The paradox of anisotropy in electric logging: A simple proof and extensions to other physics domains

ABSTRACTResistivity measurements in vertical wells through horizontally laminated formations suffer the paradox of anisotropy. In a borehole with negligible diameter, the measurement will only read the horizontal resistivity parallel to the laminae: It will be completely blind to the vertical resistivity perpendicular to the laminae, even though the source and sensor electrodes are vertically aligned. Coulomb’s law in anisotropic media explains this counterintuitive phenomenon. The anisotropy changes the Pythagorean distance to a new, anisotropic distance, which includes the inverse conductivity tensor. The mixed units of this anisotropic distance are reconciled in Coulomb’s law, whose normalization replaces the electric conductivity by the square root of the conductivity-tensor determinant. The special case of horizontal laminae and vertically aligned source and observation points simplifies Coulomb’s law in anisotropic media. The vertical conductivity can be extracted from the anisotropic distance as mu...