Induction Logging Response Research Articles

Machine-Learning-Based Convolution Method for Fast Forward Modeling of Induction Log

We built a convolution model using machine learning (ML) to calculate induction log responses for one-dimensional (1D) earth models. Compared to analytical forward modeling, the convolution model is extremely fast. ML-based convolution finds accurate induction tool responses from an earth model with layer resistivity and layer boundaries. For a unit induction tool 2C40, the 101-point, 50-ft window convolution model works satisfactorily for a well deviation angle of 60.

Numerical Simulation, Response Analysis, and Physical Experiment of Induction Logging in an Inclined Fractured Formation

The majority of previous studies of induction logging in fractured formations have typically focused on the variation in spatial resistivity distribution around the wellbore. However, the effect of cumulative charge on the boundaries of an inclined fracture with a distinct morphology on induction logging has rarely been discussed. In this work, the 3-D finite element method (3-D FEM) is adopted to investigate the dual-induction logging response of inclined fractures in formations of different resistivities considering the volume contribution and charge contribution. First, induction logging theory is studied, and a 3-D fractured formation model considering cumulative charges is established. Second, the induction logging responses under various fracture resistivities, apertures, dip angles, and formation resistivities are simulated and analyzed. Subsequently, the corresponding logging response characteristics are studied, and the different response laws of inclined fractures in high- and low-resistivity formations are systematically summarized. Third, a 1:5 miniaturized dual-induction logging instrument was designed and manufactured. Furthermore, two devices were built to measure the induction logging response of the inclined fracture (simulated by nanoscale silver film or metal wire mesh) in low- and high-resistivity formations. The results of the physical experiments are consistent with the numerical results, validating the accuracy of the numerical simulation. Both the results reveal that due to the existence of cumulative charge layers on the upper and lower inclined fracture boundaries, the induction logging response rules of the conductive fractures in high-resistivity formations differ from those in low-resistivity formations, and hence, they should be treated differently.

Modeling of triaxial induction logging responses in multilayered anisotropic formations

Triaxial induction logging tools have been widely applied to formation characterization due to their sensitivity to electric anisotropy. To model triaxial induction logs in multilayered general anisotropic formations, where the anisotropy can be arbitrary, an analytical method is applied to compute the tool responses. For the analytical method, Maxwell’s equations in the spectral domain are written into a compact first-order differential equation. The equation is solved to obtain the spectral-domain fields, which are transformed to the spatial domain through the inverse Fourier transform. The singular issue for the tool located in highly deviated wells is handled by subtracting the singular term in the spectral domain. The singularity treatment makes the integrands in the inverse Fourier transform decay faster, thus making the infinite integration computation faster. Formations with isotropic, transversely isotropic, biaxially anisotropic, and general anisotropic conductivity are modeled and compared to investigate the effects of anisotropy on the tool responses. For a tool in a general anisotropic formation, all of the [Formula: see text] components are nonzero. For a tool in a vertical well in transversely isotropic and biaxially anisotropic formations, only the diagonal components of [Formula: see text] are nonzero. For a tool located in a deviated well, the effects of tool deviation and electric anisotropy are coupled. The diagonal components are more sensitive to the electric anisotropy than the off-diagonal components, and the off-diagonal ones can clearly indicate bed boundaries.

Influential factors of inclined fracture induction logging based on numerical simulation and physical experiment

When the fracture normal forms an angle relative to the borehole axis, the induction logging response to the fractured formation consists of two contributions: one is the volume contribution caused by the eddy current, and the other is the charge contribution caused by the cumulative charges on the upper and lower boundaries of the fracture; both contributions are nonlinear. Researching the distribution of surface charges and eddy current in inclined fractured formations is of great significance for logging analysts and geologists to fully understand induction logging laws in such setting. However, there is insufficient research on this aspect. In this work, first, an electromagnetic field calculation method under the condition of existing cumulative charges at the interfaces was developed. Second, an induction logging numerical simulation model of inclined fractures based on the three-dimensional (3D) finite element method (FEM) was established, and the distribution laws of cumulative charges and eddy current in fractures under different dip angles were summarised. Third, we used the annular conductive region in the fracture as the object of our research, and analysed the influence law of different conductive regions distribution on induction logging. Finally, we carried out physical experiments to verify the numerical results and achieved good agreement between the two results. Hence, it proves the validity of the methods and conclusions.

Research on triaxial array induction logging response in inclined anisotropic formation

The logging response calculation and fast inversion are the key and difficulties for logging data interpretation and formation parameter estimation of the triaxial array induction logging. We transform the formation conductivity from media coordinate to instrument coordinate based on coordinate transformation theory. This paper examines anisotropic response of the triaxial array induction in inclined anisotropic formation by finite element method. The greater the anisotropy coefficient is, the more obvious the curve separation. The apparent conductivity decreases with the increase of dip angle for the coaxial system, but the coplanar system increases. The research results show that the apparent conductivity of the coplanar system appears as obviously positive and negative horns on the horizontal interfaces, which is generated by the accumulated surface charge at the boundary of the layer.

Numerical Simulation and Analysis of Multicomponent Induction Logging Response in Anisotropic Formation

In this article, the response characteristics of multicomponent induction logging (MCIL) tool in anisotropic formation are analyzed. To solve the 1-D MCIL problems, we use a planar layered anisotropic medium (PLAM) Green's function (PLAMGF) method which is derived from the layered-medium Green's function (LMGF) and combined with MCIL excitation source and anisotropic formation environment, and then use it to investigate the equivalence of the MCIL tool responses in two models, i.e. the microscopic thin interbed formation and macroscopic anisotropic formation, and also analyze the effects of shoulder bed and formation dip on MCIL tools. In addition, in order to solve the complex 3-D problems, such as the effects of borehole, tool eccentricity and invasion on the tool response, we use the finite element method (FEM) based on a proposed new meshing scheme to simulate and analyze. This new meshing scheme includes two aspects: the use of cylindrical hexahedron element and the flexible meshing strategy combined with domain decomposition. Finally, some key detection performances of coaxial coils and coplanar coils of MCIL tool are studied. Numerical experiments are conducted to prove the effectiveness of our proposed method and provide the detailed results of the aforementioned aspects.

Accelerated Computation of Triaxial Induction Tool Response for Arbitrarily Deviated Wells in Planar-Stratified Transversely Isotropic Formations

In this letter, a new formulation for calculating an induction logging tool response in a multilayered transversely isotropic (also called uniaxial anisotropic) dipping formation is proposed. Using transmission line theory, only five independent integrals are necessary to form all the components of the dyadic Green’s function in order to calculate the tool response, resulting in a more succinct and efficient formulation. The fast Hankel transform (FHT) is then employed to calculate these spectral integrals efficiently. In special cases, such as horizontal and vertical wells where the direct FHT fails, alternative approaches are proposed. A singularity extraction is implemented to handle the divergent integrals for a horizontal well, whereas the double-exponential integration rule is implemented for a vertical well. In order to speed up the computation, the spectral transmission line currents and voltages are precalculated and then stored in memory, and parallel computing using OpenMP is employed. Numerical examples demonstrate a speedup factor of more than two orders of magnitude compared with other reported results based solely on the FHT.

Numerical Calculation of Induction Logging Response in Cased Hole

Numerical Calculation of Induction Logging Response in Cased Hole

Study on the effect of fractures on conventional induction log based on physical and numerical simulation

Fractures are important storage area and migration pathway in fractured reservoir. Abnormal conventional induction logging response has been found in fracture zone of reservoir, but few scholars have studied the conventional induction log response of fractured reservoirs. In this paper, the effect of fractures on conventional induction log has been studied based on physical and numerical simulation. Firstly, a physical model has been built to simulate the influence of fractures on conventional log. Secondly, the geometric factor theory has been used to find the minimum fracture width that the conventional induction log could identify, and the minimum interval between two adjacent fractures that the conventional induction log could distinguish. Thirdly, a fracture evaluation method based on numerical simulation was proposed, and a complex model containing surrounding rock and fractures was established to test the possibility of evaluating the fracture parameters using this method. Finally field data were used to test this method. From the results in this paper, we conclude that fractures influence conventional induction log significantly and fractures could be detected and evaluated by the conventional induction log.

An approximate 3D inversion method for inversion of single-well induction-logging responses

Electromagnetic logging is a technique used to probe differences in electric conductivity around a measurement device. Electromagnetic logging while drilling can contribute to proactive geosteering to improve well placement in reservoirs because such measurements can serve as an indication for the structure of the surrounding geology. We have developed a novel way of predicting the 3D conductivity distribution around a drilling tool, based on the contrast-source inversion method, in which we have replaced the full integral-equation approach by the single spherical scatterer (SSS) approximation. The approximation took into account the dominant features of the diffusive electromagnetic field. This allowed for a substantial gain in computational speed and storage of the inversion method for reconstruction of the conductivity distribution. In view of the limited range of the electromagnetic probing, the overall reconstruction can be segmented in several local windows. This reduced the computational speed requirements and the storage requirements dramatically, while safeguarding the overall 3D character of reconstruction. We have synthesized 3D electromagnetic logging data using synthetic models and conductivity maps from a hydrocarbon North Sea reservoir model. Reconstructions were made for multiple source frequencies, and the results were compared with the results obtained from the Born approximation. We have observed that reconstruction based on the SSS approximation was superior to the one based on the Born approximation. Our algorithm helped us determine the feasibility of producing reconstructions of reservoir sections in a short time frame, which allows for real-time decision making during drilling operations.

A fast algorithm to forward model the triaxial induction logging response in layered transversely isotropic dipping formations

We have developed a new analytic method to compute the response of triaxial induction logging tools in layered transverse isotropic (TI) dipping formations without borehole and invasion zones. A triaxial induction tool is composed of one or more sets of three orthogonal transmitter coils and three orthogonal receiver coils arranged along the tool axis. The tool axis may intercept a formation with dip, azimuthal, and rotation angles. Based on the electromagnetic (EM) field in planarly layered media, formulations of the EM fields generated by these three transmitting coils are derived. A fast-forward-modeling algorithm to compute the responses of an induction tool was developed. Compared with the algorithm using the coefficient propagator method previously developed, the proposed algorithm calculates the generalized reflection and transmission coefficients only once for each layer regardless of the position of the transmitters, and therefore, it greatly speeds up the computation. The simulation code solves for all nine components of the magnetic field in a TI layered medium. Results of this new method were developed for several formation models, which agreed very well with the published data. For a dipping or nondipping case in a 28-layer formation, the proposed method was about 37 times faster, which was more appropriate to be used in an inversion algorithm.

An approximate 3D computational method for real-time computation of induction logging responses

Over many years, induction logging systems have been used to create well formation logs. The major drawback for the utilization of these tools is the long simulation time for a single forward computation. We proposed an efficient computational method based on a contrast-type of integral-equation formulation, in which we applied an approximation for the 3D electromagnetic field. We assumed that the dominant contribution in the integral equation is obtained by the contribution around the singularity of Green’s kernel. It is expected that the approximation yields reliable results when the (homogeneous) background conductivity around the logging tool is close to the actual conductivity at the location of the tool. We have developed a data-driven method to determine this background conductivity from the dominant part of the measured coaxial magnetic fields, which are mainly influenced by the conductivity at the tool sensors. For a synthetic model, the results were compared to the ones of a rigorous solution of the integral equation and show a good simulation response to small-scale variations in the medium. Further, the method was used to simulate the response of a realistic reservoir model. Such a model is created by a geological modeling program. We concluded that our approximate method was able to improve the approximation results in highly heterogeneous structures compared to the Born approximation and provide an effective medium-gradient around the tool. Our method, based on the wavefield approximation, also estimates the error, and hence yields a warning when the method becomes unreliable.

Forward modelling of multi-component induction logging tools in layered anisotropic dipping formations

Multi-component induction logging provides great assistance in the exploration of thinly laminated reservoirs. The 1D parametric inversion following an adaptive borehole correction is the key step in the data processing of multi-component induction logging responses. To make the inversion process reasonably fast, an efficient forward modelling method is necessary. In this paper, a modelling method has been developed to simulate the multi-component induction tools in deviated wells drilled in layered anisotropic formations. With the introduction of generalized reflection coefficients, the analytic expressions of magnetic field in the form of a Sommerfeld integral were derived. The fast numerical computation of the integral has been completed by using the fast Fourier–Hankel transform and fast Hankel transform methods. The latter is so time efficient that it is competent enough for real-time multi-parameter inversion. In this paper, some simulated results have been presented and they are in excellent agreement with the finite difference method code's solution.

Efficient and Reliable Simulation of Multicomponent Induction Logging Response in Horizontally Stratified Inhomogeneous TI Formations by Numerical Mode Matching Method

An efficient and reliable numerical mode matching (NMM) method is developed to simulate the responses of multicomponent induction logging (MCIL) in the horizontally layered inhomogeneous transversally isotropic (TI) formations with the borehole and the invasion zone. The computation of MCIL responses is entirely transformed into three axially symmetrical problems on the Fourier harmonic components of the electromagnetic (EM) fields. The two extra singular differential operators about the horizontal EM components are introduced to describe the influence of the accumulation charge at the cylindrical interfaces on the EM fields excited by the horizontal magnetic dipoles. The first and second kinds of homogeneous boundary conditions of the different harmonic components of horizontal electrical fields near the borehole axis are derived to ensure that the EM problems become well posed in the cylindrical coordinate system. Then, the NMM is applied to solve the three axisymmetric problems with the extra singular differential operators. The numerical results demonstrate that the proposed method can accurately and efficiently simulate the MCIL responses in the horizontally layered TI formations with the large contrast resistivity.

A fast integral equation solver for 3D induction well logging in formations with large conductivity contrasts

ABSTRACTSimulation of induction logging responses in formations with large conductivity contrasts is an important but challenging problem due to the singularity of a linear system caused by large contrasts. Also, three‐dimensional (3D) analysis of complex geophysical structures usually encounters high computational demands. In this paper, a pre‐corrected fast Fourier transform (pFFT)‐accelerated integral equation method is applied to overcome these difficulties. In the approach, the entire formation is included in the solution domain. The volume integral equation is set up in the region based on the fact that the total field is the summation of the excitation field and the secondary field. The emitted field by the transmitter coil (treated as a magnetic dipole) is regarded as the excitation of the system. Then the method of moments (MoM) is used to solve the integral equation. To reduce the high computational requirements of the MoM, the pFFT method is used to speed up the solution of the matrix equation and reduce the memory requirement as well. The resultant method is capable of computing induction logging problems involving large and complex formations. For problems with high conductivity contrasts, the solution of the matrix equation usually converges very slow or even fails to converge due to the large condition number of the coefficient matrix. To overcome this difficulty, an incomplete LU pre‐conditioner is used to significantly speed up the convergence of the matrix equation, thus further reducing the computation time. Numerical results show that the present method is efficient and flexible for 3D simulation of induction logging and is specifically superior for problems with high conductivity contrasts.

Adaptive Regularization Iterative Inversion of Array Multicomponent Induction Well Logging Datum in a Horizontally Stratified Inhomogeneous TI Formation

An adaptive regularization iterative inversion of array multicomponent induction well logging datum is established to simultaneously reconstruct the horizontal and vertical conductivities of both invasion zone and origin formation, invasion radius, and the interface depth of each bed in the horizontally stratified inhomogeneous transversally isotropic (TI) formation. Applying numerical mode matching method, we can obtain a much compact semianalytic expression of the electromagnetic tensor Green's functions by magnetic current source in the inhomogeneous TI formation. Then, using the perturbation principles, an efficient computation of Fréchet derivatives of the multicomponent induction logging response is set up with respect to all the model parameters. After that, the combination of Morozev's discrepancy principle with Cholesky's decomposition is applied to adaptively select regularization factor during inversion so that stabilization of inversion solution is assured as well as realization of best fit of the input data with the modeling logs. Finally, the numerical tests validate the algorithm.

Multi-component induction logging response in large dielectric formation

Wireline induction logs operating at a frequency of some tens of kHz are used to detect the resistivity of hydrocarbon reservoir. The dielectric constant of typical formation is usually smaller than that of pure water and its effect is negligible for induction log. However some peculiar rocks including organic matter named source rock have large dielectric permittivities caused by polarization, which can lead the traditional induction logging to have an nusual log response. Multi-component induction logging is new log method and has been developed in nearly ten years, which can provide more information about formation than traditional axial instrument and has large potential applications. In this paper, we study the effect of large dielectric constant on multi-component induction logging response and extract the information about resistivity and dielectric permittivity through the different combinations, there by realizing the formation parameters reconstruction and intuitive recognition for the hydrocarbon source rock.

VECTOR FINITE ELEMENT ANALYSIS OF MULTICOMPONENT INDUCTION RESPONSE IN ANISOTROPIC FORMATIONS

Multicomponent induction logging responses are simu- lated by using hierarchical mixed order vector finite element method (FEM). In order to modeling three orthogonal magnetic dipoles, we adopt the method that the total field is separated into incident field and secondary field, and only the secondary field is computed by FEM. In addition, two techniques are applied to improve the modeling accu- racy and computational efficiency: 1) Hierarchical mixed order vector basis functions are applied to FEM. Different order basis functions are used in different elements in accordance with the changing speed of the field. The mixed order scheme reduces greatly the number of unknowns without reducing accuracy, and can attain much higher com- putational efficiency. 2) The systemof the FEM equations is solved by Distributed-SuperLU, and the results of multiple measure points can be got simultaneously. The FEM result is validated against volume integral equation method and the approach of planar layered media Green's functions, and the comparisons show very good agreement. Fi- nally, the multicomponent induction response in anisotropic formations involving eccentric tools and dipping beds is included to demonstrate the flexibility of the method.

A weak-anisotropy approximation to multicomponent induction responses in cross-bedded formations

The multicomponent induction logging response to a cross-bedded formation has been modeled under a weak-anisotropy approximation. With the approximation, a cross-bedded formation can be modeled as a transversely isotropic (TI) medium. The validity of the approximation has been tested for the main (coplanar and coaxial) components of the induction response. The conditions for the weak-anisotropy approximation to be valid depend on the component of the response. For the coplanar components, the approximation is valid for an anisotropy ratio up to 2 if the relative dipping angle between the cross-bedded formation and the borehole axis is below [Formula: see text]. For the coaxial component, the approximation reduces to a previously established result that the apparent resistivity for such a component is the geometric average of the resistivities, parallel and perpendicular to the bedding planes of an anisotropic formation, respectively, if the borehole is ignored. Hence, the approximation holds for the coaxial component regardless of the anisotropy ratio.

Application of the Staggered Grid Finite Difference Method to the Computation of 3D Induction Logging Response

AbstractThe finite difference method based on staggered grids is applied to simulate the induction logging response in 3D media. The invariance property of the Krylov subspace is used to solve the resulting large sparse complex symmetric linear system. To improve iteration convergence the pseudo inverse of the coefficient matrix is employed when constructing the Krylov subspace instead of the coefficient matrix itself. In each iteration four 3D Poisson equations are computed to get a new Lanczos vector with the incomplete Cholesky decomposition conjugate method. Generally the desirable solution is achieved with the iteration less than 20 times. Also, a new material averaging formula is proposed to get a reasonable average of the conductivity, which guarantees the conservation of the electric current.