A Critical Review of Limited-Entry Liner (LEL) Technology for Unconventional Oil and Gas: A Case Study of Tight Carbonate Reservoirs

Abstract

Limited-Entry Liner (LEL) technology has emerged as a transformative solution for enhancing hydrocarbon recovery in unconventional reservoirs while addressing challenges in carbon sequestration. This review examines the role of LEL in optimizing acid stimulation, hydraulic fracturing and production optimization, focusing on its ability to improve fluid distribution uniformity in horizontal wells through precision-engineered orifices. By integrating theoretical models, experimental studies, and field applications, we highlight LEL’s potential to mitigate the heel–toe effect and reservoir heterogeneity, thereby maximizing stimulation efficiency. Based on a comprehensive review of existing literature, this study identifies critical limitations in current LEL models—such as oversimplified annular flow dynamics, semi-empirical treatment of wormhole propagation, and a lack of quantitative design guidance—and aims to bridge these gaps through integrated multiphysics modeling and machine learning-driven optimization. Furthermore, we explore its adaptability for controlled CO₂ injection in geological storage, offering a sustainable approach to energy transition. This work provides a comprehensive yet accessible overview of LEL’s significance in both energy production and environmental sustainability.

1. Introduction

The development of unconventional oil and gas resources has revolutionized global energy markets, but the inherent challenges of low-permeability formations and reservoir heterogeneity continue to hinder efficient hydrocarbon recovery [1,2]. Among the innovative solutions to these challenges, Limited-Entry Liner (LEL) technology has emerged as a pivotal advancement for optimizing reservoir stimulation. By utilizing precision-engineered orifices, LEL ensures uniform fluid distribution in horizontal wellbores, addressing critical issues such as the heel–toe effect and uneven acid placement during matrix acidizing [3,4].

Beyond its applications in hydrocarbon production, LEL also holds promise for carbon sequestration, offering precise control over CO₂ injection in heterogeneous formations. Despite its potential, research on LEL remains fragmented, with limited comprehensive reviews available. This paper synthesizes advancements in LEL technology, from its foundational fluid dynamics and acidizing mechanisms to its design methodologies and environmental applications, aiming to provide a consolidated resource for researchers and practitioners in the field.

Based on a comprehensive review of the existing literature, current research still exhibits shortcomings in the areas of LEL flow modeling, acidizing-related calculations, and LEL design. Regarding flow calculations, LEL flow models often employ nodal network-based computational methods, with insufficient in-depth research on flow details such as annular flow. For acidizing calculations, most existing papers utilize semi-empirical formulas for treatment, lacking analytical-level explanations and further interpretation of acidizing effects under the unique flow distribution of LEL (concentrated acid entry points). Whether for flow within the pipe or wormhole formation in the formation, while newer models have taken a step forward in considering formation heterogeneity and anisotropy, there is still a lack of models that better align with complex engineering realities. At the LEL design level, authors have proposed the impact of certain structural parameters of LEL (such as orifice density) on the fluid production profile, but quantitative guidance is lacking. Additionally, the application of optimization algorithms, such as machine learning, is highly insufficient.

To provide a clear conceptual roadmap of this review, Figure 1 presents a structured flowchart outlining the logical progression and key thematic sections discussed herein. Beginning with the fundamental challenges inherent in unconventional reservoir development—such as ultra-low permeability, pronounced heterogeneity, and the heel–toe effect—the diagram illustrates how Limited-Entry Liner technology emerges as an integrated solution. The core technical review is divided into three pillars: flow dynamics within LEL systems (Section 3), acidizing mechanisms and wormhole modeling (Section 4), and LEL design methodologies (Section 5). Building upon this foundation, Section 6 explores the prospective application of LEL technology beyond hydrocarbon production, specifically envisioning its potential role in carbon sequestration. This review concludes by synthesizing key findings, acknowledging current limitations, and identifying promising future research directions (Section 7).

2. Challenges in Unconventional Energy Development

2.1. Problem of Unconventional Oil and Gas

Unconventional Oil and Gas is defined as a series of hydrocarbon resources that differ from conventional reservoirs in terms of their occurrence state or extraction methods.

Unconventional Natural Gas refers to natural gas resources trapped in geological formations that require advanced extraction techniques, such as hydraulic fracturing and horizontal drilling, to be economically viable. The primary types include shale gas, tight gas, coalbed methane (CBM), and gas hydrates, each with distinct geological and production characteristics. The development of unconventional gas—particularly shale gas—has significantly reshaped global energy markets [5].

Unconventional Oil refers to hydrocarbon resources that cannot be extracted using conventional drilling and pumping methods due to their unique geological and physical properties. These resources require advanced technologies [6] such as thermal processing, hydraulic fracturing, or chemical conversion for economic recovery. Unlike conventional oil, which flows naturally through permeable reservoirs, unconventional oil is typically trapped in low-permeability formations or exists in highly viscous forms (e.g., bitumen, kerogen). Major types of unconventional oil include: Tight Oil, Shale Oil, Oil Sands, and Extra Heavy Oil. Due to their complex extraction processes, unconventional oil sources are generally more expensive and environmentally intensive than conventional oil. However, they play an increasingly vital role in global energy supply as conventional reserves decline [7].

As shown in Figure 2, unconventional oil and gas resources have broad development prospects but are challenging to exploit. Both unconventional oil and gas—particularly tight oil and gas—are typically hosted in low permeability reservoirs, requiring acidizing or hydraulic fracturing [1]. Unconventional reservoirs display significant spatial variations in lithology, porosity, and fluid saturation, even at millimeter scales. For instance, shale formations often consist of thin, interbedded organic-rich and inorganic layers, causing uneven fracture propagation during stimulation. Heterogeneity complicates “sweet spot” identification and leads to inconsistent production across wells [2].

In addition, complex fluid properties and technical limitations in extraction also profoundly impact the development of unconventional oil and gas. These low-permeability, heterogeneous formations pose significant challenges to production, necessitating specialized techniques such as hydraulic fracturing and acid stimulation [8,9].

In summary, the development of unconventional oil and gas resources is constrained by inherent geological complexities, including ultra-low permeability, pronounced heterogeneity, and challenging fluid dynamics. While technological advancements like horizontal drilling and hydraulic fracturing have enabled commercial production, these reservoirs continue to demand innovative approaches to overcome inefficiencies in fluid flow and stimulation coverage. The unique characteristics of unconventional formations underscore the need for tailored solutions to maximize recovery while addressing economic and operational constraints.

2.2. Challenges in Acid Stimulation Technology

Acid stimulation is a key well-stimulation technique in unconventional oil and gas development [10]. By injecting acidizing fluids (e.g., hydrochloric acid, organic acids) into the reservoir, it dissolves near-wellbore mineral blockages or fillings in natural fractures, thereby enhancing reservoir permeability and improving hydrocarbon flow capacity. This technology is widely applied in carbonate and sandstone reservoirs, demonstrating particularly significant efficacy in low-permeability unconventional reservoirs.

However, conventional acidizing operations face several challenges, among which the “heel–toe effect” and formation heterogeneity are the most critical. The heel–toe effect refers to the tendency of acid fluids to preferentially enter the near-wellbore heel section in long horizontal wells, resulting in insufficient stimulation at the toe section.

Formation heterogeneity, primarily due to spatial variations in reservoir properties (e.g., porosity, permeability), leads to uneven acid distribution and highly variable localized stimulation outcomes. High-permeability zones absorb more acid, leaving low-permeability regions untreated [11]. In anisotropic formations, the connectivity of pores and fractures varies significantly with direction. This directional flow may leave certain zones inadequately treated, compromising the effectiveness of acid stimulation [12].

The chemical properties of both the acid and the rock also significantly influence the effectiveness of acid stimulation [13]. Carbonate reservoirs pose significant challenges for acidizing due to their inherent heterogeneity in composition, porosity, and permeability. Unlike homogeneous formations, carbonates often exhibit complex pore structures, varying mineralogy (e.g., calcite vs. dolomite), and natural fractures, leading to non-uniform acid distribution and unpredictable wormhole propagation [14].

To address these challenges, Coiled Tubing (CT), Inflow Control Device (ICD), and Limited-Entry Liner (LEL) [3,4,15] technologies have been developed. LEL, sometimes called Controlled Acid Jetting (CAJ) or Smart Liner (SL), employs a meticulously designed non-uniform distribution of orifices along the liner. Figure 3 illustrates the general downhole configuration of an LEL system: the entire horizontal wellbore is divided into multiple segments, each equipped with a distinct orifice density, and adjacent segments are hydraulically isolated by mechanical packers. Figure 3 also depicts the acid-flow path during a matrix-acidizing treatment: acid is pumped inside the liner, experiences a pressure drop as it jets through the orifices, enters the annular space between the liner and the formation, and finally invades the reservoir rock to create wormholes.

The emergence of LEL has provided a superior solution for acid stimulation in unconventional oil and gas reservoirs. However, the specific structural design of LEL is closely related to actual formation conditions. Although LEL technology dates back to the 1980s, research on LEL remains limited, and relevant information is often only available in conference papers from petroleum industry events. Therefore, this review compiles and summarizes a series of studies related to LEL, aiming to provide readers with insights into LEL flow behavior, acid stimulation, and design methodologies.

3. Flow Dynamics in LEL Systems

As illustrated in Figure 4, based on the structural characteristics of LEL, the flow dynamics within LEL systems are governed by a complex interplay of fluid mechanics across four critical domains: pipe flow, orifice flow, annular flow and reservoir seepage. Accurate modeling of these components is essential for optimizing acid stimulation, as they dictate pressure distribution, fluid placement efficiency, and ultimately, treatment effectiveness.

Pipe friction determines the hydraulic resistance along the liner, orifice flow regulates localized pressure drops to ensure uniform fluid diversion, and reservoir seepage governs the interaction between injected fluids and heterogeneous formations. These flow-related studies form the foundation for LEL flow field prediction and design. This section synthesizes theoretical and empirical advancements in flow modeling, highlighting their pivotal role in translating LEL concepts into computational models.

3.1. Heel–Toe Effect and Frictional Pressure Loss

Pipe friction (Frictional pressure loss) refers to the resistance encountered by fluids flowing through pipes due to viscous shear forces between the fluid and pipe walls. In long horizontal wells, frictional pressure loss along the wellbore cannot be neglected, resulting in significant pressure variations from the heel to the toe. This leads to a highly uneven overall pressure distribution in the pipeline, known as the heel–toe effect. Accurate calculation of frictional pressure losses is essential for optimizing orifice designs and ensuring uniform fluid distribution across long horizontal wellbores. As shown in Table 1, over the years, researchers have developed various models to characterize pipe friction under different flow conditions, including the influence of drag-reducing agents (DRAs) and pipe roughness.

The pipe friction is calculated using the following fundamental equation [16,17]:

$${\displaystyle \frac{\mathrm{d}{\mathrm{P}}_{\mathrm{f}\mathrm{r}\mathrm{i}\mathrm{c}}}{\mathrm{d}\mathrm{x}}}=-{\displaystyle \frac{8\mathrm{f}\times \rho \times {\mathrm{Q}}^2}{D^5}}$$

(1)

where $f$ is the Fanning friction factor, $\rho$ is fluid density, Q is flow rate, and $D$ is pipe diameter.

Table 1. Summary of Pipe Friction Studies.

Authors	Year	Study Types	Main Findings
Virk [18]	1971	Experimental Study	Demonstrated drag reduction in rough pipes using polymer solutions. Found that onset wall shear stress is independent of pipe roughness and polymer concentration.
Virk [19]	1975	Theoretical Model	Proposed a drag reduction model for turbulent flow in smooth pipes, introducing the concept of maximum drag reduction asymptote.
Mogensen [20]	2007	Numerical Model	Extended Virk’s model to include pipe roughness and drag-reducing agents (DRA), enabling accurate friction predictions for CAJ completions.
Mogensen [21]	2014	Workflow Development	Developed a workflow for modeling fluid displacement in wellbores using commercial simulators, incorporating DRA effects and dynamic friction calculations.
Hansen [17]	2002	Field Application	Demonstrated pipe friction effects in CAJ for long reservoir sections (>14,000 ft), optimizing orifice spacing to balance frictional pressure losses.

Table 1. Summary of Pipe Friction Studies.

Authors	Year	Study Types	Main Findings
Virk [18]	1971	Experimental Study	Demonstrated drag reduction in rough pipes using polymer solutions. Found that onset wall shear stress is independent of pipe roughness and polymer concentration.
Virk [19]	1975	Theoretical Model	Proposed a drag reduction model for turbulent flow in smooth pipes, introducing the concept of maximum drag reduction asymptote.
Mogensen [20]	2007	Numerical Model	Extended Virk’s model to include pipe roughness and drag-reducing agents (DRA), enabling accurate friction predictions for CAJ completions.
Mogensen [21]	2014	Workflow Development	Developed a workflow for modeling fluid displacement in wellbores using commercial simulators, incorporating DRA effects and dynamic friction calculations.
Hansen [17]	2002	Field Application	Demonstrated pipe friction effects in CAJ for long reservoir sections (>14,000 ft), optimizing orifice spacing to balance frictional pressure losses.

For Newtonian fluids in rough pipes, the friction factor $f$ can be determined using the Colebrook–White equation [16]:

$$\sqrt{{\displaystyle \frac{1}{f}}}=-4\times {\log }_{10}({\displaystyle \frac{1.26}{Re\times \sqrt{f}}}+{\displaystyle \frac{\epsilon }{3.7\times D}})$$

(2)

where Re is the Reynolds number and $\epsilon$ is pipe roughness. This model is widely used but assumes no drag-reducing effects. To solve this implicit equation, numerical methods such as the Newton-Raphson iteration are typically required. In this formulation, the Reynolds number (Re) is defined as [16]:

$$\mathrm{R}\mathrm{e}={\displaystyle \frac{4}{\mathsf{\pi }}}\times {\displaystyle \frac{\rho \times \mathrm{Q}}{D\times \mathsf{\mu }}}$$

(3)

where μ is fluid viscosity.

Virk [18,19] pioneered the study of DRAs and proposed a model to quantify their impact on friction factors. Their work introduced a shift parameter δ to account for DRA concentration:

$$\sqrt{{\displaystyle \frac{1}{f}}}=(4+\delta ){\log }_{10}(Re\times \sqrt{f})-0.4-\delta {\log }_{10}(\sqrt{2}\times D\times w*)$$

(4)

$$\mathsf{\delta }=\mathsf{\alpha }\times {\mathrm{C}}_{\mathrm{D}\mathrm{R}\mathrm{A}}^{\mathsf{\beta }}$$

(5)

where $w*$ is the onset wave number for drag reduction, α and β are constants. Virk also defined the maximum drag reduction asymptote [19]:

$$\sqrt{{\displaystyle \frac{1}{f}}}=19{\log }_{10}\left(Re\sqrt{f}\right)-32.4$$

(6)

In Virk’s model, the critical wall-shear stress (${\tau }_w^{*}$) is the single parameter that marks the onset of the transition from ordinary Newtonian turbulence to the polymer-dominated (drag-reducing) regime. When $\mathsf{\tau }>{\tau }_w^{*}$, the velocity profile acquires an effective slip, and the friction factor departs onto the polymeric-regime curve, ultimately bounded by the Virk maximum-drag-reduction asymptote. This transition is a shift in the friction factor trend due to the onset of drag reduction caused by polymer additives.

Mogensen et al. [20,21] extended Virk’s work by integrating the Colebrook–White equation with DRA effects, yielding a unified formula for rough pipes:

$$\sqrt{{\displaystyle \frac{1}{f}}}=-4\times {\log }_{10}({\displaystyle \frac{1.26}{Re\sqrt{f}}}+{\displaystyle \frac{\epsilon }{3.7D}})-\delta {\log }_{10}({\displaystyle \frac{1.26}{Re\sqrt{f}}})-\delta {\log }_{10}\left(\sqrt{2}Dw*\right)$$

(7)

This model accounts for both pipe roughness and DRA concentration, enabling accurate friction predictions across a wide range of flow regimes. Mogensen et al. validated this approach against field data, demonstrating its utility in optimizing LEL designs.

In summary, the evolution of pipe friction models—from classical Colebrook–White to unified DRA-aware formulations—has been critical for designing LEL completions.

3.2. Orifice Flow and Throttling Effect

The orifices in LEL constitute another critical factor influencing pressure distribution. During acid injection operations, the fluid exiting through orifices creates substantial instantaneous pressure drops. As presented in Table 2, the researchers investigated the orifice friction from various aspects, including experimental and numerical simulation studies.

The local resistance calculation formula for orifices is as follows [22]:

$$∆{\mathrm{P}}_{\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{e}}=-{\displaystyle \frac{0.2369\mathsf{\rho }\times {\mathrm{Q}}_{\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{e}}^2}{C_D^2\times D_{hole}^4}}$$

(8)

where ${\mathrm{Q}}_{\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{e}}$ is flow rate through the orifice, $D_{hole}$ is the orifice diameter, $C_D$ is orifice discharge coefficient.

Table 2. Summary of Orifice Friction Studies.

Authors	Year	Study Types	Main Focus
Lord [23]	1994	Experimental Study	Perforation friction pressure for water, linear polymer solutions, and crosslinked gels
Crump & Conway [24]	1988	Experimental Study	Perforation erosion effects on pressure drop
El-Rabba [25]	1999	Experimental Study	Discharge coefficient for polymer solutions and slurries
Willingham [26]	1993	Experimental Study	Sand bridging and dynamic changes in discharge coefficient
McLemore [27]	2013	Experimental Study	Discharge coefficient for orifices in curved pipes
Wang [28]	2022	Numerical Study	First fully coupled 3-D fracture-to-reservoir simulator; demonstrated that perforation density/diameter must be inversely scaled to (permeability × minimum horizontal stress) within each cluster
Kebert [29]	2023	Numerical Study	Modeled dynamic perforation erosion throughout a fracturing stage; identified two erosion phases

Table 2. Summary of Orifice Friction Studies.

Authors	Year	Study Types	Main Focus
Lord [23]	1994	Experimental Study	Perforation friction pressure for water, linear polymer solutions, and crosslinked gels
Crump & Conway [24]	1988	Experimental Study	Perforation erosion effects on pressure drop
El-Rabba [25]	1999	Experimental Study	Discharge coefficient for polymer solutions and slurries
Willingham [26]	1993	Experimental Study	Sand bridging and dynamic changes in discharge coefficient
McLemore [27]	2013	Experimental Study	Discharge coefficient for orifices in curved pipes
Wang [28]	2022	Numerical Study	First fully coupled 3-D fracture-to-reservoir simulator; demonstrated that perforation density/diameter must be inversely scaled to (permeability × minimum horizontal stress) within each cluster
Kebert [29]	2023	Numerical Study	Modeled dynamic perforation erosion throughout a fracturing stage; identified two erosion phases

The throttling effect of a perforation refers to the significant pressure drop generated when fluid passes through the restricted orifice. The sudden contraction sharply reduces the flow area, forcing the fluid to accelerate, and—by Bernoulli’s principle—the resulting increase in kinetic energy is balanced by a corresponding decrease in pressure. The discharge coefficient ($C_D$) is one of the key indicators reflecting the magnitude of this throttling effect. As shown in Figure 5, the discharge coefficient reflects the drag reduction effect of the orifice and is related to factors such as the size and shape of the orifice. Therefore, many studies related to orifice friction resistance take $C_D$ as the starting point. The industry currently widely employs the sharp-edge orifice equation—developed for Newtonian fluids (such as water)—to estimate orifice friction. However, the discharge coefficient in this equation actually varies depending on fluid type, orifice size, and flow conditions.

The study by Lord et al. [23] systematically investigated the friction characteristics of different fluids (water, linear polymer solutions, and crosslinked gels) passing through orifices of varying sizes (1/4, 3/8, and 1/2 inches) using a large-scale fracturing flow simulation apparatus. The result shows that the discharge coefficient decreases significantly as the perforation size decreases, and there are clear differences between fluid types. This indicates that perforation pressure loss is influenced not only by fluid density but also closely related to fluid viscosity and perforation size. For crosslinked fluids, the results show a slight decrease in the discharge coefficient as perforation size increases, and the coefficients for crosslinked fluids are generally lower than those for non-crosslinked fluids. This further confirms the presence of an excess entry loss at the perforation entrance for crosslinked fluids, caused by elongational deformation due to their high viscoelasticity.

Crump and Conway [24] established foundational insights through systematic laboratory experiments. They designed three sets of experiments (including drilled and shaped-charge orifices) and quantified the increase in from an initial value of 0.56 to over 0.9 due to orifice erosion by pumping slurries of varying concentrations. Their work revealed that erosion rates intensify with higher sand concentrations (>12 lb/gal), smaller initial orifice diameters (<0.3 in), and greater pressure differentials (>500 psi). During LEL operation, the acid continuously erodes the perforation edge, changing its shape and thus the discharge coefficient, as illustrated in Figure 5. Higher sand content reduces flow efficiency through the perforations, increasing pressure drop. This illustrates the direct impact of proppant concentration on perforation performance during fracturing. Additionally, they proposed optimization strategies for limited-entry fracturing designs, such as using larger orifices and dynamically adjusting pumping rates. However, the model’s inability to account for multi-parameter interactions—particularly fluid rheology and proppant size distribution—limited its predictive accuracy in field conditions.

El-Rabba et al. [25] experimentally investigated the discharge coefficient $C_D$ for linear polymer solutions, titanium/borate-crosslinked gels, and proppant-laden fracturing slurries, developing corresponding correlation models. Their research revealed that $C_D$ depends not only on fluid apparent viscosity and orifice diameter but is also significantly influenced by proppant concentration, particle size, and flow conditions. For clean fluids, the authors established predictive $C_D$ models based on fluid viscosity and orifice size. In contrast, for slurries, erosion effects cause dynamic changes in $C_D$, with its variation closely linked to the kinetic energy of proppant (flow rate and cumulative proppant mass). Additionally, the study demonstrated that larger proppant (e.g., 12/20 mesh) induces more severe orifice erosion, while slurries with intermediate proppant concentrations exhibit the highest erosion rates.

Willingham, Tan, and Norman [26] conducted a systematic study of the orifice friction pressure behavior of fracturing fluid slurries through innovative experiments using transparent and high-pressure wellbore models. Their work stands out for combining visual observation with quantitative analysis, uncovering critical conditions for proppant bridging and dynamic changes in, which were previously overlooked in conventional theories. They found that the ratio of orifice diameter to proppant size (Dperf/Dproppant) is the key parameter controlling sand bridging (requiring a ratio > 5), while fluid viscosity had a much smaller impact on orifice pressure drop than expected—a conclusion that challenged the prevailing assumption about the advantages of high-viscosity fluids in proppant transport. By comparing experiments with tungsten carbide orifice materials, the authors were the first to quantify the effect of material hardness on erosion and developed a dynamic prediction model based on cumulative sand volume. Their proposed field correction method translates laboratory data into actionable orifice design and real-time pressure monitoring procedures, significantly improving the accuracy of bottomhole pressure calculations in limited-entry fracturing.

McLemore et al. [27] investigated the discharge coefficient for orifices cut into round pipes, focusing on the influence of pipe curvature on orifice area and flow characteristics. Through experimental and theoretical analyses, the authors compared four different definitions of orifice area and found that it significantly decreases as the ratio of orifice diameter to pipe diameter (d/D) increases, while it slightly increases with a reduction in the head-to-orifice-diameter ratio (h/d). Photographic observations of the vena contracta revealed that pipe curvature induces lateral flow contraction, further explaining the variations in. Ultimately, the authors proposed an empirical formula based on the elliptical area [27]:

$$C_D=0.610-0.458{({\displaystyle \frac{D_{hole}}{D}})}^{0.647}+0.309{({\displaystyle \frac{h}{d}})}^{-0.169}$$

(9)

where $D_{hole}$ is the diameter of the orifice, D is the diameter of the pipe, h is the head above the center of an orifice.

This model achieved a coefficient of determination R² of 88.8% and a root mean square error (RMSE) of 0.028, providing a more accurate prediction of flow rates for perforated risers.

Mary S. Van Domelen and colleagues present a focused investigation into the optimization of Controlled Acid Jetting (CAJ) stimulations in tight chalk reservoirs, specifically within the Danish Central Graben. Unlike conventional fracturing studies, their work emphasizes matrix acidizing under limited-entry conditions, where orifice pressure drop and the discharge coefficient critically influence acid distribution across ultra-long horizontal wellbores (up to 20,000 ft) [30].

ElGibaly and Osman [31] developed a novel artificial neural network (ANN)-based model to predict orifice friction in limited-entry hydraulic fracturing, addressing the critical challenge of orifice erosion caused by proppant-laden slurry. Leveraging real-world field data from 47 fracturing treatments, the authors employed a machine learning approach to capture complex, nonlinear interactions among key parameters—such as proppant mass, injection rate, and fluid rheology—that traditional empirical and physical models often overlook. Their optimized ANN architecture, featuring 17 hidden neurons and trained via backpropagation, demonstrated superior predictive accuracy (R = 0.95, MSE = 8.38 × 10⁻⁴) compared to six existing correlations, outperforming them in statistical metrics and robustness. The study highlights the transformative potential of machine learning in enhancing fracture design by providing reliable post-erosion friction estimates, ultimately improving fluid diversion efficiency. Future work, as noted by the authors, could further refine the model’s stability through advanced bifurcation analysis, underscoring the dynamic adaptability of ANN-based solutions in petroleum engineering challenges.

Wang et al. [28] had presented a coupled workflow that simultaneously predicted 3-D fracture geometry in each heterogeneous layer and had fed the resulting conductivity field directly into a black-oil reservoir simulator. This closed-loop approach had exposed two new insights:

• Perforation density and diameter had been shown to require inverse scaling with the product of permeability and minimum horizontal stress within each cluster, yielding a 10–20% uplift that single-objective optimizations had missed.
• In their PengLai case study, high-permeability layers had initially been observed to “steal” slurry, yet the coupled model had predicted an early screen-out that diverted fluid to low-quality zones—an effect that standalone fracture simulators had neglected.

Recent CFD modeling by Kebert et al. [29] provides unprecedented insight into the dynamic erosion of perforations—orifices in LEL completions—during hydraulic fracturing. Using pseudo-transient simulations calibrated with field imaging, the authors identified two distinct erosion phases: an early “perforation rounding” phase dominated by the total number of proppant particles passing through each orifice, and a later “stable growth” phase governed by particle inertia (Stokes number) and the instantaneous pressure drop across the orifice. Their results demonstrate that smaller initial orifice diameters and proppant-ramping schedules significantly mitigate erosion, preserving the designed throttling pressure and sustaining uniform fluid distribution throughout the treatment. This work underscores the need to couple orifice diameter selection with real-time erosion monitoring to maintain LEL performance under high-rate, proppant-laden slurry conditions.

The evolution of orifice friction modeling reflects a paradigm shift from isolated parameter studies to integrated system analyses. Early physical experiments by Crump and Willingham successfully identified key erosion mechanisms but were constrained by laboratory-scale simplifications. The ANN approach bridges this gap by capturing complex field realities, though it demands extensive calibration datasets. Future advancements should focus on real-time Cd monitoring through distributed acoustic sensing (DAS) and the development of erosion-resistant orifice materials that maintain designed pressure drops throughout treatment durations. This progression from empirical observations to data-driven modeling has transformed orifice friction from an unpredictable variable to a manageable design parameter in limited-entry operations.

3.3. Near-Wellbore Coupling

The study of annular flow in LEL completions is critical for optimizing stimulation treatments in long horizontal wells, particularly in heterogeneous carbonate reservoirs. The annular region between the liner and wellbore plays a pivotal role in fluid placement during acidizing or other injection processes. Understanding the flow dynamics in this space is essential to ensure uniform fluid distribution, mitigate heel–toe effects, and maximize stimulation efficiency. However, the complex interplay of turbulent jets from orifices, multi-directional flow, and fluid mixing poses significant modeling challenges, especially when scaling up from laboratory experiments to field-scale applications [32].

Annular flow refers to fluid movement in the annular space between two concentric cylinders, such as between a casing and tubing in oil and gas wells. In Mogensen et al. [21]’s paper, relevant frictional formulas for annular flow were also proposed.

The total pressure drop in annular flow ${∆P}_{annular}$ consists of three main components [33]:

$${∆P}_{annular}={∆P}_{friction}+{∆P}_{hydrostatic}+{∆P}_{acceleration}$$

(10)

where ${∆P}_{friction}$ is frictional pressure drop (caused by fluid shear against pipe walls). ${∆P}_{hydrostatic}$ is hydrostatic pressure drop (due to fluid density and elevation changes). ${∆P}_{acceleration}$ is acceleration pressure drop (resulting from flow velocity variations).

The frictional component depends on flow regime (laminar or turbulent) and fluid rheology (Newtonian or non-Newtonian).

For fluids like drilling mud, the frictional pressure drop is calculated as follows [33]:

$${∆\mathrm{P}}_{\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}}=\mathrm{L}\times ({\displaystyle \frac{\mathrm{P}\mathrm{V}\times {\mathrm{v}}_{\mathrm{a}}}{1000({\mathrm{D}}_{\mathrm{o}}^2-{\mathrm{D}}_{\mathrm{i}}^2)}}+{\displaystyle \frac{\mathrm{Y}\mathrm{P}}{200({\mathrm{D}}_{\mathrm{o}}-{\mathrm{D}}_{\mathrm{i}})}})$$

(11)

$${\mathrm{v}}_{\mathrm{a}}={\displaystyle \frac{0.011914\mathrm{Q}}{{\mathrm{D}}_{\mathrm{o}}^2-{\mathrm{D}}_{\mathrm{i}}^2}}$$

(12)

where $\mathrm{P}\mathrm{V}$ is plastic viscosity, $\mathrm{Y}\mathrm{P}$ is yield point, ${\mathrm{v}}_{\mathrm{a}}$ is annular velocity, ${\mathrm{D}}_{\mathrm{o}} \mathrm{and} {\mathrm{D}}_{\mathrm{i}}$ are outer and inner diameters of the annulus.

For Newtonian fluids, the pressure drop is [33]:

$$∆{\mathrm{P}}_{\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}}={\displaystyle \frac{11.4015\mathrm{f}\times \mathsf{\rho }\times \mathrm{L}\times \mathrm{Q}}{{({\mathrm{D}}_{\mathrm{o}}^2-{\mathrm{D}}_{\mathrm{i}}^2)}^2({\mathrm{D}}_{\mathrm{o}}-{\mathrm{D}}_{\mathrm{i}})}}$$

(13)

Fanning friction factor, and annular Reynolds number ${Re}_a$ are determined by [33]:

$$\sqrt{{\displaystyle \frac{1}{f}}}=-4\times {\log }_{10}({\displaystyle \frac{1.26}{{Re}_a\times \sqrt{f}}}+{\displaystyle \frac{\epsilon }{3.7\times (D_o-D_i)}})$$

(14)

$${Re}_a={\displaystyle \frac{15916\times \frac{2}{3}\times \rho \times Q}{\mu \times (D_o+D_i)}}$$

(15)

By accurately modeling annular flow, engineers can optimize wellbore operations, reduce energy losses, and improve treatment efficiency. The integration of DRA effects further enhances pressure management in high-rate injection scenarios, such as matrix acidizing and hydraulic fracturing.

Mayer’s research addresses these challenges through an integrated approach combining 1D computational modeling, large-scale experiments, and detailed CFD simulations. The experimental setup features a 27-ft transparent wellbore section with a perforated aluminum liner, designed at a 1:1 scale to replicate field conditions. Fluids are pumped through the liner orifices to displace annular fluid, with high-speed imaging capturing the displacement process. The study focuses on turbulent flow regimes (6–15 gpm per orifice) and examines configurations ranging from single (1 × 6 mm²) to multiple orifices (2 × 4 mm²). Complementary CFD simulations using ANSYS Fluent (v13.0) resolve the turbulent jet structures and mixing zones with high precision, employing 6 million grid points to validate the experimental observations [34].

Key findings from the study demonstrate that annular flow rapidly transitions from a complex 3D jet near the orifice to a uniform 1D flow within approximately 80 cm downstream, even for a single orifice. This homogenization occurs due to turbulent momentum transfer, which dominates interfacial diffusion between fluids. The CFD simulations show excellent agreement with experimental pressure drop measurements, with only a 13% deviation in discharge coefficients. The results confirm that the 1D modeling assumption is valid for typical stimulation conditions, as turbulent mixing ensures uniform annular displacement. However, the study also highlights the extended mixing zones created by limited-entry liners, which may impact the efficiency of fluid displacement. These insights provide a foundation for optimizing orifice designs and completion strategies, ensuring effective stimulation in long horizontal wells while reducing operational uncertainties.

3.4. Reservoir Seepage

The study of reservoir seepage in LEL systems is critical for addressing the challenges posed by heterogeneous formations, where uneven inflow profiles and premature water breakthrough can severely compromise production efficiency.

For reservoir flow, Darcy’s law serves as the fundamental governing equation, while accounting for factors such as formation permeability heterogeneity and multiphase flow. In acidizing processes, reservoir flow must additionally consider permeability alterations caused by wormhole formation (which will be discussed in detail in later sections). For LEL production operations, designers are particularly concerned with how formation heterogeneity affects production profiles (i.e., the flow rate distribution across different segments of the LEL completion). As shown in Figure 6, the reservoir permeability exhibits significant heterogeneity along the wellbore, leading to excessively high inflow velocities in high-permeability zones and premature water breakthrough.

Wang et al. [35] proposed an orifice optimization method for horizontal wells based on genetic algorithms to address the issue of uneven production profiles caused by permeability heterogeneity in reservoirs. The study employed a coupled reservoir-wellbore flow model to analyze the impact of permeability distribution along the wellbore on inflow velocity. By optimizing orifice density—using high orifice density in low-permeability zones and low orifice density in high-permeability zones—the production profile was effectively balanced. After optimization, the inflow velocity decreased in high-permeability zones and increased in low-permeability zones, resulting in a production profile closer to the ideal state, thereby delaying water breakthrough and improving recovery efficiency. This model provides a critical theoretical foundation for optimizing completion parameters in horizontal wells.

The coupling between reservoir flow and wellbore hydraulics is primarily addressed from the perspectives of flow continuity and pressure continuity. For instance, Zhou et al. [36,37,38] established a coupled model. The coupling of reservoir flow and wellbore hydraulics in horizontal wells is achieved through an iterative, pressure-flux feedback mechanism that resolves the interdependence between near-wellbore inflow and axial wellbore flow.

At each orifice cluster, the reservoir delivers influx $Q_i$ based on local drawdown $p_{res}-p_{wi}$, where $p_{wi}$ is the wellbore pressure at the i-the orifice. This influx is governed by the reservoir’s effective permeability and orifice geometry, with heterogeneity introducing spatial variability in inflow capacity. Simultaneously, the wellbore acts as a dynamic conduit: as fluid enters through orifices, cumulative flow increases toward the heel, driving frictional pressure losses that elevate $p_{wi}$ downstream.

The coupling is implemented numerically via sequential iteration:

• Reservoir Step: For an initial guess of wellbore pressures $p_{wi}$, solve the inflow problem $Q_i=J_i(p_{res}-p_{wi})$, where $J_i$ incorporates local permeability and orifice density.
• Wellbore Step: Update $p_{wi}$ along the wellbore by integrating the variable-mass flow equation, accounting for friction and acceleration effects from the influx $Q_i$.
• Convergence: Repeat until $p_{wi}$ and $Q_i$ stabilize, ensuring consistency between reservoir deliverability and wellbore hydraulics.

Key implications of this coupling include self-reinforcing inflow skewness, where high-permeability zones initially attract more flow but the consequent wellbore pressure increase may suppress subsequent influx, thus requiring orifice optimization. Additionally, segmented effects emerge in heterogeneous reservoirs as the coupling must account for abrupt transitions in $J_i$, demanding adaptive discretization near permeability boundaries for accurate resolution.

This bidirectional coupling ensures that completion designs (e.g., orifice density) account for both reservoir heterogeneity and wellbore flow constraints, moving beyond decoupled approximations.

Marett and Landman’s model uniquely integrates skin factor decomposition and multiphase flow corrections to enhance the reservoir-wellbore coupling. The total skin factor $s_t$ is partitioned into wellbore damage $s_w$ from mud invasion (based on k$/k_{wd}$) and orifice damage $s_p$, with the latter further differentiated for orifices inside $L_d$ and outside $L_u$ the damaged zone via an equivalent radius to adjust inflow efficiency. For multiphase flow, the model employs the Beggs–Brill correlation to compute gas–liquid frictional pressure drop and introduces a flow loss coefficient (K = 0.1) to account for orifice inflow disturbances, reducing production overestimation in high-GOR scenarios. This approach bridges theoretical predictions with field observations by quantifying permeability heterogeneity through measurable damage parameters and dynamically coupling multiphase wellbore hydraulics with reservoir deliverability [39,40].

In conclusion, reservoir seepage studies in LEL systems underscore the importance of dynamic, data-driven orifice strategies to mitigate heterogeneity-induced challenges. Marett and Landman’s model, alongside Wang et al.’s optimization techniques, exemplifies how advanced methodologies can translate permeability maps and flow dynamics into actionable completion designs, paving the way for more efficient and sustainable hydrocarbon recovery. Future advancements may further refine these models by incorporating real-time monitoring data or machine learning to enhance predictive accuracy under complex reservoir conditions.

4. Acidizing Process of LEL

4.1. Wormhole

The formation of wormholes during carbonate acidizing is a critical phenomenon that directly impacts the efficiency of LEL completions. As shown in Figure 7, wormholes are highly conductive channels [41] created by the dissolution of carbonate rock when reactive fluids, such as hydrochloric acid, interact with the formation. A wormhole signifies that the rock has been dissolved by acid, resulting in increased permeability and decreased porosity, thereby facilitating the flow of fluids such as acid and oil. For LEL systems, understanding wormhole dynamics is essential to design orifice strategies that ensure uniform acid distribution along horizontal wellbores, mitigating the heel–toe effect and addressing reservoir heterogeneity.

As shown in Figure 8, research by Fredd and Fogler [42,43,44] systematically investigated the relationship between PVbt and dissolution structures under varying injection conditions. Their work identified five distinct wormhole patterns, each associated with specific PVbt ranges and flow regimes:

Face dissolution: Occurs at very low injection rates, where acid is consumed near the injection face, resulting in high PVbt and negligible penetration.

Conical wormholes: Characterized by tapered channels due to limited acid penetration, exhibiting intermediate PVbt values.

Dominant wormholes: The most efficient pattern, featuring a single, narrow channel with minimal PVbt, achieved at optimal injection rates.

Ramified wormholes: Form at higher flow rates, displaying branched structures and increased PVbt due to dispersed acid consumption.

Uniform dissolution: Occurs at excessive flow rates, where acid is uniformly distributed without forming effective channels, leading to the highest PVbt.

Fredd’s studies further demonstrated that the Damköhler number (Da), which balances reaction and convection rates, governs the transition between these dissolution patterns. An optimum Da value (~0.29) was identified, corresponding to the formation of dominant wormholes and the lowest PVbt. This insight is particularly valuable for designing LEL systems, as it highlights the importance of controlling injection rates to achieve efficient stimulation.

The practical implications of PVbt and wormhole morphology are significant for LEL applications. By leveraging PVbt curves, engineers can tailor orifice designs and injection rates to target dominant wormhole formation, ensuring uniform stimulation across heterogeneous formations. For instance, high-permeability zones may require restricted orifices to avoid premature breakthrough, while low-permeability regions benefit from increased orifice density to enhance wormhole initiation. Thus, understanding PVbt and its relationship with wormhole patterns is essential for optimizing acidizing treatments and maximizing reservoir contact.

As shown in Figure 9, injection velocity profoundly influences PVbt and wormhole morphology. At low flow rates, acid predominantly reacts near the wellbore, resulting in face dissolution and high PVbt. As the flow rate increases, the balance between advection and reaction shifts, promoting the formation of dominant wormholes with significantly reduced PVbt. This optimal flow rate represents the most efficient acid utilization, where convection delivers acid to the wormhole tip while diffusion and reaction kinetics are balanced [45]. However, exceeding this optimal rate leads to ramified or uniform dissolution, as excessive advection prevents localized acid-rock interaction, increasing PVbt and reducing stimulation effectiveness. For LEL applications, this underscores the need to calibrate orifice designs and injection rates to achieve optimal flow rate and dominant wormhole conditions, ensuring uniform stimulation across heterogeneous formations [46].

As shown in

Table 3. Influencing factors of wormhole growth.

Influencing Factors	Authors	Main Findings
Flow Rate	Fredd [44]	Determines dissolution patterns: low rates cause face dissolution, optimal rates form dominant wormholes, and high rates lead to ramified/uniform dissolution
Reaction Rate	Fredd [45]	Governed by acid-rock kinetics. Faster reactions (e.g., HCl) require higher flow rates to balance advection-diffusion-reaction. Retarded acids (e.g., emulsified/organic acids) enable deeper penetration
Acid Concentration	Dong [46]	Higher concentrations reduce PVbt (pore volume to breakthrough) but have diminishing returns above 15% wt. HCl. Optimal flow rate increases with concentration
Temperature	Fredd [43]	Increases reaction kinetics: higher temperatures raise PVbt and shift optimal flow rates right in limestones but may reduce PVbt in dolomites. Exothermic heat has minor impact
Core Dimensions	Dong [47]	Longer cores (>6 inches) stabilize optimal interstitial velocity. Larger diameters reduce PVbt due to reduced radial acid loss
Heterogeneity/ Porosity	Zakaria [48]	Vugs and pore structure alter flow paths. Higher flowing fraction (accessible porosity) increases PVbt. Heterogeneous rocks (e.g., vuggy) form wormholes faster

, the optimal injection rate is influenced by factors such as acid concentration, acid type, rock type, temperature, and other conditions [47,48,49].

The insights from PVbt studies directly translate to LEL optimization. By leveraging PVbt curves, engineers can tailor orifice densities and injection rates to target dominant wormhole formation, minimizing acid waste and maximizing reservoir contact.

4.2. Semi-Empirical Model for Wormhole

Extensive research had previously been devoted to the development of quantitative models for wormhole growth [50,51,52].

Buijse and Glasbergen [52] developed a semi-empirical model to predict wormhole growth during carbonate acidizing, offering a practical tool for field engineers. The model captures the essential physics of wormhole propagation while incorporating empirical parameters derived from laboratory or field data. The wormhole growth rate $V_{wh}$ is expressed as a function of the interstitial acid velocity $V_i$:

$$V_{wh}=W_{eff}·V_i^{2/3}·B\left(V_i\right)$$

(16)

where $W_{eff}$ is the wormhole efficiency factor, and $B\left(V_i\right)$ is a function accounting for the compact dissolution regime at low injection rates [52]:

$$B\left(V_i\right)={(1-exp(-W_B·V_i^2\left)\right)}^2$$

(17)

The model hinges on two key parameters, $W_{eff}$ and $W_B$, which are determined experimentally. $W_{eff}$ reflects the overall efficiency of wormhole formation, while $W_B$ governs the transition from compact dissolution to wormholing. These parameters can be derived from core flow tests or field data, making the model adaptable to diverse acid-rock systems.

The breakthrough pore volume PVbt, a critical metric for acidizing efficiency, is inversely related to the wormhole growth rate [52]:

$${PV}_{bt}={\displaystyle \frac{V_i}{V_{wh}}}={\displaystyle \frac{V_i^{1/3}}{W_{eff}·B\left(V_i\right)}}$$

(18)

This relationship highlights the dependence of PVbt on injection velocity. At low $V_i$, acid spending near the wellbore results in compact dissolution and high PVbt. As $V_i$ increases, wormhole formation becomes more efficient, reaching an optimum where PVbt is minimized. Beyond this optimum, excessive injection rates lead to ramified wormholes and reduced efficiency.

The model also addresses radial flow conditions, where the interstitial velocity decreases with distance from the wellbore. Here, the wormhole growth rate diminishes as the acid front advances, reflecting the impact of radial fluid loss. This behavior is consistent with field observations, where deeper wormhole penetration requires higher injection rates to maintain efficiency.

In summary, Buijse and Glasbergen’s semi-empirical model provides a robust framework for optimizing carbonate acidizing treatments. By linking wormhole growth to measurable parameters, it bridges laboratory insights with field applications, enabling engineers to design treatments that maximize stimulation efficiency while minimizing acid usage. The model underscores the importance of injection rate control and adapts to both linear and radial flow geometries, making it a versatile tool for acidizing design.

Furui et al. [53] made significant improvements to Buijse and Glasbergen’s semi-empirical wormhole growth model, addressing the limitations of traditional models in predicting wormhole penetration depth and completion skin factors. By incorporating core size effects and a fluid loss limitation coefficient (γ), they refined the calculation method for interstitial velocity at the wormhole tip, making it more consistent with actual acid stimulation conditions. The authors proposed a modified wormhole growth rate equation, which retained the core parameters of optimal injection velocity and pore volume to breakthrough from Buijse’s model while validating the inverse relationship between wormhole tip velocity and core diameter through 3D finite element simulations.

In terms of numerical simulation, Furui et al. developed an innovative 3D finite element model to study wormhole growth characteristics under different flow geometries. For radial flow, they simulated a 60° symmetric wormhole network and found that the decay rate of interstitial velocity at the wormhole tip was significantly slower than that predicted by conventional radial flow theory. For spherical flow, the authors assumed an icosahedral symmetry in wormhole distribution and, through FEM simulations, confirmed that velocity decayed only as 1/r_wh, rather than the 1/r_wh^2 relationship predicted by classical spherical flow models. These simulation results not only validated the accuracy of their semi-analytical correlations but also revealed the dynamic mechanisms of fluid competition in multi-wormhole systems. This provided a theoretical foundation for understanding the highly efficient propagation of wormholes at field scales.

4.3. Analytical Model for Wormhole

Currently, scholars have developed various mathematical models to describe the diffusion and flow of acid in fractured carbonate porous media. Typical models include: the capillary model [54], the network model [55], and the fractal model [56].

Panga et al. [57] present a two-scale continuum model that simulates wormhole formation during the acid stimulation of carbonate rocks. The model couples macroscopic (Darcy-scale) and pore-scale physics to describe reactive dissolution, capturing the interplay between fluid flow, transport, and chemical reactions in porous media.

The model solves the following governing equations at the Darcy scale:

Darcy’s Law: This expression relates the Darcy (volume-averaged) velocity U to the pressure gradient through the permeability tensor K, providing the macroscopic momentum balance required to couple pore-scale flow with reservoir-scale pressure fields.

$$\mathbf{U}=-{\displaystyle \frac{1}{\mathsf{\mu }}}\mathbf{K}·\nabla \mathrm{P}$$

(19)

where $\mathbf{U}$ is Darcy velocity vector, $\mathbf{K}$ is permeability tensor, P is pressure.

Continuity Equation (mass conservation): Equation (20) enforces overall mass balance by stating that any temporal change in porosity must be exactly compensated by the divergence of the Darcy velocity field, thereby tracking how dissolution alters pore volume and, in turn, the flow capacity.

$${\displaystyle \frac{\partial \phi }{\partial t^{\prime }}}+\mathsf{\nabla }·\mathit{U}=0$$

(20)

Species Transport Equation (acid concentration): It balances the accumulation, convection, and dispersion of the acid species against the mass-transfer-limited reaction term, explicitly linking the cup-mixing concentration Cf in the flowing fluid to the surface concentration Cs at the dissolving solid interface.

$${\displaystyle \frac{\partial \phi C_f}{\partial t^{\prime }}}+\mathsf{\nabla }·(\mathit{U}{C}_f)=\mathsf{\nabla }·(\phi D_e·\mathsf{\nabla }{C}_f)-k_c{a}_s(C_f-C_s)$$

(21)

where $C_f$ is cup-mixing acid concentration in the fluid phase, $C_s$ is acid concentration at the solid-fluid interface, $D_e$ is effective dispersion tensor, $k_c$ is local mass transfer coefficient, $a_s$ is interfacial area per unit volume.

Reaction Kinetics at the Pore Surface: For first-order kinetics, the surface reaction rate is simply the product of the intrinsic rate constant ks and the surface concentration Cs, providing the sink term required to model continuous carbonate dissolution.

$${\mathrm{k}}_{\mathrm{c}}({\mathrm{C}}_{\mathrm{f}}-{\mathrm{C}}_{\mathrm{s}})=\mathrm{R}\left({\mathrm{C}}_{\mathrm{s}}\right)$$

(22)

where $\mathrm{R}\left({\mathrm{C}}_{\mathrm{s}}\right)$ is surface reaction rate (for first-order kinetics, $\mathrm{R}\left({\mathrm{C}}_{\mathrm{s}}\right)={\mathrm{k}}_{\mathrm{s}}{\mathrm{C}}_{\mathrm{s}}$).

Porosity Evolution Equation: This equation quantifies how local porosity increases over time as solid mass is dissolved, with the rate proportional to both the reaction rate and the amount of solid dissolved per mole of acid reacted (α*).

$${\displaystyle \frac{\partial \phi }{\partial {\mathrm{t}}^{\prime }}}={\displaystyle \frac{\mathrm{R}\left({\mathrm{C}}_{\mathrm{s}}\right){\mathrm{a}}_{\mathrm{s}}\mathsf{\alpha }*}{{\mathsf{\rho }}_{\mathrm{s}}}}$$

(23)

where $\mathsf{\alpha }*$ is dissolving power (grams of solid dissolved per mole of acid reacted) ${\mathsf{\rho }}_{\mathrm{s}}$ is solid density.

The model links pore-scale changes to macroscopic properties via empirical relations:

Permeability-Porosity Relationship: An empirical power-law correlation that translates the dynamically evolving porosity into an updated permeability tensor, thereby allowing the dissolution-induced changes in flow paths to be captured in the Darcy-scale mode.

$${\displaystyle \frac{\mathrm{K}}{{\mathrm{K}}_0}}={\displaystyle \frac{\phi }{{\phi }_0}}{({\displaystyle \frac{\phi (1-{\phi }_0)}{{\phi }_0(1-\phi )}})}^{2\mathsf{\beta }*}$$

(24)

where ${\mathrm{K}}_0$, ${\phi }_0$ are initial permeability and porosity, $\mathsf{\beta }*$ is empirical parameter.

Average Pore Radius Evolution: It links the instantaneous pore radius to the current permeability and porosity, providing the geometric information needed to update mass-transfer coefficients and dispersion tensors as the pore structure enlarges.

$${\displaystyle \frac{{\mathrm{r}}_{\mathrm{p}}}{{\mathrm{r}}_0}}=\sqrt{{\displaystyle \frac{\mathrm{K}{\phi }_0}{{\mathrm{K}}_0\phi }}}$$

(25)

where ${\mathrm{r}}_{\mathrm{p}}$ is pore radius.

Interfacial Area Evolution: By expressing the reactive surface area per unit volume in terms of the evolving pore radius and porosity, this relation ensures that the rate of acid consumption declines appropriately as pores enlarge and the specific surface area diminishes.

$${\displaystyle \frac{{\mathrm{a}}_{\mathrm{v}}}{{\mathrm{a}}_0}}={\displaystyle \frac{\mathsf{\varphi }{\mathrm{r}}_0}{{\mathsf{\varphi }}_0{\mathrm{r}}_{\mathrm{p}}}}$$

(26)

where ${\mathrm{a}}_{\mathrm{v}}$ is interfacial area available for reaction per unit volume.

Compared to Buijse and Glasbergen’s semi-empirical formula, Panga’s model provides more in-depth insights by explicitly accounting for the acid reaction process. Additionally, by adopting a two-scale approach with Darcy-scale modeling as the primary focus, this framework significantly reduces the computational complexity of numerical simulations. As a result, Panga’s two-scale model has become widely adopted by researchers for numerical modeling of wormholes, serving as a valuable tool for investigating potential factors influencing wormhole growth patterns.

4.4. Skin Factor

The skin factor is a dimensionless parameter used in petroleum engineering to quantify the additional pressure drop near a wellbore caused by formation damage or stimulation. A negative skin factor indicates stimulation, while a positive one signifies damage. It plays a critical role in evaluating well performance and optimizing production strategies [58,59].

In their study, Schwalbert et al. [60] investigated the impact of anisotropic wormhole networks on well productivity in carbonate reservoirs. Schwalbert et al. [61] simulated the acidification process through the two-scale continuum model and found that the wormholes extended longer in the direction of high permeability, forming an elliptical/ellipsoidal network, as shown in Figure 10. They focused on both openhole and limited-entry completions, demonstrating that traditional isotropic assumptions can overestimate well performance in anisotropic formations. Their work combined numerical simulations with analytical derivations to propose more accurate skin-factor equations.

Traditionally, the Hawkins formula [62], assumes isotropic, radial wormhole growth:

$$\mathrm{S}=-\mathrm{l}\mathrm{n}({\displaystyle \frac{{\mathrm{r}}_{\mathrm{w}\mathrm{h}}}{{\mathrm{r}}_{\mathrm{w}}}})$$

(27)

where ${\mathrm{r}}_{\mathrm{w}\mathrm{h}}$ is the wormhole radius, ${\mathrm{r}}_{\mathrm{w}}$ is the wellbore radius.

However, the authors highlighted its limitations in anisotropic formations and introduced revised formulas. For openhole completions, they recommended, incorporating anisotropy ratio ${\mathrm{I}}_{\mathrm{a}\mathrm{n}\mathrm{i}}$. For limited-entry completions, they derived Equations accounting for ellipsoidal wormhole networks and partial-completion effects. These equations provide conservative estimates, ensuring realistic productivity predictions.

$$\mathrm{S}=-\mathrm{l}\mathrm{n}\{{\displaystyle \frac{1}{{\mathrm{I}}_{\mathrm{a}\mathrm{n}\mathrm{i}}+1}}[{\displaystyle \frac{{\mathrm{r}}_{\mathrm{w}\mathrm{h}\mathrm{H}}}{{\mathrm{r}}_{\mathrm{w}}}}+\sqrt{{({\displaystyle \frac{{\mathrm{r}}_{\mathrm{w}\mathrm{h}\mathrm{H}}}{{\mathrm{r}}_{\mathrm{w}}}})}^2+{{\mathrm{I}}_{\mathrm{a}\mathrm{n}\mathrm{i}}}^2-1}]\}$$

(28)

where ${\mathrm{r}}_{\mathrm{w}\mathrm{h}\mathrm{H}}$ is the wormhole radius in the Horizontal direction.

The authors conducted a comprehensive sensitivity analysis on the skin factor to quantify the error introduced by assuming isotropic wormhole networks in anisotropic formations.

The overestimation error in well productivity due to assuming a circular wormhole network (instead of an elliptical one) was systematically evaluated under various reservoir and completion conditions. Key observations include:

The error increases significantly with higher reservoir anisotropy ratios ${\mathrm{I}}_{\mathrm{a}\mathrm{n}\mathrm{i}}$; it becomes negligible for near-isotropic formations but reaches substantial levels (e.g., ~30%) when ${\mathrm{I}}_{\mathrm{a}\mathrm{n}\mathrm{i}}$ = 10.

Larger stimulation treatments (i.e., greater wormhole radius lead to larger errors.

The error decreases with larger well spacing and increases with formation thickness until it plateaus.

Furthermore, the author highlighted the importance of stimulation coverage in limited-entry completions. The skin factor is shown to be a function of rather than directly dependent on anisotropy when the stimulated volume per foot is constant. The results demonstrate that:

There exists an optimal stimulation coverage range of approximately 60–70%, where the skin factor is minimized.

For coverages below this range, the skin factor increases due to partial-completion effects.

The curves for different anisotropy ratios overlap closely, indicating that the relationship between and skin factor is universal across isotropic and anisotropic formations.

These findings underscore the necessity of using the proposed anisotropic skin factor equations for accurate productivity prediction and acidizing design in carbonate reservoirs with permeability anisotropy.

4.5. LEL with Viscoelastic Diverting Acid

Viscoelastic Diverting Acid (VEDA) is a surfactant-based, self-gelling acid system that provides temporary chemical diversion during acid stimulation treatments. Unlike conventional acids, VEDA exhibits unique rheological behavior: its viscosity remains low during initial injection, allowing it to pass easily through LEL orifices, but increases significantly as the acid reacts with carbonate rock and its pH rises. This viscosity build-up forms a temporary, viscoelastic barrier in high-permeability or already-stimulated zones, diverting subsequent acid fluid toward lower-permeability or under-treated regions. Once the acid is largely consumed, the viscosity drops sharply, restoring injectivity and allowing continued treatment without permanent formation damage.

When combined with a Limited-Entry Liner, VEDA adds a chemical diversion mechanism to the mechanical diversion provided by the orifice-based completion. This dual-diversion approach is particularly effective in highly heterogeneous carbonate reservoirs, where permeability variations can lead to uneven acid distribution. The viscosity development profile of VEDA is concentration-dependent: systems with reduced surfactant (VES) concentration exhibit an earlier viscosity peak and quicker breakdown, reducing the risk of excessive near-wellbore restriction while still achieving effective diversion.

Elhadidy et al. [63] restimulated a slotted-liner horizontal well in Egypt’s ARX tight carbonate after an initial treatment failed due to coiled-tubing sticking. A low-rate (0.75 bbl/min) VEDA pill with fibers was pumped to chemically divert acid to bypassed sections, followed by 15% HCl at high rate. Productivity index jumped from 3 to 10 BBL/psi and remained stable, showing VEDA’s value when mechanical access is limited.

Morrow et al. [64] documented the first field application of VEDA in an LEL water-injector onshore UAE, where permeability varied six- to fourteen-fold along the lateral. Pre-job modeling showed that a VEDA pre-flush would improve acid placement at the low-perm toe. During execution, VEDA passed 4 mm LEL orifices, built > 600 psi diversion pressure, and quadrupled injectivity. Post-stimulation logs revealed toe uptake rising from zero to 370 bbl/d, confirming that VEDA effectively complements LEL diversion.

A JPT summary [65] of Morrow et al.’s article reinforced these findings. Limiting VEDA to 25% of total acid volume—below the usual 30–40%—and using low VES concentrations mitigated formation damage while still achieving uniform stimulation. The combined LEL-VEDA approach is now seen as a robust, low-risk option for acidizing highly variable carbonates.

5. LEL Design Methodology

5.1. Design Strategy

In the preceding introduction, it can be understood that the design of LEL must first account for the calculation of key flow dynamics, including pipe flow, orifice flow, and reservoir seepage. In LEL design, key structural parameters must be considered, including hole number, diameter, spacing, liner segment length, discharge coefficient, and tubing inner diameter/roughness. Additionally, acid intensity (acid volume per unit length) is a critical metric linking injection rate to stimulation outcomes. Depending on the design objective—uniform acid distribution, uniform wormhole length, or uniform skin factor—the allocation of acid intensity varies significantly and must be dynamically optimized based on reservoir heterogeneity and anisotropy.

Liu [66,67] propose three progressively advanced design approaches for LEL completion and acidizing to address the impact of reservoir heterogeneity on acid stimulation effectiveness.

The first design approach aims at “uniform acid distribution”, which is suitable for reservoirs with relatively homogeneous properties. Uniform acid distribution means that the flow rate of acid solution per unit length in each section is consistent, which is defined by Equation (29). By deriving analytical formulas for LEL orifice distribution, this method ensures even acid placement along the horizontal wellbore, thereby mitigating the “heel–toe effect”. However, this strategy overlooks variations in reservoir porosity and acid flow velocity, potentially leading to uneven stimulation outcomes.

$$V_{intensity}={\displaystyle \frac{Q_{inj}·t_{inj}}{L_s}}$$

(29)

where $V_{intensity}$ is the acid intensity, $Q_{inj}$ is the acid flow rate of a certain segment, ${\mathrm{t}}_{\mathrm{i}\mathrm{n}\mathrm{j}}$ is the injection time, $L_s$ is the length of a segment.

The second design approach targets “uniform wormhole length”, further accounting for inter-segment heterogeneity, such as porosity variations and differences in acid penetration capability. By adjusting acid intensity and orifice distribution for each segment, it ensures consistent wormhole length, thereby improving the uniformity of stimulation. Nevertheless, this method still assumes isotropic reservoir permeability and does not consider vertical-to-horizontal permeability contrasts.

The third design approach pursues “uniform post-acidizing skin factor”, building upon the second approach by incorporating the influence of permeability anisotropy. Using an improved skin factor calculation model, it optimizes orifice spacing and acidizing parameters to achieve consistent post-stimulation skin factors across all segments, enabling more precise reservoir stimulation.

These three design approaches form a progressive framework—from simple uniform acid distribution to accounting for inter-segment heterogeneity, and finally integrating permeability anisotropy—thereby systematically enhancing the scientific rigor and applicability of the design. They provide a comprehensive solution for optimizing production enhancement in ultra-long horizontal wells in carbonate reservoirs.

Based on field data from over 500 well deployments and NETool sensitivity simulations, Mogensen et al. [68] proposed a set of LEL design guidelines tailored for heterogeneous reservoirs. The study highlights the following key points:

• The optimal packer spacing is 800–1000 ft. Using an excessive number of packers (e.g., 20) may reduce efficiency.
• The distance between adjacent LEL holes should not exceed 80 ft. Using smaller hole sizes (e.g., 2 mm or 3 mm) is recommended to improve acid jetting performance.
• Wellbore inclination and fluid density differences (e.g., CO₂ vs. water) significantly affect injection profiles. For instance, in a downward-sloping well, CO₂ tends to accumulate at the heel, requiring higher injection rates to reach the toe.
• In reservoirs with an average permeability below 5 mD, a uniform outflow profile should not be enforced, as it may lead to excessive choking in high-permeability zones. Instead, two constraints are recommended: a minimum jet velocity ≥ 15 m/s and a minimum acid coverage ≥ 0.3 bbl/ft.

In short, Mogensen’s decision tree links permeability contrast, hole size, hole spacing, and packer position into a rapid, field-ready workflow: assess the K ratio, lock in the dual thresholds, then pick hole sizes and packer spacing—delivering an LEL design that is both fracture-safe and acid-efficient.

5.2. Transient Computational Models

Mogensen et al. [21] present an Eclipse-based workflow developed for LEL and ICDs. The liner and annulus are broken into 1-to-1 segments. Each hole is represented by an orifice segment branching off the main tubing/liner string and reconnecting to the annulus, creating an explicit looped flow path. For every unique (fluid, segment-geometry) tuple, a friction pressure is pre-computed with an external non-Newtonian pipe-flow routine (laminar, turbulent or drag-reduced). The results are stored as 2-D VFP tables. Because Eclipse cannot internally update the permeability field while simultaneously honoring the dynamically swapped VFP tables, the overall algorithm is wrapped in a higher-level loop. The workflow therefore treats the wellbore as a lookup-table reactor: flow physics is frozen into tables, and the simulator’s only task is to advance tracers and trigger the appropriate table at the right moment. This design keeps the reservoir solver untouched while still capturing the transient physics of acid displacement and wormhole growth. Buijse and Glasbergen’s semi-analytical wormhole model incorporating Damköhler, Peclet and acid-capacity numbers is solved at every time step to yield wormhole length and an updated skin based on Hawkins’ formula; the skin is then passed back to the reservoir inflow equation.

Sau et al. [69] presents the CMA-Pro (Carbonate Matrix Acidizing and Productivity) model, a sophisticated computational tool designed to simulate transient fluid flow and acid stimulation in extended-reach horizontal wells with complex completions such as LEL and inflow control devices (ICDs). The model couples wellbore-annulus-reservoir flow dynamics with dynamic wormhole growth calculations to optimize stimulation design. Built on a 1D flow assumption with radial symmetry, it simplifies the 3D Navier–Stokes equations into axial flow equations while tracking multiple fluids (e.g., acid, water, and diverters) and accounting for homogeneous mixing at fluid interfaces.

As shown in Figure 11, the model employs a finite volume method to discretize the wellbore and annulus (default cell length: 5 ft), solving mass conservation equations for fluid transport, axial momentum equations for frictional pressure drop (using Darcy-Weisbach and hydraulic diameter concepts), and radial momentum equations for pressure loss across liner orifices and reservoir inflow.

Field-tested in Abu Dhabi’s Pilot Well 5 (11,500-ft lateral), the model enabled optimized LEL design with non-uniform 4 mm orifices (fewer near the heel, increasing toward the toe), achieving uniform acid distribution at 20–25 bpm injection rates. It also demonstrated robustness against fluid loss zones when combined with annular packers. Calibration with step-rate tests and PLT data confirmed predictive accuracy. While the 1D approach efficiently handles ultra-long wells (>25,000 ft), its simplification of localized 3D effects (e.g., turbulent mixing near orifices) necessitates experimental/CFD validation. By integrating wellbore hydraulics, wormhole propagation, and skin evolution, CMA-Pro provides a reliable digital framework for completion and stimulation design in carbonate reservoirs.

Both models target the same physical problem—transient acid placement in long open-hole laterals fitted with limited-entry liners—but differ fundamentally in approach. Mogensen couples a commercial reservoir simulator (Eclipse) to external scripting and VFP tables, achieving flexibility at the cost of heavy pre-processing and an intricate time-step logic to suppress numerical diffusion. Sau’s CMA-Pro builds the transient wellbore and near-wellbore physics directly into the solver, eliminating external tables and allowing fully implicit coupling of flow, wormhole growth and skin evolution. Consequently, CMA-Pro runs faster and handles larger systems, whereas the Eclipse workflow remains attractive when full-field Eclipse models already exist.

5.3. Model for Heterogeneous Reservoirs

Asheim & Oudeman [70] proposed an orifice density–pressure drop analytical model based on classical flow theory, which balances the pressure drop between the wellbore and the reservoir by adjusting orifice density to achieve uniform inflow in homogeneous reservoirs. Their method is computationally efficient but exhibits limited adaptability to reservoir heterogeneity.

Wang et al. [41] addressed strongly heterogeneous reservoirs by introducing a genetic algorithm to optimize zonal orifice density. They mapped permeability variations to orifice density differences (reducing density in high-permeability zones and increasing it in low-permeability zones) and incorporated discrete constraints of completion parameters (charge type, phasing angle) to achieve global optimization.

Wang et al. [71] further quantified the impact of heterogeneity by proposing an improved s-k skin factor model. This model couples anisotropic reservoir flow with wellbore variable-mass flow using the source function method and solves for orifice density via backward substitution, defining engineering limits for pressure adjustment ranges and rounding errors.

The commonality among these three approaches lies in suppressing excessive flow in high-permeability zones through differentiated orifice density. The differences are as follows: the first two methods target homogeneous and segmented heterogeneous reservoirs, respectively, while the latter extends applicability to complex geological conditions through anisotropic corrections, providing comprehensive theoretical and algorithmic support for LEL design.

6. Potential Application Scenarios of LEL

Carbon capture and storage (CCS) has become a critical strategy for mitigating climate change, yet effective geological storage faces significant technical hurdles. A primary challenge is ensuring uniform CO₂ distribution in subsurface formations, as reservoir heterogeneity often leads to preferential flow paths, inefficient storage utilization, and potential leakage risks. Traditional injection methods struggle to control CO₂ migration in complex formations, particularly in depleted hydrocarbon reservoirs and deep saline aquifers, where permeability contrasts can cause uneven plume development. Additionally, repurposing existing wells for storage introduces complications, as legacy completions may not be optimized for large-scale CO₂ injection. These limitations highlight the need for adaptive well completion technologies that can dynamically regulate injection profiles while maintaining long-term storage integrity [72].

As shown in Figure 12, carbon sequestration technologies are diverse, primarily including geological sequestration [73,74], ocean sequestration [75], biological sequestration, and mineral sequestration [76], among other methods. Most of these methods involve the process of injecting CO₂ into reservoirs, making LEL-assisted carbon sequestration technology a viable option.

The LEL system, originally developed for acid stimulation in oil and gas wells, shows significant potential for enhancing CO₂ storage efficiency. By utilizing engineered orifice designs, LEL ensures controlled pressure drops along the wellbore, enabling balanced fluid distribution even in highly heterogeneous reservoirs. Key advantages include precise injection control through non-uniform orifice spacing, which mitigates the “heel–toe effect” in horizontal wells and promotes uniform CO₂ saturation. Additionally, LEL can be optimized for reservoir-specific conditions by adjusting orifice density—enhancing flow in low-permeability zones while restricting it in high-permeability areas to prevent early breakthrough. The system also offers a cost-effective solution by repurposing abandoned wells for CO₂ storage, reducing the need for new infrastructure. While further research is required to assess LEL’s long-term durability under high-pressure and corrosive CO₂ conditions, its combination of flow control, monitoring capabilities, and adaptable design positions it as a promising tool for safer and more efficient carbon sequestration in the future.

As mentioned earlier, most carbon sequestration methods involve injecting CO₂ into geological formations, which can lead to challenges such as high injection pressures due to low permeability, uneven CO₂ distribution in heterogeneous formations, and fracture propagation in unstable strata. Long-term leakage monitoring is often required. This section analyzes key geological factors affecting CO₂ storage and explores how LEL technology can address these challenges [77].

The vertical-to-horizontal permeability ratio (K_v/K_h) significantly impacts CO₂ migration and storage. Studies show that increasing K_v/K_h (from 0.1 to 1) enhances CO₂ storage capacity by up to 69.96%, primarily by improving vertical migration and reducing near-wellbore pressure buildup. Higher K_v/K_h also promotes CO₂ diffusion, increasing adsorption (up to 97.96%) and dissolution storage (59.72%). However, it accelerates the conversion of structural trapping into dissolution and adsorption, affecting long-term stability [78].

Heterogeneous wettability alters CO₂-brine-rock interactions, influencing trapping efficiency and plume migration. Strongly water-wet regions achieve higher residual CO₂ saturation (e.g., 14.9%) compared to weakly water-wet or CO₂ wet regions (e.g., 8.7%). Wettability variations can reduce trapping capacity by up to 35% and create localized high-saturation channels, increasing leakage risks [79].

Water saturation affects CO₂ injection efficiency. High water saturation reduces gas-phase permeability, limiting CO₂ displacement and storage capacity. It may also slow dissolution rates or cause water-blocking, hindering CO₂ migration. Optimizing injection parameters (e.g., pressure) can mitigate these effects [80].

Injection strategies are critical for CO₂ storage efficiency. Late-stage injection maximizes pore-space utilization, while early injection risks premature mixing. Horizontal wells enhance contact area but require homogeneous formations. Tailored strategies balance reservoir repressurization and flow control to ensure long-term containment [81].

7. Conclusions

Table 4. Key Parameters for LEL Research and Design.

Category	Parameter/Symbol	Physical Significance and Design Impact
Pipe Friction	Fanning friction factor, f	Controls pipe pressure loss; determines heel–toe effect severity
Orifice Flow	Discharge coefficient, $C_D$	Governs orifice pressure drop
Acidizing	Pore volume to breakthrough, ${PV}_{bt}$	Measures acidizing efficiency and acid consumption; determines stage acid volume
Acidizing	Skin factor, S	Overall stimulation effectiveness; used for LEL design
LEL Design	Number of orifice, $n_{hole}$	One of the key design parameters of an LEL; governing the fluid-distribution profile along the liner.
LEL Design	Acid Coverage, cov	Average acid injection rate per meter of liner, a key indicator of LEL performance.

summarizes the typical parameters that must be carefully considered during LEL research and design. These parameters are directly linked to the mechanisms or outcomes of LEL flow control; prudent adjustment of their values is essential to achieve better acidizing performance or an optimized inflow profile. In addition to those listed, other factors—such as formation permeability and the total acid volume injected through the LEL—also influence overall effectiveness.

The development and application of Limited-Entry Liner technology have significantly advanced the efficiency of unconventional energy resource extraction and opened new avenues for carbon sequestration. This review has synthesized the foundational principles, challenges, and innovations associated with LEL systems, highlighting their transformative potential in addressing key issues in both hydrocarbon production and environmental sustainability.

• Enhanced Stimulation Efficiency: LEL technology mitigates the heel–toe effect and reservoir heterogeneity by ensuring uniform acid distribution in horizontal wellbores. Through engineered orifice designs, LEL optimizes wormhole formation, maximizing near-wellbore permeability and hydrocarbon recovery.
• Flow Dynamics and Modeling: The interplay of pipe friction, orifice friction, and annular flow governs LEL performance. Advanced models, including semi-empirical and two-scale continuum approaches, provide robust tools for predicting fluid behavior and optimizing orifice strategies.
• Acidizing Mechanisms: Understanding wormhole patterns and breakthrough pore volumes (PVbt) is critical for effective stimulation. The balance between injection rates and reaction kinetics determines the formation of dominant wormholes, which are essential for efficient acidizing.
• Design Methodologies: From uniform acid distribution to skin factor optimization, LEL design has evolved to address reservoir heterogeneity and anisotropy. Transient models and coupling techniques integrate wellbore hydraulics with reservoir dynamics, enabling tailored completions for diverse geological conditions.
• Carbon Sequestration Potential: LEL systems offer promising solutions for CO₂ storage by controlling injection profiles in heterogeneous formations. Their ability to regulate flow and prevent preferential pathways enhances storage efficiency and long-term containment.

LEL is a new completion method that has demonstrated significant effectiveness in acid stimulation, fracturing, and flow control production. For carbon sequestration technology, LEL also holds promise as a potential flow control solution. In the future, further clarification of LEL’s flow and acidizing mechanisms is anticipated, along with the development of more accurate and user-friendly design software.