Geofluids

Volume 2018, Article ID 1532868, 21 pages

https://doi.org/10.1155/2018/1532868

## Integration of an Iterative Update of Sparse Geologic Dictionaries with ES-MDA for History Matching of Channelized Reservoirs

^{1}Severe Storm Research Center, Ewha Womans University, 52 Ewhayeodae-gil, Seodaemun-gu, Seoul 03760, Republic of Korea^{2}Department of Climate and Energy Systems Engineering, Ewha Womans University, 52 Ewhayeodae-gil, Seodaemun-gu, Seoul 03760, Republic of Korea^{3}Petroleum and Marine Research Division, Korea Institute of Geoscience and Mineral Resources, 124 Gwahak-ro, Gajeong-dong, Yuseong-gu, Daejeon 34132, Republic of Korea^{4}Department of Energy Resources Engineering, Seoul National University, 599 Gwanak-gu, Seoul 151-744, Republic of Korea

Correspondence should be addressed to Baehyun Min; rk.ca.ahwe@10nimhb

Received 9 March 2018; Revised 11 May 2018; Accepted 23 May 2018; Published 29 July 2018

Academic Editor: Meijing Zhang

Copyright © 2018 Sungil Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This study couples an iterative sparse coding in a transformed space with an ensemble smoother with multiple data assimilation (ES-MDA) for providing a set of geologically plausible models that preserve the non-Gaussian distribution of lithofacies in a channelized reservoir. Discrete cosine transform (DCT) of sand-shale facies is followed by the repetition of K-singular value decomposition (K-SVD) in order to construct sparse geologic dictionaries that archive geologic features of the channelized reservoir such as pattern and continuity. Integration of ES-MDA, DCT, and K-SVD is conducted in a complementary way as the initially static dictionaries are updated with dynamic data in each assimilation of ES-MDA. This update of dictionaries allows the coupled algorithm to yield an ensemble well conditioned to static and dynamic data at affordable computational costs. Applications of the proposed algorithm to history matching of two channelized gas reservoirs show that the hybridization of DCT and iterative K-SVD enhances the matching performance of gas rate, water rate, bottomhole pressure, and channel properties with geological plausibility.

#### 1. Introduction

Calibration of a subsurface system is an essential process to forecast fluid behaviors in a variety of geoenvironments such as aquifers, geothermal reservoirs, and petroleum reservoirs. History matching is an inverse process to find reservoir model parameters honoring observations by integration of static (e.g., core, logging, and seismic) and dynamic data (e.g., oil and gas rate, water cut, bottomhole pressure, and subsidence/uplift) [1]. Ensemble-based data assimilation approaches have been successfully utilized for history matching to provide subsurface models that are well conditioned to observations. For example, the ensemble Kalman filter (EnKF) [2–5], ensemble smoother (ES) [6, 7], and ES with multiple data assimilation (ES-MDA) [8–10]. However, the ensemble-based data assimilation approaches have difficulty in preserving non-Gaussian distributions of model parameters such as lithofacies [11–14]. In the ensemble-based data assimilation approaches, model parameters lose the non-Gaussianity of their original distributions that are initially constrained and the distributions of the model parameters get close to Gaussian ones.

Shin et al. [14] and Zhou et al. [15] suggested using normal score transform in the ensemble-based data assimilation approaches to preserve non-Gaussian distributions of model parameters. Non-Gaussian model parameters are transformed into Gaussian model parameters using normal score transform, and then finally, updated model parameters are backtransformed. Moreover, transformation can take advantage of parameterization if the number of essential transformed parameters is smaller than the number of original parameters in terms of saving computational cost and figuring out main features of parameters. For example, the discrete cosine transform (DCT) [16–21], fast Fourier transform [22, 23], grid connectivity transform [24], level set [13, 25], and sparse geologic dictionaries [26, 27] have been applied to reservoir characterization. In particular, Fourier transform-based methods such as DCT are capable of capturing essential traits such as main shapes and patterns of a facies channel reservoir [16, 17] but reveal a deficiency in describing a crisp contrast among different facies because of data loss from inverse transformation [28].

Sparse coding refers to the process of computing representation coefficients based on the given signal and dictionaries [29]. In sparse coding, the dictionaries indicate groups of features capable of brief expressions to represent various phenomena in the environment [30]. In geological modeling, sparse geologic dictionaries are used to represent models with sparse linear combinations of basis vectors that are essential geologic features of a reservoir [29]. Extracting essential geologic features and reducing the number of reservoir variables can be accomplished using sparse coding, thereby facilitating ensemble-based history matching [27]. Aharon et al. [29] showed the efficacy of K-singular value decomposition (K-SVD) resulting from the accelerated convergence for image reconstruction, which led Sana et al. [31] to build an archive of essential geologic features called sparse geologic dictionaries from thousands of static reservoir models using K-SVD and calibrate reservoir models with the dictionaries using EnKF. One drawback of K-SVD is its large size of sparse geologic dictionaries. References showed that sparse coding with a transformation of parameter space could reduce both computational complexity and costs that are simultaneously required for model calibration [26, 27, 32]. In this study, we note that the previous works have not considered the quality of sparse geologic dictionaries. The quality indicates how well reservoir models can be properly reconstructed by prototypes of dictionaries. Also, we expect an improvement in the history-matching performance by enhancing the quality of sparse geologic dictionaries.

This study proposes a hybridized ES-MDA algorithm that implements sparse coding in a transformed space to outperform previous history-matching methods by providing more accurate reconstructions of highly non-Gaussian model parameters. The proposed algorithm transforms multimodal facies into coefficients of discrete cosine functions using DCT. Invoking DCT is followed by iterating K-SVD for updating sparse geologic dictionaries. In each assimilation of ES-MDA, the combination of DCT and iterative K-SVD is performed to update the dictionaries and improve the quality of reservoir models. For brevity, the proposed algorithm with updated sparse geologic dictionaries is called ES-MDA-DCT-i-K-SVD in this paper. The performance of ES-MDA-DCT-i-K-SVD is tested with applications for channelized gas reservoirs and is compared with those of four ES-MDA algorithms: conventional ES-MDA, ES-MDA coupled with DCT (called ES-MDA-DCT in this paper), ES-MDA coupled with K-SVD (called ES-MDA-K-SVD in this paper), and ES-MDA coupled with DCT and K-SVD (called ES-MDA-DCT-K-SVD in this paper).

#### 2. Methodology

The novelty of the proposed algorithm ES-MDA-DCT-i-K-SVD is the integration of ES-MDA (Section 2.1), the dimensionality reduction of the parameter space using DCT (Section 2.2), and construction (Section 2.3) and updating (Section 2.4) of geologic dictionaries using sparse coding in the reduced space. Section 2.5 describes how the methods operate in the framework of ES-MDA-DCT-i-K-SVD in a complementary manner.

##### 2.1. ES-MDA

The goal of history matching can be formulated as where is the objective function of history matching and is the state vector composed of reservoir variables (e.g., permeability and facies).

The typical form of for ensemble-based history matching is presented as [33] where is the state vector before update and the superscript refers to background; is the covariance matrix of ; is the observed responses; is the dynamic vector composed of simulated responses obtained by running a reservoir simulator for the state vector ; and is the covariance matrix of observation error. Note that the right-hand side of (2) is the addition of background and observation error terms [33]. Because can contain any unknown variables such as facies indexes, coefficients of discrete cosine functions or dictionary coefficients depending on the type of algorithms were used in this study.

can be used to derive the update equation for as [8, 33]
where the subscript refers to the *i*th ensemble member; is the cross-covariance matrix of and ; is the autocovariance matrix of ; is the coefficient to inflate , which is the covariance matrix of the observed data measurement error [8]; is the observation data perturbed by the inflated observed data measurement error; and is the ensemble size (i.e., number of reservoir models in the ensemble). Conventionally, ensemble-based history matching updates reservoir models simultaneously. In (3), refers to Kalman gain , which is computed with regularization by SVD using 99.9% of the total energy in singular values [8].

The main difference between ES and ES-MDA is the update process of the state vector . ES updates the state vector of each ensemble member using observation data measured at all time steps (Emerick and Reynolds, 2011). Compared to the single assimilation of ES, ES-MDA assimilates every state vector times using an inflated covariance matrix of measurement error [8, 9]. Here, is the number of assimilations in ES-MDA.

Definitions of and are as follows: where is the mean of state vectors and is the mean of dynamic vectors.

In ES-MDA, is constrained to

In ES, and due to its single assimilation.

The perturbed observation data shown in (3) is computed as

The second term on the right-hand side of (6) is the perturbation term, which reflects the uncertainty associated with data measurement, processing, and interpretation. Stochastic characteristics of are reflected by . is the random error matrix to observations, which is generated with a mean of zero and a standard deviation of , where is the number of time steps in observations.

##### 2.2. Extraction of Geologic Features Using Discrete Cosine Transform

Discrete cosine transform (DCT) has been utilized as an image-processing tool for characterization of channelized reservoirs due to the periodicity of cosine functions [34]. DCT converts parameters into coefficients of discrete cosine functions. The coefficients are sorted in descending order from the top left, capturing the overall trend of channel patterns, to bottom right, delineating details in channel patterns. Previous studies have shown that non-Gaussian channel patterns can be reproduced sufficiently via inverse transform of essential DCT coefficients [18, 28, 35]. Updating the truncated DCT coefficients can yield a calibrated model set. Another advantage of DCT is the improvement in computational efficiency resulting from data compression, which is effective in constructing sparse geologic dictionaries described in Section 2.3.

Figure 1 illustrates how to extract geologic features from an image of a target channelized reservoir using a truncated DCT and reproduce the target reservoir through an inverse DCT (IDCT). Two images in the first row represent the physical state of sand and shale facies in the target reservoir, that is, the original image on the left and the reproduced image on the right. Let and denote the number of gridblocks of the reservoir model and the number of essential DCT coefficients, respectively. Applying DCT to the original 75 by 75 image yields an image composed of DCT coefficients, as shown in the bottom-left corner of Figure 1. In the bottom-right corner of Figure 1, filtering the coefficient state selects 465 components in the dotted triangle as essential ones. It seems that this small number of components is sufficient to restore the original image of the physical state (i.e., channel patterns) when comparing the two subfigures in the first row.