Probabilistic Modeling and Interpretation of Inaccessible Pore Volume in Polymer Flooding

Abstract

Inaccessible pore volume fraction (IPV) significantly affects polymer transport and retention in porous media during enhanced oil recovery. Conventional methods typically estimate IPV using deterministic or empirical models. These approaches often overlook the randomness in pore and polymer size distributions. This study introduces a probabilistic framework that redefines IPV as a stochastic outcome of size exclusion interactions between polymer molecules and pore throats. Ten mathematically equivalent formulations were developed based on the expectation or event probability logic, and from both polymer- and pore-centered perspectives. All models were analytically verified for consistency. Case studies using representative pore and polymer size distributions (0.1–20 μm and 1–5 μm) confirm that the models yield consistent IPV values across formulations. Sensitivity analysis shows that the results respond to the key parameters, such as the exclusion threshold. The results were computed from probability distributions, weighted based on exclusion rules derived from absolute size values. For instance, increasing the exclusion parameter from 3.0 to 5.0 led to a sharp rise in IPV from 0.0367 to 0.3713. Fundamentally, this framework offers a new perspective. It redefines IPVF as a derived probabilistic quantity governed by physically meaningful size distributions, rather than a fixed empirical input. By decoupling estimation from raw size data and emphasizing distribution-driven computation, the method improves robustness and interpretability and enables integration into uncertainty-aware simulators and data-driven workflows.

1. Introduction

Polymer flooding is widely recognized as a crucial enhanced oil recovery (EOR) technique that improves macroscopic sweep efficiency and moderates fluid mobility within petroleum reservoirs. A fundamental variable that governs its effectiveness is the inaccessible pore volume fraction (IPVF), defined as the portion of pore space that polymer molecules are unable to penetrate due to geometric constraints. Representative and physically meaningful IPVF estimation is critical for polymer selection, injectivity prediction, and simulation-based optimization of flooding strategies [1,2,3,4,5].

The transport of polymer molecules in porous media is intrinsically variable, driven by the polydispersity of polymer chains and the geometric heterogeneity of pore structures. Larger polymer molecules may occasionally traverse wider throats, while smaller ones might still be excluded depending on local pore geometry. This exclusion behavior does not follow a strict deterministic rule; rather, it emerges from a probabilistic interaction between polymer size and pore size distributions. As such, the IPVF is not merely an experimentally calibrated coefficient, but a statistical property rooted in physical variability.

Despite this intrinsic variability, the IPVF has traditionally been estimated using either empirical geometric exclusion models or experimental techniques such as core-flooding. In the latter, polymer solutions are injected through reservoir core samples, and tracer breakthrough curves, pressure drop profiles, or concentration measurements are used to infer polymer retention.

Table 1. Advantages and disadvantages of various IPVF determination methods. Adapted from [6].

Types		Advantages	Disadvantages	References
Tracer-based core flooding	Single-slug method	Simple implementation with a single polymer flood cycle	Potential inaccuracies with unfavorable mobility ratios	[7,8,9,10,11,12]
	Double-slug method	Highly reliable due to repeated flood cycles	Time-intensive due to multiple injection steps	[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]
Tracer-free core flooding	Half concentration method	Streamlined process without tracer necessity	Accuracy compromised in fractured media or large dead volumes	[35,36,37]
Numerical simulation		Avoids laboratory-based determination	Computationally demanding, requires extensive data calibration	[38,39,40,41]
Empirical model		Minimal experimental data needed (one polymer flood cycle and RRF data)	Lack of independent verification	[39,42]
Deterministic exclusion method		Straightforward analytical method	Oversimplification by using a singular polymer size value	[43,44]
Cutoff model		Convenient and user-friendly	Arbitrary nature affecting reliability	[45]

summarizes the strengths and limitations of commonly used IPVF estimation methods. While informative, these techniques capture only a single realization of a highly variable process. There is no guarantee that the resulting IPVF value reflects the statistical mean, the most probable value, or even a representative outcome. This raises important questions about the generality and interpretability of conventional approaches. Furthermore, such methods inherently rely on fixed geometric thresholds or isolated measurements, making them vulnerable to sample-specific variability and poorly suited for uncertainty-aware reservoir simulation. These limitations highlight the need for a methodological shift—from deterministic or semi-empirical calibration to probabilistic modeling that directly incorporates the distributional nature of pore and polymer sizes.

From this perspective, the challenge of estimating the IPVF lies not in numerical precision but in capturing representative behavior. Modeling efforts based solely on deterministic fitting may obscure the underlying stochastic mechanisms of pore accessibility and limit their generalizability across formations and polymer types. Prior efforts in stochastic modeling of transport in porous media, such as the Langevin equation [46], random walk models [47], or macroscopic stochastic dynamics [48], have made significant contributions to understanding post-access dispersion and velocity variation. However, they rarely address the geometric gatekeeping stage that precedes transport. Notably, earlier advances in temporal Markov process modeling have advanced ensemble transport analysis in high-variance conductivity fields, offering valuable tools for quantifying uncertainty and heterogeneity-driven dispersion [49]. However, these models typically operate after pore access is assumed and do not explicitly address the probabilistic nature of the exclusion process that defines the IPVF. In contrast, the present study introduces a stochastic framework that targets this exclusion phase directly, bridging a conceptual gap in pore-scale modeling. The proposed approach emphasizes probabilistic reasoning and distribution-based exclusion logic, enabling physically consistent estimation of the IPVF. This strategy parallels other recent efforts in materials and structural systems, where distribution-informed optimization has been applied to balance thermal and mechanical performance, as in the case of fiber-reinforced adobe composites [50]. Although differing in application, both approaches demonstrate the value of statistically grounded design frameworks in managing complex trade-offs governed by microstructural variability.

This study addresses this gap by proposing a probabilistic framework that redefines the IPVF as a stochastic outcome of particle–pore interactions governed by joint size distributions. Ten mathematically equivalent models were developed using two logical modes (statistical expectation and event probability) and two physical perspectives (polymer-centered and pore-centered). These models accommodated both discrete and continuous inputs and were analytically shown to be internally consistent. By replacing deterministic cutoffs with distribution-based logic, this framework enhances interpretability, supports uncertainty quantification, and facilitates integration with data-driven simulation platforms and machine learning workflows. It provides a statistically robust foundation for modeling size exclusion effects across heterogeneous systems. It also positions the method for use in uncertainty-aware reservoir simulators and modern decision support systems.

Figure 1 outlines the overall modeling logic, from size distribution inputs to probabilistic model families and applications. The remainder of this manuscript is organized as follows. Section 2 introduces the exclusion rule and probabilistic formulation. Section 3 presents the ten equivalent models. Section 4 applies them to synthetic cases, evaluates sensitivity and consistency, and interprets the results and computational implications. Section 5 concludes the study, and Section 6 outlines directions for future research.

2. Probabilistic Foundation

2.1. Randomness in Pore and Polymer Sizes

Sedimentary and geological processes naturally exhibit stochastic characteristics, resulting in reservoir rocks with a wide range of pore sizes (Figure 2). Pore size is a positive random variable typically represented by a probability density function (PDF) (Figure 3) or a probability mass function (PMF). Pore size distributions, often obtained from tests such as mercury intrusion porosimetry, nitrogen physisorption, small-angle neutron scattering, and ultra-small-angle neutron scattering, can be expressed as probability density curves or cumulative distribution curves [17,51,52].

Similarly, polymerization is a stochastic process, producing polymer molecules with various chain lengths or molecular weights (Figure 2). These molecules are described by their hydrodynamic size, which is also a positive random variable with a corresponding probability distribution (Figure 3). A polymer molecule’s hydrodynamic size is directly related to its molecular weight [53,54,55]. Higher molecular weight yields larger hydrodynamic sizes. Each polymer solution has a unique hydrodynamic size distribution due to varied chain lengths from polymerization [56].

Since the IPVF ranges between 0 and 1, it can be mathematically viewed either as the volume probability of pores being inaccessible to polymer particles or the number probability of polymer molecules unable to pass through the pores. Volume probability is the ratio of the inaccessible pore volume to the total pore volume, and the number probability is the ratio of the size-excluded polymer molecules to the total number of molecules. This transformation recasts the IPVF estimation as a problem of probability computation.

Importantly, these probability distributions remain unchanged under uniform scaling of size variables. That is, regardless of whether pore and polymer sizes are expressed in microns or scaled down by two orders of magnitude, the statistical structure of the distribution—and therefore the computed IPVF—remains invariant. This property decouples modeling results from the physical magnitudes of input sizes, reinforcing the probabilistic nature of the IPVF as a distribution-driven rather than magnitude-driven property.

2.2. Size Exclusion Rule

Despite this scale invariance, the physical basis of the IPVF requires a mechanism to distinguish which pores are accessible or not to given polymer sizes. Polymer entry is controlled by the size exclusion ratio (pore diameter/molecule diameter), not by the absolute size. Let two random variables $X$ and $Y$ represent the pore size (diameter) and the molecule size (diameter), respectively. Mathematically, the rule is expressed as $X-aY\le 0$, where $a$ is a variable proportionality constant, typically ranging from 3 to 5 to accommodate various scenarios with or without polymer adsorption [57,58,59,60,61,62]. Understanding this rule is vital for the IPVF estimation, as it influences which pores are considered inaccessible to polymer molecules. The size exclusion rule is integrated into the IPVF calculation models.

This condition is the only place where the absolute size values directly appear in the model; everywhere else, the IPVF is derived from probabilities. In essence, the exclusion rule determines which size combinations contribute to exclusion, but all calculations—expectations or probabilities—are performed on the distributions of $X$ and $Y$, not on their absolute values.

2.3. Probabilistic Interpretation of the IPVF

In traditional practice, the IPVF is treated as a deterministic parameter. However, from a probabilistic standpoint, it is more accurately understood as a stochastic descriptor of size exclusion interactions. This study adopts two mathematical interpretations and two physical perspectives to formalize this view.

2.3.1. Mathematical Standpoint

From a mathematical standpoint, the IPVF can be interpreted in two equivalent ways.

• Expectation-based interpretation

The IPVF is defined as the expected likelihood of exclusion across all particles or pores. Each element has a local probability of exclusion, which is aggregated as a weighted average across the system. Mathematically, this is expressed as the expectation of a probability function derived from the size exclusion condition $X\le aY$.

• Event probability interpretation

The IPVF is defined as the probability that a randomly selected polymer–pore pair satisfies the exclusion condition $X\le aY$. Here, we construct a new random variable $Z=X-aY$ and define the IPVF as the probability P (Z ≤ 0), corresponding to the joint probability that exclusion occurs.

These two interpretations form the theoretical basis for the subsequent modeling formulations, which will later be shown to yield consistent numerical results. These formulations are mathematically equivalent but differ in reasoning paths—one averaging local behavior, the other assessing global event frequency.

2.3.2. Physical Standpoint

From a physical perspective, these two interpretations can also be understood from the following perspectives:

• The polymer particle perspective

Fixing a polymer size $Y=y$ and examining the probability that it will be excluded by pores of varying size. This involves integrating or summating over the pore distribution conditioned on $x\le ay$.

• The pore perspective

Fixing a pore size $X=x$ and evaluating the probability that it will exclude various polymer sizes. This corresponds to evaluating $y\ge x/a$, again, integrated or summed over the polymer distribution.

Together, these formulations constitute a unified framework in which the IPVF is no longer a fixed value but a physically and statistically meaningful outcome of random structure–structure interactions.

3. Model Formulations

3.1. Modeling Logic and Variable Setup

This section presents ten mathematically equivalent probabilistic models for evaluating the IPVF, grounded in the exclusion rule and driven by size distributions. The central innovation of this study is not in producing a specific numerical outcome, but in establishing a fundamentally new probabilistic perspective for characterizing size exclusion phenomena in porous media. This methodology enables a shift in how polymer–pore interactions are conceptualized—transforming the IPVF from a fixed empirical parameter into a probabilistically defined quantity that is both physically meaningful and computationally robust.

Let the pore and polymer molecule diameters be denoted by random variables $X$ and $Y$, respectively. These variables are governed by number probability density functions (nPDFs) or probability mass functions (nPMFs), depending on whether the inputs are continuous or discrete. To account for volume contributions, number-based probabilities for pores were converted into volume-based weighting functions.

Critically, this modeling framework is scale-invariant: the absolute magnitudes of $X$ and $Y$ do not affect the resulting IPVF values, provided their distributions are normalized. That is, shrinking all size values by a factor of 100 preserves the probabilistic structure and results in identical IPVF outcomes. This reinforces that the IPVF is driven by distributional shape, not raw size.

The only point where absolute size enters the modeling process is via the size exclusion rule, which determines whether a given polymer–pore pair is excluded (i.e., when $X\le aY$, for a constant $a$). All subsequent calculations are performed on the probabilistic representations of $X$ and $Y$, meaning that integral/summation operations target probabilities—not raw sizes. In other words, the absolute size is used only to filter which probability terms enter the calculation.

3.1.1. Assumptions and Variable Definitions

In developing the model for estimating the IPVF, we made the following key assumptions and defined the necessary variables:

Cylindrical pore assumption: The pore space within a test sample of rock core consists of parallel cylindrical tubes with varying diameters, each maintaining a specific diameter and the same length $L$. The cylindrical pore assumption provides a mathematically and physically justifiable framework for analyzing porous media. This simplification makes the model tractable and easier to work with, allowing for the IPVF estimation without the complexity of irregular pore geometries. While it may not capture all complexities of real-world pore networks, it serves as a foundational model that can be refined or extended for more complex analyses.

Independence of entry processes: The entry processes of polymer molecules into pores were assumed to be independent random events. This means the exclusion of one molecule from a pore did not affect the exclusion of another molecule.

Pore and molecule size ranges: Let $X$ (pore diameter) and $Y$ (molecule diameter) fall within the ranges $[{\mathrm{L}}_{\mathrm{X}},{\mathrm{U}}_{\mathrm{X}}]$ and $[{\mathrm{L}}_{\mathrm{Y}},{\mathrm{U}}_{\mathrm{Y}}]$ respectively. For continuous variables, the number probability density functions (nPDFs) of pores with size corresponding to $x$ and molecules with size corresponding to $y$ are $f_X\left(x\right)$ and $f_Y\left(y\right)$; for discrete variables, the number probability mass functions (nPMFs) for pore size and molecule size are $p_X\left(x\right)$ and $p_Y\left(y\right)$. Both nPDFs and nPMFs represent the probability associated with the frequency of pores or molecules of specific sizes. These distributions are essential for evaluating the IPVF, whether in discrete or continuous form.

3.1.2. Conversion of Number Probability to Volume Probability

Since the IPVF is related to pore volume fraction, but typically only nPDFs or nPMFs are known, we needed to convert those functions of pore sizes into the corresponding volume probability distributions.

Continuous variables:

For a pore with diameter $x$, the volume of the pore is as follows:

$$v\left(x\right)=\mathsf{\pi }{\left({\displaystyle \frac{x}{2}}\right)}^{2}L$$

(1)

The volume of each pore is weighted by its number probability density $f_X\left(x\right)$. The total volume of all pores, weighted by their number probability density, is as follows:

$${\int }_{{\mathrm{L}}_{\mathrm{X}}}^{{\mathrm{U}}_{\mathrm{X}}}{f}_X\left(x\right)v\left(x\right)\mathrm{d}x={\int }_{{\mathrm{L}}_{\mathrm{X}}}^{{\mathrm{U}}_{\mathrm{X}}}{f}_X\left(x\right)\left[\mathsf{\pi }{\left(x/2\right)}^{2}L\right]\mathrm{d}x$$

(2)

Thus, the volume probability density function (vPDF) for pore diameter $x$ is as follows:

$$f_V\left[v\left(x\right)\right]={\displaystyle \frac{f_X\left(x\right)\left[\mathsf{\pi }{\left(x/2\right)}^{2}L\right]}{{\int }_{{\mathrm{L}}_{\mathrm{X}}}^{{\mathrm{U}}_{\mathrm{X}}}{f}_X\left(x\right)\left[\mathsf{\pi }{\left(x/2\right)}^{2}L\right]\mathrm{d}x}}$$

(3)

The simplified equation is as follows:

$$f_V\left[v\left(x\right)\right]={\displaystyle \frac{f_X\left(x\right)\left(x^2\right)}{{\int }_{{\mathrm{L}}_{\mathrm{X}}}^{{\mathrm{U}}_{\mathrm{X}}}{f}_X\left(x\right)\left(x^2\right)\mathrm{d}x}}={\displaystyle \frac{x^2}{\mathrm{E}\left(x^2\right)}}{f}_X\left(x\right)$$

(4)

where $\mathrm{E}\left(x^2\right)$ is the mathematical expectation of $x^2$.

It should be noted that $f_V\left[v\left(x\right)\right]$ is expressed as a function of $x$ rather than $v$. The reason is that the size exclusion rule involves $x$ when the IPVF is to be calculated.

This expression indicates that the vPDF is a function of the pore diameter $x$, not the volume $v$, because the size exclusion rule depends on $x$ when calculating the IPVF.

Discrete variables:

For discrete variables, the volume of each pore is, similarly, as follows:

$$v\left(x\right)=\mathsf{\pi }{\left({\displaystyle \frac{x}{2}}\right)}^{2}L$$

(5)

The total volume, weighted by the number probability mass function $p_X\left(x\right)$, is as follows:

$${\sum }_{{\mathrm{L}}_{\mathrm{X}}}^{{\mathrm{U}}_{\mathrm{X}}}{p}_X\left(x\right)\left[\mathsf{\pi }{\left({\displaystyle \frac{x}{2}}\right)}^{2}L\right]$$

(6)

Thus, the volume probability mass function (vPMF) for pore diameter $x$ is as follows:

$$p_V\left[v\left(x\right)\right]={\displaystyle \frac{p_X\left(x\right)\left[\mathsf{\pi }{\left(\frac{x}{2}\right)}^{2}L\right]}{{\sum }_{{\mathrm{L}}_{\mathrm{X}}}^{{\mathrm{U}}_{\mathrm{X}}}{p}_X\left(x\right)\left[\mathsf{\pi }{\left(\frac{x}{2}\right)}^{2}L\right]}}$$

(7)

The simplified equation is as follows:

$$p_V\left[v\left(x\right)\right]={\displaystyle \frac{p_X\left(x\right)\left(x^2\right)}{{\sum }_{{\mathrm{L}}_{\mathrm{X}}}^{{\mathrm{U}}_{\mathrm{X}}}{p}_X\left(x\right)\left(x^2\right)}}={\displaystyle \frac{x^2}{\mathrm{E}\left(x^2\right)}}{p}_X\left(x\right)$$

(8)

Again, $p_V\left[v\left(x\right)\right]$ is expressed as a function of $x$, as the size exclusion rule depends on $x$.

3.2. Expectation-Based Models

Each IPVF value measured in an experiment is essentially a realization of a random variable. According to the law of large numbers, the average of many such measurements approximates the expected value. This section defines the IPVF as this expectation and presents two formulations based on different physical perspectives.

3.2.1. From the Perspective of Pores

According to the exclusion rule $X/a\le Y$, a pore of diameter $x$ excludes all polymer molecules larger than $x/a$. The fraction of polymer molecules excluded by a fixed pore size $x$ for a continuous case for the nPDF is as follows:

$$g\left(x\right)={\int }_{X/a}^{{\mathrm{U}}_{\mathrm{Y}}}{f}_Y\left(y\right)\mathrm{d}y$$

(9)

This is then weighted by the volume-based probability density $f_V\left[v\left(x\right)\right]$, as derived in Section 3.1.2. The expected IPVF is as follows:

$$\mathrm{I}\mathrm{P}\mathrm{V}\mathrm{F}=\mathrm{E}\left[g\left(x\right)\right]={\int }_{{\mathrm{L}}_{\mathrm{X}}}^{{\mathrm{U}}_{\mathrm{X}}}{f}_V\left[v\left(x\right)\right]g\left(x\right)\mathrm{d}x$$

(10)

Figure 4a,b visualizes the inner integral and the weighting factor. This two-step integration reflects the fact that exclusion depends probabilistically on polymer size, and the volumetric contribution of each pore must be proportionally represented.

In Figure 4a, when the pore diameter $X$ takes the value $x$, the number probability of polymer molecules with size $Y\ge {\displaystyle \frac{X}{a}}$ is given by $g\left(x\right)={\int }_{{\displaystyle \frac{x}{a}}}^{{\mathrm{U}}_{\mathrm{Y}}}{f}_Y\left(y\right)dy$, represented by the shaded area in this figure. This term represents the probability that a randomly selected polymer molecule (with diameter y) is too large to enter a pore of diameter x.

In Figure 4b, the corresponding volume probability density is $f_V\left[v\left(x\right)\right]$. The volume probability associated with this pore diameter is given by $f_V\left[v\left(x\right)\right]\mathrm{d}x$, which is represented by the area of the shaded region in this figure. This term acts as a weighting factor, giving more weight to larger pores, as volume scales with diameter. The integration process reflects that not all pores contribute equally to the overall flow or storage—larger pores play a more significant role in volume-based calculations such as the IPVF.

Therefore, the IPVF corresponding to pore diameter $x$ is $g\left(x\right)f_V\left[v\left(x\right)\right]\mathrm{d}x$. Integrating this expression over the range of $X$ from $[{\mathrm{L}}_{\mathrm{X}},{\mathrm{U}}_{\mathrm{X}}]$ results in the mathematical expectation of $g\left(x\right)$, which represents the overall IPVF for all pores.

The discrete case for the nPMF for inaccessible pores is as follows:

$$g\left(x\right)={\sum }_{j:y_j\ge x/a}^{\mathrm{N}}{p}_Y\left(y_j\right)=\left(1-{\sum }_{j=1}^{j:y_j\le x/a}{p}_Y\left(y_j\right)\right)$$

(11)

where $i$ and $j$ represent discrete pore and molecule diameter levels, respectively.

The IPVF, as the mathematical expectation of $g\left(x\right)$, is as follows:

$$\mathrm{I}\mathrm{P}\mathrm{V}\mathrm{F}=E\left[g\left(x\right)\right]={\sum }_{i=1}^{\mathrm{M}}{p}_V\left[v\left(x_i\right)\right]g\left(x_i\right)$$

(12)

where $\mathrm{M}$ is the number of pore diameter levels, with $x_1={\mathrm{L}}_{\mathrm{X}}$, and $x_{\mathrm{M}}={\mathrm{U}}_{\mathrm{X}}$; $\mathrm{N}$ is the number of unique molecule diameter values, with $y_1={\mathrm{L}}_{\mathrm{Y}}$, and $y_{\mathrm{N}}={\mathrm{U}}_{\mathrm{Y}}$.

3.2.2. From the Perspective of Molecules

Here, a molecule of diameter $y$ is excluded by all pores with diameter less than $ay$. In a continuous case for the vPDF for a retained molecule, the volume fraction of inaccessible pores for a fixed molecule size is as follows:

$$h\left(y\right)={\int }_{{\mathrm{L}}_{\mathrm{X}}}^{ay}{f}_V\left[v\left(x\right)\right]\mathrm{d}x$$

(13)

This is weighted by the number probability density $f_Y\left(y\right)$. Thus, the expected IPVF is as follows:

$$\mathrm{I}\mathrm{P}\mathrm{V}\mathrm{F}=\mathrm{E}\left[h\left(y\right)\right]={\int }_{{\mathrm{L}}_{\mathrm{Y}}}^{{\mathrm{U}}_{\mathrm{Y}}}{f}_Y\left(y\right)h\left(y\right)\mathrm{d}y$$

(14)

Figure 5a,b illustrates this double integration from the molecule-centered perspective.

In Figure 5a, when the polymer molecule diameter $Y$ takes the value $y$, the volume probability of pores with size $X\le \mathrm{a}Y$ is given by $h\left(y\right)={\int }_{{\mathrm{L}}_{\mathrm{X}}}^{ay}{f}_V\left[v\left(x\right)\right]\mathrm{d}x$, which represents the volume probability of inaccessible pores. The IPVF corresponding to polymer molecule diameter $y$ is $h\left(y\right)f_Y\left(y\right)\mathrm{d}y$.

In Figure 5b, the corresponding number probability density is $f_Y\left(y\right)$, and the number probability for this diameter is represented by $f_Y\left(y\right)\mathrm{d}y$, shown as the shaded area in this figure. This serves as a weighting factor, ensuring that more frequent molecule sizes contribute more significantly to the final IPVF estimate.

Integrating this expression over the range of $Y$ from $[{\mathrm{L}}_{\mathrm{Y}},{\mathrm{U}}_{\mathrm{Y}}]$ results in the mathematical expectation of $h\left(y\right)$, which provides the bulk IPVF for all polymer molecules.

The discrete case for the vPMF for a retained molecule is as follows:

$$h\left(y\right)={\sum }_{i:x_i\le a\mathrm{y}}^{\mathrm{M}}{p}_V\left[v\left(x_i\right)\right]=\left(1-{\sum }_{i=1}^{i:x_i\ge a\mathrm{y}}{p}_V\left[v\left(x_i\right)\right]\right)$$

(15)

The IPVF, as the mathematical expectation of $h\left(y\right)$, is as follows:

$$\mathrm{I}\mathrm{P}\mathrm{V}\mathrm{F}=\mathrm{E}\left[h\left(y\right)\right]={\sum }_{j=1}^{\mathrm{N}}{p}_Y\left(y_j\right)h\left(y_j\right)$$

(16)

3.3. Probability-Based Models

An alternative yet equivalent formulation of the IPVF is to define it as the probability of a random exclusion event, that is, the probability that a randomly selected pore–polymer pair violates the exclusion condition $X\le \mathrm{a}Y$. This interpretation directly evaluates the likelihood that any given interaction results in exclusion.

This approach differs from expectation-based models in logic but not in result. While expectation-based models evaluate local exclusion rates and then weight them by occurrence probabilities, the event probability model integrates the exclusion condition over the joint distribution space.

3.3.1. Function Definition

$Z$ defined as a linear function of $X$ and $Y$ is as follows:

$$Z=X-aY$$

(17)

The principles of the probability theory allow us to derive the PDF of a linear function of two independent random variables. Based on these principles, we can calculate the PDF of Equation (17) for continuous variables $X$ and $Y$.

3.3.2. Continuous Cases

For continuous random variables with joint independence, the PDF of $Z$ is expressed as follows.

When $X$ is the integration variable:

$$f_Z\left(z\right)={\displaystyle \frac{1}{a}}{\int }_{\mathrm{m}\mathrm{a}\mathrm{x}({\mathrm{L}}_{\mathrm{X}},z+\mathrm{a}{\mathrm{L}}_{\mathrm{y}})}^{\mathrm{m}\mathrm{i}\mathrm{n}({\mathrm{U}}_{\mathrm{X}},z+\mathrm{a}{\mathrm{U}}_{\mathrm{Y}})}{f}_V\left[v\left(x\right)\right]f_Y\left({\displaystyle \frac{x-z}{a}}\right)\mathrm{d}x$$

(18a)

When $Y$ is the integration variable:

$$f_Z\left(z\right)={\int }_{\mathrm{m}\mathrm{a}\mathrm{x}({\mathrm{L}}_{\mathrm{y}},{\displaystyle \frac{{\mathrm{L}}_{\mathrm{X}}-z}{a}})}^{\mathrm{m}\mathrm{i}\mathrm{n}({\mathrm{U}}_{\mathrm{Y}},{\displaystyle \frac{{\mathrm{U}}_{\mathrm{X}}-z}{a}})}{f}_Y\left(y\right)f_V\left[v\left(z+ay\right)\right]\mathrm{d}y$$

(18b)

The cumulative distribution function (CDF) for $Z$ is as follows:

$$F_Z\left(z\right)=\mathrm{P}\left(Z\le z\right)={\int }_{{\mathrm{L}}_{\mathrm{X}}-\mathrm{a}{\mathrm{U}}_{\mathrm{Y}}}^z{f}_Z\left(z\right)\mathrm{d}z$$

(19a)

Alternatively, it can be computed in two ways:

$$F_Z\left(z\right)=\mathrm{P}\left(Z\le z\right)={\int }_{{\mathrm{L}}_{\mathrm{Y}}}^{{\mathrm{U}}_{\mathrm{Y}}}{f}_Y\left(y\right)\left({\int }_{{\mathrm{L}}_{\mathrm{X}}}^{z+ay}{f}_V\left[v\left(x\right)\right]\mathrm{d}x\right)\mathrm{d}y,$$

(19b)

or

$$F_Z\left(z\right)=\mathrm{P}\left(Z\le z\right)={\int }_{{\mathrm{L}}_{\mathrm{X}}}^{{\mathrm{U}}_{\mathrm{X}}}{f}_V[v\left(x\right)]\left({\int }_{{\displaystyle \frac{x-z}{a}}}^{{\mathrm{U}}_{\mathrm{Y}}}{f}_Y\left(y\right)\mathrm{d}y\right)\mathrm{d}x$$

(19c)

The IPVF is represented as the volume probability $P\left(Z=X-aY\le 0\right)$, which is equivalent to the CDF evaluated at $Z\le 0$. By setting $Z=0$ in Equations (19a)–(19c) and maintaining $a$ unchanged, we derived models for the IPVF.

For continuous variables, the IPVF model varies depending on the integration variables, bounds, and the order of integration. The following equations represent these variations:

$$\mathrm{I}\mathrm{P}\mathrm{V}\mathrm{F}=F_Z\left(0\right)=P\left(Z\le 0\right)={\int }_{{\mathrm{L}}_{\mathrm{X}}}^{{\mathrm{U}}_{\mathrm{X}}}{f}_V[v\left(x\right)]\left({\int }_{{\displaystyle \frac{x}{a}}}^{{\mathrm{U}}_{\mathrm{Y}}}{f}_Y\left(y\right)\mathrm{d}y\right)\mathrm{d}x,$$

(20a)

$$\mathrm{I}\mathrm{P}\mathrm{V}\mathrm{F}=F_Z\left(0\right)=\mathrm{P}\left(Z\le 0\right)={\int }_{{\mathrm{L}}_{\mathrm{y}}}^{{\mathrm{U}}_{\mathrm{Y}}}{f}_Y\left(y\right)\left({\int }_{{\mathrm{L}}_{\mathrm{X}}}^{ay}{f}_V\left[v\left(x\right)\right]\mathrm{d}x\right)\mathrm{d}y,$$

(20b)

$$\mathrm{I}\mathrm{P}\mathrm{V}\mathrm{F}=F_Z\left(0\right)={\int }_{{\mathrm{L}}_{\mathrm{X}}-\mathrm{a}{\mathrm{U}}_{\mathrm{Y}}}^0{f}_Z\left(z\right)\mathrm{d}z={\displaystyle \frac{1}{a}}{\int }_{{\mathrm{L}}_{\mathrm{X}}-\mathrm{a}{\mathrm{U}}_{\mathrm{Y}}}^0\left({\int }_{\mathrm{m}\mathrm{a}\mathrm{x}({\mathrm{L}}_{\mathrm{X}},z+\mathrm{a}{\mathrm{L}}_{\mathrm{y}})}^{\mathrm{m}\mathrm{i}\mathrm{n}({\mathrm{U}}_{\mathrm{X}},z+\mathrm{a}{\mathrm{U}}_{\mathrm{Y}})}{f}_V[v\left(x\right)]f_Y\left({\displaystyle \frac{x-z}{a}}\right)\mathrm{d}x\right)\mathrm{d}z,$$

(20c)

$$\mathrm{I}\mathrm{P}\mathrm{V}\mathrm{F}=F_Z\left(0\right)={\int }_{{\mathrm{L}}_{\mathrm{X}}-\mathrm{a}{\mathrm{U}}_{\mathrm{Y}}}^0{f}_Z\left(z\right)\mathrm{d}z={\int }_{{\mathrm{L}}_{\mathrm{X}}-\mathrm{a}{\mathrm{U}}_{\mathrm{Y}}}^0\left({\int }_{\mathrm{m}\mathrm{a}\mathrm{x}({\mathrm{L}}_{\mathrm{y}},{\displaystyle \frac{{\mathrm{L}}_{\mathrm{X}}-z}{a}})}^{\mathrm{m}\mathrm{i}\mathrm{n}({\mathrm{U}}_{\mathrm{Y}},{\displaystyle \frac{{\mathrm{U}}_{\mathrm{X}}-z}{a}})}{f}_Y\left(y\right)f_V\left[v\left(z+ay\right)\right]\mathrm{d}y\right)\mathrm{d}z$$

(20d)

These four expressions are mathematically equivalent due to the symmetry of integration and independence of variables. In all cases, the exclusion condition $Z=X-aY\le 0$ determines the integration limits, but the integrands are purely probability densities. Therefore, this formulation, similarly to those in Section 3.2, remains invariant to size scaling.

3.3.3. Discrete Cases

When $X$ and $Y$ are modeled as discrete variables with PMFs $p_V\left[v\left(x_i\right)\right]$ and $p_Y\left(y_i\right)$, the IPVF is either of the following:

$$p_Z\left(z\right)=P\left(Z=X-aY=z\right)={\sum }_{i=1}^M{p}_V\left[v\left(x_i\right)\right]p_Y\left({\displaystyle \frac{x_i-z}{a}}\right),$$

(21a)

Or equivalently:

$$p_Z\left(z\right)=P\left(Z=X-aY=z\right)={\sum }_{i=1}^N{p}_Y\left(y_i\right)p_V\left[v\left(z+a{y}_i\right)\right]$$

(21b)

The CDF for $Z$ in the discrete scenario is either of the following:

$$F_Z\left(z\right)=P\left(Z\le z\right)={\sum }_{i=1}^{\mathrm{M}}\left[p_V[v\left(x_i\right)]{\sum }_{j:y_j\ge {\displaystyle \frac{x_i-z}{a}}}^{\mathrm{N}}{p}_Y\left(y_j\right)\right]={\sum }_{i=1}^{\mathrm{M}}\left[p_V[v\left(x_i\right)]\left(1-{\sum }_{j=1}^{j:y_j\le {\displaystyle \frac{x_i-z}{a}}}{p}_Y\left(y_j\right)\right)\right],$$

(22a)

or:

$$F_Z\left(z\right)=P\left(Z\le z\right)={\sum }_{j=1}^{\mathrm{N}}\left[p_Y\left(y_j\right){\sum }_{i:x_i\le a{y}_j+z}^{\mathrm{M}}{p}_V\left[v\left(x_i\right)\right]\right]={\sum }_{j=1}^{\mathrm{N}}\left[p_Y\left(y_j\right)\left(1-{\sum }_{i=1}^{i:x_i\ge a{y}_j+z}{p}_V\left[v\left(x_i\right)\right]\right)\right]$$

(22b)

The discrete IPVF model varies depending on the integration variables, bounds, and the order of summation. The following equations represent these variations:

$$\mathrm{I}\mathrm{P}\mathrm{V}\mathrm{F}=F_Z\left(0\right)=P\left(Z\le 0\right)={\sum }_{i=1}^{\mathrm{M}}\left[p_V[v\left(x_i\right)]\left(1-{\sum }_{j=1}^{j:y_j\le x_i/a}{p}_Y\left(y_j\right)\right)\right],$$

(23a)

or:

$$\mathrm{I}\mathrm{P}\mathrm{V}\mathrm{F}=F_Z\left(0\right)=P\left(Z\le 0\right)={\sum }_{j=1}^{\mathrm{N}}\left[p_Y\left(y_j\right)\left(1-{\sum }_{i=1}^{i:x_i\ge a{y}_j}{p}_V[v\left(x_i\right)]\right)\right]$$

(23b)

These formulations emphasize that exclusion is determined by the relative sizes of polymers and pores, but the calculation is governed entirely by the probability weights.

3.4. Mathematical Equivalence

This section demonstrates the mathematical equivalence between the expectation-based models (Section 3.2) and the probability-based models (Section 3.3) for estimating the IPVF. Although the ten models differ in form—expectation vs. event probability, pore- vs. polymer-centered, discrete vs. continuous—they are mathematically equivalent.

This equivalence is rooted in foundational principles of the probability theory. All models define the same exclusion event—namely, that a randomly chosen pore–polymer pair satisfies the exclusion condition $X\le \mathrm{a}Y$. Because the variables $X$ and $Y$ are assumed independent, their joint probability distributions can be expressed as products of marginals. Whether integration proceeds from the pore or polymer side, or whether expectation or event probability is used, the same region of the joint probability space is evaluated. By Fubini’s theorem, integration over the same domain yields equivalent results regardless of the order.

The expectation-based model estimates the IPVF as the weighted average of exclusion probabilities, using either pore volume or molecule frequency as the weighting factor. In contrast, the probability-based model defines a new variable $Z=X-aY$ and calculates the IPVF as the cumulative probability $P\left(Z\le 0\right)$, representing the event that a molecule is excluded by a pore.

Though approached from different angles, both modeling strategies reduce to the same integral structure, confirming their mathematical equivalence. This equivalence is not only theoretical—it ensures that the two methods are interchangeable, offering practical flexibility. In implementation, one approach may be more convenient than the other depending on the available data structure (e.g., PMFs vs. PDFs) or computational tools.

In short, the consistency of results across both approaches validates the robustness of the probabilistic framework and supports its application in various computational and reservoir simulation scenarios.

To reinforce this equivalence, we summarize the ten probabilistic models developed in this study. These are organized by input type (continuous vs. discrete) and formulation logic (expectation-based vs. probability-based). As shown in

Table 2. Model classification and pairing.

Formulation Logic	Expectation-BASED		Probability-Based
Input Types	Continuous	Discrete	Continuous	Discrete
Equations	Equations (10) and (14)	Equations (12) and (16)	Equations (20a)–(20d)	Equations (23a) and (23b)
Pairs	Equation (10) ≡ Equation (20a)		${\int }_{{\mathrm{L}}_{\mathrm{X}}}^{{\mathrm{U}}_{\mathrm{X}}}{f}_V[v\left(x\right)]\left({\int }_{{\displaystyle \frac{x}{a}}}^{{\mathrm{U}}_{\mathrm{Y}}}{f}_Y\left(y\right)\mathrm{d}y\right)\mathrm{d}x$
	Equation (14) ≡ Equation (20b)		${\int }_{{\mathrm{L}}_{\mathrm{y}}}^{{\mathrm{U}}_{\mathrm{Y}}}{f}_Y\left(y\right)\left({\int }_{{\mathrm{L}}_{\mathrm{X}}}^{ay}{f}_V[v\left(x\right)]\mathrm{d}x\right)\mathrm{d}y$
	Equation (12) ≡ Equation (23a)		${\sum }_{i=1}^{\mathrm{M}}\left[p_V[v\left(x_i\right)]\left(1-{\sum }_{j=1}^{j:y_j\le x_i/a}{p}_Y\left(y_j\right)\right)\right]$
	Equation (16) ≡ Equation (23b)		${\sum }_{j=1}^{\mathrm{N}}\left[p_Y\left(y_j\right)\left(1-{\sum }_{i=1}^{i:x_i\ge a{y}_j}{p}_V[v\left(x_i\right)]\right)\right]$

, several pairs of equations, derived independently from different perspectives, yield identical expressions. This confirms the internal consistency and completeness of the probabilistic framework.

4. Numerical Evaluation and Model Interpretation

4.1. Synthetic Case Design and Input Configuration

To evaluate the internal consistency, numerical stability, and scale-invariance of the proposed probabilistic models, nine synthetic test cases were constructed. These cases were designed not to replicate specific experimental conditions but to facilitate structured analysis using analytically defined inputs. Specifically, we examined (1) consistency across model formulations; (2) robustness under variation in input distribution shapes and sampling intervals; and (3) sensitivity to the exclusion parameter $a$.

Although direct benchmarking against experimental data would be ideal, most published studies did not provide detailed pore and polymer size distributions. As such, this study relied on synthetic distributions that are analytically defined and systematically varied.

For a real-valued random variable with a continuous normal distribution, the general form of its PDF is as follows:

$$\mathrm{\varnothing }\left(x\right)={\displaystyle \frac{1}{\sqrt{2\mathsf{\pi }}\mathsf{\sigma }}}{\mathrm{e}}^{-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{x-\mathsf{\mu }}{\mathsf{\sigma }}}\right)}^2}$$

(24)

Its CDF is as follows:

$$\Phi \left(b\right)={\int }_{-\infty }^{\mathrm{b}}\mathrm{\varnothing }\left(x\right)\mathrm{d}x$$

(25)

Each case treated pore diameter $X$ and polymer size $Y$ as independent random variables drawn from truncated normal distributions. The distributions were truncated over physically realistic ranges to ensure valid support: $[0.1,20]$ μm for $X$ and $[1.0,5.0]$ μm for polymers. These intervals were chosen to reflect typical finite bounds observed in real-world porous media and polymer systems. In practical characterization, such size ranges often arise due to physical limitations of pore geometry and molecular configuration.

For the truncated discrete normal distribution for pore size and molecule size (Cases #1, #2, #3 and #4), the PMF is as follows:

$$p_X\left(x_i\right)={\displaystyle \frac{\Phi \left(x_i\right)-\Phi \left(x_{i-1}\right)}{\Phi \left({\mathrm{U}}_{\mathrm{X}}\right)-\Phi \left({\mathrm{L}}_{\mathrm{X}}\right)}}$$

(26)

For a truncated continuous normal distribution for pore size and molecule size (Cases #5, #6, #7, #8 and #9), the PDF is as follows:

$$f_X\left(x\right)={\displaystyle \frac{\mathrm{\varnothing }\left(x\right)}{\Phi \left({\mathrm{U}}_{\mathrm{X}}\right)-\Phi \left({\mathrm{L}}_{\mathrm{X}}\right)}}$$

(27)

In the discrete cases (Cases #1–#4), the truncated distributions were sampled at regular intervals. For $X$, the mean and the standard deviation were fixed at ${\mu }_X$ = 12.0 μm and ${\sigma }_X$ = 2.0 μm, with sampling intervals of 0.10 or 0.20 μm. For $Y$, the mean was fixed at ${\mu }_Y$ = 2.0 μm, the standard deviation—at ${\sigma }_Y$ = 1.0 μm, and sampling intervals were set to either 0.05 or 0.10 μm.

In the continuous cases (Cases #5–#9), $X$ was assigned with ${\mu }_X$ = 12.0 μm and ${\sigma }_X$ = 2.0 μm. $Y$ varied across combinations of ${\mu }_Y$ = 1.5, 2.0, 2.5 μm and ${\sigma }_Y$ = 0.5, 1.0, 1.5 μm to probe sensitivity to distribution shape.

In both continuous and discrete cases, the probability distributions were normalized after truncation to maintain the total probability equal to one.

These parameter variations allowed us to isolate the influence of both distribution shape and sampling resolution on the IPVF estimation. Specifically, changing the mean and the standard deviation of $Y$ helped evaluate model sensitivity to polymer size distribution characteristics, while testing different discretization intervals enabled analysis of numerical stability and resolution dependence. These setups formed the basis for the comparative results discussed in the next section.

Table 3. Features of synthetic cases.

Cases	Types of Variables	$\mathit{X}\in \left[0.1,20\right]$ μm		$\mathit{Y}\in [1,5]$ μm
#1	Discrete distribution Truncated normal distribution	${\sigma }_X$ = 2.0 ${\mu }_X$ = 12.0	Interval = 0.10	${\sigma }_Y$ = 1.0 ${\mu }_Y$ = 2.0	Interval = 0.05
#2					Interval = 0.10
#3			Interval = 0.20		Interval = 0.05
#4					Interval = 0.10
#5	Continuous distribution Truncated normal distribution	${\sigma }_X$ = 2.0 ${\mu }_X$ = 12.0		${\sigma }_Y=1.0$; ${\mu }_Y=2.0$
#6				${\sigma }_Y=0.5$; ${\mu }_Y=2.0$
#7				${\sigma }_Y=1.5$; ${\mu }_Y=2.0$
#8				${\sigma }_Y=1.0$; ${\mu }_Y=1.5$
#9				${\sigma }_Y=1.0$; ${\mu }_Y=2.5$

summarizes the distribution parameters and sampling intervals used in all the nine cases. Figure 6 visualizes the corresponding input distributions. Panels (a)–(d) represent discrete distributions used in Cases #1–#4, each with different sampling intervals. Panels (e)–(i) show the continuous distributions for Cases #5–#9, highlighting variations in distribution width and central tendency across test scenarios.

4.2. Model Results, Sensitivity, and Practical Insights

Table 4. IPVF values calculated for the discrete cases (${\sigma }_{\mathrm{X}}$ = 2.0, ${\mu }_{\mathrm{X}}$ = 12.0, ${\sigma }_{\mathrm{Y}}$ = 1.0, ${\mu }_{\mathrm{Y}}$ = 2.0).

Cases	Sampling Intervals		a = 3.0		a = 5.0
	X	Y	Equations (12)/(23a)	Equations (16)/(23b)	Equations (12)/(23a)	Equations (16)/(23b)
#1	0.10	0.05	0.0367	0.0367	0.3713	0.3713
#2	0.10	0.10	0.0392	0.0392	0.3832	0.3832
#3	0.20	0.05	0.0356	0.0356	0.3676	0.3676
#4	0.20	0.10	0.0380	0.0380	0.3794	0.3794

,

Table 5. IPVF values calculated for the continuous cases with the same ${\mu }_2$ (${\sigma }_{\mathrm{X}}$ = 2.0, ${\mu }_{\mathrm{X}}$ = 12.0).

Cases	${\mathit{\sigma }}_{\mathit{Y}}$	${\mathit{\mu }}_{\mathit{Y}}$	a = 3.0				a = 5.0
			Equations (10)/(20a)	Equations (14)/(20b)	Equation (20c)	Equation (20d)	Equations (10)/(20a)	Equations (14)/(20b)	Equation (20c)	Equation (20d)
#5	1.0	2.0	0.0359	0.0358 *	0.0359	0.0359	0.3685	0.3685	0.3685	0.3685
#6	0.5	2.0	0.0034	0.0034	0.0034	0.0034	0.2065	0.2065	0.2065	0.2065
#7	1.5	2.0	0.0907	0.0894 *	0.0907	0.0907	0.4740	0.4740	0.4740	0.4740

* The source of differences in numerical value lies in numerical integration.

and

Table 6. IPVF values calculated for the continuous cases with the same ${\sigma }_2$ (${\sigma }_{\mathrm{X}}$ = 2.0, ${\mu }_{\mathrm{X}}$ = 12.0).

Cases	${\mathit{\sigma }}_{\mathit{Y}}$	${\mathit{\mu }}_{\mathit{Y}}$	a = 3.0				a = 5.0
			Equations (10)/(20a)	Equations (14)/(20b)	Equation (20c)	Equation (20d)	Equations (10)/(20a)	Equations (14)/(20b)	Equation (20c)	Equation (20d)
#5	1.0	2.0	0.0359	0.0358 *	0.0359	0.0359	0.3685	0.3685	0.3685	0.3685
#8	1.0	1.5	0.0160	0.0160	0.0160	0.0160	0.2437	0.2437	0.2437	0.2437
#9	1.0	2.5	0.0746	0.0742 *	0.0746	0.0746	0.5209	0.5209	0.5209	0.5209

* The source of differences in numerical value lies in numerical integration.

report the IPVF values computed for the nine synthetic cases using different probabilistic models. The results confirm that all the ten model formulations introduced in Section 3 produce consistent outputs under matched input conditions. These outputs also provide a clear demonstration of the framework’s internal consistency, numerical robustness, and scale invariance.

4.2.1. Validation of Model Consistency (Discrete)

In Table 4, discrete models (Equations (12), (16), (23a) and (23b)) were applied to Cases #1–#4, which differed in sampling intervals for both pore diameter $X$ and polymer size $Y$. Each case was evaluated under two exclusion parameters: a = 3.0 and a = 5.0. Across all cases and equations, the computed IPVF values agreed exactly to four decimal places, confirming the consistency among the mathematically equivalent discrete formulations.

Minor variations in the IPVF values (ranging from 0.0356 to 0.0392 for a = 3.0 and from 0.3676 to 0.3832 for a = 5.0) were due entirely to changes in sampling intervals and input distribution configurations, not to modeling inconsistencies. Importantly, these results confirm that resolution changes—particularly between 0.1 μm and 0.2 μm—have negligible influence on the IPVF when input distributions are properly normalized.

4.2.2. Validation of Model Consistency (Continuous)

Table 5 and Table 6 summarize the IPVF estimates for Cases #5–#9, evaluated using the continuous versions of the models (Equations (10), (14), and (20a)–(20d)). As expected, all the formulations yielded nearly identical results for each test scenario. Minor differences (marked with an asterisk) are attributed to numerical integration rather than theoretical discrepancies.

For example, in Case #5, all four equations returned an IPVF value of approximately 0.0359 with a = 3.0 and 0.3685 with a = 5.0, confirming consistency across model forms.

The observed match across different formulations, again, validates the mathematical equivalence discussed in Section 3.4. This agreement demonstrates that users may select among these models based on data availability and computational convenience without sacrificing accuracy.

4.2.3. Sensitivity to Input Distribution and Exclusion Threshold

The probabilistic IPVF framework was designed to capture how the statistical structure of polymer and pore size distributions influences exclusion outcomes. To assess this sensitivity, we systematically varied both the mean (${\mu }_Y$) and the standard deviation (${\sigma }_Y$) of the polymer size distribution while keeping pore characteristics constant.

Table 5 presents the results for Cases #5–#7, where the mean polymer size was fixed and the standard deviation varied from 0.5 to 1.5 μm. The IPVF increased with greater ${\sigma }_Y$, as wider distributions led to a larger fraction of molecules exceeding the exclusion threshold defined by $X\le \mathrm{a}Y$. For example, with a = 3.0, the IPVF increased from 0.0034 to 0.0907, and from 0.2065 to 0.4740 for a = 5.0. This confirms the framework’s ability to reflect the physical reality that broader molecular distributions result in more frequent size-based exclusion.

Table 6 extends the analysis to variations in the mean polymer size while keeping the standard deviation fixed. As ${\mu }_Y$ increased from 1.5 to 2.5 μm, the computed IPVF increased correspondingly—again demonstrating the expected behavior. The change was particularly notable at higher exclusion thresholds: for a = 5.0, the IPVF increased from 0.2437 to 0.5209 across the three cases. These results show that the models responded consistently to physically relevant shifts in size distribution parameters, further validating the statistical basis of the exclusion rule.

Figure 6a–d further illustrates how discretizing the polymer size distribution at different sampling intervals affects the structure of the resulting probability mass functions (PMFs). Although both sets of distributions were derived from the same underlying truncated normal form, the coarser sampling interval of 0.10 μm in panel (d) introduced more pronounced stepwise changes and a reduced smoothness compared to the finer 0.05 μm resolution in panel (a). These visual differences reflect how resolution choices can alter the apparent shape of input distributions, particularly near exclusion thresholds.

While the computed IPVF values across these cases remained numerically consistent—differing by less than 0.005 in our tests—this example highlights a key practical consideration. Coarser discretization may inadvertently introduce local probability spikes that affect exclusion logic, especially when polymer and pore sizes are close in magnitude. In extreme cases, this may lead to non-monotonic variations in the IPVF when exclusion parameters are varied or when distribution overlaps become sensitive to bin placement (i.e., how size intervals are defined and aligned during discretization). Thus, although the framework is robust to moderate resolution changes, careful selection of sampling intervals is recommended to preserve the fidelity of input distributions in discrete modeling contexts.

4.2.4. Physical Interpretation of Simulation Results

Beyond numerical agreement, the simulation results also offered physically meaningful insights into the nature of polymer transport in porous media. First, both the central tendency and spread of the polymer size distribution were shown to substantially influence the IPVF. This reinforces the premise that polymer–pore interactions must be considered in terms of joint geometric compatibility, not as independent attributes.

Second, the exclusion parameter $a$ serves as a governing control on the strength of size-based screening. Larger values of $a$ raise the entry threshold, thereby excluding a broader portion of the polymer population and leading to higher IPVF values. This behavior aligns with expectations derived from pore-scale filtration principles and geometric exclusion theory.

Importantly, these physical interpretations hold consistently across all the probabilistic formulations tested—whether expectation-based or event-based, and whether centered on pores or polymer molecules. The results confirm that the IPVF can be understood as a volume-weighted probability of exclusion, not as a deterministic function of size. This distinction strengthens the probabilistic framework’s relevance for modeling transport under uncertainty, where size distributions—not single values—govern accessibility outcomes.

To further illustrate this effect at the field scale, simulation results using a CMG STARS template indicated that when the IPVF is zero, polymer flooding improved oil recovery by 20.1%. However, when the IPVF increased to 0.5, the recovery improvement dropped to 17.3%, highlighting the adverse impact of limited pore accessibility on polymer effectiveness.

4.2.5. Practical Implementation and Model Recommendations

The probabilistic framework is adaptable to various input formats and application settings. When polymer and pore size distributions are only available as empirical datasets, discrete models—particularly those in Equations (12), (16), and (23a,b)—offer simplicity and numerical stability, provided the sampling intervals are chosen with care. It is advisable to maintain similar resolution for both pore and polymer variables to avoid discretization bias.

When analytical distributions are available, continuous models are preferable due to their independence from sampling resolution. The exclusion parameter $a$, which defines the geometric cutoff for accessibility, plays a critical role in shaping the IPVF outcomes. While this study used values in the typical range from 3.0 to 5.0, a mid-range value of a = 4.0 is recommended as a practical default. The precise physical basis for this parameter—whether from polymer conformation, pore topology, mechanics, or chemical interactions—remains an important topic for future research. Overall, the framework’s theoretical rigor, numerical robustness, and interpretability support its integration into practical polymer flooding design, optimization, and uncertainty quantification workflows.

These results collectively demonstrate that the proposed probabilistic framework is not only mathematically rigorous and physically interpretable, but also robust under various input conditions and adaptable for practical implementation. Its consistent performance across formulations, resolution levels, and parameter variations supports its integration into modern polymer flooding simulations and uncertainty-aware reservoir modeling workflows.

5. Conclusions

This study introduces a probabilistic modeling framework for evaluating the inaccessible pore volume fraction (IPVF) in polymer flooding, treating it as a statistical consequence of size exclusion between polymer molecules and pore throats. The approach departs from traditional deterministic or semi-empirical models by emphasizing probability distributions over absolute size values.

Ten mathematically equivalent formulations were developed, derived from expectation-based and event-probability logic, and viewed from both pore- and polymer-centered perspectives. These formulations were analytically shown to be consistent and were numerically validated across nine synthetic test cases.

The models demonstrated strong internal consistency, producing identical results across all formulations under matched input distributions. They remained stable across both discrete and continuous representations, as well as under variations in sampling intervals. Sensitivity analyses confirmed the influence of the polymer size distribution’s mean and standard deviation and the exclusion parameter on the IPVF outcomes. Importantly, the results were invariant under uniform size scaling, reinforcing that model outputs are governed by distribution shape rather than absolute dimensions.

This framework offers several advantages. It decouples the IPVF from raw measurements, provides an interpretable physical insight, and supports integration into data-driven and uncertainty-aware reservoir modeling workflows. More broadly, it establishes a conceptual shift—representing the IPVF as a volume-weighted probability of exclusion rather than a deterministic threshold—which may have implications for modeling other particle–pore transport phenomena in porous media.

6. Future Work

Future efforts should focus on extending and applying the proposed framework in more complex and realistic scenarios. The key directions include:

• Validating model predictions using experimental datasets with measured pore and polymer size distributions;
• Investigating the physical basis and calibration of the exclusion factor;
• Extending the framework to hierarchical or network-based pore systems;
• Coupling with additional transport mechanisms such as adsorption or rheological effects;
• Integrating probabilistic IPVF formulations into machine learning workflows for predictive modeling.