Determination of Klinkenberg and higher-order correction tensors for slip flow in porous media

An efficient method is reported to determine correction tensors at successive orders in the Knudsen number (i.e., the Klinkenberg and higher corrections) to approximate the apparent permeability tensor characterizing one-phase, creeping, Newtonian, isothermal slip flow in homogeneous porous media. It is shown that the Klinkenberg correction tensor can be obtained from the solution of the same ancillary (closure) problem that is required to compute the intrinsic permeability tensor. More generally, correction tensors up to the (2M - 1)th order are shown to be obtained from the solution of the first M closure problems, instead of the 2M ones suggested by the upscaling procedure and associated closure scheme. Moreover, it is demonstrated that all the correction tensors are symmetric, the odd and even order ones being respectively positive and negative. In particular, this indicates that the apparent permeability tensor at the first order in the Knudsen number is symmetric positive. The model is validated by analytical solutions in the simple cases of flow in parallel plates and bundle of parallel cylindrical tubes and by numerical simulations performed in a model two-dimensional structure. An improvement in the apparent permeability prediction is shown using a Pade approximant.