The Leikoupo carbonate reservoirs in the West Sichuan gas field generally exhibit low porosity and permeability characteristics. After acidizing, the fluid flowback rate is only 46%, and a large amount of residual acid remains in the reservoir, causing damage to the pores. In this study, the rheological tests were first carried out to reveal the rheological characteristics of fresh and residual acid with different thickener contents. Then, the change in core permeability before and after acid flooding was tested by core-flood tests, and the effects of acid on pore structure in various scales were analyzed in nuclear magnetic resonance (NMR) tests. The research results show that the average viscosity of residual acid increased to about 157.14% of the fresh acid and exhibited large viscosity fluctuations, which may be caused by the formation of aggregates combined with thickener and insoluble. The change rate of core permeability (both improvement and damage) decreased with the increase in thickener content, and the effect was more significant when the thickener content changed from 0.6% to 0.8%. Under different thickener addition amounts, fresh acid mainly improved the core permeability. After core-flooding, micropores and mesopores (51.26–632.15 μm) were dissolved and enlarged, forming mesopores and macropores (632.15–3597.73 μm); when the thickener content increased to 0.8%, the solubility of acid decreased, and the pore volume increment and maximum pore size both decreased. On the other hand, residual acid mainly damages the core, thickener molecules in the acid occupying mesopore, leading to an overall shift in pore distribution from mesopores (145.24–335.53 μm) to micropores (20.59–126.32 μm), with the thickener content increased to 0.8%, the damage volume of mesopore increased by 723.90%. Based on the research results, the reduced thickener content was initially applied to the West Sichuan gas field, achieving a better acidizing effect.