DSperse Framework: Modeling Dispersivity in Porous Media

Updated 8 February 2026

DSperse Framework is a suite of computational platforms for modeling dispersivity in materials, integrating image segmentation, DNS, and closure modeling.
It enables accurate digital twins of porous media by upscaling transport properties with errors below 2% in key dispersivity predictions.
The framework supports diverse applications including porous media simulation, Stokes flow suspensions, and zero-knowledge machine learning verification.

The acronym “DSperse” denotes several technical frameworks and computational platforms for modeling, simulation, or verification in distinct scientific domains. Below, the principal DSperse Frameworks as documented in the literature are described, with focus on multiscale dispersivity modeling in porous media, but also referencing frameworks for suspensions in Stokes flow, dilute suspension multiphase flows, and zero-knowledge machine learning. Each instance is technically specific and non-overlapping.

1. DSperse Framework for Dispersivity in Digital Twins of Multiscale Porous Materials

The DSperse Framework—“Dispersivity calculation in digital twins of multiscale porous materials using the micro–continuum approach”—is a computational pipeline for predicting effective transport properties (notably the full dispersivity tensor) in hierarchical porous media where pore-scale heterogeneity spans several decades (Maes et al., 2024). The core methodology integrates multi-resolution imaging, direct numerical simulation (DNS), and closure modeling to generate digital twins with accurate upscaled transport coefficients across regimes.

Key Pipeline Components

Image preprocessing and segmentation: Multiscale 3D images (typically micro-CT) of the material are segmented into fully resolved solid ( $\epsilon=0$ ), fully resolved pore ( $\epsilon=1$ ) and “unresolved” sub-domains ( $0 < \epsilon < 1$ ).
High-resolution closure characterization: For each unresolved subdomain class, independent DNS is performed to obtain local porosity $\epsilon$ , permeability $k$ , and dispersivity behavior $D_{e,ij}(Pe)$ over a range of Péclet numbers.
Model fitting: DNS output is fit to closed-form expressions (e.g., $D_{e,ij}(Pe) \simeq (D/\tau)[1 + \beta Pe^\alpha]$ ) parameterized by $\epsilon$ and $k$ .
Digital twin assembly: The low-resolution image is assigned spatially varying fields $\epsilon(x)$ , $k(x)$ , and $D^*_{ij}(x)$ , enabling seamless multiscale simulation.
Full-domain simulation: A coupled Darcy–Brinkman–Stokes (DBS) plus closure-transport solver computes the flow and upscaled transport in a single run using the above fields.

Mathematical Formulation

Darcy–Brinkman–Stokes equations for the volumetrically averaged velocity $u_D$ and pressure $p_f$ , with variable permeability $k$ enforcing phase-dependent flow regimes:

$\nabla \cdot u_D = 0$

$\frac{1}{\epsilon} \nabla \cdot \left( \frac{u_D \otimes u_D}{\epsilon} \right) = - \nabla p_f + \nu \nabla^2 u_D - \nu k^{-1} u_D + S_{uf}$

Volume-averaged advection-diffusion for species concentration:

$\frac{\partial(\epsilon c_f)}{\partial t} + \nabla \cdot (\epsilon u_f c_f) = \nabla \cdot (\epsilon D^* \nabla c_f)$

Upscaled (macro) transport leverages volume averaging and closure on a representative volume element (REV), requiring solution of a local closure problem for the “closure variable” $\bar{f}$ and yielding the symmetric dispersivity tensor $D^*_{ij}$ :

$D^{*}_{ij}(Pe) \approx (D/\tau)[1 + \beta_{ij} Pe^{\alpha_{ij}}]$

Péclet and length scaling: $Pe = \langle u \rangle^f L / D$ , with $L = \sqrt{\lambda K / \phi}$ .

2. Multi-stage Computational Implementation

Implementation is based on OpenFOAM/GeoChemFoam, with several solver utilities:

simpleDBSFoam: Extends icoFoam with Darcy term for DBS equations.
dispersionFoam: Solves the closure problem for dispersivity.
processDispersion: Post-processing utility for $D^*$ extraction.

Typical workflow: segment image → acquire high-res REVs → DNS-based closure → fit dispersivity model → assemble digital twin → run full domain simulation.

Spatial discretization combines first-order linear upwind (convection) and second-order limited-corrected schemes (diffusion). Pressure–velocity coupling uses SIMPLE, and subdomain transitions are seamless via cellwise variation of $\epsilon$ and $k$ .

3. Validation, Model Accuracy, and Numerical Results

Benchmarks demonstrate:

Agreement between micro-continuum model and DNS for both longitudinal and transverse dispersivities ( $<2\%$ error for $D_e/D$ over $Pe = 0.01$ –$100$).
Quantitative matching for full transport problems (e.g. 2D micromodel slug test).
Computational efficiency: direct upscaling to digital twins with arbitrary heterogeneity.
In hierarchical ceramic foams and microporous carbonate rocks, closure-derived dispersivity can be represented by parameterized models with a small number of subdomain classes (e.g., 3–4 sufficing for macro-dispersivity up to $Pe=100$ ).

Sample table (from micro-continuum vs. DNS validation):

Pe	DNS x– $D_e/D$	Micro-cont x– $D_e/D$	DNS y– $D_e/D$	Micro-cont y– $D_e/D$
0.01	1.20	1.19	1.00	1.00
1.0	79	78	8.40	8.39
100	$2.90\times10^3$	$2.85\times10^3$	100	101

4. Application Case Studies

Hierarchical Ceramic Foams

Pore-scale DNS on foam microstructures, then macro-scale simulation.
Péclet–dispersivity model fitted:
- $D_{ex}=0.38D(1+70\,Pe^{1.35})$
- $D_{ey}=0.38D(1+4.6\,Pe^1)$ for $Pe<1$ , with empirical exponents for higher $Pe$ .
Lattice type and fiber porosity affect the overall dispersivity.

Microporous Carbonate Rocks (Estaillades)

Segmentation into 14 phases; DNS on phase REVs.
Macro-scale model condenses phases, maintaining $<10\%$ error in $D_e$ and $<5\%$ in permeability.
Three microporous classes sufficient for accurate effective dispersivity.

5. Assumptions, Limitations, and Future Work

Assumptions:

Local isotropy and alignment of dispersion with mean flow in unresolved pores.
Validity of closure-derived $D_e(Pe, \epsilon, k)$ over the relevant parameter space.

Limitations:

Accuracy is contingent on the representativity and heterogeneity of the selected REVs, and the sampling of $Pe$ .
Ragged spatial heterogeneity may diminish upscaling accuracy.

Future Directions:

Machine-learning–driven regression of $D_e(\epsilon, k, Pe, ...)$ from large DNS libraries.
Extensions to multiphase/multicomponent flow or non-equilibrium transport.

Though outside the porous media context, the DSperse designation is also applied to:

Scalable simulation of particulate Stokes suspensions: A computational platform for dense/active suspensions with fast collision resolution via linear complementarity and quadratic programming, supporting generic geometry and hydrodynamics (Yan et al., 2019).
Eulerian dilute suspension multiphase flow: A framework for high-fidelity simulation of dilute, small-Stokes-number particle-laden flows with two-way coupling and a DNS/LES approach validated on dam-break test cases (Berselli et al., 2014).
Zero-knowledge targeted ML verification: A slice-based, risk-driven framework for scalable, modular zero-knowledge machine learning inference, enabling targeted verification of prioritized model slices and integrating proof engines such as EZKL and JSTProve (Ivanov et al., 9 Aug 2025).

Each “DSperse” instance is domain-specific and technically distinct. No generic algorithmic, mathematical, or software substrate is shared between these frameworks beyond the high-level modularity and performance-driven design.

References:

"Dispersivity calculation in digital twins of multiscale porous materials using the micro-continuum approach" (Maes et al., 2024)
"A scalable computational platform for particulate Stokes suspensions" (Yan et al., 2019)
"Disperse two-phase flows, with applications to geophysical problems" (Berselli et al., 2014)
"DSperse: A Framework for Targeted Verification in Zero-Knowledge Machine Learning" (Ivanov et al., 9 Aug 2025)