Carbonate reservoir water injection development prediction method and system based on dual-domain neural operator

Abstract

The application discloses a carbonate reservoir water injection development prediction method and system based on a dual-domain neural operator, relates to the technical field of carbonate reservoir water injection development prediction, and comprises the following steps: obtaining an original input tensor from a reservoir reservoir space model; extracting low-frequency global features and high-frequency local features of the original input tensor from a high-dimensional feature space through a dual-domain feature extraction framework; performing adaptive fusion weight calculation based on an energy range, fusing the low-frequency global features and the high-frequency local features according to the calculated fusion weight, and obtaining fused features; processing the fused features through a feature decoding layer, outputting the processing result of the fused features through an output layer, and obtaining a numerical simulation result of the carbonate reservoir; obtaining a development prediction yield of the carbonate reservoir according to the numerical simulation result; and fundamentally solving the technical problem that the precision and efficiency of the prior art are difficult to balance under extreme heterogeneous conditions.

Description

Technical Field

[0001] This invention relates to the field of prediction technology for water injection development in carbonate oil reservoirs, and more specifically, to a method and system for prediction of water injection development in carbonate oil reservoirs based on dual-domain neural operators. Background Technology

[0002] Carbonate reservoirs occupy a central position in global oil and gas exploration and development, contributing over 60% of global reserves. However, this enormous resource potential is accompanied by relatively high development challenges. The fundamental reason lies in the extremely complex pore structure and strong heterogeneity of carbonate rocks, leading to variations in reservoir permeability across different regions. This inherent complexity makes it difficult to elucidate the flow patterns of fluids within the reservoir, making accurate modeling and efficient prediction of production dynamics in these reservoirs a significant technical challenge. To improve the accuracy and reliability of production dynamic prediction in carbonate reservoirs, reservoir numerical simulation technology has been widely applied in development scheme design. By establishing geological and reservoir models, solving partial differential governing equations based on finite difference or finite element methods, and utilizing computer simulation, the production behavior and recovery rate of the reservoir under different development schemes are predicted. Currently, the methods employed include:

[0003] 1. Traditional numerical simulation methods, such as commercial software like CMG-IMEX, Eclipse, and tNavigator, solve multiphase flow equations through mesh discretization;

[0004] 2. The agent models based on deep learning that have emerged in recent years:

[0005] 1) Fully connected neural network (FNN) agent, which takes well control parameters and rock properties as input to directly predict production dynamics; 2) Convolutional neural network (CNN) agent, which can handle spatially distributed permeability fields; 3) Recurrent neural network (RNN) agent, which is good at handling time series.

[0006] 3. Neural operator methods:

[0007] 1) Fourier Neural Operator (FNO), which achieves global feature extraction through frequency domain convolution and has high computational efficiency; 2) DeepONet, which learns operator mapping based on a branch-backbone architecture; 3) Physical Information Neural Network (PINN).

[0008] 4. The latest neural network architecture:

[0009] 1) Kolmogorov-Arnold Network (KAN), a novel architecture proposed in 2024, uses a learnable B-spline activation function and has excellent local approximation ability; 2) Hybrid architecture attempts, simple FNO+MLP or FNO+KAN in series / parallel.

[0010] The main shortcomings of the above methods are:

[0011] 1. The contradiction between computational efficiency and accuracy: Traditional numerical methods are highly accurate but slow (on the order of hours), while machine learning agents are fast but lack accuracy (error > 5%), and cannot simultaneously meet the requirements of real-time performance (< 100 ms) and high accuracy (< 3%).

[0012] 2. Poor handling capability for extreme heterogeneity: The permeability of carbonate reservoirs can vary in the range of 0.01-1000 mD. When processing such a large contrast, existing FNO methods suffer from spectral truncation, resulting in the loss of local details; while local methods (such as KAN) are difficult to capture the global propagation characteristics of the pressure field.

[0013] 3. Lack of physical constraints: Purely data-driven methods may produce predictions that violate physical laws, such as mass non-conservation and non-physical oscillations of pressure fields; even PINN-type methods are difficult to apply physical constraints within millisecond-level inference time.

[0014] 4. Limited generalization ability: When the number of wells, well locations, or permeability distribution deviates from the training range, the prediction accuracy drops sharply; existing methods usually need to be retrained or fine-tuned for each new scenario.

[0015] 5. Insufficient interpretability: Black box models are difficult to explain the physical rationality of prediction results, making them difficult for engineers to trust and adopt; they also lack quantitative analysis of the contributions of features at different scales.

[0016] 6. Deployment difficulties: Complex models require high-performance GPU support, making it difficult to run them in real time on edge computing devices in oil fields. Model compression often comes at the cost of sacrificing accuracy.

[0017] Therefore, this application is hereby submitted. Summary of the Invention

[0018] The purpose of this invention is to provide a method and system for predicting water injection development in carbonate reservoirs based on dual-domain neural operators. This method can achieve millisecond-level high-precision prediction while ensuring physical rationality, so as to support real-time water injection optimization and digital twin applications in carbonate reservoirs. It also fundamentally solves the technical problem that existing technologies cannot balance accuracy and efficiency under extreme heterogeneous conditions.

[0019] The above-mentioned technical objective of the present invention is achieved through the following technical solution:

[0020] Firstly, this application provides a method for predicting water injection development in carbonate oil reservoirs based on dual-domain neural operators, including the following specific steps:

[0021] S1. Establish a reservoir space model corresponding to the carbonate reservoir and obtain the original input tensor from the reservoir space model.

[0022] S2, the original input tensor is upgraded to a high-dimensional feature space through a learnable linear transformation, and the low-frequency global features and high-frequency local features of the original input tensor are extracted from the high-dimensional feature space through a dual-domain feature extraction framework;

[0023] S3, based on the energy range, adaptive fusion weight calculation is performed, and the low-frequency global features and high-frequency local features are fused according to the calculated fusion weight to obtain the fused features;

[0024] S4 processes the fused features through the feature decoding layer and outputs the processing results of the fused features through the output layer to obtain the numerical simulation results of the carbonate reservoir.

[0025] S5. The predicted production of carbonate reservoirs is obtained based on the numerical simulation results, which include the pressure field and saturation field of carbonate reservoirs.

[0026] Based on the above technical solution, the present invention can be further improved as follows.

[0027] Furthermore, the aforementioned dual-domain feature extraction framework includes a global spectral branch and a local sparse branch, wherein:

[0028] The global spectral branch uses an enhanced Fourier neural operator to capture low-frequency global features;

[0029] High-frequency local features are captured by Kolmogorov-Arnold based on B-splines in the local sparse branch method.

[0030] Furthermore, the aforementioned low-frequency global features are specifically as follows:

[0031] ;

[0032] In the formula, The first in the global spectrum branch The output features of the layer, superscript This serves as the identifier for the global spectrum branch; It is the ReLU activation function. The linear transformation matrix representing the residual connection. Indicates the first in the global spectrum branch The output characteristics, and These represent the 3D Fast Fourier Transform and Inverse Fourier Transform, respectively. Represented as learnable frequency domain weights, The parallel 3DU-Net module contains 3 downsampling layers and 3 upsampling layers to enhance local feature extraction;

[0033] High-frequency local features, specifically:

[0034] ;

[0035] In the formula, Indicates the first sparse branch in a localized branch The first layer Output characteristics of each neuron Indicates the connection of the first in a locally sparse branch. Layer The first neuron to the second The first layer B-spline functions of 1 neuron Indicates the first sparse branch in a local sparse branch The first layer Output characteristics of each neuron Indicates the first Number of neurons in a layer, superscript X This serves as an identifier for locally sparse branches.

[0036] Furthermore, the aforementioned fusion features are specifically as follows:

[0037] ,in:

[0038] ;

[0039] In the formula, Indicates fusion characteristics, This represents the output of the global spectrum branch. This represents the output of a locally sparse branch. This represents the fusion weights obtained through calculation. This represents the sigmoid activation function. This represents a learnable weight vector. This represents the learnable bias parameters. It represents the energy norm, used to measure the combined energy of the pressure field and the saturation field.

[0040] Furthermore, steps S2-S4 above are executed using the constructed network model, and the total loss function of the network model is specifically as follows:

[0041] ;

[0042] In the formula, Represented as the total loss function, This represents the data fitting loss. This represents the weak-form energy canonical loss. This represents the strong-form residual regularization loss. , , This indicates the weight of each loss.

[0043] Furthermore, the specific data fitting loss mentioned above is as follows:

[0044] ;

[0045] In the formula, This represents the number of training samples for the network model. Indicates the first The predicted pressure field for each sample Indicates the first The reference pressure field for each sample, Indicates the first Predicted saturation field for each sample, Indicates the first The reference saturation field for each sample, It is an L2 norm;

[0046] The weak-form energy canonical loss is as follows:

[0047] ;

[0048] In the formula, The total number of grid points in the reservoir spatial model represents the reservoir computational domain. The set of sampling points in any discretization scheme, ,in, For the first The spatial location of a discrete point h The feature scale representing the discretization; For grid points ( i , j , k The permeability value at point () is given, where the partial derivative is calculated using automatic differentiation. This represents the pressure field predicted by the neural network.

[0049] The strong residual regularization loss is as follows:

[0050] ,in:

[0051] , ;

[0052] In the formula, This represents the total number of grid points in the reservoir spatial model. This represents the water saturation field predicted by the neural network. Represents grid points (i , j , k The source and sink terms at () indicate that the injection well location is positive and the production well location is negative. For Darcy operator, Let be the water phase flow function. The average porosity, For total flow, A For cross-sectional area, The unit vector in the main flow direction The relative permeability of the aqueous phase. The relative permeability of the oil phase. The viscosity of the aqueous phase is... This represents the viscosity of the oil phase.

[0053] Furthermore, the above methods also include:

[0054] The importance of each B-spline node in the local sparse branch is calculated based on a three-factor algorithm, and B-spline nodes and their connections with importance less than a threshold are pruned. The three factors include gradient contribution, activation sparsity, and physical correlation. Node importance is specifically defined as follows:

[0055] ,in:

[0056] , , ;

[0057] In the formula, For the importance of nodes, For gradient contribution, To activate sparsity, Physical correlation, The number of training steps to be evaluated. For the first q A B-spline function, The number of non-zero activations. Total number of activations This is the distance from the node to the nearest well or high-permeability boundary. This is the critical distance. This represents the grid spacing.

[0058] Furthermore, the above methods also include:

[0059] The sampling probability of each sample in the training sample is weighted according to the penetration contrast. The specific weights of each sample after weighting are as follows:

[0060] ;

[0061] In the formula, The weights are the weights of one of the samples. and These represent the maximum and minimum permeability values ​​in the sample, respectively.

[0062] Furthermore, the predicted production of carbonate reservoirs, obtained from the numerical simulation results, is as follows:

[0063] ,in:

[0064] , ;

[0065] In the formula, To develop projected production, the cumulative oil production is obtained through time integration; Indicates the yield of the oil phase. Indicates the relative permeability of the oil phase. Indicates the viscosity of the oil phase. Indicates the output value index, This represents the pressure value of the grid containing the well, extracted from the output field of the proxy model. This indicates the bottom hole flowing pressure of the production well. Indicates water saturation.

[0066] Secondly, this application provides a carbonate reservoir water injection development prediction system based on dual-domain neural operators, applicable to any of the carbonate reservoir water injection development prediction methods based on dual-domain neural operators in the first aspect, including:

[0067] The input quantity acquisition module is used to establish a reservoir space model corresponding to a carbonate reservoir and to obtain the original input tensors from the reservoir space model.

[0068] The feature extraction module is used to increase the dimension of the original input tensor to a high-dimensional feature space through a learnable linear transformation, and extract the low-frequency global features and high-frequency local features of the original input tensor from the high-dimensional feature space through a dual-domain feature extraction framework.

[0069] The feature fusion module is used to perform adaptive fusion weight calculation based on energy range. It fuses low-frequency global features and high-frequency local features according to the calculated fusion weight to obtain fused features.

[0070] The decoding output module is used to process the fused features through the feature decoding layer and output the processing results of the fused features through the output layer to obtain the numerical simulation results of the carbonate reservoir.

[0071] The production prediction module is used to obtain the predicted production of carbonate reservoirs based on numerical simulation results, which include the pressure field and saturation field of the carbonate reservoirs.

[0072] Thirdly, this application provides an electronic device including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the method of any one of the first aspects.

[0073] Fourthly, this application provides a non-transitory computer-readable storage medium that stores computer instructions that cause a computer to perform any of the methods in the first aspect.

[0074] Compared with the prior art, the present invention has at least the following beneficial effects:

[0075] Extremely high prediction accuracy: The FNO branch accurately captures global pressure propagation, while the Sparse-KAN branch precisely describes the nonlinear flow near the wellhead and the local features of the high-contrast region. Through the orthogonal fusion of dual-domain features, the relative error of the pressure field is less than 2.1%, the tracking error of the water phase saturation front is within 3.8%, and the mass conservation deviation is controlled below 0.9%.

[0076] Millisecond-level inference speed: On standard GPU hardware (NVIDIA RTX 4090), a single full-field inference can be achieved in just 11 milliseconds, a significant speedup compared to commercial simulators. This speed improvement enables minute-level production forecasting and well control decisions.

[0077] Excellent generalization ability: For unseen well network configurations, the model maintains an error of less than 4% when the number of wells changes by ±3; under a permeability perturbation of ±40%, the error increase is limited to within 1.7%; the orthogonal gating mechanism enables the model to adaptively adjust feature weights according to new scenarios.

[0078] Extremely low data requirements: convergence can be achieved with only 500 training samples collected from historical production data. Compared with the thousands of samples usually required by traditional neural operators, the data efficiency is improved by 4-8 times, which is of great significance for practical oilfield applications.

[0079] Physical consistency guarantee: Double regularization ensures that the prediction results satisfy the mass conservation law and Darcy's law, avoiding non-physical predictions of pure data-driven methods; the pressure field maintains the proper smoothness, and the shift of the saturation front peak conforms to the laws of convection physics.

[0080] It is highly interpretable, and the energy norm provides a physically meaningful measure of feature importance, allowing engineers to understand the contribution of features at different scales; the physics-guided pruning strategy ensures that key information is preserved at well locations and permeability boundaries.

[0081] Strong industrial applicability: This method can be deployed on oilfield logging interpretation software platforms or edge computing units. With fast inference speed and good generalization performance, this method can support the digital transformation of mature carbonate oilfields and achieve real-time optimization of water injection development. Attached Figure Description

[0082] The accompanying drawings, which are included to provide a further understanding of embodiments of the invention and form part of this application, do not constitute a limitation thereof. In the drawings:

[0083] Figure 1 This is a diagram illustrating the overall architecture of the network model algorithm scheme in this embodiment of the invention.

[0084] Figure 2 This is a diagram showing the permeability field distribution and five-point well network configuration in an embodiment of the present invention;

[0085] Figure 3 This is a diagram of the bottom flow pressure control scheme for the production well in an embodiment of the present invention;

[0086] Figure 4 The figures show the distribution of the saturation field and pressure field at the final moment in this embodiment of the invention. In the figure, (a) is the distribution of the saturation field at the final moment, and (b) is the distribution of the pressure field at the final moment.

[0087] Figure 5 The figures show the prediction error distribution diagrams of the saturation field and the pressure field in an embodiment of the present invention. In the figure, (a) is the prediction error distribution diagram of the saturation field and (b) is the prediction error distribution diagram of the pressure field.

[0088] Figure 6 This is a comparison chart of the cumulative oil production curves of each production well in the embodiments of the present invention;

[0089] Figure 7 This is a comparison chart of the oil production rate curves of various production wells in the embodiments of the present invention;

[0090] Figure 8 This is a flowchart of the prediction method in an embodiment of the present invention;

[0091] Figure 9 This is a schematic diagram of the connection of the prediction system in an embodiment of the present invention. Detailed Implementation

[0092] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations.

[0093] Therefore, the following detailed description of the embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.

[0094] It should be noted that similar labels and letters in the following figures indicate similar items. Therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures.

[0095] In the description of the embodiments of the present invention, "multiple" means at least two.

[0096] Example 1: To address the issues of current models requiring large amounts of input data, such as rock physical parameters, fluid properties, and well network layout, and the simulation process demanding significant computational resources and time, while numerical simulation can provide relatively accurate prediction results, its high computational cost and long computation cycle limit its application breadth and efficiency, making it difficult to meet real-time decision-making needs; this example provides a prediction method for water injection development in carbonate oil reservoirs based on dual-domain neural operators, such as... Figure 8 As shown, the specific steps include the following:

[0097] S1. Establish a reservoir space model corresponding to the carbonate reservoir and obtain the original input tensor from the reservoir space model.

[0098] The model parameters for the established reservoir spatial model can be:

[0099] Reservoir type: porous carbonate rock with extreme heterogeneity; reservoir depth: 2800-3200m; average porosity: 12%; permeability range: 1-10000 mD (spanning 4 orders of magnitude); crude oil viscosity: 2.5 cp; formation water viscosity: 0.5 cp; initial reservoir pressure: 30 MPa; initial water saturation: 0.2; reservoir size: 500m × 500m × 50m.

[0100] Furthermore, the reservoir spatial model was discretized using a Cartesian grid with dimensions of 10m × 10m × 5m, resulting in a total of 50 × 50 × 10 = 25,000 grid cells; each grid cell... Includes the following information:

[0101] Center point physical coordinates: ,in ;

[0102] Static attribute: Permeability (Unit: mD), porosity (dimensionless);

[0103] Dynamic attribute: Pressure (Unit: MPa), water saturation (Dimensionless).

[0104] Furthermore, to adapt to the neural network architecture requirements of this scheme and ensure the generality of the method, the physical grid can be mapped to a standardized 40×40×40 computational grid. This design is based on the following considerations:

[0105] 1) Maintain isotropy to avoid directional bias (the original mesh has a large difference in resolution between the Z direction and the XY direction);

[0106] 2) Compatible with the computational efficiency requirements of the Fourier transform (close to a power of 2);

[0107] 3) Achieve the best precision-efficiency balance on existing GPU hardware;

[0108] 4) Enables reservoir models of different sizes and resolutions to use the same neural network architecture.

[0109] The mapping can employ three-dimensional spline interpolation techniques to ensure that the continuity and smoothness of the physical field are maintained.

[0110] Specifically, for the design of the well pattern in the reservoir spatial model, please refer to the appendix. Figure 2 The well network is configured using a five-point method (1 injection and 4 production); all wells are vertical wells, with completion sections covering the entire reservoir thickness; well spacing: 250m; injection wells: constant injection rate control; production wells: stepped bottomhole flowing pressure control scheme (see appendix). Figure 3 Production time span: 1000 days.

[0111] The established reservoir spatial model is shown in [reference]. Figure 2 Its model shows the permeability in the x-direction and the location of oil and water wells, as follows: Figure 2 As shown, the central injection well I1 is located in the center of the reservoir, and the four production wells P1-P4 are symmetrically distributed in the four corners; the permeability field exhibits typical carbonate rock heterogeneity characteristics, with high-permeability bands (yellow area, about 10,000 mD) and low-permeability matrix (blue area, about 1 mD) alternating, and the color bars show the logarithmic distribution of the permeability field; Figure 3 The study demonstrated the bottom-hole flowing pressure control strategy for four production wells, employing a stepped pressure reduction scheme to optimize production capacity.

[0112] Specifically, the original input tensor can be represented as , The meaning is the real number field, which can be learned through linear transformations. Upgrading to a higher-dimensional feature space can be represented as: ;

[0113] in,

[0114] For the weight matrix of increased dimension, For feature dimensions; The bias vector; the original input tensor can contain 6 types of data, that is, input through 6 channels, and the 6 input channels are respectively (or data types): permeability field (Unit: mD), Production Well BHP Control (Unit: MPa) Injection well flow control (Unit: m³ / d), Binary tensor of well location (1 indicates well location, 0 indicates non-well location), initial pressure conditions (Unit: MPa) Initial conditions for saturation (Dimensionless, range [0, 1]).

[0115] The method further includes: weighting the sampling probability of each sample in the training sample according to the penetration contrast, wherein the weight of each sample after weighting is as follows:

[0116] ;

[0117] In the formula, The weights are the weights of one of the samples. and These represent the maximum and minimum permeability values ​​in the sample, respectively.

[0118] Specifically, the sampling probability is weighted based on the penetration rate of the sample. This design has the following main objectives:

[0119] 1) Addressing Class Imbalance: In real-world oilfield data, samples with extreme permeability ratios are relatively few, yet these are precisely the most challenging. Weighting allows the model to see these samples more frequently. For samples with a permeability ratio of 1000:1, the weights are... For samples with a penetration ratio of 10:1, the weight is 2, which means that the extreme samples are twice as likely to be sampled as the normal samples.

[0120] 2) Accelerated convergence in difficult cases: Traditional FNO already performs well on homogeneous samples and does not require overtraining; the KAN branch, on the other hand, needs to learn more about how to handle local singularities.

[0121] S2 elevates the original input tensor to a high-dimensional feature space through a learnable linear transformation. Then, a dual-domain feature extraction framework is used to extract low-frequency global features and high-frequency local features of the original input tensor from the high-dimensional feature space. (See the schematic diagram of the dual-domain feature extraction framework.) Figure 1 .

[0122] Among them, the Fourier neural operator is combined with the Kolmogorov-Arnold network in a parallel manner, such as Figure 1 As shown, these are responsible for extracting low-frequency global pressure modes and high-frequency local flow characteristics, respectively. This design is based on a deep understanding of the physical properties of carbonate reservoirs: the pressure field has global propagation and needs to be captured by frequency domain methods; while wellhead singularities and high-contrast regions require description by local adaptive basis functions. This architecture is not only applicable to reservoir simulation but can also be extended to other multi-scale physical field problems.

[0123] Optionally, the above dual-domain feature extraction framework includes a global spectral branch and a local sparse branch, wherein:

[0124] The global spectral branch uses an enhanced Fourier neural operator to capture low-frequency global features, which can be represented as:

[0125] ;

[0126] In the formula, The first in the global spectrum branch The output features of the layer, superscript This serves as the identifier for the global spectrum branch; It is the ReLU activation function. The linear transformation matrix representing the residual connection. Indicates the first in the global spectrum branch The output characteristics, and These represent the 3D Fast Fourier Transform and Inverse Fourier Transform, respectively. Represented as learnable frequency domain weights, This indicates a parallel 3DU-Net module, consisting of 3 downsampling layers and 3 upsampling layers, used to enhance local feature extraction.

[0127] Specifically, the local sparse branch captures high-frequency local features based on the Kolmogorov-Arnold method using B-splines, where the high-frequency local features can be represented as:

[0128] ;

[0129] In the formula, Indicates the first sparse branch in a local sparse branch The first layer Output characteristics of each neuron Indicates the connection of the first in a locally sparse branch. Layer The first neuron to the second The first layer B-spline functions of 1 neuron Indicates the first sparse branch in a local sparse branch The first layer Output characteristics of each neuron Indicates the first Number of neurons in a layer, superscript X This serves as an identifier for locally sparse branches.

[0130] The activation function of each KAN neuron in the locally sparse branch can be defined as: ;

[0131] in: Let B-spline basis functions be defined on the node sequence. In the above, k is the order of the B-spline, which is usually preferably 3, but can be adjusted between 2 and 5 depending on the specific application. Let be the learnable coefficients of the i-th basis function of the q-th neuron.

[0132] S3, based on the energy range, performs adaptive fusion weight calculation, and then fuses the low-frequency global features and high-frequency local features according to the calculated fusion weight to obtain the fused features.

[0133] The aforementioned fusion features are achieved through an orthogonal gating fusion mechanism. In the adaptive fusion weight calculation based on energy range, the energy norm is defined as: for the joint representation of the pressure field and the saturation field... Define the energy norm respectively:

[0134] ;

[0135] Where Ω represents the reservoir computational domain. The pressure gradient norm weighted by penetration rate. It is the inverse of the permeability tensor. =0.1 is the weighting coefficient for the saturation term. This is the gradient operator.

[0136] Optionally, the above-mentioned fusion features are specifically as follows:

[0137] ,in:

[0138] ;

[0139] In the formula, Indicates fusion characteristics, This represents the output of the global spectrum branch. This represents the output of a locally sparse branch. This represents the fusion weights obtained through calculation. This represents the sigmoid activation function. This represents a learnable weight vector. This represents the learnable bias parameters. It represents the energy norm, used to measure the combined energy of the pressure field and the saturation field.

[0140] In this method, the ratio of physical energy norms is used as a measure of feature importance to achieve adaptive fusion. Compared with fixed weights or purely learned weights, this mechanism has a clear physical meaning: the energy norm reflects the contribution of features at different scales to the overall flow, the gating mechanism prevents gradient interference, and ensures that the two branches can independently optimize the feature scales they are good at.

[0141] S4 processes the fused features through the feature decoding layer and outputs the processing results of the fused features through the output layer to obtain the numerical simulation results of the carbonate reservoir.

[0142] Optionally, steps S2-S4 above are executed using the constructed network model, and the total loss function of the network model is as follows:

[0143] ;

[0144] In the formula, Represented as the total loss function, This represents the data fitting loss. This represents the weak-form energy canonical loss. This represents the strong-form residual regularization loss. , , This indicates the weight of each loss.

[0145] Optionally, the data fitting loss described above is specifically as follows:

[0146] ;

[0147] In the formula, This represents the number of training samples for the network model. Indicates the first The predicted pressure field for each sample Indicates the first The reference pressure field for each sample, Indicates the first Predicted saturation field for each sample, Indicates the first The reference saturation field for each sample, It is an L2 norm.

[0148] The weak-form energy canonical loss is specifically as follows:

[0149] ;

[0150] In the formula, The total number of grid points in the reservoir spatial model represents the reservoir computational domain. The set of sampling points in any discretization scheme, ,in, For the first The spatial location of a discrete point h The feature scale representing the discretization; For grid points ( i , j , k The permeability value at point () is given, where the partial derivative is calculated using automatic differentiation. This represents the pressure field predicted by the neural network.

[0151] Specifically, the strong residual regularization loss is as follows:

[0152] ,in:

[0153] , ;

[0154] In the formula, This represents the total number of grid points in the reservoir spatial model. This represents the water saturation field predicted by the neural network. Represents grid points ( i , j , k The source and sink terms at () indicate that the injection well location is positive and the production well location is negative. For Darcy operator, Let be the water phase flow function. The average porosity, For total flow, A For cross-sectional area, The unit vector in the main flow direction The relative permeability of the aqueous phase. The relative permeability of the oil phase. The viscosity of the aqueous phase is... This represents the viscosity of the oil phase.

[0155] Specifically, by simultaneously applying integral-form energy constraints (weak form) and point-state Darcy residual constraints (strong form), an optimal balance is achieved between training efficiency and physical accuracy. Weak regularization ensures the overall smoothness of the pressure field, while strong regularization guarantees local compliance with conservation laws. The weights are dynamically adjusted through course learning, accelerating convergence while ensuring eventual physical consistency.

[0156] During model training, the training dataset can be collected from historical production data, consisting of 500 training samples. Each sample includes: a three-dimensional permeability field (obtained through geostatistical interpolation); well control history (records of actual injection and production parameters); production response (measured pressure and saturation distribution); and a time snapshot (saving the state every 10 days).

[0157] The architecture of the network model described above is configured as follows:

[0158] 1) Input layer, input dimensions (batch_size, 6, 40, 40, 40), 6 input channels as described above.

[0159] 11) Dual-branch feature extraction: FNO branch (9-layer U-Fourier block): Feature dimension: 36; Number of Fourier patterns: 10; Parallel U-Net depth: 3 layers;

[0160] 12) Sparse-KAN branch (7 hidden layers + 1 output layer): Hidden layer neurons: [36, 32, 28, 24, 20, 16, 12]; B-spline order: 3; Initial number of nodes: 30 / layer.

[0161] 2) Feature Decoding and Output Layer:

[0162] 1. After gating, fuse the feature dimensions (batch_size, 36, 40, 40, 40);

[0163] 2. Hidden layers (3 layers):

[0164] Hidden layer 1: Conv3D(36, 64, kernel_size=3, padding=1) + BatchNorm3D + ReLU;

[0165] Hidden layer 2: Conv3D(64, 32, kernel_size=3, padding=1) + BatchNorm3D + ReLU;

[0166] Hidden layer 3: Conv3D(32, 16, kernel_size=3, padding=1) + BatchNorm3D + ReLU;

[0167] 3) Output layer: Conv3D(16, 2, kernel_size=1), output dimensions: (batch_size, 2, 40, 40, 40); 2 output channels: pressure field, saturation field.

[0168] Furthermore, the training environment and process are as follows:

[0169] Hardware platform:

[0170] CPU: AMD EPYC 9454 48-core processor; Memory: 1.5TB; GPU: 4×NVIDIA RTX 4090; Optimizer: Adam (learning rate 1e-3, first 300 rounds) + L-BFGS (last 200 rounds);

[0171] Batch size: 16; Number of training epochs: 500; Parallelism strategy: data parallelism, 4 GPUs training simultaneously; Total training time: 6.8 hours.

[0172] The training phase can be further divided into three stages, as follows:

[0173] The first stage focuses on data fitting (e.g., rounds 1-100): ;

[0174] The second stage, the balancing stage (e.g., rounds 101-300): ;

[0175] The third stage involves strengthening physical constraints (e.g., rounds 301-500): .

[0176] By employing heterogeneity-weighted sampling and a curriculum learning strategy, the data requirements were significantly reduced. Only 500 training samples were needed to achieve the accuracy required for engineering applications, which is of great value in situations where real-world oilfield data is scarce. This framework provides a reference for other data-constrained industrial applications.

[0177] Optionally, the above methods also include:

[0178] The importance of each B-spline node in the local sparse branch is calculated based on a three-factor algorithm, and B-spline nodes and their connections with importance less than a threshold are pruned. The three factors include gradient contribution, activation sparsity, and physical correlation. Node importance is specifically defined as follows:

[0179] ,in:

[0180] , , ;

[0181] In the formula, For the importance of nodes, For gradient contribution, To activate sparsity, Physical correlation, The number of training steps to be evaluated. For the first q A B-spline function, The number of non-zero activations. Total number of activations This is the distance from the node to the nearest well or high-permeability boundary. This is the critical distance. This represents the grid spacing.

[0182] Specifically, the FNO branch is already responsible for global smoothing features, so the KAN branch can be pruned more aggressively, focusing on key local regions. The pruned sparse KAN complements the dense FNO, avoiding feature redundancy. For the KAN network, the number of B-spline nodes directly determines the number of model parameters. Removing redundant nodes helps reduce model capacity, avoids the model learning overly complex mapping relationships and falling into overfitting, and accelerates inference speed, reduces GPU memory usage, and facilitates deployment on edge devices.

[0183] The three-factor pruning strategy, which comprehensively considers gradient contribution, activation sparsity, and physical distance, ensures that key information is preserved while compressing the model. In particular, the physical correlation factor ensures that nodes near well locations and permeability boundaries are not erroneously pruned, which is crucial for maintaining prediction accuracy.

[0184] Among them, the three factors mentioned above have the following characteristics:

[0185] Wellhead feature protection: the closer the node is to the well, The larger the value, the less likely it is to be sheared; this ensures that the model can accurately capture the pressure drop funnel and flow singularities at the wellhead.

[0186] Permeability boundary protection: At the boundary between high and low permeability, the flow pattern changes abruptly, and nodes at these locations are also protected.

[0187] Far-field simplification: Nodes far from critical regions are easier to prune because the flow in these regions is relatively smooth and does not require dense B-spline nodes.

[0188] S5. The predicted production of carbonate reservoirs is obtained based on the numerical simulation results, which include the pressure field and saturation field of carbonate reservoirs.

[0189] Specifically, the predicted production of carbonate reservoirs obtained from the numerical simulation results is as follows:

[0190] ,in:

[0191] , ;

[0192] In the formula, To develop projected production, the cumulative oil production is obtained through time integration; Indicates the yield of the oil phase. Indicates the relative permeability of the oil phase. Indicates the viscosity of the oil phase. Indicates the output value index, This represents the pressure value of the grid containing the well, extracted from the output field of the proxy model. This indicates the bottom hole flowing pressure of the production well. Indicates water saturation.

[0193] Since the network outputs the pressure and saturation field distribution of the entire reservoir, the production rate of each well needs to be calculated through the following steps: For wells located in... The production wells extract their corresponding pressures from a 40×40×40 grid using nearest neighbor interpolation. and saturation The output index is calculated using the Peaceman well model. .

[0194] Where K is the permeability of the grid where the well is located, h is the thickness of the perforated section (50m in this example), and μ is the fluid viscosity; Where is the wellbore radius; The sweep radius of a well is typically calculated using the following formula:

[0195] .

[0196] Furthermore, by performing two-phase flow calculations, the yields of the oil phase and the water phase are respectively: .

[0197] The relative penetration rate is calculated using the Corey model: The cumulative oil production can then be obtained by integrating the oil phase over time. Where Nt is the number of time snapshots, =10 days as the time step.

[0198] Specifically, when calculating the error index, the relative error of the pressure field and the relative error of the saturation field are first defined. The relative error of the pressure field and the relative error of the saturation field at the i-th time step are: , Where F refers to the commercial simulator, i.e., the full-order model (FOM); R refers to the surrogate model, also known as the reduced-order model (ROM); the oil production error between the commercial simulator (i.e., the full-order model) and the surrogate model is defined as the average relative error of the integral of the absolute value of the difference in oil production between each well calculated by the full-order model and the surrogate model at each stage over time, and the relative error is:

[0199] ;

[0200] in, For production well marking; In this embodiment, the number of production wells is 4; It is the total simulation time; and These are the oil production figures for the full-order model and the surrogate model, respectively.

[0201] For the method used in this embodiment, when performing prediction accuracy analysis, please refer to... Figure 4 , Figure 4 (a) shows the distribution of the saturation field at the final moment. Figure 4 (b) shows the distribution of the pressure field at the final moment. Figure 4 The results show the predicted saturation and pressure fields at 1000 days. It can be seen that the injection water front (the yellow area with saturation of 0.6-0.8) exhibits an asymmetric propagation mode controlled by the permeability field, and the water phase propagates faster in the high-permeability channel. The pressure field shows a smooth transition from the injection well (high pressure 18 MPa) to the production well (low pressure 11-12 MPa). Figure 5 The error distribution plot reveals the spatial distribution characteristics of the prediction accuracy. Figure 5 (a) shows the distribution of prediction errors for the saturation field. Figure 5 The middle (b) diagram shows the distribution of the pressure field prediction error. The saturation error is mainly concentrated at the water front (with an error of 0.08). This is because the steep gradient at the front is difficult to capture accurately. The overall pressure field error is small and evenly distributed. The maximum error occurs near the wellhead (about 18% relative error), but the absolute error is still within an acceptable range.

[0202] for Figure 2 The spatial model was established, and the prediction accuracy of each production well in the model is shown in Table 1:

[0203] Table 1

[0204] Producer Pressure error Oil error P1 1.65% 2.83% P2 2.09% 3.78% P3 1.84% 2.70% P4 1.31% 2.40%

[0205] Table 1 quantifies the prediction accuracy of each production well; the relative error of pressure prediction ranges from 1.31% to 2.09%, and the error of oil production prediction ranges from 2.40% to 3.78%; well P3 performed best, possibly because it is located in a relatively homogeneous region; the average performance on the independent test set (100 unseen samples) is as follows: relative L2 error of pressure field: 2.08% ± 0.28%; tracking error of water phase saturation front: 3.72% ± 0.41%; mass conservation bias: 0.87% ± 0.12%; cumulative oil production prediction error: 2.45% ± 0.33%; water cut prediction error: 3.21% ± 0.38%.

[0206] Further production dynamics forecasts are attached. Figure 6 and attached Figure 7 ;in, Figure 6 The comparison between the predicted (dashed line) and reference values ​​(solid line) of the cumulative oil production curves for each production well is shown; well P3, due to its location in a high-permeability channel, has the highest cumulative oil production (approximately 7.8 × 10⁻⁶). 4 (cubic meters), while well P4 had the lowest production. The predicted curves for all wells were in high agreement with the reference values, demonstrating the model's ability to accurately capture long-term production dynamics; Figure 7 The model shows the change in instantaneous oil production rate. The initial oil production rate was high (600 m³ / d in well P3), followed by a rapid decline due to water phase breakthrough. The model accurately predicted this nonlinear decline trend, including the production fluctuations in well P3 around 200 days, which were caused by uneven water front advancement.

[0207] Using the hardware provided in this embodiment, the computational efficiency is as follows: single full-field inference time: 10.8 ms (single GPU RTX4090); batch inference (32 samples): 142 ms; speedup compared to commercial simulators: 21.3 times; GPU utilization: 87%; VRAM usage: 3.2 GB; generalization performance verification: well number variation test, test range: 2-7 wells (original training was 5 wells); average error with ±3 well variations: 3.86%; maximum error occurred in the 7-well configuration: 4.12%. Permeability perturbation test: perturbation range: ±40% random perturbation; pressure field error increase: 1.52%; saturation field error increase: 1.68%; overall error increase: within 1.7%; well location offset test; offset range: ±50m random offset; average error increase: 2.24%; maximum error: 3.87% (extreme offset case).

[0208] The comparison results between the method provided in this embodiment and other methods are shown in Table 2:

[0209] Table 2

[0210] method <![CDATA[Pressure L2 Error (%)]]> Saturation error (%) Mass conservation deviation (%) Inference time (ms) Number of training samples This invention 2.08 3.72 0.87 11 500 Single FNO 4.52 9.13 2.31 8 2000 Single KAN 3.94 5.87 1.56 38 1500 PINN 3.21 6.45 0.43 156 800 Business Simulator - - <0.1 235，000 -

[0211] Table 2 comprehensively compares the performance of this invention with existing methods. This invention achieves inference speed second only to single-FNO while maintaining the highest prediction accuracy. Of particular note is that this method requires only 500 training samples, far fewer than other deep learning methods, which is significant for situations where real-world oilfield data is scarce. Although PINN performs best in terms of mass conservation (0.43%), its inference time (156 ms) limits its real-time application. While commercial simulators achieve near-perfect mass conservation (<0.1%), their 235-second computation time prevents them from being used for real-time optimization.

[0212] In this embodiment, a dual-domain feature extraction framework is constructed that can capture global pressure propagation and accurately describe local flow singularities; a physics-driven adaptive fusion mechanism is designed to intelligently adjust feature weights according to the flow characteristics of different regions; at the same time, a dual physical regularization strategy is introduced to ensure the physical rationality of the prediction results while ensuring computational efficiency; and intelligent compression of the model is realized, enabling it to achieve millisecond-level inference speed on edge devices.

[0213] Example 2: This application provides a carbonate reservoir water injection development prediction system based on dual-domain neural operators, applied to the carbonate reservoir water injection development prediction method based on dual-domain neural operators in Example 1, such as... Figure 9 As shown, it includes:

[0214] The input quantity acquisition module is used to establish a reservoir space model corresponding to a carbonate reservoir and to obtain the original input tensors from the reservoir space model.

[0215] The feature extraction module is used to increase the dimension of the original input tensor to a high-dimensional feature space through a learnable linear transformation, and extract the low-frequency global features and high-frequency local features of the original input tensor from the high-dimensional feature space through a dual-domain feature extraction framework.

[0216] The feature fusion module is used to perform adaptive fusion weight calculation based on energy range. It fuses low-frequency global features and high-frequency local features according to the calculated fusion weight to obtain fused features.

[0217] The decoding output module is used to process the fused features through the feature decoding layer and output the processing results of the fused features through the output layer to obtain the numerical simulation results of the carbonate reservoir.

[0218] The production prediction module is used to obtain the predicted production of carbonate reservoirs based on numerical simulation results, which include the pressure field and saturation field of the carbonate reservoirs.

[0219] Example 3: This application provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the method of Example 1.

[0220] Example 4: This application provides a non-transitory computer-readable storage medium that stores computer instructions that cause a computer to execute the method of Example 1.

[0221] Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can take the form of a computer program product embodied on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0222] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0223] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0224] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0225] Those skilled in the art will understand that all or part of the steps in the above facts and methods can be implemented by a program instructing related hardware. The program or the program described therein can be stored in a computer-readable storage medium. When the program is executed, it includes the following steps: at this time, the corresponding method steps are introduced. The storage medium can be ROM / RAM, magnetic disk, optical disk, etc.

[0226] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for predicting water injection development in carbonate oil reservoirs based on dual-domain neural operators, characterized in that, The specific steps include the following: S1. Establish a reservoir space model corresponding to the carbonate reservoir and obtain the original input tensor from the reservoir space model. S2, the original input tensor is upscaled to a high-dimensional feature space through a learnable linear transformation, and low-frequency global features and high-frequency local features of the original input tensor are extracted from the high-dimensional feature space using a dual-domain feature extraction framework; the dual-domain feature extraction framework includes a global spectral branch and a local sparse branch, wherein: The global spectral branch uses an enhanced Fourier neural operator to capture low-frequency global features; The local sparse branch is based on Kolmogorov-Arnold capture of B-splines to obtain high-frequency local features; S3, adaptive fusion weight calculation is performed based on the energy norm. The low-frequency global features and the high-frequency local features are fused according to the calculated fusion weights to obtain fused features; the fused features are specifically: ,in: ； In the formula, Indicates fusion characteristics, This represents the output of the global spectrum branch. This represents the output of a locally sparse branch. This represents the fusion weights obtained through calculation. This represents the sigmoid activation function. This represents a learnable weight vector. This represents the learnable bias parameters. It represents the energy norm, used to measure the combined energy of the pressure field and the saturation field; S4, the fused features are processed through the feature decoding layer, and the processing results of the fused features are output through the output layer to obtain the numerical simulation results of the carbonate reservoir; S5. Based on the numerical simulation results, the predicted production of carbonate reservoirs is obtained. The numerical simulation results include the pressure field and saturation field of the carbonate reservoirs.

2. The method for predicting water injection development in carbonate oil reservoirs based on dual-domain neural operators according to claim 1, characterized in that, The low-frequency global features are specifically: ； In the formula, The first in the global spectrum branch The output features of the layer, superscript This serves as the identifier for the global spectrum branch; It is the ReLU activation function. The linear transformation matrix representing the residual connection. Indicates the first in the global spectrum branch The output characteristics, and These represent the 3D Fast Fourier Transform and Inverse Fourier Transform, respectively. Represented as learnable frequency domain weights, This indicates a parallel 3DU-Net module, consisting of 3 downsampling layers and 3 upsampling layers, used to enhance local feature extraction.

3. The method for predicting water injection development in carbonate oil reservoirs based on dual-domain neural operators according to claim 1, characterized in that, The high-frequency local features are specifically: ； In the formula, Indicates the first sparse branch in a local sparse branch The first layer Output characteristics of each neuron Indicates the connection of the first in a locally sparse branch. Layer The first neuron to the second The first layer B-spline functions of 1 neuron Indicates the first sparse branch in a local sparse branch The first layer Output characteristics of each neuron Indicates the first The number of neurons in a layer, with the superscript X indicating a local sparse branch.

4. The method for predicting water injection development in carbonate oil reservoirs based on dual-domain neural operators according to any one of claims 1-3, characterized in that, Steps S2-S4 are executed using the constructed network model, and the total loss function of the network model is specifically: ； In the formula, Represented as the total loss function, This represents the data fitting loss. This represents the weak-form energy canonical loss. This represents the strong-form residual regularization loss. , , This indicates the weight of each loss.

5. The method for predicting water injection development in carbonate oil reservoirs based on dual-domain neural operators according to claim 4, characterized in that, The data fitting loss is specifically as follows: ； In the formula, This represents the number of training samples for the network model. Indicates the first The predicted pressure field for each sample Indicates the first The reference pressure field for each sample, Indicates the first Predicted saturation field for each sample, Indicates the first The reference saturation field for each sample, It is an L2 norm.

6. The method for predicting water injection development in carbonate oil reservoirs based on dual-domain neural operators according to claim 5, characterized in that, The weak energy canonical loss is specifically as follows: ； In the formula, The total number of grid points in the reservoir spatial model represents the reservoir computational domain. The set of sampling points in any discretization scheme, ,in, For the first The spatial location of each discrete point, where h represents the feature scale of discretization; Let be the permeability value at grid point (i, j, k), where the partial derivative is calculated through automatic differentiation. This represents the pressure field predicted by the neural network.

7. The method for predicting water injection development in carbonate oil reservoirs based on dual-domain neural operators according to claim 4, characterized in that, The strongly-formed residual regularization loss is specifically as follows: ,in: ， ； In the formula, This represents the total number of grid points in the reservoir spatial model. This represents the water saturation field predicted by the neural network. This represents the source and sink terms at grid point (i, j, k), with positive values ​​for injection wells and negative values ​​for production wells. For Darcy operator, Let be the water phase flow function. The average porosity, Let A be the total flow rate and A be the cross-sectional area. The unit vector in the main flow direction The relative permeability of the aqueous phase. The relative permeability of the oil phase. The viscosity of the aqueous phase is... This represents the viscosity of the oil phase.

8. The method for predicting water injection development in carbonate oil reservoirs based on dual-domain neural operators according to claim 1, characterized in that, The method further includes: The importance of each B-spline node in the local sparse branch is calculated based on a three-factor algorithm, and B-spline nodes and their connections with importance less than a threshold are pruned. The three factors include gradient contribution, activation sparsity, and physical correlation. Specifically, the node importance is: ； In the formula, For the importance of nodes, For gradient contribution, To activate sparsity, This is a physical correlation.

9. The method for predicting water injection development in carbonate oil reservoirs based on dual-domain neural operators according to claim 8, characterized in that, The gradient contribution is specifically as follows: ； In the formula, The number of training steps to be evaluated. The q-th B-spline function.

10. The method for predicting water injection development in carbonate oil reservoirs based on dual-domain neural operators according to claim 8, characterized in that, The activation sparsity specifically refers to: ； In the formula, The number of non-zero activations. This represents the total number of activations.

11. The method for predicting water injection development in carbonate oil reservoirs based on dual-domain neural operators according to claim 8, characterized in that, The physical correlation is specifically as follows: ； In the formula, This is the distance from the node to the nearest well or high-permeability boundary. This is the critical distance. This represents the grid spacing.

12. The method for predicting water injection development in carbonate oil reservoirs based on dual-domain neural operators according to claim 4, characterized in that, The method further includes: The sampling probability of each sample in the training sample is weighted according to the penetration contrast. The specific weights of each sample after weighting are as follows: ； In the formula, The weights are the weights of one of the samples. and These represent the maximum and minimum permeability values ​​in the sample, respectively.

13. The method for predicting water injection development in carbonate oil reservoirs based on dual-domain neural operators according to claim 1, characterized in that, The predicted production of carbonate reservoirs based on the numerical simulation results is as follows: ,in: ， ； In the formula, To develop projected production, the cumulative oil production is obtained through time integration; Indicates the yield of the oil phase. Indicates the relative permeability of the oil phase. Indicates the viscosity of the oil phase. Indicates the output value index, This represents the pressure value of the grid containing the well, extracted from the output field of the proxy model. This indicates the bottom hole flowing pressure of the production well. Indicates water saturation.

14. A prediction system for water injection development in carbonate oil reservoirs based on dual-domain neural operators, characterized in that, include: The input quantity acquisition module is used to establish a reservoir space model corresponding to a carbonate reservoir and to obtain the original input tensors from the reservoir space model. The feature extraction module is used to upscale the original input tensor to a high-dimensional feature space through a learnable linear transformation, and then extract low-frequency global features and high-frequency local features of the original input tensor from the high-dimensional feature space using a dual-domain feature extraction framework. The dual-domain feature extraction framework includes a global spectral branch and a local sparse branch, wherein: The global spectral branch uses an enhanced Fourier neural operator to capture low-frequency global features; The local sparse branch is based on Kolmogorov-Arnold capture of B-splines to obtain high-frequency local features; The feature fusion module is used to perform adaptive fusion weight calculation based on the energy norm, and to fuse the low-frequency global features and the high-frequency local features according to the calculated fusion weights to obtain fused features; the fused features are specifically: ,in: ； In the formula, Indicates fusion characteristics, This represents the output of the global spectrum branch. This represents the output of a locally sparse branch. This represents the fusion weights obtained through calculation. This represents the sigmoid activation function. This represents a learnable weight vector. This represents the learnable bias parameters. It represents the energy norm, used to measure the combined energy of the pressure field and the saturation field; The decoding output module is used to process the fused features through the feature decoding layer and output the processing results of the fused features through the output layer to obtain the numerical simulation results of the carbonate reservoir. The production prediction module is used to obtain the predicted production of carbonate reservoirs based on the numerical simulation results, which include the pressure field and saturation field of the carbonate reservoirs.

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