A nonlinear effective slip interface law for transport phenomena between a fracture flow and a porous medium

We present modeling of an incompressible viscous flow through a fracture adjacent to a porous medium. We consider a fast stationary flow, predominantly tangential to the porous medium. Slow flow in such setting can be described by the Beavers-Joseph-Saffman slip. For fast flows, a nonlinear filtration law in the porous medium and a non- linear interface law are expected. In this paper we rigorously derive a quadratic effective slip interface law which holds for a range of Reynolds numbers and fracture widths. The porous medium flow is described by the Darcys law. The result shows that the interface slip law can be nonlinear, independently of the regime for the bulk flow. Since most of the interface and boundary slip laws are obtained via upscaling of complex systems, the result indicates that studying the inviscid limits for the Navier-Stokes equations with linear slip law at the boundary should be rethought.


Introduction
Coupling between a fast viscous incompressible fracture flow and an adjacent filtration through porous medium occurs in a wide range of industrial processes and natural phenomena. The classical approach is to model the fracture flow using the lubrication approximation and to replace it by an interface condition. Subsequently, it is coupled with a porous medium flow, described for small Reynolds numbers by the Darcy's law and by the Forchheimer's law in the case of large Reynolds' number.
Study of the coupling between slow viscous incompressible fracture flow and a porous medium was undertaken in [3] and [4]. For the critical fracture width, the interface condition linked to the Reynolds' equation from lubrication was found.
To describe a contact between a porous medium and a large fracture with the width significantly larger than the pore size, the following effective slip interface law was established in the seminal work by Beavers and Joseph [2], where α BJ is a dimensionless parameter depending on the geometrical structure of the porous mediumand K is the scalar permeability. v τ is the tangential velocity and n is the unit normal exterior to the fluid region. Note that in the original version of the law (1), v τ was replaced by the difference between v τ and the tangential Darcy velocity at the interface. In [18], Saffman remarked that the tangential Darcy velocity at the interface is of order O(K). Then, the slip law without the tangential Darcy velocity at the interface (1) became generally accepted. The rigorous derivation of the law by Beavers and Joseph through a homogenization limit and by constructing the interface boundary layer was done by Jäger and colleges in [10], [11] and [12]. The pressure jump at the interface was studied analytically in [16] and using numerical simulations in [6]. For the review of the results we refer to [13], [17] and [7].
Sahraoui and Kaviany investigated in [19] a flow at the interface between a fracture and a porous medium by direct numerical simulations. The interest of this work was in the interface laws in presence of large Reynolds' numbers. The interface slip behavior in that case turned out to be complex. It was concluded that the flow inertia effects appear independently from the bulk nonlinear filtration in the porous medium. If ε is a characteristic nondimensional pore size, then for longitudinal Reynolds' numbers of order O(1/ε), numerical simulations indicate that the slip law ceases to be linear. The inertia forces at the interface become significant for Reynolds' numbers of order O(0.1/ε). Then, the slip coefficient α BJ increases. For the bulk porous medium flow, the nonlinear effects become visible only for Reynolds' numbers greater than O(3/ε).
Those observations led to a conclusion that α BJ depends on the Reynolds' number, [14] and [9]. Similar conclusion is in [15].
However, it seems that a linear slip law, even with the slip coefficient depending on Reynolds' number, is not enough for an accurate approximation and that a nonlinear slip law should be derived. We will justify it by constructing rigorously an accurate approximation to the velocity field and showing that it leads to a quadratic slip law.
In the present paper we aim to identify a setting corresponding to a nonlinear slip law. We show that for a range of values of Reynolds' number and fracture width, the homogenization leads to a nonlinear interface law, even though the bulk filtration remains of the Darcy type. To streamline the presentation, we focus on a mathematical model in a simple setting. We consider a constant driving force, present only in the fracture and, for simplicity, impose periodic longitudinal boundary conditions for the velocity and for the pressure. Such simplification allows to avoid handling the pressure field and the outer boundary layers. The general case of nonstationary flows with physical boundary conditions and forcing terms will be considered in forthcoming papers.
The paper is organized as follows: In section 2, we define the problem as a stationary incompressible Navier-Stokes flow with Reynolds' number of the order ε −γ and the fracture width of the order ε δ . Assuming a relation between γ and δ, allows us to obtain an approximation which satisfies a nonlinear slip law (11), while keeping a linear filtration in a porous medium. In section 3 we construct the approximation and prove that it provides a higher order approximation to the original problem.
2 Main result

Geometry
We consider a two dimensional periodic porous medium Ω 2 = (0, 1) × (−1, 0) with a periodic arrangement of the pores. The formal description goes along the following lines: First, we define the geometrical structure inside the unit cell Y = (0, 1) 2 . Let Y s (the solid part) be a closed strictly included subset ofȲ , and Y F = Y \Y s (the fluid part). Then, we introduce a periodic repetition of Y s all over R 2 and set Y k s = Y s + k, k ∈ Z 2 . Obviously, the resulting set E s = k∈Z 2 Y k s is a closed subset of R 2 and E F = R 2 \E s in an open set in R 2 . We suppose that Y s has a smooth boundary. Consequently, E F is connected and E s is not. Finally, we notice that Ω 2 is covered with a regular mesh of size ε, each cell being a cube i is homeomorphic to Y , by linear homeomorphism Π ε i , being composed of translation and a homothety of ratio 1/ε.

Position of the problem and the nonlinear slip law
Let 0 < γ < 3/2 and let F be a constant. In Ø ε we study the following stationary Navier-Stokes equation Remark 1. We skip here a discussion of modeling aspects. We only mention that ε γ stands for the inverse of Reynolds' number and that the small fracture width ε δ prevents creation of the Prandtl's boundary layer.
In order to simplify calculations we take a constant F . It corresponds to a pressure drop. Additionally, we assume it only in the fracture Ø ε,δ 1 . Let The variational form of problem (2)-(4) reads: Theory of the stationary Navier-Stokes equations with homogeneous boundary conditions results in existence of the least one smooth velocity field v ε ∈ W ε , div v ε = 0 in Ω ε , which solves (6) for every ϕ ∈ W ε , div ϕ = 0 in Ω ε . The construction of the pressure field goes through De Rham's theorem. For more details we refer to the classical Temam's book [22]. Now we make assumptions on the parameters δ and γ.
Then, the following estimate holds allows justifying a nonlinear interface law. Contrary to the classical situation, when Saffman's modification of the linear slip law by Beavers and Joseph (see [2] and [18]) is used, the nonlinear interface laws are rarely derived in the literature. However, they are supposed to be appropriate for fast flows.
and for the average over the pore face on Σ Next, for the shear stress we have After averaging over Σ with respect to y 1 , we obtain The above formula results in Saffman' version of the law by Beavers and Joseph, if only the first term at the right hand-side is taken into consideration. For small η, we obtain a significant deviation of the law by Beavers and Joseph from [18] and [2]. We are not aware of any rigorous derivation of a nonlinear interface law for the unconfined fluid flow coupled to the porous media flow.

Rigorous justification of the nonlinear slip law, generalizing the law by Beavers and Joseph
In this section we extend the justification of the law of Beavers and Joseph from [11] to the case of nonlinear laminar flows. In the proofs we apply the following variant of Poincaré's inequality:

The impermeable interface approximation
Intuitively, the main flow is in the fracture Ø ε,δ 1 . Following the approach from [11] we study the problem Therefore, as in [11] and [13], for the lowest order approximation {v 0 , p 0 } we impose on the interface the no-slip condition v 0 = 0 on Σ.
Such choice leads to a cut-off of the shear and it introduces an error. A solution of problem (16)- (19) is the classic Poiseuille flow in Ø ε,δ 1 , satisfying the no-slip condition at Σ. It is given by Concerning the normal derivative of the tangential velocity on Σ, we obtain ∂v 0 We extend v 0 to Ø 2 by setting v 0 = 0 for −1 ≤ x 2 < 0. p 0 is extended by 0 to Ø 2 . The question is in which sense this solution approximates the solution {v ε , p ε } of the original problem (2)-(4).
A direct consequence of the weak formulation (6) is that the difference v ε − v 0 satisfies the following variational equation It leads to the following result, which is a generalization of the result proved in [11]: Proposition 5. Let us assume that (H1)-(H2) are satisfied. Let {v ε , p ε } be a solution of (2)-(4) and {v 0 , p 0 } defined by (21). Then, it holds for ε ≤ ε 0 Applying Lemma 4 and formula (22) yield Using hypothesis (H1) and above estimates lead to We apply once more Lemma 4 and (24) follows.
This provides the uniform a priori estimates for {v ε , p ε }. Moreover, we have found that the viscous flow in Ø ε,δ 1 corresponding to an impermeable wall is an O(ε 2δ−γ+1/2 ) L 2 -approximation for v ε . The slip law, generalizing Beavers and Joseph's law, should correspond to the next order velocity correction. Since the Darcy velocity is of order O(ε δ−γ+3/2 ), we may justify Saffman's observation that the bulk filtration effects are negligible at this stage.

Justification of the nonlinear slip law
At the interface Σ the approximation from Subsection 3.1 leads to the shear stress jump equal to The shear stress jump requires construction of the corresponding boundary layer.
Since we are studying an incompressible flow, it is useful to recall properties of the conserved averages.
β bl ( x ε ) is extended by zero to Ø 2 \ Ω ε . Let H be Heaviside's function. Then for every q ≥ 1 we have (50) Hence, our correction is not concentrated around the interface and there are some nonzero stabilization constants. We will see that these constants are closely linked with our effective interface law.
As in [10] stabilization of β 0,ε towards a nonzero constant velocity C bl 1 e 1 , at the upper boundary, generates a counterflow. It is given by the two dimensional Couette flow d = C bl 1 x 2 ε δ e 1 . Now, after [10], we expected that the approximation for the velocity reads Concerning the pressure, there are additional complications due to the stabilization of the boundary layer pressure to C bl ω , when y 2 → +∞. Consequently, ω bl,ε − H(x 2 )C bl and we should take into account the pressure stabilization effect. At the flat interface Σ, the normal component of the normal stress reduces to the pressure field. Subtraction of the stabilization pressure constant at infinity leads to the pressure jump on Σ and the pressure approximation is For the rigorous justification of the pressure approximation, leading to the pressure jump law, we refer to [16] . Numerical experiments, justifying independently the pressure jump are in [6].
We now make the velocity calculations rigorous. Let us define the errors in velocity and in the pressure: Remark 9. Rigorous argument, showing that U ε is of order O(ε 2−γ ), allows justifying Saffman's modification of the Beavers and Joseph law (see [2] and [18]): On the interface Σ we obtain After averaging over Σ with respect to y 1 , we obtain the Saffman version of the law by Beavers and Joseph where u ef f 1 is the average of v 1 (ε) over the characteristic pore opening at the naturally permeable wall. The higher order terms are neglected. Nevertheless, for γ close to 1 the Beavers and Joseph slip law isn't satisfactory any more.

Next, the variational equation for
Note that U ε is divergence free and the approximation satisfies the outer boundary conditions. In analogy with Proposition 4, pages 1120-1121, from [11] we have Theorem 10. Let us suppose the hypotheses (H1)-(H2) and let U ε and P ε be defined by (53). Then, the following estimates hold Proof. We test (55) by U ε . Since div U ε = 0, P ε is eliminated from the equality. Next, arguing as in the proof of Proposition 5, we see that under assumptions (H1)-(H2) the viscous terms controls the inertia terms. Therefore, it remains to estimate the forcing term and the interface term, coming from the counterflow. We have Since ∇ y β bl decays exponentially in y 2 and the functions of x 2 behave as x 2 ε −δ for small x 2 , we obtain and the leading part in the first two terms of (v(ε)∇)v(ε) is Similarly, after integration by parts in Ω ε,δ 1 and using that β bl is divergence free, we obtain the same order of ε estimate as (57) for

Consequently, it results in
Applying Lemma 4 yields the estimate (56).
Still the shear jump at the interface dominates inertia due to the counterflow. Correcting the shear jump term − Σ εϕ 1 F 2 C bl 1 dS is as above. The only difference is that instead of ε δ we have ε and F/2 is replaced by −F C bl 1 /2. We eliminate it by modifying slightly the velocity and pressure corrections: Corollary 11. Let assumptions (H1)-(H3) hold, and U ε , P ε be defined by (53). Let Then, the following estimate holds The new shear stress jump term generated by correction (60) is given by − Σ ε 2−δ ϕ 1 F 2 (C bl 1 ) 2 dS. Then, the corresponding estimate (59) in the proof of Theorem 10 takes the form Due to hypothesis (H3), we have 5/2 − δ > 3δ − 2γ + 3/2 and the new error terms are less important than the leading inertia terms.