We present a theoretical analysis of the anisotropy of the aperture correlation and effective transmissivity in fractures generated by sliding/shear between identical self-affine surfaces. Expressions are derived for the aperture correlation function in different directions, highlighting longer persistence and a larger integral scale normal to the shear than parallel to it. A stochastic perturbation analysis of the Reynolds equation leads to an effective transmissivity tensor with higher transmissivity normal to the shear. The hydraulic anisotropy depends on the coefficient of variation of the aperture field and the Hurst exponent. Numerical simulations are presented to evaluate the above theoretical predictions.