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December 27, 2023 Page 6-1 <br />Agapito Associates, Inc. <br />6 NUMERICAL MODELING <br />Numerical modeling has been utilized in this assessment for the following two purposes: (1) to <br />confirm the results of the empirical pillar stability assessment and (2) to estimate the maximum <br />vertical surface subsidence following pillar extraction in the mains. The following sections provide <br />the details of the modeling assessments. <br />6.1 Pillar Stability Analysis <br />To confirm the empirical pillar results, Agapito performed a seven-step LaModel analysis for the <br />pillar extraction shown in Figures 6-1 to 6-7. For the modeling scenarios, the seven zones shown <br />in Figure 4-4 were modeled for stress redistribution during the proposed pillar extraction retreat. <br />These zones were selected to show the highest anticipated stress conditions during retreat. <br />LaModel is a nonlinear, boundary-element, displacement-discontinuity code for estimating stress, <br />displacement, and yielding in tabular deposits such as coal. It can handle in-seam materials based <br />on both linear and nonlinear mechanical (stress-strain) behaviors. The program performs an <br />iterative procedure to solve a set of equations representing the stress-strain state of each element <br />in a grid, emulating the mine geometry until a steady-state equilibrium is reached. Following an <br />MSHA-recommended confined core approach to pillar strength, element properties are arranged <br />so that the weakest elements are adjacent to the mine opening, with element strengths increasing <br />into the solid coal. Strain-softening elements with increasing peak and residual strengths are <br />employed to approximate elastic-plastic behaviors observed in pillars and provide close agreement <br />with classical empirical pillar design methods. <br />The model input parameters were based on a previous calibration undertaken at the mine from <br />in-mine instrumentation results (Agapito 2011). The calibration study, based on E Seam <br />instrumentation, resulted in the LaModel input parameters shown in Table 6-1. It was concluded <br />that a 1,180 psi in-situ coal strength was appropriately conservative for the E Seam, although the <br />true in-situ strength may be higher. <br />Table 6-1. LAMODEL Parameters from <br />Instrumentation Calibration <br />Parameter Value <br />Mining height (ft) 10.0 <br />In-situ coal strength, (psi) 1,180 <br />Young's Modulus (E) (coal) (psi) 477,000 <br />Poisson's ratio (ν) (coal) 0.34 <br />Elastic Modulus (E) (rock mass) (psi) 1,840,000 <br />Poisson's ratio (ν) (rock mass) 0.22 <br />Lamination thickness (ft) 160 <br />Gob stiffness (initial/final) (psi) 35,000/350,000 <br /> <br />