Robust model parameter uncertainty quantification method

Abstract

The invention provides a robust model parameter uncertainty quantification method. The method comprises the following steps: constructing a plurality of priori implementation subsets of a priori implementation set; calculating an expected range of the priori implementation subset in the data space; evaluating all expectations in the expectation range to obtain a plurality of posterior expectations and corresponding subsets; obtaining an upper limit and a lower limit of the posterior expectations from the plurality of posterior expectations; the uncertainty of the geological parameter is measured based on the upper and lower limits of the posterior expectation. According to the method, in an expected range calculation stage, the expected range is calculated in a data space only based on a priori realized subset, and forward modeling does not need to be repeatedly operated along with parameter updating, so that the calculation amount is greatly reduced. In addition, by obtaining a plurality of posterior expectations and related subsets thereof, the data value is further mined, and the uncertainty information of the geological parameters is captured more comprehensively and accurately.

Description

Technical Field

[0001] The present invention relates to the technical field of data processing, and specifically provides a robust model parameter uncertainty quantification method. Background Art

[0002] The inference of uncertainty parameters of Darcy flow in porous media is very important for groundwater and reservoir engineering. Reservoir geological parameters, such as permeability and porosity, are usually highly heterogeneous, but these parameters can often only be measured at sparse well locations through cores and well logging, resulting in very high uncertainty in geological models. Therefore, quantifying uncertainty in subsurface flow using history matching is of great significance for evaluating and predicting reservoir performance. Traditional methods are generally based on gradient, heuristic or ensemble methods to obtain maximum a posteriori or maximum likelihood estimates (MLE). However, for high-dimensional problems, these methods are computationally expensive because a large number of forward simulations are required as the physical property parameters are iteratively updated. Recently, a highly efficient data space inversion (DSI) method has been proposed. History matching based on DSI does not require iterative forward modeling. The purpose of DSI is not to invert physical parameters, but to directly calculate the posterior dynamic response using a priori dynamic response and likelihood function. History matching can be achieved based on DSI, but the posterior information of physical property parameters cannot be obtained. Therefore, a new robust model parameter uncertainty quantification scheme is needed in this field to solve the above problems. Summary of the invention

[0003] In order to overcome the above-mentioned defects, the present invention is proposed to provide a solution or at least partially solve the problem of being unable to obtain a posteriori information of physical property parameters.

[0004] The present invention provides a robust method for quantifying model parameter uncertainty, comprising: constructing a plurality of a priori realization subsets of a priori realization set; calculating an expected range of the a priori realization subsets in a data space; evaluating all expectations in the expected range to obtain a plurality of posterior expectations and corresponding subsets; obtaining upper and lower limits of the posterior expectations from the plurality of posterior expectations; and quantifying the uncertainty of a geological parameter based on the upper and lower limits of the posterior expectations.

[0005] In a technical solution of the above-mentioned robust model parameter uncertainty quantification method, before calculating the expected range of the prior realization subset in the data space, it includes: performing forward simulation on each prior realization in the prior realization set to obtain the dynamic response of each realization; calculating the expectation of the dynamic response of each realization 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.

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

[0007] In a technical solution of the above-mentioned robust model parameter uncertainty quantification method, all expectations in the expected range are evaluated to obtain a number of posterior expectations and corresponding subsets, including: obtaining an evaluation result based on the likelihood function, actual observation data evaluation and the possibility of each subset expectation; obtaining the subset expectation and the corresponding subset with maximum likelihood from the evaluation result as the obtained posterior expectation and its related subset.

[0008] In a technical solution of the above-mentioned 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: subtracting the upper and lower limits of the posterior expectation to obtain a difference; 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.

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

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

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

[0012] The above one or more technical solutions of the present invention have at least one or more of the following beneficial effects: In the technical solution for implementing the present invention, the present invention provides a robust method for quantifying model parameter uncertainty, including: constructing several a priori realization subsets of the a priori realization set; calculating the expected range of the a priori realization subset in the data space; evaluating all expectations in the expected range to obtain several posterior expectations and corresponding subsets; obtaining the upper and lower limits of the posterior expectations from the several posterior expectations; and measuring the uncertainty of the geological parameters based on the upper and lower limits of the posterior expectations. Compared with the prior art, the beneficial effect of the robust method for quantifying model parameter uncertainty provided by the present invention is that in the stage of calculating the expected range, the method only calculates the expected range in the data space based on the a priori subset, and does not need to repeatedly run the forward simulation as the parameters are updated, which greatly reduces the amount of calculation. In addition, by obtaining multiple posterior expectations and their related subsets, the value of the data is further mined, and the uncertainty information of the geological parameters is captured more comprehensively and accurately.

[0013] Furthermore, firstly, in this method, only one forward simulation run is required for the prior realization, and then the expected range is calculated in the data space based on the subset of the prior realization, without repeated forward operation, which effectively reduces the computational complexity and resource consumption. Secondly, by using the maximum likelihood estimation, the prior information and the actual observation data are cleverly integrated to accurately determine the posterior expectation in the data space, and the realization subset in the parameter space is obtained simultaneously, which provides a rich and reliable data basis for in-depth analysis. Furthermore, the generated posterior subset is the result of a series of rigorous analysis and screening, which represents a parameter combination that is relatively more in line with the actual situation and the corresponding result situation. When these posterior subsets are used to predict the future behavior of the system, they show unique advantages compared with other methods. It does not need to simulate many situations on a large scale, but only simulates a small number of non-repeated realizations in the posterior subset (that is, those highly representative different situations), and can obtain more effective prediction results. In this way, not only the accuracy of the prediction is effectively ensured, but also the computing resources and time cost required in the prediction stage are greatly reduced, greatly improving the overall efficiency and benefits. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] The disclosure of the present invention will become more easily understood with reference to the accompanying drawings. It is easy for those skilled in the art to understand that these drawings are only for illustrative purposes and are not intended to limit the scope of protection of the present invention. In addition, similar numbers in the figures are used to represent similar components, among which: Figure 1 is a schematic flow chart of main steps of a robust model parameter uncertainty quantification method according to an embodiment of the present invention; Figure 2(a) is a schematic diagram of the actual situation of the permeability field for generating 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; Figure 3 is a schematic diagram of traffic flow simulation using a priori and a posteriori according to an embodiment of the present invention; Figure 4 is a comparison graph of negative log-likelihood of a subset of the posterior realization of ESMDA and NEI according to an embodiment of the present invention. DETAILED DESCRIPTION

[0015] Some embodiments of the present invention are described below with reference to the accompanying drawings. It should be understood by those skilled in the art that these embodiments are only used to explain the technical principles of the present invention and are not intended to limit the protection scope of the present invention. Embodiment 1

[0016] See attached Figure 1 , Figure 1 FIG. 1 is a flow chart of 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 an embodiment of the present invention mainly includes the following steps S1 to S5.

[0017] Step S1, construct several a priori realization subsets of the a priori realization set. In this embodiment, the a priori realization set can be regarded as a series of possible situation sets determined based on existing knowledge, experience or preliminary assumptions before analysis. These situations are usually represented by different combinations of parameter values, for example, in a geological model, the a priori realization set contains various possible geological conditions constituted by different permeability, porosity and other parameter values. Divide the a priori realization set into several a priori realization subsets. There can be many bases for division, such as classification according to parameter value range, geological regional characteristics, similar physical properties, etc. For example, those a priori realizations with permeability in a certain interval range and porosity in a corresponding specific range can be classified into a subset, and multiple different subsets can be constructed in this way, which is convenient for subsequent processing and analysis of each subset separately.

[0018] Step S2, calculate the expected range of the a priori realization subset in the data space. In this embodiment, according to the various specific realization situations contained in the subset (simulation results corresponding to different parameter values, etc.), the corresponding statistical calculation method is used to obtain the expected possible range of each subset in the data space.

[0019] Step S3, evaluate all expectations in the expected range to obtain several a posteriori expectations and corresponding subsets. In this embodiment, based on the expected ranges of each subset, further evaluate all expectations contained in these ranges. After evaluation and screening, several a posteriori expectations are determined. These a posteriori expectations are expected values ​​that are relatively more likely to represent the actual situation after combining the prior information and the current actual considerations. At the same time, the corresponding subsets that generate these a posteriori expectations (that is, part of the previously divided a priori realization subsets) are the relevant subsets. Step S4, obtaining the upper limit and lower limit of the posterior expectation from the plurality of posterior expectations. In this embodiment, among the plurality of posterior expectations obtained, the maximum value is found as the upper limit of the posterior expectation, and the minimum value is found as the lower limit of the posterior expectation. These two boundary values ​​define an interval range in which the posterior expectation is located.

[0020] Step S5, quantify the uncertainty of the geological parameters based on the upper and lower limits of the posterior expectation. In this embodiment, the uncertainty of the geological parameters is measured based on the interval range determined by the upper and lower limits. The larger the interval range, the higher the uncertainty of the parameter, that is, its true value may vary within a wider interval; conversely, the smaller the interval range, the lower the uncertainty of the parameter, and its true value is closer to a relatively clear range.

[0021] In one embodiment, step S2, before calculating the expected range of the a priori realization subset in the data space, includes: performing forward simulation on each a priori realization in the a priori realization set to obtain the dynamic response of each realization; calculating the expectation of the dynamic response of each realization 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.

[0022] For example, for a given set of geological parameter combinations with a certain permeability and a certain porosity, the Darcy flow model is used to simulate the dynamic conditions of the fluid flow rate, pressure change, etc. under such parameter conditions. The simulated flow rate and pressure change data are the dynamic responses corresponding to the realization. By performing such operations on each element in the prior realization set, the corresponding dynamic responses under all different parameter settings can be obtained. Calculating the expectation of these flow values ​​(such as simply averaging these flow values) can obtain a value that represents the average performance of the subset as a whole in terms of fluid flow. Put all the expected values ​​calculated on each subset together, and find the maximum value as the upper limit and the minimum value as the lower limit. Define an interval with the upper and lower limits, and this interval is the expected range.

[0023] In one embodiment, before obtaining the expected range based on the upper limit and the lower limit, it also includes: judging whether the actual measured data is within the range formed by the upper limit and the lower limit; if the actual measured data is not within the range formed by the upper limit and the lower limit, increasing the prior knowledge until the actual measured data is within the range formed by the upper limit and the lower limit.

[0024] In this embodiment, the increase in prior knowledge can be achieved by expanding the parameter range. For example, if the Darcy flow in porous media is studied, key parameters such as permeability and porosity can be considered to expand their possible value range when the actual measured data exceeds the expected range. In addition to permeability and porosity, parameters such as viscosity and density of the fluid may also be involved. For these parameters, their reasonable value range and distribution assumptions should also be re-examined. More influencing factors can also be included: through detailed geological exploration data, the location, scale, permeability changes of faults, the shape and direction of folds, and their changes in porosity can be added 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 laws should be taken into account. In addition, the physical model can also be improved. For example, in some cases, it may be necessary to consider non-Darcy flow effects. When the flow rate 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, etc.) can be taken into consideration, and under what conditions a more complex model is needed to describe fluid flow according to the actual situation, such model improvement can be used as a new priori understanding. If the system under study involves multiphase flow (such as oil-gas-water three-phase flow) or chemical reactions (such as the dissolution and precipitation of solutes in groundwater), it is necessary to increase the consideration of these complex processes in the priori understanding. For example, in the process of oil reservoir exploitation, the dynamic effects of the interaction between oil, gas, and water on permeability and porosity, as well as how mineral precipitation or dissolution caused by chemical reactions changes the pore structure of porous media, thereby affecting the Darcy flow parameters, are considered, and the theoretical and empirical understanding of these complex physical and chemical processes is integrated into the priori realization set.

[0025] In one embodiment, step S3, evaluating all expectations in the expected range to obtain a plurality of posterior expectations and corresponding subsets, comprises: Step S31, based on the likelihood function, the actual observed data evaluation and the likelihood of each subset expectation, obtain an evaluation result. Step S32, obtain the subset expectation with maximum likelihood and the corresponding subset from the evaluation result as the obtained posterior expectation and its related subset.

[0026] In this embodiment, the actual observation data is the key information reflecting the real state of the system under study, such as groundwater level data at different time points in a certain area and water flow data at different locations obtained through field measurements. After substituting these real data and each subset expectation into the likelihood function, a specific probability value can be obtained through calculation. This probability value is a quantitative reflection of the possibility of the subset expectation. By performing such operations on all subset expectations in the expected range, a set of evaluation results can be obtained, each of which corresponds to the degree of fit between a subset expectation and the actual observation data (i.e., the probability of the actual observation data appearing).

[0027] Alternatively, in an alternative approach, the evaluation result can also be obtained based on the Bayesian Information Criterion. Specifically, for each subset expectation, the likelihood function is first calculated, and then the number of parameters and the number of samples of the model are determined, and the BIC value is calculated by substituting 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.

[0028] Alternatively, in an alternative approach, the evaluation result can also be obtained based on the Akaike Information Criterion. Specifically, similar to BIC, for each model setting corresponding to the subset expectation, the likelihood function is first calculated, the number of parameters is determined, and then the AIC value is substituted into the formula. The smaller the AIC value, the better the model corresponding to the subset expectation, and the higher its probability.

[0029] Alternatively, in an alternative approach, the evaluation results can also be obtained based on the cross-validation method. Specifically, for each subset expectation, the actual observed data is divided into a training set and a validation set. The training set data is used to build a model based on the subset expectation (for example, if the effect of geological parameters on fluid flow is studied, a fluid flow model is built based on the geological parameters in the subset expectation), and then the validation set data is used to evaluate the prediction accuracy of the model. The prediction error of the model on the validation set can be measured by indicators such as mean square error (MSE) and mean absolute error (MAE). The smaller the prediction error, the better the model performance corresponding to the subset expectation, and the higher its probability.

[0030] Of course, the methods for obtaining evaluation results are not limited to the above methods, as long as the evaluation results can be obtained.

[0031] In one embodiment, step S5, the process of quantifying the uncertainty of the geological parameter based on the upper and lower limits of the posterior expectation includes: Step S51, subtract the upper limit and the lower limit of the posterior expectation to obtain a difference; 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.

[0032] In one embodiment, step S31, based on the likelihood function, the actual observed data is evaluated with respect to the likelihood of each subset expectation, and the process of obtaining the evaluation result includes: for each subset expectation, the actual observed data and the subset expectation are substituted into the likelihood function for calculation, and each calculated probability value is the likelihood evaluation result with respect to each subset expectation.

[0033] In one embodiment, step S32, the process of obtaining the subset expectation and the corresponding subset with maximum likelihood from the evaluation result as the obtained posterior expectation and the related subset thereof includes: Sort each evaluation result; according to actual needs or a pre-set number, select the first n subset expectations with maximum likelihood after sorting and their corresponding subsets. The selected subset expectations are regarded as the posterior expectations, and their corresponding subsets are the relevant subsets.

[0034] In one embodiment, the method further includes: obtaining a posterior subset; selecting representative non-repetitive realizations from the posterior subset for simulation to obtain a linear expectation of the posterior subset; based on the linear expectation of the posterior subset, obtaining a nonlinear expectation of the posterior subset; wherein the upper and lower limits of the linear expectation of the posterior subset are the nonlinear expectation; and predicting reservoir geological parameters based on the nonlinear expectation.

[0035] In the following example, the uncertain parameter is the heterogeneous permeability field, the observed data are 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 of the permeability field to flow rate, which is approximated by the single-phase and two-phase Darcy flow discretization schemes.

[0036] Two-dimensional 100m*100m permeability field The inversion of the permeability field is performed with a grid size of 10*10 given the observed flow. The permeability field is heterogeneous, isotropic, and uncertain. All other parameters remain unchanged, as shown in Table 1. A total of 50 prior realizations of the permeability field were established through sequential Gaussian simulation (SGS), among which the permeability The unit is The actual permeability field that generates the observed data is Figure 2 As shown in (a). The pressure field at the end of the simulation using the true permeability field is as follows Figure 2 As shown in (b). The forward simulation has 180 time steps. sec. Assume that the noise in the observed data follows a multivariate Gaussian distribution , so it seems to obey Assume that the noise covariance matrix is a diagonal matrix, the standard deviation , which is about 1.7% of the initial flow.

[0037] Table 1 , The permeability field inverted by this method is compared with that by the ensemble smoother for multiple data assimilation (ESMDA), which is an efficient parameter inversion method widely used in subsurface seepage.

[0038] For this method, 50 a priori realizations were forward modeled, and no further forward modeling was required during the inversion process. Assuming that each subset contains no more than three realizations, a total of Subsets to calculate the expectation , where the flow rate . The expectations for the subsets with fewer realizations are calculated first, because they cover a wider range and the evaluation with fewer realizations is more efficient. Next, the 50 subsets with the highest likelihood become the posterior subsets. There are 25 non-repeating realizations in the union of the posterior subsets. For forecasting using this method, further simulations are performed on the 25 realizations and the corresponding flow expectations on the posterior subsets are calculated. The nonlinear expectations are then the upper and lower bounds on the expected flow.

[0039] For ESMDA, since the distribution of h is close to Gaussian distribution, h is inversely calculated according to The permeability k is recovered. The set of 50 realizations is updated for 6 iterations until convergence to produce the posterior realizations. For prediction using ESMDA, all 50 posterior realizations are simulated.

[0040] The inference results of NEI and ESMDA are very close, such as Figure 3 As shown, Figure 3 The simulated traffic using prior and posterior realizations is shown. The posterior traffic of NEI in this method is the expectation of the posterior subset. Figure 4 The negative log-likelihood of the posterior implementation of ESMDA and the proposed method NEI on the subsets was compared, which means that the inference results using the proposed method have a slightly higher likelihood. In addition, the proposed method NEI is more computationally efficient using 50 forward simulation runs, while ESMDA requires an additional 300 forward runs. The CPU time for ESMDA is about 26 seconds and the CPU time for the proposed method NEI is about 6 seconds, of which 2.5 seconds are used for the forward simulation and 3.5 seconds are used to calculate the expectation on the subset (Table 2). The calculations for all test cases were performed on a desktop workstation with an i9 CPU.

[0041] Table 2 , So far, the technical solutions of the present invention have been described in conjunction with the preferred embodiments shown in the accompanying drawings. However, it is easy for those skilled in the art to understand that the protection scope of the present invention is obviously not limited to these specific embodiments. Without departing from the principle of the present invention, those skilled in the art can make equivalent changes or substitutions to the original technical features, and the technical solutions after these changes or substitutions will fall within the protection scope of the present invention.

Claims

1. A robust method for quantifying model parameter uncertainty, characterized in that: include: Construct several a priori realization subsets of the a priori realization set; Calculating the expected range of the a priori realization subset in the data space; Evaluate all expectations in the expected range to obtain several posterior expectations and corresponding subsets; Obtaining an upper limit and a lower limit of the posterior expectation from a plurality of posterior expectations; The uncertainty of the geological parameters is quantified based on the upper and lower bounds of the posterior expectations.

2. The method according to claim 1, characterized in that Calculating the expected range of the a priori realization subset in the data space includes: Performing forward simulation on each prior realization in the prior realization set to obtain a dynamic response of each realization; Calculate the expectation of the dynamic response of each realization on the corresponding subset; The upper and lower limits of all expectations are obtained, and the expected range is obtained based on the upper and lower limits.

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

4. According to the method of claim 1, the process of evaluating all expectations in the expected range to obtain a plurality of posterior expectations and corresponding subsets comprises: Based on the likelihood function, the actual observed data evaluation and the expected likelihood of each subset, the evaluation results are obtained; The subset expectation and the corresponding subset with maximum likelihood are obtained from the evaluation results as the obtained posterior expectation and its related subset.

5. The method according to claim 4, characterized in that The process of quantifying the uncertainty of geological parameters based on the upper and lower bounds of the posterior expectations includes: Subtracting the upper limit and the lower limit of the posterior expectation to obtain a difference; The larger the difference is, the higher the uncertainty of the geological parameter distribution is; the smaller the difference is, the lower the uncertainty of the geological parameter distribution is.

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

7. The method according to claim 4, characterized in that The process of obtaining the subset expectation and the corresponding subset with maximum likelihood from the evaluation result as the obtained posterior expectation and the related subset thereof includes: Rank each evaluation result; According to actual needs or a preset number, the first n subset expectations with maximum likelihood after sorting and their corresponding subsets are selected, and the selected subset expectations are regarded as posterior expectations, and their corresponding subsets are the relevant subsets.

8. The method according to any one of claims 1 to 7, characterized in that: The method further comprises: Selecting a representative non-repeated realization from the posterior subset for simulation to obtain a linear expectation of the posterior subset; Based on the linear expectation of the posterior subset, a 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 expectation; The reservoir geological parameters are predicted based on the nonlinear expectation.

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