A Novel Intake Inflow Performance Relationship for Optimizing Pump Setting Depth in Low-Permeability Oil Wells

Abstract

The optimization of pump setting depth in low-permeability oil wells remains a persistent challenge, as conventional inflow performance relationship (IPR) curves fail to capture the coupled effects of downhole pump intake depth and reservoir productivity. To address this limitation, this study proposes a novel Low-Permeability Intake Inflow Performance Relationship (LIIPR) framework. The method establishes a theoretical link between pump depth and production by integrating low-permeability reservoir inflow models with multiphase wellbore flow calculations. On this basis, a series of derivative concepts and analytical tools are introduced, including (i) a three-zone classification of inflow curves to distinguish effective, inefficient, and abnormal production regimes; (ii) a multi-pump-depth analysis to determine the feasible range and optimal boundaries of pump setting depth; and (iii) a three-dimensional deep-pumping limit map that couples inflow and outflow dynamics through nodal analysis, providing a comprehensive criterion for system optimization. The proposed LIIPR methodology enables accurate identification of optimal pump depth and intake pressure conditions, overcoming the ambiguity of traditional IPR-based approaches. Unlike previous IPR- or EIPR-based methods, LIIPR introduces for the first time a unified inflow–outflow coupling framework that quantitatively links pump intake depth with well productivity. This integration represents a novel theoretical and computational advance for deep-pumping optimization in low-permeability reservoirs. Applications for field cases in Shengli Oilfield confirm the theoretical findings and demonstrate the practical potential of the method for guiding efficient deep pumping operations in low-permeability reservoirs.

1. Introduction

Low-permeability reservoirs play a critical role in global oil and gas development [1]. They account for a substantial proportion of reserves and present significant challenges to efficient recovery, making them an important target for enhancing oil recovery and ensuring energy security [2,3,4]. Compared with conventional reservoirs, low-permeability formations are characterized by complex pore structures, poor fluid flow capacity, and strong stress sensitivity [5,6,7], which often lead to inadequate fluid supply and limited well productivity [8]. In field practice, rod pumping is the most widely used artificial lift method for such reservoirs, and the selection of pump setting depth directly determines the pressure drawdown and well productivity [9,10]. Optimizing the pump depth not only enhances production but also reduces energy consumption and operating costs, thereby holding central importance in production optimization of low-permeability oil wells [11]. In field practice, the pump setting depth is usually determined based on on-site experience and production history, which lacks a clear theoretical basis and quantitative support.

The conventional inflow performance relationship (IPR) curve has long served as an important tool for well productivity analysis, effectively describing the relationship between bottomhole flowing pressure and production rate, and it has been widely applied in conventional reservoirs [12,13]. However, this method fails to capture the critical influence of pump setting depth on pressure drawdown and productivity, which limits its applicability in deep pumping optimization for low-permeability wells [14]. To address this, Caicedo (2015) [15] introduced the concept of the Enriched IPR (EIPR), in which the reference point is shifted from the bottomhole to the pump intake, thereby linking pump depth to production capacity. While this approach provided a new perspective for pump depth optimization, its application was restricted to conventional solution-gas drive reservoirs and lacked a systematic computational and analytical framework [16,17,18]. In low-permeability reservoirs, common phenomena such as stress sensitivity, threshold pressure gradient, and solution gas exsolution exert significant impacts on productivity, making both conventional IPR and EIPR inadequate for accurately capturing the coupled relationship between pump depth and production [19]. This limitation hinders their effective use in optimizing deep pumping operations for low-permeability wells.

To address these challenges, this study introduces a novel framework termed the Low-Permeability Intake Inflow Performance Relationship (LIIPR). The proposed method theoretically integrates productivity prediction models for low-permeability reservoirs with multiphase wellbore flow calculations, thereby establishing a quantitative link between pump setting depth, pressure drawdown, and well productivity. Building on this foundation, several derivative analytical tools are developed: (i) a three-zone classification of inflow curves to identify effective, inefficient, and abnormal production regimes during pressure drawdown; (ii) a multi-pump-depth analysis that compares LIIPR curves under different pump depths to determine the feasible range and boundaries of pump setting depth; and (iii) a three-dimensional deep pumping limit map, which incorporates nodal analysis to unify inflow and outflow dynamics within a comprehensive framework, providing systematic criteria for optimizing pump depth and operating conditions. Compared with conventional methods, the LIIPR approach not only explicitly illustrates the influence of pump depth on productivity but also offers a more accurate and systematic theoretical tool for optimizing deep pumping operations in low-permeability oil wells. In contrast to previous studies that treated inflow and outflow as separate processes, the LIIPR framework uniquely integrates them within a single theoretical structure. This novelty enables a more rigorous quantification of how pump setting depth governs productivity and provides a missing bridge between reservoir inflow models and wellbore hydraulics that has not been explicitly formulated in earlier works.

This study introduces the concept of the Low-Permeability Intake Inflow Performance Relationship (LIIPR) and its derivative analytical methods, establishing a systematic framework for optimizing pump setting depth. Validation with field data from representative wells demonstrates that the proposed approach can accurately identify the optimal pump depth and intake pressure conditions, while avoiding the uncertainties inherent in traditional methods. The LIIPR thus provides both a novel theoretical basis and a practical reference for deep pumping optimization in low-permeability oil wells.

2. Methodology

Introduction of LIIPR and Fundamental Principles. The conventional inflow performance relationship (IPR) curve, referenced to bottomhole flowing pressure, has been widely applied for productivity analysis in both conventional and low-permeability reservoirs [20,21,22]. For low-permeability formations, numerous models have been developed that incorporate effects such as stress sensitivity, threshold pressure gradient, and solution gas exsolution [23,24], which enable the IPR to characterize reservoir-specific flow behaviors. However, a common limitation of all IPR formulations is that the reference point remains fixed at the bottomhole, making it incapable of directly capturing the influence of pump setting depth on production [25,26,27].

To overcome this limitation, Caicedo (2015) introduced the concept of the Enriched IPR (EIPR), in which the reference point is shifted from the bottomhole to the pump intake, thereby linking pump depth with productivity. While this approach provided a new perspective, it was primarily based on the Vogel-type IPR assumption and restricted to conventional solution-gas drive reservoirs, lacking a systematic computational framework and thus offering limited applicability to low-permeability wells [28].

Building on this idea, the present study proposes the concept of the Low-Permeability Intake Inflow Performance Relationship (LIIPR). This framework integrates low-permeability IPR models with multiphase wellbore flow calculations. By retaining the ability of low-permeability IPRs to capture complex effects such as stress sensitivity and threshold pressure gradients, while simultaneously shifting the reference point to the pump intake, LIIPR establishes a quantitative link among pump setting depth, pressure drawdown, and productivity. As illustrated in Figure 1, LIIPR curves under multiple pump depths provide a clear and systematic depiction of how pump depth influences well productivity, thereby extending and complementing the capabilities of both IPR and EIPR in low-permeability reservoir conditions.

Computational Procedure of LIIPR in Low-Permeability Wells. The computation of LIIPR essentially couples productivity prediction models for low-permeability reservoirs with multiphase wellbore flow calculations, using the pump intake as the reference point to generate an inflow performance curve that reflects the influence of pump setting depth. The procedure consists of three main steps:

First, an appropriate productivity model must be selected according to reservoir characteristics. While different productivity equations and multiphase flow correlations will yield different LIIPR curves, the computational steps remain consistent. As an example, the stress-sensitive fractured well model proposed by S. Hamed Tabatabaie et al. [29]. introduces a permeability modulus to describe near-wellbore stress sensitivity, making it suitable for productivity prediction in fractured low-permeability reservoirs. This model is further adopted here for its ability to capture fracture–matrix interaction and stress-dependent flow behavior representative of the studied carbonate formation. The governing equations can be expressed as:

$$q\left(t\right)={\displaystyle \frac{k_{F}wh}{B\mu }}{\displaystyle \frac{p_i-p_w}{\sqrt{\alpha }}}\tanh \left({\displaystyle \frac{X_f}{\sqrt{\alpha }}}\right)$$

(1)

$$\alpha =\sqrt{\pi {\eta }_{M}t}{\displaystyle \frac{k_{F}w}{k_M}}$$

(2)

$$\eta ={\displaystyle \frac{k_M}{{\varphi }_M\mu c}}$$

(3)

$$k_F\left(P_W\right)=k_{Fi}{e}^{\gamma \left(p_w-p_i\right)}$$

(4)

The parameters are defined as follows:

• $k_F$, $k_M$: fracture and matrix permeability, mD;
• $w$: fracture width, m;
• $h$: reservoir thickness, m;
• $B$: formation volume factor, dimensionless;
• $\mu$: oil viscosity, mPa·s;
• $\eta$: reservoir diffusivity, m²/s;
• $X_f$: fracture half-length, m;
• c: total compressibility, MPa⁻¹;
• $\gamma$: permeability modulus, MPa⁻¹;
• $p_{wf}$, $p_e$: reservoir pressure and bottomhole flowing pressure, MPa.

Compared with the Vogel equation, this model more realistically captures productivity variations in low-permeability reservoirs under stress-sensitive conditions, thus serving as an essential basis for the LIIPR.

Second, a series of characteristic points (production rate–bottomhole pressure pairs) are selected from the IPR curve, and multiphase wellbore flow calculations are performed starting from the bottomhole. Using classical correlations such as Orkiszewski’s method [30], the pressure drop from the bottomhole to the pump intake is computed point by point, thereby yielding the corresponding pump intake pressure. Repeating this process generates a new production rate–pump intake pressure curve, namely the LIIPR curve. As illustrated in Figure 2, the workflow can be summarized as a mapping process: productivity model → IPR curve → multiphase flow calculation → LIIPR curve. It should be noted that in this formulation we assume quasi-steady flow and do not explicitly account for strong non-Darcy effects or phase transition phenomena under extreme conditions, which may limit the model’s accuracy in highly dynamic regimes.

Finally, by varying the pump setting depth, a family of LIIPR curves can be obtained. As the pump depth increases, the curves systematically shift within the coordinate system, reflecting the combined impact of pump depth on production and intake pressure. As shown in Figure 3, the comparison of multiple LIIPR curves in a single coordinate system provides direct insights into the productivity differences under different pump depths, thereby laying the foundation for subsequent multi-pump-depth analysis and the construction of the three-dimensional deep-pumping limit map.

Derivative Analytical Methods of LIIPR. The Low-Permeability Intake Inflow Performance Relationship (LIIPR) not only establishes a new theoretical link between pump setting depth and well productivity but also enables the development of several derivative methods with direct engineering implications. These methods extend LIIPR from a single curve description into a systematic toolkit for production optimization.

Three-Zone Classification. During the production process of low-permeability wells, the productivity response to increasing pressure drawdown typically exhibits staged characteristics rather than linear growth. Based on LIIPR curves, the inflow performance can be divided into three zones:

• Effective zone: a regime where pressure drawdown and productivity remain strongly correlated, indicating efficient fluid supply and optimal production conditions.
• Inefficient zone: a regime where further increases in drawdown result in diminishing productivity gains, reflecting declining system efficiency.
• Abnormal zone: a regime where LIIPR curves may bend backward or display inflection points, with productivity decreasing despite higher drawdown, often caused by stress sensitivity, liquid loading, or gas lock phenomena.

As illustrated in Figure 4, a typical LIIPR curve can be segmented into these three zones by identifying slope changes and inflection points. This classification provides a straightforward criterion for field engineers to avoid inefficient or abnormal operating conditions and to maintain wells in the effective regime.

Multi-Pump-Depth Analysis. In practice, pump setting depth directly governs intake pressure and drawdown distribution, thereby affecting well productivity [31]. A single LIIPR curve reflects only one pump depth, while comparing multiple LIIPR curves under different depths reveals the systematic impact of pump placement.

As shown in Figure 5, increasing the pump depth shifts the LIIPR curve downward and to the right, meaning that at the same production rate, a lower intake pressure is required. This trend indicates that deeper pump settings reduce flow resistance and enhance drawdown, thereby improving productivity. However, the benefit is not unlimited; beyond a certain depth, the productivity gain diminishes and may even reverse due to decreased pump efficiency or fluid accumulation. Thus, multi-pump-depth analysis enables the identification of feasible ranges and optimal boundaries of pump setting depth, providing a reliable basis for design and operation of deep pumping wells.

Three-Dimensional Deep-Pumping Limit Map. Although LIIPR and multi-pump-depth comparisons offer valuable insights, they remain limited in representing the integrated wellbore–reservoir system. To address this, the concept of a three-dimensional deep-pumping limit map is introduced by incorporating nodal analysis. This framework constructs a 3D coordinate system involving pump setting depth, intake pressure, and production rate, thereby offering a comprehensive optimization criterion.

As depicted in Figure 6, the map consists of three types of characteristic curves:

• Pump-depth curves, describing how LIIPR curves shift with varying pump depths and reflecting productivity trends;
• Intake–pressure curves, showing the variation in pump intake pressure with production rate, representing the coupling between flow losses and inflow performance;
• Coordination curves, formed by the intersections of LIIPR curves with wellbore outflow curves, identifying the stable operating points of the system.

The red line in Figure 6 represents the connection points between the wellbore and the reservoir under different pump depths. The blue line is the two-dimensional projection of the three-dimensional curve, which facilitates data reading.The three-dimensional deep-pumping limit map provides a direct and systematic depiction of the dynamic relationship among pump depth, intake pressure, and productivity. It serves as a powerful tool for determining the optimal pump setting depth, appropriate submergence, and stable operating conditions. Compared with conventional two-dimensional analysis, this method is better suited to address the complex inflow–outflow coupling in low-permeability reservoirs and offers stronger engineering guidance.

3. Results and Discussion

Case Description and Data Source. To validate the proposed Low-Permeability Intake Inflow Performance Relationship (LIIPR) method, a representative low-permeability well from the Linnan block of Shengli Oilfield was selected as the study case. The reservoir is a sandstone formation with relatively low porosity and permeability, exhibiting strong stress sensitivity and heterogeneity. The well is produced using rod pumping, featuring typical characteristics of deep pumping in low-permeability reservoirs, and thus provides a suitable case for demonstrating the applicability of the LIIPR method.

According to field test results, the productivity characteristics of this well conform to the following productivity equation:

$$q={\displaystyle \frac{2\pi kh}{B\mu m_o\ln \left(\frac{r_e}{r_w}\right)}}\left(1-e^A\right)$$

(5)

$$A=m_o\left(p_i-p_w\right)$$

(6)

$$m_o=a\left(p_i-p_w\right)+b$$

(7)

The parameters are defined as follows:

• $k$: reservoir permeability, mD;
• $h$: reservoir thickness, m;
• $B$: formation volume factor, dimensionless;
• $\mu$: oil viscosity, mPa·s;
• $r_e$: drainage radius, m;
• $r_w$: wellbore radius, m;
• $p_i$: initial reservoir pressure, MPa;
• $p_w$: bottomhole flowing pressure, MPa;
• $a$, $b$: stress sensitivity constants, dimensionless.

Based on this equation, the IPR curve of the study well can be constructed, as shown in Figure 7, which serves as the basis for subsequent LIIPR curve development.

In addition, the reservoir and geometric parameters of the study well are summarized in

Table 1. Main reservoir and geometric parameters of the study well.

Parameter	Value	Unit
Initial reservoir pressure	14	MPa
Reservoir thickness	3	m
Drainage radius	300	m
Wellbore radius	0.124	m
Permeability	150	mD
Stress sensitivity constant (a)	0.03	–
Stress sensitivity constant (b)	0.001	–
Oil viscosity	15	mPa·s
Well depth	3000	m

, which serve as input for constructing the IPR curve. The initial operating regime is provided in

Table 2. Initial operating parameters of the study well.

Parameter	Value	Unit
Stroke length	2.7	m
Strokes per minute (SPM)	6	rpm
Rod cross-sectional area	0.001	m2
Tubing cross-sectional area	0.003019	m2
Wellhead pressure	0.5	MPa

, including stroke length, strokes per minute, rod and tubing cross-sectional areas, and wellhead pressure. Together, these parameters form the essential inputs for LIIPR curve computation and subsequent field validation.

Three-Zone Classification Analysis. Based on the IPR curve of the study well (see Figure 7), the productivity response to pressure drawdown can be clearly divided into distinct zones. As the bottomhole flowing pressure decreases, production does not increase monotonically but instead exhibits stage-dependent behavior. For this well, two evident zones are identified—an effective zone and an abnormal zone—while no typical inefficient zone is observed.

As illustrated in Figure 8, when the bottomhole pressure is within the range of 8–14 MPa, production increases significantly with drawdown, representing the effective zone. In this regime, the system operates efficiently, and production enhancement measures are highly effective. The maximum production rate, approximately 6.4 m³/d, occurs when the bottomhole pressure is slightly above 8 MPa. When the bottomhole pressure falls below 8 MPa, production begins to decline rather than increase, indicating the onset of the abnormal zone. This phenomenon is commonly attributed to stress sensitivity, liquid accumulation, or gas locking effects in low-permeability reservoirs.

From an operational perspective, it is essential to maintain the bottomhole pressure within the effective zone and as close as possible to the maximum productivity point (slightly above 8 MPa). For instance, when the bottomhole pressure exceeds 8 MPa and the production rate has not yet reached the maximum, measures such as increasing stroke length and strokes per minute or lowering the pump setting depth can be adopted to reduce bottomhole pressure and enlarge the pressure drawdown. Conversely, if the bottomhole pressure drops below 8 MPa, reducing stroke length and frequency or raising the pump setting depth is recommended to avoid abnormal declines in production.

Multi-Pump-Depth Analysis. Following the basic concept of LIIPR construction, a series of characteristic points were evenly selected along the IPR curve of the study well. For each specified pump setting depth, multiphase wellbore flow calculations were performed to map the bottomhole pressure–rate pairs to the pump intake, thereby generating LIIPR curves under different depths. The resulting family of curves is shown in Figure 9.

Figure 9 presents the IPR curve together with four LIIPR curves corresponding to pump setting depths of 1800 m, 2100 m, 2400 m, and 2700 m. Several important conclusions can be drawn:

• When the pump setting depth is less than 2100 m, the LIIPR curve exhibits no inflection point (maximum value). In this case, no clear maximum production rate can be achieved regardless of how the operating regime is adjusted (e.g., stroke length or strokes per minute). Therefore, a pump setting depth greater than 2100 m is recommended.
• When the pump setting depth exceeds 2400 m, further increasing the depth results in almost no improvement in the maximum value (inflection point) of the LIIPR curve. Since the production gain becomes saturated while rod string costs and failure risks increase, it is not advisable to set the pump deeper than 2400 m.

Considering both technical performance and economic constraints, the recommended feasible range of pump setting depth for the study well is 2100–2400 m, with a depth close to 2400 m offering near-optimal production without incurring significant additional costs.

Three-Dimensional Deep-Pumping Limit Map Analysis. Building on the multi-pump-depth analysis, nodal analysis was further introduced to extend the family of LIIPR curves into a three-dimensional deep-pumping limit map, thereby providing a more intuitive representation of the coupled relationships among pump setting depth, pump intake pressure, and production rate.

As shown in Figure 10, the pink–yellow inflow surface represents the spatial extension of the multi-pump-depth LIIPR curves. Compared with the discrete set of LIIPR curves, this surface exhibits greater continuity and clarity, allowing the prediction of production for any given pump depth and intake pressure, and directly visualizing the influence of pump depth on the position and shape of LIIPR curves. The green–yellow outflow surface reflects the wellbore outflow performance under different pump depths, representing the effect of pump placement on lifting efficiency. The intersection of these two surfaces forms a coordination curve, whose projections onto the production–intake pressure plane and the production–pump depth plane yield the intake pressure curve and the pump depth curve, respectively. From this analysis, when the production reaches its maximum, the corresponding submergence is 129 m, with an optimal pump setting depth of 2300 m. Under the current stroke length and strokes per minute, this configuration is recommended as the optimal operating point. The inflow and outflow surfaces were generated by interpolating multi-pump-depth LIIPR curves and coupling them with multiphase wellbore flow models through iterative nodal analysis, which ensures mass and pressure balance at the pump intake. The predicted optimal pump depth of 2300 m and submergence of 129 m show good agreement with the field-tested configuration, confirming the reliability of the proposed algorithm.

To further evaluate the applicability of the deep-pumping limit map, multiple field operating regimes were tested by varying stroke length and strokes per minute, as summarized in

Table 3. Comparison of multiple production systems for the study well.

Stroke Length (m)	Strokes per Minute (rpm)	Maximum Production (m3/d)	Recommended Submergence (m)	Recommended Pump Depth (m)
2.7	6	7.2	129	2300
3	5	7.1	145	2300
3.5	3	6.9	238	2400
4.6	2.3	6.8	218	2400
4.6	1.8	6.8	300	2500
4.8	1.3	6.4	441	2500

. The results show that different operating regimes lead to variations in maximum production, recommended submergence, and optimal pump depth. Among them, the combination of stroke length 2.7 m and 6 rpm achieved the highest production rate of 7.2 m³/d, with a recommended submergence of 129 m and an optimal pump depth of 2300 m. This operating regime is therefore recommended as the optimal production scheme for the study well.

Discussion. The analysis of the study well using the IPR curve and the extended LIIPR framework demonstrates significant advantages of the proposed method in pump setting depth optimization and production regime design. First, the three-zone classification based on the IPR curve quantitatively identifies the ranges of effective and abnormal zones. It clearly indicates that when the bottomhole pressure falls below 8 MPa, production declines rather than increases, providing a reliable criterion for avoiding abnormal operating conditions. Compared with traditional empirical judgments, this method is both more intuitive and quantitative.

Second, the multi-pump-depth analysis shows that increasing pump depth substantially improves intake pressure and productivity, but beyond a certain threshold (2400 m), the production gain becomes saturated. Further deepening the pump leads to higher rod string costs and operational risks. Accordingly, the feasible range of pump depth was determined to be 2100–2400 m, with an optimal depth recommended at 2300–2400 m. This highlights the ability of LIIPR to balance productivity enhancement with engineering economics.

Finally, the three-dimensional deep-pumping limit map extends the multi-pump-depth LIIPR curves into a continuous inflow surface and couples it with the outflow surface, forming a coordination curve. This method not only accurately identifies the optimal pump depth (2300 m) and submergence (129 m) but also provides systematic criteria under different stroke lengths and pumping speeds. Compared with two-dimensional analyses, the three-dimensional map is more intuitive and comprehensive, simultaneously revealing the multi-dimensional relationships among pump depth, intake pressure, and productivity, and offering stronger theoretical support for optimizing production regimes under complex conditions. Although optimizing pump setting depth may potentially reduce operating costs through improved energy efficiency, establishing a generalized quantitative relationship between depth and cost remains challenging due to site-specific geological and operational factors. Future work will focus on developing data-driven methods to better evaluate such economic effects.

To further verify the general applicability of the LIIPR framework, pump setting depth optimization was conducted for multiple wells in the Linpan block of Shengli Oilfield, and the results are summarized in

Table 4. Summary of well examples in Linpan.

	Well Example 1	Well Example 2	Well Example 3	Well Example 4	Well Example 5
First Adjustment	2001.39	1769.42	1799.42	1809.7	2003.73
Second Adjustment	2206.5	2100.2	1914.52	2110.6	2301.2
Third Adjustment	2386.6	--	--	2106.1	--
Fourth Adjustment	--	--	--	2006.25	--
Recommended Setting Depth	2300	1800–2100	2500	2500	2500

.

Overall, the LIIPR framework not only compensates for the limitations of EIPR but also demonstrates enhanced applicability and scalability in engineering practice. Its key advantages include: (i) the ability to identify abnormal zones of productivity decline in low-permeability reservoirs; (ii) the capability to determine feasible ranges and optimal values of pump depth; and (iii) the provision of a systematic and intuitive three-dimensional tool for production optimization. These results confirm that LIIPR holds significant potential for deep-pumping optimization in low-permeability oil wells.

4. Conclusions

This study proposes the concept and computational framework of the Low-Permeability Intake Inflow Performance Relationship (LIIPR) for optimizing pump setting depth in low-permeability oil wells. By integrating low-permeability IPR models with multiphase wellbore flow calculations, LIIPR establishes a quantitative link among pump depth, pressure drawdown, and production, and further extends into a series of derivative analytical tools. The main conclusions are as follows:

• The three-zone classification of the IPR curve enables the identification of effective and abnormal operating regimes, providing clear criteria to avoid undesirable productivity decline.
• Multi-pump-depth analysis reveals the systematic influence of pump setting depth on well performance, indicating that pump depth has a feasible range, while excessively shallow or deep settings reduce efficiency.
• The three-dimensional deep-pumping limit map unifies inflow and outflow processes within a single framework, offering an intuitive and systematic criterion for production optimization under complex field conditions.

In summary, the LIIPR not only compensates for the limitations of conventional EIPR but also demonstrates strong applicability and scalability in engineering practice. Future work may incorporate gas–liquid coupling and dynamic reservoir–wellbore interactions to extend the applicability of LIIPR to a wider variety of low-permeability well conditions.

The originality of this work lies in establishing a unified LIIPR framework that quantitatively couples low-permeability inflow characteristics with multiphase wellbore flow dynamics, a relationship that had not been previously defined. This innovation provides a new theoretical foundation and practical tool for deep-pumping optimization and offers clear improvements over traditional IPR and EIPR methods.