We present a Lagrangian approach to study the clogging evolution of a one-dimensional homogeneous micro-fractured medium by deposition of one chemical species. Semi-analytical expressions are determined by means of the mass balance equations of fluid and solute in which the chemical and hydrodynamic disequilibrium makes an important dynamic contribution to the clogging mechanism. We describe some aspects of the coupling between the fluid transport and the chemical reaction with a qualitative analysis of these expressions by comparing the clogging evolution as a function of the imposed boundary conditions. In particular, a feedback effect underlies the solute transport if the pressure is imposed. Its analysis reveals a competition between local scales and large scales amplified by the variation rate of the porosity, so that we have to distinguish the clogging process at short times and at long times. By means of the Lagrangian formulation, we are able to find an accurate analytical expression of the hydraulic conductivity of the overall medium. But it is necessary to use the successive approximation method to obtain quantitative results for the other quantities. Applications to heterogeneous media modeled by capillary tubes networks have displayed similar hydrodynamic behavior compared to homogeneous media if the deposition kinetics is not too sensitive to heterogeneities.