Izvestiya, Physics of the Solid Earth

, Volume 54, Issue 2, pp 359–371

# One Solution of the Forward Problem of DC Resistivity Well Logging by the Method of Volume Integral Equations with Allowance for Induced Polarization

Article

## Abstract

For theoretically studying the intensity of the influence exerted by the polarization of the rocks on the results of direct current (DC) well logging, a solution is suggested for the direct inner problem of the DC electric logging in the polarizable model of plane-layered medium containing a heterogeneity by the example of the three-layer model of the hosting medium. Initially, the solution is presented in the form of a traditional vector volume-integral equation of the second kind (IE2) for the electric current density vector. The vector IE2 is solved by the modified iteration–dissipation method. By the transformations, the initial IE2 is reduced to the equation with the contraction integral operator for an axisymmetric model of electrical well-logging of the three-layer polarizable medium intersected by an infinitely long circular cylinder. The latter simulates the borehole with a zone of penetration where the sought vector consists of the radial Jr and Jz axial (relative to the cylinder’s axis) components. The decomposition of the obtained vector IE2 into scalar components and the discretization in the coordinates r and z lead to a heterogeneous system of linear algebraic equations with a block matrix of the coefficients representing 2x2 matrices whose elements are the triple integrals of the mixed derivatives of the second-order Green’s function with respect to the parameters r, z, r', and z'. With the use of the analytical transformations and standard integrals, the integrals over the areas of the partition cells and azimuthal coordinate are reduced to single integrals (with respect to the variable t = cos ϕ on the interval [−1, 1]) calculated by the Gauss method for numerical integration. For estimating the effective coefficient of polarization of the complex medium, it is suggested to use the Siegel–Komarov formula.

## References

1. Dmitriev, V.I. and Zakharov, E.V., Integral’nye uravneniya v kraevykh zadachakh elektrodinamiki (Integral Equations in the Boundary Problems of Electrodynamics), Moscow: Mosk. univ., 1987.Google Scholar
2. Gradshtein, I.S. and Ryzhik, I.M., Tablitsy integralov, summ, ryadov i proizvedenii (Table of Integrals, Series, and Products), Moscow: Nauka, 1970.Google Scholar
3. Komarov, V.A., Basic principles of using the method of induced polarization in prospecting for ore, in Metodika i tekhnika razvedki (Methods and Techniques for Prospecting), Leningrad: ONTI VITR, 1960, no. 23, pp. 77–86.Google Scholar
4. Kormil’tsev, V.V. and Ratushnyak, A.N., Modelirovanie geofizicheskikh polei pri pomoshchi ob”emnykh vektornykh integral’nykh uravnenii (Modeling the Geophysical Fields with the Use of Volume Vector Integral Equations), Yekaterinburg: Institut Geofiziki Ural’skogo otdeleniya RAN, 1999.Google Scholar
5. Pankratov, O.V., Avdeev, D.B., and Kuvshinov, A.V., Electromagnetic field scattering in the heterogeneous medium: forward problem solution, Fiz. Zemli, 1995, no. 3, pp. 17–25.Google Scholar
6. Seigel, H., Mathematical formulation and type curves for induced polarization. Geophysics, 1959, vol. 24, no. 3, pp. 547–565.
7. Singer, B.Sh., Method for solution of Maxwell’s equations in non-uniform media: Geophys. J. Int., 1995, Geophys. J. Int., pp. 590–598.Google Scholar
8. Tikhonov, A.N. and Samarskii, A.A., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: MGU, 1999.Google Scholar