What if a difficult-to-measure soil property could be estimated from a map, a fitted water-retention curve, and one saturated permeability measurement? This study builds that route for Malan loess across China’s Loess Plateau.

Unsaturated permeability coefficients are often used to solve geotechnical problems associated with unsaturated soils, but they are very difficult to measure. Their value can instead be predicted by using soil-water characteristic curves.

The Van Genuchten model is widely used because it is smooth and provides good fitting, so it has the most research data. The study collects four Van Genuchten parameters for Malan loess to examine their spatial distribution. Trend surface analysis is used to clarify regional-scale distribution and variation patterns in ArcGIS.

Linear regression then fits the relationship between suction and water content in three regions divided by loess properties and particle gradation. After measuring a sample’s saturated permeability coefficient, the method uses that relationship and the trend-surface contour map to predict its unsaturated permeability coefficient.

Verification shows small differences at lower saturation and slightly larger deviations at higher saturation, all within the acceptable range. Permeability coefficient, also called hydraulic conductivity, represents water seepage velocity through different media under a unit hydraulic gradient.

The unsaturated value changes continuously with water content and matrix suction. Because its variation can span several orders of magnitude, accurately measuring the unsaturated permeability coefficient is difficult. That difficulty has motivated extensive research on prediction methods.

One framework divided calculation models into three categories: empirical models, statistical models, and macroscopic models. Soil-water characteristic curves describe the relationship between water content and matrix suction. Water content can be volumetric moisture content, gravimetric moisture content, or degree of saturation.

Matrix suction can also be called capillary pressure, and it can be represented by pore air pressure minus pore water pressure. The drying and wetting curves are sigmoidal. On the drying curve, the air-entry value and residual suction value divide the curve into a boundary effect zone, transition zone, and residual zone.

Figure one shows a typical soil–water characteristic curve, relating saturation to matric suction, or capillary pressure. As suction increases along the logarithmic horizontal axis, the curve passes through a boundary-effect zone, a transition zone, and a residual zone, with points labeled G one and G two marking notable locations.

This framework matters because SWCCs describe how soil water content changes with suction and support prediction approaches based on soil properties and grain-size distributions. Direct measurement methods have limitations: some instruments cannot cover the full suction range, while others are time-consuming, require special testing environments, or have expensive testing costs.

Prediction approaches include database mining from similar soils, prediction from the grain-size distribution curve, and correlations linking soil-water characteristic-curve parameters to soil properties. The grain-size approach translates grain-size distribution into pore-size distribution and relates it to the soil-water characteristic curve through capillary theory.

The Van Genuchten model accounts for an inflection point, giving it greater flexibility across a wider suction range and allowing it to capture the sigmoidal shape of typical curves more effectively. Its smooth transitions at the air-entry value and its approach toward the residual suction value are also captured more effectively.

The parameter m reduces flexibility but simplifies optimization and permits a closed-form hydraulic-conductivity solution. Loess is widely distributed in arid and semi-arid regions, covering about ten percent of the world’s land area. In China, the Loess Plateau extends over six point four times ten to the fifth square kilometers and accounts for over seven percent of the country’s land area.

Because Malan loess is relatively loose and engineering construction commonly occurs in it, loess engineering properties are generally studied in Malan loess. The study collects the four Van Genuchten parameters of Malan-loess soil-water characteristic curves tested by researchers across the Loess Plateau.

It uses trend surface analysis and linear regression to study their regional distribution. The study also proposes a new method for calculating the unsaturated permeability coefficient of loess through the connection between Van Genuchten parameters and unsaturated permeability coefficient.

Soil-water characteristic curves for Malan loess are treated as key to solving engineering problems involving loess and water. The study therefore gathers and summarizes Van Genuchten model parameters from Malan-loess curves in the Chinese Loess Plateau. Table one reports van Genuchten model parameter statistics across named areas, with source references for each entry.

It lists theta sub s, theta sub r, alpha, and n; missing values are marked with dashes, such as n for Xingxian and Yanglin, and theta sub r for Xining. This matters because the table makes the parameter inputs and their provenance explicit for the different study areas.

The zone averages show clear regularity. In Zone Two, the average saturated and residual water contents are both larger than in the other two zones. The average alpha value is lower in Zone One than in Zones Three and Two.

The average n value is largest in Zone One, second in Zone Two, and smallest in Zone Three. Table three reports the van Genuchten model parameters for Zones I, II, and III, including theta s, theta r, alpha, and n, with values presented as ranges and central values.

The authors note that Zone II has the largest average theta s and theta r, Zone I has the smallest average alpha, and the average n decreases from Zone I to Zone III. These zone-specific parameters provide the basis for statistically analyzing differences in soil hydraulic behavior across the study area.

Trend-surface analysis constructs an equation that forms a plane or a more complex spatial surface, allowing a geological variable to be represented across space. It separates regional variation, or trends, from local variation, or abnormalities.

The method can remove the regional variation component and highlight the local variation component, supporting the study of abnormal points and their causes. The radial basis function handles multivariate function approximation by expanding a spatial surface and applying that surface to describe data.

It is described as efficient, simple to operate, easy to program, and independent of grids. Universal Kriging represents regional-variable variability with deterministic, relevant, and random parts. For non-stationary variables, the deterministic part varies through space as a drift or tendency.

The corresponding best linear estimation process is called the universal Kriging method. In ArcGIS, the radial basis function is used for measured-value contour graphs because it handles multiple problems, large scattered datasets, and approximation.

Universal Kriging is used for predicted-value contour graphs and predicted standard-error graphs. Universal Kriging also allows measurement errors to exist, which is relevant when producing the prediction maps and their standard-error maps.

Figure three maps the spatial behavior of the VG model parameter alpha across the Chinese Loess Plateau. Panel A shows the fitted trend surface, panel B the measured-value contours, panel C the predicted contours, and panel D the prediction standard error, allowing the authors to compare the mapped pattern with prediction reliability.

The paper notes that most errors are below zero point zero one seven, while the Luoyang area reaches zero point zero three, indicating generally accurate predictions with a localized higher-error area. Figure four maps the VG model parameter n across the Chinese Loess Plateau using a trend surface, measured values, predictions, and prediction standard error.

The authors report higher values in Zone I, followed by Zone II and Zone III; measured values reach two point zero three to two point one one around Heifangtai and Dongxiang, while Qishan is one point zero two to one point one four. The error map indicates generally small deviations, with the largest values in Xining, at zero point one five to zero point one seven.

Figure five examines the VG-model parameter theta-s across the Chinese Loess Plateau in four views: a spatial trend, measured contours, predicted contours, and prediction standard error. The maps show a concentrated higher-value area around Tongchuan, Heyang, and Luochuan in Shaanxi, with the predicted pattern broadly reflecting the measured distribution.

The error map indicates generally small deviations across much of Shaanxi, western Shanxi, and the Xining–Lanzhou belt, while larger errors appear around Yinchuan and Hohhot. Figure six maps the VG-model parameter theta r across the Chinese Loess Plateau through four views: its directional trend, measured values, predicted values, and prediction standard error.

The authors report a southeast-to-northwest pattern, with measured values highest in parts of Zone Two and lowest in northern Shaanxi, while the prediction map shows a similar spatial distribution. The error map indicates how closely predictions correspond to measurements, making this figure useful for assessing both regional variation and prediction reliability.

For alpha, the trend-analysis graph and prediction-value contour graph provide the trend pattern across the Chinese Loess Plateau. The measured-value contour graph illustrates alpha’s spatial distribution. The prediction-standard-error graph reflects the accuracy of predicted alpha values and abnormal data situations.

Together, the four graphs provide the regional distribution of alpha. The same analysis method is used to obtain the regional distribution of parameter n across the Chinese Loess Plateau. For saturated water content, the trend is higher in the southeast and lower in the northwest.

It decreases from west to east and then gradually increases, and overall is higher in the east than the west. The trend also decreases gradually from north to south before increasing slowly, and overall is higher in the south than in the north.

The study combines previous research on soil-water characteristic curves with the prediction map to analyze soil-water characteristic-curve properties on the Loess Plateau. The unsaturated permeability coefficient for a particular location is deduced by applying the hydraulic-conductivity equation and combining it with statistical trend data.

To obtain the unsaturated permeability coefficient for a sample, first read alpha, n, saturated water content, and residual water content at the location from the prediction map. Then select one of formulas fourteen through sixteen according to the zone and calculate matrix suction.

Next, calculate m through n and substitute it into formula thirteen to obtain effective saturation. After measuring saturated permeability coefficient k-s and using formula twelve, the unsaturated permeability coefficient k is obtained.

Measured unsaturated permeability coefficients from other researchers, covering different zones, are used to verify the prediction accuracy of the method. The verification data are shown in Table Four. The proposed method predicts unsaturated permeability coefficients in three research areas.

The predicted results are compared with experimental test data in Figures Ten through Twelve and Table Six. Figure ten compares predicted and test unsaturated permeability coefficients for Groups A, B, and C in Zone Three, Yangling, across soil saturation.

The curves show that permeability changes sharply with saturation before approaching values near zero, while the predicted and experimental markers generally track one another. The authors report that prediction deviations are basically controlled within plus or minus four point seven seven times ten to the power of minus seven centimeters per second, supporting the model’s overall accuracy.

Figure eleven compares predicted and test unsaturated permeability coefficients across saturation for three groups in Zone Two, Yan’an. The chart shows generally corresponding predicted and measured curves, while the authors report prediction deviations basically controlled within plus or minus two point one seven times ten to the power of minus eight centimeters per second.

In the lower-saturation section, the reported deviation is plus or minus one point zero six times ten to the power of minus eight centimeters per second, supporting the figure’s role in evaluating prediction accuracy. The method combines regional Van Genuchten parameter maps with zone-specific linear fitting to estimate unsaturated permeability.

Its deviations are very small at lower saturation and slightly larger at higher saturation, while remaining within the acceptable range.