Open AccessArticle

Physics-Informed Generative Adversarial Network Solution to Buckley–Leverett Equation

Xianlin Ma

^1,2,*,

Chengde Li

¹,

Jie Zhan

^1,2,* and

Yupeng Zhuang

College of Petroleum Engineering, Xi’an Shiyou University, Xi’an 710065, China

Engineering Research Center of Development and Management for Low to Extra-Low Permeability Oil & Gas Reservoirs in Western China, Ministry of Education, Xi’an Shiyou University, Xi’an 710065, China

Authors to whom correspondence should be addressed.

Mathematics 2024, 12(23), 3833; https://doi.org/10.3390/math12233833

Submission received: 7 November 2024 / Revised: 29 November 2024 / Accepted: 3 December 2024 / Published: 4 December 2024

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Figure 1
A reservoir model illustrating water frontal advance using the B-L equation. "> Figure 2
Water saturation profiles at different hours. "> Figure 3
Determination of water shock front saturation by Welge’s approach. "> Figure 4
Schematic representation of PIG-GAN for solving B-L equation. "> Figure 5
Water saturation profiles obtained using Welge’s graphic method. (a) Distribution of water saturation; (b) solutions of B-L equation at different moments. "> Figure 6
Data-driven GAN solutions with different numbers of training samples (500, 2000, 4000), (dashed red) and analytical solution (solid blue) at three different moments. "> Figure 7
Comparison between PIG-GAN (dashed red) and analytical solution (solid blue) at three different moments. "> Figure 8
Comparison between DI-GAN (dashed red) and analytical solution (solid blue) at three different moments. "> Figure 9
Comparison between the PIG-GAN (dashed red) results with the inferred parameters and analytical solutions (solid blue). "> Figure 10
Distribution of initial water saturation. "> Figure 11
Comparison between PIG-GAN (dashed red) results with inferred parameters and analytical solutions (solid blue) under noisy training data. ">

Versions Notes

Abstract

Efficient and economical hydrocarbon extraction relies on a clear understanding of fluid flow dynamics in subsurface reservoirs, where multiphase flow in porous media poses complex modeling challenges. Traditional numerical methods for solving the governing partial differential equations (PDEs) provide effective solutions but struggle with the high computational demands required for accurately capturing fine-scale flow dynamics. In response, this study introduces a physics-informed generative adversarial network (GAN) framework for addressing the Buckley–Leverett (B-L) equation with non-convex flux functions. The proposed framework consists of two novel configurations: a Physics-Informed Generator GAN (PIG-GAN) and Dual-Informed GAN (DI-GAN), both of which are rigorously tested in forward and inverse problem settings for the B-L equation. We assess model performance under noisy data conditions to evaluate robustness. Our results demonstrate that these GAN-based models effectively capture the B-L shock front, enhancing predictive accuracy while embedding fluid flow equations to ensure model interpretability. This approach offers a significant advancement in modeling complex subsurface environments, providing an efficient alternative to traditional methods in fluid dynamics applications.

Keywords:

PINNs; Buckley–Leverett equation; generative adversarial networks; waterflooding

MSC:

76S05

1. Introduction

The efficient extraction of hydrocarbons from subsurface reservoirs and the geological storage of CO₂ rely on a solid understanding of multiphase fluid dynamics in porous media [1,2,3,4]. These fluid movements are commonly governed by partial differential equations (PDEs) that describe fluid mechanics and phase interactions. Traditional approaches to solving these PDEs such as finite-difference, finite-volume, and finite-element methods achieve high accuracy but require substantial computational resources, especially for fine-scale simulations that capture the complex flow dynamics involved [5,6].

To enhance computational efficiency and enable real-time predictions, researchers are increasingly turning to deep learning (DL) as an alternative modeling tool. By integrating deep neural networks with traditional approaches, DL methods can both handle complex data relationships and offer predictive capabilities in real time [7,8,9]. For instance, Zhong et al. [10] employed a generative adversarial network (GAN) to predict reservoir pressure and fluid saturation during waterflooding, realizing significant computational savings. Other examples include Vikara et al.’s use of LSTM networks for predicting multiphase production in unconventional wells [11] and Fan et al.’s temporal convolutional network with gated recurrent units (TCN-GRUs), which improves oilfield production forecasting accuracy [12]. DL-based approaches excel at handling non-linear data and dynamic reservoir conditions, making them suitable for real-time applications. Alakeely and Horne [13] further applied variational autoencoders (VAEs) to estimate oil and water production across diverse reservoir conditions, while Wen et al. [14] developed a CO₂ plume migration model that provides a scalable and faster alternative to evaluate geological CO₂ storage.

Despite their advantages, DL models often lack interpretability compared to traditional numerical methods and require large datasets for accurate predictions. To address these issues, Physics-Informed Neural Networks (PINNs) incorporate the PDEs within the loss function of neural networks, using automatic differentiation to enable the model to learn directly from both data and physics-based constraints [15,16]. Raissi et al. [17] initiated this approach, leading to the development of various PINN adaptations, such as physics-guided and physics-constrained neural networks, as well as physics-informed attention mechanisms [18,19,20].

PINNs have successfully solved PDEs including flow, seismic wave, heat transfer, and diffusion equations [21,22,23,24]. For multiphase flow, Almajid et al. [25] applied PINNs to solve the Buckley–Leverett (B-L) equation for two-phase flow, while Fuks and Tchelepi [26] evaluated PINNs in three fluid displacement scenarios: concave, convex, and non-convex flux. They found that the PINNs underperformed on non-convex cases, though the addition of a second-order diffusive term significantly improved the model’s accuracy [27,28]. However, solving hyperbolic PDEs like the B-L equation remains challenging due to the formation of shock fronts with discontinuous gradients.

To address these challenges, this study proposes a novel GAN-based approach for solving the B-L equation. By incorporating the B-L equation constraints within the framework, two GAN configurations, a Physics-Informed Generator GAN (PIG-GAN) and Dual-Informed GAN (DI-GAN), are assessed for both forward and inverse problems, particularly in noisy data scenarios. Through two case studies, these models demonstrate the potential of GAN-based physics-informed networks for accurately predicting fluid flow dynamics in complex subsurface reservoirs.

2. Mathematical Formulation

2.1. Buckley–Leverett Equation

The B-L equation [29] is a hyperbolic partial differential equation (PDE) essential for understanding two-phase flow in porous media, particularly when water displaces oil in a 1D reservoir. This model, named after Ralph Buckley and Edward Leverett, characterizes the spatial and temporal evolution of water saturation as water is injected into the oil reservoir, as depicted in Figure 1.

The following Equation (1) is derived from mass conservation principles and Darcy’s law using fractional flow theory and is based on key assumptions: (1) linear and horizontal flow, (2) injection of water into an oil reservoir, (3) incompressibility of both oil and water, (4) immiscibility between oil and water, and (5) negligible effects of gravity and capillary pressure [30].

\frac{\partial S_{w}}{\partial t} + \frac{q_{T}}{ϕ A} \frac{\partial f_{w} (S_{w})}{\partial x} = 0

(1)

S_{w}

is the water saturation, t is time, x is the spatial coordinate,

q_{T}

is the total flow rate,

ϕ

is the porosity of a porous medium, A is the cross-sectional area, and L is the formation length.

The associated initial and boundary conditions can be written as follows:

\{\begin{matrix} S_{w} (x, 0) = S_{w c} \\ S_{w} (0, t) = S_{i n j} = 1 - S_{o r} \end{matrix}

(2)

S_{w c}

is the irreducible or connate water saturation,

S_{i n j}

is the injection water saturation, and

S_{o r}

is the residual oil saturation.

In Equation (1), the fractional flow function of water

f_{w} (S_{w})

is defined as the ratio of water mobility (

λ_{w}

) to that of total mobility (

λ_{T}

) of water and oil, and is written as follows:

f_{w} = \frac{λ_{w}}{λ_{T}} = \frac{1}{1 + \frac{k_{r o} μ_{w}}{k_{r w} μ_{o}}}

(3)

k_{r i}

and

μ_{i}

are the relative permeability and viscosity of phase

i

, respectively. We use Corey-type relative permeability functions [31]. Specifically, the relative permeability functions of water and oil phases are written as follows:

k_{r w} = k_{r w, m a x} {(\frac{S_{w} - S_{w c}}{1 - S_{w c} - S_{o r}})}^{n_{w}}

(4)

k_{r o} = k_{r o, m a x} {(\frac{1 - S_{w} - S_{o r}}{1 - S_{w c} - S_{o r}})}^{n_{o}}

(5)

k_{r i, m a x}

is the maximum relative permeability value of phase

i

, and

n_{i}

is the relative permeability exponent of phase

i

2.2. Analytical Solution of B-L Equation

A defining feature of the B-L solution is the formation of a shock front. At this front, water saturation abruptly transitions from irreducible water saturation to a sharp-front saturation (illustrated in Figure 2), which progresses through the reservoir at a constant rate, pushing oil ahead. The water saturation profile behind this shock front changes smoothly from the injection saturation level to the sharp-front saturation level.

To analytically determine the position of this shock front, Welge’s graphical method [32] and the Method of Characteristics [33] are commonly used, and the position of the shock front can be expressed as follows:

x_{f} = \frac{q t}{A ϕ} {(\frac{d f_{w}}{d S_{w}})}_{f}

(6)

Welge’s method involves a fractional flow curve representing the fraction of water flow as a function of water saturation within the reservoir, as shown in Figure 3. The key step in Welge’s approach is drawing a tangent line to this curve from the irreducible water saturation point on the horizontal axis, which enables the identification of the shock position. The fractional flow function typical in this context is a non-convex, S-shaped curve, as shown in Figure 3. This type of function is widely encountered in fractional flow analyses and results in a characteristic discontinuity that is central to understanding the behavior of two-phase flows in porous media.

3. Physics-Informed Generative Adversarial Network

Generative Adversarial Networks (GANs), a type of deep learning model, involve two neural networks, a generator and a discriminator, that train adversarially. The generator creates synthetic data samples that mimic real data, while the discriminator evaluates and differentiates these samples from actual data [34]. To incorporate physical constraints, a physics-informed GAN (PIG-GAN) embeds physical laws and constraints of initial and boundary conditions into its loss function, enabling the generation of data aligned with known physical laws. In our proposed approach, we use a GAN to solve the B-L equation through two configurations. The first configuration designs the generator as a Physics-Informed Neural Network (PINN), ensuring generated outputs adhere to the B-L theory. Here, the discriminator discerns between these generated solutions and actual water saturation observations.

3.1. Physics-Informed Generator (PIG)-GAN

In the PIG-GAN model, shown in Figure 4, the generator integrates boundary and initial conditions alongside collocation points to produce water saturation values constrained by the B-L equation. This structure not only enforces alignment with the B-L theory but also enables the discriminator to assess these outputs against actual data. The generator’s loss function includes a physical constraint term derived from the B-L equation, which helps ensure the generated data remain physically plausible, promoting greater generalization. Specifically, these constraints consider collocation points and boundary/initial condition points, represented, respectively, as red dots and blue crosses in Figure 4, allowing the model to adhere closely to the B-L dynamics.

The generator’s loss function is defined as follows:

L_{g e n e r a t o r} = L_{G} + λ_{P D E} L_{P D E}

(7)

λ_{P D E}

is the weight of physics loss functions; the loss functions

L_{G}

and

L_{P D E}

are given as follows.

The loss function

L_{G}

of the generator is defined as follows:

L_{G} = - \frac{1}{M} \sum_{i = 1}^{M} [l o g (D ({\hat{S}}_{w} (x_{i}, t_{i})))]

(8)

M is the number of data fed into the PIG-GAN in each training batch;

D ({\hat{S}}_{w} (x_{i}, t_{i}))

is the probability that the discriminator determines the generated water saturation

{\hat{S}}_{w} (x_{i}, t_{i})

by the generator as real water saturation

S_{w} (x_{i}, t_{i})

The physical constraints

L_{P D E}

are represented as follows:

L_{P D E} = L_{B C} + L_{I C} + L_{B L}

(9)

The loss

L_{B C}

corresponds to the boundary data:

L_{B C} = \frac{1}{N_{B C}} \sum_{i = 1}^{N_{B C}} {[{\hat{S}}_{w} (0, t_{i}) - S_{i n j}]}^{2}

(10)

The loss

L_{I C}

corresponds to the initial data:

L_{I C} = \frac{1}{N_{I C}} \sum_{i = 1}^{N_{I C}} {[{\hat{S}}_{w} (x_{i}, 0) - S_{w c}]}^{2}

(11)

The loss

L_{B L}

on the B-L equation residual is the following:

L_{B L} = \frac{1}{N_{B L}} \sum_{i = 1}^{N_{B L}} {[R (x_{i}, t_{i})]}^{2}

(12)

The residual of the B-L equation

R (x_{i}, t_{i})

is defined as follows:

R = \frac{\partial {\hat{S}}_{w}}{\partial t} + \frac{q_{T}}{ϕ A} \frac{\partial f_{w} ({\hat{S}}_{w})}{\partial x}

(13)

N_{B L}

N_{I C}

, and

N_{B C}

are the number of collocation points and number of boundary/initial condition points, respectively.

To enhance accuracy, the PIG-GAN applies automatic differentiation for the spatial–temporal partial derivatives of water saturation, avoiding the approximation errors common in finite-difference schemes. Unlike traditional methods, which require a strict grid layout, the PIG-GAN operates on neural network outputs and can use mesh-free approaches, such as Latin hypercube sampling for collocation point selection. This enables efficient and robust handling of PDEs even with noisy input data, enhancing model performance on complex fluid dynamics without the grid-block limitations of conventional numerical solvers.

The loss function of discriminator is expressed as follows:

L_{d i s c r i m i n a t o r} = - \frac{1}{2 M} \sum_{i = 1}^{M} [l o g (D (S_{w} (x_{i}, t_{i})) + l o g (1 - D ({\hat{S}}_{w} (x_{i}, t_{i})))]

(14)

D (S_{w} (x_{i}, t_{i})

is the probability that the discriminator determines the real data as real data;

1 - D ({\hat{S}}_{w} (x_{i}, t_{i}))

is the probability that the predicted water saturation from the generator by the discriminator data is judged as fake data.

The PIG-GAN is trained by solving the min–max problem inherent to the GAN framework:

\min_{G} \max_{D} L (D, G) = L_{d i s c r i m i n a t o r} + L_{g e n e r a t o r}

(15)

Here, the discriminator (D) maximizes the objective by distinguishing accurate solutions from incorrect ones, while the generator (G) minimizes the objective by generating solutions that satisfy physical constraints.

3.2. Dual-Informed Generative Adversarial Network (DI-GAN)

The PIG-GAN model aims to generate water saturation predictions that align with the B-L equation, ensuring that the generated data adheres to fundamental flow mechanisms. However, PIG-GAN’s effectiveness is limited by its inability to fully leverage the adversarial optimization process typical of the GAN framework for minimizing complex physics-based loss functions. This shortfall arises because the PIG-GAN’s discriminator lacks physics-based supervision, restricting it from evaluating unlabeled instances based on physical accuracy and diminishing the model’s potential for realistic, physics-informed outputs.

To address this gap, the Dual-Informed GAN (DI-GAN), inspired by the work of Daw et al. [35], introduces physics-based supervision into both the generator and discriminator. This approach enables the DI-GAN to generate samples that not only appear realistic but also conform to underlying physical laws. By incorporating B-L equation constraints into the discriminator, DI-GAN can assess the physical plausibility of the generator’s outputs. The discriminator now evaluates the residuals of the B-L equation in both generated and actual data, improving the overall robustness and accuracy of predictions. Thus, DI-GAN surpasses PIG-GAN by ensuring that both generator and discriminator work in tandem to adhere to physical constraints, making it a more comprehensive and physics-compliant model for complex fluid dynamics.

DI-GAN retains the binary cross-entropy loss function framework, with the generator loss function defined by Equation (7), and a modified discriminator loss function defined in Equation (16).

\begin{matrix} L_{d i s c r i m i n a t o r} = & - \frac{1}{2 N} \sum_{i = 1}^{N} [l o g (D (S_{w} (x_{i}, t_{i})) + l o g (1 - D ({\hat{S}}_{w} (x_{i}, t_{i})))] \\ - \frac{1}{2 N_{B L}} \sum_{i = 1}^{N_{B L}} [l o g (D (R (x_{i}, t_{i}))) + l o g (1 - D (R (x_{i}, t_{i})))] \end{matrix}

(16)

D (R (x_{i}, t_{i}))

represents the probability that the B-L equation residuals are identified as accurate, while

1 - D (R (x_{i}, t_{i}))

denotes the probability that the discriminator classifies the generated data residuals as inaccurate.

3.3. Evaluation Metrics

To assess the predictive accuracy of the models, three standard evaluation metrics are used: Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and Mean Squared Error (MSE). These metrics are defined as follows:

M A E = \frac{1}{N} \sum_{i = 1}^{N} |{\hat{y}}_{i} - y_{i}|

(17)

M A P E = \frac{1}{N} \sum_{i = 1}^{N} |\frac{{\hat{y}}_{i} - y_{i}}{y_{i}}| \times 100 %

(18)

M S E = \frac{1}{N} \sum_{i = 1}^{N} {(y_{i} - {\hat{y}}_{i})}^{2}

(19)

N

represents the total number of data samples,

y_{i}

denotes the true value at the i-th position, and

{\hat{y}}_{i}

represents the predicted value at the i-th position.

4. Results and Discussion

4.1. Problem Setup

To facilitate solving the B-L equation, Equation (1) can be transformed into a dimensionless form as follows:

\frac{\partial S_{w}}{\partial t_{D}} + \frac{\partial f_{w}}{\partial x_{D}} = 0

(20)

In Equation (20), the dimensionless time,

t_{D}

, and

x_{D}

are introduced and defined as follows:

t_{D} = \frac{q_{T}}{ϕ A L} t x_{D} = \frac{x}{L}

(21)

Using the dimensionless time and distance, Equation (13) becomes the following:

R = \frac{\partial {\hat{S}}_{w}}{\partial t_{D}} + \frac{\partial f_{w} ({\hat{S}}_{w})}{\partial x_{D}}

(22)

Following the work by Fuks and Tchelepi [27,28], we incorporate a diffusive term into Equation (22), yielding the following revised equation:

R = \frac{\partial {\hat{S}}_{w}}{\partial t_{D}} + \frac{\partial f_{w} ({\hat{S}}_{w})}{\partial x_{D}} - γ \frac{\partial^{2} {\hat{S}}_{w}}{\partial x_{D}^{2}}

(23)

Here,

γ

is a regularization parameter chosen to be 0.001. The addition of this diffusive term helps improve the model performance.

For this study, a 1D homogeneous reservoir model extending 100 m in the x-direction was used, and the specific formation and fluid parameters can be found in Table 1.

Welge’s graphical method, implemented in MATLAB 2020a by Wu [36], was employed to analytically solve the B-L equation with the parameters listed in Table 1. Figure 5 illustrates the 1D water saturation profiles obtained.

4.2. Network Architecture

To investigate the effectiveness of the PIG-GAN and DI-GAN models, we used the consistent network structure parameters detailed in Table 2.

The generator in each model comprises 10 hidden layers, while the discriminator includes 7 hidden layers, with each hidden layer containing 20 neurons. Both networks apply the tanh activation function and use the Adam optimizer.

4.3. Forward Problem

Our objective in the forward problem is to model the spatial and temporal evolution of water saturation accurately. Initially, a purely data-driven GAN model was applied, generating water saturation profiles. The PIG-GAN and DI-GAN frameworks were subsequently used, adding B-L equation constraints, enabling a comparative analysis of each model’s predictive accuracy.

4.3.1. PIG-GAN Model

Using the hyperparameter values from Table 2, we trained the data-driven GAN model. Three sets of samples, numbering 500, 2000, and 4000, were randomly taken from the water saturation distribution shown in Figure 5. The resulting GAN-generated profiles are presented in Figure 6 at the dimensionless moments

t_{D}

= 0.4, 0.55, 0.70, revealing that the prediction accuracy improves with more training data, as demonstrated by closer alignment with Welge’s analytical solution. The data-driven model failed to capture the position of shock front even with 4000 training samples, although it could predict the saturation distribution behind the shock front.

Further, using the PIG-GAN, we trained the model with 100 boundary and 50 initial data points, and 10,000 collocation points sampled from the domain interior

\{(x_{D}, t_{D}) | 0 \leq x_{D}, t_{D} \leq 1\}

. The resulting profiles in Figure 7 show strong alignment with the analytical solutions at the shock front with only 150 training samples. The PIG-GAN accurately reproduces water saturation values in regions preceding the shock front, where saturation consistently remains at zero, as well as capturing the shock front’s position effectively.

4.3.2. DI-GAN Model

In training DI-GAN, we utilized the architecture described in Section 3.2, mirroring the same network layer structure and sampling approach as the PIG-GAN’s. After 15,000 training epochs, Figure 8 compares the DI-GAN’s predicted solutions against the analytical solutions. The DI-GAN model consistently achieves lower error rates, with metrics such as MSE, MAPE, and MAE outlined in Table 3, largely due to the dual physics-based supervision of both generator and discriminator. This dual supervision allows the DI-GAN to effectively capture non-linear solution features, resulting in more accurate performance compared to the data-driven and PIG-GAN approaches.

4.4. Inverse Problem

The objective of the inverse problem is to infer the parameters of the B-L equation, such as the irreducible water saturation (

S_{w c}

) and residual oil saturation (

S_{o r}

). This is particularly valuable in reservoir engineering, where uncertainties are often high due to complex geology and fluid dynamics. By estimating these parameters with reduced uncertainty, better-informed decisions can be made for oil-field development.

In multi-parameter inversion, relying solely on equation constraints may lead to unrealistic outcomes. Additional regularization helps ensure the inversion results remain reasonable. Consequently, the generator’s loss function includes inversion parameter errors, as shown in Equation (24):

L_{g e n e r a t o r} = L_{G} + λ_{P D E} L_{P D E} + λ_{i n v} [{({\tilde{S}}_{w c} - S_{w c})}^{2} + {({\tilde{S}}_{o r} - S_{o r})}^{2}]

(24)

{\tilde{S}}_{w c}

and

{\tilde{S}}_{o r}

represent the inverted parameters, and

[{({\tilde{S}}_{w c} - S_{w c})}^{2} + {({\tilde{S}}_{o r} - S_{o r})}^{2}]

is the error between the estimated and their real values.

λ_{i n v}

is the weight of the corresponding loss terms.

Furthermore, adjusting the weight

λ_{i n v}

significantly impacts the outcome; a larger

λ_{i n v}

increases sensitivity to loss, accelerating convergence but potentially leading to suboptimal local solutions, whereas a smaller

λ_{i n v}

slows convergence and raises computational costs. After extensive testing, the optimal

λ_{i n v}

value was found to be two in this study.

As shown in Equation (24), the PINNs offer an advantage in inverse problems by embedding physical laws directly into the training process, which reduces computational costs by optimizing for unknown parameters in a single model, unlike traditional methods that require repeated forward simulations.

4.4.1. Case 1: Noise-Free Training Data

For this study, the initial parameter values for irreducible water saturation (

{\tilde{S}}_{w c}

) and residual oil saturation (

{\tilde{S}}_{o r}

) were randomized within the range [0, 1], consistent with the physical constraints for fluid saturation. After 15,000 training epochs, Table 4 shows that the model’s inferred values for these parameters are closely aligned with the actual values used to generate the analytical solutions.

Figure 9 illustrates a comparison between the PIG-GAN’s predictions with the inferred parameters and the analytical solutions. The PIG-GAN effectively captures the shock front.

4.4.2. Case 2: Noisy Training Data

To further evaluate robustness, random Gaussian noise

N (μ = 0, σ^{2} = 0.03)

was added to the initial water saturation, keeping other parameters constant. Figure 10 shows the absolute initial saturation distribution, and Table 5 compares the inferred values of

{\tilde{S}}_{w c}

and

{\tilde{S}}_{o r}

and their errors. Despite the noise, the PIG-GAN produced robust estimates. The model’s training on the underlying flow mechanics of the B-L equation improves its resilience to noisy inputs, making it a promising tool for practical reservoir modeling.

In Figure 11, the PIG-GAN model predictions are compared with the analytical solutions. While the PIG-GAN model generally aligns well with the analytical solution, some discrepancies appear at the shock front, where predictions exhibit a slight transition zone. This zone indicates a degree of smoothing, which deviates from the strict discontinuity expected in the analytical profiles.

5. Conclusions

Traditional numerical methods for solving the B-L equation often introduce truncation errors that cause a diffused shock front, hindering precise modeling of abrupt changes in the water saturation. Physics-informed GANs provide a promising alternative by applying deep learning techniques to solve both forward and inverse B-L problems. This study introduces two specialized GAN models: a Physics-Informed Generator GAN (PIG-GAN) and a Dual-Informed GAN (DI-GAN). Both models have shown significant accuracy in predicting the temporal evolution of water saturation profiles, using initial and boundary condition data to guide the learning process. This approach allows for capturing the shock front more sharply compared to traditional methods, as both models reduce error accumulation and integrate physical constraints by using automatic differentiation.

In the forward problem scenario, both PIG-GAN and DI-GAN exhibit a high degree of accuracy in the water saturation predictions. Notably, the DI-GAN outperforms the PIG-GAN in capturing the shock front, a critical aspect of modeling two-phase flow in porous media. In the inverse problems, where certain B-L parameters remain unmeasured, the PIG-GAN model displays robustness and accuracy even under initial condition corruption. The results underscore the model’s capability to handle parameter estimation challenges while maintaining resilience when faced with data imperfections. The study has demonstrated that the physics-informed GANs can provide a viable alternative to traditional numerical methods in accurately modeling subsurface fluid dynamics.

Future research will focus on generalizing the proposed framework to handle three-phase flow problems, which could significantly enhance its applicability. This requires incorporating additional governing equations and interfacial dynamics into the physics-informed constraints of the GAN architecture. Improving the framework for inverse problems like history matching and uncertainty quantification in complex reservoirs remains a promising direction. These advancements could position physics-informed GANs as efficient and accurate alternatives to traditional numerical methods in multiphase fluid dynamics.

Author Contributions

Conceptualization, X.M. and J.Z.; methodology, X.M.; validation, C.L. and Y.Z.; formal analysis, X.M.; investigation, X.M. and J.Z.; data curation, C.L. and Y.Z.; writing—original draft preparation, X.M.; writing—review and editing, X.M. and J.Z.; visualization, C.L. and Y.Z.; funding acquisition, X.M. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Natural Science Foundation of China (51934005).

Data Availability Statement

Data are unavailable due to commercial restraints.

Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Figure 1. A reservoir model illustrating water frontal advance using the B-L equation.

Figure 2. Water saturation profiles at different hours.

Figure 3. Determination of water shock front saturation by Welge’s approach.

Figure 4. Schematic representation of PIG-GAN for solving B-L equation.

Figure 5. Water saturation profiles obtained using Welge’s graphic method. (a) Distribution of water saturation; (b) solutions of B-L equation at different moments.

Figure 6. Data-driven GAN solutions with different numbers of training samples (500, 2000, 4000), (dashed red) and analytical solution (solid blue) at three different moments.

Figure 7. Comparison between PIG-GAN (dashed red) and analytical solution (solid blue) at three different moments.

Figure 8. Comparison between DI-GAN (dashed red) and analytical solution (solid blue) at three different moments.

Figure 9. Comparison between the PIG-GAN (dashed red) results with the inferred parameters and analytical solutions (solid blue).

Figure 10. Distribution of initial water saturation.

Figure 11. Comparison between PIG-GAN (dashed red) results with inferred parameters and analytical solutions (solid blue) under noisy training data.

Table 1. Parameters used for fractional flow.

Parameter	Value	Unit
Cross-section area, $A$	1	m²
Length of formation, $L$	100	m
Injection rate, $q_{t}$	10⁻⁴	m³/s
Injection time, $t$	24	hours
Water viscosity, $μ_{w}$	1.0	mPa·s
Oil viscosity, $μ_{o}$	5.0	mPa·s
Residual oil saturation, $S_{o r}$	0.0	-
Connate water saturation, $S_{w c}$	0.0	-
Maximum relative permeability of water, $k_{r w, m a x}$	0.80	-
Maximum relative permeability of oil, $k_{r o, m a x}$	0.80	-
Relative permeability exponent of water, $n_{w}$	2.0	-
Relative permeability exponent of oil, $n_{o}$	2.0	-

Table 2. Hyper-parameters of GAN.

Networks	Parameter	Value
Generator	Input dimension	3
	Output dimension	1
	Number of hidden layers	10
	Number of neurons each layer	20
	Activation function	tanh
	Optimizer	Adam
Discriminator	Input dimension	3
	Output dimension	1
	Number of hidden layers	7
	Number of neurons each layer	20
	Activation function	tanh
	Optimizer	Adam

Table 3. Comparison among three models.

MODEL	MSE	MAPE	MAE
Pure data-driven	0.00676	9.161	0.03449
PIG-GAN	0.00397	4.957	0.01375
DI-GAN	0.00220	2.463	0.01022

Table 4. Inferred values of the parameters

{\tilde{S}}_{w c}

and

{\tilde{S}}_{o r}

and their errors.

Table 4. Inferred values of the parameters

{\tilde{S}}_{w c}

and

{\tilde{S}}_{o r}

and their errors.

Parameter	Actual Value	Inverted Value	MSE
${\tilde{S}}_{w c}$	0.000	0.0047	0.0000221
${\tilde{S}}_{o r}$	0.000	0.0197	0.0003204

Table 5. Inferred values of the parameters

{\tilde{S}}_{w c}

and

{\tilde{S}}_{o r}

and their errors.

Table 5. Inferred values of the parameters

{\tilde{S}}_{w c}

and

{\tilde{S}}_{o r}

and their errors.

Parameter	Real	Prediction	MSE
${\tilde{S}}_{w c}$	0.0000	0.0052	0.0000270
${\tilde{S}}_{o r}$	0.0000	0.0029	0.0000084

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Ma, X.; Li, C.; Zhan, J.; Zhuang, Y. Physics-Informed Generative Adversarial Network Solution to Buckley–Leverett Equation. Mathematics 2024, 12, 3833. https://doi.org/10.3390/math12233833

AMA Style

Ma X, Li C, Zhan J, Zhuang Y. Physics-Informed Generative Adversarial Network Solution to Buckley–Leverett Equation. Mathematics. 2024; 12(23):3833. https://doi.org/10.3390/math12233833

Chicago/Turabian Style

Ma, Xianlin, Chengde Li, Jie Zhan, and Yupeng Zhuang. 2024. "Physics-Informed Generative Adversarial Network Solution to Buckley–Leverett Equation" Mathematics 12, no. 23: 3833. https://doi.org/10.3390/math12233833

APA Style

Ma, X., Li, C., Zhan, J., & Zhuang, Y. (2024). Physics-Informed Generative Adversarial Network Solution to Buckley–Leverett Equation. Mathematics, 12(23), 3833. https://doi.org/10.3390/math12233833

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Physics-Informed Generative Adversarial Network Solution to Buckley–Leverett Equation

Abstract

1. Introduction

2. Mathematical Formulation

2.1. Buckley–Leverett Equation

2.2. Analytical Solution of B-L Equation

3. Physics-Informed Generative Adversarial Network

3.1. Physics-Informed Generator (PIG)-GAN

3.2. Dual-Informed Generative Adversarial Network (DI-GAN)

3.3. Evaluation Metrics

4. Results and Discussion

4.1. Problem Setup

4.2. Network Architecture

4.3. Forward Problem

4.3.1. PIG-GAN Model

4.3.2. DI-GAN Model

4.4. Inverse Problem

4.4.1. Case 1: Noise-Free Training Data

4.4.2. Case 2: Noisy Training Data

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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