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Slip flow in porous media is encountered in many applications involving gas flow (when Knudsen effects become significant) or even liquid flow when an effective boundary condition at the pore walls replaces no-slip flow over rough surfaces [1]. The macroscopic model describing flow with slip effects in homogeneous porous media takes the form of Darcy’s law in which the effective (or apparent) permeability coefficient is composed of the intrinsic permeability complemented by a slip correction that can be decomposed into a series of corrective coefficients at the successive orders in the dimensionless slip length [2]. The intrinsic permeability and slip corrective terms are tensors that are obtained from the solution of ancillary (closure) problems formally derived from upscaling the pore-scale flow model. These closure problems are sequentially coupled at the successive orders in the dimensionless slip length. In this work, it is shown that the first order slip correction, known as the Klinkenberg correction, can be equally obtained from the solution of the 0th order ancillary problem that provides the intrinsic permeability without any extra computation. More generally, it is demonstrated that the correction terms up to the (2M − 1)th order are obtained from the solution of the first M ancillary problems, yielding a speed-up of a factor of 2 [3]. Properties (symmetry, positiveness) of the slip correction tensors at the successive orders are reported. It is shown that they are all 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 (Darcy-Klinkenberg) is symmetric positive. An accurate estimate of the apparent permeability tensor is further shown using a Padé approximant. Illustrative results demonstrate the efficiency of the macroscopic model and the method of determination of the effective coefficients. Extension of the approach to slip flow in a fracture relying on the Reynolds equations is also mentioned, showing results analogous to those in the porous media case [4].
[1] Zampogna, G.A, Magnaudet, J. and Bottaro, A. 2019, Generalized slip condition over rough surfaces, J. Fluid Mech., 858, 407-436.
[2] Lasseux, D., Valdés-Parada, F.J. and Porter, M.L. 2016, An improved macroscale model for gas slip flow in porous media, J. Fluid Mech., 805, 118-146.
[3] Lasseux, D., Zaouter, T. and Valdés-Parada, F.J. 2023, Determination of Klinkenberg and higher-order correction tensors for slip flow in porous media, Phys. Rev. Fluids, 8, 053401.
[4] Zaouter, T., Valdés-Parada, F.J., Prat, M. and Lasseux, D. 2023, Effective transmissivity for slip flow in a fracture, J. Fluid Mech., 969, A9.
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