Methods to Mitigate Loss of Variance Due to Sampling Errors in

Methods to Mitigate Loss of Variance Due to Sampling Errors in Ensemble Data Assimilation with Non-local Model Parameters Johann M. Lacerdaa , Alexandre A. Emerickb , Adolfo P. Piresc a

Petrobras and UENF b Petrobras c UENF

Abstract Ensemble data assimilation methods are among the most successful techniques for assisted history matching. However, these methods suffer from sampling errors caused by the limited number of ensemble members employed in practical applications. A typical consequence of sampling errors is an excessive loss of ensemble variance. In practice, the posterior ensemble tends to underestimate the uncertainty range in the parameter values and production predictions. Distance-based localization is the standard method for mitigating sampling errors in ensemble data assimilation. A properly designed localization matrix works remarkably well to improve the estimate of gridblock properties such as porosity and permeability. However, field history-matching problems also include non-local model parameters, i.e., parameters with no spatial location. Examples of non-local parameters include relative permeability curves, fluid contacts, global property multipliers, among others. In these cases, we cannot use distance-based localization. This paper presents an investigation on several methods proposed in the literature that can be applied to non-local model parameters to mitigate erroneous loss of ensemble variance. We use a small synthetic history-matching problem to evaluate the performance of the methods. For this problem, we were able to compute a reference solution with a very large ensemble. We compare the methods mainly in terms of the preservation of the ensemble variance without compromising the ability of matching the observed data. We also analyse the robustness and difficulty to select proper configurations for each method. Keywords: Ensemble data assimilation, history matching, ensemble smoother, Preprint submitted to Journal of Petroleum Science and Engineering

August 3, 2018

sampling errors, localization, covariance inflation

1

1. Introduction

2

Ensemble-based methods proved to be useful techniques to solve history-matching

3

problems. In spite of their success in several field applications reported in the lit-

4

erature (Skjervheim et al., 2007; Bianco et al., 2007; Evensen et al., 2007; Haugen

5

et al., 2008; Cominelli et al., 2009; Zhang and Oliver, 2011; Emerick and Reynolds,

6

2011a, 2013b; Chen and Oliver, 2014; Emerick, 2016; Luo et al., 2018b), some chal-

7

lenges remain. Perhaps the main concerns are related to sampling errors. Because

8

of the computational cost of reservoir simulations, the data assimilation in reser-

9

voir engineering is done with small-sized ensembles, typically on the order of 100

10

members. Unfortunately, the use of small ensembles causes inaccurate estimation

11

of covariance matrices introducing spurious correlations between predicted data and

12

model parameters. The consequence of spurious correlation is typically manifested

13

as significant underestimation of posterior variances (Aanonsen et al., 2009). Even

14

an absolute collapse of ensemble can arise in some extreme cases. It is interesting to

15

note that sampling errors can be exacerbated by nonlinearity in the model. Raanes

16

et al. (2018) illustrate this fact using a simple univariate problem with a linear and a

17

nonlinear model, both cases with the same posterior (Gaussian) distribution. They

18

showed that for a linear model the sampling errors are quickly attenuated, and the

19

ensemble statistics converge to the exact ones. However, for the nonlinear model

20

the sampling errors are chronic.

21

Localization (Houtekamer and Mitchell, 2001) is a widely used method to mit-

22

igate problems due to spurious correlations and has the additional advantage of

23

increasing the degrees of freedom to assimilate data. Most of localization methods

24

rely on distances, which means that they can be applied when both, model variables

25

and observations, have associated spatial locations. Although it is not necessar-

26

ily obvious to specify an appropriate size for the localization region, the number

27

of publications with successful applications is quite extensive; see, e.g., (Chen and

28

Oliver, 2010b, 2014; Emerick and Reynolds, 2011a,b; Emerick, 2016). There are two

29

main localization procedures in the literature: Schur-product localization and do2

30

main localization. Sakov and Bertino (2011) compared Schur product and domain

31

localization and concluded that both methods produce similar results for typical

32

geophysical data assimilation problems. However, Schur-product localization may

33

be more accurate for “strong” (accurate measurements) data assimilation problems.

34

Nevertheless, the performance of both methods is highly dependent of the choice of

35

the localization region.

36

However, important parameters in history matching such as permeability and

37

porosity multipliers, relative permeability curves, analytical aquifer parameters, rock

38

and fluid compressibilities, water-oil and gas-oil depth contacts may not have asso-

39

ciated spatial relation with typical observations such as oil, water and gas rates,

40

pressure measurements and 4D seismic. For these cases, a non-distance dependent

41

localization scheme is necessary. Here, we refer to these parameters as “non-local”

42

parameters1 .

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There exists localization schemes designed for general covariance structures with-

44

out making the assumption that covariance depends on the distance between model

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parameters and data. However, these methods have been rarely applied to history-

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matching problems. Perhaps, the first work on a non-distance localization scheme

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is the hierarchical filter from Anderson (2007a). In this method, an ensemble of Ne

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members is divided into Nh sub-ensembles of size Ne /Nh , which are used to correct

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the regression coefficients (entries of the Kalman gain matrices). Vall`es and Næv-

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dal (2009) tested the hierarchical filter in the Brugge case (Peters et al., 2010) and

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concluded that it improved the diversity in the updated models compared to the

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standard ensemble Kalman filter (EnKF). Zhang and Oliver (2010) proposed that

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instead of Nh ensembles of Ne members, we can resample a single ensemble with

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replacement to generate Nb bootstrapped ensembles. They tested the method in

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a synthetic history matching problem and concluded that it improved the results 1

The terminology localization was introduced after (Houtekamer and Mitchell, 2001) based on

the idea of limiting the spatial location of the updates, removing long-distance correlations. Hence, for non-local model parameters the terminology localization seems less appropriate. Nevertheless, here we still use the term localization for historical reasons and because it is well diffused in the literature.

3

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obtained by EnKF with a small ensemble (Ne = 30) for the estimation of grid-

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block permeabilities. Furrer and Bengtsson (2007) present a simple expression that

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can be used as a non-distance localization scheme. They minimized term-by-term

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the norm of the difference between the true forecast covariance matrix and the lo-

60

calized estimate ignoring the positive-definiteness constraint. The final expression

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depends on the true covariance. The authors suggest to replace the true covariance

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by the ensemble estimate to construct a non-distance localization function, which is

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computationally inexpensive and straightforward to implement. Bishop and Hodyss

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(2007) proposed to obtain the localization matrix by raising each element of a sam-

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ple correlation matrix of model variables to a certain power. To the best of our

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knowledge, there are no history-matching applications of non-distance localizations

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schemes from Furrer and Bengtsson (2007) and Bishop and Hodyss (2007) reported

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in the literature. More recently, Luo et al. (2018) proposed a localization scheme

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that does not rely on the spatial locations and update model parameters using only

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the observations that have relatively high correlations with them. They tested the

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method to update gridblock porosity and permeability in the presence of seismic

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data showing that the proposed method avoided ensemble collapse.

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Another procedure that is quite common in the literature to compensate the un-

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derestimation of posterior variances in ensemble data assimilation is covariance infla-

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tion (Anderson and Anderson, 1999). Covariance inflation is often used in oceanogra-

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phy and numerical weather prediction applications (Anderson, 2007b; Hamill et al.,

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2001; Li et al., 2009), but has rarely been applied in reservoir history-matching prob-

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lems. Perhaps one of the few works using covariance inflation for history matching

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was presented by Chen and Oliver (2010a). They concluded that covariance infla-

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tion was useful to compensate for the insufficiency of the ensemble variability for the

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Brugge case. In covariance inflation, the forecast ensemble is replaced by another

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one with the same mean but with an inflated variance. The inflation factor is a

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number slightly greater than one, but its choice is problem dependent. Typically a

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higher inflation factor is required for smaller ensembles.

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Ensemble methods such as EnKF and ensemble smoother use a “perturbed ob-

86

servation scheme,” which means that a random vector is added to the measurements 4

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before the analysis. However, it has been shown that the perturbed observations is

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an additional source sampling errors (Anderson, 2001; Whitaker and Hamill, 2002;

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Tippett et al., 2003). This fact motivated the development of deterministic or

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square-root schemes which avoid the need of perturbed observations. In spite of

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being standard methods in areas such as oceanography and numerical weather pre-

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diction, deterministic schemes have been rarely used in reservoir history matching.

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A review on the main square-root schemes can be found in (Tippett et al., 2003;

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Sakov and Oke, 2008a). Among these methods, the deterministic EnKF (DEnKF)

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(Sakov and Oke, 2008b) has the practical advantage of having an implementation

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very close to the standard EnKF. In fact, both methods compute the same Kalman

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gain, allowing Schur product localization. Besides avoiding the need of perturbed

98

observations, DEnKF has a “built-in” covariance inflation, which makes the method

99

less prone to variance underestimation than EnKF.

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Despite of the aforementioned works on non-distance localization schemes and

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covariance inflation, our experience is that these methods are less effective than

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distance-based localization in eliminating sampling errors in gridblock properties

103

such as porosity and permeability. Emerick (2012, Chap. 4) compared some of these

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methods and concluded that only distance-based localization was able to prevent a

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severe variance reduction in two synthetic history matching problems. Most of the

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previous works, including the comparisons presented in (Emerick, 2012), consider

107

only the problem of localization of gridblock parameters. However, as previously

108

mentioned, there are several types of non-local model parameters in real-life reservoir

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history-matching problems, in which case the development of robust non-distance

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localization schemes is highly desirable. Surprisingly, to the best of our knowledge,

111

publications about this problem in the history-matching literature are non-existent.

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This work attempts to contribute to fill this lacuna. We present a comparison

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among five non-distance localization methods from literature, covariance inflation

114

and a deterministic scheme to address the problem of excessive loss of variance.

115

Moreover, we also propose a simple sensitivity analysis procedure which can be used

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to select or improve the localization coefficients for non-local parameters.

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This paper is organized as follows: the next section briefly reviews the base data 5

118

assimilation method used in this paper. The section after that describes the test

119

case proposed to compare the methods. Then, we present a separate section for each

120

method investigated. In each of these sections, we provide a brief description of the

121

method followed by the corresponding results. After that, we present a section with

122

a discussion of the overall performance of the methods. The last section of the paper

123

presents the conclusions.

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2. Ensemble Data Assimilation

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2.1. Ensemble Smoother with Multiple Data Assimilation

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Recent papers indicate that iterative forms of the ensemble smoother (ES) are

127

better suited for reservoir history matching than the standard sequential data assim-

128

ilation from EnKF (Chen and Oliver, 2012, 2013; Emerick and Reynolds, 2013a; Luo

129

et al., 2015; Stordal and Elsheikh, 2015; Le et al., 2016; Ma et al., 2017). Among the

130

iterative ES, the ensemble smoother with multiple data assimilation (ES-MDA) (Em-

131

erick and Reynolds, 2013a) has been widely used in history-matching applications.

132

In this paper, we use the standard ES-MDA, although the methods investigated here

133

could be used with other iterative smoothers.

134

ES-MDA was introduced in (Emerick and Reynolds, 2013a) with the objective

135

to improve the history-matching results obtained by ES. In the ES-MDA, the same

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data are assimilated multiple times with an inflated covariance of the measurement

137

errors. The ES-MDA analysis equation for a vector of model parameters m ∈ RNm

138

can be written as

mk+1 = mkj + Ckmd Ckdd + αk Ce j


−1

dobs + ekj − g mkj





,

(1)

139

for j = 1, . . . , Ne , with Ne denoting the ensemble size. In the above equation, the

140

superscript k stands for the data assimilation index; dobs ∈ RNd is the vector of

141

observations, ekj ∈ RNd is the vector of random perturbations which is obtained by

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143

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sampling N (0, αk Ce ), with Ce ∈ RNd ×Nd denoting the data-error covariance matrix
and αk denoting the ES-MDA inflation coefficient. g mkj ∈ RNd is the vector of predicted data. Ckmd ∈ RNm ×Nd is the matrix containing the cross-covariance values

6

145

between model parameters and predicted data and Ckdd ∈ RNd ×Nd is the covariance

146

matrix of predicted data. Both matrices are estimated using the ensemble, i.e.,

Ckmd 147

Ne   >

1 X g mkj − g (mk ) g mkj − g (mk ) , Ne − 1 j=1

Ne 1 X = mk Ne j=1 j

and g (mk ) =

150

151

(3)

where

mk

149

(2)

and

Ckdd = 148

Ne   >
1 X k k k k mj − m = g mj − g (m ) , Ne − 1 j=1

(4)

Ne
1 X g mkj . Ne j=1

(5)

In the standard implementation of ES-MDA, the inflation coefficient αk > 1 is a applied a pre-defined number of times, Na , and the set {αk }N k=1 must satisfy the conP a −1 dition N k=1 αk = 1. This condition was derived by Emerick and Reynolds (2013a)

152

to ensure that ES-MDA is consistent with the standard ES for linear-Gaussian prob-

153

lems. The same condition can also be derived directly from Bayes’ rule as discussed

154

in (Stordal and Elsheikh, 2015; Emerick, 2016). Here, we use the standard choice of

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αk = Na = 4 (Emerick and Reynolds, 2013a; Emerick, 2016).

156

Defining the Kalman gain matrix as

Kk = Ckmd Ckdd + αk Ce 157


−1

,

(6)

we can rewrite Eq. 1 in a compact form

mk+1 = mkj + Kk dobs + ekj − g mkj j





.

(7)

158

In this paper, the matrix inversion in Eq. 6 is done using truncated singular

159

value decomposition (TSVD). The number of singular values retained in the TSVD is

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defined by keeping 99% of the sum of the singular values. It is important to note that

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TSVD acts as a regularization of the inversion helping to stabilize the Kalman gain. 7

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This procedure is known to help to reduce undesirable variance loss in the ensemble,

163

especially in the case of a large number of redundant measurements. However, in

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our experience, TSVD alone is not enough to avoid variance underestimation and

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some sort of localization or inflation scheme are required in realistic settings. On

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a side note, we usually apply TSVD before the construction of the matrix Ckdd ,

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because it results in a more efficient method when the number of data points is

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larger. This procedure is known as subspace inversion (Evensen, 2004). Nevertheless,

169

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here because we consider a relatively small test case, we apply TSVD directly to
Ckdd + αk Ce rescaled by the data variance to avoid loss of relevant information,

171

as discussed in (Wang et al., 2010).

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2.2. Sampling Errors

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The limited size of the ensembles employed in practice introduces sampling errors

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which generates spurious correlations between data and model parameters. As a

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result, we observe incorrect changes in model parameters resulting in excessive loss

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of variance. In the extreme cases, even a collapse of the ensemble may be observed.

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Hamill et al. (2001) investigated the effect of sampling errors in a hypothetical

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problem with two parameters and one observation. Here, we present a shortened

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version of their result to highlight some points that are important in the analysis of

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the results presented in the next sections.

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Assume that we want to estimate the porosity and permeability of a rock sample,

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m = [φ κ]> , given a single observation of porosity with variance σe2 . Assume that

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there is a prior correlation between porosity, φ, and permeability, κ, and that the

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actual prior covariance matrix has the form  Cm = 

σφ2

ρσφ σκ

ρσφ σκ

σκ2

 ,

(8)

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where σφ and σκ are the prior standard deviation of porosity and permeability,

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respectively, and ρ is the prior correlation coefficient between φ and κ. Suppose

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e m where the variances are that instead of Cm , we have an inaccurate version of C

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correct but the cross-covariances are corrupted with an additive sampling error cε ,

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i.e., 8

 em =  C

σφ2

ρσφ σκ + cε

ρσφ σκ + cε

σκ2

 .

(9)

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For a given observation φobs , it is straightforward to show that the maximum a

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posteriori (MAP) (Tarantola, 2005) estimate of κ is

κmap = κpr +

ρσφ σκ (φobs − φpr ) σφ2 + σe2

(10)

if we use the correct prior covariance and κ emap = κpr +

ρσφ σκ + cε σφ2 + σe2

! (φobs − φpr )

(11)

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e m . In the above expressions, φpr and κpr are if we use the corrupted covariance C

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the prior estimates of φ and κ, respectively. We want to analyse the error in the

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estimate of κ. It follows that the expected squared error has the form   ρ2 σφ2 σκ2 E (κ − κ emap )2 = σκ2 − 2 σφ + σe2

 1−

cε ρσφ σκ

2 ! .

(12)

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The derivation of (12) is straightforward and follows from the direct application of

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the definitions of expectation and variance. Hamill et al. (2001) interpreted the term

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cε ρσφ σκ

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points: (i) if

199

clearly an inconsistency (there is a degradation of the estimate of κ by assimilating

200

data); (ii) If the actual correlation coefficient between φ and κ is small, it is more

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likely that we have a degradation in our estimates due to the sampling error.

as a “relative error” in the covariance. From Eq. 12, we can emphasize two cε ρσφ σκ

> 1 the variance of κ increases by assimilating data, which is

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The second point is particularly important because weak correlations are harder

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to estimate with limited samples. In order to illustrate this last statement, we con-

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ducted a small numerical experiment to estimate coefficient between two parameters,

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φ and κ, based on an ensemble of Ne = 50 members. We consider two cases where

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the true correlations are ρ = 0 and ρ = 0.8 and repeat the sampling 100 times.

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Fig. 1 shows the histograms of the estimates of ρ, which clearly indicates a large

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variance for the case with ρ = 0.

9

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2.3. Localization

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Localization is the standard technique to reduce spurious correlations caused by

211

sampling problems in ensemble-based methods. Localization is done by making the

212

Schur (Hadamard) product (element-wise product of matrices) between a correlation

213

matrix and a covariance matrix. This Schur product has the property of increasing

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the degrees of freedom available to assimilate data (Aanonsen et al., 2009). Here,

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we apply the Schur product directly to the Kalman gain matrix because this leads

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to an efficient implementation for the case with a large number of measurements

217

(Emerick, 2016). This procedure corresponds to our standard implementation for

218

practical applications. In this case, the ES-MDA analysis equation can be written

219

as

mk+1 = mkj + R ◦ Kk j




dobs + ekj − g mkj





.

(13)

220

where ◦ denotes the Schur product and R ∈ RNm ×Nd is the so-called localization

221

matrix.

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In the typical application of ES-MDA to update porosity and permeability fields,

223

the components of the matrix R are computed assuming a correlation function with

224

compact support with argument defined by the distance between the location of each

225

datum, usually the position of a well, and the parameter location, the position of

226

a gridblock. Hence, the name localization. For non-local parameters, on the other

227

hand, we cannot define distances, but the overall idea of applying the Schur product

228

is still valid. In this case, we need to derive alternative schemes to compute the

229

entries of the matrix R without relaying in the Euclidian distance between data

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and parameters. It is worth noting that here we emphasize the case of non-local

231

parameters because it is a very common problem in history matching. However,

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the same situation occurs with non-local data, for example, if one is interested in

233

assimilating data from the total field production as opposed to assimilate data on a

234

well-by-well basis.

10

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3. Test Problem

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The test problem consists of a modified version of the well-known PUNQ-S3 case

237

(Floris et al., 2001). The PUNQ-S3 model contains 19 × 28 × 5 gridblocks, of which

238

1761 blocks are active (Fig. 2). The reservoir is supported by a strong analytical

239

aquifer on West and South sides. There are six oil producing wells under oil-rate

240

control operating during the historical period. We consider a production history

241

with a first year of extended well testing, followed by a three-years shut-in period

242

and then four more years of production. There are five infill-drilling wells starting

243

production in the forecast period. The observed data corresponds to measurements

244

of water (WPR) and gas (GPR) production rates and bottom-hole pressure (BHP).

245

The observations were corrupted with a Gaussian noise with zero mean and standard

246

deviation corresponding to 10% of the data value for rate data and 1% for BHP data.

247

In the original PUNQ-S3 case the history matching parameters are the poros-

248

ity and the horizontal and vertical permeability distribution. However, because

249

our goal is to evaluate the performance of techniques for non-local model param-

250

eters, we consider a different set of parameters: five porosity multipliers (MULT-

251

POR1, ..., MULTPOR5) and five permeability multipliers (EXP MULTIPERM1, ...,

252

EXP MULTIPERM5), one for each layer of the model. The parameters for perme-

253

ability multipliers correspond to the exponent of a base-10 power. For example, the

254

permeability multiplier applied to the first layer of the model is 10EXP MULTIPERM1 .

255

We also included one parameter defining the rock compressibility (CCPOR), two

256

parameters to compute variations in the water-oil (DELTA DWOC) and gas-oil

257

(DELTA DGOC) contact depths and two exponents defining the analytical aquifer

258

radius (EXP AQRADIUS1 and EXP AQRADIUS2). Table 1 shows the prior distri-

259

butions for these parameters including the true values used to generate the synthetic

260

production history. Note that we intentionally selected the prior mean for the param-

261

eters EXP AQRADIUS1, EXP AQRADIUS2, DELTA DWOC and DELTA DGOC

262

biased compared to the true values. The parameters were selected such that we

263

have a wide spectrum of data sensitivity. For example, the selected measurements

264

are highly sensitive to permeability multipliers and fluid contacts, while rock com-

265

pressibility and aquifer radius have a moderate sensitivity and porosity multiplies are 11

266

weakly sensitive. Therefore, we expect different levels of variance reduction in these

267

parameters. Moreover we added five insensitive parameters, labeled as DUMMY1,

268

..., DUMMY5. Because these parameters are not related to data, we expect any

269

variance reduction to be due to sampling errors. Hereafter, we refer to the sensitive

270

parameters as “actual” and the five insensitive as “dummy” parameters.

271

In order to compare the methods, we consider ten independently sampled prior

272

ensembles with 100 members each. All cases use the same prior ensembles with

273

ES-MDA with αk = Na = 4 for k = 1, . . . , Na . We consider the following metrics to

274

compare the methods:

275

(i) Average normalized data-mismatch objective function: Ne 1 X Od (m) = Od (mj ) Ne j=1

276

(14)

where

Od (mj ) =

1 (dobs − g(mj ))> C−1 e (dobs − g(mj )) . 2Nd

(15)

277

Here, based on our experience with the test case and the arguments presented

278

in Oliver et al. (2008, Chap. 8), we assume that a good data match correspond to a

279

value of Od (m) approximately equal to or less than one. The total number of data

280

points in the test problem is Nd = 2784.

281

(ii) Root-mean-square error (RMSE): v u 2 Nm  u 1 X mi − mtrue,i t RMSE = , Nm i=1 σm,i

(16)

282

where mtrue,i is the true value for the ith model parameter and mi is the correspond-

283

ing ensemble mean. σm,i is the prior standard deviation of the ith model parameter.

284

(iii) Sum of normalized variance (SNV): Nm 1 X var[m0i ] SNV = , Nm i=1 var[mi ]

12

(17)

285

where var[m0i ] and var[mi ] denote the variance of the ith posterior and prior model

286

parameters, respectively. The SNV is used as an approximate measure of the re-

287

duction in uncertainty (Oliver et al., 2008). Note that SNV = 0 means ensemble

288

collapse and SNV = 1 means no uncertainty reduction.

289

As final remark on our test problem, we note that even though sampling errors

290

are the main cause of underestimation of the posterior variances, in practice, we

291

may observe similar problems caused by neglecting or poor treatment of model and

292

measurement errors. These cases are not addressed in this paper. Nevertheless, it

293

is important to note that because we used a controlled synthetic problem, we have

294

no model error and the correct level of measurement errors is known. Therefore, we

295

only have the sampling errors problem.

296

3.1. Reference Case

297

The sampling errors are caused by the limited size of the ensemble. Conceptually,

298

increasing the size of the ensemble mitigates this problem, however, computational

299

efficiency limits the size of the ensemble in practical applications. Nevertheless, for a

300

small problem such as the PUNQ-S3 case it is computationally feasible to consider a

301

very large ensemble. Therefore, we consider a case with standard ES-MDA without

302

localization and an ensemble of Ne = 50000 members as reference for comparisons.

303

The last row of the Table 2 presents the values of Od (m), RMSE and SNV obtained

304

by the reference case. The results in this table show that for the actual model

305

parameters, we observe an average reduction in the ensemble variance of 23%. For

306

the dummy parameters the SNV is 1.00, indicating that the selected size for the

307

ensemble is enough to mitigate significant effects of sampling errors. The value of

308

Od (m) is 0.8, which is less than one, indicating a good data match according to our

309

criterium. The value of the RMSE is 0.89.

310

3.2. No Localization Case

311

We initially consider the data assimilations results with standard ES-MDA with-

312

out any localization or inflation method. Table 2 summarizes the results for this

313

case. The results in this table indicates good data matches with Od (m) inferior to

314

the reference case. The average RMSE is 1.19, which is 34% higher than reference 13

315

value. This result shows a reduction in the ability of the data assimilation to identify

316

the true values of the parameters. However, the main differences are in the values

317

of SNV. For the actual model parameters the average value of SNV is 0.10, which is

318

less than half of the reference result. For the dummy parameters the average SNV

319

is only 0.43. These results show a significant variance underestimation, evidencing

320

the need of localization or inflation.

321

3.3. Pseudo-optimal Localization

322

Furrer and Bengtsson (2007) derived a localization function for general covari-

323

ance structures without relaying on distance between variables. They minimized

324

term-by-term the norm of the difference between the true covariance and the local-

325

ized estimate ignoring the positive-definiteness constraint. The resulting expression

326

depends on the true covariance

ri,j =

c2i,j


, c2i,j + c2i,j + ci,i cj,j /Ne

(18)

327

where ri,j is the (i, j)th entry of the localization matrix R and ci,j is the corre-

328

sponding covariance value. For the cases of interest in this paper, ci,j represents the

329

covariance between the ith model parameter and the jth predicted data, i.e., ci,j is

330

an entry of the matrix Cmd . Furrer and Bengtsson (2007) proposed to replace ci,j by

331

the corresponding ensemble estimate, forming a simple and computationally inex-

332

pensive non-distance dependent localization procedure. In the following, we refer to

333

this procedure as POL, which stands for pseudo-optimal localizaiton. The authors

334

also suggested that sparseness can be introduced by zeroing small values of ci,j .

335

Although the definition of small value for ci,j is problem-dependent, a reasonable

336

choice is to set rij = 0 if √ |ci,j | <  ci,i cj,j ,

(19)

337

where  > 0 is small user-defined threshold. Our tests indicate that the choice of 

338

has a significant impact in the overall performance of the method.

339

Table 3 summarizes the results of POL method with  = 10−3 and  = 10−1 .

340

The results for the case  = 10−3 are more stable in terms of the data match. Note 14

341

that the case with  = 10−1 resulted in large values of Od (m) for the fifth and eighth

342

ensembles. In terms of RMSE, both cases resulted in average values smaller than the

343

reference case. The most interesting results are the SNV. The case with  = 10−1

344

resulted in an average SNV of 0.98 for the dummy parameters, which is very close

345

to the reference. However, the average SNV of the actual parameters is considerable

346

higher than the reference. The case with  = 10−3 resulted in a better value for the

347

average SNV for the actual parameters, but only 0.77 for the dummy parameters.

348

Even though the mean SNV of dummy parameters for the case  = 10−1 is close

349

to the unity, it is important to note that for some ensembles, we obtained SNV

350

larger then one, which can be explained by the discussion in the Section 2.2 when

351

the relative error in the covariance is larger than one. Overall, we concluded that

352

 = 10−3 is a better choice for our test problem. We also tested the cases with  = 0

353

and  = 10−2 . Fig. 3 presents boxplots of SNV for different values  indicating that

354

the method is relatively stable for  ≤ 10−2 .

355

ES-MDA can be considered an iterative method in the sense that the analysis

356

equation is applied multiple times. Before each iteration, the covariance matrices

357

are updated with the current ensemble. Therefore, it is reasonable apply Eq. 18

358

before each iteration to update the localization coefficients. However, our tests indi-

359

cate that it is better to compute the localization matrix R based only on the prior

360

ensemble. The results shown in Table 3 and Fig. 3 were obtained computing R

361

using the prior ensemble. It is not completely clear the reason for this behaviour.

362

However, one possible explanation is that the application of the analysis builds cor-

363

relations in the ensemble members. This effect is referred to as inbreeding in the

364

literature (Houtekamer and Mitchell, 1998; van Leeuwen, 1999). Fig. 4 illustrates

365

this effect showing cross-plots between the parameter DUMMY5 and the predicted

366

BHP from well PRO-4 at time 2921 days for each iteration. The estimated correla-

367

tion coefficient is very small for the prior ensemble. However, after the first iteration,

368

a spurious correlation between DUMMY and BHP appears, which affects the com-

369

putation of the localization coefficient. Moreover, the variance of the predicted data

370

tends to reduce after iterations, this is a natural consequence of the improvement

371

of the data match. Note that the horizontal axes of the plots in Fig. 4 are different 15

372

and that the variance of BHP clearly reduces. It is conceivable that this fact also

373

impacts the calculation of the localization coefficient with Eq. 18. The data variance

374

corresponds to the entry cj,j in this equation. Hence, small values of cj,j tend to

375

increase the value of rj,j .

376

3.4. Bootstrap Localization

377

Zhang and Oliver (2010) presented a modification of the hierarchical filter of

378

Anderson (2007a), where instead of Nh ensembles of Ne members, a single ensemble

379

is resampled with replacement to generate Nb bootstrapped ensembles. This modi-

380

fication has the advantage of avoiding the cost of running additional ensembles. In

381

this method, the Nb bootstrapped ensembles are used only to compute the confi-

382

dence factors; the data assimilation is done with the original ensemble. They use

383

the bootstrapped ensembles to generate Nb Kalman gains, K∗n , and estimate the

384

2 population variance, σK , as i,j

2 σK i,j

Nb 1 X = ([K∗n ]i,j − [K]i,j )2 Nb n=1

(20)

385

where [K∗n ]i,j and [K]i,j denote the (i, j)th entry of the nth bootstrapped and original

386

Kalman gain matrices, respectively.

387

Zhang and Oliver (2010) followed the same procedure presented in Anderson

388

(2007a) to derive an expression for confidence factors for the Kalman gain, which

389

are used as entries of the localization matrix R. Because the resulting procedure

390

sometimes generates negative values, Zhang and Oliver (2010) proposed to add a

391

regularization term and derived the following expression for the confidence factors

ri,j =

1+

2 βi,j

1 , (1 + 1/σr2 )

(21)

392

2 2 where βi,j = σK /[K]2i,j and σr is a weighting factor for regularization of the estimate i,j

393

of ri,j . Here, we refer to this method as bootstrap localization (BL).

394

We tested BL method using different values for σr with Nb = 50 and Nb = 100.

395

We selected the case with Nb = 50 and σr = 0.6, which is the same value suggested

396

by Zhang and Oliver (2010). Table 4 summarizes the results indicating that BL

397

method was able to improve the average values of SNV for both, actual and dummy 16

398

parameters. The mean RMSE value is relatively close to the reference. However,

399

BL failed to mach data for two out of the ten ensembles, showing a possible lack of

400

robustness in the method. Note that the SNV of the two ensembles with failed data

401

match are much higher, increasing the mean values reported in Table 4. Without

402

these two ensemble the mean values of SNV are 0.24 and 0.78 for the actual and

403

dummy parameters, respectively.

404

3.5. SENCORP

405

Another localization scheme available in the literature is the Smoothed Ensemble

406

Correlations Raised to a Power (SENCORP) proposed by Bishop and Hodyss (2007).

407

In this method, spurious correlations are attenuated by raising them to a power. In

408

the following, we summarize the main steps of the SENCORP method adapted for

409

our problem and notation. A complete description of each step can be found in the

410

original paper (Bishop and Hodyss, 2007).

411

412

1. Define the augmented model vector y ∈ RNy , where Ny = Nm + Nd , by including the predicted data, i.e.,  y=

m g(m)

 ,

(22)

413

and standardize it by subtracting the mean and dividing its elements by the

414

corresponding standard deviation, i.e.,

e= y

h

y1 −y1 σy,1

where

···

yNy −yNy σy,Ny

i>

Ne 1 X yi = yi,j Ne j=1

and σy,i 415

y2 −y2 σy,2

v u Ne u 1 X =t (yi,j − yi )2 . Ne j=1

(23)

(24)

(25)

2. Compute the ensemble estimate of the covariance matrix N

e 1 X ej y ej> . Cye = y Ne − 1 j=1

17

(26)

416

Note that for a general history-matching problem, the vector y may include

417

rock properties for all reservoir gridblocks. In this case, the matrix Cye may

418

be very large, which may limit the application of the method. The reason

419

for using the augmented vector is to allow the matrix product of step 4. For

420

the problems of interest of this paper, however, we have only a few non-local

421

parameters. Hence, the construction of Cye is not an issue. Also note that the

422

resulting matrix can be divided into four sub-matrices  Cye = 

Cm e

Cm e ed

Cde m Cde de e

 .

(27)

423

For Kalman gain localization, we need only the elements referent to the sub-

424

matrix Cm e . However, we need to carry the entire Cy e to perform the matrix ed

425

products of step 4. At this step, Bishop and Hodyss (2007) also applies a

426

spatial smoothing procedure to this estimated covariance. However, it is not

427

obvious how to define this smoothing procedure for our non-local parameters,

428

so we do not apply this step in our test case.

429

430

3. Raise the matrix to the element-wise power n by a sequence of n Schur products, i.e., C◦n e ◦ Cy e ◦ · · · ◦ Cy e. e = Cy y | {z }

(28)

n times

431

4. Raise the resulting matrix to the power q by a sequence of multiplications C◦n e y


q

◦n ◦n = C◦n e × Cy e × · · · × Cy e . y | {z }

(29)

q times



432

433

5. Renormalize the matrix C◦n e y

q

using

^ q
q −1/2 ◦n Cye = S−1/2 C◦n S (30) e y h q i where S = diag C◦n . According to Bishop and Hodyss (2007), the mae y

434

trix product has an smoothing effect, but it boosts both actual and spurious

435

correlations. For this reason, they introduced a final element-wise product

436

(step 6) to attenuate the spurious correlations. 18

437

q ^ 6. Raise the matrix C◦n to the element-wise power p, i.e., e y "

q ^ C◦n e y

#◦p

q ^ q q ^ ^ ◦n ◦n = C◦n ◦ C ◦ · · · ◦ C . e e e y y y | {z }

(31)

p times

"

438

439

440

q ^ ◦n The matrix Cye

#◦p is used for Schur product localization. Again, because

here we"use only #Kalman gain localization, we need only the Nm × Nd sub^q ◦p matrix C◦n to construct the localization matrix R. e ed m

441

The number of matrix element-wise products and matrix multiplications used

442

in SENCORP are defined by the integers n, q and p. The optimal values of these

443

parameters must be determined through experimentation. The final element-wise

444

product parameter p should be chosen to ensure that the SENCORP elements are

445

positive or zero in order to keep the signs of correlations. The fact that we have

446

three free parameters to select, n, q and p, gives a potential flexibility to the method,

447

but it also makes harder to make a robust selection. We tested 28 combinations for

448

the values n, q and p. Based on our tests (not reported here), we selected n = 2,

449

q = 2 and p = 1.

450

Table 5 summarizes the results obtained with SENCORP. The method failed

451

to match data in one ensemble. The mean RMSE is 0.74, which is smaller than

452

the reference case, showing a good performance on revealing the true parameter

453

values. The mean SNV of the actual parameters is 0.27, which is larger than the

454

reference indicating an excessive localization. For the dummy parameters, on the

455

other hand, the mean SNV is 0.76, which is clearly an improvement compared to ES-

456

MDA without localization, but is still smaller than the reference. The localization

457

coefficients with SENCORP were recalculated every MDA iteration. We also tried a

458

case with localization computed based only on the prior ensemble, but this procedure

459

did not improve the results and several ensembles failed to match data properly.

460

3.5.1. Simplified SENCORP

461

462

Bannister (2015) proposed to use a simplified version of SENCORP by applying only steps 1–3, in which case we have only one free parameter to select n, i.e., 19

R = Cm e ed


◦n

.

(32)

463

With a single free parameter, the method becomes easier to tune. Moreover, we do

464

not need to construct the entire Ny ×Ny matrix Cye , which is an advantage compared

465

to the original SENCORP. Hereafter, we refer to this procedure a S-SENCORP.

466

Table 6 shows that S-SENCORP resulted in excessively large values for the SNV of

467

the actual parameter. For the dummy parameters that mean SNV is 0.94, which

468

is close to the correct value. However, because of the values of SNV for the actual

469

parameters, we conclude that this method introduced an excessive localization. This

470

is also in concordance with the fact that the values of Od (m) obtained are larger

471

than one, indicating poor data matches.

472

3.6. Correlation-Based Localization

473

Luo et al. (2018) proposed a localization method that does not rely on the dis-

474

tance between variables. They estimate correlation coefficient between the ith model

475

parameter and jth predicted data, ρi,j as 

 gj (mn ) − gj (m) n=1 mi,n − mi r =r 2 P 2 .  PNe  Ne n=1 mi,n − mi n=1 gj (mn ) − gj (m) PNe 

ρi,j

476

(33)

Based on the value of ρi,j , Luo et al. (2018) select the localization coefficients using

ri,j

  1, if |ρ | > θ i,j =  0, otherwise

(34)

477

where θ > 0 is a threshold value. Luo et al. (2018) described two special cases in

478

the use of Eq. 33 to avoid division by zero: (i) if all values of mi in the ensemble are

479

identical, we set ρi,j = 0, which corresponds to ri,j = 0. (ii) Similarly, if all values of

480

gj (m) in the ensemble are the same, set ρi,j = 0 and ri,j = 0. Luo et al. (2018) also

481

present a procedure for selecting θ for the case with grid-based model parameters

482

and seismic data points. Luo et al. (2018) suggested that a different threshold value

483

could be used for each type of model parameter. For non-local parameters, the

484

authors mention that Eqs. 33 and 34 can still be used, but the choice of θ must be 20

485

provided by the user. We considered a single and constant value of θ for each data

486

assimilation run. We tested the method updating the localization matrix every ES-

487

MDA iteration and using a constant matrix computed based on the prior ensemble.

488

The second procedure resulted in better results, similarly to what it was observed

489

in the POL method. The same observation was reported in Luo et al. (2018). Here,

490

we refer to this method as CBL (correlation-based localization).

491

We run some experiments to determine the best θ in our test problem and the

492

results are summarized in Fig. 5. For θ = 0.1 or higher, we observed a failure on

493

matching data indicating excessive localization. For θ = 0.01 the results approached

494

the case without localization. Our experiments indicate that θ = 0.05 was the best

495

choice. Table 7 sumarizes the results for θ = 0.05. The method failed to match data

496

for the second ensemble. The mean SNV value of actual parameters is close to the

497

reference, but a large variation is observed among the ensembles indicating a lack of

498

robustness in the method. Note, for example that the SNV of the actual parameters

499

is 0.39 for the seventh ensemble, but only 0.18 for the ninth. Moreover, the SNV

500

values for the dummy parameters are larger than one for almost all ensembles, which

501

shows that the method has difficulties dealing with weakly or uncorrelated model

502

parameters.

503

3.7. Covariance Inflation

504

Covariance inflation (CI) (Anderson and Anderson, 1999) is typically applied

505

with EnKF and other square-root filters to compensate for the erroneous variance

506

loss due to sampling error in the numerical weather prediction literature. Here, we

507

apply an inflation factor, γ > 1, after each iteration of ES-MDA. The idea is to

508

inflate the variance without changing the mean, which is accomplished using

mkinf,j

= γk



mkj

−

mk



+ mk ,

(35)

509

for j = 1, . . . , Ne , where the superscript k denotes the ES-MDA data assimilation

510

index. The choice of the inflation factor is problem dependent. There are several

511

works in the literature proposing covariance inflation schemes; see, e.g., (Kotsuki

512

et al., 2015) and references therein. Here, we use the scheme proposed by Evensen 21

513

(2009) adapted for use with ES-MDA. In this procedure, before each ES-MDA iter-

514

ation, we augment the vector of model parameters with a vector of white noise, i.e.,

515

zj ∼ N (0, I). After each ES-MDA update, we compute the inflation factor as

γk = where σz,i

v u u =t

1 PNz

1 Nz

i=1

(36)

σz,i

N

e 1 X (zi,j − zi )2 . Ne − 1 j=1

(37)

516

We applied this covariance inflation scheme in the test case with Nz = 100. Fig. 6

517

shows boxplots of the inflation factors for the four ES-MDA iterations. The inflation

518

factor varies between 1.06 to 1.29. The largest inflations occur after the second ES-

519

MDA iteration. Table 8 shows the results of covariance inflation indicating a clear

520

improvement in the values of SNV compared to the case with no localization without

521

compromising the quality of the data match or the RMSE. The final SNV of the

522

dummy parameters is larger than one for half of the ensembles, but the average

523

is 1.02, which is very close to the correct value. Compared to the other methods

524

investigated in this paper, covariance inflation with the scheme from (Evensen, 2009)

525

has the advantage of not requiring multiple tests to tune the method.

526

3.8. Deterministic ES-MDA

527

Deterministic or square-root schemes avoid the need to perturb the observation

528

vector before analysis. The deterministic EnKF of Sakov and Oke (2008b) is not

529

exactly an square scheme but it was proposed with the same objective, i.e., avoid

530

the sampling errors caused by the process of perturbing the observations. Recently

531

the DEnKF scheme was combined with ES-MDA forming the method DES-MDA

532

(Emerick, 2018). DES-MDA computes the same Kalman gain of ES-MDA (Eq. 6),

533

which is used to update the ensemble mean with

k+1

m

k

k

=m +K



dobs −

g(mk )



.

(38)

534

Unlike the standard ES-MDA, DES-MDA updates the matrix with the ensemble

535

deviations from the mean directly using 22

1 ek ∆Mk+1 = ∆Mk − K ∆Dk , 2

(39)

where ∆M =

h

m1 − m · · ·

mNe − m

i

(40)

and ∆D =

h

g(m1 ) − g(m) · · ·

g(mNe ) − g(m)

i

.

(41)

After that, the ensemble members are computed using Mk+1 = ∆Mk+1 + M

k+1

,

(42)

536

where M = [m · · · m]. Sakov and Oke (2008b) showed that the use of Eq. 39

537

incurs in an extra positive semi-definite term in the estimated posterior covariance.

538

They interpreted this term as an implicit covariance inflation in the method, which

539

may alleviate the underestimation posterior variances.

540

We applied DES-MDA with no localization with the same configuration of ES-

541

MDA, i.e, αk = Na = 4. Table 9 summarizes the results of DES-MDA. The mean

542

SNV obtained are 0.19 and 0.50 for actual and dummy parameters, respectively.

543

These values represent improvements compared to the standard ES-MDA without

544

localization, but we still observe a significant underestimation of posterior variances

545

compared to the reference case. In terms of Od (m) and RMSE the results of DES-

546

MDA are in reasonable agreement with the reference.

547

3.9. Localization Based on Sensitivity Analysis

548

Here, we introduce another procedure to mitigate sampling errors for non-local

549

parameters based on simple one-at-a-time sensitivity analysis (SA). The idea is to

550

run two reservoir simulations for each model parameter, one for the parameter at its

551

minimum value and another at its maximum value. All other parameters are fixed

552

at a base (or mean) value. Than, we compute the sensitivity coefficients using  si,j =

gj (mi,max ) − gj (mi,min ) σe,j

2 ,

(43)

553

where si,j corresponds to the sensitivity coefficient for the ith model parameters

554

with respect to the jth predicted data point, gj (mi,max ) and gj (mi,min ) denote the jth 23

555

predicted data from the ith model parameter at the maximum and minimum values,

556

respectively, σe,j denotes the standard deviation of the jth observed data point.

557

Based on the value of si,j , we compute the corresponding entry in the localization

558

matrix, R, as

ri,j 559

  1, if s > θ i,j =  0, otherwise

(44)

where θ > 0 is a threshold value.

560

SA is similar to CBL method in the sense that generates a matrix R with only

561

zeros and ones. This procedure is able to identify insensitive parameters, such as

562

the dummy parameters used in the test case, regardless the choice of the threshold

563

θ. Nevertheless, the value of θ impacts in the overall performance of the method.

564

Large values of θ imposes more localization. We tested different values of θ and the

565

results are summarized in Fig. 7. The results in this figure indicate that θ = 10−1

566

is a good choice. Smaller values of θ resulted in too small posterior variance in the

567

actual parameters. For θ = 1, the scheme imposed too much localization resulting

568

in too large value of SNV and in a poor data match (the average objective function

569

was 16.5).

570

Table 10 shows the results for SA localization. The method failed to match data

571

for one ensemble. The SNV of the actual parameters are lower than the reference,

572

indicating insufficient localization, although the results are better than the case

573

without localization. As expected, this procedure correctly identified the insensitive

574

parameters, preserving the variance.

575

The SA scheme is very effective to remove insensitive parameters, but it is not

576

able to compute different localization coefficients for parameters with different levels

577

of sensitivity, because the procedure returns only binary values. This suggests that

578

the SA scheme can be combined with another method, such as the POL to form a

579

more effective localization scheme. In this case, the localization matrix is the result

580

of the Schur product between the two localization matrices, i.e.,

R = RSA ◦ RPOL ,

24

(45)

581

where RSA is the localization matrix resulted from the SA scheme, preferable with

582

a small threshold used only to remove very weak sensitive parameters and RPOL

583

is the localization matrix resulted from POL method. We tested this procedure

584

considering  = 10−3 (threshold for POL) and θ = 10−4 (threshold for SA) and the

585

results are summarized in Table 11. We selected a small threshold for SA to ensure

586

that the method will remove only the data points with very small sensitivities with

587

respect to the parameters. Overall, the results are close to the ones obtained with

588

POL method, but the SNV of the dummy parameters are preserved.

589

Compared to the other methods, SA has the disadvantage of requiring additional

590

reservoir simulations, two for each parameter, which may limit the application of the

591

method. It is worth mentioning, however, that these simulations are independent in

592

the sense that they can be executed simultaneously, reducing the total time of the

593

process if the simulations are executed in a cluster of computers.

594

4. Discussion

595

Table 12 presents the average values of SNV for each parameter computed with

596

the ten ensembles considering the best configurations found for each method. For

597

comparisons, the second column of Table 12 shows the SNV obtained by the refer-

598

ence case. Clearly the standard ES-MDA without localization or inflation resulted

599

in underestimation of posterior variance for all parameters. Overall the methods

600

investigated in this paper minimized this problem, but in different levels and, in

601

some cases, the posterior variances were overestimated. Among the parameters, the

602

permeability multipliers are the most influential in the history matching. We observe

603

a severe variance reduction for these parameters in the reference case, with excep-

604

tion for the multiplier of the second layer, in which case the reference SNV is 0.16.

605

The same behaviour was obtained with all methods investigated. Fig. 8 presents

606

boxplots for the parameter EXP MULTIPERM5 showing that all methods resulted

607

in narrow posterior distributions very close to the correct value for the parameter.

608

The methods S-SENCORP and DES-MDA results in slightly overestimated poste-

609

rior distributions for this parameter, but the results seem acceptable. Note that the

610

scale of the boxplots showing the prior is different from the boxplots showing the 25

611

posterior.

612

The results in Table 12 show a different behaviour for the porosity multipliers.

613

The predicted data are clearly less sensitive to these parameters resulting in smaller

614

reductions in the posterior variance in the reference case. Among all methods, co-

615

variance inflation obtained the values of SNV closest to the reference; POL, SA and

616

DES-MDA also obtained improvements compared to the case without localization.

617

The other methods presented a tendency to overestimate the posterior variances for

618

porosity multipliers. One exception, however, is for the porosity multiplier of the

619

second layer. This is the least influential actual parameter in the history match-

620

ing. The SNV of the reference case is 0.86. All methods underestimated this value.

621

Fig. 9 presents the boxplots for this parameter showing that significantly different

622

posterior distributions were obtained for this parameter when we repeated the data

623

assimilation with ten different priors. The two versions of the SENCORP method

624

presented a slightly better performance in this specific aspect. Unfortunately the

625

same methods overestimated the variances substantially for the remaining porosity

626

multipliers, which indicates an excessive localization. For the other actual parame-

627

ters, all methods resulted in reasonable improvements in term of SNV, although we

628

observe some overestimation, specially for the parameter EXP AQRADIUS2.

629

For the dummy parameters, all methods increased the SNV compared the case

630

with no localization. However, the methods POL, BL, both SENCORP implemen-

631

tations and DES-MDA still underestimated the reference value. Covariance inflation

632

resulted in an slightly overestimation, while SA was able to correctly identify that

633

these parameters have no influence in the prediction. CBL resulted is severe overes-

634

timation of the SNV for the dummy parameters. It is worth noting that the results

635

of CBL methods were obtained with a single threshold value (same value for actual

636

and dummy parameters). This value was selected based on the results presented in

637

Fig. 5. However, as noted by (Luo et al., 2018), one could use different truncation

638

thresholds for different types of parameters. For example, based on Fig. 5, we could

639

select θ = 0.05 for the actual parameters and θ ≥ 0.3 for the dummy. In this case,

640

the method would correctly remove (localize) the updates in all dummy parame-

641

ters resulting in SNV = 1. Of course, tuning the method can be very expensive in 26

642

practice and we typically do not known the correct level of variance for comparisons.

643

Fig. 10 shows the boxplots of predicted increment in cumulative oil production,

644

∆Np , for a total production period of 16.5 years. Besides the six original wells,

645

the values of ∆Np include the production of five infill-drilling wells (wells X1 to

646

X5 in Fig. 2). This figure shows the case without localization underestimates the

647

uncertainty in ∆Np . All methods considered in this paper increased the uncertainty

648

range in ∆Np , but in some cases the range was overestimated. In particular, we

649

note that BL resulted in poor estimation of the distribution of ∆Np for two out

650

of ten ensembles. These two ensemble correspond to the ones with failure in the

651

data match, as indicated in Table 4. S-SENCORP was the method with the poorest

652

estimation of the posterior distribution of ∆Np . The range between the 25th and

653

75th percentiles (size of the boxes) is clearly overestimated and the medians (red

654

line in each box) are biased towards smaller values of ∆Np .

655

Fig. 11 shows the Kalman gain for two parameters EXP MULTPERM5 and

656

DUMMY3 with respect to BHP at well PRO-4 and WPR at well PRO-11, respec-

657

tively. These parameters and wells were selected to illustrate some aspects of the

658

behaviour of the methods. In both cases, we show the Kalman gain obtained with

659

the reference (Ne = 50000) and the first ensemble (Ne = 100) for each method.

660

First, we note in Fig. 11a that the Kalman gain predicted with the ensemble of

661

Ne = 100 is systematically smaller than the reference. In this case, localization

662

cannot improve the Kalman gain. In fact, the best in this case would be to use a lo-

663

calization coefficient of one. Among the methods tested, we note that POL, BL and

664

both SENCORP implementations reduced even more the Kalman gain. This effect

665

was more pronounced for BL and S-SENCORP. SA and CBL were able to correctly

666

identify that the localization coefficients should be one. Note that Fig. 11 does not

667

include the case with covariance inflation nor DES-MDA because both methods use

668

the same Kalman gain of the case without localization. The case POL+SA is not

669

shown but it is essentially the same as POL alone. Fig. 11b shows the same type of

670

plot but for a DUMMY parameter. Ideally, we would like to obtain a Kalman gain

671

of zero for this case. It is interesting to observe that even an ensemble with 50000

672

members is not enough to completely eliminate spurious correlations and some small 27

673

nonzero values of the Kalman gain were obtained. This figure shows that all meth-

674

ods reduced the Kalman gain. For both SENCORP implementations and SA the

675

localized Kalman gain was exactly zero. CBL removed most of the spurious Kalman

676

gain, but failed to remove the values between the time steps 70 and 80.

677

5. Conclusions

678

Sampling errors introduced by the limited ensemble size are one of the main limi-

679

tations of ensemble data assimilation in history-matching applications. For gridblock

680

properties, such as porosity and permeability, distance-based localization is the stan-

681

dard method to address this limitation. However, there is very little discussion in

682

the literature on how to mitigate sampling errors for non-local parameters. In con-

683

trast, some of these parameters, such as relative permeability curves and property

684

multipliers are among the most used in real-life history-matching cases. This paper

685

addressed this problem presenting a systematic comparison among the main methods

686

in the literature to alleviate the negative effects of sampling errors. These meth-

687

ods were compared in a synthetic problem and based on the results the following

688

conclusions can be stated:

689

• Sampling errors caused significant reduction in the posterior ensemble variance

690

in the test case evidencing the need of some strategy to address this problem.

691

• Overall all methods investigated in this paper were able to reduce the vari-

692

ance underestimation, but in different levels and, in some cases, at a cost of

693

compromising the ability of matching data properly.

694

• Most of these methods require the selection of internal parameters with impor-

695

tant impact in the performance of the method. In some cases, these parame-

696

ters can be difficult to select, which may limit the application of the method

697

in large-scale problems where repeating the data assimilation multiple times

698

is not feasible.

699

• POL showed a consistent improvement in the data assimilation results com-

700

pared to the case without localization or inflation. The method seems relatively 28

701

robust with respect to the choice of the truncation threshold as long as the se-

702

lected value is not too large. Our tests indicate that  ≤ 10−2 is a good choice.

703

The method is simple to implement adding no significant computational cost

704

to the data assimilation.

705

• BL also showed improvements in preserving the posterior variances, but the

706

method seems to be less robust as we observed failure to match data in some

707

ensembles. The method is relatively easy to tune.

708

• Both SENCORP implementations showed a tendency of introducing exces-

709

sive dumping (localization) in the Kalman gain resulting in a tendency to

710

overestimate the posterior variances. The original SENCORP was the most

711

difficult method to tune. We observed a strong impact of the choice of the

712

tuning parameters (integer exponents) of the method. The simplified version

713

of the method is easier to tune, but its performance was inferior to the original

714

method.

715

• CBL also showed to be very dependent on the choice of the tuning parameter

716

(truncation level in the correlations). The results indicate that the method is

717

not robust as we observed significant variations in the results when we repeated

718

the data assimilation with different prior ensembles.

719

• Covariance inflation was the method with best performance in the test prob-

720

lem. The method is very simple to apply adding no relevant computational

721

cost in the data assimilation. Even though the selection of the inflation fac-

722

tor impacts the performance of the method, the adaptive scheme of (Evensen,

723

2009) worked remarkably well is our test case.

724

• DES-MDA improved the results of standard ES-MDA in terms of preserving

725

variance without compromising the data match. However, the method was not

726

enough to avoid a significant level of variance underestimation.

727

• We introduced a simple sensitivity analysis to select localization coefficients for

728

non-local parameters. The procedure is effective to localize weakly sensitive

729

data and can be used in conjunct with other localization schemes. 29

730

731

Acknowledgement The authors would like to thank Petrobras for supporting this research and for

732

the permission to publish this paper.

733

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36

Table 1: Prior distributions of the actual and dummy parameters

Parameter

Mean

True

Distribution

Min.

Max.

MULTPOR1

1.00

1.00

Triangle

0.50

1.50

MULTPOR2

1.00

1.00

Triangle

0.50

1.50

MULTPOR3

1.00

1.00

Triangle

0.50

1.50

MULTPOR4

1.00

1.00

Triangle

0.50

1.50

MULTPOR5

1.00

1.00

Triangle

0.50

1.50

EXP MULTIPERM1

0.00

0.00

Triangle

−1.00

1.00

EXP MULTIPERM2

0.00

0.00

Triangle

−1.00

1.00

EXP MULTIPERM3

0.00

0.00

Triangle

−1.00

1.00

EXP MULTIPERM4

0.00

0.00

Triangle

−1.00

1.00

EXP MULTIPERM5

0.00

0.00

Triangle

−1.00

1.00

CCPOR

5 × 10−6

5 × 10−6

Triangle

1 × 10−6

5 × 10−5

EXP AQRADIUS1

1.00

1.74

Triangle

0.00

2.00

EXP AQRADIUS2

1.00

1.75

Triangle

0.00

2.00

DELTA DWOC

0.00

1.00

Triangle

−3.00

3.00

DELTA DGOC

0.00

−1.00

Triangle

−3.00

3.00

DUMMY1

0.00

-

Normal

−4.00

4.00

DUMMY2

0.00

-

Normal

−4.00

4.00

DUMMY3

0.00

-

Normal

−4.00

4.00

DUMMY4

0.00

-

Normal

−4.00

4.00

DUMMY5

0.00

-

Normal

−4.00

4.00

37

Table 2: Results for the data assimilation without localization and inflation Ensemble

Od (m) (prior)

Od (m) (post)

RMSE

SNV (actual)

SNV (dummy)

1

551.7

0.3

0.97

0.11

0.44

2

589.2

0.4

1.21

0.10

0.46

3

719.9

0.4

1.19

0.11

0.45

4

645.9

0.4

0.96

0.10

0.43

5

802.6

0.4

1.25

0.09

0.43

6

613.9

0.3

1.39

0.10

0.45

7

721.2

0.5

1.36

0.11

0.41

8

1148.1

0.3

0.99

0.09

0.42

9

1236.7

0.3

1.20

0.10

0.39

10

810.4

0.7

1.32

0.11

0.39

Mean

784.0

0.4

1.19

0.10

0.43

Reference

695.6

0.8

0.89

0.23

1.00

38

Table 3: Results for the POL method  = 10−3 Ensemble

Od (m)

RMSE

(post)

 = 10−1

SNV

SNV

Od (m)

(actual)

(dummy)

(post)

RMSE

SNV

SNV

(actual)

(dummy)

1

0.4

0.87

0.20

0.79

0.5

0.82

0.27

1.08

2

0.6

0.71

0.22

0.80

0.4

0.64

0.30

0.89

3

0.5

0.74

0.22

0.77

0.9

0.64

0.30

0.97

4

0.4

0.70

0.23

0.73

0.5

0.76

0.34

0.82

5

0.6

1.06

0.24

0.87

2.1

1.02

0.35

1.20

6

0.4

0.98

0.26

0.70

0.4

0.91

0.33

0.89

7

0.5

0.93

0.26

0.78

0.5

0.81

0.36

0.92

8

0.4

0.81

0.27

0.74

6.1

0.77

0.40

0.85

9

0.5

1.07

0.21

0.80

0.5

0.99

0.29

1.02

10

0.6

0.80

0.27

0.78

0.7

0.80

0.41

1.16

Mean

0.5

0.87

0.24

0.77

1.3

0.82

0.34

0.98

Reference

0.8

0.89

0.23

1.00

0.8

0.89

0.23

1.00

39

Table 4: Results for the BL method Ensemble

Od (m) (post)

RMSE

SNV (actual)

SNV (dummy)

1

1.1

0.93

0.27

0.98

2

0.8

0.78

0.28

0.71

3

0.6

0.78

0.22

0.72

4

0.5

1.00

0.24

0.74

5

3.1

0.80

0.40

1.18

6

0.4

1.04

0.17

0.67

7

0.9

1.06

0.26

0.73

8

0.5

0.78

0.22

0.82

9

0.7

1.24

0.23

0.90

10

25.3

0.71

0.41

1.09

Mean

3.4

0.91

0.27

0.85

Reference

0.8

0.89

0.23

1.00

40

Table 5: Results for the SENCORP method Ensemble

Od (m) (post)

RMSE

SNV (actual)

SNV (dummy)

1

0.5

0.89

0.26

0.73

2

0.4

0.61

0.26

0.79

3

3.8

0.74

0.26

0.80

4

0.5

0.64

0.27

0.75

5

0.5

0.75

0.27

0.78

6

0.5

0.81

0.29

0.72

7

0.5

0.86

0.28

0.74

8

0.5

0.69

0.28

0.71

9

0.4

0.75

0.26

0.78

10

0.4

0.67

0.27

0.76

Mean

0.8

0.74

0.27

0.76

Reference

0.8

0.89

0.23

1.00

41

Table 6: Results for the S-SENCORP method Ensemble

Od (m) (post)

RMSE

SNV (actual)

SNV (dummy)

1

1.0

0.79

0.44

0.94

2

1.3

0.61

0.46

0.94

3

0.9

0.68

0.48

0.96

4

1.7

0.72

0.46

0.92

5

1.3

0.71

0.48

0.95

6

1.7

0.78

0.49

0.92

7

1.2

0.79

0.46

0.93

8

2.4

0.76

0.49

0.92

9

1.7

0.78

0.48

0.96

10

2.4

0.74

0.48

0.94

Mean

1.6

0.74

0.47

0.94

Reference

0.8

0.89

0.23

1.00

42

Table 7: Results for the CBL method (θ = 0.05) Ensemble

Od (m) (post)

RMSE

SNV (actual)

SNV (dummy)

1

0.7

0.97

0.20

1.08

2

5.9

0.79

0.21

0.98

3

0.6

0.89

0.26

1.17

4

0.7

0.93

0.22

1.09

5

1.6

1.00

0.22

1.50

6

0.7

0.97

0.29

1.18

7

0.6

0.63

0.39

1.82

8

0.4

0.86

0.29

1.10

9

0.4

1.05

0.18

1.60

10

0.5

1.04

0.26

1.05

Mean

1.2

0.91

0.25

1.26

Reference

0.8

0.89

0.23

1.00

43

Table 8: Results for the covariance inflation Ensemble

Od (m) (post)

RMSE

SNV (actual)

SNV (dummy)

1

0.4

0.88

0.23

1.00

2

0.4

0.85

0.20

1.01

3

0.5

0.73

0.25

1.05

4

0.6

0.82

0.19

1.06

5

0.4

0.90

0.22

1.11

6

0.4

1.01

0.22

0.99

7

0.7

1.05

0.24

0.91

8

0.6

0.64

0.22

0.98

9

0.6

0.81

0.24

0.94

10

0.7

0.85

0.26

1.10

Mean

0.5

0.85

0.23

1.02

Reference

0.8

0.89

0.23

1.00

44

Table 9: Results for DES-MDA Ensemble

Od (m) (post)

RMSE

SNV (actual)

SNV (dummy)

1

0.5

0.83

0.19

0.51

2

0.8

0.88

0.17

0.51

3

0.8

0.87

0.20

0.51

4

0.8

0.78

0.19

0.52

5

1.2

1.04

0.19

0.53

6

1.1

0.97

0.20

0.48

7

0.8

1.03

0.19

0.45

8

0.6

0.77

0.19

0.47

9

0.5

1.02

0.17

0.51

10

1.0

0.98

0.18

0.48

Mean

0.8

0.92

0.19

0.50

Reference

0.8

0.89

0.23

1.00

45

Table 10: Results for the SA localization (θ = 10−1 ) Ensemble

Od (m) (post)

RMSE

SNV (actual)

SNV (dummy)

1

0.4

0.85

0.17

1.00

2

0.4

0.93

0.16

1.00

3

0.5

1.06

0.19

1.00

4

0.5

0.97

0.18

1.00

5

0.4

1.13

0.17

1.00

6

4.7

0.71

0.29

1.00

7

0.6

1.14

0.20

1.00

8

0.4

0.94

0.17

1.00

9

0.5

1.07

0.24

1.00

10

0.6

1.18

0.19

1.00

Mean

0.9

1.00

0.20

1.00

Reference

0.8

0.89

0.23

1.00

46

Table 11: Results for the SA localization (θ = 10−3 ) combined with POL method ( = 10−4 ) Ensemble

Od (m) (post)

RMSE

SNV (actual)

SNV (dummy)

1

0.4

0.84

0.20

1.00

2

0.8

0.76

0.22

1.00

3

0.6

0.78

0.24

1.00

4

0.4

0.61

0.23

1.00

5

0.6

1.08

0.23

1.00

6

0.4

1.01

0.28

1.00

7

0.5

0.84

0.28

1.00

8

0.4

0.77

0.29

1.00

9

0.4

1.15

0.22

1.00

10

0.6

0.88

0.25

1.00

Mean

0.5

0.87

0.24

1.00

Reference

0.8

0.89

0.23

1.00

47

48

1.00

1.00

DUMMY5

1.00

DUMMY2

1.00

1.00

DUMMY1

DUMMY3

0.12

DELTA DGOC

DUMMY4

0.37

0.20

DELTA DWOC

0.20

EXP AQRADIUS2

0.09

0.14

CCPOR

EXP AQRADIUS1

0.39

0.44

0.42

0.43

0.44

0.06

0.10

0.22

0.10

0.00

0.00

0.00

0.07

0.00

0.00

EXP MULTIPERM3

0.00

0.16

EXP MULTIPERM2

0.01

EXP MULTIPERM5

0.01

EXP MULTIPERM1

0.10

0.09

0.17

0.36

0.20

No local.

EXP MULTIPERM4

0.21

0.25

MULTPOR4

MULTPOR5

0.86

0.43

MULTPOR2

0.42

MULTPOR1

MULTPOR3

Reference

Parameter

0.79

0.84

0.73

0.77

0.73

0.14

0.16

0.45

0.14

0.20

0.00

0.00

0.00

0.20

0.01

0.33

0.27

0.53

0.62

0.52

POL

0.85

0.90

0.87

0.80

0.86

0.17

0.21

0.43

0.25

0.20

0.00

0.00

0.00

0.29

0.04

0.33

0.43

0.53

0.64

0.52

BL

0.74

0.76

0.75

0.78

0.76

0.15

0.21

0.51

0.17

0.24

0.00

0.00

0.00

0.27

0.02

0.36

0.31

0.54

0.70

0.55

SENCORP

0.93

0.95

0.94

0.94

0.93

0.37

0.37

0.86

0.29

0.45

0.01

0.00

0.01

0.66

0.08

0.64

0.68

0.85

0.91

0.88

S-SENCORP

Table 12: Mean SNV

1.19

1.38

1.68

1.10

0.93

0.13

0.14

0.45

0.13

0.18

0.00

0.00

0.00

0.22

0.01

0.30

0.32

0.61

0.65

0.63

CBL

0.97

1.04

1.06

1.05

0.96

0.12

0.19

0.48

0.19

0.15

0.00

0.00

0.00

0.18

0.01

0.25

0.20

0.43

0.72

0.47

CI

0.50

0.51

0.51

0.48

0.49

0.10

0.16

0.34

0.20

0.14

0.01

0.01

0.01

0.18

0.04

0.20

0.25

0.35

0.37

0.28

DES-MDA

1.00

1.00

1.00

1.00

1.00

0.08

0.21

0.48

0.13

0.14

0.00

0.00

0.00

0.14

0.01

0.27

0.25

0.41

0.54

0.31

SA

1.00

1.00

1.00

1.00

1.00

0.14

0.17

0.48

0.14

0.20

0.00

0.00

0.00

0.22

0.01

0.30

0.28

0.56

0.63

0.54

POL+SA

0.6

Relative Frequency

0.5

U = 0 (std. dev. = 0.157) U = 0.8 (std. dev. = 0.040)

0.4 0.3 0.2 0.1 0 -0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Correlation Coefficient Figure 1: Histograms of estimated correlation coefficient for two cases: uncorrelated variables (ρ = 0) and strongly correlated variables (ρ = 0.8).

49

Figure 2: PUNQ-S3 model (Floris et al., 2001). The red contour indicates the position of the gas-oil contact and the blue contour the position of the oil-water contact. The black circles shows the oil producing wells operating during the historical period. The white circles shows the positions of infill-drilling wells.

50

1.2 0.4

1.1 SNV(dummy)

SNV(actual)

0.35

0.3

0.25

1 0.9 0.8 0.7

0.2 0

0.001

0.01

0

0.1

ε

0.001

0.01

0.1

ε

(a) Actual

(b) Dummy

Figure 3: Boxplot of SNV obtained by POL with different values of the threshold.

51

4

3

3

2

2

2

1

1

1

0

DUMMY5

4

3

DUMMY5

DUMMY5

4

0

0

−1

−1

−1

−2

−2

−2

−3

−3

−3

ρ=−0.018082 −4 1000

1500

2000

ρ=0.24486

2500 3000 BHP PRO−4

3500

4000

−4 1800

2200

2400 2600 BHP PRO−4

ρ=−0.20896 2800

3000

4

4

3

3

2

2

1

1

0

−4 2400

2450

−1

−2

−2

−3

2500

2550 2600 2650 BHP PRO−4

2700

2750

(c) Iteration #2

0

−1

−3 ρ=0.0012971

−4 2400

3200

(b) Iteration #1

DUMMY5

DUMMY5

(a) Prior ensemble

2000

2450

2500

2550 2600 2650 BHP PRO−4

ρ=0.055806 2700

2750

2800

(d) Iteration #3

−4 2400

2450

2500

2550 2600 2650 BHP PRO−4

2700

2750

2800

(e) Iteration #4

Figure 4: Correlations between the parameter DUMMY5 and the predicted BHP for welll PRO-4 at time 2921 days.

52

2800

0.6

2

SNV(dummy)

SNV(actual)

0.5 0.4 0.3

1.5

1

0.2 0.5

0.1 0.01

0.05

0.10

0.15

0.20

0.25

0.30

0.01

θ

0.05

0.10

0.15

0.20

0.25

0.30

θ

(a) Actual

(b) Dummy

Figure 5: Boxplot of SNV obtained by CBL with different values of the threshold.

53

1.35

1.3

1.25

γ

1.2

1.15

1.1

1.05

1 1

2 3 ES−MDA iteration

Figure 6: Covariance inflation factors.

54

4

0.7

SNV(actual)

0.6 0.5 0.4 0.3 0.2 0.1 0.001

0.01

0.10

1.0

θ

Figure 7: Boxplot of SNV for the actual model parameters obtained with SA localization with different values of θ.

55

1

0.2

0.2

0.5

0.1

0.1

0

0

0

−0.5

−0.1

−0.1

−1

R

1

2

3

4

5

6

7

8

−0.2

9 10

R

(a) Prior

1

2

3

4

5

6

7

8

−0.2

9 10

0.2

0.1

0.1

0.1

0

0

0

−0.1

−0.1

−0.1

1

2

3

4

5

6

7

8

−0.2

9 10

R

1

(d) BL

2

3

4

5

6

7

8

−0.2

9 10

0.2

0.1

0.1

0.1

0

0

0

−0.1

−0.1

−0.1

1

2

3

4

5

6

7

8

−0.2

9 10

(g) CBL

R

1

2

3

4

5

1

0.2

0.1

0.1

0

0

−0.1

−0.1

R

7

8

−0.2

9 10

R

1

(h) Covariance inflation

0.2

−0.2

6

1

2

3

4

5

6

7

8

9 10

−0.2

(j) SA

R

1

3

4

5

6

7

8

9 10

2

3

4

5

6

7

8

9 10

8

9 10

(f) S-SENCORP

0.2

R

R

(e) SENCORP

0.2

−0.2

2

(c) POL

0.2

R

1

(b) No localization

0.2

−0.2

R

2

3

2

3

4

5

6

7

(i) DES-MDA

4

5

6

7

8

9 10

(k) POL+SA

Figure 8: Boxplots of EXP MULTIPERM5. The label R in each plot stands for reference case and the numbers represent each ensemble. The horizontal black line in each plot indicates the true value.

56

1.5

1.5

1.5

1

1

1

0.5

R

1

2

3

4

5

6

7

8

0.5

9 10

R

(a) Prior

1

2

3

4

5

6

7

8

0.5

9 10

1.5

1

1

1

1

2

3

4

5

6

7

8

0.5

9 10

R

1

(d) BL

2

3

4

5

6

7

8

0.5

9 10

1.5

1

1

1

1

2

3

4

5

6

7

8

0.5

9 10

R

1

2

3

4

5

6

7

8

(h) Covariance inflation

1.5

1.5

1

1

R

1

2

3

4

5

6

7

8

9 10

0.5

(j) SA

R

1

2

0.5

9 10

(g) CBL

0.5

3

4

5

6

7

8

9 10

1

2

3

4

5

6

7

8

9 10

8

9 10

(f) S-SENCORP

1.5

R

R

(e) SENCORP

1.5

0.5

2

(c) POL

1.5

R

1

(b) No localization

1.5

0.5

R

3

R

1

2

3

4

5

6

7

(i) DES-MDA

4

5

6

7

8

9 10

(k) POL+SA

Figure 9: Boxplots of MULTPOR2. The label R in each plot stands for reference case and the numbers represent each ensemble. The horizontal black line in each plot indicates the true value.

57

7

4

7

x 10

2.8

7

x 10

2.8

3.5

2.7

2.7

3

2.6

2.6

2.5

2.5

2.5

2

2.4

2.4

1.5

2.3

2.3

1

R

1

2

3

4

5

6

7

8

2.2

9 10

R

1

(a) Prior

3

4

5

6

7

8

2.2

9 10

7

x 10

2.8

2.8 2.7

2.6

2.6

2.6

2.5

2.5

2.5

2.4

2.4

2.4

2.3

2.3

2.3

1

2

3

4

5

6

7

8

2.2

9 10

R

1

(d) BL

2

3

4

5

6

7

8

2.2

9 10

2.8

2.8

2.7

2.7

2.6

2.6

2.6

2.5

2.5

2.5

2.4

2.4

2.4

2.3

2.3

2.3

R

1

2

3

4

5

6

7

8

2.2

9 10

(g) CBL

R

1

2

3

4

5

2.8 2.7

2.6

2.6

2.5

2.5

2.4

2.4

2.3

2.3 R

1

6

7

8

9 10

1

2

3

4

5

6

7

8

9 10

6

7

8

2.2

9 10

8

9 10

x 10

R

1

2

3

4

5

6

7

(i) DES-MDA

7

x 10

2.7

2.2

R

(h) Covariance inflation

7

2.8

5

7

x 10

2.7

2.2

4

(f) S-SENCORP

7

x 10

3

x 10

(e) SENCORP

7

2.8

2

7

x 10

2.7

R

1

(c) POL

2.7

2.2

R

(b) No localization

7

2.8

2

x 10

2

3

4

5

6

7

8

9 10

2.2

(j) SA

x 10

R

1

2

3

4

5

6

7

8

9 10

(k) POL+SA

Figure 10: Boxplots of ∆Np . The label R in each plot stands for reference case and the numbers represent each ensemble. The horizontal black line in each plot indicates the true value.

58

−6

x 10

Kalman Gain

5 4

−4

x 10 4

Reference No Localization POL BL SENCORP Simp. SENCORP CBL SA

Reference No Localization POL BL SENCORP Simp. SENCORP CBL SA

3 Kalman Gain

6

3 2

2 1 0 −1

1

−2

0 0

20

40

60 time

80

100

50

(a) EXP MULTPERMI5 and BHP of PRO-4

60

70

80 90 time

100

110

(b) DUMMY3 and WPR of PRO-11

Figure 11: Estimated Kalman gain for the first ensemble at the first iteration of ES-MDA.

59