Computation of the Permeability Tensor of Non-Periodic Anisotropic Porous Media from 3D Images

Abstract

The direct proportionality between the flow rate and the pressure gradient of creeping flows was experimentally discovered by H. Darcy in the 19th century and theoretically justified a couple of decades ago using upscaling methods such as volume averaging or homogenization. X-ray computed micro-tomography (CMT) and pore-scale numerical simulations are increasingly used to estimate the permeability of porous media. However, the most general case of non-periodic anisotropic porous media still needs to be completely numerically defined. Pore-scale numerical methods can be split into two families. The first family is based on a direct resolution of the flow solving the Navier–Stokes equations under the assumption of creeping flow. The second one relies on the resolution of an indirect problem—such as the closure problem derived from the volume averaging theory. They are known to provide the same results in the case of periodic isotropic media or when dealing with representative element volumes. To address the most general case of non-periodic anisotropic porous media, we have identified four possible numerical approaches for the first family and two for the second. We have compared and analyzed them on three-dimensional generated geometries of increasing complexity, based on sphere and cylinder arrangements. Only one, belonging to the first family, has been proved to remain rigorously correct in the most general case. This has been successfully applied to a high-resolution 3D CMT of Carcarb, a carbon fiber preform used in the thermal protection systems of space vehicles. The study concludes with a detailed analysis of the flow behavior (streamlines and vorticity). A quantitative technique based on a vorticity criterion to determine the characteristic length of the material is proposed. Once the characterized length is known, the critical Reynolds number can be estimated and the physical limit of the creeping regime identified.

Appendices

Appendix 1: Remarks on the Volume Averaging Method

This appendix has the only purpose to introduce the concepts of the volume averaging method used in this article. Further details can be found in the literature Whitaker (2013).

The volume averaging is a technique used to derive continuum-macroscopic equations for multiphase systems. In this way, the complexity of a porous medium is replaced with an equivalent porous-continuum model in which each point is characterized by the properties of a representative elementary volume (REV) centered on it. Within this, REV variables can be averaged. Two different definitions of averages have been adopted: the phase average which in this article has been used for the pressure term

$$\begin{aligned} P = \dfrac{1}{V_{f}} \int _{V_{f}} p dV \end{aligned}$$

(17)

and the intrinsic average used for the velocity field

$$\begin{aligned} \varvec{U} = \dfrac{1}{V_{REV}} \int _{V_{f}} \varvec{u} dV \end{aligned}$$

(18)

Inside the REV, each variable can be decomposed (Gray’s decomposition Gray (1975)) as the summation of its average plus a deviation contribution. For the pressure field, this decomposition is as follows

$$\begin{aligned} p = P + \tilde{p} \end{aligned}$$

(19)

Appendix 2: KclosureSolver

KclosureSolver is a solver implemented in PATO to solve the closure problem Eq. (20). By following the work of Anguy and Bernanrd, (1994) Anguy et al. (1994), transients terms have been added to the system to improve its stability. The desired solution is taken at the steady state. An artificial compressibility, c, and an artificial viscosity coefficient, $\mu _{art}$, are also introduced. The modified transient problem is as follows

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{\partial \underline{\underline{\varvec{d}}}}{\partial t} + \nabla \varvec{e} - \nabla ^2 ( \mu _{art} \underline{\underline{\varvec{d}}}) = \underline{\underline{\varvec{I}}} \;in \;V \vspace{1mm} \\ \dfrac{\partial \varvec{e}}{\partial t} + c^2 \nabla \cdot \underline{\underline{\varvec{d}}} = 0 \;in \;V \\ BC1: \underline{\underline{\varvec{d}}} = 0 \;at\; A_{fs} \\ BC2: \underline{\underline{\varvec{d}}}(\varvec{r}+l_i) = \underline{\underline{\varvec{d}}}(\varvec{r}), \varvec{e(\varvec{r}+l_i)} = \varvec{e(\varvec{r})} \;i=1,2,3 \end{array}\right. } \end{aligned}$$

(20)

The two equations are solved sequentially. Each equation can be iterated in a loop. The time integration is carried out through an implicit scheme.

Appendix 3: Mesh Convergence Test Cases

In Sect. 3, numerical simulations on six test cases are presented. The mesh refinement has been selected after a convergence analysis. By considering the 1-sphere unit cell with porosity $\epsilon = 0.55$, results of the convergence study with strategy A1 are presented in Table 6.

Table 6 Mesh convergence analysis with strategy A1 for the 1-sphere unit cell case defined in Sect. 3 with constant porosity $\epsilon = 0.55$

The error% is evaluated taking into account the permeability estimation of two consecutive simulations as follows

$$\begin{aligned} error\% = 100 \dfrac{|K_{Finer\_Mesh}-K_{Coarser\_Mesh}|}{K_{Finer\_Mesh}} \end{aligned}$$

(21)

The quantity $V_{ratio}$ is the ratio between the volume of the biggest cell in the domain and the domain itself. From the results in the table, we can see that the convergence of the mesh is immediately achieved since the error is always decreasing by increasing the refinement. This trend, however, should stop when the tolerances of the simulation are reached. That is what happens in the table for the most refined meshes. The resolution of the mesh with 233039-cells can be then selected for the analysis in Sect. 3.