Pore-scale dynamics of one phase flow commonly involves adherence (no-slip) boundary conditions at the fluid/solid interface. However, improved modeling such as flows at moderate Knudsen numbers (i.e. for values below 0.1), or homogenization of rock matrix roughness, may require slip conditions [1,2]. It turns out that a lack of knowledge on the rock matrix wall, built by X-Ray micro tomography, leads to the same type of slip conditions v = β 2 (I nn)
(∇v + ∇vT )
n where v is the pore-scale velocity, n is the unit normal vector oriented towards the fluid and β is twice the slip-length. The macroscopic model corresponding to steady one-phase flow in the creeping regime (Stokes equations) at the pore-scale with the slip condition was derived in [1, 2]. The macroscopic momentum equation corresponds to Darcy’s law in which the permeability tensor is slip-dependent. This leads to a computation of an apparent permeability that depends on β and that we denote Kβ . This apparent permeability can be expanded in a power series of a Knudsen number, the zeroth-order term identifying to the intrinsic permeability with no slip and the higher order terms to slip-correction tensors, the first one generalizing, for an ideal gas, the classical Klinkenberg correction. All the tensors are given by the solution of coupled closure problems at the successive orders [2]. While this expansion has been addressed in [1] and [2] and used in the case of synthetic geometries, it has been used in large three-dimensional real geometries in [3] in the context of isotropic permeability, that is to say Kβ = κβ I where κβ is a scalar. In this last case, the expansion κβ = κ0 + βλ1 + β2 2 λ2 + O(β3) is shown to describe the uncertainty on the permeability values that results from the gray scale uncertainty generated by micro-tomography. The coefficients λk , capturing this uncertainty estimation, are given by the solutions of the Stokes-like closure problems reported in [2] by formal expansion (from averaged equations) and re-established in [3] by asymptotic analysis (two-scale homogenization). They satisfy a non-homogeneous Dirichlet boundary condition (prescribed velocity) at the pore walls, whose value involves the immediate lower order slip momentum. While the first order of this expansion has been detailed in [3], its second order estimation is provided for the first time in this presentation for operational 3D geometries: we apply our results to the high resolution rock sample studied in [4] and show that the second order brings a significant improvement of the uncertainty estimation on the absolute permeability. These results in a real geometry confirm what was anticipated in [2] for simplified 2D geometries and are relevant as they show the importance of slip at the macroscale.