Formation resistivity measurement based on a transient ...

Journal of Geophysics and Engineering

Journal of Geophysics and Engineering (2020) 17, 222–230 doi:10.1093/jge/gxz100

Formation resistivitymeasurement based on atransient electromagnetic field excited by an acousticprobe in an open-hole

Xiaofei Sheng1, Jianguo Shen1,* and Yongjin Shen2

1 School of Microelectronics, Tianjin University, No. 92, Weijin Road, Nankai District, Tianjin, 300072, China2 Research and Development Center, Beijing Huahui Shengshi Energy Technology Co., Ltd, Beijing, China

*Corresponding author: Jianguo Shen. E-mail: [email protected]

Received 2 April 2019, revised 8 October 2019Accepted for publication 28 October 2019

AbstractThe monopole probe commonly used in acoustic logging can generate vibrations in its naturalfrequency under the excitation of the pulse signal, and excite transient electromagnetic (TEM)signals of the same frequency. The acoustic probe receives both acoustic and TEM field signals.The TEM field propagates and attenuates in conductive well fluid and formations, carryingformation resistivity information that could be used for formation evaluation. Based on theaxisymmetry of the TEM field in an open-hole, theoretical calculation of the TEM fieldcomponents in an open-hole model was performed. The results revealed that TEM responsesignals decay rapidly with time, and electric field intensity along the well axis Ez is approximatelylinear with formation resistivity. On this basis, we proposed a newmethod for measuringformation resistivity in an open-hole, which could supplement conventional logging methods.Also, it does not affect signal processing of acoustic logging and only applies the TEM signal todetermine formation resistivity. The newmethod could accomplish a more comprehensivepetroleum formation evaluation, which is of great significance to the integrated design of the welllogging instrument.

Keywords: TEM field, monopole probe, resistivity, acoustic logging

1. Introduction

Acoustic loggingmeasurement (developedwell before 1965)was originally designed for determining the compressionalvelocity of rocks surrounding a wellbore. Today, it plays animportant role in a wide variety of geological, geophysicaland engineering applications (Cheng et al. 1992; Ellis &Singer 2007; Zhao et al. 2016; Yao et al. 2019). Duringacoustic logging, a pulse of pressure must be applied to theformation surrounding the wellbore, which is based on themagnetostrictive behavior or piezoelectric effect of certainmaterials (Serra&Serra 2004; Serra 2008).Meanwhile, thereare two kinds of transducer used in acoustic logging. Thefirst type is the magnetostrictive transducer, which responds

to changes in the magnetic field by a volume change of ferro-magnetic material; while the second type is the piezoelectrictransducer, which corresponds to an applied electric field bya volume change of ceramic materials. When the frequencyof the appliedmagnetic field or electric field is identical to thenatural frequency of the transducer, the transducer will vi-brate according to its natural frequency, leading to the largestvibration amplitude. At the same time, sound waves with asimilar frequency will be emitted at both ends of the trans-ducer (Chu 1987; Zhang et al. 2009). Depending on the ex-citation mode, several types of acoustic source exist. Amongthem, monopole source acts as an omnidirectional pressuresource and is widely used in acoustic logging. It creates a

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Journal of Geophysics and Engineering (2020) 17, 222–230 Sheng et al.

Figure 1. Waveforms measured using the array acoustic logging tools. Note that the longitudinal coordinate represents source-receiver spacing.

compressional wave pulse in the borehole fluid, which prop-agates out into the formation, exciting both compressionaland shear waves in the formation (Haldorsen et al. 2006).

Despite the advances in tool engineering and charac-terisation, transmitter technology and better understandingabout the acoustic logging signals benefiting from computermodelling jointly promote the emergence of new technolo-gies and transmitters (such as shear wave slowness measure-ments, dipole sources and sonic scanner tool), themonopolereceivers and transmitters still deserve our attention (Closeet al. 2009). In this paper, we consider the monopole trans-ducer, which is usually a piezoelectric tube, as the sourceof acoustic signal. Under the excitation of the pulse exci-tation signal, the piezoelectric tube will vibrate and emitacoustic signals. At the same time, a sudden change in thevoltage between the inner and outer surfaces of the piezo-electric tube will excite the TEM field in the surround-ing medium (Nabighian 1987), which is widely used inthe fields of mineral, coal, groundwater exploration, open-hole and cased-hole logging (Xue et al. 2012, 2015, 2018;Commer et al. 2015; Xi et al. 2016; Chen et al. 2017;Khan et al. 2018; Rasmussen et al. 2018; Xue et al. 2018;Sheng et al. 2019). The TEM field propagates in the well(Song 2012), and the corresponding TEM response (shownin figure 1) is received by the acoustic receiving probes.This TEM signal, which arrives before the acoustic log-ging signal in the response waveform, is usually elimi-nated because it is an interference signal in acoustic log-ging (Rader 1982). In terms of signal composition, theTEM response received is mainly induced by the displace-ment current and conduction current. Since the influence

of the dielectric constant on the electromagnetic response isrelatively small within the working frequency band of acous-tic logging, the TEM response in an open-hole is primarilyaffected by formation resistivity, i.e. the TEM response car-ries information about formation resistivity.

In the process of open-hole logging, suppose that forma-tion resistivity can be extracted from TEM responses (ob-tained by the acoustic logging tool); it would enrich themeans of formation resistivity measurement in open-holewells and significantly affect the integration of the instru-ment. By combining it with important information about theformation gathered from acoustic signals (such as slownessandporosity), it is possible to gain amore comprehensive un-derstanding regarding the target formation. In this paper, weexamine the feasibility of this idea experimentally and theo-retically.

The paper is organised as follows. In Section 2, we use theon-site acoustic logging data to present the source and char-acteristics of TEM induction signals in acoustic logging. InSection 3, we calculate theTEMfield components excited bypiezoelectric tube in the open-hole. In Section 4, by changingsource-receiver spacing and formation resistivity, the char-acteristics of the responses are studied and the relationshipbetween response signal and formation resistivity is also ex-plored. Detailed analysis and discussion are performed inSection 5. Our findings are concluded in Section 6.

2. Electrical signals before the acoustic logging signals

In this section, the characteristics of the raw single-receivervariable density plot in acoustic logging are studied. In

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Figure 2. Processing result of open-hole logging data obtained by a digital acoustic logging tool. (a) Slowness-time-coherence (STC) plot, (b) slownessand (c) raw single-receiver variable density plot.

addition, the initial reasons for our research on this subjectare given.

Briefly, the electrical signals (which arrive before theacoustic signal in the response waveform) are the TEM re-sponse signals excited by the piezoelectric tube. Figure 2

illustrates the processing result of the open-hole logging dataobtained using the digital acoustic logging tool. Figure 2apresents the slowness-time-coherence (STC) plot obtainedusing the phasemethod; figure 2b shows the slowness curve;and figure 2c displays a raw single-receiver variable density

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Figure 3. Schematic diagram of the open-hole model.

plot. Apparently, figure 2 reflects that the physical parame-ters of the formation at different depths are different. Cor-respondingly, the amplitude of the TEM signal varies withdepth (refer to figure 2c). Hence, we deduce that the TEMsignal recorded by the acoustic logging tool is related to for-mation information. Since the frequency of acoustic loggingis within a frequency range of about 20 kHz, the effect ofthe dielectric constant can be neglected. Thus, the inductionsignal should be related to the resistivity of the formation(Zhang1984;Kaufman&Dashevsky2003).As such,weper-form the studyof extracting formation resistivity informationfrom TEM signals recorded in acoustic logging in currentresearch.

3. Theoretical calculation of TEMfield components:H

𝜽, Er and Ez

In this section, the TEM field components excited by piezo-electric tube in an open-hole are calculated using the open-hole model (figure 3). The cylindrical coordinate system(r, 𝜃, z) is adopted in our calculation. Specifically, medium1 represents the piezoelectric tube; medium 2 represents thewell liquid andmedium3 represents the formation. The elec-tromagnetic field excited by the piezoelectric tube is axisym-metric; while the electric field and magnetic field along thecircumference are identical. When the excitation occurs, ahigh-voltage pulse signal of a certain frequency is applied onthe inner andouter surfaces of the piezoelectric tube.Thepo-tential changes between the inner and outer surfaces, caus-ing the acoustic vibration of the piezoelectric tube. Since the

potentials of all points on the outer surface of the piezo-electric tube are identical, an infinite number of cylindricalequipotential surfaces are formed in the well fluid. Thereis a potential difference between any two equipotential sur-faces, thus forming the electric field strength pointing to in r-and z-directions. Themagnetic field induced by the changingelectric field is along the circumference direction, perpendic-ular to the electric field direction.

The current merely flows between the inner and outerwalls of the piezoelectric tube and does not flow into thewell fluid. The TEM field in the well fluid is generated by thevariable magnetic field. This leads to the conductive currentpropagating along the radial r-direction and axial z-direction,thus we have∇ ⋅

E = 0. Following the basic principle of fieldtheory (Pollack & Stump 2005), we write

E as:

E = ∇ ×

A, (1)

where

A is the vector potential. It describes the TEM fieldexcited by the piezoelectric tube.

According to Maxwell’s equation, ∇ ×

H = 𝜎

E + 𝜀𝜕E𝜕t.

For a sinusoidal time dependence ei𝜔t

∇ ×

H = (𝜎 + i𝜀𝜔)

E = (𝜎 + i𝜀𝜔) ∇ ×

A

= ∇ ×[(𝜎 + i𝜀𝜔)

A]. (2)

From equation (2), we have:

H = (𝜎 + i𝜀𝜔)

A. (3)

Due to the axisymmetry of the electromagnetic field, wedefine

A ashavingonly a single component in the𝜃-direction,i.e.

A = A𝜃

u𝜃 , to describe the TEM field, where u𝜃 is theunit vector in the𝜃-direction.Thus, themagnetic field causedby the conduction current has only one component in the𝜃-direction, i.e.

H𝜃 = (𝜎 + i𝜀𝜔) A𝜃. (4)

By substituting

A = A𝜃

u𝜃 into equation (1), we obtain:

E = ∇ ×

A = −𝜕A𝜃

𝜕zer +

(𝜕A𝜃

𝜕r+

A𝜃

r

)ez. (5)

Then

Er = −𝜕A𝜃

𝜕z, (6)

Ez =𝜕A𝜃

𝜕r+

A𝜃

r. (7)

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Following previous studies (Song 2012; Wu et al. 2017),the electric potential of the well liquid can be expressed as:

Φ𝜃 (r, z) = ∫∞

−∞ ∫∞

−∞[V(𝜔)][B(kz,𝜔)K1(l2r)

+C(kz,𝜔)I1(l2r)]ei(kzz−𝜔t)dkzd𝜔, (8)

where l22 = kz2 − 𝛾22, 𝛾2 = i𝜔𝜇(𝜎2 + i𝜀2𝜔), 𝜎2 and 𝜀2 are

the conductivity and dielectric constant of the well liquid,respectively. V(𝜔) is the spectrum of the excitation source,B(kz , 𝜔) and C(kz , 𝜔) are generalised reflection coeffi-cients related to resistivity and permeability of the formationsurrounding the wellbore. The first term of the integrand inΦ𝜃(r, z) represents the outward propagating directive wavesexcited by the piezoelectric tube; its second term representsall the reflected waves, which is also referred to as a gener-alised reflection wave due to the interfaces of the media.

The electric vector potential of the formation outside theopen-hole can be expressed as (Song 2012;Wu et al. 2017):

Ψ𝜃 (r, z) = ∫∞

−∞ ∫∞

−∞[V(𝜔)][D(kz,𝜔)K1(l3r)]e

i(kzz−𝜔t)dkzd𝜔, (9)

where l32 = kz2 − 𝛾32, 𝛾3 = i𝜔𝜇3(𝜎3 + i𝜀3𝜔) and 𝜎3 and 𝜀3

are the conductivity and dielectric constant of the formation,respectively.

Under the tangentially continuous boundary conditions(Ez2 = Ez3 ,H𝜃2 = H𝜃3) and the normal continuous bound-ary condition (𝜀1Er1 = 𝜀2Er2), the coefficients B(kz, 𝜔),C(kz, 𝜔) andD(kz, 𝜔) can be obtained by solving the equa-tions. Thus, the integral solutions of TEM field componentsinside and outside the wellbore can be obtained. By solvingthe double integration in equations (8) and (9) using thereal-axis integration method (Sheng et al. 2019), the TEMfield components at different times and source-receiver spac-ings are acquired.

In our calculation, the corresponding excitationwaveform

spectrum V(𝜔) is the Gauss function e−(f−f0)2

p2 , whose mainfrequency is 25 kHz (see figure 4). The Gauss coefficient isp= 5.

4. Theoretical calculation results

Based on the theoretical calculationmethodmentioned Sec-tion 3, the TEM field components excited by the piezoelec-tric tube in open-hole are calculated (with given calcula-tion parameters) and their characteristics are studied. Therelationship between TEM field signal and formation resis-tivity is explored by changing source-receiver spacing andformation resistivity.

Figure 4. Spectrum of excitation signal used in theoretical research.

4.1. TEM field components H𝜽, Er and Ez excited by

piezoelectric tube in open-hole

Figure5adisplays the calculatedTEMfield components (H𝜃 ,Er and Ez) in the open-hole at 1.2 m source-receiver spac-ing. The calculation parameters are: 𝜎2 = 1 S m−1, 𝜀2 = 1,𝜇2 = 1; 𝜎3 = 5 S m−1, 𝜀3 = 1 and 𝜇3 = 1. Despite the sim-ilar parameters, our results reveal that the amplitudes andshapes of Ez, Er and H𝜃 waveforms are obviously different.Figure 5 parts b, c and d are the waveforms of H𝜃 , Er andEz at different source-receiver spacings. It can be seen thatwith an increase of source-receiver spacing, the amplitudesof H𝜃 , Er and Ez gradually decrease, in which the shapes ofH𝜃 and Ez remain unchanged, while the shape of Er changesconsiderably. Notably, the waveforms shift upward succes-sively to a certain offset with the increase of source-receiverspacing for the convenience of observation (see figure 5b,c and c).

4.2. Relationship between TEM field components (H𝜽, Er

and Ez) and formation resistivity

To study the relationship between the TEM field compo-nents and formation resistivity, we calculate the TEM fieldcomponents using different formation resistivity values. Thewaveforms of H𝜃 , Er and Ez calculated using different for-mation resistivity values at a 1.2 m source-receiver spacingare shown in figure 6a, b and c, respectively; while the wave-forms of Ez at a 2.0 m source-receiver spacing are presentedin figure 6d. In figure 6, the corresponding resistivity val-ues in descending order are 15, 14, 13, 12, … , 2, 1, 1/2,1/3, 1/4, … , and 1/25Ωm−1, respectively. From figure 6aand b, it is obvious that the waveforms of H𝜃 and Er barelychange with the formation resistivity. From figure 6c and d,the waveform of Ez changes obviously with formation resis-tivity. The larger the formation resistivity value, the larger the

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Figure 5. The waveforms of TEM field components (H𝜃 , Er and Ez) at different source-receiver spacings.

amplitude. Therefore, the relationship between Ez and for-mation resistivity is further explored to investigate the feasi-bility of extracting formation resistivity information from Ez.

Figure 7 illustrates the relationship between formationresistivity and the maximum value of Ez. In figure 7a, thesource-receiver spacing is 1.2 m where an obvious linear re-lationship is demonstrated; in figure 7b, the source-receiverspacing is 2.0 m where an approximately linear relationshipis demonstrated.

5. Analysis and discussion

The monopole acoustic probe commonly used in acousticlogging can generate acoustic vibration and excite the TEMfield in the surrounding spacewhen thepulse signal is excited.The TEM signal received by the acoustic receiving probe isthe transient electric field signal, which is excited by themag-netic field coupling in the borehole. It is different from the

direct current response where the current flows directly intothe fluid and formation.

There are two propagation modes of the TEM field in anopen-hole, which are caused by the displacement current andconduction current, respectively. These two modes can bedescribed using the Helmholtz equation of electric field

Eand magnetic field

H expressed in equations (10) and (11):

∇2

E + (𝜇𝜀𝜔2 − i𝜇𝜎𝜔)

E = 0, (10)

and

∇2

H + (𝜇𝜀𝜔2 − i𝜇𝜎𝜔)

H = 0. (11)

The propagation velocity of the first mode, which iscaused by displacement current, is 1∕

√𝜀𝜇. It is the propa-

gation velocity of the electromagnetic wave in the medium.The velocity of the second mode, which is caused by the

conduction current, is 2√

f𝜋𝜎𝜇. If the conductivity of the

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Figure 6. Variation of waveforms of TEM field components at different formation resistivity values: (a)H𝜃 at 1.2 m source-receiver spacing, (b) Er at1.2m source-receiver spacing, (c)Ez at 1.2m source-receiver spacing and (d)Ez at 2.0m source-receiver spacing (from bottom to top, the correspondingresistivity values are 15, 14, 13, 12,… , 2, 1, 1/2, 1/3,… , and 1/25Ωm−1, respectively.).

media is 1 S m−1, the propagation velocity is 2√

f𝜋𝜎𝜇

=

2√

𝜋25 ×103

1 × 4𝜋×10−7= 5 × 105 m s−1. The superposition of

these two propagation modes forms the TEM signals re-ceived in the open-hole acoustic logging data. Since thevelocities of these two modes are far beyond the acoustic ve-locity in thewell liquid (about1.5×103 mS−1), theTEMsig-nal appears at the front part of the received waveform, com-pletely separated from the acoustic signals.

The results of TEM field components show that the am-plitude of Ez is sensitive to the change of formation resis-tivity, while H𝜃 and Er are insensitive. This phenomenoncould be explained by using the boundary conditions ofelectromagnetic fields at dielectric surfaces. Since the op-erating frequency in acoustic logging is within the fre-quency range of about 20 kHz, the TEM field componentscalculated here aremainly caused by the conduction current.

In the well liquid, the electric field intensity

E can be de-composed into tangential component Ez and normal com-ponent Er, where Er plays the dominant role. Following

j= 𝜎

E = 𝜎(Ezuz + Er

ur), the conduction current mainlyconcentrates in the r-direction and is incident to the forma-tion obliquely at the wellbore. The resistivities of the forma-tion and well fluid are different, so the direction of the cur-rent is changed when it enters the formation. If 𝜎2 < 𝜎3, theamplitude of Er decreases, while Ez remains unchanged, the

E in the formation will be more inclined to the z-axis. Thus,the current flowing along the z-axis in the formation is largerthan that in the well fluid. If 𝜎2 > 𝜎3, the amplitude of Er in-creases, Ez remains unchanged, the

E in the formation willbe more inclined to the r-axis. In both cases, the tangentialcomponents of the

Eonboth sides of thewellbore are contin-uous (i.e.Ez2 = Ez3). If the formation remains unchanged on

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Figure 7. Relationship between the formation resistivity and the maximum value of Ez for: (a) 1.2 m source-receiver spacing and (b) 2.0 m source-receiver spacing.

Figure 8. Schematic diagramof horizontal formationwith the parameters𝜎4, 𝜀4, 𝜇4.

the z-axis, the induced electromotive force (EMF) obtained,which is the integration of Ez2 along the z-direction, remainsunchanged at short source-receiver spacing. Nonetheless, ifthe formation changes, for the simplest case, the current flow-ing along the z-direction frommedium 3 to medium 4 (as il-lustrated in figure 8), thingswill be different.On the interfacebetween media 3 and 4, the current density is continuous inthe normal direction (i.e. Jz3 = Jz4, we have 𝜎3 Ez3 = 𝜎4Ez4,

Ez3∕𝜌3 = Ez4∕𝜌4, Ez4 = (Ez3∕𝜌3)𝜌4, Ez2 = Ez4), indicatingthat the electric field intensity Ez2 in the well fluid is pro-portional to the resistivity of medium 4. The induced EMFobtained along the z-direction is directly affected by the re-sistivity of the medium 4 when it is at the short source-receiver spacing, causing the amplitude change of the TEMsignal (shown in figure 2c). However, with an increase ofsource-receiver spacing, the linear relationship between Ezand formation resistivity weakens due to other influencefactors.

As the acoustic receiving probes are arranged along thez-direction in the open-hole, the received EMF is directly re-lated to Ez and the formation resistivity. Therefore, when theformation resistivity changes, the amplitude of the electro-magnetic induction signals arrive before the acoustic signalwill change significantly. It should be noted that the resultsshown in figure 2 are measured using a digital acoustic log-ging tool, whose mechanical parts and the shell are all metal.The signals are transmitted through a shielded line and thelogging tool is grounded at multiple locations. Hence, theelectromagnetic signal has been attenuated severely. Never-theless, we can see that the electromagnetic induction signalobviously changes with the formation. It can be inferred thatthe actual electromagnetic induction signal is relatively large.

While the frequency band of the electromagnetic field ex-cited by the acoustic probe is similar to that of inductionlogging, the electromagnetic field excited by the acousticprobe is perpendicular to that excited by the transmitter coilused in induction logging (because of their different excita-tionmodes). Ourmethodmeasures the formation resistivityalong the z-direction, and induction loggingmeasures forma-tion resistivity along the circumferential direction. The EMF

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of induction logging is a circular closed eddy current, whilein our method, the measured EMF is along the z-direction.

6. Conclusion

By analysing the acoustic logging data, we found that: (i) inacoustic logging, a monopole probe can generate a relativelystrong TEM field under the excitation of a periodic pulsesignal; (ii) in the response waveform of acoustic logging,the TEM response signals, which arrive before the acous-tic response signal can be clearly observed and (iii) if theformation changes, the amplitude of TEM response signalwill change accordingly. Based on these findings, this papercalculated the TEM field components (H𝜃 , Er and Ez) in theopen-hole and studied the relationship between the TEMfield components and formation resistivity. The results re-vealed that Ez is most closely related to formation resistivityand its amplitude is approximately proportional to formationresistivity at short source-receiver spacings. In this sense, theformation resistivity can be obtained by measuring the EMFreceived along the z-direction. On this basis, we proposed anewmethod for measuring formation resistivity in the open-hole, which is different from induction logging and laterallogging. The formation resistivity measured by this methodis along the z-direction. It is a new method of formationresistivity measurement, which is different from the conven-tional logging methods (e.g. induction logging methods andresistivity logging methods). Furthermore, it can measureimportant information about the formation obtained fromacoustic signals and formation resistivity simultaneouslyusing one instrument. In this way, it should benefit the com-prehensive detection of formation characteristics, and havea great significance in improving instrument integration.

Acknowledgement

This research was supported by the National Key Research andDevelopment Program of China (no. 2016YFC0802008).

Conflict of interest statement. None declared.

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