XLB Direct-Image Permeability Benchmark

This report documents a controlled verification study of voids against a voxel-scale lattice-Boltzmann reference solved with XLB. The purpose is not to replace pore-network modeling with LBM, but to quantify how closely the current extracted-network workflow tracks a higher-fidelity direct-image calculation on the same segmented geometry.

The reproducible artifact for this report is notebook notebooks/13_mwe_synthetic_volume_xlb_benchmark.ipynb.

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Goal

The benchmark answers the following question:

Given the same binary segmented volume, how different is the absolute permeability predicted by:

1. direct-image LBM on the voxel geometry, and
2. voids after snow2 extraction and pore-network flow solution?

That is the scientifically relevant comparison for voids, because the main approximation is not only the linear solver, but the reduction from voxel geometry to an extracted pore-throat network.

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Governing Formulations

voids PNM Workflow

For each binary image:

1. snow2 extracts a pore network from the segmented volume.
2. The network is pruned to the axis-spanning subnetwork.
3. voids solves steady incompressible single-phase flow on that graph.

The voids pressure solve is the graph-Laplacian system

\[ \mathbf{A}\,\mathbf{p} = \mathbf{b}, \]

with throat fluxes

\[ q_t = g_t (p_i - p_j), \]

where \(g_t\) is the hydraulic conductance of throat \(t\).

After solving the pore pressures, voids converts the inlet flow rate to apparent permeability using Darcy's law:

\[ K = \frac{|Q|\,\mu\,L}{A\,|\Delta p|}. \]

For this benchmark, the voids side uses:

• conductance_model = "valvatne_blunt"
• solver = "direct"
• \(\mu = 1.0 \times 10^{-3}\) Pa s

The valvatne_blunt model is the complete single-phase conduit closure implemented in voids. It prefers available shape-factor and conduit-length information when present, but it is not a full reproduction of Valvatne-Blunt multiphase physics.

XLB Direct-Image LBM Workflow

The reference solve is run directly on the segmented binary image, without network extraction.

The current voids adapter uses XLB's incompressible Navier-Stokes lattice-Boltzmann stepper. On each lattice direction \(i\), the distribution function update is

\[ f_i(\mathbf{x} + \mathbf{c}_i \Delta t, t + \Delta t) = f_i(\mathbf{x}, t) - \omega \left[f_i(\mathbf{x}, t) - f_i^{\mathrm{eq}}(\rho, \mathbf{u})\right]. \]

The macroscopic fields are recovered from the moments

\[ \rho = \sum_i f_i, \qquad \rho \mathbf{u} = \sum_i \mathbf{c}_i f_i, \]

and the standard quadratic equilibrium is

\[ f_i^{\mathrm{eq}} = w_i \rho \left[ 1 + \frac{\mathbf{c}_i \cdot \mathbf{u}}{c_s^2} + \frac{(\mathbf{c}_i \cdot \mathbf{u})^2}{2 c_s^4} - \frac{\mathbf{u} \cdot \mathbf{u}}{2 c_s^2} \right]. \]

In lattice units,

\[ \nu_{\mathrm{lu}} = c_s^2 \left(\tau - \frac{1}{2}\right), \qquad \omega = \tau^{-1}, \qquad p_{\mathrm{lu}} = c_s^2 \rho. \]

This last relation is the key point behind the current boundary-condition implementation. In this isothermal LBM formulation, prescribing density at the inlet and outlet is the standard way to prescribe pressure. The public voids adapter now therefore accepts pressure BCs or a pressure drop and converts them internally to the equivalent densities expected by the XLB boundary operator.

To couple the same physical pressure drop to both PNM and XLB, the benchmark uses the standard unit conversion

\[ \nu_{\mathrm{phys}} = \frac{\mu_{\mathrm{phys}}}{\rho_{\mathrm{phys}}}, \qquad \Delta t_{\mathrm{phys}} = \frac{ \nu_{\mathrm{lu}}\,\Delta x_{\mathrm{phys}}^2 }{ \nu_{\mathrm{phys}} }, \]

followed by

\[ \Delta p_{\mathrm{lu}} = \Delta p_{\mathrm{phys}} \frac{ \Delta t_{\mathrm{phys}}^2 }{ \rho_{\mathrm{phys}}\,\Delta x_{\mathrm{phys}}^2 }. \]

That is the consistency condition used in the current benchmark wrapper: for each comparison mode, voids and XLB use the same imposed physical \(\Delta p\), while XLB still applies it through the equivalent densities.

For the current benchmark formulation, the absolute pressure offset is only a gauge choice. The public high-level benchmark wrappers therefore now accept a preferred physical input delta_p, with optional pin and pout retained only for cases where an absolute pressure reference should also be recorded. In practice, delta_p = 1 Pa, pin = 1 Pa / pout = 0 Pa, and delta_p = 1 Pa with pin = 101326 Pa / pout = 101325 Pa are all the same current permeability-driving condition.

For the current benchmark implementation, this means:

• D3Q19 lattice for 3-D cases
• single-relaxation-time BGK collision
• pull streaming
• lattice viscosity \(\nu_{\mathrm{lu}} = 0.10\)
• pressure boundary conditions, converted internally to equivalent densities
• half-way bounce-back on solid voxels
• sealed side walls orthogonal to the flow axis
• six fluid buffer cells before and after the sample so pressure BCs act on clean planar reservoir faces rather than directly on a perforated porous face

The benchmark uses the sample-averaged superficial axial velocity from the original sample domain, not from the padded reservoir region.

Permeability is then recovered from lattice units as

\[ K_{\mathrm{phys}} = \frac{ \nu_{\mathrm{lu}}\,U_{\mathrm{lu}}\,L_{\mathrm{lu}}\,\Delta x_{\mathrm{phys}}^2 }{ \Delta p_{\mathrm{lu}} }, \]

with

\[ \Delta p_{\mathrm{lu}} = p_{\mathrm{in,lu}} - p_{\mathrm{out,lu}}. \]

This means the XLB side is used here as a permeability reference. It is not being interpreted as a fully pressure-calibrated physical transient model.

Steady Stokes-Limit Mode

The continuum target for creeping flow in the pore space is the steady Stokes system

\[ -\nabla p + \mu \nabla^2 \mathbf{u} = \mathbf{0}, \qquad \nabla \cdot \mathbf{u} = 0 \quad \text{in the void space}, \]

with no-slip walls on the solid boundary.

The installed XLB package used here does not expose a separate Stokes-only operator. For that reason, voids now provides XLBOptions.steady_stokes_defaults(), which still uses the same incompressible-Navier-Stokes LBM stepper but applies:

• a smaller shared pressure drop
• tighter steady-state controls
• explicit low-inertia diagnostics:
• xlb_mach_max
• xlb_re_voxel_max

Scientifically, the correct statement is therefore:

steady_stokes_limit is an LBM approximation to the steady creeping-flow limit, not a distinct upstream Stokes discretization inside XLB.

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Why The Two Methods Differ

Even when both workflows are implemented correctly, they solve different representations of the same sample.

Aspect	voids	XLB
Geometry	Extracted pore-throat network	Original voxel image
Unknowns	One pressure unknown per pore	Distribution functions and velocity field per voxel
Transport law	Network conductance closure	Mesoscopic lattice Boltzmann dynamics
Solid treatment	Encoded indirectly through extracted geometry	Explicit bounce-back on solid voxels
Main approximation	Image-to-network reduction and conductance closure	Voxel staircasing, finite lattice resolution, BC discretization

Therefore, mismatch between voids and XLB is not automatically a voids bug. It can come from:

• loss of geometric information during extraction
• limitations of the selected pore-network conductance model
• voxel-resolution effects in the LBM reference
• boundary-condition sensitivity on small samples

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Synthetic Benchmark Setup

All cases in this report use:

• binary spanning volumes generated with generate_spanning_blobs_matrix
• shape (24, 24, 24)
• flow axis x
• voxel size 2.0e-6 m
• fluid viscosity 1.0e-3 Pa s
• fluid density 1.0e3 kg m^-3
• preferred benchmark input delta_p = 25/3 Pa ≈ 8.333 Pa
• preferred steady Stokes-limit input delta_p = 5/3 Pa ≈ 1.667 Pa
• the report uses the default benchmark gauge choice pout = 0 Pa, pin = delta_p; any common pressure offset would be equivalent for this current incompressible benchmark
• XLB benchmark options:
• formulation = "incompressible_navier_stokes"
• max_steps = 3000
• min_steps = 400
• check_interval = 50
• steady_rtol = 1.0e-3
• Δp_lu = 3.333e-4
• inlet_outlet_buffer_cells = 6
• XLB steady Stokes-limit options:
• formulation = "steady_stokes_limit"
• max_steps = 3000
• min_steps = 800
• check_interval = 50
• steady_rtol = 5.0e-4
• Δp_lu = 6.667e-5
• inlet_outlet_buffer_cells = 6

The 15-case verification set spans five porosity levels and three blobiness levels per porosity:

Case	Target porosity	Blobiness	Seed used
phi028_b12	0.28	1.2	101
phi028_b14	0.28	1.4	135
phi028_b15	0.28	1.5	155
phi032_b12	0.32	1.2	271
phi032_b14	0.32	1.4	305
phi032_b15	0.32	1.5	322
phi036_b12	0.36	1.2	441
phi036_b14	0.36	1.4	475
phi036_b15	0.36	1.5	492
phi040_b12	0.40	1.2	611
phi040_b14	0.40	1.4	645
phi040_b15	0.40	1.5	662
phi044_b12	0.44	1.2	781
phi044_b14	0.44	1.4	815
phi044_b15	0.44	1.5	832

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Figures

Left: voids permeability against XLB permeability with the one-to-one line. Right: per-case relative difference.

Porosity-permeability trend for the 15-case set. This is useful for checking whether voids follows the same macroscopic trend as the direct-image XLB reference even when the pointwise agreement is imperfect.

Permeability scatter and per-case error after rerunning all 15 cases with the lower-driving steady Stokes-limit configuration.

Porosity-permeability trend in the steady Stokes limit. This checks whether the direct-image LBM reference changes materially when the forcing is reduced.

Direct comparison between benchmark-mode XLB permeability and steady Stokes-limit XLB permeability across the full 15-case set.

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Results

The full CSV generated by the notebook is available here: xlb_15_case_results.csv.

The full steady Stokes-limit CSV is available here: xlb_stokes_15_case_results.csv.

The representative Stokes-limit rerun is available here: xlb_stokes_limit_representative.csv.

Case	Image porosity	K_voids [m^2]	K_xlb [m^2]	K_voids / K_xlb	Rel. diff. [%]	XLB steps	Converged
phi028_b12	0.280020	1.161e-14	1.563e-14	0.743	25.71	450	yes
phi028_b14	0.280020	1.598e-14	2.174e-14	0.735	26.50	400	yes
phi028_b15	0.280020	5.235e-14	2.198e-14	2.381	58.01	400	yes
phi032_b12	0.320023	2.308e-14	2.531e-14	0.912	8.81	2750	yes
phi032_b14	0.320023	5.138e-14	3.216e-14	1.598	37.41	1200	yes
phi032_b15	0.320023	6.371e-14	2.690e-14	2.368	57.77	1100	yes
phi036_b12	0.360026	4.851e-14	4.265e-14	1.138	12.09	1050	yes
phi036_b14	0.360026	4.214e-14	3.482e-14	1.210	17.37	2300	yes
phi036_b15	0.360026	3.517e-14	3.733e-14	0.942	5.77	450	yes
phi040_b12	0.400029	1.561e-13	4.789e-14	3.259	69.31	1850	yes
phi040_b14	0.400029	3.443e-14	4.983e-14	0.691	30.92	2250	yes
phi040_b15	0.400029	8.599e-14	4.612e-14	1.864	46.36	500	yes
phi044_b12	0.440032	1.799e-13	6.046e-14	2.976	66.40	2300	yes
phi044_b14	0.440032	2.143e-13	6.444e-14	3.325	69.93	400	yes
phi044_b15	0.440032	9.731e-15	6.948e-14	0.140	85.99	400	yes

Summary statistics for this 15-case set:

• mean relative difference: 41.22 %
• median relative difference: 37.41 %
• mean factor gap: 2.24 x
• median factor gap: 1.60 x
• worst factor gap: 7.14 x

Full 15-Case Steady Stokes-Limit Results

Case	Image porosity	K_voids [m^2]	K_xlb,stokes [m^2]	K_voids / K_xlb,stokes	Rel. diff. [%]	XLB steps	Converged
phi028_b12	0.280020	1.161e-14	1.565e-14	0.742	25.81	800	yes
phi028_b14	0.280020	1.598e-14	2.159e-14	0.740	25.99	1100	yes
phi028_b15	0.280020	5.235e-14	2.223e-14	2.354	57.52	1250	yes
phi032_b12	0.320023	2.308e-14	2.719e-14	0.849	15.14	1800	yes
phi032_b14	0.320023	5.138e-14	3.194e-14	1.609	37.84	950	yes
phi032_b15	0.320023	6.371e-14	2.692e-14	2.366	57.74	1200	yes
phi036_b12	0.360026	4.851e-14	4.386e-14	1.106	9.59	1550	yes
phi036_b14	0.360026	4.214e-14	3.721e-14	1.133	11.70	850	yes
phi036_b15	0.360026	3.517e-14	3.818e-14	0.921	7.89	1350	yes
phi040_b12	0.400029	1.561e-13	5.025e-14	3.106	67.80	800	yes
phi040_b14	0.400029	3.443e-14	5.276e-14	0.653	34.75	1150	yes
phi040_b15	0.400029	8.599e-14	4.623e-14	1.860	46.23	850	yes
phi044_b12	0.440032	1.799e-13	6.221e-14	2.892	65.43	1500	yes
phi044_b14	0.440032	2.143e-13	6.439e-14	3.328	69.95	800	yes
phi044_b15	0.440032	9.731e-15	6.775e-14	0.144	85.64	1150	yes

Summary statistics for the steady Stokes-limit 15-case set:

• mean relative difference: 41.27 %
• median relative difference: 37.84 %
• mean factor gap: 2.21 x
• median factor gap: 1.62 x
• worst factor gap: 6.96 x
• maximum Mach_max: 3.83e-5
• maximum Re_voxel,max: 2.21e-4

Benchmark-Mode Versus Steady Stokes-Limit Sensitivity

Across the full 15-case set, the direct-image XLB benchmark and the lower-drive steady Stokes-limit rerun differed by:

• mean relative difference: 2.58 %
• median relative difference: 2.30 %
• worst relative difference: 7.46 %

This is the key result for the “is the benchmark already Stokes-like?” question. For these synthetic cases, the answer is still broadly yes, but less strongly than the earlier draft suggested: reducing the imposed pressure drop changed the direct-image permeability by a few percent on average while still lowering the velocity-scale diagnostics substantially.

Representative Steady Stokes-Limit Comparison

The notebook also reruns the representative case phi036_b14 with XLBOptions.steady_stokes_defaults() and reports:

Mode	K_xlb [m^2]	Steps	Converged	Δp_phys [Pa]	Δp_lu	Mach_max	Re_voxel,max	Rel. diff. vs benchmark [%]
Benchmark	3.482e-14	2300	yes	8.333	3.333e-4	1.240e-4	7.143e-4	0.00
Steady Stokes limit	3.721e-14	850	yes	1.667	6.667e-5	2.804e-5	1.619e-4	6.86

For this representative geometry, reducing the shared physical pressure drop by a factor of five lowered both Mach_max and Re_voxel,max by about a factor of four to five while changing the XLB permeability by 6.86 %. That still supports the creeping-flow interpretation, but it is not small enough to treat benchmark mode and steady-Stokes-limit mode as numerically interchangeable on every morphology.

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Interpretation

These results support the following conclusions:

1. voids and XLB can agree quite closely on some cases. In this set, phi036_b15 differs by only 5.77 %.
2. The mismatch can still be substantial on other morphologies, reaching a factor gap of about 7.14 x for phi044_b15.
3. The porosity-permeability trend is broadly similar between the methods, but the pointwise spread remains too large to treat the current extracted-network workflow as interchangeable with a direct-image LBM reference.
4. The new steady-Stokes-limit mode makes the physical interpretation cleaner, but it should not be described as a separate solver backend unless XLB exposes an actual Stokes-only operator in a future release.

The practical interpretation is that the current voids image-to-network workflow is plausible, but still morphology-sensitive relative to XLB. That is exactly the kind of signal a verification benchmark should reveal.

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Limits Of This Verification

This report is intentionally narrow. It does not establish universal agreement between voids and LBM.

Important limits and assumptions:

• the cases are small synthetic spanning volumes, not real rock images
• side walls are sealed in the XLB benchmark; periodic transverse boundaries are not used here
• voids is compared against one specific network conductance closure: valvatne_blunt
• the XLB permeability conversion is a lattice-unit permeability mapping, not a full dimensional calibration of the transient flow field
• the steady-Stokes-limit mode is an interpretation of the same LBM operator in a low-inertia regime, not a separate Stokes discretization