Effective transmissivity for slip flow in a fracture

Abstract

A simple efficient method is presented for the determination of the intrinsic transmissivity tensor, as well as the intrinsic correction tensors at successive orders in the dimensionless slip parameter, that predicts the effective transmissivity tensorial coefficient for steady, one-phase, isothermal, creeping flow of a Newtonian fluid with slip boundary condition in a rough fracture. It is demonstrated that the solution of the first $N$ ancillary closure problems provides the slip correction tensors up to the $2N-1$ order, hence reducing the computational requirements by a factor of ${\sim }2$ compared with the conventional approach. In particular, the first-order correction tensor (i.e. a Klinkenberg-like tensor) can be obtained by solving the closure problem required for the computation of the intrinsic transmissivity tensor. In addition, symmetry and definiteness (positiveness or negativeness) properties of the individual tensors are analysed. It is shown that a Padé approximant, built on the correction tensors at the first three orders, outperforms the predictions for the effective transmissivity tensor. The new approach is illustrated and validated with numerical examples on model rough fractures.