Whole-space inversion using phase correction method for multi-frequency dielectric array logging tool

A method of evaluating an earth formation intersected by a borehole, the method comprising:

• exciting an upper transmitter and a lower transmitter on a carrier in the borehole at at least one frequency;
• receiving signals at a plurality of receivers between the upper transmitter and the lower transmitter on the carrier responsive to the excitation;
• estimating a phase shift greater than 2π radians attributable to only a volume of interest of the earth formation adjacent to a section of the carrier between a pair of receivers of the plurality of receivers to mitigate effects of phase wrapping using:
• a first relative phase value comprising a measured phase shift between signals received at a first receiver of the pair of receivers and signals received at a second receiver responsive to excitation of the upper transmitter;
• and
• a second relative phase value comprising a measured phase shift between signals received at a second receiver of the pair of receivers and signals received at a first receiver responsive to excitation of the lower transmitter.

An apparatus for evaluating an earth formation intersected by a borehole, the apparatus comprising:

• a carrier configured to convey an upper transmitter, a lower transmitter, and a plurality of receivers between the upper transmitter and the lower transmitter in the borehole;
• and
• at least one processor configured to execute a method, the method comprising:
• exciting an upper transmitter and a lower transmitter in the borehole at at least one frequency;
• receiving signals at a plurality of receivers between the upper transmitter and the lower transmitter on the carrier responsive to the excitation, wherein a section of the carrier includes a plurality of receivers;
• and
• estimating a phase shift greater than 2π radians attributable to only a volume of interest of the formation adjacent to a section of the carrier corresponding to the plurality of receivers to mitigate effects of phase wrapping by estimating phase shifts attributable only to portions of the volume of interest adjacent to corresponding portions of the section between a plurality of pairs of receivers of the plurality of receivers;
• wherein each pair of the plurality of pairs of receivers is positioned at a corresponding distance that is no more than half of a shortest wavelength of at least one wavelength of the at least one frequency associated with the received signals.

An apparatus for evaluating an earth formation intersected by a borehole, the apparatus comprising:

• a carrier configured to convey an upper transmitter, a lower transmitter, and a plurality of receivers between the upper transmitter and the lower transmitter in the borehole;
• and
• at least one processor configured to execute a method, the method comprising:
• exciting the upper transmitter and the lower transmitter in the borehole at at least one frequency;
• receiving signals at a plurality of receivers responsive to the excitation;
• and
• estimating a phase shift greater than 2π radians attributable to only a volume of interest of the earth formation adjacent to a section of the carrier between a pair of receivers of the plurality of receivers to mitigate effects of phase wrapping using:
• a first relative phase value comprising a measured phase shift between signals received at a first receiver of the pair of receivers and signals received at a second receiver responsive to excitation of the upper transmitter;
• and
• a second relative phase value comprising a measured phase shift between signals received at a second receiver of the pair of receivers and signals received at a first receiver responsive to excitation of the lower transmitter.

A method of evaluating an earth formation intersected by a borehole, the method comprising:

• exciting an upper transmitter and a lower transmitter on a carrier in the borehole at at least one frequency;
• receiving signals at a plurality of receivers between the upper transmitter and the lower transmitter on the carrier responsive to the excitation, wherein a section of the carrier includes a plurality of receivers;
• estimating a phase shift greater than 2π radians attributable to only a volume of interest of the earth formation adjacent to a section of the carrier corresponding to the plurality of receivers to mitigate effects of phase wrapping by estimating phase shifts attributable only to portions of the volume of interest adjacent to corresponding portions of the section between pairs of receivers of the plurality of receivers;
• wherein the upper transmitter, the lower transmitter and each of the plurality of receivers has a corresponding unknown internal electronic equipment phase shift error.