Key Takeaways & Executive Findings
- •• Existing analytical solutions for stress calculation using HI strain cells are limited to double-layer media and are inaccurate for three-layer media, leading to significant errors. • A numerical simulation method using FLAC3D was developed to obtain constitutive matrices relating strain measurements to stress tensors in a three-layer medium. • An artificial neural network surrogate model was trained on 764 numerical models, achieving high accuracy (R2=0.999, MSE=1.254) for predicting constitutive matrices. • The surrogate model was validated through numerical simulations, analytical solutions, and laboratory experiments, and successfully applied to correct stress calculations in a coal mine roadway roof.
Abstract
Accurate acquisition of the rock stress is crucial for various rock engineering applications. The hollow inclusion (HI) technique is widely used for measuring in-situ rock stress. This technique calculates the stress tensor by measuring strain using an HI strain cell. However, existing analytical solutions for stress calculation based on an HI strain cell in a double-layer medium are not applicable when an HI strain cell is used in a three-layer medium, leading to erroneous stress calculations. To address this issue, this paper presents a method for calculating stress tensors in a three-layer medium using numerical simulations, specifically by obtaining a constitutive matrix that relates strain measurements to stress tensors in a three-layer medium. Furthermore, using Latin hypercube sampling (LHS) and orthogonal experimental design strategies, 764 groups of numerical models encompassing various stress measurement scenarios have been established and calculated using FLAC3D software. Finally, a surrogate model based on artificial neural network (ANN) was developed to predict constitutive matrices, achieving a goodness of fit (R2) of 0.999 and a mean squared error (MSE) of 1.254. A software program has been developed from this surrogate model for ease of use in practical engineering applications. The method's accuracy was verified through numerical simulations, analytical solution and laboratory experiment, demonstrating its effectiveness in calculating stress in a three-layer medium. The surrogate model was applied to calculate mining-induced stress in the roadway roof rock of a coal mine, a typical case for stress measurement in a three-layer medium. Errors in stress calculations arising from the use of existing analytical solutions were corrected. The study also highlights the significant errors associated with using double-layer analytical solutions in a three-layer medium, which could lead to inappropriate engineering design.
1. Introduction
Understanding the state of stress is important for a variety of rock mechanics applications, such as geotechnical engineering design, surrounding rock failure and stability analyses, earthquake potential evaluation, and hydraulic fracturing analysis [1–3]. Stress measurements in field is the primary method for obtaining the state of stress. It mainly includes over-coring stress relief, hydraulic fracturing, acoustic emission, strain/stress recovery, borehole breakout, and geophysical methods [4–6]. Over-coring stress relief method is a borehole relief technique that has been developed for nearly 60 years and is considered a relatively mature in-situ stress measurement method [7].
Depending on the measurement principles and the physical quantities being measured as well as their locations, the over-coring stress relief method can be divided into three types: the borehole wall strain measurement method [8], the borehole diametrical deformation method [9], and the borehole bottom deformation method [10]. The borehole wall strain measurement method can measure the full stress tensor with a single borehole and is widely used in underground engineering. In contrast, the other two methods require measurements from boreholes intersecting in three different directions to obtain the full stress components [11]. The borehole wall strain measurement method calculates the stress tensor by measuring strain with a triaxial cell during the stress relief process. There are two types of strain cells commonly used for borehole wall strain measurement. One technique is the CSIR triaxial strain cell, where strain gauges are directly attached to the borehole wall. However, this method is difficult to implement, requires high integrity of the rock mass, and has a low success rate [12]. The other is the CSIRO triaxial strain cell [13], where strain gauges are embedded in a thin epoxy resin tube. During stress measurement, the strain gauge is bonded to the borehole wall via epoxy resin, and the strain is measured during over-coring. The CSIRO cell is more robust and easier to install, making it widely used in practice.
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Changkun Qin, Wusheng Zhao, Weizhong Chen, Peiyao Xie, Shuai Zhou (2025). A surrogate model for estimating rock stress by a hollow inclusion strain cell in a three-layer medium. SinoTechIntel Verified Research. https://doi.org/10.1016/j.ijmst.2025.02.003
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Frequently Asked Questions
What is the hollow inclusion (HI) strain cell technique?
The hollow inclusion (HI) strain cell technique is a method for measuring in-situ rock stress. It involves embedding strain gauges in a hollow epoxy cylinder that is bonded to the borehole wall. During over-coring, the rock relieves stress, causing deformation that is measured by the strain gauges. These strain measurements are then used to calculate the full stress tensor.
Why are existing analytical solutions not applicable for three-layer media?
Existing analytical solutions for HI strain cells were developed for a double-layer medium (e.g., rock and epoxy). In a three-layer medium, such as when a hollow inclusion cell is used in a borehole with a grout layer, the stress distribution is more complex, and the double-layer assumptions lead to significant errors in stress calculations.
How does the surrogate model work?
The surrogate model is based on an artificial neural network (ANN) trained on 764 numerical simulations performed with FLAC3D. It takes input parameters such as material properties and geometry, and outputs the constitutive matrix that relates measured strains to the stress tensor in a three-layer medium. This allows for quick and accurate stress calculation without running complex simulations each time.
What is the accuracy of the surrogate model?
The surrogate model achieved a goodness of fit (R2) of 0.999 and a mean squared error (MSE) of 1.254, indicating very high accuracy in predicting the constitutive matrices.
How was the method validated?
The method was validated through numerical simulations, comparison with analytical solutions, and laboratory experiments. It was also applied to a real-world case of mining-induced stress measurement in a coal mine roadway roof, where it corrected errors from using double-layer analytical solutions.
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