In　this　paper,　we　conduct　a　convergence　rate　analysis　of　the　augmented　Lagrangian　method　with　a　practical　relative　error　criterion　designed　in　Eckstein　and　Silva　[Mathematical　Programming,　141,　319　348　(2013)]　for　convex　nonlinear　programming　problems.　We　show　that　under　a　mild　local　error　bound　condition,　this　method　admits　locally　a　Q-linear　rate　of　convergence.　More　importantly,　we　show　that　the　modulus　of　the　convergence　rate　is　inversely　proportional　to　the　penalty　parameter.　That　is,　an　asymptotically　superlinear　convergence　is　obtained　if　the　penalty　parameter　used　in　the　algorithm　is　increasing　to　infinity,　or　an　arbitrarily　Q-linear　rate　of　convergence　can　be　guaranteed　if　the　penalty　parameter　is　fixed　but　it　is　sufficiently　large.　Besides,　as　a　byproduct,　the　convergence,　as　well　as　the　convergence　rate,　of　the　distance　from　the　primal　sequence　to　the　solution　set　of　the　problem　is　obtained.