Data-Enhanced Variable Start-Up Pressure Gradient Modeling for Production Prediction in Unconventional Reservoirs

Abstract

Unconventional reservoirs are critical for future energy supply, but present major challenges for predictions of production due to their ultra-low permeability, strong pressure sensitivity, and non-Darcy flow. Mechanistically grounded physics-based models depend on uncertain parameters derived from laboratory tests or empirical correlations, limiting their field reliability. A data-enhanced variable start-up pressure gradient framework is developed herein, integrating flow physics with physics-informed neural networks (PINNs), surrogate models, and Bayesian optimization. The framework adaptively refines key parameters to represent spatial and temporal variability in reservoir behavior. Validation with field production data shows significantly improved accuracy and robustness compared to baseline physics-based and purely data-driven approaches. Sensitivity and uncertainty analyses confirm the physical consistency of the corrected parameters and the model’s stable predictive performance under perturbations. Comparative results demonstrate that the data-enhanced model outperforms conventional models in accuracy, generalization, and interpretability. This study provides a unified and scalable approach that bridges physics and data, offering a reliable tool for prediction, real-time adaptation, and decision support in unconventional reservoir development.

1. Introduction

Unconventional oil and gas reservoirs have become effective substitutes for conventional resources and are critical to ensuring sustainable global energy supply [1]. The rapid progress of horizontal drilling and hydraulic fracturing technologies has enabled the large-scale development of unconventional reservoirs, making them essential for sustaining the future energy supply. Unconventional reservoirs are characterized by distinct geological and petrophysical properties. Typical features include porosities of generally less than 10%, pore throat diameters smaller than 1 μm, and air permeabilities below 1 mD [2]. Due to such characteristics, unconventional reservoirs cannot sustain common industrial production rates. Commercial exploitation requires large-scale hydraulic fracturing, after which the productivity of fractured reservoirs can only be approximated by their equivalent low- or ultra-low-permeability systems. While productivity models for low- and ultra-low-permeability formations offer partial guidance, these models fail to fully account for the fracture–matrix interactions and multi-scale transport processes in unconventional reservoirs.

Production prediction for unconventional reservoirs remains a major challenge due to their complex flow physics. Their fluid transport is dominated by non-Darcy effects, particularly the presence of a start-up pressure gradient and pressure-sensitive permeability, both of which evolve dynamically with reservoir pressure and flow conditions [3]. The concept of the start-up pressure gradient was first introduced in the 1980s to explain anomalous flow behavior in low-permeability media [4] and has since been refined into variable-gradient models that incorporate pressure-dependent permeability and stress-dependent effects [5,6]. In ultra-low-permeability formations, low porosity, minimal permeability, and high flow resistance impede the development of effective displacement systems [6,7]. As reservoir pressure declines during production, the start-up pressure gradient typically increases, further restricting fluid mobility [8,9]. Although the physical significance of the start-up gradient [10] and pressure sensitivity level [11] is well established, their field-scale implementation often depends on oversimplified empirical correlations. A common limitation is the direct scaling of laboratory-measured gradients to field conditions without sufficient calibration [12,13]. Field identification of the start-up pressure gradient commonly relies on pressure-transient analysis or rate-decline diagnostics, while laboratory approaches include steady-state and unsteady-state core flooding experiments under controlled stress conditions [14,15,16]. Due to reservoir heterogeneity and fracture complexity, translating lab-derived values to field scales remains uncertain [17,18].

Conventional reservoir simulators often prioritize computational efficiency by adopting simplified physics such as constant start-up gradients, homogeneous permeability fields, or idealized fracture networks. While these assumptions reduce model complexity and runtime, they compromise physical realism, especially in highly heterogeneous unconventional systems. In contrast, data-enhanced physics-informed models aim to preserve essential flow physics while leveraging field data for calibration, thereby achieving a better balance between accuracy and practicality. In recent years, data-driven frameworks have been increasingly adopted for reservoir modeling and have demonstrated promising results in history matching and parameter estimation [19,20,21]. Building on this progress, physics-informed neural networks (PINNs) have emerged as a more robust approach by embedding the governing physical equations directly into the learning process. This integration ensures that predictions remain consistent with fundamental flow laws while still adapting to field observations. Recent applications of PINNs in reservoir engineering, including permeability field inversion, dynamic parameter calibration, and production forecasting in unconventional systems, have shown improved accuracy and generalization compared to purely data-driven methods [22,23,24]. These advances highlight the potential of PINNs to better capture the coupled effects of non-Darcy flow and pressure sensitivity in real-world reservoirs.

To address these limitations, this paper proposes a data-enhanced variable start-up pressure gradient modeling framework. The framework integrates physics-based flow equations with machine learning techniques, including physics-informed neural networks, regional surrogate models, and Bayesian optimization. By dynamically refining parameter fields and accommodating spatial variation in flow regimes, the proposed method enhances the reliability of production prediction in ultra-low-permeability systems and provides a unified approach for modeling a wide range of unconventional reservoirs.

2. Physics-Based Modeling in Unconventional Reservoirs

Fractured unconventional reservoirs display nonlinear flow behaviors that challenge conventional modeling. Non-Darcy effects, pressure-sensitive permeability, and start-up pressure gradients interact under stress and depletion, altering pore structure and increasing flow resistance. As a result, the start-up pressure gradient becomes variable, evolving with pressure and position. Capturing these coupled effects is essential for realistic productivity prediction and forms the basis for developing physics-based formulations tailored to unconventional reservoir conditions.

A unified physics-based production model is first established under clear assumptions. The model incorporates non-Darcy flow, pressure-dependent permeability, variable start-up pressure gradient, and skin effects as fundamental governing laws. These mechanisms are not treated as optional or modular components. They are regarded as intrinsic elements of flow physics in unconventional reservoirs. The resulting model provides a consistent theoretical foundation for subsequent data-enhanced modeling. It ensures that further calibration and prediction remain anchored in physically realistic behavior.

2.1. Flow Mechanisms in Ultra-Low-Permeability Systems

2.1.1. Start-Up Pressure Gradient

Bear introduced the concept of the initial start-up pressure gradient through core-scale experiments, aiming to define the lower limit of Darcy’s law in low-permeability and ultra-low-permeability media [11]. For steady, single-phase, one-dimensional laminar flow, Darcy’s law [25] is expressed as follows:

$${\displaystyle \frac{dP}{dl}}={\displaystyle \frac{q\mu }{A}}{K}^{-1}$$

(1)

where P is pressure of fluid flow through the core, MPa; l is length of fluid flow through the core, cm; q is fluid volume through the core, cm³; A is cross-sectional area of the core, cm²; and K is permeability of the core, 10⁻³ μm².

In flow tests, if the pressure gradient–flow curve has a non-zero intercept, the intercept defines the start-up pressure gradient. This indicates that flow initiates only when the applied pressure exceeds a threshold, reflecting non-Darcy behavior in ultra-low-permeability media.

$$\lambda ={\displaystyle \frac{\mathrm{d}P}{\mathrm{d}l}}-{\displaystyle \frac{\mathrm{q}\mu }{\mathrm{A}}}{K}^{-1}$$

(2)

where $\lambda$ is pressure gradient, MPa/m.

For homogeneous, single-phase, steady, constant-temperature, and laminar planar radial flow, the pressure gradient is described by

$${\displaystyle \frac{\mathrm{d}P}{\mathrm{d}r}}={\displaystyle \frac{\mathrm{q}}{2\pi \mathrm{r}\mathrm{h}}}{\mu K}^{-1}={\mu K}^{-1}v$$

(3)

where r is radius of fluid flow, m; h is thickness of the layer, m; v is velocity of fluid flow, m.

Under Darcy flow conditions, the pressure gradient is linearly proportional to velocity. In high-velocity regions near the wellbore, this relationship becomes nonlinear (Figure 1). Forchheimer proposed the following expression to account for this [26]:

$${\displaystyle \frac{\mathrm{d}P}{\mathrm{d}r}}={\mu K}^{-1}v+{\rho \beta v}^2$$

(4)

where β is Forchheimer coefficient.

The first term represents Darcy flow, dominant in regions far from the wellbore, while the second term captures non-Darcy behavior near the wellbore, where velocity increases due to reduced seepage area. The pressure gradient thus varies with radius, making the start-up pressure gradient a radial-dependent function. From this, the start-up pressure gradient can be defined as follows:

$$\lambda ={\displaystyle \frac{\mathrm{d}P}{\mathrm{d}l}}-{\mu K}^{-1}v-{\rho \beta v}^2=-\mu v{K}^{-1}+{\displaystyle \frac{\mathrm{d}P}{\mathrm{d}l}}-{\rho \beta v}^2$$

(5)

Xu proposed using the pressure difference–flow rate method to identify the transition between linear and nonlinear flow regimes in low-permeability cores and determine the corresponding start-up pressure gradient [27].

Based on core experiments (Figure 2), a relation between permeability and the equivalent start-up pressure gradient is established as follows:

$${\lambda }_e=a·K^{-1}+b$$

(6)

where ${\lambda }_e$ is equivalent start-up pressure gradient, MPa/m; K is permeability, 10⁻³ μm²; a is the start-up pressure gradient coefficient; and b is fitting coefficient.

The equivalent start-up pressure gradient λ_e represents an effective, block-scale parameter that averages the intrinsic spatial and temporal variability of the true start-up pressure gradient over a given reservoir unit. By expressing λ_e as a function of permeability through coefficients a and b, Equation (6) provides a practical means to capture the dominant influence of formation heterogeneity on non-Darcy flow behavior. The coefficient a reflects the sensitivity of the start-up pressure gradient to permeability reduction, and b accounts for baseline flow resistance under high-permeability conditions. This parametric form enables straightforward quantification from core data, and offers a low-dimensional, physically interpretable structure.

2.1.2. Pressure-Sensitive Permeability

The pressure-sensitive effect plays a critical role in fluid flow within unconventional reservoirs [28]. As reservoir pressure declines, permeability decreases due to compaction of the pore structure. This phenomenon was first verified by Fatt through core experiments [12], confirming that permeability is not constant, but pressure-dependent (Figure 3). In large-scale fractured unconventional reservoirs, the pressure-sensitive effect is even more pronounced than in conventional low-permeability reservoirs, making it a dominant factor in production performance [29].

The pressure sensitivity coefficient M is defined as follows:

$$M={\displaystyle \frac{\mathrm{d}\mathrm{K}}{\mathrm{d}\mathrm{P}}}{K}^{-1}$$

(7)

By separating variables and integrating with respect to pressure, the permeability–pressure relationship becomes

$$K=K_i{e}^{-\mathrm{M}(P_i-P)}$$

(8)

where K_i is the initial permeability,10⁻³ μm²; P is the current formation pressure, MPa; P_i is the original formation pressure, MPa; and M is the pressure-sensitive effect coefficient, MPa⁻¹.

Experimental data from low- and ultra-low-permeability core samples indicate that the value of M decreases with increasing permeability (Figure 4). This inverse relationship reflects that more compact rock structures are more sensitive to pressure changes. As a result, accurate modeling of reservoir behavior requires incorporating the pressure-sensitive effect to represent the evolution of permeability over time.

2.1.3. Variable Start-Up Pressure Gradient

The pressure-sensitive effect causes a continuous decline in permeability as the formation pressure decreases. Since the start-up pressure gradient is inversely proportional to permeability, this leads to a corresponding increase in the gradient required to initiate flow. The start-up pressure gradient thus becomes a dynamic quantity that evolves with pressure during reservoir depletion.

By combining the exponential permeability–pressure relationship with the empirical form of the start-up pressure gradient, the equivalent gradient under pressure-sensitive conditions can be expressed as follows:

$${\lambda }_e=a·{K_i}^{-1}{e}^{M(P_i-P)}+b$$

(9)

where ${\lambda }_e$ is equivalent start-up pressure gradient, MPa/m; K is permeability,10⁻³ μm²; a is the start-up pressure gradient coefficient; and b is fitting coefficient.

Equation (9) reflects the joint influence of the stress-induced permeability reduction and its effect on flow initiation. It provides a more realistic representation of fluid flow resistance under variable reservoir conditions and serves as the physical basis for constructing predictive models in ultra-low-permeability reservoirs. This expression reflects the combined influence of permeability reduction and flow initiation threshold under changing pressure conditions.

To capture spatial and temporal variability in actual reservoir conditions, Equation (10) is extended to a distributed form:

$${\lambda }_e\left(r,t\right)=a·{K_i}^{-1}\left(r\right)e^{M\left(P_i\right(r)-P(r,t\left)\right)}+b$$

(10)

where λ_e(r,t) is equivalent start-up pressure gradient at radius r and time t, MPa/m; K_i(r) is initial permeability at radius r, 10⁻³ μm²; P_i(r) is initial pressure at radius r, MPa; P(r,t) is pressure at radius r and time t, MPa; a,b can be learned from core data or production history; and M is pressure-sensitive coefficient, MPa⁻¹, calibrated using laboratory or field data.

This generalized formulation allows the variable start-up pressure gradient to be embedded into region-specific flow models. In high-gradient zones near the wellbore, where nonlinearity dominates, parameters such as a, M, and K_i can be estimated using surrogate models trained on laboratory core data or field measurements. This hybrid approach enables physically grounded and dynamically adaptable flow modeling in complex unconventional reservoirs.

2.2. Mathematical Formulation of Physics-Based Models

2.2.1. Pressure Distribution

Pressure and flow field propagation under radial conditions are strongly influenced by the pressure-sensitive effect and non-Darcy flow behavior. Consider a reservoir with constant outer boundary pressure and steady-state radial flow toward a production well. The formation is assumed to be homogeneous and isotropic. Capillary pressure and gravity effects are neglected. Due to mathematical constraints, the start-up pressure gradient is treated as a constant within the solution domain, and the equivalent start-up pressure gradient is used instead of a radius-dependent formulation.

Based on these assumptions, the governing differential equation for radial flow becomes

$${\displaystyle \frac{1}{\mathrm{d}r}}[r{\displaystyle \frac{K}{\mu }}\left({\displaystyle \frac{\mathrm{d}P}{\mathrm{d}r}}-{\lambda }_e\right)]=0$$

(11)

with boundary conditions

$${\left.P\right|}_{r=r_w}=P_w$$

(12)

$${\left.P\right|}_{r=r_e}=P_e$$

(13)

where P_w is bottom flow pressure; MPa; P_e is outer boundary pressure, MPa; r is the radial distance, m; r_w is the well radius, m; and r_e is supply radius, m;

By integrating the pressure-sensitive permeability model and the equivalent start-up pressure gradient expression,

$$P=P_i+{\displaystyle \frac{1}{\mathrm{M}}}\ln ({\mathit{C}}_1{\displaystyle \frac{\mathrm{M}\mu }{K_i}}\mathit{ln}r+{\mathit{C}}_{2}M+{\displaystyle \frac{\mathrm{M}a}{K_i}}r)$$

(14)

constants C₁ and C₂ are determined by applying the following boundary conditions:

$$C_1={\displaystyle \frac{K_i[1-e^{-M\left(p_{\mathrm{e}}-p_{\mathrm{w}}\right)}]-\mathrm{M}a\left(r_{\mathrm{e}}-r_{\mathrm{w}}\right)}{M\mu \ln \frac{r_{\mathrm{e}}}{r_{\mathrm{w}}}}}$$

(15)

$$C_2={\displaystyle \frac{1}{M}}{e}^{-M\left(p_{\mathrm{e}}-p_{\mathrm{w}}\right)}-{\displaystyle \frac{1}{M{K}_0}}{\displaystyle \frac{K_i[1-e^{-M\left(p_{\mathrm{e}}-p_{\mathrm{w}}\right)}]-\mathrm{M}a\left(r_{\mathrm{e}}-r_{\mathrm{w}}\right)}{\ln \frac{r_{\mathrm{e}}}{r_{\mathrm{w}}}}}\ln r_{\mathrm{w}}-{\displaystyle \frac{a}{K_0}}{r}_{\mathrm{w}}$$

(16)

The pressure distribution model considering the variable start-up pressure gradient is

$$P(r)=P_e+{\displaystyle \frac{1}{\mathrm{M}}}\ln [{\displaystyle \frac{1-e^{-M\left(p_{\mathrm{e}}-p_{\mathrm{w}}\right)}}{\ln \frac{r_{\mathrm{e}}}{r_{\mathrm{w}}}}}\ln {\displaystyle \frac{r}{r_{\mathrm{w}}}}-{\displaystyle \frac{\mathrm{M}a\left(r_{\mathrm{e}}-r_{\mathrm{w}}\right)}{K_i\ln \frac{r_{\mathrm{e}}}{r_{\mathrm{w}}}}}+{\displaystyle \frac{\mathrm{M}a\left(\mathrm{r}-r_{\mathrm{w}}\right)}{K_i}}+e^{-M\left(p_{\mathrm{e}}-p_{\mathrm{w}}\right)}]$$

(17)

2.2.2. Permeability Evolution

The spatial distribution of permeability can be derived by substituting the pressure distribution into the following pressure-sensitive equation:

$$K(r)=K_i+K_i[{\displaystyle \frac{1-e^{-M\left(p_{\mathrm{e}}-p_{\mathrm{w}}\right)}}{\ln \frac{r_{\mathrm{e}}}{r_{\mathrm{w}}}}}\ln {\displaystyle \frac{r}{r_{\mathrm{w}}}}-{\displaystyle \frac{aM\left(r_{\mathrm{e}}-r_{\mathrm{w}}\right)}{K_i\ln \frac{r_{\mathrm{e}}}{r_{\mathrm{w}}}}}\ln {\displaystyle \frac{r}{r_{\mathrm{w}}}}+{\displaystyle \frac{aM\left(\mathrm{r}-r_{\mathrm{w}}\right)}{K_i}}+e^{-M\left(p_{\mathrm{e}}-p_{\mathrm{w}}\right)}-1]$$

(18)

This expression describes how permeability degrades nonlinearly from the outer zone to the wellbore due to pressure drawdown and stress sensitivity.

2.2.3. Pressure Gradient

Given the pressure sensitivity, the pressure gradient also varies dynamically. The corresponding start-up pressure gradient is calculated as follows:

$$\lambda (r)={\displaystyle \frac{a}{K_i}}{[{\displaystyle \frac{1-e^{-M\left(p_{\mathrm{e}}-p_{\mathrm{w}}\right)}}{\ln \frac{r_{\mathrm{e}}}{r_{\mathrm{w}}}}}\ln {\displaystyle \frac{r}{r_{\mathrm{w}}}}-{\displaystyle \frac{aM\left(r_{\mathrm{e}}-r_{\mathrm{w}}\right)}{K_i\ln \frac{r_{\mathrm{e}}}{r_{\mathrm{w}}}}}\ln {\displaystyle \frac{r}{r_{\mathrm{w}}}}+{\displaystyle \frac{aM\left(\mathrm{r}-r_{\mathrm{w}}\right)}{K_i}}+e^{-M\left(p_{\mathrm{e}}-p_{\mathrm{w}}\right)}]}^{-1}$$

(19)

This formulation provides a dynamic coupling of pressure, permeability, and flow resistance throughout the reservoir. It forms the analytical foundation of the physics-based baseline model.

2.3. Physics-Based Production Models

2.3.1. Fluid Volume

Assuming steady-state, single-phase flow in a homogeneous, isotropic circular reservoir, the radial flow equation with a start-up pressure gradient is expressed as follows [30]:

$$q={\displaystyle \frac{2\pi rhK}{\mu B}}·({\displaystyle \frac{\mathrm{d}p}{\mathrm{d}r}}-{\lambda }_e)$$

(20)

where q is fluid volume, m³/d; r is radial distance, m; h is thickness of formation, m; K is permeability of formation, 10⁻³ μm²; μ is viscosity of formation fluid, mPa·s; B is volume coefficient of formation fluid, ${\displaystyle \frac{\mathrm{d}p}{\mathrm{d}r}}$ is pressure gradient, MPa/m; and ${\lambda }_e$ is equivalent start-up pressure gradient, MPa/m.

Considering variable start-up pressure gradient, the volumetric flowrate is given by

$$q={\displaystyle \frac{2\pi h{K}_i}{\mu B}}·{\displaystyle \frac{\left[1-e^{-M\left(p_{\mathrm{e}}-p_{\mathrm{w}}\right)}\right]-aM\left(r_{\mathrm{e}}-r_{\mathrm{w}}\right)}{M·\left[\ln \left(\frac{r_{\mathrm{e}}}{r_{\mathrm{w}}}\right)+S\right]}}$$

(21)

where S is skin factor.

2.3.2. Productivity

The productivity index J relates the production rate to the pressure drawdown and is a key performance indicator for reservoir deliverability [31]. Incorporating variable start-up pressure gradient, the generalized productivity index is given by

$$J_{M\lambda }={\displaystyle \frac{2\pi h{K}_i}{\mu B}}·{\displaystyle \frac{{(p_{\mathrm{e}}-p_{\mathrm{w}})}^{1-M}+a(M-1){·K_0}^{-1}·(r_{\mathrm{e}}-r_{\mathrm{w}})}{\left(p_{\mathrm{e}}-p_{\mathrm{w}}\right)·\left(1-M\right)\left[\ln \left(\frac{r_{\mathrm{e}}}{r_{\mathrm{w}}}\right)+S\right]}}$$

(22)

The model provides essential inputs for reservoir performance prediction and serves as the baseline for constructing the data-enhanced framework.

3. Data-Enhanced Production Prediction in Unconventional Reservoirs

Physics-based models form the foundation for production prediction in unconventional reservoirs but are insufficient for capturing complex reservoir behavior [32,33]. Their structural rigidity and reliance on uncertain parameters such as the pressure gradient λ, pressure-sensitive coefficient M, empirical constants a, b, and skin factor S are typically derived from laboratory conditions or historical assumptions, which may not reflect in situ reservoir behavior. These parameters are difficult to measure accurately and often lack sufficient field validation. To address these limitations, a data-enhanced framework based on Physics-Informed Neural Networks (PINNs) is proposed to enable adaptive correction and improve predictive robustness [34].

3.1. Physics-Informed Neural Networks for Production Prediction

A data-enhanced modeling framework is developed by integrating PINNs with mechanistic models. PINNs incorporate physical laws directly into the learning process, enabling the correction of spatial non-stationarity and improving production prediction performance even under sparse or noisy data conditions.

The framework consists of input features including reservoir parameters and test parameters. These variables evolve according to nonlinear governing equations that account for non-Darcy flow, pressure-sensitive effects, and variable start-up pressure gradients. These surrogates correct spatial variability and non-stationary effects, enhancing production prediction performance. The network outputs key target variables, such as pressure distribution P(r,t), fluid volume rate q(t), and productivity index J(t), ensuring consistency with governing equations while adapting to field data through learning-based corrections.

Unlike conventional numerical simulators that rely on local grid refinement to capture steep gradients, the PINN-based approach does not discretize the domain. Instead, it enforces physical laws globally through the PDE residual across all collocation points. This shifts the focus from local numerical accuracy to global physics–data consistency; the solution is constrained by the governing equations while being anchored to observed production data, leading to robust predictions without mesh-dependent artifacts.

To ensure the surrogate model is both expressive and computationally efficient within the PINN framework for production prediction, the neural network architecture is selected through a capacity–physics–data trade-off protocol that directly accounts for the physical complexity. The architecture starts with a single hidden layer. Its width follows the geometric-pyramid rule, resulting in 40 to 60 neurons for the five-input and three-output setting. The inputs include reservoir and test parameters. The outputs are pressure distribution P(r,t), fluid rate q(t), and productivity index J(t). The architecture is refined iteratively. The number of neurons is increased by 10 only if the PDE residual remains above 1 percent of the data loss for more than 2000 epochs, or if the validation R² improves by less than 2 percent. A second hidden layer is added only when sharp saturation fronts or strong spatial non-stationarity appear in the predicted fields, as a single layer cannot capture these features adequately. Throughout this process, the total number of trainable parameters is kept no greater than the number of available field data points. This prevents overfitting and ensures stable training under the sparse and noisy data conditions typical of unconventional reservoirs. The resulting PINN can accurately represent the nonlinear physics of non-Darcy flow, pressure-sensitive permeability, and variable start-up pressure gradients. It also remains efficient and robust in correcting spatial variability through learning as shown in Figure 5.

3.2. PINN-Enhanced Production Calculation Models

PINNs are adopted to enhance physical models by learning residual patterns, correcting spatial variability, and adapting to field data.

The governing pressure propagation equation in radial flow under variable start-up pressure gradients λ_e(r,t) is as follows:

$${\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left[{\displaystyle \frac{rK(r,t)}{\mu }}·\left({\displaystyle \frac{\mathrm{d}p\left(r,t\right)}{\mathrm{d}r}}-{\lambda }_e(r,t)\right)\right]=0$$

(23)

where P(r,t) is pressure at radius r and time t, MPa; K(r,t) is time- and space-dependent permeability, 10⁻³ μm²; and λ_e(r,t) is equivalent start-up pressure gradient, MPa/m.

Equation (23) uses a steady-state form as a simplified physical constraint in the PINN framework. This helps maintain numerical stability while preserving physical meaning. The model accounts for the complex, time-varying behavior of real reservoirs including transient flow, fracture effects, and fluid heterogeneity through data-driven parameters and other effective terms that are learned from field data.

In the data-enhanced framework, the governing partial differential equation (PDE) is embedded as a soft constraint within the PINNs, allowing the network to learn corrective parameters that reflect deviations from ideal physical behavior. The model infers spatially and temporally varying coefficients the pressure-sensitive factor M^∗(r,t), the start-up pressure gradient coefficient a^∗(r,t), b^∗(r,t) and skin factor S^∗(r,t) based on observed production responses instead of solving the equations explicitly. The corrected coefficients are formulated as follows:

$$a^{\ast }\left(r,t\right)=a_p\left(r\right)+{∆a}_{PINN}\left(r,t\right)$$

(24)

$$b^{\ast }\left(r,t\right)=b_p\left(r\right)+{∆b}_{PINN}\left(r,t\right)$$

(25)

$$M^{\ast }\left(r,t\right)=M_p\left(r\right)+{∆M}_{PINN}\left(r,t\right)$$

(26)

$$S^{\ast }\left(r,t\right)=S_p\left(r\right)+{∆S}_{PINN}\left(r,t\right)$$

(27)

where a_p, b_p, M_p, S_p are initial estimates from physics-based or empirical models; and Δa_PINN, Δb_PINN, ΔM_PINN, ΔS_PINN are learned correction terms from the neural network.

The correction terms Δa_PINN, Δb_PINN, ΔM_PINN, and ΔS_PINN are not meant to represent specific physical mechanisms. Instead, they adjust the initial physics-based estimates to better match field data. These terms are learned under the guidance of the governing PDE, so they reflect the combined effects of unresolved processes such as stress-dependent permeability or fracture changes while staying within a physically consistent framework. They act as data-driven error corrections, not arbitrary black-box fits.

These learned parameters are incorporated into the physically derived expressions, and the corrected equivalent start-up pressure gradient can be expressed as follows:

$${\lambda }_e^{\ast }\left(r,t\right)=a^{\ast }\left(r,t\right)·{K_i}^{-1}\left(r\right)e^{M^{\ast }\left(r,t\right)\left(P_i\right(r)-P(r,t\left)\right)}+b^{\ast }\left(r,t\right)$$

(28)

where λ_e^∗(r,t) is the corrected equivalent start-up pressure gradient dynamically inferred by the network.

The pressure field is represented in a distributed form as follows:

$$P^{\ast }(r,t)=P_e+{\displaystyle \frac{1}{M^{\ast }\left(r,t\right)}}\ln [{\displaystyle \frac{1-e^{-M^{\ast }\left(r,t\right)\left(p_{\mathrm{e}}-p_{\mathrm{w}\left(t\right)}\right)}}{\ln \frac{r_{\mathrm{e}}}{r_{\mathrm{w}}}}}\ln {\displaystyle \frac{r}{r_{\mathrm{w}}}}-{\displaystyle \frac{M^{\ast }\left(r,t\right)a^{\ast }\left(r,t\right)\left(r_{\mathrm{e}}-r_{\mathrm{w}}\right)}{K_i\ln \frac{r_{\mathrm{e}}}{r_{\mathrm{w}}}}}+{\displaystyle \frac{M^{\ast }\left(r,t\right)a^{\ast }\left(r,t\right)\left(\mathrm{r}-r_{\mathrm{w}}\right)}{K_i}}+e^{-M^{\ast }\left(r,t\right)\left(p_{\mathrm{e}}-p_{\mathrm{w}\left(t\right)}\right)}]$$

(29)

The permeability field corrected by the learned PINN parameters becomes

$$K^{\ast }(r,t)=K_i(r)+K_i(r)[{\displaystyle \frac{1-e^{-M^{\ast }(r,t)\left(p_{\mathrm{e}}-p_{\mathrm{w}(t)}\right)}}{\ln \frac{r_{\mathrm{e}}}{r_{\mathrm{w}}}}}\ln {\displaystyle \frac{r}{r_{\mathrm{w}}}}-{\displaystyle \frac{a^{\ast }(r,t)M^{\ast }(r,t)\left(r_{\mathrm{e}}-r_{\mathrm{w}}\right)}{K_i\ln \frac{r_{\mathrm{e}}}{r_{\mathrm{w}}}}}\ln {\displaystyle \frac{r}{r_{\mathrm{w}}}}+{\displaystyle \frac{a^{\ast }(r,t)M^{\ast }(r,t)\left(\mathrm{r}-r_{\mathrm{w}}\right)}{K_i}}+e^{-M^{\ast }(r,t)\left(p_{\mathrm{e}}-p_{\mathrm{w}(t)}\right)}-1]$$

(30)

The fluid volume rate inferred under the data-enhanced regime is expressed as follows:

$$q^{\ast }\left(t\right)={\displaystyle \frac{2\pi h{K}_i\left(r\right)}{\mu B}}·{\displaystyle \frac{\left[1-e^{-M^{\ast }\left(r,t\right)\left(p_{\mathrm{e}}-p_{\mathrm{w}\left(t\right)}\right)}\right]-a^{\ast }\left(r,t\right)M^{\ast }\left(r,t\right)\left(r_{\mathrm{e}}-r_{\mathrm{w}}\right)}{M^{\ast }\left(r,t\right)\left[\ln \left(\frac{r_{\mathrm{e}}}{r_{\mathrm{w}}}\right)+S^{\ast }\left(r,t\right)\right]}}$$

(31)

The corrected dynamic productivity index J^*(t) is defined as follows:

$$J^{\ast }\left(t\right)={\displaystyle \frac{2\pi h{K}_i\left(r\right)}{\mu B}}·{\displaystyle \frac{{(p_{\mathrm{e}}-p_{\mathrm{w}\left(t\right)})}^{1-M^{\ast }\left(r,t\right)}+a^{\ast }(r,t)(M^{\ast }(r,t)-1){·K_0}^{-1}(r_{\mathrm{e}}-r_{\mathrm{w}})}{\left(p_{\mathrm{e}}-p_{\mathrm{w}}\left(t\right)\right)·\left(1-M^{\ast }\left(r,t\right)\right)\left[\ln \left(\frac{r_{\mathrm{e}}}{r_{\mathrm{w}}}\right)+S^{\ast }\left(r,t\right)\right]}}$$

(32)

These augmented formulations retain physical interpretability while benefiting from the data-driven adaptability of PINNs, enabling improved prediction of pressure, permeability, fluid rate, and productivity index under realistic, heterogeneous unconventional reservoirs.

3.3. Composite Loss Function Design and Training Strategy

The network architecture consists of fully connected layers with inputs comprising variables associated with spatial coordinates r and t, and outputs P(r,t), q(t), and optionally J(t). The training process minimizes a composite loss function:

$$L=L_d+{\omega }_1{L}_p+{\omega }_2{L}_b+{\omega }_3{L}_r$$

(33)

where L_d is data mismatch loss; L_p is physics loss; L_r is rate loss; and ω_i is tunable weight.

The data mismatch loss L_d minimizes the mean squared error (MSE) between predicted values (pressure or production rate) and actual observations:

$$L_d={\displaystyle \frac{1}{N_d}}{\sum }_{i=1}^{N_d}{(y_i^{\ast }-y_i^0)}^2$$

(34)

where N_d is number of data points used for observation-based loss; y_i^* is predicted value at the i-th data point; and y_i⁰ is observed/measured value at the i-th data point.

The physics loss L_p enforces the residual minimization of the governing equation (Equation (32)) at selected collocation points using automatic differentiation:

$$L_p={\displaystyle \frac{1}{N_p}}{\sum }_{i=1}^{N_p}{\left(R_j\right)}^2$$

(35)

$$R_j={\displaystyle \frac{1}{r_j}}{\displaystyle \frac{\partial }{\partial r}}\left[{\displaystyle \frac{r_{j}K(r_j,t_j)}{\mu }}·\left({\displaystyle \frac{\mathrm{d}p\left(r_j,t_j\right)}{\mathrm{d}r}}-{\lambda }_e(r_j,t_j)\right)\right]$$

(36)

where N_p is number of collocation points for evaluating physical residuals; R_j is residual of PDE at collocation point; r_j, t_j is spatial and temporal coordinates of the j-th collocation point; K(r_j,t_j) is permeability at point (r_j,t_j), either predicted or from the physical model; p(r_j,t_j) is predicted pressure by the neural network; and λ_e(r_j,t_j) is equivalent start-up pressure gradient.

The boundary loss L_b enforces pressure constraints at inner and outer boundaries (r = r_w, r_e):

$$L_b={\displaystyle \frac{1}{N_b}}{\sum }_{k=1}^{N_b}{\left(p^{\ast }\right(r_k,t_k)-p_b(r_k,t_k\left)\right)}^2$$

(37)

where N_b is number of boundary condition points; p(r_k,t_k) is predicted pressure at point (r_k,t_k); and p_b(r_k,t_k) is boundary condition pressure at point (r_k,t_k), as p(r_w,t) = p_w(t), p(r_e,t) = p_e.

The loss rate L_r ensures the consistency of network predictions with measured production rate q(t):

$$L_r={\displaystyle \frac{1}{N_r}}{\sum }_{m=1}^{N_r}{(q_m^{\ast }-q_m^0)}^2$$

(38)

and productivity index J(t)

$$L_r={\displaystyle \frac{1}{N_r}}{\sum }_{m=1}^{N_r}{(J_m^{\ast }-J_m^0)}^2$$

(39)

where N_r is number of data points for production rate or productivity index; q_m^* is predicted fluid rate at time step m; q_m⁰ is observed/measured fluid rate at time step m; J_m^* is predicted productivity index at time step m; and J_m⁰ is observed/measured productivity index at time step m.

The weights ω₁, ω₂, and ω₃ are adaptively tuned via gradient-based or uncertainty-aware heuristics, balancing physical constraints and data fidelity. An emphasis on L_p can be prioritized when data are limited. The combined use of real and synthetic data ensures robust, physically consistent generalization under complex reservoir conditions.

$${\omega }_i={\displaystyle \frac{1}{Var\left(L_i\right)+\epsilon }}$$

(40)

where ω_i is tunable weight for loss component i, balancing contributions of physics, data, and boundary fidelity; Var(L_i) is variance of loss component L_i, used in uncertainty-aware adaptive weighting; and ϵ is small regularization constant to avoid division by zero.

This composite loss design enables the network to jointly learn from noisy field data and physically consistent priors, ensuring model generalizability, stability, and robustness under complex reservoir conditions.

4. Evaluation and Interpretation of Data-Enhanced Model

4.1. Evaluation of Data-Enhanced Model

4.1.1. Validation Using Field Production Data

To evaluate the reliability and applicability of the proposed data-enhanced model, a comprehensive validation was performed using production data from a representative reservoir block. The validation process included dynamic response variables such as monthly production, cumulative production, formation pressure, and permeability, allowing for a multi-dimensional assessment of model fidelity.

Figure 6 shows actual field cumulative production data and model predictions spanning 2000–2024. All curves show monotonic growth: gradual during 2000–2010, accelerated from 2010 to 2018, and slower during 2018–2024. Model predictions diverge significantly post-2003. The Darcy flow model severely overestimates production due to neglected sub-Darcy effects, projecting 4.26 × 10⁴ tons in 2024 with a 32.7% overestimation. The pressure-sensitivity model predicts 3.98 × 10⁴ tons with 24.0% deviation. The start-up pressure gradient model predicts 3.70 × 10⁴ tons with 15.3% deviation. The variable start-up pressure gradient model reduces deviation to 10.3%.

The data-enhanced model integrates multiphysics mechanisms with field data, achieving the closest 2024 prediction of 3.28 × 10⁴ tons versus the actual 3.21 × 10⁴ tons, limiting deviation to 2.2%.

Analysis confirms substantial prediction errors from traditional Darcy models in low-permeability reservoirs. Effective prediction requires integration of multiphysics mechanisms, including start-up pressure gradients and pressure-sensitivity effects with field data calibration. The data-enhanced model demonstrates superior long-term predictive capability, providing reliable decision support for field development.

The data clearly illustrate the evolution of errors in different reservoir prediction models versus actual production from 2000 to 2024 as shown in Figure 7. The traditional Darcy flow model consistently overestimates production, with errors decreasing from 62.75% in 2003 to 33.00% in 2024 while maintaining the highest deviation. The model considering pressure-sensitive effect reduced errors from 51.13% to 24.10%. Introducing start-up pressure gradient further enhanced accuracy, converging errors from 41.57% to 15.39%. The variable start-up pressure gradient model showed superior performance, systematically reducing errors from 32.50% to 10.34%, proving its adaptability to complex reservoirs. The data-enhanced model achieved significant improvement, with an initial error rate of 9.75% in 2003 that decreased to 2.15% in 2024, maintaining sub-2.5% errors after 2021. All models exhibited decreasing errors over time, with rapid improvement during 2003–2012 transitioning to stable convergence post-2012. In 2024, the data-enhanced model demonstrated absolute superiority, with a 2.15% error rate versus the variable start-up pressure gradient, validating the significant advantage of data fusion techniques under physical constraints.

Figure 8 presents formation pressure distribution across a 250 m well spacing. Field pressure data at 0 m and 250 m were used for boundary calibration and parameter tuning, making the 125 m pressure value an independent validation benchmark. Field measurements show 2.6 MPa at injection well 0 m, 8.66 MPa at midpoint 125 m, and 12.5 MPa at production well 250 m. All models exhibit monotonic pressure increases, converging precisely at 12.5 MPa. Significant mid-spacing deviations emerge: Darcy flow predicts 11.73 MPa and pressure-sensitive model 11.65 MPa, both exceeding measured values. Conversely, start-up pressure gradient, variable start-up, and data-enhanced models yield 8.85 MPa, 8.79 MPa, and 8.8 MPa, respectively, closely matching field data. This confirms non-Darcy flow models accurately capture critical mid-spacing behavior, with the data-enhanced model demonstrating superior profile consistency.

Based on the comparison of formation pressure errors at 125 m well space, the Darcy flow model exhibits an error of 34.53%, while the pressure-sensitive effect model shows an error of 35.45%, indicating systematic deviations in classical theories (Figure 9). The introduction of the start-up pressure gradient drastically reduces the error to 2.20%, and the variable start-up pressure gradient model further optimizes it to 1.65%. The Data-enhanced model achieves the lowest error of 1.52%, representing an improvement over traditional models.

Permeability predictions across models exhibit significant divergence with increasing well space (Figure 10); the Darcy flow model maintains a constant 4.5 × 10⁻³ μm², disregarding permeability dynamics; the pressure-sensitive effect model shows gradual elevation but overestimates permeability at 125 m with 4.38 × 10⁻³ μm² versus the actual 4.1 × 10⁻³ μm²; and the variable start-up pressure gradient and data-enhanced models demonstrate exceptional congruence, precisely aligning with field measurements.

Field test data at 0 m and 250 m were used for model boundary calibration. Permeability values at 125 m were independently evaluated for prediction accuracy. Darcy flow showed the highest error rate at 9.76%, while pressure-sensitive modeling achieved 7.68%. The variable start-up pressure gradient model yielded perfect matching at 0.00%, and the data-enhanced model registered a −0.03% error rate as shown in Figure 11. Both advanced models maintained errors below 0.03%, significantly outperforming conventional approaches. The deviation in the data-enhanced model stems from its global optimization strategy, confirming its practical reliability in reservoir engineering applications.

4.1.2. Comparative Assessment of Modeling Paradigms

To evaluate the robustness and applicability of the proposed framework, we compared three categories of models. Physics-based models rely on fixed assumptions and predefined correlations. Data-enhanced models incorporate physics-informed neural networks to iteratively refine parameter fields under physical constraints. Pure data-driven models, specifically Gaussian Process Regression (GPR) and Deep Neural Networks (DNN), depend solely on observational data without prior physical knowledge. All models were tested on identical datasets and assessed for predictive accuracy, physical consistency, generalization to new wells, and data requirements.

The analysis reveals that physics-based models provide interpretability and low data demands but exhibit high RMSE values and poor adaptability in heterogeneous reservoirs. Pure data-driven models achieve moderate accuracy but demonstrate limited robustness due to their dependence on large datasets and weak extrapolation in new conditions (Figure 12). In contrast, the data-enhanced model achieves the optimal balance, delivering the lowest RMSE, strong generalization, and moderate data requirements while maintaining physical consistency. This demonstrates the clear advantage of integrating physical principles with data-driven learning for reliable and transferable unconventional reservoir production prediction.

While the current comparison focuses on representative modeling paradigms, the proposed data-enhanced model shares a key feature with other hybrid approaches such as physically constrained Gaussian processes, both embedding physical laws into data-driven frameworks. However, the PINN-based approach offers distinct practical advantages; it scales more effectively to the high-dimensional and nonlinear systems typical of unconventional reservoirs, supports end-to-end training with heterogeneous data sources, and enforces physical constraints directly through the loss function without requiring an explicit covariance structure. These characteristics make the data-enhanced model a highly promising, and potentially optimal, choice among hybrid strategies for complex, data-scarce prediction.

4.1.3. Benchmarking Against Commercial Simulators and Recent Data-Driven Models

To quantitatively evaluate the performance of our proposed data-enhanced model, we benchmarked it against two reference approaches, including a commercial numerical reservoir simulator that represents industry-standard tools such as CMG or ECLIPSE, and the pure data-driven Vector Autoregressive (VAR) model [35]. The validation was conducted on a well group of a reservoir and spans the period from 1970 to 2018, using the same test period and the same error metrics as reported in the original study.

As shown in

Table 1. Comparison of mean relative error (%) in daily oil rate prediction.

Well ID	Commercial Simulator	VAR Model	Data-Enhanced Model
X50-5-1	12.80	3.65	2.10
X50-7-2	5.40	1.26	0.95
X51-5-1	8.70	6.30	3.80
X50-6-3	7.20	2.90	1.75
X51-6-2	6.50	2.40	1.40
Mean	8.12	3.30	2.00

, the commercial simulator exhibits the highest errors across all wells with a mean of 8.12%, and it shows particularly poor performance in X50-5-1 with an error rate of 12.80%. This reflects the challenges of numerical simulation in complex reservoirs where uncertainties in geological modeling, relative permeability, and grid discretization can degrade history-matching and prediction accuracy. The VAR model significantly reduces prediction error with an error rate of 3.30% by leveraging statistical dependencies in the historical injection–production time series. The VAR model is purely linear and data-driven; it has limited ability to generalize to wells with weak injector–producer correlation, as shown by the relatively high error rate of 6.30% for X51-5-1. The data-enhanced model achieves the lowest mean error rate of 2.00% and outperforms across all wells. It reduces the error rate for well X51-5-1 by 2.5% compared to the VAR model, which demonstrates enhanced robustness in low-correlation scenarios as production prediction.

As shown in

Table 2. Benchmarking of production prediction performance.

Model/Method	Mechanism	Daily Production Error (%) (Range)	Cumulative Production Error (%) (Range)
Commercial Simulator	Physics-based Numerical Simulation	8.12 (5.40–12.80)	17.9 (14.6–22.5)
VAR Model	Pure Data-Driven (Linear Flow)	3.30 (1.26–6.30)	14.8 (10.7–20.4)
Data-enhanced model	Physics-Constrained + Data-Driven (Variable Start-Up Pressure Gradient)	2.00 (0.95–3.80)	2.8 (2.1–3.6)

, the VAR model improves short-term rate prediction; its cumulative error rate remains at 14.8%, indicating error accumulation over time, which is a common issue in autoregressive models. The commercial simulator shows larger cumulative deviations. The data-enhanced model maintains low errors in both metrics with a daily error rate of 2.00% and a cumulative error rate of 2.8%, confirming its temporal consistency and physical fidelity. Overall, the commercial simulator is hampered by parametric uncertainty, and the VAR model is limited by linearity and correlation dependence. The data-enhanced model establishes a new standard of performance by combining the interpretability and generalization of physics-based modeling with the flexibility and data-adaptivity of machine learning.

4.1.4. Limitations of the Data-Enhanced Model

The data-enhanced model performs well but still has clear limitations. First, the model needs reliable field data for calibration. Validation using field production data shows that it uses measured values at well locations to adjust parameters. If such data are missing or inaccurate, the model’s predictions may become unreliable. Second, the model does not eliminate errors completely, as shown in benchmarking against commercial simulators and recent data-driven models, and the model does not fully reproduce real field behavior. Third, in some cases, the improvement over advanced physics-based models is small, suggesting that complex data-driven methods are not always necessary when the physics are well represented. Fourth, the model depends on correctly specified physical laws. In reservoirs where key flow mechanisms are unknown or not included in the model structure, its performance may suffer.

In summary, the data-enhanced model is effective but not perfect. Its accuracy depends on data quality, the completeness of physical assumptions, and the reservoir conditions. It works best when field data are available and the dominant flow physics are properly defined.

4.2. Parameter Consistency and Robustness Analysis

4.2.1. Physical Consistency of Data-Enhanced Parameters

The physical consistency of the data-enhanced model is assessed by analyzing the spatial pattern of the start-up pressure gradient resulting from the joint influence of the data-enhanced parameters M(r,t), a(r,t), b(r,t), and S(r,t), which are refined through the PINN framework. These parameters originate from initial estimates a_p, b_p, M_p, and S_p, derived from physics-based models, empirical correlations, or core tests. This analysis emphasizes parameter space comparison and physical response consistency.

Figure 13 demonstrates consistency achieved through data-enhanced parameter optimization. The PINN-constrained learning framework systematically refines the parameter fields to accurately capture the coupled pressure sensitivity-startup gradient dynamics. This correction yields a spatially heterogeneous gradient profile that aligns with physical phenomena and engineering characteristics, near-well stress-sensitive zones show 15–25% gradient elevation due to M(r,t) amplification, while high-permeability layers exhibit 18–22% flow resistance reduction from optimized a(r,t), b(r,t) creating low-gradient corridors with 0.028–0.032 MPa/m. Models considering the start-up pressure gradient as a constant gradient oversimplify the flow dynamics with the uniform value as 0.036 MPa/m, and models considering variable start-up pressure gradients produce monotonic decreases; the optimized solution captures non-monotonic variations. The characteristic profiles of high, near-well and low, off-well was validated against core data, showing 20–30% higher gradients near the wellbore, and seismic permeability maps. S(r,t) further amplify the distal contrasts between near-well and off-well, explaining the production-logging-identified flow characteristics.

This spatial heterogeneity confirms the physical consistency of the parameter corrections. The gradient distribution directly reflects reservoir architecture and flow constraints, explaining 92% of historical production variance while providing actionable insights for stimulation targeting and well spacing optimization. By capturing the coupling between pressure sensitivity and startup gradients, the data-enhanced parameters represent the multiphysics governing low-permeability reservoirs.

Figure 14 shows the spatial heterogeneity of the start-up pressure gradient in two dimensions. The map indicates asymmetric, high-gradient zones near the wellbore with gradual attenuation toward the reservoir, forming preferential flow corridors and local flow barriers. These patterns reflect reservoir-scale stress distribution and lithological contrasts that cannot be captured by one-dimensional profiles. The alignment of high-gradient clusters with known heterogeneity zones demonstrates the model’s ability to combine multiphysics effects with geological structure. This two-dimensional view validates the corrected parameter fields and confirms their capacity to reproduce realistic drainage geometry, strengthening confidence in the data-enhanced model as a physically consistent representation of reservoir behavior.

4.2.2. Sensitivity Analysis

Sensitivity and uncertainty analyses were performed to quantify the influence of M(r,t), a(r,t), b(r,t), and S(r,t) on production predictions after refinement through the data-enhanced framework. Each parameter was perturbed by ±10% from its data-enhanced optimized value while holding the remaining parameters constant.

As shown in Figure 15, a(r,t) exerts the strongest influence, particularly on the initiation of sustained flow and the early-to-mid production profile. M(r,t) follows closely, with perturbations producing substantial deviations throughout the production period, especially during the early stage when pressure sensitivity dominates. S(r,t) shows a relatively weaker but persistent effect, steadily modifying production rates due to its control on near-wellbore flow resistance. b(r,t) has a minor impact, as perturbations result in negligible deviations from the baseline.

The analysis indicates that a(r,t) requires the most careful calibration, followed by M(r,t) and S(r,t); b(r,t) proves to be the most robust. These findings confirm that the data-enhanced model maintains stable and physically consistent predictive capability under significant parameter perturbations.

4.2.3. Uncertainty Analysis

In reservoir engineering, parameter uncertainty arises from measurement errors and subsurface heterogeneity. While sensitivity analysis clarifies the role of individual parameters, a comprehensive assessment requires us to quantify the combined impact of parameter variations on production forecasts. To achieve this, a Monte Carlo simulation with 1000 realizations was performed, where all four parameters were perturbed simultaneously within ±12% uniform distributions. Results are compared against deterministic perturbations of ±10% for M(r,t), a(r,t), b(r,t), and −10% for S(r,t), allowing direct distinction between single-parameter effects and probabilistic multi-parameter uncertainty.

Figure 16 illustrates the outcomes. The gray band represents the P10–P90 envelope, covering 80% of simulated trajectories. This envelope captures realistic variability expected under normal field conditions. Superimposed individual perturbation curves highlight distinct parameter impacts. The a(r,t) parameter exhibits the strongest effect, with deviations extending beyond the P90 bound in the later production stage, underscoring its critical role in long-term performance prediction and the need for precise calibration. M(r,t) produces relatively modest deviations, largely contained within the envelope, suggesting greater robustness to uncertainty. b(r,t) parameter exerts noticeable influence during intermediate periods as 20–40 months, S(r,t) primarily governs early production rates, consistent with its near-wellbore control.

Certain deterministic perturbations exceed the probabilistic envelope. This outcome is statistically consistent, since the Monte Carlo envelope represents an 80% confidence interval, whereas deterministic ±10% perturbations assume extreme but unlikely parameter realizations. Taken together, the analysis highlights that a(r,t) should be prioritized in measurement and monitoring, while uncertainty in M(r,t), b(r,t), and S(r,t) can be tolerated without severely compromising predictive reliability. This provides practical guidance for uncertainty management in production prediction and decision-making.

4.3. Field Deployment and Real-Time Adaptation

To translate the data-enhanced model into practical use, it is essential to establish a standardized workflow for its deployment in field operations. The key requirement is to ensure that model parameters are consistently derived, updated, and assimilated from diverse data sources. These parameters originate from core analysis, well logging, production monitoring, and dynamic performance evaluation. Outputs from commercial reservoir simulators can be incorporated into the workflow, providing supplementary inputs where direct measurements are unavailable.

The standardized deployment workflow of the data-enhanced model in an unconventional reservoir is as shown in Figure 17. The process begins with multi-source data acquisition, incorporating core tests, field monitoring, production logs, and numerical simulation outputs. These heterogeneous datasets provide the foundation for parameter initialization, where baseline estimates of M(r,t), a(r,t), b(r,t), and S(r,t) are derived and integrated into the model. Online updating modules continuously refine the parameter fields through Bayesian or PINN-based assimilation loops, ensuring consistency with real-time operational data streams. This adaptive mechanism allows the model to capture evolving reservoir dynamics under production disturbances or geological variability. The updated predictions are incorporated into decision-making applications, including production prediction, well-spacing optimization, and drawdown management. By establishing a structured and repeatable workflow, the framework transitions the data-enhanced model from theoretical optimization to a practical tool for field operations. This standardization ensures reproducibility, transparency, and scalability in reservoir management practices.

This standardized workflow establishes a clear protocol for applying the data-enhanced model in field operations. By linking multi-source data, parameter initialization, real-time updating, and decision-driven applications, the framework ensures that reservoir predictions remain both physically consistent and operationally relevant. The process provides a practical guideline for unconventional reservoir management, enabling reproducible prediction, adaptive optimization, and more informed development strategies.

5. Conclusions

This study developed and validated a data-enhanced variable start-up pressure gradient model to improve production prediction in unconventional reservoirs. Building upon both physics-based mechanisms and data-driven optimization, the work provides a comprehensive framework that advances theoretical understanding and practical applicability. The key conclusions are summarized as follows:

(1)

From physics-based modeling, the study clarified the combined effects of non-Darcy flow, start-up pressure gradient, and pressure-sensitive permeability in ultra-low-permeability systems. By deriving the variable start-up pressure gradient, the research established a mechanistic representation of coupled flow behaviors. This formulation enabled the construction of mathematical and production models that preserve the fundamental physical mechanisms of unconventional reservoirs.

(2)

The data-enhanced model incorporating physics-informed neural networks was proposed to overcome the limitations of purely physics-based or purely data-driven approaches. By embedding prior physical models into the learning process, the framework achieved improved accuracy, robustness, and generalizability in production prediction.

(3)

Systematic evaluation confirmed the data-enhanced model’s strong consistency with field production data, reliable parameter corrections, and robust performance under sensitivity and uncertainty. A standardized workflow for field deployment was developed, demonstrating its potential for real-time adaptation and decision support in unconventional reservoir management.