Effects of tool eccentricity on wave dispersion properties in borehole
acoustic logging while drilling
Yibing Zheng and M. Nafi Toksöz
Earth Resources Laboratory
Dept. of Earth, Atmospheric, and Planetary Sciences
Massachusetts Institute of Technology
Cambridge, MA 02139
Abstract
In this paper, a finite element approach is applied to study the dispersion properties of non-leaky
acoustic waves inside boreholes with off-centered LWD (logging while drilling) tools. Both soft and hard
formation cases are studied with focus on phase velocity dispersions of Stoneley, dipole and quadrupole
modes. When an LWD tool is off-centered, the dispersion curve of the Stoneley mode shifts to the lower
phase velocity compared with that of a centered-tool condition. Meanwhile, the dipole dispersion curve
splits into two curves with different phase velocities. This splitting phenomenon is also observed with the
quadrupole mode. The splitting and shifting of the dipole mode are greater than those of the quadrupole
mode. These new observations are important for interpreting acoustic LWD data accurately.
1
Introduction
Acoustic logging while drilling (LWD) becomes a widely used practice in petroleum industry. The structure
of an LWD tool is quite different from that of a traditional wireline tool. An acoustic LWD tool is assembled
into the drill collar. Its size is much larger than the size of a wireline tool. Drilling mud is circulated through
a hole at its center. Therefore, an acoustic LWD tool occupies a major portion of the borehole cross-section.
The fluid annulus between the tool and the borehole wall is thin, as shown in Figure 1. When the tool is
off-centered slightly, the percentage change of the thickness of the fluid annulus can be large. This suggests
that the issue of the tool eccentricity may be more important in LWD condition than in wireline logging. It is
still unclear how an off-centered LWD tool will affect the acoustic measurements. This problem is addressed
in this study.
Most early analysis and modeling of acoustic logging while drilling are based on a concentric condition of
the tool and the borehole (Tang et al., 2002; Tang and Cheng, 2004). This geometry is axisymmetric, and it
is convenient to use the cylindrical coordinate system. The analytical method has been applied successfully
in the study of acoustic responses of a borehole with a centered LWD tool, in both soft and hard formations
(Rao and Vandiver, 1999; Rao et al., 1999). However, it becomes a complicated and tedious job for the
analytical method to deal with off-centered tools, since the boundary conditions need to be formulated using
an infinite series of high order Bessel functions.
The finite difference time domain (FDTD) method has been used to investigate the effects of tool eccentricity on borehole acoustic measurements (Huang, 2003). In the FDTD approach, the acoustic sources
and receivers are positioned as those are located on a real LWD tool. Wavelet functions are applied to the
acoustic sources, and the wave propagation inside the borehole is simulated in the time domain. Wave forms
from the array of receivers are collected and analyzed to get the acoustic dispersion curves, the same procedure used in real logging data analysis. The acoustic measurements with monopole, dipole and quadrupole
sources are studied. When the tool is off-centered, multipole modes are observed. The way of identifying
each acoustic mode in the FDTD simulation is to overlay the theoretical dispersion curves on the calculated
curves. However, the dispersion curves with an off-centered tool may differ from those of the concentric tool.
1
Formation
Formation
Inner Fluid
Borehole
Borehole
Fluid
Fluid
Wireline Tool
LWD Tool
a
b
Figure 1: Configurations of (a) a wireline tool and (b) an LWD tool
In this study, we use the ability of the finite element method (FEM) in solving wave eigenvalue problems to
directly calculate dispersion curves of non-leaky modes inside boreholes with centered or off-centered LWD
tools. The wave equations are solved in the frequency-wavenumber domain, where for each wavenumber
along the borehole axis, its corresponding eigen-frequencies are calculated to obtain dispersion curves. In
the following sections, the theory and methods used in the finite element modeling will be discussed in detail.
This approach is validated by the comparison of both analytical and numerical results of a centered tool. The
effects of tool eccentricity on acoustic dispersion properties are studied for both soft and hard formations.
2
Method
The borehole is modeled as an acoustic waveguide extending to infinity along the borehole axis z. All
properties of the formation, the fluid and the tool are assumed to be invariable along the z-axis. The
surrounding formation extends to infinity away from the borehole. This model has four layers. From the
outmost to the center, the layers are the formation, the borehole fluid (drilling mud), the LWD tool collar,
and the inner fluid (drilling mud) inside the tool.
In this paper, we concentrate our study on the non-attenuated guided modes (trapped modes), whose
energy is fully trapped without any loss. The phase velocities of these modes along the direction of the
borehole axis are smaller than the formation shear velocity.
To solve the acoustic wave propagation inside a borehole, the typical analytical approach is to develop
wave equations in each layer. We can then match the boundary conditions at the interfaces between the
layers, and apply the condition that the wave energy is zero at infinity. The acoustic dispersion relations
are obtained when non-zero wave solutions are acquired. The real wavenumbers represent non-attenuated
guided waves. The complex wavenumbers are related to leaky modes.
To solve this problem by the finite element method, we start with wave equations in frequency domain.
Wave equation for an acoustic wave in a fluid can be written as
vf2 ∇2 p + ω 2 p = 0,
(1)
where p is the pressure, vf is the wave velocity in the fluid, and ω is the angular frequency. Combining it
with the boundary conditions, we can generate the FEM formulation as
[K]{p̂} − ω 2 [M]{p̂} = {0},
(2)
where [K] and [M] are the stiffness and mass matrices in the finite element method. {} denotes a single
column matrix. {p̂} represent the amplitudes of acoustic field at the element nodes. Therefore, in the FEM,
it becomes a general eigenvalue problem (Zienkiewicz and Taylor, 2000).
2
2.1
Weak forms of wave equations in FEM
The geometry of our problem contains four fluid and solid layers. In the fluid phases, the pressure, p, is used
as the primary variable in FEM formulation, while in the solid phases, the wave equation is formulated with
the displacement vector, ~us (with three components). The wave equations in the fluid and solid phases are
1 ∂2p
− ∇2 p = 0,
vf2 ∂t2
(3)
∂ 2 ~us
− ∇ · (C : E) = 0,
(4)
∂t2
where C is the elasticity tenser, E is the strain tensor, and ρs is the density of the solid. The subscripts f
and s denote the fluid and the solid.
At the interface between a solid and a fluid, the normal particle velocities on both sides must be continuous, that is,
~u˙ f · n = ~u˙ s · n,
(5)
ρs
where n is the normal unit vector of the interface. In the fluid side, the equation of motion is
ρf
∂ 2 ~uf
= −∇p.
∂t2
(6)
Taking the time derivative of the velocity boundary condition, Eq. (5), and substituting Eq. (6) into it, we
get a boundary condition
1
1 ∂p
¨ s · n.
= −~u
(7)
∇p · n =
ρf
ρf ∂n
There is another pressure boundary condition at the interface, which states that the normal force on each
side must be continuous, so that
~tn = n · T = −pn,
(8)
where T is the stress in the solid.
To derive the FEM formulation for the wave equations, we use the variational principles (Zienkiewicz
and Taylor, 2000). For the fluid, we can write the wave equation, Eq. (3), in its weak form
"
#
Z
1
1 ∂2p
2
δΠf =
δp
− ∇ p dΩ = 0.
(9)
ρf vf2 ∂t2
Ωf
Integrating it by parts and substituting the boundary condition Eq. (7) in it lead to
Z
Z
Z
1
1
1
T
(∇δp) ∇pdΩ +
(δp)p̈dΩ −
δp∇p · nf dΓ
2
Ω f ρf
Ωf ρf vf
Γ ρf
Z
Z
Z
1
1
¨ s · nf )dΓ
=
(∇δp)T ∇pdΩ +
(δp)p̈dΩ
+
δp(~u
2
Ω f ρf
Ωf ρf vf
Γ
= 0,
where Ωf is the fluid volume, and Γ is the area of the interface.
Similarly, for the solid phase, the weak form of its wave equation can be written as
Z
Z
Z
¨ s dΩ − δ~us · ~tn dΓ = 0,
(Sδ~us )T : C : (S~us )dΩ +
ρs δ~us · ~u
s
Ωs
Ωs
(10)
(11)
Γ
where S is the strain operator as
E = S~u.
(12)
ns = −nf ,
(13)
Since at the solid-fluid interface,
3
the boundary condition Eq. (8) becomes
~tns = −pns = pnf .
(14)
With this boundary condition, Eq. (11) has the form as
Z
Z
Z
¨ s dΩ − (δ~us · nf )pdΓ = 0.
(Sδ~us )T : C : (S~us )dΩ +
ρs δ~us · ~u
Ωs
2.2
Ωs
(15)
Γ
Discrete coupled system
Layers of the fluids and solids in this model are coupled by the boundary conditions. In the weak forms, the
coupling terms are the integrals over the interfaces.
To discretize the system, for the fluid phase, the acoustic pressure is approximated as
p ≈ p̂ = Np p̃.
(16)
While the displacement vector in the solid phase is approximated as
~us ≈ ûs = Nu ũ.
(17)
Here p̃ and ũ are the nodal pressures and displacements in fluid and solid layers, and Np and Nu are the
appropriate shape functions.
Applying the discretization to the weak form of the wave equation in the solid, Eq. (15), we have the
FEM formulation as
¨ − Qp̃ = 0,
(18)
Ks ũ + Ms ũ
where
Z
Ks
=
(SNu )T C(SNu )dΩ,
(19)
ρs NTu Nu dΩ,
(20)
Ωs
Z
Ms
=
Ωs
Z
Q =
NTu nf Np dΓ.
(21)
Γ
The the solid phase is coupled with the fluid phase by the boundary condition at the interface. The coefficient
of the coupling is Q.
Similarly, when the discretization is applied to the fluid wave equation, Eq. (10), the FEM formulation
becomes
¨ + QT ũ
¨ = 0,
Kf p̃ + Mp̃
(22)
where
Z
Kf
=
Ωf
Z
Mf
=
Ωf
1
(∇Np )T (∇Np )dΩ,
ρf
(23)
1
NT Np dΩ.
ρf vf2 p
(24)
The FEM formulation for the total coupled system is a set of equations combining Eqs. (18) and (22), as
Ms
0
ũ
Ks −Q
ũ
2
= 0,
(25)
−ω
0 Kf
p̃
p̃
QT Mf
which is formulated in frequency domain. This is a general eigenvalue problem. The eigenvalues are related
to the natural resonance frequencies, ω 2 . The global stiffness and mass matrices for this coupled system are
Ks −Q
K =
,
(26)
0 Kf
Ms
0
M =
.
(27)
QT Mf
4
The above example is a simple two-layer coupled system. The model of a logging tool inside a borehole
consists of two fluid layers and two solid layers. There are three coupling interfaces. The borehole model is
an expansion of the two-layer coupled system.
2.3
Wavenumber domain
Since the properties of the tool, formation and fluid are assumed to be invariable along the borehole axis
z, and we are considering only the non-attenuated modes, wave solutions can be expanded along the z-axis
using the space Fourier transform. At a certain frequency ω, the space Fourier transforms of the acoustic
pressure and the particle displacement have the forms of
Z +∞
P(x, y, kz , ω) =
p(x, y, z, ω)e−ikz z dz,
(28)
−∞
Z +∞
1
p(x, y, z, ω) =
P(x, y, kz , ω)eikz z dkz ,
2π −∞
Z +∞
~
~u(x, y, z, ω)e−ikz z dz,
U(x,
y, kz , ω) =
(29)
(30)
−∞
~u(x, y, z, ω)
=
1
2π
Z
+∞
~
U(x,
y, kz , ω)eikz z dkz .
(31)
−∞
In the discrete finite element system, the acoustic pressure and the displacement are approximated as the
functions of nodal field values as in Eqs. (16) and (17). With the inverse Fourier transform, these approximate
functions are
Z +∞
1
Np (x, y)P̃(kz , ω)eikz z dkz ,
(32)
p̂(x, y, z, ω) =
2π −∞
Z +∞
1
û(x, y, z, ω) =
Nu (x, y)Ũ(kz , ω)eikz z dkz
(33)
2π −∞
While the shape functions, Np and Nu , in Eqs. (16) and (17) are three dimensional, the space Fourier
transform reduces them to two dimensional. The current problem can now be solved using two dimensional
finite element mesh.
Triangular elements are used to discretize the whole 2D model. For a fluid element, the column vector
of nodal pressures, P̃(kz , ω) in Eq.(32) is


 P1 (kz , ω) 
P2 (kz , ω)
P̃(kz , ω) =
.
(34)


P3 (kz , ω)
The shape function Np within a fluid element is a single row matrix, as
Np (x, y) = [N1 (x, y) N2 (x, y) N3 (x, y)].
The column vector for the displacements at element nodes has 9

U1x (kz , ω)




U1y (kz , ω)




U1z (kz , ω)



 U2x (kz , ω)

U2y (kz , ω)
Ũ(kz , ω) =


U2z (kz , ω)




U

3x (kz , ω)



U

3y (kz , ω)


U3z (kz , ω)
5
components,














.













(35)
(36)
While the shape function Nu for a solid element is a 3×9 matrix, as
Nu (x, y) =


N1 (x, y)
0
0
N2 (x, y)
0
0
N3 (x, y)
0
0

.
0
N1 (x, y)
0
0
N2 (x, y)
0
0
N3 (x, y)
0
0
0
N1 (x, y)
0
0
N2 (x, y)
0
0
N3 (x, y)
(37)
In the Cartesian coordinate system, the strain operator S is


∂
0 
 ∂x 0


 0
∂
0 


∂y



∂ 
 0
0 ∂z 

S=


∂
∂ 
 0
∂z
∂y 



∂ 
 ∂
0 ∂x

 ∂z


∂
∂
0
∂y
∂x
(38)
Therefore, the strain Ê(x, y, z, w) is given by
Ê(x, y, z, w)
= Sû
=
where
1
2π








B0 = 







Z
+∞
[B0 + ikz B1 ]Ũ(kz , ω)eikz z dkz ,
(39)
−∞
∂
∂x
0
0
0
0
∂
∂y

0







B1 = 







0
0
0
1
0

0 

∂
0 

∂y


0
0 
 Nu (x, y),

∂ 
0 ∂y


∂ 
0 ∂x 

∂
0
∂x

0 0 

0 0 



0 1 
 Nu (x, y),

1 0 


0 0 


0 0
0
After the substitution of Eqs. (33) and (39) to the weak wave equation of the solid, Eq. (15), the first term,
which includes the strain term, becomes
Z
Z
Z
Z +∞
1
(Sδ~us )T : C : (S~us )dΩ =
dx
dy
dz
4π 2
Ωs
−∞
Z +∞
Z +∞
0
×
[(B0 + ikz0 B1 )Ũ(kz0 , ω)]T eikz z dkz0 C
(B0 + ikz B1 )Ũ(kz , ω)eikz z dkz .
(40)
−∞
−∞
6
When it is integrated over z, we have
Z
+∞
0
ei(kz +kz )z dz = 2πδ(kz + kz0 ).
(41)
−∞
Ellefsen et al. (1991) detailed the integral of Eq. (40), which can finally be written as
Z
Z +∞
1
(Sδ~us )T : C : (S~us )dΩ =
δ ŨH (kz , ω)Ks Ũ(kz , ω)dkz ,
2π −∞
Ωs
where, the stiffness matrix of the solid is
Z
Z
Ks = dx dy[BT0 CB0 + kz2 BT1 CB1 + ikz (BT0 CB1 − BT1 CB0 )].
Similarly, we can get other matrices as
Z
Z
Ms =
dx dyρs NTu (x, y)Nu (x, y),
"
#
T
T
Z
Z
∂
∂
∂
∂
1
2 T
Kf =
dx dy
Np
Np +
Np
Np + kz Np Np ,
ρf
∂x
∂x
∂y
∂y
Z
Z
1
Mf =
dx dy
NT (x, y)Np (x, y),
ρf vf2 p
Z
Q =
dΓNTu (x, y)nf Np (x, y).
(42)
(43)
(44)
(45)
(46)
(47)
Γ
2.4
Infinite elements
Since the whole model extends to infinity, infinite elements are used as the outmost elements to simulate the
unbounded domain. The 2D finite element mesh without infinite elements is generated as a rectangular box,
whose edges are parallel to the x and y axes, as shown in Figure 2. The edges of this mesh should be far
enough from the center so that the wave energy is already weak at the edges. Each infinite element has two
edges parallel to the axes and extending to infinity. Figure 3 shows one of the infinite elements that extends
to infinity along the positive x direction. For the non-leaky modes, there is no acoustic wave propagating
away from the borehole, and the field decays to zero at infinity. One can approximate the attenuation of the
field to an exponential decay, as u(x, y) = u(y) exp(−αx) for an infinite element extending to +x infinity,
where u is the particle displacement, and α is the decay constant. Inside such an infinite element, the
displacement is approximated using the field values at the only two nodes,


u1x 







u1y 




H1 0
0 H2 0
0


u1z


0 H1 0
0 H2 0
û =
,
(48)
 u2x 

0
0 H1 0
0 H2 



u2y 






u2z
where the shape functions are
H1
=
H2
=
y − y2 −α(x−x1 )
e
,
y 1 − y2
y1 − y −α(x−x1 )
e
y 1 − y2
(49)
(50)
since x1 = x2 .
In the calculation of the stiffness and mass matrices of the infinite elements using Eqs. (43) and (44),
the integration limits over x should extends to infinity. Since the shape functions H1 and H2 are used, the
integrals still have finite values.
The infinite element at the corner of the mesh has only one node and its two edges are parallel to x and
y axes respectively. The exponential field decay is assumed along both axes.
7
Figure 2: The finite element mesh with a borehole and an LWD tool. Elements are very dense inside the
borehole.
8
8
Node 2
Inner
Element
Infinite element
8
Exp(- x)
Node 1
Figure 3: An infinite element
Borehole
LWD Tool
a
b
c
Figure 4: Three tool positions: (a) centered, and off-centered by (b) 1/3 and (c) 2/3 of the original annulus
thickness
2.5
Eigenvalue problem solver
The assembly of the local matrices of all elements in the four layers leads to the global stiffness and mass
matrices, K and M, as in Eqs. (26) and (27). They contain the terms from the solid and liquid phases, the
coupling terms at the fluid-solid interfaces, and the infinite element terms at the outmost boundaries. K is
complex since Ks has complex components as in Eq. (43).
To calculate the acoustic dispersion curves, we first specify a certain wavenumber kz . K and M are
functions of kz but not of frequency, so that they are fully determined. When the general eigenvalue problem
Eq. (25) is solved, its eigenvalues are correspondent to mode frequencies. The Arnoldi method is used to solve
this large-scale eigenvalue problem to find a few fundamental frequencies (Lehoucq et al., 1997). Although
the stiffness matrix is complex, the eigen-frequencies we are looking for are all real, which are related to
non-leaky modes.
3
Modeling and results
The finite element eigenvalue method is used to study the acoustic dispersion properties of various trapped
modes (non-leaky modes) inside a borehole with an off-centered LWD tool. We focus on the phase velocity
dispersion of the Stoneley (monopole), dipole and quadrupole modes in both soft and hard formation cases.
First we calculated the dispersion curves with a centered LWD tool to compare the results with the
analytical solutions. Then the tool is moved away from the center by one-third and two-thirds of the
thickness of the fluid annulus, as illustrated in Figure 4. The numerical results will tell us how and how
much the dispersion curves are going to change. The borehole and tool dimensions are listed in Table 1, and
the material properties are given in Table 2.
9
Table 1: Borehole and LWD tool dimensions
Borehole ID
8.75 in
Tool OD
7.25 in
Tool ID
1.9 in
Table 2: Material properties
Density, ρ (kg/m3 )
P-Wave Velocity, Vp , (m/s)
S-Wave Velocity, Vs , (m/s)
3.1
3.1.1
Soft formation
2200
3000
1200
Hard formation
2200
3000
2000
Tool
7800
5860
3130
Fluid
1000
1500
Soft formation
Centered LWD tool
For this centered tool case, the analytical method has successfully calculated the acoustic dispersion curves
(Rao et al., 1999). Figure 5 shows both the numerical and analytical results. We see good agreement between
these two sets of results, which suggests that our finite element method is able to get very accurate solutions.
Figure 5 shows only the Stoneley (monopole) mode, and the first dipole mode and the first quadrupole mode.
These three modes are of the most interests, and have been used in real acoustic measurement.
3.1.2
Off-centered tool
We first study the condition that the LWD tool is off-centered by 0.25 in along the x-axis, which is one-third
of the original fluid annulus when the tool is centered. The dispersion curves from the numerical calculation
are shown in Figure 6. An interesting phenomenon is that when the tool is off-centered, the dispersion curve
of the dipole mode splits into two curves with different phase velocities. The splitting is observed with the
quadrupole mode as well. However, the separation of the two quadrupole dispersion curves is much smaller
than that of the two dipole dispersion curves. This curve splitting phenomenon is due to the breakdown of
the axisymmetric condition of the whole system.
When solving the eigenvalue problem of Eq. (25), we get not only the eigenvalues, which are the square
of the angular frequencies, but also the eigenvectors, which are the acoustic fields associated with the corresponding eigen-frequencies. The acoustic field inside the borehole fluid is of the most interest, because it is
the field that we want to measure in well logging.
Figure 7 shows the acoustic pressure field of the Stoneley mode inside the fluid annulus. In this mode, the
fluid particle vibrates in phase, thus the field values are all positive in Figure 7. Unlike the Stoneley mode
with a centered tool, which has axisymmetric pressure distribution, the highest pressure of the Stoneley
mode here is at the small annulus side. The axisymmetry of the Stoneley mode is broken down when the
tool is off-centered.
Figures 8 and 9 show the acoustic field of the two split dipole modes under the same tool eccentricity.
Although the axisymmetric condition no longer exists, the whole system is still symmetric with respect
to the x-axis. The acoustic fields must be either symmetric or anti-symmetric with respect to the x-axis.
Therefore, there exist only two split dipole modes. The mode whose acoustic pressure anti-nodes are located
at the x-axis is called symmetric dipole mode in this paper, since it is symmetric to x-axis. The symmetric
dipole mode has a higher phase velocity. The other mode whose pressure nodes are at the x-axis is called
anti-symmetric dipole mode, which is anti-symmetric with respect to the x-axis and has the lower phase
velocity. The pressure anti-nodes of the anti-symmetric mode are not located exactly on the y-axis, but
move a little toward the small annulus direction, the same direction to which the tool moves.
The two split quadrupole modes have the similar properties as the dipole modes. Figures 10 and 11 show
the acoustic fields in the annulus of the quadrupole modes. Again, the mode that is symmetric to x-axis
10
is called symmetric quadrupole mode, which has a higher phase velocity. The other one is anti-symmetric
mode having a lower phase velocity.
When the LWD tool is off-centered by 0.5 in, which is two-thirds of the original annulus thickness, the
splitting of two dipole modes becomes even larger, as shown in Figure 12. The phase velocity difference
between the two split dipole modes is about 5% at around 5 kHz. The splitting of the quadrupole modes
also becomes larger, but is still very small compared with the splitting of the dipole modes.
Here we are going to compare the shifting of the dispersion curves for each mode when the tool moves
from the borehole center to off-centered positions.
Figure 13 is the comparison of the dispersion curves of the Stoneley modes. The further the tool is
off-centered, the further the Stoneley dispersion curve shifts to the lower phase velocity. When the tool is
off-centered by 0.5 in or two-thirds of the original annulus thickness, the Stoneley phase velocity is lowered
by 10% at 5 kHz. The shift of the Stoneley phase velocity is smaller at lower frequency (f ≤ 2 kHz).
Figure 14 shows the comparison of the dipole dispersion curves. When the tool is off-centered less than
0.5 in, the phase velocity of the symmetric dipole mode is almost the same as the dipole velocity for the
centered tool. However, the phase velocity of the anti-symmetric dipole mode shifts to a lower velocity. The
shifting of the anti-symmetric dipole mode is only about 5% at 5 kHz even the tool is off-centered by 0.5 in.
Within this tool eccentricity range, the effect of the tool positions on the dispersion of the dipole mode is
not significant.
The shifting of the anti-symmetric quadrupole mode is even smaller than that of the dipole mode, as
shown in Figure 15, where all quadrupole modes of the three tool positions are compared.
3.2
Hard formation
The shear velocity of the hard formation is larger than the acoustic speed of the borehole fluid. We only
compare the Stoneley, dipole and quadrupole modes when the tool is centered and off-centered. The results
are shown in Figure 16 to 18. Similar mode splitting and shifting phenomena are observed for the hard
formation as well. While the size of phase velocity shifting is also similar to that of the soft formation.
4
Conclusions and discussion
Our finite element method has been applied successfully to study the effects of tool eccentricity on dispersion
properties of trapped modes in borehole acoustic logging while drilling. We observe the mode splitting
phenomenon in dipole and quadrupole modes when a tool is off-centered, due to the breakdown of the
axisymmetry. Another significant and interesting phenomenon is the shifting of the dispersion curves. Among
the Stoneley, dipole and quadrupole modes, the Stoneley mode moves towards the lower phase velocity the
most. These results are very important for correct interpretation of acoustic logging data.
This method can also be used in the study of dispersion characteristics of irregular boreholes, and
anisotropic formation cases, as long as the borehole shape and material properties are invariable along the
borehole axis.
The finite difference time domain method is capable of simulating real conditions of acoustic logging,
with the acoustic source and receiver positions specified the same as real LWD tools. After analyzing the
wave forms from all the receivers, one can get the dispersion curves from the FDTD simulation. The finite
element method can help the FDTD modeling to identify each mode, since the FDTD method cannot tell
this by itself alone. The FDTD method is able to get all the modes that are exited, as well as the mode
having the strongest energy. With the help of both the finite element method and finite difference modeling,
we can better understand the effects and phenomena associated with eccentric tools in logging while drilling.
5
Acknowledgments
This study was supported by the MIT Borehole Acoustics and Logging Consortium, and by the Founding
Members of the Earth Resources Laboratory at MIT.
11
References
Ellefsen, K. J., Cheng, C. H., and Toksöz, M. N. (1991). Effects of anisotropy upon the normal modes in a
borehole. J. Acoust. Soc. Am., 89.
Huang, X. (2003). Effects of Tool Positions on Borehole Acoustic Measurements: A Stretched Grid Finite
Difference Approach. PhD thesis, MIT.
Lehoucq, R. B., Sorensen, D. C., and Yang, C. (1997). ARPACK Users’ Guide.
Rao, V. N. R., Burns, D. R., and Toksöz, M. N. (1999). Models in LWD applications. In Borehole Acoustics
and Logging and Reservoir Delineation Consortia - Annual Report.
Rao, V. N. R. and Vandiver, J. K. (1999). Acoustics of fluid-filled boreholes with pipe: Guided propagation
and radiation. J. Acoust. Soc. Am, 105(6):3057–3066.
Tang, X. M. and Cheng, C. H. A. (2004). Quantitative Borehole Acoustic Methods. Elsevier.
Tang, X. M., T., W., and Patterson, D. (2002). Multipole acoustic logging while drilling. In SEG expanded
abstract.
Zienkiewicz, O. C. and Taylor, R. L. (2000). The Finite Element Method, volume 1. Butlerworth-Heinemann,
Oxford, 5th edition.
12
Soft formation, Centered tool, Vs=1200 m/s
1300
1200
1100
Phase Velocity (m/s)
1000
900
800
700
Quadrupole (numerical)
Quadrupole (analytical)
Dipole (numerical)
Dipole (analytical)
Monopole (numerical)
Monopole (analytical)
600
500
400
300
0
2
4
6
8
10
12
14
16
18
20
Frequency (KHz)
Figure 5: Comparison of FEM and analytical solutions of the dispersion curves with a centered LWD tool
13
Soft formation, Off−centered 0.25" (1/3), Vs=1200 m/s
1300
1200
1100
Phase Velocity (m/s)
1000
900
800
700
600
Quadrupole, Symmetric
Quadrupole, Anti−symmetric
Dipole, Symmetric
Dipole, Anti−symmetric
Monopole (Stoneley)
500
400
300
0
2
4
6
8
10
12
14
16
18
20
Frequency (KHz)
Figure 6: Dispersion curves with an off-centered LWD tool (off-centered by 0.25 in)
Figure 7: Acoustic fields of the Stoneley mode in the fluid annulus with an off-centered tool
14
Figure 8: Acoustic fields of the anti-symmetric dipole mode in the fluid annulus with an off-centered tool
Figure 9: Acoustic fields of the symmetric dipole mode in the fluid annulus with an off-centered tool
15
Figure 10: Acoustic fields of the anti-symmetric quadrupole mode in the fluid annulus with an off-centered
tool
Figure 11: Acoustic fields of the symmetric quadrupole mode in the fluid annulus with an off-centered tool
16
Soft formation, Off−centered 0.5" (2/3), Vs=1200 m/s
1300
1200
1100
Phase Velocity (m/s)
1000
900
800
Quadrupole, Symmetric
Quadrupole, Anti−symmetric
Dipole, Symmetric
Dipole, Anti−symmetric
Monopole (Stoneley)
700
600
500
400
300
0
2
4
6
8
10
12
14
16
18
20
Frequency (KHz)
Figure 12: Dispersion curves with an off-centered LWD tool (off-centered by 0.5 in)
Soft formation, Centered and Off−centered tools, Monopole (Stoneley) mode, Vs=1200 m/s
1300
1200
1100
Phase Velocity (m/s)
1000
900
800
700
600
Centered Tool
Off−centered by 0.25" (1/3)
Off−centered by 0.5" (2/3)
500
400
300
0
2
4
6
8
10
12
14
16
18
20
Frequency (KHz)
Figure 13: Comparison of the Stoneley modes of a soft formation borehole with centered and off-centered
tools
17
Soft formation, Centered and Off−centered tool, Dipole mode, Vs=1200 m/s
1300
1200
1100
Phase Velocity (m/s)
1000
900
800
700
Centered Tool
Off−centered by 0.25" (1/3), Symmetric
Off−centered by 0.25" (1/3), Anti−symmetric
Off−centered by 0.5" (2/3), Symmetric
Off−centered by 0.5" (2/3), Anti−symmetric
600
500
400
300
0
2
4
6
8
10
12
14
16
18
20
Frequency (KHz)
Figure 14: Comparison of the dipole modes of a soft formation borehole with centered and off-centered tools
Soft formation, Centered and Off−centered tool, Quadrupole mode, Vs=1200 m/s
1300
1200
1100
Phase Velocity (m/s)
1000
900
800
Centered Tool
Off−centered by 0.25" (1/3), Symmetric
Off−centered by 0.25" (1/3), Anti−symmetric
Off−centered by 0.5" (2/3), Symmetric
Off−centered by 0.5" (2/3), Anti−symmetric
700
600
500
400
300
0
2
4
6
8
10
12
14
16
18
20
Frequency (KHz)
Figure 15: Comparison of the quadrupole modes of a soft formation borehole with centered and off-centered
tools
18
Hard formation, Stoneley mode, Vs=2000 m/s, Vf=1500 m/s
2200
2000
1800
Phase Velocity (m/s)
1600
1400
1200
1000
800
Centered
Offcentered (0.25" 1/3)
Offcentered (0.5" 2/3)
600
400
200
0
5
10
15
20
25
Frequency (kHz)
Figure 16: Comparison of the Stoneley modes of a hard formation borehole with centered and off-centered
tools
Hard formation, Dipole modes, Vs=2000 m/s, Vf=1500 m/s
2200
2000
1800
Phase Velocity (m/s)
1600
1400
1200
1000
800
Centered
Offcentered (0.25" 1/3), Symmetric
Offcentered (0.25" 1/3), Anti−symmetric
Offcentered (0.5" 2/3), Symmetric
Offcentered (0.5" 2/3), Anti−symmetric
600
400
200
0
5
10
15
20
25
Frequency (kHz)
Figure 17: Comparison of the dipole modes of a hard formation borehole with centered and off-centered
tools
19
Hard formation, Quadrupole mode, Vs=2000 m/s, Vf=1500 m/s
2200
2000
Phase Velocity (m/s)
1800
1600
1400
1200
1000
800
Centered
Offcentered (0.25", 1/3), Symmetric
Offcentered (0.25", 1/3), Anti−Symmetric
Offcentered (0.5", 2/3), Symmetric
Offcentered (0.5", 2/3), Anit−Symmetric
600
400
200
0
5
10
15
Frequency (KHz)
20
25
Figure 18: Comparison of the quadrupole modes of a hard formation borehole with centered and off-centered
tools
20