⚡ electrical engineering

Scaling Laws for Rainfall Infiltration in Unsaturated Soils: Derivation and Validation Using Centrifuge Model Tests

This study derives and experimentally validates scaling laws for rainfall infiltration in unsaturated soils using centrifuge model tests, confirming that wetting-front velocity, saturated hydraulic conductivity, and infiltration time scale as 1:1, 1:1, and N²:1 respectively, while establishing the Gardner equation as a critical criterion for maintaining suction similarity under centrifugal acceleration.

Original authors: Xiaochuan Liu, Gaofei Kong, Junxi He, Haijie He, Yunfang Wu, Chao Zhang

Published 2026-09-21

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CC BY 4.0

Original authors: Xiaochuan Liu, Gaofei Kong, Junxi He, Haijie He, Yunfang Wu, Chao Zhang

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a heavy rainstorm falling on dry, cracked earth. The water does not soak in all at once; instead, it pushes a distinct boundary downward, a sharp line where the dry soil meets the wet. This moving line is called a wetting front. As it travels, it changes the soil's behavior, turning a dusty, porous material into a heavy, saturated sponge. Engineers need to predict exactly how fast this front moves and how deep the water will go, because if the ground becomes too wet too quickly, it can trigger landslides or cause foundations to fail. To study these dangerous events safely, scientists often build small-scale models of the ground and test them in giant centrifuges. These machines spin the models at high speeds to create a force that mimics the weight of a much larger, real-world landscape. However, a critical question has lingered: do the rules that govern water moving through a tiny, spinning model actually match the rules for the massive, real earth?

A team of researchers from universities in China set out to answer this by testing the specific rules, or scaling laws, that connect a small model to the real world. They focused on unsaturated soil, which is ground that contains both air and water, and how it reacts to rainfall. The team built a horizontal layer of loess, a type of silty soil common in northern China, inside a specialized container. They placed this container on a large centrifuge at Zhejiang University and spun it to create forces equivalent to twenty, forty, and sixty times the normal pull of gravity. While the soil spun, they used a custom-built rainfall simulator to pour water onto the surface, creating a constant pool of water three centimeters deep. To track what happened inside, they embedded tiny sensors at different depths to measure how the water content changed and how the soil's suction, or its natural grip on water, evolved as the rain soaked in.

The researchers compared their measurements against a classic theory known as the Green-Ampt model, which treats the wetting front as a distinct, horizontal line moving through uniform soil. Their experiments confirmed that the speed at which this wetting front travels is directly tied to the strength of the gravity force. When the centrifuge spun at forty times normal gravity, the wetting front moved roughly twice as fast as it did at twenty times gravity, and three times as fast as it did at sixty times gravity. This simple, predictable relationship held true for the time it took the water to reach specific depths as well; as the infiltration depth increased, the ratio of arrival times between different gravity levels approached the inverse ratio of the gravity levels. For instance, if a process takes one second in the model under high gravity, it would take roughly four seconds in the real world under normal gravity if the gravity level was four times higher, though the exactness of this ratio improved with greater depth and time. The study also found that the soil's ability to let water flow through it, once it was fully wet, increased in direct proportion to the gravity force applied.

A significant part of the investigation involved a more subtle factor: suction similarity. In unsaturated soil, suction is the force that holds water in the tiny pores, acting like a vacuum that keeps the ground dry. The researchers needed to know if the spinning force of the centrifuge would accidentally change this suction, making the model soil behave differently than the real soil. They used a mathematical relationship known as the Gardner equation to predict how much the spinning would increase the soil's suction. Their results showed that at lower spinning speeds, specifically twenty and forty times gravity, the predicted increase in suction was smaller than the soil's natural starting suction. In these cases, the soil's internal grip on water remained unchanged, and the model was a perfect mirror of reality. However, at sixty times gravity, the predicted suction increase was larger than the soil's natural grip. Here, the spinning force actually pulled more water out of the soil, causing the measured suction to increase by 7.8–8.5 kPa compared with the initial value. Despite this change, the speed of the wetting front still followed the expected rules, suggesting that even when the soil dries significantly under extreme force, the fundamental timing of the infiltration remains reliable.

The study concludes that the mathematical rules used to translate small-scale centrifuge tests to real-world scenarios are robust for rainfall infiltration. The speed of the wetting front, the time it takes to infiltrate, and the soil's ability to conduct water all scale predictably with gravity. The researchers also validated a clear threshold for when these tests remain accurate: as long as the force of the spin does not exceed the soil's natural ability to hold water, the model will faithfully represent the prototype. This work provides engineers with a firmer foundation for using centrifuge models to predict how rain will move through the ground, offering a reliable way to assess the stability of slopes and foundations before a single drop of rain falls on a real construction site.

Technical Summary: Scaling Laws for Rainfall Infiltration in Unsaturated Soils

Problem Statement
Centrifuge model testing is a standard method for reproducing in-situ stress states in soil mechanics, particularly for foundation settlement studies. However, the application of centrifuge modeling to rainfall infiltration in unsaturated soils faces a critical gap: the scaling laws governing this specific process have not been fully validated by systematic experimental evidence. While previous studies have established scaling ratios for transient seepage in earth dams, finger flow, and capillary rise, rainfall infiltration presents distinct hydraulic boundary conditions and wetting-front evolution processes. Furthermore, the issue of "suction similarity" remains a key challenge; centrifugal acceleration can alter the initial water content and suction of model soils, potentially breaking the similarity between the model and the prototype. There is currently a lack of a clear, experimentally validated criterion to determine when suction similarity is maintained under hypergravity.

Methodology
This study employs a dual approach combining theoretical derivation and experimental validation:

Theoretical Derivation: The authors derived scaling laws for rainfall infiltration based on the classical Green–Ampt model and the first similarity theorem. The derivation assumes a homogeneous soil with a distinct horizontal wetting front, constant suction at the front, and one-dimensional vertical flow. The model considers the scaling of geometric dimensions, hydraulic heads, and time.
Suction Similarity Criterion: The study utilizes the Gardner equation to establish a criterion for suction similarity. The hypothesis is that suction similarity is maintained only if the centrifugal acceleration does not induce a suction greater than the soil's initial suction. If the calculated centrifugal suction exceeds the initial suction, the soil dries, and similarity is lost.
Experimental Validation: Centrifuge model tests were conducted on unsaturated loess foundations using the ZJU-400 centrifuge at Zhejiang University.
- Soil: Compacted loess (clayey silt) with a dry density of 1.45 g/cm³ and an initial water content of approximately 27.55%.
- Conditions: Tests were performed at three different g-levels: 20g, 40g, and 60g.
- Setup: A horizontal soil foundation (50 cm × 40 cm × 28 cm) was subjected to a constant water head of 3 cm to simulate rainfall infiltration.
- Instrumentation: Miniature TDR probes monitored volumetric water content, and miniature tensiometers measured matric suction at three depths (5.5 cm, 13.5 cm, and 21.5 cm).
- Procedure: The centrifuge was accelerated to the target g-level, and rainfall was simulated until steady-state infiltration was observed. Data was recorded during acceleration and the constant-head infiltration phase.

Key Contributions

Derivation of Specific Scaling Laws: The study provides a rigorous derivation of scaling laws specifically for rainfall infiltration, distinguishing it from general unsaturated seepage problems.
Experimental Validation: It offers systematic experimental evidence validating these derived laws across multiple g-levels, addressing the previous lack of direct validation for rainfall infiltration.
Evaluation of the Gardner Criterion: The paper experimentally tests the Gardner equation as a predictive tool for suction similarity, defining the threshold where centrifugal forces begin to alter the initial hydraulic state of the soil.

Results

Scaling Laws Validation:
- Wetting-Front Migration Velocity ( $v_f$ ): The experimental results confirmed a scaling ratio of $1:N$ (Model:Prototype). The velocity in the model was $N$ times faster than in the prototype.
- Saturated Hydraulic Conductivity ( $k_s$ ): The scaling ratio for saturated hydraulic conductivity in the wetted zone was found to be $1:N$ , consistent with theoretical predictions.
- Infiltration Time ( $t$ ): The time required for the wetting front to reach a specific depth scaled as $N^2:1$ . The time in the model was $1/N^2$ of the prototype time.
- Wetting-Front Depth ( $z_f$ ) and Suction Head ( $s_f$ ): These maintained a $1:1$ scaling ratio, indicating geometric and suction head similarity.
Suction Similarity Findings:
- At 20g and 40g, the suction calculated via the Gardner equation was lower than the measured initial soil suction. Consequently, the centrifugal force did not significantly alter the initial suction (changes were less than 1.1 kPa), and suction similarity was maintained.
- At 60g, the calculated centrifugal suction exceeded the initial soil suction. The measured suction increased by 7.8–8.5 kPa, indicating that the initial hydraulic state was altered and suction similarity was broken.
- Despite the loss of suction similarity at 60g, the scaling laws for wetting-front migration velocity remained valid, suggesting that the observed suction increase did not introduce significant errors in the velocity scaling for this specific range.

Significance and Claims
The paper claims that the derived scaling laws ( $v_f \propto N$ , $t \propto 1/N^2$ , $k_s \propto N$ ) are robust and agree well with experimental data for rainfall infiltration in unsaturated soils. The study validates the Gardner equation as an effective criterion for evaluating suction similarity, establishing that similarity holds when the induced centrifugal suction is less than the initial soil suction.

The authors modestly note that while the scaling laws held true even when suction similarity was violated at 60g, further research is required to determine the allowable limit of suction difference between the model and prototype that would not cause significant errors in other scaling parameters. The work provides a theoretical and experimental foundation for more reliable centrifuge modeling of rainfall-induced geotechnical problems, such as slope instability.

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