Determination of dispersive resistive and dielectric properties of earth formations using multi-frequency measurements

An apparatus configured for determining geo-electrical parameters of an earth formation the apparatus comprising:

• a carrier (11) configured for conveyance in a borehole (12) and having disposed thereon a multi-frequency electromagnetic propagation instrument (201, 300) with at least one transmitter (T1, T2) and at least one receiver (R1, R2);
• and
• at least one computing unit (15, 20, 29), including at least one processor (17), configured for:
• (a) operating the multi-frequency electromagnetic propagation instrument at at least three frequencies by energizing the at least one transmitter of the instrument to induce currents in the formation, the formation possessing dispersive resistivity and dispersive dielectric constant, and receiving electromagnetic signals induced by the formation;
• (b) generating a 2-dimensional grid of a plurality of values of formation resistivity in a first selected range and a plurality of values of dielectric constant in a second selected range, with the first dimension of the grid being a resistivity of the formation and the second dimension of the grid being a dielectric constant of the formation;
• (c) using at least one computing unit to:
• i. generate a lookup table of synthetic signals of the propagation instrument for a set of models whose formation resistivities and dielectric constants are defined in nodes of said 2-dimensional grid and a set of operating frequencies;
• and
• ii. perform a 2-dimensional interpolation of the synthetic signals from the lookup table, where the first dimension is resistivity and the second dimension is dielectric constant;
• (d) defining first polynomial functions interpolating a logarithm of the dispersive resistivity of the formation and second polynomial functions interpolating a logarithm of the dispersive dielectric constant of the formation;
• (e) generating a multi-dimensional grid comprising a first number of first polynomial coefficients of said first polynomial functions and a second number of second polynomial coefficients of said second polynomial functions;
• (f) using said first polynomial functions to perform interpolation of the dispersive resistivity using the first number of first polynomial coefficients describing resistivity values from said multi-dimensional grid;
• (g) using said second polynomial functions to perform interpolation of the dispersive dielectric constant using the second number of second polynomial coefficients from said multi-dimensional grid;
• and
• (h) performing a multi-parametric scan over the sets of at least 2 resistivity parameters and sets of at least 3 dielectric constant parameters using the multi-dimensional grid, 2-dimensional interpolation, first polynomial functions, and second polynomial functions to find a combination of first polynomial coefficients and second polynomial coefficients which describe dispersive resistivity and dielectric constant providing a best fit between measured and synthetic data.

A method of determining geological parameters of earth formation material, the method comprising:

• (a) operating a multi-frequency electromagnetic propagation instrument at at least three frequencies by energizing at least one transmitter of the instrument to induce currents in the material, the material possessing dispersive resistivity and dispersive dielectric constant, and receiving electromagnetic signals induced by the material;
• (b) generating a 2-dimensional grid of a plurality of values of material resistivity in a first selected range and a plurality of values of dielectric constant in a second selected range, with the first dimension of the grid being a resistivity of the material and the second dimension of the grid being a dielectric constant of the material;
• (c) using at least one computing unit to:
• i. generate a lookup table of synthetic signals of the propagation instrument for a set of models whose formation resistivities and dielectric constants are defined in nodes of said 2-dimensional grid and a set of operating frequencies;
• and
• ii. perform a 2-dimensional interpolation of the synthetic signals from the lookup table, where the first dimension is resistivity and the second dimension is dielectric constant;
• (d) defining first polynomial functions interpolating a logarithm of the dispersive resistivity of the material and second polynomial functions interpolating a logarithm of the dispersive dielectric constant of the material;
• (e) generating a multi-dimensional grid comprising a first number of first polynomial coefficients of said first polynomial functions and a second number of second polynomial coefficients of said second polynomial functions;
• (f) using said first polynomial functions to perform interpolation of the dispersive resistivity using the first number of first polynomial coefficients describing resistivity values from the said multi-dimensional grid;
• (g) using said second polynomial functions to perform interpolation of the dispersive dielectric constant using the second number of second polynomial coefficients from said multi-dimensional grid;
• and
• (h) performing a multi-parametric scan over the sets of at least 2 resistivity parameters and sets of at least 3 dielectric constant parameters using the multi-dimensional grid, 2-dimensional interpolation, first polynomial functions, and second polynomial functions to find a combination of first polynomial coefficients and second polynomial coefficients which describe dispersive resistivity and dielectric constant providing a best fit between measured and synthetic data.