A New Productivity Evaluation Method for Horizontal Wells in Offshore Low-Permeability Reservoir Based on Modified Theoretical Model

Abstract

In the early stages of offshore low-permeability oil field development, it is crucial to ascertain the productivity of production wells to select high-production, high-quality reservoirs, which affects the design of the development plan. Therefore, accurate evaluation of well productivity is essential. Drill Stem Testing (DST) is the only way to obtain the true productivity of offshore reservoirs, but conducting DST in offshore oilfields is extremely costly. This article introduces a novel productivity evaluation method for horizontal wells in offshore low-permeability reservoirs based on an improved theoretical model, which relieves the limitations of traditional methods. Firstly, a new horizontal well productivity evaluation theoretical model is derived, with the consideration of the effects of the threshold pressure gradient, stress sensitivity, skin factor, and formation heterogeneity on fluid flow in low-permeability reservoirs. Then, the productivity profiles are classified based on differences in the permeability distribution of horizontal well sections. Thirdly, the productivity evaluation equation is modified by calculating correction coefficients to maximize the model’s accuracy. Based on the overdetermined equation concepts and existing DST productivity data, the derived correction coefficients in this paper are x₁ = 3.3182, x₂ = 0.7720, and x₃ = 1.0327. Finally, the proposed method is successfully applied in an offshore low-permeability reservoir with nine horizontal wells, increasing the productivity evaluation accuracy from 65.80% to 96.82% compared with the traditional Production Index (PI) method. This technology provides a novel approach to evaluating the productivity of horizontal wells in offshore low-permeability reservoirs.

1. Introduction

Offshore oilfields are pivotal in the global energy landscape, holding about 30% of the world’s oil reserves [1,2,3]. These fields contribute significantly to oil production, providing approximately 25–30% of the global output [4,5]. Their development is essential for diversifying energy sources and ensuring energy security [6,7,8,9]. The extraction of offshore oil involves advanced technologies and methodologies, pushing the boundaries of engineering and geoscience [10,11,12].

Well productivity prediction is crucial in optimizing oil and gas extraction, including offshore oilfield developments, where it informs decisions on optimization strategies, significantly impacting the economic viability and environmental safety of these complex, high-investment operations [13,14,15,16]. Accurate prediction models are key to maximizing offshore oil and gas resource utilization while minimizing ecological risks.

Numerous researchers have developed analytical and semi-analytical methods to predict horizontal well productivity, focusing on enhancing accuracy and reliability. These methods integrate complex reservoir characteristics, flow dynamics, and technological advancements to optimize horizontal well productivity prediction efficiency. Wu et al. [17] used molecular dynamics simulation to investigate the microscopic mechanisms of CO2 enhancing shale oil recovery, establishing molecular behavior models and achieving an improved understanding of oil displacement in nanopores. Al-Rbeawi and Artun [18] presented a comprehensive analysis of pressure behavior, flow regimes, and the productivity index in oil and gas reservoirs exploited via horizontal wells with fishbone-type completion technology. Dong et al. [19] developed a semi-analytical method combining various established principles and methods for optimizing the injection–production schedule to maximize oil recovery through the gas and water bidirectional displacement process. Al-Kabbawi [20] presented simplified productivity index correlations for horizontal wells in closed reservoirs, based on an infinite-conductivity model. These correlations, developed using nonlinear regression on numerous cases, accurately calculate various pressure drop measures in pseudo-steady state conditions. Jreou [21] compared horizontal well productivity in the Burgan oilfield by using established methods for steady and pseudo-steady states. Xiong et al. [22] presented a two-dimensional heterogeneous structure model to predict oil production from horizontal wells with complex fracture networks, which incorporated non-Darcy fluid flow and porous media deformation characteristics. Jia et al. [23] developed a model to predict the performance of horizontal oil wells in areas with bottom water and limited water bodies by using mathematical methods to calculate productivity considering various factors. Ke et al. [24] discussed the complexity of seepage in horizontal producers, especially under certain pressure conditions, leading to oil and gas two-phase seepage. It introduced a transient calculation model for this two-phase flow, focusing on fluid properties, seepage, and coupling inflow performance with wellbore flow. Bi et al. [25] presented a 3D unsteady productivity prediction model for horizontal wells, incorporating reservoir anisotropy and perforation skin. Using the finite volume method and an improved Peaceman model, it studied the impact of various parameters on well productivity. Wu et al. [26] used potential theory to establish seepage models for multiple-fractured horizontal wells, optimizing fracture parameters and improving productivity in low-permeability gas reservoirs.

Several researchers have studied the impact of threshold pressure gradient and stress sensitivity on productivity analysis and evaluation, highlighting the necessity of incorporating these parameters [27,28,29]. For instance, productivity analysis in tight gas reservoirs using a gas–water two-phase flow model has shown that increasing water saturation leads to a higher threshold pressure gradient, significantly affecting gas well productivity [30]. Another study established a novel model for low-permeability and tight reservoirs, demonstrating that stress sensitivity, including the effect of effective stress on permeability and rock compressibility, substantially influences well performance [31]. Additionally, research on multistage fractured horizontal wells in triple-porosity reservoirs has considered stress-dependent natural fracture permeability, further emphasizing the importance of stress sensitivity [32]. The dynamic pseudo threshold pressure gradient has also been studied, showing that it increases with effective stress, impacting fluid flow and well productivity in low-permeability and tight reservoirs [33]. These studies collectively underscore the critical role of threshold pressure gradient and stress sensitivity in accurately predicting productivity in low-permeability and tight reservoirs.

These methods often rely on simplifying assumptions that may not accurately represent complex geological formations and fluid interactions, potentially leading to less reliable predictions. Additionally, they can be less adaptable to heterogeneous reservoirs, where variability in properties like porosity and permeability is high. Furthermore, semi-analytical methods, which combine analytical solutions with numerical techniques, might require extensive computational resources, especially when dealing with large or complicated reservoir models.

A lot of researchers have extensively explored physical experiment methods for predicting oil well productivity [34,35,36,37]. Huang et al. [38] delved into the complex issue of interlayer interference in multilayer commingled production, particularly in offshore heavy oil reservoirs. Dasheng Q. and Bailin P. [39] outlined a study focused on evaluating the production potential of different sand control techniques in producers. Kadeethum et al. [40] employed a full factorial experimental design approach to investigate the hydromechanical effects on well productivity in fractured porous media. Zhang et al. [35] details a study focused on addressing the challenges of water breakthrough and water plugging in tight oil reservoirs by employing a physical simulation method based on the parallel connection of double cores and a large-scale physical simulation system. Duan et al. [41] proposed a study focused on understanding the composition of recovered shale gas and its impact on long-term gas production trends based on both experimental and mathematical modeling approaches.

Physical experiment methods in oil well productivity prediction present several drawbacks. Primarily, they can be prohibitively expensive due to the need for specialized equipment and materials. Additionally, these methods often face scalability challenges, as replicating the exact subsurface conditions at a smaller scale can be complex and sometimes inaccurate. Furthermore, physical experiments can be time-consuming, limiting the speed at which data can be gathered and analyzed, which may delay decision-making processes in fast-paced development environments.

Extensive research has been conducted on numerical simulation methods for oil well productivity prediction, with scholars delving deeply into various models and algorithms [42,43,44]. Hu et al. [45] proposed a study focused on predicting the productivity of horizontal wells based on the utilization of a power law exponential decline model and analyzing their influencing factors, particularly in the context of tight oil reservoirs. Sun et al. [46] discussed a study that delves into the effects of superheated steam on oil well productivity, with a focus on the contribution of physical heating compared to chemical reactions using a numerical method. Li et al. [47] developed a unique coupling scheme integrating full-field and near-wellbore poromechanics to model productivity index degradation over time, assess operational strategies, and optimize well productivity. Roque and Araújo [48] presented a study on optimizing well placement in oil reservoirs using a novel proxy function and reservoir simulation to identify areas with high potential for oil recovery. Zhang et al. [49] presented a mathematical model for simulating the productivity of multistage fractured horizontal wells in tight oil reservoirs. The model integrated fluid flow and porous media deformation, utilizing the finite element method for numerical solutions. Jing et al. [50] proposed a study about developing and verifying a productivity prediction model, which addressed the gap in model verification in the field of fishbone multilateral wells, contributing to more reliable productivity predictions and practical applications in oil and gas field development. Maschio and Schiozer [51] proposed a two-stage process to address the complexity of production forecasting, particularly for short-term forecasts in oil fields. This approach isolated factors affecting well productivity over time. The numerical simulation methods need substantial computational costs and can be time intensive. Additionally, the accuracy of simulations is contingent on the quality of input data; inaccuracies in initial conditions can lead to erroneous predictions.

Accurate productivity evaluation is essential for optimizing well performance and reservoir management, especially in offshore low-permeability reservoirs where operational costs are high, and geological complexity poses significant challenges. Traditional productivity evaluation techniques, such as Drill Stem Testing (DST) and the conventional Production Index (PI) method, often fall short due to their reliance on simplified assumptions. For instance, DST, while providing reliable productivity data, is prohibitively expensive for offshore applications. Similarly, the PI method typically overlooks critical reservoir characteristics such as stress sensitivity, threshold pressure gradient, and heterogeneity, leading to inaccuracies in predicting well performance. These limitations highlight the need for more comprehensive and cost-effective evaluation methods that incorporate the unique dynamics of low-permeability reservoirs.

This study proposes a theoretical model for evaluating the productivity of horizontal wells in offshore low-permeability reservoirs based on an improved theoretical model. This model takes into account the effects of reservoir heterogeneity and fluid flow characteristics on the productivity of production wells and uses actual DST data to calibrate the horizontal well productivity model. The structure of this article is as follows: Section 2 mainly introduces the workflow of the method proposed in this article. Section 3 details the derivation and solution of the low-permeability reservoir horizontal well productivity model, which considers reservoir heterogeneity and fluid flow characteristics. Section 4 shows how to calibrate the productivity model based on real DST productivity data and the concept of overdetermined equations. Section 5 presents the application of the proposed method in an actual reservoir and demonstrates its prediction accuracy. Finally, the discussion and conclusions are presented in Section 6.

2. Workflow

The aim of this study is to establish a robust workflow for the productivity evaluation of low-permeability reservoirs, which can accurately and quantitatively portray production profiles of heterogeneous reservoirs at the micro-scale. This workflow, shown in Figure 1, consists of two key components.

The first part involves creating a nonlinear seepage mathematical model specifically designed for offshore low-permeability reservoirs. This model incorporates critical factors, including longitudinal heterogeneity, threshold pressure gradient, stress sensitivity, and skin coefficient, to comprehensively describe fluid flow behavior. Using this model, the productivity profile of a single well is calculated and compared against DST data. If a significant error is observed, it indicates potential inaccuracies in the well log interpretation of permeability, necessitating the use of well test permeability for correction. The productivity correction equation used in this step is derived from equations in Section 3, which integrates these reservoir properties.

The second part focuses on classifying the productivity profile based on variations in horizontal well permeability. This classification allows the application of a set of overdetermined equations, as represented by Equation (19), to solve for correction coefficients corresponding to each permeability category. These coefficients are optimized to minimize the discrepancy between calculated and observed productivity values, effectively enhancing the predictive accuracy of the model.

3. Establishment of a Horizontal Well Productivity Function for Offshore Low-Permeability Reservoirs

Calculation of well productivity using the conventional PI method typically relies on the average permeability of the reservoir. This approach, while straightforward, neglects the critical influence of longitudinal heterogeneity in low-permeability reservoirs, leading to inaccuracies in productivity evaluation. To address this limitation, Figure 2 presents schematic diagrams of homogeneous and heterogeneous models.

For the fluid seepage characteristics of offshore low-permeability reservoirs, the flow of fluid in low-permeability reservoirs is affected by the threshold pressure gradient and stress sensitivity; while considering the strong heterogeneity in the longitudinal direction, due to the similarity between the differential equations of incompressible fluid flow in the subsurface through the porous medium and that of the charge flow through the conductor material, the production of different small segments in the layer is approximated to parallel circuits, and the total production is equal to the sum of the production of the different small segments, as shown in Equation (6):

$$V={\displaystyle \frac{k}{\mu }}\left({\displaystyle \frac{\partial P}{\partial r}}-G\right)$$

(1)

where $V$ is velocity, m/s; $G$ is the threshold pressure gradient, MPa/m; r is the radial distance, m; µ is the fluid viscosity, cp; k is the permeability at each measurement point, mD; p is the formation pressure, MPa.

Considering permeability stress sensitivity:

$$k=k_i{e}^{-{\alpha }_k\left(P_i-P\right)}$$

(2)

Constant pressure outer boundary condition:

P(r = r_e) = P_i

(3)

where r_e is the supply radius, m.

$$V={\displaystyle \frac{Q}{S}}={\displaystyle \frac{Q}{2\pi rh}}=-{\displaystyle \frac{k}{\mu }}\left({\displaystyle \frac{\partial P}{\partial r}}-G\right)=-{\displaystyle \frac{k_i{e}^{-{\alpha }_k\left(P_i-P\right)}}{\mu }}\left({\displaystyle \frac{\partial P}{\partial r}}-G\right)$$

(4)

$${\displaystyle \frac{Q_i\mu }{2\pi r{h}_i{k}_i}}=e^{-{\alpha }_{ki}\left(P_i-P\right)}\left({\displaystyle \frac{dP}{dr}}-G_i\right)$$

(5)

$${\displaystyle \sum _{i=1}^n\frac{Q_i\mu }{2\pi r{h}_i{k}_i}}={\displaystyle \sum _{i=1}^n\left[e^{-{\alpha }_{ki}\left(P_i-P\right)}\left(\frac{dP}{dr}-G_i\right)\right]}$$

(6)

where Q_i is the surface crude oil production at each measurement point, m³/d; µ is the fluid viscosity, cp; r is the radial distance, m; h_i is the thickness between two measurement points, m; k_i is the permeability at each measurement point, mD; α_ki is the stress sensitivity coefficient at each measurement point, 1/MPa; G_i is the threshold pressure gradient computed at each measurement point, MPa/m; P_i is the original formation pressure, MPa; p is the formation pressure, MPa.

For Equation (6), the pressure conversion variable H is introduced:

$$H=e^{{\alpha }_{ki}p}$$

(7)

Then, Equation (6) can be transformed into Equations (8) and (9):

$${\displaystyle \sum _{i=1}^n\frac{Q_i\mu }{2\pi h_i{k}_i}\frac{H_i}{r}}={\displaystyle \sum _{i=1}^n(\frac{1}{{\alpha }_{ki}}\cdot \frac{dH}{dr}-G_i{H}_i)}$$

(8)

$${\displaystyle \frac{dH}{dr}}-{\displaystyle \sum _{i=1}^n{\alpha }_{ki}{G}_i{H}_i}={\displaystyle \frac{1}{2}}\cdot {\displaystyle \sum _{i=1}^n\frac{Q_i\mu H_i}{\pi h_i{k}_i}\cdot \frac{{\alpha }_{ki}}{r}}$$

(9)

The solution to Equation (9) is:

$$H(r)={\displaystyle \sum _{i=1}^n{e}^{{\alpha }_{ki}{G}_{i}r}}\left\{{\displaystyle \sum _{i=1}^n\frac{1}{2}\cdot \frac{Q_i\mu H_i}{\pi h_i{k}_i}\cdot \frac{{\alpha }_{ki}}{r}\cdot [\ln r-{\alpha }_{ki}{G}_{i}r]+C}\right\}$$

(10)

The boundary conditions are:

$$H(r_i)={\displaystyle \sum _{i=1}^n{H}_i}$$

(11)

A further transformation of Equation (10) can be obtained:

$$H(r)={\displaystyle \sum _{i=1}^n\left\{H_i\cdot \exp [{\alpha }_{ki}{G}_i(r-r_e)]\right\}}-{\displaystyle \sum _{i=1}^n\exp \left({\alpha }_{ki}{G}_{i}r\right)}\cdot {\displaystyle \frac{1}{2}}\cdot {\displaystyle \sum _{i=1}^n\left\{\frac{Q_i\mu H_i}{\pi h_i{k}_i}\cdot \frac{{\alpha }_{ki}}{r}[\ln \frac{r_e}{r}-{\alpha }_{ki}{G}_i(r_e-r)]\right\}}$$

(12)

where r_e is the supply radius, m.

The pressure distribution at any point of the formation can be obtained from Equation (13):

$$p_i-p={\displaystyle \sum _{i=1}^n\left\{-\frac{1}{{\alpha }_{ki}}\ln \left\{1-\frac{1}{2}\cdot {\displaystyle \sum _{i=1}^n\frac{Q_i{\alpha }_{ki}\mu }{\pi h_i{k}_i}{e}^{{\alpha }_{ki}{G}_{i}r}}\cdot [\ln \frac{r_e}{r}-{\alpha }_{ki}{G}_i(r_e-r)]\right\}\right\}}+{\displaystyle \sum _{i=1}^n{G}_i(r_e-r)}$$

(13)

In Equation (13), if α_kG << α_k, and G << 1, then it can be further simplified to Equation (14):

$$p_i-p={\displaystyle \sum _{i=1}^n\left\{-\frac{1}{{\alpha }_{ki}}\ln [1-\frac{1}{2}\cdot {\displaystyle \sum _{i=1}^n\frac{Q_i{\alpha }_{ki}\mu }{\pi h_i{k}_i}{e}^{{\alpha }_{ki}{G}_{i}r}\cdot \ln \frac{r_e}{r}}]\right\}}+{\displaystyle \sum _{i=1}^n{G}_i(r_e-r)}$$

(14)

In Equation (13), if α_kG << α_k, and G << 1 are not taken into account, the production Q of the horizontal well can be solved:

$$Q={\displaystyle \sum _{i=1}^n{Q}_i}={\displaystyle \sum _{i=1}^n\frac{2\pi k_i{h}_i}{{\alpha }_{ki}\mu }\cdot \frac{1-e^{-{\alpha }_{ki}[p_i-p_w-G_i(r_e-r_w)]}}{e^{{\alpha }_{ki}{G}_i{r}_e}[\ln \frac{r_e}{r}-{\alpha }_{ki}{G}_i(r_e-r_w)]+S}}$$

(15)

where Q is the nonlinear seepage productivity of an offshore low-permeability reservoir considering longitudinal heterogeneity, threshold pressure gradients, stress sensitivity, and skin effects.

The proposed theoretical model introduces a novel approach by dividing the horizontal section of the well into multiple segments to calculate the segmented productivity. This method allows for a comprehensive consideration of reservoir heterogeneity. Additionally, the model incorporates the threshold pressure gradient and stress sensitivity, which are critical factors in low-permeability reservoirs. By accounting for these parameters, the model better captures the characteristics of subsurface fluid flow, thereby enabling more accurate productivity predictions. This enhanced accuracy is particularly beneficial for optimizing well performance and reservoir management in low-permeability formations.

4. Horizontal Well Productivity Classification and Determination of Interference Factors

In this section, the discrepancies observed between the theoretical productivity predicted by the newly derived horizontal well productivity formula and the actual data obtained from DST operations are addressed.

Table 1. Errors between the productivity formula results and DST results.

Calculation Methods	Well No.	Calculated Productivity (m3/d)	Productivity by DST (m3/d)	Error (%)	Average Error (%)
Horizontal well productivity formula (Joshi formula)	1	258.5	210.9	22.6	44.3
	2	180.1	123.5	45.8
	3	289.5	220.6	31.2
	4	455.7	290.2	57.0
	5	151.1	74.6	102.5
	6	388.8	284.3	36.8
	7	310.1	236.9	30.9
	8	254.4	176.8	43.9
	9	225.6	385.3	41.4
	10	190.8	145.6	31.0
Horizontal well productivity formula (The proposed formula)	1	246.7	210.9	17.0	20.2
	2	148.8	123.5	20.5
	3	260.1	220.6	17.9
	4	350.5	290.2	20.8
	5	101.5	74.6	36.1
	6	304.3	284.3	7.0
	7	287.8	236.9	21.5
	8	225.6	176.8	27.6
	9	430.2	385.3	11.7
	10	178.2	145.6	22.4

presents the errors between the calculated productivity results and the DST test values. Recognizing the significant variability of permeability along the horizontal wellbore, as depicted by well logging curves, the productivity profile is categorized into distinct permeability classes.

This categorization enables a more precise representation of reservoir heterogeneity by isolating productivity contributions from different permeability ranges. Following this, an overdetermined system of equations is employed to deduce corrective coefficients specific to each permeability class. These coefficients are derived by minimizing the discrepancies between the theoretical and observed productivity values, effectively calibrating the formula to align with empirical data.

The proposed approach enhances the predictive accuracy of the horizontal well productivity formula by accounting for critical factors such as the longitudinal heterogeneity of the reservoir, threshold pressure gradients, stress sensitivity, and skin effects. This detailed methodology not only bridges the gap between theoretical predictions and real-world observations but also provides a robust framework for evaluating the initial productivity of horizontal wells in low-permeability reservoirs with improved precision. By systematically addressing and correcting the errors, the method significantly contributes to the reliability and practical applicability of productivity evaluation models in complex reservoir conditions.

4.1. Establishment and Classification of Productivity Profiles for Horizontal Wells

The above-established nonlinear productivity formula for horizontal wells, which considers reservoir vertical heterogeneity, initiation pressure gradient, and stress sensitivity, is utilized to calculate the productivity profile of offshore low-permeability reservoirs.

The static parameters required by the formula such as depth, permeability, and porosity are obtained using well logging data, and the parameters such as threshold pressure gradient, stress sensitivity, and skin factor are obtained by regression relations from real experimental cores in the oilfield.

Subdividing the horizontal well productivity into micro-scale (single point scale) can better reflect the non-homogeneity of the reservoir and quantitatively characterize the output of different permeability levels, and the theoretical productivity of the horizontal well can be obtained by summing the micro-scale productivity.

Figure 3 depicts key data pertaining to a horizontal well W-1 in a low-permeability reservoir located in the eastern South China Sea. Specifically, Figure 3a shows the permeability profile as a relationship of depth, with a primary distribution range from 5 to 25 mD. This profile is plotted with permeability on the horizontal axis and depth on the vertical axis, revealing the variability of permeability at different strata within the well.

Figure 3b illustrates the productivity curve of the same well, which has been calculated using the productivity formula derived in Section 3 of this paper. This curve represents the calculated single-point productivity, predominantly spanning between 0 to 2.5 m³/day, against the well depth on the vertical axis. The alignment of the productivity with specific depths and permeability indicates the expected productivity at various intervals along the wellbore.

Both graphs are presented with depth on the vertical axis, ensuring a direct comparison between the permeability and the resultant calculated productivity at each depth level. The juxtaposition of the two curves allows for an assessment of how permeability variations influence the productivity of the horizontal well in the context of the low-permeability reservoir under study.

The classification of the horizontal well productivity profile is a critical step in our analysis for several strategic reasons. Firstly, this classification allows us to elucidate the contribution of different permeability segments to the overall well productivity. Such differentiation is crucial for understanding the heterogeneity of the reservoir and for optimizing well performance.

Secondly, the categorization serves as a foundation for subsequent steps where an overdetermined system of equations will be employed. This system aims to fit the productivity formula to the actual production data obtained from DST. The prediction method can be fine-tuned to reflect the practical realities of well performance more accurately.

The criteria for this classification have been established based on the extensive research conducted by reservoir engineers on low-permeability reservoirs in the eastern South China Sea region over the years. Their empirical and analytical work has yielded a robust framework for assessing permeability and its impact on productivity.

The specific indices for classification are as follows: Class A parts of the profile are those with permeability less than 5 mD, Class B parts of the profile have permeability ranging from 5 to 20 mD, and Class C parts of the profile are characterized by permeability greater than 20 mD. This stratification reflects a significant gradation in reservoir quality and guides the allocation of resources for development and intervention programs.

4.2. Productivity Prediction Optimization Based on Overdetermined System

In this study, determining productivity correction coefficients post-classification of the productivity profile is essential for two primary reasons. First, it aligns the calculated values from our productivity formula with actual data from DST operations, ensuring our model accurately reflects actual conditions. Second, these coefficients enable the prediction of initial productivity for new wells in similar low-permeability reservoirs, offering valuable insights for reservoir management and development planning.

To accomplish this, an overdetermined system of equations is employed, which is particularly suited for situations where the dataset exceeds the number of unknowns. This approach allows for the derivation of statistically significant correction coefficients, bridging the gap between theoretical models and practical application, thereby enhancing the reliability of productivity forecasts for new horizontal drilling wells.

In the realms of mathematics and engineering, an overdetermined system is constituted by a set of linear equations that surpasses the quantity of unknowns within the system in number. Such systems typically arise from modeling real-world problems, especially in contexts where parameter estimation, data fitting, or system identification are imperative.

An overdetermined system is typically represented as Ax = b, where A is an m × n coefficient matrix, x is a column vector of n unknowns, and b is a column vector of m measurements or observations. In the overdetermined scenario, m (the number of equations) is greater than n (the number of unknowns), indicating that the system is overly constrained. Since these equations are generally not all satisfiable simultaneously, a solution that is ‘best’ in some sense is sought.

The principle of solving overdetermined systems rests on minimizing the discrepancy between the predicted and actual observations, typically under the sum of squares. This is achieved through the least squares method, aiming to minimize the residual sum of squares—that is, to minimize ||Ax − b||², where (||·||) denotes the Euclidean norm of a vector. In the least squares method, the optimal solution $\widehat{x}$ can be obtained as follows:

$$\widehat{x}={\left(A^{T}A\right)}^{-1}{A}^{T}b$$

(16)

Here, A^T is the transpose of A, and (A^TA)⁻¹ is the inverse matrix of A^TA. However, due to potential numerical stability issues, computing (A^TA)⁻¹ directly is often not recommended in practice. Instead, numerically stable methods like Singular Value Decomposition (SVD) can be used to compute the pseudoinverse A⁺ of A, thereby obtaining the optimal solution:

$$\widehat{x}=A^{+}b$$

(17)

With the pseudoinverse, the solution to the above least squares problem becomes:

$$\widehat{x}={\left(A^{T}A\right)}^{+}{A}^{T}b$$

(18)

where (A^TA)⁺ represents the pseudoinverse of A^TA. When the columns of A are linearly independent, (A^TA)⁺ is equivalent to (A^TA)⁻¹, and this solution realizes the minimization of the sum of the squares of the residuals, providing a solution vector $\widehat{x}$ that is closest to the actual data in the sense of squared errors. Through this method, a statistically reasonable solution can be found even in the face of overly constrained systems.

In this study, the productivity profile of horizontal wells is classified into three permeability categories: 0–5 mD, 5–20 mD, and greater than 20 mD. For each category, correction coefficients (x₁, x₂, and x₃) are derived, corresponding to the respective productivity indices (q_i1, q_i2, and q_i3). These coefficients are crucial for aligning the calculated productivity, as per the formula derived in Section 3, with the actual production capacities Q_i of the ith well observed during DST operations. The formulation, represented in Equation (19), sets the foundation for adjusting theoretical outputs to match real-world data, thereby enhancing the model’s predictive accuracy for estimating the initial productivity of new wells in similar geological settings.

$$Q_i=q_{i1}{x}_1+q_{i2}{x}_2+q_{i3}{x}_3$$

(19)

Compiling production data from multiple wells in the oilfield to establish a dataset enables the construction of an overdetermined system’s coefficient matrix, as outlined in Equation (20).

$$\left[\begin{array}{c}{Q}_1\\ Q_2\\ \dots \\ Q_i\end{array}\right]=\left[\begin{array}{ccc}{q}_{11}& q_{12}& q_{13}\\ q_{21}& q_{22}& q_{23}\\ \dots & \dots & \dots \\ q_{i1}& q_{i2}& q_{i3}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]$$

(20)

Q_i is the actual production capacities of the ith well observed during DST operations; x₁, x₂, and x₃ correspond to productivity indices q_i1, q_i2, and q_i3, respectively.

5. Case Study

5.1. Reservoir Background

The target area A Reservoir of this study is located in the Zhujiangkou Basin in the eastern part of the South China Sea. The Zhujiangkou Basin is a Mesozoic and Cenozoic continental margin rift basin. Its formation and evolution are associated with the subduction collision of the Pacific Plate toward the Eurasian Plate, the collision between the Indian Plate and the Eurasian Plate, and the South China Sea’s expansion. The Zhujiangkou Basin can be divided from north to south into five structural units: the northern step-fault zone, the northern depression zone, the central uplift zone, the southern depression zone, and the southern uplift zone. The well connection profile of the A Reservoir is shown in Figure 4.

The water depth of the A Reservoir is approximately 145 m, with the reservoir buried at depths ranging from −3536 m to −4272 m. This indicates significant stratigraphic amplitude variation and deeper burial of the oil layer. The reservoir type of the A reservoir is characterized as a low-permeability clastic reservoir, with average porosity ranging from 10.2% to 13.7% and average permeability ranging from 0.2 mD to 87.4 mD. This classifies it as a medium-porosity, low-permeability oil reservoir, where reservoir fractures are underdeveloped.

The A reservoir is primarily structural. The primary drive mechanisms are the edge water drive and solution gas drive. Analysis of surface crude oil properties and Pressure-Volume-Temperature (PVT) analysis of the A Reservoir reveal that the crude oil characteristics are favorable, exhibiting light, low viscosity, and low sulfur content.

5.2. Determining Correction Coefficients by Overdetermined Equations

This study has compiled a dataset comprising nine production wells from five low-permeability reservoirs that have undergone DST testing. Logging measured depth, permeability, and porosity data were derived from log curves with a resolution of 0.1 m. The initiation pressure gradient and stress sensitivity parameters were calculated based on a comprehensive analysis of a large amount of true low-permeability core data from the eastern South China Sea. There is an inverse proportional relationship between the initiation pressure gradient and permeability: λ = 0.07/K, and a linear relationship between stress sensitivity and porosity: α_i = −5.63 × 10⁻⁵ · φ + 1.04 × 10⁻³. Utilizing the permeability and porosity interpreted from logging, the stress sensitivity and initiation pressure gradient for each logging point were calculated. Actual productivity data were obtained from DST test values, and the calculated productivity was derived using the horizontal well productivity formula developed in this study (as shown in Equation (15)). The overview of this dataset is shown in

Table 2. Dataset description of one horizontal well.

	Measured Depth (m)	Permeability (mD)	Porosity (%)	Stress Sensitivity Factor	Threshold Pressure Gradient	Qi (m3/d)
Average	3975.94	15.23	9.90 × 10−2	1.03 × 10−3	7.33 × 10−1	3.95 × 10−1
Max value	4007.88	72.22	1.46 × 10−1	1.04 × 10−3	7	2.05 × 100
Min value	3944.00	0.01	1.83 × 10−2	1.03 × 10−3	9.69 × 10−4	0
Amount	512

, with the dataset containing over 4500 lines.

Based on this dataset, 15 equations were established, and MATLAB programming was used to solve for the correction coefficients of the horizontal well productivity equation applicable to offshore low-permeability reservoirs in the eastern South China Sea area through the least squares method. The correction coefficients are x₁ = 3.3182, x₂ = 0.7720, x₃ = 1.0327. The productivity prediction equation can be corrected as $Q_i=3.3182q_{i1}+0.7720q_{i2}+1.0327q_{i3}$.

5.3. Prediction Accuracy Comparison

After determining the productivity correction coefficients for three categories through an overdetermined system of equations, this method was applied to predict the productivity of nine horizontal wells (W1 to W9) not included in the model training dataset to verify its accuracy. The accuracy comparison between the method proposed in this study and the traditional PI method is illustrated in Figure 5 and Figure 6 and

Table 3. Prediction results of the proposed method and PI method.

	W1	W2	W3	W4	W5	W6	W7	W8	W9
Drill Stem Test (m3/d)	102.3	156.3	103.1	146.5	145.2	297.8	203.6	164	60
The Proposed Method (m3/d)	102	161.5	107.6	142.6	153.1	294.6	195.6	166.2	63.7
PI Method (m3/d)	149.5	220.6	72.5	90.5	112.1	220.8	143.2	220.5	36.1
Accuracy of proposed method (%)	99.71	96.67	95.64	97.34	94.56	98.93	96.07	98.66	93.83
Accuracy of PI method (%)	53.86	58.86	70.32	61.77	77.20	74.14	70.33	65.55	60.17

. The proposed method achieved a productivity prediction accuracy ranging from 93.83% to 99.71%, with an average accuracy of 96.82%. In contrast, the PI method showed a significantly lower prediction accuracy, ranging from 53.86% to 77.20%, with an average accuracy of 65.80%.

The significant improvement in prediction accuracy can be attributed to the comprehensive consideration of critical reservoir characteristics, such as longitudinal heterogeneity, threshold pressure gradients, stress sensitivity, and skin effects, within the proposed model. The correction coefficients derived through the overdetermined equations effectively calibrated the theoretical formula, aligning it closely with empirical data and enhancing its predictive performance.

This high accuracy translates into notable practical benefits for offshore reservoir management. By reducing the reliance on costly DST operations, the proposed method lowers operational expenses and expedites decision-making processes. Furthermore, its precise predictions enable operators to optimize well placement, design, and production strategies, maximizing resource utilization and minimizing environmental impact. The reduced error margin also ensures more reliable financial forecasting and risk mitigation, providing a robust framework for sustainable and efficient reservoir management in complex offshore settings.

6. Conclusions and Recommendations

This study conducted a comprehensive evaluation of productivity in nine horizontal wells from offshore low-permeability reservoirs in the eastern South China Sea region. The proposed methodology, which integrates critical reservoir characteristics, significantly improved prediction accuracy compared to traditional methods. The specific contributions and implications of this study are summarized as follows:

• By incorporating key factors such as the threshold pressure gradient, stress sensitivity, skin factor, and formation heterogeneity into a nonlinear seepage mathematical model, this study derived a robust formula for evaluating horizontal well productivity. This formula enables a detailed depiction of productivity profiles at the resolution level of logging curves, providing a more accurate representation of reservoir behavior.
• Based on the differences in permeability distribution of horizontal wells, the productivity of individual wells was classified. The introduction of overdetermined equation concepts allowed the derivation of correction coefficients suitable for the productivity evaluation equation of these horizontal wells: x₁ = 3.3182, x₂ = 0.7720, x₃ = 1.0327. This approach resolved the issue of significant discrepancies between the horizontal well productivity formula and DST productivity data.
• A case study on offshore low-permeability reservoirs in the eastern South China Sea region, through evaluating the productivity of nine horizontal wells in the study area, demonstrated that the method of this study significantly improved the accuracy of productivity evaluation. Compared to the traditional PI method, the accuracy increased from 65.80% to 96.82%.

The enhanced accuracy and practical applicability of the proposed method provide a valuable tool for efficient reservoir management, particularly in complex offshore environments where precise productivity predictions are critical.