Numerical simulation acceleration method based on physical knowledge driven machine learning operator network

Abstract

The application discloses a numerical simulation acceleration method based on a physical knowledge driven machine learning operator network, wherein the machine learning operator network specifically adopts a Fourier neural operator network; the method comprises the following steps: constructing a flow equivalent mathematical model of a seepage parameter field; designing a flow equivalent simulation acceleration method based on the Fourier neural operator network; designing a Fourier neural operator network training strategy based on physical knowledge driving; sampling and constructing a permeability field data set, and training and testing the Fourier neural operator network; and deploying the trained Fourier neural operator network online to assist in real-time flow equivalent calculation. The application can realize label-free training without a numerical simulator, and can quickly perform equivalent processing on different original grid permeabilities through the trained model, and the equivalent calculation result meets physical constraints and has higher equivalent flow simulation solving precision.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of seepage numerical simulation, and particularly relates to a numerical simulation acceleration method based on a physical knowledge driven machine learning operator network. BACKGROUND

[0002] In history fitting, scheme optimization and uncertainty evaluation, numerical simulation needs to be repeatedly performed to analyze the seepage process in a heterogeneous reservoir. The simulation calculation cost on the original grid is high and low in efficiency. In order to reduce the calculation amount, the industry usually converts the original grid model into a post-processing grid model by using "flow equivalence". A "global flow equivalence method" solves a global single-phase flow problem on the original grid to calculate equivalent conductivity and well index, so that the equivalent model has a similar global flow state to the original model under a given boundary condition. However, the single processing of the global equivalence calculation method has a calculation complexity of the same order of magnitude as the original grid simulation; when facing a large number of random permeability fields, large-scale linear equation solving must be repeatedly performed on the original grid, and the calculation cost is huge.

[0003] In order to improve the efficiency of equivalent calculation, some researches currently replace part of the calculation process by establishing a proxy model, including using a neural network to establish an "original grid parameter field to post-processing grid parameter field" end-to-end proxy model, and using a neural network to establish a "original grid parameter field to original grid flow field" numerical simulation proxy model. However, the existing data-driven proxy model relies on large-scale high-quality data samples for training, and once the boundary condition or well pattern arrangement changes, a large number of numerical simulations must be performed to generate data labels. The existing data-physical hybrid driving method still has the problem of insufficient physical consistency. Such a method often tends to minimize empirical errors, lacks strict explicit constraints on Darcy's equation and conservation laws, and is easy to produce non-conservative or numerical instability equivalent parameters. At the same time, the existing physical constraint method relies on automatic differentiation to construct a residual; in a strongly heterogeneous medium, high-order gradient sensitivity often leads to error amplification and unstable training process. In addition, the network generalization ability of the existing proxy assisted equivalent calculation method is insufficient; although the common neural network method can learn the "field to field" mapping, it is insufficient to extract the main mode, resulting in inaccurate reasoning and prediction of unknown parameter fields and decreased equivalent processing precision. SUMMARY

[0004] To solve the above problems, the application provides a numerical simulation acceleration method based on a physical knowledge driven machine learning operator network, which is wholely independent of a numerical simulator; flux continuity and source-sink balance relationship are strictly maintained by discretizing control equations through a finite volume method, so as to avoid instability and high cost of automatic differentiation in high-order spatial derivative calculation; a Fourier neural operator network is combined with linear transformation and frequency domain low-pass filtering in each hidden layer, so as to effectively capture general patterns and improve the capturing ability of the model to a multi-scale heterogeneous field; physical consistency is ensured by strictly following physical constraints in neural network training and strictly calculating equivalent conductivities and well exponents of post-processing grids according to equivalent analytical formulas in post-processing; under the same computing resources, the application can realize parallel fast flow equivalent processing for batch permeability fields, has the same prediction accuracy as the original model, and has significantly improved computing efficiency compared with existing flow equivalent methods.

[0005] The technical scheme of the application is as follows:

[0006] A numerical simulation acceleration method based on a physical knowledge driven machine learning operator network, wherein the machine learning operator network specifically adopts a Fourier neural operator network; the method comprises the following steps:

[0007] Step 1, constructing a flow equivalent mathematical model of a seepage parameter field;

[0008] Step 2, designing a flow equivalent simulation acceleration method based on a Fourier neural operator network;

[0009] Step 3, designing a Fourier neural operator network training strategy based on physical knowledge driving;

[0010] Step 4, sampling and constructing a permeability field data set, training the Fourier neural operator network, and performing flow equivalent calculation and effect test of the Fourier neural operator network assisted acceleration based on the flow equivalent mathematical model of step 1;

[0011] Step 5, deploying the trained Fourier neural operator network online to assist in real-time flow equivalent calculation.

[0012] Further, in step 1, the flow equivalent mathematical model comprises control equations, boundary conditions, discretization formats of the control equations and flow equivalent equations; and the specific construction process is as follows:

[0013] Step 1.1, constructing control equations:

[0014] ;

[0015] ;

[0016] wherein, is the flow velocity; is the spatial position coordinate; is the permeability; is the fluid viscosity; is the pressure field; denotes the source-sink term; is the gradient operator;

[0017] Step 1.2, define the boundary condition as the Neumann boundary condition of the flow velocity being 0:

[0018] ;

[0019] wherein, is the Neumann boundary position; is the unit normal vector of the boundary position;

[0020] Step 1.3, construct the discrete format of the control equation:

[0021] ;

[0022] wherein, and are the serial numbers of two different grids, respectively; denotes the adjacent cell of the th grid; denotes the control volume of the th grid; are the pressure fields of the th grid and the th grid, respectively; is the conductivity between the adjacent th grid and the th grid, and the calculation formula is:

[0023] ;

[0024] wherein, and are the normal distances from the center of the th grid and the center of the th grid to the contact surface, respectively; is the area of the contact surface between the adjacent th grid and the th grid; is the harmonic average permeability between the adjacent th grid and the th grid;

[0025] The source-sink term is calculated by using the Pitsman formula:

[0026] ​;

[0027] wherein, is the bottom hole pressure; is the well index, calculated by:

[0028] ;

[0029] wherein, is the equivalent reservoir radius; is the wellbore radius; is the skin factor; is the payzone thickness;

[0030] Step 1.4, constructing the flow equivalence equation between the post-processing grid and the original grid; the flow equivalence equation includes the equivalent conductivity and the equivalent well index, and the calculation formulas are respectively:

[0031] ;

[0032] ;

[0033] wherein, and are the serial numbers of two different post-processing grids, respectively; is the equivalent conductivity between the adjacent th post-processing grid and the th post-processing grid; is the contact surface between the adjacent th post-processing grid and the th post-processing grid; is the equivalent well index of the th post-processing grid; is the well index of the th grid; is the control volume of the th post-processing grid; and are the equivalent pressure fields of the th post-processing grid and the th post-processing grid, respectively, and the calculation formula is:

[0034] ;

[0035] ;

[0036] wherein, the control volume of the th post-processing grid; is the control volume of the th grid.

[0037] Further, the specific process of step 2 is:

[0038] Step 2.1, the mapping formula of constructing the Fourier neural operator network is:

[0039] ;

[0040] Wherein, is the permeability field input into the input layer of the Fourier neural operator network; , , is the different pressure fields output by the hidden layer of the Fourier neural operator network; is the number of hidden layers of the Fourier neural operator network; is the pressure field output by the output layer of the Fourier neural operator network; is a connection symbol; a full connection layer is used between the input layer and the hidden layer for dimensionality increasing transformation; a full connection layer is used between the hidden layer and the output layer for dimensionality decreasing transformation; the hidden layer and the hidden layer are calculated by the following formula:

[0041] ;

[0042] Wherein, is the hidden layer number of the Fourier neural operator network; is the pressure field input into the hidden layer, which is also the pressure field output by the hidden layer; is the pressure field output by the hidden layer; is the Fourier transform; is the inverse Fourier transform; is the low-pass filter in the frequency domain; is a linear transformation; is a nonlinear rectifier unit activation function;

[0043] Step 2.2, referring to the flow equivalent equation, define the flow equivalent analytical formula based on the Fourier neural operator network, which includes the equivalent conductivity and the equivalent well index; the calculation formula is:

[0044] ;

[0045] ;

[0046] Wherein, is the equivalent conductivity between the adjacent post-processing grid and the post-processing grid output by the Fourier neural operator network; is the equivalent well index output by the Fourier neural operator network The equivalent well index of the post-processing grid; 、 They are the outputs of the Fourier neural operator network. grid, The pressure field of a grid; 、 They are the outputs of the Fourier neural operator network. The post-processing grid and The equivalent pressure field of a post-processing grid is calculated by the following formulas:

[0047] ;

[0048] .

[0049] Furthermore, in step 3, the physical knowledge-driven Fourier neural operator network training strategy is trained by a physical constraint loss function based on the finite volume method; the physical constraint loss function based on the finite volume method is:

[0050] ;

[0051] in, is the loss value; is the number of grid cells in the original grid model; The governing equation is The residual of a grid is calculated as:

[0052] .

[0053] Furthermore, the specific process of step 4 is as follows:

[0054] Step 4.1: Set various parameters, use the spectral domain method to sample a permeability field dataset that obeys a log-normal distribution, and divide it into a training set and a test set;

[0055] Step 4.2: Based on the physical constraint loss function constructed in step 3, the Fourier neural operator network is trained using the training set. The trained Fourier neural operator network is used to assist in accelerating the flow equivalent calculation, thereby obtaining a neural operator network agent-assisted flow equivalent model.

[0056] Step 4.3: Use the trained Fourier neural operator network to assist in flow equivalent calculation on the test set to evaluate the equivalent calculation effect assisted by the Fourier neural operator network.

[0057] Furthermore, the specific process of step 4.1 is: setting the size of the grid to ,in yes The number of grids in the direction, is The number of grids in the direction; set the log permeability field Subject to normal distribution; specifically, a spectral domain method based on Monte Carlo superposition is used to sample the permeability field data set subject to lognormal distribution; the sample number of the generated data set, the proportion of the training set, the proportion of the test set, the bottom hole pressure of the injection well and the bottom hole pressure of the production well are set in advance.

[0058] Further, the specific process of step 4.2 is:

[0059] The Adam optimizer is used to train the neural network parameters on the training set; the small batch gradient descent is adopted, and the loss function of the small batch gradient descent is :

[0060] ;

[0061] Wherein, is the batch size; is the sequence number of the batch; is the neural network parameter, and the update formula is:

[0062] ;

[0063] Wherein, , are the neural network parameters at the first , iteration; is the learning rate; is the momentum bias correction estimate at the first iteration; is the second moment bias correction estimate at the first iteration; is a small constant greater than zero; the gradient descent training is executed times in total.

[0064] Further, the specific process of step 4.3 is: input the original grid permeability of different permeability distribution in the test set into the trained Fourier neural operator network in the form of a multi-dimensional matrix, and output the original grid pressure value; based on the flow equivalent analytical formula of step 2.2, the equivalent conductivity and equivalent well index of the post-processing grid are calculated; based on the cumulative injection volume, cumulative liquid production, and total water cut production evaluation index, the equivalent calculation effect is evaluated.

[0065] Further, the step 5 specific process is: deploying the Fourier neural operator network trained with physical knowledge online, collecting the latest permeability of the original grid in real time, inputting the trained Fourier neural operator network, outputting the pressure value of the original grid in real time, and based on the equivalent analytical formula of flow, synchronously calculating the equivalent conductivity and equivalent well index of the post-processing grid after real-time output.

[0066] The beneficial technical effects brought by the present application: the present application can equivalent process the different original grid permeability into post-processing grid permeability through the trained model, the equivalent calculation result conforms to the physical knowledge constraint, has higher equivalent flow simulation solving precision. Under the same parameter setting and calculation resource, the present application can realize batch parallel equivalent processing for the permeability field, has the same prediction precision as the original model, and has significantly improved calculation efficiency compared with the existing flow equivalent method. BRIEF DESCRIPTION OF DRAWINGS

[0067] Figure 1 The flowchart of the present application based on the physical knowledge driven machine learning operator network numerical simulation acceleration method.

[0068] Figure 2 The physical knowledge driven agent training and agent assisted model equivalent processing algorithm structure diagram provided by the present application.

[0069] Figure 3 The definition of numerical simulation size, grid division and well distribution schematic diagram provided by the embodiment of the present application.

[0070] Figure 4 The logarithmic permeability field schematic diagram of different permeability distribution based on the spectral domain method sampling provided by the embodiment of the present application.

[0071] Figure 5 The loss reduction curve of the training process of the neural network agent model provided by the embodiment of the present application.

[0072] Figure 6 The single-phase flow pressure field of the original non-equivalent processing model provided by the embodiment of the present application.

[0073] Figure 7 The single-phase flow pressure field of the flow equivalent model based on numerical simulation provided by the embodiment of the present application.

[0074] Figure 8 The single-phase flow pressure field of the neural operator network agent assisted flow equivalent model provided by the embodiment of the present application.

[0075] Figure 9 The two-phase flow pressure field of the original non-equivalent processing model provided by the embodiment of the present application.

[0076] Figure 10The two-phase flow pressure field of the flow equivalent model based on numerical simulation provided by the embodiment of the present application.

[0077] Figure 11 The two-phase flow pressure field of the flow equivalent model assisted by the neural operator network agent provided by the embodiment of the present application.

[0078] Figure 12 The two-phase flow water saturation field of the original non-equivalent processing model provided by the embodiment of the present application.

[0079] Figure 13 The two-phase flow water saturation field of the flow equivalent model based on numerical simulation provided by the embodiment of the present application.

[0080] Figure 14 The two-phase flow water saturation field of the flow equivalent model assisted by the neural operator network agent provided by the embodiment of the present application.

[0081] Figure 15 The two-phase flow cumulative water injection amount change curve comparison chart before and after equivalent processing provided by the embodiment of the present application.

[0082] Figure 16 The two-phase flow cumulative liquid production amount change curve comparison chart before and after equivalent processing provided by the embodiment of the present application.

[0083] Figure 17 The two-phase flow water cut change curve comparison chart before and after equivalent processing provided by the embodiment of the present application. DETAILED DESCRIPTION

[0084] The present application will be further described in detail below in combination with the accompanying drawings and specific embodiments:

[0085] As shown in Figure 1 and Figure 2 , the present application proposes a numerical simulation acceleration method based on a physical knowledge driven machine learning operator network, wherein the physical knowledge driven of the present application refers to that in the flow equivalent mathematical model, the equivalent calculation of the model is driven by the flow equivalent based on the physical equation, which makes the equivalent result conform to the physical law. The machine learning operator network specifically adopts a Fourier neural operator network; the method of the present application specifically includes the following steps:

[0086] Step 1, constructing a flow equivalent mathematical model of a seepage parameter field; the flow equivalent mathematical model of the seepage parameter field is a single-phase flow flow equivalent mathematical model based on Darcy's law, including a control equation, a boundary condition, a discretization format of the control equation, and a flow equivalent equation, and the specific construction process is as follows:

[0087] Step 1.1, constructing a control equation of the flow equivalent mathematical model:

[0088] ;

[0089] ;

[0090] in, is the flow velocity; is the spatial position coordinate; is the permeability; is the fluid viscosity; It is a pressure field; represents source and sink terms; is the gradient operator;

[0091] Step 1.2: Define the boundary conditions of the flow equivalent mathematical model as the Newman boundary conditions with a flow velocity of 0:

[0092] ;

[0093] in, It is the Newman boundary location; is the unit normal vector at the boundary position;

[0094] Step 1.3: Construct the discretization format of the control equation:

[0095] ;

[0096] in, and are the serial numbers of two different grids; Indicates the adjacent cells of a grid; Indicates the The control volume of the grid; 、 Respectively grid, The pressure field of a grid; It is the adjacent The grid and The conductivity between the grids is calculated as follows:

[0097] ;

[0098] in, and They are The grid center and the The normal distance from the center of each grid to the contact surface; It is the adjacent The grid and The area of ​​the contact surface between the grids; It is the adjacent The grid and The harmonic mean permeability among the grids;

[0099] The source and sink terms are calculated using the Peaceman formula:

[0100] ;

[0101] in, is the bottom hole pressure; is the well index, calculated as:

[0102] ;

[0103] in, is the equivalent reservoir radius; is the wellbore radius; is the skin factor; is the thickness of the perforated stratum;

[0104] Step 1.4: Construct the flow equivalent equation between the post-processed grid and the original grid. The flow equivalent processing method can make the equivalent model and the non-equivalent model have good physical consistency. The flow equivalent equation includes the equivalent conductivity and the equivalent well index. The calculation formulas of the equivalent conductivity and the equivalent well index are:

[0105] ;

[0106] ;

[0107] in, and are the serial numbers of two different post-processing grids; It is the adjacent The post-processing grid and Equivalent conductivity between post-processing grids; It is the adjacent The post-processing grid and The contact surface between the post-processing meshes; It is The equivalent well index of the post-processing grid; For the Well index of each grid; and They are The post-processing grid and The equivalent pressure field of a post-processing grid is calculated as follows:

[0108] ;

[0109] ;

[0110] wherein, and are control volumes of the first post-processing grid and the second post-processing grid, respectively; are control volumes of the first post-processing grid and the second post-processing grid, respectively; are control volumes of the first post-processing grid and the second post-processing grid, respectively; are control volumes of the first post-processing grid and the second post-processing grid, respectively. are control volumes of the first post-processing grid and the second post-processing grid, respectively.

[0111] Step 2, design a flow equivalent simulation acceleration method based on a Fourier neural operator network; the Fourier neural operator network is used to construct a high generalization neural network proxy model for mapping a permeability field to a pressure field, so as to avoid repeated pressure field solving based on numerical simulation and accelerate equivalent calculation; the specific process is as follows:

[0112] Step 2.1, the mapping formula of the Fourier neural operator network is:

[0113] ;

[0114] wherein, is a permeability field input into an input layer of the Fourier neural operator network; , , are different pressure fields output by a hidden layer of the Fourier neural operator network; is a number of hidden layers of the Fourier neural operator network; is a pressure field output by an output layer of the Fourier neural operator network; is a connection symbol; dimension increasing transformation is performed by using a full connection layer; dimension decreasing transformation is performed by using a full connection layer; arbitrary hidden layer mapping is calculated by using the following formula:

[0115] ;

[0116] wherein, is a hidden layer sequence number of the Fourier neural operator network; is a pressure field input into the first hidden layer and a pressure field output by the first hidden layer; is a pressure field output by the first hidden layer; is a pressure field output by the first hidden layer; is a pressure field output by the first hidden layer; is a Fourier transform; is an inverse Fourier transform; is a low-pass filter in a frequency domain; is a linear transformation; is a nonlinear rectifier unit activation function;

[0117] ​Step 2.2, referring to the flow equivalent equation, define the flow equivalent analytical formula based on the Fourier neural operator network. The flow equivalent analytical formula also includes the equivalent conductivity and equivalent well index; the calculation formula is:

[0118] ;

[0119] ;

[0120] in, The adjacent first output of the Fourier neural operator network The post-processing grid and Equivalent conductivity between post-processing grids; The output of the Fourier neural operator network The equivalent well index of the post-processing grid; 、 They are the outputs of the Fourier neural operator network. grid, The pressure field of a grid; 、 The calculation is performed by a highly generalized neural network, which can avoid repeated numerical simulation calculations, and the process can be processed in batches; 、 They are the outputs of the Fourier neural operator network. The post-processing grid and The equivalent pressure field of the post-processing grid is calculated by the following formulas:

[0121] ;

[0122] ;

[0123] Step 3. Design a Fourier neural operator network training strategy driven by physical knowledge; the physical knowledge-driven Fourier neural operator network training strategy refers to training through a physical constraint loss function based on the finite volume method to achieve physical consistency embedding; this training strategy eliminates the need to generate "label data" through numerical simulation (this label data-based approach is data-driven, which is different from the method of the present invention), and ensures that the output of the Fourier neural operator network conforms to physical laws.

[0124] The physical constraint loss function based on the finite volume method is:

[0125] ;

[0126] in, is the loss value; is the number of grid cells in the original grid model; The governing equation is The residual of a grid is calculated as:

[0127] ;

[0128] pass Physically driven training can make the output of the neural network have good physical consistency;

[0129] Step 4: Sampling and constructing a permeability field data set, training the Fourier neural operator network, and performing flow equivalent calculation and effect testing assisted by the Fourier neural operator network based on the flow equivalent mathematical model of step 1. The specific process is as follows:

[0130] Step 4.1: Set various parameters and use the spectral domain method to sample a permeability field dataset that obeys a log-normal distribution and divide it into a training set and a test set. The specific process is as follows:

[0131] Set the grid size to ,in yes The number of grids in the direction, yes The number of grids in the direction;

[0132] Set the log permeability field It obeys the normal distribution with a mean of , the variance is ;

[0133] Specifically, a spectral domain method based on Monte Carlo wave synthesis is used to sample a permeability field dataset that follows a log-normal distribution. Monte Carlo wave synthesis obtains the energy spectral density by Fourier transforming the variogram. A radial spectral cumulative distribution function is established in the spectral domain, and the frequency modulus is randomly sampled and inversely transformed. Random directions and phases are sampled. Finally, wave superposition, normalization, and exponential transformation are performed to obtain the permeability field. This method can be implemented using readily available tools. The variogram model is:

[0134] ;

[0135] in, is the range value; is the gold nugget value; is the spatial lag distance; is the variation function value; is an exponential function with base e;

[0136] Set the number of samples in the generated dataset to indivual;

[0137] Set the proportion of the training set , the proportion of the test set ;

[0138] Set the bottom hole pressure of the injection well as , and the bottom hole pressure of the production well as ;

[0139] Step 4.2, based on the physical constraint loss function constructed in step 3, the Fourier neural operator network is trained using the training set, and the trained Fourier neural operator network is used to assist the equivalent flow calculation, so as to obtain the neural operator network proxy assisted flow equivalent model;

[0140] The training parameters are set as follows:

[0141] The Adam optimizer is used to train the neural network parameters on the training set; the learning rate is set as ; the small batch gradient descent is adopted, and the loss function of the small batch gradient descent is :

[0142] ;

[0143] Wherein, is the batch size; is the batch number; is the neural network parameter, and its update formula is:

[0144] ;

[0145] Wherein, , are the neural network parameters at the , th iteration; is the learning rate; is the momentum bias correction estimate at the th iteration; is the second moment bias correction estimate at the th iteration; is a small constant greater than zero; a total of gradient descent training is performed;

[0146] Step 4.3, flow equivalent calculation assisted by the trained Fourier neural operator network on the test set to evaluate the equivalent calculation effect under the assistance of the Fourier neural operator network; the specific process is: inputting the original grid permeability of different permeability distributions in the test set in the form of a multi-dimensional matrix into the trained Fourier neural operator network, and outputting the original grid pressure value; calculating the equivalent conductivity and well index of the post-processing grid based on the flow equivalent analytical formula of step 2.2; and evaluating the equivalent calculation effect based on the cumulative injection volume, cumulative liquid production, and total water cut production evaluation indicators. The specific process is:

[0147] Test the loss of simulation accuracy of single-phase flow before and after the equivalent processing of the Fourier neural operator network, and test the loss of numerical simulation accuracy of slightly compressible oil-water two-phase flow problems before and after the equivalent processing of the Fourier neural operator network; select the original non-equivalent processing model, the flow equivalent model based on numerical simulation, and the neural operator network proxy assisted flow equivalent model established by the present application for testing and comparison, and compare the simulation results of the three models for cumulative injection volume, cumulative liquid production, and total water cut in three production evaluation indicators.

[0148] Among them, the original non-equivalent processing model refers to the original model without any processing. This model is usually constructed with very fine grids, and has high calculation accuracy, but when directly used for numerical simulation, the simulation speed is very slow, and it is the model to be compared in the method of the present application.

[0149] The flow equivalent model based on numerical simulation (processed by the existing system method) refers to a model obtained by solving the pressure based on numerical simulation and then performing flow equivalent calculation. This model reduces the number of grids and is faster in simulation when used for numerical simulation. It has very close accuracy to the non-equivalent model. However, the equivalent process needs to repeatedly perform numerical simulation of the “original non-equivalent processing model”, and the equivalent calculation speed is very slow.

[0150] The neural operator network proxy assisted flow equivalent model has the same accuracy as the “numerical simulation flow equivalent model”, and does not need to repeatedly perform numerical simulation calculation, and the equivalent calculation speed is very slow. And also reduces the number of grids, and is very fast in simulation when used for numerical simulation.

[0151] Cumulative injection volume The calculation formula is:

[0152] ;

[0153] Among them, is time; is the time index; is the initial time; is the time step; is all injection wells; The injection rate of the injection well in the first grid is calculated according to the following formula:

[0154] The calculation formula of the cumulative liquid production is as follows:

[0155]

[0156] The total production well is represented by W. The production well liquid production rate in the first grid is represented by q. The calculation formula of the total water cut is as follows:

[0157]

[0158]

[0159] The production well water production rate in the first grid is represented by W. Step 5, deploying the trained Fourier neural operator network online to assist in real-time flow equivalent calculation. The specific process is as follows: deploying the trained Fourier neural operator network driven by physical knowledge online, collecting the latest permeability of the original grid in real time, inputting the trained Fourier neural operator network, outputting the pressure value of the original grid in real time, and based on the flow equivalent analytical formula, synchronously calculating the equivalent conductivity and equivalent well index of the post-processing grid after real-time output.

[0160] The physical-driven agent-assisted seepage parameter field flow equivalent method provided in the embodiment can be independent of the training, prediction and equivalent calculation of the whole process of the numerical simulator; the finite volume method is used to discretize the control equation to strictly maintain the flux continuity and source-sink balance relationship, avoid the instability and high cost of automatic differentiation in high-order spatial derivative calculation; the Fourier neural operator network is combined with linear transformation and frequency domain low-pass filtering in each hidden layer to effectively capture general patterns and improve the model's ability to capture multi-scale heterogeneous fields; the neural network training strictly follows the physical constraints, and the equivalent conductivity and well index of the post-processing grid are calculated in the post-processing according to the flow equivalent analytical formula, thereby ensuring the physical consistency; under the same computing resources, the present application can replace the numerical simulation part in the existing equivalent calculation process, and can also realize parallel processing for batch permeability fields, has significantly improved computing efficiency compared with the existing flow equivalent method, and has a prediction accuracy comparable to the original model.

[0161] The physical-driven agent-assisted seepage parameter field flow equivalent method provided in the embodiment can be independent of the training, prediction and equivalent calculation of the whole process of the numerical simulator; the finite volume method is used to discretize the control equation to strictly maintain the flux continuity and source-sink balance relationship, avoid the instability and high cost of automatic differentiation in high-order spatial derivative calculation; the Fourier neural operator network is combined with linear transformation and frequency domain low-pass filtering in each hidden layer to effectively capture general patterns and improve the model's ability to capture multi-scale heterogeneous fields; the neural network training strictly follows the physical constraints, and the equivalent conductivity and well index of the post-processing grid are calculated in the post-processing according to the flow equivalent analytical formula, thereby ensuring the physical consistency; under the same computing resources, the present application can replace the numerical simulation part in the existing equivalent calculation process, and can also realize parallel processing for batch permeability fields, has significantly improved computing efficiency compared with the existing flow equivalent method, and has a prediction accuracy comparable to the original model.

[0162] ​​​​​​​​To prove the feasibility of the present application, a specific embodiment of single-phase flow in heterogeneous porous media based on Darcy equation is given, and the specific process of proxy model construction, auxiliary equivalent processing, equivalent calculation effect evaluation of physical driving is as follows:

[0163] Firstly, a numerical simulation model is constructed; the size, grid division and well distribution are as shown in Figure 3 The size of the model is 675m*675m, the grid division is 45*45 Cartesian grid, and the size parameter of each grid is 15m*15m; all the boundaries are closed Neumann boundary; there are 7 injection wells I1, I2, I3, I4, I5, I6 and I7, and 18 production wells P1, P2, P3, P4, P5, P6, P7, P8, P9, P10, P11, P12, P13, P14, P15, P16, P17 and P18, and the overall well pattern is arranged by five-point method and the injection-production mode is fixed flow pressure.

[0164] Then, the permeability distribution obtained by sampling by spectral domain method is as shown in Figure 4 The sampled log permeability field is subject to normal distribution with mean of 0 and variance of 0.3; the nugget value of the variogram model is 0, the range value is 75m, the generated permeability field range is 0.0028mD to 153.34mD, and the permeability difference range is 118.39 to 14172.09; a total of 200 different permeability field samples are sampled, of which 100 samples are taken as the training set, and the remaining 100 samples are taken as the test set.

[0165] Then, the Fourier neural operator network is set, including one input layer, three hidden layers and one output layer, and the mapping sequence is input layer, first hidden layer, second hidden layer, third hidden layer and output layer; the neural network is trained on the training set, the learning rate of Adam optimizer is set to 0.001, the batch size of small batch gradient descent is set to 2, and a total of 10000 generations are trained, and the log loss value curve in the training process is as shown in Figure 5 .

[0166] Finally, global flow simulation is performed on the original unequivalent processing model; global flow equivalent calculation is performed with the aid of numerical simulation, and the flow equivalent model based on numerical simulation is obtained as the control group; the trained Fourier neural operator network model is batch inferences on the test set, the batch pressure field of the original grid is obtained, the post-processing calculation is performed according to the flow equivalent analytical formula, and the neural operator network proxy assisted flow equivalent model is obtained.

[0167] In this embodiment, single-phase flow simulation is performed on the original non-equivalent processing model, the flow equivalent model based on numerical simulation, and the neural operator network proxy assisted flow equivalent model. The comparison of the simulated single-phase flow pressure fields before and after equivalent processing is shown in Figure 6 、 Figure 7 、 Figure 8 From Figure 6 、 Figure 7 、 Figure 8 It can be seen that the pressure field distribution of single-phase flow before and after equivalent processing is similar, and has good single-phase flow pressure prediction consistency.

[0168] Two-phase flow simulation is performed on the original non-equivalent processing model, the flow equivalent model based on numerical simulation, and the neural operator network proxy assisted flow equivalent model. The comparison of the simulated two-phase flow pressure fields before and after equivalent processing is shown in Figure 9 、 Figure 10 、 Figure 11 The comparison of the simulated two-phase flow saturation fields before and after equivalent processing is shown in Figure 12 、 Figure 13 、 Figure 14 The comparison of the simulated two-phase flow cumulative water injection curves before and after equivalent processing is shown in Figure 15 The comparison of the simulated two-phase flow cumulative liquid production curves before and after equivalent processing is shown in Figure 16 The comparison of the simulated two-phase flow water cut curves before and after equivalent processing is shown in Figure 17 From Figure 12 、 Figure 13 、 Figure 14 It can be seen that the saturation field distribution of two-phase flow before and after equivalent processing is similar, and has good two-phase flow saturation prediction consistency, proving that the neural operator network proxy assisted flow equivalent model has comparable prediction accuracy with the original model. From Figure 15 It can be seen that the cumulative water injection curves of two-phase flow before and after equivalent processing are close, and have good injection flow prediction consistency, proving that the neural operator network proxy assisted flow equivalent model has comparable prediction accuracy with the original model; among them, the results of the neural operator network proxy assisted flow equivalent model are more consistent with the original non-equivalent processing model, proving that the neural operator network proxy assisted flow equivalent model has higher equivalent flow simulation solving accuracy. From Figure 16 It can be seen that the cumulative liquid production curves of two-phase flow before and after equivalent processing are close, and have good production flow prediction consistency, proving that the neural operator network proxy assisted flow equivalent model has comparable prediction accuracy with the original model; among them, the results of the neural operator network proxy assisted flow equivalent model are more consistent with the original non-equivalent processing model, proving that the neural operator network proxy assisted flow equivalent model has higher equivalent flow simulation solving accuracy. From Figure 17It can be seen that the water content curves of two-phase flow before and after equivalent treatment are close, and have good consistency of two-phase flow prediction, which proves that the neural operator network agent assisted flow equivalent model has equivalent prediction accuracy as the original model; among them, the neural operator network agent assisted flow equivalent model result is more consistent with the original non-equivalent treatment model, which proves that the neural operator network agent assisted flow equivalent model has higher equivalent flow simulation solving accuracy.

[0169] The time consumption comparison of the flow equivalent calculation based on numerical simulation and the neural operator network agent assisted flow equivalent calculation is shown in Table 1.

[0170] Table 1 Time consumption comparison result table

[0171] .

[0172] As can be seen from Table 1, the neural operator network agent assisted flow equivalent has significantly less calculation time consumption under 500 times and 1000 times equivalent calculation, which proves that the neural operator network agent assisted flow equivalent method has significantly improved calculation efficiency compared with the existing flow equivalent method.

[0173] Of course, the above description is not a limitation on the present application, and the present application is not limited to the above examples, and the changes, modifications, additions or replacements made by the person skilled in the art within the essential scope of the present application also belong to the protection scope of the present application.

Claims

1. A numerical simulation acceleration method based on a physics knowledge driven machine learning operator network, characterized by, The machine learning operator network specifically adopts a Fourier neural operator network; the method comprises the following steps: Step 1, constructing a flow equivalent mathematical model of the seepage parameter field; Step 2, designing a flow equivalent simulation acceleration method based on the Fourier neural operator network; Step 3, designing a Fourier neural operator network training strategy driven by physical knowledge; Step 4, sampling and constructing a permeability field dataset, training the Fourier neural operator network, and performing flow equivalent calculation and effect test based on the flow equivalent mathematical model of step 1; Step 5, deploying the trained Fourier neural operator network online to assist real-time flow equivalent calculation; In step 1, the flow equivalent mathematical model includes control equations, boundary conditions, discretization format of control equations, and flow equivalent equations; the specific construction process is as follows: Step 1.1, constructing control equations: ； ； wherein, is the flow velocity; is the spatial position coordinate; is the permeability; is the fluid viscosity; is the pressure field; denotes the source-sink term; is the gradient operator; Step 1.2, defining the boundary condition as a Neumann boundary condition with a flow rate of 0: ； wherein is a Neumann boundary position; is a unit normal vector of the boundary position; Step 1.3, constructing the discretization format of the control equation: ； where, and are the serial numbers of two different grids, respectively; represents the adjacent cell of the grid; represents the control volume of the grid; , are the pressure fields of the grid and the grid, respectively; is the conductivity between the adjacent grid and the grid, and the calculation formula is: ； wherein, and are the normal distance from the center of the grid to the center of the grid to the contact surface; is the area of the contact surface between the adjacent grid and the grid; is the harmonic average permeability between the adjacent grid and the grid. The source and sink term is calculated using the Pismann formula: ； wherein is the bottom hole pressure; is the well index, calculated as: ； wherein, is the equivalent reservoir radius; is the wellbore radius; is the skin factor; is the pay zone thickness; Step 1.4, constructing the flow equivalent equation of the post-processing grid and the original grid; the flow equivalent equation includes the equivalent conductivity and the equivalent well index, and the calculation formulas are as follows: ； ； in, and are the serial numbers of two different post-processing grids; It is the adjacent The post-processing grid and Equivalent conductivity between post-processing grids; It is the adjacent The post-processing grid and The contact surface between the post-processing meshes; It is The equivalent well index of the post-processing grid; For the Well index of each grid; For the A control volume for the post-processing grid; and They are The post-processing grid and The equivalent pressure field of a post-processing grid is calculated as follows: ； ； in, No. A control volume for the post-processing grid; It is The control volume of the grid; The specific process of step 2 is as follows: Step 2.1, constructing the mapping formula of the Fourier neural operator network: ； wherein, is the permeability field input to the Fourier Neural Operator network input layer; , , are different pressure fields output by the Fourier Neural Operator network hidden layers; is the number of hidden layers of the Fourier Neural Operator network; is the pressure field output by the Fourier Neural Operator network output layer; is a connection symbol; a fully connected layer is used between the input layer and the hidden layer for dimensionality increasing transformation; a fully connected layer is used between the hidden layer and the output layer for dimensionality decreasing transformation; the hidden layer and the hidden layer are calculated using the following formula: ； wherein, is the hidden layer number of the Fourier Neural Operator network; is the pressure field input to the th hidden layer; is the pressure field output from the th hidden layer; is the pressure field output from the is the Fourier Transform; is the Inverse Fourier Transform; is the low-pass filter in the frequency domain; is the linear transformation; is the nonlinear rectified unit activation function; Step 2.2, referring to the flow equivalent equation, defining the flow equivalent analytical formula based on the Fourier neural operator network, which includes the equivalent conductivity and the equivalent well index; the calculation formula is as follows: ； ； in, The adjacent first output of the Fourier neural operator network The post-processing grid and Equivalent conductivity between post-processing grids; The output of the Fourier neural operator network The equivalent well index of the post-processing grid; 、 They are the outputs of the Fourier neural operator network. grid, The pressure field of a grid; 、 They are the outputs of the Fourier neural operator network. The post-processing grid and The equivalent pressure field of a post-processing grid is calculated by the following formulas: ； ； In step 3, the Fourier neural operator network training strategy driven by physical knowledge is trained by using a physical constraint loss function based on the finite volume method; the physical constraint loss function based on the finite volume method is as follows: ； wherein, is a loss value; is the number of grid cells of the original mesh model; is the residual of the control equation at the grid, which is calculated as: 。 2. The method of claim 1, wherein, The specific process of step 4 is as follows: Step 4.1, set the parameters, sample the permeability field dataset conforming to the lognormal distribution using the spectral domain method, and divide it into a training set and a test set; Step 4.2, based on the physical constraint loss function constructed in step 3, train the Fourier neural operator network using the training set, and use the trained Fourier neural operator network to assist in flow equivalent calculation, thereby obtaining a neural operator network proxy assisted flow equivalent model; Step 4.3, use the trained Fourier neural operator network to assist in flow equivalent calculation on the test set, and evaluate the equivalent calculation effect assisted by the Fourier neural operator network.

3. The method of claim 2, wherein, The specific process of step 4.1 is: setting the size of the grid as wherein is the number of grids in the direction of is the number of grids in the direction of Setting a log permeability field The permeability field data set obeying the lognormal distribution is sampled by using a spectral domain method based on Monte Carlo superposition; the sample number of the generated data set, the proportion of the training set, the proportion of the test set, the bottom hole pressure of the injection well, and the bottom hole pressure of the production well are set in advance.

4. The method of claim 2, wherein, The specific process of step 4.2 is as follows: training the neural network parameters using an Adam optimizer on the training set; employing mini-batch gradient descent, the loss function for the mini-batch gradient descent being is: ； wherein, is the batch size; is the sequence number of the batch; is the neural network parameter, and the update formula is: ； in, 、 Respectively 、 Neural network parameters at iterations; is the learning rate; No. The bias-corrected estimate of momentum at iteration ; It is The bias-corrected estimate of the second moment at iteration ; is a small constant greater than zero; the total execution Gradient descent training for 1 epoch.

5. The method of claim 4, wherein, The specific process of step 4.3 is as follows: input the original grid permeability of different permeability distributions in the test set into the trained Fourier neural operator network in the form of a multi-dimensional matrix, and output the original grid pressure value; calculate the equivalent conductivity and equivalent well index of the post-processing grid based on the flow equivalent analytical formula of step 2.2; evaluate the equivalent calculation effect based on the cumulative injection volume, cumulative liquid production, and total water cut production evaluation indicators.

6. The method of claim 5, wherein, The specific process of the step 5 is as follows: deploying a Fourier neural operator network trained by physical knowledge online, collecting the latest permeability of the original grid in real time, inputting the Fourier neural operator network trained, outputting the pressure value of the original grid in real time, and synchronously calculating the equivalent conductivity and equivalent well index of the post-processing grid based on the equivalent analytical formula of the flow.

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