The effects of tool eccentricity and formation anisotropy on resistivity logs: forward modeling

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ScienceDirect Russian Geology and Geophysics 57 (2016) 1477–1484 www.elsevier.com/locate/rgg

The effects of tool eccentricity and formation anisotropy on resistivity logs: forward modeling A.D. Karinskiy *, D.S. Daev Sergo Ordzhonikidze Russian State Geological Prospecting University, ul. Miklukho-Maklaya 23, Moscow, 117997, Russia Received 28 October 2015; accepted 29 January 2015

Abstract The theory of resistivity logging (RL) originally stems from the Fock–Stefãnescu forward problem solution for the stationary electric field E in a piecewise homogeneous isotropic medium, with a single boundary corresponding to the surface of a cylinder unlimited by height. The cylinder simulates a borehole filled with drilling mud of resistivity ρ = ρb, which penetrates a formation with resistivity of rocks ρ = ρr. The primary field E is produced by the charge of a current electrode A placed on the cylinder axis. In this paper the forward problem for the field E is investigated for the electrode A at an arbitrary point off the axis of a borehole embedded in a transversely isotropic formation, with an anisotropy axis parallel to the borehole axis. The solution of forvard problem is used in algorithms and respective software for processing resistivity logs affected by electrode eccentricity and formation anisotropy. The changes caused to apparent resistivity by the two effects are estimated in percent for axial and lateral electrode dispositions. © 2016, V.S. Sobolev IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved. Keywords: resistivity logging; electrodes; borehole wall; anisotropy; eccentricity; forward modeling

Introduction The theory of resistivity logging (RL) and approaches to data interpretation originally stem from the forward solution for the stationary electric field E in an isotropic medium with a single cylindrical boundary. A cylinder of an unlimited height simulates a borehole filled with drilling mud of the resistivity ρ = ρb which penetrates a formation with uniform resistivity of rocks ρ = ρr. The primary field E is produced by a charge of a current electrode A placed on the cylinder axis. The solution to the problem, named the Fock–Stefãnescu forward problem (Alpin, 1971), is applicable to the processing of RL acquired with probe length much shorter than the thickness of the borehole traveled impenetrable formation. The approach to the solution was used further for the case of an electrode placed on the symmetry axis of a piecewise homogeneous isotropic medium with several coaxial cylindrical boundaries. Shared borders of these areas can simulate borehole, penetration zone (filtrate washing liquid), and unmodified part of permeable formation. However, transmitter and receiver electrodes used in real resistivity logging are more often placed on the borehole wall,

* Corresponding author. E-mail address: [email protected] (A.D. Karinskiy)

off the axis, even in deep vertical holes, and the effect of this eccentricity on RL data remains poorly studied. Some of our results of such studies for RL probe used in the logging practice were reported at the conference “New Ideas in Geosciences” (Daev and Karinskiy, 2013). Resistivity logs are also subject to effects of anisotropy as the patterns of resistivity (ρ) in many formations depend on the direction. Resistivity anisotropy of terrigenous sedimentary formations is due to alternation of thin lithologically differing layers of siltstones and mudstones or sandstones and mudstones. Such formations are most often simulated in terms of axial anisotropy, with their resistivity consisting of two components: parallel to the anisotropy axis or normal to bedding (ρn) and anisotropy-orthogonal, along any orthogonal  ρn / ρt . to n direction t (ρt); the anisotropy coefficient is λ = √ Parameters ρn and ρt are also called “vertical” resistivity and “horizontal” resistivity. In usual conditions, at horizontal or nearly-horizontal bedding, the axis n is close to vertical, and the formation anisotropy can be assumed to align with the borehole axis. According to previous forward modeling for these conditions, the ρt value has a major impact on RL results if the RL probe is oriented parallel to the anisotropy axis (the anisotropy paradox).

1068-7971/$ - see front matter D 201 6, V.S. So bolev IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.rgg.201 + 6.09.004

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With the recent advance in the computing facilities (increased computation speed, RAM size, etc.), as well as improving methods of calculations, it becomes possible to use 3D forward modeling and faithful models of real conditions. Now it is possible to determine the impact of environmental parameters on the results of the electrical and electromagnetic methods of well research models for approaching the actual measurement conditions (e.g., Epov et al., 2007, 2015). In this study, forward modeling is applied to the stationary electric field E, using a 1-D two-layer model of a borehole in an anisotropic formation of infinite thickness, with a current electrode set off the axis of the borehole, on its wall. A comparision of the calculated double-layer curves for gradient probes and potential probes RL is performed for the location of electrodes on the model axis and the wall of the borehole. The effect of resistivity anisotropy on the apparent electrical resistivity is estimated for electrodes on the borehole wall.

is L = √  l 2 + z 2 , where l 2 = rA2 + r 2 − 2⋅r⋅rA⋅cos ϕ. Expressed via Weber’s integral 2 ∞ 1 = ∫ K0 (ml) cos (mz) dm z2 π 0  √ l 2 + (where K0(ml) is the zero-order McDonald function), the potential Uprim becomes U prim =

2C ∞  K  m√ r 2 + rA2 − 2⋅r⋅rA⋅cos ϕ  cos (mz) dm.  π ∫0 0 

In accordance with the addition theorem for the Bessel functions,

Forward solution The forward problem for the stationary electric field E was solved using a model (Fig. 1a) consisting of a borehole (isotropic area of space V1: cylinder of unlimited height; ρb = ρ1). Current electrode A is located in the V1 area of space at the distance rA from the symmetry axis Z of the medium model. “External” area of space V2 is anisotropic with the anisotropy axis n parallel to the axis Z of the borehole, and with resistivities ρn, and ρt. The field E is given by sec E = Eprim + Esec = −grad U, U1,2 = U prim + U1,2 ,

Fig. 1b for locations of point A in the plane z = 0 and the projection M′ of point M on this plane. C The primary field potential is U prim = = L ρ1 I C ,C= . The distance between A and M 4π  √ (x − xA)2 + y 2 + z2

(1)

sec are the potentials of the total (E) and where U1,2 and U1,2 secondary (Esec) fields, respectively, in the space areas V1 and V2. The primary field Eprim with the potential Uprim is generated by the charge eA of the electrode A which emits the current I. This charge in the borehole with ρ1 is eA = ε0⋅ρ1⋅I,

where ε0 ≈ 10–9/(36 π) F/m is the electric constant. The secondary field Esec results from induced surface charges at the boundary S between the areas V1 and V2 (Fig. 1a) and volume charges in the area V2, at ρn ≠ ρt and the anisotropy coefficient λ ≠ 1. Unlike the isotropic medium, the volume charges can arise in a homogeneous anisotropic medium if the component of the field E changes in the direction of the anisotropy axis n (Kaufman and Keller, 1989). The distribution of the induced charges in the anisotropic medium at different ways of E excitation was discussed by Karinskiy (2010). The Cartesian (x, y, z) and cylindrical (r, ϕ, z) coordinates have the common axis Z and the origin 0; the azimuth angle ϕ is counted from the X direction; the current electrode A lies in the plane z = 0 on the axis X, at the distance rA from the origin of coordinates; the receiver electrode M has arbitrary coordinates {x, y, z}, {r, ϕ, z } (Fig. 1). In this configuration, the coordinates opoints A and M are related as xA = rA, yA = 0, zA=0, ϕA = 0; xM = x = r⋅cosϕ, yM = y = r⋅sinϕ, zM = z. See

 r 2 + rA2 − 2⋅r⋅rA⋅cos ϕ  = K0 (mrA) I0 (mr) K0 m√   ∞

+ 2 ∑ Kn (mrA) In (mr) cos (nϕ) n=1

∞

=2

∑

Kn (mrA) In (mr) cos (nϕ),

n=0(0.5)

where In(ζ) and Kn(ζ) are, respectively, the modified Bessel function and the n-order McDonald function of the argument ζ. In the right-hand side of the latter equation, n = 0(0.5) in the series Σ means that the term corresponding to n = 0 is multiplied by 0.5. Thus, the potential of the primary field Eprim at the point M is U prim =

∞ ∞  ρ1 I   (mr ) (mr) (nϕ) cos I K  cos (mz) dm.(2) A n 2 ∫ ∑ n π 0 n=0(0.5)   

The boundary-value problem corresponding to the model of Fig. 1 was formulated taking into account the current continuity equation div j = 0 valid for the stationary electric field, where j is the current density vector. In the isotropic domain V1, j(1) = E(1)/ρ1 = –grad U1/ρ1, according to the Ohm law; then div (grad U1/ρ1) = 0. In a homogenous isotropic medium (ρ1 = const), the Laplace equation div gradU1 = ∇ 2U1 = 0 is valid for the potentials U1 (at L ≠ 0), Uprim, and U1sec = U1 − U prim (throughout the domain V1, in which ∇ 2U1sec = 0). In the anisotropic domain V2, the scalar components of the 1 vector j in the cylindrical coordinates are jr(2) = Er(2) = ρt 1 1 ∂U2 1 (2) 1 ∂U2 − , , jϕ(2) = Eϕ(2) = − jz(2) = E = ρt ρn z ρt ∂r r⋅ρt ∂ϕ 1 ∂U2 − , wherefrom we can derive the differential equation ρn ∂z

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for the potential U2. In addition to the equations for U1 and U2, the boundary-value problem includes the infinity conditions, the condition at the singular point A, and the conjugation conditions at the boundary S where the coordinate r = a = d/2 (Fig. 1a). The potential U and the component jr normal to the S boundary are continuous at this boundary. Finally, the boundary-value problem for the potential U of the field E is: 2 2  1 ∂  ∂U1  1 ∂ U1 ∂ U1  + = 0; r + 2 r ∂r  ∂r  r ∂ϕ 2  ∂z 2  2 2 2 2 λ ∂  ∂U2  λ ∂ U2 ∂ U2 + = 0; r > a: r + r ∂r  ∂r  r 2 ∂ϕ 2 ∂z 2  L → ∞: U1 → 0, U2 → 0;   L → 0: U1 → ∞ as U prim;  1 ∂U2 1 ∂U1  = .  r = a: (a) U2 = U1, (b) ρt ∂r ρ1 ∂r 

1) at r < a: 2) at 3) at 4) at 5) at

(3)

The forward solutions for the field E that satisfies equations (3) for U1 and U2 are unique, in accordance with the uniqueness theorem (Alpin et al., 1985). Note that U1 and U2 must be even functions with respect to the argument z and even periodic functions with respect to ϕ, according to the problem conditions and to the even symmetry of the field U relative to the planes y = 0 and z = 0 (at zA = 0, ϕA = 0). The solutions to the differential equations for the potentials U1,2 in (3) are obtained using the approach similar to that of Svetov (2008) for the isotropic model. The second equation in system (3) is solved by separation of variables. Assume that the particular solution to the equation for U2 can be expressed as a product of three independent functions of r, ϕ, z: U2 = R (r)⋅Φ (ϕ)⋅Z (z).

(4)

At condition (4), it follows from the second equation in system (3) that 2

2

2

2

λ ⋅Φ⋅Z d  dR  λ ⋅R⋅Z d Φ d Z + R⋅Φ 2 = 0. + r r dr  dr  r 2 dϕ 2 dz Dividing all terms of the equation by the dot product R⋅Φ⋅Z and moving the last term of the equation to the rightλ2 d  dR  1 d 2Z λ2 d 2Φ =− hand side, we obtain . r +  2 2 Z dz 2 R⋅r dr  dr  Φ⋅r dϕ The left-hand side of the latter equation depends on the arguments r and ϕ, while the right-hand side depends on z. This becomes possible when the two sides are equated to the same arbitrary constant, for instance, m2 ≥ 0. Then,  1 d  dR  1 d 2Φ  1 d 2Z = m 2. = − λ2  r +   2 2 Z dz 2  R⋅r dr  dr  Φ⋅r dϕ  Therefore, the function Z satisfies the second-order differential d 2Z = −m 2⋅Z and is given by equation dz 2 Z = a1 cos (mz) + a2 sin (mz), where a1 and a2 are constants.

(5)

Fig. 1. Model of a medium and placement of electrodes A and M.

 1 d  dR  1 d 2Φ  =m2 From the equation λ2  r +  2 2 R⋅r dr dr Φ⋅r dϕ     2 2 1 d Φ m 1 d  dR  + = it follows that or r R⋅r dr  dr  Φ⋅r 2 dϕ 2  λ  2

1 d 2Φ r d  dR   m⋅r  . The latter equation means r −  =−  Φ dϕ 2 R dr  dr   λ  that two mutually independent functions are equal. The leftor right-hand sides of this equation can be equated to an arbitrary constant n2 ≥ 0. One of the resulting equations is d 2Φ / dϕ 2 = −n 2 Φ, which is solved as Φ = b1 cos (nϕ) + b2 sin (nϕ),

(6)

where b1 and b2 are constants. Another equation is 2

2 r d  dR   m⋅r  d 2R dR  m⋅r  − = n 2 or r 2 2 + r − + r     R dr  dr   λ  dr  λ  dr  n 2 R = 0. Straightforward derivations, with ζ = m⋅r /λ, lead to  a modified Bessel equation (Abramowitz and Stegun, 1972):

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d 2R

dR  2 +ζ − ζ + n 2 R = 0, with the modified Bessel 2 dζ  dζ function In(m⋅r/λ) and the McDonald function Kn(m⋅r/λ) of order n (n is an integer) of the real argument m⋅r/λ as its particular solutions. Their linear combination makes the solution ζ2

R = c1 In (m⋅r / λ) + c2 Kn (m⋅r / λ),

(7)

where c1 and c2 are constants. Note that Kn(ζ) → ∞ at ζ → 0 and In(ζ) → ∞ at ζ → ∞. Taking into account that a < r < ∞ in the domain V2, it is assumed that c1 = 0 in (7) satisfies condition 3 in equations (3). Since the potential U is an even function of the arguments ϕ and z in the chosen coordinates (see above), a2 = 0 and b2 = 0 in equations (5) and (6). According to (4), the particular solution for U2 is U2 (m, n) = D (m, n) Kn (m⋅r / λ) cos (nϕ)⋅ cos (mz), where D (m, n) = a1 ⋅ b1 ⋅ c2. Knowing that the potential U is a periodic function of ϕ, the function U2(m) can be represented as a Fourier series ∞

∑

U2 (m) =

n=0(0.5)

D (m, n)⋅Kn (m⋅r / λ) cos (nϕ) cos (mz).  

Ez(1) =

∞

 ∞  m  ∑ In (mr) Kn (mrA) 2 ∫ π 0 n=0(0.5) 

ρ1 I

 + A cos (nϕ) sin (mz) dm.   

(9)

Developing numerical algorithms was more difficult than obtaining an analytical forward solution. According to (8) and (9), the equations for the potentials U1 and U2 and the components Ez(1) of the field E are integrals (in the sense of the principal value). The integrand functions are infinite series, and their terms contain the modified Bessel function In and the McDonald function Kn of an arbitrary integer order n, as well as trigonometric functions. The functions In and Kn can be calculated using equations from (Abramowitz and Stegun, 1972). The improper integrals (in the sense of the principal value) were calculated numerically, at alternating dependence of the integrand function on the integration variable m (cos (mz) in (8) and sin (mz) in (9)). For this we used the Euler transform (Dashevsky, 1982), who drew our attention to its efficiency.

With m being any nonnegative constant, the general solu∞

tion for U2 is found as the integral U2 = ∫ U2 (m) dm, i.e.,

Results of numerical modeling for gradient probe and potential probe resistivity logging

0

∞

  ∞   U2 = ∫  ∑ D (m, n)⋅Kn (m⋅r / λ) cos (nϕ) cos (mz) dm.    0 n=0(0.5)  In the domain V1, 0 < r < a, while λ = 1. Therefore, equation 1 in system (3) for U1 can be solved as   ∞   U1 = ∫  ∑ F (m, n)⋅In (m⋅r) cos (nϕ) cos (mz) dm.    0 n=0(0.5)  ∞

However, in order to satisfy condition 4 in (3) and simplify the definition of unknown factors, we representthe expression for U1 and U2 as follows U1 =

U2 =

ρ1 I π2 ρ1 I π2

∞

∫ 0

∞

∫ 0

 ∞      ∑ In (mr) Kn (mrA) + A cos (nϕ) cos (mz) dm;   n=0(0.5)      (8)  ∞       ∑ B⋅Kn (m⋅r / λ) cos (nϕ) cos (mz) dm,   n=0(0.5)     

where the coefficients A = A(m, n) and B = B(m, n) depend on the parameters of the model in Fig. 1a and the coordinates rA, but are obviously independent of the coordinates r, ϕ, z of the measuring electrode M. The coefficients A and B were found using two conjugation conditions at the borehole boundary (r = a) in the boundary-value problem (3). At E = –grad U, the scalar component Ez = –∂U/∂z in the domain V1 from the first equation in (8) is

This section presents numerical simulation results obtained on the basis of expressions (8) and (9). The comparison of the simulation results for signal of gradient probe and potential probe RL when measuring and current electrodes are located on the borehole wall, either on its axis. In the latter case, the basis for the calculations was the Fock–Stefãnescu forward problem solution. Figures 2 and 4 show the model for which calculations and curves gradient-sensing probe and the potential probe RL were conducted. The curves are given for the following cases: (1) Current electrodes and measuring point lie on the axis of symmetry of the isotropic model (dash-dot line). (2) The electrodes are located on the borehole wall, on a line parallel to the borehole axis (solid line). Thus, ρn = ρt = ρr, λ = 1, r = rA = a, and ϕ = 0 (Figs. 1 and 3). Electrodes position is the same as in the previous case, but the environment is anisotropic: ρn = 4ρt and λ = 2 (dashed line). For limit unipolar gradient probe (current electrode B∞) the distance between the electrodes foster MN → 0, and the measured signal is proportional to the component Ez(O) in the center O of MN segment if the orientation of the probe is parallel to the axis Z. For this probe apparent resistivity 2 ρa = ρgp a = [4πL |Ez (O)|] / I, where L = AO is the probe length. The relative effect δ caused by the probe eccentricity (offset from the hole axis to the wall) was estimated for the isotropic cases (ρn = ρt = ρr) as wall − ρaxis δ = [(ρwall a a ) / ρa ]⋅100%,

(10)

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Fig. 2. Model of the medium and gradient probe (a) and two-layer resistivity curves (b).

Fig. 3. Dependences δ (a) and ∆ (b) of the L/d ratio. Gradient probe.

where ρwall and ρaxis are the ρa values for the RL probe on a a the borehole wall and axis, respectively. This effect (δ) is negative, while its magnitude may reach hundreds of percent, for RL probes at L/d < 1–1.5

and ρr/ρb << 1, and δ ≈ 50% at ρr/ρb >> 1 and L/d < 0.2–0.3 (Fig. 3). When these inequalities can neglect the influence of the curvature of the borehole wall, and the model is approximately corresponds to the case when the probe in the electrical

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Fig. 4. Model of the medium and potential probe (a) and two-layer resistivity curves (b).

resistivity method is located on the plane boundary of homogeneous conductive half-space (with ρ = ρb) with an insulator. In this case, ρa is known to be twice greater than in an unbounded infinite uniform medium with ρ = ρb. In the lateral logging sounding practice, the length of the shortest probe is greater than the borehole diameter. The magnitude |δ| does not exceed 5% if L/d > 2–3 at ρr/ρb > 1 or if L/d > 5–6 at ρr/ρb < 1 (Fig. 3a). This actually means (for a two-layer model) that the eccentricity, at ρr/ρb > 1, does not cause a significant effect on the data of medium and long probes. At the same time, it can strongly affect the data of short probes (few decimeters) at a large borehole diameter, especially at ρr/ρb < 1. For instance, δ ≈ 15% at ρr/ρb = 0.2 and L/d = 2–2.5. The effect of resistivity anisotropy of rocks (ρt, ρn) on the logging results was studied by forward modeling for the stationary field E in an unbounded anisotropic medium. The apparent resistivity ρa = ρt if the electrodes lie on a straight line oriented along the anisotropy axis n (anisotropy paradox). When the probe is located on the borehole axis, the value of ρn influences on ρa, but this effect is minor. Quantitatevily,

the degree of relative influence of anisotropy can be estimated by using the expression is is ∆ = [(ρan a = ρa ) / ρa ]⋅100%,

(11)

an where ρis a and ρa are the ρa values at ρn = ρt and ρn ≠ ρt, respectively. Our earlier calculation results based on solutions for axially symmetric model “borehole–formations” showed that, e.g., at λ = 2 and wide ranges of L/d and ρt/ρb, the ∆ value is not more than 35% (Karinskiy, 2004). Note that ∆ can exceed 60% in a three-layer model, which includes decrease in resistivity of invasion zone at λ = 1.5–2 (Karinskiy, 2005). The solutions (8) and (9) of forward problem and the modeling results (solid and dash lines in Fig. 2) can be used also to estimate the effect of probe eccentricity on ∆. Its value is from –10% to +30% at ρt/ρb from 0.1 to 200 and λ = 2. The values |∆|, as a rule, are the maxima if the relationship L/d corresponds to the extremes of the resistivity curves. If L << d or L << d , ∆ ≈ 0. Based on simulation results, we can conclude that the eccentricity of the RL gradient probe does not significantly influence on ρgp a value (Figs. 2 and 3).

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Fig. 5. Dependences δ (a) and ∆ (b) of the L/d ratio. Potential probe.

Influence of ρn and λ on ρgp a is small. With the probe on the borehole wall, the main influence on ρgp a has ρt parameter, as well as ρb and L/d. Figure 4 shows the curves of sounding for the potential probe RL logging (electrodes B∞ and N∞). As shown by the results obtained by mathematical modeling, eccentricity of this probe has an other effect on the apparent resistivity ρa = ρpp a than on ρgp a . In particular, the influence of eccentricity calculated according to the equation (10) is δ ≥ 5% if L/d < 3–4 at ρr/ρb << 1 (Fig. 5a). The influence of the anisotropy of the medium (for a fixed value λ and the arrangement of the electrodes on the borehole wall) on the value of ρpp a is slightly smaller than on the value of ρgp a . Modeling data showed that (for the same change of ρt/ρb and λ = 2) ∆ value varies from –8% to + 23% for potential probe (Fig. 5b). As already noted, for the gradient probe at λ = 2, ∆ changes from –10% to + 30% (Fig. 3b). For both types of probes, the anisotropy effect tends to zero (∆ → 0 at L/d → ∞).

Conclusions The algorithms based on forward solutions (8), (9) for the stationary electric field E in the model of Fig. 1 were used for the numerical simulation of resistivity logging, under typical conditions, when the electrodes are situated on the wall of the borehole instead its axis. The solution was obtained for the conditions when the environment can be transverselyisotropic (uniaxial-anisotropic) with the n-axis anisotropy collinear to the axis of the borehole. Mathematical modeling results for the currently used in resistivity logging (RL) gradient probes and potential probes have shown the following. Under typical conditions, when the length of the RL probes several times exceeds the diameter of the borehole, the displacement of electrodes from the axis of the borehole on its wall does not substantially impact on

the RL results (Fig. 2, 3a, 4, and 5a). Such a displacement of the electrodes are not significantly alter the effect of the anisotropy parameters of the borehole surrounding environpp ment. The values of ρgp a and ρa depend primarily on the longitudinal resistivity ρt, as well as on ρb (Figs. 2, 3b, 4, and 5b). On the basis of the solution (8) mathematical modeling programs have been developed not only for the currently used two types of RL probes (gradient probe and potential probe). This simulation was carried out (for not yet applied at well logging) the dipole–dipole equatorial RL probe (DEP) (Karinskiy and Kaurkin, 2013, 2015). According to mathematical and physical modeling results, the anisotropy coefficient λ can influence significantly the EDA apparent resistivity values under certain conditions. This result is consistent with those reported by Dashevsky et al. (1999).

References Abramowitz, M., Stegun, I.A. (Eds.), 1972. Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards. Applied Mathematics Series, Washington–New York. Alpin, L.M., 1971. Practical Training in the Field Theory [in Russian]. Nedra, Moscow. Alpin, L.M., Daev, D.S., Karinskiy, A.D., 1985. The Field Theory: Application to Exploration Geophysics [in Russian]. Nedra, Moscow. Daev, D.S., Karinskiy, A.D., 2013. Forward modeling in the resistivity logging theory for the cases of array eccentricity and formation anisotropy, in: New Ideas in Geosciences, Proc. KHI Int. Conf., Vol. 1, Moscow, p. 379. Dashevsky, Yu.A., 1982. Euler transform for modeling stationary and harmonic EM fields in layered formations, in: Electromagnetic Methods of Geophysical Surveys [in Russian]. IGiG, Novosibirsk, pp. 78–88. Dashevsky, Yu.A., Polozov, S.V., Epov, M.I., Martynov, A.A., Surodina, I.V., 1999. DC resistivity logging arrays, with non-axisymmetrical transmitters, for isotropic and anisotropic formations, in: DC and Induction Logging of Petroleum Wells [in Russian]. Izd. SO RAN, NIC OIGGM, Novosibirsk, pp. 130–145. Epov, M.I., Shurina, E.P., Nechaev, O.V., 2007. 3D forward modeling of vector field for induction logging problems. Russian Geology and Geophysics (Geologiya i Geofizika) 48 (9), 770–774 (989–995).

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Epov, M.I., Glinskikh, V.N., Sukhorukova, K.V., Nikitenko, M.N., Eremin, V.N., 2015. Forward modeling and inversion of LWD induction data. Russian Geology and Geophysics (Geologiya i Geofizika) 56 (8), 1194–1200 (1520–1529). Karinskiy, A.D., 2004. The AC field of cable and AB line in anisotropic media. EAGO Journal, Geofizika, No. 1, 40–48. Karinskiy, A.D., 2005. Electromagnetic field of different sources in a cylindrically layered model of an anisotropic medium. EAGO Journal, Geofizika, No. 6, 46–54. Karinskiy, A.D., 2010. Induced charges in micro- and macro-anisotropic media and their effect on the electric field in anisotropic formations. EAGO Journal, Geofizika, No. 2, 37–48.

Karinskiy, A.D., Kaurkin, M.D., 2013. Mathematical and physical modeling of dipole resistivity logging arrays in isotropic and anisotropic models. Geofizika, No. 4, 36–42. Karinskiy, A.D., Kaurkin, M.D., 2015. Physical (laboratory) and mathematical modeling of equatorial dipole arrays for resistivity logging: substantiation of the method for estimating transverse resistivity of anisotropic formations. Karotazhnik 248 (2), 34–49. Kaufman, A.A., Keller, G.V. 1989. Induction Logging. Methods in Geochemistry and Geophysics. Elsevier, Amsterdam. Svetov, B.S., 2008. Fundamentals of Resistivity Surveys [in Russian]. LKI, Moscow.

Editorial responsibility: M.I. Epov