Acidic fluid flow in geologic formations leads to mineral dissolution and, under certain circumstances, to localized dissolution forming a dendritic pattern, known as wormhole. Such patterns of conduits and caves are often observed in karstic aquifer and deliberately engineered in oil and gas well stimulation with acid injection. Two different kinds of instability are thought to cause wormholes. One is heterogeneous property in rocks. The other is a process itself such as reaction infiltration or viscous fingering. To assess each one properly, we need to separate one from the other. While most numerical studies rely on randomly seeded material heterogeneities to induce wormholes, Daccord and Lenormand (1987a) demonstrated that even water injection into a homogeneous plaster can form wormholes. Here, we focus on the process instability driven by reaction-infiltration in homogeneous materials. We apply a phase-field approach, which diffuses a sharp interface in a continuous manner and show that it is capable of simulating wormhole without random seeds by accounting for the energy expenditure in the dissolution topology. We verified the model against the sharp interface counterpart in one-dimensional simulations. We then performed the two-dimensional simulations to qualitatively validate wormhole formation and growth patterns against acid injection experiments on carbonate rocks under radial flow conditions. Our simulation results indicate that the injected acid is rapidly consumed near the acid entry point at low injection rates while the live acid becomes available at the tip of the dissolved cavity under high rates and thus wormhole starts to grow resulting in much faster acid breakthrough.