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Abstract

Resistivity-tool response is affected by the borehole mud, washouts, shoulder beds, thin beds, invasion, and bed dip, and in addition, every type of resistivity tool also has its inherent physical limits in measurement accuracy and resolution. Such limited vertical resolution and accuracy in detecting Rt by a specific resistivity tool may only be resolved by tool-response modeling if the minimum changes in tool response are greater than the specified accuracy of that tool. Because of the limitations imposed by tool physics, the uncertainty in resistivity-tool-response inversion is an issue even with an ideal data set measured in perfect borehole conditions. A clear understanding of how the smallest change in input (for instance, bed thickness or Rt in 1-D modeling) will produce a detectable change in output (the computed resistivity-tool response) is critical in the evaluation of the inversion results and uncertainty analysis. Using results from benchmark modeling, this chapter illustrates sensitivities and limitations of resistivity-tool-response modeling and potential errors in hydrocarbon-saturation calculation.

Overview

Introduction

Resistivity-tool response is affected by the borehole mud, washouts, shoulder beds, thin beds, invasion, and bed dip, and in addition, every type of resistivity tool also has its inherent physical limits in measurement accuracy and resolution. Such limited vertical resolution and accuracy in detecting Rt by a specific resistivity tool may only be resolved by tool-response modeling if the minimum changes in tool response are greater than the specified accuracy of that tool. Because of the limitations imposed by tool physics, the uncertainty in resistivity-tool-response inversion is an issue even with an ideal data set measured in perfect borehole conditions. A clear understanding of how the smallest change in input (for instance, bed thickness or Rt in 1-D modeling) will produce a detectable change in output (the computed resistivity-tool response) is critical in the evaluation of the inversion results and uncertainty analysis. Using results from benchmark modeling, this chapter illustrates sensitivities and limitations of resistivity-tool-response modeling and potential errors in hydrocarbon-saturation calculation.

Terminology and Definitions

It is necessary to address terminology and definitions for a metrological description of logging measurements. The general terms and definitions introduced by the International Organization for Standardization (IOS), “Guide to the Expression of Uncertainty in Measurement” (1995), are adapted in this chapter.

Rt: True Value

True Value (of a quantity) is defined as “a value consistent with the definition of a given particular quantity” (Annex B. 2. 3, p. 32).

Note that True Value is a value that would be obtained by a perfect measurement, and hence, it is by nature indeterminate, such as Rt measured by a resistivity instrument. In addition, the indefinite article “a” rather than the definite article “the” is used in conjunction with True Value because there may be many values consistent with the definition of a given particular quantity.

Conventional True Value

Conventional True Value (of a quantity) is defined as “a value attributed to a particular quantity and accepted, sometimes by convention, as having an uncertainty appropriate for a given purpose” (Annex B. 2. 4., p. 23).

Note that a conventional True Value is sometimes called an assigned value, or a best estimate of the value. An Rt applied in resistivity forward modeling is an assigned value, and an Rt profile resulting from the inverse-modeling process may be taken as the best estimate of the value. In addition, in forward-tool-response modeling Rt can be a True Value, but in the inverse-modeling process Rt can only be a conventional True Value.”

Resolution

Resolution is defined here as the ability to localize an event through a window (i.e., a half width of the major lobe [a local maximum bounded by identifiable nulls]).

Note that qualitatively, resolution is about the ability to separate two features that are close together. In the logging industry, resolution is commonly regarded as a tool’s vertical resolution, which is defined and discussed for laterolog and induction instruments in Chapter 2 (also refer to Theys, 1999).

Sensitivity

Sensitivity is defined here as the smallest change in a quantity that a sensor is able to perceive.

Note that an instrument can have excellent sensitivity and yet poor accuracy. For example, the FMI tool is very sensitive to relative formation resistivity changes, but very poor in detecting Rt.

In technical areas other than well logging, one may find that resolution is defined as the smallest change in input that will produce a detectable change in output. The definitions above are more consistent with logging industry usage.

Accuracy — The essential information for a log

Accuracy of a measurement is rigorously defined by the ISO. The definition given in the “Guide to the Expression of Uncertainty in Measurement” (ISO, 1995), is the closeness of the agreement between the result of a measurement and the true value of the measurand (Annex B.2.14, p. 33). Note that accuracy will be a qualitative concept and relative term when the above definition is used. The key problem in this definition is that it is impossible to determine True Value of the measurand (quantity being measured).

In all practical situations, measurements are affected by imperfections of the measurement system that give rise to errors in the result, even in an ideal environment. For instance, because of mechanical stability, thermal noise, and electronic noise, induction-tool sonde error is inherent even when the tool is suspended in free space (a zero-conductivity environment). For practical purposes, then, accuracy can be defined as a function of error.

Accuracy is defined here as the total error compared with conventional (or conditional) True Value, or the degree of freedom from error. Note that such error is often expressed as the sum of the squares of the maximum limit, or overall root-mean-square (RMS), which is composed of random and systematic errors. This is the essential information for a logging tool. For resistivity tools, the overall RMS error is usually defined through experiments with homogeneous background assumptions. In forward modeling, Rt can be considered as True Value, and the tool response is unique when Rm, Rxo, borehole diameter, depth of invasion, and bed boundaries are given. Hence, accuracy can be clearly defined by the difference between the measured tool response and given Rt. In inverse modeling, Rt may not be unique, but the overall RMS error of a tool can be used as guidance as to when to stop iterations and also can be used to minimize the differences between the computed tool response and field log.

The repeatability of measurements, defined as the closeness of the agreement between the results of successive measurements of the same measurand, is also called precision and is often expressed by the mean squared deviation of a set of measurements from the average value. Repeatability and precision should not be confused with accuracy.

Sensitivity of Tool Response

Bed-thickness effect

Figure 6.1 illustrates the sensitivity of ILD tool response when bed thickness is changed slightly. In the left track, interbedded 6-ft (1.8-m)-thick resistive and conductive beds with 1 and 20 ohm m resistivities are shown as a red solid line. Corresponding ILD response based on 1-D modeling is shown with a red dashed line. When the bed thickness is changed by ±5% (i.e. from 6 ft to 6.3 ft [1.8 to 1.9 m] and from 6 ft to 5.7 ft [1.8 to 1.7 m] [shown in blue and green respectively]), the relative changes of the ILD response are about ±15% in resistive beds, but the response in the conductive beds remains almost unchanged. The relative differences are plotted in the right track; they are calculated as the ILD response from the 6 ft (1.8 m) ±5% models divided by the 6 ft (1.8 m) model.

Figure 6.1.

Sensitivity of induction response to ±5% changes of bed thickness in a 6-ft (1.8-m)-thick interbedded-formation model. 1 ft = 0.3 m.

Figure 6.1.

Sensitivity of induction response to ±5% changes of bed thickness in a 6-ft (1.8-m)-thick interbedded-formation model. 1 ft = 0.3 m.

It is clear from this example that ILD response is much more sensitive to thickness changes in resistive beds than conductive beds. The accurate definition of resistive-bed thickness is therefore critical in deriving Rt of the resistive beds in an inversion process.

Resistivity effect

Figure 6.2 shows the sensitivity of ILD tool response to changes in formation resistivity. In the left track, interbedded 6-ft (1.8-m)-thick resistive and conductive beds with 1 and 20 ohm m resistivity are shown as a squared red solid line. The corresponding ILD response from 1-D modeling is shown as a red dashed line. When the resistivity of each bed is changed by ±5%, from 20 to 21 ohm m and 1 to 1.05 ohm m, or from 20 to 19 ohm m and 1 to 0.95 ohm m, (shown in blue and green respectively), the relative changes of the ILD responses are about ±5% in conductive beds with negligible change in resistive beds, as shown on the right track. The ILD responses remain almost unchanged in resistive beds, as seen in the red-, blue-, and green-dashed lines, which basically overlay each other, on the left track.

Figure 6.2.

Sensitivity of induction response to ±5% changes of bed resistivity in a 6-ft (1.8-m)-thick interbedded-formation model. 1 ft = 0.3m.

Figure 6.2.

Sensitivity of induction response to ±5% changes of bed resistivity in a 6-ft (1.8-m)-thick interbedded-formation model. 1 ft = 0.3m.

Apparently, the ILD response is much more sensitive to relative changes of resistivity in conductive beds than in resistive beds. This is why the induction measurement is sometimes referred to as a conductivity measurement, because the induction measurement is much more sensitive and accurate in conductive formations than in highly resistive formations.

Note that these sensitivity studies on the ILD response were conducted on interbedded formations with a specific resistivity contrast (1 to 20 ohm m) and with bed thickness (6 ft [1.8 m]) close to the ILD vertical resolution. Induction-log response depends in complex and nonlinear ways on the resistivity contrast and bed thickness. The results shown here should therefore not be over-generalized. Similar sensitivity evaluation of tool responses should be conducted for each specific case to increase the resistivity-log-modelers’ and/or -interpreters’ confidence.

Limitations Due to Tool Resolution and Accuracy

Vertical resolution, sampling theorem, and aliasing

With the advent of computerized logging units, the continuous alternating voltage and current signals of resistivity measurements are converted to a series of values at fixed intervals, and recorded in a discrete format as a function of depth (spatial sampling). At constant logging speed, there is a direct relationship between constant sample rate and measurement depth.

The resistivity measurements are related to the propagation of an electromagnetic field and can be considered instantaneous when compared to logging speeds up to 20,000 ft (6100 m)/h (Theys, 1999). Therefore, the foremost important factor in designing the sample rate of a resistivity tool is the vertical resolution of the tool, and then, the sampling theorem determines the minimum sample rate.

The sampling theorem states that a band-limited function can be completely represented by equally spaced discrete data if there are two or more samples per cycle for the highest frequency (the Nyquist frequency) present.

This remarkable theorem, when applied to logging, reveals that no information is lost by discrete sampling, provided that the sample frequency is greater than twice the highest frequency component in the formation resistivity profile being sampled by a resistivity tool. In addition, the minimum sampling allows for a complete recovery of the information logged by a resistivity tool within its physical vertical resolution limit. Nothing will be gained by using finer sampling than the minimum sampling. Lastly, the sampling theorem defines that the sampling interval must be no more than half the vertical resolution of the logging tool.

For instance, the sampling theorem requires that the minimum sample rate must be 3 in. (7.6 cm) or less for the 6-in. (15.2 cm) array of the AIT measurement. Because logging tools have different vertical resolutions, from a fraction of an in. (cm) (e.g., FMI) to several feet (meters) (induction), a variety of sample rates are utilized in modern logging data acquisition systems. The most commonly used sampling interval for standard logging suites is 6 in. (15.2 cm), and some array induction and high resolution neutron-density logs use 3 in. (7.6 cm).

Aliasing refers to ambiguity in spatial frequency resulting from the discrete-sampling process.

Aliasing will occur when there are fewer than two samples per cycle; hence, an input signal at one frequency yields the same sample output values as a signal at a different frequency. Figure 6.3 illustrates sample aliasing. The formation-resistivity model is shown by green rectangles. Resistivity-tool response is shown as blue dotted curves. Spatial samples are shown by red arrows. In Figure 6.3a, there are two samples per cycle of the tool response (the log), which satisfies the minimum-sampling rate defined by the sampling theorem to recover completely the log information from the discrete digital samples. Therefore, the formation model may be inverted from the digital log. In Figure 6.3b, the same spatial sampling rate was kept and the same sample values are obtained compared with Figure 6.3a, but the spatial-frequency information contained in the log could not be recovered from the digital samples because there are less than two samples per cycle of the log and the formation model.

Figure 6.3.

Illustration of sample aliasing. (a) There are two samples per cycle of the log, satisfying the sampling theorem. (b) There are not enough samples to correctly reconstruct the log.

Figure 6.3.

Illustration of sample aliasing. (a) There are two samples per cycle of the log, satisfying the sampling theorem. (b) There are not enough samples to correctly reconstruct the log.

Sample aliasing is not likely to be an issue for deep-reading resistivity tools, because the sampling interval for deep resistivity logs (for standard tools, 0.5 ft [15.2 cm]; for array tools, 0.25 ft [7.6 cm]) is usually much smaller than the vertical resolution of the tool. The vertical resolution of the DLL is about 2 ft (0.6 cm), DIT-D/E is about 6 to 8 ft (1.8 to 2.4 m), and AIT forms 1, 2, and 4 ft (0.3, 0.6, and 1.2 m) resolution by post-processing. The tool vertical resolution is usually the factor that limits the spatial frequency information from deep-reading resistivity logs, as discussed in Chapter 3 and illustrated in Figure 3.1.

Sampling theorem also defines the highest spatial frequency information (e.g., minimum bed thickness) that a log can potentially carry. For example, any attempt to define and invert beds thinner than 1 ft (0.3 m) with a log sampled at 0.5 ft (15.2 cm) intervals will result in high uncertainty, not only because the 1 ft (0.3 m) bed thickness may be below the vertical resolution of the tool, but also because of the sampling rate.

Aliasing due to digital sampling may occur in MSFL data that is recorded at the standard 0.5 ft (15.2 cm) sample interval. The stated vertical resolution of the MSFL is about 2–3 in. (5–8 cm) which is half of the standard sample rate. These conditions are illustrated schematically in Figure 6.3b.

Sample aliasing becomes an important issue when OBDT (0.011-house dipmeter tool), EPT, FMI, or Pe data recorded with a higher sampling rate are simply resampled in 0.5 ft (15.2 cm) intervals (decimated) without proper filters. The resulting sequence may not contain the same information on both the bedding frequency and apparent values of the property (Ra, for instance) compared with the original recorded data. A proper filter must be applied when decimating high-sampling-rate data to derive bed boundaries for forward and inverse modeling.

A short sampling interval, allowing many resistivity log measurements over the same formation, may seem desirable. However, shorter sampling intervals require reduced logging speeds because of the constraint that the measurement’s duration controls the measurement’s accuracy. This is especially true with nuclear logging tools.

Resolution-related non-uniqueness

It has been emphasized throughout this volume that a high-resolution log or core image is necessary to define bed thickness for inversion. Figure 6.4 shows a resolution-related non-uniqueness problem that can arise if the inversion is performed with only a deep induction curve. In the left track, the formation earth model (red line shaded in light blue) is composed of 1-ft (0.3-m)-thick interbedded layers with 1-20 ohm m resistivity contrast. In the right track, the earth model is a 20-ft (6.1-m)-thick 2 ohm m formation with 1 ohm m shoulder beds.

Figure 6.4.

Identical ILD responses from two completely different earth models, resulting from limited vertical resolution of the ILD tool. 1 ft = 0.3 m.

Figure 6.4.

Identical ILD responses from two completely different earth models, resulting from limited vertical resolution of the ILD tool. 1 ft = 0.3 m.

The two earth models shown in Figure 6.4 produce nearly identical ILD responses (blue dotted lines). The earth model in the left track is below the ILD blind frequency.

If inversion is performed based only on the ILD response, one can obtain at least these two earth models, which are both valid as far as the tool-response modeling is concerned.

To resolve the resolution-related non-uniqueness problems when performing deep-resistivity-inverse modeling, high-resolution data must be used to constrain the beds of the earth model.

Accuracy-related non-uniqueness

Sensitivity and accuracy of the tool response will also introduce non-uniqueness in the inversion process. Figures 6.5 through 6.8 show ILD responses from center beds of increasing thickness (1, 2, 3, and 5 ft [0.3, 0.6, 0.9, and 1.5 m] respectively) with 10, 100, and 1000 ohm m resistivity, all surrounded by shoulder beds of 1 ohm m resistivity.

Figure 6.5.

Deep induction (ILD) responses from a 1-ft (0.3 m)-thick center bed with varying resistivity of 10 (pink), 100 (purple), 1000 (green) ohm m. 1 ft = 0.3 m.

Figure 6.5.

Deep induction (ILD) responses from a 1-ft (0.3 m)-thick center bed with varying resistivity of 10 (pink), 100 (purple), 1000 (green) ohm m. 1 ft = 0.3 m.

Figure 6.6.

Deep induction (ILD) responses from a 2-ft (0.6-m)-thick center bed with varying resistivity of 10 (pink), 100 (blue), 1000 (green) ohm m. 1 ft = 0.3 m.

Figure 6.6.

Deep induction (ILD) responses from a 2-ft (0.6-m)-thick center bed with varying resistivity of 10 (pink), 100 (blue), 1000 (green) ohm m. 1 ft = 0.3 m.

Figure 6.7.

Deep induction (ILD) responses from a 3-ft (0.9-m)-thick center bed with varying resistivity of 10 (pink), 100 (blue), 1000 (green) ohm m. 1 ft = 0.3 m.

Figure 6.7.

Deep induction (ILD) responses from a 3-ft (0.9-m)-thick center bed with varying resistivity of 10 (pink), 100 (blue), 1000 (green) ohm m. 1 ft = 0.3 m.

Figure 6.8.

Deep induction (ILD) responses from a 5-ft (1.5-m)-thick center bed with varying resistivity of 10 (pink), 100 (purple), 1000 (green) ohm m. 1 ft=0.3 m.

Figure 6.8.

Deep induction (ILD) responses from a 5-ft (1.5-m)-thick center bed with varying resistivity of 10 (pink), 100 (purple), 1000 (green) ohm m. 1 ft=0.3 m.

No difference is observable between the ILD logs from the 100 and the 1000 ohm m beds in the 1-ft (0.3 m) and 2-ft (0.6-m)-thick-bed cases (Figures 6.5, 6.6). The bed thickness of the center bed cannot be identified correctly from the ILD logs alone. (For example, the inflection points do not correctly identify the boundaries.)

The differences between ILD logs from the 10, 100, and 1000 ohm m models start to be observable as the bed thickness is increased to 3 ft (0.3 m) and are clearly distinguishable in the 5-ft (1.5-m)-thick-bed case (Figures 6.7, and 6.8).

In the inversion process, the non-uniqueness issue arises when the tool responses (the computed logs) become insensitive to changes in the formation model. In Figures 6.5 and 6.6, there are hundreds of Rt values between 100 and 1000 ohm m that will result in the same ILD responses, and hence, Rt cannot be uniquely determined via tool response modeling and inversion.

In Figures 6.7 and 6.8, Rt may be determined through tool-response modeling if the bed boundaries are carefully constrained. As shown in Figure 6.1, the ILD tool-response changes are more sensitive to bed-thickness changes than to resistivity changes in the resistive bed.

In general, the inflection points of the ILD log are not an accurate locator of bed boundaries. For these illustrations, bed boundaries are close to ILD inflection points only in the 3-ft (0.9-m)-thick-bed bed case (Figure 6.7).

Figure 6.9 shows bed thickness correction chart Rcor-9 (Schlumberger Log Interpretation Charts, 2005), where Rs, RIDP, and RIDPCOR stand for shoulder-bed, phasor-induction, and bed-thickness-corrected Phasor-Induction resistivities, respectively. The lack of sensitivity and limited accuracy of the induction responses, as shown in Figures 6.5 to 6.7, are the reason why the ratio of the RIDPCOR/RS goes toward infinity for a resistive center bed as the bed thickness becomes less than 6 ft (1.8 m). Inversely, the RIDPCOR/RS goes to zero for a conductive center bed, but the bed-thickness effect is less severe.

Figure 6.9.

Schlumberger’s bed-thickness-effect correction chart Rcor-9 in Schlumberger Log Interpretation Charts prior to 2005), where Rs, RIDP, and RIDPCO stand for shoulder-bed, phasor-induction and bed-thickness-corrected Phasor-Induction resistivity, respectively.

Figure 6.9.

Schlumberger’s bed-thickness-effect correction chart Rcor-9 in Schlumberger Log Interpretation Charts prior to 2005), where Rs, RIDP, and RIDPCO stand for shoulder-bed, phasor-induction and bed-thickness-corrected Phasor-Induction resistivity, respectively.

Error in conductivity relative to error in resistivity for induction tools

Logging-tool-accuracy specifications can be analyzed quantitatively to address the associated non-uniqueness in the inversion process. As stated earlier in this chapter, accuracy is often expressed in terms of the sum of the squares of the maximum values which is composed of random and systematic errors, or overall root-mean-square (RMS): 

formula

Sometimes RMS is also expressed in terms of the standard deviation σ in the form of 

formula

It is documented that “The overall RMS error for AIT logs is the same as for the DIT-E logs, 2% or ±0.75 mS/m, whichever is higher.” (Anderson and Barber, 1996, p. 32). This RMS error is defined at the 95% confidence level (i.e., 2σ), and the accuracy is defined for the 4 ft (1.2 m) vertical resolution and 60 in. (152 cm) radial depth of investigation curve of AIT tool in a homogeneous resistivity background, with no bed(s) or resistivity contrasts, and with no borehole effect.

Firstly, there is a conductivity value C at which 2% of it equals ±0.75mS/m, which is C = 37.5 mS/m. If the AIT/DIT is run in a homogeneous formation with conductivity lower than this value (C ≤ 37.5 mS/m), then ±0.75 mS/m defines the measurement error. In a homogeneous formation with conductivity C higher than this value (C ≥37.5 mS/m), then 2% of C determines the measurement error, since it is higher.

Secondly, to understand how large this error is in different resistivity ranges, the RMS error defined in the conductivity domain needs to be translated to the resistivity domain by 

formula

where ΔC is the 2% or ±0.75 mS/m specified above.

Figure 6.10 applies Equation 6-3 to graphically illustrate the induction error (2% or ± 0.75 mS/m, whichever is higher), for both conductivity and resistivity. The top bar shows resistivity variation from 0.1 to 1000 ohm m, and the corresponding reciprocal conductivity from 10,000 to 1 mS/m. The middle bar shows the absolute (ΔC) and relative (ΔC/C) error in conductivity, and the bottom bar shows the absolute (ΔR)and relative (ΔR/R) error in resistivity. Note that, for a given level of resistivity, the relative errors in conductivity and resistivity are the same.

Figure 6.10.

Accuracy analysis of AIT and DIT-E Induction tools. The vertical dashed arrow shows the point where 2% error is equal to ± 0.75mS/m over the 0.1–1000 ohm m resistivity or 10,000–1 mS/m conductivity ranges. Note the constant 2% relative error from 26.7 ohm m, or 37.5 mS/m, to the left, and the constant ±0.75 mS/m absolute error from 26.7 ohm m, or 37.5 mS/m, to the right of the horizontal shaded arrow.

Figure 6.10.

Accuracy analysis of AIT and DIT-E Induction tools. The vertical dashed arrow shows the point where 2% error is equal to ± 0.75mS/m over the 0.1–1000 ohm m resistivity or 10,000–1 mS/m conductivity ranges. Note the constant 2% relative error from 26.7 ohm m, or 37.5 mS/m, to the left, and the constant ±0.75 mS/m absolute error from 26.7 ohm m, or 37.5 mS/m, to the right of the horizontal shaded arrow.

When the AIT/DIT is run in a homogeneous formation of resistivity 100 ohm m, the absolute error is 7.5 ohm m and the relative error is 7.5%. When the AIT/DIT is run in a homogeneous formation of resistivity 1,000 ohm m, the absolute error is 750 ohm m and the relative error is 75%.

Such huge potential errors in the induction measurement when logged in a high-resistivity formation will limit inversion accuracy and result in high uncertainty when the computed tool response is matched against the field induction logs.

Impact of tool accuracy in conductive beds vs. in resistive beds

Limited vertical resolution and accuracy of the induction measurements may yield non-unique inversion results and high uncertainty; these effects are further illustrated in the following benchmark studies. Figure 6.11 shows three ILD responses from one-ft (0.3 m)-thick interbedded sand and shale sequences with resistivity contrasts of 1 to 10, 1 to 100, and 1 to 1000 ohm m.

Figure 6.11.

Three ILD responses from 1-ft-thick interbedded sand and shale sequences with three different resistivity contrasts, 1–10, 1–100, and 1–1000 ohm m. 1 ft = 0.3 m.

Figure 6.11.

Three ILD responses from 1-ft-thick interbedded sand and shale sequences with three different resistivity contrasts, 1–10, 1–100, and 1–1000 ohm m. 1 ft = 0.3 m.

Note that the ILD response corresponding to the 1-to-10 ohm m contrast model is distinguishable from the other two higher-contrast models, and the ILD response for the 1–100-ohm m-contrast model cannot be distinguished from response for the 1–1000-ohm m-contrast model.

It is well documented that ILD response can be closely approximated in thinly bedded laminar formations by the parallel conductivity model: 

formula

where Ca is the apparent conductivity measured by the induction tool, Csh and Csd are the formation shale and sand conductivity, and Vsh and Vsd are the volume fractions of the shale and sand, respectively. Using Equation 6-4, the exercise in Figure 6.11 is summarized in Table 6.1.

Table 6.1.

Parallel conductivity approximation of the induction response

Rsh (ohm m)Rsd (ohm m)Ra (ohm m)Ca (mS/m)
1101.818550
11001.98505
110001.998500.5
Rsh (ohm m)Rsd (ohm m)Ra (ohm m)Ca (mS/m)
1101.818550
11001.98505
110001.998500.5

Note that the difference between the ILD responses from the 1-to-100 ohm m-contrast model and 1-to-1000-ohm m-contrast model is only 4.5 mS/m. This difference is less than half the absolute error of the induction sonde defined in Figure 6.10 (i.e., 500 mS/m × 2% = 10 mS/m). Thus, these two cases cannot be distinguished by the induction measurement.

If the sonde error is propagated using Equation 6-4 for the uncertainty in finding Rt of sand (e.g., Rsd or CSD), the ΔCSD can be expressed as 

formula

Assuming that VSH and VSD are exactly 50% with no error, and that the formation shale conductivity, CSH, is 1000 mS/m with no error, and ΔCA from ILD log is about 505 × 2% = ±10.1 mS/m, then from Equation 6-5, we get ΔCSD ≌ |ΔCa/VSD| = ± 20.2 mS/m. To derive sand conductivity CSD through inversion, this error implies that any conductivity in the range 0 ≤ CSD ≤ 30.2 mS/m (where the Rsd value is greater than 33.1 ohm m and less than infinity) may result in identical ILD responses. Therefore, Rsd is not determinable within this range because of the limited accuracy of the tool response. In addition, as shown in the first term of Equation 6-5, the absolute error of the sand conductivity is inversely proportional to sand volume fraction. The smaller the sand volume fraction, the larger the error in sand conductivity.

In practice, uncertainties in all the other terms of Equation 6-5 also contribute to the ambiguity in determining sand resistivity. Equation 6-5 can be rewritten for relative change as 

formula

Assume that VSH and VSD are exactly 50%, that there is no error in VSH, VSD, and Ca, and that Ca from the ILD log is about 505 mS/m, then, ΔCSD/CSD ≌ 100|ΔCSH/VSH|, that is, a 1% relative change in shale conductivity will introduce a 100% relative change in sand resistivity. Strictly speaking in terms of mathematics, the differential approximation (Equation 6-5 and 6-6) may only be valid when the absolute error is small compared with the conventional true value in measurement. Figure 6.12 illustrates this result with two identical ILD responses from a 1–100-ohm m-contrast model and a 1.1–9.91-ohm m-contrast model.

Figure 6.12.

Identical ILD responses from a 1–100-ohm m-contrast model and a 1.1–9.91-ohm m-contrast model. 1 ft = 0.3 m.

Figure 6.12.

Identical ILD responses from a 1–100-ohm m-contrast model and a 1.1–9.91-ohm m-contrast model. 1 ft = 0.3 m.

Indeed, sand Rt cannot be determined by resistivity-tool-response modeling in such cases. However, it is critical to understand these vertical-resolution and accuracy effects when modeling low-resistivity and low-contrast pay zones, and such pay zones should not be missed simply because of resistivity-response limitations. A statistical approach such as volumetric laminated sand analysis (VLSA) (Passey et al., 2006) is an alternative approach in these cases.

Ambiguity due to miscalibrated tools or postprocessing

In some cases, the inversion process cannot reduce the difference between the field log and the computed tool-response below a set minimum threshold. For example, a field induction log may read a resistivity over 1000 ohm m in some zones, but computed tool-response cannot reach that high even if the resistivity of the zone is defined as infinite in the model (Anderson and Barber, 1988).

In such a case it may be found that tool calibration was purposely adjusted so that the field induction log can read as high as 1000 ohm m. Sometimes and in some places, such an adjustment has been an accepted practice by both service companies and oil companies because it made the induction read “right” in zones with very low water-saturation, and had relatively little effect in lower-resistivity zones.

Such miscalibration was acceptable because induction tools respond to conductivity. Increasing ILD response from 500 to 1000 ohm m is effectively decreasing ILD from 2 to 1 mS/m, a net change of −1 mS/m. When this 1 mS/m decrease is translated to the lower resistivity zones where ILD should read about 50 ohm m (20 mS/m), the miscalibration yields an ILD reading of 52.6 ohm m (corresponding to 19 mS/m). Therefore, while the 2.6 ohm m difference in 50 ohm m zones is barely noticeable when plotted on a log scale, the 500 ohm m difference in a 1000 ohm m zone is presumed corrected.

When the field induction log shows resistivity higher than 200 ohm m, the log is usually enhanced by a signal-processing technique such as skin-effect boosting or deconvolution. Such postprocessing may not only boost the resistivity readings, but may also introduce some spurious responses that a computed log, generated from given assumptions and an idealized environment, may never match. Unfortunately, information regarding what tools can really measure and the limits of the boosting by signal processing have not been clear in the available literature from logging-service companies. This situation became even fuzzier when array-induction tools and software focusing were introduced. These factors may add further ambiguity to the process of forward modeling and inversion.

General specifications of induction and lateral tools

When performing resistivity-tool-response modeling, a thorough understanding of the tool calibration, sensitivity, and accuracy is essential to constrain the final inversion results. The following specifications on induction and laterolog measurements are derived from available literature, and may serve as qualitative guidance in understanding the tool-response accuracy and limits for modeling and interpretation.

Table 6.2.

Resistivity tool specifications

ParameterDual/Array Induction (DIT-D/E, AIT-B/H))Dual Laterolog (DLT-D/E)
Temperature rating350° F (175° C)350° F (175° C)
Pressure rating20 kpsi (13.8 kPa)20 kpsi (13.8 kPa)
Max. logging speed10,000 ft/hr (3048 m/hr)10,000 ft/hr (3048m/hr)
Sampling interval6 or 3 in. (15.2 or 7.6 cm)6 or 3 in. (15.2 or 7.6 cm)
Vertical resolutionIDPH/ILD: 7~8 ft (2.1~2.4 m)LLD: 2 ft (0.6 m)
IMPH/ILM: 5~6 ft (1.5~1.8 m)LLS: 2 ft (0.6 m)
Depth of investigationIDPH/ILD: 5 ft (1.5m)LLD: 6 ft (1.8 m)
IMPH/ILM: 2.5 ft (0.8 m)LLS: 2~4 ft (0.6~1.2 m)
Accuracy0.75~1 mS/m, or 2~5%, whichever is higher5%, linear in the measurement range
Formation-resistivity-measurement-range limit0.1~150 ohm m, less accurate in high-resistivity formation1~1000 ohm m, less accurate in low-resistivity formation
ParameterDual/Array Induction (DIT-D/E, AIT-B/H))Dual Laterolog (DLT-D/E)
Temperature rating350° F (175° C)350° F (175° C)
Pressure rating20 kpsi (13.8 kPa)20 kpsi (13.8 kPa)
Max. logging speed10,000 ft/hr (3048 m/hr)10,000 ft/hr (3048m/hr)
Sampling interval6 or 3 in. (15.2 or 7.6 cm)6 or 3 in. (15.2 or 7.6 cm)
Vertical resolutionIDPH/ILD: 7~8 ft (2.1~2.4 m)LLD: 2 ft (0.6 m)
IMPH/ILM: 5~6 ft (1.5~1.8 m)LLS: 2 ft (0.6 m)
Depth of investigationIDPH/ILD: 5 ft (1.5m)LLD: 6 ft (1.8 m)
IMPH/ILM: 2.5 ft (0.8 m)LLS: 2~4 ft (0.6~1.2 m)
Accuracy0.75~1 mS/m, or 2~5%, whichever is higher5%, linear in the measurement range
Formation-resistivity-measurement-range limit0.1~150 ohm m, less accurate in high-resistivity formation1~1000 ohm m, less accurate in low-resistivity formation

In general, these accuracy values can be used as a reference when considering how close the match between the field log and computed tool-response needs to be and when to stop the iterative inversion process.

When the field log values exceed the tool’s physical measurement range limit, tool-response modeling with the field log must be carried out with caution. A reasonable match and conversion between the computed tool-response and field log may often not be achievable, and the field log may need to be reprocessed or recalibrated.

Limitations Due to the Modeling Code

Modeling code accuracy

Some commercial resistivity-tool-response-modeling codes disguise computational simplifications, such as the geometrical factor, a linear convolution filter, or equivalent circuit-based approximations as the exact electromagnetic-equation-based solutions for rigorous resistivity-tool-response modeling. It is often claimed that these codes run hundreds of times faster than others without explaining how and why.

These codes may work in some simple environments, but they may result in significantly different tool responses in complex formation, borehole, and invasion geometries. Resistivity-log interpreters may be deceived into believing such mathematical simplifications represent valid tool responses founded on tool physics.

Commercial codes must be carefully evaluated for accuracy before they can be used for the application of resistivity-tool-response-modeling for formation evaluation. The nature of a commercial code must be understood, and rationalizations, such as competitiveness and confidentiality, should not restrain the necessary technical evaluation.

Assumption of the earth models: 1-D, 2-D, and 3-D

As discussed in the first three chapters, 1-D, 2-D, and 3-D earth models are the idealized geometries under which a tool response is simulated. In a 1-D earth model it is assumed that resistivity only varies as a function of the measured depth, with no borehole and no invasion effects. In a 2-D earth model, it is assumed that the tool is centered in the borehole, the borehole is cylindrical, and invasion is axisymmetrical following a step profile.

In many cases, these idealized geometries are adequate to describe the logging conditions where field logs are obtained from real resistivity tools. In a smooth and regular borehole drilled with oil-based mud, for instance, the induction-tool response may often satisfy 1-D assumptions. In a few cases, the computed log, based on these 1-D or 2-D earth-model assumptions, cannot be matched to the field-log response that is obtained in a more complex 3-D geometry. Limitations of the tool-response modeling due to the simple 1-D or 2-D earth model may be encountered when simulating field logs obtained from a spiral borehole, with annular invasion, with a tool that is run “slick” (without standoff or centralizer in deviated borehole), etc.

Sensitivity of Water Saturation to Errors in Resistivity

Water saturation error due to resistivity error

One obvious reason that resistivity-tool-response modeling is applied in formation evaluation is to reduce uncertainty in water-saturation (Sw) estimation. It is important to understand how resistivity errors translate to errors in computed water saturation. Assuming that the standard Archie equation is the transform for resistivity to water saturation, 

formula

Where ϕ is porosity and m is a constant cementation factor for a given rock type, n is the saturation exponent and a is a constant. Taking the total derivative of Equation 6-7, the total absolute error formula for Sw can be expressed as: 

formula

Instead of total absolute error as shown in Equation (6-8), another common expression for the absolute error is the RMS, summation of the standard deviation errors in the following form: 

formula

The relative error for Sw can be expressed as: 

formula

To address the error in Sw resulting from the error in resistivity, we assumed that all other sources of error in Equation 6-9 could be eliminated, and then: 

formula

Table 6.3 shows the results for the inherent error of the DIT-E/AIT, assuming the conductivity error is 2% or ±0.75 mS/m, whichever is higher. Note that for Rt = 100 ohm m and n = 2, there is 3.75% inherent error in Sw due to inherent error in the induction sonde. In comparison, for the laterolog at Rt = 100 ohm m and n = 2, the inherent error in Sw is 2.5% based on the laterolog specification of 5% resistivity error between 1 and 1000 ohm m.

Table 6.3.

Approximate relative errors derived from Equation 6-10

Rt Value (ohm m)Relative Tool Error (%)Average Relative Error in Sw (%)Approximate Relative Error in Sw (%)
n = 2n = 1.5n = 2n = 2.5
0.12.01.01.331.00.8
1.02.01.01.331.00.8
10.02.01.01.331.00.8
26.72.01.01.331.00.8
100.07.53.755.03.753.0
1000.07.541.1550.03.7530.0
Rt Value (ohm m)Relative Tool Error (%)Average Relative Error in Sw (%)Approximate Relative Error in Sw (%)
n = 2n = 1.5n = 2n = 2.5
0.12.01.01.331.00.8
1.02.01.01.331.00.8
10.02.01.01.331.00.8
26.72.01.01.331.00.8
100.07.53.755.03.753.0
1000.07.541.1550.03.7530.0

Figure 6.13 shows the approximate errors derived from Equation 6-10 for n = 2, n = 1.5, and n = 2.5 in green, blue and magenta respectively. The differential error approximation, Equation 6-10, is nominally valid only when the absolute error, ΔRt, is small compared with the conventional true value, Rt. For comparison, the explicitly calculated exact error for n = 2 is shown in Figure 6.13 in red (labeled Av. Rel. Er. Sw). The approximate and exact errors are the same for Rt up to 100 ohm m, and differ by only about 4% at 1000 ohm m.

Figure 6.13.

Plots of error in water saturation resulting from error inherent in induction sonde, based on the Archie equation. Green, purple, and orange values are approximations calculated from Equation 6-10. Red values represent an exact error calculation.

Figure 6.13.

Plots of error in water saturation resulting from error inherent in induction sonde, based on the Archie equation. Green, purple, and orange values are approximations calculated from Equation 6-10. Red values represent an exact error calculation.

Effect of resistivity modeling on water saturation

Assuming that a field resistivity log (Ra) is used with the standard Archie equation for the resistivity to Sw calculation, then the apparent water saturation (Swa) can be expressed as 

formula

and then the ratio of Swa to the true water saturation (Swt) using the modeled resistivity as the true resistivity (Rt) is 

formula

Equation 6-12 can be used for sensitivity and uncertainty analysis, and Figure 6.14 shows the graphic presentation of this relation. Note that the ratio (Swa/Swt) is the nth root of the (Rt/Ra) ratio. For example, when Rt/Ra is 10, Swa/Swt is about 3.16, and when Rt/Ra is increased 10 times to 100, Swa/Swt only becomes 10.

Figure 6.14.

Sensitivity of Swa/Swt as a function of Rt/Ra for n=1.5 (blue), n=2 (red), and n=2.5 (green).

Figure 6.14.

Sensitivity of Swa/Swt as a function of Rt/Ra for n=1.5 (blue), n=2 (red), and n=2.5 (green).

This kind of sensitivity analysis should be used, along with appropriate benchmark modeling results, mainly to understand the impact of resistivity modeling in water saturation estimation, and to constrain how high Rt should be in the inversion process, especially when the tool response is insensitive to changes in Rt, such as in thin-bed cases.

Other sources of error, such as in porosity, m, and n, can also be analyzed in a similar way. Figure 6.15 shows such a sensitivity analysis by a spider chart where parameters are varied multiplicatively.

Figure 6.15.

Sensitivity of total water saturation to multiplicative variations in the Archie equation parameters.

Figure 6.15.

Sensitivity of total water saturation to multiplicative variations in the Archie equation parameters.

Figures & Tables

Figure 6.1.

Sensitivity of induction response to ±5% changes of bed thickness in a 6-ft (1.8-m)-thick interbedded-formation model. 1 ft = 0.3 m.

Figure 6.1.

Sensitivity of induction response to ±5% changes of bed thickness in a 6-ft (1.8-m)-thick interbedded-formation model. 1 ft = 0.3 m.

Figure 6.2.

Sensitivity of induction response to ±5% changes of bed resistivity in a 6-ft (1.8-m)-thick interbedded-formation model. 1 ft = 0.3m.

Figure 6.2.

Sensitivity of induction response to ±5% changes of bed resistivity in a 6-ft (1.8-m)-thick interbedded-formation model. 1 ft = 0.3m.

Figure 6.3.

Illustration of sample aliasing. (a) There are two samples per cycle of the log, satisfying the sampling theorem. (b) There are not enough samples to correctly reconstruct the log.

Figure 6.3.

Illustration of sample aliasing. (a) There are two samples per cycle of the log, satisfying the sampling theorem. (b) There are not enough samples to correctly reconstruct the log.

Figure 6.4.

Identical ILD responses from two completely different earth models, resulting from limited vertical resolution of the ILD tool. 1 ft = 0.3 m.

Figure 6.4.

Identical ILD responses from two completely different earth models, resulting from limited vertical resolution of the ILD tool. 1 ft = 0.3 m.

Figure 6.5.

Deep induction (ILD) responses from a 1-ft (0.3 m)-thick center bed with varying resistivity of 10 (pink), 100 (purple), 1000 (green) ohm m. 1 ft = 0.3 m.

Figure 6.5.

Deep induction (ILD) responses from a 1-ft (0.3 m)-thick center bed with varying resistivity of 10 (pink), 100 (purple), 1000 (green) ohm m. 1 ft = 0.3 m.

Figure 6.6.

Deep induction (ILD) responses from a 2-ft (0.6-m)-thick center bed with varying resistivity of 10 (pink), 100 (blue), 1000 (green) ohm m. 1 ft = 0.3 m.

Figure 6.6.

Deep induction (ILD) responses from a 2-ft (0.6-m)-thick center bed with varying resistivity of 10 (pink), 100 (blue), 1000 (green) ohm m. 1 ft = 0.3 m.

Figure 6.7.

Deep induction (ILD) responses from a 3-ft (0.9-m)-thick center bed with varying resistivity of 10 (pink), 100 (blue), 1000 (green) ohm m. 1 ft = 0.3 m.

Figure 6.7.

Deep induction (ILD) responses from a 3-ft (0.9-m)-thick center bed with varying resistivity of 10 (pink), 100 (blue), 1000 (green) ohm m. 1 ft = 0.3 m.

Figure 6.8.

Deep induction (ILD) responses from a 5-ft (1.5-m)-thick center bed with varying resistivity of 10 (pink), 100 (purple), 1000 (green) ohm m. 1 ft=0.3 m.

Figure 6.8.

Deep induction (ILD) responses from a 5-ft (1.5-m)-thick center bed with varying resistivity of 10 (pink), 100 (purple), 1000 (green) ohm m. 1 ft=0.3 m.

Figure 6.9.

Schlumberger’s bed-thickness-effect correction chart Rcor-9 in Schlumberger Log Interpretation Charts prior to 2005), where Rs, RIDP, and RIDPCO stand for shoulder-bed, phasor-induction and bed-thickness-corrected Phasor-Induction resistivity, respectively.

Figure 6.9.

Schlumberger’s bed-thickness-effect correction chart Rcor-9 in Schlumberger Log Interpretation Charts prior to 2005), where Rs, RIDP, and RIDPCO stand for shoulder-bed, phasor-induction and bed-thickness-corrected Phasor-Induction resistivity, respectively.

Figure 6.10.

Accuracy analysis of AIT and DIT-E Induction tools. The vertical dashed arrow shows the point where 2% error is equal to ± 0.75mS/m over the 0.1–1000 ohm m resistivity or 10,000–1 mS/m conductivity ranges. Note the constant 2% relative error from 26.7 ohm m, or 37.5 mS/m, to the left, and the constant ±0.75 mS/m absolute error from 26.7 ohm m, or 37.5 mS/m, to the right of the horizontal shaded arrow.

Figure 6.10.

Accuracy analysis of AIT and DIT-E Induction tools. The vertical dashed arrow shows the point where 2% error is equal to ± 0.75mS/m over the 0.1–1000 ohm m resistivity or 10,000–1 mS/m conductivity ranges. Note the constant 2% relative error from 26.7 ohm m, or 37.5 mS/m, to the left, and the constant ±0.75 mS/m absolute error from 26.7 ohm m, or 37.5 mS/m, to the right of the horizontal shaded arrow.

Figure 6.11.

Three ILD responses from 1-ft-thick interbedded sand and shale sequences with three different resistivity contrasts, 1–10, 1–100, and 1–1000 ohm m. 1 ft = 0.3 m.

Figure 6.11.

Three ILD responses from 1-ft-thick interbedded sand and shale sequences with three different resistivity contrasts, 1–10, 1–100, and 1–1000 ohm m. 1 ft = 0.3 m.

Figure 6.12.

Identical ILD responses from a 1–100-ohm m-contrast model and a 1.1–9.91-ohm m-contrast model. 1 ft = 0.3 m.

Figure 6.12.

Identical ILD responses from a 1–100-ohm m-contrast model and a 1.1–9.91-ohm m-contrast model. 1 ft = 0.3 m.

Figure 6.13.

Plots of error in water saturation resulting from error inherent in induction sonde, based on the Archie equation. Green, purple, and orange values are approximations calculated from Equation 6-10. Red values represent an exact error calculation.

Figure 6.13.

Plots of error in water saturation resulting from error inherent in induction sonde, based on the Archie equation. Green, purple, and orange values are approximations calculated from Equation 6-10. Red values represent an exact error calculation.

Figure 6.14.

Sensitivity of Swa/Swt as a function of Rt/Ra for n=1.5 (blue), n=2 (red), and n=2.5 (green).

Figure 6.14.

Sensitivity of Swa/Swt as a function of Rt/Ra for n=1.5 (blue), n=2 (red), and n=2.5 (green).

Figure 6.15.

Sensitivity of total water saturation to multiplicative variations in the Archie equation parameters.

Figure 6.15.

Sensitivity of total water saturation to multiplicative variations in the Archie equation parameters.

Table 6.1.

Parallel conductivity approximation of the induction response

Rsh (ohm m)Rsd (ohm m)Ra (ohm m)Ca (mS/m)
1101.818550
11001.98505
110001.998500.5
Rsh (ohm m)Rsd (ohm m)Ra (ohm m)Ca (mS/m)
1101.818550
11001.98505
110001.998500.5
Table 6.2.

Resistivity tool specifications

ParameterDual/Array Induction (DIT-D/E, AIT-B/H))Dual Laterolog (DLT-D/E)
Temperature rating350° F (175° C)350° F (175° C)
Pressure rating20 kpsi (13.8 kPa)20 kpsi (13.8 kPa)
Max. logging speed10,000 ft/hr (3048 m/hr)10,000 ft/hr (3048m/hr)
Sampling interval6 or 3 in. (15.2 or 7.6 cm)6 or 3 in. (15.2 or 7.6 cm)
Vertical resolutionIDPH/ILD: 7~8 ft (2.1~2.4 m)LLD: 2 ft (0.6 m)
IMPH/ILM: 5~6 ft (1.5~1.8 m)LLS: 2 ft (0.6 m)
Depth of investigationIDPH/ILD: 5 ft (1.5m)LLD: 6 ft (1.8 m)
IMPH/ILM: 2.5 ft (0.8 m)LLS: 2~4 ft (0.6~1.2 m)
Accuracy0.75~1 mS/m, or 2~5%, whichever is higher5%, linear in the measurement range
Formation-resistivity-measurement-range limit0.1~150 ohm m, less accurate in high-resistivity formation1~1000 ohm m, less accurate in low-resistivity formation
ParameterDual/Array Induction (DIT-D/E, AIT-B/H))Dual Laterolog (DLT-D/E)
Temperature rating350° F (175° C)350° F (175° C)
Pressure rating20 kpsi (13.8 kPa)20 kpsi (13.8 kPa)
Max. logging speed10,000 ft/hr (3048 m/hr)10,000 ft/hr (3048m/hr)
Sampling interval6 or 3 in. (15.2 or 7.6 cm)6 or 3 in. (15.2 or 7.6 cm)
Vertical resolutionIDPH/ILD: 7~8 ft (2.1~2.4 m)LLD: 2 ft (0.6 m)
IMPH/ILM: 5~6 ft (1.5~1.8 m)LLS: 2 ft (0.6 m)
Depth of investigationIDPH/ILD: 5 ft (1.5m)LLD: 6 ft (1.8 m)
IMPH/ILM: 2.5 ft (0.8 m)LLS: 2~4 ft (0.6~1.2 m)
Accuracy0.75~1 mS/m, or 2~5%, whichever is higher5%, linear in the measurement range
Formation-resistivity-measurement-range limit0.1~150 ohm m, less accurate in high-resistivity formation1~1000 ohm m, less accurate in low-resistivity formation
Table 6.3.

Approximate relative errors derived from Equation 6-10

Rt Value (ohm m)Relative Tool Error (%)Average Relative Error in Sw (%)Approximate Relative Error in Sw (%)
n = 2n = 1.5n = 2n = 2.5
0.12.01.01.331.00.8
1.02.01.01.331.00.8
10.02.01.01.331.00.8
26.72.01.01.331.00.8
100.07.53.755.03.753.0
1000.07.541.1550.03.7530.0
Rt Value (ohm m)Relative Tool Error (%)Average Relative Error in Sw (%)Approximate Relative Error in Sw (%)
n = 2n = 1.5n = 2n = 2.5
0.12.01.01.331.00.8
1.02.01.01.331.00.8
10.02.01.01.331.00.8
26.72.01.01.331.00.8
100.07.53.755.03.753.0
1000.07.541.1550.03.7530.0

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