PSI - Issue 43

V.I. Golubev et al. / Procedia Structural Integrity 43 (2023) 29–34 V.I. Golubev et al. / Structural Integrity Procedia 00 (2022) 000 – 000

32

4

{ XΔX = Δ ΔX = +

Finally, the system on X is

2 1 〈 | 3 3+1 | − 1〉 . | 3 3+1 | − 1〉 = Δ ,

(27)

X + 1 2 〈

So,

(28)

and for ≥ | 3 3 +1 | we obtain

X =

2 1 + +2Δ | 3 1 3+1 | .

(29)

Substituting initial notations, we derive that X = | 3 3 +1 | 1+2Δ Finite difference equations on 3 +1 may be written in the form of 3 +1 ΔX + 3 ΔT = 3 , or 1+2 | 3 3+1 | , Δ = Δ . The final form of the correction formulas with the second order of approximation is 3 +1 = | 3 3 +1 | 1+2ΔP 1+2 | 3 3+1 | 3 −ΔT 3 ΔP , ΔP = √ ( Σ 2 − 2 ΔT) − ΔT( S 2 − 2 ΔT) , 3 +1 = 3 − 3 ΔX ΔT = 3 −ΔT 3 ΔP .

(30)

(31)

(32)

(33)

(34)

ΔT = 2√ 1 3 3 〈

√ 3 3 | 3 3 | − 1〉 , Σ = √ 3 3 ,

(35)

S = √ 3 3 , T = √ 3 3 , = 1,2.

(36) In this work, we rely on the grid-characteristic method on the wide stencil, as the explicit elastic solver, that guarantees the stability of the scheme for ℎ < 1 , where is the P-wave velocity. The usage of the polynomial interpolation of the third power provides the grid-characteristic scheme of the third spatial and time approximation (Golubev et al., 2021b; Golubev et al., 2022a). 3. Simulation results The problem of the seismic wave propagation in an inhomogeneous fractured geological model based on the well known Marmousi2 model in a two-dimensional case is considered. The total width of the model is 17000 meters, and the total height is 3500 meters. The geometry and the mechanical properties of the background layered Marmousi2 structure are taken according with (Martin et al., 2006). A single fractured inclusion is inserted into this elastic model. An example of the model is presented on Fig. 1.

Fig. 1. The elastic two-dimensional Marmousi2 model with the fractured inclusion (black color). Density values are represented in the gray scale.