A robust method for quantifying model parameter uncertainty

Abstract

This invention provides a robust method for quantifying model parameter uncertainty, comprising: constructing several subsets of prior realizations; calculating the expected range of the prior realization subsets in the data space; evaluating all expectations within the expected range to obtain several posterior expectations and their corresponding subsets; obtaining the upper and lower bounds of the posterior expectations from the several posterior expectations; and quantifying the uncertainty of geological parameters based on the upper and lower bounds of the posterior expectations. In the expected range calculation stage, this method calculates the expected range in the data space only based on the subsets of prior realizations, eliminating the need to repeatedly run forward simulations as parameters are updated, thus significantly reducing computational load. Furthermore, by obtaining multiple posterior expectations and their related subsets, the data value is further mined, and the uncertainty information of geological parameters is captured more comprehensively and accurately.

Description

Technical Field

[0001] This invention relates to the field of data processing technology, specifically providing a robust method for quantifying the uncertainty of model parameters. Background Technology

[0002] Inferring Darcy flow uncertainty parameters in porous media is crucial for groundwater and reservoir engineering. Reservoir geological parameters, such as permeability and porosity, are typically highly heterogeneous, but these parameters can often only be measured at sparse well locations using core samples and logging, leading to very high uncertainty in geological models. Therefore, quantifying the uncertainty of subsurface flow using history fitting is significant for evaluating and predicting reservoir dynamics. Traditional methods generally obtain maximum a posteriori or maximum likelihood estimates (MLE) based on gradient, heuristic, or ensemble methods. However, for high-dimensional problems, these methods are computationally expensive because iterative updates of physical properties require extensive forward simulations. Recently, an efficient data spatial inversion (DSI) method has been proposed. History fitting based on DSI does not require iterative forward simulation. The purpose of DSI is not to invert physical parameters, but to directly calculate the posterior dynamic response using prior dynamic responses and likelihood functions. While DSI can achieve history fitting, it cannot obtain posterior information on physical properties. Therefore, a new robust model parameter uncertainty quantification scheme is needed to address this issue. Summary of the Invention

[0003] To overcome the above-mentioned shortcomings, the present invention is proposed to provide a solution, or at least a partial solution, to the problem of the inability to obtain a posteriori information on physical property parameters.

[0004] This invention provides a robust method for quantifying the uncertainty of model parameters, comprising: constructing several subsets of prior realizations of a prior realization set; calculating the expected range of the prior realization subsets in the data space; evaluating all expectations in the expected range to obtain several posterior expectations and their corresponding subsets; obtaining the upper and lower bounds of the posterior expectations from the several posterior expectations; and quantifying the uncertainty of geological parameters based on the upper and lower bounds of the posterior expectations.

[0005] In one technical solution of the robust model parameter uncertainty quantification method described above, the process before calculating the expected range of the prior realization subset in the data space includes: performing forward simulation on each prior realization in the prior realization set to obtain the dynamic response of each realization; calculating the expected dynamic response of each realization on the corresponding subset; obtaining the upper and lower bounds of all expectations, and obtaining the expected range based on the upper and lower bounds.

[0006] In one technical solution of the robust model parameter uncertainty quantification method mentioned above, before obtaining the expected range based on the upper and lower limits, it also includes: determining whether the actual measurement data is within the range formed by the upper and lower limits; if the actual measurement data is not within the range formed by the upper and lower limits, then adding prior knowledge until the actual measurement data is within the range formed by the upper and lower limits.

[0007] In one technical solution of the robust model parameter uncertainty quantification method mentioned above, the process of evaluating all expectations within the expected range to obtain several posterior expectations and corresponding subsets includes: evaluating the probability of each subset expectation based on the likelihood function and actual observation data to obtain the evaluation result; and obtaining the subset expectation with maximum likelihood and the corresponding subset from the evaluation result as the obtained posterior expectations and their related subsets.

[0008] In one technical solution of the aforementioned robust model parameter uncertainty quantification method, the process of quantifying the uncertainty of geological parameters based on the upper and lower limits of the posterior expectation includes: taking the difference between the upper and lower limits of the posterior expectation to obtain the difference value; the larger the difference value, the higher the uncertainty of the geological parameter distribution; the smaller the difference value, the lower the uncertainty of the geological parameter distribution.

[0009] In one technical solution of the aforementioned robust model parameter uncertainty quantification method, the process of obtaining the evaluation result based on the likelihood function, the actual observation data, and the probability of each subset expectation includes: for each subset expectation, substituting the actual observation data and the subset expectation into the likelihood function for calculation, and the calculated probability value is the probability evaluation result of each subset expectation.

[0010] In one technical solution of the robust model parameter uncertainty quantification method described above, the process of obtaining the subset expectations with maximum likelihood and the corresponding subsets from the evaluation results as the obtained posterior expectations and their related subsets includes: sorting each evaluation result; selecting the top n subset expectations with maximum likelihood and their corresponding subsets after sorting, according to actual needs or a pre-set number, and the selected subset expectations are regarded as posterior expectations, and their corresponding subsets are the related subsets.

[0011] In one technical solution of the aforementioned robust model parameter uncertainty quantification method, the method further includes obtaining a posterior subset; selecting representative non-repeating implementations from the posterior subset for simulation to obtain the linear expectation of the posterior subset; obtaining the nonlinear expectation of the posterior subset based on the linear expectation; wherein the upper and lower limits of the linear expectation of the posterior subset are nonlinear expectations; and predicting reservoir geological parameters based on the nonlinear expectations.

[0012] The above-described technical solutions of the present invention have at least one or more of the following beneficial effects:

[0013] In implementing the technical solution of this invention, a robust method for quantifying model parameter uncertainty is provided, comprising: constructing several prior realization subsets of a prior realization set; calculating the expected range of the prior realization subsets in the data space; evaluating all expectations within the expected range to obtain several posterior expectations and their corresponding subsets; obtaining the upper and lower bounds of the posterior expectations from the several posterior expectations; and quantifying the uncertainty of geological parameters based on the upper and lower bounds of the posterior expectations. Compared with the prior art, the beneficial effects of the robust method for quantifying model parameter uncertainty provided by this invention are as follows: In the stage of calculating the expected range, this method only calculates the expected range in the data space based on the prior subsets, without needing to repeatedly run forward simulations as parameters are updated, thus greatly reducing the computational load. Furthermore, by obtaining multiple posterior expectations and their related subsets, the value of the data is further mined, and the uncertainty information of geological parameters is captured more comprehensively and accurately.

[0014] Furthermore, firstly, this method only requires one forward simulation run on the prior realizations, followed by the calculation of the expected range in the data space based on a subset of the prior realizations. This eliminates the need for repeated forward runs, effectively reducing computational complexity and resource consumption. Secondly, by employing maximum likelihood estimation, it cleverly integrates prior information with actual observation data to accurately determine the posterior expectation in the data space and simultaneously acquires the realization subset in the parameter space, providing a rich and reliable data foundation for in-depth analysis. Thirdly, the generated posterior subset is the result of a series of rigorous analyses and selections, representing parameter combinations and corresponding outcomes that are more closely aligned with actual conditions. When using these posterior subsets to predict the future behavior of the system, it exhibits unique advantages over other methods. It does not require large-scale simulations of numerous scenarios; simulations are only performed on a small number of non-repeating realizations (i.e., those highly representative different cases) within the posterior subset, yielding relatively effective prediction results. This not only strongly ensures the accuracy of the predictions but also significantly reduces the computational resources and time costs required in the prediction phase, greatly improving overall efficiency and effectiveness. Attached Figure Description

[0015] The disclosure of this invention will become more readily understood with reference to the accompanying drawings. It will be readily understood by those skilled in the art that these drawings are for illustrative purposes only and are not intended to limit the scope of protection of this invention. Furthermore, similar numbers in the drawings are used to denote similar components, wherein:

[0016] Figure 1 This is a schematic flowchart of the main steps of a robust model parameter uncertainty quantification method according to an embodiment of the present invention;

[0017] Figure 2(a) is a schematic diagram of the actual permeability field used to generate observation data according to an embodiment of the present invention; (b) is a schematic diagram of the pressure field at the end of the simulation using the actual permeability field.

[0018] Figure 3 This is a schematic diagram of simulated traffic flow using prior and posterior methods according to an embodiment of the present invention;

[0019] Figure 4 This is a comparison diagram of the negative log-likelihood of the posterior implementation of ESMDA and a subset of NEI according to an embodiment of the present invention. Detailed Implementation

[0020] Some embodiments of the present invention will now be described with reference to the accompanying drawings. Those skilled in the art should understand that these embodiments are merely illustrative of the technical principles of the present invention and are not intended to limit the scope of protection of the present invention. Example 1

[0021] See appendix Figure 1 , Figure 1 This is a schematic flowchart illustrating the main steps of a robust model parameter uncertainty quantification method according to an embodiment of the present invention. Figure 1 As shown, a robust model parameter uncertainty quantification method in this embodiment of the invention mainly includes the following steps S1-S5.

[0022] Step S1: Construct several subsets of the prior realization set. In this embodiment, the prior realization set can be viewed as a set of possible scenarios determined based on existing knowledge, experience, or preliminary assumptions before analysis. These scenarios are usually represented by different combinations of parameter values. For example, in a geological model, the prior realization set includes various possible geological condition settings composed of different values ​​of parameters such as permeability and porosity. The prior realization set is divided into several subsets. There are various criteria for this division, such as classification according to parameter value range, geological region characteristics, or similar physical properties. For example, prior realizations with permeability within a certain range and porosity within a corresponding specific range can be grouped into one subset. In this way, multiple different subsets can be constructed, facilitating subsequent processing and analysis of each subset separately.

[0023] Step S2: Calculate the expected range of the prior realization subset in the data space. In this embodiment, based on the various specific realizations contained in the subset (simulation results corresponding to different parameter values, etc.), the corresponding statistical calculation methods are used to obtain the expected range of each subset in the data space.

[0024] Step S3: Evaluate all expectations within the expected range to obtain several posterior expectations and corresponding subsets. In this embodiment, based on the already obtained expected ranges for each subset, all expectations contained within these ranges are further evaluated. After evaluation and filtering, several posterior expectations are determined. These posterior expectations are expected values ​​that are more likely to represent the true situation after combining prior information and current practical considerations. Simultaneously, the subsets that generate these posterior expectations (that is, a part of the previously defined prior realization subsets) are the relevant subsets.

[0025] Step S4: Obtain the upper and lower bounds of the posterior expectation from several posterior expectations. In this embodiment, among the obtained multiple posterior expectations, the maximum value is identified as the upper bound of the posterior expectation, and the minimum value is identified as the lower bound of the posterior expectation. These two boundary values ​​define a range within which the posterior expectation lies.

[0026] Step S5: Quantify the uncertainty of geological parameters based on the upper and lower bounds of the posterior expectation. In this embodiment, the degree of uncertainty of geological parameters is measured based on the range determined by the upper and lower bounds. The larger the range, the higher the uncertainty of the parameter, that is, its true value may vary within a wider range; conversely, the smaller the range, the lower the uncertainty of the parameter, and its true value is closer to a relatively clear range.

[0027] In one embodiment, before step S2, calculating the expected range of the prior implementation subset in the data space, the method includes: performing forward simulation on each prior implementation in the prior implementation set to obtain the dynamic response of each implementation; calculating the expected dynamic response of each implementation on the corresponding subset; obtaining the upper and lower bounds of all expectations, and obtaining the expected range based on the upper and lower bounds.

[0028] For example, given a combination of geological parameters such as a certain permeability and a certain porosity, the Darcy flow model can be used to simulate the dynamic changes in fluid velocity and pressure under these parameter conditions. The simulated velocity and pressure data represent the dynamic response corresponding to the given realization. By performing this operation on each element in the prior realization set, the dynamic responses corresponding to all different parameter settings can be obtained. Calculating the expected value of these flow rates (e.g., by simply averaging these flow rates) yields a value representing the average performance of the subset in terms of fluid flow. All the expected values ​​calculated for each subset are then grouped together, and the maximum value is identified as the upper limit, and the minimum value as the lower limit. An interval is defined by the upper and lower limits; this interval is the expected range.

[0029] In one embodiment, before obtaining the desired range based on the upper and lower limits, the method further includes: determining whether the actual measurement data is within the range formed by the upper and lower limits; if the actual measurement data is not within the range formed by the upper and lower limits, then adding prior knowledge until the actual measurement data is within the range formed by the upper and lower limits.

[0030] In this embodiment, increasing prior knowledge can be achieved by expanding the parameter range. For example, if the study involves Darcy flow in porous media, key parameters such as permeability and porosity can have their possible value ranges expanded when actual measurement data exceeds the expected range. Besides permeability and porosity, parameters such as fluid viscosity and density may also be involved. For these parameters, their reasonable value ranges and distribution assumptions should also be re-examined. More influencing factors can also be incorporated: detailed geological exploration data can be used to add factors such as the location, scale, and permeability variations of faults, the shape and orientation of folds, and their impact on porosity to the prior knowledge. Groundwater flow may also be affected by external factors such as tides and surface water recharge. When increasing prior knowledge, these external driving factors and their changing patterns should be considered. Furthermore, physical models can be improved. For example, in some cases, non-Darcy flow effects may need to be considered; when the flow velocity is high or the pore size is small, Darcy's law may no longer be fully applicable. At this point, non-Darcy flow-related models (such as the Forchheimer equation) can be incorporated into the consideration. Based on the actual situation, it can be determined under what conditions a more complex model is needed to describe fluid flow, and this model improvement can be considered as new a priori knowledge. If the system under study involves multiphase flow (such as oil-gas-water three-phase flow) or involves chemical reactions (such as the dissolution and precipitation of solutes in groundwater), these complex processes need to be considered in the a priori knowledge. For example, in oil reservoir development, the dynamic impact of the interaction between the oil, gas, and water phases on permeability and porosity, and how mineral precipitation or dissolution caused by chemical reactions alters the pore structure of porous media, thus affecting Darcy flow parameters, should be considered. Theoretical and empirical knowledge of these complex physicochemical processes should be integrated into the a priori realization set.

[0031] In one embodiment, step S3, evaluating all expectations within the expected range to obtain several posterior expectations and their corresponding subsets, includes:

[0032] Step S31: Based on the likelihood function and actual observed data, evaluate the probability of each subset expectation to obtain the evaluation result. Step S32: From the evaluation result, obtain the subset expectation with maximum likelihood and the corresponding subset as the obtained posterior expectation and its related subset.

[0033] In this embodiment, actual observation data is key information reflecting the true state of the system under study, such as groundwater level data at different times in a certain area and water flow data at different locations obtained through field measurements. Substituting these real data and the expectation of each subset into the likelihood function, a specific probability value can be obtained through calculation. This probability value is a quantitative representation of the likelihood of the expected subset. By performing this operation on all subset expectations within the expected range, a set of evaluation results can be obtained, with each result corresponding to the degree of fit between a subset expectation and the actual observation data (i.e., the probability of the actual observation data occurring).

[0034] Alternatively, an alternative approach is to obtain the evaluation results based on the Bayesian information criterion. Specifically, for each subset expectation, the likelihood function is first calculated, then the number of model parameters and the number of samples are determined, and the BIC value is calculated by substituting these values ​​into the formula. The smaller the BIC value, the more likely the model corresponding to the subset expectation is to be a better model; that is, the higher the probability of this subset expectation.

[0035] Alternatively, an alternative approach is to obtain the evaluation results based on the Akaike Information Criterion. Specifically, similar to the BIC, for each subset's expected model specification, the likelihood function is first calculated to determine the number of parameters, and then the AIC value is calculated using the formula. The smaller the AIC value, the better the model corresponding to that subset's expected model, and the higher its probability.

[0036] Alternatively, an alternative approach is to use cross-validation to obtain the evaluation results. Specifically, for each subset expectation, the actual observation data is divided into a training set and a validation set. A model is built based on the subset expectations using the training set data (e.g., if studying the impact of geological parameters on fluid flow, a fluid flow model is built based on the geological parameters in the subset expectations). Then, the model's prediction accuracy is evaluated using the validation set data. Metrics such as mean squared error (MSE) and mean absolute error (MAE) can be used to measure the model's prediction error on the validation set. The smaller the prediction error, the better the model performance corresponding to that subset expectation, and the higher its probability.

[0037] Of course, the methods for obtaining evaluation results are not limited to the above-mentioned ones; any method that can obtain evaluation results is acceptable.

[0038] In one embodiment, step S5, the process of quantifying the uncertainty of geological parameters based on the upper and lower bounds of posterior expectation, includes:

[0039] Step S51: Subtract the upper and lower limits of the posterior expectation to obtain the difference value;

[0040] Step S52: The larger the difference, the higher the uncertainty of the geological parameter distribution; the smaller the difference, the lower the uncertainty of the geological parameter distribution.

[0041] In one embodiment, step S31, which evaluates the probability of each subset expectation based on the likelihood function and actual observation data, and obtains the evaluation result, includes: for each subset expectation, substituting the actual observation data and the subset expectation into the likelihood function for calculation, and the calculated probability value is the probability evaluation result of each subset expectation.

[0042] In one embodiment, step S32, obtaining the subset expectation with maximum likelihood and the corresponding subset from the evaluation result as the obtained posterior expectation and its related subset, includes:

[0043] Sort each evaluation result; based on actual needs or a pre-set quantity, select the top n subsets with the highest likelihood and their corresponding subsets. The selected subset expectations are regarded as posterior expectations, and their corresponding subsets are the relevant subsets.

[0044] In one embodiment, the method further includes: obtaining a posterior subset; selecting representative non-repeating implementations from the posterior subset for simulation to obtain the linear expectation of the posterior subset; obtaining the nonlinear expectation of the posterior subset based on the linear expectation; wherein the upper and lower limits of the linear expectation of the posterior subset are nonlinear expectations; and predicting reservoir geological parameters based on the nonlinear expectations.

[0045] In the following example, the uncertain parameter is a heterogeneous permeability field, the observed data is the flow rate data of the production well at each time step under a fixed bottom hole pressure, and the forward model is a nonlinear mapping from the permeability field to the flow rate, which is approximated by the single-phase and two-phase Darcy flow discretization scheme.

[0046] Two-dimensional 100m*100m permeability field The inversion was performed using a 10x10 grid, given the observed flow rate. The permeability field exhibits heterogeneity, isotropy, and uncertainty. All other parameters remained constant, as shown in Table 1. Fifty prior realizations of the permeability field were established using Sequential Gaussian Simulation (SGS), among which permeability... The unit is The actual situation of the permeability field generated from the observation data is as follows: Figure 2 As shown in (a). The pressure field at the end of the simulation using the actual permeability field is as follows. Figure 2 As shown in (b). The forward simulation has 180 time steps. Seconds. Assume the noise in the observed data follows a multivariate Gaussian distribution. Therefore, it seems to obey Assume the noise covariance matrix is... It is a diagonal matrix with standard deviation. This is approximately 1.7% of the initial flow.

[0047] Table 1

[0048] ,

[0049] Comparing this method with the ensemble smoother of multiple data assimilation (ESMDA) method for inverting permeability fields, ESMDA is a highly efficient parametric inversion method widely used in subsurface seepage.

[0050] For this method, forward simulation was performed on 50 prior implementations, while no further forward simulation is required during the inversion process. It is assumed that each subset contains no more than three implementations, and a total of [number missing] implementations are constructed. A subset is used to calculate the expectation. , of which flow rate First, the expectation of the subset with fewer implementations is calculated because they cover a wider range, and evaluations of fewer implementations are more efficient. Next, the 50 subsets with the highest likelihood are selected as the posterior subset. The union of the posterior subsets contains 25 non-repeating implementations. For predictions using this method, further simulations are performed on these 25 implementations, and the corresponding traffic expectations on the posterior subset are calculated. The nonlinear expectation then represents the upper and lower bounds of the expected traffic.

[0051] For ESMDA, since the distribution of h is close to a Gaussian distribution, h is inversely calculated based on... The penetration rate k was recovered. The set of 50 implementations was updated for 6 iterations until convergence yielded a posterior implementation. For prediction using ESMDA, all 50 posterior implementations were simulated.

[0052] The inference results of NEI and ESMDA obtained by this method are very close, such as Figure 3 As shown, Figure 3 This demonstrates the simulated flow using prior and posterior perspectives. The posterior flow of our method NEI is the expectation of a subset of the posterior. Figure 4 The negative log-likelihood of the posterior implementation of ESMDA and a subset of the NEI from our proposed method was compared, indicating that the inference results using our method have a slightly higher likelihood. Furthermore, our NEI method is computationally more efficient with 50 forward simulation runs, while ESMDA requires an additional 300 forward runs. ESMDA's CPU time is approximately 26 seconds, while our NEI method's CPU time is approximately 6 seconds, with 2.5 seconds dedicated to forward simulation and 3.5 seconds to calculating the expectation on the subset (Table 2). All test case computations were performed on a desktop workstation with an i9 CPU.

[0053] Table 2

[0054] ,

[0055] The technical solution of the present invention has been described above with reference to the preferred embodiments shown in the accompanying drawings. However, it will be readily understood by those skilled in the art that the scope of protection of the present invention is obviously not limited to these specific embodiments. Without departing from the principles of the present invention, those skilled in the art can make equivalent changes or substitutions to the original technical features, and the technical solutions resulting from these changes or substitutions will all fall within the scope of protection of the present invention.

Claims

1. A method for predicting reservoir geological parameters, characterized in that, include: A prior realization subset of reservoir geological parameters is constructed based on parameter value range, geological region characteristics, and similar physical properties; the prior realization set of reservoir geological parameters includes various geological conditions composed of different reservoir geological parameters; Calculate the expected range of the prior realization subset of the reservoir geological parameters in the data space; The evaluation process involves assessing all expectations within the stated expectation range to obtain posterior expectations and their corresponding subsets. Specifically, this includes: evaluating the likelihood of each subset expectation based on the likelihood function and actual observation data to obtain an evaluation result; extracting the subset expectations with maximum likelihood and their corresponding subsets from the evaluation result as the obtained posterior expectations and their related subsets; obtaining the upper and lower limits of the posterior expectations from the posterior expectations; and quantifying the uncertainty of reservoir geological parameters based on the upper and lower limits of the posterior expectations. Representative non-repeating realizations are selected from the posterior subset of reservoir geological parameters for simulation to obtain the linear expectation of the posterior subset; based on the linear expectation of the posterior subset, the nonlinear expectation of the posterior subset is obtained; wherein the upper and lower limits of the linear expectation of the posterior subset are the nonlinear expectations; based on the nonlinear expectations, the reservoir geological parameters are predicted to obtain the prediction results. Before calculating the expected range of the prior realized subset of the reservoir geological parameters in the data space, the process includes: performing a forward simulation on each prior realized in the prior realized set of the reservoir geological parameters based on the Darcy flow model to obtain the dynamic response of each realized; calculating the expected dynamic response of each realized on the corresponding subset; obtaining the upper and lower limits of all expectations, and obtaining the expected range based on the upper and lower limits.

2. The method according to claim 1, characterized in that, Before obtaining the expected range based on the upper and lower limits, the following steps are also included: Determine whether the actual measured data is within the range formed by the upper and lower limits; If the actual measurement data is not within the range formed by the upper and lower limits, then the prior knowledge is increased until the actual measurement data is within the range formed by the upper and lower limits.

3. The method according to claim 2, characterized in that, The process of quantifying the uncertainty of geological parameters based on the upper and lower limits of the posterior expectation includes: The difference between the upper and lower bounds of the posterior expectation is obtained; the larger the difference, the higher the uncertainty of the geological parameter distribution; the smaller the difference, the lower the uncertainty of the geological parameter distribution.

4. The method according to claim 1, characterized in that, The process of obtaining the evaluation results based on the likelihood function, actual observation data, and the expected probability of each subset includes: For each subset expectation, the actual observed data and the subset expectation are substituted into the likelihood function for calculation. Each calculated probability value is the probability assessment result of each subset expectation.

5. The method according to claim 1, characterized in that, The process of obtaining the subset expectation with maximum likelihood and the corresponding subset as the obtained posterior expectation and its related subset from the evaluation results includes: Sort each evaluation result; Based on actual needs or a pre-set quantity, select the top n sorted subsets with maximum likelihood expectations and their corresponding subsets. The selected subset expectations are regarded as posterior expectations, and their corresponding subsets are the relevant subsets.

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