⚡ electrical engineering

Analysis of Casing Loads in Coalbed Methane Horizontal Wells during Multi-Stage Hydraulic Fracturing

This study develops a high-precision, composite induced-stress prediction model for deep coalbed methane horizontal wells in the Ordos Basin by integrating a 3D geomechanical model with a modified analytical approach optimized via the sparrow search algorithm, ultimately providing a validated framework for optimizing fracturing parameters and ensuring casing integrity during multi-stage hydraulic fracturing.

Original authors: Qiang Miao, Zhili Zhang, Zenglong Wang, Jinliang Han, Yipu Chen, Gan Yang, Kanhua Su, Meng Li, Mei Kuang

Published 2026-08-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Qiang Miao, Zhili Zhang, Zenglong Wang, Jinliang Han, Yipu Chen, Gan Yang, Kanhua Su, Meng Li, Mei Kuang

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 the Earth's crust as a giant, multi-layered cake, but instead of frosting and sponge, it's made of rock, sand, and ancient coal. Deep inside this cake, trapped in the coal layers, is a valuable gas called coalbed methane. To get this gas out, engineers drill long, horizontal tunnels deep underground and then use a technique called "hydraulic fracturing." Think of this like injecting a super-strong, high-pressure slushie into the coal to crack it open, creating a network of tiny tunnels so the gas can flow freely. However, there's a catch: the pipes (called "casing") that line these tunnels are like the straws holding up the cake. When the slushie hits the coal, it doesn't just crack the rock; it creates a massive, invisible shockwave of stress that pushes and squeezes the pipe from the outside. If the pipe gets squeezed too hard, it can bend, crumple, or break, ruining the whole operation. This is a big problem because these pipes are buried miles underground, and if they fail, the gas is lost, and the well is damaged.

This paper dives into a specific, tricky version of this problem: what happens when you try to crack open deep coal seams (over 1,500 meters down) using a "multi-stage" approach? Instead of cracking the whole tunnel at once, engineers do it in sections, one after another, like popping a row of balloons. Each time they pop a balloon (fracture a stage), the stress wave travels through the rock and hits the next section of the pipe. The deeper the coal, the stranger the rock behaves, and the more complex these stress waves become. The big question is: how do we predict exactly how much the pipe will be squeezed so we don't accidentally crush it?

The researchers, led by Qiang Miao and Zhili Zhang, decided to build a super-accurate digital map of the underground world to solve this. They focused on a real gas field in the Ordos Basin in China, looking at a specific coal layer called the Taiyuan Formation, which sits about 2,899 meters deep. First, they built a 3D geomechanical model—a virtual twin of the underground rock, coal, and pipes. They fed this model with real data from drilling logs, rock samples, and even wide-area electromagnetic monitoring (which is like using a giant MRI to see where the cracks actually formed). They checked their model against reality and found it was spot-on, matching the actual fracture shapes and sizes with over 92% accuracy.

Once they had a reliable map, they tackled the math. The old ways of calculating stress were like using a simple ruler to measure a squiggly line; they assumed the rock was uniform and the stress traveled perfectly straight. But deep coal is messy and full of tiny cracks (cleats) that soak up the fracturing fluid, making the stress behave differently. The team realized the old math was overestimating the pressure by about 27% and underestimating how quickly the stress faded away as it traveled. To fix this, they introduced two "correction knobs" (mathematical coefficients) to tweak the formulas. They then used a smart computer algorithm called the "Sparrow Search Algorithm" (imagine a flock of birds looking for the best spot to land) to find the perfect settings for these knobs. This new, corrected model predicted the stress with much higher precision, achieving a match rate of 93.7% with their computer simulations.

With their new, improved stress calculator, they ran thousands of simulations to see how different drilling choices affected the pipe. They tested variables like how much fluid they pumped (treatment scale), how fast they pumped it (injection rate), how far apart the cracks were (fracture spacing), and how many cracks they made (number of stages). They discovered a few key patterns:

The "Tipping Point": The stress on the pipe builds up quickly at first but then levels off. After about the 7th stage of fracturing, adding more cracks doesn't squeeze the pipe much harder because the stress waves from the distant cracks have faded away.
The Speed Trap: Pumping faster creates a bigger squeeze. If they pumped at 24 cubic meters per minute, the pipe felt about 18% more stress than at a slower 16 cubic meters per minute.
The Spacing Rule: Putting the cracks closer together (50 meters apart) made the stress pile up much more than spacing them out (90 meters apart).
The Volume Myth: Surprisingly, just pumping more total fluid (increasing the scale from 2,000 to 6,000 cubic meters) didn't change the stress as much as changing the speed or spacing did.

To make this useful for real engineers, the team created a "reference sheet" (a regression model and prediction charts). This tool lets engineers plug in their planned numbers and instantly see how much stress the pipe will face. They tested this tool on a real well (Well Case 3). The model predicted the pipe would start to bend (yield) around the 6th fracturing stage. In the real world, the pipe actually started showing signs of bending after the 4th stage. While there was a two-stage difference, the researchers explained this is likely because their model used average numbers and didn't account for the wild, real-time pressure spikes or the cooling effect of the cold fluid, which can make the pipe more brittle. Despite the small gap, the tool was close enough to be very helpful.

In the end, the paper suggests that to keep these deep wells safe, engineers should focus less on the total amount of fluid they use and more on controlling how fast they pump and how far apart they space their cracks. By slowing down the pump and spreading out the cracks, they can significantly reduce the risk of crushing the pipe, ensuring the gas can flow safely for years to come.

Technical Summary: Analysis of Casing Loads in Coalbed Methane Horizontal Wells during Multi-Stage Hydraulic Fracturing

Problem Statement
Deep coalbed methane (CBM) resources, particularly those in the Ordos Basin of China, present significant strategic value but pose unique engineering challenges. During multi-stage hydraulic fracturing of deep horizontal wells, fracture propagation induces complex stress redistribution that superimposes on the original in-situ stress field. This results in abnormal increases in casing stress, elevating the risks of yielding, deformation, and failure. Existing theoretical models and engineering charts, largely derived from shale gas reservoirs or shallow coal seams, suffer from three primary limitations when applied to deep CBM: (1) insufficient quantitative sensitivity analysis of key operational parameters (e.g., injection rate, fracture spacing); (2) limited applicability due to the failure to account for high in-situ stress, strong heterogeneity, and complex fracture networks specific to deep coal; and (3) inadequate model refinement, as traditional analytical models rely on simplifying assumptions regarding net-pressure attenuation and stress interference ranges that do not hold in deep coal environments.

Methodology
The study focuses on the Taiyuan Formation coal seam in the Daniudi Gas Field, Ordos Basin, employing a multi-step integrated approach:

3D Geomechanical Modeling: A high-precision three-dimensional geomechanical model was constructed using geostatistical methods, integrating stratigraphic data, logging curves, rock mechanics experiments, and in-situ stress tests. The model reliability was verified against wide-area electromagnetic monitoring data, achieving a fracture-affected area matching degree exceeding 92%.
Modified Induced-Stress Model: The authors developed a composite induced-stress prediction model by modifying the classical analytical solution for single-fracture induced stress. Two correction coefficients were introduced:
- Fracture Net-Pressure Correction Coefficient ( $A$ ): To account for enhanced fluid leakoff due to the cleat system in coal seams.
- Stress Attenuation Distance Correction Coefficient ( $B$ ): To address the distance-dependent attenuation of stress propagation in heterogeneous coal media.
Parameter Inversion: Five intelligent optimization algorithms (Particle Swarm Optimization, Sparrow Search Algorithm, Simulated Annealing, Genetic Algorithm, and Nonlinear Least Squares) were compared to invert the correction coefficients. The Sparrow Search Algorithm (SSA) was selected for its superior convergence and accuracy.
Multi-Stage Superposition: A composite stress model was established by coupling treatment scale, injection rate, fracture spacing, and stage number using the principle of linear superposition.
Finite Element Simulation and Response Surface Methodology (RSM): A 3D finite element model of the casing–cement sheath–formation system was built in ABAQUS. A Box–Behnken design (BBD) with 25 numerical simulation schemes was used to develop a quadratic regression model for casing stress prediction.
Field Validation: The predictive model was validated against field data from Well Case 3, comparing predicted yielding stages with actual casing inner diameter variations measured by caliper logs.

Key Contributions and Results

Model Accuracy and Optimization: The SSA algorithm achieved the best parameter inversion performance ( $R^2 = 0.9371$ , RMSE = 0.3761). The optimal correction coefficients were determined to be $A = 0.7330$ and $B = 0.9238$ . These values indicate that the effective net pressure in deep coal seams is approximately 26.7% lower than theoretical PKN model predictions due to fluid leakoff, and stress propagation distance is slightly attenuated compared to homogeneous models.
Stress Evolution Trends: The composite induced stress increases rapidly during the initial fracturing stages and stabilizes after approximately the seventh stage. Higher injection rates and smaller fracture spacing were found to significantly increase peak composite induced stress. Specifically, a 24 m³/min injection rate with 50 m spacing created the highest risk conditions.
Regression Model and Sensitivity: A quadratic regression model was established linking operational parameters to maximum equivalent casing stress. The model demonstrated high statistical significance ( $P < 0.0001$ $P < 0.0001$ ) and fitting accuracy ( $R^2 = 0.9826$ $R^{2} = 0.9826$ ). Sensitivity analysis revealed the following ranking of influence on casing stress:
1. Fracturing stage number
2. Injection rate
3. Fracture spacing
4. Treatment scale
Field Validation: In the validation case (Well Case 3), the model predicted casing yielding at the 6th fracturing stage, while field caliper data indicated an abrupt inner-diameter change (yielding) occurred after the 4th stage. The authors attribute the two-stage deviation to factors such as the model's use of average internal pressure (ignoring dynamic fluctuations), the simplification of fractures as symmetric single entities (ignoring complex networks), spatial heterogeneity of rock parameters, and the exclusion of thermal stress effects from low-temperature fracturing fluids.

Significance and Claims
The paper claims to provide theoretical and technical support for optimizing fracturing parameters and protecting casing integrity in deep coalbed methane reservoirs. By addressing the specific heterogeneity and stress conditions of deep coal seams, the study offers a refined prediction tool that overcomes the limitations of existing shale-based models. The generated casing stress prediction charts and regression equations are presented as practical tools for engineers to adjust injection rates and fracture spacing to mitigate casing failure risks. The study emphasizes that while the model shows reasonable accuracy, it serves as a guide for parameter optimization rather than a perfect predictor of dynamic field events, acknowledging the inherent complexities of deep coal reservoirs.

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