# Improving the influence function method to take ground topographic variations into account in mining subsidence prediction

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## Underground extraction zone and ground mining subsidence

Mining subsidence causes the appearance of a bowl-shape subsidence trough (also named subsidence basin) on the ground surface. The subsidence trough has a three-dimensional geometry that depends on many factors related to the mining method, the geometry of the extraction zone, the underground geology and its mechanical properties, the surface topography, and so on.
As in Figure 1, which illustrates half of a vertical section of the subsidence trough, several parameters of the mining zone are relevant in subsidence studies, including the depth, thickness, and length. With these parameters, the location and the shape of the extraction zone can be determined.
The curves of vertical subsidence (or vertical displacement), horizontal displacement, slope, horizontal strain, and curvature, which can be used to quantitatively represent the mining subsidence, are shown in Figure 1. For each surface point, the vertical subsidence and horizontal displacement can be considered as the movement distance from the original position to the position after subsidence in the vertical and horizontal directions, respectively. These quantities often can be obtained by in situ measurements. The slope is defined as the first derivative of the vertical subsidence, and the curvature is defined as the first derivative of the slope (i.e. the second derivative of the vertical subsidence). The horizontal strain is the first derivative of the horizontal displacement.
The maximum vertical subsidence (Sm), angle of influence ( ), and angle of break ( ) are commonly concerned subsidence parameters as shown in Figure 1, and sometimes are known in a particular mining zone. In Lorraine, for instance, the maximum vertical subsidence value is about 20 – 40% of the mining thickness (Deck 2002). The influence angle is used to describe the edge of the subsidence trough, while the break angle is employed to depict the location of the maximum positive horizontal strain (in the tension area). Usually, the angles of influence and break are 10 – 40° and -10 – 20° (Deck 2010, Saeidi 2009, 2015), respectively.
To distinguish the mining extent, three terminologies are used, including critical area, subcritical area, and supercritical area (NCB 1975). Critical area refers to an area of working, which causes the complete subsidence (i.e. the maximum subsidence under a certain mining condition) of one point on the surface; subcritical area and supercritical area are the areas of working smaller and greater than a critical area. Subsidence caused by the critical, subcritical, and supercritical mining areas can be termed as critical, subcritical, and supercritical subsidence, respectively.

### Mining subsidence in Lorraine

France was one of the major mining nations until the second half of the 20th century, especially rich in coal and iron minerals. But the exploitations in France are totally stopped now.
In Lorraine, the iron basin was intensively exploited from the late 19th century. Until the 1960s, 63 million tons of minette, which consists of iron ore of sedimentary origin, were extracted, benefiting from an increase in industrial production. After that, the production fell due to international competition. The last exploitation was closed in 1997. In Lorraine, the excavated area was around 1300 km2, and more than 3 billion tons of iron ore were extracted (DDE, 2005).
Two major mining technologies, both based on the room and pillar method, were employed: in the first, after excavating the rooms, the remaining pillars were also excavated starting at the farthest point from the stope access so that the overburden collapsed, usually leading to a surface subsidence but eliminating almost any residual risk; while in the second, the rooms and pillars were left in place to serve as a long-term ground support, especially under urban zones where any ground movement is not desired. With the second method, a sufficient number of pillars (with a sufficient size) must be left in order to ensure the stability of the extracted zones. In many cases, protective pillars were kept under urban areas to prevent any risk of subsidence (Geoderis, 2000). As a result, now in Lorraine, there are a lot of abandoned extraction zones supported by pillars at different depths due to old exploitations; some of them being relatively very close to the surface (few meters). In terms of risk analysis, these extraction zones can be considered as hazard zones. As mentioned in many researches carried out by the GISOS1, it is exactly the excavations with the method of abandoned rooms and pillars that cause subsidence problems today. The subsidence process is shown as in Figure 2: due to the fact that pillars can no longer withstand the weight of the overburden after a quite long time and fall to rupture, the overlying layers may gradually settle due to the instability of the remaining pillars, then the subsidence appears on the surface and the buildings (or other surface features) suffer destructions.
Figure 2. Subsidence due to the rupture of the pillars when using the room and pillar method: (a) excavation using the room and pillar method; (b) the rupture of the pillars, the collapse of the extraction zone, and the settlement of the overlying strata; 90% of the subsidence occurs in a few hours or days; (c) the final subsidence trough becomes stable in a few months Some photographs of subsidence examples in Lorraine are shown in Figure 3. In these photographs, the securities and functions of farmlands, buildings, and roads are affected by subsidence.

#### Simplify the mining conditions

(1) Simplify the overlying strata:
How subsidence is affected by topography is a complex problem. There are many types of rocks, and their relative or absolute characteristics may affect the subsidence intensity and profile. For studying the relationship between subsidence and topography, we only considered the case of one continuous stratum of rock formation above the mined layer without taking into account any discontinuity through the system that could affect the transmission of movement from the mined zone to the surface.
(2) Simplify the surface shape:
For testing purposes, the surface condition is simplified. The problem is complicated if the dip angle and dip direction of the surface vary (e.g. a hilly ground surface). Therefore, we checked the new implemented method against a simple topographic variation of the ground, i.e. a global slope over the whole mined area. A global slope means that the surface is dipping in one direction, at one angle. Later, the subsidence under the condition of varying surface slope will also be studied to verify the achievements. The simplification of the overlying strata and surface conditions must be considered before complicated configurations can be approached.

Numerical simulation models

Using the surface and overlying stratum conditions described above, the subsidence laws are determined from a finite difference modeling calculation (FLAC 2D).
The numerical simulation models are similar to the model illustrated in Figure 12. They consist of three strata: one horizontal floor, one horizontal ore body (part of which will be mined), and one roof with variable global dip angle. All strata are isotropic and have the properties mentioned in Table 3. Properties for the ore body and floor come from iron mines in Lorraine (Fougeron et al. 2005). For the roof, the Young’s modulus has been divided by 1000 to increase the subsidence while keeping the material elastic. Reducing the Young’s modulus is not a problem because we are interested in the shape of the vertical subsidence and horizontal displacement more than their magnitudes (which are then adjusted to fit field data). The properties used here to generate a subsidence profile approximately correspond to an influence angle of 45° when the surface is flat. The profile shape can be adjusted to any field influence angle.
In all our numerical simulation models, the horizontal displacement is prevented on both the left and right sides, the vertical displacement is fixed at the bottom, and the top is free. Initial stress field corresponding to gravity loading is given at the start.
Before excavation, the model is solved to achieve equilibrium (we consider the maximum velocity less than 10-7 m as balance). This phase leads to a little adjustment of the given initial stress field. Then all displacements and rotations (which are actually very small) are reset to zero so that the next phase exhibits the displacements induced by the mining excavation only. After that, part of the ore body is excavated and the model is solved until a new equilibrium is reached. Then, the displacements of the top surface are exported for analysis. For each model, the computational time, which depends on the size of the model and the performance of the computer, is around 10 – 30 minutes.
These calculations must be understood as a tool for designing a new influence function, not as a tool to study directly the subsidence of any particular geometry. Moreover, only 2D calculations are used here but the resulting influence function will operate in 3D on almost any kind of surface with varying topography. Therefore, the computational effort is expected to be far less than using 3D numerical models, especially when making some sensitivity studies or back analysis.

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Characteristics of the subsidence changed by the topography

Several models were set up to study the characteristics of the subsidence under simplified model conditions and rock properties when the surface is not flat. Here, three of these models are chosen to illustrate the results.
For comparison, the models are the same with different surface angles and mean depths. The length of the model is 2400 m. The extraction zone is 400 m long, 5 m thick and located in the middle of the ore body. The mean mining depth, which is the elevation difference between the center of the extraction zone and the surface point above it, is 400 or 500 m so that the subsidence remains subcritical (i.e. do not reach its maximum value). The surface dip angle is 0° or 15°, and the surface dips to the negative direction of the x-axis.
The subsidence data (as listed in Table 39 in Annex 1) are achieved by three numerical calculations with FLAC 2D. Generally, we can monitor the positions of surface points before and after mining, then the vertical and horizontal subsidence data can be derived as the differences of them. Figure 13 shows the variation of the vertical subsidence, horizontal displacement, slope and horizontal strain at the ground surface for different surface dip angles and mean mining depths. The vertical and horizontal subsidence curves in Figure 13(a) are obtained directly by the subsidence data (in Table 39). The slope and the horizontal strain in Figure 13(b) are respectively computed as the derivative of the vertical subsidence and the horizontal displacement; they are often concerned in mining damage studies (for example, damage assessment of building upon a mine). The maximum and minimum values of these subsidence curves are listed in Table 4.

Improving the influence function method

The improvements presented in this section are based on the results of the previously described simplified numerical simulations. They have been introduced into a subsidence computation code developed in our laboratory (Deck 2002, Saeidi et al. 2009, 2010 and 2013).
As indicated above, the influence functions are used to simulate element subsidence. By studying the characteristics of element mining subsidence using numerical simulations, we tried to find new asymmetrical influence functions to describe element subsidence, wherein the surface angle and mean depth are used as parameters to integrate topography into the influence function method.

Element mining subsidence

With the simplified surface shape and given rock properties, a small part of the ore body is mined to compute the element subsidence. As mentioned above, the numerical models are the same, but the surface angle and mean depth vary for each. The element mining zone is always located in the center of the ore body, and the top surface slopes to the left side in each model.
To understand the characteristics of element subsidence, two series of simulations were performed: one with varying surface slope angle and a fixed mean mining depth (Figure 14), the other with varying mean mining depth and a fixed surface slope angle (Figure 15).

Chapter 1: State of the art in mining subsidence and building damage assessment
1.1 Mining subsidence caused by underground excavation
1.1.1 Underground extraction zone and ground mining subsidence
1.1.2 Mining subsidence in Lorraine
1.1.3 Methods of mining subsidence calculation
1.2 Building damage caused by mining subsidence
1.2.1 Behavior of building affected by subsidence
1.2.2 Building damage evaluation
Chapter 2: Improving the influence function method to take ground topographic variations into account in mining subsidence prediction
2.1 The influence function method
2.1.1 A widely used method
2.1.2 Principles of the influence function method
2.1.3 Characteristics of subsidence in flat terrain due to horizontal underground mining
2.2 Topography influence on subsidence
2.2.1 Data sources
2.2.2 Characteristics of the subsidence changed by the topography
2.3 Improving the influence function method
2.3.1 Element mining subsidence
2.3.2 Asymmetrical influence function
2.3.3 Full-scale subsidence
2.4 The usage of the developed code
2.4.1 Corrections from field data
2.4.2 The methodology of the developed code
2.5 Application cases
2.5.1 Case study 1
2.5.2 Case study 2
2.6 Conclusions
Chapter 3: Introducing structural mechanics into building damage assessment under mining subsidence
3.1 Plane framed structural model
3.1.1 The choice of the plane framed model
3.1.2 Structural model and its components
3.2 The choice of the matrix displacement method
3.3 Principle of the matrix displacement method for the analysis of a plane framed structure
3.3.1 Preparation of a structural model and the input data lists
3.3.2 Force-displacement relations of an element (mainly after Bao and Gong 2006, Leet et al. 2011)
3.3.3 Force-displacement relations of the structure
3.3.4 Output data
3.3.5 Verification
3.4 Building damage evaluation
3.4.1 Building damage evaluation depending on the internal forces
3.4.2 Kinematic analysis
3.5 Conclusions
Chapter 4: Case study – damage evaluation of Joeuf city due to mining subsidence
4.1 Overview of the city of Joeuf
4.2 Modelling of the mining subsidence in Joeuf
4.2.1 The topography in Joeuf
4.2.2 The iron mines under Joeuf
4.2.3 Mining subsidence computation
4.3 Definition of structural models for the buildings in Joeuf
4.3.1 Investigations about the buildings in Joeuf
4.3.2 Standardization of structural models
4.4 Damage evaluation of the buildings in Joeuf due to mining subsidence
4.4.1 The structural models with the influence of subsidence
4.4.2 The internal force criteria for building damage evaluation
4.4.3 Damage evaluation results in Joeuf
4.5 Conclusions
General conclusions and perspectives
Conclusions générales et perspectives
References

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