Geology Reference
In-Depth Information
10.3.3.1
General displacement approach.
The work described here is completely new and original. The general
displacement equations governing elasticity in transversely isotropic media
without attenuation are given in Love (1944) - again, as we have demonstrated
previously, displacement formulations are ideal for acoustic source modeling.
The influence of porosity and coupled matrix flow can be considered, but for
simplicity, we will not do so here, focusing only on a simple limit for illustrative
purposes. If x and y represent coordinates in the cross-plane of the borehole and
z is axial, and if u(x,y,z,t), v(x,y,z,t) and w(x,y,z,t) represent displacements in
the x, y and z directions, respectively, and is density, the following coupled,
linear, second-order transient partial differential equations are obtained.
Horizontal u(z,x,y,t) equation
2
u/ t
2
= A
2
u/ x
2
+ N
2
u/ y
2
+ L
2
u/ z
2
(10.27a)
2
v/ x y + (F + L)
2
w/ x z
+ (A - 2N)
Horizontal v(z,x,y,t) equation
2
v/ t
2
= N
2
v/ x
2
+ A
2
v/ y
2
+ L
2
v/ z
2
(10.27b)
2
u/ x y + (F + L)
2
w/ y z
+ (A - 2N)
Vertical w(z,x,y,t) equation
2
w/ t
2
= L
2
w/ x
2
+ L
2
w/ y
2
+ C
2
w/ z
2
(10.27c)
+ ( F + L)
2
u/ x z + ( F + L)
2
v/ y z
The constants A, C, F, L and N are Love' s constants; they reduce to A = C
= + 2 , L = N = and F = in the limit of vanishing anisotropy, where and
are the Lame coefficients. Also, the linear wave operator in the first line of
Equation 10.27a contains “u” only, while the second contains mixed derivatives
in “v” and “w” (similarly, the first line of Equation 10.27b contains “v” only,
while the second contains mixed u and w derivative terms, and the first line in
Equation 10.27c contains “w” only, while the second contains mixed derivatives
in u and v) - we do not have three independent (classical) wave equations in
each of u, v and w, but instead, highly coupled elastic vibration modes as is well
known in seismic modeling. White (1983) gives measured values for the above
constants in different formations and also solves the Cartesian form of the
displacement formulation using numerical Fourier transform techniques. Others
have solved these equations on Cartesian meshes using velocity-stress
formulations on staggered grids - one must anticipate, however, numerical noise
originating from boundary condition implementation at curved borehole walls.
In our modeling, we found borehole geophysics extremely challenging because
it allowed us to expand and combine all of the ideas developed previously in the
context of a single application. We now describe the solution strategy
developed to address the Department of Energy sponsored project.
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