This disclosure relates to using contraction mapping to efficiently determine a dielectric constant and/or an electrical resistivity of a geological formation.
This section is intended to introduce the reader to various aspects of art that may be related to various aspects of the present techniques, which are described and/or claimed below. This discussion is believed to be helpful in providing the reader with background information to facilitate a better understanding of the various aspects of the present disclosure. Accordingly, it should be understood that these statements are to be read in this light, and not as an admission of any kind.
Identifying the composition of a geological formation can provide information about the likely presence or absence of hydrocarbons in a subsurface geological formation. As such, many downhole tools have been developed that aim to analyze the geological formation from within a wellbore. Electromagnetic logging tools are one type of downhole tool that may be used to measure properties of the geological formation. By measuring the way that propagation radio waves travel through the geological formation, the composition of the geological formation may be ascertained at various depths through the well. Indeed, the attenuation and phase shift of the propagation radio waves through the geological formation are dependent on electromagnetic properties of the materials that compose the geological formation—namely, electrical resistivity and dielectric constant, among others. These electromagnetic values may correspond to the composition of the geological formation.
In many cases, the dielectric constant of the geological formation may be determined due to a relationship to the resistivity of the geological formation, using an empirical formula based on measurements of a large number of rock samples, mainly sandstone and carbonate. For shale formations, however, the relationship is not so straightforward. Instead, dielectric inversion may be used to obtain the formation resistivity and dielectric constant independently in shale. This inversion may involve an iterative minimization procedure to minimize a discrepancy between the measured electromagnetic data and simulated data for models related to particular formation compositions with a forward modeling program or with a two-dimensional lookup table. In this way, through multiple iterations, values of electrical resistivity and dielectric constant may sometimes be found.
But the approach noted above is not without its drawbacks. Indeed, the iterative minimization procedure to determine electrical resistivity and dielectric constant may sometimes fail to converge—that is, to sufficiently minimize the discrepancies between the measured and simulated electromagnetic data—thereby producing no solution values or producing values that may not be reliable. The approach may also be heavily dependent on the initial guess of the proper values of electrical resistivity and dielectric constant, meaning that the ability to converge to a solution or the number of iterations to converge to the solution may vary depending on the initial guess of these values.