# A novel one-domain approach for modeling flow in a fluid-porous system including inertia and slip effects

Abstract: A new one-domain approach is developed in this work yielding an operational average description of one-phase flow in the classical Beavers and Joseph configuration including a porous medium topped by a fluid channel. The model is derived by considering three distinct regions: the homogeneous part of the porous domain, the inter-region, and the free fluid region. The development is carried out including inertial flow and slip effects at the solid–fluid interfaces. Applying an averaging procedure to the pore-scale equations, a unified macroscopic momentum equation, applicable everywhere in the system and having a Darcy form, is derived. The position-dependent apparent permeability tensor in this model is predicted from the solution of two coupled closure problems in the inter-region and in the homogeneous part of the porous medium. The performance of the model is assessed through in silico validations in different flow situations showing excellent agreement between the average flow fields obtained from direct numerical simulations of the pore-scale equations in the entire system and the prediction of the one-domain approach. Furthermore, validation with experimental data is also presented for creeping flow under no-slip conditions. In addition to the fact that the model is general from the point of view of the flow situations it encompasses, it is also simple and novel, hence providing a practical and interesting alternative to models proposed so far using one- or two-domain approaches.

## Summary (3 min read)

### Introduction

- The model is derived by considering three distinct regions: the homogeneous part of the porous domain, the inter-region, and the free fluid region.
- The development is carried out including inertial flow and slip effects at the solid–fluid interfaces.
- 1,2 Applications of such a configuration are numerous, ranging from hydrology to chemical engineering.

### B. The g x inter-region

- This region is the transition zone where the average (or macroscopic) velocity experiences abrupt changes.
- Modeling flow in this region has been addressed using a generalized transport equation (or penalization approaches7,8) as explained below, or even by direct numerical simulations.
- Both approaches have strengths and limitations, which are briefly discussed in the following paragraphs.
- In fact, it was shown in this reference that the one-domain approach is necessary in the derivation of the jump conditions using the volume averaging method,18 justifying its archival value.
- The structure of the average models in the three regions is shown to involve a Darcylike momentum equation, which can be condensed into a single one-equation model with a position-dependent apparent permeability tensor.

### II. MICROSCALE MODEL AND DEVELOPMENT OF THE ONE-DOMAIN APPROACH

- The configuration under consideration is the one represented in Fig. 1.
- Before developing the one-domain macroscale model, the underlying boundary value flow problem must be formulated at the microscale.
- Finally, the macroscopic momentum transport equation is derived in each region (Secs. II C–II E).

### B. Average mass balance equation

- The development starts with the derivation of the macroscopic mass balance equation in each region.
- In this domain, it is reasonable to decompose the fluid pressure gradient according to36 rpx ¼ rhpxibx þr~px; (6) ~px being the pressure deviations, and regard rhpxi b x as a constant within the REV.
- These authors solved the closure problem given in Eq. (9) in a variety of flow situations.
- Note that, for the particular problem envisaged by Beavers and Joseph,1 there are no inertial nor slip effects and the tensor Hx is the intrinsic permeability tensor.
- To conclude this section, the average velocity must now be derived by applying the superficial averaging operator to Eq. (19).

### E. Analysis in the fluid-porous medium inter-region (g x region)

- This transition zone is comprised between z ¼ zx (below which the homogeneous porous medium begins) and z ¼ zg (above which the free fluid region commences) as sketched in Fig.
- The solution of the closure problem in the inter-region is the step that is computationally the more demanding.
- The numerical requirements are considerably smaller than those needed for the solution of the pore-scale problem in the entire three-region domain.

### III. RESULTS

- The objective of this section is to present some numerical results on model structures in order to validate the model derived above and illustrate its performance.
- In the following paragraphs, the numerical results are presented for specific flow conditions.
- The numerical tests were performed considering two types of porous structures.
- To this end, the closure problems given in Eqs. (9) and (25) were numerically solved in a coupled manner in a single unit cell for the x-region and in the entire g-x region, respectively.
- It was verified that taking zg ¼ r0 and zx ¼ 11‘c satisfied the above criteria for all the cases reported in the present section.

### A. Creeping flow regime

- The first test is carried out considering the simple situation in which the flow remains in the creeping regime under no-slip conditions (i.e., for n1k ¼ n2k ¼ 0).
- The value of the intrinsic permeability, obtained from Eq. (12) is Hxxx ¼ 0:019 47‘2c , which is in perfect agreement with the estimate from the analytical expression provided by Chai et al.38.
- It is now of interest to investigate the influence of the porous medium geometry and of the averaging domain size, r0, on the predictions of the average velocity in the inter-region.
- The velocity profiles resulting from the ODA and the DNS are reported in Fig. 6, where, once again, excellent agreement is found.
- This is due to the fact that the flow remains unidirectional and non-inertial in the free fluid region for these values of Re.

### C. Creeping slip-flow

- The final case study corresponds to slip flow under non-inertial conditions, i.e., Re¼ 0.
- More specifically, four values of nk are examined, with the largest one remaining Oð0:1Þ in order to avoid using the governing flow equations beyond their range of validity (i.e., out of the slip regime).
- 41,42 The use of the same slip coefficient in both the porous medium and at the top wall of the free-fluid region is justified by the fact that the characteristic length of the latter region is not much larger than the one in the porous medium in the particular configuration considered here.
- Nevertheless, since the boundary condition given in Eq. (1c) may also be conceived as an effective boundary condition over rough surfaces, it is certainly possible to encounter physical situations in which n1k ¼ n2k when the contrast of the characteristic lengths is more pronounced.
- Finally, it is important to mention that the predictions of the average velocity from the DNS and the ODA are in excellent agreement in all the physical situations considered here.

### D. Comparison with experimental data

- This configuration is similar to the one used by Beavers and Joseph with the difference that the porous medium is made of an array of inline micropillars having a square cross section of 240lm 240lm.
- The velocity fields obtained with this instrument are line averages in the y-direction of the sketch shown in Fig. 1 (or equivalently, the z-direction in the work by Terzis et al.).
- These results were subsequently averaged along the x-direction in zones near the entrance, the outlet and in the middle of the system as shown in Fig.
- The experimental results taken at the middle of the system are now compared to the predictions of the model resulting from the ODA presented above.
- Notice that due to the shape of the experimental system, i.e., the wall effects in the y-direction, the velocity profiles tend to reach a constant value in the free fluid region sufficiently far away from the upper wall and the porous medium surface.

### IV. CONCLUSIONS

- A new formulation for describing steady, Newtonian, and incompressible flow between a porous medium and a fluid was derived.
- The predictive capabilities of the model were validated through in silico experiments by comparing the average velocity profiles resulting from the new ODA with pore-scale DNS, finding excellent agreement (i.e., with relative percent error values with respect to the DNS smaller than 0.1%).
- The microstructure plays a key role in the magnitude and shape of the flow field not only in the homogeneous part of the porous medium, but also in the inter-region.
- When the slip condition is imposed only in the porous medium, the velocity in the free fluid region experiences the most drastic changes.
- From the above, it is concluded that the new one-domain approach developed here is quite practical as it represents accurately data from both direct numerical simulations and experiments at a reasonable computational cost.

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