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Summary Understanding the transport and retention of elastic nanogel and microgel particles in porous media has been a significant research subject for decades, essential to the application of enhanced oil recovery (EOR). However, a lack of dynamic adsorption and desorption studies, in which the kinetics in porous media are seldom investigated, hinders the design and application of polymer nanogel in underground porous media. In this work, we visualized and quantified the transport and dynamic adsorption of polymer nanogel in 3D glass micromodels that were manufactured by packing glass beads in capillaries. Calibrating the linearity of fluorescence intensity to concentration, we calculated the adsorption kinetics at concentrations of 0.1 wt%, 0.2 wt%, and 0.3 wt% and flow rates of 0.01 mL/h, 0.02 mL/h, and 0.03 mL/h. In addition to time, concentration, and flow rate, the experimental results showed that dynamic adsorption is also a function of transport distance, which is due to the different adsorption abilities of particles. We also found that the uneven adsorption distribution can be attenuated by decreasing nanogel concentration or increasing flow rate. The work provides a new method to obtain adsorption and desorption kinetics and adsorption profile of submicron particles in porous media at flowing conditions through microfluidics. 

Abstract In this paper, we propose and analyze a finite-element method of variational data assimilation for a second-order parabolic interface equation on a two-dimensional bounded domain. The Tikhonov regularization plays a key role in translating the data assimilation problem into an optimization problem. Then the existence, uniqueness and stability are analyzed for the solution of the optimization problem. We utilize the finite-element method for spatial discretization and backward Euler method for the temporal discretization. Then based on the Lagrange multiplier idea, we derive the optimality systems for both the continuous and the discrete data assimilation problems for the second-order parabolic interface equation. The convergence and the optimal error estimate are proved with the recovery of Galerkin orthogonality. Moreover, three iterative methods, which decouple the optimality system and significantly save computational cost, are developed to solve the discrete time evolution optimality system. Finally, numerical results are provided to validate the proposed method.