Sofia Davydycheva

An efficient finite-difference scheme for electromagnetic logging in 3D anisotropic inhomogeneous media

We consider a problem of computing the electromagnetic field in 3D anisotropic media for electromagnetic logging. The proposed finite-difference scheme for Maxwell equations has the following new features based on some recent and not so recent developments in numerical analysis: coercivity (i.e., the complete discrete analogy of all continuous equations in every grid cell, even for nondiagonal conductivity tensors), a special conductivity averaging that does not require the grid to be small compared to layering or fractures, and a spectrally optimal grid refinement minimizing the error at the receiver locations and optimizing the approximation of the boundary conditions at infinity. All of these features significantly reduce the grid size and accelerate the computation of electromagnetic logs in 3D geometries without sacrificing accuracy.

Electrical-prospecting method for hydrocarbon search using the induced-polarization effect

We propose a method of surface and marine electrical prospecting using controlled-source excitation. The method is designed to detect hydrocarbon deposits at depths of a few kilometers and to map their boundaries. The technique is based on imaging the induced-polarization (IP) parameters of the geologic formation. We use the fact that, because of the imaginary part of the electric conductivity, polarized media support wave propagation processes whose nature is similar to displacement currents induced by the dielectric permittivity. However, unlike displacement currents, these processes reveal themselves at much lower frequencies and, therefore, at greater depths. It is established that the ratio of the second and the first differences of the electric potential does not decay after the current turn-off in polarized media, whereas it decays quickly if the IP effect is absent. Thus, the IP response can be observed directly and separated from the electromagnetic (EM) response. We use a vertical focusing of the electric current to decrease the effect of laterally adjacent formations to apply a 1D layered model in a 3D environment. This method obtained promising results in several regions of Russia.

A 3D parametric inversion algorithm for triaxial induction data

We develop a parametric inversion algorithm to determine simultaneously the horizontal and vertical resistivities of both the formation and invasion zones, invasion radius, bed boundary upper location and thickness, and relative dip angle from electromagnetic triaxial induction logging data. This is a full 3D inverse scattering problem in transversally isotropic media. To acquire sufficient sensitivity to invert for all of these parameters, we collect the data using a multicomponent, multispacing induction array. For each transmitter-receiver spacing this multicomponent tool has sets of three orthogonal transmitter and receiver coils. At each logging point single-frequency data are collected at multiple spacings to obtain information at different depths of investigation. This inversion problem is solved iteratively with a constrained regularized Gauss-Newton minimization scheme. As documented in the literature, the main computational bottleneck when solving this full 3D inverse problem is the CPU time associated with constructing the Jacobian matrix. In this study, to achieve the inversion results within a reasonable computational time, we implement a dual grid approach wherein the Jacobian matrix is computed using a very coarse optimal grid. Furthermore, to regularize the inversion process we use the so-called multiplicative regularization technique. This technique automatically determines the regularization parameter. Synthetic data tests indicate the developed inversion algorithm is robust in extracting formation and invasion anisotropic resistivities, invasion radii, bed boundary locations, relative dip, and azimuth angle from multispacing, multicomponent induction logging data.

Hybrid finite-difference integral equation solver for 3D frequency domain anisotropic electromagnetic problems

The modeling of the controlled-source electromagnetic (CSEM) and single-well and crosswell electromagnetic (EM) configurations requires fine gridding to take into account the 3D nature of the geometries encountered in these applications that include geological structures with complicated shapes and exhibiting large variations in conductivities such as the seafloor bathymetry, the land topography, and targets with complex geometries and large contrasts in conductivities. Such problems significantly increase the computational cost of the conventional finite-difference (FD) approaches mainly due to the large condition numbers of the corresponding linear systems. To handle these problems, we employ a volume integral equation (IE) approach to arrive at an effective preconditioning operator for our FD solver. We refer to this new hybrid algorithm as the finite-difference integral equation method (FDIE). This FDIE preconditioning operator is divergence free and is based on a magnetic field formulation. Similar to the Lippman-Schwinger IE method, this scheme allows us to use a background elimination approach to reduce the computational domain, resulting in a smaller size stiffness matrix. Furthermore, it yields a linear system whose condition number is close to that of the conventional Lippman-Schwinger IE approach, significantly reducing the condition number of the stiffness matrix of the FD solver. Moreover, the FD framework allows us to substitute convolution operations by the inversion of banded matrices, which significantly reduces the computational cost per iteration of the hybrid method compared to the standard IE approaches. Also, well-established FD homogenization and optimal gridding algorithms make the FDIE more appropriate for the discretization of strongly inhomogeneous media. Some numerical studies are presented to illustrate the accuracy and effectiveness of the presented solver for CSEM, single-well, and crosswell EM applications.

Focused-source electromagnetic survey versus standard CSEM: 3D modeling in complex geometries

A novel focused-source electromagnetic FSEM method focusestheEMfieldintheverticaldirectiontoprovidedeepreadingresistivitydata.FSEMoffersbetterspatialresolution and greater depth of investigation than the conventional controlled-source electromagnetic CSEM method for land and marine EM surveys. We have proven the high efficiency of FSEM by analyzing 3D models of various complex geologic formations in the presence of seafloor bathymetry, shallow resistive gas-hydrate overburdens, and secondary gas reservoirs formed above deeper oil reservoirs. Combining the powerofourfocusingtechniquewiththepowerofour3Dnumerical modeling method, we have developed exceptionally challenging test cases to conclude that FSEM automatically cancels unwanted shallow effects and allows simple visual interpretationofdeepreservoirresponses.Inaddition,FSEM is insensitive to imperfections in the setup geometry. We achievetheseadvantagesusingapropercombinationofmeasurementsacquiredinthereceiverexcitedbytransmitterssituated at different space points. The method is promising in anisotropicformationsaswell.

Separation of azimuthal effects for new-generation resistivity logging tools — Part I

Symmetrization/antisymmetrization of tensor resistivity measurements and data rotation technique enable separation of the formation response from the tool eccentricity effect in the borehole. Similar principles of data processing can be applied to tensor measurements acquired by both wireline and logging-while-drilling tools of the new generation. I show how to directly determine the bed boundary positions and the formation anisotropy azimuth and how to perform visual interpretation of raw tool data in the presence of the tool eccentricity. I study the tool behavior in conductive water-based mud boreholes — the situation that requires much more complicated numerical modeling than the case of resistive oil-based mud boreholes. I show when and how the tool eccentricity effect can be separated from the formation response. The separation technique can accelerate and improve existing methods of formation interpretation.

3D modeling of new-generation (1999–2010) resistivity logging tools

Electromagnetic (EM) logging is an im-portant method of formation evaluation because of its sensitivity to resistivity, which is a function of fluid saturation and fluid properties. Conventional induction and propagation resistivity logging tools excite the geological formation by axial magnetic dipole transmitter(s). Axial re-ceivers measure the formation response, reflecting the medium resistivity. Such axial measurement has no azimuthal sensitivity and is insensitive to the anisotropy and many other details of the formation.

Finite-difference solution of the three-dimensional electromagnetic problem using divergence-free preconditioners

SUMMARY The modelling of the marine electromagnetic (EM) problem requires fine gridding to account for the seafloor bathymetry and to model complicated targets. This makes the computational cost of the problem large by using conventional Finite-Difference (FD) solvers. To circumvent these problems, we employ a volume Integral Equation (IE) approach to arrive at a preconditioning operator. Such an approach significantly reduces the condition number and the size of the stiffness matrix. The preconditioner is divergence free and is based on a magnetic field formulation. The cost of constructing the preconditioning operator is much less than the one used to construct the operator in a standard IE approach. Further a homogenization technique that allows grids to be non-conformal to the inhomogeneity interfaces is used. The optimal grid technique is used to extend the boundaries of the simulation domain to infinity. Some numerical results are presented in order to illustrate the accuracy and the effectiveness of the method.

Focused-source EM survey versus time-domain and frequency-domain CSEM

A new focused-source electromagnetic (FSEM) method exploits the idea of focusing the EM field in a vertical direction to provide deep resistivity data. It allows directing the EM energy into the formation, similar to acoustic beam used in seismic exploration, enabling higher spatial resolution and greater depth of investigation, than the conventional controlled-source EM (CSEM) method.

Modeling of electromagnetic logs in a layered, biaxially anisotropic medium

A fast 1D electromagnetic (EM) modeling method has been developed and tested. It allows simulating triaxial responses of induction and propagation resistivity logging tools for both logging-while-drilling and wireline applications. An important new feature of the method is its ability to model resistivity tool responses in 1D biaxial anisotropic medium, whose anisotropy tensor has up to three different principal values. This feature is particularly useful to evaluate fractured formations. A few possible implementations of the method have been discussed. The modeling code has been extensively tested versus three other independent modeling methods. The new method is used to demonstrate a high sensitivity of transverse and cross-couplings of triaxial tensor measurement to all three principal values of the conductivity tensor.