  We study the effective conductivity $\sigma_e$ for a random wire problem on
the $d$-dimensional cubic lattice ${\mathbb Z}^d, d \geq 2$ in the case when
random conductivities on bonds are independent identically distributed random
variables. We give exact expressions for the expansion of the effective
conductivity in terms of the moments of the disorder parameter up to the 5th
order. In the 2D case using the duality symmetry we also derive the 6th order
expansion. We compare our results with the Bruggeman approximation and show
that in the 2D case it coincides with the exact solution up to the terms of 4th
order but deviates from it for the higher order terms.
