Basics of Resistivity Tools

Published:January 01, 2011
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
Electrodetype (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 lowfrequency 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 electrodetype 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 electrodetype 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 A_{0} on the left and B on the right.
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:
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 A_{o} with area S_{o} carrying a current I_{o} 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 A_{o}. The electrode A_{o} may be called the main electrode, A’ is called the guard electrode, and current I_{o} will be focused. The resistivity now can be measured more accurately than before by:
where K_{o} 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 A_{1}, A_{0}, A_{1}’ and return electrode B with coaxial cylindrical diameter D_{A} and D_{B}, between which a voltage V is applied. Then, the resistivity is given by:
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:
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
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, ∇^{2}V = 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 2D, produced by a current ring source located at r=r’ and z=z’, is given by:
Applying superposition and reciprocity principles to the transferimpedance functions, when all current ring sources are activated, the voltage produced on the measurement electrode can be written as:
where Z_{ij} is the transfer impedance between the current emitting electrode A_{i} and measurement electrode M_{j}. Detailed solutions of Equations 27 and 28 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 A_{0} are two longguard electrodes, A_{1} and A_{2}, which are electrically connected to each other. A current I_{0} is emitted from the A_{0} electrode whose potential U_{0} is fixed. The current from the guard electrodes A_{1} and A_{2} is automatically adjusted to maintain A_{1} and A_{2} at the fixed potential U_{0}; 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 I_{0} is proportionate to formation conductivity.
Considering that the total tool length is much greater than the main electrode length (L >> L_{0}), and that the length of the main electrode is much greater than the diameter of the main electrode (L_{0} >>d_{A}), the electrode potential V, at an arbitrary point (r,z) in the formation is approximated from Equation 25:
Assuming that the LL3 recording point is at z = 0, r_{0}= d_{A}/2, then Equation 29 can be rewritten as:
Again, by Ohm’s law, Equations 26, the apparent resistivity seen by LL3 tool can be expressed as
where K = 2πL_{0}/(lnL − lnd_{A}). The apparent resistivity seen by the LL3 is proportional to the ratio V(r_{0},0)/I_{0} 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:
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, A_{0}, and three pairs of electrodes, M_{1} and M_{2}, M_{1}′, and M_{2}′, and A_{1} and A_{2} (Figure 2.3). The electrodes of each pair are symmetrically located with respect to A_{0} and electrically connected to each other. A constant current I_{0} is emitted from A_{0} and currents from the bucking electrodes A_{1} and A_{2} are automatically adjusted so that the two pairs of monitoring electrodes, M_{1} and M_{2}, and M_{1}′ and M_{2}′, 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 I_{0}, this potential (u) varies directly with formation resistivity in the form Ra = K(U/I_{0}), where K is the LL7 tool coefficient and can be expressed as
where the arrows signify distance. Since the potential difference between the M_{1}M_{2} and the M_{1}′M_{2}′ pairs is maintained at zero, no current from A_{0} is flowing in the hole between M_{1}M_{1}′ or between M_{2}M_{2}′ (i.e., the current from the A_{0} 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 I_{0} 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.
where K_{SFL} is SFL tool coefficient, ΔU_{M0O1} is the potential drop between the measurement electrode M_{o} (M’_{o}) and the midpoint O_{1} between the monitoring electrodes M_{1}M_{2} (M_{1}’M_{2}’).
The SFL had too large a volume of investigation to serve well as an R_{xo} 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)
Deepreading 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 LaterologMicroSFL 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.
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 currentbeam thicknesses, but have different foci for different depths of investigation. The long electrode pair A_{2} and A_{2}′ are used as bucking (or guard) electrodes for the deep measurement (LLD) to form the nominal beam and force the beam current I_{o} 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 A_{2} and A_{2}′ are used as current return electrodes, which cause the emitted current I_{o} 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 27), 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 DLSB and DLSE electrode configurations.
Highresolution laterolog array (HRLA)
An array laterolog supplying five depths of investigation, called the HRLA highresolution 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 currentreturn 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 drillpipe effects when the tool is deployed in drillpipeconveyed mode. (Previously the drill pipe could act as a currentreturning electrode, hence shortcircuiting 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 (A_{0}), and six bucking current electrodes on either side, plus twelve monitoring electrodes, for a total of 25 electrodes. The A_{0} electrode emits a survey current I_{0}, and the axially segmented buckingcurrent electrodes allow focusing of the survey current into the formation with a variable depth of investigation. The larger the number of buckingcurrent electrodes around A_{0} 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.
The fundamental building block of this array tool is like the LL3. Raw primary measurements are made by laterologstyle 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 resolutionmatched and depthaligned. The shallowest mode, RLA0, is mostly sensitive to the borehole and may be used to estimate the mud resistivity. The apparent resistivities RLA_{1} through RLA_{5} are all sensitive to the formation, becoming progressively deeper in investigation (see Chapter 3 for HRLAbenchmarkresponse examples).
Induction Tool Basics
R and X signals of induction (eddycurrent type) tool
The induction instrument was originally developed to measure formation resistivity in boreholes containing oilbased muds and in airdrilled boreholes where electrode devices could not work (Doll, 1949). For all induction tools, an alternating highfrequency 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 timevarying 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 N_{R} and N_{T} turns, current in transmitter I_{T} with frequency ω magnetic permeability of the formation μ, propagation constant k = (iωμσ)^{1/2}, where σ is the formation conductivity, and transmitterreceiver spacing L, the voltage is computed as
The voltage is calibrated to formation conductivity using a factor K, which is the tool coefficient defined as
The voltage given by Equation 215 is a complex number, meaning it is phaseshifted away from the phase of the transmitter current I_{T}. 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 215 and 216 yield the components of the complex conductivity signal, σ_{R} and σ_{X} signals:
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 Taylorseries expansion in powers of L/δ,
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 218, 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 zeroconductivity medium such as air. The third term on the right represents the conductivitydependent skin effect, with the real and imaginary parts being equivalent. After the mutual inductance is removed through careful buckingcoil design or digital data filtering and processing, the X signal provides a firstorder approximation of the skin effect.
Doll’s geometrical factor theory
Doll (1949) formulated an approximate model for inductiontool response, termed geometrical factor theory, which is quantitatively valid only where conductivity is zero. However, the geometrical factor theory defines an inductionresponse 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 twocoil system (one transmitter and one receiver, refer to Figure 2.7, g can be expressed as:
where A is the angle between lines ρ_{T} and ρ_{R}. Equation 219 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 YZ 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, zeroconductivity background. Refer to Moran (1982), and Gianzero and Anderson (1982) for geometrical factors with skin effect and finite conductivity considered.
6FF40 deepinduction 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 deepinduction (ILD) array, and its successor the dual induction tool (DIT) (Tanguy, 1967) became an industry standard for 30 years.
where g_{mn} is the geometric factor for the m^{th} transmitter and n^{th} receiver pair defined by Equation (219). The terms r, ρ_{T}, and ρ_{R} are defined in Figure 2.7 and L_{mn} is the distance between the m^{th} transmitter and the n^{th} receiver.
Integrated radial g factor and depth of investigation
Taking the integral of g(r,z) in Equation 220 with respect to z, the radial geometrical factor is defined as:
Solutions of Equation 221 do not lead to simple analytical expressions, but to first and secondorder elliptical functions with numerical values that can be found in tables of functions. The physical meaning of g_{r}(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 g_{r}(r) with respect to r, the integrated radial response function G_{r}(R) is defined as:
Traditionally, the induction tool’s depth of investigation is defined as the median point on the integrated radial response function G_{r}(r).
Vertical g factor and vertical resolution
Taking the integral of g(r,z) in Equation 220 with respect to r, the vertical geometrical factor is defined as:
Traditionally, the induction tool’s vertical resolution is defined as the width between the 5^{th} and 95^{th} percentile of the vertical geometrical factor g_{z}(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.
Deconvolution and skineffect correction on dual induction
The focused measurements made by a dual induction, multiarray 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 toolD (DITD) (prior to the phasor phaserinduction SFL tool and AIT array induction imager tools) are known as threepoint deconvolution, to reduce the shoulderbed effect and to improve the vertical resolution, and skineffect boosting, to correct the nonlinear responses of the induction tools.
Threepoint deconvolution
Doll (1965) introduced a threepointdeconvolution 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 σ_{R} in Equations 217 and 218 and applying a threestation windowing filter at ± 78 in. (198 cm) about a given logging station, and can be expressed as:
where σ_{D} is the threepoint deconvolution of σ_{R}. The distance of ± 78 in. (198 cm) was selected by inspection of the vertical geometrical factor for the 6FF40. The weights a_{0} and a_{1} were determined empirically for several values of shoulderbed resistivity and only applied to ILD curve, the weights a_{0} and a_{1} under different shoulderbed resistivity (SBR) values are listed in Table2.2 (Anderson and Barber, 1999).
Resistivity (ohm m)  1  10  100  1000 

Skin Depth δ  140 in. (356 cm)  443 in. (1125 cm)  1400 in. (3556 cm)  4429 in. (11,250 cm) 
Resistivity (ohm m)  1  10  100  1000 

Skin Depth δ  140 in. (356 cm)  443 in. (1125 cm)  1400 in. (3556 cm)  4429 in. (11,250 cm) 
SBR (ohm m)  a_{0}  a_{1} 

0.25  1.00  0.00 
0.5  1.06  0.03 
1.0  1.10  0.05 
2.0  1.16  0.08 
4.0  1.20  0.10 
SBR (ohm m)  a_{0}  a_{1} 

0.25  1.00  0.00 
0.5  1.06  0.03 
1.0  1.10  0.05 
2.0  1.16  0.08 
4.0  1.20  0.10 
Skineffect boosting
Skineffect 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 threepoint deconvolution. As pointed out by Moran and Kunz (1962), the raw conductivity signal σ_{R} (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:
and Re is the real part of the term following it. The difference between the true formation conductivity (σ) and the measured apparent conductivity (σ_{a}) is attributed to the skin effect. Such difference is greater when the formation has higher conductivity as shown in Figure 2.9.
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 225, and the fitting function used is:
Where η was chosen such that the formula gives the correct σ_{a} near 2 mS/m, and β is an array constant. In practice, the value of σ_{D} (z) in equation 224 is used to substitute σ_{R} in equation 226 for ILD or 6FF40, and σ_{R} 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).
Tool  η  β 

6FF40  1.0739  0.000135 
ILD  1.0899  0.000135 
ILM  1.0494  0.000030 
Tool  η  β 

6FF40  1.0739  0.000135 
ILD  1.0899  0.000135 
ILM  1.0494  0.000030 
Artifacts caused by deconvolution and skineffect boosting on ILD
In some cases, especially with high resistivitycontrast between the shoulder bed and target bed, threepoint deconvolution will introduce spurious infiniteresistivity spikes in the log (Barber, 1985; Shen, 1989). Figure 2.10 shows a computed ILD raw response (R_RAW), response after the threepointdeconvolution correction (R_DEC), and response after the threepointdeconvolution and skineffectboosting (R_DEC_SKIN) for a synthetic formation with bed thickness varying from 1 to 15 ft (0.3 to 4.6 m) and 1100 ohm m resistivity contrast. (This specific synthetic earth model is referred to as a chirp formation.)
The appearance of spikes in the 6, 7 and 8ft (1.83, 2.13, and 2.44m) thick beds is partly due to the coil configuration of the deepfocused induction sonde, and the threepoint deconvolution amplifies the magnitudes of the spikes. In fact, the deconvolved conductivity in the 100 ohm m 7ft (2m) 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 225 and the fitting function used in Equation 226 is for tool response in a homogeneous and infinitely thick medium, the skineffectcorrection algorithm is always applied to the entire length of the conductivity curve, not just to thick beds.
Note that the skineffect 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 threepointdeconvolution 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 skineffect 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 resistivitylog inversion against raw resistivity measurements instead of logs processed through these deconvolution and skineffectboosting 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 firstorder 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 Xsignal 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 nonlineardeconvolution technique that corrects the induction log in real time for shoulderbed effects and improves the thinbed 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:
where σ_{P}(z) is the resulting conductivity by Phasor processing, σ_{R}(n) is the n^{th} sample of the Rsignal, σ_{X}(n) is the n^{th} sample of the Xsignal, α(σ_{X}) is the magnitude fitting function and is identified as the nonlinear term. h(zn) is the deconvolution filter and b(zn) is the Xsignal fitting filter, and both are linear finiteimpulseresponse filters.
The essence of Phasor processing is to add the skineffect signal derived from the X signal to the deconvolved R signal to simultaneously produce resolutionenhanced and skineffectcorrected 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 multiplearray 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 TR 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, AITB, in 1990 (Hunka et al., 1990), and a shorter AITH 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 AITB 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 AITH 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).
Borehole signal correction
The first postprocessing step is to correct all raw array signals for borehole effects. The boreholecorrection 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 1D 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 σ_{f} is always solved, and two of the other borehole parameters (borehole radius r, mud conductivity σ_{m}, or the tool position x with respect to the borehole wall [standoff]) must be input from external measurements; the remaining one can be computed in the optimization routine.
Software focusing
Software focusing combines several boreholecorrected 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 boreholecorrected simple array raw measurements. The process can be expressed as:
Where σ_{log}(z) is the postprocessed AIT log, σ_{a}^{(n)}(zz′) is the measured log from the n^{th} subarray, 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.
The weights w_{n}(z′) in Equation 228 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 twodimensional responses. Figure 2.12 shows the integrated radial and the vertical responses of the AIT logs. In summary, such postprocessing or softwarefocusing 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 twocoil 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 hardwarefocused 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 hardwarefocusing tools is that the downhole hardwarefocused measurement may result in higher signaltonoiseratio data, and better measurement accuracy in severe borehole environments and highcontrast formations, than singlearray measurements. This is possible because the hardwarefocused multicoil system defines where the signal comes from better than the singlecoil system (i.e., it forces as large a portion of the signal as possible 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 DITE and the array induction AIT tools for higher and improved vertical resolution. However, the accuracy of the tool response in highcontrast 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 1Dbenchmark 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 (DITD) 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 3point deconvolution and skineffectboosting section of this chapter, the anomalous streaks occur in the 7 and 6ft (2.1 and 1.8m)thick beds partly because of the coil configuration of the DITD induction sonde, and the threepoint 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).
As shown in Figure 2.13b and c, both the Phasor processing and AIT postprocessing derived curves can be used to resolve beds thinner than 5 ft (1.5 m) better than traditional DITD logs. However, the DITE and AITB 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 2ft (0.6m) vertical resolution) are attributed to the fact that the assumptions in the linearfiniteimpulseresponsefitting filters, h(zn) and b(zn) in Equation 227, 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 w_{n}(z’ ) in Equation 228 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 2D 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 90in. (25, 51, 76, 152, and 229m) depthofinvestigation curves should not be interpreted as representing resistive invasion profiles solely based on arrayinduction data. This is especially true in resistive beds with highcontrast and lowresistivity shoulder beds.
Figures & Tables
Resistivity (ohm m)  1  10  100  1000 

Skin Depth δ  140 in. (356 cm)  443 in. (1125 cm)  1400 in. (3556 cm)  4429 in. (11,250 cm) 
Resistivity (ohm m)  1  10  100  1000 

Skin Depth δ  140 in. (356 cm)  443 in. (1125 cm)  1400 in. (3556 cm)  4429 in. (11,250 cm) 
SBR (ohm m)  a_{0}  a_{1} 

0.25  1.00  0.00 
0.5  1.06  0.03 
1.0  1.10  0.05 
2.0  1.16  0.08 
4.0  1.20  0.10 
SBR (ohm m)  a_{0}  a_{1} 

0.25  1.00  0.00 
0.5  1.06  0.03 
1.0  1.10  0.05 
2.0  1.16  0.08 
4.0  1.20  0.10 
Tool  η  β 

6FF40  1.0739  0.000135 
ILD  1.0899  0.000135 
ILM  1.0494  0.000030 
Tool  η  β 

6FF40  1.0739  0.000135 
ILD  1.0899  0.000135 
ILM  1.0494  0.000030 
Contents
Application of ResistivityToolResponse Modeling for Formation Evaluation
Understanding resistivitytool response and resistivitylog interpretation for formation evaluation is vital for the matching of the reconstructed deepreading resistivity logs with the field log curves. AAPG Archie 2 introduces the fundamental concepts required. Resistivityloggingtool physics and measurement accuracy are reviewed, and forward and inversemodeling resistivitytool responses are introduced. In the case studies presented, welldeviation, shoulderbed, bedthickness, borehole, mudresistivity, and invasion effects on restivitylog responses are discussed. This volume has been written for geoscientists and engineers working with and interpreting resistivity logs, petrophysicists and reservoir engineers integrating resistivitybased and capillarypressurebased quantitative calculation of formation water saturation, and formation evaluation specialists.