Fast Laplace solver approach to pore-scale permeability

We introduce a powerful and easily implemented method to calculate the permeability of porous media at the pore scale using an approximation based on the Poiseulle equation to calculate permeability to fluid flow with a Laplace solver. The method consists of calculating the Euclidean distance map of the fluid phase to assign local conductivities and lends itself naturally to the treatment of multiscale problems. We compare with analytical solutions as well as experimental measurements and lattice Boltzmann calculations of permeability for Fontainebleau sandstone. The solver is significantly more stable than the lattice Boltzmann approach, uses less memory, and is significantly faster. Permeabilities are in excellent agreement over a wide range of porosities.

Figure 1

Analytical solution of the Laplace equation for parallel flows and illustration of the three steps. Step 1: $S(m,\alpha )$ is drawn for $m=5$. Step 2: $T(M,n,\alpha )$ is illustrated three times for $M=6$ and $n=1$, 2, 3. Step 3: $U(M,N,\alpha )$ is illustrated for $M=6$ and $N=3$.

Figure 2

Comparison between the analytical solution (17) (thick solid lines) and the analytical solution (14) for the Laplace equation for various values of the parameter $\alpha$ (thin solid lines). Data are for $\alpha =0.1$, 0.2, ..., 0.9. The broken lines correspond to $\alpha =0.5$. (a) Plane channel, (b) circular channel; the dotted line corresponds to the numerical solution by a classical code.

Figure 3

Rectangular channels. Comparison between the analytical solution (18) (thick solid line) and the analytical solution (14) for the Laplace equation for various values of the parameter $\alpha$ (thin solid lines). Data are for $\alpha =0.1$, 0.2, ..., 0.9. The thick broken lines correspond to $\alpha =0.5$. The various subfigures are for various discretizations: $\ell /a=6$ (a, d), 10 (b, e), 40 (c, f); the right column depicts small $L$ behavior.

Figure 4

The dimensionless permeability $K^{\prime }=K/{\ell }^2$ for square and rectangular channels as a function of the aspect ratio $r=L/\ell$. Data are for analytical solution (18) (solid line), Laplace approximation (16) (broken line), interpolation between the square and the plane channel (20) (dotted line).

Figure 5

Comparison between the Laplace and the lattice Boltzmann dimensionless permeabilities for 2D and 3D sinusoidal channels. $\alpha =0.5.$ (a) 2D channels; data are for $K_{xx}^{\prime }$ ($\circ$), $K_{yy}^{\prime }$ ($\square$); $\eta =0.2$ (filled dots), 0.4 (filled squares). (b) 3D channels; data are for $\eta =0.2$ (empty dots), 0.4 (filled dots).

Figure 6

Subsample of a tight sandstone measured by Focussed Ion Beam/Scanning Electron Microscope (FIB/SEM). It was displayed in Fig. 8(a) of Ref. [26] under the name sech1.

Figure 7

Comparison of numerically derived and experimentally measured permeability for Fontainebleau sandstone. The accuracy of the numerical calculations (in particular $k_{\mathrm{LB}}$) breaks down at permeabilities below $k\approx a^2$, where $a=5.68\mu \mathrm{m}$ is the resolution of the image. (a) $k_{\mathrm{EDT}}$ and $k_{\mathrm{LB}}$ as function of porosity against experiment, $a=5.68\mu \mathrm{m}$; (b) distribution of critical diameters for $a=5.68\mu \mathrm{m}$; (c) $k_{\mathrm{EDT}}$ vs $k_{\mathrm{LB}}$ at $a=5.68\mu \mathrm{m}$; (d) $k_{\mathrm{EDT}}$ vs $k_{\mathrm{LB}}$ at $a=2.84\mu \mathrm{m}$.

Figure 8

Cross sections through cubic subdomains of segmented tomograms of four outcrop sandstone samples: (a) Berea sandstone (${750}^3$ voxels, $a=2.33\mu \mathrm{m}$); (b) Bentheimer sandstone (${1200}^3$ voxels, $a=5.39\mu \mathrm{m}$); (c) Castlegate sandstone (${900}^3$ voxels, $a=3.36\mu \mathrm{m}$); (d) Leopard sandstone (${900}^3$ voxels, $a=2.15\mu \mathrm{m}$). White and light gray indicate denser phases (resolved pore space is black).

Figure 9

Comparison of permeability convergence behavior between Laplace approximation and LBM approach for the four outcrop sandstone samples: (a) Berea sandstone, (b) Bentheimer sandstone, (c) Castlegate sandstone, (d) Leopard sandstone, (e–h) Fontainebleau sandstone (${480}^3$, $a=5.68\mu \mathrm{m}$; FB08, FB13, FB15, FB22).