Calculation of tensorial flow properties on pore level: Exploring the influence of boundary conditions on the permeability of three-dimensional stochastic reconstructions

While it is well known that permeability is a tensorial property, it is usually reported as a scalar property or only diagonal values are reported. However, experimental evaluation of tensorial flow properties is problematic. Pore-scale modeling using three-dimensional (3D) images of porous media with subsequent upscaling to a continuum scale (homogenization) is a valuable alternative. In this study, we explore the influence of different types of boundary conditions on the external walls of the representative modeling domain along the applied pressure gradient on the magnitude and orientation of the computed permeability tensor. To implement periodic flow boundary conditions, we utilized stochastic reconstruction methodology to create statistically similar (to real porous media structures) geometrically periodic 3D structures. Stochastic reconstructions are similar to encapsulation of the porous media into statistically similar geometrically periodic one with the same permeability tensor. Seven main boundary conditions (BC) were implemented: closed walls, periodic flow, slip on the walls, linear pressure, translation, symmetry, and immersion. The different combinations of BCs amounted to a total number of 15 BC variations. All these BCs significantly influenced the resulting tensorial permeabilities, including both magnitude and orientation. Periodic boundary conditions produced the most physical flow patterns, while other classical BCs either suppressed crucial transversal flows or resulted in unphysical currents. Our results are crucial to performing flow properties upscaling and will be relevant to computing not only single-phase but also multiphase flow properties. Moreover, other calculation of physical properties such as some mechanical, transport, or heat conduction properties may benefit from the technique described in this study.

Figure 1

General scheme of the image processing, 3D stochastic reconstructions, and preparation of the pore geometries for pore-scale simulations (pores are shown in black).

Figure 2

Schematic diagrams of all boundary conditions explored shown in 2D for clarity: (a) the flow in real subsurface system containing the modeling domain, (b) CW BC, (c) Per BC, (d) Slip BC, (e) Free BC, (f) Trans BC, (g) Sym BC, and (h) Subim BC (a more-detailed description of each boundary condition is presented in the text).

Figure 3

Preparation of larger 3D images from original stochastic replicas by translation and symmetry for (a) shale and (b) soil samples (pores are in black).

Figure 4

The rendering of the pore geometry in 3D and directional two-point probability and lineal correlation functions for 3D stochastic replicas of (a) shale and (b) soil. The $y$ and $z$ directions are similar to original images shown in Fig. 1, while the $x$ direction is the one deduced as their average (according to correlation function statistics as shown on the right side).

Figure 5

The meshes used to perform simulations with Comsol for (a) shale and (b) soil samples.

Figure 6

Visualization of tensorial permeabilities for the shale sample using 3D ellipsoids (color shading of ellipsoids does not represent any values and is solely for visual purposes): (a) CW, (b) Per, (c) Slip, (d) Free, (e) Sym (CW, subvolume), and (f) Subim (one pixel) boundary conditions. Note that coordinate system positioning and magnitude are not the same for each subplot due to visibility issues.

Figure 7

Visualization of simulated pressure (left) and velocity (right) fields under different boundary conditions: (a) CW, (b) Per, (c) Free, (d) Sym (CW, subvolume).

Figure 8

General scheme of the flow patterns observed in the soil and shale reconstructed samples based on the following boundary conditions: (a) CW, (b) Per, (c) Slip, (d) Free, (e) Sym, and (f) Subim. This scheme does not follow soil and shale sample flow lines closely but provides the general difference between those under the conditions investigated.

Figure 9

Permeability REV analysis for (a) shale and (b) soil samples. The analysis is based on permeability tensor evaluation (based on Per boundary conditions) in terms of eigenvectors along the $x$, $y$, and $z$ major directions ($y$ axis) plotted versus increasing volume of the cubical domain ($x$ axis). In addition to eigenvalues, a visual representation of the full permeability tensor is shown for each domain size to facilitate the comparison.