## Upscaling of a class of nonlinear parabolic equations for the flow transport in heterogeneous porous media.(English)Zbl 1090.76058

Summary: We develop an upscaling method for the nonlinear parabolic equation $\partial_tb(u_\varepsilon)-\nabla\cdot\bigl({\mathbf g}^\varepsilon(x, u_\varepsilon)+{\mathbf a}^\varepsilon (x, u_\varepsilon)\nabla u_\varepsilon\bigr) =f(x,t),$ which stems from the description of flow transport in porous media. Our direct motivation is the Richards equation which models the flow transport in unsaturated porous media. We provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are periodic. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain the asymptotic structure of the solutions. Numerical experiments are carricd out for Richards equaton of exponential model with periodic and randomly generated log-normal permeability to demonstrate the efficiency and accuracy of the proposed method.

### MSC:

 76M50 Homogenization applied to problems in fluid mechanics 76S05 Flows in porous media; filtration; seepage 76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure

### Keywords:

convergence; homogenization; Richards equaton
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