We study a reactive immiscible incompressible two-phase flow in a periodic double porosity reservoirs. The mathematical model is given by a coupled system of two-phase flow equations and an energy balance equation. The model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy-Muskat and the capillary pressure laws containing the source terms corresponding to the chemical reactions in the reservoir. The medium is made of two superimposed continua, a connected fracture system and an ε-periodic system of disjoint matrix blocks. We assume that the permeability of the fissures is of order one, while the permeability of the blocks is of order ε². By the method of formal asymptotic expansions, we derive the global behavior of the model by passing to the limit as ε → 0 and obtain the global model of the reactive flow. It is shown that the homogenized model can be represented as the usual equations of a reactive immiscible incompressible two-phase flow except for the addition of new source terms calculated by a solution to a local problem in the matrix block. These source terms exhibit the nonlocal behavior of the model with respect to the time variable. The non-locality in time of the reaction source terms in the case of gas producing reaction can lead to the instability of stationary reaction front propagation in the porous medium and development of self-oscillations. The results of the numerical simulation of the reactive immiscible incompressible two-phase flow are presented.