Laminated formations exhibit anisotropy in several petrophysically relevant forms, notably electric resistivity and fluid-flow permeability. Resistivity anisotropy has qualitatively been known for a long time as a phenomenon in shales and laminates. Since the early 1990s it has been quantitatively measured with 2-MHz propagation-resistivity tools used while drilling directional wells.
The anisotropy provides different resistivities parallel and perpendicular to the laminates, also called horizontal resistivity Rh and vertical resistivity Rv. Maxwells equations require that the vertical resistivity is the volumetric average of the resistivities, while the horizontal resistivity is the inverse of the volumetric average of the conductivities in the individual laminate components. This averaging implies that the vertical resistivity is always larger than the horizontal resistivity.
The 2-MHz resistivity measurements use propagating electromagnetic waves. The waves can propagate only due to the formation dielectric properties. In the formation, the dielectric permittivity is combined with the conductivity to a complex-valued dielectric constant. In laminated formations, the anisotropic properties are determined in terms of the volumetric averages of these complex-valued dielectric constants.
The rules of complex arithmetic lead to some surprising insights as the lamination anisotropy for propagation-resistivity measurements is evaluated. There are realistic formation models where the horizontal conductivity is less than the vertical conductivity ? or the vertical resistivity is less than the horizontal resistivity. Similarly, there are cases where the horizontal dielectric permittivity is less than the vertical dielectric permittivity, contrary to the insights for the purely dielectric laminate. It is possible that a 1-ohm-m shale and a 1-ohm-m sandstone with different dielectric permittivities show an anisotropy in the resistivity, even though their resistivities are identical.
The combination of dielectric permittivity and electric conductivity into a complex-valued dielectric constant is a frequency-dependent function. The anisotropy averaging procedure will give different horizontal and vertical resistivities for the 2-MHz propagation, the 400-kHz propagation, and the ~20-kHz wireline induction tools. Such differences may already have been observed as resistivity dispersion.
This counterintuitive anisotropy behavior of propagation-resistivity measurements makes a saturation interpretation from the resistivities more challenging. It is not possible to simply use the readily available vertical or horizontal resistivity; instead, the laminates must be analyzed according to their volumetric composition.