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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. 

In this article, we consider a phase field model with different densities and viscosities for the coupled two-phase porous media flow and two-phase free flow, as well as the corresponding numerical simulation. This model consists of three parts: a Cahn–Hilliard–Darcy system with different densities/viscosities describing the porous media flow in matrix, a Cahn–Hilliard–Navier–Stokes system with different densities/viscosities describing the free fluid in conduit, and seven interface conditions coupling the flows in the matrix and the conduit. Based on the separate Cahn–Hilliard equations in the porous media region and the free flow region, a weak formulation is proposed to incorporate the two-phase systems of the two regions and the seven interface conditions between them, and the corresponding energy law is proved for the model. A fully decoupled numerical scheme, including the novel decoupling of the Cahn–Hilliard equations through the four phase interface conditions, is developed to solve this coupled nonlinear phase field model. An energy-law preservation is analyzed for the temporal semi-discretization scheme. Furthermore, a fully discretized Galerkin finite element method is proposed. Six numerical examples are provided to demonstrate the accuracy, discrete energy law, and applicability of the proposed fully decoupled scheme.