Fuzzy Cloud Model Brings Sharper Early Warning to Deforming Tunnels

Deep beneath the mountains of China, some of the world’s longest tunnels are quietly being squeezed, buckled, and cracked by the immense pressures of weak rock and high stress. Engineers call the phenomenon large deformation, and it is one of the most stubborn threats in modern underground construction: primary supports split, steel arches distort, tunnel inverts fail, and crews are forced into repeated rework that stretches schedules and inflates costs. A new study published in Environmental Earth Sciences by Shibin Yao and Jian Zhou of Central South University and Peixi Yang of the University of New South Wales offers a fresh way to grade this risk, replacing rigid thresholds with a fuzzy, cloud-based framework that mirrors how geologists actually think about uncertainty.

The core problem the researchers set out to solve is deceptively simple to state. Tunnel large deformation is not triggered by any single factor. It emerges from the coupled effects of rock mass strength and integrity, groundwater action, discontinuity geometry, and construction disturbance. Yet conventional grading methods draw hard lines between deformation grades, treating a sample just above or below a threshold as fundamentally different even when the underlying conditions are nearly identical. When grade boundaries are blurred and indicator definitions vary between projects, these threshold-based approaches lose both accuracy and interpretability, particularly when a model trained on one tunnel is applied to another.

The team’s answer is built on cloud model theory, a mathematical framework designed to express the mapping between qualitative concepts and quantitative values while simultaneously capturing randomness and fuzziness. A basic cloud model describes any grade concept with three numerical characteristics: expectation, which marks the center of the concept; entropy, which reflects its dispersion and boundary uncertainty; and hyper-entropy, which describes how much that entropy itself fluctuates. A forward cloud generator then scatters thousands of cloud drops across the domain, each drop carrying a quantitative position and a membership degree between zero and one, producing a visual distribution that shows exactly where a grade concept is certain and where it dissolves into ambiguity.

What makes the new framework distinctive is its asymmetry. In real tunnels, grade intervals are rarely equal in length, and transitions toward the risk side, where deformation grows more severe, tend to be more sensitive than transitions in the safe direction. A single symmetric entropy cannot represent that physics. The researchers therefore introduced separate left and right entropies for each indicator and grade, derived from the spacing between adjacent grade centers, so the membership function decays at different rates on either side of the grade center. At the extreme outer grades, a saturation rule fixes membership at one once an indicator enters the endpoint zone, keeping the boundary response continuous and consistent with engineering semantics.

From these asymmetric clouds, the method extracts membership intervals rather than single values. By proportionally scaling the entropy to 0.9 and 1.1 times its nominal value, sensitivity analysis showed a stable intermediate range, the team constructs upper and lower envelopes of membership degree for every indicator at every grade. The bandwidth between those envelopes quantifies how dispersed and variable a sample is near a boundary, effectively softening the sensitivity that plagues hard-threshold grading. When an indicator value sits close to a grade interface, its support for neighboring grades rises simultaneously, which is precisely what field engineers observe in progressively evolving deformation rather than abrupt jumps.

Weighting the thirteen indicators in the system posed its own dilemma. Subjective weights, derived here from an expert judgment matrix built on the Saaty 1-to-9 scale and checked for logical consistency, capture hard-won engineering experience but carry randomness. Objective weights, computed statistically, have a firmer mathematical basis yet can drift from engineering intent. Rather than choosing one, the team computed four competing objective schemes, the entropy weight method, CRITIC, PCA, and TOPSIS, each emphasizing a different notion of importance, from information dispersion to inter-indicator correlation to variance structure to distance from ideal solutions. The objective scheme achieving the highest grading accuracy was selected, and then fused with the subjective weights in equal measure to form comprehensive weights.

The framework was validated on three tunnels with markedly different geology: the Telmo Tunnel in Sichuan, a 7.827-kilometer alignment contributing forty data groups; the Huangjiagou Tunnel in Hubei, 6.514 kilometers long with twenty data groups; and the Muzhailing twin-tube tunnel in Gansu, with tubes of 15.226 and 15.168 kilometers and thirteen data groups dominated by deep-burial squeezing conditions. Using indicators such as uniaxial compressive strength, rock quality designation, groundwater inflow, strike-tunnel axis angle, and discontinuity condition, the method achieved grading accuracies of 88 percent at Telmo, 90 percent at Huangjiagou, and 85 percent at Muzhailing, with a pooled accuracy of 88 percent across all seventy-three samples.

Perhaps the most striking result is how comprehensively the fused weighting outperformed every single scheme. At Telmo, accuracies for the entropy, CRITIC, PCA, TOPSIS, and purely subjective schemes ranged from 55 to 83 percent, all below the 88 percent of the comprehensive weights. At Muzhailing, where samples were few and boundaries most sensitive, the gap widened dramatically: objective schemes managed only 39 to 62 percent, while the fused weights reached 85 percent. Across the three tunnels, accuracy fluctuation shrank to just five percentage points under the comprehensive scheme, compared with up to 46 points for individual objective methods, evidence that the fusion strategy delivers both accuracy and cross-project stability.

Equally important is what the method does with its own mistakes. Misclassified samples clustered at adjacent-grade transitions, notably near the Grade IV to V and Grade III to IV boundaries, and the team showed these are not random failures but meaningful signals. In one representative Telmo case, the discontinuity orientation indicator supported Grade IV while rock strength leaned toward Grade III, placing the sample squarely in a transition zone where risk escalation was driven by discontinuity geometry rather than weak rock. The practical implication is direct: pre-reinforcing segments with unfavorable discontinuity orientations and adjusting support closure timing offers more improvement potential than trying to further strengthen the rock mass.

Aggregating the three projects produced a global ranking of all thirteen indicators, with uniaxial compressive strength at 0.2019 and rock quality designation at 0.1623 standing far above the rest, confirming that surrounding-rock strength and integrity are the fundamental controls on deformation grades. Strike-tunnel axis angle at 0.1241 and groundwater inflow at 0.0928 followed, with support closure time at 0.0900 highlighting the role of construction sequencing. The authors caution that the entropy perturbation factors remain empirical, the validation dataset is modest, and the global ranking reflects the cases included so far. Even so, the framework offers something tunnel engineering has lacked: a unified, interpretable, and transferable tool that turns fuzzy boundaries from a liability into a source of actionable early-warning guidance.

Subject of Research: A cloud-model-based fuzzy evaluation framework for grading nonlinear large deformation in tunnels

Article Title: A multi-dimensional cloud-model-based fuzzy evaluation method for grading nonlinear tunnel large deformation

Article References: Yao, S., Yang, P., & Zhou, J. (2026). A multi-dimensional cloud-model-based fuzzy evaluation method for grading nonlinear tunnel large deformation. Environmental Earth Sciences, 85(16), Article 413. https://doi.org/10.1007/s12665-026-13108-8

Image Credits: AI Generated

DOI: 10.1007/s12665-026-13108-8

Keywords: tunnel large deformation, cloud model, fuzzy evaluation, asymmetric membership, subjective-objective weighting, geotechnical engineering, weak rock, groundwater, discontinuities, risk grading, early warning, Muzhailing Tunnel