Many natural physical processes exhibit a multiscale behavior, that is the same processes allow different mathematical description on different scales. The microscale description is usually based on fundamental conservation laws that form a closed system of ordinary differential equations (ODEs) or partial differential equations (PDEs) but the numerical discretization of these equations may produce a system of ODEs with an enormous number of unknowns. Furthermore, a time integration of the microscale equations usually requires time steps that are smaller than the observation time by many orders of magnitude. A direct solution of these ODEs can be extremely expensive. This necessitates development of advanced algorithms for model (or dimension) reduction. Often, we are only interested in the average behavior of the system rather than the exact solution of the ODEs. Here we developed a novel dimension reduction method that gives an approximate solution of the ODEs and gives an accurate prediction of the average behavior. The method relies on a computational closure of averaged evolution balance equations. The computational closure is achieved via short bursts of microscale simulation. The dimension reduction model was used to simulate flow and transport with mixing controlled reactions and mineral precipitation. A pore-scale model was used as a microscale model. A good agreement between microscale simulations and the accelerated microscale simulations confirms the accuracy and computational efficiency of the dimension reduction model. The method significant accelerates microscale simulations, while providing accurate approximation of the solution and accurate prediction of the average behavior of the system.