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Joint inversion of induction and galvanic logging data in axisymmetric geological models
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ScienceDirect Russian Geology and Geophysics 58 (2017) 752–762 www.elsevier.com/locate/rgg
Joint inversion of induction and galvanic logging data in axisymmetric geological models I.V. Mikhaylov a,b,*, V.N. Glinskikh a,b, M.N. Nikitenko a, I.V. Surodina a,c a
c
A.A. Trofimuk Institute of Petroleum Geology and Geophysics, Siberian Branch of the Russian Academy of Sciences, pr. Akademika Koptyuga 3, Novosibirsk, 630090, Russia b Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch of the Russian Academy of Sciences, pr. Akademika Lavrent’eva 6, Novosibirsk, 630090, Russia Received 15 March 2016; accepted 1 September 2016
Abstract We have developed a computational algorithm for joint inversion of array induction and galvanic data in 2D models. It is based on a finite-difference solver and nonlinear minimization and is designed to develop realistic geoelectric models of complex fluid-saturated formations. The algorithm is tested and verified on noisy synthetic induction and galvanic data. The obtained 2D inversion results are compared with those corresponding to the traditional 1D radial approach. The developed algorithm was used to conduct joint 2D inversion of VEMKZ and BKZ logs from the Fedorovskoe and Vostochno-Surgutskoe oil fields in the E–W striking Ob’ River area. © 2017, V.S. Sobolev IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved. Keywords: joint inversion; electrical logging; VEMKZ; BKZ; axisymmetric model; electrical resistivity
Introduction In recent years, the global development and operation of new types of hydrocarbon reservoirs with complex geological structure has led to a significant complication and extension of the range of well logging problems. This, in turn, necessitates the improvement of hardware systems and methodical support for processing and interpretation of well logging data. Among the new import-substituting technologies for logging oil and gas wells is the SKL hardware system, which, along with the main commonly used geophysical methods, incorporates induction and galvanic logging methods (Epov et al., 2010). Using this system saves tripping time, and simultaneous measurements eliminate the need to match log depths. Electrical logging methods have found application at all stages of construction of oil and gas wells—from geosteering to reaming (Epov et al., 2015; Kayurov et al., 2014). The development of new hardware systems for exploration of oil and gas deposits leads to the necessity to design software and algorithmic means aimed at increasing the reliability of quantitative determination of parameters of heterogeneous
* Corresponding author. E-mail address: [email protected] (I.V. Mikhaylov)
reservoirs. Improving the accuracy of evaluating the reservoir saturation from electrical logging is particularly important in this case. One way to do this is to employ realistic interpretation models. These models should include a description of the complex spatial distribution of the electrophysical parameters of the geological model and take into account various physical effects. However, their employment is only possible with the use of numerical simulation and inversion when solving multidimensional problems of computational electrodynamics. Traditionally, the reservoir saturation pattern is determined from induction and galvanic logging data based on analysis of the radial electrical resistivity profile determined within the scope of a 1D cylindrically layered model (Epov and Nikitenko, 1993). The radial resistivity distribution pattern is due to the displacement of formation fluids by mud filtrate and the change in salinity in the near-wellbore zone during formation drilling. However, when interpreting in formations of limited thickness, characterized by an inhomogeneous invaded zone and depth-variable oil content, it is necessary consider the effect of overlaying and underlying rocks. In addition, the presence of conductive shale and high-resistivity (compacted sand and carbonate) thin layers commonly found in reservoirs leads to significant inaccuracies in determining the resistivity
1068-7971/$ - see front matter D 201 7, V.S. So bolev IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.rgg.201 + 6.09.032
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and subsequent unreliable estimates of fluid saturation. To avoid this, it is necessary to use an axisymmetric interpretation model which takes into account both the radial and vertical resistivity distributions (Epov et al., 2013; Mikhaylov and Glinskikh, 2015; Nikitenko et al., 2015). Integration of well logging methods is widely used to improve their efficiency in exploration of complex oil and gas reservoirs (Kneller and Potapov, 1992; Mezzatesta et al., 1994; Rabinovich and Tabarovsky, 2001; Shanjun et al., 2007; Yang, 2001). Due to the different sensitivity and resolution of induction and galvanic logging measurements, the integration of these methods improves the reliability of resistivity determinations. Joint inversion of electrical logging data provides an integrated geoelectric model. The construction of a consistent interpretation model narrows the region of ambiguity (or equivalence) in determining the model parameters, which is due to the different physical principles of field excitation and signal measurement and the different influence of the parts of a medium on the recorded signals. It is known that conductive regions have a greater influence on induction signals, whereas resistive ones on galvanic signals. The use of an integrated interpretation model to examine different physical processes provides more accurate evaluation of the properties of reservoirs. For example, an integrated geoelectric and hydrodynamic model of the near-wellbore zone has been widely applied for an integrated petrophysical interpretation of electrical logging data. Subsequently, the need for further consideration of geomechanical properties has led to the development of a multidisciplinary approach based on an integrated electro-hydrodynamic and geomechanical model (Yeltsov et al., 2014). In addition, joint inversion of induction and galvanic logging data makes it possible to study the various effects involved in the propagation and interaction of electromagnetic fields with the medium. That is, it becomes possible to study in detail the frequency dispersion of the resistivity and electrical macroanisotropy of rocks, which have recently attracted considerable interest, in particular, for deviated and horizontal wells (Epov et al., 2010; Nikitenko et al., 2016). Their consideration in well logging data interpretation necessitates the use of specialized software. There are various approaches to the solution of computational problems of electrical logging using axisymmetric geological models. Forward problems have been solved by numerical-analytical and approximate methods (Epov and Nikitenko, 1993; Glinskikh et al., 2013a,b, 2014; Tabarovsky and Rabinovich, 1998). These methods have found widespread use as they allow developing fast algorithms for data processing and inversion. Due to the complication of electrical logging models and the rapid development of computational methods for electrodynamic problems, grid methods are preferred. Of these, finite-difference and finite-element methods are most widely used to simulate electromagnetic fields in inhomogeneous media (Epov et al., 2007; Surodina and Epov, 2012). However, solutions of multidimensional problems in full statement for processing and interpreting large volumes of field data are very resource intensive and have therefore found only limited use. Significant advances in this
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area have been made possible through the development of computer information technologies and multiprocessor computation systems. It has become feasible to use computationally intensive operations for practical purposes of data processing and interpretation, which will significantly improve the accuracy of determination of geoelectric parameters and the reliability of the results of interpretation based on the data of a suite of well logging tools. This paper presents an algorithm for joint numerical inversion of high-frequency electromagnetic logging (VEMKZ) and lateral logging (BKZ) data on axisymmetric geological models and the results of its application on synthetic and field data. It should be noted that the developed algorithm can be easily adapted to the numerical inversion of data of electrical logging tools with induction, galvanic or mixed induction-galvanic types of signal generation and measurement.
Computational inversion algorithm The basic interpretation model is a 2D axisymmetric geoelectric model which describes the penetration of a horizontally layered section by a vertical well with the formation of invaded near-wellbore zones (Fig. 1). The axisymmetric model includes a sequence of layers with plane parallel horizontal boundaries penetrated by a vertical cylindrical well. In the near-wellbore zone of each of the layers, there may be two zones—an invaded zone and a low-resistivity annulus— and in some cases, an additional flushed zone. They are separated from each other, from the borehole, and the formation by coaxial-cylindrical boundaries. Each of the regions of the geoelectric model is characterized by its own resistivity. The developed joint inversion algorithm is based on grid finite difference algorithms for numerical simulation of induction and galvanic data for media with axial symmetry (Surodina and Epov, 2012). Numerical solutions of forward problems after finite difference discretization are reduced to systems of linear algebraic equations (SLAE) with large sparse matrices, which are solved using well-known direct and iterative methods. The main features of the numerical simulation algorithms are as follows. Numerical 2D simulation of galvanic logging data reduces to the solution of the Poisson equation. Using a conservative difference scheme reduces the problem to a SLAE, which is symmetrized and then solved by the conjugate gradient method. Due to the ill-conditioning of the SLAE, it is preconditioned using Hotelling’s algorithm (Labutin and Surodina, 2013). The solution of the forward 2D problem of induction logging reduces to the solution of the Helmholtz equation. The obtained SLAE is solved using the COCR iterative method (Sogabe and Zhang, 2007), also with preconditioning. The developed algorithms for solving forward problems have been carefully verified, tested, and validated on a large number of realistic geoelectric models.
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Fig. 1. Interpretation geoelectric model with axial symmetry.
The solution of the inverse 2D problem reduces to the reconstruction of the spatial resistivity distribution of the rocks from measured log responses. The 1D inversion results obtained for the cylindrically layered model from the data of a suite of methods or an individual method are effectively used as the starting model. The numerical inversion is performed in two stages. At the first stage, 1D inversion with reconstruction of the radial resistivity profile from the wellbore wall into the formation is carried out at the layer interval, with a preliminary manual or automated determination of the position of horizontal boundaries through the analysis of derivatives. At the second stage, the resistivity distribution and the positions of the horizontal boundaries and cylindrical zones are refined using a 2D model. The numerical inversion of field data involves minimizing the standard deviation of measured data from synthetic data by changing the model parameters using a given algorithm. The numerical inversion algorithm is based on the downhill simplex method. It quickly selects the right search direction and significantly reduces the minimized function as early as at initial iterations. The inversion algorithm can use arbitrary combinations of signals from a complete set of measurements in the case of, e.g., the absence of signals or their low-quality. In the inversion model, the parameters can be fixed or constrained. This makes it possible to refine the values of individual parameters if reinversion is necessary, and reduce the number of fitted parameters when the survey interval contains thick homogeneous layers whose parameters have been already determined with good accuracy using 1D inversion.
Testing the inversion algorithm on noisy synthetic electrical logs To test the performance of the developed numerical inversion algorithm, we conducted a comparative analysis of 2D and 1D joint inversions of noisy synthetic electrical logs calculated using the finite difference method. Realistic geoelectric models of terrigenous oil and gas reservoirs of the E–W striking Ob’ River area in West Siberia, complicated by thin layers of shales and carbonates, are
penetrated by boreholes with fresh drilling mud (radius 0.108 m, resistivity 2 Ohm m). The permeable intervals include an invaded zone and a low-resistivity annulus in the oiland water-saturated formations with movable oil and water (Antonov et al., 2012). The drilling mud invasion pattern is due to the resistivity change in the near-wellbore zone associated with salinity redistribution between the formation water and mud filtrate. The signals of five main VEMKZ probes (DF05, DF07, DF10, DF14, and DF20) and four BKZ probes (A0.4M0.1N, A1.0M0.1N, A2.0M0.5N, and A4.0M0.5N) were analyzed. Synthetic logs were made noisy with the use of a normally distributed random variable described by a real measurement error model. Figure 2a, b shows the results of numerical inversion of noisy synthetic electrical logs for the model of a gas-oil-water reservoir overlain by shales and underlain by carbonate rocks. The reservoir is complicated by thin (0.3 m) interlayers. Thus, between the gas- (2.0–3.5 m) and oil-saturated (3.8–5.3 m) formations there is a shale interlayer (3.5–3.8 m) and between the oil- and water-saturated (5.6–7.1 m) portions is a carbonate interlayer (5.3–5.6 m), which is typical of sections in West Siberia. A conductive invaded zone is present in the permeable gas-saturated formation, and a resistive invaded zone in the other two formations. Figure 2a shows synthetic VEMKZ logs (left) and synthetic BKZ logs (right) calculated via 2D simulation. Here and below, the solid lines correspond to the noisy signals in the starting model and the dashed lines to the signals in the model obtained from the results of 1D inversion in the cylindrically layered model. Figure 2b shows the distributions of the resistivity (left) and cylindrical and horizontal boundaries (right) in the model considered. Here and below, the solid lines correspond to the starting model and the dashed lines to the results of 1D inversion in the cylindrically layered model. Analyzing the noisy electrical logs, we note the following (Fig. 2a). The behavior of VEMKZ logs differs significantly from that of BKZ logs. VEMKZ signals are characterized by higher information content when performing a visual analysis, and a higher degree of sectional layering. The shale interlayer is clearly identified in all VEMKZ logs, while in BKZ logs its presence is detected only by the A0.4M0.1N and A2.0M0.5N probes. The carbonate interlayer, in contrast, is manifested in the logs of all BKZ probes. As for VEMKZ, this high-resistivity object in the section can be identified only presumably, from the data of the short DF05 probe. Below the high-contrast boundary of the water-saturated formation and the underlying carbonate rocks, asymptotic values are reached faster for the logs of the long VEMKZ probes than for the BKZ logs. This is due, in particular, to the significant influence of the borehole on BKZ reponses. We analyze the results of the joint 1D inversion of the VEMKZ and BKZ data obtained at the first stage of the numerical interpretation (Fig. 2a, b). Note that the reconstructed values of many geoelectric parameters differ significantly from the true values. The positions of the horizontal
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Fig. 2. Synthetic VEMKZ and BKZ logs (a) for the model of a gas-oil-water reservoir with thin shale and carbonate interlayers (b). Solid lines correspond to the starting model, and dashed lines to the model obtained by 1D inversion. Here and below: F is a formation; A is a low-resistivity annulus; IZ is an invaded zone; FZ is a flushed zone.
boundaries shown by a dashed line on the right of Fig. 2b were determined using an automatic boundary detection algorithm. As can be seen, the differences between their coordinates and the true values reach 0.3 m. For better agreement between the field and simulated sounding curves in the shale and carbonate interlayers, it was required to
introduce invaded zones which were absent in the starting model. The obtained resistivity of the shale interlayer differs from the model value by a factor of 1.6, and the resistivity of the carbonate interlayer differs from it by a factor of 14. As regards the permeable formations, the largest deviations from the starting model are observed in the gas- and oil-saturated
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Fig. 3. Synthetic VEMKZ and BKZ logs (a) for the model of a gas-oil-water reservoir with thin shale and carbonate interlayers (b). Solid lines corresponds to the starting model, and dashed lines to the model obtained by joint 2D inversion.
intervals. In the first of them (2.0–3.5 m), with a resistivity of 50 Ohm m, the fitted resistivity is 9.4 Ohm m, which corresponds to oil and water saturation. In this case, instead of a conductive invaded zone with a radius of 0.2 m and a resistivity of 35 Ohm m, a wide resistive invaded zone (0.7 m, a resistivity of 27 Ohm m) is obtained. The reason for this
incorrect result is the strong influence of low-resistivity 1.5 m thick shale interlayer over the top and under the bottom of the layer considered. In the oil-saturated reservoir (3.8–5.3 m) with a resistivity of 15 Ohm m, the resistivity found from the results of the 1D approach is 7.9 Ohm m; the radius of the invaded zone differs from the true value by a factor of 1.5.
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Fig. 4. Synthetic VEMKZ and BKZ logs (a) for the model of a gas-oil-water reservoir in mudstone complicated by a thin carbonate interlayer (b). Solid lines correspond to the starting model, and dashed lines to the model obtained by 1D inversion.
This is largely related to the high resistivity contrast of the reservoir considered and the underlying and overlying rocks. At the second stage, we carried out a joint 2D inversion of VEMKZ and BKZ data (Fig. 3a, b). The results of the 1D inversion were used as the starting model. Resistivities of the host rocks were fixed and were not further fitted. All other
electrophysical and geometrical parameters of the section (resistivities of the formations and invaded zones, positions of the radial and horizontal boundaries) were refined during the 2D inversion considered. Separately, we note that the invaded zones in the shale and carbonate interlayers did not provide a satisfactory match between same-name logs and were ex-
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Fig. 5. Synthetic VEMKZ and BKZ logs (a) for the model of a gas-oil-water reservoir in mudstone complicated by a thin carbonate interlayer (b ). Solid lines correspond to the starting model, and dashed lines to the model obtained by joint 2D inversion.
cluded, with their corresponding resistivities being redistributed among adjacent layers. As a result, there was excellent agreement between the practial and fitted logs and parameters of the medium. Among the electrophysical parameters, the resistivity of the carbonate interlayer was reconstructed with the greatest error (13%), and among the geometrical parame-
ters, the radii of the invaded zones in the gas- and oil-saturated formations (errors of 15 and 10%, respectively) were reconstructed the least accurately due to the equivalence of the signals. Figure 4a, b shows the results of numerical inversion of noisy synthetic electrical logs for the model of an oil-water
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Fig. 6. VEMKZ and BKZ logs (a) and the result of the joint 2D inversion (b) in the AS5–6 formation interval of the Fedorovskoe field. Solid lines correspond to field logs, and dashed lines to synthetic logs for the fitted model.
reservoir in mudstone complicated by a thin carbonate interlayer. In the model considered, the underlying mudstone is overlain by a water-saturated formation (5.3–6.8 m), which is overlain by an interval containing movable oil and water (3.8–5.3 m). Above it is an oil-saturated formation (2.0–
3.5 m), which is separated from the latter by a thin (3.5–3.8 m) carbonate interlayer and overlain by mudstone. A resistive invaded zone is present in all permeable formations, and a low-resistivity annulus is observed in the oil- and water-saturated formation.
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Fig. 7. VEMKZ and BKZ logs of the SKL hardware system (a) and the result of joint 2D inversion (b) on the YuS2 formation interval of the Vostochno-Surgutskoe field. Solid lines correspond to field logs, and dashed lines to synthetic logs for the fitted model.
Analysis of the logs shows that the high-resistivity carbonate interlayer is clearly identified in BKZ logs and is almost not detected from VEMKZ responses (Fig. 4a). The converse is true for the 3.8–5.3 m oil- and water-saturated interval with a low-resistivity annulus, to which VEMKZ is highly sensitive.
It can be seen from Fig. 4b that the result of the joint 1D inversion of VEMKZ and BKZ is insufficiently accurate. In particular, in the carbonate interlayer, an erroneous invaded zone is introduced. Nevertheless, except for the position of the lower horizontal boundary, the parameters of the invaded zone, low-resistivity annulus, and the uninvaded zone of the
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oil- and water-saturated formation are determined fairly accurately. During the joint 2D inversion, all parameters, except for the fixed resistivity of host mudstone, were fitted with high accuracy, as shown in Fig. 5a, b. The reconstruction error was the greatest for the resistivity of the thin carbonate interlayer (11.7%), the radius of the invaded zone, and the resistivity of the uninvaded zone of the oil-saturated formation (4.5%). The latter is due to the low resistivity contrast in this formation. Thus, using noisy synthetic VEMKZ and BKZ signals for typical reservoir models, it has been shown that the developed algorithm for numerical inversion of well logging suite data provides a consistent geoelectric model that takes into account the 2D resistivity distribution of complex geological formations, which improves the accuracy of evaluation of their fluid saturation. Joint numerical inversion of field data with the construction of geoelectric models of terrigenous reservoirs The algorithm was tested on VEMKZ and BKZ field logs from dozens of wells in various fields of the Surgut Arch of the E–W striking Ob’ River area, where substantial oil accumulations are confined to Lower Cretaceous (AS, BS) and Jurassic (YuS) rocks. Consider as an example the AS5-6 formation interval of the Fedorovskoe field, which is one of the largest multilayer fields. The interval is penetrated by a borehole containing clay drilling mud with a resistivity of 2.7 Ohm⋅m. The joint 2D inversion of VEMKZ and BKZ responses (Fig. 6a) provided a realistic geoelectric model for a geological section of complex structure (Fig. 6b). An oil- and water-saturated reservoir (38.0–53.7 m) complicated by a high-resistivity carbonate interlayer (47.8–49.6 m) is identified. The reservoir is overlain by shale deposits and underlain by a thick water-saturated formation AS7-8 (55.3–80.0 m, the figure shows the upper part), which is separated from the latter by a clayish interval (53.7–55.3 m). The readings of the long VEMKZ probes increase monotonically in the depth range 41.0–47.8 m, indicating an increase in water saturation. In the upper part of the reservoir (38.7–45.8 m) is a low-resistivity annulus, reflecting its saturation with movable oil and formation water. Below, at depths of 45.8–47.8 and 49.6–52.5 m, are water-saturated reservoir zones. The verification of the 1D inversion results using 2D simulation does not provide good agreement between the field and synthetic VEMKZ and BKZ logs in the survey interval. The model parameters were refined by joint 2D inversion. It was found that the spatial resistivity distributions obtained using the 1D and 2D approaches differ significantly. The resistivities of high-contrast formations are satisfactorily fitted, as a rule, after a preliminary clarification of the position of the horizontal boundaries. In conclusion, we turn to Fig. 7, which shows the results of applying the algorithm to VEMKZ and BKZ field data recorded using the SKL hardware system in the Vostochno-
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Surgutskoe field. The borehole was drilled with clay mud (resistivity 2.8 Ohm m). Since the data were recorded by the high-precision equipment, preliminary filtering and smoothing were not needed. The interval considered (85.0–91.0 m, Fig. 7a) is the Middle Jurassic sandy horizon YuS2, with a complex geological structure. In particular, the reservoir is partially carbonized, which is especially pronounced in logs at depths of 86.3–87.0 and 89.5–90.0 m. In view of the considerable vertical heterogeneity of the interval, most of it was divided into layers about 0.5 m thick, in which the inversion was performed on a layer-by-layer basis. The joint 2D inversion resulted in a geoelectric model of the reservoir in mudstone, whose high resistivity indicates oil saturation. Unlike in the previous examples, a flushed high-resistivity zone of small radius was introduced at all intervals, if necessary. Note that the logs of the short probes, primarily BKZ, were fitted slightly worse. This may be due to the formation of caverns, which changed the nominal radius of the well and complicated the resistivity distribution directly near the wellbore wall. As regards the computational performance, the use of the sequential algorithms for solving the forward 2D problem is rather resource-intensive. At the same time, graphics processors hugely increase the speed, allowing near real-time 2D inversion. Thus, the numerical interpretation of VEMKZ and BKZ field logs from the Lower and Middle Jurassic intervals of the Fedorovskoe and Vostochno-Surgutskoe fields demonstrated the possibility of obtaining a consistent geoelectric model of complex hydrocarbon reservoirs from electrical logging suite data. Conclusions The reliability of the determination of geoelectric parameters of complex fluid-saturated formations is improved by combining induction and galvanic logging measurements within the framework of an integrated consistent interpretation model. A computational algorithm for joint 2D numerical inversion of electrical logging data was developed and software implemented. The algorithm is based on numerical solutions of forward problems using the finite difference method and nonlinear minimization using the downhill simplex method. A stepwise numerical inversion scheme is used, during which geoelectric parameters are determined in cylindrically-layered and 2D models. The algorithm was tested on noisy synthetic and field VEMKZ and BKZ logs from the Fedorovskoe and Vostochno-Surgutskoe oil fields of the E–W striking Ob’ River area, including those recorded by the SKL hardware system. It is shown that the algorithm improves the accuracy of interpretation of electrical logs compared to the widely used 1D approach. The algorithm can be easily adapted to the numerical interpretation of data acquired with other induction, galvanic, and mixed induction-galvanic logging tools.
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This work was supported by the Government of the Novosibirsk region.
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19, 120–126, http://interval.louisiana.edu/reliable-computing-journal/volume-19/reliable-computing-19-pp-120-126.pdf. Mezzatesta, A.G., Payton, C.C., Strack, K.M., Tabarovsky, L.A., 1994. Simultaneous Inversion of Galvanic and Induction Logging Measurements to Improve Resolution, Proceedings of the SPE Latin America/Caribbean Petroleum Engineering Conference (Buenos Aires, Argentina, 27–29 April 1994). Paper SPE 26976, https://www.onepetro.org/conference-paper/SPE26976-MS. Mikhaylov, I.V., Glinskikh, V.N., 2015. Features of VIKIZ signals and their numerical inversion in the vicinity of shale-sandstone interfaces. Karotazhnik, No. 1 (247), 24–34. Nikitenko, M.N., Surodina, I.V., Mikhaylov, I.V., Glinskikh, V.N., Suhorukova, C.V., 2015. Formation Evaluation via 2D Processing of Induction and Galvanic Logging Data Using High-Performance Computing. Extended Abstracts of the 77th EAGE Conference & Exhibition (Madrid, Spain, 1–4 June 2015). Paper Tu N107 15, http://dx.doi.org/10.3997/22144609.201412646. Nikitenko, M.N., Suhorukova, C.V., Mikhaylov, I.V., Glinskikh, V.N., Surodina, I.V., 2016. Effective Inversion of Electromagnetic and Electrical logging Data from Vertical and Horizontal Oil and Gas Wells. Extended Abstracts of the 78th EAGE Conference & Exhibition (Vienna, Austria, 30 May–2 June 2016). Paper Tu STZ2 09, http://dx.doi.org/10.3997/22144609.201600893. Rabinovich, M., Tabarovsky, L., 2001. Enhanced Anisotropy from Joint Processing of Multicomponent and Multi-Array Induction Tools. Transactions of the SPWLA 42nd Annual Logging Symposium (Houston, Texas, United States, 17–20 June 2001). Paper HH, https://www.onepetro.org/ conference-paper/SPWLA-2001-HH. Shanjun, L., Richard, C.L., Navarro, D., Changxi, L., Cancan, Z., 2007. Joint Inversion Of SP and Dual Laterlog/Dual Induction Data in 2D Formation. Transactions of the SPWLA 48th Annual Logging Symposium (Austin, Texas, United States, 3–6 June 2007). Paper ZZ, https://www.onepetro. org/conference-paper/SPWLA-2007-ZZ. Sogabe, T., Zhang, S., 2007. A COCR method for solving complex symmetric linear systems. Journal of Computational and Applied Mathematics 199, 297–303. Surodina, I.V., Epov, M.I., 2012. High-frequency induction data affected by biopolymer-based drilling fluids. Russian Geology and Geophysics (Geologiya i Geofizika) 53 (8), 817–822 (1062–1069). Tabarovsky, L.A., Rabinovich, M.B., 1998. Real time 2D inversion of induction logging data. Journal of Applied Geophysics 38 (4), 251–275. Yang, W., 2001. Determining Resistivity Anisotropy by Joint Lateral and Induction Logs, Transactions of the SPWLA 42nd Annual Logging Symposium (Houston, Texas, United States, 17–20 June 2001). Paper CC, https://www.onepetro.org/conference-paper/SPWLA-2001-CC. Yeltsov, I.N., Nazarova, L.A., Nazarov, L.A., Nesterova, G.V., Sobolev, A.Yu, Epov, M.I., 2014. Geomechanics and fluid flow effects on electric well logs: multiphysics modeling, Russian Geology and Geophysics (Geologiya i Geofizika) 55 (5–6), 775–783 (978–990).
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Joint inversion of induction and galvanic logging data in axisymmetric geological models I.V. Mikhaylov a,b,*, V.N. Glinskikh a,b, M.N. Nikitenko a, I.V. Surodina a,c a
c
A.A. Trofimuk Institute of Petroleum Geology and Geophysics, Siberian Branch of the Russian Academy of Sciences, pr. Akademika Koptyuga 3, Novosibirsk, 630090, Russia b Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch of the Russian Academy of Sciences, pr. Akademika Lavrent’eva 6, Novosibirsk, 630090, Russia Received 15 March 2016; accepted 1 September 2016
Abstract We have developed a computational algorithm for joint inversion of array induction and galvanic data in 2D models. It is based on a finite-difference solver and nonlinear minimization and is designed to develop realistic geoelectric models of complex fluid-saturated formations. The algorithm is tested and verified on noisy synthetic induction and galvanic data. The obtained 2D inversion results are compared with those corresponding to the traditional 1D radial approach. The developed algorithm was used to conduct joint 2D inversion of VEMKZ and BKZ logs from the Fedorovskoe and Vostochno-Surgutskoe oil fields in the E–W striking Ob’ River area. © 2017, V.S. Sobolev IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved. Keywords: joint inversion; electrical logging; VEMKZ; BKZ; axisymmetric model; electrical resistivity
Introduction In recent years, the global development and operation of new types of hydrocarbon reservoirs with complex geological structure has led to a significant complication and extension of the range of well logging problems. This, in turn, necessitates the improvement of hardware systems and methodical support for processing and interpretation of well logging data. Among the new import-substituting technologies for logging oil and gas wells is the SKL hardware system, which, along with the main commonly used geophysical methods, incorporates induction and galvanic logging methods (Epov et al., 2010). Using this system saves tripping time, and simultaneous measurements eliminate the need to match log depths. Electrical logging methods have found application at all stages of construction of oil and gas wells—from geosteering to reaming (Epov et al., 2015; Kayurov et al., 2014). The development of new hardware systems for exploration of oil and gas deposits leads to the necessity to design software and algorithmic means aimed at increasing the reliability of quantitative determination of parameters of heterogeneous
* Corresponding author. E-mail address: [email protected] (I.V. Mikhaylov)
reservoirs. Improving the accuracy of evaluating the reservoir saturation from electrical logging is particularly important in this case. One way to do this is to employ realistic interpretation models. These models should include a description of the complex spatial distribution of the electrophysical parameters of the geological model and take into account various physical effects. However, their employment is only possible with the use of numerical simulation and inversion when solving multidimensional problems of computational electrodynamics. Traditionally, the reservoir saturation pattern is determined from induction and galvanic logging data based on analysis of the radial electrical resistivity profile determined within the scope of a 1D cylindrically layered model (Epov and Nikitenko, 1993). The radial resistivity distribution pattern is due to the displacement of formation fluids by mud filtrate and the change in salinity in the near-wellbore zone during formation drilling. However, when interpreting in formations of limited thickness, characterized by an inhomogeneous invaded zone and depth-variable oil content, it is necessary consider the effect of overlaying and underlying rocks. In addition, the presence of conductive shale and high-resistivity (compacted sand and carbonate) thin layers commonly found in reservoirs leads to significant inaccuracies in determining the resistivity
1068-7971/$ - see front matter D 201 7, V.S. So bolev IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.rgg.201 + 6.09.032
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and subsequent unreliable estimates of fluid saturation. To avoid this, it is necessary to use an axisymmetric interpretation model which takes into account both the radial and vertical resistivity distributions (Epov et al., 2013; Mikhaylov and Glinskikh, 2015; Nikitenko et al., 2015). Integration of well logging methods is widely used to improve their efficiency in exploration of complex oil and gas reservoirs (Kneller and Potapov, 1992; Mezzatesta et al., 1994; Rabinovich and Tabarovsky, 2001; Shanjun et al., 2007; Yang, 2001). Due to the different sensitivity and resolution of induction and galvanic logging measurements, the integration of these methods improves the reliability of resistivity determinations. Joint inversion of electrical logging data provides an integrated geoelectric model. The construction of a consistent interpretation model narrows the region of ambiguity (or equivalence) in determining the model parameters, which is due to the different physical principles of field excitation and signal measurement and the different influence of the parts of a medium on the recorded signals. It is known that conductive regions have a greater influence on induction signals, whereas resistive ones on galvanic signals. The use of an integrated interpretation model to examine different physical processes provides more accurate evaluation of the properties of reservoirs. For example, an integrated geoelectric and hydrodynamic model of the near-wellbore zone has been widely applied for an integrated petrophysical interpretation of electrical logging data. Subsequently, the need for further consideration of geomechanical properties has led to the development of a multidisciplinary approach based on an integrated electro-hydrodynamic and geomechanical model (Yeltsov et al., 2014). In addition, joint inversion of induction and galvanic logging data makes it possible to study the various effects involved in the propagation and interaction of electromagnetic fields with the medium. That is, it becomes possible to study in detail the frequency dispersion of the resistivity and electrical macroanisotropy of rocks, which have recently attracted considerable interest, in particular, for deviated and horizontal wells (Epov et al., 2010; Nikitenko et al., 2016). Their consideration in well logging data interpretation necessitates the use of specialized software. There are various approaches to the solution of computational problems of electrical logging using axisymmetric geological models. Forward problems have been solved by numerical-analytical and approximate methods (Epov and Nikitenko, 1993; Glinskikh et al., 2013a,b, 2014; Tabarovsky and Rabinovich, 1998). These methods have found widespread use as they allow developing fast algorithms for data processing and inversion. Due to the complication of electrical logging models and the rapid development of computational methods for electrodynamic problems, grid methods are preferred. Of these, finite-difference and finite-element methods are most widely used to simulate electromagnetic fields in inhomogeneous media (Epov et al., 2007; Surodina and Epov, 2012). However, solutions of multidimensional problems in full statement for processing and interpreting large volumes of field data are very resource intensive and have therefore found only limited use. Significant advances in this
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area have been made possible through the development of computer information technologies and multiprocessor computation systems. It has become feasible to use computationally intensive operations for practical purposes of data processing and interpretation, which will significantly improve the accuracy of determination of geoelectric parameters and the reliability of the results of interpretation based on the data of a suite of well logging tools. This paper presents an algorithm for joint numerical inversion of high-frequency electromagnetic logging (VEMKZ) and lateral logging (BKZ) data on axisymmetric geological models and the results of its application on synthetic and field data. It should be noted that the developed algorithm can be easily adapted to the numerical inversion of data of electrical logging tools with induction, galvanic or mixed induction-galvanic types of signal generation and measurement.
Computational inversion algorithm The basic interpretation model is a 2D axisymmetric geoelectric model which describes the penetration of a horizontally layered section by a vertical well with the formation of invaded near-wellbore zones (Fig. 1). The axisymmetric model includes a sequence of layers with plane parallel horizontal boundaries penetrated by a vertical cylindrical well. In the near-wellbore zone of each of the layers, there may be two zones—an invaded zone and a low-resistivity annulus— and in some cases, an additional flushed zone. They are separated from each other, from the borehole, and the formation by coaxial-cylindrical boundaries. Each of the regions of the geoelectric model is characterized by its own resistivity. The developed joint inversion algorithm is based on grid finite difference algorithms for numerical simulation of induction and galvanic data for media with axial symmetry (Surodina and Epov, 2012). Numerical solutions of forward problems after finite difference discretization are reduced to systems of linear algebraic equations (SLAE) with large sparse matrices, which are solved using well-known direct and iterative methods. The main features of the numerical simulation algorithms are as follows. Numerical 2D simulation of galvanic logging data reduces to the solution of the Poisson equation. Using a conservative difference scheme reduces the problem to a SLAE, which is symmetrized and then solved by the conjugate gradient method. Due to the ill-conditioning of the SLAE, it is preconditioned using Hotelling’s algorithm (Labutin and Surodina, 2013). The solution of the forward 2D problem of induction logging reduces to the solution of the Helmholtz equation. The obtained SLAE is solved using the COCR iterative method (Sogabe and Zhang, 2007), also with preconditioning. The developed algorithms for solving forward problems have been carefully verified, tested, and validated on a large number of realistic geoelectric models.
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Fig. 1. Interpretation geoelectric model with axial symmetry.
The solution of the inverse 2D problem reduces to the reconstruction of the spatial resistivity distribution of the rocks from measured log responses. The 1D inversion results obtained for the cylindrically layered model from the data of a suite of methods or an individual method are effectively used as the starting model. The numerical inversion is performed in two stages. At the first stage, 1D inversion with reconstruction of the radial resistivity profile from the wellbore wall into the formation is carried out at the layer interval, with a preliminary manual or automated determination of the position of horizontal boundaries through the analysis of derivatives. At the second stage, the resistivity distribution and the positions of the horizontal boundaries and cylindrical zones are refined using a 2D model. The numerical inversion of field data involves minimizing the standard deviation of measured data from synthetic data by changing the model parameters using a given algorithm. The numerical inversion algorithm is based on the downhill simplex method. It quickly selects the right search direction and significantly reduces the minimized function as early as at initial iterations. The inversion algorithm can use arbitrary combinations of signals from a complete set of measurements in the case of, e.g., the absence of signals or their low-quality. In the inversion model, the parameters can be fixed or constrained. This makes it possible to refine the values of individual parameters if reinversion is necessary, and reduce the number of fitted parameters when the survey interval contains thick homogeneous layers whose parameters have been already determined with good accuracy using 1D inversion.
Testing the inversion algorithm on noisy synthetic electrical logs To test the performance of the developed numerical inversion algorithm, we conducted a comparative analysis of 2D and 1D joint inversions of noisy synthetic electrical logs calculated using the finite difference method. Realistic geoelectric models of terrigenous oil and gas reservoirs of the E–W striking Ob’ River area in West Siberia, complicated by thin layers of shales and carbonates, are
penetrated by boreholes with fresh drilling mud (radius 0.108 m, resistivity 2 Ohm m). The permeable intervals include an invaded zone and a low-resistivity annulus in the oiland water-saturated formations with movable oil and water (Antonov et al., 2012). The drilling mud invasion pattern is due to the resistivity change in the near-wellbore zone associated with salinity redistribution between the formation water and mud filtrate. The signals of five main VEMKZ probes (DF05, DF07, DF10, DF14, and DF20) and four BKZ probes (A0.4M0.1N, A1.0M0.1N, A2.0M0.5N, and A4.0M0.5N) were analyzed. Synthetic logs were made noisy with the use of a normally distributed random variable described by a real measurement error model. Figure 2a, b shows the results of numerical inversion of noisy synthetic electrical logs for the model of a gas-oil-water reservoir overlain by shales and underlain by carbonate rocks. The reservoir is complicated by thin (0.3 m) interlayers. Thus, between the gas- (2.0–3.5 m) and oil-saturated (3.8–5.3 m) formations there is a shale interlayer (3.5–3.8 m) and between the oil- and water-saturated (5.6–7.1 m) portions is a carbonate interlayer (5.3–5.6 m), which is typical of sections in West Siberia. A conductive invaded zone is present in the permeable gas-saturated formation, and a resistive invaded zone in the other two formations. Figure 2a shows synthetic VEMKZ logs (left) and synthetic BKZ logs (right) calculated via 2D simulation. Here and below, the solid lines correspond to the noisy signals in the starting model and the dashed lines to the signals in the model obtained from the results of 1D inversion in the cylindrically layered model. Figure 2b shows the distributions of the resistivity (left) and cylindrical and horizontal boundaries (right) in the model considered. Here and below, the solid lines correspond to the starting model and the dashed lines to the results of 1D inversion in the cylindrically layered model. Analyzing the noisy electrical logs, we note the following (Fig. 2a). The behavior of VEMKZ logs differs significantly from that of BKZ logs. VEMKZ signals are characterized by higher information content when performing a visual analysis, and a higher degree of sectional layering. The shale interlayer is clearly identified in all VEMKZ logs, while in BKZ logs its presence is detected only by the A0.4M0.1N and A2.0M0.5N probes. The carbonate interlayer, in contrast, is manifested in the logs of all BKZ probes. As for VEMKZ, this high-resistivity object in the section can be identified only presumably, from the data of the short DF05 probe. Below the high-contrast boundary of the water-saturated formation and the underlying carbonate rocks, asymptotic values are reached faster for the logs of the long VEMKZ probes than for the BKZ logs. This is due, in particular, to the significant influence of the borehole on BKZ reponses. We analyze the results of the joint 1D inversion of the VEMKZ and BKZ data obtained at the first stage of the numerical interpretation (Fig. 2a, b). Note that the reconstructed values of many geoelectric parameters differ significantly from the true values. The positions of the horizontal
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Fig. 2. Synthetic VEMKZ and BKZ logs (a) for the model of a gas-oil-water reservoir with thin shale and carbonate interlayers (b). Solid lines correspond to the starting model, and dashed lines to the model obtained by 1D inversion. Here and below: F is a formation; A is a low-resistivity annulus; IZ is an invaded zone; FZ is a flushed zone.
boundaries shown by a dashed line on the right of Fig. 2b were determined using an automatic boundary detection algorithm. As can be seen, the differences between their coordinates and the true values reach 0.3 m. For better agreement between the field and simulated sounding curves in the shale and carbonate interlayers, it was required to
introduce invaded zones which were absent in the starting model. The obtained resistivity of the shale interlayer differs from the model value by a factor of 1.6, and the resistivity of the carbonate interlayer differs from it by a factor of 14. As regards the permeable formations, the largest deviations from the starting model are observed in the gas- and oil-saturated
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Fig. 3. Synthetic VEMKZ and BKZ logs (a) for the model of a gas-oil-water reservoir with thin shale and carbonate interlayers (b). Solid lines corresponds to the starting model, and dashed lines to the model obtained by joint 2D inversion.
intervals. In the first of them (2.0–3.5 m), with a resistivity of 50 Ohm m, the fitted resistivity is 9.4 Ohm m, which corresponds to oil and water saturation. In this case, instead of a conductive invaded zone with a radius of 0.2 m and a resistivity of 35 Ohm m, a wide resistive invaded zone (0.7 m, a resistivity of 27 Ohm m) is obtained. The reason for this
incorrect result is the strong influence of low-resistivity 1.5 m thick shale interlayer over the top and under the bottom of the layer considered. In the oil-saturated reservoir (3.8–5.3 m) with a resistivity of 15 Ohm m, the resistivity found from the results of the 1D approach is 7.9 Ohm m; the radius of the invaded zone differs from the true value by a factor of 1.5.
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Fig. 4. Synthetic VEMKZ and BKZ logs (a) for the model of a gas-oil-water reservoir in mudstone complicated by a thin carbonate interlayer (b). Solid lines correspond to the starting model, and dashed lines to the model obtained by 1D inversion.
This is largely related to the high resistivity contrast of the reservoir considered and the underlying and overlying rocks. At the second stage, we carried out a joint 2D inversion of VEMKZ and BKZ data (Fig. 3a, b). The results of the 1D inversion were used as the starting model. Resistivities of the host rocks were fixed and were not further fitted. All other
electrophysical and geometrical parameters of the section (resistivities of the formations and invaded zones, positions of the radial and horizontal boundaries) were refined during the 2D inversion considered. Separately, we note that the invaded zones in the shale and carbonate interlayers did not provide a satisfactory match between same-name logs and were ex-
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Fig. 5. Synthetic VEMKZ and BKZ logs (a) for the model of a gas-oil-water reservoir in mudstone complicated by a thin carbonate interlayer (b ). Solid lines correspond to the starting model, and dashed lines to the model obtained by joint 2D inversion.
cluded, with their corresponding resistivities being redistributed among adjacent layers. As a result, there was excellent agreement between the practial and fitted logs and parameters of the medium. Among the electrophysical parameters, the resistivity of the carbonate interlayer was reconstructed with the greatest error (13%), and among the geometrical parame-
ters, the radii of the invaded zones in the gas- and oil-saturated formations (errors of 15 and 10%, respectively) were reconstructed the least accurately due to the equivalence of the signals. Figure 4a, b shows the results of numerical inversion of noisy synthetic electrical logs for the model of an oil-water
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Fig. 6. VEMKZ and BKZ logs (a) and the result of the joint 2D inversion (b) in the AS5–6 formation interval of the Fedorovskoe field. Solid lines correspond to field logs, and dashed lines to synthetic logs for the fitted model.
reservoir in mudstone complicated by a thin carbonate interlayer. In the model considered, the underlying mudstone is overlain by a water-saturated formation (5.3–6.8 m), which is overlain by an interval containing movable oil and water (3.8–5.3 m). Above it is an oil-saturated formation (2.0–
3.5 m), which is separated from the latter by a thin (3.5–3.8 m) carbonate interlayer and overlain by mudstone. A resistive invaded zone is present in all permeable formations, and a low-resistivity annulus is observed in the oil- and water-saturated formation.
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Fig. 7. VEMKZ and BKZ logs of the SKL hardware system (a) and the result of joint 2D inversion (b) on the YuS2 formation interval of the Vostochno-Surgutskoe field. Solid lines correspond to field logs, and dashed lines to synthetic logs for the fitted model.
Analysis of the logs shows that the high-resistivity carbonate interlayer is clearly identified in BKZ logs and is almost not detected from VEMKZ responses (Fig. 4a). The converse is true for the 3.8–5.3 m oil- and water-saturated interval with a low-resistivity annulus, to which VEMKZ is highly sensitive.
It can be seen from Fig. 4b that the result of the joint 1D inversion of VEMKZ and BKZ is insufficiently accurate. In particular, in the carbonate interlayer, an erroneous invaded zone is introduced. Nevertheless, except for the position of the lower horizontal boundary, the parameters of the invaded zone, low-resistivity annulus, and the uninvaded zone of the
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oil- and water-saturated formation are determined fairly accurately. During the joint 2D inversion, all parameters, except for the fixed resistivity of host mudstone, were fitted with high accuracy, as shown in Fig. 5a, b. The reconstruction error was the greatest for the resistivity of the thin carbonate interlayer (11.7%), the radius of the invaded zone, and the resistivity of the uninvaded zone of the oil-saturated formation (4.5%). The latter is due to the low resistivity contrast in this formation. Thus, using noisy synthetic VEMKZ and BKZ signals for typical reservoir models, it has been shown that the developed algorithm for numerical inversion of well logging suite data provides a consistent geoelectric model that takes into account the 2D resistivity distribution of complex geological formations, which improves the accuracy of evaluation of their fluid saturation. Joint numerical inversion of field data with the construction of geoelectric models of terrigenous reservoirs The algorithm was tested on VEMKZ and BKZ field logs from dozens of wells in various fields of the Surgut Arch of the E–W striking Ob’ River area, where substantial oil accumulations are confined to Lower Cretaceous (AS, BS) and Jurassic (YuS) rocks. Consider as an example the AS5-6 formation interval of the Fedorovskoe field, which is one of the largest multilayer fields. The interval is penetrated by a borehole containing clay drilling mud with a resistivity of 2.7 Ohm⋅m. The joint 2D inversion of VEMKZ and BKZ responses (Fig. 6a) provided a realistic geoelectric model for a geological section of complex structure (Fig. 6b). An oil- and water-saturated reservoir (38.0–53.7 m) complicated by a high-resistivity carbonate interlayer (47.8–49.6 m) is identified. The reservoir is overlain by shale deposits and underlain by a thick water-saturated formation AS7-8 (55.3–80.0 m, the figure shows the upper part), which is separated from the latter by a clayish interval (53.7–55.3 m). The readings of the long VEMKZ probes increase monotonically in the depth range 41.0–47.8 m, indicating an increase in water saturation. In the upper part of the reservoir (38.7–45.8 m) is a low-resistivity annulus, reflecting its saturation with movable oil and formation water. Below, at depths of 45.8–47.8 and 49.6–52.5 m, are water-saturated reservoir zones. The verification of the 1D inversion results using 2D simulation does not provide good agreement between the field and synthetic VEMKZ and BKZ logs in the survey interval. The model parameters were refined by joint 2D inversion. It was found that the spatial resistivity distributions obtained using the 1D and 2D approaches differ significantly. The resistivities of high-contrast formations are satisfactorily fitted, as a rule, after a preliminary clarification of the position of the horizontal boundaries. In conclusion, we turn to Fig. 7, which shows the results of applying the algorithm to VEMKZ and BKZ field data recorded using the SKL hardware system in the Vostochno-
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Surgutskoe field. The borehole was drilled with clay mud (resistivity 2.8 Ohm m). Since the data were recorded by the high-precision equipment, preliminary filtering and smoothing were not needed. The interval considered (85.0–91.0 m, Fig. 7a) is the Middle Jurassic sandy horizon YuS2, with a complex geological structure. In particular, the reservoir is partially carbonized, which is especially pronounced in logs at depths of 86.3–87.0 and 89.5–90.0 m. In view of the considerable vertical heterogeneity of the interval, most of it was divided into layers about 0.5 m thick, in which the inversion was performed on a layer-by-layer basis. The joint 2D inversion resulted in a geoelectric model of the reservoir in mudstone, whose high resistivity indicates oil saturation. Unlike in the previous examples, a flushed high-resistivity zone of small radius was introduced at all intervals, if necessary. Note that the logs of the short probes, primarily BKZ, were fitted slightly worse. This may be due to the formation of caverns, which changed the nominal radius of the well and complicated the resistivity distribution directly near the wellbore wall. As regards the computational performance, the use of the sequential algorithms for solving the forward 2D problem is rather resource-intensive. At the same time, graphics processors hugely increase the speed, allowing near real-time 2D inversion. Thus, the numerical interpretation of VEMKZ and BKZ field logs from the Lower and Middle Jurassic intervals of the Fedorovskoe and Vostochno-Surgutskoe fields demonstrated the possibility of obtaining a consistent geoelectric model of complex hydrocarbon reservoirs from electrical logging suite data. Conclusions The reliability of the determination of geoelectric parameters of complex fluid-saturated formations is improved by combining induction and galvanic logging measurements within the framework of an integrated consistent interpretation model. A computational algorithm for joint 2D numerical inversion of electrical logging data was developed and software implemented. The algorithm is based on numerical solutions of forward problems using the finite difference method and nonlinear minimization using the downhill simplex method. A stepwise numerical inversion scheme is used, during which geoelectric parameters are determined in cylindrically-layered and 2D models. The algorithm was tested on noisy synthetic and field VEMKZ and BKZ logs from the Fedorovskoe and Vostochno-Surgutskoe oil fields of the E–W striking Ob’ River area, including those recorded by the SKL hardware system. It is shown that the algorithm improves the accuracy of interpretation of electrical logs compared to the widely used 1D approach. The algorithm can be easily adapted to the numerical interpretation of data acquired with other induction, galvanic, and mixed induction-galvanic logging tools.
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This work was supported by the Government of the Novosibirsk region.
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Editorial responsibility: M.I. Epov