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Abstract

Resistivity logs use electrode arrays or coil arrays for focusing. The accuracy of the measurement is enhanced by designing the array to focus into the zone of interest.

The most significant difference between laterolog and induction devices is that laterologs put the borehole fluid, the invaded zone, and the undisturbed zone in series in the radial direction of the borehole as regards current flow, whereas the induction devices put these regions in parallel. If the injected currents are constant, the measured potentials for laterolog tools increase proportionally with resistivity. The eddy current induced in the formation and the electric potential measured by induction devices will increase as the conductivity increases (i.e., as the resistivity decreases). This chapter briefly describes laterolog and induction tool response theories.

Overview

Introduction

Resistivity logs use electrode arrays or coil arrays for focusing. The accuracy of the measurement is enhanced by designing the array to focus into the zone of interest.

The most significant difference between laterolog and induction devices is that laterologs put the borehole fluid, the invaded zone, and the undisturbed zone in series in the radial direction of the borehole as regards current flow, whereas the induction devices put these regions in parallel. If the injected currents are constant, the measured potentials for laterolog tools increase proportionally with resistivity. The eddy current induced in the formation and the electric potential measured by induction devices will increase as the conductivity increases (i.e., as the resistivity decreases). This chapter briefly describes laterolog and induction tool response theories.

Laterolog (Galvanic) Tool Basics

Electrode-type (galvanic) tool basics

For all laterologs, current is injected into the formation from electrodes via a conducting borehole fluid. The current is accompanied by a potential distribution in the borehole dependent on the formation resistivity and the configuration of the injection electrodes. In principle, laterologs can use direct current, but in practice low-frequency alternating current, on the order of a few hertz to a hundred hertz, is used. This is because the electronic circuitry attached to the electrodes makes use of electronic conduction while the borehole fluid and formation fluids conduct via ionic transfer, which generates a static potential because of an electrochemical reaction occurring at the electrodes. Frequencies up to 1kHz would be desirable to reduce the effect of these static potentials, but other factors, such as minimizing the Delaware gradient and Groningen effect require the frequency to be as low as a few hertz. The Delaware gradient is the electrostatic field gradient existing at the position of the nucleus in atoms, depending on the effective atomic number (Barnes and Smith, 1954), and the Groninger effect refers to error in measurement of the resistivity of a formation by an electrode-type tone when a thick layer of higher resistivity lies over a more conductive formation whose resistivity is to be evaluated (Ellis and Singer, 2007).

The focusing principle of the electrode-type tool is based on the use of guard electrodes. This principle is illustrated first with two simple setups for a laboratory measurement of resistivity. Figure 2.1a shows an edge view of a circular disc of rock sandwiched between two circular disc electrodes, A and B, both of area S. Figure 2.1b shows the same disc of rock sandwiched between concentric disc electrodes A’ and A0 on the left and B on the right.

Figure 2.1.

Schematic of resistivity measurement using (a) unguarded and (b) guarded planar electrodes.

Figure 2.1.

Schematic of resistivity measurement using (a) unguarded and (b) guarded planar electrodes.

Ohm’s law

In Figure 2.1a, if a voltage V is applied to the disc electrodes, generating a current I, one can estimate resistivity R from Ohm’s Law: 

formula

For the rock sample of thickness L, the resistivity R of the rock can be approximated by measuring V and I, and the electrodes’ geometrical constant (S/L). This is only approximate, because the current density in the rock is not uniform and the current lines do not all have uniform length L.

In Figure 2.1b, the disc electrode A is split into a small central disc Ao with area So carrying a current Io surrounded by a concentric disc A’ held at the same potential V. This results in nearly constant current density and orthogonal current lines radiating out from Ao. The electrode Ao may be called the main electrode, A’ is called the guard electrode, and current Io will be focused. The resistivity now can be measured more accurately than before by: 

formula

where Ko is known as the tool coefficient. This use of guard electrodes for current focusing is the basic concept of the laterolog.

To bring these considerations a little closer to the situation encountered in borehole logging, the planar electrodes need to be replaced by cylindrical electrodes as depicted in Figure 2.2. Assume a cylindrical rock of interest is shaped in contact with the tool electrodes A1, A0, A1’ and return electrode B with coaxial cylindrical diameter DA and DB, between which a voltage V is applied. Then, the resistivity is given by: 

formula

Figure 2.2.

Schematic of resistivity measurement in a borehole by modification of the laboratory apparatus in Figure 2.1b using three cylindrical coaxial guarded electrodes, A0, A1 and A′1. This is the basic laterolog 3 (LL3) tool design, except in the real LL3 tool, the current-return electrode B is located far away from the tool’s current-emitting electrodes.

Figure 2.2.

Schematic of resistivity measurement in a borehole by modification of the laboratory apparatus in Figure 2.1b using three cylindrical coaxial guarded electrodes, A0, A1 and A′1. This is the basic laterolog 3 (LL3) tool design, except in the real LL3 tool, the current-return electrode B is located far away from the tool’s current-emitting electrodes.

The essential difference is that the outer electrode B cannot, of course, be used in borehole geometry (i.e., the current return electrode B must either be placed in the borehole or remotely at the surface). Instead of the total current from emitting electrodes, a small element with area S is considered. Again, by Ohm’s law, a voltage drop can be written as: 

formula

where R, the formation resistivity, and the beam cross section S are both a function of distance l along the current emanating lines. Integrating over l from the surface of the electrodes A to B, for instance, yields 

formula
 
formula

This is how the tool measures apparent resistivity, Ra, as a function of tool geometry and spatial distribution of true formation resistivity (Moran and Chemali, 1985).

Laplace’s equation

To compute a laterolog response numerically for a given electrode array and distribution of resistivities involves the solution of the governing equation for the electrical potentials, ∇2V = 0, Laplace’s equation. Assuming that resistivity variations are continuous in the radial direction, and heterogeneous in depth, the expression for the potential, V(r,z) in 2-D, produced by a current ring source located at r=r’ and z=z’, is given by: 

formula

Applying superposition and reciprocity principles to the transfer-impedance functions, when all current ring sources are activated, the voltage produced on the measurement electrode can be written as: 

formula

where Zij is the transfer impedance between the current emitting electrode Ai and measurement electrode Mj. Detailed solutions of Equations 2-7 and 2-8 are laborious and the mathematics involved is beyond the scope of this volume (see Gianzero et al., 1985).

Laterolog 3 and laterolog 7

As shown in Figure 2.3, the actual laterolog 3 (LL3) tool uses bucking currents to focus the measuring current into a horizontal sheet penetrating into the formation. Symmetrically placed on either side of the central electrode A0 are two long-guard electrodes, A1 and A2, which are electrically connected to each other. A current I0 is emitted from the A0 electrode whose potential U0 is fixed. The current from the guard electrodes A1 and A2 is automatically adjusted to maintain A1 and A2 at the fixed potential U0; this variable focusing current is termed the bucking current. All electrodes of the sonde are thus held at the same constant potential, and the magnitude of the current I0 is proportionate to formation conductivity.

Figure 2.3.

Schematic of electrode-type (galvanic) tools, laterolog 7, laterolog 3, and SFL (spherically focused log) emitting current patterns (Schlumberger, 1989). 01 and 02 are insulators.

Figure 2.3.

Schematic of electrode-type (galvanic) tools, laterolog 7, laterolog 3, and SFL (spherically focused log) emitting current patterns (Schlumberger, 1989). 01 and 02 are insulators.

Considering that the total tool length is much greater than the main electrode length (L >> L0), and that the length of the main electrode is much greater than the diameter of the main electrode (L0 >>dA), the electrode potential V, at an arbitrary point (r,z) in the formation is approximated from Equation 2-5: 

formula

Assuming that the LL3 recording point is at z = 0, r0= dA/2, then Equation 2-9 can be rewritten as: 

formula

Again, by Ohm’s law, Equations 2-6, the apparent resistivity seen by LL3 tool can be expressed as 

formula

where K = 2πL0/(lnL − lndA). The apparent resistivity seen by the LL3 is proportional to the ratio V(r0,0)/I0 that is the total resistance rg sensed by the LL3 tool. In a borehole environment, the total resistance rg can be expressed by a series summation: 

formula

where rm, rmc, rxo, ri, and rt, stand for resistance of borehole mud, mud cake, flushed zone, invaded zone, and true formation, respectively. This is how the laterolog puts the resistivity of the borehole fluid, mud cake, invaded zone, and uninvaded zone in series in the radial direction of the borehole as regards current flow.

The Latrolog 7 (LL7) device comprises a center electrode, A0, and three pairs of electrodes, M1 and M2, M1′, and M2′, and A1 and A2 (Figure 2.3). The electrodes of each pair are symmetrically located with respect to A0 and electrically connected to each other. A constant current I0 is emitted from A0 and currents from the bucking electrodes A1 and A2 are automatically adjusted so that the two pairs of monitoring electrodes, M1 and M2, and M1′ and M2′, are forced to have the same potential.

The potential drop is measured between the monitoring electrodes and a remote electrode. With constant main electrode current I0, this potential (u) varies directly with formation resistivity in the form Ra = K(U/I0), where K is the LL7 tool coefficient and can be expressed as 

formula

where the arrows signify distance. Since the potential difference between the M1-M2 and the M1′-M2′ pairs is maintained at zero, no current from A0 is flowing in the hole between M1-M1′ or between M2-M2′ (i.e., the current from the A0 must penetrate horizontally into the formations). Figure 2.3 shows the schematic distribution of current lines.

Vertical Resolution of LL3 and LL7

The thickness of the I0 current sheet or vertical resolution, is approximately 32 in. (81 cm) for the LL7, and 12 in. (30 cm) for the LL3.

Spherically focused log and microspherical focused log

The SFL spherically focused resistivity log tool was originally designed to provide a relatively shal low investigation to evaluate the effects of invasion on deep resistivity measurements. Whereas the LL7 and LL3 systems attempt to focus the current into planar disks, the SFL system establishes essentially constant potential shells around the current electrode, to preserve the spherical potential distribution in the formation over a wide range of wellbore variables. To accomplish this, the SFL device was composed of a bucking current system, to prevent current flowing in the borehole and establish the equipotential spheres, and the survey current system, to flow through the volume of investigation. The intensity of the survey current is inversely proportional to formation resistivity.

The apparent resistivity measured by a spherically focused tool can be expressed as 

formula

where KSFL is SFL tool coefficient, ΔUM0O1 is the potential drop between the measurement electrode Mo (M’o) and the mid-point O1 between the monitoring electrodes M1M2 (M1’M2’).

The SFL had too large a volume of investigation to serve well as an Rxo measurement tool, hence, a Micro Spherically Focused log tool was introduced (Suau et al., 1972). The MSFL has a similar electrode configuration with smaller dimensions for a smaller volume of investigation. All MSFL electrodes are imbedded in a pad which is pushed against the borehole wall.

Dual laterolog tool (DLL)

Deep-reading galvanic resistivity tools have been designed for measuring resistivity beyond the invaded zone, but no single measurement has yet overcome the effects of the borehole and invaded zone entirely. One solution was to measure the resistivity with several arrays having different depths of investigation to correct for borehole and invasion effects on Rt. Such a solution resulted in the development of the DLL Dual Laterolog-Micro-SFL resistivity tool with simultaneous recordings of three different depths of investigation (Suau et al., 1972). Figure 2.4 is a schematic of the tool, showing the electrode array and current distributions for deep and shallow laterologs.

Figure 2.4.

Schematic of the Dual Laterolog, tool electrodes array (left), and current beam distribution of LLD (laterolog deep resistivity) and LLS (laterolog shallow resistivity) (Schlumberger, 1989). The Rxo part shown in the figure is the MSFL pad. 1 ft = 0.3 m; 1 in. = 2.5 cm.

Figure 2.4.

Schematic of the Dual Laterolog, tool electrodes array (left), and current beam distribution of LLD (laterolog deep resistivity) and LLS (laterolog shallow resistivity) (Schlumberger, 1989). The Rxo part shown in the figure is the MSFL pad. 1 ft = 0.3 m; 1 in. = 2.5 cm.

Vertical resolution and depth of investigation of DLL

The nominal beam thickness defines the vertical resolution that is approximately 2 ft (0.6 m) for both laterolog deep resistivity (LLD) and laterolog shallow resistivity (LLS).

The deep and shallow devices use the same electrodes and have similar current-beam thicknesses, but have different foci for different depths of investigation. The long electrode pair A2 and A2′ are used as bucking (or guard) electrodes for the deep measurement (LLD) to form the nominal beam and force the beam current Io to flow into the formation, thereby achieving relatively deep depths of investigation. The distance between the extreme ends of the guard electrodes of the DLL is approximately 28 ft (8.5 m).

For the shallow measurement (LLS), the same electrodes A2 and A2′ are used as current return electrodes, which cause the emitted current Io to diverge more quickly in the formation. The tool thus responds more sensitively to regions affected by invasion. Computer simulation of DLL tool response can be realized by the solutions of the Laplace equation (Equation 2-7), with boundary conditions of V = constant at electrode surface and infinite boundary, and ∂V/∂R = 0 on the insulating surface and the borehole axis. Figure 2.5 shows the Schlumberger dual laterolog models DLS-B and DLS-E electrode configurations.

Figure 2.5.

Schematics of the Schlumberger models DLS-B and DLS-E dual laterolog electrode configurations.

Figure 2.5.

Schematics of the Schlumberger models DLS-B and DLS-E dual laterolog electrode configurations.

High-resolution laterolog array (HRLA)

An array laterolog supplying five depths of investigation, called the HRLA high-resolution laterolog array tool was designed with the aim of a better vertical resolution for thin beds, and a better differentiated set of measurements for inversion (Smits et al., 1998). This design also eliminated the surface current-return electrode (i.e., the bridle, and the need for a deep mode). All measurements use bucking currents returning to the tool rather than to the surface, eliminating the Groningen and drill-pipe effects when the tool is deployed in drill-pipe-conveyed mode. (Previously the drill pipe could act as a current-returning electrode, hence short-circuiting the bucking current returning to the surface and making the deep mode appear shallower.)

A schematic representation of the HRLA is shown in Figure 2.6. The tool is composed of a central current electrode (A0), and six bucking current electrodes on either side, plus twelve monitoring electrodes, for a total of 25 electrodes. The A0 electrode emits a survey current I0, and the axially segmented bucking-current electrodes allow focusing of the survey current into the formation with a variable depth of investigation. The larger the number of bucking-current electrodes around A0 that are kept at the same potential, the deeper the measurement. The remaining outer electrodes are at zero potential and act as the current return. The six monitoring electrodes ensure additional accuracy of the equipotential conditions close to the center of the tool.

Figure 2-6.

Schematic of the HRLA high resolution laterolog array electrodes configuration and electric-potential profiles. The yellow sections are insulated (Smits et al., 1998).

Figure 2-6.

Schematic of the HRLA high resolution laterolog array electrodes configuration and electric-potential profiles. The yellow sections are insulated (Smits et al., 1998).

The fundamental building block of this array tool is like the LL3. Raw primary measurements are made by laterolog-style hardware focusing, rather than an unfocused array of normal or lateral devices. Additional focusing (i.e., the imposition of the equipotential conditions on the array) is implemented in software focusing (i.e., computed focusing). The focused currents are computed as linear combinations of primary measurement modes, measured simultaneously at different signal frequencies. The weighting factor for each primary mode in the linear combination is computed such that all the focusing conditions are satisfied.

The result is six focused modes with varying depths of investigation that are intrinsically resolution-matched and depth-aligned. The shallowest mode, RLA0, is mostly sensitive to the borehole and may be used to estimate the mud resistivity. The apparent resistivities RLA1 through RLA5 are all sensitive to the formation, becoming progressively deeper in investigation (see Chapter 3 for HRLA-benchmark-response examples).

Induction Tool Basics

R and X signals of induction (eddy-current type) tool

The induction instrument was originally developed to measure formation resistivity in boreholes containing oil-based muds and in air-drilled boreholes where electrode devices could not work (Doll, 1949). For all induction tools, an alternating high-frequency current of constant intensity is sent through a transmitter coil, which generates electromagnetic fields around the borehole and formation. The alternating magnetic field induces an eddy current in the formation. The eddy current flows in circular ground loops coaxial with the transmitter coil, which in turn creates a time-varying magnetic flux that induces a voltage in the receiver coil.

The intensity of the eddy current is dependent on the formation conductivity. Hence, the induced voltage in the receiver coil is directly proportional to the measured formation conductivity rather than to resistivity, even though the induction measurement is referred to as a resistivity measurement. Moreover, the contributions to the total conductivity signal from various individual regions of the formation combine additively, because currents generated by the coaxial coils all flow parallel to each other. The voltage in the receiver coil of a simple induction tool can be shown to be a vector voltage for which the magnitude and phase are functions of the conductivity of the surrounding formation (Moran and Kunz, 1962). Assuming a receiver and transmitter of radius a with NR and NT turns, current in transmitter IT with frequency ω magnetic permeability of the formation μ, propagation constant k = (iωμσ)1/2, where σ is the formation conductivity, and transmitter-receiver spacing L, the voltage is computed as 

formula

The voltage is calibrated to formation conductivity using a factor K, which is the tool coefficient defined as 

formula

The voltage given by Equation 2-15 is a complex number, meaning it is phase-shifted away from the phase of the transmitter current IT. The real and imaginary components are referred to as the resistive (R) signal and reactive (X) signal in analogy to the usage in alternating current circuit theory.

Equations 2-15 and 2-16 yield the components of the complex conductivity signal, σR and σX signals: 

formula

which allows the calculation of the actual response of an induction sonde in a homogeneous medium. An approximation to the exact solution can be obtained by a Taylor-series expansion in powers of L/δ, 

formula

Where σ is the formation conductivity, and δ is the skin depth of the electromagnetic wave emitted by the transmitter and defined as k = [2/(ωμσ)]1/2. By definition, the skin depth 8 gives the order of magnitude of the penetration of the electromagnetic field into the formation. For the conventional 6FF40 induction tool operated at 20 kHz, and assuming a magnetic permeability of 4π × 10-7 H/m, the following table shows the depth that the induction tool can sense.

In Equation 2-18, the first term on the right is simply a linear term in formation conductivity, validating that an induction device measures parallel conductivity. The second term on the right is independent of formation conductivity and is the direct mutual inductance between the transmitter and receiver that exists even in a zero-conductivity medium such as air. The third term on the right represents the conductivity-dependent skin effect, with the real and imaginary parts being equivalent. After the mutual inductance is removed through careful bucking-coil design or digital data filtering and processing, the X signal provides a first-order approximation of the skin effect.

Doll’s geometrical factor theory

Doll (1949) formulated an approximate model for induction-tool response, termed geometrical factor theory, which is quantitatively valid only where conductivity is zero. However, the geometrical factor theory defines an induction-response function that explains the signal source and describes the induction tool’s depth of investigation and vertical resolution.

From a unit ground loop of radius r, and situated at an altitude z with respect to the origin located at the middle point between transmitter and receiver, Doll derived the geometrical factor g that depends exclusively on the geometry. For a two-coil system (one transmitter and one receiver, refer to Figure 2.7, g can be expressed as: 

formula

Figure 2.7.

Schematic of the geometrical factor for the different regions of the earth around a two-coil system. The circumference in red-dashed line having T-R for diameter corresponds to the maximum possible value for g (sin A = sin 90° = 1), which is used as reference. The colored areas correspond to more than 50% of the maximum g covered by blue, between 35 and 50% by green, between 2 and 35% by yellow, respectively. dxdydz = unit volume.

Figure 2.7.

Schematic of the geometrical factor for the different regions of the earth around a two-coil system. The circumference in red-dashed line having T-R for diameter corresponds to the maximum possible value for g (sin A = sin 90° = 1), which is used as reference. The colored areas correspond to more than 50% of the maximum g covered by blue, between 35 and 50% by green, between 2 and 35% by yellow, respectively. dxdydz = unit volume.

where A is the angle between lines ρT and ρR. Equation 2-19 shows that all the unit loops for which the angle A has the same sine, have the same geometrical factor and, therefore, contribute the same amount to the total signal provided homogeneous conductivity. All the unit loops having the same g are on a circumference that passes through the center of the transmitter T and receiver R.

In Figure 2.7, a cross section of the ground in the Y-Z plane containing the axis of the borehole illustrates a certain number of the circumferences corresponding to a given value of g. This is how the geometrical factor theory defines an induction response function that answers the question of signal source.

Note that the above is only valid for g’s in homogeneous, zero-conductivity background. Refer to Moran (1982), and Gianzero and Anderson (1982) for geometrical factors with skin effect and finite conductivity considered.

6FF40 deep-induction multicoil array’s geometrical factor

The 6FF40 induction array was introduced in 1959 (Tanguy, 1962). This array was licensed and run with minor variations by most service companies, and the 6FF40 deep-induction (ILD) array, and its successor the dual induction tool (DIT) (Tanguy, 1967) became an industry standard for 30 years.

For a multi-coil array system, the geometrical factor g can be expressed as 

formula

where gmn is the geometric factor for the mth transmitter and nth receiver pair defined by Equation (2-19). The terms r, ρT, and ρR are defined in Figure 2.7 and Lmn is the distance between the mth transmitter and the nth receiver.

Integrated radial g factor and depth of investigation

Taking the integral of g(r,z) in Equation 2-20 with respect to z, the radial geometrical factor is defined as: 

formula

Solutions of Equation 2-21 do not lead to simple analytical expressions, but to first- and second-order elliptical functions with numerical values that can be found in tables of functions. The physical meaning of gr(r) is the relative contribution of a unit cylindrical shell with radius r to the measured conductivity (i.e., the radial investigation characteristic).

Taking the integral of gr(r) with respect to r, the integrated radial response function Gr(R) is defined as: 

formula

Traditionally, the induction tool’s depth of investigation is defined as the median point on the integrated radial response function Gr(r).

Vertical g factor and vertical resolution

Taking the integral of g(r,z) in Equation 2-20 with respect to r, the vertical geometrical factor is defined as: 

formula

Traditionally, the induction tool’s vertical resolution is defined as the width between the 5th and 95th percentile of the vertical geometrical factor gz(z).

In designing the 6FF40 array, the depth of investigation was optimized using the radial g factor, and the vertical resolution was determined using the vertical g factor. Figure 2.8 is the schematic of the 6FF40 g factor, with about 40 to 60 in. (100 to 150 cm) depth of investigation and 5 ft (1.5 m) vertical resolution.

Figure 2.8.

Schematic of the 6FF40 array’s g factors. Note the depth of investigation is defined as the median (or 50%) point on the integrated radial response function Gr(r), and the vertical resolution is defined as the width between the 5th and 95th percentile of vertical geometrical factor gz(z).

Figure 2.8.

Schematic of the 6FF40 array’s g factors. Note the depth of investigation is defined as the median (or 50%) point on the integrated radial response function Gr(r), and the vertical resolution is defined as the width between the 5th and 95th percentile of vertical geometrical factor gz(z).

Deconvolution and skin-effect correction on dual induction

The focused measurements made by a dual induction, multi-array coil system respond to a relatively large volume of the formation. Hence, the logs over beds of interest can still be affected by adjacent beds, the borehole, invasion, and a nonlinear response, especially in conductive formations. Over the years, a series of correction algorithms has been derived to compensate for these environmental and propagation effects.

The two most common algorithms applied to dual induction tool-D (DIT-D) (prior to the phasor phaser-induction SFL tool and AIT array induction imager tools) are known as three-point deconvolution, to reduce the shoulder-bed effect and to improve the vertical resolution, and skin-effect boosting, to correct the nonlinear responses of the induction tools.

Three-point deconvolution

Doll (1965) introduced a three-point-deconvolution algorithm by giving greater proportional weight to the signal measured at the sonde center than to signals measured above and below the center point.

In practice, the algorithm involves memorizing the raw conductivity signal σR in Equations 2-17 and 2-18 and applying a three-station windowing filter at ± 78 in. (198 cm) about a given logging station, and can be expressed as: 

formula

where σD is the three-point deconvolution of σR. The distance of ± 78 in. (198 cm) was selected by inspection of the vertical geometrical factor for the 6FF40. The weights a0 and a1 were determined empirically for several values of shoulder-bed resistivity and only applied to ILD curve, the weights a0 and a1 under different shoulder-bed resistivity (SBR) values are listed in Table-2.2 (Anderson and Barber, 1999).

Table 2.1.

Electromagnetic wave penetration depth as a function of formation resistivity

Resistivity (ohm m)1101001000
Skin Depth δ140 in. (356 cm)443 in. (1125 cm)1400 in. (3556 cm)4429 in. (11,250 cm)
Resistivity (ohm m)1101001000
Skin Depth δ140 in. (356 cm)443 in. (1125 cm)1400 in. (3556 cm)4429 in. (11,250 cm)
Table 2.2.

ILD 3-point deconvolution coefficients

SBR (ohm m)a0a1
0.251.000.00
0.51.060.03
1.01.100.05
2.01.160.08
4.01.200.10
SBR (ohm m)a0a1
0.251.000.00
0.51.060.03
1.01.100.05
2.01.160.08
4.01.200.10

Skin-effect boosting

Skin-effect boosting is the amplification of the raw signal to compensate for nonlinear losses resulting from the EM wave propagation. The correction is applied to both the deep (ILD) and medium (ILM) responses after the three-point deconvolution. As pointed out by Moran and Kunz (1962), the raw conductivity signal σR (real part of σ) is always lower than the true conductivity even when the induction tool is in a thick bed. For a multicoil array in a thick bed, it can be shown that: 

formula

and Re is the real part of the term following it. The difference between the true formation conductivity (σ) and the measured apparent conductivity (σa) is attributed to the skin effect. Such difference is greater when the formation has higher conductivity as shown in Figure 2.9.

Figure 2.9.

The difference between the true formation conductivity σt and the measured apparent conductivity σa is attributed to the skin effect. Note the difference is greater when the formation is more conductive.

Figure 2.9.

The difference between the true formation conductivity σt and the measured apparent conductivity σa is attributed to the skin effect. Note the difference is greater when the formation is more conductive.

The skin effect is corrected by an algorithm that is an approximate fit based on the tool response in a homogeneous medium as in Equation 2-25, and the fitting function used is: 

formula

Where η was chosen such that the formula gives the correct σa near 2 mS/m, and β is an array constant. In practice, the value of σD (z) in equation 2-24 is used to substitute σR in equation 2-26 for ILD or 6FF40, and σR is directly used for ILM of the DIT tools. In Table 2.3, η and β for different tools are listed (Anderson and Barber, 1999).

Table 2.3.

Skin-effect boosting coefficient

Toolηβ
6FF401.07390.000135
ILD1.08990.000135
ILM1.04940.000030
Toolηβ
6FF401.07390.000135
ILD1.08990.000135
ILM1.04940.000030

Artifacts caused by deconvolution and skin-effect boosting on ILD

In some cases, especially with high resistivity-contrast between the shoulder bed and target bed, three-point deconvolution will introduce spurious infinite-resistivity spikes in the log (Barber, 1985; Shen, 1989). Figure 2.10 shows a computed ILD raw response (R_RAW), response after the three-point-deconvolution correction (R_DEC), and response after the three-point-deconvolution and skin-effect-boosting (R_DEC_SKIN) for a synthetic formation with bed thickness varying from 1 to 15 ft (0.3 to 4.6 m) and 1-100 ohm m resistivity contrast. (This specific synthetic earth model is referred to as a chirp formation.)

Figure 2.10.

ILD raw response (R_RAW), response after the three-point deconvolution correction (R_DEC), and response after the three-point deconvolution and skin-effect boosting (R_DEC_SKIN) in a synthetic chirp formation with bed thickness varying from 1 to 15 ft (0.3 to 4.6 m) and 1-100 ohm m resistivity contrast (red solid). 1 ft = 0.3 m.

Figure 2.10.

ILD raw response (R_RAW), response after the three-point deconvolution correction (R_DEC), and response after the three-point deconvolution and skin-effect boosting (R_DEC_SKIN) in a synthetic chirp formation with bed thickness varying from 1 to 15 ft (0.3 to 4.6 m) and 1-100 ohm m resistivity contrast (red solid). 1 ft = 0.3 m.

The appearance of spikes in the 6-, 7- and 8-ft (1.83-, 2.13-, and 2.44-m) thick beds is partly due to the coil configuration of the deep-focused induction sonde, and the three-point deconvolution amplifies the magnitudes of the spikes. In fact, the deconvolved conductivity in the 100 ohm m 7-ft (2-m) thick bed becomes negative in conductivity space. The negative conductivity is represented as infinite resistivity (i.e., zero conductivity).

Although the assumption in deriving Equation 2-25 and the fitting function used in Equation 2-26 is for tool response in a homogeneous and infinitely thick medium, the skin-effect-correction algorithm is always applied to the entire length of the conductivity curve, not just to thick beds.

Note that the skin-effect corrected curve (R_DEC_SKIN ≈ 0.98 ohm m) is indeed close to the true formation resistivity Rt in those 1 ohm m conductive beds compared with the raw response (R_RAW ≈ 1.38 ohm m) and the response after the three-point-deconvolution correction (R_DEC ≈ 1.33 ohm m). The absolute difference is only 0.36 ohm m, but is visually exaggerated by the logarithmic scale.

The skin-effect correction also reduces the resistivity readings in the resistive beds, for instance, from R_DEC ≈ 35.03 ohm m down to R_DEC_SKIN ≈ 33.32 ohm m at depth 187.5’ . Although the absolute difference is only 1.71 ohm m, which is 4.75 times greater than the difference in the conductive beds, it is not apparent in the logarithmic scale that is commonly applied to the resistivity log display.

It is always desirable to run resistivity-log inversion against raw resistivity measurements instead of logs processed through these deconvolution and skin-effect-boosting algorithms; but this was not possible until the raw data, R and X signals, were recorded and provided to customers by service companies.

Phasor induction tool and nonlinear deconvolution

The basic theory of induction logging describes the nonlinear response due to skin effect. After removing the direct mutual inductance term between the transmitter and receiver, the X signal pro vides a first-order approximation of the skin effect (Moran, 1964). Furthermore, Doll’s geometrical factor theory, which is only valid in zero conductivity, was later extended to the case where skin effect is significant (Gianzero and Anderson, 1982; Moran, 1982).

However, “we cannot measure the reactive X-signal with sufficient precision and stability due to lack of control of the direct coupling term, which is of the same phase. Whether or not this objection remains valid today is open to question” (Moran, 1982, p. 6). This statement by Moran may still be a subject of debate today because the X signal from a geological formation can be hundreds of times smaller than the quadrature signal through direct coupling, especially in a resistive formation.

It was claimed that the key to the development of the Phasor* tool was a nonlinear-deconvolution technique that corrects the induction log in real time for shoulder-bed effects and improves the thin-bed resolution over the full range of formation conductivity (Schaefer et al., 1984). The introduction of digital telemetry to well logging opened up new data channels, and the Phasor Dual Induction Tool (DITE) incorporated measurement of the X signals from deep and medium arrays (Barber, 1985). A kernel algorithm for the nonlinear Phasor deconvolution can be expressed as: 

formula

where σP(z) is the resulting conductivity by Phasor processing, σR(n) is the nth sample of the R-signal, σX(n) is the nth sample of the X-signal, α(σX) is the magnitude fitting function and is identified as the nonlinear term. h(z-n) is the deconvolution filter and b(z-n) is the X-signal fitting filter, and both are linear finite-impulse-response filters.

The essence of Phasor processing is to add the skin-effect signal derived from the X signal to the deconvolved R signal to simultaneously produce resolution-enhanced and skin-effect-corrected logs. Phasor logs are produced in three resolutions: standard (8 ft [2.4 m] IDPH, 5 ft [1.5 m] [IMPH], 3 ft [0.9 m] [IDER, IMER], and 2 ft [0.6 m] [IDVR and IMVR]).

Array induction tools

The concept of multiple-array induction measurements may not be new (Poupon, 1957), but limitations in returning data to the surface through the logging cable prevented practical application until the introduction of digital telemetry to well logging. BPB Instrument Ltd. first published papers on its digital induction tool that had one transmitter and four receivers with T-R spacing equal to 20, 30, 40, and 60 in. (50, 76, 102, and 152 cm). They also illustrated weighted summation in the construction of deep and medium responses by linear filtering (Martin et al., 1984; Elkington and Patel, 1985).

Schlumberger introduced its standard array induction tool, AIT-B, in 1990 (Hunka et al., 1990), and a shorter AIT-H tool for their Platform Express integrated wireline tool combination in 1995 (Barber et al., 1995). Both tools have one transmitter and eight pairs of receiver arrays. The AIT-B tool operates simultaneously at three frequencies (low, medium, and high). R and X signals are acquired from each array at one or two frequencies suitable for that array spacing, resulting in 28 data channels. The AIT-H tool operates at a single frequency, similar to the 6FF40 tool, between 20 and 30 kHz, and R and X signals are acquired from each array resulting in 16 data channels. Refer to Figure 2.11 for schematic representations of these tools. All measurements are simultaneously acquired every 3 in. (7.6 cm) of depth (i.e., the sample rate is 4/ft [13/m], and 15 resistivity logs are formed through postprocessing).

Figure 2.11.

Schematic of the DIT and AIT transmitter and receiver coil configurations and operating frequencies.

Figure 2.11.

Schematic of the DIT and AIT transmitter and receiver coil configurations and operating frequencies.

Borehole signal correction

The first postprocessing step is to correct all raw array signals for borehole effects. The borehole-correction algorithm minimizes the difference between the modeled and actual logs from the four shortest arrays (Grove and Minerbo, 1991).

The theoretical model used to compute the AIT response for different borehole conditions assumes that the borehole is an infinite right circular cylinder in a homogeneous formation with the sonde either centered or eccentered in the borehole (i.e., a 1-D radial model).

In practice, a data table was generated using the modeling code, and a polynomial representation of the data table was compared to the actual short array measurements.

At each depth level, formation conductivity σf is always solved, and two of the other borehole parameters (borehole radius r, mud conductivity σm, or the tool position x with respect to the borehole wall [stand-off]) must be input from external measurements; the remaining one can be computed in the optimization routine.

Software focusing

Software focusing combines several borehole-corrected simple array measurements to achieve optimized responses with enhanced radial and vertical resolutions. The optimization process used for AIT logs is an extension to two dimensions of the deconvolution first proposed by Doll (1965) and implemented in Phasor processing as nonlinear deconvolution, as discussed in the previous section on the phasor tool (Barber, 1984, 1985, 1989; Barber and Rosthal, 1991).

The AIT logs are formed as weighted sums of the borehole-corrected simple array raw measurements. The process can be expressed as: 

formula

Where σlog(z) is the postprocessed AIT log, σa(n)(z-z′) is the measured log from the nth sub-array, and N is the total number of measurement channels. This process results in a number of response functions that are weighted sums of the response functions of each individual array channel n (R and X signals at appropriate frequencies from all arrays). These response functions are summarized in Figure 2.12.

Figure 2.12.

(a) Five integrated radial response functions and (b) three vertical responses functions of the AIT logs (after Anderson and Barber, 1996, used by permission of the Society of Petrophysicists and Well Log Analysts). 1 in. = 2.5 cm; 1 ft = 0.3 m.

Figure 2.12.

(a) Five integrated radial response functions and (b) three vertical responses functions of the AIT logs (after Anderson and Barber, 1996, used by permission of the Society of Petrophysicists and Well Log Analysts). 1 in. = 2.5 cm; 1 ft = 0.3 m.

The weights wn(z′) in Equation 2-28 are functions of the effective background conductivity, and they were computed for 13 conductivity levels (0, 10, 20, 50, 100, 200, 500, 1000, 2000, 3500, 5000, 7000, and 10,000 mS/m). In practice, the weights used in postprocessing are interpolated from the table values nearest the effective background value at the measurement point.

The radial depths are defined by the median depth of investigation of the integrated radial response function G(r), and five depths (10, 20, 30, 60, and 90 in. [25, 51, 76, 152, and 229 cm]) are chosen to represent the information content on the raw array measurements. Three vertical resolution widths (1, 2, and 4 ft [0.3, 0.6, and 1.2 m]) are available with increasingly robust two-dimensional responses. Figure 2.12 shows the integrated radial and the vertical responses of the AIT logs. In summary, such postprocessing or software-focusing produces 15 logs in five depths of investigation and three vertical resolutions.

Hardware focusing vs. software focusing

The introduction of auxiliary coils on a basic two-coil sonde to maximize the signal coming from a particular earth region of interest is referred to as focusing. For induction tools, focusing can be achieved by the superposition of two or more coil responses to either subtract responses from unwanted regions (e.g., borehole, caves, and mud cake), or add responses from desired regions.

There have been two ways to achieve focusing. One is to form multiple transmitter coil responses and to gather multiple receiver coil signals downhole (i.e., these multiple transmitters and receivers are hardware connected, and one composite hardware-focused signal is output from the sonde and sent uphole). The second way is to gather signal from each receiver individually, and send them all uphole for postprocessing and analysis in the uphole computer unit. This is software focusing.

The designers of the early induction tools (6FF40 and DITB, and DITD tools) had to choose hardware focusing by default because the wireline was not capable of transferring large amounts of data in real time (Anderson and Barber, 1996).

One of the most important features of hardware-focusing tools is that the downhole hardware-focused measurement may result in higher signal-to-noise-ratio data, and better measurement accuracy in severe borehole environments and high-contrast formations, than single-array measurements. This is possible because the hardware-focused multicoil system defines where the signal comes from better than the single-coil system (i.e., it forces as large a portion of the signal as pos-sible to come from a particular region of interest that is beyond the borehole and invaded zones).

With today’s digital telemetry, software focusing by postprocessing and analysis have become a reality for induction measurements and have led to the development of the DIT-E and the array induction AIT tools for higher and improved vertical resolution. However, the accuracy of the tool response in high-contrast formations with software focusing appears to be problematic, as the following example illustrates.

Figure 2.13 shows the dual induction, phasor induction, and array induction 1-D-benchmark responses in the synthetic chirp formation with 1 to 100 ohm m (2.7 to 0.3 m) resistivity contrast and bed thickness varying from 10 to 1 ft (2.7 to 0.3 m). The standard dual induction (DIT-D) response shown in Figure 2.13a appears to be reasonable in detecting Rt from a resistive bed thicker than 8 ft (2.4 m). As discussed in the 3-point deconvolution and skin-effect-boosting section of this chapter, the anomalous streaks occur in the 7- and 6-ft (2.1- and 1.8-m)-thick beds partly because of the coil configuration of the DIT-D induction sonde, and the three-point deconvolution amplifies the magnitudes of the horns. Both the ILD and ILM fail to resolve Rt of the resistive beds thinner than 5 ft (1.5 m), and the ILD cannot discriminate beds thinner than 2 ft (0.6 m).

Figure 2.13.

The dual-induction (a), phasor-induction (b), and array-induction (c) 1-D benchmark responses under a chirp formation with 1 to 100 ohm m resistivity contrast and varying bed thickness from 1 to 10 ft. (0.3 to 3 m). 1 ft = 0.3 m.

Figure 2.13.

The dual-induction (a), phasor-induction (b), and array-induction (c) 1-D benchmark responses under a chirp formation with 1 to 100 ohm m resistivity contrast and varying bed thickness from 1 to 10 ft. (0.3 to 3 m). 1 ft = 0.3 m.

As shown in Figure 2.13b and c, both the Phasor processing and AIT post-processing derived curves can be used to resolve beds thinner than 5 ft (1.5 m) better than traditional DIT-D logs. However, the DIT-E and AIT-B responses apparently have problems resolving Rt for the resistive beds thicker than 5 ft (1.5 m) in such high resistivity contrast formations.

The resistivity overshoots in IDVR and IMVR (very enhanced resistivity Phasor processing with 2-ft (0.6-m) vertical resolution) are attributed to the fact that the assumptions in the linear-finite-impulse-response-fitting filters, h(z-n) and b(z-n) in Equation 2-27, do not apply to the formation in question.

The Born approximation is the process of taking the incident field in place of the total field as the driving field at each point in the scatterer. It is originally introduced in the scattering theory, particularly in quantum mechanics. It is the perturbation method applied to scattering by an extended body, and it is accurate if the scattered field is small, compared to the incident field, in the scatterer. When there is large contrast between the formation resistivities, such as in this instance 1 to 100, the scattered EM field is no longer small compared to the incident field, the weighting function wn(z’ ) in Equation 2-28 and the Born approximation in AIT postprocessing are no longer valid for the formation with these levels of resistivity contrast. Hence, the sets of AIT logs formed by the weighted deconvolution summation software focusing will inevitably exhibit nonphysical horns and overshoots (i.e., apparent resistivity varying from 10 to 2,000 ohm m and noninterpretable sequences of resistivity curves). Further studies of the AIT responses in 2-D reveals that the five different depth of investigation curves (10, 20, 30, 60, and 90 in. [25, 51, 76, 152, and 229 cm]) cannot be used to indicate the radial resistivity variation caused by the borehole and resistive invasion in the 1 to 10 ohm m resistivity contrast chirp formation.

The example in Figure 2.13 illustrates why the separation among the 10-, 20-, 30-, 60-, and 90-in. (25-, 51-, 76-, 152-, and 229-m) depth-of-investigation curves should not be interpreted as representing resistive invasion profiles solely based on array-induction data. This is especially true in resistive beds with high-contrast and low-resistivity shoulder beds.

Figures & Tables

Figure 2.1.

Schematic of resistivity measurement using (a) unguarded and (b) guarded planar electrodes.

Figure 2.1.

Schematic of resistivity measurement using (a) unguarded and (b) guarded planar electrodes.

Figure 2.2.

Schematic of resistivity measurement in a borehole by modification of the laboratory apparatus in Figure 2.1b using three cylindrical coaxial guarded electrodes, A0, A1 and A′1. This is the basic laterolog 3 (LL3) tool design, except in the real LL3 tool, the current-return electrode B is located far away from the tool’s current-emitting electrodes.

Figure 2.2.

Schematic of resistivity measurement in a borehole by modification of the laboratory apparatus in Figure 2.1b using three cylindrical coaxial guarded electrodes, A0, A1 and A′1. This is the basic laterolog 3 (LL3) tool design, except in the real LL3 tool, the current-return electrode B is located far away from the tool’s current-emitting electrodes.

Figure 2.3.

Schematic of electrode-type (galvanic) tools, laterolog 7, laterolog 3, and SFL (spherically focused log) emitting current patterns (Schlumberger, 1989). 01 and 02 are insulators.

Figure 2.3.

Schematic of electrode-type (galvanic) tools, laterolog 7, laterolog 3, and SFL (spherically focused log) emitting current patterns (Schlumberger, 1989). 01 and 02 are insulators.

Figure 2.4.

Schematic of the Dual Laterolog, tool electrodes array (left), and current beam distribution of LLD (laterolog deep resistivity) and LLS (laterolog shallow resistivity) (Schlumberger, 1989). The Rxo part shown in the figure is the MSFL pad. 1 ft = 0.3 m; 1 in. = 2.5 cm.

Figure 2.4.

Schematic of the Dual Laterolog, tool electrodes array (left), and current beam distribution of LLD (laterolog deep resistivity) and LLS (laterolog shallow resistivity) (Schlumberger, 1989). The Rxo part shown in the figure is the MSFL pad. 1 ft = 0.3 m; 1 in. = 2.5 cm.

Figure 2.5.

Schematics of the Schlumberger models DLS-B and DLS-E dual laterolog electrode configurations.

Figure 2.5.

Schematics of the Schlumberger models DLS-B and DLS-E dual laterolog electrode configurations.

Figure 2-6.

Schematic of the HRLA high resolution laterolog array electrodes configuration and electric-potential profiles. The yellow sections are insulated (Smits et al., 1998).

Figure 2-6.

Schematic of the HRLA high resolution laterolog array electrodes configuration and electric-potential profiles. The yellow sections are insulated (Smits et al., 1998).

Figure 2.7.

Schematic of the geometrical factor for the different regions of the earth around a two-coil system. The circumference in red-dashed line having T-R for diameter corresponds to the maximum possible value for g (sin A = sin 90° = 1), which is used as reference. The colored areas correspond to more than 50% of the maximum g covered by blue, between 35 and 50% by green, between 2 and 35% by yellow, respectively. dxdydz = unit volume.

Figure 2.7.

Schematic of the geometrical factor for the different regions of the earth around a two-coil system. The circumference in red-dashed line having T-R for diameter corresponds to the maximum possible value for g (sin A = sin 90° = 1), which is used as reference. The colored areas correspond to more than 50% of the maximum g covered by blue, between 35 and 50% by green, between 2 and 35% by yellow, respectively. dxdydz = unit volume.

Figure 2.8.

Schematic of the 6FF40 array’s g factors. Note the depth of investigation is defined as the median (or 50%) point on the integrated radial response function Gr(r), and the vertical resolution is defined as the width between the 5th and 95th percentile of vertical geometrical factor gz(z).

Figure 2.8.

Schematic of the 6FF40 array’s g factors. Note the depth of investigation is defined as the median (or 50%) point on the integrated radial response function Gr(r), and the vertical resolution is defined as the width between the 5th and 95th percentile of vertical geometrical factor gz(z).

Figure 2.9.

The difference between the true formation conductivity σt and the measured apparent conductivity σa is attributed to the skin effect. Note the difference is greater when the formation is more conductive.

Figure 2.9.

The difference between the true formation conductivity σt and the measured apparent conductivity σa is attributed to the skin effect. Note the difference is greater when the formation is more conductive.

Figure 2.10.

ILD raw response (R_RAW), response after the three-point deconvolution correction (R_DEC), and response after the three-point deconvolution and skin-effect boosting (R_DEC_SKIN) in a synthetic chirp formation with bed thickness varying from 1 to 15 ft (0.3 to 4.6 m) and 1-100 ohm m resistivity contrast (red solid). 1 ft = 0.3 m.

Figure 2.10.

ILD raw response (R_RAW), response after the three-point deconvolution correction (R_DEC), and response after the three-point deconvolution and skin-effect boosting (R_DEC_SKIN) in a synthetic chirp formation with bed thickness varying from 1 to 15 ft (0.3 to 4.6 m) and 1-100 ohm m resistivity contrast (red solid). 1 ft = 0.3 m.

Figure 2.11.

Schematic of the DIT and AIT transmitter and receiver coil configurations and operating frequencies.

Figure 2.11.

Schematic of the DIT and AIT transmitter and receiver coil configurations and operating frequencies.

Figure 2.12.

(a) Five integrated radial response functions and (b) three vertical responses functions of the AIT logs (after Anderson and Barber, 1996, used by permission of the Society of Petrophysicists and Well Log Analysts). 1 in. = 2.5 cm; 1 ft = 0.3 m.

Figure 2.12.

(a) Five integrated radial response functions and (b) three vertical responses functions of the AIT logs (after Anderson and Barber, 1996, used by permission of the Society of Petrophysicists and Well Log Analysts). 1 in. = 2.5 cm; 1 ft = 0.3 m.

Figure 2.13.

The dual-induction (a), phasor-induction (b), and array-induction (c) 1-D benchmark responses under a chirp formation with 1 to 100 ohm m resistivity contrast and varying bed thickness from 1 to 10 ft. (0.3 to 3 m). 1 ft = 0.3 m.

Figure 2.13.

The dual-induction (a), phasor-induction (b), and array-induction (c) 1-D benchmark responses under a chirp formation with 1 to 100 ohm m resistivity contrast and varying bed thickness from 1 to 10 ft. (0.3 to 3 m). 1 ft = 0.3 m.

Table 2.1.

Electromagnetic wave penetration depth as a function of formation resistivity

Resistivity (ohm m)1101001000
Skin Depth δ140 in. (356 cm)443 in. (1125 cm)1400 in. (3556 cm)4429 in. (11,250 cm)
Resistivity (ohm m)1101001000
Skin Depth δ140 in. (356 cm)443 in. (1125 cm)1400 in. (3556 cm)4429 in. (11,250 cm)
Table 2.2.

ILD 3-point deconvolution coefficients

SBR (ohm m)a0a1
0.251.000.00
0.51.060.03
1.01.100.05
2.01.160.08
4.01.200.10
SBR (ohm m)a0a1
0.251.000.00
0.51.060.03
1.01.100.05
2.01.160.08
4.01.200.10
Table 2.3.

Skin-effect boosting coefficient

Toolηβ
6FF401.07390.000135
ILD1.08990.000135
ILM1.04940.000030
Toolηβ
6FF401.07390.000135
ILD1.08990.000135
ILM1.04940.000030

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