Construction of prior models for ES-MDA by a deep neural network with a stacked autoencoder for predicting reservoir production

Journal of Petroleum Science and Engineering 187 (2020) 106800

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Construction of prior models for ES-MDA by a deep neural network with a stacked autoencoder for predicting reservoir production Jaejun Kim a, Sungil Kim b, Changhyup Park c, Kyungbook Lee b, * a

Department of Energy Systems Engineering, Seoul National University, Seoul, 08826, South Korea Petroleum and Marine Research Division, Korea Institute of Geoscience and Mineral Resources, Daejeon, 34132, South Korea c Department of Energy and Resources Engineering, Kangwon National University, Chuncheon, Kangwon 24341, South Korea b

A R T I C L E I N F O

A B S T R A C T

Keywords: Deep neural network (DNN) Stacked autoencoder (SAE) Ensemble smoother with multiple data assimilation (ES-MDA) History matching Egg model

The design of prior models has continued to receive research attention because of their importance for ensemblebased methods. There are two approaches to ensemble design: selection and regeneration. Both strategies try to utilize dynamic data in a qualified prior model. The regeneration approach has a higher degree of freedom than the selection approach because the latter is selected among given initial models. However, previous regeneration methods are still sensitive to prior models because these methods create new models based on the priors’ sta­ tistics. In this study, a deep neural network (DNN) was implemented to build new prior models. If there is a data pair for a reservoir model and its production, the production data and model parameters become the input and output data, respectively, for training the DNN. The trained DNN can generate new prior models according to the observed production history. For the output layer of the DNN, the main information of the permeability field is extracted by a stacked autoencoder (SAE) to improve the DNN performance. New prior models from DNN with SAE become the qualified prior set for ensemble-based methods because they already reflect observed dynamic data. For validation, an ensemble smoother with multiple data assimilation (ES-MDA) was applied to an Egg model. The proposed method (i.e., prior models designed with DNN-SAE) gave reliable posterior permeability fields and future reservoir performances. Compared with the two control groups (i.e., prior models and prior models with DNN), the proposed method mitigated the overshooting problem and found channel connectivity in the reference field. In a comparison with reliable posterior models, the simulation results of the proposed method not only matched observed production data but also reliably predicted future production. This study is an example of the successful application of a machine learning algorithm to history matching.

1. Introduction Building a reliable reservoir model is important for making a field development plan and evaluating a reservoir’s future performance. Before dynamic data are obtained from a target reservoir, static data are used with geostatistical methods such as sequential Gaussian simulation to model the reservoir. Here, “dynamic data” refers to time series data such as the oil production rate and bottom-hole pressure, and “static data” means constant data such as well logging and core data. For example, four-dimensional seismic data can be classified as dynamic, even though the first three dimensions are static. After the reservoir is modeled with available static data, it is simulated to predict production. When dynamic data are obtained from the target reservoir, there is usually a misfit between the predicted and true productions. History

matching involves using the observed dynamic data to modify the reservoir model to minimize the difference. The prior (or initial) models are generated from static geostatistical data, and they are modified by history matching to be the posterior (or updated) models. Many researchers have considered ensemble-based methods for history matching, such as the ensemble Kalman filter (EnKF) and ensemble smoother (ES) (Aanonsen et al., 2009; Oliver and Chen, 2011; Jung et al., 2018). These methods have several advantages compared to other gradient or non-gradient optimization techniques. First, the un­ certainty range for future production of a target reservoir is easy to predict because the concept of an ensemble is used. Second, posterior models are generated based on a relatively small number of reservoir simulations, especially with ES. The iterative form of ES has been pro­ posed by several researchers to secure reliable history matching results

* Corresponding author. E-mail address: [email protected] (K. Lee). https://doi.org/10.1016/j.petrol.2019.106800 Received 14 July 2019; Received in revised form 1 December 2019; Accepted 10 December 2019 Available online 16 December 2019 0920-4105/© 2019 Elsevier B.V. All rights reserved.

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Journal of Petroleum Science and Engineering 187 (2020) 106800

Fig. 1. Workflow for ensemble-based methods and the proposed method: (a) ensemble-based methods with prior models; (b) generation of prior models with DNN to replace prior models in (a); and (c) generation of prior models with DNN-SAE to replace prior models in (a). The blue and green colors represent the reservoir simulation and history matching steps, respectively. The red color indicates feature extraction from prior reservoir models. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

Therefore, the regeneration scheme goes one step further than the se­ lection scheme to generate input data for geostatistical methods. Kim et al. (2017) generated 400 prior models for a reservoir simulation and selected 50 qualified models based on the misfit from the observed production data. Based on the mean of permeability of the 50 models, the grids with the top 5% and bottom 5% permeabilities were used as pseudo-hard data for constructing new prior models through geo­ statistical methods. Lee et al. (2017) used the mean of selected models as pseudo-soft data (i.e., the facies probability) for geostatistical tech­ niques to build a trend of new facies models. This concept was iteratively applied in Lee et al. (2019). New reservoir models can provide reason­ able prediction results compared to prior models because they already reflect dynamic data through pseudo-static data. Therefore, this ensemble design concept can improve the performance of ensemble-based methods. Previous researches on the regeneration approach depended on the results of the selection scheme. Recently, Ahn et al. (2018) developed a novel regeneration scheme that uses an artificial neural network (ANN). A data pair of reservoir model and simulated production was used to train the ANN as output and input layers, respectively. Here, the data used for the input and output layers were opposite to those used to create a proxy model that can replace the full reservoir simulation. To improve the results of the ANN and reduce the size of the network, they introduced a stacked autoencoder (SAE) (see Subsection 2.4) to extract the main information for the reservoir model; this feature was used for the output layer instead of the reservoir model itself. In previous studies, the SAE has been successfully applied to reservoir models. Canchumuni et al. (2017) replaced prior models with encoded parameters in the ensemble smoother with multiple data assimilation (ES-MDA) (see Subsection 2.5), and Lee et al. (2018) used encoded parameters as a criterion of the dissimilarity between models. In this study, a new workflow was developed for ensemble-based methods in terms of the initial ensemble design. The deep learning-

with a small number of reservoir simulation (Chen and Oliver, 2012; Emerick and Reynolds, 2013; Luo et al., 2015; Ma and Bi, 2019). Several researchers have stressed the importance of creating proper initial ensembles because a posterior ensemble is sensitive to a prior ensemble (Aanonsen et al., 2009; Jafarpour and McLaughlin, 2009; Kang et al., 2016; Lee et al., 2016; Skjervheim and Evensen, 2011). If prior ensembles differ significantly from the target reservoir, this can cause overshooting and ensemble collapse. “Overshooting” means that the posterior model has very large or small values that do not fit a physical phenomenon. “Ensemble collapse” refers to the situation when hundreds of posterior models converge into a single model, which makes the uncertainty impossible to assess. These phenomena become worse if reservoir parameters do not follow the normal distribution. Previous researches have proposed two approaches for designing reliable prior models: selection and regeneration. In the selection scheme, after hundreds of possible reservoir models are generated, appropriate models based on dynamic data are selected. Various spatial correlation data (e.g., multiple training images for multipoint simula­ tion) are used to broaden the uncertainty of prior reservoir models (Scheidt and Caers, 2009; Lee et al., 2019). After qualified models are selected, they are used in ensemble-based methods as input models. Jung et al. (2017) generated 400 initial models and selected 50 qualified models based on the relative error between the true production and simulated production. Kang et al. (2017) selected 100 among 400 models to improve the EnKF results for two cases: a two-dimensional channelized reservoir and the PUNQ-S3 benchmark field. The selec­ tion scheme not only reduces the number of simulations but also im­ proves the history matching results. However, these studies still relied on prior models because the models are chosen from a pool of prior models. In the regeneration scheme, reservoir models are rebuilt with both the given static data and pseudo-static data. The latter are created by qualified prior models through the previous selection scheme. 2

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Journal of Petroleum Science and Engineering 187 (2020) 106800

Fig. 2. Example procedure for the proposed method: (a) prior reservoir models are obtained from the standard Egg model; (b) the permeability model is encoded by the SAE to extract features; (c) the oil production rate and encoded permeability data are used as the input and output layers, respectively, in the DNN-based inverse modeling; (d) prior models with DNN-SAE are built by the trained DNN in (c) with observed dynamic data as the input layer; (e) posterior models with DNN-SAE are obtained with an ensemble-based method; and (f) future production is estimated by the posterior models with DNN-SAE.

based model generation algorithm from Ahn et al. (2018) was adopted to generate prior models for ensemble-based methods. For the Egg benchmark field, three ensemble design cases were compared: prior models (no modification), new prior models with a deep neural network (DNN), and new prior models with the proposed method (DNN-SAE). The rest of the paper is organized as follows. Subsection 2.1 provides a detailed workflow of the proposed method, and Subsection 2.2 ex­ plains the Egg model. Subsections 2.3–2.5 deal with specific method­ ologies such as the DNN, SAE, and ES-MDA. Subsection 3.1 demonstrates the limitation of the ensemble-based method through the first ensemble design case. Subsection 3.2 discusses how to regenerate prior models with DNN and DNN-SAE, and Subsection 3.3 shows the improvements in the results of the ES-MDA with them. Section 4 con­ cludes the paper.

2. Methodology 2.1. Procedure of the proposed method Fig. 1 shows the workflows for an ensemble-based method and the proposed method. The key difference between the two methods is the prior reservoir models. Ensemble-based methods use prior models that are generated by integrating static data (Fig. 1(a)), while the proposed method uses new prior models that are generated by the DNN-based inverse modeling (Figs. 1(b) and (c)). For the DNN model, the input layer in Figs. 1(b) and (c) is the same as the result of the reservoir simulation. However, there is a difference in the output layer for the DNN: in the prior models with DNN, reservoir models are used as the output layer themselves (Fig. 1(b)); in the prior models with DNN-SAE,

Fig. 3. First layer of the Egg model: (a) permeability distribution of the reference model and (b) its histogram.

3

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Journal of Petroleum Science and Engineering 187 (2020) 106800

ensemble-based method using the prior models with DNN-SAE. The first part is to create a new prior model for reliable history matching, while the second is the same as typical ensemble-based methods. As shown in Fig. 2(a), 100 prior reservoir models are obtained from the standard Egg model (see Subsection 2.2). Forward simulation is implemented for all reservoir models to obtain production data, and the permeability of the reservoir models is transformed into a vector. The size of the vector is the same as the number of grid cells of the reservoir model and is encoded by the SAE for feature extraction (Fig. 2(b)). The production profiles of the reservoir models are standardized for numerical stability before the DNN is constructed. The encoded (or featured) data and oil production rate for each production well are used in the output and input layers, respectively, for the DNN-based inverse modeling (Fig. 2 (c)). After the DNN is trained, new feature vectors can be generated when the observed production data are used for the input layer in the trained DNN. These encoded data are decoded to the permeability model by the SAE (Fig. 2(d)). The prior models with DNN-SAE become the new reservoir models for an ensemble-based method to obtain the posterior models with DNN-SAE (Fig. 2(e)). Then, future productions of the reference model can be estimated by simulating the posterior models with DNN-SAE (Fig. 2(f)).

Table 1 Petrophysical parameters for reservoir simulation. Parameters

Values

Initial conditions

Pressure [bar] Oil–water contact depth, [m]

400 at 4000 m 5000 0.2 0.1 1 at 400 bar

Oil/water Rock Oil Water Oil Water

1.00E-05 0 5 1 900 1000

Porosity [fraction] Initial water saturation [fraction] Oil/water formation volume factor [rm3 /sm3 ] Compressibility [1/bar] Viscosity of fluid [cP] Density of fluid ½kg= m3 � Well constraint

Injection well

Surface rate target ½sm3 =day�

79.5

Production well

BHP (upper limit) [bar] BHP (lower limit) [bar]

410 395

encoded parameters are utilized for the output layer (Fig. 1(c)). After the prior models are generated with DNN or with DNN-SAE, they are used for the workflow of ensemble-based methods in Fig. 1(a). In this study, we developed the integrated code for DNN, SAE, and ES-MDA in the framework of MATLAB. Fig. 2 depicts an example of the proposed method in Fig. 1(c). The proposed method consists of two parts: (1) the DNN-based inverse modeling with data encoding and decoding by the SAE, and (2) the

2.2. Benchmark field: Egg model The Egg model was used in this study; this is widely used as a benchmark for reservoir simulation, optimization of reservoir

Fig. 4. Oil and water production rates of the reference model for the four production wells: (a) PROD1, (b) PROD2, (c) PROD3, and (d) PROD4. 4

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Journal of Petroleum Science and Engineering 187 (2020) 106800

Table 3 Parameters of the DNN-based inverse model. Value

Parameters Input neuron Output neuron Hidden neuron Active function

development plans, and history matching (Jansen et al., 2014; Wang et al., 2016; Jung et al., 2017). A set of 101 benchmark fields, i.e. the reference field and a hundred of initial models, can be obtained from public domain (Jansen, 2013) and the model dimension was 60 � 60 � 7. In this study, only the top layer of the Egg model was used to verify the proposed method in the case of two dimensions. There were 3600 grids in total and 2491 active cells. Fig. 3(a) displays the permeability dis­ tribution of the reference model with the well locations, and Fig. 3(b) shows its histogram. The reference model showed high-permeability channels with a low-permeability background, and each initial model had different channel patterns. Table 1 presents the petrophysical properties of the Egg model, which had eight injection wells and four production wells. For the in­ jection wells, the water injection rate target was set to 79.5 sm3 =day, and the upper limit of the bottom-hole pressure was 410 bar. The lower limit of the bottom-hole pressure for the production wells was set to 395 bar. Fig. 4 shows the oil and water production profiles of the reference model up to 3600 days. Water production was not observed in the initial production period because the initial water saturation was set to 0.1 as same as the irreducible water saturation. The oil production rate sharply decreased after water production was observed, and the water break­ through time of each production well differed from each other. There­ fore, reliably predicting the water breakthrough time of production wells is essential.

(1) where Nwell , Nens , and Nobs are the number of production wells, ensemble members, and observed time steps, respectively. In this study, there are four production wells, an ensemble of 100 reservoir models, and 24 time steps (from 30 to 720 days at the interval of 30 days). dobs i;k means the observed history data for ith production well at kth time step. di;j;k stands for the simulated production data for ith production well on jth reali­ zation at kth time step. The input data were normalized for numerical stability as follows:

2.3. DNN-based inverse modeling The ANN interprets the nonlinear relationship between static and dynamic data in petroleum engineering based on supervised training, which assumes that a pair of input and output dataset is available (Min et al., 2011; Ahn et al., 2018; Ma et al., 2018). The DNN can be defined as having more than two hidden layers between the input and output layers. Each layer consists of neurons and they are connected to each other by weights. Each output of neuron is estimated by applying acti­ vation function such as sigmoid function to the summation of the product of the neuron in the previous layer and the weight between the neurons. The objective of DNN is to match the calculated outputs of neurons in output layer with the given output values by adjusting weights and bias parameters. After the parameters are optimized for training dataset, the trained DNN model, i.e. the optimized parameters,

Ik ðtÞ ¼ 2 �

3

5

Matching error (the averaged MAPE, %) Computing time (sec)

26.3 267

24.8 648

24.5 1349

Pk ðtÞ Pmin ðtÞ Pmax ðtÞ Pmin ðtÞ

1

(2)

where Ik ðtÞ is the input data of the kth model at time t and Pk ðtÞ is the production rate of the kth model at time t. The subscripts “max” and “min” represent the maximum and the minimum production rates at time t, respectively at time t. The input data were normalized for numerical stability as follows: Table 3 summarizes the hyperparameters of the DNN. In detail, the input neurons consisted of the oil production rate every 30 days for PROD1, PROD2, PROD3, and PROD4 up to 720 days. There were 2491 output neurons, which was the number of active grids of the reservoir as shown in Fig. 3(a). The three hidden layers each had 500 neurons. The sigmoid function was used as an activation function between the layers. When the trained DNN was used to generate 100 new reservoir models, a random observation error of 0–10% was added for the observed pro­ duction data in the input layer. It is an important procedure because the

Table 2 Sensitivity analysis on the number of hidden layers for the DNN-based inverse model. 1

24 � 4 ¼ 96 2491 100 500 Sigmoid function

is verified for test dataset. It is important to prevent the DNN model from overfitting training dataset because the trained model has to apply to test data that is not used for training. The performance of DNN is sensitive to the structure of neural network, e.g. the number of hidden layers and the number of neurons for each layer, and training condition, e.g. learning rate and optimization method. These factors are called hyperparameters. Fig. 5 illustrates the DNN framework in this study, which had three hidden layers (H1 ; H2 ; H3 ). For inverse modeling based on the DNN, the input layer (I) and output layer (O) consisted of the production profiles and reservoir properties, respectively. In this research, sensitivity analysis was carried out for the number of hidden layers: single, three, and five hidden layers. The three layers are chosen for further study because the three-layer case showed much improved performance than the single-layer case and similar performance with the five-layer case (Table 2). Note that matching error is estimated by the averaged mean absolute percentage error (MAPE) between the observed oil production rates from the reference field (Eq. (1)) and the simulated oil production rates from the generated fields after the trained DNN-based inverse model is applied to the observed data. In other words, the averaged MAPE is not estimated for train, test, or validation sets. � � ! � obs � Nwell X Nobs �di;j;k Nens X di;k � 1 1 1 X the averaged MAPE ¼ � 100 obs di;k Nwell Nens Nobs i¼1 j¼1 k¼1 ! � � Nobs � Nens X dj;k dkobs � 1 1 X MAPE ¼ � 100 Nens Nobs j¼1 k¼1 dkobs

Fig. 5. Architecture of the DNN-based inverse model. The output layer has 2491 and 100 neurons in the DNN case and DNN-SAE case, respectively.

Number of hidden layers

Prior models with DNN Prior models with DNN-SAE

5

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Journal of Petroleum Science and Engineering 187 (2020) 106800

Fig. 6. Architecture of the SAE. Encoding extracts the main information from the original data, and decoding restores the dimensions of the original data in the input layer from the main information.

Fig. 6 shows the SAE scheme used in this study, where I 2 RX ; H1 2 RY ; H2 2 RZ ; H3 2 RY , and O 2 RX represent the input, hidden, and output layers, respectively. R is the data space for each layer, and the superscripts X, Y, and Z indicate the number of neurons. Because the SAE was used to extract the main information of the permeability model in this study, Z was smaller than Y, and Y was smaller than X. As shown in Fig. 6, the SAE procedure is divided into the encoding (Eq. (3)) and decoding processes (Eq. (4)) (Zabalza et al., 2016):

observed production data are obtained by a reservoir simulator, which is typically subject to modeling errors. 2.4. DNN-SAE For a typical neural network, the backpropagation algorithm reduces the error between neurons in the output layer and the true information by modifying the weights and bias. However, as the number of layers and neurons in each layer increases, the backpropagation algorithm becomes inefficient, and its accuracy decreases (Hirose et al., 1991). To improve the training performance, the SAE has been proposed for the dimensional compression of data (Erhan et al., 2010; Liou et al., 2014). The SAE is unsupervised pre-training that improves the calculation ef­ ficiency and performance of a neural network by extracting features from the original data (Bengio et al., 2007). In this study, after the SAE was used to encode the reservoir model, the encoded data were used in the output layer for the DNN in Fig. 5 instead of the permeability model itself. The principle of the SAE is to encode neurons in the input layer to neurons in the hidden layers and restore them to neurons in the output layer. The SAE consists of two parts as shown in Fig. 6: encoding and decoding. For encoding, the information in the input layer is compressed into main features of the information in the hidden layer. In case of decoding, the compressed features are restored to the original dimension in output layer. The SAE typically has an odd number of layers and shows a symmetrical architecture with respect to the middle hidden layer. During training, the weight and bias are optimized to minimize the difference between the information in the input layer and the restored neurons in the output layer.

Value

Input neuron ðIÞ

2491 ðXÞ

Hidden neuron 1 ðH1 Þ

600 ðYÞ

Hidden neuron 2 ðH2 Þ

100 ðZÞ

Hidden neuron 3 ðH3 Þ

600 ðYÞ

Output neuron ðOÞ

2491 ðXÞ

Active function

Logistic sigmoid function

(3)

H3 ¼ f ðWH3 H2 þ bH3 Þ; O ¼ f ðWO H3 þ bO Þ

(4)

where W represents the weight matrix and b is the bias vector. f is the activation function of the SAE, and the logistic sigmoid function was used in this study. Table 4 summarizes the hyperparameters of the SAE used. There were 2491 input neurons, which were the active cells of the reservoir, and the hidden layers 1 and 3 each had 600 neurons. The hidden layer 2 had 100 neurons, which became the output layer for the DNN-SAE in­ verse model shown in Fig. 5. After the inverse modeling was imple­ mented by the DNN, the predicted feature vector with 100 neurons was decoded in the hidden layer 3 and sent to the output layer, as shown in Fig. 6. Then, a permeability model with 2491 grids was obtained that could be used for ensemble-based methods as the prior models with DNN-SAE, as shown in Fig. 1(c). For the split of training, validation, and test sets, a sensitivity anal­ ysis was conducted for the number of training data. For a basic case, the 100 prior models were divided into three types: supervised learning (training), validation, and test; 80% of the data pairs were used for training, 10% were used for validation, and the other 10% were used for the test. During the analysis, only the number of training data is changed

Table 4 Parameters of the SAE for the encoded model in the output layer of the DNN-based inverse model. Parameters

H1 ¼ f ðWH1 I þ bH1 Þ; H2 ¼ f ðWH2 H1 þ bH2 Þ

Table 5 Sensitivity analysis on the number of training data. Number of training data Matching error (the averaged MAPE, %) Computing time (sec)

6

DNN

DNN-SAE

80

60

40

80

60

40

36.1

36.5

37.3

24.8

26.4

27.6

4467

4290

3991

648

640

611

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Journal of Petroleum Science and Engineering 187 (2020) 106800

Fig. 7. Oil production rates observed at the production wells: (a) prior models and (b) posterior models by the ES-MDA.

from 80 to 60 and 40 (Table 5). Consequently, a matching error in­ creases rapidly as the number of training data decreases. In contrast, the effect of time saving was negligible. Therefore, we used all 80 models for training date instead of excluding available data. Note that the worst DNN-SAE case gives better the averaged MAPE and computing cost than those from the best DNN case.

ES-MDA have been modified into many variations and improved with auxiliary methods (Aanonsen et al., 2009; Oliver and Chen, 2011; Jung et al., 2018). In many cases of petroleum engineering, the target pa­ rameters are the permeability, porosity, and rock type while the observed data are the production and pressure history (Lee et al., 2016; Kim et al., 2018). In this study, the ES-MDA was used to update per­ meabilities according to oil production data. The ES-MDA minimizes the following objective function for history matching: �T � �T � JðmÞ ¼ m mb M 1 m mb þ dobs d D 1 dobs d ; d ¼ fðmÞ

2.5. ES-MDA Ensemble-based methods calibrate target static parameters corre­ sponding to observed data based on the Bayesian conditional probability (Evensen, 1994). Ensemble-based methods such as the EnKF, ES, and

(5)

7

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Journal of Petroleum Science and Engineering 187 (2020) 106800

Fig. 8. Mean permeability and its histogram for (a) the prior and (b) posterior models by the ES-MDA. The red-dotted histogram is for the reference model. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

the mean of all dynamic vectors. The ES-MDA has one special parameter αp , which satisfies the following condition:

where mb represents model parameters before an update and the su­ perscript b indicates the background. M is the covariance matrix be­ tween reservoir models. dobs is the observed history data. The expression d ¼ fðmÞ represents simulation responses obtained by running the reservoir simulation f with model parameters m. D is the covariance matrix of the observation error due to measurement error. To minimize

XNa 1 p¼1

�

1

diunc

� di for ​ i ¼ 1; :::; Nens ; p ¼ 1; :::; Na

(6)

where the subscript i and p stand for the index numbers of the ith real­ ization and of the pth iteration, respectively. Nens and Na are the ensemble size and the number of iteration steps, respectively. Cmd is the cross-covariance matrix between model parameters m and dynamic vector d, and Cdd is the auto-covariance matrix of the dynamic vector d. αp is the inflation coefficient for CD , which indicates the covariance matrix of the measurement error in the observed data. dunc is the observed data considering uncertainty based on the inflated observed data measurement error. The specific term of Cmd ðCdd þ αp CD Þ called the Kalman gain. Cmd and Cdd are calculated as follows:

Cmd ¼

Cdd ¼

Nens X

1 Nens -1

ðmi

1

3.1. Overshooting problem of the ES-MDA with the prior models (standard Egg model)

is

mÞðdi

dÞT

(7)

i¼1

ðdi

dÞðdi

dÞT

(8)

(9)

3. Results and discussion

The ES-MDA was used for history matching in the presence of geological uncertainty (i.e., the spatial distribution of the absolute permeability). The mean value of the absolute permeability of the reference field was approximately 660 mD (natural-log-transformed value of 6.5), while the mean of the 100 prior reservoir models had 900 mD (natural-log-transformed value of 6.8). The posterior models were the results of the ES-MDA with these prior models. Fig. 7 depicts the oil production rates at the production wells during history matching (up to 720 days). Fig. 7(a) shows the oil rates of prior realizations, and Fig. 7(b) describes the oil rates of posterior models by

i¼1

1 XNens Nens

1

¼1

In this study, four assimilations, i.e. Na is equal to 4, were applied to calibrate the reservoir models, and αp was set to 4 for every single assimilation of the ES-MDA. In some cases, αp can be gradually decreased step by step because this induces a large variation in prior models and obtains a narrow variation (Emerick and Reynolds, 2012). The overall performance of history matching can be improved by con­ trolling this parameter (Rafiee and Reynolds, 2017; Evensen, 2018; Emerick, 2019).

JðmÞ, ∂JðmÞ ∂m should be zero, and the updated m is derived as follows:

mi ¼ mbi þ Cmd Cdd þ αp CD

αp

where m is the mean of model parameters from all realizations and d is 8

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Journal of Petroleum Science and Engineering 187 (2020) 106800

Fig. 9. Oil production rates of 100 prior models generated by (a) DNN and (b) DNN-SAE.

the ES-MDA. The posterior models reduced the uncertainty range of the prior models, but the history matching performance was not satisfac­ tory. The oil rates of the posterior models failed to cover the true response for all wells, and the mean trajectories of the posterior models did not follow the true profile. Before 200 days, the trajectories of the posterior models reliably explained the true profile because the water breakthrough had little influence on the oil rates. However, after the water breakthrough, the uncertainty of the oil production increased because the decreased oil rate may have given wrong information for updating the permeability. The decreased dunc in Eq. (6) after water production could reduce the permeability to decrease the simulated oil rate, even though early water

production indicates a connection with high permeability between the injection and production wells (Lee et al., 2014). The poor performance of the ES-MDA could be due to the lack of essential dynamic data, small size of the prior domain, and unrealistic prior models. The water pro­ duction affected the oil rates but was not used for data assimilation. Also, a hundred models may be too few to properly explain the geological uncertainty for the spatial distribution of the permeability. Fig. 8 compares the permeability distributions of the prior models and posterior models; the grid values are the averaged permeability of 100 models. Fig. 8(a) depicts the averaged permeability and the histo­ gram of the averaged values for the prior models, and Fig. 8(b) shows those of the posterior models. Note that each permeability model has 9

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Journal of Petroleum Science and Engineering 187 (2020) 106800

Table 6 Summary of the MAPEs for oil and water productions in the three prior datasets during the history matching period (up to 720 days). Parameters

Production well

Prior models

Prior models with DNN

Prior models with DNN-SAE

Oil production rate

PROD1 PROD2 PROD3 PROD4 Average PROD1 PROD2 PROD3 PROD4 Average

33.3% 29.0% 42.1% 27.9% 33.1% 59.1% 56.0% 115.5% 311.6% 135.6%

21.2% 34.7% 46.4% 40.8% 35.8% 47.6% 43.9% 143.1% 444.3% 169.7%

23.9% 22.0% 28.8% 20.8% 23.9% 34.4% 45.6% 42.9% 63.0% 46.5%

Water production rate

Table 7 Comparison of simulation costs for training machine learning algorithms. Procedure

DNN

DNN-SAE

SAE

Computing time (sec)

4467

648

984

instead of the given prior models from the standard Egg model. The prior models with DNN and with DNN-SAE already considered the production history of the oil rates up to 720 days (i.e., history matching period) as input neurons. Fig. 9 shows the oil production rates of the prior models constructed with (a) DNN and (b) DNN-SAE. The prior models with DNN reduced the uncertainty range of the prior models in Fig. 7(a), but they still had a huge mismatch with the true profiles. The prior models with DNN-SAE decreased the difference between the mean trajectory and true response because they widely covered the true profile (Fig. 9(b)). Even though the mean trajectory for PROD3 did not match well with the true profile, it could be modified by the following ES-MDA because the band of oil rates tended to include the true values with proper uncertainty. Table 6 summarizes the errors of the three prior sets. The averaged MAPE with all production wells up to 720 days for the prior models with DNN were 35.8% for the oil rate and 169.7% for the water rate; these were slightly larger than those for the existing prior population. The prior models with DNN-SAE reduced the averaged MAPE to 23.9% for the oil rate and 46.5% for the water rate. Thus, DNN-SAE can construct more reliable prior models than DNN. Of note is the reduced error for the water rate, which decreased from 135.6% (prior models) to 46.5% (prior models with DNN-SAE) even though the water production rate was not used in any process. The smaller errors for the profiles of the oil and water rates confirm that DNN-SAE improves the prior models to match the reference field.

Gaussian distribution even though the histogram of the mean field shows almost a single value nearby the average permeability in Fig. 8(a). The overshooting problem was severe for the posterior models with ESMDA. The prior models had only two distinct values (see the histogram in Fig. 8(a)), while the posterior models showed many outliers, which is a typical problem for ensemble-based data methods. These results revealed that the prior reservoir models are not adequate for the ESMDA to explain the spatial distribution of the permeability and to secure reliable posterior models. New prior reservoir models need to be constructed with high reliability not only to overcome the overshooting problem but also to improve the ES-MDA performance. 3.2. Construction of new prior models for ES-MDA: with DNN vs. with DNN-SAE In this study, DNN or DNN-SAE was used to construct prior models

Fig. 10. Mean permeability and its histogram for (a) the prior models with DNN and (b) the prior models with DNN-SAE. The red-dotted histogram is for the reference model. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.) 10

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Fig. 11. Oil production rates from the posterior models after the ES-MDA with the (a) prior models, (b) prior models with DNN, and (c) prior models with DNN-SAE. The vertical broken line indicates the end of the history matching period (720 days).

Fig. 10 illustrates the spatial distribution of the mean permeability of 100 prior models and its histogram (a) with DNN and (b) with DNN-SAE. The mean for the prior models with DNN was similar to the mean for the prior models in Fig. 8(a), while the mean for the prior models with DNN-

SAE showed a wider histogram and approached the permeability dis­ tribution of the reference field. Especially, only the prior models with DNN-SAE in Fig. 10(b) mimicked the low-permeability area near PROD3 and PROD4 in the reference field (Fig. 3(a)). This area caused a late 11

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Journal of Petroleum Science and Engineering 187 (2020) 106800

Fig. 12. Water production rates from the posterior models after the ES-MDA with the (a) prior models, (b) prior models with DNN, and (c) prior models with DNNSAE. Note that the water rates were not used for history matching.

decrease in the oil production rate for the two wells, and Fig. 9(b) shows the proper production profiles compared to the true production. This resulted in the low error values for the oil and water productions of PROD3 and PROD4 given in Table 6.

The importance of feature extraction is demonstrated here. The DNN used 96 neurons (oil rates observed at the production wells) in the input layer and 2491 neurons (permeability of active cells) in the output layer, while DNN-SAE applied the encoded dataset for the output layer (i.e., 12

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model. Table 7 shows a comparison of simulation costs for the DNN, DNN-SAE, and SAE. The DNN-SAE took much less computing time for training than the DNN because the number of neurons in the output layer for the DNN-SAE, 100, is much less than that of the DNN, 2491. However, the DNN-SAE requires additional trained SAE model to extract features from reservoir models. Computing time for the SAE is relatively high, but the summation of simulation cost for the SAE and DNN-SAE (1532 seconds) is still about 36.5% of the simulation cost for the DNN (4467 seconds). Therefore, the DNN-SAE case can generate much better reliable prior models with reasonable simulation cost.

Table 8 Summary of the MAPEs during the prediction period (from 721 days to 3600 days). Parameters

Posterior models

Posterior models with DNN

Posterior models with DNN-SAE

Oil production rate

15.9% 107.6% 14.6% 56.7% 48.7% 19.8% 32.7% 11.2% 48.8% 28.1% 8.4%

27.7% 37.3% 49.0% 30.4% 36.1% 54.2% 37.7% 21.1% 58.9% 43.0% 16.1%

10.3% 12.1% 15.8% 9.2% 11.9% 8.5% 4.4% 14.5% 10.4% 9.5% 1.3%

29.0%

42.7%

4.0%

PROD1 PROD2 PROD3 PROD4 Average Water PROD1 production PROD2 rate PROD3 PROD4 Average Total oil volume produced until 3600 days Total water volume produced until 3600 days

3.3. History matching by the ES-MDA: posterior models with DNN vs. posterior models with DNN-SAE The history matching performance by the ES-MDA was evaluated in terms of matching history and forecasting unknown production rates. The three prior sets for the ES-MDA were compared: (1) the prior models (base case), (2) the prior models with DNN (comparison case), and (3) the prior models with DNN-SAE (proposed case). As explained in Sub­ section 3.1, the prior models on the ES-MDA showed the overshooting problem and high errors for the production, even during the history matching period. Fig. 11 compares the ES-MDA results for the three cases. The pos­ terior models with DNN could not give a reliable prediction (Fig. 11(b)) with even worse results than the posterior models with the base case. The prior models in Fig. 7(a) covered the true production, but the prior models with DNN in Fig. 9(a) failed to include the observed data. Although the uncertainty range was reduced by the DNN, the ES-DMA results were worse because the reduced range did not properly repre­ sent the actual production. In contrast, the posterior models with DNN-SAE gave a reliable performance as shown in Fig. 11(c) because the prior models with DNNSAE kept the true production within the band of gray lines with reduced

100 neurons by the SAE from the 2491 neurons) with the same 96 input neurons. Thus, DNN-SAE can mitigate the nonlinear correlation between the input and output layers and give a more reliable trained model (Ahn et al., 2018). In short, DNN failed to improve the prior models, while DNN-SAE generated more reliable prior models for the following ES-MDA. Even, the prior models with DNN showed worse matching than the prior models (Table 6) because it is difficult to find the proper values of all grid, 2491, in the output layer with only 96 production data in the input layer. Even though the wide uncertainty ranges in the prior models (Fig. 7(a)) were reduced after the DNN application (Fig. 9(a)), simula­ tion results from the prior models with DNN cannot include the true production data anymore. That is why the DNN case had higher the MAPEs than ones from the prior models in Table 6. It is a critical issue for simulation cost to train a machine learning

Fig. 13. Comparison of the cumulative oil (upper) and water (lower) productions according to the (a) posterior models, (b) posterior models with DNN, and (c) posterior models with DNN-SAE. The vertical broken line in the upper figures indicates the end of the history matching period (720 days). The water rates were not used for history matching. 13

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Fig. 14. Mean permeability and its histogram for the (a) posterior models with DNN and (b) posterior models with DNN-SAE. The red-dotted histogram is for the reference model. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

Fig. 15. Standard deviation map from the posterior models after the ES-MDA with the (a) prior models, (b) prior models with DNN, and (c) prior models with DNN-SAE.

uncertainty. Considering that the standard ES-MDA was applied in the three cases, these different results indicate that the reliability of the prior models is essential to the overall performance of the ES-MDA. The predicted water rates were examined to verify the robustness of the proposed workflow (i.e., the prior models with DNN-SAE for the ESMDA). If reliable posterior models are obtained by the ES-MDA, the water rates can be predicted even though such data were not used for history matching. Fig. 12 shows the water rates observed at the four production wells for the three posterior sets. Both the base (posterior models) and comparison (posterior models with DNN) cases failed to predict the true responses completely (Figs. 12(a) and (b)), while the posterior models with DNN-SAE matched the trend of the true produc­ tion, especially PROD2 and PROD4. (Fig. 12(c)). The predicted water production is important information for designing the size of surface

facilities that process such water. Therefore, the prior models with DNNSAE had a positive effect on the ensemble-based history matching. Table 8 summarizes the MAPEs during the prediction period (from 721 days to 3600 days) for the three posterior sets. The proposed workflow had the smallest averaged MAPE of 11.9% for the oil rates and 9.5% for the water rates. Compared with the posterior models, the posterior models with DNN decreased the averaged MAPE from 48.7% to 36.1% for the oil rates but increased the averaged MAPE from 28.1% to 43.0% for the water rates. Fig. 13 demonstrated the cumulative oil and water productions and, like Figs. 11 and 12, confirmed the positive influence of the prior models with DNN-SAE on history matching. Both the posterior models and posterior models with DNN showed a large difference between the predicted and true production. However, the proposed case provided 14

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acceptable predictions on the cumulative volumes for both oil and water. Table 8 presents a numerical analysis of the cumulative pro­ duction; the proposed case had much improved results compared to the other two cases. Fig. 14 illustrates the mean of the permeability distribution and its histogram for the posterior models with DNN and DNN-SAE. The pos­ terior models with DNN did not mitigate the overshooting problem in the posterior models of Fig. 8(b). Moreover, Fig. 14(a) shows an unre­ alistic mosaic permeability distribution with no connection. However, the proposed case in Fig. 14(b) fixed the overshooting problem and mimicked the Gaussian distribution in the reference field of Fig. 3(b). Compared with the reference field in Fig. 3(a), the posterior models with DNN-SAE in Fig. 14(a) had a similar permeability distribution. Figs. 11(b) and (c) seem that the ensemble collapse problem has occurred because the bands of gray curves are too narrow. If the problem happens, a hundred of posterior models by the ES-MDA become a single model. However, some production responses, e.g. PROD2 in Figs. 12(b) and (c), still have a certain uncertainty range. Also, when standard de­ viation map for the 100 posterior models by the ES-MDA was examined, the DNN and DNN-SAE cases still have variance for each grid (Fig. 15). The proposed case (prior models with DNN-SAE) can help improve the overall history matching performance, as presented in Figs. 11–15. It achieved the MAPEs of less than 5% for the cumulative oil and water volumes in Table 8. Prior models with high reliability can mitigate the overshooting problem of ensemble-based history matching. Data encoding by the machine learning algorithm, SAE, plays a key role in the construction of new reservoir models with the neural network-based inverse modeling scheme.

Author contributions Jaejun Kim: Conceived and designed the analysis, Collected the data, Contributed data or analysis tools, Performed the analysis, Wrote the paper. Sungil Kim: Contributed data or analysis tools, Performed the analysis, Wrote the paper. Changhyup Park: Conceived and designed the analysis, Performed the analysis, Wrote the paper. Kyungbook Lee: Conceived and designed the analysis, Contributed data or analysis tools, Performed the analysis, Wrote the paper. Acknowledgments This study was supported as a project of the Korea Institute of Geo­ science and Mineral Resources (GP2017-024) and Ministry of Trade, Industry and Energy (20172510102090 (NP2017-021), 20162510102040). Additionally, the research facilities were provided by the Institute of Engineering Research at Seoul National University. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.petrol.2019.106800. References Aanonsen, S.I., Nævdal, G., Oliver, D.S., Reynolds, A.C., Vall� es, B., 2009. The ensemble Kalman filter in reservoir engineering—a review. SPE J. 14 (3), 393–412. https:// doi.org/10.2118/117274-PA. Ahn, S., Park, C., Kim, J., Kang, J.M., 2018. Data-driven inverse modeling with a pretrained neural network at heterogeneous channel reservoirs. J. Pet. Sci. Eng. 170, 785–796. https://doi.org/10.1016/j.petrol.2018.06.084. Bengio, Y., Lamblin, P., Popovici, D., Larochelle, H., 2007. Greedy layer-wise training of deep networks. In: Advances in Neural Information Processing Systems. MIT Press, Cambridge, MA. Canchumuni, S.A., Emerick, A.A., Pacheco, M.A., 2017. Integration of ensemble data assimilation and deep learning for history matching facies models. In: Proc. OTC Brasil, 24–26 October, Rio de Janeiro, Brazil. OTC-28015-MS. Chen, Y., Oliver, D.S., 2012. Ensemble randomized maximum likelihood method as an iterative ensemble smoother. Math. Geosci. 44 (1), 1–26. https://doi.org/10.1007/ s11004-011-9376-z. Evensen, G., 1994. Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res. 99 (C5), 10143–10162. https://doi.org/10.1029/94JC00572. Evensen, G., 2018. Analysis of iterative ensemble smoothers for solving inverse problems. Comput. Geosci. 22 (3), 885–908. https://doi.org/10.1007/s10596-0189731-y. Emerick, A.A., 2019. Analysis of geometric selection of the data-error covariance inflation for ES-MDA. J. Pet. Sci. Eng. 182, 106168. https://doi.org/10.1016/j. petrol.2019.06.032. Emerick, A.A., Reynolds, A.C., 2012. History matching time-lapse seismic data using the ensemble Kalman filter with multiple data assimilations. Comput. Geosci. 16 (3), 639–659. https://doi.org/10.1007/s10596-012-9275-5. Emerick, A.A., Reynolds, A.C., 2013. Ensemble smoother with multiple data assimilation. Comput. Geosci.-UK. 55, 3–15. https://doi.org/10.1016/j.cageo.2012.03.011. Erhan, D., Bengio, Y., Courville, A., Manzagol, P.A., Vincent, P., Bengio, S., 2010. Why does unsupervised pre-training help deep learning? J. Mach. Learn. Res. 11, 625–660. Hirose, Y., Yamashita, K., Hijiya, S., 1991. Back-propagation algorithm which varies the number of hidden units. Neural Netw. 4 (1), 61–66. https://doi.org/10.1016/08936080(91)90032-Z. Jafarpour, B., McLaughlin, D.B., 2009. Estimating channelized-reservoir permeabilities with the ensemble Kalman filter: the importance of ensemble design. SPE J. 14 (2), 374–388. https://doi.org/10.2118/108941-PA. Jansen, J.D., Fonseca, R.M., Kahrobaei, S., Siraj, M.M., van Essen, G.M., van den Hof, P. M.J., 2014. The egg model – a geological ensemble for reservoir simulation. Geosci. Data J. 1 (2), 192–195. https://doi.org/10.1002/gdj3.21. Jansen, J.D., 2013. The Egg Model - Data Files. TU Delft. Dataset. https://doi.org/ 10.4121/uuid:916c86cd-3558-4672-829a-105c62985ab2. Jung, H., Jo, H., Kim, S., Lee, K., Choe, J., 2017. Recursive update of channel information for reliable history matching of channel reservoirs using EnKF with DCT. J. Pet. Sci. Eng. 154, 19–37. https://doi.org/10.1016/j.petrol.2017.04.016. Jung, S., Lee, K., Park, C., Choe, J., 2018. Ensemble-based data assimilation in reservoir characterization: a review. Energies 11 (2), 445. https://doi.org/10.3390/ en11020445. Kang, B., Lee, K., Choe, J., 2016. Improvement of ensemble smoother with SVD-assisted sampling scheme. J. Pet. Sci. Eng. 141, 114–124. https://doi.org/10.1016/j. petrol.2016.01.015.

4. Conclusions This study focused on a novel preprocessing method for the input models of the ES-MDA to realize reliable history matching. A deep learning algorithm was implemented to build reliable prior models by integrating dynamic data. The proposed method coupled the DNN al­ gorithm with the SAE for feature extraction to train a pair of reservoir models (output) and simulated productions (input). The DNN-based inverse modeling can be efficiently applied to the ES-MDA because the both methods require hundreds of permeability models and their simulation results. When the DNN was used to generate prior models without the SAE, the input and output layers had 96 and 2491 neurons, respectively. After the SAE was used to successfully compress the original permeability field, the output layer had 100 neurons. Because the number of pa­ rameters in the DNN was dramatically reduced, the nonlinearity be­ tween the input and output layers was alleviated in DNN-SAE. The two sets of prior models from DNN and DNN-SAE reduced the uncertainty range of the production, but only the prior models with DNN-SAE included the true production in the uncertainty range. As a result, the posterior models with DNN performed even worse than the both prior models with DNN and posterior models without DNN in terms of the overshooting problem. However, the posterior models with DNNSAE identified the vertical connectivity of high permeability in the reference Egg model without the overshooting problem. The proposed case also properly mimicked the histogram of the reference field and gave a reasonable prediction for the water production, which was not used for history matching. This research realized the successful coupling of a machine learning algorithm in the field of petroleum engineering. The importance of feature extraction for neural networks (Lee et al., 2019) was demon­ strated, and considering dynamic data for generating the prior models was shown to help mitigate the overshooting problem. Because the DNN and SAE frameworks need to set the sizes of the hidden neurons and layers, design optimization of them remains a challenge for future study.

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