New Insight of Nanosheet Enhanced Oil Recovery Modeling: Structural Disjoining Pressure and Profile Control Technique Simulation

Abstract

This study presents a novel Enhanced Oil Recovery (EOR) method using Smart Black Nanocards (SLNs) to mitigate the environmental impact of conventional thermal recovery, especially under global warming. Unlike prior studies focusing on wettability alteration via adsorption, this research innovatively models ‘oil film detachment’ in a reservoir simulator to achieve wettability alteration. Using the CMG-STARS (2020) simulator, this study highlights SLNs’ superior performance over traditional chemical EOR and spherical nanoparticles by reducing residual oil saturation and shifting wettability toward water-wet conditions. The structural disjoining pressure (SDP) of SLNs reaches 20.99 × 10³ Pa, 16.5 times higher than spherical particles with an 18.5 nm diameter. Supported by the Percus–Yevick (PY) theory, the numerical model achieves high accuracy in production history matching, with oil recovery and water cut fitting within precision error ranges of 0.02 and 0.05, respectively. This research advances chemical EOR technologies and offers an environmentally sustainable, efficient recovery strategy for low-permeability and heavy oil reservoirs, serving as a promising alternative to thermal methods.

1. Introduction

In the context of global warming, selecting environmentally sustainable Enhanced Oil Recovery (EOR) technologies has become increasingly critical. Conventional thermal recovery methods often exacerbate greenhouse gas emissions, presenting an urgent need for alternative solutions. Nanofluid-based EOR has emerged as a promising approach, offering the potential to enhance oil recovery by increasing surface activity and reducing oil viscosity in non-thermal recovery processes [1]. This is particularly relevant for ultra-low-permeability reservoirs, where conventional chemical flooding agents are ineffective [2,3,4,5]. However, existing studies lack a unified framework for simulating nanofluid behavior in reservoirs. Addressing this gap, our study provides a novel perspective on nanofluid modeling and its practical applications.

An oil film forms when oil droplets spread across the rock surface [6], creating challenges for oil displacement due to its lipophilic nature. This adhesion alters the sign of capillary pressure, transforming the waterflooding direction into resistance, and contributes to a boundary layer effect that increases the flow capacity of oil and water phases [7]. The interaction between polar substances on the oil droplet surface and the rock surface’s van der Waals and electrostatic forces underpins this behavior, making wettability alteration a key focus of EOR research [8,9,10,11,12]. Altering the wettability of rock surfaces has become a significant topic in EOR research, spanning both the 20th and 21st centuries.

Numerous studies confirm that nanofluids can alter wettability and enhance flow capacity in porous media through three main mechanisms [5,13,14,15,16]:

• Adsorption: Nanoparticles adsorb onto the reservoir surface, exposing hydrophilic groups to the water phase, modifying rock wettability.
• IFT Reduction: Surface-active nanoparticles lower IFT between oil and water, reducing capillary resistance and enhancing water-wet conditions.
• SDP: Nanoparticles self-assemble into wedge-shaped structures in the COBR three-phase contact zone, generating forces that detach oil films and mitigate oil-wet characteristics.

While the first two mechanisms align closely with surfactant behavior, the third represents a unique attribute of nanoparticles, particularly nanosheets like Smart Black Nanocards (SLNs). SLNs are an innovative nanosheet material independently developed by our team for EOR applications. These materials have amphiphilic properties derived from the functional modification of molybdenum disulfide nanosheets. SLNs, approximately 60 nm in size and 1.2~1.5 nm thick, exhibit curling behavior under hydrodynamic conditions, reducing their apparent size [17,18,19]. Unlike surfactants, SLNs have minimal adsorption on reservoir rocks, with adsorption rates 100~1000 times lower than conventional systems [17]. Moreover, SLNs do not significantly reduce the IFT between oil and water. At 0.005 wt%, they reduce IFT to only 10⁻¹ mN/m, far from the ultra-low IFT levels (10⁻³ mN/m) traditionally required for effective oil displacement. SP/ASP and nanofluids, representing different mechanisms for altering wettability, exhibit varying performance parameters during the oil displacement process, as shown in

Table 1. Comparison of performance parameters: SP/ASP vs. nanofluids in wettability alteration.

Type	SP/ASP	Nanofluid
IFT	➢ Concentration: ≥0.1 wt% ➢ Additive: 0.8~1.2 wt% alkali or co-surfactant ➢ IFT: ≤10−3 mN/m	➢ Concentration: ≤0.05 wt% ➢ Additive: no or little surfactant added ➢ IFT: 10−1~101 mN/m
Wettability Alteration	➢ Principle: adsorption ➢ Adsorption: hydrophobic hydrocarbon chains adsorb on active substances on the rock surface, with the head groups exposed to the water phase	➢ Principle: oil film detachment (SDP) ➢ SDP: a wedge film is formed at the junction of three phases to generate SDP to peel off the oil film
Critical Capillary Number	10−2~10−1	10−6~10−5
Formation

.

In our prior research, we conducted experimental and field studies to evaluate SLNs’ performance in EOR. These studies demonstrated that SLNs could significantly reduce residual oil saturation without relying on high capillary numbers. For instance, field-scale experiments showed that nanofluids efficiently enhanced recovery without requiring ultra-low IFT, a finding supported by Qu’s laboratory experiments on MoS₂ nanofluids [7]. Zhang reached a similar conclusion in their study, where they found that, when using SiO₂ nanofluid for oil displacement, a higher displacement rate and capillary number led to a reduction in the effectiveness of nanoparticle-based oil displacement [20]. However, while these studies highlighted the potential of SLNs in oil displacement, they did not incorporate numerical simulations, leaving a significant gap in understanding their interactions with reservoir fluids.

The classical DLVO theory suggests that interactions between confined surfaces are the result of the combined effects of electrostatic forces and van der Waals dispersion forces [21]. However, the emergence of new studies on micro–nano confinement has revealed additional microscopic forces at the surface, leading to the inapplicability of the DLVO theory for thin films and confined particles. The PY theory, which models particle interactions in fluids, offers a more reliable approach in this context. The PY theory accounts for structural forces that are not captured by the DLVO theory, particularly in high-density systems, where repulsive forces give rise to oscillatory interactions. Unlike the DLVO theory, which is primarily applicable to larger colloidal particle systems, the PY theory is better suited to describe interactions in systems involving smaller particles, such as the nanoparticles used in this study. This makes the PY theory a more appropriate framework for explaining how nanoparticles in nanofluids influence reservoir wettability through the generation of structural forces that modulate the oil–water–rock interactions [22]. Extending PY theory, this study hypothesizes that SLNs alter wettability through SDP and the ‘peeling force’ generated at the three-phase contact area.

This study marks a significant advancement by introducing SDP and FIE as critical parameters within the CMG-STARS reservoir simulator, enabling the precise emulation of oil displacement driven by sheet-like nanofluids. While past research has focused on wettability alteration and IFT reduction, our work pioneers the incorporation of SDP and FIE to capture the nuanced interactions between nanosheets and reservoir fluids. Using this methodology, we successfully mimic oil displacement behaviors across varying permeability ranges and nanofluid concentrations, validating FIE as a robust interpolation factor. This innovative framework achieves precise oil displacement history matching, with an accuracy of ±0.02 in oil recovery and ±0.05 in water cut. Additionally, it enhances our understanding of how sheet-like nanofluids influence phase permeability and flow dynamics, providing a foundation for advanced experimental and field-scale applications in EOR processes, particularly for low-permeability and heavy oil reservoirs.

2. Preparation and Calculation

This study is organized into two main sections. In Section 1, we will analyze the sensitivity of the simulator, while in Section 2, we will simulate the coreflooding process based on laboratory experiments on relative permeability using the simulator. The methodology flowchart is provided in Figure 1.

2.1. Structural Disjoining Pressure

Nanofluids are defined as a dispersion system in which a liquid suspends nanoparticles, with these nanoparticles serving as the dispersed phase and the suspending medium referred to as the dispersant. When nanoparticles are introduced into the dispersant, sedimentation and agglomeration can occur within the system. The components of this dispersion system exhibit vastly different geometric scales, including bubbles or droplets that are much larger than the nanoscale colloidal particles or micelles in dispersion, while the nanoscale colloids themselves are significantly larger than the atoms or molecules of the liquid solvent in which they are dispersed. In this context, the surfaces of larger particles impose slit-like constraints on the system of smaller particles.

The stability of fluid films is influenced by various forces acting between these films. With advancing research, additional types of microscopic forces have been identified on surfaces, generally termed structural and dissipative forces. The origin of these forces lies in the difference in normal stresses applied to the large particle ‘film’ and ‘bulk’ materials. When the distance between particle surfaces is sufficiently large—that is, the thickness of the large particle film is considerable, at the scale of several confined particle diameters—the pressure in the ‘film’ and ‘bulk’ regions becomes equal, rendering the structural force negligible. As the film thickness reduces to less than a few confined particle diameters, the normal pressure component in the confined area (film side) begins to shift, leading to the development of a structural force opposing the ‘bulk’ side. Essentially, this structural force arises from particle stratification within the film. According to Trokhymchuk’s theoretical analysis of this model, the structural force per unit area between films decreases with increasing gap distance, combining a decreasing cubic function (for $0<H<d$) and an oscillatory function (for $H\ge d$). When $0<H<d$, meaning the effective diameter of the small spherical particles is less than the slit length, the structural force and energy needed to separate large spherical bodies are defined by cubic functions, as presented in Equations (1) and (2) [22]:

$$\mathsf{\Pi }\left(H\right)=\rho kT{\displaystyle \frac{1+\varnothing +{\varnothing }^2-{\varnothing }^3}{{\left(1-\varnothing \right)}^3}}$$

(1)

$$W\left(H\right)=\rho kT{\displaystyle \frac{1+\varnothing +{\varnothing }^2-{\varnothing }^3}{{\left(1-\varnothing \right)}^3}}\left(d-H\right)-2\sigma$$

(2)

where, $\mathsf{\Pi }\left(H\right)$ is referred to as the SDP, measured in dyn/cm², $W\left(H\right)$ is known as the FIE, measured in dyn/cm, $\rho$ represents the particle number density, in particles/cm³, $k$ is the Boltzmann constant, equal to 1.380649 × 10⁻²³ J/K, $T$ denotes the temperature, in Kelvin, and $\mathrm{\varnothing }$ denotes the volume fraction of spherical particles.

When $H\ge d$, meaning the effective diameter of arranged small spherical particles is greater than the length of the slit, the expressions for the SDP and the FIE required to separate large spherical bodies are given by oscillatory Equations (3) and (4), respectively [22]:

$$\mathsf{\Pi }\left(H\right)={\mathsf{\Pi }}_0\cos \left(\omega H+{\phi }_1\right)e^{-\kappa H}+{\mathsf{\Pi }}_1{e}^{-\delta \left(H-d\right)}$$

(3)

$$W\left(H\right)=W_0\cos \left(\omega H+{\phi }_2\right)e^{-\kappa H}+W_1{e}^{-\delta \left(H-d\right)}$$

(4)

Here, ${\mathsf{\Pi }}_0\cos \left(\omega H+{\phi }_1\right)e^{-\kappa H}$ and $W_0\cos \left(\omega H+{\phi }_2\right)e^{-\kappa H}$ represent the asymptotic behavior as $H$ tends to infinity, denoted as ${\mathsf{\Pi }}^{PY}\left(H\to \infty \right)$ and $W^{PY}\left(H\to \infty \right)$ respectively. The subscript ‘PY’ indicates that the SDP and FIE are the unit area interaction results approximated by the PY theory. ${\phi }_1$, ${\phi }_2$, ${\mathsf{\Pi }}_0$, $W_0$, $\omega$ and $\kappa$ are parameters contributing to the calculation of SDP and FIE. ${\mathsf{\Pi }}_1{e}^{-\delta \left(H-d\right)}$ and $W_1{e}^{-\delta \left(H-d\right)}$ represent the correction terms for SDP and FIE, where ${\mathsf{\Pi }}_1$ and $W_1$ are correction amplitude coefficients, and δ is the correction term exponent.

The ‘peeling force’ generated by SLNs in the three-phase contact region of the COBR system can be utilized to further interpret the PY theory. Within this system, crude oil (oil film) and the rock surface can be conceptualized as two large particles with infinite radii, while the nanoparticles have effective diameters considerably smaller than the ‘diameters’ of the crude oil and rock surfaces. Driven by interfacial activity, nanoparticles are spontaneously transported to the confined region formed between the crude oil and rock surfaces, creating a restricted fluid zone in this area (see Figure 2). Wasan refers to this region as the ‘wedge film’ [23].

According to the PY theory, the magnitude of SDP is influenced by the volume fraction and particle size of the nanofluids. Kondiparty examined the behavior of wedge films under varying volume percentages and nanoparticle sizes and conducted simplified calculations on the magnitude of SDP [24]. As nanoparticle size decreases, the wedge region accessible to nanoparticles becomes thinner, resulting in increased SDP and FIE. Notably, a higher volume fraction of nanofluids does not always yield optimal results. In Kondiparty’s experiments [24], an initial nanofluid volume fraction of 30.0 vol% led to wedge formation. However, increasing the volume fraction to 31.4 vol% caused the spreading coefficient of crude oil on the rock surface to reach zero, indicating the detachment of the oil film. Upon further increasing the volume fraction, the ability of nanofluids to restrict oil film spreading on the rock surface was diminished.

The appropriate thickness of nanoparticles is critical in influencing the permeability of crude oil. A compilation and fitting of various studies on SDP across different nanoparticle sizes reveal an exponential increase in SDP as particle size decreases. Kumar reached a similar conclusion, indicating that smaller particles lead to higher SDP and more significant meniscus displacement in nanofluids with a fixed volume fraction [25].

The reduction in nanoparticle size enables them to penetrate thinner sections of the wedge film, which primarily contributes to the notable increase in SDP. As illustrated in Figure 3, a top-view observation of nanofluid spreading across the rock surface reveals two distinct contact lines due to droplet contraction. During contact angle measurements, these two contact lines become visible: the outer contact line marking the boundary of the oil droplet, and the inner contact line. The micro-wedge region forms between these two lines due to nanofluid penetration [16,26]. Consequently, when measuring contact angles in a COBR system with nanofluid-induced wettability alteration, the measurement only reflects the phase behavior at the outer contact line. Although methods such as electron microscopy, fluorescence microscopy, X-ray imaging, and micro-CT have been employed to capture contact lines and wedge regions, and the Frumkin-Dejaguin approach has been used to estimate contact angles at the inner contact line [27], an accurate technique for directly measuring the inner contact line angle is still unavailable.

Typically, spherical nanoparticles require a minimum concentration of 0.8 to 1.0 wt% to form wedge films, though some studies suggest that they can do so at even lower concentrations [28,29]. While most SDP research has focused on spherical nanoparticles, our group’s studies and field applications of SLNs over recent years reveal that, in comparison to spherical particles, SLNs can form wedge films at lower concentrations, penetrate further within the wedge region, and deliver superior EOR effects [17,18]. Consequently, both the morphology of the wedge region and the contact angle at the inner contact line are influenced by nanoparticle shape. Figure 4 illustrates the dispersion states of two types of nanomaterials in a wedge film. Due to the ‘sheet-like’ structure and flexibility of SLNs, they can bend during flow, reducing their effective radius, which allows them to assemble closer to the vertex in the three-phase contact region than spherical nanoparticles. Using Equations (1) and (3), the SDP distribution for SLN fluids with an effective diameter of 1.3 to 1.5 nm was calculated and compared to that of 18.5 nm spherical nanoparticles [24]. SLNs with thinner layers exhibit an SDP at the vertex that is 16.5 times greater than that of spherical nanoparticles (Figure 5).

This phenomenon can also be interpreted through the potential difference generated in the wedge film region and the theory of the diffuse double layer, as the nanofluid’s behavior of peeling the oil film is spontaneous. When nanoparticles with opposite charges converge at the three-phase contact region of COBR, they form a wedge film that creates an SDP gradient. This gradient effectively removes the oil film. The thickness of the wedge film is influenced by the arrangement of charged nanoparticles. With more layers, the wedge film holds more charges, enhancing its capacity to neutralize charges on the rock and crude oil surfaces, leading to a higher degree of apparent electrical neutrality. When fewer layers are arranged, the positive charges of the nanoparticles are insufficient to neutralize the negative charges carried by crude oil and rock, resulting in the emergence of two mechanisms:

• With the rock surface as the fixed layer, positively charged nanoparticles and negatively charged surfactants on the crude oil are attracted around the fixed layer, forming a diffuse double layer that generates electrostatic repulsion, ultimately separating the crude oil from the rock surface.
• Due to the potential difference, charges naturally flow from high to low potential regions. As the oil film detaches, the three-phase contact area expands continuously in the direction perpendicular to the SDP. Charged nanoparticles from highly stratified regions migrate toward the vertex of the three-phase contact area until the oil film is completely removed. SLNs, which exhibit stronger stratification effects, are more likely to generate potential differences, confirming that even low-concentration SLNs can form wedge films that effectively peel off the oil films.

SDP is the key mechanism in SLNs’ oil displacement. Using Equations (2) and (4), the FIE at $H=d$ was calculated for various SLNs mass concentrations (see Figure 6). As the SLNs mass concentration increases, FIE initially rises and then plateaus. When the SLNs mass concentration exceeds 0.005 wt%, further increases in FIE within the wedge film become negligible. Note that FIE units can be converted to N/m.

2.2. Interpolation Factor Selection

In the CMG-STARS simulator, interpolation functions play a crucial role in precisely representing changes in relative permeability and capillary pressure curves [30]. Typically, two sets of relative permeability curves are used to capture the reservoir’s initial state (minimum) and final state (maximum). The interpolation function computes an interpolation factor based on the current injection conditions, producing an ‘intermediate state’ relative permeability curve that lies between the initial and final state curves. The interpolation factor equation in the simulator is provided in Equation (5):

$$\omega ={\left({\displaystyle \frac{{DTRAP}_{W,N}^{Intermediate}-{DTRAP}_{W,N}^{Initial}}{{DTRAP}_{W,N}^{Final}-{DTRAP}_{W,N}^{Initial}}}\right)}^{n_{w,o}}$$

(5)

Here, $\omega$ is the interpolation factor of the interpolation function, and $DTRAP$ is the phase interpolation parameter, where W and N denote the water phase and non-water phase, respectively. In the simulator, DTRAP_W and DTRAP_N are set as two input points by default. Intermediate, Initial and Final refer to the intermediate state (current injection condition parameters), the initial state, and the final state relative permeability curves, respectively, $n_{w,o}$ is the curvature index, for the water phase and oil phase, default value is 1.

Depending on the variation of $\omega$, the intermediate state’s relative permeability and capillary pressure curves can be calculated using the interpolation formula, the weighted result of the initial and final state curves. The interpolation formulas are shown in Equation (6):

$$K_{rl}=\omega \times K_{rl}^{Final}+\left(1-\omega \right)K_{rl}^{Initial}$$

(6)

In the Equation, $K_{rl}$ represents the relative permeability calculated using the Brooks–Corey method, where $l=o,w$ denotes the oil phase and water phase, respectively. Initial and Final refer to the initial and final state relative permeability curves. It is observed that when $\omega =1$, i.e., ${DTRAP}_{W,N}^{Current}={DTRAP}_{W,N}^{Final}$, and $K_{rl}=K_{rl}^{Final}$, the interpolation result is the same as the final (maximum) relative permeability curves. When $\omega =0$, i.e., ${DTRAP}_{W,N}^{Current}={DTRAP}_{W,N}^{Initial}$, and $K_{rl}=K_{rl}^{Initial}$, the interpolation result is the same as the initial (minimum) relative permeability curves.

Through changes in the interpolation factor, the relative permeability curves adjust, influencing the final simulation results. To determine the interpolation factor, CMG-STARS provides three interpolation modes:

• Without the IFT table, DTRAP represents the lowest and highest levels of the solute molar fraction.
• When the IFT table is provided, DTRAP represents the lowest and highest levels of the logarithm of the capillary number.
• When the foam interpolation option is selected, DTRAP corresponds to the lowest and highest levels of the foam mobility factor (FM).

In current CMG-STARS numerical simulation studies, only these three interpolation methods are typically used for curve interpolation, which imposes various limitations on the simulator’s application. The simulator provides users with essential functions while concealing implementation details, allowing users to call necessary functions as needed. The interpolation factor, $\omega$, is a dimensionless parameter representing the linear variation and curve shifting between the highest and lowest levels through a weighted method.

Performing a sensitivity analysis on the interpolation factor $\omega$ in the CMG-STARS simulator, while including the IFT table, enhances the flexibility of interpolation methods, aiming to reduce or eliminate the theoretical impact of IFT in simulations. With a typical oil–water IFT value set at 30 mN/m, an ultra-low IFT of 10⁻³ mN/m significantly improves oil displacement efficiency. When capillary number interpolation is used, the simulator calculates the capillary number corresponding to the current injection concentration based on the IFT. The capillary number equation is shown as Equation (7):

$$N_C={\displaystyle \frac{{\mu }_w\times V_w}{{\sigma }_{o-w}}}$$

(7)

Here, $N_C$ is the capillary number, which is dimensionless, ${\mu }_w$ is the displacing phase viscosity, mPa·s, $V_w$ is the displacing phase linear velocity, cm/s, ${\sigma }_{o-w}$ is the oil–water IFT, mN/m.

After unit conversion to ensure dimensional consistency, the dimensionless quantity $N_C$ is obtained. Taking the logarithm of $N_C$ yields the required input parameter, DTRAP. In the subsequent analysis, the viscosity and linear velocity of the displacing phase are kept constant, while the IFT table is varied to examine its significant effects on interpolation.

The simulation includes seven sets of oil–water IFT tables. While varying the oil–water IFT, the interpolation factor $\omega$ remains constant across the initial, final, and intermediate states. The data configurations for the seven oil–water IFT tables are presented in

Table 2. IFT between oil and water with different levels.

Concentration (wt%)	IFT Between Oil and Water (mN/m)
	1	2	3	4	5	6	7
0	28.31	28.31	283.1	2.831	0.2831	0.02831	28310
0.0025	0.025295	2.5295	25.295	0.25295	0.00025295	0.00002529	2529.5
0.005	0.018964	1.8964	18.964	0.18964	0.00018964	0.00001896	1896.4
0.0075	0.01172	1.172	11.72	0.1172	0.0001172	0.00001172	1172
0.01	0.008902	0.8902	8.902	0.08902	0.00008902	0.00000890	890.2

.

Although Tests 1, 5 and 6, as well as Tests 2, 3, 4 and 7, exhibit different oil–water IFTs, the final interpolation factor $\omega$ remains consistent, measured at 0.90622503 and 0.781395815, respectively. The injection rate for the model is maintained at 0.3 mL/min, and the aqueous viscosity is 0.873958 mPa·s.

Capillary number calculations are performed for the various tests to obtain the logarithmic values of capillary numbers corresponding to the maximum, minimum, and medium IFT. The interpolation factor $\omega$ is computed according to Equation (1).

Simulation results for pressure, recovery factor, and water cut for Tests 1, 5, and 6 are illustrated in Figure 7, while the results for Tests 2, 3, 4, and 7 are presented in Figure 8.

Based on the simulation results shown in Figure 6 and Figure 7, where the minimum, medium, and maximum correspond to system concentrations of 0 wt%, 0.005 wt%, and 0.01 wt%, respectively, it can be observed that Tests 1, 5, and 6, as well as Tests 2, 3, 4, and 7, exhibit identical historical trends in core flooding simulations, regardless of whether the oil–water IFT values corresponding to the system concentrations are the same or whether the settings adhere to theoretical expectations.

Table 3. Recovery factor of different test groups.

No.	$\mathit{\omega }$	Recovery Factor
		Maximum	Medium	Minimum
1	0.906	74.07	70.59	39.66
5		74.07	70.59	39.66
6		74.07	70.59	39.66
2	0.781	74.39	71.10	43.24
3		74.39	71.10	43.24
4		74.39	71.10	43.24
7		74.39	71.10	43.24

summarizes the calculation of DTRAP for Tests 1–7 based on the highest, intermediate, and lowest levels, along with the interpolation factor $\omega$ and the corresponding recovery factor results. An interesting observation arises from calculating DTRAP, $\omega$, and the corresponding recovery factors across Tests 1–7: despite significant variations in capillary numbers and oil–water IFT values, even when the set values are inconsistent with the current understanding of oil–water IFT theory, the resulting recovery factor values are entirely consistent as long as the interpolation factor $\omega$ remains consistent.

Furthermore, based on the information conveyed in Figure 6 and Figure 7, the historical trends in oil displacement during the numerical simulation process are identical. This indicates that the oil–water IFT table in the simulator is solely involved in calculating the interpolation factor $\omega$ without any other related effects. The numerical values of oil–water IFT do not significantly influence the interpolation process of this model, suggesting that they can be substituted to represent the effects of other interpolation methods.

Therefore, the scope of the second interpolation method in the simulator has been expanded. FIE is chosen as the sensitive parameter for the SLN oil displacement mechanism, with its computation and interpolation factor equations defined by Equations (8) and (9). DTRAP is redefined as Augmentation DTRAP (DTRAP*).

Equation for ${DTRAP}^{*}$.

$${DTRAP}^{*}=\mathrm{l}\mathrm{g}\left({\displaystyle \frac{{\mu }_w\times V_w}{\delta }}\right)$$

(8)

Equation for $\omega$:

$$\omega ={\left({\displaystyle \frac{{{DTRAP}^{*}}_{W,N}^{Intermediate}-{{DTRAP}^{*}}_{W,N}^{Initial}}{{{DTRAP}^{*}}_{W,N}^{Final}-{{DTRAP}^{*}}_{W,N}^{Initial}}}\right)}^{n_{w,o}}$$

(9)

where $\delta$ is the sensitivity parameter of the oil displacement mechanism, and is only applicable to physical quantities measured in ‘N/m’. Based on the δ, the maximum and minimum of ${DTRAP}^{*}$ are determined.

3. Model Establishment

The core data for the SLN small-scale mechanistic model is derived from Yan’s low-permeability core SLN flooding experiments in 2022 [31]. The specific steps of the experiment are as follows.

• Dry the core sample and measure its dry weight. Determine gas permeability before subjecting the core to vacuum evacuation.
• Saturate the core with formation water, measure the wet weight, and calculate porosity.
• Perform a waterflood experiment at a constant flow rate. Once pressure stabilizes, calculate the water-phase permeability.
• Saturate the core with a silicone oil and kerosene mixture (1:1 by volume) and age it under reservoir temperature conditions for 7 days to induce oil-wet conditions.
• Saturate the core with crude oil, age it under reservoir temperature conditions for 7 days, and determine the effective oil-phase permeability under irreducible water saturation.
• Conduct waterflooding at a constant flow rate, recording cumulative oil production, cumulative liquid production, and the pressure differential at various time intervals. Record the water breakthrough time precisely, and, after breakthrough, increase recording frequency. As oil production declines, extend the recording intervals appropriately. Terminate the experiment when the water cut at the outlet exceeds 99.95%.
• Clean the core to remove oil, dry it, and replace the displacing fluid with an SLN solution. Repeat steps 1 through 6.
• Analyze the experimental data using the J.B.N. method to derive oil–water relative permeability curves and oil–SLN relative permeability curves.

The crude oil used in the experiments is from the Daqing Oilfield in China, with a viscosity of 20 mPa·s at 75 °C and a flow rate of 0.3 mL/min. The viscosity of the SLN fluid is 0.874 mPa·s. The relative permeability curves from Yan’s experiments were digitized, with the relative permeability curve under waterflooding conditions selected as the first set for numerical simulation. In contrast, the relative permeability curve under SLN fluid flooding conditions was used as the second set. To ensure convergence and stability in the CMG-STARS simulator, the relative permeability curves were optimized using the Brooks–Corey method.

3.1. Fluid Model

(1)

SDP

After conducting sensitivity tests on the oil–water IFT table and calculating for SLN fluid at different mass concentrations, the SDP of SLN fluid was modeled using the CMG-STARS simulator. Since the sensitivity parameters of the simulator for oil displacement mechanisms must be in the form of ‘N/m’ physical quantities, the FIE of SLN fluid was adopted as the sensitivity parameter for oil displacement mechanisms. At this point, Equations (8) and (9) can be rewritten as Equations (10) and (11). FIE^* is the logarithm of FIE.

Equation for ${\mathrm{F}\mathrm{I}\mathrm{E}}^{\mathrm{*}}$.

$${\mathrm{F}\mathrm{I}\mathrm{E}}^{\mathrm{*}}=\mathrm{l}\mathrm{g}\left({\displaystyle \frac{{\mu }_w\times V_w}{\mathrm{F}\mathrm{I}\mathrm{E}}}\right)$$

(10)

Equation for $\omega$:

$$\omega ={\left({\displaystyle \frac{{{\mathrm{F}\mathrm{I}\mathrm{E}}^{\mathrm{*}}}_{W,N}^{Intermediate}-{{\mathrm{F}\mathrm{I}\mathrm{E}}^{\mathrm{*}}}_{W,N}^{Initial}}{{{\mathrm{F}\mathrm{I}\mathrm{E}}^{\mathrm{*}}}_{W,N}^{Final}-{{\mathrm{F}\mathrm{I}\mathrm{E}}^{\mathrm{*}}}_{W,N}^{Initial}}}\right)}^{n_{w,o}}$$

(11)

where ${\mathrm{F}\mathrm{I}\mathrm{E}}^{\mathrm{*}}$ is the interpolation parameter ${DTRAP}^{\mathrm{*}}$. ${{\mathrm{F}\mathrm{I}\mathrm{E}}^{\mathrm{*}}}_{W,N}^{Intermediate}$, ${{\mathrm{F}\mathrm{I}\mathrm{E}}^{\mathrm{*}}}_{W,N}^{Initial}$, and ${{\mathrm{F}\mathrm{I}\mathrm{E}}^{\mathrm{*}}}_{W,N}^{Final}$ represent the current, minimum, and maximum of the logarithmic (FIE) of the SLNs, respectively. The calculated FIE at different concentrations of SLNs is shown in

Table 4. FIE at different mass concentrations of SLNs.

Mass Concentration (wt%)	FIE (Wd2/kT)
0.0010	0.005
0.0025	0.157
0.0040	0.914
0.0050	2.110
0.0075	2.102
0.0100	2.130

.

(2)

Settling performance

Existing numerical simulation studies typically increase channel flow resistance by raising fluid viscosity and reducing reservoir porosity. However, viscosity tests conducted on SLN fluid at different temperatures revealed that SLN fluid concentrations ranging from 0.001 to 0.01 wt% exhibited viscosities comparable to that of water at the same temperature. Additionally, viscosity tests on the crude oil displaced by SLN fluid in core flooding experiments showed no increase in the viscosity of the oil phase. Therefore, increasing fluid viscosity to enhance flow resistance within favorable channels is not applicable to SLN fluid.

SLN is a solid lamellar nanomaterial suspended in the aqueous phase, which settles to a certain extent after the SLN fluid suspension stands for some time. Within natural porous media reservoirs, SLNs are trapped by pore structures, reducing porosity and limiting the flow capacity of channels. Due to hydrodynamic forces, flexible SLNs undergo bending and deformation, transitioning from a trapped state to a free state and continuing to move along the oil–water interface. However, the self-profile control and flooding capability of SLN fluid has its limits. Laboratory physical simulation experiments have shown that SLNs demonstrate good self-profile control and flooding capability in medium to low permeability reservoirs, though these findings have not yet been published.

In contrast, the self-profile control and flooding capability in heterogeneous reservoirs with higher average permeability is generally lower. This is because high-average-permeability reservoirs often have larger pore sizes, making SLNs less likely to be trapped. The combined effects of pore size and the boundary layer in smaller pores contribute to the more frequent occurrence of displacement phenomena in medium to low permeability reservoirs, as depicted in Figure 9.

This study is a qualitative analysis based solely on the experimental observations of SLN fluid. The goal is to provide an initial exploration into the numerical simulation of SLN fluid, and quantitatively describing the flow-limiting factors of SLN fluid falls outside the scope of this paper. Future research will primarily focus on experiment-based approaches to precisely quantify the flow-limiting factors.

(3)

Emulsification performance

An O/W emulsion can be formed by thoroughly mixing SLN fluid with crude oil in a certain proportion. Studies show that the emulsifying ability of SLN fluid with crude oil varies depending on core permeability, which in turn influences the particle size of the resulting emulsion. Smaller-sized O/W emulsions exhibit properties closer to the water phase, with viscosities resembling that of water. As a result, in low-permeability reservoirs, emulsions tend to have smaller particle sizes, leading to a greater reduction in crude oil viscosity and a more significant enhancement in the flow capability of both the oil and water phases within the porous medium. On the other hand, in high-permeability regions, where pore sizes are larger than those in low-permeability zones, the emulsion droplets tend to be larger. While this also improves the flow capability of the oil and water phases within the porous medium, the enhancement is less pronounced compared to low-permeability areas.

The emulsification performance of SLN is defined by the reaction Equations (12) and (13):

$$\mathrm{S}\mathrm{L}\mathrm{N}\left(S\right)+H_{2}O\leftrightarrow \mathrm{S}\mathrm{L}\mathrm{N}\left(L\right)$$

(12)

$$\mathrm{S}\mathrm{L}\mathrm{N}\left(L\right)+Oil\leftrightarrow \mathrm{S}\mathrm{L}\mathrm{N}\left(E\right)$$

(13)

where S, L, and E represent the solid, fluid, and emulsion phase, respectively.

Reactions (12) and (13) are reversible reactions. Reaction (12) represents the formation of an SLN fluid suspension in the reservoir, driven by hydrodynamic effects and the dispersion stability of SLNs. Since SLNs can settle and become mechanically trapped in the porous media, the reverse reaction rate is set to 1/1000 of the forward reaction rate, based on unpublished laboratory settling evaluations. Reaction (13) involves emulsifying a specific amount of SLN fluid with crude oil to form an O/W emulsion. However, due to the inherent properties of SLNs, the emulsion may undergo demulsification over time. As such, the reverse reaction rate is set to 1/10,000 of the forward reaction rate, based on unpublished laboratory evaluations of emulsification and demulsification. Since this study focuses on exploring the numerical simulation of SLN fluid, the reaction orders for the SLN fluid emulsification process are set to 1. Furthermore, the forward and reverse chemical reaction rates, as well as the activation energy required for the reactions, are not considered in this study. This research qualitatively characterizes the emulsification ability of SLN fluid under different reservoir permeabilities. Future studies will aim to quantitatively assess the emulsification ability of SLN fluid based on laboratory physical experiments.

3.2. Core Model

This study employs a 1-D homogeneous model and a 2-D heterogeneous model to evaluate the effectiveness of SLNs under different reservoir conditions.

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1-D homogeneous model

The core samples RF-1 to RF-4 from Yan’s study [31] were selected for modeling. The cross-sectional area of the cores was calculated using the equivalent diameter method, transforming them into 1-D rectangular prism models. The cores have a length of 30 cm and an equivalent diameter of 2.22 cm (see Figure 10). Additional parameters relevant to the mechanistic model are provided in

Table 5. Core data of 1-D mechanism model.

No.	Porosity (%)	Permeability (mD)	Pore Volume (cm3)
RF-1	17.33	20.0	25.52
RF-2	18.97	45.0	27.93
RF-3	17.09	15.0	25.17
RF-4	16.38	5.0	24.12

. It is important to note that the permeability at this stage refers to gas permeability.

The model dimensions, grid size, and well events are provided in

Table 6. 1-D mechanism model data.

Grid Type	Cartesian
Direction	Δx	Δy	Δz
Dimensions	30	1	1
Size (for each grid)	1 cm	2.22 cm	2.22 cm
Well events	Position	Constraints	Comments
Injector	(1, 1, 1)	0.3 cm3/min	Constant-rate injection
Producer	(30, 1, 1)	-	-

. The injector was operated in constant-rate injection mode, with the injection rate matching Yan’s experimental setup [31]. The model was validated and successfully passed the grid refinement analysis test.

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2-D heterogeneous model

Based on the 1-D mechanistic model, a dual-layer 2-D heterogeneous model was developed. The porosity and permeability of the low-permeability (low-perm) layer were kept consistent with those of the RF-3, while the thief-zone layer featured higher porosity and permeability levels (see Figure 11). The foundational parameters of the 2-D heterogeneous model are provided in

Table 7. Core data of 2-D mechanism model.

Layer	Porosity (%)	Permeability (mD)	Pore Volume (cm3)
Low-perm	17.09	15.0	12.59
Thief zone	20.00	100.0	14.73

.

The model dimensions, grid size, and well events are provided in

Table 8. 2-D mechanism model data.

Grid Type	Cartesian
Direction	Δx	Δy	Δz
Dimensions	30	1	2
Size (for each grid)	1 cm	2.22 cm	2.22 cm
Well events	Position	Constraints	Comments
Injector	(1, 1, 1)~(1, 1, 2)	0.3 cm3/min	Constant-rate injection
Producer	(30, 1, 1)~(30, 1, 2)	-	-

. The completion method involved open-hole completions with perforations in each layer. The injector operated in a constant-rate injection mode.

4. Validations and Results

4.1. 1-D Oil Displacement Matching

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Dynamic oil displacement matching

RF-1 to RF-4 are used to match oil recovery and water cut under different SLN fluid mass concentrations and reservoir permeabilities. The experimental curves are digitized, with error bars set at 0.02 for oil recovery and 0.05 for water cut. The oil displacement process using SLN fluid is numerically simulated and compared. The simulator accurately captures the breakthrough time and oil displacement history of the core, all within the error bar range, as shown in Figure 12, Figure 13, Figure 14 and Figure 15.

It is worth mentioning that SLN fluid demonstrates excellent water control and oil recovery enhancement effects. Compared to waterflooding, SLN fluid at concentrations of 0.005 wt% and 0.01 wt% can temporarily delay the breakthrough time. However, during the breakthrough process, the water cut in SLN fluid flooding does not directly increase to 100%. Instead, it reaches a plateau at around 60%. The sudden appearance of this plateau during the significant increase in water cut indicates that the SLN fluid has reached new residual oil zones. After forming a wedge film at the oil–brine–rock interface, the SLN fluid generates intense interfacial disjoining pressure, which separates the oil film from the rock surface. As the wedge film forms, the separated oil film accumulates to create an oil plug, increasing oil saturation and decreasing the water cut at the outlet.

The oil recovery mechanism of SLN fluid flooding differs from that of waterflooding. As reservoir permeability decreases, the degree of oil recovery initially increases and then decreases, indicating the presence of an optimal permeability window for SLN fluid flooding. This window correlates with the morphology and SDP of SLNs. When the reservoir permeability is too low, the proportion of small-sized pore throats increases, which are incompatible with the size of the SLNs, preventing them from entering the pores for oil displacement and resulting in lower oil recovery. Conversely, when permeability is relatively high, the average pore throat size is larger, causing SLNs to layer more easily than in smaller pore throats. As a result, forming an SDP capable of separating the oil film becomes more challenging, and the effective wedge ‘shoveling oil’ process cannot be achieved.

The water control capability of SLN fluid with the same mass concentration varies in cores with different permeabilities. After injecting 0.2 PV, the greatest reduction in water cut by SLN fluid occurs in the 15 mD core, while the lowest reduction is observed in the 5 mD core. This trend is consistent with the pattern of oil recovery degrees demonstrated in cores with different permeabilities (Figure 15).

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Relative permeability curves

Relative permeability curves are closely related to the dynamics of oil displacement, with factors such as capillary number [32,33], wettability [34,35], temperature [36,37], and pressure [38,39], thereby affecting the entire oil displacement process. To correspond with the previous research, the changes in relative permeability curves are studied due to the different mass concentrations of SLN fluid and permeabilities.

Before the injection of the SLN fluid, the core was initially oil-wet. After the injection of the SLN fluid, the intersection point shifted to the right. As the concentration of SLNs increases, the rightward shift of the intersection point becomes more pronounced. At an SLN mass concentration of 0.001 wt%, the intersection point of the oil–water two-phase system is slightly below 0.5, indicating that the core’s wettability has not been altered. This aligns with the argument that, at 0.001 wt%, SLNs cannot form a wedge film. However, as the mass concentration of SLNs increases to 0.005 wt%, the intersection point shifts obviously to the right, signifying core wettability alteration and a substantial increase in oil-phase permeability (Figure 16). This suggests that the SLN fluid changes the reservoir’s initial surface wettability, increasing the hydrophilicity of the core and enhancing oil displacement efficiency. The curvature of the relative permeability curves also changes. For this core, the water-phase relative permeability curve became more curved, and the relative permeability values at the same water saturation decreased, while those for the oil phase increased to some degree. This indicates that the SLN fluid had the effect of restricting water-phase flow, effectively reducing the water breakthrough rate and water cut and enhancing the flow capacity of the oil phase, thereby increasing the oil cut at the outlet.

Oil–water front oil displacement efficiency.

$$E_{Di}={\displaystyle \frac{S_{wf}-S_{wc}}{S_{wc}}}$$

(14)

Average water saturation at the oil–water front:

$$\overline{S_w}=S_{wc}+{\displaystyle \frac{1}{f^{\prime }\left(S_{wf}\right)}}$$

(15)

where $E_{Di}$ is the efficiency of oil displacement at the front, as a fraction, $S_{wf}$ is the water saturation at the oil–water front, as a decimal, and $S_{wc}$ represents the irreducible water saturation, as a decimal.

According to Equations (14) and (15), the waterflooding efficiency at the front is only 8.57%, with an average water saturation of only 0.394. This indicates that the oil–water transition zone has a relatively high oil saturation, poor stability at the oil–water front, and a poor displacement effect on the crude oil in the pore space. When the mass concentration of SLN fluid increases to 0.005 wt%, the average water saturation of the oil–water front significantly increases, rising by 16.5%. The front displacement efficiency also significantly improves, increasing by 37.58%. This indicates that, with the increase in SLN fluid mass concentration, the stability of the displacement front is effectively enhanced, and the effectiveness of it in displacing oil in porous media is also strengthened. Compared to the effect of 0.005 wt% SLN fluid, further increasing the SLN fluid mass concentration to 0.01 wt% results in only a 0.3 percentage point increase in the average water saturation of the front and a mere 4.4 percentage point increase in front displacement efficiency, suggesting that, when the mass concentration of SLN fluid exceeds 0.005 wt%, increasing the fluid concentration contributes less to the displacement effect (Figure 17).

For different reservoir permeabilities, the horizontal and vertical comparisons of the water- and oil-phase shape factors before and after SLN fluid flooding are shown in Figure 18a,b. For waterflooding, the water-phase shape factor changes minimally, gradually increasing as permeability increases. Conversely, the oil-phase shape factor shows a linear decrease, indicating that the water-phase permeability changes minimally with increasing permeability, while the oil-phase permeability significantly improves.

During SLN fluid flooding, the water-phase shape factors uniformly increase at the same permeability level, with peak values occurring between 5 and 20 mD. This suggests that SLN fluid demonstrates optimal water control capability within this permeability range. Conversely, the oil-phase shape factors decrease uniformly, with the most noticeable reduction observed within 5~20 mD, indicating that SLN fluid substantially enhances the oil flow capability in low-permeability reservoirs, resulting in notable incremental production benefits.

With insufficient SLN fluid concentration to create a wedge film at the three-phase contact area, the water-phase shape factor is similar to that seen in waterflooding, unable to delay water breakthrough or improve water control (Figure 19a).

When the SLN fluid concentration is sufficient to form a stable wedge film structure at the three-phase contact area, the shape factor of the water phase maintains a high level, indicating effective profile control by SLN within this range. After SLN fluid forms a wedge film, it significantly enhances the mobilization and flow capacity of crude oil. The formation of a wedge film by SLN significantly enhances crude oil mobilization and flow capacity. It facilitates oil film peeling and wettability alteration, causing oil films to coalesce into slug-like segments for enhanced displacement. The effectiveness plateaus after reaching a concentration of 0.005 wt%, indicating that increasing concentration increases the platform period of water content growth. When the concentration exceeds 0.005 wt%, the platform does not show significant growth (Figure 19b).

4.2. 2-D Oil Displacement Simulation

After waterflooding with 1.33 PV, different concentrations of SLN fluid were injected. The oil recovery and water cut of different concentrations of SLN fluid in the 2-D heterogeneous model are illustrated in Figure 20.

Figure 20 and Figure 21 show the variation in the water saturation field, where deeper red indicates higher oil saturation and deeper blue indicates higher water saturation. The presence of a thief zone limits waterflooding recovery to 21.0%, as inject water channeling occurs quickly through dominant channels, creating a thief layer that deteriorates the oil displacement in low-permeability layers, as shown in Figure 21a.

The water cut rapidly rises and eventually approaches 100%. After injecting SLN fluid, the recovery rate increases to some extent. The recovery with a mass concentration of 0.0025 wt% SLN fluid reaches 25.66%, 4.66% higher than the waterflooding stage. This is because the SLN fluid barely reaches the critical mass concentration for forming the wedge film, resulting in a small SDP, and the disjoining effect of the oil film is not ideal. In addition, some SLNs will also be mechanically trapped by the porous medium, resulting in fewer SLNs in the bulk phase. The superposition effect of the two causes the low-concentration SLNs to have a little increase in recovery, and the self-profile control and flooding and oil displacement phenomenon are not prominent.

When the mass concentration of SLN fluid is increased to 0.005 wt%, the ultimate oil recovery rate reaches 41.12%, marking a substantial 20.12% increase compared to waterflooding. After injecting the SLN fluid, the water cut curve shows a noticeable decrease. Following the decrease in water cut, there is a ‘concave’ segment between 1.5 PV and 1.9 PV in the water cut curve. This is the result of the limiting effect on the dominant channel water flow, causing the subsequent liquid to automatically flow to the low-permeability channels, initiating the oil in the low-permeability layer, as shown in Figure 21c. Compared to Figure 21a, under the same injection volume, injecting 0.005 wt% SLN fluid reduces the water saturation in the low-permeability layer and does not wash the oil in the dominant channels as effectively as waterflooding. When the total injection volume reaches 4 PV, the 0.005 wt% SLN fluid forms a nearly ‘wedge-shaped’ propulsive oil displacement pattern in the heterogeneous model with a permeability contrast of 66.7 times.

When the mass concentration of SLN fluid is increased to 0.01 wt%, the ultimate oil recovery rate reaches 40.43%, marking a 19.43% increase compared to waterflooding, which is slightly less than the enhancement potential of 0.005 wt% SLN fluid flooding. This is attributed to the intensified layering effect of SLN fluid at higher concentrations, leading to a significant reduction in the structural separation pressure gradient. The SLN fluid only temporarily restricts the flow in the high-permeability layers, as shown in Figure 21e. As SLNs are mechanically trapped, the bulk-phase concentration slightly decreases, and the SDP gradually strengthens, resulting in a subsequent oil displacement history similar to 0.005 wt% SLN fluid flooding. Therefore, when the total injection volume is 4 PV, the water saturation distribution is similar, but the overall oil-washing effect on the core is mediocre.

5. Conclusions

This study successfully developed a fluid model to characterize the SDP, settling, and emulsification behaviors of SLN fluid. The calculations revealed that the SDP of nanoparticles with SLNs significantly surpasses that of spherical particles, with an SDP at the three-phase contact area approximately 16.5 times higher than spherical particles. However, further refinement in measurement accuracy is still required. Additionally, stratification and layering were identified as the primary factors contributing to the notable reduction in SDP.

The core flooding results, modeled under various permeability conditions and SLN mass concentrations, showed a close match, validating the feasibility of using FIE as an interpolation factor for SLN fluid. Furthermore, this study provided valuable insights into the relative permeability curves, flow behavior, and effects of SLNs under diverse conditions, highlighting the self-profile control and Enhanced Oil Recovery capabilities of SLNs. These findings offer essential guidance and theoretical foundations for future experimental research and field applications in the numerical simulation of nanofluids.

6. Research Gaps and Future Recommendations

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Nanosheet Interpolation Factor Optimization

Further investigation is essential to enhance the selection of interpolation factors, particularly focusing on the nanosheet structure–size relationship, SDP/FIE dynamics, and settling behaviors. A deeper understanding of these interactions will be instrumental in improving the accuracy of SLN-based reservoir models.

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Incorporation of Nanosheet Mechanisms in Simulation Models

Current Enhanced Oil Recovery (EOR) simulations lack the detailed integration of SLN-specific oil displacement mechanisms, including their influence on flow dynamics and relative permeability. Integrating these mechanisms is essential for enhancing the predictive precision of numerical simulations.

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Optimization Strategies for Low-Permeability Reservoirs

In low-permeability reservoirs, optimizing SLN injection parameters—such as concentration, slug size, and shut-in time—is critical to maximizing EOR efficiency. Employing multivariate regression and response surface analysis can help identify the most effective operational strategies.

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Experimental Verification and Field Application

Future work should prioritize the experimental validation of SDP- and FIE-based models, progressing towards field-scale trials. This will allow for a comprehensive assessment of SLN technology’s practical potential in real-world reservoir environments.