Viscoelastic surfactant (VES) gelled aqueous fluids containing a VES, an internal breaker, and optionally a viscosity enhancer are useful as diverting fluid for directing placement of an acid into a subterranean formation, where the acid is injected subsequent to introducing the VES gelled fluid. These VES-based diverting fluids have faster and more complete clean-up than polymer-based diverting fluids. The viscosity enhancers may include pyroelectric particles and/or piezoelectric particles. The VES gelled fluid may optionally contain a fluid loss agent which increases the viscosity of the fluid and/or facilitates development of an external viscous VES fluid layer (e.g. a pseudo-filter cake) on the formation face. The VES gelled fluid may also optionally contain an agent that stabilizes the viscosity of the fluid, for instance at high temperatures, such as MgO, Mg(OH).sub.2, CaO, Ca(OH).sub.2, NaOH, and the like.