Nonlinear flow through double porosity media in variable exponent Sobolev spaces

We studied the asymptotic behavior of the solution of a nonlinear parabolic equation with nonstandard growth in a εε-periodic fractured medium, where εε is the parameter that characterizes the scale of the microstructure tending to zero. We consider a double porosity type model describing the flow of a compressible fluid in a heterogeneous anisotropic porous medium obeying the nonlinear Darcy law. We assume that the permeability ratio of matrix blocks to fractures is of order εpε(x)εpε(x), where pεpε is a continuous positive function. We obtained the convergence of the solution and a macroscopic model of the problem was constructed using the notion of two-scale convergence combined with the variational homogenization method in the framework of Sobolev spaces with variable exponents.