Pore-Morphology Method: Geometric Insights

• Pore-morphology method is a geometry-driven approach that uses binary images and distance transforms to extract pore-scale characteristics.
• It employs techniques like morphological opening, watershed segmentation, and maximal-ball extraction to derive capillary pressure, pore-body metrics, and network connectivity.
• Applications include efficient capillary modeling, representative volume identification, and generative reconstruction of 3-D pore architectures.

The pore-morphology method is a geometry-driven family of techniques in which pore space morphology, rather than only direct field solution, is used to derive pore-scale or bulk properties. In the porous-media literature, the term is used most prominently for image-based methods that operate on binary or segmented pore representations through morphological binary operations, distance maps, watershed segmentation, maximal balls, and related constructs in order to calculate capillary pressure-saturation relations, locate pore bodies and throats, identify representative volume elements, or generate statistically equivalent pore architectures (Alonso-Marroquin et al., 13 Jan 2025).

1. Terminological scope and core idea

The literature does not use “Pore-Morphology Method” for a single algorithm. Instead, the label covers several technically distinct but conceptually related formulations. In each case, pore geometry is treated as the primary source of information, and the method seeks to map morphology to capillarity, connectivity, representativeness, or macroscopic transport. This suggests that the common denominator is not a specific numerical scheme but a morphology-centered representation of the pore space.

Usage in the literature	Core operation	Typical output
Capillary PMM	“morphological binary operations”	“capillary pressure-saturation relations”
Curvature-based pore-space characterization	“curvature of the distance map”	“pore body and throat centers” and pore-space partitioning
Maximal-ball / medial-axis extraction	“Euclidean Distance Transform (EDT)” and “maximal balls”	“pore-throat network” and “medial-axis network”
RVE identification	“morphology and spatial distribution of pores”	“representative volume element”
Generative pore reconstruction	“property-constrained” or “distance-based morphological statistics”	representative pore-scale or 3-D pore images

In capillarity-oriented work, the method is presented as “a computationally efficient method to calculate the capillary pressure-saturation relations of immiscible multiphase flow on two-dimensional pore morphologies,” and as an extension that “includes wetting angle and trapped mechanism of the displaced fluid, and calculation of material properties by density functional theory” (Alonso-Marroquin et al., 13 Jan 2025). In network-extraction and image-analysis work, the same general orientation appears as distance-transform, watershed, maximal-ball, or Hessian-curvature methods that identify pore bodies, throats, and skeletons directly from segmented images (Ben-Noah et al., 2024).

2. Morphological representations and mathematical primitives

A recurring starting point is a binary or segmented image of the medium in which the pore space is explicitly represented. One capillarity formulation applies morphological opening to a pore subimage $I_p$ using a spherical or circular structuring element $B_r$, written $I_p \circ B_r$, and varies $r$ from zero to the maximum inscribed radius $R_p$. This yields a sequence of ganglion configurations in the pore and a corresponding sequence of $(\kappa_b,S_b)$ data points, where $S_b$ is saturation and $\kappa_b$ is interface curvature (Laku et al., 8 Apr 2026). In this setting, the method is fundamentally a mapping from local geometry to local curvature-saturation behavior.

Distance-transform formulations begin from the Euclidean Distance Transform of the void space. In maximal-ball extraction, each void voxel is associated with the radius of the largest sphere that can be inscribed there, and maximal spheres are retained to define pore bodies and throats (Barsi-Andreeta et al., 2019). In curvature-based characterization, the signed distance map is differentiated to form the Hessian matrix; local maxima and minima of the determinant map of the Hessian are then used to locate pore body and throat centers, while eigenvalues of the Hessian guide watershed or medial-axis partitioning (Ben-Noah et al., 2024). The explicit claim is that this strategy “eliminates the common problem of saddle-induced over-partitioning shared by all traditional marker-based watershed methods” and determines “the skeleton of the pore space without the need for morphological reconstruction” (Ben-Noah et al., 2024).

A third morphological primitive is statistical similarity rather than binary morphology in the strict sense. Representative-volume identification in noisy Micro-CT data has been formulated from “voxel intensity pattern along 1D rows,” with similarity measured by the Pearson correlation coefficient between row segments. Repetitiveness $R(\varepsilon,L)$ is then used to define a representative elementary size along each Cartesian direction, and the representative volume element is the parallelepiped $(\mathrm{RES}_x,\mathrm{RES}_y,\mathrm{RES}_z)$ (Grigoriev et al., 2020). This formulation is notable because it “works directly on noisy, grayscale Micro-CT reconstructions” and does not require prior computation of porosity or permeability.

These variants share an operational principle: pore morphology is encoded either explicitly, through binary structuring operations and inscribed spheres, or implicitly, through curvature or similarity statistics. The significance is methodological. The governing object is the pore space itself, not merely an effective property derived after substantial homogenization.

3. Capillarity, curvature, and saturation

The capillarity-centered pore-morphology method is the formulation most directly associated with capillary pressure-saturation relations. In the two-dimensional setting, it has been described as using “only morphological binary operations,” which makes it “more efficient than well-established high-resolution voxel dynamics methods such as Lattice Boltzmann Methods and Level-set computational fluid dynamics” (Alonso-Marroquin et al., 13 Jan 2025). The same work states that, apart from pore morphology, “only the material parameters related to contact angle (wettability) and interfacial tension are required to connect the pore-saturation relation and pore throat distribution,” and that the effects of “interfacial tension, wettability, sample size, and pore throat distribution” are investigated for “entry pressure and residual saturation” (Alonso-Marroquin et al., 13 Jan 2025).

In an image-based pore-network implementation, capillary pressure is written as

$B_r$0

and, for a 2D micromodel with gap thickness $B_r$1 and zero contact angle,

$B_r$2

Here the pore-morphology method provides the in-plane curvature through morphological opening, with $B_r$3 for structuring-element radius $B_r$4 (Laku et al., 8 Apr 2026). The resulting pore-wise $B_r$5 relationship is then fitted separately for three dynamic pore types: singleton pore, terminal pore, and bridge pore. The super-critical branch is PMM-based in all three cases, while the sub-critical branch depends on connectivity. This construction allows ganglia that span multiple pores to be represented without introducing a single global capillary pressure-volume curve for the entire ganglion (Laku et al., 8 Apr 2026).

One consequence is that capillary events can be posed as local stability criteria. In the same framework, snap-off, invasion, and retraction are triggered by comparisons between throat entry or snap-off pressures and pore capillary pressures derived from the PMM-based $B_r$6 curves. The method thereby couples capillarity, topology change, and transport. In the Ostwald-ripening application, pore-wise $B_r$7 also enters Henry’s law, so that morphology controls local equilibrium concentration and therefore diffusive mass transfer between ganglia (Laku et al., 8 Apr 2026).

The broader significance is that pore geometry is used to compute curvature directly, rather than to define an idealized pore shape class. This is why the image-based pore-network model described in the ripening work “requires no idealization of pore shapes, as the effect on capillarity is encoded locally in curvature-saturation curves computed via the pore-morphology method” (Laku et al., 8 Apr 2026).

4. Pore bodies, throats, skeletons, and representative volumes

In network extraction, the pore-morphology method is concerned less with capillary pressure than with structural decomposition of the pore space. The maximal-ball formulation begins by computing the Euclidean Distance Transform over the void space, retaining only maximal spheres that are not fully included in larger spheres. A hierarchical tagging process then classifies local maxima as pore centers and identifies throat spheres when sphere families collide. From this procedure one obtains pore radius, pore volume, throat radius, throat length, and connectivity, and can construct both a classical pore-throat network and a one-voxel-wide medial-axis network through shortest-path analysis on the sphere graph (Barsi-Andreeta et al., 2019).

The curvature-of-distance-map method refines this by shifting attention from the distance map itself to its second derivatives. Pore body centers are associated with local maxima of the distance map, whereas throat centers are found from “local maxima and minima of the determinant map of the Hessian matrix of the DM.” The partitioning step then uses “the eigenvalues of the Hessian, rather than the DM,” and can be implemented by either watershed or medial-axis transforms (Ben-Noah et al., 2024). Because the approach avoids saddle-induced over-partitioning, it is explicitly positioned as an efficient route to pore-space skeletonization and to pore-network construction from binary images.

Representative-volume identification constitutes another morphology-based branch. The method described for noisy Micro-CT data uses only “morphology and spatial distribution of pores,” not physical properties such as porosity. Similarity between row segments is defined through the absolute Pearson correlation coefficient, and a normalized repetitiveness curve determines the smallest length at which the morphology is representative along each axis (Grigoriev et al., 2020). The method is designed to be “flexible and does not overestimate the volume size in the case of anisotropic samples,” and the resulting volume can then be used either for property computation after filtering and binarization or for selecting filtering parameters in a denoising workflow (Grigoriev et al., 2020).

Taken together, these formulations show that pore-morphology methods are not restricted to one downstream task. They can define the pore-throat graph itself, the medial axis used for flow simulation, or the representative support on which physical properties should be computed. This suggests that morphology functions not only as an input to transport models but also as a criterion for the validity of the computational domain.

5. Reconstruction and generation of pore architectures

Recent work extends the pore-morphology method into stochastic and generative reconstruction. One route is direct 3-D generation from a single 2-D cross-sectional SEM image of an isotropic porous membrane. The method extracts “distance-based morphological statistics” from the pore and solid phases, builds a “multi-scale morphological descriptor,” and uses Bayesian optimization to tune a stochastic 3-D generator until the central slice of the generated volume matches the SEM-derived descriptor (Danalou et al., 8 Aug 2025). Validation against X-ray tomography showed “excellent agreement in structural metrics,” while the SEM-based reconstruction achieved “superior resolution in resolving fine pore features” (Danalou et al., 8 Aug 2025). The explicit limitation is that the method is formulated for “isotropic porous membrane structures,” and “further work is needed for 3-D structure generation of anisotropic membranes” (Danalou et al., 8 Aug 2025).

A second route is deep generative modeling. “PCP-GAN: Property-Constrained Pore-scale image reconstruction via conditional Generative Adversarial Networks” is presented as “a multi-conditional Generative Adversarial Network (cGAN) framework that generates representative pore-scale images with precisely controlled properties” (Sadeghkhani et al., 22 Oct 2025). In the reported carbonate case, the model was conditioned simultaneously on porosity and depth, achieved “exceptional porosity control ($B_r$8) across all formations with mean absolute errors of 0.0099-0.0197,” and preserved “average pore radius, specific surface area, and tortuosity” within acceptable tolerances (Sadeghkhani et al., 22 Oct 2025). The generated images also showed “dual-constraint errors of 1.9-11.3% compared to 36.4-578% for randomly extracted real sub-images” when both porosity and permeability representativeness were considered (Sadeghkhani et al., 22 Oct 2025).

These reconstruction methods differ from classical maximal-ball or watershed workflows in their use of learned or stochastic generators, but they remain morphology-centered. Their objective is not only visual plausibility; it is representativeness with respect to pore-network characteristics, geological context, or transport-relevant descriptors. A plausible implication is that the phrase “pore-morphology method” has broadened from a label for mathematical morphology on binary images to a wider class of methods that generate or constrain morphology by explicit pore-space statistics.

6. Broader measurement frameworks, limitations, and common misconceptions

The morphology-centered perspective also appears in measurement frameworks that do not employ binary morphological operators in the classical sense. In multi-modal X-ray imaging with Hartmann masks, scanning the correlation length and measuring mask-visibility reduction provides “average pore size, relative pore fraction, fractal dimension, and Hurst exponent,” and, because the mask is two-dimensional, the method also resolves “pore size anisotropy” (Zakharova et al., 2021). In q-space NMR diffraction, the pore space function

$B_r$9

is reconstructed from $I_p \circ B_r$0 through phase retrieval based on the hybrid input-output algorithm, the error reduction algorithm, and a dynamically adapting support via shrinkwrap (Demberg et al., 2021). These are not identical to the image-morphology workflows used in porous-media simulation, but they are still morphology methods in the sense that they infer pore shape from morphology-sensitive observables.

A common misconception is that pore morphology alone collapses to porosity. Work on air flow through uncompressed and compressed paper sheets shows otherwise. A pore network model built from $I_p \circ B_r$1-CT data becomes significantly more predictive when combined with “porosity, surface area, median pore radius, and geodesic tortuosity,” and the study explicitly states that “a high correlation among descriptors does not necessarily imply redundancy in a combined prediction” (Gräfensteiner et al., 12 Jun 2025). This is a direct warning against reducing morphology to a single scalar descriptor even when pairwise correlations are strong.

Another misconception is that all pore-morphology methods are intrinsically three-dimensional and broadly transferable. Several of the cited methods are explicitly 2D: the capillary pressure-saturation formulation of (Alonso-Marroquin et al., 13 Jan 2025) is for “two-dimensional pore morphologies,” and the ripening model treats 2D micromodels with a constant out-of-plane curvature term (Alonso-Marroquin et al., 13 Jan 2025, Laku et al., 8 Apr 2026). Conversely, some reconstruction methods are 2D-to-3D only under isotropy assumptions, and generative models such as PCP-GAN are “formation-specific” and trained on a limited set of depths (Sadeghkhani et al., 22 Oct 2025, Danalou et al., 8 Aug 2025). These constraints do not invalidate the methods, but they delimit the regime in which morphology can be translated into property predictions without major bias.

The central limitation across formulations is therefore not morphology per se, but the relation between representation and physics. Binary morphology is efficient, but may omit viscous effects or dynamic interfacial phenomena. Distance-map curvature is efficient for partitioning, but depends on segmentation quality and filtering scale. Generative models can preserve descriptors, but may require formation-specific training. Measurement-based methods can recover pore size and anisotropy, but not necessarily a pore-throat graph. The literature accordingly presents the pore-morphology method not as a replacement for all high-fidelity simulation or imaging, but as a technically diverse class of morphology-centered tools for extracting, predicting, or reconstructing pore-space behavior from geometry.