SPE-173214-MS

An Adaptive Ensemble Smoother with Multiple Data Assimilation forAssisted History Matching

Duc H. Le, U. of Tulsa; Alexandre A. Emerick, Petrobras; Albert C. Reynolds, U. of Tulsa

Copyright 2015, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE Reservoir Simulation Symposium held in Houston, Texas, USA, 2325 February 2015.

This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contentsof the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflectany position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the writtenconsent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations maynot be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract

Recently, Emerick and Reynolds introduced the ensemble smoother with multiple data assimilations(ES-MDA) for assisted history matching. Via computational examples, they demonstrated that ES-MDAprovides both a better data match and a better quantification of uncertainty than is obtained with theensemble Kalman filter (EnKF). However, like EnKF, ES-MDA can experience near ensemble collapseand results in too many extreme values of rock property fields for complex problems. These negativeeffects can be avoided by a judicious choice of the ES-MDA inflation factors but, prior to this work, theoptimal inflation factors could only be determined by trial and error. Here, we provide two automaticprocedures for choosing the inflation factor for the next data assimilation step adaptively as the historymatch proceeds. Both methods are motivated by knowledge of regularization procedures; the first isintuitive and heuristical; the second is motivated by existing theory on the regularization of least-squaresinverse problems. We illustrate that the adaptive ES-MDA algorithms are superior to the originalES-MDA algorithm by history-matching three-phase flow production data for a complicated syntheticproblem where the reservoir model parameters include the porosity, horizontal and vertical permeabilityfields, depths of the initial fluid contacts and the parameters of power-law permeability curves.

IntroductionThe ensemble smoother with multiple data assimilation (ES-MDA) was introduced by Emerick andReynolds (2012, 2013a) to improve the ensemble Kalman filter (EnKF) (Evensen, 1994) for problemswhere the relation between the model (vector of reservoir model parameters) and data predicted from theforward model (here, the reservoir simulator) is highly nonlinear. The EnKF has two major shortcomings:(1) because EnKF updates both model parameter and states, it can introduce inconsistencies betweenupdated reservoir model parameters and updated states (here, primary variables solved for by thesimulator) (Thulin et al., 2007; Wang et al., 2010) and (2) it can produce an unacceptable data match.Unlike the EnKF, which is a sequential data assimilation technique which updates both model parametersand states of the system, the ensemble smoother (ES) assimilates all data simultaneously and updates onlyreservoir model parameters. Because the ES does not update states, there is no inconsistency between theupdated reservoir model parameters and the corresponding primary simulation variables, which are simplycomputed by running the simulator from time zero with the updated model parameters. However, when

history matching production data, ES can provide an even worse data match than is obtained with EnKF(Emerick and Reynolds, 2013c).

As discussed in Emerick and Reynolds (2012), ES-MDA was motivated by the analogy between ESand one Gauss-Newton iteration (Reynolds et al., 2006). Simply put, ES-MDA assimilates the same dataset Na times (Na 1) using the same form of the updating (analysis) formula used for the ensemblesmoother. However, at the th data assimilation step, the measurement error covariance matrix, CD, isreplaced by CD where 1. The s are chosen to harmonically sum to one to ensure that, for thelinear Gaussian case, ES-MDA produces a theoretically rigorous sampling of the posterior probabilitydensity function as the ensemble size goes to infinity. Iterative ensemble-based methods have been usedpreviously to obtain a better data match than is obtained with EnKF or ES, but at least for the simplereservoir example considered in Emerick and Reynolds (2013c), the performance of ES-MDA was farsuperior to the other iterative ensemble-based methods tested, not only in the data match obtained, butmore importantly, in terms of uncertainty quantification. Specifically, ES-MDA produced the bestapproximation of the posterior marginal probability density functions (pdfs) for the reservoir modelparameters and the best approximation for the posterior probability distribution for predicted future rateswhere the correct posterior pdfs were generated using Markov chain Monte Carlo (MCMC), whichtheoretically produces a correct sampling of the posterior pdf for the vector of model parameters. Wehasten to add that the comparison paper of Emerick and Reynolds (2013c) did not consider two recentensemble-based iterative methods, namely a Levenberg-Marquardt form of an iterative ensemblesmoother (Chen and Oliver, 2013) and a regularizing Levenberg-Marquardt ensemble smoother (RLM-ES) (Iglesias, 2014a, b).

Even though ES-MDA works well for the problems considered (Emerick and Reynolds, 2012, 2013a,c), as well as for field cases (Emerick and Reynolds, 2013b), there still exist two drawbacks in usingES-MDA. The first drawback is that the users have to determine the number of data assimilations a priori.In general, four to ten iterations appear to be sufficient, but there has been no clear guidance on how tochoose the optimal number of iterations (data assimilations). The problem with this approach is that, if thedata assimilation does not lead to satisfactory results, the whole process has to be repeated with a differentnumber of iterations. The second drawback is that it is not clear how to choose the optimal inflationfactors. Equal inflation factors were used for most cases (Emerick and Reynolds, 2012, 2013a) becausethis choice worked well. The two drawbacks of ES-MDA may not be apparent when dealing withrelatively easy history-matching problems, including field cases, where the choice of four equal inflationfactors often works well. For the problem considered in this paper, however, a simple implementation ofES-MDA provides a reasonable data match but results in severe overshooting, or roughness, in the finalpermeability and porosity fields. This roughness arises from the ill conditioning of the matrix that mustbe inverted at each ES-MDA iteration and the relatively large step size used at early iterations which leadsto a large change (over correction) in the rock property fields at early iterations. This over correctionoccurs because ES is similar to one Gauss-Newton iteration with a full step (stepsize unity) (Reynolds etal., 2006) whereas a smaller step size should be used at early iterations. One way to avoid this overcorrection is to restrict or damp the change in model parameters at early iterations which can be done byusing large inflation factors at early iterations. (Gao and Reynolds, 2006) has provided heuristical waysto do this when using the randomized maximum likelihood for Bayesian inversion with a full rank priorcovariance matrix. Here, however, we focus on ES-MDA and seek a systematic way to adaptivelydetermine inflation factors and the number of required iterations to mitigate the effects of ill conditioningand control overshooting.

In this paper we investigate two ways to choose the inflation factors. The first is based on a simplecriterion which restricts the change in the vector of model parameters and is a heuristical implementationof the restricted step method. The second is based on the regularizing Levenberg-Marquardt algorithm(Hanke, 1997, 2010; Iglesias and Dawson, 2013). As discussed later, the second algorithm is somewhat

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similar to the RLM-ES algorithm of Iglesias (2014a) which is not particular surprising as both his workand ours were directly motivated by the results of Hanke (1997) and Iglesias and Dawson (2013). The twoadaptive ES-MDA algorithms of this paper were first presented in May 2014 to the industrial sponsors ofTUPREP (Le and Reynolds, 2014) and was briefly presented without much technical detail by Reynolds(2014).

The paper is organized as follows: in the next section, we explain the methodology for two adaptiveES-MDA methods. Then our proposed methods are applied to a test case based on the PUNQ-S3 problem.In the last section, we present the conclusions.

Methodology

Throughout this paper, we assume that the history-matching problem is equivalent to minimizing anobjective function which consists of the sum of a data mismatch term and a regularization term which isusually, but not necessarily, based on a prior geostatistical model (Oliver et al., 2008). The objectivefunction is most often expressed as

(1)

where m is the vector of model parameters; mpr is the prior mean the model; g(m) is the data predictedfrom the forward model (reservoir simulator); dobs is the vector of observed data; CD is the matrix ofmeasurement errors and CM is the prior covariance matrix of model parameters.One classic way to minimize an objective function is to use the iterative Gauss-Newton procedure

which can be written as

(2)

where is the iteration index, is the step size that can be determined by a line search in the searchdirection (the terms in braces), G (pmg(m)

T)T is the sensitivity matrix at m. The result of thisminimization problem is called the MAP (maximum a posteriori) estimate because it represents the mostlikely posterior model. However, in history matching, the MAP estimate is often insufficient because itdoes not offer a way to quantify the uncertainty in the posterior model. To solve this problem, Kitanidis(1995) and Oliver et al. (1996) independently introduced the Randomized Maximize Likelihood (RML)scheme to sample from the posterior model. In the application of RML, we replace m by a sample, muc,j, from N(mpr, CM) and replace dobs by a sample, duc, j, from N(dobs, CD) in the objective function of Eq.1 to obtain

(3)

and then minimize the modified objective function, Oj(m), to obtain a sample of the posterior pdf. Byrepeating the process with a set of different samples of muc and duc, we generate an approximate sampleof the posterior pdf. RML provides a theoretically rigorous sampling of the posterior pdf only in the linearGaussian case (Oliver et al., 2008) but often appears to provide a reasonable sampling of the posterior pdfobtained by integrating production data into a prior geostatistical model (Emerick and Reynolds, 2013c).If the Gauss-Newton method is used for minimization of the modified objective function, then

(4)

where , j is the step size G (pmg(m)T)T denotes the sensitivity matrix evaluated at m, j. Note that

with RML, the optimization of each Oj(m) is done independently. However, if one (1) performs a singleiteration for each j, j 1, 2, . . . Ne, (2) starts the jth optimization with an initial guess given by

SPE-173214-MS 3

, (3) uses a unit step size, and (4) uses the same average sensitivity matrix, G , to replace

every G, j, then setting 0 in Eq. 5 and using the notation gives

(5)

Based on the results of Reynolds et al. (2006), the covariances in the ensemble update of Eq. 10 belowsatisfy the following approximations:

(6)

and then it is easy to show (Reynolds et al., 2006) that one iteration of ES is similar to oneGauss-Newton iteration provided conditions (1)-(4) listed above hold. Emerick and Reynolds (2012,2013a) use this analogy to propose the ES-MDA algorithm, which repeats the ES procedure several timeson the same observed data. Since the ES-MDA algorithm is motivated by the Gauss-Newton algorithm,it is natural to adopt ideas from previous works on how to control over correction with Gauss-Newton(Hanke, 1997, 2010; Gao and Reynolds, 2006; Tavakoli and Reynolds, 2010, 2011; Iglesias and Dawson,2013) to improve the ES-MDA algorithm. The following subsections discuss two adaptive ES-MDAalgorithms developed using this past knowledge as motivation.

ES-MDA-RSOne important lesson from previous works on Gauss-Newton is that excessive modification of the

model parameters at each iteration is not ideal and often makes the model susceptible to overshooting (Liet al., 2003; Gao and Reynolds, 2006; Tavakoli and Reynolds, 2010). One way to alleviate this issue isto increase the inflation factor so that the change in model parameters at each iteration is below areasonable threshold. We propose to use two standard deviations of the prior model as the limit of howmuch the model parameters can change in one iteration. This choice is motivated by the fact that 95% ofthe samples from a Gaussian distribution is within two standard deviations of the mean. In addition, webelieve that the inflation factor should be a positive function of the data mismatch because, intuitively, asmaller correction (greater inflation factor) should be used when the data mismatch is bigger and viceversa. This idea is similar to Gao and Reynolds (2006) where the authors use the magnitude of theobjective function to determine the damping factor. Combining the two ideas, we propose an adaptiveensemble smoother algorithm called ES-MDA-RS where RS stands for restricted step. The algorithm isgiven below.

FOR each iteration:

1. Run the ensemble from time zero.2. Calculate the average normalized objective function

(7)

where the normalized objective function ONd, j is defined by(8)

3. Set as the initial guess for the inflation factor, where is the index of the currentiteration. As discussed above, this choice is motivated by the proposal that the inflation factorshould be based on the value of the objective function. This serves as the minimum value of theinflation factor for the current iteration.

4. For each ensemble member, perturb the observation vector using(9)

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where Zj ~ N(0, INd).5. Update the ensemble using

(10)

where , is the cross-covariance matrix between the vector of model parameters, mf, and thevector of predicted data, df; is the auto-covariance matrix of predicted data.

6. For each ensemble member j and for each component i of the jth vector of model parameters:

Calculate the change from the previous iteration to the current iteration(11)

Compare mj, i with the prior standard deviation of the ith model parameter, i. If mji is greater than 2i for any model component of any ensemble member, discard the

updated ensemble calculated in step 5, double and redo steps 4, 5 and 6 until no moreviolation occurs.

7. Calculate the sum of the inverse inflation factors up to the current iteration,(12)

If 1, terminate the algorithm. If 1, increase so that is exactly 1, discard the updated ensemble calculated in step

5 and redo steps 4 and 5 to calculate the final model parameters, then terminate the algorithm. If 1, the algorithm continues to the next iteration.

END FOR In the standard ES-MDA (Emerick and Reynolds, 2012, 2013a), the number of iterations(Na) and the inflation factors (s) are set before doing any computations and steps 2 and 3 of thealgorithm are deleted.

ES-MDA-RLMInsight on how to improve ES-MDA can also be drawn from the work of Iglesias and Dawson (2013).

Although their work focuses on generating a regularized solution of a pure least-squares problem, itprovides insight of value here. Specifically, Iglesias and Dawson (2013) seek to minimize the objectivefunction given by

(13)

Note that this objective function is similar to that in Eq. 1 except the regularization term provided bythe prior model is missing. Iglesias and Dawson replace g(m) in Eq. 13 by the approximation

(14)

where G is the sensitivity matrix defined above and and then replace the minimizationof Od(m) by the linear inverse problem: find w m - m

which minimizes

(15)

Minimizing the objective function in Eq. 13 is generally ill-posed, in which case, we requireregularization to define a unique solution. Iglesias and Dawson (2013) use a regularizing Levenberg-Marquardt scheme, or more specifically, the regularizing scheme of Hanke (1997, 2010), where Tikonovregularization is applied to the linear inverse problem of Ea. 15. In this case, m1 is obtained as

SPE-173214-MS 5

(16)

where is the regularization parameter. In essence, the scheme in Eq. 16 regularizes the linear inverseproblem of Eq. 15 by penalizing the change, m1, in the model parameters over the iteration. Thesolution to the regularized minimization problem of Eq. 16 is

(17)

As noted previously, the covariances in the ensemble update of Eq. 10 satisfy the approximations ofEq.6. Thus, Eq. 17 has the same structure as the ES-MDA update equation (Eq. 10). In particular, theregularization parameter a in ES-MDA has the same effect as the regularization parameter in Eq. 17.Therefore, we can adopt Hankes recommendation for choosing the regularizing parameter to our choiceof the ES-MDA inflation factor (also denoted by a). According to the ideas of Hanke (also see Iglesiasand Dawson (2013) on regularizing Levenberg-Marquardt), should satisfy

(18)

for some with 0 1. As discussed later, finding according to this criterion requires that wespecify a value of and there is no firm theoretical guidance for how to choose . However, with thischoice of , Iglesias and Dawson (2013) prove that if the noise level approaches zero, the regularized leastsquares solution converges to an m* such that g(m) g(m*). Given the aforementioned equivalentstructure of Eq. 17 and the ES-MDA update equation (Eq. 10), we can by simple analogy extend Eq.18to ES-MDA to obtain a criterion for choosing the inflation factor at each iteration . In particular, wepropose choosing so that

(19)

From Eq. 19, we can see that larger p requires larger , which will lead to more iterations, but larger also invokes more damping, which should reduce overshooting (over correction) with ES-MDA andresult in smoother and more geologically reasonable property fields. Iglesias and Dawson (2013)performed experiments on a 2D reservoir case with values of ranging from 0.65 to 0.95 and showed thatthe final data misfit is smaller for larger values of p but at the cost of more iterations. The authors thensuggested using [0.7, 0.8] for the best compromise between efficiency and accuracy. However, withthis choice of , the value a required to satisfy the criterion is unreasonably large (greater than 10100) inthe example problem considered here. Because of this issue, for the example shown here, we use 0.2in our implementation of the new adaptive ES-MDA method, which was motivated by the work of Iglesiasand Dawson (2013). Here, this adaptive ES-MDA algorithm is denoted by ES-MDA-RLM where RLMstands for regularizing Levenberg-Marquardt. This algorithm is given below.

FOR each iteration:

1. Run the ensemble from time zero.2. Calculate the average normalized objective function

(20)

where the normalized objective function ONd, j is defined by(21)

3. Set as the initial guess for the inflation factor, where is the index of the currentiteration.

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4. For each ensemble member, perturb the observation vector using(22)

where Zj ~ N(0, INd).5. For each ensemble member j:

Check the condition in Eq. 19, which is repeated below for continuity in presentation:(23)

If the condition is violated for any ensemble member, double and redo steps 4 and 5 untilno violation occurs.

6. Calculate the sum of the inverse inflation factors up to the current iteration,(24)

If 1, the algorithm ends after the completion of step 7 of this iteration. If 1, increase so that is exactly 1. The algorithm ends after the completion of step

7 of this iteration. If 1, the algorithm continues to the next iteration after the completion of step 7 of this

iteration.

7. Update the ensemble using(25)

where is the cross-covariance matrix between the vector of model parameters, mf, and thevector of predicted data, df; is the auto-covariance matrix of predicted data.

END FOR One advantage of the ES-MDA-RLM method compared to the ES-MDA-RS method is thatwe do not have to recompute the analysis equation (Eqs. 10 and 25) for every new guess of a. Instead,we only have to recompute the right-hand side of Eq. 19, which is much faster because Nd Nm in allcases of interest to us. This advantage becomes more apparent when the number of model parameters ison the order of hundreds of thousands or more.

ES-MDA-RLM with an Alternate Stopping RuleIglesias and Dawson (2013) provide a stopping rule for the regularized Levenberg-Marquardt algo-

rithm. The stopping rule is as follows,

(26)

where the theory dictates that must be chosen so that and , where Nd is the number of

observed data. Note introducing requires that we have two parameters to choose a priori, and . Iglesiasand Dawson (2013) suggest using , and we do so for the results shown in this paper. To adapt

the stopping rule to our ensemble-based methods, we replace , which is the average of thepredicted data, so that our choice of the stoping rule is

(27)

This stopping rule is hereafter referred to as the RLM stopping rule (RLM-SR) to distinguish it fromthe traditional ES-MDA termination criterion which requires we iterate until the sum of the inverse

SPE-173214-MS 7

inflation factors is unity. If we use the RLM stopping rule, the adaptive ES-MDA procedure can no longerbe proven to sample correctly in the linear-Gaussian case. One test we did on a simple linear Gaussianhistory matching problem indeed confirms that the algorithm does not sample correctly and the uncer-tainty in the posterior model depends on the value of . With the RLM stopping rule, we introduce analternate version of the ES-MDA-RLM algorithm. The algorithm is as follows:

FOR each iteration:

1. Run the ensemble from time zero.2. Check the RLM stopping rule in Eq. 27, which is repeated below for continuity in presentation,

(28)

If the stopping rule is satisfied, terminate the algorithm without proceeding to subsequent steps.3. Calculate the average normalized objective function,

(29)

where the normalized objective function ONd, j is defined by(30)

4. Set , where is the index of the current iteration, as the initial guess for the inflationfactor.

5. For each ensemble member, perturb the observation vector using(31)

where Zj ~ N(0, INd).6. For each ensemble member j:

Check the condition in Eq. 19, repeated below for continuity in presentation,(32)

If the condition is violated for any ensemble member, double , and redo steps 5 and 6 untilno more violation occurs.

7. Update the ensemble using(33)

where is the cross-covariance matrix between the vector of model parameters, mf, and thevector of predicted data, df; is the auto-covariance matrix of predicted data.

END FOR At this point, it is worth mentioning that, independently of our work, Iglesias pursues hisown adaptation of the regularizing Levenberg-Marquardt scheme to ensemble-based methods (Iglesias,2014a, b). There are four major differences between our proposed adaptive ES-MDA algorithms andIglesias algorithm:

1. We try two different termination criteria: one that comes directly from ES-MDA and guaranteesthat we sample correctly in the linear Gaussian case and one adapted from the stopping rule inIglesias and Dawson (2013). (In the latest version of his paper, Iglesias (2014a) also utilizes thestopping rule of Iglesias and Dawson (2013).)

2. We check the regularization criteria based on the worst-case ensemble member instead of basedon the ensemble average as is done by Iglesias (2014a, b).

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3. At each th ES-MDA data assimilation, we generate new perturbed data, duc, js by samplingN(dobs, CD whereas Iglesias (2014a, b) generates the duc, js only at the first iteration. Our choiceis dictated by the objective that we wish to guarantee that the method samples correctly in thelinear-Gaussian case.

4. Iglesias (2014a) uses between 0.7 and 0.9, consistent with his suggestion in Iglesias and Dawson(2013) but violates the theoretical condition that 1/ by using 1 regardless of the choiceof . For the example considered here, we find that 0.7 leads to a highly inefficient algorithm.Moreover, we always honor the theoretical condition that 1/.

Case StudyCase descriptionWe consider the three-phase (oil, gas and water), three-dimensional PUNQS3 reservoir model (Floris etal., 2001; Barker et al., 2001; Gao et al., 2006). The reservoir grid is 19 x 28 x 5; the size of each gridblockis 590.55 ft x 590.55 ft in the x and y directions but the thicknesses of the gridblocks vary. There are 1761active gridblocks in the model. There are six producers with the locations and perforated intervals thesame as in the original model (Floris et al., 2001; Barker et al., 2001); see Fig. 1. All production wells areunder oil rate control. The reservoir is supported by a strong analytical aquifer on the west and south sides.The production consists of a first year of extended well testing, followed by a 3-year shut-in period andthen 12.5 years of production. The oil production rates for each well during the first year of well testingfor each of the four 3-month periods are specified as 628.98, 1257.96, 628.98 and 314.49 STB/D.Subsequent to the 3-year shut-in period, the oil production rate at each well is set equal to 943.47 STB/D.The history-matching period consists of the first 11.5 years. The forecast period makes up the remaining5.5 years. The data to be matched include bottomhole pressure (BHP), gas/oil ratio (GOR) and water cut(WCT). To generate the observed data, we add Gaussian noise to the true production data obtained fromthe reservoir simulator. The standard deviations for the measurement error (noise) are specified as 10 psifor BHP and 3% of the true values for GOR and WCT data, where CD, the data or measurement errorcovariance matrix, is assumed to be diagonal.

The true horizontal log-permeability field is shown in Fig. 2. The unconditional log permeability andporosity fields are generated using sequential Gaussian co-simulation based on the geostatistical param-eters given in Gao et al. (2006), where no hard data are used. The lack of hard data at wells makes this

Figure 1Locations of perforations in layers 3, 4 and 5. There is no perforation in layers 1 and 2.

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a challenging problem because almost none of the prior realizations have the high permeability streakspassing through the production wells, making the predicted well data dramatically different from theobserved data.

The true GOC and OWC depths are 7726.4 ft and 7857.6 ft, respectively. In the prior realizations ofGOC and OWC depths, we deliberately introduce a bias in the prior means, setting these means equal to7746.4 ft and 7877.6 ft, respectively. The standard deviation of the GOC and OWC depths in the priorrealizations are both 20 ft.

To generate realizations of the relative permeability curves, we slightly modify the power law modelused in Wang et al. (2010) to better fit the requirements of the simulator. For example, Eclipse 100requires that the relative permeability data for water, Krw, are available from Sw Siw to Sw 1, but thekrw model used in Wang et al. (2010),

(34)

can only generate krw from Sw Siw to Sw 1 - Sorw. Eclipse will accept this incomplete input dataand will then extrapolate to obtain the missing data. The power law relative permeability models that weuse are given by

(35)

(36)

(37)

(38)

where the Stone Model I (Aziz and Settari, 1979) is used to calculate the three-phase relativepermeability from the sets of two-phase relative permeability functions.

The parameters used to generate the prior realizations of the relative permeability curves are shown inTable 1. Note that the values of the end-point saturations are fixed; we only try to match the exponentsand the coefficients.

ResultsWe perform history matching on 10 different ensembles where each ensemble has 100 ensemblemembers. Note that we repeat the history matching with multiple ensembles in order to investigate the

Figure 2True horizontal permeability field (ln k)

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consistency of the proposed methods, all of which are stochastic. We do not intend to imply that doinghistory matching with 10 ensembles of size 100 yields a better characterization of the posterior pdf thandoing history matching with one ensemble of size 1000. In fact, using one ensemble of size 1000 seemsto be give better results for this particular problem (Le and Reynolds, 2013). We test and compare theresults of five methods:

ES-MDA 8x: equal inflation factors of 8. ES-MDA 16x: equal inflation factors of 16. ES-MDA-RS: model change restriction of 2 standard deviations. On average, this algorithmfinishes in 20 iterations.

ES-MDA-RLM: 0.2. On average, ES-MDA-RLM finishes in 16 iterations using traditionalES-MDA stopping rule and finishes in 12 iterations using the RLM stopping rule.

LM-EnRML (Chen and Oliver, 2013). On average, this algorithm finishes in 25 iterations.

In the LM-EnRML method, the initial value of the multiplier is chosen to be 100, 000, which is onthe same order of magnitude as the average data mismatch calculated using Eq. 30. The parameter ,which dictates how much is changed from one iteration to the next, is 10 as recommended by Chen andOliver (2013).

Fig. 3 shows the evolution of the inflation factor for the two adaptive ES-MDA algorithms, ES-MDA-RS and ES-MDA-RLM, with the traditional ES-MDA stopping rule. It can be seen that the initialinflation factor needs to be very high in order to satisfy the regularization conditions but it decreases toa much smaller value after a few iterations. This result is consistent with previous works of Gao andReynolds (2006) and Tavakoli and Reynolds (2010) that show that controlling the change in parametersduring the first few iterations is critical in order to avoid over correction of model parameters andunrealistically rough property fields.

Fig. 4 shows the third layers log horizontal permeability for the first realization of each posteriorensemble. The true log horizontal permeability is also plotted in the same figure for the convenience ofthe readers. It can be seen that the problem of overshooting due to ill conditioning is quite severe in theresult obtained with ES-MDA 8x, i.e. we obtain a very rough permeability field which contain mostly veryhigh and very low permeability values. The problem is alleviated somewhat with ES-MDA 16x but is stillquite serious. LM-EnRML and the two adaptive ES-MDA methods show much less overshooting than isobtained with the two standard ES-MDA methods. Using the RML stopping rule, the ES-MDA-RMLalgorithm completes earlier, which results in slightly less overshooting than the other two adaptiveES-MDA methods, even though we can no longer prove that the ES-MDA-RLM2 algorithm samplescorrectly for the linear Gaussian case. As we will show later, the reduced overshooting in LM-EnRML ispartly because the algorithm does not significantly modify the initial values of the model so that theposterior models are similar to the prior models, and consequently we have virtually no overshooting.

Fig. 5 shows the ratio of the posterior standard deviation to the prior standard deviation for thegridblock log-horizontal permeabilities of layer 3. Note that the standard deviation values are calculated

Table 1Specifications of relative permeability paramaters

Mean Stdev Min Max

1.0 0.0 1.0 1.0

0.85 0.05 0.7 1.0

0.25 0.05 0.1 0.4

ew 4.75 0.9 2.5 7.0

eow 2.25 0.3 1.5 3.0

eg 7.0 1.2 4.0 10.0

eog 2.35 0.46 1.2 3.5

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Figure 3Evolution of the inflation factors in the adaptive ES-MDA methods.

Figure 4Layer 3 log horizontal permeability of the first realization of each ensemble.

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for each individual ensemble. Without a rigorous sampling method like MCMC, it is not possible to knowthe correct values of the standard deviation. All the ensemble smoother methods, including ES-MDA 8x,ES-MDA 16x, ES-MDA-RS and ES-MDA-RLM, give standard deviations that do not vary significantlyacross ensembles, which shows these methods are quite consistent. For ES-MDA-RLM with the RLMstopping rule, the values of posterior standard deviations are generally much greater than the valuesobtained with other ensemble smoother methods, which is expected because the data assimilation isterminated earlier. It is interesting to observe that the standard deviations obtained from LM-EnRML varysignificantly from one ensemble to another. For ensembles #3, 7, 8 and 10, the posterior standarddeviations obtained with LM-EnRML are not much smaller than the prior standard deviations. As shownlater, this is because the algorithm is unable to obtain a good data match for these ensembles and is thusunable to reduce the uncertainty.

Figure 5Standard deviation of the third layers log horizontal permeability.

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One metric to measure how close the posterior model is to the true model is by using the followingequation:

(39)

where DM is a diagonal matrix whose diagonal elements are the prior standard deviations of the modelparameters. R can be interpreted as the average difference between the value of a posterior modelparameter and its true value in terms of the prior standard deviations.

Figure 6Average difference between a posterior model parameter and the true value.

Figure 7Normalized objective function.

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Figure 8Posterior production data obtained with different history-matching methods.

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The plot of R obtained with different history-matching methods is shown in Fig. 6. Note that the tworegular ES-MDA methods lead to higher values of R than is obtained by using the prior model in Eq. 39due to the excessive overshooting exhibited by these two methods. The adaptive ES-MDA methods areable to alleviate this problem, resulting in smaller values of R than are obtained with the prior model.

Fig. 7 shows the box plot of the normalized objective function, which is created by combining all 10ensembles. Overall, the two adaptive ES-MDA methods, ES-MDA-RS and ES-MDA-RLM with thetraditional ES-MDA stopping rule, result in the best data match although this does not mean that thesemethods necessarily give a good approximation of the posterior pdf. On the other hand, a large value ofthe objective function suggests that the corresponds model pertains to a high value of the posterior pdfwhich often means that this model is from a low probability region. Note that the range of the objectivefunction values obtained with LM-EnRML is very large. A careful investigation of the objective functionfor each ensemble obtained with LM-EnRML reveals that the method is not consistent for this exampleproblem; it performs very well for some ensembles but performs badly for others. The ES-MDA-RLMalgorithm with the RLM stopping rule results in a bigger data mismatch than is obtained without usingthis rule; this is the expected result because this stopping rule trades a tight data match for more variationin the posterior ensemble. In fact, the stopping rule is designed to avoid over matching the data, based ona discrepancy rule (Hanke, 1997, 2010; Iglesias and Dawson, 2013; Iglesias, 2014a).

The predicted well data from 10 ensembles are shown in Fig. 8. Note that the history period consistsof the first 4, 000 days while the forecast period consists of the remaining 2, 000 days. For the historyperiod, the two adaptive ES-MDA methods clearly give better data matches than are obtained with theother methods. For the forecast period, it is interesting to see that there the history matched ensembles cansometimes lead to poor and biased predictions even though the corresponding history-match is good.

Comments and ConclusionsThe adaptive ES-MDA algorithms developed here yield an improvement over the original ES-MDAalgorithm in the sense that they provide sufficient damping to avoid over corrections to the property fieldsand improve the conditioning of the matrix that must be inverted in ES-MDA algorithms. The adaptiveES-MDA methods trade a preselection of the number of data assimilation and damping factors of theoriginal ES-MDA algorithm for a preselection of a single parameter , or a preselection of the number ofstandard deviations that we allow model parameters to change over an iteration. Although we believe theselection of a single parameter instead of several is very desirable, how to choose is not so clear.Choosing close to unity may cause the algorithms to require so many iterations that their applicationbecomes computationally infeasible for large scale problems. Thus, it may still require some trial and errorwhere we try a few iterations with 0.5 and then increase or decrease it based on the results. On theother hand, restricting the change in model parameters over an iteration to two standard deviations seemsto work well in examples that we have tried but can sometime lead to more iterations than are necessaryto obtain a good data match and smooth property fields.

Based on the example provided as well as computations not shown, the following conclusions arewarranted:

1. The adaptive ES-MDA algorithms are more robust than the original ES-MDA in that they preventover correction of the initial guesses and result in good data matches with reasonable rock propertyfields.

2. At early iterations, an exceeding large damping factor may be required. These damping factors aremuch larger than any we imagined when we developed and tested the original ES-MDA algorithm.

3. A good data match does not imply good or unbiased future predictions, although with the adaptiveES-MDA algorithms, most ensembles result in an ensemble of unbiased future predictions.

16 SPE-173214-MS

AcknowledgementsThe support of the member companies of The University of Tulsa Petroleum Reservoir ExploitationProjects (TUPREP) is gratefully acknowledged.

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An Adaptive Ensemble Smoother with Multiple Data Assimilation for Assisted History MatchingIntroductionMethodologyES-MDA-RSEND FOR

ES-MDA-RLMEND FOR

ES-MDA-RLM with an Alternate Stopping RuleEND FOR

Case StudyCase descriptionResults

Comments and Conclusions

AcknowledgementsReferences