ABSTRACT

Resistivity 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 multiplicative factor, which then cancels a corresponding term in the normalization determinant. Any electrode-resistivity measurement can then be described as superposition of point sources and sensors. The analysis of Coulomb’s law in anisotropic media carries over to other physics domains with similar, close-form solutions: Compressional waves with anisotropic bulk modulus, thermal flow with anisotropic thermal conductivity, and fluid flow in porous media with anisotropic hydraulic permeability all are generalizations of the field equation in the anisotropic medium. Each physics domain introduces its own anisotropic distance.

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