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|2.1 INTRODUCTION When total extraction of an opening of sufficient size is reached in a horizontal coal seam, the roof strata in the overburden deform continuously to reach a new equilibrium condition. The severity of deformation decreases upward toward the surface. As the downward saggings of the strata propagate and reach the surface, there will be a depression zone on the surface directly above, but extending beyond the edges of the underground opening. This is the surface subsidence basin or surface subsidence trough. The surface subsidence basin is circular in plan view, if the coal seam is horizontal and the mined-out opening is square in shape. But it is rectangular with rounded corners or elliptical if the coal seam is horizontal and the mined-out opening is a long- and thin rectangle or a short-rectangular, respectively (Fig. 2.1). Most underground openings (e.g., longwall panel) assume rectangular shape when total extraction has been completed. Theoretically the edges of the subsidence basin are the points of zero subsidence. But it is difficult to exactly locate the points of zero subsidence. Therefore in practice the points with vertical subsidence of 0.4 in. (10 mm) are used. The final subsidence basin is that which exists long after the mining has been completed, because its magnitude and shape are quite different from the dynamic subsidence basin formed while the face is moving. 2.2 CHARACTERISTICS AND TYPES OF DEFORMATION IN THE FINAL SUBSIDENCE BASIN For a horizontal coal seam, every point in the subsidence basin moves toward the center of the basin. Subsidence is maximum at the center of the basin. Any cross-section that passes through the point of maximum subsidence and either parallel to AB or CD line (Fig. 2.1) is a major cross-section along which principal directions of surface movements occur. However among those infinite numbers of major cross-sections, two specific ones are of special significance, not only because the magnitudes of surface movements are the largest, but also because they are the most easily identifiable directions, i.e., one that is parallel to the faceline at the center of the basin (CD in Fig. 2.1) and' the other is that perpendicular to the faceline but parallel to the diction of face advance (AB in Fig. 2.1). Nearly all the subsidence data obtained in the US have been derived from these two cross-sections, although some cross- sections parallel to CD but near the edges of the panel have also been included. In addition to moving horizontally toward the center of the basin, every point in the basin also subsides vertically. The magnitude of subsidence increases toward the center of the basin. Therefore surface subsidence is a three-dimensional problem and should be treated so in all cases. On all the major cross-sections, only principal subsidence and principal displacement occur. Since subsidence and displacement vary continuously in every major cross-section, three additional deformation components are de- rived, i.e., slope, curvature, and strain. On all other non-major cross-sections on the other hand the five components are accompanied by two additional components, i.e., twisting and shear strain. The seven components of the surface movement are defined as follows (Fig. 2.2): 1. Subsidence, S. On any cross-section, the vertical component of the surface movement vector is called surface subsidence. It generally points downward. But sometimes it points upward in areas ahead of the faceline or beyond the edges of the opening. In such cases it is a surface heave which is usually less than 6 in. 2. Displacement, U. On any cross-section, the horizontal component of the surface movement vector is called surface horizontal displacement. It generally points toward the center of the subsidence basin. But in steep terrain, it moves along the downdip direction 3. Slope, i. On any cross-section, the difference in surface subsidence between the two end points of a line section divided by the horizontal distance between the two points is called the surface slope of the section. 4. Curvature, K. On any cross-section, the difference in surface slope between two adjacent line sections divided by the average length of the two line sections is called the surface curvature of those two line sections. There are two types of curvature: con- vex or positive curvature and concave or negative curvature. 5. Horizontal strain, e. On any cross-section, the difference in horizontal displacement between any two points divided by the distance between the two points is called horizontal strain. If the distance between the two points is lengthening, it is tensile strain with positive sign. Conversely, if it is shortening, it is compressive strain with negative sign 6. Twisting, T. On the surface of the subsidence basin, the difference in slope between two parallel line sections divided by the distance between the two line sections is called twisting. 7. Shear strain, y. Shear strain is the changes in internal angles of a square on the surface of the subsidence basin or on any major cross-section. It is the summation of the differences in incremental (or decremental) lengths between the two opposite sides divided by the original distance between the two opposite sides. More precisely, the surface deformation indices (i.e., slope, strain, curvature, twisting and shear) are defined by derivatives of surface movement components. For simplicity, the x- and y-axes of the cartesian coordinate system are set to be parallel and perpendicular to the cross-section of interest, respectively. In such a coordinate system, slope and curvature along x direction are the first and the second derivatives of the vertical components (S) of surface movement with respect to x, respectively, or i, = ds/dx and kx = d2s/dx2. Horizontal strain along x direction is the first derivative of the component along x direction of the horizontal displacement,|