Modified Mohr-Coulomb Criterion and Risk Coefficient for Mining-Induced Fault Slip

Imagine the Earth's crust as a giant, multi-layered cake, but instead of frosting and sponge, it's made of rock, coal, and ancient faults—cracks where the ground has slipped in the past. Deep underground, miners dig tunnels to get the coal, which is like scooping out a slice of that cake. But here's the tricky part: when you remove a slice, the layers above it don't just sit there; they shift, sag, and sometimes snap. This is where things get dangerous. If a hidden crack (a fault) is nearby, the shifting weight of the rock above can push or pull on it. If the push is too strong, the crack can suddenly slip, releasing a massive burst of energy. In the mining world, this is called a "coal burst," and it's like a tiny, underground earthquake that can be devastating. To predict this, scientists use a classic rule called the Mohr-Coulomb criterion. Think of this rule as a simple math equation that says, "If the sliding force on a crack gets stronger than the friction holding it together, the crack will slip." However, this old rule was designed for static situations, like a rock sitting still. It didn't account for the chaotic, moving stress caused by a mining machine chugging forward.

This paper, written by Shi Ruiming and Wang Hongwei, tries to fix that old rule by adding the "mining effect" to the mix. They wanted to see exactly how the stress on a fault changes as a mining face gets closer, and they wanted to create a new "risk score" to tell miners when a fault is about to slip. They didn't just guess; they built a physical model using sand and lime to mimic the rock layers, and they used powerful computer simulations (FLAC3D) to run thousands of scenarios. They looked at how the angle of the fault and the pressure from the sides of the earth (lateral pressure) change the game. Their main finding is that the risk isn't just about how close the miner is to the fault; it's a complex dance between the fault's angle, the side pressure, and the distance. They discovered that high-angle faults (steep cracks) are particularly tricky because they tend to hold onto stress until it suddenly snaps, while the side pressure determines where on the fault the stress lets go. They propose a new "Fault Slip Risk Coefficient" ($R_f$) that acts like a real-time dashboard, showing miners exactly how close they are to a disaster based on these changing conditions.

The Story of the Slipping Crack

So, how do you know if a crack in the ground is going to give way? The authors started by updating the old "slip rule." In the past, scientists used a formula that compared the sliding force to the friction holding the rock together. If the slide was stronger, the rock slipped. But this was like trying to predict a car crash by only looking at the car when it's parked. It didn't account for the fact that the ground is shaking because a giant machine is digging nearby. The authors modified the formula to include the "mining effect." They realized that as the mining face moves, it creates a zone of high stress right in front of it and a zone of low stress (relief) behind it. This moving stress wave hits the fault, changing the forces on it in real-time.

To test this, they built a miniature world in a lab. They used a block of material made of fine sand, lime, and gypsum to represent the rock layers, with a "fault" running through it. They placed sensors (little pressure detectors) all over the model to watch what happened as they "mined" (cut away) the material layer by layer. They found that as they dug closer to the fault, the stress on the fault didn't just creep up; it behaved in stages. First, it was stable. Then, as they got closer, the stress started to rise slowly. Finally, when the mining face was very close (within about 70 cm in their model), the stress spiked sharply, and the fault actually slipped. The physical model showed that the roof rock would collapse in big chunks, releasing huge amounts of energy that triggered the fault to move.

But a sand model can only tell you so much. To get a deeper look, the authors turned to a computer simulation. They built a digital version of the mine, 192 meters wide and 48 meters deep, and they tested it with faults at different angles: 35°, 45°, 55°, 65°, and 75°. They also changed the "lateral pressure coefficient," which is basically a number that tells us how much the earth is squeezing the rock from the sides. They tested three levels: 0.7 (low squeeze), 1.0 (medium squeeze), and 1.3 (high squeeze).

Here is where the story gets interesting. The angle of the fault matters a lot. The simulations showed that steep faults (high dip angles) are more dangerous than shallow ones. Why? Because steep faults tend to hold onto their "normal stress" (the force pressing the two sides of the crack together) even as the mining gets close. When the mining face finally gets very near, this stress doesn't just fade away; it stays high, or even increases in some spots, while the "shear stress" (the force trying to slide the crack) jumps up and down wildly. It's like a rubber band that is stretched tight and then suddenly snapped. The authors found that for these steep faults, the shear stress can jump from its lowest point to its highest point very quickly, creating a perfect storm for a sudden slip.

The side pressure (lateral pressure coefficient) also plays a huge role in where the danger lies. When the side pressure is low (0.7), the stress release happens mostly at the top of the fault. This means the top part becomes loose, but the bottom stays tight, so the whole fault doesn't slip all at once. However, when the side pressure is high (1.0 or 1.3), the story changes. The stress release shifts to the bottom of the fault. Now, the bottom part becomes loose while the top stays tight, but more importantly, the shear stress builds up across the entire fault. The authors found that with high side pressure, the fault becomes a "stress accumulation area," meaning the whole thing is primed to slip at once. It's the difference between a door hinge that's a bit loose at the top versus a door that's been jammed shut and then suddenly forced open.

The New Risk Score

To make all this useful for real miners, the authors created a new tool: the Fault Slip Risk Coefficient ($R_f$). Imagine a dashboard in a mine control room. This number is calculated by dividing the force trying to slide the fault by the force holding it together.

• If the number is less than 1.0, the fault is safe (the holding force wins).
• If the number is exactly 1.0, the fault is on the edge of slipping.
• If the number is greater than 1.0, the fault has slipped or is about to.

The authors used their simulations to see how this number changes as the mining face moves. They found that the risk isn't constant. When the mining face is far away, a fault with a 45° angle actually shows the highest risk. But as the mining face gets closer to the fault, the risk shifts. Suddenly, the steep faults (65° and 75°) become the most dangerous. It's like a race where the lead changes depending on how close you are to the finish line.

Furthermore, the side pressure changes the race too. In the early stages of mining, a lower side pressure (0.7) actually made the risk score higher. But as the mining face got closer, a higher side pressure (1.3) caused the risk score to skyrocket across the whole fault. The authors suggest that the most dangerous scenario is a combination of a steep fault and high side pressure when the mining face is right next to the fault. In this situation, the fault is under high "shear-compression," meaning it's being squeezed and pushed to slide at the same time, making a sudden, violent slip very likely.

What This Means

The paper doesn't claim to have solved the problem of coal bursts forever. Instead, it offers a new way to look at the problem. By modifying the old slip rule to include the moving stress of mining, and by creating a dynamic risk score, the authors provide a method to see the danger as it happens. They showed that you can't just look at the fault's angle or the pressure separately; you have to watch how they interact as the mining face moves. Their simulations suggest that if you have a steep fault and high side pressure, you need to be extra careful when the mining face gets close. The risk isn't just a static number; it's a moving target that changes with every meter of coal dug.

In short, this research gives miners a better map. It tells them that the danger isn't just about how deep they are or how big the fault is, but about the specific dance between the fault's angle, the earth's side squeeze, and how close the mining machine is. By watching the "Risk Coefficient" ($R_f$) climb, they might be able to stop the machine before the ground decides to let go.