Upscaling methods for a class of convection–diffusion equations with highly oscillating coefficients

This paper investigates the upscaling method to the following parabolic equation: ∂ t c + ∇ · ( u c ) - ∇ · ( D ∇ c ) = f ( x , t ) , which stems from the application of solute transport in porous media. Because of the highly oscillating permeability of the porous media, the Darcy velocity u hence the dispersion tensor D has many scales with high contrasts. Thus, how to calculate the macro-scale equivalent coefficients of the above equation becomes the target of this paper. A new upscaling method is proposed and studied via comparing with another upscaling method which was proposed in [Z. Chen, W. Deng, H. Ye, Discrete Contin. Dyn. Syst. 13 (2005), 941–960]. The two different equivalent coefficients computing formulations are based on the solutions of two different cell (local) problems, which one utilizes the elliptic operator with terms of all orders while the other only uses the second order term. Error estimates between the equivalent coefficients and the homogenized coefficients are given under the assumption that the oscillating coefficients are periodic (which is not required by the method). Numerical experiments are carried out for the periodic coefficients to demonstrate the accuracy of the proposed method. Moreover, the upscaling method is applied to solve the solute transport in a porous medium with a random log-normal relative permeability. The results show the efficiency and accuracy of the proposed method.