Studies of the rock matrix acidizing for enhanced recovery of oil or gas have entirely focused on the fully water saturated conditions. In fact, matrix acidizing can be conducted in low-water-cut oil-production wells without pre-flushing of water or in high water production by injecting gas or oil ahead of the acid injection. These conditions yield a multiphase system, where the dynamics of acid transport and reactions can be altered by the presence of an immiscible phase. Against this backdrop, we present an investigation of the impact of initial saturation of an immiscible phase in the damaged zone on the efficiency of wormhole generation and growth in acidizing operations. We present a dimensionless two-phase reactive transport modelling tailored for studying the processes associated with rock acidizing. For a case study of acid injection into calcite with random porosity and permeability distribution, we show that an initial two-phase condition has positive feedback on the generation of wormholes. The results, however, indicate that the relative magnitude of reduction in the amount of pore volume of injected acid to produce effective wormholes depends on the mobility ratio, so that a higher mobility ratio facilitates a faster wormhole generation process. Under the conditions of modelling study presented, we demonstrate that in addition to the commonly used pair of Peclet-Damkohler regime identification, the mobility ratio of the displacing/displaced fluids, the relative permeability and phase condition need be accounted for in the analysis, if there are two-phase flow conditions across the target region of acidizing. (C) 2017 Elsevier Ltd. All rights reserved.