PaperlayerView exact revision on arXiv

Source-linked AI summary

A Query-Time Framework for Transient 2D Pore-Scale Flow Prediction and Generative Design

Yiming Wang, Jiale Zhu, Zhichen Ye, Yandong Lv, Shiqi Wang, Jinlong Liu, Yucheng Fan

arXiv:2608.22235v1cs.LG

TL;DR

Repeated transient LBM simulation across porous geometries is costly, motivating a query-time surrogate and inverse-design framework. The paper introduces CT-PoreFlow and evaluates it with LBM-verified design screening, achieving a velocity relative L2 error of 0.2248 and 72.28% conditional design success for guided GAN designs.

  • Problem

    Repeated transient LBM simulation requires independent geometry solves and multiple time states, while existing formulations lack direct non-sequential geometry-to-field prediction for new pore structures.

  • Method

    The framework constructs the QSGS-Transient-7606 benchmark and uses CT-PoreFlow to predict flow fields at requested times before screening generated structures for LBM-verified inverse design.

  • Results

    Velocity relative L2 is 0.2248 and guided conditional-GAN designs achieve 72.28% Success@20, with CT-PoreFlow improving on Time-U-Net across paired field and transport metrics.

  • Takeaways & Limitations

    The framework supports reusable query-time transient-flow prediction and surrogate-guided inverse-design screening with final LBM verification.

  • Takeaways & Limitations

    The study is limited to two-dimensional QSGS geometries with fixed generation, resolution, fluid, forcing, and orientation parameters, and does not establish universal temporal or morphological generalisation.

Abstract

from arXiv · show

Pore-scale flow governs transport and permeability behaviour in porous media engineering applications, yet repeated lattice Boltzmann method (LBM) simulation across many geometries and design queries remains costly for repeated deployment. This study formulates transient pore-scale flow prediction as a geometry-conditioned query-time operator and introduces QSGS-Transient-7606, a benchmark of 7,606 two-dimensional porous structures each paired with 30 logarithmically sampled LBM states. The proposed continuous-time pore-scale flow surrogate model (CT-PoreFlow) integrates topology-aware geometry encoding, compressed spectral mixing, and log-time conditioning with a late-time flux-calibration objective. On unseen test geometries, CT-PoreFlow achieves a velocity relative L2 of 0.2248 and a terminal permeability error of 12.81%. Frozen morphology and computed tomography image audits confirm reasonable cross-geometry robustness without fine-tuning. The surrogate is then embedded in an inverse design workflow, screening 9,216 generative adversarial network and diffusion candidates across 18 property targets prior to LBM verification. Guided GAN sampling attains 98.11% through-connectivity and 72.28% conditional design success, exceeding diffusion-based generation. The framework unifies transient flow prediction, transport-aware screening, and LBM-verified inverse design for porous media.

1. Introduction

The introduction frames transient pore-scale flow prediction as a costly, connectivity-dependent problem requiring both local boundary resolution and global transport information. It motivates an integrated framework that couples continuous-time surrogate prediction with transport-oriented porous-structure inverse design.

  • Motivation: Repeated LBM simulation is costly because each geometry requires an independent iterative solve, while transient analysis resolves multiple evolving flow states.Data-driven surrogates are presented as a way to predict new geometries and query times without repeating the full LBM solution.
  • Motivation: Transient flow is harder than converged steady flow because pressure redistribution, velocity development, and connectivity-dependent relaxation occur at different rates.Narrow throats, dead-end pores, and long inlet-to-outlet pathways can produce different relaxation behaviour despite similar porosity or local aperture.
  • Model requirements: An effective transient surrogate must preserve detailed pore boundaries and connected pathways while combining local pore-scale resolution with global transport information.Narrow throats and no-slip boundaries require accurate local reconstruction, whereas permeability and hydraulic connectivity depend on organisation across the entire domain.
  • Inverse design: Forward prediction alone is insufficient because porous-media engineering also requires identifying structures that achieve prescribed transport properties.The introduction motivates combining learned structure–property relations with generative or optimisation-based inverse design.
  • Framework contribution: The study develops an integrated framework linking transient pore-scale flow prediction with transport-oriented porous-structure design using a large-scale QSGS–LBM benchmark and CT-PoreFlow.The benchmark resolves velocity and pressure evolution from rest toward steady flow across thousands of statistically distinct porous geometries.

2. Methodology · 2.1. Pore-scale flow simulation

The methodology resolves transient, pressure-driven flow through generated two-dimensional porous structures using a D2Q9-BGK LBM solver, then derives transport observables and standardized transient field outputs. Geometry generation, connectivity screening, boundary conditions, convergence checks, and engineering definitions are fixed to support reproducible simulation and surrogate evaluation.

  • 2. Methodology: Pressure-driven flow evolves from rest toward convergence, with transient velocity and pressure fields supporting flow-rate, permeability, and hydraulic-tortuosity evaluation.A pressure difference drives seepage through heterogeneous pore networks while the resolved fields provide the basis for effective transport properties.
  • 2.1. Pore-scale flow simulation: The solver uses the D2Q9-BGK lattice Boltzmann formulation to recover weakly compressible Navier–Stokes behaviour under the study’s low-Mach regime.D2Q9 provides nine discrete particle velocities, while BGK relaxation uses a single relaxation time.
  • 2.1. Pore-scale flow simulation: Darcy velocity and permeability are computed from streamwise velocity sums across open measurement rows using the imposed positive pressure drop Δp = pin − pout > 0.Reference and predicted-field evaluations use their respective stored or solver-side measurement rows while retaining the engineering Darcy definition.
  • 2.1. Pore-scale flow simulation: QSGS generates binary microstructures with target porosity sampled across eight width-0.05 intervals spanning 0.30–0.70, followed by two 3 × 3 median-filter passes.The construction controls phase fraction and directional growth through specified axial, diagonal, and solid-core probabilities.
  • 2.1. Pore-scale flow simulation: Each candidate is retried until filtered porosity meets its assigned interval and four-neighbour connectivity links inlet to outlet before LBM simulation.Isolated cavities and cul-de-sac branches remain in the binary geometry, preserving storage and local pressure effects.
  • 2.1. Pore-scale flow simulation: The fixed implementation sets ρin = 1.00001, ρout = 1.0, τLBM = 1.0, five fluid buffer rows, and bounce-back on solid nodes.Pressure-driven seepage is imposed through density boundary conditions using the same D2Q9-BGK configuration for every geometry.
  • 2.1. Pore-scale flow simulation: The simulation starts from rest, checks convergence every 500 iterations, caps solves at 50,000 iterations, and requests 30 logarithmically scheduled fields.The implementation also requires m < 10−3, finite states, and Mamax < 0.10; converged late-time fields are copied with a steady-fill indicator.

2.2. Dataset and Geometry Representation

QSGS-Transient-7606 provides 7,606 geometry-disjoint porous structures with 30 logarithmically sampled states each. CT-PoreFlow represents each native pixel geometry through five deterministic channels encoding pore space, connectivity, aperture, and transport position.

  • Dataset: 7,606 accepted geometries span porosity 0.30–0.70 and provide 228,180 geometry–time records from 30 logarithmically scheduled states.The states span iteration 0 to 50,000.
  • Dataset: 6,078 training, 764 validation, and 764 test structures form a geometry-disjoint split, with all states and static channels kept together.Normalisation statistics are calculated from training data only.
  • Geometry representation: Five native pixel-lattice channels encode pore space, through-connected pore space, local aperture, and inlet- and outlet-referenced transport positions.The representation retains the native lattice without contour smoothing and forms X_G.
  • Geometry representation: Multi-source breadth-first traversals define connectivity and directed shortest-path distances, with graph distances divided by max(H, W) = 100 and zeroed outside M_c.Attached cul-de-sac branches remain connected when reachable from both boundary traversals.

2.3. Transient Pore-Flow Surrogate model

CT-PoreFlow predicts transient pore-scale velocity and pressure fields directly from geometry and a requested time through a local–global encoder–decoder. Its compressed spectral mixing, logarithmic time conditioning, and multi-term objective are designed to preserve pore-scale detail while supporting Darcy-scale transport evaluation.

  • Architecture: CT-PoreFlow encodes transport-oriented geometry at multiple scales, mixes global pore-network information at a compressed bottleneck, and decodes time-conditioned velocity and pressure fields.The architecture combines local detail with global representation before reconstructing the requested fields.
  • Time conditioning: The surrogate evaluates fields directly at the requested time rather than recursively advancing prior predictions.A logarithmic query coordinate provides greater resolution during the densely sampled early transient while retaining the terminal state at τ_t = 1.
  • Compressed spectral mixing: At the 6 × 6 bottleneck, compressed spectral mixing applies a two-dimensional FFT with three retained modes per direction alongside parallel local channel mixing.The spectral block acts only on the compressed representation, while convolutional skips preserve wall and throat detail.
  • Time conditioning: Time conditioning embeds the scalar query coordinate with sine and cosine functions at exponentially spaced frequencies from 1 to 10^-4, then modulates decoder features through scale and shift parameters.This introduces time dependence without appending a spatially constant time image at every resolution.
  • Training objective: The training objective combines normalized field fidelity, fluid-only gradient preservation, and late-time flux calibration for Darcy-relevant boundary behavior.Edge masking avoids penalizing the intentional hard pore-mask discontinuity while preserving gradients around narrow pore throats.

2.4. Training and validation

Training used standardized optimization and checkpoint-selection procedures, while validation and evaluation combined field-fidelity, transport-recovery, statistical, interpolation, and morphology-transfer assessments. Frozen audits tested rule-based and computed-tomography-derived geometries without fine-tuning, but did not establish broader generalization.

  • Training and validation: Training used AdamW, cosine learning-rate decay, automatic mixed precision, and gradient-norm clipping at 1.0, with validation on 256 geometries at seven time indices.Validation indices were {0, 5, 10, 15, 20, 25, 29}.
  • Training and validation: Baseline models received the same five-channel geometry tensor, with Time-FNO and DeepONet specified by fixed coordinate, spectral-block, width, mode, and basis-function configurations.Time-FNO uses width 48, 16 modes per direction, and four spectral blocks; DeepONet uses 64 basis functions per output channel.
  • Training and validation: Evaluation combined pore-masked field errors, velocity relative L2, physical speed MAE, flux, velocity, permeability, and hydraulic-tortuosity metrics.Velocity relative L2 excludes the zero initial field and applies only when the pore-restricted target norm exceeds 10^-6; with fixed H, ν, and Δp, permeability is a constant multiple of Q.
  • Training and validation: Terminal metrics contributed one paired value per geometry for n=764, with 10,000 paired geometry-level bootstrap resamples and corrected two-sided Wilcoxon signed-rank comparisons.Errors were stratified by total and connected porosity, dead-pore ratio, hydraulic tortuosity, and LBM convergence iteration.
  • Training and validation: Frozen morphology-transfer audits used independently generated rule-based geometries and computed-tomography-derived slices without fine-tuning, recalibration, or OOD-specific optimization.The computed-tomography audit isolates morphology shift and does not establish real-rock, three-dimensional, parameter, or boundary-condition generalization.

2.5. Porous structure inverse design

Porous-structure inverse design deploys CT-PoreFlow to screen generator-proposed geometries against target properties before D2Q9-BGK LBM verification. The workflow compares conditional GAN and diffusion candidates using exact morphology and connectivity audits, surrogate-predicted terminal properties, and solver-based final labels.

  • Design workflow: Inverse design uses target-conditioned candidate generation, exact topology audits, CT-PoreFlow screening, and D2Q9-BGK LBM verification.Conditional GAN and diffusion models serve as alternative candidate-generation strategies.
  • Target construction and success criteria: 18 targets combine three training porosity quartiles, three local permeability quartiles, and two local tortuosity quartiles; Success@20 requires porosity error ≤0.03, permeability error ≤20%, and tortuosity error ≤10%.QC-conditional Success@20 is computed only for candidates passing the original LBM quality-control criteria.
  • Baseline comparison: The finite-library comparison gives every method the same 764 unseen structures, while the LBM oracle uses test labels only as a nondeployable attainable bound.Random-QSGS draws 50 candidates per target, morphology-only retrieval uses three distance terms, and learned screens use predicted terminal fields.
  • Generative search: 9,216 raw generated structures comprise 4,608 masks from each generator, with exact morphology and through-connectivity evaluated for every mask.Each generator produces 256 masks for each of the 18 targets, and CT-PoreFlow predicts terminal permeability and tortuosity only for connected candidates.
  • Non-generative validation: A separate closed loop uses 120 fresh QSGS structures, with CT-PoreFlow ranking three targets and D2Q9-BGK rerunning one guided and one uniformly sampled candidate per target.The surrogate performs screening in every design experiment, while the final engineering property label comes only from the D2Q9-BGK LBM solver.

3. Results and Validation

Results show that CT-PoreFlow improves principal field and terminal transport accuracy over Time-U-Net, while benchmark analyses reveal strong topology-dependent variability and property-specific limitations. Ablations and transfer audits identify compressed global information exchange as especially important, but unseen-time and morphology tests temper claims of universal superiority.

  • Benchmark quality and variability: All 7,606 retained simulations satisfy velocity-residual, permeability-change, mass-balance, and Mach-number criteria, with only one reaching 50,000 iterations.Representative states also show distinct permeability and tortuosity among geometries with comparable total porosity.
  • Surrogate accuracy: 0.2248 velocity relative L2 error and 12.81% terminal permeability error were achieved by CT-PoreFlow, versus 0.2439 and 14.99% for Time-U-Net.CT-PoreFlow also reduced hydraulic-tortuosity error from 7.81% to 5.10% and mass-balance error from 0.0974 to 0.0839.
  • Transport-property validation: R2 = 0.988 for CT-PoreFlow permeability calibration, but tortuosity reached R2 = −0.731 despite a 2.1% median absolute percentage error.The negative R2 reflects substantial errors concentrated in a small number of extreme geometries.
  • Structure–property relationships: Connected porosity correlates with lattice permeability at rho = 0.80, whereas dead-pore ratio correlates at rho = −0.72 and hydraulic tortuosity is approximately uncorrelated with connected porosity.The accepted structures therefore span multiple conductance and path-complexity regimes rather than being determined by total porosity alone.
  • Ablation analysis: 26.78% velocity-error degradation followed removal of the FNOBlock, the largest major ablation degradation, supporting compressed spectral mixing for global pore-network information exchange.Replacing logarithmic with linear time conditioning increased velocity error by 14.40%, while using field loss alone increased it by 11.71%.
  • Generalisation audits: 0.3267 unseen-time velocity relative L2 error for CT-PoreFlow exceeded Time-U-Net’s 0.3224, while morphology-transfer errors were 0.3183 and 0.3084 for CT-PoreFlow and Time-FNO.Query-time evaluation permits arbitrary requested times, but these audits do not establish superior interpolation or universal cross-geometry performance.

4. Surrogate-Assisted Inverse Design and LBM Verification

CT-PoreFlow improved surrogate-guided candidate selection in controlled and generated-design settings, with LBM verification confirming stronger target fidelity for guided conditional-GAN designs than for random selection. Conditional diffusion offered greater morphological diversity but lower through-connectivity and target compatibility.

  • Finite-library ranking: 0.386 mean LBM-verified target distance and 83.33% Success@20 were achieved by CT-PoreFlow, versus 1.618 and 7.67% for random-QSGS.CT-PoreFlow also recorded 11.44% permeability target error and 2.80% hydraulic-tortuosity target error.
  • Fresh-QSGS screening: 77.59% descriptive reduction lowered mean LBM-verified target distance from 1.494 to 0.335 across three paired targets.All six LBM simulations satisfied the original convergence, velocity-residual, permeability-change, mass-balance, and Mach-number criteria.
  • Limitations: 17.31% permeability overshoot for the low-τ_h target illustrates that matching one transport property does not necessarily produce a simultaneous optimum in the other.The residual mismatch reflects coupling between permeability and hydraulic tortuosity through connected flow-path organisation.
  • Conditional generation: 98.11% mean through-connectivity for conditional GAN exceeded 55.36% for conditional diffusion, while diffusion explored broader binary morphology space.Mean pairwise Hamming distance was 0.127 for conditional GAN and 0.425 for conditional diffusion; both produced 100% unique thresholded masks.
  • LBM-verified selection: 0.481 mean verified distance and 72.28% Success@20 were achieved by surrogate-guided conditional-GAN selection, versus 1.160 and 22.77% for random GAN selection.Guided conditional diffusion also improved mean verified distance from 3.233 to 1.202 and Success@20 from 0.96% to 10.19%.
  • Surrogate-to-LBM calibration: 0.960 permeability R^2 across 414 QC-passing generated candidates indicated close surrogate-to-LBM agreement, while hydraulic-tortuosity R^2 was 0.449.Permeability RMSE and MAE were 0.086 and 0.054 log units; hydraulic-tortuosity RMSE and MAE were 0.102 and 0.046.

5. Limitations and future work

The study’s scope is limited by a narrowly controlled two-dimensional benchmark and filtering that under-represents the lowest-porosity regimes. Future datasets should broaden morphology coverage and include balanced near-percolation samples.

  • Current limitations: The benchmark is restricted to two-dimensional QSGS geometries with fixed growth probabilities, lattice resolution, fluid properties, pressure gradient, and flow orientation.These fixed settings define the current study scope.
  • Current limitations: 394 generation tasks exhausted the retry limit, contributing to under-representation of the lowest-porosity regimes after connectivity and convergence filtering.Filtering and retry limits reduce coverage of difficult low-porosity cases.
  • Future work: Future datasets should include balanced near-percolation samples and alternative morphology-generation rules.The passage identifies broader dataset construction as a future direction.

6. Conclusions

The study presents an LBM-verified framework that reframes transient pore-scale flow analysis as reusable query-time prediction and inverse design. CT-PoreFlow supports accurate unseen-geometry prediction and inverse-design screening, while conclusions remain bounded by a fixed two-dimensional QSGS-LBM regime and limited image auditing.

  • Framework: The framework integrates QSGS-Transient-7606, CT-PoreFlow, engineering-property recovery, conditioned candidate generation, and final LBM verification.It unifies transient prediction, transport-aware design screening, and physical verification in a reusable workflow.
  • Prediction: 0.2248 velocity relative 𝐿2 error and 12.81% terminal permeability error demonstrate CT-PoreFlow’s unseen-geometry prediction accuracy.The study reports improvement on Time-U-Net across all paired field and transport metrics under the principal comparison.
  • Inverse design: 83.33% LBM-verified Success@20 in finite-library retrieval and 72.28% for guided conditional-GAN designs establish CT-PoreFlow’s value for inverse-design screening.Both results exceed their respective random and morphology-only controls.
  • Limitations: Greater morphological diversity in the conditional diffusion model does not ensure higher connectivity or target fidelity, so diversity and design utility should remain separate criteria.The conclusions also note that the framework is established within a fixed two-dimensional QSGS-LBM regime, while image auditing uses one segmented computed tomography volume.

CRediT authorship contribution statement

The CRediT statement assigns distinct methodological, analytical, software, investigative, resource, validation, writing, and visualization contributions across seven authors.

  • Yiming Wang led methodology, software, formal analysis, data curation, original-draft writing, and visualization.
  • Jiale Zhu contributed conceptualization, methodology, supervision, and review-and-editing, while Yucheng Fan contributed conceptualization, methodology, software, formal analysis, visualization, and review-and-editing.
  • Zhichen Ye handled software, investigation, validation, and data curation; Yandong Lv handled investigation and resources; Shiqi Wang handled investigation and formal analysis.
  • Jinlong Liu contributed resources and validation.
Loading 2608.22235v1…