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Article

The Response Mechanism of Borehole Shear-Horizontal Transverse-Electric Seismoelectric Waves to Fluid Salinity

1
Department of Acoustics and Microwave Physics, College of Physics, Jilin University, Changchun 130012, China
2
College of Petroleum Engineering, Panjin Vocational and Technical College, Panjin 124010, China
3
Department of Civil and Architectural Engineering and Mechanics, University of Arizona, Tucson, AZ 85721, USA
*
Author to whom correspondence should be addressed.
Submission received: 28 March 2022 / Revised: 17 May 2022 / Accepted: 18 May 2022 / Published: 19 May 2022
(This article belongs to the Special Issue Integration of Methods in Applied Geophysics)

Abstract

:
The converted electric field in the seismoelectric effect can be used to monitor the salinity of the reservoir. Compared to some conventional excitation methods (e.g., Monopole source), the response law of borehole shear-horizontal transverse-electric (SH-TE) seismoelectric waves to fluid salinity is unique. In order to explore its physical mechanism, we study the influence of fluid salinity on borehole SH-TE wave fields in this paper. First, to analyze the effect of salinity on the electric field, we simulate the response for different salinity levels inside and outside the borehole. Then we study the wave fields in case of a radial salinity discontinuity outside the borehole, and simulate the interface response by the secant integral method. Finally, we show the feasibility of using the borehole SH-TE wavefields to estimate the salinity interface position combining the slowness-time coherence (STC) method. The results show that the electric field amplitude changes monotonously with the pore fluid salinity level. However, the borehole fluid salinity has almost no effect on the electric field. This is caused by the excitation method and the low frequency of the sound source. An interface converted electromagnetic wave response is generated when an SH wave passes through the salinity discontinuity interface. The interface position estimation examples show that the borehole SH-TE wave field is a potential method to evaluate the behavior and the location of the interface next to the borehole.

1. Introduction

Salinity is the total amount of ions per unit volume of fluid. It is an important indicator of reservoir fluid. Salinity is used to evaluate the salt content in fluids, not only for pore fluid identification but also for the evaluation of formation fluid pollution [1,2]. The salinity of the pore fluid directly affects the electrical properties of the formation. Since the converted electric field of the acoustic field in the porous media is sensitive to the electrical properties, it becomes one of the potential methods for detecting the salinity level of the pore fluid [3,4,5].
The phenomenon of acoustic waves propagating in the pore fluid to generate the coupling electromagnetic field is called the seismoelectric effect [6,7,8,9,10,11,12,13]. The electric double-layer structure and the seepage of the pore fluid are important factors for this phenomenon. The prospecting method based on seismoelectric signals receives both elastic wave and electromagnetic wave information, which is convenient for obtaining the formation’s elastic information as well as electrical information. Studies have shown that the seismoelectric signal is not only sensitive to the rock modulus, porosity, permeability, saturation, and other parameters that affect the acoustic properties but is also closely related to the electrical properties of the rock such as electrical conductivity [14,15,16,17,18,19,20,21,22,23]. This also makes researchers pay close attention to the phenomenon of acoustic-electric coupling. Research on the theory and experiment of acoustic-electric coupling has been extensively carried out, and has obtained many meaningful results [24,25,26,27,28,29,30,31,32,33,34,35,36].
Some researchers also proposed a seismoelectric logging method based on the experience of sonic logging [37]. In addition to obtaining formation elasticity and electrical information at the same time, this method also has advantages such as deep exploration. Hu et al. [38,39] were the first to simulate the seismoelectric waves in the borehole. For inducing electric fields, in addition to the accompanying electric fields that propagate with the velocity of the acoustic waves, there also exists the interface responses propagating with the speed of the electromagnetic waves in the formation. Later, they proposed a quasi-stable method for independently calculating the acoustic fields and the electric fields [40], and verified the feasibility and applicability of the method [41]. Some investigators also used the finite difference [42] and the finite element [43,44] methods to simulate the borehole seismoelectric wave fields. The results show that the wavefield characteristics obtained by multiple methods are consistent. In actual exploration, factors such as geological structures or fluid pollution will make the elastic or electrical properties of the formation inhomogeneous. The interface-converted electromagnetic wave generated by the elastic wave impinging the interface of the medium is one of the important characteristics [45,46,47]. This characteristic can be used to detect the position and nature of the formation interface or the structural characteristics of the geological body next to the borehole. Liu et al. [48] simulated the acoustoelectric coupling fields for the while-drilling condition when the formation has an interface separating two materials having different elastic properties and found that the elastic properties on both sides of the interface affect the intensity of the interface response. Ding et al. [49] simulated the borehole seismoelectric waves for the case of a formation having an interface separating fluids with two different levels of salinity. However, this article did not make an in-depth investigation of the interface response law for this formation situation. Duan et al. [5] investigated the effect of the salinity interface on the seismoelectric logging by acoustic monopole source, and the results show that the inducing electric waves have a complicated response law to salinity. The salinity interface formed between the borehole fluid and the formation has a significant effect on the induced electric fields. When using the seismoelectric effect for logging, the electromagnetic wave interface response generated by the acoustic waves hitting the medium interface is important information. Since the amplitude of the interface converted electromagnetic wave is relatively small, when the geological structure of the formation is complex, the electric field full-wave sometimes cannot fully reflect the interface response properties. The method of complex analysis can be used to solve a single component of the wave field, where the secant integral method is an effective tool for solving the interface response in the seismoelectric effect [50,51,52,53]. In addition, the component wave analysis of the seismoelectric wave fields by using the pole residue can have a deeper understanding of the propagation characteristics of the mode waves and their accompanying electric field [54]. In this article, we will discuss the influence of the interface, across which the electrical properties differ, on the seismoelectric wave fields, and focus on the response mechanism of the interface radiated electromagnetic waves. Besides, there are few studies on the use of seismoelectric wave fields to detect borehole-side interface position. Here, we propose an interface position inversion method combining slowness-time coherence (STC) [55,56,57] and the time-domain waveforms. At the same time, the feasibility of using the borehole SH-TE wavefields to invert the position of the salinity interface next to the borehole is also discussed.
The SH-TE acoustoelectric coupling waves have been widely used in shallow exploration [58,59,60,61,62,63]. The results show that when the SH waves pass through the interface of the horizontally layered formations, interface responses appear. The electromagnetic waves generated at the interface reach different receivers almost simultaneously. The differences in elasticity, consolidation, and electrical properties between the two sides of the interface are important factors that influence the strength of the interface response [64]. In addition, the high signal-to-noise ratio is also an advantage of using SH-TE waves for exploration [65]. White [66] first proposed the idea of using the pure SH waves for logging and performed some simple calculations for homogenous elastic media. A subsequent study showed that layered structure media complicates the borehole SH wave composition since the multi-order and dispersive mode waves are excited [67]. As for the borehole SH-TE waves, Cui et al. [68] simulated the SH-TE acoustoelectric coupling fields of a homogeneous porous medium. Similar to the borehole SH wave, when the nature of the formation is discontinuous, the dispersive accompanying electric fields are also excited. In addition, the electromagnetic interface response becomes complicated [53]. Then in a later work Wang et al. [52] discussed the response law of borehole SH-TE waves in partially saturated soil. Although this work also examined the influence of pore fluid salinity on the electric field, it did not investigate in detail its influence mechanism.
To further understand the seismoelectric coupling phenomenon in the reservoir and investigate the application potential of the borehole SH-TE wave fields in exploration, here we study the influence of fluid salinity on SH-TE waves in a borehole and explore the influence mechanism. The composition of the article is as follows: In Section 2, we derive the analytical expressions of the field quantities in each region. In Section 3, we study the sensitivity of the electrical properties to the salinity of the borehole fluid and the pore fluid and discuss the influence mechanism of salinity on borehole electric field responses. In Section 4, we simulate the coupling wave fields for a salinity interface near the borehole, and the secant integral is used to simulate the electromagnetic interface response under these circumstances. In addition, we also use the STC method to process the signal array to obtain formation sound velocity and arrival time information, discussing the feasibility of using the borehole SH-TE wavefields to invert the position of the borehole-side interface. Finally, we summarize the full text and discuss the limitations of the model in this article and the problems that need to be solved in the future.

2. Theoretical Background

2.1. Model and Field Expressions

Figure 1 shows the SH-TE logging geometry, a = 0.12 m is the radius of the borehole. Outside the borehole, we use a homogeneous saturated porous medium extending to infinity in the radial direction to simulate the typical sandstone formation. In order to excite and receive pure cylindrical shear waves, we take an annular shear source and place the acoustic receivers on the borehole wall. Since the electromagnetic field can radiate into the fluid through the borehole wall, we place the electromagnetic receiver in the borehole and take a radial distance of a / 2 from the borehole axis.
Next we derive the expressions of the borehole electromagnetic fields. According to Cui et al. [68], the borehole radial and tangential electric and magnetic field expressions can be expressed as Equations (1)–(4).
E r b = i η e 2 k E z b r + ω μ r H z b θ
H r b = i η e 2 k H z b r + ω ε b r E z b θ
E θ b = i η e 2 k r E z b θ ω μ H z b r
H θ b = i η e 2 k r H z b θ ω ε b E z b r
where E r b ,   E θ b and H r b , H θ b represent the electric field and the magnetic field, respectively. The subscript “b” represents the quantities in the borehole and the subscripts “r” and θ represent the radial component and the tangential component, respectively. η e is the electromagnetic radial wave number in the borehole. k is the axial wave number. ω is the circular frequency and μ = 4 π × 10 7   H / m is the vacuum permeability. E z b cannot be excited due to the shape of the source. Combining Equations (1)–(4), the frequency-wavenumber domain expressions for E θ b ^ and H r b ^ are given by Equations (5) and (6).
E θ b ^ = i η e ω μ A H I 1 η e r
H r b ^ = i η e k A H I 1 η e r
where I 1 is the first-order first-kind modified Bessel function. Combining the Helmholtz decomposition method, the field expressions of the formation are given by Equations (7)–(10).
u θ o ^ = η s h o K 1 η s h o r B s h o + η e m o K 1 η e m o r B e m o
τ r θ o ^ = G o η s h o 2 K 2 η s h o r B s h o + η e m o 2 K 2 η e m o r B e m o
E θ o ^ = β s h o η s h o K 1 η s h o r B s h o + β e m o η e m o K 1 η e m o r B e m o
H z o ^ = i ω μ β s h o η s h o 2 K 0 η s h o r B s h o + β e m o η e m o 2 K 0 η e m o r B e m o
u θ o ^ , τ r θ o ^ , and E θ o ^ represent the tangential displacement, stress, and electric field, respectively. H z o ^ is the axial magnetic field. Superscript “o” represents the quantities corresponding to the porous formation. G o represents the formation shear modulus. For the meanings of β s h o and β e m o the readers are referred to Pride et al. [10]. η j o 2 = k 2 k j o 2 , j = s h , e m are the formation’s SH and electromagnetic radial wave numbers. B s h o and B e m o are the undetermined coefficients. K 0 , K 1 , and K 2 are the second kind modified Bessel functions.

2.2. Boundary Conditions and Solutions of Equations

When a homogeneous fluid-saturated pore formation exists outside of the borehole, the boundary conditions are:
At the borehole wall (r = a):
Shear   stress   continuity :   τ r θ o = τ s b
Tan gential   electric   field   strength   continuity :   E θ o = E θ b
Axial   magnetic   field   strength   continuity :   H z o = H z b
τ s b represents the shear stress source, we take its amplitude as one-thousandth of the formation shear modulus, namely τ s b = 10 3 × G o . Substituting Equations (5)–(10) into boundary conditions (11)–(13), the matrix Equation (14) is obtained.
M i j 3 × 3 · A j 3 × 1 = B j 3 × 1
M i j 3 × 3 is the coefficient matrix. The first column corresponds to the homogeneous general solutions of the borehole electromagnetic field, the second and third columns correspond to the homogeneous general solutions of the coupling wave fields in the porous formation. A j 3 × 1 = A H , B s h o , B e m o T is the undetermined coefficient matrix. B j 3 × 1 = τ s b , 0 , 0 T is a matrix related to the source. In order to solve the borehole fluid electromagnetic fields, the undetermined coefficient A H should be solved first:
A H = D e t [ N i j ] 3 × 3 D e t [ M i j ] 3 × 3
In Equation (15), matrix N i j 3 × 3 is obtained by replacing the elements in the first column of the coefficient matrix M i j 3 × 3 with the elements of B j 3 × 1 . The expressions of each matrix element are given in Appendix A. Substituting A H into Equation (5), we can obtain a complete analytical expression of the borehole fluid tangential electric field.

3. Results and Discussion

3.1. Influence of Fluid Salinity

3.1.1. Influence of Pore Fluid Salinity

The influence of fluid salinity on the formation is reflected on its influence on the conductivity of the porous medium and the electrokinetic coupling coefficient [5,10]. Since the formation dynamic conductivity in the logging frequency band varies little with frequency, we use the low-frequency dynamic conductivity form given by Haines and Pride [69]:
σ σ f ϕ α
σ f = e 2 z ^ 2 N f b + + b
In Equations (16) and (17), σ is the conductivity of the formation, α is the tortuosity of the pore. σ f and ϕ are the pore fluid conductivity and the porosity, respectively. e = 1.602 × 10 19 C is the electronic charge and z ^ is the ion valence of the electrolyte solution. In this paper, the borehole fluid or the pore fluid is regarded as NaCl solution, so z ^ = 1 . N f = N a × 10 3 × C f is the pore fluid ion concentration, where N a = 6.022 × 10 23 M o l 1 is Avogadro constant and C f is the pore fluid salinity. b + and b are the absolute mobility of cations and anions, respectively. Both of these values are equal to 3 × 10 11 N s / m . According to Equations (16) and (17), the influence of C f on σ is due to its influence on σ f . When C f increases monotonically, the formation conductivity σ increases monotonically as well, and σ is proportional to C f .
Next, we investigate the influence of C f on the electrokinetic coupling coefficient L ω . According to Pride et al. [9], L ω is given by Equations (18) and (19).
L ω = L 0 1 i m 4 ω ω c 1 2 d ˜ Λ 2 1 i 3 2 d ˜ δ 2 1 2
L 0 = ϕ α ε f ζ η 1 2 α d ˜ Λ
L 0 is the static electrokinetic coupling, and Λ is the pore feature size. m = ϕ Λ 2 α κ 0 , where κ 0 is the absolute permeability. Research by Shi et al. [70] shows that the above equation can well describe the electrokinetic coupling coefficient of the cylindrical pores, but it overestimates the influence of the pore geometry. For any pore geometry, m could be taken as 8. η is the viscosity coefficient and ε f is the fluid permittivity. δ is the skin depth and ζ is the zeta potential.   ω c is the frequency turning point. d ˜ is the electric double layer thickness, and its value is close to Debye length d. Therefore, Debye length d is usually used to replace d ˜ . According to Equations (18) and (19), the influence of C f on L ω is reflected on its influence on the Debye length d and the zeta potential ζ . The expression of Debye length is given by Equation (20) [10].
d = ε f k B T e 2 z 2 N f
where k B and T are Boltzmann’s constant and the absolute temperature, respectively. According to Equation (24), C f influences the Debye length d by influencing the ion concentration N f . According to Pride and Morgan [8], ζ is given by Equation (21), and the value of C f directly affects the zeta potential of the electric double layer. Figure 2 shows the dependence of L ω on the frequency and pore fluid salinity.
ζ = 0.008 + 0.026 log 10 C f
As shown in Figure 2, as the frequency increases, the electrokinetic coupling coefficient decreases monotonically. It implies the electrokinetic coupling weakens with increasing frequency. As fluid salinity C f increases, the value of L ω also decreases monotonically, which indicates that the greater the pore fluid salinity, the weaker is the electrokinetic coupling effect.
Next, we compare the electric fields in the borehole for different C f to illustrate the influence of salinity on the electric field. This paper considers the typical sandstone formation for which the media parameters are given in Table 1. This set of parameters can well reflect the properties of the typical sandstone formation [39,64]. For the parameters given in Table 1, the velocity of SH wave in the formation is 2492.2 m/s, thus it is a fast formation. The acoustic source frequency is 10 kHz, and the density of the borehole fluid is 1000   kg / m 3 . The fluid salinity in this part is fixed at C b = 0.001   Mol / L .
From the above discussion and Equation (5), since C f has no effect on η e , the influence of C f on the borehole electric field is reflected on its influence on the electromagnetic response coefficient A H . Figure 3 shows the borehole electric fields for different C f , and the source distance corresponding to the waveforms in Figure 3 is z = 3 m. It should be noted that, according to the assumption of Pride et al. [9], the salinity for pore fluid should not be greater than 1 Mol/L. Here we combine the work of Duan et al. [5] to limit the fluid salinity range of 10 4 10 2   Mol / L , which is a common groundwater salinity grade range.
According to Figure 3, the general shape or composition of the electric field for different C f does not change. The complete waveform of E θ b consists of two parts—the interface response and the accompanying electric field [68]. When the salinity of the pore fluid increases, the arrival time of each component does not change but the amplitude decreases monotonically. Since the interface responses are generated at the borehole wall and propagate with the speed of the formation’s electromagnetic waves, the salinity of the pore fluid hardly influences their arrival times. Besides, the pore fluid salinity mainly influences the formation’s electrical properties but hardly influences the elastic properties. Therefore, when the pore fluid salinity changes, the SH wave velocity remains almost unchanged, thus the accompanying electric field’s starting time also remains unchanged. The variation of the amplitude is mainly caused by the influence of pore fluid salinity on the formation conductivity and the electrokinetic coupling. Between the two, the influence of fluid salinity on the electrokinetic coupling plays the major role.

3.1.2. Influence of Borehole Fluid Salinity

Salinity in the borehole mainly affects the fluid conductivity value. The relationship between the borehole fluid ion concentration N b and the borehole fluid salinity is: N b = N a × 10 3 × C b , where C b is the borehole fluid salinity. Replacing N f in Equation (17) by N b and substituting it into Equation (16), we can get the conductivity of the borehole fluid σ b . Figure 4 shows the electric fields in the borehole for different C b values; the formation parameters of this part are consistent with Table 1, and the pore fluid salinity is fixed at 0.0001 Mol/L. Similarly, the vertical source distance taken here is still z = 3 m.
As shown in Figure 4, the influence of C b on the electric field is different from that of the formation salinity level. When the fluid salinity C b varies, the waveform of E θ b hardly changes. According to Equation (9), the influence of C b . on the borehole electric field E θ b is reflected on its influence on A H and η e . Since η e = k 2 k e 2 , the change of C b directly leads to the change of k e and thus influences η e . The borehole electromagnetic wave number k e is given by Equation (22).
k e = ω 2 μ ε b 1 + i σ b ω ε b
ε b is the borehole fluid dielectric constant. Since ε b and μ are independent of salinity, the influence of C b on k e is only reflected on its influence on σ b . According to Equation (15) and the matrix elements given in Appendix A, the A H form shown in Equation (23) can be obtained.
A H = G n 22 n 33 G n 23 n 32 m 21 m 13 m 32 m 12 m 33 + m 31 m 23 m 12 m 13 m 22
m i j and n i j are the matrix elements of M i j 3 × 3 and N i j 3 × 3 respectively, i , j = 1 , 2 , 3 . The borehole fluid salinity C b influences A H only by influencing the borehole electromagnetic wave number k e . The numerator on the right side of Equation (23) does not include k e . The two matrix elements containing k e in the denominator are m 21 = i ω μ η e I 1 η e a and m 31 = I 0 η e a , where η e = k 2 k e 2 . Therefore, it is only necessary to investigate the matrix elements m 21 and m 31 to analyze the influence of C b on the electromagnetic response coefficient A H . Figure 5 is the schematic diagram of the first kind of zero-th order and first-order Bessel functions of imaginary argument, when the independent variable range is 0–0.2. From left to right are the absolute value of I 0 ( η e a ) , the absolute value of I 1 ( η e a ) , and the ratio of I 1 ( η e a ) to the independent variable.
As shown in Figure 5, when the value range of the independent variable is 0–0.2, the value of I 0 ( η e a ) hardly varies with the variation of the independent variable; it is fixed at 1. Under the conditions of this article, the magnitude range of the independent variable η e a of the Bessel function is 10 4 ~ 10 2 , which is in the independent variable range of Figure 5. Therefore, the value of the matrix element m 31 does not change with the change of the independent variable η e a . The ratio of I 1 ( η e a ) to the independent variable is fixed at 0.5, that is, I 1 η e a η e in the matrix element m 21 is a fixed value. Since the variation of the borehole electromagnetic wave number cannot change the element value of the coefficient matrix, the electromagnetic field response coefficient A H is not influenced by the salinity of the borehole fluid. According to Equation (5), in addition to the response coefficient A H , the remaining terms can be expressed in the form i ω μ I 1 η e a η e , and its value hardly influenced by η e . In summary, although the change in borehole fluid salinity can make the change of the fluid conductivity and thus influence the wave number of borehole electromagnetic waves, the variation of the wave number in the logging frequency band hardly influences the borehole electric field. As a result, the salinity level in the borehole hardly influences the borehole electric field. This is consistent with the results presented in Figure 4. However, for the conventional acoustic sources, the influence of the salinity level of the borehole fluid on the seismoelectric response is more complicated [48]. Both the salinity values of the borehole fluid and the formation affect the electric field. In addition, the ratio between the two also affects the normalized amplitude of the electric field [5]. Figure 6 compares the seismoelectric wavefields of a centered monopole source for different borehole salinity, with a source center frequency of 6 kHz. E z is the axial electric field.
When fluid salinity C b varies, the composition of E z does not change. As shown in Figure 6, the first component is the electromagnetic interface response. The following components are the accompanying electric fields. The first to arrive is the accompanying electric field of P-wave, and then the accompanying electric fields of S-wave and Stoneley wave. When the salinity in the borehole varies, the arrival time of each component is almost unchanged but the amplitude changes. Obviously, the greater the salinity, the smaller the amplitude. According to the earlier discussion, this amplitude change is mainly caused by the influence of the fluid salinity on the borehole fluid conductivity. That is to say, borehole fluid salinity has an obvious influence on the electric field, which is very different from the response law of the borehole SH-TE waves to borehole salinity. According to the above discussion, the response law of the coupling electric field of the borehole pure SH wave to borehole salinity is a special case. This is mainly due to the low frequency source and the special shape of the acoustic source.

3.2. Effect of Salinity Discontinuity Next to the Borehole

In this part, we investigate the case when the formation has an interface between two fluids having different levels of salinity. In this case, the formation elastic properties are not significantly affected while the electrical properties are changed [49]. Here we consider the elastic properties of the formation to be continuous across the interface and only the electrical properties are discontinuous. Therefore, we construct a cylindrical layered porous formation with only pore fluid salinity level being different. The salinity difference interface here is parallel to the borehole axis direction. The expressions for the field quantity in the borehole are still determined by Equations (5) and (6). The field quantity expressions for the outermost porous medium are consistent with Equations (7)–(10). For the porous medium with limited thickness (between the salinity discontinuity interface and the borehole wall), the field quantities are determined by Equations (24)–(27). For the boundary conditions and solution techniques please refer to Ref. [53].
u θ ^ = η s h K 1 η s h r B s h η s h I 1 η s h r A s h + η e m K 1 η e m r B e m η e m I 1 η e m r A e m
E θ ^ = β s h η s h K 1 η s h r B s h β s h η s h I 1 η s h r A s h + β e m η e m K 1 η e m r B e m β e m η e m I 1 η e m r A e m
τ r θ ^ = G i η s h 2 K 2 η s h r B s h + η s h 2 I 2 η s h r A s h + η e m 2 K 2 η e m r B e m + η e m 2 I 2 η e m r A e m
H z ^ = i ω μ β s h η s h 2 K 0 η s h r B s h + β s h η s h 2 I 0 η s h r A s h + β e m η e m 2 K 0 η e m r B e m + β e m η e m 2 I 0 η e m r A e m
In Equations (24)–(27), u θ ^ , E θ ^ , τ r θ ^ , and H z ^ are the tangential displacement, tangential electric field, tangential stress, and axial magnetic field, respectively for the inner porous medium. η s h 2 = k z 2 k s h 2 and η e m 2 = k z 2 k e m 2 are the radial wave numbers of the SH wave and the electromagnetic wave, respectively for the inner porous medium. Where k s h and k s h are the SH wave and electromagnetic wave numbers, respectively for the inner porous medium. A s h , B s h , A e m and B e m are the undetermined coefficients. G i is the shear modulus of the inner porous medium. I 0 and I 2 are Bessel function of second kind. We first simulate the borehole SH-TE wave fields and discuss the influence of the salinity interface on the acoustic field and electric field. Then we also simulate the interface electromagnetic waves by the secant integral method to illustrate the influence of the salinity interface on the electric field interface response. Finally, we use the slowness-time coherence method (STC) to process the acoustic field signal to obtain the formation’s SH wave velocity, combining the time-domain waveforms of the electric field to discuss the feasibility of using the borehole SH-TE waves to estimate the position of the salinity interface.

3.2.1. Wave Fields Simulation

Figure 7 shows the normalized wavefields when the formation has a salinity difference across an interface. The interface is located at d = 1.5 m away from the borehole axis. The parameters used in this part are consistent with Table 1. The salinity of the inner porous medium is 0.01 Mol/L and the outer porous medium is 0.001 Mol/L. The placement of the source and the receivers is the same as Figure 1. u θ and E θ represent the displacement field and the electric field in the borehole respectively.
As shown in Figure 7, when there is a salinity difference interface near the borehole, the acoustic field is almost unaffected. Under this circumstance, the acoustic field still has only one set of components, namely the formation’s SH wave. Unlike the previous simulation results, the electric field component becomes complicated. In addition to the accompanying electric field (component c-c in Figure 7) of the SH wave, there are two sets of components that arrive earlier (components a-a and b-b in Figure 7). These two components reach the receivers at different distances almost simultaneously, which indicates that they are the electromagnetic wave interface responses generated by the formation’s sound wave impinging the interface of the medium. As discussed earlier, component a-a is the interface response produced by SH waves impinging the borehole wall. In order to further explain the generation cause of component b-b, we calculate the secant integral of the formation’s electromagnetic wave branch point, and the interface converted electromagnetic waves are obtained. For a detailed discussion of this specific method readers are referred to Wang et al. [54]. Figure 8 shows the simulation results. In order to distinguish them from the electric field full waves obtained by the real axis integral, we denote the electromagnetic wave interface response obtained by the secant integral as E θ c u t . Figure 8a is the normalized waveforms of the interface electromagnetic wave, and Figure 8b is the waveform comparison between E θ c u t and E θ , taking z = 3 m as an example. In addition, we also compare the waveforms of E θ c u t for different interface positions in Figure 8c.
As shown in Figure 8a, the interface electromagnetic wave obtained by secant integral has only two groups of components, denoted as a-a and b-b. Unlike the case of the elastic difference interface next to the borehole [53], the interface response for the electrical difference interface is relatively simple. This is because the elastic properties of the formation are uniform, and the SH wave will not be refracted and reflected multiple times in the inner porous medium. As shown in Figure 8b, the interface electromagnetic wave response calculated by secant integral is completely consistent with the first two components of the time-domain waveform of the electric field full wave, which further illustrates that the components a-a and b-b shown in the electric field full wave (Figure 7) are the electromagnetic wave interface responses. In addition, it also shows that it is feasible to calculate interface waves by secant integral. Under the parameters given in Table 1, the theoretical SH wave velocity of the formation is 2492.2 m/s. As shown in Figure 8c, when the salinity interface goes 0.5 m away from the borehole axis, the arrival time difference of component b-b coincides with the time when the formation’s SH wave propagates 0.5 m. Therefore, component b-b in Figure 8a is generated by the SH wave passing through the salinity difference interface. In addition, we also find that the position of the salinity interface hardly affects the interface response at the borehole wall. This shows that the electromagnetic wave interface responses generated at different interfaces do not affect one another.

3.2.2. Interface Position Estimation

The above results show that it is difficult to judge the properties of the formation only by the acoustic field for the formation with different electrical properties. However, the converted electric field from the sound field is very sensitive to the level of formation salinity. Next, we discuss the feasibility of using electric field signals to estimate the position of the salinity interface. The general idea is as follows: First, the slowness-time coherence method (STC) is used to process the acoustic field array signal to obtain the time-velocity coherence diagram of the acoustic field, and get the formation’s SH wave velocity. Then combining with the time-domain waveforms of the electric field, read the time difference of the interface responses, and the multiplication of the two is the radial distance between the salinity interface and the borehole wall.
In order to extract the time and slowness (velocity) information of the signal, Kimball et al. [55] proposed the slowness-time coherence method. The expression of the coherence coefficient is:
ρ T , s = 1 m T T + T w i = 1 m A i t + s i 1 d 2 d t T T + T w i = 1 m A i t + s i 1 d 2 d t
where ρ is the coherence coefficient, T is the window opening time point, T w is the window length, s is the slowness, and d is the receiver distance. According to Equation (28), select an appropriate window length and perform slowness scanning to obtain a series of coherence coefficients, then draw the contour map of the coherence coefficient, and we can find the area with the larger coherence coefficient, and then the wave packet arrival time and slowness (speed) information can be extracted. Figure 9 is the result of STC processing on the acoustic field signal in Figure 7. In order to display the information more intuitively, we give the velocity-time coherence diagram here.
In Figure 9, one can see that the coherence coefficient of one area is significantly larger than that of other areas, which indicates that there is only one set of valid information in the signal, which are the formation’s SH waves. It further indicates that the electrical properties of the pore fluid hardly affect the properties of the acoustic field. In order to observe the inversion results of the SH wave velocity in greater detail, Figure 10 shows the fitting diagram of the relationship between the wave velocity and the coherence coefficient.
It can be seen that the relationship between wave velocity and coherence coefficient is a single-peak curve. When the wave velocity is 2512 m/s, the coherence coefficient reaches the peak value, which indicates that the wave velocity of effective wave field information is 2512 m/s. As mentioned before, under the parameters adopted in Table 1, the SH wave velocity is 2492.2 m/s, and the relative error is 0.8%. This shows that it is feasible to get the formation’s SH wave velocity by STC method. Next, we combine the electric field information given in Figure 7 to obtain the interface position. According to Figure 7, the interface responses a-a and b-b are the interface electromagnetic waves generated by the SH wave impacting the borehole wall and the salinity difference interface, respectively. Since the magnitude of the interface electromagnetic wave propagation velocity is very large and the distance between the receivers is not large, the time to propagate from the interface back to the receiver is negligible. In other words, the time difference between the interface responses corresponds to the time for the SH wave to propagate in the porous formation. We take the arrival time of the wave peak as a reference basis. According to Figure 7, the arrival time of the interface responses are 0.196 ms and 0.742 ms, respectively. Thus, the time difference between the two arrival times is 0.546 ms. The time difference is multiplied by the inverted SH wave speed, and then added to the borehole radius a = 0.12 m to obtain the position of the salinity interface d = 1.491 m. The inversion results are in good agreement with the interface position parameters used in the simulation. To further illustrate the feasibility of the method, we use the interface position or the shear moduli as variables, and carry out inversion simulations. The results are shown in Table 2. V S T is the theoretical wave velocity of the SH wave, V S R is the inverted SH wave velocity, δ t T is the theoretical arrival time difference between the interface responses, and δ t R is the arrival time of the two sets of interface responses obtained by the electric field. d T = V S T × δ t T + a is the theoretical salinity interface position,   d R = V S R × δ t R + a is the inverted salinity interface position, both are defined as the radial distance of the interface from the borehole axis. The error given in the last column is the absolute value of the relative error of the interface position inversion result. The upper table is the inversion result when the formation’s shear modulus G o is fixed while the interface position is changed. The following table shows the inversion result when the interface position is fixed while the formation’s shear modulus G o is changed.
From Table 2 we can state that irrespective of whether the interface position or the formation’s shear modulus (or SH wave velocity) is changed, the inversion results of the interface position are in good agreement with the theoretical values. The inversion errors are within 3%, and very little trend of change is observed. It also shows that this method is applicable to different formation conditions. In summary, it is feasible to obtain the location of the borehole-side electrochemical interface by the SH-TE seismoelectric coupling wave fields. At the same time, borehole SH-TE acoustic-electric coupling effect may also become a potential method for detecting groundwater pollution and the electrical continuity of the formation. However, the disadvantage of this method is that when the interface position is very close to the borehole wall, the two sets of interface responses may have waveform aliasing, resulting in inaccurate readings of the arrival time. Therefore, when the salinity interface is close to the borehole wall, how to accurately extract the arrival time of the interface response is one of the problems that needs to be solved to further improve the method.

4. Conclusions

Borehole SH wave and its coupling electric field make the waveform relatively simple and hence easy to analyze. SH-TE seismoelectric response is one of the potential logging methods. In this paper, based on the available studies on SH-TE seismoelectric responses [52,53,68], we investigate the special response law for fluid salinity and focus on the physical mechanisms. The following main conclusions and understandings are the results of this investigation:
(a)
The salinity of the pore fluid has a significant effect on the induced electric fields of the SH waves. The electrokinetic coupling effect is weakened with the increasing pore fluid salinity level. The attenuation of electromagnetic waves monotonically increases with the increase of pore fluid salinity. The borehole electric field response amplitude decreases with the increasing formation’s salinity. The monotonic change of the amplitude is an advantage of using borehole SH-TE seismoelectric waves for pore fluid salinity monitoring.
(b)
Borehole fluid salinity hardly influences the borehole electric field response. This special influence mechanism is caused by the low-frequency band and the nature of the SH acoustic source. There is no need to artificially match the borehole fluid salinity with the formation’s salinity when using the SH-TE logging method.
(c)
The SH acoustic wave does not respond to the salinity interface, but the electric field responds significantly to it. When the SH wave passes through the salinity difference interface, the electromagnetic interface responses that propagate with the electromagnetic wave speed are observed. Estimation of electrical interface position shows that the SH-TE coupling physical effect is a potential method for detecting the position of the electrical interface next to a borehole. When there is water pollution in a small area next to the well, the elastic properties of the formation next to the well may not be changed. It is effective to use the converted electric field of acoustic waves to judge the nature and location of groundwater.
(d)
The model used in this investigation is somewhat idealized. For example, exciting a pure SH wave downhole is still a challenging task. The fluid salinity on two sides of an interface is not expected to change abruptly; most of them are gradual. Besides, in normal circumstances, the earth is a horizontally layered medium. The problem of seismoelectric logging in horizontal layered porous media is one of the important issues for our future research.

Author Contributions

W.W.: Writing—original draft, Conceptualization, Methodology, Formal analysis. W.G.: Methodology, Formal analysis. J.L.: Funding acquisition, Methodology, Software. T.K.: Writing—review and editing, Z.C.: Supervision, writing—review and editing, Conceptualization, Methodology, Resources. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the Natural Science Foundation of Jilin Province of China (Grant Nos. 20210101140JC and 20180101282JC) and partially by the National Natural Science Foundation of China (Grant Nos. 42074139 and 40974069).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data are contained within the article.

Acknowledgments

We are indebted to Laurence Jouniaux for helpful comments and suggestions.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A

Matrix elements of M i j 3 × 3 and N i j 3 × 3 in Equation (15).
Equation (A1): Matrix elements of M i j 3 × 3 :
m 11 = 0 ,   m 12 = G η s h o 2 K 2 η s h o a ,     m 13 = G η e m o 2 K 2 η e m o a m 21 = i ω μ η e I 1 η e a ,   m 22 = β s h η s h o 2 K 1 η s h o a ,   m 23 = β e m η e m o 2 K 1 η e m o a m 31 = I 0 η e a ,   m 32 = i ω μ η s h o 2 β s h K 0 η s h o a ,   m 33 = i ω μ η e m o 2 β e m K 0 η e m o a
Equation (A2): Matrix elements of N i j 3 × 3 :
n 11 = τ s b ,   n 12 = m 12 ,   n 13 = m 13 n 21 = 0 ,   n 22 = m 22 ,   n 23 = m 23 n 31 = 0 ,   n 32 = m 32 ,   n 33 = m 33

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Figure 1. Schematic diagram for SH-TE logging.
Figure 1. Schematic diagram for SH-TE logging.
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Figure 2. Dependence of electrokinetic coupling on frequency and fluid salinity.
Figure 2. Dependence of electrokinetic coupling on frequency and fluid salinity.
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Figure 3. Electric fields for different levels of C f . The solid line, dashed line and dot-dashed line correspond to the salinity level C f = 0.01 , 0.001 , 0.0001   Mol / L respectively.
Figure 3. Electric fields for different levels of C f . The solid line, dashed line and dot-dashed line correspond to the salinity level C f = 0.01 , 0.001 , 0.0001   Mol / L respectively.
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Figure 4. Electric fields for different levels of borehole fluid salinity. The solid line, dashed line, and dot-dashed line correspond to the salinity C b = 0.01   Mol / L , 0.001   Mol / L , 0.0001   Mol / L , respectively.
Figure 4. Electric fields for different levels of borehole fluid salinity. The solid line, dashed line, and dot-dashed line correspond to the salinity C b = 0.01   Mol / L , 0.001   Mol / L , 0.0001   Mol / L , respectively.
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Figure 5. Diagram of the first kind modified Bessel function values. The left one is the first-kind zero-th order modified Bessel function I 0 ( η e a ) , the middle one is the first-kind first-order modified Bessel function I 1 ( η e a ) , and the right one is the ratio of I 1 ( η e a ) to the independent variable.
Figure 5. Diagram of the first kind modified Bessel function values. The left one is the first-kind zero-th order modified Bessel function I 0 ( η e a ) , the middle one is the first-kind first-order modified Bessel function I 1 ( η e a ) , and the right one is the ratio of I 1 ( η e a ) to the independent variable.
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Figure 6. Axial electric fields for different borehole fluid salinity levels. The solid line, dashed line, dotted line, dash dotted line, and double-dot dashed line correspond to the borehole fluid salinity of 0.0025, 0.005, 0.01, 0.02, and 0.04 Mol/L, respectively.
Figure 6. Axial electric fields for different borehole fluid salinity levels. The solid line, dashed line, dotted line, dash dotted line, and double-dot dashed line correspond to the borehole fluid salinity of 0.0025, 0.005, 0.01, 0.02, and 0.04 Mol/L, respectively.
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Figure 7. Borehole SH-TE acousto-electric fields when the formation has a salinity difference across an interface. The solid line is E θ , and the dashed line is u θ . The components a-a and b-b are the electromagnetic wave interface responses generated at borehole wall and salinity difference interface respectively, and the component c-c is accompanying electric field.
Figure 7. Borehole SH-TE acousto-electric fields when the formation has a salinity difference across an interface. The solid line is E θ , and the dashed line is u θ . The components a-a and b-b are the electromagnetic wave interface responses generated at borehole wall and salinity difference interface respectively, and the component c-c is accompanying electric field.
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Figure 8. Interface converted electromagnetic waves. (a) The normalized waveforms of E θ c u t . (b) Comparison of E θ and E θ c u t . (c) E θ c u t for different interface positions. The components a-a and b-b are same as in Figure 7.
Figure 8. Interface converted electromagnetic waves. (a) The normalized waveforms of E θ c u t . (b) Comparison of E θ and E θ c u t . (c) E θ c u t for different interface positions. The components a-a and b-b are same as in Figure 7.
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Figure 9. Time-velocity coherence diagram of the acoustic field array signal.
Figure 9. Time-velocity coherence diagram of the acoustic field array signal.
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Figure 10. Acoustic field velocity-coherence coefficient fitting curve.
Figure 10. Acoustic field velocity-coherence coefficient fitting curve.
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Table 1. Formation parameters.
Table 1. Formation parameters.
ParametersSymbolsUnitsValues
Porosity ϕ -0.20
Permeability κ 0 μ m 2 0.01
Frame Shear modulus G o Gpa14.41
Frame density ρ s kg / m 3 2650
Pore fluid density ρ f kg / m 3 1000
Frame permittivity ε s ε 0 4
Fluid permittivity ε f ε 0 80
Fluid viscosity η w Pa · s 0.001
Tortuosity α -2.0
Table 2. Inversion results of interface position.
Table 2. Inversion results of interface position.
G o = 14.41 Gpa,   V S T   = 2492.2 m/s
V S R (m/s) d T (m) δ t T (ms) δ t R (ms) d R (m)Error (%)
0.70.2320.2220.6793.00
25121.00.3530.3430.9802.00
1.30.4730.4641.2851.15
1.50.5530.5461.4910.60
1.70.6340.6231.6721.64
d T = 1.5 m
G o V S T (m/s) V S R (m/s) δ t T (ms) δ t R (ms) d R (m)Error (%)
13.212386.22393.10.5780.5681.4791.40
15.412577.32571.90.5350.5261.4721.86
20.512973.33032.70.4640.4531.4930.47
25.123290.53325.60.4190.4101.4831.13
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Wang, W.; Gao, W.; Liu, J.; Kundu, T.; Cui, Z. The Response Mechanism of Borehole Shear-Horizontal Transverse-Electric Seismoelectric Waves to Fluid Salinity. Appl. Sci. 2022, 12, 5132. https://0-doi-org.brum.beds.ac.uk/10.3390/app12105132

AMA Style

Wang W, Gao W, Liu J, Kundu T, Cui Z. The Response Mechanism of Borehole Shear-Horizontal Transverse-Electric Seismoelectric Waves to Fluid Salinity. Applied Sciences. 2022; 12(10):5132. https://0-doi-org.brum.beds.ac.uk/10.3390/app12105132

Chicago/Turabian Style

Wang, Weihao, Wenyang Gao, Jinxia Liu, Tribikram Kundu, and Zhiwen Cui. 2022. "The Response Mechanism of Borehole Shear-Horizontal Transverse-Electric Seismoelectric Waves to Fluid Salinity" Applied Sciences 12, no. 10: 5132. https://0-doi-org.brum.beds.ac.uk/10.3390/app12105132

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