Modeling Anisotropic Crosswell Magnetic Responses: A Magnetic-Source Integral Approach with Air-Effect Analysis

Abstract

Crosswell electromagnetic imaging serves as a pivotal method for analyzing the distribution of residual oil in oil and gas reservoirs, as well as for optimizing drilling strategies. Current challenges in crosswell electromagnetic (EM) modeling encompass large-scale discretization, with limited research addressing the effects of magnetic sources in anisotropic media or the influence of borehole air. This study introduces a novel iterative Fourier domain integral algorithm for three-dimensional (3D) magnetic-source magnetic simulation in an anisotropic medium. The proposed method employs the Fourier domain method and quasi-complete Fourier techniques to realize adaptive sampling and efficient 3D modeling. The accuracy and efficiency of the method are validated through models. Parametric analyses quantify the impact of several factors, including source depth, frequency, borehole air effects, and conductivity anisotropy on magnetic field components. For the dynamic monitoring of oil and gas reservoirs, the relationship among the magnetic field, frequency, and water saturation is discussed. Furthermore, comparative response differences between electric and magnetic sources are examined, thereby providing theoretical foundations for real-time EM imaging in anisotropic hydrocarbon reservoirs.

1. Introduction

The ground electromagnetic method is significantly constrained by detection depth and resolution in practical applications. To address these limitations, the crosswell electromagnetic method has been developed. This geophysical technique involves the emission and reception of electromagnetic wave signals in two or more boreholes, utilizing these signals for imaging and detecting the physical properties between the wells. By positioning the transmitter and receiver in deep boreholes, this method achieves substantial transmission distances and detection depths, making it widely applicable in engineering, environmental geophysical exploration, resource monitoring and exploration, coal mine safety, and oil exploration [1,2,3,4,5,6,7,8,9,10].

Numerical simulation serves as an effective method for analyzing the electromagnetic response characteristics between wells. In the initial stages of inter-well electromagnetic numerical simulation development, the lack of measurement data and the underdeveloped theoretical models led to the preference for one-dimensional (1D), two-dimensional (2D), and 2.5-dimensional axisymmetric simplified models for forward numerical simulation. However, as exploration techniques have advanced, there has been an increased demand for enhanced accuracy and efficiency in the forward modeling of complex models [11,12,13,14,15].

The numerical forward algorithms for crosswell electromagnetic response mainly include the finite difference method (FDM), the finite element method (FEM), the finite volume method (FVM), and the integral equation method (IE). The basic idea of the FDM is to replace the differential quotient in the differential equation with the difference quotient. Its advantage is simplicity and flexibility, and it is easy to implement on a computer. Some scholars have implemented three-dimensional (3D) finite difference methods in the time domain and frequency domain of anisotropic media based on cylindrical coordinate systems [16,17,18,19]. Recently, it has been widely applied in wellbore electromagnetic field simulations, real-time drilling interpretation, etc. Its parallelization implementation and boundary condition optimization strategies have further improved the calculation accuracy and efficiency [20,21]. The FEM discretizes differential equations through variational principles and weighted residual formulations, and mesh subdivisions can adopt triangles or tetrahedra, which have advantages for the electromagnetic numerical simulation of logging environments. Its development includes efficient LIN preconditioning technology under low induction number conditions, the decomposition method for non-coordinated grid areas to improve computational efficiency, and 3D well-to-ground electromagnetic modeling with high-order elements and octree grids, etc. [22,23,24,25,26,27]. The FVM discretizes computational domains into non-overlapping control volumes and enforces conservation laws at discrete levels, and it has become instrumental in 3D electromagnetic well-logging simulations within inhomogeneous media over the past two decades. The developments are in the grid, equation solvers, cylindrical coordinate systems, the simulation of ultra-deep resistivity-while-drilling logging, and analysis of the detection distance of 3D targets in anisotropic media [28,29,30,31,32,33].

The integral equation method (IE) developed rapidly in its early stage due to its advantages, such as automatically satisfying the boundary conditions and only requiring subdivision of the outliers. Born approximation, the improved Born sequence, and local nonlinear iterative algorithms, etc., have emerged, which have great advantages when the research area is large, the simulation accuracy requirement is not high, or when solving the far-field [34,35,36,37,38]. For solving large-scale 3D problems, it is currently generally believed that the conjugate gradient method with IE and its improved algorithms are more effective, such as CG-FFT and BGS-FFT combined with the fast Fourier transform algorithm, which greatly reduces the computing time and storage requirements. Zhang and Liu combined the iterationally stabilized double conjugate gradient (BiCGSTAB) technique with the fast Fourier transform (FFT) algorithm (i.e., the BGS-FFT method) to calculate the electromagnetic field of 3D non-uniform medium induction logging, demonstrating the advantages of this method in terms of computational efficiency and accuracy [39]. Abubakar and van den Berg conducted a forward simulation of the electromagnetic integral equation using the CG-FFT method [40]. Subsequent innovations include DTA-BCGS hybrid algorithms [41], pFFT-accelerated matrix solutions [42], and integral equation approximations for reservoir modeling [43]. Recent developments feature weak-form BCGS for anisotropic formations [44], BGS-FFT-optimized (biconjugate gradient stabilized fast Fourier transform) linear systems [45], and Krylov subspace-based domain decomposition methods to minimize computational costs [46]. These FFT-driven methodologies collectively improve the accuracy and efficiency in simulating the large-scale realistic reservoir responses with the IE method.

Despite advancements in numerical methods for crosswell electromagnetic modeling, challenges remain in computational efficiency, where conventional methods suffer from dense discretization requirements limiting large-scale 3D anisotropic simulations. Meanwhile, existing studies predominantly use electric sources and isotropic assumptions, magnetic-source research is limited, and there is ignorance of the borehole air impacts, which our validation shows cause >30% amplitude error in H_x fields.

To address these problems, unlike the existing crosswell electromagnetic numerical simulation methods, this study presents a novel iterative Fourier domain integral algorithm for magnetic-source 3D electromagnetic fields in anisotropic media. It leverages space–wavenumber transformations and a quasi-complete information Fourier transform method to achieve adaptive sampling strategies and efficient calculation. Through verification models, the accuracy and efficiency are validated compared with existing methods. The influences of the depth of the emission source, the operating frequency, the drilling environment (air influence), and the anisotropic parameters of electrical conductivity on the magnetic field response were further analyzed. Considering the impact of the air within the well, the measurement of the secondary magnetic field component along a survey line inside the well may exhibit an average disturbance of up to 30%. For the dynamic monitoring of oil and gas reservoirs, combined with the relationship between reservoir electrical conductivity and water saturation, the law of magnetic field response varying with water saturation and frequency is verified. The differences between electrical sources and magnetic sources and the responses of magnetic sources were compared, providing theoretical support for real-time reservoir monitoring of inter-well electromagnetic imaging.

2. Materials and Methods

2.1. Three-Dimensional Crosswell Magnetic Field Modeling Using an Iteration Method

The crosswell electromagnetic logging model is shown in Figure 1. To study the anisotropic characteristics of the formation of electrical conductivity, this paper assumes that both the transmitting coil and the receiving coil are orthogonal to three axes, and the centers of the coils coincide with the well axis. The dots represent the position points of the transmitting coil and the receiving coil during data acquisition.

The time dependence is set as e^-iωt, and the total magnetic field H(r) can be expressed in terms of the integrals [34]:

$$\mathit{H}\left(\mathit{r}\right)={\mathit{H}}_b\left(\mathit{r}\right)+{\mathit{H}}_s\left(\mathit{r}\right)$$

(1)

$${\mathit{H}}_s\left(\mathit{r}\right)={\displaystyle {\mathbf{\iiint }}_v{\mathit{G}}^{\mathit{H}}\left(\mathit{r},{\mathit{r}}^{\prime }\right)\mathit{J}\left({\mathit{r}}^{\prime }\right)\mathrm{d}{v}^{\prime }}$$

(2)

where H_b(r) represents the primary magnetic field, r = (x, y, z) are the coordinates of the observation points, H_s(r) is the secondary magnetic spatial field, $\mathit{J}\left({\mathit{r}}^{\prime }\right)=\Delta \sigma \left({\mathit{r}}^{\prime }\right)\mathit{E}\left({\mathit{r}}^{\prime }\right)$ is the equivalent current density, r’ = (x’, y’, z’) are the coordinates of the anomalous bodies, and E(r’) is the total electric field. Δσ = σ − σ_b is the conductivity contrast between the anomaly and the background, and σ is the conductivity tensor of anomalous bodies. It can be diagonalized and expressed via three principal conductivities, σ_x, σ_y, and σ_z, and three Euler angles, α_S (strike), α_D (dipping), and α_L (slant) [47]. Here we express it as σ_x/σ_y/σ_z/α_S/α_D/α_L. σ_b is the conductivity of the background. v’ denotes the volume of the anomaly, and G^H(r,r’) is the magnetic Green’s tensors.

To obtain the magnetic field, we should calculate the electric field first. The total electric field E(r) and the integral formula of the secondary electric field E_s(r) can be written as [34]:

$$\mathit{E}\left(\mathit{r}\right)={\mathit{E}}_b\left(\mathit{r}\right)+{\mathit{E}}_s\left(\mathit{r}\right)$$

(3)

$${\mathit{E}}_s\left(\mathit{r}\right)={\displaystyle {\mathbf{\iiint }}_v\mathit{G}\left(\mathit{r},{\mathit{r}}^{\prime }\right)\mathit{J}\left({\mathit{r}}^{\prime }\right)\mathrm{d}{v}^{\prime }}$$

(4)

where E_b(r) represents the primary electric field. G(r,r’) is the electric Green’s tensor.

2.2. Fourier Domain Method to Calculate the EM Field

The Fourier domain method utilizes the characteristics of the convolution, the efficiency of the Fourier transforms, the high accuracy of one-dimensional integration, and the stability of the iteration procedures to achieve the 3D EM modeling in anisotropic media. Using the horizontal 2D Fourier transforms, the convolution Equations (2) and (4) are converted to 1D integrals, as follows:

$$\tilde{H}\left(k_x,k_y,z\right)={\displaystyle \int {\tilde{\mathit{G}}}^{\mathit{H}}\left(k_x,k_y,z,z^{\prime }\right)\tilde{\mathit{J}}\left(k_x,k_y,z^{\prime }\right)d{z}^{\prime }}$$

(5)

$$\tilde{E}\left(k_x,k_y,z\right)={\displaystyle \int \tilde{\mathit{G}}\left(k_x,k_y,z,z^{\prime }\right)\tilde{\mathit{J}}\left(k_x,k_y,z^{\prime }\right)d{z}^{\prime }}$$

(6)

where ${\tilde{\mathit{H}}}_{\mathit{s}}$ and ${\tilde{\mathit{E}}}_{\mathit{s}}$ represent the secondary magnetic and electric fields in the Fourier domain, respectively; k_x and k_y are the wavenumbers in the wavenumber domain; ${\tilde{\mathit{G}}}^{\mathit{H}}$ and $\tilde{\mathit{G}}$ are the electric and magnetic Green’s tensors in the Fourier domain, respectively; and $\tilde{\mathit{J}}$ indicates the equivalent current density in the Fourier domain.

To calculate the 1D integrals, Equations (5) and (6), the quadratic function is used to fit the equivalent current density $\tilde{\mathit{J}}$ in each element. Then, we can obtain the analytical expression of the element integral combined with the exponential term in the Green’s tensors. After calculating all the wave numbers of the 1D integral, the space domain field can be obtained via an inverse Fourier transform. The 1D integral numercial method is similar to paper [48], so it is not covered here.

We adopt an iterative method with a contraction operator to achieve an accurate convergence solution for the electric field:

$${\mathit{E}}^{\left(n\right)}={\mathit{E}}_b+G\left(\mathbf{\Delta }\mathit{\sigma }\mathit{\cdot }{\mathit{E}}^{\left(n-1\right)}\right)$$

(7)

where E⁽ⁿ⁾ represents the total electric field calculated for the nth time, G(·) is a linear operator, and E⁽ⁿ⁻¹⁾ represents the value from the previous iteration (set as 0 for the first iteration); the background electric fields are chosen as the iteration initial value.

The operator G is modified based on the energy inequality according to Singer [49] and Pankratov’s theory [50,51], and then the contract operator can be written as:

$${\widehat{\mathit{E}}}^{\left(n\right)}=a\cdot \mathit{\alpha }\cdot {\mathit{E}}_b+\mathit{a}\cdot \mathit{\alpha }\cdot G\left(2\sqrt{{\sigma }_b}\mathit{\beta }\cdot {\widehat{\mathit{E}}}^{\left(n-1\right)}\right)+\mathit{\beta }\cdot \mathit{a}\cdot {\widehat{\mathit{E}}}^{\left(n-1\right)}$$

(8)

where the variable $\widehat{\mathit{E}}$ is an intermediate variable, and the remaining $\mathit{\alpha },\hspace{0.33em}\mathit{\beta },\mathit{\hspace{0.33em}}a$ can be obtained by:

$$\alpha =2{\sigma }_b{\left(2{\sigma }_b+\Delta \sigma \right)}^{-1},\hspace{0.33em}\mathit{\beta }=I-\mathit{\alpha },\hspace{0.33em}\widehat{\mathit{E}}=a\cdot \mathit{E},\hspace{0.33em}a={\displaystyle \frac{2{\sigma }_b+\Delta \sigma }{2\sqrt{{\sigma }_b}}}$$

(9)

After obtaining the electric fields, the magnetic fields can be obtained via the relationships in Equation (5).

2.3. Quasi-Complete Information Fourier Transform Method

The Fourier transform plays a crucial role in the Fourier domain algorithm, and its precision significantly affects the accuracy of the modeling response. We use a two-dimensional quasi-complete information Fourier transform method [52] to employ the Fourier transform process. Taking the two-dimensional forward Fourier transform as an example, it can be written as follows:

$$F(k_x,k_y)={\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle {\int }_{-\infty }^{\infty }f(x,y)e^{-i(k_{x}x+k_{y}y)}dxdy}}$$

(10)

where $f(x,y)$ is the field in the spatial domain, and $F(k_x,k_y)$ represents the field in the Fourier domain.

In numerical applications, the x and y direction ranges are limited, and we set the x direction range as [$x_a~x_b$] and the y direction range as [$y_a~y_b$].

$$F(k_x,k_y)={\displaystyle {\int }_{y_a}^{y_b}{\displaystyle {\int }_{x_a}^{x_b}f(x,y)e^{-i(k_{x}x+k_{y}y)}dxdy}}$$

(11)

Equation (11) can be transformed as two one-dimensional integrals:

$$F_x(k_x,y)={\displaystyle {\int }_{x_a}^{x_b}f(x,y)e^{-i{k}_{x}x}dx}$$

(12)

$$F(k_x,k_y)={\displaystyle {\int }_{y_a}^{y_b}{F}_x(k_x,y)e^{-i{k}_{y}y}dy}$$

(13)

where $F_x(k_x,y)$ is the spectrum of the Fourier domain in the x direction.

Discretize the calculation domain using either uniform or non-uniform grids along the x and y directions, dividing it into N_x × N_y elements. Equation (12) can be written as:

$$F_x(k_x,y)={\displaystyle \sum _{j=1}^{M_x}{\displaystyle {\int }_{e_j}f(x,y)e^{-i{k}_{x}x}dx}}$$

(14)

where e_j represents the jth element, the coordinates of three nodes in the jth element are (x_j1, x_j2, x_j3), and x_j2 is the midpoint, $x_{j1}+x_{j3}=2x_{j2}$. Using the quadratic shape function to fit the function f(x,y) in each element,

$$f(x,y)=N_{j1}f(x_{j1},y)+N_{j2}f(x_{j2},y)+N_{j3}f(x_{j3},y)$$

(15)

where $N_{j1}={\displaystyle \frac{\left(x_{j3}+x_{j1}-2x\right)\left(x_{j3}-x\right)}{{\left(x_{j3}-x_{j1}\right)}^2}}$, $N_{j2}=4{\displaystyle \frac{\left(x_{j3}-x\right)\left(x-x_{j1}\right)}{{\left(x_{j3}-x_{j1}\right)}^2}}$, and $N_{j3}={\displaystyle \frac{\left(2x-x_{j1}-x_{j3}\right)\left(x-x_{j1}\right)}{{\left(x_{j3}-x_{j1}\right)}^2}}$.

Substitute Equation (15) into Equation (14), and the final expression can be written as

$$F_x(k_x,y)={\displaystyle \sum _{j=1}^{M_x}{\displaystyle {\int }_{e_j}f(x,y)e^{-i{k}_{x}x}dx}}={\displaystyle \sum _{n=1}^{2M_x+1}I(x_n,k_x)\cdot f(x_n,y)}$$

(16)

where $I(x_n,k_x)$ represents the integration coefficient of the Fourier transform at the nth node along the x direction. The expressions of the coefficient in each element are provided in Appendix A.

Similarly, Equation (13) can be written as follows:

$$F(k_x,k_y)={\displaystyle \sum _{j=1}^{M_y}{\displaystyle {\int }_{e_j}{F}_x(k_x,y)e^{-i{k}_{y}y}dy}}={\displaystyle \sum _{m=1}^{2M_y+1}j(y_m,k_y)\cdot f(k_x,y_m)}$$

(17)

where $j(y_m,k_y)$ represents the integration coefficient of the Fourier transform at the mth node along the y direction. It will not be elaborated here.

The quasi-complete information Fourier transform method can adopt uniform or non-uniform grids in both spatial and wavenumber domains, which can provide flexible sampling. In the wavenumber domain, sampling can be carried out using either uniform interval sampling or logarithmic interval sampling according to paper [52]. Analyze the variation law of the electromagnetic field spectrum, and combine the uniform and logarithmic sampling rules. Encrypted sampling is carried out at wavenumbers with drastic spectrum changes, while sparse sampling is conducted at wavenumbers with gentle changes, which can effectively enhance computational accuracy and efficiency.

2.4. Flow of the Method

The computer used in this study is an Intel(R) Core(TM) i7-6700HQ CPU, with 8 cores, a main frequency of 2.60 GHz, and a memory of 16.0 GB. The programming language is Fortran, and the Fortran code is parallelized with OpenMP. The flow of the algorithm is as follows:

(1)

Initialization: n = 0, calculate the background electric field and magnetic field E_b, H_b, assign values to the scattering field of the first iteration ${\mathit{E}}_s^{\left(n\right)}=0$, and calculate the total electric field ${\mathit{E}}^{\left(n\right)}={\mathit{E}}_b+{\mathit{E}}_s^{\left(n\right)}$;

(2)

Calculate the scattering field: n = n + 1. Calculate the scattering current term $\mathit{J}=\Delta \sigma \cdot {\mathit{E}}^{\left(n\right)}$. Use the irregular Fourier transform mentioned in Section 2.3 and perform a 2D Fourier transform in the horizontal direction. Then substitute the scattering current term in the Fourier domain into Equation (6), calculate the 1D integral of Equation (6) under different wave numbers, and then perform the inverse Fourier transform to obtain the scattering electric field in the spatial domain ${\mathit{E}}_s^{\left(n\right)}$;

(3)

Update the total electric field: Calculate the total electric field ${\mathit{E}}_s^{\left(n\right)}$ using the newly calculated scattering electric field ${\mathit{E}}^{\left(n\right)}={\mathit{E}}_b+{\mathit{E}}_s^{\left(n\right)}$, and update the total electric field ${\mathit{E}}^{\left(n\right)}$ using the compression operator Equations (8) and (9).

(4)

Judge the stopping condition for the iteration $\left|{\displaystyle \frac{{\displaystyle \sum _N\left|{\mathit{E}}^{(n+1)}\right|}-{\displaystyle \sum _N\left|{\mathit{E}}^{(n)}\right|}}{{\displaystyle \sum _N\left|{\mathit{E}}^{(n+1)}\right|}}}\right|<{10}^{-4}$, where N represents the number of calculation nodes, and |·| represents the absolute value operation.${\displaystyle \sum _N\left|{\mathit{E}}^n\right|}$ and ${\displaystyle \sum _N\left|{\mathit{E}}^{n+1}\right|}$ represent the sum of the absolute value of the total electric fields of all nodes calculated for the nth, and (n + 1)th time, respectively. The iteration termination condition is that the relative errors of the sum of the values of the three components of the electric field in two adjacent iterations are all less than 10⁻⁴. When the errors do not meet the iteration termination condition, the iteration is returned to step (2). Otherwise, the iteration terminates and continues to execute.

(5)

Calculate the magnetic field: To calculate the scattering current term $\mathit{J}=\Delta \sigma \cdot {\mathit{E}}^{\left(n\right)}$, the irregular forward and reverse Fourier transforms mentioned in Section 2.3 are adopted. The two-dimensional Fourier transform in the horizontal direction is performed. Then, the scattering current term in the Fourier domain is brought into Equation (5) to calculate the 1D integral of Equation (5) under different wavenumbers. Then, the inverse Fourier transform is carried out to obtain the scattering electric field in the spatial domain ${\mathit{H}}_s^{\left(n\right)}$ and calculate the total magnetic field ${\mathit{H}}^{\left(n\right)}={\mathit{H}}_b+{\mathit{H}}_s^{\left(n\right)}$.

3. Results

3.1. Method Validation

We first verify the proposed algorithm. The model is shown in Figure 2. The conductivities of the air and underground are 10⁻⁸ S/m and 0.1 S/m, respectively. The calculation range is [−120 m, 120 m] in the x and y directions and [0 m, 300 m] in the z direction. The number of subdivision nodes is 27 × 27 × 25. The horizontal grid utilizes non-uniform spacing, specifically 2 m within the well and 25 m for the anomalous body. The vertical grid utilizes uniform spacing of 12.5 m. The transmitter coil and receiver coil are in two wells each. The center points of the two wells are symmetrical at the origin, and the distance is 200 m. The transmitter coil is in one well at a position of (−100, 0, 100). The receiver coils are deployed along the second well [0 m, 300 m] with horizontal coordinates of (100 m, 0 m). Both wells feature 4 m × 4 m cross-sections along 300 m lengths. A cubic anomaly with a size of 50 × 50 × 50 m³, centered beneath the origin with its top surface at 150 m depth, is set as isotropic/anisotropic. The frequency is 500 Hz.

When the transmitter and receiver coils are coplanar in the xz plane, only specific magnetic field components exhibit non-zero responses. X-oriented transmitters induce H_x and H_z components at the receiver, which are denoted as H_xx and H_zx, respectively. Y-oriented ones generate solely H_y, denoted as H_yy, while Z-oriented sources produce H_x and H_z components, which are represented as H_xz and H_zz, respectively. For the isotropic anomaly model, the conductivity of the anomaly is 0.004 S/m. We compare the numerical solutions between the proposed method (Fourier domain IE) and the IE method in INTEM3D software (INTEM3DQL Version 1.0) [53]. For the anisotropic anomaly model, the conductivity of the anomaly is 0.001/0.0001/0.0005/30°/60°/0° S/m. We compare the numerical solutions between the proposed method (Fourier domain IE) and COMSOL (COMSOL Multiphysics^® 6.1).

Figure 3 and Figure 4 show the comparison of the five magnetic fields with an isotropic anomaly and an anisotropic anomaly model, respectively. As can be seen, there is little difference between the different numerical solutions. For the isotropic model, the maximum relative errors between our data and those from the other two methods with five magnetic fields are 1.27, 1.35, 1.23, and 1.49%, respectively. For the anisotropic model, the maximum relative errors between our data and those from the other two methods with five magnetic fields are 1.54, 1.78, 1.89, and 1.65%, respectively. The errors are acceptable. The results demonstrate the correctness of the proposed Fourier domain IE algorithm for both 3D isotropic and anisotropic models.

Utilizing the quasi-complete information Fourier transform method necessitates an analysis of the spectral distribution. Through extensive testing, we have determined that the spectrum of the electromagnetic field is symmetrically distributed around the zero wavenumber [54]. Our sampling strategy incorporates two critical enhancements:

(1)

The implementation of symmetric positive–negative wavenumber pairing to leverage spectral symmetry;

(2)

Adaptive logarithmic sampling within the wavenumber range of (0.001~0.1), where spectral energy exhibits steep gradients (approximately 85% of the total energy), transitioning to sparse sampling in the wavenumber range of (0.1~1.5), which demonstrates gradual spectral decay (15% residual energy). The final count of wavenumbers is 101 × 101.

For the isotropic model, the forward evolution iterations of the magnetic sources in the x, y, and z directions are 312, 288, and 354 times, respectively. The total forward modeling time across the three directions is approximately 25.42 s, utilizing about 37.3 MB of memory. In the case of the anisotropic model, the forward modeling iterations of magnetic sources in the x, y, and z directions are 331, 324, and 366 times, respectively. The total time for magnetic sources in the three directions is approximately 28.14 s, occupying about 40.8 MB of memory, thereby demonstrating the convergence of the algorithm. Furthermore, we conduct a comparison with COMSOL under the parameter conditions of the anisotropic model to assess efficiency. The boundary has been expanded to five times its original size, and an unstructured mesh has been employed to achieve the desired calculation accuracy. The algorithm requires 653.7 s to execute and utilizes approximately 8.4 GB of memory, thereby highlighting the advantages of the proposed algorithm.

3.2. Different Parameters of the Coil Source and Drilling Environment

This section investigates the influence of coil source parameters, like transmitter position and frequency, and borehole air conditions, on crosswell magnetic responses. In practical applications, the primary field intensity generated directly by the source typically exceeds the secondary field induced by anomalies by 0.5–2 orders of magnitude, making it challenging to discern subtle parameter influences through total field observation. To address this limitation, our comparative analysis focuses exclusively on secondary field components obtained through numerical simulation.

The simulation adopts the validated anisotropic reservoir model established in Section 3.1. Excitation frequencies span four decades: 1, 10, 100, and 1000 Hz. Transmitter coil positioning follows a depth profile at 100, 150, 200, and 250 m within the source well at coordinates (−100, 0, 100). Receiver coils are distributed vertically from 0 to 300 m depth in the monitoring well at the surface position (100, 0). As illustrated in Figure 5, Figure 6, Figure 7 and Figure 8, the distribution patterns of secondary magnetic fields associated with anisotropic high-resistance anomalies were systematically investigated under varying conditions, including transmitter positions, frequencies, and the presence of air-filled borehole effects. The results demonstrate the following:

(1)

Transmitter position effects: H_xx component—The minimum value position in the H_xx secondary magnetic field response shifts depending on the relative depth between the transmitter and the anomalous body. When the transmitter is positioned above the anomaly, the response minimum occurs below the target zone. Conversely, when the transmitter is located below the anomaly, the minimum appears above it. This characteristic enables effective delineation of the anomaly’s vertical boundaries. H_xz component—The H_xz response (representing the H_x field generated by a z-oriented transmitter) displays a distinct minimum directly coinciding with the anomaly’s position, providing precise vertical localization. Common features in H_yy, H_zz, and H_zx—These components share similar distribution patterns, with maximum response amplitudes occurring at the anomaly depth, followed by symmetrical upward and downward decay. For the H_zx and H_xz responses, both fields exhibit an increase in amplitude with depth, peaking near the anomaly before exhibiting decay. The distinction lies in the H_zx generated by the horizontal x direction magnetic source, which induces azimuthal current loops that close within the horizontal plane, with energy primarily dissipated as radiation. Conversely, the H_xz produced by the vertical z direction magnetic source generates charges at the boundary of the scattering body when vertically propagating currents encounter conductive discontinuities, resulting in a minimum amplitude near the scattering anomaly.

(2)

Frequency-dependent response: For identical transmitter coil positions, the secondary magnetic field distributions within the borehole exhibit similar profile shapes across different frequencies. Higher frequencies produce increased magnitudes of the secondary field, with amplitude differences spanning approximately one order of magnitude at 1 Hz, 10 Hz, and 100 Hz. Furthermore, the field strength generated at 1000 Hz is marginally lower than that at 100 Hz.

(3)

Borehole air anomaly impacts: The H_zz component remains unaffected by air-filled borehole effects across all tested transmitter positions and frequencies. The H_yy, H_zx, and H_xz components show minor distortions limited to regions near the borehole’s upper and lower extremities, while their responses proximate to the anomaly remain stable. The H_xx component exhibits significant sensitivity to borehole air anomalies, particularly within the borehole environment. This phenomenon is fundamentally attributed to the propagation characteristics of the magnetic field in both the air and the geological formation. From the perspective of geometric coupling, the toroidal flux of H_xx encircles the x-axis, while air wells extend vertically in the z direction. This configuration results in significant orthogonality and radial intersection with the wellbore, thereby maximizing spatial overlap and interaction efficiency. In terms of electromagnetic characteristics, the air well operates as a high-resistivity anomaly (σ_air ≪ σ_background). While H_xx-induced eddy currents naturally attenuate in homogeneous formations, the near-zero conductivity of the well necessitates current detours, which result in significant distortion of the amplitude in proximity to the well, thereby leading to enhanced sensitivity.

Figure 5. Responses to multi-frequency excitations with the transmitting coil positioned at a depth of 100 m.

Figure 5. Responses to multi-frequency excitations with the transmitting coil positioned at a depth of 100 m.

Figure 6. Responses to multi-frequency excitations with the transmitting coil positioned at a depth of 150 m.

Figure 6. Responses to multi-frequency excitations with the transmitting coil positioned at a depth of 150 m.

Figure 7. Responses to multi-frequency excitations with the transmitting coil positioned at a depth of 200 m.

Figure 7. Responses to multi-frequency excitations with the transmitting coil positioned at a depth of 200 m.

Figure 8. Responses to multi-frequency excitations with the transmitting coil positioned at a depth of 250 m.

Figure 8. Responses to multi-frequency excitations with the transmitting coil positioned at a depth of 250 m.

3.3. Different Anisotropic Conductivity Anomaly

This section investigates the anisotropic conductivity response characteristics using the established model. With the coil source positioned at 150 m depth operating at a frequency of 100 Hz, we varied the principal axis conductivities (σ_x: 0.001~100 S/m, σ_y: 0.0001~10 S/m, σ_z: 0.0005~50 S/m) while maintaining α_S = 0°, α_D = 30°, and α_L = 60°. The corresponding secondary magnetic field responses are presented in Figure 9, Figure 10 and Figure 11.

In Figure 9, the analysis indicates that variations in σ_x (ranging from 0.001 to 100 S/m) lead to a decrease of less than 0.2 orders of magnitude in the secondary field response, suggesting limited sensitivity to changes in σ_x. The H_xx, H_yy, and H_xz components exhibit relatively higher sensitivity, particularly showing maximum variation rates in the low-conductivity regime (σ_x ranging from 0.5 to 1 S/m), with response saturation observed beyond this range. In contrast, in Figure 10 and Figure 11, σ_y and σ_z variations demonstrate more pronounced effects. Increasing σ_y (0.0001~10 S/m) produces a gradual 0.3-order magnitude response reduction without significant curve shape alteration. Notably, σ_z modification causes changes in the order of magnitude of the abnormal response and the shape of the curve. The influence of the changes in σ_y and σ_z on the secondary magnetic field is greater than that of the change in σₓ. The reason is that under the given Euler angles (α_S = 0°, α_D = 30°, α_L = 60°), for α_S = 0°, the σ_x axis coincides with the observation axis x. For α_D = 30°, the y-axis is rotated, making σ_z closer to the vertical direction. For α_L = 60°, the new z-axis is rotated, strengthening the σ_y projection in the horizontal plane. The coordinate system rotation makes the y-z plane the dominant vortex plane, making σy and σ_z the dominant parameters of the magnetic field response.

These findings indicate that crosswell magnetic surveys conducted in anisotropic media should prioritize multi-component analysis in conjunction with multi-frequency data integration to enhance sensitivity and address the limitations associated with single-parameter interpretation.

3.4. Dynamic Magnetic Response Analysis of Anisotropic Reservoirs During Waterflooding

In reservoir dynamic monitoring and simulation, water injection processes modify critical parameters such as remaining hydrocarbon saturation and formation porosity, resulting in substantial variations in electrical conductivity. Utilizing field data from an oilfield in western China [55], we develop an anisotropic reservoir model to examine the evolution of magnetic field responses during hydrocarbon extraction for dynamic surveillance. The petrophysical relationships governing the reservoir model are articulated as [55]:

$${\sigma }_{og}={\displaystyle \frac{{\sigma }_w{\phi }^p}{\tilde{a}\tilde{b}}}{\left(S_w\right)}^q$$

(18)

where σ_og denotes the anisotropic conductivity of the hydrocarbon reservoir, σ_w = 0.3 S/m represents pore-water conductivity, and the lithology coefficient $\tilde{a}$ varies within 0.4~1.5. Water saturation S_w and oil saturation S_o satisfy S_o + S_w = 1, with the saturation index q ranging from 1.5 to 2.5 (q = 2 herein). Porosity φ, governed by pore structure and cementation, defines horizontal anisotropy through:

$$\left[\begin{array}{ccc}{\phi }_h& & \\ & {\phi }_h& \\ & & {\phi }_v\end{array}\right]$$

(19)

The horizontal and vertical porosities are set at φ_h = 15% and φ_v = 10%, respectively. The porosity index (typically 1.3–2.5) is maintained at 2. Figure 12 shows the reservoir’s water saturation–electrical conductivity variations, with corresponding numerical values detailed in

Table 1. Anisotropic conductivity variations with water saturation in reservoirs [55].

Water Saturation	20%	40%	60%	80%	100%
Horizontal conductivity (S/m)	0.00027	0.00108	0.00243	0.00432	0.00675
Vertical conductivity (S/m)	0.00012	0.00048	0.00108	0.00192	0.003

. As can be seen in the figure, as the water saturation increases, the reservoir conductivity gradually decreases, approximately following a logarithmic growth pattern.

The model is illustrated in Figure 13. The conductivities of the air and underground are 10⁻⁸ S/m and 0.04 S/m, respectively. The calculation range is [−3500 m, 3500 m] in the x and y directions, and [0 m, 6000 m] in the z direction. The center points of the two wells are symmetrical about the origin, with a separation distance of 6 km. The transmitter coil is positioned in one well at coordinates of (−3000, 0, 3000). The receiving coils are arranged in the other well, spanning from 0 to 6 km, with a horizontal reference point at (3000, 0). The dimensions of both wells are 4 × 4 × 6000 m³. The horizontal grid is non-uniformly divided, with a grid spacing of 2 m within the well and 100 m within the anomaly body. The grid in the z direction is uniformly divided with a spacing of 100 m. The number of subdivision nodes is 73 × 73 × 61, and the number of wavenumbers is 201 × 201. There is an anisotropic oil reservoir dipping slab. The isotropic block is 1 km long, 1 km wide, and 2 km high. The dip angle is 45°. The anomaly body is centered directly below the origin, and the top surface has a depth of 2 km. The calculated frequencies are 0.1, 1, and 10 Hz. Both electric and magnetic dipole sources are employed, each defined with a unit moment magnitude (1 A·m for electric dipoles; 1 A·m² for magnetic dipoles).

Figure 14, Figure 15 and Figure 16 present the secondary magnetic field responses to electric and magnetic sources versus water saturation at 0.1, 1, and 10 Hz, respectively. The following observations can be made from the figures:

(1)

The amplitude of the secondary magnetic field generated by both electric and magnetic dipoles decreases with increasing water saturation. This phenomenon is attributed to the enhanced conductivity of the medium, which reduces the contrast between the reservoir conductivity σ_og and the background conductivity σ_background, thereby diminishing the effects of magnetic induction. This response characteristic demonstrates that the secondary magnetic field can serve as a sensitive indicator of water saturation changes, providing a new method for monitoring water invasion in oilfield development. Notably, as concluded in Section 3.2, the H_xx component variation patterns are different due to the influence of the air inside the drilling well, as shown in Figure 16b. This needs to be corrected in practical applications.

(2)

As the frequency increases, the magnitude of the secondary magnetic field values generated by both the electrical source and the magnetic source decreases and is determined by their attenuation. Meanwhile, the forms of secondary magnetic fields at different frequencies are different. The lower the frequency, the smoother the response curve.

(3)

The secondary magnetic fields generated by electric and magnetic dipole sources exhibit reciprocal correspondence: the H_xx, H_zz, H_zx, and H_xz components from electric sources mirror the H_xz, H_zx, H_zz, and H_x components from magnetic sources, respectively. This field symmetry substantiates the fundamental electromagnetic duality in 3D space. This symmetrical relationship can guide the design of observation systems, enhance data completeness through the combination of complementary sources, and provide cross-validation constraints for inversion algorithms. Furthermore, the magnetic field components generated by unit electric dipole sources and magnetic dipole sources reveal that the magnitude of the magnetic field produced by the magnetic source surpasses that of the electric source, highlighting the advantages of magnetic sources in monitoring between oil and gas wells.

Regarding efficiency, the anisotropic waterflooding reservoir model extends across [−3.5 km, 3.5 km] in both the x and y directions, with a wellbore dimension of 4 × 4 m². Conventional Fourier transform techniques [56] utilize uniform sampling, necessitating 3501 × 3501 × 61 uniform grid nodes, which incurs substantial computational costs. In contrast, our quasi-complete Fourier transform method employs 73 × 73 adaptive non-uniform sampling in the horizontal domain, utilizing logarithmic discretization (10⁻⁵~10⁻³) in regions of high spectral-energy gradients, transitioning to sparse sampling (10⁻³~10⁻¹). The final wavenumber count is 201 × 201. Under this configuration, the computation time for the magnetic field responses of the oil and gas reservoir model, with water saturation at three distinct frequencies at a given moment, is 360.15 s, and the memory usage is 1004.9 MB, thereby demonstrating the efficiency of the proposed algorithm.

Figure 14. Response of the secondary magnetic field for different water saturations at 0.1 Hz. (a) The electric source, (b) the magnetic source.

Figure 14. Response of the secondary magnetic field for different water saturations at 0.1 Hz. (a) The electric source, (b) the magnetic source.

Figure 15. Response of the secondary magnetic field for different water saturations at 1 Hz. (a) The electric source, (b) the magnetic source.

Figure 15. Response of the secondary magnetic field for different water saturations at 1 Hz. (a) The electric source, (b) the magnetic source.

Figure 16. Response of the secondary magnetic field for different water saturations at 10 Hz. (a) The electric source, (b) the magnetic source.

Figure 16. Response of the secondary magnetic field for different water saturations at 10 Hz. (a) The electric source, (b) the magnetic source.

4. Conclusions

This study introduces a novel iterative Fourier domain integral algorithm that leverages the Fourier domain method and the quasi-complete information Fourier transform method to achieve adaptive sampling strategies for 3D magnetic field modeling based on magnetic sources in anisotropic media. Here are the conclusions:

(1)

Validations conducted against established methodologies, including the integral equation approach and COMSOL simulations, indicate that the maximum relative errors for both isotropic and anisotropic 3D models remain below 2%, thereby affirming the accuracy of the algorithm. Compared to the efficiency of COMSOL, under the same model, COMSOL requires 653.7 s and approximately 8.4 GB of memory, whereas the proposed method requires 28.14 s, occupying about 40.8 MB of memory, thereby highlighting the advantages of the proposed algorithm.

(2)

① Higher frequencies enhance field magnitude while maintaining similar distribution patterns; ② H_xx minima shift inversely with source position relative to anomalies, whereas H_xz minima pinpoint anomaly locations; and ③ air effects negligibly impact H_zz but significantly alter H_xx responses. Practical applications should prioritize H_yy, H_zz, H_zx, and H_xz components to minimize modeling complexity when disregarding borehole air influences.

(3)

For the dynamic monitoring of oil and gas reservoirs, ① increased water saturation reduces the conductivity contrast between the anomaly and the background, and secondary magnetic field amplitudes from both electric source and magnetic source types attenuate. ② Frequency dependence: Field magnitudes decrease with rising frequency (0.1–10 Hz) while response curves smooth at lower frequencies, consistent with EM attenuation mechanisms in conductive media. ③ Source symmetry and sensitivity: Electric and magnetic sources demonstrate field component reciprocity (H_xx (ele)→H_xz (mag), H_zz (ele)→H_zx (mag), H_zx (ele)→H_zz (mag), and H_xz (ele)→H_xx (mag)), with magnetic sources exhibiting higher signal magnitude, confirming their advantage for crosswell reservoir monitoring.

These results indicate the advantages of the magnetic source in monitoring oil and gas wells and provide critical guidance for electromagnetic exploration optimization in anisotropic environments. In subsequent research, we intend to systematically examine the effects of wellbore geometry—including geometrical shape, drilling trajectory deviation, wall roughness, and non-uniform packing—on the outcomes of numerical simulations. Concurrently, we will apply the algorithm to inversion and investigate the influence of measurement uncertainties, such as noise sources, sensor errors, and instrument limitations, on the performance of the inversion algorithm.