Simulation of Monopole and Multipole Seismoelectric Logging

Abstract

In a fluid-saturated porous formation, acoustics and electromagnetic waves are coupled based on Pride seismoelectric theory. An exact treatment of the nonaxisymmetric seismoelectric field excited by acoustic multipole sources is presented. The frequency wavenumber domain representations of the acoustic field and associated seismoelectric field due to acoustic multipole sources are formulated. The full waveforms of acoustic waves and electric and magnetic fields in the time domain propagation in borehole are simulated by using discrete wave number integration, and frequency versus axial-wave number responses are presented and analyzed.

1. Introduction

The study of wave propagation in a fluid-saturated porous medium is of considerable interest in acoustics and geophysics due to its important applications in various technical and engineering processes. The investigation of wave propagation in fluid-saturated porous media was early developed by Biot [1, 2]. One of the major findings of Biot’s theory was that there is a compressional slow wave in a fluid-saturated porous medium. The first clear experimental observation of this slow wave was reported by Plona [3]. Biot predicted the slow waves should have an important bearing on electrokinetic effect [4]. This predication has been quantitatively confirmed by Pride [5] and Hu [6]. Elastic waves propagating in fluid-saturated porous media generate a movement of the ions in the pore fluid. Such movement induces an electromagnetic (EM) field. Thompson and Gist [7] have made field measurement clearly demonstrating that seismic waves can induce electromagnetic disturbances in saturated sediments. Pride [8] derived the governing equations for the coupled acoustics and electromagnetic waves in fluid-saturated porous media. Pride and Haartsen [9] analyzed the basic properties of seismoelectric waves.

The electric field induced by elastic waves is weak and attenuates in propagation. In order to detect the seismoelectric signal effectively, Haartsen and Pride [10] suggested measuring vertical electroseismic profile. Mikhailov et al. [11] measured the electric field converted from low frequency Stoneley waves in a borehole and made theoretical analysis. Seismoelectric logging method has been proposed to detect deep target formation. The advantage of seismoelectric logging is the distance both from transmitter to target formation and from the transmitter to the receiver is small, and signals can be received with relative high amplitude. Zhu et al. [12–14] made laboratory experiments and observed the seismoelectric conversion in model wells, and their experimental results confirm that seismoelectric logging could be a new borehole logging technique. Seismoelectric logging methods based on the excitation and reception of axisymmetric wave phenomena in a fluid-filled borehole embedded in permeable formations have been investigated [6, 15]. In the present paper, we consider the nonaxisymmetric seismoelectric field excited by acoustic multipole sources based on the entire Pride seismoelectric theory [9]. We theoretically formulate and numerically simulate both axisymmetric and nonaxially symmetric waves in a fluid-filled borehole surrounded by a homogeneous fluid-saturated porous formation. The frequency versus axial-wave number responses are presented, and time-domain transient waveforms of acoustic waves and electromagnetic waves propagation in borehole are simulated and compared. The current model may be an important tool in understanding and predicting borehole seismoelectric phenomena. This paper may be of interest for researchers in the field referred to above as well as experimental wave propagation in fluid-saturated porous media.

2. Theoretical Formulations

In this section we formulate a frequency wavenumber integral representation for the acoustic, electric, and magnetic fields excited by a multipole acoustic source in the borehole fluid. Consider a fluid-filled borehole of radius 𝑎 embedded in linear isotropic fluid-saturated porous medium and acoustic multipole sources in the borehole. In the borehole, the acoustic and electromagnetic fields are not coupled, but when the acoustic source in the borehole is excited, according to Pride’s theory [8, 9], the acoustic waves in porous formation will generate accompanying electric and magnetic fields, and acoustic wave induces radiating EM wave at borehole wall (discontinuous interface). We can receive electric or magnetic fields signal inside the borehole due to the boundary condition at the borehole wall.

2.1. The Acoustic Field in the Borehole Fluid

We consider acoustic multipole sources inside the borehole. The Dipole source is constructed from two point sources of opposite sign placed close together in the same horizontal plane. Equivalently, the dipole can be viewed as a point force oriented horizontally and pointed at an azimuthal angle of 𝜃₀. Similarly, a quadrupole source can be viewed as two closely spaced dipole sources pointing in opposite directions. Higher-order multipoles have similar interpretations. Following Kurkjian and Chang [16], with emphasizing the radius of a multipole source being very small and |𝜂_𝑓𝑟₀|≪1, the representation in the frequency-wavenumber domain of the displacement potential associated with 𝑛 acoustic multipole sources (𝑟>𝑟₀) is given byΦ₀𝑟,𝜃,𝑘_𝑧,𝜔=𝛾𝜀_𝑛𝐾_𝑛𝜂_𝑓𝑟cos𝑛𝜃−𝜃₀,(1) where 𝛾=𝑉₀(𝜔)(𝜂_𝑓𝑟₀/2)^𝑛/𝑛!,  𝜀_𝑛 is Neumann’s factor (𝜀_𝑛=1 for 𝑛=0; 𝜀_𝑛=2 for 𝑛=otherwise), 𝑉₀(𝜔) is the source spectrum, and 𝜂^2𝑓=𝑘^2𝑧−𝜔²/𝑐^2𝑓 is the radial wavenumber with respect to the acoustic velocity of bore fluid 𝑐_𝑓.𝐾_𝑛(𝑥) is the 𝑛th order modified Bessel function of the second kind.

The total acoustic wave field in the borehole is the sum of the source contribution and the reflected acoustic field. The radiation conditions prescribe that the reflected field should be finite at the axis of the borehole, thus the reflected waves displacement potential associated with 𝑛 acoustic multipole sources isΦ₁𝑟,𝜃,𝑘_𝑧,𝜔=𝛾𝐴_𝑛𝐼_𝑛𝜂_𝑓𝑟cos𝑛𝜃−𝜃₀,(2) where 𝐼_𝑛(𝑥) is the 𝑛th-order modified Bessel function of the second kind. Thus, the total field inside the borehole in the frequnecy-wavenumber domain is given byΦ𝑟,𝜃,𝑘_𝑧,𝜔=𝛾𝐴_𝑛𝐼_𝑛𝜂_𝑓𝑟+𝜀_𝑛𝐾_𝑛𝜂_𝑓𝑟cos𝑛𝜃−𝜃₀,(3) the reflection coefficients 𝐴_𝑛 can be derived by the boundary conditions at the borehole wall. The radial component of the displacement field in the borehole fluid is given by 𝑢_𝑟=𝜕Φ/𝜕𝑟 and the radial component of normal stress in the borehole fluid is equal to the negative of borehole fluid pressure, that is, 𝜏_𝑟𝑟=−𝑃𝑟,𝜃,𝑘_𝑧,𝜔=−𝜌_𝑓𝜔²Φ.(4)

2.2. The Electromagnetic Field in the Borehole Fluid

In cylindrical coordinate system (𝑟,𝜃,𝑧), it is convenient for us to take the z-component of vector 𝐄 and 𝐇, and they satisfy the Homheltz-type wave equations without electric current sources in the borehole,∇²𝐸_𝑧+𝑘^2𝑒𝐸_𝑧=0,(5)∇²𝐻_𝑧+𝑘^2𝑒𝐻_𝑧=0,(6) where 𝑘^2𝑒=𝜔²𝜇𝜀_𝑏,𝜀_𝑏=𝜀_𝑓(1+𝑖𝜎_𝑓/𝜔𝜀_𝑓),𝜀_𝑓 is the borehole fluid’s electrical permittivity, 𝜎_𝑓 is the borehole fluid’s electrical conductivity, and𝜇 is the magnetic permeability. For really fluid and all porous media of interest it can be assumed that 𝜇=𝜇₀. In borehole fluid and fluid-saturated porous media we take 𝜇=𝜇₀ throughout the paper.

Considering the radiation conditions, that prescribe the reflected electromagnetic field should be finite at the axis of the borehole, we can take the solutions for 𝐸_𝑧 and 𝐻_𝑧 as 𝐸_𝑧𝑟,𝜃,𝑘_𝑧,𝜔=𝛾𝐴^𝐸𝑛𝐼_𝑛𝜂_𝑓𝑟cos𝑛𝜃−𝜃₀,(7)𝐻_𝑧𝑟,𝜃,𝑘_𝑧,𝜔=𝛾𝐴^𝐻𝑛𝐼_𝑛𝜂_𝑓𝑟sin𝑛𝜃−𝜃₀,(8) where 𝜂^2𝑒=𝑘^2𝑧−𝑘^2𝑒. As we will see, in order to satisfy the boundary conditions of continuity of tangential electromagnetic fields at borehole wall 𝑟=𝑎, the solutions in general require a combination of both TE and TM fields. The modes with 𝐸_𝑧 and 𝐻_𝑧 components present are hybrid modes. Transverse field components are obtained from Maxwell’s equations, 𝐸_𝜃=𝑖𝐻𝜂^2𝑒𝜔𝜇𝜕𝜕𝐻𝑟^𝑧−𝑘𝑟^𝑧𝐸𝜕𝜃^𝑧,(9)_𝜃=−𝑖𝜂^2𝑒𝜔𝜀^𝑏𝜕𝐸_𝑧𝐸𝜕𝑟+𝑘𝑟^𝑧𝜕𝜕𝐻𝜃^𝑧,(10)_𝑟=−𝑖𝜂^2𝑒𝜔𝑟𝜇𝜕𝜕𝐻𝜃^𝑧+𝑘^𝑧𝜕𝐸_𝑧𝐻𝜕𝑟,(11)_𝑟=𝑖𝜂^2𝑒𝜔𝑟𝜀^𝑏𝜕𝜕𝐸𝜃^𝑧−𝑘^𝑧𝜕𝐻_𝑧𝜕𝑟.(12)

Reflected electromagnetic field inside the borehole in the time-space domain is given by Π(=𝑟,𝜃1,𝑧,𝑡)2𝜋^∞∞Π𝑟,𝜃,𝑘_𝑧,𝜔exp𝑖𝑘_𝑧𝑧−𝜔𝑡𝑑𝑘_𝑧𝑑𝜔;(13) here we use Π to denote any component of electric or magnetic field given by (7)–(12). The reflection coefficients 𝐴^𝐸𝑛, 𝐴^𝐻𝑛 can be derived by the boundary conditions at the borehole wall.

2.3. Seismoelectric Waves in the Porous Formation

For the problem of modeling the propagation of coupled electromagnetic and mechanical disturbances in an isotropic-porous material, Pride [8] has derived equations that control such “seismoelectric” phenomena. According to Pride’s theory, without the applied force and electric current sources existing, and assuming an time dependence exp(−𝑖𝑤𝑡), we may write the equations for the coupled electromagnetic and acoustics in macroscopically homogenous, isotropic, fluid saturated porous media as follows𝝉=𝐻∇⋅−𝝉2=𝐺_𝑏−𝜔²𝜌𝐮+𝜌_𝑓𝐰,(14)+𝐺_𝑏(∇⋅𝐮)𝐈+𝐶(∇⋅𝐰)𝐈−𝑝_𝑓∇𝐮+∇𝐮^𝑇,(15)−𝑖𝜔𝐰=−=∇𝑝𝐶_𝑓(∇⋅𝐮)+𝑀(∇⋅𝐰),(16)+𝜌_𝑓𝜔²𝐮𝑘(𝜔)𝐉=−∇𝑝+𝑝_𝑓𝜂+𝐿(𝜔)𝐄,(17)∇×∇𝐇×𝜔𝐄=²=𝐮−𝑖𝑖𝜔𝜔𝐿𝜀𝜇𝐄(𝐇𝜔+,)+𝐉,𝜎(𝜔)𝐄,(((112890)))

Those governing equations are the Biot equations for porous media acoustic along with the Maxwell equations for the electric and magnetic fields 𝐄 and 𝐇. Here 𝝉 is the bulk stress in the porous medium, 𝑝_𝑓 is the pressure in the pore fluid, 𝐮 is the displacement in the solid, and 𝐰 is the relative fliud-solid motion. The symbol 𝜌 denotes the bulk density of the porous medium, 𝜌=(1−𝜙)𝜌_𝑠+𝜙𝜌_𝑓, 𝜙is the porosity of the medium, 𝜌_𝑠 is the solid density, and 𝜌_𝑓 denotes the fliud density. Equations (17) and (18) are in the form of Darcy law and Ohm law, through which acoustic and electromagnetic fields are coupled, where 𝐉 is the electric-current density and −𝑖𝜔𝐰 is the Darcy filtration velocity. Where 𝐻, 𝐶, 𝑀, and 𝐺_𝑏 are four moduli of isotropic porous media. As the most important coefficient is set to zero, Pride’s equations will be separated into Biot’s equations for elastic field and Maxwell equations for electromagnetic field. Here 𝜎(𝜔) is the frequency-dependent electrical conductivity of the medium, 𝑘(𝜔) is the dynamic permeability, 𝜂 is the fluid dynamic viscosity, and 𝐿(𝜔) is the frequency-dependent electrokinetic coupling coefficient.The expressions for 𝜎(𝜔), 𝑘(𝜔), and 𝐿(𝜔) are given in Pride [8].

The solution to the coupled equations of seismoelectric waves motions can be obtained by separating the basic field 𝐮, 𝐰, 𝐄 into its compressional, and vertically and horizontally polarized shear components, and written in terms of scalar potential functions as𝐮=∇Φ_𝑝𝑓+Φ_𝑝𝑠+∇×Ψ_𝑠ℎ+Ψ_𝑒𝑚𝐞_𝑧𝐰=+∇∇𝛼×_𝑝𝑓∇×Γ_𝑠ℎ+Γ_𝑒𝑚𝐞_𝑧,(21)Φ_𝑝𝑓+𝛼_𝑝𝑠Φ_𝑝𝑠+∇×𝛼_𝑠ℎΨ_𝑠ℎ+𝛼_𝑒𝑚Ψ_𝑒𝑚𝐞_𝑧𝐄=+∇∇𝛽×_𝑝𝑓∇×𝛼_𝑠ℎΓ_𝑠ℎ+𝛼_𝑒𝑚Γ_𝑒𝑚𝐞_𝑧,(22)Φ_𝑝𝑓+𝛽_𝑝𝑠Φ_𝑝𝑠+∇×𝛽_𝑠ℎΨ_𝑠ℎ+𝛽_𝑒𝑚Ψ_𝑒𝑚𝐞_𝑧+∇×∇×𝛽_𝑠ℎΓ_𝑠ℎ+𝛽_𝑒𝑚Γ_𝑒𝑚𝐞_𝑧,(23) where Φ_𝑗 is the compressional wave potential, 𝐞_𝑧 is the unit vector in the 𝑧 direction, Ψ_𝑗 is the horizontal polarized shear potential, and Γ_𝑗 is vertically polarized shear wave potential. Each of these potentials satisfy Helmholtz-type wave equation. The factors 𝛼_𝑝𝑓, 𝛼_𝑝𝑠, 𝛼_𝑠ℎ, 𝛼_𝑒𝑚, 𝛽_𝑝𝑓, 𝛽_𝑝𝑠, 𝛽_𝑠ℎ, and 𝛽_𝑒𝑚 as well as the formulas of the body wave velocities can be found in [9, 15, 17].

Outside the borehole, because the radiation conditions prescribe no incoming wave existing at the position where 𝑟 tends to infinity, the solutions in the frequency-axial wavenumber with respect to an 𝑛th-order multipole source are, similar to only acoustic fields case [18, 19], Φ_𝑗𝑟,𝜃,𝑘_𝑧,𝜔=𝛾𝐵^𝑗𝑛𝐾_𝑛𝜂_𝑗𝑟cos𝑛𝜃−𝜃₀,(24)Ψ_𝑗𝑟,𝜃,𝑘_𝑧,𝜔=𝛾𝐶^𝑗𝑛𝐾_𝑛𝜂_𝑗𝑟cos𝑛𝜃−𝜃₀,(25)Γ_𝑗𝑟,𝜃,𝑘_𝑧,𝜔=𝛾𝐷^𝑗𝑛𝐾_𝑛𝜂_𝑗𝑟cos𝑛𝜃−𝜃₀,(26) where𝜂^2𝑗=𝑘^2𝑧−𝑘^2𝑗(𝑘^2𝑗=𝜔²𝑠^2𝑗) are the radical wavenumbers of the seismoelectric waves whose complex slowness 𝑠_𝑗(𝑗=𝑝𝑓,𝑝𝑠,𝑠ℎ,𝑒𝑚) expressions can be found in [9, 15, 17]. Here, 𝐵^𝑗𝑛, 𝐶^𝑗𝑛, and 𝐷^𝑗𝑛 which characterize the outgoing seismoelectric waves, are functions of the axial wavemumber 𝑘_𝑧 and the angular frequency 𝜔. Once 𝐮, 𝐰, 𝐄 are known, all other quantities such as stress tensor, pore fluid pressure, and magnetic vector can be deduced by (15)-(16) and (19). The components of seisoelectric filed in the formation can be written in terms of the potentials using (23) and (26). The detailed expressions of those formulations can be found in [17].

2.4. Boundary Conditions and Resolution

The unknown coefficients 𝐴_𝑛, 𝐴^𝐸𝑛, 𝐴^𝐻𝑛, 𝐵^𝑛𝑝𝑓, 𝐵^𝑛𝑝𝑠, 𝐶^𝑛𝑠ℎ, 𝐶^𝑛𝑒𝑚, 𝐷^𝑛𝑠ℎ, and𝐷^𝑛𝑒𝑚 which appear in the expressions for the field potentials are determined by the boundary conditions. The boundary conditions of seismoelectric waves for a cylindrical interface (at 𝑟=𝑎) between a fluid and a fluid-saturated porous media for the case of permeable interface are𝑢^𝐼𝑟=𝑢^𝑟𝐼𝐼+0𝑤=^𝑟𝐼𝐼,𝜏_{𝐼𝐼𝑟𝜃}−𝑃^𝐼=𝜏^{𝐼𝐼𝑟𝑟},0=𝜏^{𝐼𝐼𝑟𝑧};𝐸_𝐼𝜃,𝑃^𝐼=𝑝^𝑓𝐼𝐼;=𝐸^𝜃𝐼𝐼,𝐸^𝐼𝑧=𝐸^𝑧𝐼𝐼,𝐻^𝐼𝜃=𝐻^𝜃𝐼𝐼,𝐻^𝐼𝑧=𝐻^𝑧𝐼𝐼,(27) where the superscript 𝐼 and 𝐼𝐼 represent the media inside and outside the borehole, respectively. In the present paper, we do not consider the possibility of a sealed or partially sealed boundary condition. With the expressions of 𝑤_𝑟, 𝑢_𝑟, 𝜏_𝑟𝑟, 𝜏_𝑟𝑧, 𝜏_𝑟𝜃, 𝐸_𝜃,𝐸_𝑧, 𝐻_𝜃, 𝐻_𝑧 deduced from (14)–(26), (7) –(10), and 𝑢^𝐼𝑟=𝜕(Φ₀+Φ₁)/𝜕𝑟, 𝑃^𝐼=𝜌_𝑓𝜔²(Φ₀+Φ₁) boundaries (27) may be written in a matrix form as𝑀𝐴=𝐵,(28) where 𝑀 are 9×9 complex matrices whose elements are given in the [17] and 𝐴=𝐴_𝑛,𝐵^𝑛𝑝𝑓,𝐵^𝑛𝑝𝑠,𝐶^𝑛𝑠ℎ,𝐷^𝑛𝑠ℎ,𝐶^𝑛𝑒𝑚,𝐷^𝑛𝑒𝑚,𝐴^𝐸𝑛,𝐴^𝐻𝑛^𝑇,(29)𝐵=𝑏₁,𝑏₂,0,0,𝑏₅,0,0,0,0^𝑇,(30) where 𝑏₁=𝜀_𝑛[𝜂_𝑓𝐾_𝑛+1(𝜂_𝑓𝑎)−(𝑛/𝑎)𝐾_𝑛(𝜂_𝑓𝑎)], 𝑏₂=𝑏₅=−𝜌_𝑓𝜔²𝜀_𝑛𝐾_𝑛(𝜂_𝑓𝑎).

3. Simulations

In this section, we present synthetic acoustic waves and electromagnetic field in the borehole excited by acoustic monopole, dipole, and quadrupole sources. The formation in our numerical example is permeable fluid saturated porous media and the medium input parameters describing the porous formation are given in Table 1. The borehole radius 𝑎 is 0.1 m. The same medium parameters and source function as in [20]. The source is a cosine pulse [19]. If set acoustic transducer, electrode, and magnetometer in the borehole fluid, one could measure acoustic, electric and magnetic fields, respectively. For acoustic logging, only the axisymmetric modes contribute to the pressure on the axis [21]. However, for electroacoustic logging, beside axisymmetric modes, dipole modes of nonaxisymmetric modes also contribute to the electric and magnetic field on the axis [20, 22]. For example, we only select some position (0.05, 0, 𝑧) off the axis to record the full waveforms. The spectral response is calculated similar to [23].

Table 1 

Medium properties used at input for numerical calculations.

3.1. Fast Formation

Figure 1 shows spectral response and synthetic waveforms of acoustic waves and electric field for fast formation generated by an acoustic monopole source. In our calculation, the source is set to be pressure source, and the center frequency is 6 kHz. The pressure amplitude is 10 MPa measured at 1 cm away from the source center. The numerical result of borehole pressure waveforms is on the order of those by Kurkjian [16]. From Figures 1(b) and 1(d), we can find there are four wave groups in electric field 𝐸_𝑧 that is different from three in acoustic waveforms. A clear seismoelectric response A-A group for the electric field is relatively weak but visible in full waveforms, which were recorded at all receivers simultaneously. And the following groups for the electric field having apparent velocities are comparable to those of borehole acoustic waves. Because of the nonequilibrium of free charges in the pore fluid at the borehole wall, that is, at the interface, the acoustic wave generates a propagating electromagnetic wave (EM wave) [9]. The almost same arrival time of small wave components in full waveforms indicates that they propagate with higher speed than that of any acoustic waves.

(a)

(b)

(c)

(d)

Figure 1 

Spectral representation and time response of borehole pressure and electric fields generated by monopole source. (a) and (c) are spectral of pressure and electric fields, respectively. (b) and (d) are time responses of pressure and electric fields, respectively.

In the spectral representation, the response allows the identification of the pesudo-Rayleigh wave (pR.1), which has an about cut-off frequency of 7 kHz. The frequency of the excited source pulse is enough to reveal this wave mode. The excitation is stronger at low frequency for Stoneley wave without cut-off frequency. Spectral representation of electric field (Figure 1(c)) 𝐸_𝑧 shows that the borehole seismoelectric conversion dominates at low frequency. Figures 1(c) and 1(d) shows that P wave response may be identified easily in both spectral representation and waveforms of electric field. In this case the seismoelectric conversion efficiency of P wave is biggest, Stoneley wave is second.

Figure 2 shows spectral response and synthetic waveforms of acoustic waves and magnetic field 𝐻_𝑟 for fast formation generated by an acoustic dipole source. In the spectral representations of borehole pressure (Figure 2(a)), the two flexual modes obviously appear in the frequency range 0–20 kHz; however, in the spectral representations of magnetic field, the lowest mode is dominating. From Figures 2(b) and 2(d), the waveforms of borehole pressure with 3 kHz center frequency dominates by flexural guided waves; however, that of magnetic field has two groups with different velocity. Again, from Figure 2(d) we see a relatively weak but visible A-A group; it arrives simultaneously on all receivers. And the following groups having apparent velocities of acoustic flexural waves velocities of borehole acoustic waves.

(a)

(b)

(c)

(d)

Figure 2 

Spectral representation and time response of borehole pressure and magnetic fields for fast formation generated by dipole source. (a) and (c) are spectral of pressure and magnetic field respectively. (b) and (d) are time responses of pressure and magnetic field respectively.

Figure 3 shows spectral response and synthetic waveforms of acoustic waves and magnetic field 𝐻_𝑟 for fast formation generated by an acoustic quadrupole source. In the spectral representations of borehole pressure (Figure 3(a)), the two screw modes obviously appear in the range of 0–20 kHz; however, in the spectral representations of magnetic field, the lowest mode is dominating. From Figures 3(b) and 3(d), the waveforms of acoustic waves quadrupole sources with the center frequency 5 kHz mainly dominated by screw waves and that of magnetic field consists of groups having apparent velocities of acoustic screw waves. And we can find almost no EM waves in magnetic field full waveforms.

(a)

(b)

(c)

(d)

Figure 3 

Spectral representation and time response of borehole pressure and magnetic fields for fast formation generated by quadrupole source. (a) and (c) are spectral of pressure and magnetic fields respectively. (b) and (d) are time responses of pressure and magnetic fields respectively.

3.2. Slow Formation

Figure 4 shows spectral response and synthetic waveforms for acoustic waves and electric field for slow formation generated by an acoustic monopole source. From Figures 4(b) and 4(d), we can find there are three wave groups in electric field 𝐸_𝑧 different from two in acoustic waveforms. A clear seismoelectric response A-A group for the electric field 𝐸_𝑧 is relatively weak but visible in full waveforms, which were recorded at all receivers simultaneously. And the following groups for the electric field having apparent velocities are comparable to those of borehole acoustic waves.

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Figure 4 

Spectral representation and time response of borehole pressure and electric fields for slow formation generated by monopole source. (a) and (c) are spectral of pressure and electric fields, respectively. (b) and (d) are time responses of pressure and electric fields, respectively.

In the spectral representation, the pesudo-Rayleigh wave disappears and the response allows the identification of P wave and Stoneley wave. The excitation is stronger at low frequency for Stoneley wave for slow formation same as fast formation. Spectral representation of electric field (Figure 4(c)) 𝐸_𝑧 shows that the borehole seismoelectric conversion dominates at low frequency. Figures 4(c) and 4(d) shows that P wave response may be identified easily in both spectral representation and waveforms of electric field. Different from fast formation case, the seismoelectric conversion efficiency of Stoneley wave is biggest, P wave is second.

Figure 5 shows spectral response and synthetic waveforms of acoustic waves and magnetic field 𝐻_𝑟 for slow formation generated by an acoustic dipole source. In the spectral representations of acoustic (Figure 5 (a)), one flexural mode appears in the frequency range 0–20 kHz, and in the spectral representations of magnetic field (Figure 5 (b)), the flexural mode is dominating. From Figures 5(b) and 5(d), the waveforms of borehole pressure with 1 kHz center frequency dominates by flexural guided waves; however, that of magnetic field has two groups with different velocity. Again, from Figure 5(d) we see a relatively weak but visible A-A group; it arrives simultaneously on all receivers. And the following group has apparent velocities of acoustic flexural waves velocities of borehole acoustic waves. Again, it shows the borehole seismoelectric conversion efficiency is strong at low frequency range.

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Figure 5 

Spectral representation and time response of borehole pressure and magnetic fields for slow formation generated by dipole source. (a) and (c) are spectral of pressure and magnetic fields, respectively. (b) and (d) are time responses of pressure and magnetic fields, respectively.

Figure 6 shows spectral response and synthetic waveforms of acoustic waves and magnetic field 𝐻_𝑟 for slow formation generated by an acoustic quadrupole source. In the spectral representations of acoustic (Figure 6(a)), a screw mode appears in the range of 0–10 kHz; however, in the spectral representations of magnetic field, this mode is more evident. From Figures 6(b) and 6(d), the waveforms of acoustic waves excited by quadrupole source with the center frequency 3 kHz mainly dominated by screw waves and that of magnetic field consists of groups having apparent velocities of acoustic screw waves. We almost cannot find the EM wave in the full waveforms of magnetic field 𝐻_𝑟.

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Figure 6 

Spectral representation and time response of borehole pressure and magnetic fields for slow formation generated by quadrupole source. (a) and (c) are spectral of pressure and magnetic fields, respectively. (b) and (d) are time responses of pressure and magnetic fields, respectively.

4. Conclusions and Discussions

In this paper, multipole seismoelectric logging waveforms are simulated in a fluid-filled borehole embedded in permeable formations based on the entire Pride seismoelectric theory. We consider the nonaxisymmetric seismoelectric field excited by acoustic multipole sources. Expressions for the multipole seismoelectric field quantities of interest were derived from the potential formulations. The full waveforms of acoustic waves and electric and magnetic fields in the time domain propagation in borehole are simulated by using a discrete wave number integration. The results show that electric and magnetic field detectors in a borehole can detect both the EM waves and the seismoelectric field. One can measure three components of electric field and magnetic field when higher order multipole sources (𝑛≥1) are excited. The results shows that one can measure tens of millivolt electric fields as long as one selects an acoustic source with pressure amplitude being 1 MPa measuring 0.1 m from the source center. However, it is the weakness of electrical and magnetic signals in the presence of the big background noise that is one main obstacle to application of seismoelectric tools. Therefore, enhancing the signal-to-noise ratio is the key for seismoelectric logging measurement. The amplitudes of electromagnetic field should be related to formation parameters through coupling coefficients 𝐿(𝜔); the seismoelectric conversion efficiencies of the Stoneley wave, the flexural guided wave and the screw guided waves are slightly different due to borehole effect in nature. The borehole seismoelectric conversion mainly depends on the relative motion between a solid and a fluid, which is generated by different acoustic modes. The relationship between borehole seismoelectric conversion efficiencies and the formation parameters needs further research in detail.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grants nos. 40974067 and 41004044), the State Key Laboratory of Acoustics (IACAS) (Grant no. 200807) and Scientific Forefront and Interdisciplinary innovation project of Jilin University (Grant no. 200903319).