We consider an inhomogeneous Bernoulli bond percolation process on the d-dimensional integer lattice (d>1). All edge occupation probabilities are given by p except for edges lying on the first coordinate axis which are occupied with probability p'. For any fixed p<p_c, we provide a detailed analysis of the consequences of the modified bond occupation probabilities p' on the exponential rate of decay of the connectivities along the line and on the behaviour of the corresponding cluster.