13 Elastic Wave Propagation Modeling During Exploratory Drilling …
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13.3.2 Grid-Characteristic Method
A numerical solution is found using the grid-characteristic method [9–14]. We carry
out a coordinatewise splitting, and by changing variables, we reduce the system to a
system of independent scalar transport equations in Riemann invariants:
∂
w
∂t
+ i
∂
w
∂ξ
i
= 0, i = 1, 2, 3.
For each transfer equation, all nodes of the computational mesh are bypassed, and
characteristics are omitted for each node. From the time layer n, the corresponding
component of the vector
w is transferred to the time layer n + 1 by the formula:
w
n+1
k
ξ
i
= w
n
k
ξ
i − ω k τ
,
where τ is the time step.
After all the values are transferred, there is a reverse transition to the vector of the
desired values of
u.
Interpolation on unstructured and regular grids is considered. Values at each point
are found using values at grid reference points
w
r i jkl
and weights of these points
p i jkl ( r ) as
w( r ) =
i, j,k,l
p i jkl ( r )
w
r i jkl
.
The grid-characteristic method allows the most correct algorithms to be applied
at the boundaries and contact boundaries of the integration region [9, 10].
13.3.3 Boundary and Contact Boundary Conditions
The boundary condition can be written in general form as
D
u(ξ 1 , ξ 2 , ξ 3 , t + τ ) =
d,
where D is some 9 × 3 matrix,
d is a vector, and
u(ξ 1 , ξ 2 , ξ 3 , t + τ ) are the values
of the desired velocity values and components of the stress tensor at the boundary
point at the next time step.
At the boundaries of the integration region, the following boundary conditions
were used: