Laboratory Experiment and Multivariate Regression Analysis of N₂ Huff-n-Puff in Developing Carbonate Reservoirs

Abstract

Efficient development is a hot topic for carbonate reservoirs. The efficacy of conventional waterflooding diminishes over time due to challenges such as viscous fingering, fluid retention, and significant horizontal heterogeneity. N₂ huff-n-puff (N₂-HNP) operations are the essential method to enhance oil recovery after waterflooding. Taking the X Block of Fuman Oilfields in China as an example, this study leveraged Gaussian Process-Ridge Regression to investigate the impact of parameters on development performance. First, the regression model is built based on experimental data. Univariate analysis explores the effect of single parameters, while multivariate analysis provides an equation for recovery. The parameter importance evaluation ultimately shows that the number of cycles and injection pore volume are the dominant factors controlling the development performance. The results show that (a) the recovery rises to a peak before declining with depth, showing optimal production performance at the middle depths, (b) the recovery exhibits a positive but rapidly saturating correlation with injected pore volume, which suggests an optimal range of 1.0–1.2 PV, (c) the recovery shows a nonlinear response to production pressure difference and its highest sensitivity defines the recommended optimal range of 7.0–9.0 MPa, and (d) the recovery exhibits a nonlinear increase with cycle number, characterized by a rapid initial rise followed by a plateau. This study fills the gap in multivariate regression analysis of parameters for this reservoir type and offers a systematic approach for optimizing the development of similar complex reservoirs worldwide with significant scientific and practical value.

1. Introduction

Worldwide carbonate rock oil and gas resource are abundant, which are especially distributed in the Middle East, North America, Australia, West Asia, and China. Carbonate rocks constitute 20% of the global sedimentary rock area and account for approximately 50% of the residual recoverable oil and gas reserves worldwide, demonstrating their critical importance in the field of oil and gas exploration and development. , With the continuous advancement of carbonate reservoir exploration and development worldwide, fracture-cavity type carbonate reservoirs have garnered widespread attention. Carbonate reservoirs mainly store hydrocarbons within caves and transport fluid through fractures. Characterized by high complexity and poor connectivity, − these reservoirs exhibit vast spatial variations in size and diverse structural forms. In addition to pronounced heterogeneity and anisotropy, the seepage law is complex, which makes development exceptionally challenging. Consequently, they have emerged as a key research focus in the oil and gas exploration sector in recent years.

The X Block of the Fuman Oilfields in China, a single-cave carbonate reservoir predominantly controlled by strike-slip fault zones and subsequent dissolution, is situated on the northern margin of the Taklamakan Desert. The reservoir matrix shows poor pore development, marked by large dissolution fractures and cavities, with significant heterogeneity in reservoir quality. , The fault zone develops a core belt structure, which is characterized by a variety of types, varying maturity, significant asymmetry, and a fence-like distribution. In the early production stage, the significant density difference between oil and water favored a superior vertical displacement efficiency. Thus, waterflooding was adopted as the primary development method. However, as waterflooding continues, challenges such as fingering caused by oil–water viscosity contrast, fluid retention, and significant horizontal heterogeneity become more pronounced. These factors collectively undermine displacement efficiency and lead to a decline in performance. Experimental studies show that the process is hindered by factors such as localized high-water levels, poor well control, flow pathway obstruction, and insufficient hydrodynamic forces after waterflooding. Residual oil of various typesrepresented by “loft oil”forms within fracture-cavity systems. , To enhance oil recovery, considering factors including field situation, N₂-HNP will be used to flood residual oil, considering field conditions. − For guiding on-spot gas injection processes effectively, it is imperative to figure out the law how parameters influence development.

The scholars have conducted numerous experimental and numerical studies on the N₂ injection development mechanism. Studies show that N₂ rapidly enters the reservoir, alters oil distribution, , boosts gravity differentiation, replenishes formation energy, , and improves oil displacement through dissolution and viscosity reduction. , Some studies suggest that low gas injection fails to replenish formation energy, , while excessive injection can cause gas channeling which reduces oil recovery. The recovery increases with the number of throughput cycles, but the oil production gain diminishes with each cycle. − Building on multistage CO₂ displacement and transport studies from slim-tube and core-scale experiments, it was established that mass transfer, pore-scale connectivity, and transport staging collectively govern fluid migration and redistribution under flow-constrained conditions. Although contextualized within CO₂ flooding, this experimental framework and its stage-based transport analysis provide a useful analog for interpreting particle migration and deep placement behavior in fractured systems. Complementing this, recent investigations into fluid occurrence states and transport behavior in heterogeneous tight reservoirs underscore the controlling influence of pore–throat structure and permeability contrasts on flow stability and mobilization efficiency. Furthermore, experimental evidence from near-miscible flooding and threshold pressure gradient studies in low-permeability media offers critical perspectives on flow-transition mechanisms and resistance evolution under pronounced fluid–solid interactions. Overall, current research on N₂-HNP mainly focuses on single-factor influence mechanisms, lacking multifactor regression analysis.

Recent advances in computer technology have led to widespread use of machine learning in petroleum engineering, − bringing some challenges. Davoodi et al. developed a hybrid MLPNN-LSSVM model with GWO and PSO for predicting CO₂ storage and oil production, which requires high data accuracy and diversity, constrained by high application thresholds and strong data dependence. Gao et al. introduced machine learning models such as CMCM and XGBoost to optimize the Eaton method, improving pore pressure prediction accuracy. However, the model is so complex that it requires substantial computing resource and lengthy training times. Xia et al. studied gas huff-n-puff sensitivity in shale oil reservoirs using the Sobol method but overlooked nonlinear relationships and dynamic changes in real production conditions. Diwu et al. applied the Decision Tree algorithm to identify key factors of CO₂-HNP in heavy oil, but its sensitivity to outliers resulted in poor evaluation outcomes. Mansour et al. developed models to predict bubble point pressure in reservoir, which may not test on global oil data sets.

To analyze the nonlinear and coupled effects of injection parameters under limited laboratory experimental data, this study uses the Gaussian Process-Ridge Regression (GPRR) method for analyzing and evaluating complex parameters in a single-cave model. Different from conventional sensitivity analysis, response surface, and machine-learning approaches, the main contribution of this study is the construction of an integrated GPRR-based workflow for small-sample N₂-HNP experimental data. First, GPR constructs a nonlinear response surface that effectively captures the nonmonotonic and plateau-type effects of associated parameters. Second, ridge regression integrates multiple importance indicators (Spearman, Pearson, Kendall, partial correlation, and random forest importance) to mitigate multicollinearity and produce a stable composite ranking, which standard GPR alone does not provide. Compared with standard GPR alone, this framework provides a composite importance evaluation; compared with ridge regression alone, it retains the ability to characterize nonlinear coupled responses through the GPR-predicted response surface. Therefore, the GPRR framework is more suitable for the present task.

2. Experiments

2.1. Materials

Figure presents the statistical results derived from the thin-section data in this study area. The rock types are mainly grainstone, packstone, and mudstone. The reservoirs exhibit an extremely strong heterogeneity.

1.

Frequency distribution histogram of rock types.

Figure shows the results of physical property analysis. It indicates that the porosity of the core samples ranges from 0.54% to 7.46%, with a mean value of 2.17% and a median of 1.51%. The dominant peak lies within the interval of 1.8% to 4.5%. The permeability ranges from 0.0079 × 10^–3 to 391 × 10^–3 mD, with an average value of 11.31 × 10^–3 mD and a median of 0.097 mD. The main peak falls between 0.1 × 10^–3 and 1 × 10^–3 mD.

2.

Distribution histograms of measured core porosity (a) and permeability (b).

A physical model was designed for the experiments, using carbonate rock fragments as the filling medium. The model has a length of 700 mm and an inner diameter of 64.5 mm, corresponding to an internal bulk volume of 2287.22 mL. The model was designed to represent the dominant flow mechanisms of the X Block reservoir. Following the similarity principle for reservoir simulation, the model preserved the storage-flow relationship between vugs and fractures, vertical fluid redistribution, pressure-gravity interactions, and the cyclic injection-soaking-production process. In the model, vugs provide storage space and fractures act as preferential flow channels. The wettability condition was approximated by using carbonate rock fragments saturated with formation oil and compound formation water, thereby preserving the main oil-brine-rock interaction characteristics of the target carbonate reservoir. The bottom-water condition was represented by the initial vertical oil–water configuration in the lower part of the model. Boundary conditions were set as follows: (1) fixed temperature and high initial pressure; (2) constant N₂ injection rate and a soaking period; and (3) back-pressure-controlled production.

The experimental materials included formation oil from the study block (average viscosity at 60 °C: 23.6 mPa s), compound formation water (mineralization: 200,000 mg/L, with its composition shown in Table ), and high-purity nitrogen (99.9% concentration). The experiment was conducted at 60 °C and 30 MPa (above the saturation pressure of the formation oil). Under the experimental condition, the compressibility factor of nitrogen was estimated to be approximately 1.1 with low solubility.

1. Ionic Composition of Formation Brine.

ion	Ca2+	Mg2+	SO4 2–	Cl–	HCO3 –
concentration (mg/L)	11,970	1420	605	131,800	110

2.2. Experimental Apparatus

The experimental setup included a carbonate rock bottom water reservoir model, a displacement pump, a confining pressure pump, a back pressure pump, a precision temperature control box, a back pressure valve, a pressure sensor, an oil-gas separator, a gas flow meter, a camera, a light panel, and a computer, as shown in Figure . This study examined the impact of four key parameters (injection depth, production pressure difference, injection pore volume, and throughput rounds) on N₂-HNP in the vertical injection direction to clarify their effect on enhancing recovery in fracture-cavity oil reservoirs.

3.

Schematic of experimental apparatus.

2.3. Procedure

The experimental steps are as follows.

• The model was installed in the confining pressure holder, vacuum applied to the model and pipeline, saturated with experimental oil, initial reservoir conditions established, and the karst cave volume accurately calculated.

• Ring pressure was applied to the gripper to raise the model pressure to 30 MPa, while maintaining a constant temperature of 60 °C using the heating system.

• N₂ was injected at a constant flow rate of 1 mL/min. After reaching the predetermined gas volume, the inlet and outlet valves were closed, the well sealing operation was performed, and gas and oil were allowed to fully interact under reservoir conditions.

• After being well sealed, a back pressure valve was connected at the outlet to simulate bottom-hole pressure. Production was started, collected, and the produced oil and gas were measured. The experiment was stopped when the oil production rate approached zero.

• The steps were repeated for comparative experiments to systematically analyze the impact of the throughput parameters.

Table presents the details of the experimental design. The tested ranges of these parameters were determined according to the physical scale of the carbonate reservoir model, the operating window of the experimental system, and the key dynamic stages of N₂-HNP development. Injection depth (shallow, intermediate, deep) was varied to assess oil redistribution and bottom-water effects. Injection pore volume (from insufficient to excessive) spans the transition from limited contact to sweep decline and channeling. The production pressure differential was tested across drawdown levels to evaluate elastic expansion and residual oil mobilization. The cycle number covered early, transitional, and plateau stages to capture the full response evolution. Accordingly, the 55-sample matrix was configured to achieve sufficient parameter space coverage for regression analysis over the studied ranges. To assess the reproducibility, selected runs were repeated under representative conditions. The repeated tests yielded consistent variation trends with only minor deviations in the final recovery, confirming the reliability of the experimental observations under the current laboratory settings.

2. Summary of Experimental Design and Tested Parameter Levels .

experimental series	variable parameter	tested levels	fixed experimental conditions
1	injection depth	10, 20, 30, 40, 50, 60 cm	temperature = 60 °C
2	injection pore volume	0.1, 0.3, 0.5, 1.0, 2.0, 4.0, 6.0 PV	initial pressure = 30 MPa
3	production pressure differential	5, 6, 7, 8, 9, 10 MPa	N2 injection rate = 1 mL/min
4	cycles	1, 2, 3, 4, 5, 6, 7, 8, 9, 10

2.4. Experimental Results

Figure shows the relationship between cumulative oil and oil rate and time at different injection depths. In this work, the injection depth refers to the distance from the injection point to the upper part of the core. The oil rate exhibits a decrease-increase-decrease trend. The reason is that early-stage production relies on natural energy. During the middle stage, the decreasing pressure causes N₂ to exsolve and form a gas cap, thereby increasing the N₂-oil contact. However, the limited gas-cap volume leads to a renewed decline in the production rate in the late stages.

4.

Experimental results at different injection depths.

Figure displays the relationship between cumulative oil and time in different injection pore volumes. The aim is to study the impact of the single-cycle injection pore volume on the production. The experimental curves showed a monotonically increasing trend with the growth rate gradually slowing. At low injection pore volumes (e.g., 0.1 PV and 0.3 PV), oil production was relatively low, indicating that limited N₂ injection reduced contact with the oil. As the injected pore volume increased, oil production rose significantly, with the fastest growth at medium injection volumes (e.g., 1 PV and 2 PV), where the sweep efficiency was optimized. However, at higher injection volumes (e.g., 4 PV and 6 PV), the increase in oil production is not significant, showing that injecting pore volume produces a strong effect only within a certain range. Therefore, determining the optimal gas injection volume is crucial as further increases beyond a certain threshold offer limited benefits.

5.

Experimental results at different injection pore volumes.

Figure illustrates the evolution of cumulative oil production over time at different production pressure differentials. All curves exhibit a generally ascending trend, in which the rate of increase is initially rapid but gradually diminishes, eventually asymptotically approaching a steady state. This indicates that a higher production pressure differential leads to a higher displacement efficiency and higher final production.

6.

Experimental results at different production pressure differentials.

Figure shows the evolution law of recovery with cycles under different production pressure differentials. In this work, a Huff-n-puff cycle consists of injection, soaking, and depletion phases. It is clear that the recovery increases gradually with each cycle, until it stabilizes. This indicates that increasing the number of cycles facilitates the mobilization of additional oil, consequently improving oil recovery. The reduction in movable oil during the final stage leads to a slower increase in oil recovery.

7.

Experimental results at different cycles.

Based on the above experiments, a total of 55 valid experimental samples were obtained for subsequent model training and analysis. Each sample contained four input parameters, while the corresponding output parameter was the recovery factor. The experimental data set covered injection depths from 10 to 60 cm, injection pore volumes from 0.1 to 2.0 PV, production pressure differentials from 5 to 10 MPa, and cycle numbers from 1 to 10. Parameter levels were selected across low, medium, and high ranges to reflect the main variability of the process in the model.

3. Multiple Regression Analysis

3.1. Mathematic Theory

Gaussian Process Regression (GPR) is a nonparametric probabilistic model based on Bayesian theory. It treats functions as realizations of random processes and defines the mapping between input and output by modeling the similarity between functions using kernel functions. , For example, Kennedy and O’Hagan (2001) proposed a Bayesian computer model calibration method using GPR to account for multiple error sources, including parameter uncertainty, model defects, observation errors, and code uncertainty, resulting in more reliable predictions.

The input x and output y _k of the GPR model constructed in this study are shown in eqs and .

$$\mathbf{x}={[H,V_{\mathrm{i}\mathrm{n}\mathrm{j}},△p,n]}^T$$ 1

$$y_{\mathrm{k}}\in \{E_{\mathrm{R}},C_{\mathrm{R}}\}$$ 2

In eq , E _R is the recovery and C _R is the displacement efficiency. The training input matrix X and output matrix y _k are as follows:

$$\mathbf{X}=[x^{(1)},···,x^{(M)}]$$ 3

$$y_{\mathrm{k}}={[y_{\mathrm{k}}^{(1)},···,y_{\mathrm{k}}^{(M)}]}^T$$ 4

Gaussian Process Regression (GPR) is a Bayesian inference method where the function is a realization of a random process, following a multivariate Gaussian distribution at any finite-dimensional input point. Its description is shown as eq .

$$y_k(\mathbf{x})\sim \mathcal{G}\mathcal{P}(m_{\mathrm{k}}(\mathbf{x}),k(\mathbf{x},x^{\prime })),k\in \{E_{\mathrm{R}}\}$$ 5

Where m _k (x) is the mean value function and k(x,x ^′) is the kernel function.

To accurately model complex relationships and differentiate parameter influences, this study uses the squared exponential kernel function with automatic relevance determination (ARD), as shown in eq .

$$k(\mathbf{x},x^{\prime })={\sigma }_{\mathrm{f}}^2\exp [-\frac{1}{2}\sum _{d=1}^4\frac{{(x_{\mathrm{d}}-x_{\mathrm{d}}^{\prime })}^2}{{\mathcal{l}}_{\mathrm{d}}^2}]+{\sigma }_{\mathrm{n}}^2{\delta }_{\mathrm{x}{\mathrm{x}}^{\prime }}$$ 6

In this formula, the first term represents the smooth residual correlation and the second term represents the experimental measurement noise. x _d represents the d-th parameter under a specific set of experimental conditions. x _d ^′ is the value taken under another set of experimental conditions. ${\mathcal{l}}_{\mathrm{d}}$ is the influence range of each parameter, which responses variation along each input direction: smaller values imply faster variation, while larger values imply smoother responses. σ_f ² is the residual fluctuation range of the output data. σ _n ² is the experimental noise variance of the measured value. ${\delta }_{\mathrm{x}{\mathrm{x}}^{\prime }}$ is the Kronecker function which is shown in eq .

$${\delta }_{\mathrm{x}{\mathrm{x}}^{\prime }}=\{\begin{array}{l}1,ifx=x^{\prime }\\ 0,else\end{array}$$ 7

Take the partial derivative of eq to obtain eq .

$$\frac{\partial }{\partial x_d}k(\mathbf{x},x^{(i)})=k(\mathbf{x},x^{(i)})\frac{[x_{\mathrm{d}}^{(i)}-x_{\mathrm{d}}]}{{\mathcal{l}}_{\mathrm{d}}^2}$$ 8

Therefore, a covariance matrix of the training data is constructed, as shown in eq .

$$K_{\mathrm{i}\mathrm{j}}=k(x^{(i)},x^{(j)})i,j=1,2,···,M$$ 9

Select the mean function, as shown in eqs –.

$$m_{\mathrm{Q}}(\mathbf{x})=Q$$ 10

$$m_{\mathrm{E}\mathrm{R}}(\mathrm{x})=\frac{m_{\mathrm{Q}}(\mathbf{x})}{Q_{\mathrm{i}\mathrm{n}\mathrm{i}}}$$ 11

$$m_{\mathrm{C}\mathrm{R}}(\mathrm{x})=\frac{m_{\mathrm{Q}}(\mathbf{x})}{V_{\mathrm{i}\mathrm{n}\mathrm{j}}}$$ 12

In eq (3.11), Q _ini is the initial amount of oil.

When a new prediction point x is given, the predicted mean of GPR can be expressed as eq .

$${\mu }_{\mathrm{k}}(x)=m_{\mathrm{k}}(x_{*})+k_{*}^{\mathrm{T}}{K}^{-1}[y_{\mathrm{k}}-m_{\mathrm{k}}(\mathrm{X})]$$ 13

Where,

$$k_{*}=[k(x_{*},x^{(1)}),···,k(x_{*},x^{(M)})]$$

Take the partial derivative of eq to obtain eq .

$$\frac{\partial }{\partial x_{\mathrm{d}}}{\mu }_{\mathrm{k}}(x_{*})=\frac{\partial }{\partial x_{\mathrm{d}}}{m}_{\mathrm{k}}(x_{*})+\sum _{i=1}^M{\alpha }_{\mathrm{i}}^{(\mathrm{k})}k(x_{*},x^{(i)})\frac{[x_{\mathrm{d}}^{(i)}-x_{\mathrm{d}}^{*}]}{{\mathcal{l}}_{\mathrm{d}}^2}$$ 14

In the formula,

$${\alpha }^{(\mathrm{k})}=K^{-1}[y_{\mathrm{k}}-m_{\mathrm{k}}(\mathbf{X})]$$

The uncertainty of the predicted output is given by the variance formula in eq . The first term represents the covariance of the prediction point with itself, while the second term reflects the projection of the correlation between the prediction point and the training samples onto the covariance matrix.

$${\sigma }_{\mathrm{k}}^2(x_{*})=k(x_{*},x_{*})-k_{*}^T{K}^{-1}{k}_{*}$$ 15

In petroleum engineering, traditional sensitivity analysis methods, such as single-factor analysis and response surface methods, often suffer from inconsistent sensitivity scales, which make it difficult to compare the relative influence of different parameters. To overcome this limitation, this study adopts the concept of elasticity, which is widely used as a dimensionless local-sensitivity index.

Based on the standard definition of point elasticity, the local sensitivity of response Y _k to input variable X _d at a prediction point x can be expressed as shown in eq . This dimensionless index represents the relative change in Y _k caused by a relative change in X _d.

$${\eta }_{Y_{\mathrm{k}},X_{\mathrm{d}}}(x_{*})=\frac{X_{\mathrm{d}}^{*}}{Y_{\mathrm{k}}(x_{*})}\frac{\partial Y_{\mathrm{k}}(x_{*})}{\partial X_{\mathrm{d}}}$$ 16

To simultaneously characterize the local sensitivity of recovery to the four input parameters, this study defines an elasticity vector, as shown in eq .

$$\eta (x_{*})=({\eta }_{E_{\mathrm{R}},H}{\eta }_{E_{\mathrm{R}},V_{\mathrm{i}\mathrm{n}\mathrm{j}}}{\eta }_{E_{\mathrm{R}},△p}{\eta }_{E_{\mathrm{R}},n})$$ 17

Where ${\eta }_{E_{\mathrm{R}},H}$ , ${\eta }_{E_{\mathrm{R}},V_{\mathrm{i}\mathrm{n}\mathrm{j}}}$ , ${\eta }_{E_{\mathrm{R}},△p}$ , and ${\eta }_{E_{\mathrm{R}},n}$ denote the local elasticities of recovery factor with respect to dimensionless injection depth, injection pore volume, pressure differential, and cycles, respectively.

3.2. Single-Factor Analysis

For the single-factor analysis, a base case of 30 cm injection depth, 1.0 PV injected pore volume, 6 MPa production pressure differential, and one huff-n-puff cycle was used. In each sweep, only the target parameter was varied, while the other operating and experimental conditions were kept constant.

3.2.1. Analysis on Injection Depth

The analysis results of the gas injection depth are shown in Figures and . From Figure a, cumulative oil exhibits nonmonotonic variation with gas injection depth, with the elastic coefficient strongly influenced by depth. Sensitivity is high at both shallow and deep depths but is low at intermediate depths. It is attributable to the limited sweep of shallow gas injection, whereas deep injection drives the bottom water into the wellbore. From Figure b, both the recovery and displacement efficiency reach their maximum values at a gas injection depth of 30 cm. Further bootstrap-based uncertainty analysis supports this viewpoint, with a 95% bootstrap confidence interval of 32.3–35.5 cm and a bootstrap median of 34.0 cm. These results support the recommendation of the middle injection depth interval rather than a unique, deterministic depth value.

8.

Variation of cumulative oil, elasticity, recovery, and displacement efficiency with respect to injection depth: (a) cumulative oil production and elasticity and (b) recovery and displacement efficiency.

9.

Spatiotemporal distribution of cumulative oil and oil rate with respect to injection depth and time: (a) cumulative oil production heatmap, (b) oil rate heatmap, and (c) 3D response surface of cumulative oil production.

Figure presents the distribution of cumulative oil production and oil rate as functions of time and injection depth. These results highlight the coupled effects of these two factors on production behavior. From Figure a, cumulative oil increases monotonically over time with a significant nonmonotonic depth effect, which indicates that a middle injection depth optimizes sweep efficiency. Figure b shows that the oil rate reaches a peak at the middle depth during the early stage (0–10 min), followed by a rapid decline. Figure c shows that high production rates are concentrated at intermediate depths during the early to middle stages. By contrast, shallow depths (10–20 cm) exhibit only brief and moderate responses, whereas deeper depths (≥50 cm) show delayed, low-amplitude responses and fail to sustain high-rate production.

3.2.2. Analysis on Injection Pore Volume

The analysis results of the injection pore volume are shown in Figures and . The results show that as the injected pore volume increases, the oil volume continues to rise but its elasticity gradually decreases. In other words, after a certain injection volume, further injection has a diminishing effect on recovery efficiency. Figure b illustrates that the recovery increases with injection pore volume but stabilizes around 1.2 PV. However, the displacement efficiency declines after reaching its peak, indicating that the increase in displacement efficiency is limited beyond a certain injection volume. The injection pore volume significantly affects recovery and displacement efficiency up to a threshold, beyond which recovery stabilizes and efficiency declines. Increasing the injection volume results in better sweep, increased oil-gas contact, and increased pressure.

10.

Variation of cumulative oil, elasticity, recovery, and displacement efficiency with respect to injected pore volume: (a) cumulative oil production and elasticity and (b) recovery and displacement efficiency.

11.

Spatiotemporal distribution of cumulative oil and oil rate with respect to injection pore volume and time: (a) cumulative oil production heatmap, (b) oil rate heatmap, and (c) 3D response surface of cumulative oil production.

Figure illustrates how the injected pore volume affects development. In Figure a, cumulative oil volume increases monotonically over time, with a notable nonmonotonic response to injection pore volume, peaking at 1.0–1.2 PV. As the injection volume rises, the cumulative oil growth slows. Figure b,c shows a sharp early peak in oil production at 1.0–1.1 PV, followed by a rapid decline to a steady state. Low injection volumes show moderate early responses, whereas high volumes exhibit a delayed response with smaller fluctuations. To sum up, increasing injection pore volume results in better sweep, thereby increasing oil-gas contact and, most importantly, increasing pressure.

3.2.3. Analysis on Pressure Differential

The analysis results of the pressure differential are shown in Figures and . Figure a shows a significant nonlinear relationship between cumulative oil and pressure differential within the 5.0 to 10.0 MPa range. As the pressure differential increases, the cumulative oil gradually rises. Elastic analysis reveals that the pressure differential most strongly influences oil production in the medium range, indicating that the fluid displacement efficiency improves most significantly within this range. As shown in Figure b, higher pressure differential enhances displacement efficiency, thereby boosting recovery and displacement efficiency.

12.

Variation of cumulative oil, elasticity, recovery, and displacement efficiency with respect to pressure differential: (a) cumulative oil production and elasticity and (b) recovery and displacement efficiency.

13.

Spatiotemporal distribution of cumulative oil and oil rate with respect to pressure differential and time: (a) cumulative oil production heatmap, (b) oil rate heatmap, and (c) 3D response surface of cumulative oil production.

Figure illustrates how the pressure differential impacts development. Cumulative oil increases monotonically over time and the pressure differential. An early peak in oil production is observed, especially under high-pressure differentials, after which the rate rapidly declines and stabilizes at a low level. At low-pressure differences, the response is brief and moderate. According to the three-dimensional coupling analysis, increasing the pressure differential markedly strengthens the initial displacement intensity and accelerates the early flow development. In contrast, during the subsequent stages, the dynamic response diminishes only marginally as the system approaches a stable regime.

3.2.4. Analysis on Cycles

The analysis of the impact of cycles on the development is shown in Figure . As shown in Figure a, the elasticity of the cycle decreases rapidly with the number of cycles. Figure b illustrates that both recovery and displacement efficiency increase monotonically in an S-shaped trend, with most improvements occurring within the first 3–4 cycles before approaching a plateau. This is also because with each new cycle there is less oil to be produced. Parameter uncertainty analysis also supported this conclusion, yielding an inclusion probability of 72.66% within the recommended range.

14.

Variation of elasticity, recovery, and displacement efficiency with respect to cycles: (a) recovery and displacement efficiency and (b) elasticity.

Current recommended parameter ranges are derived primarily from laboratory experiments and regression analyses, pending validation against large-scale pilot production data. These ranges should, therefore, be regarded as experiment-supported optimization outcomes rather than definitive field criteria, with future pilot tests and production monitoring needed for further refinement and validation.

3.3. Multivariate Analysis

Before model training, the input parameters were normalized to eliminate dimensional differences among these parameters. The parameters were made dimensionless, as shown in eq .

$$\{\begin{array}{l}{H}_{\mathrm{D}}=\frac{H}{H_{\max }}\\ △p_{\mathrm{D}}=\frac{△p}{△p_{\max }}\end{array}$$ 18

In the formula, H _D is the dimensionless depth and △p _D is the dimensionless pressure difference. H _max is the bottom depth, m, and △p _max is the pressure of the reservoir.

Gaussian process regression considering multiparameter coupling was performed to obtain the recovery equation, as shown in eq .

$$E_{\mathrm{R}}=f(H_{\mathrm{D}},V_{\mathrm{i}\mathrm{n}\mathrm{j}},△p_D,n)=ym+\sum _{i=1}^N{w}_i\exp [-\frac{S(i)}{2}]$$ 19

where N = 55 is the number of training samples, w _i is the regression coefficient of the i-th training sample, ym is the mean value of the output variable, and S(i) represents the scaled squared distance between the input vector and the i-th standardized training vector. S(i) can be expressed as eq .

$$S(i)={\left[\frac{\frac{H_D-x{m}_1}{x{s}_1}-Xs(i)}{{\mathcal{l}}_1}\right]}^2+{\left[\frac{\frac{V_{\mathrm{i}\mathrm{n}\mathrm{j}}-x{m}_2}{x{s}_2}-Xs(i)}{{\mathcal{l}}_2}\right]}^2+{\left[\frac{\frac{n-x{m}_3}{x{s}_3}-Xs(i)}{{\mathcal{l}}_3}\right]}^2+{\left[\frac{\frac{△p_{\mathrm{D}}-x{m}_4}{x{s}_4}-Xs(i)}{{\mathcal{l}}_4}\right]}^2$$ 20

Where xm and xs are mean vector and standard deviation vector used for input standardization, respectively. $\mathcal{l}$ is the length-scale vector of the Gaussian kernel. X _S(i) is the standardized input vector of the i-th training sample. The four elements of these vectors correspond to H _D, V _inj, Δp _D, and n, respectively.

The regression coefficients and standardized training vectors are given in Table , and the normalization parameters and kernel length-scale parameters are given in Table . Among these parameters, H _D has the largest length scale, which suggests that the fitted recovery response changes relatively smoothly along the injection depth direction. In contrast, the length scales of V _inj and n are relatively close. This indicates that their effects on the fitted response surface cannot be clearly ranked solely based on the ARD length scale values. The above discussion can provide certain reference for the evaluation of importance.

3. Regression Coefficients and Standardized Input Vectors .

i	w i	X S
1	–3.662108	[-1.727814, 1.570959, −0.436369, 1.083419]
2	1.877916	[-1.721499, 1.388786, 0.837020, 0.869346]
3	2.610359	[-1.707284, −0.591695, 0.683909, 0.918806]
4	–2.542108	[-1.692582, −1.150992, −0.414648, −1.026406]
5	3.383563	[-1.687297, −1.329467, 0.007916, −0.891731]
..
51	–2.659068	[-0.066746, 1.234703, 0.416894, −0.739620]
52	0.812482	[-0.037503, 0.629556, −1.233296, 0.945041,]
53	–2.088691	[0.009365, −1.439962, −0.684179, −0.553357]
54	–5.747139	[0.176105, 0.436614, −0.7983705, 1.045349]
55	0.178759	[0.280838, −1.616739, −1.569616, 1.436521]

4. Kernel Parameters after Training.

parameters	Value
ym	0.319923797785956
xm	[34.999907, 1.150042, 5.000099, 7.499973]
xs	[14.446315, 0.491193, 2.888941, 1.444436]
$$\mathcal{l}$$	[1.7203353, 1.211790, 1.434719, 1.221742]

The recovery predictive equation is fully specified by eqs and in conjunction with the trained parameters provided in Tables and . Given that a full expansion would entail 55 kernel terms and compromise readability, the equation is reported in its concise summation form.

Figure indicates that the effects of these four parameters are coupled. Injection pore volume determines the gas supply for pressure support and oil-gas contact, while cycles govern cyclic depletion and residual oil redistribution. Insufficient pore volume renders additional cycles ineffective due to weak pressure support; moderate pore volume allows further cycles to mobilize residual oil, yet beyond an optimum, incremental recovery declines as movable oil is depleted and gas channels through fractures. The pressure differential acts jointly with pore volume and cycles. A suitable differential enhances depletion and gas expansion, but its benefit depends on a sufficient gas volume. Low pore volume makes higher differential only accelerate a limited energy release. Sufficient pore volume enables pressure conversion, although excessive differential shortens gas-oil contact and promotes fracture channeling. Thus, differential should be optimized jointly with the pore volume and cycles. Injection depth affects gas distribution. Shallow injection limits contact with lower oil, while overly deep injection may disturb the bottom water. Intermediate depth improves the sweep efficiency. Optimal performance thus requires coordinated matching of all four parameters, with pore volume and cycles ensuring pressure replenishment and mobilization and differential and depth avoiding channeling, insufficient sweep, or inefficient utilization.

15.

Analysis of multifactor.

3.4. Model Validation and Predictive Reliability

To further evaluate the predictive reliability of the proposed GPRR model, model validation was performed using an independent 80/20 training-testing split and 5-fold cross-validation. The predictive performance was quantified by using RMSE, MAE, and R². The results are summarized in Table . This indicates that the model maintained good predictive accuracy for unseen samples. The 5-fold cross-validation (5-fold CV) results were also stable.

5. Validation Performance of the GPRR Model .

index	explanation	training set	testing set	5-fold CV
RMSE	root mean square error	0.069	0.542	0.109
MAE	mean absolute error	0.055	0.237	0.086
R 2	coefficient of determination	0.99995	0.99637	0.99987

To validate the predictive performance of the proposed GPRR model, XGBoost and support vector regression (SVR) were introduced as benchmark models using the same experimental data set and input parameters. Figure indicates that the GPRR model achieves the best overall performance among the three models, with a higher coefficient of determination and lower prediction errors. This further demonstrates that the GPRR is better suited to multivariate response analysis.

16.

Performance comparison of GPRR, XGBoost, and SVR.

The predictive uncertainty, overfitting risk, and residual behavior of the GPRR model were further evaluated. The 95% prediction interval achieved a coverage of 98.24% with an average width of 1.834, indicating a slightly conservative but acceptable uncertainty estimate. Although the testing RMSE was higher than the training RMSE, the testing R² remained high at 0.996, and the 5-fold cross-validation results were stable, suggesting a reliable predictive capability on unseen samples. The residual mean was small, while the skewness and kurtosis indicated several large positive residuals, which may be attributed to the limited sample size and reservoir heterogeneity.

4. Analysis of Main Controlling Factors

To provide clearer guidance for the development of on-site nitrogen injection in oil fields, this study proposes methods based on correlation analysis to identify the key factors influencing the development effectiveness.

4.1. Comprehensive Evaluation Method

To achieve robust importance ranking under nonlinear coupling and limited samples, we developed a comprehensive evaluation method. Individual indicators, including Pearson, Spearman, Kendall, partial correlation, and random forest importance, each reflect a distinct facet of parameter influence, which results in yielding divergent rankings. Thus, ridge regression was applied to fuse these indicators into a composite score, which reduces reliance on any single criterion and improves the ranking stability.

4.1.1. Spearman Correlation

The Spearman correlation coefficient is a nonparametric statistical method for measuring the strength and direction of a monotonic relationship between two variables. , Its central idea is that if two variables exhibit a monotonic relationship, their hierarchical order should tend to align.

For a sample containing n groups of data, the calculation steps for determining the Spearman correlation coefficient between any input parameter and any output parameter are as follows: (a) Convert the raw data into grades R(U _in,i) or R(U _out,i) (from 1 to n), respectively; if identical values occur, take the average grade. (b) Calculate the grade difference d for each pair of observations: d _i = R(U _in,i) – R(U _out,i). (c) Calculate the Spearman correlation coefficient, as shown in eq .

$${\rho }_{\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{n}}=1-\frac{6\sum _{i=1}^n{d}_i^2}{n(n^2-1)}$$ 21

4.1.2. Pearson Correlation

The Pearson correlation coefficient is used to quantify the strength and direction of a linear relationship of two variables. It is a parametric test method and is sensitive to the distribution shape of the data. The basic concept is to assess whether two variables deviate from their means at the same rate by calculating the ratio of their covariance to the product of their standard deviations. If one variable deviates from its mean, and the other variable tends to deviate in the same or opposite direction at a consistent proportion, a linear relationship is considered to exist between the two variables. For a given sample of n sets of data, the Pearson correlation coefficient between an input parameter U _in and an output metric U _out can be calculated, as shown in eq .

$${\rho }_{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{n}}=\frac{\sum _{i=1}^n(U_{\mathrm{i}\mathrm{n},i}-{\mathrm{U̅}}_{in})(U_{\mathrm{o}\mathrm{u}\mathrm{t},i}-{\mathrm{U̅}}_{\mathrm{o}\mathrm{u}\mathrm{t}})}{\sqrt{\sum _{i=1}^n{(U_{\mathrm{i}\mathrm{n},i}-{\mathrm{U̅}}_{\mathrm{i}\mathrm{n}})}^2}\sqrt{\sum _{i=1}^n{(U_{\mathrm{o}\mathrm{u}\mathrm{t},i}-{\mathrm{U̅}}_{\mathrm{o}\mathrm{u}\mathrm{t}})}^2}}$$ 22

4.1.3. Kendall Correlation

The Kendall correlation coefficient is a nonparametric method for measuring correlation based on the consistency ratio of variable ordinal ranks, offering greater robustness for small panel data. For any two pairs of observations (U _in,i, U _out,i) and (U _in,j, U _out,j), if (U _in,i – U _in,j) (U _out,i – U _out,j) > 0, they are called a consistent pair; if (U _in,i – U _in,j ) (U _out,i – U _out,j) < 0, they are called an inconsistent pair. Denote the number of concordant pairs as m ₁ and the number of discordant pairs as m ₂, then the Kendall correlation coefficient can be calculated using eq .

$${\rho }_{\mathrm{K}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{l}\mathrm{l}}=\frac{m_1-m_2}{\sqrt{(m_1+m_2+l_{\mathrm{i}\mathrm{n}})(m_1+m_2+l_{\mathrm{o}\mathrm{u}\mathrm{t}})}}$$ 23

In the formula, l _in represents the number of data pairs with identical injection parameters but have different development performance metrics, while l _out denotes the number of data pairs with identical development performance metrics but differing injection parameters.

4.1.4. Partial Correlation

Partial correlation quantifies the strength of the linear association between two target variables while controlling for one or more other variables. This simultaneous control is achieved by including all variables to be controlled into a single multiple linear regression model so that the confounding effects of all control variables are eliminated.

First, fit a linear regression model to the input variable U _in, as shown in eq .

$$U_{\mathrm{i}\mathrm{n}}=\sum _{i=1}^n{\beta }_i{U}_{\mathrm{i}\mathrm{n},i}+{ϵ}_i$$ 24

Where ϵ _i is the error term and β _i is the coefficient that determines the relationship between each input variable U _in,i and the output variable U _in, which is solved via the ordinary least-squares method and achieves simultaneous control over multiple input variables. Expressed in the matrix form as U _in = Xβ + ϵ where X is the design matrix containing all control variables U _in,i, the partial regression coefficient vector is calculated as:

$$\mathrm{\beta ̂}={(X^{\mathrm{T}}X)}^{-1}{X}^{\mathrm{T}}{U}_{\mathrm{i}\mathrm{n}}$$ 25

Second, calculate the residual error e _i :

$$e_i=U_{\mathrm{i}\mathrm{n}}-{Û}_{\mathrm{i}\mathrm{n}}=U_{\mathrm{i}\mathrm{n}}-\sum _{i=1}^n{\beta }_i{U}_{\mathrm{i}\mathrm{n},\mathrm{i}}$$ 26

Due to the orthogonality property of least-squares estimation, the residual e _i is uncorrelated with every control variable included in the model, which confirms that the linear effects of all n input variables have been simultaneously eliminated from U _in.

Finally, calculate the Partial Correlation:

$${\rho }_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}=\frac{\mathrm{C}\mathrm{o}\mathrm{v}(U_{\mathrm{o}\mathrm{u}\mathrm{t}},e_{\mathrm{i}})}{\sqrt{\mathrm{V}\mathrm{a}\mathrm{r}(U_{\mathrm{o}\mathrm{u}\mathrm{t}})\times \mathrm{V}\mathrm{a}\mathrm{r}(e_{\mathrm{i}})}}$$ 27

In the equation, Cov­(U _out, e _i) is the covariance between the residual e _i of the output variable U _out and the input variable U _in after controlling for the effects of other variables, Var­(U _out) is the variance of the output variable U _out, and Var­(e _i ) is the variance of the residuals of the input variable U _in.

4.1.5. Random Forest

Random forest (RF) is an ensemble learning algorithm that improves model accuracy and robustness by constructing multiple decision trees and aggregating their predictions. It effectively captures nonlinear relationships and interactions among the parameters. This study evaluates feature importance based on an out-of-bag (OOB) error assessment.

First, for the training data set $U={\{(U_{\mathrm{i}\mathrm{n},\mathrm{i}},U_{\mathrm{o}\mathrm{u}\mathrm{t},\mathrm{i}})\}}_{i=1}^n$ , multiple decision trees are constructed using Bootstrap sampling, with each tree splitting nodes based on a randomly selected subset of features. Then, for each out-of-bag sample (U _in,i,U _out,i), the trained trees predict the output variable ${Û}_{\mathrm{o}\mathrm{u}\mathrm{t},\mathrm{i}}^{(b)}$ , as shown in eq .

$${Û}_{\mathrm{o}\mathrm{u}\mathrm{t},\mathrm{i}}^{(b)}=f_{\mathrm{b}}(U_{\mathrm{i}\mathrm{n},\mathrm{i}})$$ 28

Where, f _b represents the trained tree. Subsequently, the mean squared error for out-of-bag samples is calculated:

$$\mathrm{M}\mathrm{S}{\mathrm{E}}_{\mathrm{O}\mathrm{O}\mathrm{B}}=\frac{1}{n_{\mathrm{O}\mathrm{O}\mathrm{B}}}\sum _{i\in \mathrm{O}\mathrm{O}\mathrm{B}}{(U_{\mathrm{o}\mathrm{u}\mathrm{t},\mathrm{i}}-{Û}_{\mathrm{o}\mathrm{u}\mathrm{t},\mathrm{i}}^{(b)})}^2$$ 29

where n _OOB represents the number of samples outside the bag.

Second, for each input variable U _in,i, shuffle its values across all OOB samples to generate a new sample U _in,i ^′. The prediction error and the shuffled mean squared error are then recalculated, as shown in eqs and , respectively.

$${Û}_{\mathrm{o}\mathrm{u}\mathrm{t},\mathrm{i}}=f_{\mathrm{b}}(U_{\mathrm{i}\mathrm{n},\mathrm{i}})$$ 30

$$\mathrm{M}\mathrm{S}{\mathrm{E}}_{\mathrm{O}\mathrm{O}\mathrm{B}}^{\prime }=\frac{1}{n_{\mathrm{O}\mathrm{O}\mathrm{B}}}\sum _{i\in \mathrm{O}\mathrm{O}\mathrm{B}}{(U_{\mathrm{o}\mathrm{u}\mathrm{t},\mathrm{i}}-{Û}_{\mathrm{o}\mathrm{u}\mathrm{t},\mathrm{i}}^{\prime })}^2$$ 31

Third, for each input variable U _in,i, compute the error increment as shown in eq . This increment reflects the deterioration in the model’s predictive performance after perturbing U _in,i ^′. More important features result in larger error increments.

$$\mathrm{\Delta }\mathrm{M}\mathrm{S}{\mathrm{E}}_{\mathrm{j}}=\mathrm{M}\mathrm{S}{\mathrm{E}}_{\mathrm{O}\mathrm{O}\mathrm{B}}^{\prime }-\mathrm{M}\mathrm{S}{\mathrm{E}}_{\mathrm{O}\mathrm{O}\mathrm{B}}$$ 32

Finally, standardizing the representation of input variables is crucial:

$${\rho }_{\mathrm{R}\mathrm{F}}(U_{\mathrm{i}\mathrm{n},\mathrm{i}})=\frac{\mathrm{\Delta }\mathrm{M}\mathrm{S}{\mathrm{E}}_{\mathrm{i}}}{\sum _i\mathrm{\Delta }\mathrm{M}\mathrm{S}{\mathrm{E}}_{\mathrm{i}}}$$ 33

4.1.6. Comprehensive Evaluation Based on Ridge Regression

Ridge regression is a linear regression technique that is used to address multicollinearity issues. The steps are as follows. First, we use the acquired evaluation metrics as input feature W _in, with the target variable being the overall evaluation metric W _out.

Second, data standardization, as shown in eq .

$$z_{\mathrm{j}}=\frac{W_{\mathrm{i}\mathrm{n},\mathrm{j}}-{\mu }_{\mathrm{j}}}{{\sigma }_{\mathrm{j}}}$$ 34

Where μ _j is the mean value of this feature and σ_j is the standard deviation.

Third, construct a ridge regression model:

$$L(\gamma )=\parallel W_{\mathrm{o}\mathrm{u}\mathrm{t}}-W\parallel +\lambda {\parallel \gamma \parallel }^2$$ 35

In the equation, W _in ^′ is the standardized input feature matrix, γ is the regression coefficient, and λ is the regularization parameter.

Fourth, by adjusting the regularization parameter λ to balance model fit and complexity, the regression coefficients for each feature are calculated.

Finally, calculate evaluation ${Ŵ}_{\mathrm{o}\mathrm{u}\mathrm{t}}$ , shown in eq .

$${Ŵ}_{\mathrm{o}\mathrm{u}\mathrm{t}}=\sum _i{\gamma }_i{W}_{\mathrm{i}\mathrm{n},i}={\rho }_{\mathrm{c}\mathrm{o}\mathrm{m}}$$ 36

In the formula, ρ_com is the comprehensive evaluation results predicted by the ridge regression model.

Notably, ridge regression is employed here as an integration tool for multiple importance indicators, thereby reducing single-criterion dependence and improving ranking robustness.

4.2. Results and Discussion

Table shows the results of the comprehensive evaluation. According to the evaluation, the relative importance of the parameters follows the order: cycles > injection pore volume > production pressure differential > injection depth. These results indicate that the number of cycles is the primary controlling parameter for recovery, while the injection pore volume is the second most influential parameter. Therefore, cycle design should be prioritized, , together with the optimization of injected pore volume, in the N₂-HNP development of carbonate reservoirs.

6. Evaluation of Importance.

parameters	Spearman	Pearson	Kendall	partial	RF	comprehensive
injection depth	0.2140	0.2104	0.2187	0.1820	0.1805	0.3033
injection pore volume	0.4619	0.4541	0.4721	0.3928	0.3896	0.4277
pressure differential	0.2574	0.2530	0.2630	0.2189	0.2171	0.3251
cycles	0.8559	0.8414	0.8748	0.7279	0.7220	0.6254

The above importance ranking is not merely a statistical output but also reveals the underlying physical mechanisms governing N₂-HNP in carbonate reservoirs. More precisely, the dominant role of cycles can be explained by the repeated pressure replenishment and depletion processes during N₂-HNP. In each cycle, injected gas replenishes reservoir energy during the injection and soaking stages, while gas expansion during the production stage provides an additional driving force for oil flow. This cyclic process also increases the contact time between the gas and residual oil, which is beneficial for gas-oil interaction and residual oil redistribution. Similar staged gas displacement studies have shown that oil production is more effective before gas breakthrough, whereas the incremental recovery gradually slows after dominant gas-flow channels are formed. Therefore, repeated cycles can progressively mobilize readily movable oil and part of the capillary-confined residual oil within fractures and cavities, enhancing the overall utilization of the system. However, as readily movable oil is gradually depleted and the remaining oil becomes more difficult to mobilize, incremental recovery diminishes in later cycles.

Injection pore volume is identified as the second most important factor because it directly controls the amount of gas available for pressure maintenance, gas expansion, and oil-gas contact. A sufficient injection pore volume improves pressure support, enlarges the swept region, and promotes oil swelling and mobility improvement, all of which contribute to higher recovery. This interpretation is consistent with recent studies showing that gas injection can reduce flow resistance and improve seepage capacity when sufficient gas–oil interaction is achieved. Nevertheless, excessive gas injection tends to channel through highly conductive fractures after the main flow paths are established. In this case, additional gas mainly flows through dominant channels instead of effectively contacting unswept residual oil, thereby reducing fracture-vuggy utilization efficiency and offering limited additional production. Accordingly, the effectiveness of injected pore volume declines after reaching an optimal threshold.

5. Application

The numerical simulations of nitrogen injection huff-and-puff processes were conducted by using CMG-GEM software in order to further evaluate the effectiveness of nitrogen injection in enhancing recovery. The three-dimensional geological model was constructed and the upscaled grid system contained 55 × 60 × 300 cells. The main reservoir parameters from the geological model included an initial reservoir pressure of 78 MPa, reservoir temperature of 150 °C, rock compressibility of 1 × 10^–4 MPa^–1, porosity of 0.05, and permeability ranging from 58 to 350 mD. Figure shows oil-water and oil-gas relative permeability. The numerical model was calibrated by history matching before the nitrogen huff-and-puff prediction.

17.

Relative permeability: (a) oil-water and (b) oil-gas.

Table shows the orthogonal case design. Table is consistent with the preceding analysis, which lists the cases to representative simulation results selected from the orthogonal design. The recovery factors of the investigated schemes ranged from 34.35% to 37.96%, indicating that the recovery response varied within a relatively limited interval under the tested design conditions. The recovery factor is sensitive to the number of cycles. A reduction in the number of cycles leads to a noticeable decrease in the recovery factor, whereas further increases in operational intensity result in only marginal additional improvement.

7. Orthogonal Case Design.

parameter	level 1	level 2	level 3
injection depth (\)	shallow	middle	deep
cycles (number)	27	29	31
cumulative gas injection volume (107 m3)	6.48	6.96	7.44
pressure differential (MPa)	7.6	7.8	8.0

8. Representative Simulation Cases Selected from the Orthogonal Design.

case	injection depth (\)	cycles (number)	cumulative gas injection volume (107 m3)	pressure differential (MPa)	recovery factor (%)
#1	shallow	27	6.48	7.6	34.35
#2	middle	27	6.48	7.6	35.95
#3	deep	27	6.48	7.6	36.01
#4	shallow	29	6.96	7.8	35.20
#5	middle	29	6.96	7.8	37.62
#6	deep	29	6.96	7.8	37.67
#7	shallow	31	7.44	8.0	35.43
#8	middle	31	7.44	8.0	37.88
#9	deep	31	7.44	8.0	37.96

In contrast, injection depth exerts a comparatively minor influence as the deep-injection case only slightly outperforms the middle-depth case. This trend is consistent with the parameter-importance ranking obtained from the comprehensive evaluation, in which the number of cycles is identified as the dominant controlling factor, followed by the injection pore volume.

It should be noted that case 5 does not correspond to the maximum simulated recovery factor. For example, case 6 provides a slightly higher recovery factor than case 5, but the improvement is only 0.05%. This limited increment suggests that further increasing the operation intensity only provides a marginal recovery improvement after a certain level. Considering the limited incremental recovery and the additional economic and operational costs associated with well completion and perforation, case 5 is recommended as the preferred practical scheme. The corresponding oil saturation distributions are presented in Figure . Overall, these results support the applicability and practical usefulness of the GPRR-based framework.

18.

Oil saturation distribution of the recommended scheme.

6. Conclusions

• The proposed GPRR method demonstrates strong applicability to the N₂-HNP process in carbonate reservoirs, enabling the effective determination of optimal combinations of key parameters based on the experimental data. Numerical simulation results further validated the accuracy and reliability of the method.

• Furthermore, the comprehensive evaluation method was employed to identify the dominant controlling factors, revealing that the gas injection volume is the primary parameter governing oil recovery.

• The results indicate that the proposed method can effectively characterize and predict the N₂-HNP process, providing a feasible and promising technical pathway for enhancing oil recovery in carbonate reservoirs.

Glossary

NOMENCLATURE

E _R

recovery factor

C _R

displacement efficiency

X

training input matrix

y_k

output matrix

m _k (x)

mean value function

k(x,x^′)

kernel function

x _d

the d-th parameter under a specific set of experimental conditions

x _d ^′

the value taken under another set of experimental conditions

${\mathcal{l}}_d$

the influence range of each parameter

σ_f ²

the residual fluctuation range of the output data

σ_n ²

the experimental noise variance of the measured value

${\delta }_{\mathbf{x}{\mathbf{x}}^{\prime }}$

Kronecker function

Q _ini

initial amount of oil

H _D

dimensionless depth

△p _D

dimensionless pressure difference

H _max

bottom depth

△p _max

pressure of the reservoir

n

the number of cycles

ρ

the correlation coefficient

The authors declare no competing financial interest.