We experimentally investigate the flow of hydrolyzed polyacrylamide (HPAM) solution with and without salt in model porous media at high Weissenberg numbers (Wi > 1.0). The effect of pore shapes on the flow pattern and pressure drop is explored by using periodic arrays of circular and square pillars in aligned and staggered layouts. In the apparent shear-thinning regime, we observe stationary dead zones upstream of the pillars. In addition, we confirm that the size of stationary dead zones correlates with the level of shear-thinning, by varying the amount of salt in HPAM solution. At higher shear rates (or Wi), these dead zones are periodically washed away. We present the mechanism of this elastic instability and characterize it based on the pressure drop fluctuation spectral density.