Detection Performance Analysis of Array Dielectric Dispersion Logging Based on Sensitivity Function

Dielectric logging is a critical method for exploring and developing complex oil and gas reservoirs, such as tight reservoirs, low-resistivity contrast reservoirs, and shale oil and gas reservoirs. The sensitivity function is extended to high-frequency dielectric logging in this paper. The detection characteristics of attenuation and phase shift of an array dielectric logging tool in different modes are investigated, along with the influencing factors such as resistivity and dielectric constant. The results show the following: (1) The symmetrical coil system structure makes the sensitivity distribution symmetrically distributed, and the detection range is more focused. In the same measurement mode, the depth of investigation (DOI) becomes deeper under high resistivity formation, and the sensitivity range oscillates outward when the dielectric constant becomes greater. (2) The DOIs of different frequencies and source spacings cover the radial zone between 1 cm and 15 cm. The detection range has been enlarged to include part of the invasion zones, improving the measurement data’s dependability. (3) With the increase in the dielectric constant, the curve tends to oscillate, and this behavior makes the DOI slightly shallower. Additionally, this oscillation phenomenon is obvious when the frequency, resistivity, and dielectric constant increase, particularly in high-frequency detection mode (F2, F3).


Introduction
Dielectric logging is a significant logging technique for distinguishing between oil and water. This logging method can directly measure the water saturation of the formation, and it is especially critical in tight gas sands, where water saturation can be very low and impossible to predict using standard resistivity logs. It also provides information about rock lithology, water porosity, and clay content [1][2][3][4][5][6][7]. The majority of the early dielectric logging methods use centered mandrel measurement. The receiving coils' amplitude attenuation and phase shift reflect the change in complex permittivity [8], and the general working frequency is dozens of MHz, making the dielectric sensitivity insufficient. The Deep Electromagnetic Propagation Tool (DPT) from Schlumberger operates at a frequency of 25 MHz. The transmitting coils are equipped with a pair of near-receiving coils and a pair of far-receiving coils. The Electromagnetic Propagation Tool (EPT), introduced in 1977, embeds the transmitting and receiving antennas into the groove of the pad and measures the formation information by pushing the pad against the borehole wall. The operating frequency is increased to 1.1 GHz [9]. The High-Frequency Dielectric Logging Tool (HFD) was developed by Halliburton [10,11]. This push-type measurement method is less affected by the borehole under ideal push-type conditions. However, this kind of tool has some drawbacks, including a narrow detection range, loose pushing when the borehole is irregular, difficulty eliminating high-frequency measurement errors, and insufficient information from single-frequency measurements. Atlas introduced dual-frequency (47, 200 MHz) dielectric logging equipment after the 1980s. The 47 MHz mode uses a centered measurement approach, whereas the 200 MHz mode uses a pad placed against the borehole wall for measurement, allowing for both deep and shallow detection depths. At the same time, the flushing zone and invasion zone information is measured [12,13].
According to the development history, dielectric logging tools have been developed in multi-frequency and multi-source spacing logging modes. The Array Dielectric Logging Tool (ADT) from Schlumberger uses multi-frequency and multi-source spacing detection modes to measure formation information, overcoming the limitations of traditional dielectric logging's shallow detection depth and insufficient measurement information [14][15][16][17][18]. The Array Dielectric Logging Tool has been successfully used in the evaluation of reservoirs with fresh water and heavy oils, and because of its high vertical resolution it has been used for the evaluation of laminated reservoirs [19,20]. With the development of unconventional oil and gas resources, the application of array multi-frequency dielectric logging tools in oil fields is growing, both at home and abroad. Studying the detection sensitivity of array dielectric logging is therefore of great theoretical and practical importance [3,[21][22][23][24][25].
The sensitivity function (also known as the geometric factor or response function) has been used to characterize the spatial distribution characteristics of logging responses for the formation of electrical parameters, which can be useful for analyzing instrument detection performance and data processing [26,27]. The Doll geometric factor was first developed to describe the geometric distribution of the independent unit ring's contribution distribution to the induction logging response [28]. Gianzero, Moran, Howard, and Zhang extended the geometric factor theory's applicability to the condition of low-resistivity formation as the frequency increases [29][30][31][32]. For the calculation of the sensitivity function, McGillivray proposed the difference approximation and the Fréchet differential method [33]. Tølbøll then investigated the detection sensitivity of different dipole systems based on the Fréchet derivatives [34]. Wang et al. extended the Born geometric factor to the anisotropic formation and simulated the spatial distribution characteristics of geometric factors of multicomponent induction logging [35,36]. The operating frequency of electromagnetic logging is generally high, and resistivity is calibrated using attenuation and phase shift. Based on the geometric factor theory, Zhou et al. investigated the phase-shift response function of the EPRL instrument [37]. Xing et al. extended the Born approximation theory of electromagnetic fields to derive the response function of attenuation and phase shift to dielectric constant and conductivity, but without providing a method to quantitatively identify the detection range [38]. Deng et al. defined a sensitivity function for electromagnetic logging while drilling based on difference approximation theory, and they quantitatively described the detection ranges of different frequencies under different resistivity conditions [39]. Zhang et al. studied the attenuation and phase-shift sensitivity of logging while drilling extra-deep azimuthal resistivity measurements based on the extended Born theory [40].
However, current research on sensitivity function is generally based on induction logging or electromagnetic logging-while-drilling responses. The following problems still exist when it is applied to array dielectric logging: (a) Electromagnetic waves decay rapidly due to the strong skin effect at high frequencies, and the dielectric logging response is heavily influenced by the formation parameters; the applicability of the sensitivity function is not clear under these conditions. (b) The influence of the dielectric constant is always neglected in current research, but it is not to be ignored, as the frequency increases in array dielectric logging. (c) There are a lot of data that need to be processed in array dielectric logging curves, and it is necessary to figure out the sensitivity of each detection mode to determine the priority of the inversion parameters. In a word, finding a novel and proper method that can explain the detection characteristics of high-frequency electromagnetic logging tools is important, and clarifying the detection performance of different modes is of significance for the inversion and data processing of the array dielectric tool.
To solve the above problems, this paper extends the electromagnetic sensitivity function to array dielectric logging. It obtains a set of theories to represent the spatial distribution characteristics of attenuation and phase shift. First, the sensitivity function expression of the basic coil system is derived, and the spatial distribution characteristics of its sensitivity function in different detection modes are simulated and analyzed in a homogeneous medium under different electrical parameters. Simultaneously, the depth of investigation is quantitatively characterized by a radial integral sensitivity function, and its applicability is investigated. Finally, the vertical resolution of the array dielectric logging is investigated using various layer thickness models.

Method
As shown in Figure 1b, the array dielectric logging tool selects four operating frequencies covering the frequencies of different polarization mechanisms. F0 = 20 M, F1 = 200 M, F2 = 500 M, and F3 = 1 GHz are the specific frequencies. Additionally, symmetrically distributed two-transmitter and eight-receiver coils are employed for measurement. The spacing between each transmitting coil is approximately 1 inch (0.0254 cm), and the spacing between the TA and RA1, TB, and RB1 is 1.5 inches (0.0381 cm). Each transmitting and receiving coil set contains horizontal and vertical coils that can be taken as magnetic dipoles. When the instrument is turned on, the voltage signals of various xx and zz components are measured. For the zz component, each set of transmitting and receiving coils can be simplified to the structural combination shown in Figure 1c, and all frequency and source spacing (L) combinations are shown in Table 1. For example, F0L0 means a four-coil system operated at 20 MHz and with source spacing of [−2, −0.5, 0.5, 2] inches.  To investigate the electromagnetic propagation characteristics and detection sensitivity to the surrounding area of various dielectric scanner detection modes, as shown in Figure 1c, we set the location of the receiver at r T and the receiver at r, and the coil was regarded as a magnetic dipole. When there is an abnormal body in the formation, the complex conductivity can be equivalent to σ = σ b + δσ, σ = σ* − iωε, where σ b is the background conductivity, and δσ is the change in the abnormal body conductivity compared with the background medium. The electric field changes caused by the abnormal body can be written as follows [33,34,38]: E p is the primary field when the dipole source is located at a homogeneous background medium. E(r ) is the electric field inside the abnormal body; it can be replaced by the background field E p (r ) based on Born approximation. G(r,r ) is a green function contributed to a receiving signal from an abnormal body; it can be calculated by placing a dipole source at r and computing the response at r. The expression of the primary field and green function in the homogeneous formation can be derived analytically [35]. The voltage in the receivers can be expressed as follows:   The combination of Equation (1) and the definition of attenuation and phase shift based on a single-transmitter double-receiver coil system can be written as follows: where A and ∆θ represent the attenuation and phase shift, respectively. V A , V B can represent any two receiving signals. The geometric factor of electromagnetic logging can be obtained by deriving the logarithm of the above formula.
For the symmetrical coil system shown in Figure 1c, the sensitivity function of array dielectric logging can be expressed as follows: To analyze the depth of investigation (DOI) of each measurement mode of array dielectric logging, the radial integral sensitivity function can be obtained by integrating the sensitivity function S of dielectric logging. Under this situation, the abnormal body (V s ) is integrated into an abnormal zone that extends vertically and indefinitely; the formation becomes a radially symmetric two-layered model; the complex conductivity of the inner layer is σ + δσ, while that of the outer layer is σ; S HI corresponds to the radius of the interface between two layers (r), and it is normalized between [0, 1] when r changes from 0 to ∞; the DOI is the corresponding r when S HI meets the average line.
To verify this algorithm, we established a cylindrical two-layer model, as shown in Figure 2a, including the inner layer (σ i , ε i ) and the outer layer (σ t , ε t ). A pseudo-geometric factor G was defined due to the responses during the change in radial depth r, as shown in Equation (7), where f is the response of dielectric logging (Att or PS), f (σ i , ε i ) and f (σ t , ε t ) represent the response in a homogeneous formation with (σ i , ε i ) and (σ t , ε t ), respectively, and f (σ t , ε t; r, σ i , ε i ) is the response when both layers exist. The electromagnetic field of a magnetic dipole located in a cylindrical two-layer formation can be calculated by a pseudo-analytical solution based on a general transmission and reflection matrix [23,41]. in Equation (7), where f is the response of dielectric logging (Att or PS), f(σi, εi) and f(σt, εt) represent the response in a homogeneous formation with (σi, εi) and (σt, εt), respectively, and f(σt, εt; r, σi, εi) is the response when both layers exist. The electromagnetic field of a magnetic dipole located in a cylindrical two-layer formation can be calculated by a pseudo-analytical solution based on a general transmission and reflection matrix [23,41]. Next, we calculated the results of the Array Dielectric Logging Tool by both the radial integrated sensitivity function SHI and the pseudo-geometric factor G, as shown in Figure  3b. As for the radial integrated sensitivity function SHI, we set R = 1 Ω·m, εr = 1; meanwhile, as for the pseudo-geometric factor G, (σt, εt) = (1, 1), and (σi, εi) must be close to (σt, εt), where a minor difference of 1% is imposed that (σi, εi) = (1 + 1%)(σt, εt), because only when the electrical difference between the two layers is small enough are the results of SHI and G theoretically equivalent. As shown in Figure 3b, the results of SHI (solid lines for phase shift and dashed lines for attenuation) and G (solid cycles for phase shift and hollow cycles for attenuation) were consistent with one another in different detection modes; thus, the effectiveness of the algorithm was validated.

Results and Discussion
Figuring out the relative contribution of each formation part around the borehole to the logging response is important for the accurate evaluation of the formation parameters. The Array Dielectric Logging Tool has many detection modes with different frequencies and source spacings; the detection characteristics of each mode are simulated and dis- Next, we calculated the results of the Array Dielectric Logging Tool by both the radial integrated sensitivity function S HI and the pseudo-geometric factor G, as shown in Figure 3b. As for the radial integrated sensitivity function S HI , we set R = 1 Ω·m, ε r = 1; meanwhile, as for the pseudo-geometric factor G, (σ t , ε t ) = (1, 1), and (σ i , ε i ) must be close to (σ t , ε t ), where a minor difference of 1% is imposed that (σ i , ε i ) = (1 + 1%)(σ t , ε t ), because only when the electrical difference between the two layers is small enough are the results of S HI and G theoretically equivalent. As shown in Figure 3b, the results of S HI (solid lines for phase shift and dashed lines for attenuation) and G (solid cycles for phase shift and hollow cycles for attenuation) were consistent with one another in different detection modes; thus, the effectiveness of the algorithm was validated.

Results and Discussion
Figuring out the relative contribution of each formation part around the borehole to the logging response is important for the accurate evaluation of the formation parameters. The Array Dielectric Logging Tool has many detection modes with different frequencies and source spacings; the detection characteristics of each mode are simulated and discussed in this section, including the two-dimensional sensitivity, depth of investigation, and vertical resolution. This might be helpful for the processing of array dielectric logging data.

Two-Dimensional Sensitivity Distribution
Based on the formation model shown in Figure 1c, we simulated the sensitivity functions when the location r of the abnormal body changed; since the tool was simplified as a coil array placed along the z-axis, the electromagnetic fields E p (r ) and G(r,r ) in Equation (1) were rotationally symmetrical, leading to the rotational symmetry of the sensitivity function. Thus, we only present the results in the rOz plane. The two-dimensional sensitivity distribution of attenuation and phase shift under different detection modes are modeled below.
Set R = 1 Ω·m, ε r = 1. Figure 3 depicts the results. The sensitivity distribution is symmetric in the higher and lower ranges, and it diffuses in the radial direction. The sensitive area is within 10 cm in this case. The detection range widens radially and the detection depth is deeper in the long-source-spacing mode. When the frequency is high, the sensitivity function is more concentrated than in F3L0, and the detection depth is shallow but the sensitivity is high. At the same time, when the left and right maps are compared, the amplitude distribution is more dispersed, the detection range is larger than the phase shift, and it is highly affected by the surrounding strata. The phase shift's sensitive area is centered at the well, and the sensitivity ranges change slightly longitudinally, with a good detection resolution.
By varying the formation resistivity and dielectric constant, the distribution of the phase shift sensitivity function in the F1L0 detection mode was investigated. As shown in Figure 4, the relative dielectric constant on the left is 1, while that on the right is 40, and the resistivity from top to bottom is 0.1, 1, and 100 Ω·m, respectively. In Figure 4a,b, they look similar with different dielectric constants when the resistivity is 0.1 Ω·m. It can be seen that under low-resistivity conditions, the dielectric constant has little effect on the phase shift. When the resistivity increases, the detection range increases significantly. At the same time, under high-resistivity conditions, when the dielectric constant is large, the detection range of the phase-shift signal oscillates slightly and diffuses outward. tivity function is more concentrated than in F3L0, and the detection depth is shallow but the sensitivity is high. At the same time, when the left and right maps are compared, the amplitude distribution is more dispersed, the detection range is larger than the phase shift, and it is highly affected by the surrounding strata. The phase shift's sensitive area is centered at the well, and the sensitivity ranges change slightly longitudinally, with a good detection resolution.  the resistivity from top to bottom is 0.1, 1, and 100 Ω·m, respectively. In Figure 4a,b, they look similar with different dielectric constants when the resistivity is 0.1 Ω·m. It can be seen that under low-resistivity conditions, the dielectric constant has little effect on the phase shift. When the resistivity increases, the detection range increases significantly. At the same time, under high-resistivity conditions, when the dielectric constant is large, the detection range of the phase-shift signal oscillates slightly and diffuses outward.  At the same time, the distribution characteristics of the attenuation sensitivity in the F0L0 and F1L0 detection modes were simulated, as shown in Figures 5 and 6, respectively. Because of the huge shift in the sensitivity magnitude, the color interval in the graph is not uniform. Figure 6 demonstrates that when the resistivity and the dielectric constant increase, the attenuation sensitivity zone diffuses outward, with the diffusion being most visible at a high dielectric constant. Figure 6 shows the attenuation detection sensitivity in F1L0 mode. The dielectric constant increases with frequency, generating apparent rhythmic fluctuations, and the sensitivity distribution is not concentrated.
As illustrated in Figures 7 and 8, the sensitive areas of phase shift and attenuation in the F3L0 detection mode were simulated. Under high-frequency and low-resistivity conditions, the phase-shift signal is affected by the dielectric constant. When the resistivity increases, the dielectric constant has a greater influence, and the signal oscillates strongly. It is not appropriate at this time to describe the detection range of dielectric logging by the sensitivity function.  At the same time, the distribution characteristics of the attenuation sensitivity in the F0L0 and F1L0 detection modes were simulated, as shown in Figures 5 and 6, respectively. Because of the huge shift in the sensitivity magnitude, the color interval in the graph is not uniform. Figure 6 demonstrates that when the resistivity and the dielectric constant increase, the attenuation sensitivity zone diffuses outward, with the diffusion being most visible at a high dielectric constant. Figure 6 shows the attenuation detection sensitivity in F1L0 mode. The dielectric constant increases with frequency, generating apparent rhythmic fluctuations, and the sensitivity distribution is not concentrated.  As illustrated in Figures 7 and 8, the sensitive areas of phase shift and attenuation in the F3L0 detection mode were simulated. Under high-frequency and low-resistivity conditions, the phase-shift signal is affected by the dielectric constant. When the resistivity increases, the dielectric constant has a greater influence, and the signal oscillates strongly. It is not appropriate at this time to describe the detection range of dielectric logging by the sensitivity function.

Depth of Investigation (DOI)
Firstly, the influences of DOI at different frequencies and source spacings were simulated. The resistivity was set to 1 Ω·m and the relative dielectric constant was 1. The abscissa in Figure 9 represents the radial depth, and the ordinate represents the radial

Depth of Investigation (DOI)
Firstly, the influences of DOI at different frequencies and source spacings were simulated. The resistivity was set to 1 Ω·m and the relative dielectric constant was 1. The abscissa in Figure 9 represents the radial depth, and the ordinate represents the radial integral sensitivity function. The figure's four curves correspond to the four source spacings. The curve value is the absolute value and tends to be 1 as the radial depth increases. At the frequency F0 = 20 MHz, the phase-shift DOI was between 0.045 and 0.078 m, and the attenuation DOI was between 0.088 and 0.133 m, as determined by the analysis. The DOI increased linearly with the increase in the source spacing, and the DOI of the amplitude ratio was between 1.5 and 2 times the phase shift. The phase shift DOI was 0.018~0.031 m and the attenuation DOI was 0.032~0.051 m for F3 = 1 GHz. The DOI becomes shallower with the increase in the frequency, and the curve corner becomes obvious. The DOIs of different frequencies and source spacings cover the radial zone between 2 cm and 13 cm under this formation condition. Additionally, some detection modes share the same DOI, making the data more reliable and robust to characterize the formation accurately.
The effect of varying resistivity on the DOI was simulated, as shown in Figure 10. We set resistivity changes from 0.01 Ω·m to 10 Ω·m; the relative dielectric constant was 1, and we selected L3 as the source spacing and simulated the radial integral sensitivity functions at four frequencies. Figure 10 shows that the DOI changes significantly with the change in resistivity. In Figure 10a, the Att DOI changes from 4.3 cm to 24.5 cm when the resistivity changes from 0.01 Ω·m to 10 Ω·m. In Figure 10h, the Att DOI can still reach 8.5 cm at high frequency (F3 = 1 GHz) when the resistivity is 10 Ω·m. However, the DOI of most of the detection modes ranges from 1 to 10 cm. This outperforms typical high-frequency dielectric logging. It is no longer confined to measuring mud cakes and flushing zones around the well, and the detection range is enlarged to include some invasion zones, improving the measurement data's dependability.
We set the relative dielectric constant changes from 1 to 20. The influence of different dielectric constants on the DOI was studied under detection modes F2L0 and F3L0. Additionally, two resistivity values (0.5 Ω·m, 1 Ω·m) were chosen to be simulated. Figure 11a,b show that the PS curve changes more than the Att curve when the dielectric constant changes from 1 to 20; comparing them with panels (c) and (d), the dielectric constant has a greater influence on the DOI at a higher frequency. With the increase in the dielectric constant, the curve seems to oscillate, and this behavior makes the DOI slightly shallower. When the dielectric constant and resistivity continue to rise, the curve's oscillation becomes so intense that it is impossible to quantify its detection properties, particularly in high-frequency detection mode (F2, F3).

Vertical Resolution
A formation model with various layer thicknesses was built to investigate the vertic resolution of array dielectric logging, and the phase-shift and attenuation curves of var ous detection modes were simulated. As shown in Figure 12, the resistivity of the su rounding rock was 4 Ω·m, the relative dielectric constant was 10, the resistivity of the mi

Vertical Resolution
A formation model with various layer thicknesses was built to investigate the vertical resolution of array dielectric logging, and the phase-shift and attenuation curves of various detection modes were simulated. As shown in Figure 12, the resistivity of the surrounding rock was 4 Ω·m, the relative dielectric constant was 10, the resistivity of the middle interlayer was 1.5 Ω·m, and the relative dielectric constant was 17.5. The interlayers were constructed with thicknesses of 0.5, 1, 2, and 4 inches. The red line is the zz component curve, and the black line is the xx component curve. The figure shows that the zz component attenuation curve can distinguish a 1-inch thin layer, that the formation thickness determined by the half-=amplitude point is more accurate, and that the curve amplitude is close to the true value of formation resistivity, whereas the xx component amplitude deviates from the true value when 0.5 and 1 inch are used. The phase shift curve can better depict the thin-layer information. In summary, the coaxial component zz outperforms the coplanar component xx, and the attenuation and phase-shift signals can better provide the formation conductivity and dielectric constant information. Compared with the phase shift, the attenuation is more susceptible to the surrounding rock. Near the formation contact, the coplanar component is substantially nonlinear, and the larger the source spacing, the stronger the nonlinearity. When the longitudinal resolution of dielectric logging is combined with the spatial geometric factor distribution under different detection modes, it is clear that the vertical resolution is related to the distance between the two transmitting coils, and the longitudinal resolution of dielectric logging can reach 1 inch. component a enuation curve can distinguish a 1-inch thin layer, that the formation thickness determined by the half-=amplitude point is more accurate, and that the curve amplitude is close to the true value of formation resistivity, whereas the xx component amplitude deviates from the true value when 0.5 and 1 inch are used. The phase shift curve can be er depict the thin-layer information. In summary, the coaxial component zz outperforms the coplanar component xx, and the a enuation and phase-shift signals can be er provide the formation conductivity and dielectric constant information. Compared with the phase shift, the a enuation is more susceptible to the surrounding rock. Near the formation contact, the coplanar component is substantially nonlinear, and the larger the source spacing, the stronger the nonlinearity. When the longitudinal resolution of dielectric logging is combined with the spatial geometric factor distribution under different detection modes, it is clear that the vertical resolution is related to the distance between the two transmi ing coils, and the longitudinal resolution of dielectric logging can reach 1 inch.

Conclusions
This paper investigated the relative contribution of dielectric constant and resistivity for array dielectric logging responses based on sensitivity function, with the following findings: (1) The DOIs of different frequencies and source spacings cover the radial zone between 1 cm and 15 cm. This outperforms typical high-frequency dielectric logging. It is no longer confined to measuring mud cakes and flushing zones around the well, and the

Conclusions
This paper investigated the relative contribution of dielectric constant and resistivity for array dielectric logging responses based on sensitivity function, with the following findings: (1) The DOIs of different frequencies and source spacings cover the radial zone between 1 cm and 15 cm. This outperforms typical high-frequency dielectric logging. It is no longer confined to measuring mud cakes and flushing zones around the well, and the detection range is enlarged to include some invasion zones, improving the measurement data's dependability. (2) With the increase in the dielectric constant, the curve tends to oscillate, and this behavior makes the DOI slightly shallower. When the dielectric constant and resistivity continue to rise, the curve's oscillation becomes so intense that it is impossible to quantify its detection properties, particularly in high-frequency detection mode (F2, F3). (3) The vertical resolution is related to the spacing between the two transmitting coils and can reach 1 inch, which proves to be able to evaluate the laminated formation.
The detection performance analysis based on several detection modes has an impact on the inversion and data processing of the array dielectric logging measurements. The highsensitivity properties of different detection modes can be studied based on the detection range to reduce the amount of calculation.