Van Genuchten-Mualem Model Research Articles

Inverse analysis of soil hydraulic parameters of layered soil profiles using physics‐informed neural networks with unsaturated water flow models

AbstractInformation about the spatial distribution of soil hydraulic parameters is necessary for the accurate prediction of soil water flow and the coupled movement of chemicals and heat at the field scale using a process‐based model. Physics‐informed neural networks (PINNs), which can provide physical constraints in deep learning to obtain a mesh‐free solution, can be used to inversely estimate soil hydraulic parameters from less and noisy training data. Previous studies using PINNs have successfully estimated soil hydraulic parameters for homogeneous soil but estimating such parameters of layered soil profiles where the interface depth and the parameters are unknown still has some difficulties. The objective of this study was to develop PINNs to inversely estimate the distribution of soil hydraulic parameters, such as saturated hydraulic conductivity and α and n of the Mualem–van Genuchten model directly within layered soil profiles by predicting changes in the pressure head from training data based on simulation results at given depths during infiltration. The impact of factors affecting PINNs performance, such as the weights assigned to each component of the loss function, time range used in error computations, and number of samples used to assess the physical constraint, was investigated. By assigning a larger weight to the physical constraint and excluding the earlier stage of infiltration in the loss function, the changes in the pressure head and the three soil hydraulic parameter distributions within the layered soil were successfully estimated. The developed PINNs can be further applied to more complex soils and can be improved.

Inverse modeling of frequency domain-based one-dimensional soil water flow in layered soils

Soils are heterogeneous at multiple scales. Estimates of spatially varying saturated hydraulic conductivity (Ks) of soils dictate the resolution of modeling water flux based on the Richardson-Richards equation (RRE), but direct measurements of Ks everywhere in a field are practically impossible. As a result, this study develops an inverse approach to estimate spatially varying Ks utilizing soil moisture observations driven by periodic weather variability at different frequencies. Using a numerical solution to vertical, one-dimensional RRE based on Gardner-Kozeny (GK) model in a heterogeneous soil with periodic flux, we investigate the spatial cross-correlation between Ks and observations (the amplitude or phase shift) of soil moisture fluctuation at different frequencies without high computational cost in the conventional time-domain model. These correlation patterns vary with the spatial location and frequency, suggesting that soil moisture fluctuations at different frequencies at one monitoring location carry non-redundant information about the heterogeneous distribution of Ks. Guided by cross-correlation maps, several inverse experiments reveal the effectiveness of multi-frequency data of soil moisture fluctuation for mapping the heterogeneous distribution of Ks. Monte Carlo simulations suggest that multi-frequency data can provide non-redundant and useful information about the estimation of heterogeneous Ks. Synthetic data based on a commonly-applied nonlinear van Genuchten-Mualem (VG) model is then tested in our inversion algorithm. The results indicate that the spatial pattern of the inverted GK Ks is very similar to that of the VG Ks, although these two parameters are inherently not identical since they are used in two different models. In addition, the inverted GK Ks can successfully predict the soil moisture fluctuation of frequencies that were not used in the inversion. Despite having no field experiments to validate this model, we believe this is the first attempt to conduct frequency-domain inverse modeling in vadose zone hydrology using soil moisture fluctuation.

Implementation of the Brunswick model system into the Hydrus software suite

AbstractThe Brunswick modular framework for modeling unsaturated soil hydraulic properties (SHP) over the full moisture range was implemented in the Hydrus suite. Users can now additionally choose between four different variants of the Brunswick model: (i) van Genuchten–Mualem (VGM), (ii) Brooks–Corey, (iii) Kosugi, and (iv) modified van Genuchten. For demonstration purposes, simulation results for two different setups, (i) bare soil evaporation and (ii) root water uptake, are presented, along with a comparison of the original VGM model and its Brunswick variant. Results show that the original VGM model underestimates the simulated cumulative evaporation and cumulative transpiration due to the inconsistent representation of the SHP in the dry moisture range. We also implemented a two‐step hydro‐PTF (pedotransfer function) into the Hydrus suite that converts the parameters of the original VGM model (from Rosetta) to the corresponding Brunswick variant. In that way, physically comprehensive simulations are ensured if no data on SHP are directly available, but information on physical soil properties (e.g., texture and bulk density) exists.

Estimating soil hydraulic properties from oven-dry to full saturation using shortwave infrared imaging and inverse modeling

To minimize uncertainty related to soil processes in extreme events, we need accurate soil hydraulic properties across the entire range of soil water content. However, conventional methods are time-consuming and limited to specific ranges. To estimate soil hydraulic properties throughout the entire range, we conducted inverse modeling using upward infiltration experiments, where a shortwave infrared imaging camera was used to obtain high-resolution soil moisture data in space and time. Because the commonly used van Genuchten–Mualemmodel is unsuitable for describing soil hydraulic properties for dry conditions, we tested an alternative model, the Peters-Durner-Iden model, which considers both capillary and film water. The inverse modeling successfully estimated soil hydraulic properties for sandy loam and loam soils, and we demonstrated that the Peters-Durner-Iden model captured soil moisture dynamics better than the van Genuchten–Mualemmodel for dry conditions. However, both models could not adequately describe the soil moisture data for the other soils. The direct observation of the water flow via shortwave infrared images clarified that the reduced success was because of violating the one-dimensional flow assumption for coarse-textured soils and the micro-heterogeneity in soil hydraulic properties for soils with fine silt and clay materials.

Analytical solution for steady vertical flux through unsaturated soils based on van Genuchten-Mualem model

Various analytical solutions for steady-state vertical flux through unsaturated soils exist in the literature. These solutions often assume specific hydraulic conductivity forms such as exponential form, rational power form, or power-law form such as the Brooks-Corey model. The van Genuchten-Mualem model is widely used in practical applications of flow through unsaturated soils. It is among the most complicated hydraulic conductivity forms for which no analytical solutions are available. This paper deals with the development of an analytical solution for steady-state vertical flux using van Genuchten-Mualem model. The analytical solution is obtained using Maclaurin series expansions and the multinomial theorem. The general form expresses the depth as a function of pressure head. The analytical solution can be used to model both infiltration (positive flux rate) and evaporation (negative flux rate). The derived analytical solution is compared with numerical results and shows excellent agreement. It can also be used to identify the hydraulic parameters analytically, eliminating the need of a numerical inverse procedure. As far as the author’s knowledge, the proposed solution is the first analytical solution dealing with van Genuchten-Mualem model, which is directly derived from the flow equation without any predefined mathematical form of the pressure head.

Hydraulic Characterization of Green Roof Substrates by Evaporation Experiments

Green roofs can be a valid solution for stormwater management in urban environments. The objective of this study was to develop a laboratory procedure for the hydraulic characterization of artificial substrates, used in the realization of green roofs, based on transient evaporation and steady-state unit hydraulic gradient (UHG) experiments. The retention, θ(h), and hydraulic conductivity, K(h), curves of two commercial substrates Terra Mediterranea® (TMT) and AgriTERRAM® (ATV) and a specifically developed substrate made by mixing peat, compost and sandy loam soil (MIX) were investigated. The unimodal van Genuchten–Mualem (VGM) hydraulic functions obtained by the direct evaporation method with different choices of the fitting parameters were compared with UHG measurements of K(h) conducted close to saturation. A numerical inversion of the transient evaporation experiments performed by Hydrus-1D software was also conducted, assuming that the hydraulic properties could be expressed either by unimodal or bimodal VGM models. The results indicated that an appropriate a priori choice of the residual water content parameter improved the estimation of the water retention curve. Moreover, the water retention data estimated from the direct evaporation method were not statistically different from those obtained with the inverse Hydrus-1D. The unsaturated hydraulic conductivity estimations obtained by the direct and inverse methods were highly correlated and the use of the bimodal VGM model improved the estimation of K(h) in the wet range. The numerical inversion of laboratory evaporation data with the hydraulic characteristics expressed by the bimodal VGM model proved to be a reliable and effective procedure for hydraulic characterization of artificial substrates, thus improving the reliability of simulated water fluxes in green roofs.

Approximation of van Genuchten Parameter Ranges from Hydraulic Conductivity Data.

The use of retention function and relative conductivity function is essential for the calculation of flow in a variably saturated media using the Richards equation. A widely used mathematical model for this is the Mualem-van Genuchten model which requires the shape parameters and . These, however, are difficult to obtain. When data is scarce, and are often taken from literature and may deviate largely from actual values. The current study presents a novel mathematical model for the approximation of and and for the further estimation of realistic value ranges, which may be used as parameter space, for example, for the calibration of a numerical model. The model was developed for cases where data is scarce and only values of saturated hydraulic conductivity are available. It is based on a large data set from literature and it is demonstrated that the model estimates mean values from that data set with a good accuracy. In order to show the applicability of the model, a second data set has been compiled anew (provided as Supporting Information). The model is incorporated into the current version of the freeware computer program HYPAGS, which enables easy usage.

Soil water retention and hydraulic conductivity measured in a wide saturation range

Abstract. Soil hydraulic properties (SHPs), particularly soil water retention capacity and hydraulic conductivity of unsaturated soils, are among the key properties that determine the hydrological functioning of terrestrial systems. Some large collections of SHPs, such as the UNSODA and HYPRES databases, have already existed for more than 2 decades. They have provided an essential basis for many studies related to the critical zone. Today, sample-based SHPs can be determined in a wider saturation range and with higher resolution by combining some recently developed laboratory methods. We provide 572 high-quality SHP data sets from undisturbed, mostly central European samples covering a wide range of soil texture, bulk density and organic carbon content. A consistent and rigorous quality filtering ensures that only trustworthy data sets are included. The data collection contains (i) SHP data, which consist of soil water retention and hydraulic conductivity data, determined by the evaporation method and supplemented by retention data obtained by the dewpoint method and saturated conductivity measurements; (ii) basic soil data, which consist of particle size distribution determined by sedimentation analysis and wet sieving, bulk density and organic carbon content; and (iii) metadata, which include the coordinates of the sampling locations. In addition, for each data set, we provide soil hydraulic parameters for the widely used van Genuchten–Mualem model and for the more advanced Peters–Durner–Iden model. The data were originally collected to develop and test SHP models and associated pedotransfer functions. However, we expect that they will be very valuable for various other purposes such as simulation studies or correlation analyses of different soil properties to study their causal relationships. The data are available at https://doi.org/10.5880/fidgeo.2023.012 (Hohenbrink et al., 2023).

Development and functional evaluation of pedotransfer functions for estimating soil hydraulic properties in Portuguese soils: Implications for soil water dynamics

Accurate knowledge of soil hydraulic properties is essential for effectively simulating soil water dynamics in field plots. However, determining these properties directly can be challenging, especially when a large amount of data is required. Therefore, the primary objective of this study was to develop four hierarchical pedotransfer functions (PTFs) to estimate the soil hydraulic parameters of the Mualem-van Genuchten model. The PTFs that were developed ranged from utilizing a limited set of variables to those comprising a more comprehensive set of variables. These included particle size distribution, soil bulk density, soil depth, and soil water contents at matric potentials of −33 and − 1580 kPa. The functional performance of these PTFs was also assessed in simulations of the soil water balance in ten agricultural fields over two growing seasons using the HYDRUS-1D model. The simpler PTFs exhibited moderate accuracy in estimating soil water contents at different matric potentials, while the more complex PTFs with a greater number of variables yielded increasing accuracy. However, estimation errors were more noticeable in the drier range of the soil water retention curve. The inclusion of bulk density as a predictor only affected the wetter range of the same curve. Estimation of log Ks improved from the simpler to the more complex PTFs, but the statistical indicators were consistently modest. The functional performance of the PTFs showed increased errors in simulating soil water contents in the ten study sites compared to reference simulations. The more complex PTFs with the greatest number of variables provided the best predictions in most cases. The PTFs primarily affected flow conditions in the soil profile, resulting in increased soil water storage and impacting predictions of deep percolation the most. Water availability in the soil profile then influenced actual transpiration, alleviating stress in some cases but causing waterlogging in one situation. The study emphasizes the cautious use of the PTFs and the need for additional information on clay-rich soils and hydraulic conductivity data to enhance their performance.

Estimating Soil Hydraulic Parameters during Ponding Infiltration Using a Hybrid Algorithm

Accurate inversion of soil hydraulic parameters based on the van Genuchten–Mualem model has received much attention in soil science research. Herein, a hybrid algorithm method using particle swarm optimization and vector-evaluated genetic algorithm was used to invert the parameters θs, α, n, and Ks, with the objective functions of infiltration rate, cumulative infiltration, and soil water content. Then, numerical experiments were conducted on four typical soils at three initial water content levels (20, 40, and 60% effective saturation) to verify the accuracy of the inverse method. The results showed that the inversed soil water retention and conductivity curves were approximately the same as the real curves, with the root mean square errors of 0.00101–0.00192 cm3·cm−3, 0.00800–0.02519 cm3·cm−3, respectively, and both the Nash-Sutcliffe coefficients were approximately 1.0. Additionally, laboratory experiments were also performed to compare with the inversed parameters for verification, within small root mean squared errors and approximately 1.0 Nash–Sutcliffe coefficients. Furthermore, the method can also achieve acceptably accurate parameter inversion even with substantial measurement errors included in the cumulative infiltration, initial water content, and final water content. Thus, the method is effective and robust and found to be practical in field experiments.

Tracing lateral subsurface flow in layered soils by undisturbed monolith sampling, targeted laboratory experiments, and model‐based analysis

AbstractLateral subsurface flow (LSF) is a phenomenon frequently occurring in the field induced by local water saturation along horizon boundaries under nonequilibrium conditions. However, observations of LSF in undisturbed soils under controlled irrigation in the laboratory are limited but needed for model improvement, prediction, and quantification of LSF. We present a method for extracting an undisturbed soil monolith along a soil horizon boundary and introduce an experimental setup for the measurement of LSF and an irrigation device for simulating rainfall. An experimental test run was simulated using HYDRUS 2D. Water infiltrating into the monolith and flowing either laterally along the horizon boundary or vertically through the bottom horizon could be separately captured by suction discs at the side and the bottom. Thus, a clear distinction between lateral and vertical flow was possible. Pressure heads and water contents were recorded by tensiometers and frequency domain reflectometry (FDR) sensors distributed across the monolith in a regular two‐dimensional, vertical, cross‐sectional pattern. Sensor readings indicated the presence of nonequilibrium conditions within the monolith. Modeling results could reproduce the lateral and vertical outflow of the monolith under constant irrigation, thus showing that water flow within the monolith under steady‐state conditions can be explained by the Richards equation and the van Genuchten–Mualem model. The presented method can be used to improve and verify models designed for the prediction of the onset of LSF including that induced by local nonequilibrium conditions.

Experimental study on gas permeability of cement-based materials

Gas permeability is an important property associated with the durability of cement-based materials. In this study, an experimental method based on applying a quasi-stationary flow is adopted to evaluate the gas permeability of cement-based materials. The results indicate that the intrinsic permeability of dried cement-based materials is affected by the addition of limestone powder, fly ash, and aggregate. The Katz-Thompson equation could predict the intrinsic permeability by using the capillary porosity, average diameter, and pore tortuosity. The intrinsic permeability of partially saturated cement-based materials decreases when the water saturation increases. The Van Genuchten-Mualem model could describe the relative gas permeability of partially saturated cement-based materials, and the model parameters n value between 1 and 2 while m value ranging from 0.64 to 0.69 for cement and mortar. Moreover, the power function relationship between the intrinsic permeability and Klinkenberg constant of cement-based materials has also been corroborated. Finally, a prediction model of gas permeability of cement-based materials under arbitrary water saturation and pressure gradient is proposed.

SPATIAL VARIABILITY OF HYDRO-RELATED PHYSICAL PROPERTIES OF AL-RASHEED LOAM

A field study was conducted on 2 hectare area at Al- Rashid district, south of Baghdad to analyze the spatial variability of saturated hydraulic conductivity (Ks), initial infiltration rate (IR), Porosity (F) and bulk density (BD). Based on measured BD values Rosetta software was used in this study to estimate water retention parameters, water content at θ33 and θ1500 kPa and unsaturated hydraulic conductivity at 33 kPa(k33), 100 kPa(k100) and 1500 kPa(k1500) according to Van Genuchten-Mualem model. Measured and predicted data were analyzed both statistically and geostatistically and, the results showed a strong to moderate spatial dependence for the studied characteristics. The spatial correlation values (r2) were obtained with a spherical model for Ks, θ33, θ1500, k33, k100 and k1500, an exponential model for IR, and a Gaussian model for F and BD. Ks increased significantly with increasing of IR (r2 = 0.49**) and decreased with increasing of F and BD, IR increased with decreasing of BD (r2 = -0.326*) and BD increased with increasing of F (r2 = 0.989 **). In general the spatial distribution was moderately skewed (-0.5 to 0.5) for the studied characteristics with pronounced kurtosis (>2.5). The limit distance for the search radius to estimating spatial dependency varied from 29.9 m for BD to 105 m for IR.

Physical-hydric properties of Oxisols as influenced by soil structure and clay mineralogy

The soil water retention curve (SWRC) and soil hydraulic conductivity (K) are important soil hydraulic properties. In Oxisols, the degree of microaggregation depends on the amount of iron and aluminum oxides, kaolinite and biological activity engaged in modifying the pore size distribution (PSD) of the soil porous system. The objectives of this study were (i) to characterize four Brazilian Oxisols in terms of SWRC and PSD, using undisturbed (structural SWRCs) and disturbed (textural SWRCs) soil samples; (ii) to apply the van Genuchten-Mualem (vGM) model to the prediction of K and (iii) to correlate the classes of porosity and saturated hydraulic conductivity (Ksat) with the physical, chemical and mineralogical properties. Four soils were studied: Xantic Kandiustox, Rhodic Haplustox, Anionic Acrustox and Typic Hapludox. The soil samples were collected from the Bw horizon of each soil. The structural SWRCs were determined by Haine’s funnels and Richards’ chambers, whereas the textural SWRCs were determined by dew point meter. Ksat was evaluated by constant head permeameter and, additionally, soil chemical and mineralogical analyses were prepared. It was concluded that: (i) the four soils showed bimodal PSD; (ii) the four soils presented the maximum frequency of equivalent radius in the class of macropores (structural SWRCs) with rmax varying between 43.37 and 88,98 μm, and in the class of cryptopores (textural SWRCs) with rmax varying between 0.012 and 0.014 µm; (iii) the higher the Ksat, the higher the K predicted by the vGM model over the entire soil water content range of the structural SWRCs, except for Xanthic Kandiustox and Typic Hapludox in which this phenomenon occurred closer to saturation; (iv) the K values estimated by the vGM model from textural SWRCs were very low, varying approximately from 10−25 to 10−11mm h−1 for the four soils; (v) the degree of microaggregation of Rhodic Haplustox and Anionic Acrustox resulting from the high levels of Fe and Al oxides, leading to the formation of greater inter-aggregate space, had an influence on the high Ksat and showed that the mineralogy of the clay fraction was decisive in the physical-hydric behavior of these soils.

Determining the unsaturated hydraulic conductivity of remoulded loess with filter paper method and soil column seepage test

Loess is very widely distributed, and the unsaturated hydraulic conductivity of loess is related to many engineering issues. To determine the unsaturated hydraulic conductivity of remoulded loess more conveniently and at a lower cost, filter paper test and soil column seepage test were carried out. The results indicate that in the one-dimensional soil column seepage process, the unsaturated hydraulic conductivity of loess increases with the increase of the volumetric water content, and as the seepage time continues, the unsaturated hydraulic conductivity of loess from the top to the depth of 40 cm gradually becomes uniform. The changes in the microstructure indicate that the collapsible settlement will occur during the seepage process, which will change the pore structure of loess, thereby reducing the unsaturated hydraulic conductivity of the underlying loess. Compared with the experimental results, the soil hydraulic conductivity curve (SHCC) obtained by the van Genuchten–Mualem model (VG–M model) underestimates the magnitude of unsaturated hydraulic conductivity in the part with a low volumetric water content (< 20%). and the Childs and Collis-George model (CCG model) has more consistent results with the experimental results, because it is based on more segments of the soil–water characteristic curve (SWCC).

Non-isothermal water retention curve model for adsorption

Non-isothermal water retention curve (WRC) in high suction range is a fundamental and interpretative tool to describe the behavior of bentonite and is crucial to ensure the safety and effective operation of high-level nuclear waste repository. In this study, an adsorption equation at given water content is proposed according to Kelvin equation, Clausius-Clapeyron equation and the adsorption enthalpy equation. Then, the proposed adsorption equation is incorporated into van Genuchten-Mualem model to establish a non-isothermal WRC model for adsorption. Four series of non-isothermal WRC data in a temperature range (293∼353 K) and high suction range (>1 MPa) from four different soils are employed to illustrate the application and performances of the proposed model. The proposed WRC model provides good agreement between the calibrated/predicted and measured non-isothermal WRCs.

Testing the EPIC Richards submodel for simulating soil water dynamics under different bottom boundary conditions

AbstractMost biogeochemical models simulate water dynamics using the tipping bucket approach, which has been often found to be too simplistic to represent vadose zone dynamics adequately under shallow groundwater conditions. Recently, a solution to the Richards equation using the Mualem–van Genuchten model (Rich‐vGM) has been added into the EPIC (Environmental Policy Integrated Climate) model to address this shortfall. Its performance was tested using lysimeters operating under free drainage (FD) and at a shallow water table (60‐ [WT60] and 120‐cm depth [WT120]). Model accuracy was also compared with the upgraded tipping bucket‐based method implemented into EPIC (the variable saturation hydraulic conductivity method [VSHC]). Soil water content (SWC) data were split into calibration and validation subsets. Model evaluation also included annual evapotranspiration (ET), percolation (PRK), and upward water movements to assess underlying soil water balance factors. The submodels provided accurate and similar results upon comparison with SWC measures under FD (Nash–Sutcliffe coefficient [NSE] = 0.26 and 0.61 using VSHC and Rich‐vGM, respectively). The Rich‐vGM model accurately reproduced observed SWC and ET (e.g., NSE = 0.70 and percentage bias [PBIAS] = −3.7% for WT120, respectively) although it slightly overestimated PRK (PBIAS = 47.8%, on average). Instead, VSHC proved unable to correctly simulate shallow groundwater conditions (e.g., NSE = −1.85 for WT60 SWC). Under shallow groundwater conditions, the Rich‐vGM method is recommended, despite the additional data required and the need to define the bottom boundary conditions according to water table fluctuations. In conclusion, the Richards solver introduced and tested in EPIC improved the model's ability to represent complex biophysical and biogeochemical processes in terrestrial ecosystems associated with the hydrological balance.

Effects of pore water distributions on P-wave velocity–water saturation relations in partially saturated sandstones

SUMMARY We have developed a new method for introducing the effects of pore water distribution into the interpretation of P-wave velocity (VP) dispersion in a partially saturated rock. Pore water in an unsaturated rock is distributed as patches, and mesoscopic-scale wave-induced fluid flow (WIFF) from saturated patches to empty patches is known as a key mechanism for interpreting P-wave dispersion. Mesoscopic WIFF depends on permeability and the scale of the saturated patch, thus evaluating their influences is important for estimating the relationship between water saturation (${S_{\rm{w}}}$) and VP. The simplest and conventional law for simulating the ${S_{\rm{w}}} - {V_{{P}}}$ relation considering the mesoscopic WIFF is White's model, which describes the patch size changes proportional to the saturation. However, White's model assumes homogeneity in the size and distribution of pores, and does not consider the effects of pore water distribution in a partially saturated rock. In the rock, pore water is distributed heterogeneously according to the pore radius, causing non-linear changes in the permeability and size of the saturated patches against the water saturation. In this study, we modified White's model by introducing relative permeability and a new coefficient describing the nonlinear change of the patch size into the conventional version, and applied the modified model to better interpret our experimental ${S_{\rm{w}}} - {V_{{P}}}$ relations, which were measured by an ultrasonic technique at 200 and 500 kHz for two types of Berea sandstone samples with different permeabilities as their water saturation was decreased by evaporative drying. The relative permeabilities were determined by applying the Mualem–van Genuchten model into the capillary pressure curves from mercury intrusion porosimetry. Furthermore, we proposed a calculation method for the new coefficient using tortuosity, which corresponds to the pore water connection. The modified White's model could reproduce the experimental ${S_{\rm{w}}} - {V_{{P}}}$ relations better than other conventional models. Consequently, our modification, considering the effect of pore water distribution, would be useful for more quantitative interpretation of P-wave velocity and attenuation under coexisting multifluid conditions.

Approximate analytical solution and parameter estimation for one‐dimensional horizontal absorption based on the van Genuchten–Mualem model

AbstractThe van Genuchten–Mualem (VGM) model is used to describe the absorption of water by soil. The model is highly nonlinear, so developing a simple analytical or approximate method to predict the movement of soil water via the VGM model then estimating the model parameters is necessary. We present a approximate functional extremum solution (FES) for the problem of one‐dimensional horizontal absorption (1DHA) of water in soil under constant saturated boundary conditions and uniform initial conditions based on the least action and the variational principles. Simple linear relationships between the infiltration characteristics (distribution of soil water content, cumulative infiltration, infiltration rate, wetting front distance, and infiltration time) and the VGM model parameters were obtained. We developed a simple approximate analytical method for estimating the VGM model parameters by testing 1DHA. Numerical data simulated by HYDRUS‐1D were compared with the analytical results and used to verify our method's feasibility. The results indicated that the FES could accurately predict the infiltration characteristics, with relative errors between the approximate analytical and numerical solutions of 0.8 to 8.5% for the distribution of soil water content, 0.1 to 0.8% for cumulative infiltration, 1.1 to 6.4% for infiltration rate, and 0.1 to 6.9% for wetting front distance. Estimates of the parameters indicated that errors ranged from 3.2 to 28.3% for α (inverse of the mean pore diameter) and 0.4 to 8.0% for n (pore size distribution index). The proposed methods can be used to predict 1DHA and to estimate α and n in the VGM model.

Fitting process-dependence performance of the van Genuchten soil water retention model to simulate the soil water flow

Determining the parameters of the soil water retention curve (SWRC) is essential to model the water flow in soil and investigate the water and nutrients transport around the roots of plants and ensure optimal management of irrigation. The unweighted least squares regression (ULS) is the most common approach applied to fit the SWRC functions to the observed data-points to optimize their parameters. The main purpose of this study was to evaluate the ULS and the weighted least squares regression (WLS) fitting process of SWRC and their impact on simulations of soil water flow. In this regard, the measured SWRC data in six parallel samples from given depth were fitted to the SWRC equations to optimize their parameters, through either the WLS or the ULS. In the WLS approach, the weights were calculated as the inverse of the variance in each pressure head of different iterations (six repetitions in this study). The results showed that although WLS generally resulted in an increased error in estimating the SWRC in the van Genuchten-Mualem model (RMSE = 0.027 and 0.043 cm3 cm−3 in the ULS and WLS methods, respectively), it improved the accuracy of the estimations at lower water contents (dry-end), as compared to the ULS. However, the efficiency of the ULS and WLS in estimating the SWRC was different from that in simulating the soil water flow. The water flow in the soil was more accurately simulated using the hydraulic parameters obtained by the WLS (with RMSE = 0.029 and 0.026 cm3 cm-3 at x = 0 and x = 20 cm, respectively, in the WLS, and RMSE = 0.062 and 0.065 cm3 cm-3 at x = 0 and x = 20 cm, respectively, in the ULS), especially at the low water range. While there was an important difference between the ULS and WLS in terms of estimating the SWRC and water flow simulations, other dispersion scenarios should be evaluated in future studies to compute the error of the fitting processes at low-pressure heads.