an implicit finite volume scheme that is coupled to the system of shallow-water equations solved in one dimension using an explicit finite-volume scheme. The coupled model is validated on several situations of flows through and over earthen embankments with different constitutive materials. [ABSTRACT FROM AUTHOR]

Ponding at the soil surface exerts profound impacts on infiltration. However, the effects of ponding depth on infiltration, especially the development of a saturated zone below the soil surface, have yet to be considered in present infiltration models. A new general Green‐Ampt model solution (GAMS) was derived for a one‐dimensional vertical infiltration problem under a uniform initial moisture distribution with ponding on its surface. An expression was included in the new solution for simulating the saturated layer developed below the soil surface as long as the pressure head at the surface is sufficiently high to saturate the soil. The GAMS simulates the infiltration processes closer to the numerical solution by HYDRUS‐1D than the traditional and the recently improved Green‐Ampt model. Moreover, an inversion method to improve the estimates of soil hydraulic parameters from one‐dimensional vertical infiltration experiments that is based on the GAMS was suggested. The effect of ponding depth (hp), initial soil moisture content, soil texture, and hydraulic soil properties (saturated hydraulic conductivity Ks, water‐entry suction hd and shape coefficient n of soil water retention curve) in the saturated zone was also evaluated. The results indicate that the saturated zone length increased at a comparable rate with the unsaturated wetted zone length during infiltration. Generally, a larger saturated zone was found for soils with higher initial soil moisture contents, coarser texture, higher Ks values, greater n, and lower −hd. Our findings reveal that including the saturated zone in the infiltration model yields a better estimate of the soil hydraulic parameters. The proposed GAMS model can improve irrigation design and rainfall‐runoff simulations. Key Points: An implicit expression was proposed for the development of saturated zone during infiltrationThe newly proposed formula improved the simulation accuracy in all stages of ponded infiltrationThe new approach eliminated the time‐dependency of the estimated soil hydraulic properties [ABSTRACT FROM AUTHOR]

The authors propose an algorithm for finding the optimal source capacity for the two-dimensional quasi-linear Richards equation in a rectangular domain. The Kirchhoff transformation with scaling of coordinates and capacities of buried sources is used, which allows formulating a dimensionless problem. The existence of a solution to the problem of optimizing moisture transfer in an unsaturated porous medium is substantiated. The task of this study is to find the capacity of sources buried in a porous medium, such that the humidity distribution at the final instant of time is close to the given indicators or the objective function. The numerical solution approximates the optimal values of the sources. [ABSTRACT FROM AUTHOR]

Srivastava and Yeh (1991, https://doi.org/10.1029/90WR02772) derived an exact solution to the linearized Richards equation (LRE) for two‐layer medium infiltration using the Laplace transform (LT) method with a particular initial condition assumed, making the most pioneering contribution to the derivation of exact solutions to the layered‐medium LRE (i.e., ES‐LMLREs). However, the LT method is unsuitable for deriving an ES‐LMLRE that considers either an arbitrary initial condition or an arbitrary number of layers, or both, preventing further progress in developing ES‐LMLREs. Adopting a new solution strategy, namely a conjunctive use of the variable separation method and the transfer matrix method, we develop a novel exact layered‐medium‐LRE infiltration solution, overcoming the above difficulties. First, the proposed solution is successfully validated against the Srivastava‐Yeh solution. As a feature‐demonstration example, a layered‐medium water absorption process is simulated, and our solution well captures how the heterogeneity of hydraulic parameters affects the dynamics of this process. Moreover, the proposed solution is a valuable benchmark for related numerical models. Key Points: An exact solution for transient infiltration in layered media that enjoys flexibility in initial condition and layer number is presentedThe new analytical solution can analyze the moisture absorption process from the water table in initially dry layered soilsThe proposed solution can serve as a valuable benchmark for related numerical models [ABSTRACT FROM AUTHOR]

An algorithm for finding the optimal source power for the two-dimensional quasi-linear Richards equation for a rectangular region is proposed. The Kirchhoff transformation with scaling of coordinates and powers of buried sources is used, which allows formulating adimensionless problem. The existence of the solution for the problem of optimizing moisturetransfer in an unsaturated porous medium is substantiated. The task of this study is to find thepower of sources buried in a porous medium — such that at the final moment of time thedistribution of humidity will be close to the given indicators or the objective function. Thenumerical solution leads to an approximation of the optimal мalues of the sources. [ABSTRACT FROM AUTHOR]

The computational cost of approximating the Richards equation for water flow in unsaturated porous media is a major challenge, especially for tasks that require repetitive simulations. Data-driven modeling offers a faster and more efficient way to estimate soil moisture dynamics, significantly reducing computational costs. Typically, data-driven models use one-dimensional vectors to represent soil moisture at specific points or as a time series. However, an alternative approach is to use images that capture the distribution of porous media characteristics as input, allowing for the estimation of the two-dimensional soil moisture distribution using a single model. This approach, known as image-to-image regression, provides a more explicit consideration of heterogeneity in the porous domain but faces challenges due to increased input–output dimensionality. Deep neural networks (DNNs) provide a solution to tackle the challenge of high dimensionality. Particularly, encoder–decoder convolutional neural networks (ED-CNNs) are highly suitable for addressing this problem. In this study, we aim to assess the precision of ED-CNNs in predicting soil moisture distribution based on porous media characteristics and also investigate their effectiveness as an optimizer for inverse modeling. The study introduces several novelties, including the application of ED-CNNs to forward and inverse modeling of water flow in unsaturated porous media, performance evaluation using numerical model-generated and laboratory experimental data, and the incorporation of image stacking to account for transient moisture distribution. A drainage experiment conducted on a sandbox flow tank filled with monodisperse quartz sand was employed as the test case. Monte Carlo simulation with a numerical model was employed to generate data for training and validation of the ED-CNN. Additionally, the ED-CNN optimizer was validated using images obtained through non-intrusive photographic imaging. The results show that the developed ED-CNN model provides accurate approximations, addressing the high-dimensionality problem of image-to-image regression. The data-driven model predicted soil moisture with an R2 score of over 91%, while the ED-CNN optimizer achieved an R2 score of over 89%. The study highlights the potential of ED-CNNs as reliable and efficient tools for both forward and inverse modeling in the analysis of unsaturated flow. [ABSTRACT FROM AUTHOR]

This paper focuses on the application of an efficient technique, namely, the fractional natural decomposition method (FNDM). The numerical solutions of the model containing the water transport in unsaturated porous media, called Richards equation, are extracted. This model is used to describe the non-locality behaviors which cannot be modeled under the framework of classical calculus. To demonstrate the effectiveness and efficiency of the scheme used, two cases with time-fractional problems are considered in detail. The numerical stimulation is presented with results accessible in the literature, and corresponding consequences are captured with different values of parameters of fractional order. The attained consequences confirm that the projected algorithm is easy to implement and very effective to examine the behavior of nonlinear models. The reliable algorithm applied in this paper can be used to generate easily computable solutions for the considered problems in the form of rapidly convergent series. [ABSTRACT FROM AUTHOR]

The paper studies the accuracy of modeling moisture transport under the conditions of sprinkler irrigation using evapotranspiration assessment methods that take into account the soil moisture conditions. Appropriate modifications of the PenmanMonteith and the Priestley-Taylor models are considered. Moisture transport modeling is performed using the Richards equation in its integer- and fractionalorder forms. Parameters identification is performed by the particle swarm optimization algorithm based on the readings of suction pressure sensors. Results for the two periods of 11 and 50 days demonstrate the possibility of up to ~20% increase in the simulation accuracy by using a modified Priestley-Taylor model when the maintained range of moisture content in the root layer is 70%-100% of field capacity. When irrigation maintained the range of 80%-100% of field capacity, moisture content consideration within evapotranspiration assessment models did not enhance simulation accuracy. This confirms the independence of evapotranspiration from soil moisture content at its levels above 80% of field capacity as in this case actual evapotranspiration reaches a level close to the potential one. Scenario modeling of the entire growing season with the subsequent estimation of crop (maize) yield showed that irrigation regimes generated using evapotranspiration models, which take into account soil moisture data, potentially provide higher yields at lower water supply. [ABSTRACT FROM AUTHOR]

The Richards equation is a nonlinear degenerate advection diffusion equation that models flow in saturated/unsaturated porous media, it's crucially important for prediction of disasters when heavy rain attacks. Efficient and precise linearized numerical schemes are necessary, but there is few study related it, and the numerical theory is incomplete because of the degeneracy and strong nonlinearity. In this paper, we establish a linearized Crank–Nicolson finite difference scheme which is a three-level scheme with almost second-order accuracy. In stability analysis, we develop a creative technique to overcome the degeneracy by adding a small positive perturbation ϵ. We also propose the error estimates by applying Young's inequality and prove the convergence order is approximate to second-order. Numerical examples are also provided to verify our main results and show the relationship between the computational error and ϵ is linear. [ABSTRACT FROM AUTHOR]

An analytical study of the problem of water infiltration into a homogeneous unsaturated soil showed that, in the case of periodical water recharge through the soil surface, the character of the flow will tend to uniform with the depth. The stabilization of the flow was found to be due to the effect of two factors: capillary dissipation and nonlinearity. The role of each factor was studied by constructing appropriate exact solutions. An estimate was proposed for the depth at which flow variations become insignificant. This estimate takes into account the joint effect of both these factors. An explicit expression for it contains the hydraulic characteristics of the soil and the main characteristics of the surface recharge regime. The pumping effect was also studied and it was showed that, at some assumptions regarding the soil hydraulic characteristics, it manifests itself in that the time-averaged water content in the top soil layers is less than that at greater depth. [ABSTRACT FROM AUTHOR]

In hydrology and soil sciences, infiltration is the process by which water on the ground surface enters the soil, and is described mathematically by Richard's equation. The present paper applies the reduced differential transform method to find the approximate analytical solution of Richard’s equation describing infiltration phenomena in porous media. Some standard cases of Richard’s equation are discussed to demonstrate the effectiveness and reliability of the method. Comparing approximate analytical solutions obtained by RDTM with exact solutions shows that the proposed method is reliable and accurate and can be applied for solving practical scientific and technological problems. The results obtained are also compared with the analytical solution obtained by some well-known methods available in the literature. The proposed approach does not need any linearization, discretization, or perturbation parameters to obtain the solution for non-linear PDE, and its direct applicability reduces numerical computation. Convergence analysis and error estimation of the approximate solution of Richard’s equation is also addressed in this research.

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The viscous flow of two immiscible fluids in a porous medium on the Darcy scale is governed by a system of nonlinear parabolic equations. If infinite mobility of one phase can be assumed (e.g., in soil layers in contact with the atmosphere) the system can be substituted by the scalar Richards model. Thus, the porous medium domain may be partitioned into disjoint subdomains where either the full two‐phase or the simplified Richards model dynamics are valid. Extending the previously considered one‐model situations we suggest coupling conditions for this hybrid model approach. Based on an Euler implicit discretization, a linear iterative (L‐type) domain decomposition scheme is proposed, and proved to be convergent. The theoretical findings are verified by a comparative numerical study that in particular confirms the efficiency of the hybrid ansatz as compared to full two‐phase model computations. [ABSTRACT FROM AUTHOR]