Borehole Electromagnetics

In 1999, in response to the request of the CEMI sponsors, we initiated a project on 3-D induction logging in anisotropic media. It is now one of the first priority projects of the CEMI. During last few years we have been working on developing an interpretation technique for tensor induction well logging (TIWL) in anisotropic formations. We examine the response of a tri-axial electromagnetic induction well logging instrument in anisotropic media with deviated well. This instrument detects three components of magnetic field due to each of three transmitters, which represent the nine components of the induction tensor.

Figure 1. Schematic view of the tensor induction well logging (TIWL) instrument in the deviated well.

By theoretically analyzing the tri-axial induction instrument for its response to the magnetic field components induced in the conductive medium, we derived low frequency approximations for the quadrature components of our induction tensor. Based on this analysis, we found that by measuring the quadrature components of the induction tensor in a deviated borehole, the conductivity anisotropy of the media can be resolved from the instrument response, which can be used as the basis for a tensor logging instrument response interpretation Zhdanov et al., 2001; Foundations of tensor induction well logging.

We examined the basic principles of tensor induction well logging by numerical simulation of tensor induction data in layered anisotropic formations. The modeling of the tensor induction tool response in layered anisotropic media was based on an Electromagnetic Green's Tensors Library GT3D, developed by CEMI (EM Green's tensor library for horizontally layered anisotropic medium). This library makes the computing the Green’s tensors in any layered anisotropic formations as simple as calling the sine or cosine functions routine. The GT3D software is available to CEMI sponsors. We have applied this software for modeling and inversion of the typical TIWL data in a deviated well in an anisotropic medium (Modeling of tensor induction well logging in a deviated well in an anisotropic medium).

We have developed a method for TIWL interpretation in the horizontally layered formation. The goal of interpretation is to find the layer's interfaces and conductivities. However, the solution of this problem meets a lot of difficulties even for the case of the conventional induction logging in isotropic layered formation. The problem is that the traditional inversion methods use the smooth models of the conductivity distribution along the borehole to provide a stable and reliable solution. However, in the layered formation the conductivity changes sharply when we cross the layer's boundaries. In this case a smooth model does not represent well the real physical properties of the medium. We propose use in the inversion of the induction logging data the new approach of focusing inversion developed by CEMI. This approach is based on using specially selected stabilizing functional, which minimizes the area where strong model parameter variations and discontinuity occur. The method recovers the sharp boundaries between the different anisotropic geoelectrical layers and reconstructs both the horizontal and vertical resistivity profiles. The developed algorithm was tested by interpreting the synthetic TIWL data collected by a typical practical tensor induction tool in a deviated well in the layered anisotropic formations.

As an example, we consider the so-called Oklahoma benchmark model (Barber et al., 1999), which is widely used for testing modeling codes in well logging (Figure 2, left panel). The original isotropic Oklahoma model consists of 27 high resistivity contrast layers with varying thicknesses from 0.3 m up to infinity (the unbounded half-space). We extend this model to anisotropic one, with the horizontal resistivities equal to the resistivities of the original Oklahoma model, and with the different vertical resistivities in 13 of 27 layers. The configuration of an elementary tensor induction tool with tri-axial transmitting and receiving magnetic dipoles is shown in Figure 1. The two-coil probe spacing is 1 m, the moments of all tool component subarrays are equal to 1 Am², and the operational frequency is 20 KHz. The deviated borehole line is tilted at the constant dip angle of 30 degrees with respect to the vertical axes. The synthetic tensor induction well logging (TIWL) data have been computed for the anisotropic Oklahoma model using the GT3D software.

Figures 2-3 show the results of TIWL data interpretation for the anisotropic Oklahoma model using focusing inversion. Figure 2 presents a vertical profile of the horizontal resistivity obtained with the focusing inversion algorithm (shown by green line in the left panel). This result shows good restoration of the blocky resistivity profiles in anisotropic formations. The horizontal resistivity images match precisely the piecewise geometry and electric properties of the layered structure.

Figure 2. The solid black line on the left shows a vertical profile of the horizontal resistivity for Oklahoma benchmark model (after Barber et al., 1999), while the green line presents the same profile obtained with the focusing inversion algorithm. The blue and red curves on the right present the in-phase and quadrature components of the observed vertical magnetic field respectively (solid lines) due to vertical magnetic dipole transmitter. The predicted data computed for the inverse model, shown in green on the left, are plotted by the dotted lines on the right.

Figure 3 presents the inversion result for the vertical to horizontal resistivity ratio (Rv/Rh) for Oklahoma benchmark model, which provides a reasonable estimation of the true vertical resistivity profile. However, the images of vertical resistivity cross-section are slightly contaminated by the jumps happened in the vicinities of the high-contrast layer boundaries. At the same time, the predicted data fit the observed data extremely well, as one can see in the right-hand parts of Figures 2 and 3.

Figure 3. The solid black line on the left shows a vertical profile of the vertical to horizontal resistivity ratio (Rv/Rh) for Oklahoma benchmark model while the green line presents the same profile obtained with the focusing inversion algorithm. The blue and red curves on the right present the in-phase and quadrature components of the observed horizontal magnetic field respectively (solid lines) due to horizontal magnetic dipole transmitter. The predicted data computed for the inverse model, shown in green on the left, are plotted by the dotted lines on the right.

The tri-axial induction instrument was originally introduced to resolve anisotropic properties of the formation. However, this tool may find another important application. It can be used for the directional probing of the rock formations surrounding the borehole. This application may be extremely important in reservoir evaluation and monitoring of the oil/water contact. The conventional induction logging instrument (with the transmitter and receiver oriented along the borehole) usually provides an image of the resistivity distribution that is axially symmetrical with respect to the borehole axis. The tensor (tri-axial) induction instrument has a directional sensitivity, which allows finding the correct location of the 3-D resistive or conductive targets from the single-hole data.

We have developed a rapid technique for the 3-D inversion of the tensor induction data, based on the novel localized quasi-linear (LQL) approximation of the electromagnetic field. Our study demonstrates that the LQL approximation can be effectively used for 3-D imaging from a single borehole. The main advantage of this technique is that the LQL approximation makes it possible to run inversion at once for all transmitter and receiver positions. As a result, the LQL inversion is extremely fast and can be done in real time (within a few minutes), which is important for practical well logging applications. This demonstrates that the method can be used as the basis for a fast 3-D imaging in single hole imaging. We have also validated the efficiency of this method for a model typical in reservoir monitoring applications.

Figure 4, left panel, shows a model of one resistive layer with a resistivity of 100 Ohm-m and one conductive layer with a resistivity of 10 Ohm-m located on one side of the borehole. The background resistivity is 20 Ohm-m (so that the anomalous conductivity is of -0.04 S/m for the resistive layer and of 0.05 S/m for the conductive layer). The instrument is moving in the vertical direction with the observation points located every one meter. The operating frequencies are 1, 10, and 100 KHz. In this Figure, the stars highlight the transmitter locations and the crosses show the receiver locations.

Figure 4. A model of one resistive layer with a resistivity of 100 Ohm-m and one conductive layer with a resistivity of 10 Ohm-m located on one side of the borehole (left panel). The result of 3-D focusing inversion of the TIWL data (right panel).

The synthetic magnetic data for this model were computed using integral equation forward modeling code INTEM3D developed by the CEMI consortium (Hursan and Zhdanov, 2003). The data were contaminated by 2% random noise. The result of 3-D focusing inversion for this model is shown in Figure 4, right panel. The conductive and resistive layers appear in the right positions in this image.

The final model consists of a resistive layer with a resistivity of 100 Ohm-m and a conductive layer with a resistivity of 10 Ohm-m located on the opposite sides of the borehole (Figure 5, left panel). The background resistivity is 20 Ohm-m. The synthetic TIWL data for this model was inverted using our inversion code. Figure 5, right panel, presents a volume image of the inversion result obtained for this synthetic data. One can see an excellent reconstruction of the true model.

Figure 5. A model of one resistive layer with a resistivity of 100 Ohm-m and one conductive layer with a resistivity of 10 Ohm-m located on the opposite sides of the borehole (left panel). The result of 3-D focusing inversion of the TIWL data (right panel).

Note that our inversion algorithm is extremely fast and can be done in real time (within a few minutes), which is important for practical well logging applications. This demonstrates that the method can be used as the basis for a fast 3-D imaging in single hole imaging.

Hursan, G. and M. S. Zhdanov, 2002, Contraction integral equation method in 3-D electromagnetic modeling: Radio Science, 37, No. 6, 1089.

Zhdanov, M. S., Kennedy, W. D., and E. Peksen, 2001, Foundations of tensor induction well logging: Petrophysics, 42, 588-610.

Cheryauka, A. B., Sato, M., and M. S. Zhdanov, 2001, Induction logging with directional coil polarizations: modeling and resolution analysis, Petrophysics, 42, 227-236.

Zhdanov, M. S., Kennedy, W. D., Cheryauka, A., and E. Peksen, 2001, Principles of the Tensor Induction Well-Logging in a Deviated Well in an Anisotropic Medium, 42nd Annual SPWLA Symposium, Houston.

Cheryauka, A. B. and M. S. Zhdanov, 2001, Fast modeling of a tensor induction tool response in a horizontal well in inhomogeneous anisotropic formations, 42nd SPWLA Symposium, Houston.

Peksen, E. and M. S. Zhdanov, 2002, Apparent resistivity correction for tensor induction well logging in a deviated well in an anisotropic medium: Petrophysics, 44, 196-204.

Zhdanov, M. S., Cheryauka, A. B., and E. Peksen, 2003, Sharp boundary inversion of the tensor induction well-logging data: 44th Annual SPWLA Symposium, Galveston, Texas. PP.