Bounds for the resistivity anisotropy in thinly-laminated sand-shale

I consider the electric resistivity anisotropy of a thinly laminated sand-shale formation, for the case when both the sandstone and shale are electrically anisotropic. Simple upper and lower bounds will be derived for the possible maximal and minimal values for the coefficient of resistivity anisotropy in such a formation, and it is shown that the fulfillment of these bounds guarantee the existence of physically meaningful (i.e., real and positive) solutions of the system of equations expressing the horizontal and vertical shale and sand resistivities ρ _sh-h, ρ _sh-v, ρ _sd-h, ρ _sd-v in terms of the measured horizontal and vertical effective resistivities ρ _h and ρ _v, the specific anisotropics λ _sh and λ _sd, and the relative shale and sand volumes V _sh and V _sd. The main result of the paper is expressed by inequalities which give lower and upper bounds for the co-efficient of resistivity anisotropy of rocks built up of alternating parallel layers of inherently anisotropic sand and shale. The lower bound can be used in the numerical or graphical interpretation of triaxial induction logs to exclude such "non- physical" cases when the graphical or numerical solutions would result in negative, or complex-valued specific resistivities, or in unrealistic formation anisotropics for which λ = √ρ _v/ρ _h < 1.