Issue 63

L. Nazarova et alii, Frattura ed Integrità Strutturale, 63 (2023) 13-25; DOI: 10.3221/IGF-ESIS.63.02

 



V σ

A B

ασ

(5)

( )

exp(

)

0

0

where A 0 =2698 m/s, B 0 =484 m/s, α =0.0779 1/MPa [47]. Fig. 4 depicts the velocities V in the sub-panel at different distances between WZ and the roadway (solid lines— b =8 m, dashed lines— b =6 m), calculated by model (1)–(4) with regard to (5) and at the mean normal stress σ =(1+ ν )( σ xx + σ yy )/3 as per the plane strain conditions. Apparently, the velocity fields in the vicinity of WZ have a noticeable difference detectable using seismic tomography if the first break times are determined with an absolute accuracy of no less than 0.1 ms (spatial resolution 0.2 m).

Figure 4: Isolines of P- wave velocity V (m/s) in coal bed at different locations of WZ. We formulate the inverse boundary value as follows: at the known vertical stresses σ yy ( x , H )= F y ( x ) in the coal bed roof, we find the shear stresses σ xy ( x , H )= F x ( x ) if the distribution of P-wave velocities in the illuminated domain I (Fig. 3) is known at a given shooting scheme. The roof and floor sites where σ xy ( x , H ) ≈ 0 are associated with WZ. This formulation is allowable since the distribution of the abutment pressure in subhorizontal coal mining can be found a priori (based on the configuration of mined-out space and rock density), while the conditions at the coal-bed–host rock interface are usually unknown [5, 30, 48]. Let us generate input data for the inverse problem. For a certain value of b and the above parameters of model (1)–(4) in the region R ={| x | ≤ L , | y | ≤ H } (Fig. 2), we calculate the stresses and then, using (5), the distribution of P-wave velocities V 0 ( x , y ). In the domain I ={| x| ≤ L , 1 ≤ y ≤ 3}, we define a shooting scheme that includes five sources S q ( q =1,…,5) and receivers T p ( p =1,…,5) at the vertical boundaries of the sub-panel (red circles in Fig. 2). Based on V 0 ( x , y ), we calculate the first break times 0 pq t to be later on imposed with the multiplicative noise with the magnitude δ =0.1 ms

  *

0 (1 )

t

pq δξ t ,

pq

where ξ is a random variable uniformly distributed over [-1,1]. Using the original code [49], we perform tomography by * { } pq t , find the distribution of the velocities V * ( x , y ) in the illuminated domain I (Fig. 5) and, finally, from (5), obtain the input data

B

 1 ( , ) ln

0

.

(6)

σ x y

*

 A V x y ( , )

α

0

*

Figure 5: Synthesized P-wave velocity V * distribution. In the domain G of the finite element grid, we select a fragment corresponding to the extraction sub-panel R (Fig. 2). For each element of the set     1 1 2 1 , , ..., } m m m {x L x x L , we set the conditions at ∂ R for (1)–(3)

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