An Analytical-Numerical Method for Generating Full Waveform Synthetic Acoustic and VSP Logs in Multilayered Formations Intercepted by Wells, 14-R9564

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Principal Investigators
Jorge O. Parra
Sarah H. Gonzales
Chris L. Hackert
Peicheng Xu

Inclusive Dates:  08/01/05 – 01/31/07

Background - Borehole acoustic (sonic) logging has become an indispensable tool for petroleum reservoir exploration, reserve estimation, well completion, and hydrocarbon production. Unlike surface seismic, borehole sonic allows us to carry out reservoir characterization at the scale of about one foot. To make use of the rich information provided by sonic logs, we must have a solid understanding of the acoustic-seismic wavefield in formations with depth dependent properties and boreholes with irregular caliper (illustration). Such wavefields can be quite complicated because of the scattering and resonance induced by formation inhomogeneity and the changing patterns of borehole multiples and geometrical spreading due to a varying caliper. Bouchon and his colleagues developed a boundary element method based solution for sonic waves associated with layered formations or irregular boreholes [Bouchon M. and Schmitt, D. P., 1989, Full-wave acoustic logging in an irregular borehole: Geophysics, 54(6), 758 – 765; Bouchon, M., 1993, A numerical simulation of the acoustic and elastic wavefields radiated by a source on a fluid-filled borehole embedded in a layered medium: Geophysics, 58(4), 475 – 481; Dong, W., Bouchon, M., and Toksoz, M. N., 1995, Borehole seismic-source radiation in layered isotropic and anisotropic media: Boundary element modeling: Geophysics, 60(3), 735 – 747.] Unfortunately, his model does not deal with attenuation and has not handled a very long borehole or a large number of layers. Besides, his discrete wavenumber method for the calculation of Green functions is not adequately accurate and efficient.

Approach - The objective of our work is to develop a more robust analytical-numerical technique and algorithm for generating synthetic full waveform sonic logs using more realistic formation and borehole models, namely boreholes with depth-dependent caliper in viscoelastic anisotropic formations consisting of dozens of layers. The algorithm will be very useful for interpreting and processing observed acoustic logs. We use the boundary integral equation (BIE) method to solve this borehole acoustic model. The BIE method reduces the three-dimensional partial differential equations in the full space to two-dimensional integral equations on the surface of the borehole wall: r=r(z), 0<θ<2π, -<z<, with θ being the azimuthal angle. The BIE method is more efficient than the finite difference method and the finite element method, especially when at least one dimension of the geometry is large and/or a large number of source-receiver locations are involved. Furthermore, it is more convenient to include attenuation and frequency dependent parameters in the BIE approach. The BIE formulation is exact. For monopole sources in horizontally layered formations, the model becomes axisymmetric and the q dependency of the BIEs can be removed.

In the conventional BIE approach, such as Bouchon's method, the two-dimensional (θ-z) boundary surface is discretized by boundary elements. Each integrand in the BIEs contains products of known coefficients and unknown displacement or pressure components at the boundary nodes. These coefficients are the Green functions in the layered medium and the associated stress tensors, calculated at the Gaussian points on the boundary elements and produced by unit forces and pressure at the boundary nodes. They are given in the form of radial wavenumber (kρ) integrals. Finally, the BIEs are reduced to a system of linear equations of the displacements and pressure on the boundary nodes, from which we can obtain the entire wavefield both in the borehole and formation. This approach is equivalent to the one developed by Bouchon. A major difference is that we use the modified Clenshaw-Curtis (MCC) integration method [Parra, J. O., Sturdivant V. R., and Xu, P.-C, 1993, Interwell seismic transmission and reflection through a dipping low-velocity layer: J. Acoust. Soc. Am., 93(4), 1954 - 1969], instead of discrete wavenumber method, to calculate the Green functions and stress tensor. The former proved to be superior to the latter both in accuracy and efficiency [Dravinski, M. and Mossessian T. K., 1988, On evaluation of the Green functions for harmonic line loads in a viscoelastic half space: Int. J. Num. Methods Eng., 26, 823-841.

In a novel alternative approach, the Transformed Boundary Integral Equation (TBIE) method, we apply an integral transform with exp(ikzz) to the BIEs on the borehole wall surface Xu, P.-C. and Parra, J. O., 2003, Effects of single vertical fluid-filled fractures on full waveform dipole sonic logs: Geophysics, 68(2), 487 - 496] .The integral transform replaces the z-dependence by the kz dependence. Instead of discretizing the boundary by boundary elements in both z and θ, the mesh is reduced to one-dimensional (in θ only) for each kz. As a tradeoff, we need an inverse integral transform with exp(-ikzz) to obtain the solution of the original BIEs. The integral transform and its inverse is direct integration in z and kz and is more advantageous than solving the original BIEs with a large dimension in z. The integration in both z and kz can be extremely efficient using the MCC integration method.

Accomplishments - We first took the conventional BIE approach. We derived all BIE equations and expressions for the borehole sonic model in both Cartesian and cylindrical forms. We implemented algorithms for the adaptive MCC integration of kρ- integrals, Gaussian quadrature of θ-integrals, adaptive z-meshes, bookkeeping of the Greenís functions and stress tensor, and their θ-integrals, assembling influence matrices of BIEs, equation solver for full, complex, nonsymmetric matrices, various testing options, and benchmark and reference solutions. We re-structured, expanded, modified and combined previous BIE based codes to construct the computer code for this borehole sonic model. We conducted systematic verification tests of the code against the result calculated by a wavenumber integral (WNI) method for a uniform borehole in a uniform medium. After an intensive debugging process, the BIE result now agrees with that by the WNI reasonably well. However, some discrepancies persist. They may indicate a numerical problem caused by long z-meshes. In attempts to avoid this problem, we have switched to the TBIE approach. We derived the additional equations and expressions needed for the TBIEs. We implemented algorithms for applying the MCC integration to z- and kz- integrals and for the bookkeeping of these integrals. The new, TBIE based code for the borehole sonic model will soon be completed and tested.

Geometry of a fluid-filled borehole with varying caliper in a multilayered formation.

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