Page 127 - Machine Learning for Subsurface Characterization

104 Machine learning for subsurface characterization

property is causal in nature; it is mathematically represented as a complex func-
tion of the operating frequency because of the phase difference between the
material response and the applied EM field. Complex dielectric permittivity
∗ ∗
(ε ) and complex conductivity (σ ) measurements in the frequency range of
10 Hz to 1 GHz are widely used for the oil and gas, hydrological, and several
∗ ∗
geophysical applications. ε and σ are fundamental material parameters that
influence the transport and storage of electromagnetic (EM) energy in geolog-
0
ical materials. ε is a measure of storage of EM energy due to charge separation
00
and accumulation (also, referred to as polarization), ε is a measure of dissipa-
tion of EM energy due to friction between polarized structures, and σ is a mea-
sure of dissipation of EM energy due to charge transport in a material in
response to an external EM field. Consequently, for porous geomaterials and
geological formations that are not very conductive, σ* ¼ – iωε* ¼ σ – iω(ε +
0
00 00 0
iε ) ¼ (σ + ωε ) – iωε ¼
σ eff – iωε eff , where subscript ’eff’ denotes effective.
Dielectric permittivity exhibits frequency dependence because various
polarization phenomena are dominant in various frequency ranges and the
effects of polarization lags the applied EM field, that is, polarization does
not change instantaneously with the applied EM field. For measurements at
∗
low frequencies, the imaginary component of ε is much lower than the real
∗
component of ε . As the frequency increases, the imaginary component of ε ∗
∗
increases, whereas the real component of ε decreases. For fluid-filled porous
geomaterials, frequency dispersion is due to various polarization phenomena,
each dominant over certain frequency range. Polarization is due to charge sep-
aration and subsequent charge accumulation in the presence of externally
applied EM field. Few examples of polarization phenomena in fluid-filled geo-
materials are orientation/dipolar polarization of dipoles in fluid, Maxwell-
Wagner polarization at brine-matrix interface, interfacial polarization of
conductive minerals, membrane or double-layer polarization of clays and sur-
face charge-bearing grains, and concentration polarization due to differential
mobilities of ions present in fluid. Each polarization mechanism is dominant
within a distinct frequency range.
As a consequence of the various polarization phenomena, frequency-
∗
dependent ε is sensitive to petrophysical properties, such as water saturation,
connate water salinity, porosity, tortuosity, wettability, clay content, conductive
mineral content, grain sizes, grain texture, and pore size distribution. Multifre-
quency effective conductivity and permittivity (in the range of megahertz to
gigahertz) of a subsurface geological formation are acquired using the wireline
dielectric dispersion (DD) logging tool that is run in an open-hole well intersect-
ing the geological formation of interest [1]. DD logging tool has EM transmit-
ters and receivers typically placed on a pad that pushes against the borehole wall
to make firm contact with the formation for reliable DD log acquisition. Pad-
based transmitters send EM waves of known magnitude and phase generally at
four discrete frequencies in the range of 10 MHz to 1 GHz. EM waves travel
through the fluid-filled porous formation and reach the pad-based receivers,