In this paper, an efficient modified two-grid block-centered finite difference algorithm with second-order backward difference formula is presented for coupled nonlinear reaction-diffusion equations of the solutes in the groundwater flow. The main steps of the modified two-grid algorithm include dealing with a small-scale nonlinear problem on the coarse grid with a large time step size and then dealing with a large-scale linear problem on the fine grid with a smaller time step size base on the second-order backward difference formula. Moreover, the theoretical analysis illustrates that the modified two-grid algorithm is able to maintain second convergence order in space-time domain. Several numerical experiments including a practical example are carried out to support the theoretical results and verify that the modified two-grid block-centered finite difference with second-order backward difference formula algorithm can achieve high efficiency while maintaining the second-order accuracy, meanwhile, has a certain applicable value in actual problem.