Determining Formation Conductivity with Propagation Measurements

Properties of a geological formation, such as dielectric constant and conductivity, may be determined by a propagation well log data acquired by a propagation well logging tool based at least in part on a relative longitudinal position of two or more receivers of the propagation well logging tool. In some embodiments, the relative longitudinal position of the at least two receivers is based at least in part on a first distance between a first receiver of the at least two receivers and a transmitter of the propagation well logging tool and a second distance between a second receiver of the at least two receivers and the transmitter.

BACKGROUND

This disclosure relates to identifying properties of a geological formation using a downhole electromagnetic measurement. More specifically, this disclosure relates to techniques for determining a formation conductivity from propagation measurements.

Producing hydrocarbons from a wellbore drilled into a geological formation is a remarkably complex endeavor. In many cases, decisions involved in hydrocarbon exploration and production may be informed by measurements from well logging tools (e.g., downhole well logging tools) that are conveyed deep into the wellbore. The measurements may be used to infer properties or characteristics of the geological formation surrounding the wellbore. One example of such downhole well logging tools are propagation well logging tools. However, conventional processing methods for handling the measurements may lead to relatively high processing times as well as memory requirements for storing the processed measurements.

SUMMARY

One embodiment of the present disclosure relates to a method. The method includes acquiring, via a processor, propagation measurements in a wellbore through a geological formation using one or more propagation downhole well logging tools having at least two receivers. The method also includes converting, via the processor, the propagation measurements to two apparent conductivities based at least in part on a frequency associated with the propagation measurements, a relative longitudinal position of the at least two receivers, and a phase shift measurement and an attenuation measurement.

DETAILED DESCRIPTION

In the present context, the term “about” or “approximately” is intended to mean that the values indicated are not exact and the actual value may vary from those indicated in a manner that does not materially alter the operation concerned. For example, the term “about” or “approximately” as used herein is intended to convey a suitable value that is within a particular tolerance (e.g., ±10%, ±5%, ±1%, ±0.5%), as would be understood by one skilled in the art.

As mentioned above, oil and gas exploration organizations may make certain oil and gas production decisions, such as determining where to drill, based at least in part on well log data. More specifically, a well logging tool acquires well logging measurements, which may be processed (e.g., normalized, de-noised, provided as inputs to a model, etc.) by a suitable computing device to generate the well log data. As referred to herein, “well log data” is a measurement or a property derived from measurements versus depth or time, or both, of one or more properties (e.g., resistivity, conductivity, dip and azimuth, and the like) in or around a wellbore, and thus, may be used to identify a location within the wellbore that corresponds to an area of interest (e.g., hydrocarbons, an organic deposit, a “bed” or layer of sedimentary rock, or stratum, and the like). At least in some instances, the well log data may be transformed into one or more visual representations (e.g., a well log) that are presented as hard copies or on an electronic display, where each visual representation of the one or more visual representations may depict the well log data resulting from the well logging measurements.

One type of well logging measurement that may be used to inform the oil and gas production decisions are propagation well logging measurements. In general, propagation well logging measurements may be acquired using one or more propagation well logging tools that each include a number of transmitter coils and receiver coils. As should be understood by one of ordinary skill in the art, a propagation well logging tool may operate at a frequency (e.g., approximately 100 kHz, 200 kHz, 300 kHz, 400 kHz, 1000 kHz, 2000 kHz, and the like), which may facilitate the determination of certain properties of a geological formation, such as the resistivity and dielectric properties. It should be noted that measuring the resistivity and dielectric properties may enable differentiation between certain components that are present within the geological formation such as water, oil, and gas. However, while the ability to measure at these frequencies may be advantageous for differentiating between these components (e.g., water and oil) based at least in part on the resistivity and dielectric properties, propagation well logging measurements at these frequencies (e.g., measured by a receiver of the propagation well logging tool) may be adversely affected due to the skin effect.

Conventional propagation well logging tools may process the phase shift and attenuated propagation measurements using an inversion in order to generate conductivity and/or resistivity well logs. Existing processes for inverting the measurements (e.g., the phase shift and attenuate measurements) may be computationally expensive (e.g., having high memory usage), and thus take a relatively large amount of time to process the measurements to generate the well logs, which are used for informing oil and gas productive decisions.

Accordingly, the present disclosure relates to techniques for processing propagation well logging measurements to generate an apparent conductivity and/or apparent resistivity based at least in part on the propagation well logging measurements without using an inversion or a resistivity transform. In particular, the disclosed techniques for processing the propagation well logging measurements may utilize a tool constant that is based on a relative longitudinal position of at least two of the receivers of the propagation well logging tool. The disclosed tool constant and the apparent conductivity and/or the apparent resistivity may be used to generate apparent conductivity and/or apparent resistivity well logs. In some embodiments, the apparent conductivity may be a phase shift apparent conductivity and/or an attenuation apparent conductivity. As discussed in further detail with regard toFIGS.3and4, the phase shift apparent conductivity may provide a measure of formation conductivity and the attenuation apparent conductivity may provide a measure of the skin effect. In some embodiments, the phase shift apparent conductivity may be corrected for the skin effect using the attenuation apparent conductivity to generate an improved apparent conductivity that may be representative of the true formation conductivity.

With this in mind,FIG.1illustrates a well logging system10that may employ the systems and methods of this disclosure. The well logging system10may be used to convey a propagation well logging tool12through a geological formation14via a wellbore16. The propagation well logging tool12may be conveyed on a cable18via a logging winch system20. Although the logging winch system20is schematically shown inFIG.1as a mobile logging winch system carried by a truck, the logging winch system20may be substantially fixed (e.g., a long-term installation that is substantially permanent or modular). Any suitable cable18for well logging may be used. The cable18may be spooled and unspooled on a drum22and an auxiliary power source24may provide energy to the logging winch system20and/or the propagation well logging tool12.

Moreover, although the propagation well logging tool12is described as being a wireline downhole tool, it should be appreciated that any suitable conveyance may be used. For example, the propagation well logging tool12may instead be conveyed as a logging-while-drilling (LWD) downhole tool as part of a bottom hole assembly (BHA) of a drill string, conveyed on a slickline or via coiled tubing, and so forth. For the purposes of this disclosure, the propagation well logging tool12may be any suitable measurement downhole tool that acquires propagation logging measurements through depths of the wellbore16.

Many types of propagation well logging tools12may acquire propagation well logging measurements in the wellbore16. These include, for example, a compensated dual resistivity (CDR) downhole tool, an array resistivity compensated (ARC) downhole tool, PERISCOPE, and the like. The propagation well logging tool12may provide propagation logging measurements26to a data processing system28via any suitable telemetry (e.g., via electrical signals pulsed through the geological formation14or via mud pulse telemetry). The data processing system28may process the propagation logging measurements26to identify a horizontal conductivity and/or horizontal resistivity, a vertical conductivity and/or vertical resistivity, a dip and an azimuth at various depths of the geological formation14in the wellbore16.

To this end, the data processing system28thus may be any electronic data processing system that can be used to carry out the systems and methods of this disclosure. For example, the data processing system28may include a processor30, which may execute instructions stored in memory32and/or storage34. As such, the memory32and/or the storage34of the data processing system28may be any suitable article of manufacture that can store the instructions. The memory32and/or the storage34may be ROM memory, random-access memory (RAM), flash memory, an optical storage medium, or a hard disk drive, to name a few examples. A display36, which may be any suitable electronic display, may provide a visualization, a well log, or other indication of properties in the geological formation14or the wellbore16using the propagation logging measurements26.

FIG.2illustrates a method40of various processes that may be performed based at least in part on analysis of well logs, in accordance with aspects of the present disclosure. A location of hydrocarbon deposits within a geological formation may be identified (process block42) based at least in part on well log measurements. In some embodiments, the well log measurements may be analyzed to generate a map or profile that illustrates regions of interest with the geological formation.

Based on the identified locations and properties of the hydrocarbon deposits, certain downhole operations on positions or parts of the geological formation14may be performed (process block44). That is, hydrocarbon exploration organizations may use the locations of the hydrocarbon deposits to determine locations in the wellbore to isolate for extracting liquid, frack, and/or drill into the Earth. As such, the hydrocarbon exploration organizations may use the locations and properties of the hydrocarbon deposits and the associated overburdens to determine a path along which to drill into the Earth, how to drill into the Earth, and the like.

After exploration equipment has been placed within the geological formation14, the hydrocarbons that are stored in the hydrocarbon deposits may be produced (block46) via natural flowing wells, artificial lift wells, and the like. Further, the produced hydrocarbons may be transported (block48) to refineries and the like via transport vehicles, pipelines, and the like. Further still, the produced hydrocarbons may be processed (block50) according to various refining procedures to develop different products using the hydrocarbons.

It should be noted that the processes discussed with regard to the method40may include other suitable processes that may be based at least in part on the locations and properties of hydrocarbon deposits as indicated in the seismic data acquired via one or more seismic survey. As such, it should be understood that the processes described above are not intended to depict an exhaustive list of processes that may be performed after determining the locations and properties of hydrocarbon deposits within the geological formation.

With the foregoing in mind,FIG.3shows an illustrated embodiment of a propagation well logging tool12that is a multi-axial EM downhole tool with mutually orthogonal and collocated transmitter and receiver coils. As shown in the illustrated embodiment, the propagation well logging tool12includes three transmitters52, three first receivers54, and three second receivers56. Generally speaking, the three transmitters52induce electric eddy current in the formation that flow parallel to orthogonal planes oriented with their normals in the X (e.g., along the X-axis58), Y (e.g., along the Y-axis60), and Z directions (e.g., along the Z-axis62, referred to herein as the “downhole tool axis”, “tool axis” or “longitudinal tool axis”), which are defined by the directions of the magnetic dipole moments of each of the three transmitter coils. As such, the propagation well logging tool12shown inFIG.3may measure all nine orthogonal couplings to determine formation resistivity and resistivity anisotropy as well as formation dip. While the illustrated embodiment of the propagation well logging tool12is a triaxial EM downhole tool (e.g., each receiver of the first receivers54and the second receivers56shown inFIG.3are along X-axis58, Y-axis60, and Z-axis62), the number of axes that include receivers is not limited to three, but maybe two or more.

As shown in the illustrated embodiment, the first receivers54are disposed at a longitudinal distance64from the transmitters52, and the second receivers56are disposed at a longitudinal distance66from the transmitters52. As discussed in more detail below regarding the discussion ofFIGS.4and5, the longitudinal distances64and66may be used as a tool constant, which may facilitate calculation of the apparent conductivity and/or apparent resistivity.

The illustrated example of the propagation well logging tool12is shown communicatively coupled to the data processing system28. As discussed herein, the propagation well logging tool12(e.g., multi-axial well logging tool) may acquire measurements within a wellbore16of the geological formation14. The processor30of the data processing system28may receive these measurements. The memory32may store information such as control software, frequency, configuration data, etc. The memory32may include a volatile memory, such as random access memory (RAM), and/or a nonvolatile memory, such as read-only memory (ROM). The memory32may store a variety of information and may be used for various purposes. For example, the memory32may store processor-executable instructions including firmware or software for the processor30to execute. In some embodiments, the memory32is a tangible, non-transitory, machine-readable-medium that may store machine-readable instructions for the processor30to execute. The memory32may include ROM, flash memory, a hard drive, or any other suitable optical, magnetic, or solid-state storage medium, or a combination thereof. The memory32may store data, instructions, and any other suitable data.

As shown in the illustrated embodiment, the propagation well logging tool12is within a geological formation14having conductivity and permittivity with a traverse isotropy. In the illustrated embodiment, the downhole tool axis (e.g., Z-axis62) of the triaxial propagation well logging tool12is aligned to the normal of the lamination planes68(e.g., the interfaces of geological layers). Each lamination (e.g., geological layer), a few of which are shown (e.g.,68a,68b,68c, and68d), of the geological formation14has a conductivity and/or a permittivity that is approximately homogeneous along X-axis58and the Y-axis60, while the conductivity and/or permittivity may vary along the Z-axis62. In the current embodiment, all the light-colored laminations (e.g.68aand68c) share the same conductivity, and all the dark-colored laminations (e.g.68band68d) share the same conductivity. Therefore, the conductivity, σ, and the permittivity, ∈, of all laminations as a whole may be represented by the equations:

where σhand σvare the horizontal conductivities (e.g., along the X-axis58and the Y-axis60) and the vertical conductivities (e.g., along the Z-axis62), respectively, and εhand εvare the horizontal permittivity and the vertical permittivity, respectively. The general directions of the horizontal conductivities and vertical conductivities are shown in the axis70. The unit vectors {circumflex over (x)}, ŷ, and {circumflex over (z)} correspond to X-axis58, Y-axis60, and Z-axis62. As referred to herein, the transmitters52and the receivers (e.g., first receivers54and second receivers56) that are generally aligned with the X-axis58and the Y-axis60are referred to as being coplanar (e.g., coplanar transmitters and/or coplanar receivers) and the transmitters52and the receivers (e.g., first receivers54and second receivers56) that are generally aligned with the Z-axis62are referred to as being coaxial or coaxial receivers (e.g., coaxial transmitters and/or coaxial receivers).

With the foregoing in mind,FIG.4is a flow diagram of a process80for generating one or more apparent conductivity well logs based at least in part on propagation measurements acquired by the propagation well logging tool12. Although described in a particular order, which represents a particular embodiment, it should be noted that the process80may be performed in any suitable order. Additionally, embodiments of the process80may omit process blocks and/or include additional process blocks. Moreover, in some embodiments, the process80may be implemented at least in part by executing instructions stored in a tangible, non-transitory, computer-readable medium, such as memory32, using processing circuitry, such as processor30implemented in the data processing system28

Generally, the process80includes acquiring (process block82) propagation measurements associated with a geological formation using one or more propagation well logging tools having at least two receivers (e.g., a first receiver54and a second receiver56). The process80also includes converting (process block84) the propagation measurements to apparent conductivity measurements based at least in part on a frequency associated with the propagation measurement, a relatively longitudinal position of the receivers (e.g., a first receiver54and a second receiver56), and a phase shift measurement and/or an attenuation measurement. Further, the process80includes generating (process block86) one or more apparent conductivity well logs based at least in part on the apparent conductivity measurements. While process80is discussed above with respect to apparent conductivity measurements, it should be noted that the apparent conductivity may also be represented as an apparent resistivity measurement.

In process block82, the data processing system28(e.g., processor30) may receive and/or acquire propagation measurements from a propagation well logging tool12. In some embodiments, acquiring the propagation measurements from the propagation well logging tool12may include the processor30sending suitable control signals to the propagation well logging tool12to begin acquiring the propagation measurements. As discussed herein, the propagation measurements may include phase shift measurements and/or attenuation measurements.

In process block84, the processor30may convert the propagation measurements to apparent conductivity measurements based on a frequency (e.g., an operating frequency of the propagation well logging tool12such as approximately 100 kHz, 200 kHz, 400 kHz, 1000 kHz, 2000 kHz, 2 MHz, and the like). It should be noted that converting the propagation measurements may depend on the orientation of the receivers (e.g., a first receiver54and a second receiver56) and the transmitter(s)52of the propagation well logging tool12. That is, in some embodiments, the propagation measurements may be acquired by a coaxial propagation well logging tool, a coplanar propagation well logging tool, a triaxial propagation well logging tool, and the like, as discussed in further detail below. In any case, the processor30may convert the propagation measurements based on a relative longitudinal position of the receivers (e.g., a first receiver54and a second receiver56) as discussed in further detail below (e.g., with regards to equation 31). Then, in process block86, the processor30may generating apparent conductivity well logs based on the apparent conductivity measurements.

The discussion below provides an example for converting the propagation measurements to apparent conductivity, as described above with regards to process block84. For a coaxial propagation downhole tool (e.g., having transmitter coil and receiver coils that are oriented along the Z-axis62, the voltages induced in the two receiver coils (e.g., the first receiver54and the second receiver56) are:

And for a coplanar propagation downhole tool (e.g., having transmitter coil and receiver coils oriented along the X-axis58or Y-axis60), the voltages induced in the two receiver coils are:

For both Eqns. (3) and (4), I is the current in the transmitter coil, ω is the angular frequency, ω=2πf where f is the frequency of current I. NTand NRj, j=1, 2 are the numbers of turns of transmitter and receiver coils, and ATand ARj, j=1, 2 are their areas. i is the imaginary unit, i=√{square root over (−1)}; Lj, j=1, 2 is the distance between transmitter and receiver coils. It is assumed that receiver2is further away than receiver1from the transmitter, i.e. L1<L2. khand kvare wavenumbers corresponding to σhand εh, and σvand εv, respectively, given by:

In some embodiments, khand kvmay be expressed in terms of a complex number:

In certain conventional propagation processing techniques, the ratio of voltages at the two receivers (e.g., first receiver54and second receiver56) is acquired and converted to phase shift and attenuation. The voltage ratio measurement can compensate for transmitter gains. A composite downhole tool using two or more transmitter coils may provide compensation for receiver gains and borehole rugosity. In the following, low-frequency asymptotic expressions of phase shift and attenuation are derived for an elemental downhole tool. Low-frequency asymptotic expressions of a fully compensated downhole well logging tool consisting of multiple transmitters and receivers are then given by means of the superposition of those of elemental downhole tools.

Expansion of the Voltage Ratio of an Elemental Coaxial Propagation Tool

As discussed above, certain technique for using propagation measurements utilize the ratio of the voltages measured by the receivers. For the co-axial propagation well logging tool, the logarithm of the ratio of the voltages at the receivers are, in accordance with equation 3:

Denoting the moment of the receivers by MRj=NRjARjand j=1, 2, then:

Without loss of generality, assuming that MR1=MR2, then:

Eqn. 11 may be expanded in powers of khL using Taylor's expansion for the ln z function:

Using Eqn. 12, the third term on the right-hand side of Eqn. 10 may be written as:

And noting that:

The first term on the right-hand side of Eqn. (21) is attributed to the geometrical decay of EM wave in the air and can be removed from the two sides. Then:

It is noted that the first order term −ikhΔL in Eqn. (16) is cancelled out due to the first term of Eqn. (16). When the frequency is low

The above asymptotic form suggests that the attenuation and phase shift measurements can be converted to apparent conductivity as is normally done for induction measurements. To this end, let:

Using the polar form of the two complex voltages, i.e.:

the attenuation and phase shift in Eqn. (26) can be written as:

According to Eqns. (25) and (26):

A downhole tool constant may be defined for the coaxial propagation well logging tool:

then the apparent conductivity of the coaxial propagation downhole tool is:

The asymptotic forms in the above are opposite to those of apparent conductivity for the coaxial induction downhole tool. Alternatively, the apparent conductivity can be defined as:

The asymptotic forms of the second definition for the apparent conductivity are the same as those of the coaxial induction downhole tool. Both forms of the definition indicate that the phase shift apparent conductivity σa,zzPSis a measure of conductivity and the attenuation apparent conductivity σa,zzATa measure of permittivity or dielectric constant when the frequency is low.

Expansion of the Voltage Ratio of an Elemental Coplanar Propagation Downhole Tool

The logarithm of the ratio of voltages at the two coplanar receivers is:

As with the ratio for the coaxial propagation downhole tool, assuming that NR1AR1=NR2AR2, or MR1=MR2, then:

In some embodiments, an error calibration may be performed to account for the shield and tool geometry that may improve the measurements, in at least some instances. The error calibration may include determining the attenuation and/or phase shift measurements for the tool in air (e.g., by acquiring an additional propagation measurement at the surface above the geological formation, in a lab, and the like) and/or under ambient conditions and then modifying the propagation measurements obtained downhole accordingly, such as via a background subtraction.

According to Eqn. (9), the third part of Eqn. (37) can be expanded as:

Recall the expansion given in Eqn. (17), when

Substituting Eqns. (43)-(45) into Eqn. (39), and rearranging the results, it can be shown that:

It is noted that the first order term −ikhΔL in Eqn. (39) is cancelled out due to the first term of Eqn. (43). Moreover, the second order term of khin Eqn. (43) is also cancelled out due to the first term of Eqn. (44). Therefore, the only second term is that of kv. As with Eqn. (22), the first term on the right-hand side of Eqn. (46) is the geometrical decay of EM wave in the air and can be removed from the two sides. Then:

When the frequency is low:

As with Eqn. (25), the asymptotic form in the above suggests that the attenuation and phase shift measurements can be converted to apparent conductivity measurements. In a similar manner, let:

Using the polar form of the two complex voltages, i.e.:

The attenuation and phase shift in Eqn. (51) can be written as:

According to Eqn. (50):

Define the following downhole tool constant for the coplanar propagation downhole tool:

Accordingly, the apparent conductivity of the coplanar propagation downhole tool is:

The asymptotic forms in the above are opposite to those of apparent conductivity for the coplanar induction downhole tool. As with the coaxial voltage ratio, an alternative to the apparent conductivity of Eqn. (57) may be given by:

The asymptotic forms of the second definition for the apparent conductivity are the same as those of the coplanar induction downhole tool. Both forms of the definition indicate that the phase shift apparent conductivity σa,xxPSis a measure of conductivity and the attenuation apparent conductivity σa,xxATa measure of permittivity or dielectric constant when the frequency is low.

Apparent Conductivity of a Fully Compensated Propagation Downhole Tool

A fully compensated propagation downhole tool may compensate for both the transmitter and receiver gains. In one example, the z-axis of the downhole tool coordinates coincides with the downhole tool axis (e.g., Z-axis62), and there are in total NTtransmitters, some of which are below R1and R2, and the rest of which are above R1and R2. The positive direction of the Z-axis is pointed from R1to R2. If Tl, the l-the transmitter is on the lower side of the two receivers, the logarithmic voltage ratio of R1and R2is:

If Tlis on the upper side of the receivers, the voltage ratio is:

Eqns. (63) and (64) can be combined into one equation such that:

where γl=sgn(LTl,R2−LTl,R1), with the symbol “sgn” designating the signum function. Here, LTl,R1and LTl,R2are the spacings of the first receiver54, R1, and the second receiver56, R2, from transmitter Tl. A compensated voltage ratio of the downhole tool is the weighted average of all the individual voltage ratios:

The terms related to receiver gains can be separated out, yielding:

For a fully compensated downhole tool:

The above compensation condition must be satisfied so that the receiver gains are eliminated from the combined logarithmic voltage ratio given in Eqn. (67). With this in mind, Eqn. (67) becomes:

For the l-th transmitter, the voltage ratio can be written as:

In the above equation, ATTl0is the air signal of the l-th transmitter, given by:

It has been shown in the above that the air-signal-corrected voltage ratio can be converted to the apparent conductivity as follows:

In the above, KTlis the downhole tool constant for the elemental voltage ratio of transmitter Tl. For an elemental coaxial downhole tool:

For an elemental coplanar downhole tool:

Recall that when the frequency is low, the apparent conductivity of an elemental voltage ratio approaches the true conductivity of the formation, i.e.:

Here, for a coaxial measurement, σ=σh, ε=εh; for a coplanar measurement, σ=σv, ε=εv, assuming the downhole tool axis is perpendicular to the lamination plane. Using the above asymptotic form in Eqn. (75), then:

Eqn. (77) suggests that the apparent conductivity for the combined phase shift and attenuation measurements can be defined as:

As with the apparent conductivity of an elemental voltage ratio, σa,Cof the compensated downhole tool approaches the true conductivity of the formation,

when the frequency is relatively low.

Skin Effect Correction

FIG.5is a flow diagram of a process90for determining the apparent dielectric constant and/or a skin effect corrected apparent conductivity based at least in part on propagation measurements acquired by a propagation well logging tool12. Although described in a particular order, which represents a particular embodiment, it should be noted that the process90may be performed in any suitable order. Additionally, embodiments of the process90may omit process blocks and/or include additional process blocks. Moreover, in some embodiments, the process90may be implemented at least in part by executing instructions stored in a tangible, non-transitory, computer-readable medium, such as memory32, using processing circuitry, such as processor30implemented in the data processing system28.

Generally, the process90includes acquiring (process block92) propagation measurements associated with a geological formation using one or more propagation well logging tools having at least two receivers (e.g., a first receiver54and a second receiver56). The process80also includes converting (process block94) the propagation measurements to apparent conductivity measurements based at least in part on a frequency associated with the propagation measurement, a relatively longitudinal position of the receivers, and a phase shift measurement and/or an attenuation apparent measurement, and the apparent conductivity includes a phase shift apparent conductivity and an attenuation apparent conductivity. Further, the process90includes determining (process block96) the skin effect corrected apparent conductivity based the apparent conductivity. Further still, the process90includes determining (process block98) the dielectric constant based on the apparent conductivity. While process90is discussed above with respect to apparent conductivity measurements, it should be noted that the apparent conductivity may also be represented as an apparent resistivity measurement.

In process block92, the data processing system28(e.g., processor30) may receive and/or acquire propagation measurements from a propagation well logging tool12. In some embodiments, acquiring the propagation measurements from the propagation well logging tool12may include the processor30sending suitable control signals to the propagation well logging tool12to begin acquiring the propagation measurements. As discussed herein, the propagation measurements may include phase shift measurements and/or attenuation measurements.

In process block94, the processor30may covert the propagation measurements to apparent conductivity measurements based on a frequency (e.g., an operating frequency such as approximately 100 kHz, 200 kHz, 400 kHz, 1000 kHz, 2000 kHz, 2 MHz, and the like). It should be noted that converting the propagation measurements may depend on the orientation of the receivers (e.g., first receiver54and second receiver56) and the transmitter(s)52of the propagation well logging tool12. That is, in some embodiments, the propagation measurements may be acquired by a coaxial propagation well logging tool, a coplanar propagation well logging tool, a triaxial propagation well logging tool, and the like, as discussed in further detail below. In any case, the processor30may convert the propagation measurements based on a relative longitudinal position of the receivers (e.g., first receiver54and second receiver56) as discussed in further detail below (e.g., with regards to equation 31). Then, in process block96, the processor30may determine the skin effect corrected apparent conductivity based on the apparent conductivity, as discussed in more detail below.

Substituting Eqn. (22) in Eqn. (35), the following equation ensues

When the displacement current is negligible, namely, when ωεh<<σh, Eqn. (80) reduces to

In component form:

It should be noted that the leading term of σa,zzATis identical to the second term of σa,zzPSin terms of magnitude but the sign is opposite. This relationship can be used to correct the skin effect on σa,zzPSby simply adding σa,zzATto σa,zzPS, namely:

The above may be true for a coaxial elemental propagation downhole tool.

In a generally similar manner, for a coplanar elemental propagation downhole tool, it can be shown that when the displacement current is negligible, the following equation may hold true:

In component form:

As with σa,zz, the leading term of σa,zzATis identical to the second term of σa,zzPSin terms of magnitude but the sign is opposite. Likewise, this relationship can be used to correct the skin effect on σa,xxPSby simply adding σa,xxATto σa,xxPS, namely:

The apparent conductivity of a fully compensated downhole tool is a linear superposition of those of the underlying elemental downhole tools. Therefore, the skin effect correction schemes given in Eqns. (86) and (90) may hold true for a fully compensated downhole tool.

Apparent Dielectric Constant

Additionally or alternatively, in process block98, the processor30may determine the apparent dielectric constant based on the apparent conductivity as discussed in more detail below.

For example, the apparent dielectric constants may be defined as follows:

The above two expressions are for a coaxial and coplanar elemental downhole tool, respectively, and are obtained using the second form of apparent conductivity for both tools. In a highly resistive formation (e.g., above a resistivity threshold, such as where ωεh>>σhand/or ωεv>>σv) apparent dielectric constants given in Eqns. (91) and (92) may provide an estimate of horizontal and vertical dielectric constants, respectively.
A Multi-Array Propagation Tool with a Uniform Tool Constant

The tool constants given in Eqns. (31) and (56) for are a function of spacing and frequency. Therefore, the tool constants may be used for optimizing tool design. In one application, the tool constants for all arrays of a multi-array propagation tool consisting of multiple elemental arrays may be selected such that:

where j is the index for the arrays, M is the total number of arrays, and C is a constant number. The subscript α of Kααjis the orientation of the transmitter and receiver coils, α=x,z. Such a selection ensures that the measurements from all arrays have the same skin effect. It should be noted that this property may be important when using the spread of logs from multiple arrays for a quick invasion evaluation.

Numerical Results

FIG.6shows three panels (e.g., panel100, panel102, and panel104) that depict apparent conductivity (e.g., axis106) versus true conductivity (e.g., axis108) for a 28-in coaxial elemental propagation tool having two receivers where L1=25 in., L2=31 in, ΔL=6 in., and an operating frequency of 100 kHz. The lines (e.g., line110, line112, and line114) are related to the true conductivity, and the lines (e.g., line116, line118, and line120) are related to the phase-shift apparent conductivity, the attenuation apparent conductivity, and the skin effect-corrected phase shift apparent conductivity, respectively. Here and in the following, the true conductivity of an isotropic formation is denoted by σt, σt=σh=σv.FIG.7shows three panels (e.g., panel122, panel124, and panel126) that depict apparent conductivity (e.g., axis128) versus true conductivity (e.g., axis130) for a 28-in coaxial elemental propagation tool where L1=25 in., L2=31 in, ΔL=6 in., and at an operating frequency of 400 kHz. The lines (e.g., line132, line134, and line136) are related to the true conductivity and the lines (e.g. line140, line142, and line144) are related to the phase-shift apparent conductivity, the attenuation apparent conductivity, and the skin effect-corrected phase shift apparent conductivity, respectively.FIG.8shows three panels (e.g., panel144, panel146, and panel148) that depict apparent conductivity (e.g., axis150) versus true conductivity (e.g., axis152) for a 28-in coaxial elemental propagation tool where L1=25 in., L2=31 in, ΔL=6 in., and at an operating frequency of 2000 kHz. The lines (e.g., line154, line156, and line158) are related to the true conductivity and the lines (e.g., line160, line162, and line164) are related to the phase-shift apparent conductivity, the attenuation apparent conductivity, and the skin effect-corrected phase shift apparent conductivity, respectively.

As shown in the graphs depicted inFIGS.6-8, for a given frequency, the phase-shift apparent conductivity (e.g., corresponding to line116, line140, and line160) approaches the true conductivity (e.g., corresponding to line110, line132, and line154) for relatively small values of the true conductivity and deviate the true conductivity (e.g., corresponding to line110, line132, and line154) for relatively large values of the true conductivity. Moreover, the attenuation apparent conductivity (e.g., corresponding to line118, line142, and line162) approaches zero when the true conductivity (e.g., corresponding to line112, line134, and line156) is relatively small and the attenuation apparent conductivity increases monotonically when the true conductivity increases. Further, the difference between the skin effect corrected phase shift apparent conductivity (e.g., corresponding to line120, line144, and line164) and true conductivity (e.g., corresponding to line114, line136, and line158) is smaller than for the phase-shift apparent conductivity (e.g., corresponding to line116, line140, and line160).

FIG.9shows three panels (e.g., panel166, panel168, and panel170) that depict apparent conductivity (e.g., axis172) versus vertical conductivity (e.g., axis174) for a 28-in coplanar elemental propagation tool having two receivers where L1=25 in., L2=31 in, ΔL=6 in., and an operating frequency of 100 kHz. The lines (e.g., line176, line178, and line180) are related to the horizontal conductivity, the line182, line184, and line186are related to the vertical conductivity, and the line188, line190, and line192are related to the phase-shift apparent conductivity, the attenuation apparent conductivity, and the skin effect-corrected phase shift apparent conductivity, respectively.FIG.10shows three panels (e.g., panel194, panel196, and panel198) that depict apparent conductivity (e.g., axis200) versus vertical conductivity (e.g., axis202) for a 28-in coplanar elemental propagation tool having two receivers where L1=25 in., L2=31 in, ΔL=6 in., and an operating frequency of 400 kHz. The lines (e.g., line204, line206, and line208) are related to the horizontal conductivity, the lines (e.g., line210, line212, and line214) are related to the vertical conductivity, and the lines (e.g., line216, line218, and line220) are related to the phase-shift apparent conductivity, the attenuation apparent conductivity, and the skin effect-corrected phase shift apparent conductivity, respectively.FIG.11shows three panels (e.g., panel222, panel224, and panel226) that depict apparent conductivity (e.g., axis228) versus vertical conductivity (e.g., axis230) for a 28-in coplanar elemental propagation tool having two receivers where L1=25 in., L2=31 in, ΔL=6 in., and an operating frequency of 2000 kHz. The lines (e.g., line232, line234, and line236) are related to the horizontal conductivity, the lines (e.g., line238, line240, and line242) are related to the vertical conductivity, and the lines (e.g., line244, line246, and line248) are related to the phase-shift apparent conductivity, the attenuation apparent conductivity, and the skin effect-corrected phase shift apparent conductivity, respectively.

As shown in the graphs depicted inFIGS.9-11, for a given frequency, the phase shift apparent conductivity (e.g., corresponding to line188, line216, and line244) approach the true vertical conductivity (e.g., the vertical conductivity corresponding to line182, line210, and line238) when the conductivity is relatively small and the phase shift apparent conductivity deviates gradually from the true conductivity when the conductivity is relatively large. However, the deviation increases more rapidly than that for the coaxial tool as shown inFIGS.6-8. The attenuation apparent conductivity (e.g., corresponding to line190, line218, and line246) approaches zero when the true vertical conductivity (e.g., the vertical conductivity corresponding to line182, line210, and line238) is relatively small. When the true conductivity (e.g., the vertical conductivity corresponding to line184, line212, and line240) increases, the attenuation apparent conductivity (e.g., corresponding to line190, line218, and line246) increase monotonically and then deviate for relatively high values of conductivity. The threshold (e.g., at data point250) for the rollover is shown inFIG.11, and is inversely proportional to the operating frequency. The same tendency is true with the phase shift apparent conductivity (e.g., corresponding to line188, line216, and line244). For example, the threshold for the apparent conductivity of line188ofFIG.9is at a higher conductivity than the threshold for the apparent conductivity of line216ofFIG.10.

FIG.12shows two panels (e.g., panel252and panel254) that depict a phase shift apparent conductivity and an apparent dielectric constant determined based at least in part on an attenuation apparent conductivity for a coaxial propagation well logging tool operating at 100 kHz.FIG.13shows two panels (e.g., panel256and panel258) that depicts a phase shift apparent conductivity and an apparent dielectric constant determined based at least in part on an attenuation apparent conductivity for a coplanar propagation well logging tool operating at 100 kHz, in accordance with aspects of the present disclosure.

In particular,FIGS.12and13show apparent dielectric constants εr,a,zzand εr,a,xxfor two elemental propagation tools in a homogeneous formation. Here, εr,a,zz=−σa,zzAT/ωε0, εr,a,xx=−σa,xxAT/ωε0. The frequency is set to 100 kHz in the simulation. The conductivities of the formation are fixed to σh=2 mS/m and σv=1 mS/m, respectively in the simulation. Results show that εr,a,zz≈εh,r, εr,a,xx≈εv,rover the whole range of εh,rand εv,ras expected. Moreover, when εh,rand εv,rdecreases, σa,zzPS≈σhand σa,xxPS≈σv.

FIG.14shows three panels (e.g., panel260, panel262, and panel264) that each depict apparent conductivity well logs based at least in part on measurements acquired by a fully compensated coaxial propagation well logging tool operating at 100 kHz within a 4-layer formation.FIG.15shows three panels (e.g., panel266, panel268, and panel270) that each depict apparent conductivity well logs based at least in part on measurements acquired by a fully compensated coplanar propagation well logging tool operating at 100 kHz within a 4-layer formation.

In particular,FIGS.14and15show apparent conductivities σa,zzand σa,xxof a fully compensated triaxial propagation tool in a 4-layer formation as identified by blue and green square logs. The triaxial tool is the same as CDR in terms of geometry. The operating frequency is 100 kHz. In this example, the triaxial propagation tool consists of two transmitters (T1and T2) and two receivers (R1and R2). T1is placed below R1. T2is above R2. The spacings between T1and R1and R2are 25 in and 31 in, respectively. The spacings between T2and R1and R2are 31 in and 25 in, respectively. For the coaxial measurement, the phase shift apparent conductivity curve σa,zzPSfollows σhclosely. The attenuation apparent conductivity σa,zzATvaries slowly with depth as a result of large depth of investigation. It should be noted that, the skin effect-corrected phase shift apparent conductivity curve, namely σa,zzPS+σa,zzATis closer to σhthan σa,zzPSwithout skin effect correction. For the coplanar measurements, results show that the phase shift apparent conductivity curve σa,xxPSappears to follow σvmore closely than σh. After the skin effect correction, the phase shift apparent conductivity curve, namely σa,xxPS+σa,xxATis closer to σv. It should be noted that there are spikes occurring on the σa,xxPScurve as a result of induced current crossing bed boundaries. As far as the root cause is concerned, this phenomenon is the same as polarization horns that have been observed on coplanar couplings of a triaxial induction tool, or responses of coaxial array induction and propagation tools at high relative dip. In the simulation, it is assumed that the tool axis is perpendicular to the bedding planes. The dielectric constants of the model are given following the empirical relationship for ARC tool.

FIG.16shows three panels (e.g., panel272, panel274, and panel278) that each depict apparent conductivity well logs based at least in part on measurements acquired by an array resistivity compensated propagation well logging tool operating at 400 kHz within a multi-layer formation (e.g., having greater than 4 layers), in accordance with aspects of the present disclosure.FIG.17shows three panels (e.g., panel280, panel282, and panel284) that each depict apparent conductivity well logs based at least in part on measurements acquired by an array resistivity compensated propagation well logging tool operating at 2 MHz within a multi-layer formation (e.g., having greater than 4 layers), in accordance with aspects of the present disclosure.

More specifically,FIGS.16and17show apparent conductivity σa,zzof ARC5 (e.g., an array resistivity compensated (ARC) well logging tool by Schlumberger) 400 kHz and 2 MHz, respectively in a multilayer formation identified by the dark green square log. Shown in the left track (e.g., panels272and280) is phase shift apparent conductivity σa,zzPS, and in the middle track (e.g., panels274and282) is attenuation apparent conductivity σa,zzAT. The skin effect-corrected phase shift apparent conductivity curve, namely σa,zzPS+σa,zzATdisplayed in the right track (e.g., panels278and284). For a given frequency, the shallow arrays follow more closely the true formation conductivity because of small shoulder bed effect and mild skin effect. σa,zzPSis apparently a better measure of true formation conductivity than σa,zzAT. After the skin effect correction, the phase shift apparent conductivity curves, namely those identified by σa,zzPS+σa,zzATin the right track (e.g., panels278and284) are closer to the true formation conductivity. Moreover, the spread of the five curves is smaller than those in the left track (e.g., panels272and280). Between two frequencies, the logs at 2 MHz exhibit stronger skin effect than their counterparts at 400 kHz. In the simulation, it is assumed that the tool axis is perpendicular to the bedding planes. The dielectric constants of the model are given following the empirical relationship for ARC well logging tool.

As mentioned above, the techniques of the present disclosure may be used for designing a multi-array propagation well-logging tool having a uniform tool constant.FIG.18is a flow diagram of a method of manufacture290for building a multi-array propagation well logging tool with a uniform tool constant for at least a portion of the arrays (e.g., a combination of a transmitter and two receivers in different longitudinal positions). The method of manufacture290includes arranging a first transmitter of a first array in a first longitudinal position, a first receiver of the first array in a second longitudinal position, and a second receiver of the first array in a third longitudinal position (block292). The method of manufacture290also includes arranging a second transmitter of a second array in a fourth longitudinal position, a third receiver of the second array in a fifth longitudinal position, and a fourth receiver in a sixth longitudinal position (block294). The method of manufacture290also includes selecting a first operating frequency of the first array and a second operating frequency of the second array such that a first tool constant for the first array is approximately equal to a second tool constant for the second array (block296). At least in some embodiments, the first tool constant and/or the second tool constant may be modified based on the longitudinal positions of the transmitter(s) and receiver(s) such that a predetermined resolution and/or depth of investigation may be achieved.

Accordingly, the present disclosure relates to techniques for processing propagation well logging measurements to generate an apparent conductivity and/or apparent resistivity based at least in part on the propagation well logging measurements without using an inversion or a resistivity transform. In particular, the disclosed techniques for processing the propagation well logging measurements may utilize a tool constant that is based on a relative longitudinal position of at least two of the receivers of the propagation well logging tool. Using the disclosed tool constant, apparent conductivity and/or apparent resistivity may be computed, which may be used to generate apparent conductivity and/or apparent resistivity well logs. In some embodiments, the apparent conductivity may be a phase shift apparent conductivity and/or an attenuation apparent conductivity. As discussed herein, the phase shift apparent conductivity may provide a measure of formation conductivity and the attenuation apparent conductivity may provide a measure of the skin effect. In some embodiments, the phase shift apparent conductivity may be corrected for the skin effect using the attenuation apparent conductivity to generate an improved apparent conductivity that may be representative of the true formation conductivity.