Effects of tool eccentricity on wave dispersion properties in borehole acoustic logging while drilling

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

Borehole Formation Borehole Formation Inner Fluid Fluid Wireline Tool Fluid 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 v 2 f 2 p + ω 2 p = 0, (1) where p is the pressure, v f 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]{ˆp} ω 2 [M]{ˆp} = {0}, (2) where [K] and [M] are the stiffness and mass matrices in the finite element method. {} denotes a single column matrix. { ˆp} 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, u s (with three components). The wave equations in the fluid and solid phases are 1 v 2 f 2 p t 2 2 p = 0, (3) 2 u s ρ s (C : E) = 0, (4) t2 where C is the elasticity tenser, E is the strain tensor, and ρ s 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) where n is the normal unit vector of the interface. In the fluid side, the equation of motion is ρ f 2 u f t 2 = p. (6) Taking the time derivative of the velocity boundary condition, Eq. (5), and substituting Eq. (6) into it, we get a boundary condition 1 p n = 1 p ρ f ρ f n = u s n. (7) There is another pressure boundary condition at the interface, which states that the normal force on each side must be continuous, so that t n = 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 [ δπ f = δp 1 1 Ω f ρ f v 2 f 2 p t 2 2 p ] dω = 0. (9) Integrating it by parts and substituting the boundary condition Eq. (7) in it lead to 1 ( δp) T 1 1 pdω + Ω f ρ f Ω f ρ f vf 2 (δp) pdω δp p n f dγ Γ ρ f 1 = ( δp) T 1 pdω + Ω f ρ f Ω f ρ f vf 2 (δp) pdω + δp( u s n f )dγ (10) Γ = 0, where Ω f 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 (Sδ u s ) T : C : (S u s )dω + ρ s δ u s u s dω δ u s t ns dγ = 0, (11) Ω s Ω s Γ where S is the strain operator as Since at the solid-fluid interface, E = S u. (12) n s = n f, (13) 3

the boundary condition Eq. (8) becomes t ns = pn s = pn f. (14) With this boundary condition, Eq. (11) has the form as (Sδ u s ) T : C : (S u s )dω + ρ s δ u s u s dω (δ u s n f )pdγ = 0. (15) Ω s Ω s Γ 2.2 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 While the displacement vector in the solid phase is approximated as p ˆp = N p p. (16) u s û s = N u ũ. (17) Here p and ũ are the nodal pressures and displacements in fluid and solid layers, and N p and N u 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 K s ũ + M s ũ Q p = 0, (18) where K s = M s = Q = (SN u ) T C(SN u )dω, (19) Ω s ρ s N T u N u dω, (20) Ω s N T u n f N p 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 K f p + M p + Q T ũ = 0, (22) where K f = M f = Ω f 1 ρ f ( N p ) T ( N p )dω, (23) Ω f 1 ρ f v 2 f N T p N p dω. (24) The FEM formulation for the total coupled system is a set of equations combining Eqs. (18) and (22), as [ ] { } [ ] { } Ks Q ũ ω 0 K f p 2 Ms 0 ũ Q T = 0, (25) M f p 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 K f [ ] Ms 0 M =. (27) Q T M f 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 P(x, y, k z, ω) = + p(x, y, z, ω) = 1 2π U(x, y, k z, ω) = u(x, y, z, ω) = 1 2π + + + p(x, y, z, ω)e ikzz dz, (28) P(x, y, k z, ω)e ikzz dk z, (29) u(x, y, z, ω)e ikzz dz, (30) U(x, y, k z, ω)e ikzz dk z. (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 ˆp(x, y, z, ω) = 1 2π û(x, y, z, ω) = 1 2π + + N p (x, y) P(k z, ω)e ikzz dk z, (32) N u (x, y)ũ(k z, ω)e ikzz dk z (33) While the shape functions, N p and N u, 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, P(k z, ω) in Eq.(32) is P(k z, ω) = P 1 (k z, ω) P 2 (k z, ω) P 3 (k z, ω) The shape function N p within a fluid element is a single row matrix, as. (34) N p (x, y) = [N 1 (x, y) N 2 (x, y) N 3 (x, y)]. (35) The column vector for the displacements at element nodes has 9 components, U 1x (k z, ω) U 1y (k z, ω) U 1z (k z, ω) U 2x (k z, ω) Ũ(k z, ω) = U 2y (k z, ω). (36) U 2z (k z, ω) U 3x (k z, ω) U 3y (k z, ω) U 3z (k z, ω) 5

While the shape function N u for a solid element is a 3 9 matrix, as N u (x, y) = N 1(x, y) 0 0 N 2 (x, y) 0 0 N 3 (x, y) 0 0 0 N 1 (x, y) 0 0 N 2 (x, y) 0 0 N 3 (x, y) 0 0 0 N 1 (x, y) 0 0 N 2 (x, y) 0 0 N 3 (x, y) In the Cartesian coordinate system, the strain operator S is x 0 0 0 y 0 0 0 S = z 0 z y z 0 x Therefore, the strain Ê(x, y, z, w) is given by where Ê(x, y, z, w) = Sû = 1 2π B 0 = y + x 0 0 0 y 0 0 0 0 x 0. (37) (38) [B 0 + ik z B 1 ]Ũ(k z, ω)e ikzz dk z, (39) 0 0 y 0 0 x y x 0 N u (x, y), 0 0 0 0 0 0 0 0 1 B 1 = N u (x, y), 0 1 0 1 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 (Sδ u s ) T : C : (S u s )dω = 1 + Ω s 4π 2 dx dy dz { + [(B 0 + ik zb 1 )Ũ(k z, ω)] T e ik z z dk z 6 } C { + (B 0 + ik z B 1 )Ũ(k z, ω)e ikzz dk z }. (40)

When it is integrated over z, we have + e i(kz+k z )z dz = 2πδ(k z + k z). (41) Ellefsen et al. (1991) detailed the integral of Eq. (40), which can finally be written as (Sδ u s ) T : C : (S u s )dω = 1 + Ω s 2π δũh (k z, ω)k s Ũ(k z, ω)dk z, (42) where, the stiffness matrix of the solid is K s = dx dy[b T 0 CB 0 + kzb 2 T 1 CB 1 + ik z (B T 0 CB 1 B T 1 CB 0 )]. (43) Similarly, we can get other matrices as M s = dx dyρ s N T u (x, y)n u (x, y), (44) [ ( K f = dx dy 1 ) T ( ) ] T ρ f x N p x N p + y N p y N p + kzn 2 T p N p, (45) M f = Q = 2.4 Infinite elements Γ dx dy 1 ρ f vf 2 N T p (x, y)n p (x, y), (46) dγn T u (x, y)n f N p (x, y). (47) 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, where the shape functions are û = H 1 0 0 H 2 0 0 0 H 1 0 0 H 2 0 0 0 H 1 0 0 H 2 u 1x u 1y u 1z u 2x u 2y u 2z, (48) H 1 = y y 2 e α(x x1), y 1 y 2 (49) H 2 = y 1 y e α(x x1) y 1 y 2 (50) since x 1 = x 2. 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 H 1 and H 2 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

Node 2 8 Inner Element Infinite element Exp(- x) Node 1 8 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 K s has complex components as in Eq. (43). To calculate the acoustic dispersion curves, we first specify a certain wavenumber k z. K and M are functions of k z 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 Tool OD Tool ID 8.75 in 7.25 in 1.9 in Table 2: Material properties Soft formation Hard formation Tool Fluid Density, ρ (kg/m 3 ) 2200 2200 7800 1000 P-Wave Velocity, V p, (m/s) 3000 3000 5860 1500 S-Wave Velocity, V s, (m/s) 1200 2000 3130 3.1 Soft formation 3.1.1 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

1300 Soft formation, Centered tool, Vs=1200 m/s 1200 1100 Phase Velocity (m/s) 1000 900 800 700 600 500 400 Quadrupole (numerical) Quadrupole (analytical) Dipole (numerical) Dipole (analytical) Monopole (numerical) Monopole (analytical) 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

1300 Soft formation, Off centered 0.25" (1/3), Vs=1200 m/s 1200 1100 1000 Phase Velocity (m/s) 900 800 700 600 500 400 Quadrupole, Symmetric Quadrupole, Anti symmetric Dipole, Symmetric Dipole, Anti symmetric Monopole (Stoneley) 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

1300 Soft formation, Off centered 0.5" (2/3), Vs=1200 m/s 1200 1100 Phase Velocity (m/s) 1000 900 800 700 600 500 Quadrupole, Symmetric Quadrupole, Anti symmetric Dipole, Symmetric Dipole, Anti symmetric Monopole (Stoneley) 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 1000 Phase Velocity (m/s) 900 800 700 600 500 400 Centered Tool Off centered by 0.25" (1/3) Off centered by 0.5" (2/3) 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

1300 Soft formation, Centered and Off centered tool, Dipole mode, Vs=1200 m/s 1200 1100 Phase Velocity (m/s) 1000 900 800 700 600 500 400 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 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 1300 Soft formation, Centered and Off centered tool, Quadrupole mode, Vs=1200 m/s 1200 1100 Phase Velocity (m/s) 1000 900 800 700 600 500 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 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

2200 Hard formation, Stoneley mode, Vs=2000 m/s, Vf=1500 m/s 2000 1800 1600 Phase Velocity (m/s) 1400 1200 1000 800 600 400 Centered Offcentered (0.25" 1/3) Offcentered (0.5" 2/3) 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 2200 Hard formation, Dipole modes, Vs=2000 m/s, Vf=1500 m/s 2000 1800 1600 Phase Velocity (m/s) 1400 1200 1000 800 600 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 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

2200 Hard formation, Quadrupole mode, Vs=2000 m/s, Vf=1500 m/s 2000 1800 Phase Velocity (m/s) 1600 1400 1200 1000 800 600 400 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 200 0 5 10 15 20 25 Frequency (KHz) Figure 18: Comparison of the quadrupole modes of a hard formation borehole with centered and off-centered tools 20