GEOPHYSICS, VOL. 67, NO. 2 (MARCH-APRIL 2002); P. 413–426, 10 FIGS., 2 TABLES. 10.1190/1.1468601
Three-dimensional induction logging problems, Part I: An integral equation solution and model comparisons
Dmitry B. Avdeev∗ , Alexei V. Kuvshinov∗ , Oleg V. Pankratov∗ , and Gregory A. Newman‡
ABSTRACT
responses requires the numerical solution of Maxwell’s equations in three dimensions along with independent tests to validate the solution approach and its accuracy. In this paper, we compare two independent 3-D frequencydomain solutions for the problem, based on finite differences and the integral equation technique. Specific examples in the quasi-static limit are studied, including a 45◦ deviated borehole that intersects formation bed boundaries as well as cases where the bedding exhibits transverse anisotropy. All comparisons made in this paper show very good agreement and demonstrate, for the first time, verifiable induction log responses in the presence of deviated boreholes. We also show that responses arising from deviated boreholes can be significant and must be accounted for properly when interpreting induction logs.
A 3-D frequency-domain solution based on a volume integral equation approach has been implemented to simulate induction log responses. In our treatment of the problem, we assume that the electrical properties of the bedding as well as the borehole and invasion zones can exhibit transverse anisotropy. The solution process uses a Krylov subspace iteration to solve the scattering equation, which is based on the modified iterative dissipative method. Internal consistency checks and comparisons with mode matching and finite-difference solutions for vertical borehole models demonstrate the accuracy of the solution. There are no known analytical solutions for induction log responses arising from deviated boreholes intersecting horizontal bed boundaries. To simulate such
INTRODUCTION
tion. However, we use Krylov subspace iteration, which converges much faster than the Neumann series iteration. In this paper we first present the governing itegral equations as applied to transverse anisotropic media and discuss the computational loads of the numerical solution along with the meshing required to ensure its accuracy for induction logging simulations. This is followed by self-consistency checks and checks performed for vertical-borehole models. We then present a simulation of induction log responses in the presence of deviated boreholes intersecting horizontal bed boundaries. These simulations include comparisons with the finitedifference solution of Newman and Alumbaugh (2002), which also uses iterative Krylov subspace methods in the solution process along with several preconditioning options. We refer the interested reader to Part II, appearing in this issue, which describes specific details of this solution (Newman and Alumbaugh, 2002). To our knowledge there has been no comparison of independent solutions for the deviated wellbore
Advances in both computers and numerical methods now make it possible to calculate the responses of full 3-D induction logging models that include anisotropic bedding and deviated boreholes with invasion. Encouraging examples of such 3-D simulations using the spectral Lanczos decomposition method have been reported by Druskin et al. (1999) and van der Horst et al. (1999). Here we investigate an integral equation approach for solving 3-D induction logging problems. This approach combines the modified iterative dissipative method with a Krylov subspace iteration. The modified iterative dissipative method was originally presented by Singer (1995) for a quasistatic field and for isotropic formations. Later, it was extended to formations with displacement currents and anisotropy (Pankratov et al., 1995, 1997; Singer and Fainberg, 1995, 1997). In this method, the conventional scattering equation is modified so it can be solved by a convergent Neumann series itera-
Manuscript received by the Editor August 25, 1999; revised manuscript received July 2, 2001. ∗ Geoelectromagnetic Research Institute, Russian Academy of Sciences, 142190 Troitsk, Moscow region, Russia. E-mail:
[email protected]. ‡Sandia National Laboratories, Department 6116, MS 0750, 1515 Eubank SE, Albuquerque, New Mexico 87185-0750. E-mail:
[email protected]. ° c 2002 Society of Exploration Geophysicists. All rights reserved. 413
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problem. Such comparisons are needed because the accurate simulation of 3-D induction logs, including the borehole, is a nontrivial calculation because of the fine meshes required for the problem. Thus, independent solutions are needed to verify the calculations and to provide critical information on proper mesh design. These comparisons are also made for a number of frequencies in the quasi-static range and for various borehole/formation resistivity contrasts, as well as for cases where the formation exhibits transverse anisotropy.
zones. The z-axis is chosen to be normal to bedding. The axis is inclined at a dip angle α relative to the axis of the borehole, which is assumed to be known and constant. No distinction is made between the dip and the deviation angles. Assuming a time-harmonic dependence of e−iωt , the electric E(x, y, z, ω) and magnetic H(x, y, z, ω) fields in the model satisfy Maxwell’s equations:
INTEGRAL EQUATION APPROACH
Formulation We consider the 3-D earth model shown in Figure 1. It includes layered bedding and a deviated borehole with invasion
∇ × H = ζ (x, y, z, ω)E,
(1)
∇ × E = iωµ(z)H + mδ(r − rT ).
(2)
Here, ζ (x, y, z, ω) denotes generalized conductivity, which is defined by the conductivity σ (x, y, z, ω) and√dielectric permittivity ε(x, y, z, ω) as ζ = σ − iωε, where i = −1. The form of the generalized conductivity means that both displacement and ohmic currents are included in the formulation. Moreover, we assume that ζ is now a complex-valued 3 × 3 diagonal tensor, ζ = diag(ζx x , ζ yy , ζzz ). The magnetic permeability µ(z) is a positive real-valued function of depth z. For the sake of clarity and without the loss of generality, we assume that the model is excited by an arbitrarily polarized magnetic dipole source of moment m positioned at point rT . Thus, the moment and position of the source are designated by mδ(r − rT ), where δ is the Dirac delta function. Hereafter, dependencies on position, r = (x, y, z), and ω are omitted but remain implied where they are not essential. Following the volume integral equation approach, the solution of equations (1) and (2) is found as follows. Maxwell’s equations for the scattered field.—Within the scattering volume V S , (r S ∈ V S ), which comprises the circularly cylindrical wellbore with invasion zones, we determine the background electric field Eb (r S ). For any point r, this field, Eb , is defined as the electric field excited by the transmitting dipole mδ(r − rT ) in a transversely anisotropic (TI) 1-D background medium that is described by a generalized conductivity tensor, ζ b (z, ω) = diag(ζbτ , ζbτ , ζbz ). In other words, the field Eb satisfies Maxwell’s equations:
∇ × Hb = ζ b (z, ω)Eb ,
(3)
∇ × E = iωµ(z)H + mδ(r − rT ).
(4)
b
b
The solution of equations (3) and (4) is written as
Z
Eb (r S ) =
FIG. 1. Example of a 3-D induction logging problem. The model includes a number of beds where the resistivity ρt , dielectric permittivity εt , and thickness h t describe bed t. The borehole is described by resistivity ρm , dielectric permittivity εm and radius rm along with a number of invasion zones, which for the jth invasion zone include the resistivity ρxoj , (r 0 ), dielectric permittivity εxoj (r 0 ) and radii rxoj . The borehole is deviated at an angle α with respect to the direction normal to the bedding, and the induction tool transmitters/receivers are multidirectional magnetic dipoles. The position and orientation of the tool are arbitrary within the well.
vS
0 0 0 em Gem b (r S , r )mδ(r − rT ) dν = Gb (r S , rT )m,
(5)
0 where Gem b (r, r ) is a 3 × 3 dyadic for the magnetic-to-electric tensor Green’s function of the background bedding. Explicit forms of the 3 × 3 dyadic Green’s tensors used here are presented in Appendix A of Avdeev et al., (1997), or they can be readily developed from there. Subtracting equations (3) and (4) from equations (1) and (2), we write Maxwell’s equations for the scattered fields, Es = E − Eb and Hs = H − Hb , as
∇ × Hs = ζ(x, y, z, ω)Es + js ,
(6)
∇ × E = iωµ(z)H ,
(7)
s
s
where
js = (ζ − ζ b )Eb .
(8)
3-D Integral Equation Solution for Induction Logging
The modified scattering equation.—To derive the modified scattering equation we introduce a transversely anisotropic (TI) 1-D reference medium, described by a generalized conductivity tensor ζ o (z, ω) = diag(ζoτ , ζoτ , ζoz ), where τ denotes transverse or horizontal conductivity, and rewrite equations (6) and (7) as
∇ × Hs = ζ o (z, ω)Es + jq ,
(9)
∇ × Es = iωµ(z)Hs ,
(10)
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with any initial guess χ(1) (r). Moreover this iteration procedure when using χo (r) as an initial guess produces
Z χ(r) = χo (r) + K(r, r0)R(r0 )χo (r0 ) dv 0 M v Z + K(r, r0 )R(r0 ) M v ¶ µZ K(r0 , r00 )R(r00 )χo (r00 ) dv 00 dv 0 + · · · , (17) × vM
which is identical in form to the series expansion given in equation (13).
where
jq = js + (ζ − ζ o )Es .
(11)
The conductivities ζoτ (z, ω) and ζoz (z, ω) can be chosen as any complex-valued functions of z with positive real part Re ζoτ (z, ω) > 0 and Re ζoz (z, ω) > 0. Equations (9) and (10) lead to the conventional scattering equation with respect to the unknown field Es (r):
Z
E (r) = Eo (r) + s
vM
0 0 Gee o (r, r )(ζ(r )
0
0
0
− ζ o (z ))E (r ) dv , s
(12) R 0 s 0 0 ee 0 where the free term Eo (r) = vS Gee o (r, r ) j (r ) dv , Go (r, r ) is the 3 × 3 dyadic for the electric-to-electric Green’s function of the 1-D reference medium, and V M is volume where ζ(r0 ) − ζ o (z 0 ) differs from zero. Now a formal solution of equation (12) can be expressed as an infinite Neumann series, starting from Eo (r) as the zero-order approximation to Es (r): Z 0 0 0 0 0 Es (r) = Eo (r) + Gee o (r, r )(ζ(r ) − ζ o (z ))Eo (r ) dv M v µZ Z ee 0 0 0 0 00 + Go (r, r )(ζ(r ) − ζ o (z )) Gee o (r , r ) vM vM ¶ 00 00 00 00 × (ζ(r ) − ζ o (z ))Eo (r ) dv dv 0 + · · · . (13)
However, convergence of such a series is not guaranteed, and for a scattered field very different from Eo (r) this series does not converge at all. To achieve a convergent series, we modify equation (12) to
Z
χ(r) = χo (r) +
vM
K(r, r0 )R(r0 )χ(r0 ) dv 0 .
(14)
Derivation of equation (14) and definitions of χ and the tensors K and R are given in the Appendix. Equation (14) now possesses a contracting integral kernel (cf., Pankratov et al., 1995, 1997), where
Krylov subspace iteration.—In the limit as n → ∞ and χ(n) →χ, equation (16) satisfies the operator equation
Aχ = χo ,
(18)
where A = 1 − M, 1 is the identity operator, and
Z
Mχ =
vM
K(r, r0 )R(r0 )χ(r0 ) dv 0 .
(19)
Because the Krylov subspace iteration represents χ using a basis given by the individual terms in equation (17) but does this in an optimal way by limiting the size of the subspace, or in effect the number of terms in equation (17), it should be far more efficient than the simple Neumann iteration scheme of equation (16). Moreover, the computational work needed to construct this subspace of size m is nearly equivalent to summing the first m terms in the Neumann series. Now we can choose the reference conductivity ζ o (z) to coincide with that of the background ζ b (z). The merit of this choice is that the integrations in equation (19) are performed exactly over the scattering volume V S , since this follows from the expression for R in equation (A-7). On the other hand, were we to choose ζ o (z) to differ from ζ b (z), as prescribed in Singer (1995) and Pankratov et al. (1995), then we would have to integrate over a larger volume V M , V M ⊇ V S . The reward for the latter choice is that it can significantly accelerate the convergence of the Krylov subspace iteration. Our stopping criteria for the Krylov iteration is based on driving the relative residual, given by
r (m) =
kχo − A · χ(m) k , kχo k
(20)
for any choice of a generalized conductivity tensor ζ o (z) that R describes the reference medium. Here, kχk, = vM |χ(r)|2 dv.
down to a predetermined level that is dependent upon the resistivity contrast between the scatterer and the host. Because the operator A is complex and asymmetric, the Krylov iteration we have chosen to implement is the generalized biconjugate gradient method (Zhang, 1997). To stabilize the erratic convergence behavior of this method we also incorporate the quasi-minimal residual smoothing of Zhou and Walker (1994) into the iteration scheme. Hereinafter we referred to the combined method as GPBiCG-Z/W-QMR.
Neumann series iteration.—The property given by equation (15) allows one to find the solution of equation (14) by iterating
Scattered fields determination.—To determine the scattered electric Es (r) and magnetic Hs (r) fields for any point r, we rewrite Maxwell’s equations in equations (6) and (7) as
¯¯ Z ¯¯ ¯¯ ¯¯
¯¯ ¯¯ K(r, r )R(r )χ(r ) dv ¯¯ < kχk, ∀χ M 0
0
0
0 ¯¯
(15)
v
Z
χ(n+1) (r) = χo (r) +
vM
K(r, r0 )R(r0 )χ(n) (r0 ) dv 0 , n = 1, 2, . . . (16)
∇ × Hs = ζ b (z, ω)Es + j,
(21)
∇ × E = iωµ(z)H ,
(22)
s
s
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where
j = (ζ − ζ b )(E + E ). b
s
(23)
Since χ(r) is known from the solution of equation (18) and is given within the scattering volume V S , (χ(r): r ∈ V M {V M ⊇ V S }), we have determined from equation (A-3) that
¢ ¡ Es (r S ) = (ζ(r S ) + ζ ∗o (z S ))−1 2ν(z S )χ(r S ) − js (r S ) . (24)
Substituting this expression and equation (5) into equation (23), we can then determine j(r S ). The scattered electric Es (r) and magnetic Hs (r) fields for any r are then readily found from the solution of Maxwell’s equations [equations (21) and (22)] using
Z
E (r) = s
vS
Z Hs (r) =
vS
0 0 0 Gee b (r, r ) j(r ) dv ,
(25)
0 0 0 Gme b (r, r ) j(r ) dv ,
(26)
me 0 0 where Gee b (r, r ) and Gb (r, r ) are the respective 3 × 3 dyadic electric-to-electric and electric-to-magnetic tensor Green’s functions of the background model based on the generalized conductivity ζ b (z). The total fields are obtained by adding the appropriate background fields to the scattered fields.
Discretization and computational loads Let us briefly describe the discretization of the numerical solution with an example of the wellbore model shown in Figure 2. This model includes a central bed with shoulders, a circularly cylindrical borehole, and a pistonlike invasion zone in the center bed. The borehole is tilted at a deviation angle α. The figure presents a side view of the modeling volume V M that comprises the borehole, the invasion zone, and some part of the back-
ground bedding. To discretize the model, we adopt a numerical mesh as follows. We consider V M as a combination of Nz identical rectangular parallelepipeds that align in such a way that each is shifted with respect to the previous one so that the entire combination is tilted at an angle α. Next we discretize each parallelepiped by N x × N y rectangular prisms of dimensions dx × d y × dz . Here, dz is the vertical size of the parallelepiped, and the faces of each prism are parallel to the coordinate planes of the bedding in the Oxyz coordinate system. Thus, we discretize the whole tilted volume V M by N x × N y × Nz rectangular prisms with dimensions given by
dx = D y / cos(α), d y = D y , dz = Dz cos(α), when α ≤ 45◦ , dx = Dz sin(α), d y = D y , dz = 2D y / sin(α), when α > 45◦ , (27) where the grid constants D y and Dz are selected to ensure accurate results. Expressions in equation (27) reflect the fact that in the (x, y)-plane section both the circularly cylindrical borehole and the invasion zone are ellipses. To preserve the geometry of the borehole and invasion zone, we prescribe to each rectangular prism a generalized conductivity value determined from the average value of the true generalized conductivity distribution within the prism. Note also that if a bed boundary intersects one of the Nz rectangular parallelepipeds (which comprise volume V M ), we subdivide this parallelepiped into two nonshifted parallelepipeds. Thus, our discretization implies that the 3-D mesh is equal in the horizontal direction but in general can be variable along the vertical axis. Next, both the Neumann series and GPBiCG-Z/W-QMR iterations require matrix–vector products equivalent to the volume integrations of the type given in equation (19). Here, we refer the reader to Avdeev et al. (1997) for details on how these integrations are carried out. In accordance with that paper, the estimates of the number of operations M and memory sizes S needed for our solution are given by
M ≈ O(N x N y Nz (l Nz + (l + Nz ) log2 N x log2 N y )), ¢ ¡ S ≈ O N x N y Nz2 bytes.
(28) (29)
In equation (28), l represents the number of the matrix–vector products needed to produce a solution χ that yields a relative residual that falls below a preselected value. We also observe from equation (28) that M depends on the horizontal dimensions N x and N y as O(N x N y log N x log N y ) and depends quadratically on its vertical size, Nz . In contrast, 3-D IE solutions that use direct matrix inversion methods give a cubic dependence, O(N x3 N y3 Nz3 ), (see Newman et al., 1986; Wannamaker, 1991; Xiong and Tripp, 1995). Comparison of Neumann series and Krylov subspace iteration
FIG. 2. A model discretized with rectangular prisms. The modeling volume V M is composed of seven parallelepipeds of heights dz i , i = 1, . . . , 7. In this example the modeling volume V M is wider than the scattering volume V S . This means the reference generalized conductivity ζo (z, ω) is chosen to differ from the background conductivity, ζb (z, ω). Angles φ and θ describe the polarization of the transmitting dipole.
To examine to what extent the Krylov iteration outperforms the Neumann series, we select a model of a vertical borehole residing in a uniform medium. Here, the background and mud resistivities are set to 1000 and 0.02 ohm-m, where the diameter of the wellbore is 20 cm and the transmitter–receiver offset L is set at 20 cm. The transmitter is a magnetic dipole, coaxial with the borehole, and the receiver samples the coaxial magnetic field at a frequency of 160 kHz.
3-D Integral Equation Solution for Induction Logging
The left plot in Figure 3 illustrates the relative residual, r (m) , of the GPBiCG-Z/W-QMR and Neumann series solutions with respect to the iteration counter, l = 2m. (Apart from a Neumann series iterate that requires one matrix–vector product, a GPBiCG-Z/W-QMR iterate requires two matrix– vector products.) Both solutions demonstrate exponential decay of the relative residual, r (m) ≈ 10−m/λ , as m → ∞. Here, λ is a factor that depends upon the iteration used, where λGPBiCG < λNS . It is readily seen from this plot that the relative residual of the GPBiCG-Z/W-QMR solution drops about 5.5 times faster than that of the Neumann series solution. Other plots in Figure 3 show the real, σ R (m) , and imaginary, σ X (m) , parts of the apparent conductivity σ (m) = σ R (m) + iσ X (m) = 4πL(iωµo )−1 1Hz (m) . Here, 1Hz (m) is the coaxial magnetic field calculated with the direct-coupled field, Hz{vacuum} = (2πL 3 )−1 (1 + iω/c)e−iωL/c , removed, where c is the speed of light in a vacuum. These plots demonstrate that the Krylov subspace iteration converges nine times faster than the Neumann series iteration to the solution, which is computed from the mode matching solution of Chew et al. (1984). The computational loads and mesh sizes needed for the IE solution for this particular example appear in Table 1 as model 1.
Table 1. Model 1 2a 2b 2c 3 4 5 6 7 8
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VERIFICATION OF SOLUTION
Self-consistency check We see from Figure 2 that the mesh we used is oriented with respect to the bedding. This means the numerical approximation of the borehole depends upon the deviation angle, α. To confirm the internal consistency of our solution for deviated boreholes, we simulate responses of the borehole residing in a uniform background for different deviation angles. These models are shown in Figure 4a . Theoretically, the responses calculated for these models should coincide among themselves, since there is no bedding in the models. These models consider a magnetic dipole transmitter and a number of magnetic dipole receivers located on the well axis where, in this example and the examples that follow, the moment of the transmitter is set at 1 A · m2 . Figures 4b and 4c demonstrate the 10-kHz responses for borehole mud resistivities of 10 and 0.8 ohm-m, respectively, where the formation resistivity is set at 100 ohm-m. Results are presented for three deviation angles: 0, 45, and 70◦ . As expected, we see excellent agreement between the responses calculated for the various angles where the maximum discrepancy is
