Quantification of the impact of ensemble size on the quality of an ensemble gradient using principles of hypothesis testing

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Delft University of Technology

Quantification of the impact of ensemble size on the quality of an ensemble gradient using

principles of hypothesis testing

Fonseca, R. M.; Kahrobaei, S. S.; Van-Gastel, L. J T; Leeuwenburgh, O.; Jansen, J. D.

Publication date 2015

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Society of Petroleum Engineers - SPE Reservoir Simulation Symposium 2015

Citation (APA)

Fonseca, R. M., Kahrobaei, S. S., Van-Gastel, L. J. T., Leeuwenburgh, O., & Jansen, J. D. (2015). Quantification of the impact of ensemble size on the quality of an ensemble gradient using principles of hypothesis testing. In Society of Petroleum Engineers - SPE Reservoir Simulation Symposium 2015 (Vol. 2, pp. 804-828). [SPE-173236-MS] Society of Petroleum Engineers.

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SPE 173236-MS

Quantification of the Impact of Ensemble Size on the Quality of an Ensemble

Gradient Using Principles of Hypothesis Testing

R.M. Fonseca, S.S. Kahrobaei, L.J.T.van Gastel, Delft University of Technology (TU Delft), O.Leeuwenburgh, TNO, and J.D. Jansen, TU Delft

Copyright 2015, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE Reservoir Simulation Symposium held in Houston, Texas, USA, 23–25 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). Contents of 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 reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent 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 may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract

With an increase in the number of applications of ensemble optimization (EnOpt) for production optimization, the theoretical understanding of the gradient quality has received little attention. An important factor that influences the quality of the gradient estimate is the number of samples. In this study we use principles from statistical hypothesis testing to quantify the number of samples needed to estimate an ensemble gradient that is comparable in quality to an accurate adjoint gradient. We develop a methodology to estimate the necessary ensemble size to obtain an approximate gradient that is within a predefined angle compared to the adjoint gradient, with a predefined statistical confidence. The method is first applied to the Rosenbrock function (a standard optimization test problem), for a single realization, and subsequently for a case with uncertainty, represented by multiple realizations (robust optimization). The maximum allowed error applied in both experiments is a 10° angle between the directions of the EnOpt gradient and the exact gradient. For the single-realization case we need, depending on the perturbation size, 900, 5 and 3 samples to estimate a “good” gradient with 95% confidence at 50 points in the optimization space for 50 different random sequences. For the robust case, the conventional EnOpt approach is to couple one model realization with one control sample, which leads to a computationally efficient technique to estimate a mean gradient. However, our results show that in order to be 95% confident the original one-to-one model realization to control sample ratio formulation is not sufficient. To achieve the required confidence requires a ratio of 1:1100, i.e. each model realization is paired with 1100 control samples using the original formulation. However, using a modified formulation we need a ratio of 1:10 to stay within the maximum allowed error for 95% of the points in space, though a 1:1 ratio is sufficient for 85% of the points. We also tested our methodology on a reservoir case for deterministic and robust cases, where we observe similar trends in the results. Our results provide insight into the necessary number of samples required for EnOpt, in particular for robust optimization, to achieve a gradient comparable to an adjoint gradient.

Introduction

Multiple studies have shown the successful application of various optimization algorithms to maximize hydrocarbon recovery or net present value (NPV) over the producing life of a hydrocarbon reservoir. For such problems, gradient-based techniques, in terms of accuracy and computational efficiency, are the most successful and widely applied. The adjoint method provides the most accurate gradient and is computationally the most efficient method. However, the adjoint method requires access to a reservoir simulator's source code which for commercial simulators is practically impossible. Additionally it requires a considerable amount of time to implement and is not very flexible in adaptation to different control types. These limitations of the adjoint method have led to the development of alternative gradient-based techniques. One such alternative technique, Ensemble Optimization (EnOpt), inspired by the Ensemble Kalman Filter (EnKF) method was first introduced by Lorentzen et al. (2006) and Nwaozo (2006).

Chen (2008) proposed the now standard formulation of the EnOpt method which uses an ensemble of randomly perturbed control vectors to approximate a gradient of the objective function with respect to some specific controls. The major advantages of EnOpt are its ease of implementation, flexibility to adapt to different control types and ability to be used with any reservoir simulator. The major drawback of this method, relative to the adjoint method, is its computational inefficiency and inaccuracy of the gradient approximation. Nonetheless, recently many studies such as Chen et al. (2009), Chen and Oliver (2010), and Leeuwenburgh et al. (2010) have demostrated the applicability of EnOpt for large-scale production optimization

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problems. Most of these papers have focused on deterministic optimization problems starting from a single reservoir model. However, in reality the geological and reservoir modeling process is fraught with uncertainties since a reservoir is modeled using uncertain interpretations based on uncertain data sources such as seismics, well logs etc. Incorporating these uncertainties into the optimization framework is vital to achieve results of any practical significance.

Van Essen et al. (2009) introduced a ‘robust optimization’ methodology in conjunction with the adjoint method to include the effect of uncertainties into the optimization framework. They used an ensemble of equi-probable reservoir models with differing geology and maximized the expectation of the objective function over this ensemble of models. Chen (2008) introduced this robust optimization concept within the ensemble optimization framework. They proposed the use of an ensemble of controls of equal size as the ensemble of geological models. Coupling of one member from the control ensemble with one member of the geological ensemble, a mean gradient can be approximated with the EnOpt formulation. This formulation, while computationally very attractive for robust optimization, has received scant attention with respect to its theoretical understanding. Recently Fonseca et al. (2014) demonstrated a case wherein the original formulation for ensemble-based robust optimization leads to inferior results and suggested a modified gradient formulation.

For EnOpt the two main inputs which influence the quality of the approximate gradient are the covariance matrix used to create the ensemble of perturbed controls and the number of control samples created, i.e. the ensemble size. The effect of the covariance matrix has been investigated recently in Fonseca et al. (2013) and a theoretical foundation for the use of a varying covariance matrix has been provided in Stordal et al. (2014). Sarma and Chen (2014) have investigated the applicability of different sampling techniques to improve the quality of a gradient estimate. However none of those studies have performed a detailed investigation into the effect of ensemble size on the estimated ensemble gradient quality.

In this study we aim to quantify the ensemble size required to approximate a gradient comparable to the adjoint gradient especially for robust optimization problems using principles from hypothesis testing and statistical analysis. In this paper we first provide an introduction of the hypothesis testing methodology and the different test statistics used. This will be followed by a detailed set of experiments on the widely used Rosenbrock function for cases with and without model uncertainty. Finally we test the proposed methodology on a medium-sized reservoir model, again with and without geological uncertainty

Theory

The two most commonly used objective functions for production optimization are ultimate recovery or an economic objective such as Net Present Value (NPV). In this work we chose the objective function J to be the NPV, defined in the usual fashion as

, , ,

1 ( ) ( ) ( ) , (1 )k t K o k o wp k wp wi k wi k t k q r q r q r t b J                       

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where qo,k is the oil production rate in bbl/day, qwp,k is the water production rate in bbl/day, qwi,k is the water injection rate in bbl/day, ro is the price of oil produced in $/bbl, rwp is the cost of water produced in $/bbl, rwi is the cost of water injected in $/bbl, tk is the difference between consecutive time steps in days, b is the discount factor expressed as a fraction per year, tk is the cumulative time in days corresponding to time step k, and t is the reference time period for discounting, typically one year. Gradient-based optimization requires the gradient (dJ d )T

g u which is used within an optimization algorithm to iteratively

optimize the objective function. For a detailed description of various available optimization algorithms see, e.g., Nocedal and Wright (2006). Usually the elements of the control vector are required to stay within upper and lower bounds, and different approaches for such bound control problems are available. Moreover, in addition to these constraints on the inputs, there may be constraints on the outputs of the simulator, which are much more difficult to handle. However, in this paper we are only considering the quality of the gradients under the presence of simple bound constraints.

Ensemble Optimization (EnOpt)

Ensemble-Based Deterministic Formulation

In this section we outline the standard formulation of the EnOpt algorithm as proposed by Chen et al. (2009). We take u to be a single control vector containing all the control variables to be optimized. This vector has length N equal to the product of the controllable well parameters (number of well settings like bottom hole pressures, rates or valve settings) and the number of control time steps. Chen et al. (2009) sample the initial mean control vector from a Gaussian distribution while, at later iteration steps the final control vector of the previous iteration is taken as the mean control. However the initial controls can also be chosen by the user, as will be done in our experiments.

1 2  . T N u u u   u (2)

To estimate the EnOpt gradient, a multivariate, Gaussian distributed ensemble {u1, u2, …, uM} is generated with a distribution mean u and a predefined distribution covariance matrix C where M is the ensemble size. During the iterative optimization process, u is updated until convergence, whereas C is, traditionally, kept constant. [An alternative procedure, in which C is

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updated during the optimization process, is decribed in Fonseca et al. (2013)]. In our implementation of EnOpt the ensemble members ui, i = 1, 2, …, M, are created using

1 2 , i i u  u C z (3) with 1 1 M i i M  

u u . (4)

We use a Cholesky decomposition to calculate C , and draw z1 2

i from a univariate Gaussian distribution. To estimate the gradient, a mean-shifted ensemble matrix is defined as

1 2

.

    M

U u u u u u u (5)

[Note that in earlier publications we used the transposed version of U. We modified our notation to bring it in line with that of textbooks such as Conn et al. (2009).] A mean-shifted objective function vector is defined as

1 2 , T M J J J J J J       j  (6)

where the expectation of the objective function is given by

1 1 M . i i J J M  

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The approximate gradient as proposed by Chen (2008) and Chen et al. (2009) is given by

1 , uu uJ gC c (8) where 1 ( ) 1   T uu M C UU (9) and 1 ( ) 1   uJ M c Uj (10)

are ensemble (sample) covariance and cross-covariance matrices respectively. (Note that cuJ is a one-dimensional matrix, i.e. a vector.) For the usual case where M < N, matrix Cuu is rank-deficient, and Chen (2008) and Chen et al. (2009) therefore propose not to use expression (8) but, instead, to use

1 = , uu uu uJ uJ g C C cc (11) or . uu uJ   g C c (12)

Alternatively, the pre-multiplication in equation (12) can be performed with C , leading to .

uJ

 

g Cc (13)

All three expressions (11), (12) and (13) can be interpreted as modified (regularized or smoothed) approximate gradients. In the present paper we use a straight gradient, i.e. expression (8), computed as the underdetermined least squares solution

† ( T) 

g UU Uj , (14)

where the superscript † indicates the Moore-Penrose pseudo inverse, which is conveniently computed using a singular value decomposition (SVD); see, e.g., Strang (2006). Moreover, we use smoothed and double-smoothed versions of equation (14):

† ( )  T   g C UU Uj , (15) † ( )   T   g CC UU Uj , (16)

Equation (14) was also described in Dehdari and Oliver (2012), while Do and Reynolds (2013) recently demonstrated that it is akin to what is known as a ‘Simplex gradient’ in, e.g., Conn et al. (2009). Do and Reynolds (2013) also provided theoretical connections between various ensemble methods such as simulataneous perturbation stochastic approximation (SPSA),

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Simplex gradient, EnOpt etc. Moreover, they proposed a modification to the gradient formulation which uses the current control vector u and the corresponding objective function value J to calculate the control and objective function anomalies

U and j: 1 2 ,          M U u u u u u u (17) 1 2 ,         M T J J J J J J j (18)

where the superscript ℓ is the optimization iteration counter.

Equations (11-16) can all be used to estimate a gradient-based on either the original [equations (5) and (6)] or the modified [equations (17) and (18)] formulations. Thus we can estimate as many as twelve different gradient formulations for deterministic cases. Further varieties will emege when considering robust optimization.

Ensemble-Based Robust Formulation

Chen (2008) proposed a computationally efficient robust ensemble algorithm in which she used an ensemble of controls of equal size as the ensemble of geological models. Coupling one member from the control ensemble with one member of the geological ensemble a mean gradient is approximated; see Chen (2008) for details. Therefore the ensemble size M of the controls is equal to the number of geological models, i.e. a 1:1 ratio. Hence only M simulation runs are needed to approximate the ‘robust’ gradient of the objective function. Recently Stordal et al. (2014) reached a similar conclusion starting from a different mathematical viewpoint. However the theoretical understanding of using this 1:1 ratio is still incomplete. As an alternative to this formulation, Fonseca et al. (2014) propose a modified formulation for the robust gradient which no longer uses the mean-shifted control samples and objective values, equations (5) and (6). Instead, in equation (5) the control sample mean u is replaced by the control vector of the current iteration step, u:

1 2 ,

 

     

M

U u u u u u u (19)

The new formulation replacing equation (6) is

1 1 2 2 ,          T M M J J J J J J j (20)

Note that equation (19) is identical to equation (17) as used in the deterministic modified expression of Do and Reynolds (2013), but that equation (20) is different from equation (18). This modified gradient formulation [based on equations (19) and (20)] will also be tested in our set of experiments. It behaves distinctly different compared to the original robust formulation [based on equations (5) and (6)]. First, because the subtractions in the objective function values in equation (20) are with respect to the individual objective function values 

i

J and not with respect to the mean. Second, because for bound-constrained control problems, u and u may be shifted with respect to each other. Thirdly, because the effect of outliers,

which may strongly influence the mean value our least-squares approach to estimate the gradient, is reduced in the modified formulation.

Note: all the different gradient formulations (11-16) for deterministic optimization are also applicable to the robust case. Together with the robust modified formulation [equations (19) and (20)] this leads to a total of 18 potential robust gradient formulations for the 1:1 ratio (i.e. one control perturbation for each geological realization) approach. However, another distinction can be made if we use other ratios. E.g., Raniolo et al. (2013) suggest the use of 20 control perturbations for every model realization. For every model realization, using the 1:20 ratio, they estimate an individual gradient, whereafter they take the mean of the individual gradients to obtain the robust gradient. This formulation will hereafter be referred to as the ‘Mean of Individual Gradients’ (MIG). Alternatively, one can combine all the controls and objective function anomalies to estimate a single robust gradient, i.e. not estimate individual gradients for every model realization. This approach will hereafter be referred to as the ‘Hotch-Potch Gradient’ (HPG). This additional disctinction leads to a total of 30 potential formulations [2 times 18 minus 6 because for the MIG approach there is no difference between using equations (18) and (20)].

Adjoint method

The adjoint method has been investigated extensively for use in data assimilation and production optimization. Detailed derivations for the production optimization case can be found in, e.g., Brouwer and Jansen (2004), Sarma et al. (2005), Kraaijevanger et al. (2007) and Jansen (2011). The adjoint method is the most accurate and computationally efficient method for computing a gradient. Computation of the gradient only requires one forward simulation and one fast backward computation. Therefore the number of simulation runs is independent of the number of controls. However for robust optimization using the adjoint requires running the forward and backward simulation for every geological realization, thus to compute the robust gradient, the same number of simulation runs will be performed as required for the robust EnOpt gradient using the 1:1 ratio. For our experiments we assume that the adjoint gradient is the exact gradient, which the EnOpt method

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tries to approximate. In this study the adjoint module available in the Shell in-house simulator was used (Kraaijevanger et al. 2007).

Hypothesis testing

We use principles from hypothesis testing to validate the research goal of this paper, namely to test if the approximate EnOpt gradient is comparable in quality to the adjoint gradient. To be able to determine the difference in gradients we compute the angle between them by using the dot product:

. cos( ) adj ens .

adj ens

  g g

g g (21)

Another measure that describes the difference in direction of two vectors is the length of the difference between the normalized adjoint and EnOpt gradients, defined as

.   adjens ens adj g g g g (22)

To eliminate the effect of a difference in gradient magnitude between both methods the gradient vectors are made into unit vectors by dividing them by their norm. When  goes to zero, the two gradients will point in the same direction, just like if the angle goes to zero. These two test parameters can be used to cross-validate each other, because a difference of 10◦ corresponds to a dimensionless length difference of 0.175. Thus the two equivalent null hypotheses used are

0: 10 , : 0.175 . H     (23) The statistical inference method used is based on pre-defined confidence intervals for the testing parameters defined above.

Confidence intervals

Creating a confidence interval is a method to define a range at and the certainty that the true value of an estimated parameter lies within it, based on the knowledge of the sampling distribution (Dekking et al., 2005). In our numerical experiments we create a dataset of our parameters, given in equations (21) and (22). The parameter of interest  is the maximum allowable deviation of the EnOpt gradient with regard to the exact or adjoint gradient. As it is virtually impossible to achieve a 100% confidence, we apply a confidence level of  = 0.95. The general definition of a confidence interval assumes a two-sided interval, i.e. an upper and a lower limit. However it is also possible to have a one-sided interval. As the test parameters used for the numerical experiments are absolute values of deviations we only want to find the confidence interval of the maximum deviation, thus the upper limit. Using a one-sided interval, the confidence interval is just the integral of the probability distribution, i.e. the cumulative density function (CDF).

Beta distribution

The EnOpt method samples random points from a normal distribution with a user-defined standard deviation. The distribution of the test parameters ( and ) are, however, not normally distributed due to the non-linearity of the objective function and the function for the test statistics. In order to determine a confidence interval the distribution of the underlying parameters needs to be known. The two parameters  and  are , by definition, ratios of two different gradients, and the beta distribution is suitable to fit data sets that are ratios. The beta distribution forms a class of continuous probability distributions, parameterized by two shape parameters a and b that define the shape of the distribution. AbouRizk et al. (1994) outline several methods to determine these shape parameters, of which we have chosen to use the maximum likelihood estimator. AbouRizk et al. (1994) also demonstrate that there is virtually no difference in the results when using different fitting methods. Note that the beta distribution is always bounded in the interval [0,1]. However, in this study, because we use the cosine function, our data is bounded between [-1 1]. Thus in order to use the beta distribution for angles greater than 90 degrees (i.e. cos() < 0) we simply reset the angle to 90 degrees (i.e. cos() =0).

Methodology using Beta Distributions

The methodology proposed in this work to either accept or falsify our null hypothesis and estimate the necessary ensemble size to accept our null hypothesis is as follows:

 Sample points in the control space and compute the adjoint and EnOpt gradients at each point.  Estimate the fitting parameters a and b of the beta distribution using the data.

 Plot the CDF for the beta distribution and compute the confidence interval for the predefined error.  Repeat for varying ensemble sizes.

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We have chosen the 95% confidence interval to test our methodology, however any different confidence interval can be chosen within the same workflow. Varying the desired confidence interval will automatically vary the ensemble size needed to accept our hypothesis. In essence an ensemble size is found that gives an accurate gradient approximation in a number of times equal to the confidence interval, i.e. if the numerical experiment would be repated many times, 95% of the experiments would result in an approximate gradient within the error margin compared to the true (adjoint) gradient. For a more detailed explanation of confidence intervals see, e.g. Dekking et al. (2005).

Methodology with Traditional Hypothesis Testing Principles

In addition to using the beta distribution to test our null hypothesis we can also use a traditional hypothesis testing approach to either accept or falsify our hypothesis and estimate the necessary ensemble size as follows:

 Sample points in the control space and compute the adjoint and EnOpt gradients at each point.  Count the number of points that satisfy the null hypothesis

 Compute the confidence interval for the predefined maximum allowable error.  Repeat for varying ensemble sizes until the confidence interval is achieved.

In this paper we compare the results of the two methodologies for the different test statistics in the following section.

Numerical Example Rosenbrock Function

The methodology is first applied to the non-linear Rosenbrock function named after the mathematician who first used it to demonstrate his optimization algorithm, see Rosenbrock (1960). This analytical function has since been used as a standard test case in mathematical optimization. The Rosenbrock function consists of a curved narrow valley which most algorithms have little difficulty finding. However once found, the difficulty lies in finding the global optimum which is situated inside the valley. Equation (24) is a slightly altered version, as it is multiplied by -1, making it a maximization problem opposed to a minimization problem:

2 2 2

1 2 2 1 1

( , ) 100( ) (1 )

J u u   uu  u . (24)

The Rosenbrock function has an optimal solution J=0 at (u1,u2) = [1 1]. This point lies on a long curved ridge; see Fig. 1 for a contour plot. Since it is an analytical function it is possible to compute the exact gradient. The red dots in Fig. 1 are 50 randomly distributed points in space which will be used to test our methodology. All the numerical experiments will be carried out over the same set of points for different scenarios. Since we are working with approximate gradient techniques the effect of different random sequences also needs to be accounted for. Therefore, all the experiments are repeated for 50 different random sequences. The results presented below are the mean values for the 50 points in space over the 50 different random sequences.

Fig. 1: Contour plot of Rosenbrock function given by equation (24). Red dots are 50 points randomly distributed in space.

Deterministic Case

We first test the two hypothesis testing methodologies which consist of calculating the angle between the gradient estimated by EnOpt and the exact gradient for different ensemble sizes. The points are uniformly distributed so as to capture the effect of the spatial variability in the objective space on the gradient quality, with many points that are on the ridge or on

u 1 u2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 1.5 2 2.5 3

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m a Normal d a fast and accu

value for c1 m e beta distribut nstead of the ed to the resu lculating an an ed in this work bation sizes. F e 50 points usi s for the same ve) we achieve ence interval al hypothesis

ere for an ens e for an ensem strongly affec s. We observe nterval. For si e need an ense been investiga roduced to mim 2 2 1 2 1 (c u u )  distribution, w urate way to mainly affects

tion for varying

angle as the ults shown abo

ngle or the  k we would ne ig. 4a (left-sid ing different e e 50 points wi e a 100% conf with an ensem testing metho semble size of mble size of 3 a

ted by the per e that, for larg ignificantly sm emble size equ

ated before, so mic geologica

2

sin( ) (1c

  

where the stand test the vario s the steepness

g ensemble siz

data. We obs ove with exac norm with the eed approxim de plot) depict ensemble sizes ith  = 0.001. fidence interva mble size of 3 d as well as u 5 or higher th nd higher the h rturbation size ge perturbatio maller perturb ual to n+1, jus o uncertainty al uncertainty. 2 1) . u dard deviation us ‘robust’ En s of the object zes for = 0.1. serve (results ctly the same he same data s mately 900 sam ts, similar to t s with  = 0.0 . We observe al while for  3 samples (Fig using the  nor

he hypothesis hypothesis is s e which in tur on sizes, we n bation sizes w st like if we w in the Rosenb . n reflects the EnOpt gradien tive space, wh SPE 173236-not shown) v trend in the d should lead to mples when us the curves sho 01. Fig. 4b (rig that for  = 0  = 0.001, i.e. g. 4b, red curv rm. is satisfied at satisfied. rn determines need significan we would need were to estima brock function (2 magnitude of t formulation hile c2 rotates -MS very data. the sing own ght-0.01 one ve). t all the ntly d an ate a n is 25) f the s to the

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SP spa Th Fig unc Fig unc In wit Ch ma ens exp we ens me sat on sig hav Fig (blu Th we me wh E 173236-MS ace. Fig. 5 an his was done to

g. 5: Contour p certainty. g. 6: Contour p certainty. view of the r th the tradition hen (2008) fir athematical re sembles for bo periments the ere reasoned t semble size o ean angle sign tisfy our null h equations (19 gnificantly bet

ve satisfied ou

g. 7: Comparis ue) and modifi

hese results ar e usually use a ean angle of 4 hen using the 1

nd Fig. 6 show o investigate t

plot of five rea

plot of five rea

results from th nal hypothesis rst proposed

asoning for th oth the model e original 1:1 to be the lack f model realiz nificantly redu hypothesis wh 9) and (20) (re tter. We obser ut null hypoth

son of the tren ed form (red) u e based on th an ensemble 40 degrees wh 1:1 ratio for ro w five realizat the impact of u alizations of th alizations of th he determinis s approach to the idea of th he applicabilit ls and consequ formulation i k of a good qu zations and, c uces for an in hen using the o ed curve) whi rve that for an hesis and achie

nd in the mean using the comp

he case represe size of 100 m hile the modifi obust optimiza

tions each for uncertainty on

he Rosenbroc

e Rosenbrock

tic case, for t determine the he 1:1 ratio e ty of this 1:1 r uently the con in many insta uality gradien consequently, ncreasing ens original formu ch still retains n ensemble siz eved the 95% n angle over th putationally att

enting the hig models to capt fied formulatio ation, the mod

cases which n the gradient

k function giv

function give

the robust cas e ensemble siz explained in ratio provided ntrols. In realit ances indeed p nt. Fig. 7 illus also control r semble size. H ulation (blue c s the computa ze of 1000 w confidence in he 50 spatial p tractive 1:1 rat ghest uncertain ture the uncer on achieves a dified formula are representa estimate. ven by equatio n by equation se we only us ze required to the theory se d by Chen (20 ty, however, w produces resu strates the eff realizations. W However, even curve). On the ational attracti we have a mea nterval.

points with inc io. nty in the mo rtainty. For th mean angle o ation based on ative of low a on (25) for a ca (25) for a cas e the angles a estimate a go ection to estim 08) is only ap we usually wo ults of insuffic fect of using t We observe, in n for an ense e other hand th iveness of the n angle of app creasing ensem dels. In our re hat size, the or of 7 degrees. W n equations (19

and high unce

case mimicking

se mimicking a

as our testing ood quality ‘ro

mate a ‘robus pplicable if we ork with 100 m cient quality. the 1:1 ratio f n line with th emble size of he modified fo original form proximately 4

mble sizes for

reservoir optim riginal formu We recommen 9) and (20) be rtainty scenar g a low degree a higher degree parameter al obust’ gradien st’ gradient. T e have very la models and in The poor res for an increas e theory, that 1000 we do formulation ba mulation perfor 4 degrees and r the original fo mization prob ulation achieve nd therefore t e applied. 9 rios. e of e of ong nt. The arge our sults sing the not ased rms d we orm lem es a that,

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10 stu the mo for aga ori sca rea bet com . Fig for des Fig the all the rea sho mo of me the Effect of Hig The results i udies, e.g. Ran e use of higher odel and cont rmulation usin ainst the ense iginal formula ale). In both p alizations. Onc tter gradient e mputational c g. 8: (a). Illustr rmulation. (b) Il Effect of Un The effect of sired confiden g. 9(a) (left-si e lowest uncer the results thu e modified fo alizations was

own) that for odified formul uncertainty, t ean angle less e necessary co gher Ratios in Fig. 7 were niolo et al. (20 r ratios (1:20 trol perturbat ng a perturbat emble size of ation, and Fig. plots we observ

ce again, the m estimates can

osts of using t

rates the effec llustrates the e

ncertainty

f uncertainty i nce interval. F ide plot) depic rtainty scenar us far, that the ormulation gr s kept constan r the original

lation the grad the original fo than 10 degr onfidence inter

e for the case 013) and Li et etc.) to find b tions on the tion size equa model realiza 8(b) which d ve that an incr modified form be achieved w these ratios m ct of using hig effect when usi

is investigated Fig. 9 is an ill cts the results io for both th e modified for radient estima nt at 100 for t formulation dient estimate ormulation w rees. Note: if t rval. representativ al. (2013), did etter gradient quality of th al to 0.01. Fi ation with the displays the res

reased ratio g mulation outpe while using hi must also be ac

gher ratios on ing the modifie

d in conjunctio lustration of t

for the highe e original and rmulation resu ate is less se his exercise a using a large e improved for ould need a r the mean ang

ve of the highe d not achieve estimates. W he gradient e ig. 8 consists e curves repre sults using the gives better gra erforms the o igher ratios. N ccounted for, e n the gradient ed formulation on with the ra the impact tha est uncertainty d modified for ults in signific ensitive to th and a perturba er perturbatio r smaller pertu ratio of 1:20 gle is less than

est model unc results of pra e investigate h estimate. Note of two plots, esenting the ra e modified for adient estimat riginal formul Note though th especially for quality when with the HPG f atios needed to at uncertainty y case while F rmulations. In cantly better g he effect of u ation size of 0 on size resulte urbation sizes or 1:50 i.e. 2 n 10 degrees i certainty using ctical value w here the impac e: the results Fig. 8(a) wh atio used for t rmulation (not tes for the diff lation also for hat while it is large-scale hi

using the orig formulation. N o satisfy the n has on the qu Fig. 9(b) (righ n both plots w gradients than uncertainty. T 0.01 was used ed in better m s. We observe 000 or 5000 it does not gua

g the 1:1 ratio with the 1:1 rat ct of varying t are obtained hich is the me the gradient e te the differen ferent ensemb r larger ratios better to use igh-dimension ginal formulat ote the differe

null hypothesi uality of the g ht-side plot) d we observe, in n the original f The ensemble d. We also no mean angles, e that, dependi function eval uarantee that w SPE 173236-o. Some previ tio and sugges the ratio betw d with the H ean angle plot estimate with nce in the vert ble sizes of mo

. Thus in gen higher ratios, nal problems.

ion with the H nt vertical scal s and achieve gradient estim depicts results accordance w formulation. A e size of mo ticed (results whereas for ing on the deg luations to fin we have achie -MS ious sted ween HPG tted the tical odel eral the HPG les. the mate. for with Also odel not the gree nd a eved

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SP Fig Illu ang con tha ori als for for Fig unc hyp we dim ‘ro the pro Fig rat dif the the E 173236-MS g. 9: (a) shows ustrates the res

gles for the hig

Hypothesis T Fig. 10 illus nfidence inter at for a smalle iginal formula so observe tha r the highest u rmulation for a g. 10: Hypothes certainty case pothesis and a Mean of Ind The results p e would requi mensional con obust’ gradien e adjoint form oblem we shou g. 2. Using th io of 1:5 to e fference betwe e ratio needed e same trend is s the effect of sults for the lo gh uncertainty

Test Results

strates the ra rval for the di er ensemble s ation while fo at the required uncertainty sce

an ensemble s

sis testing resu with the HPG achieve the des

dividual Grad presented in F re ratios as h ntrol problem nt when using mulation and al uld need 3 fun he MIG formu estimate a me een the origin compared to s observed for

a high degree owest uncertain

case are highe

atio necessary fferent gradie size of model or the same en d ratio decreas enario. For th size of 10 mod

ults using the G formulation. sired confidenc dients (MIG) Figs. 8-10, wh high as 1:1100 these results a ratio other t lso by Raniol nction evaluat ulation we obs ean angle less

al and modifie the results wi r the low unce

e of uncertaint nty case. Note er than the low

y to satisfy th ent formulation l realizations, nsemble size, ses with an in e lowest unce del realization traditional met A lower ratio ce interval.

ich are obtain 0 to satisfy th are complete than 1:1. Note o et al. (2013 tions to estima serve in Fig. s than 10 degr ed formulation ith the HPG fo ertainty case. ty on the mea e: for both thes w uncertainty ca

he null hypot ns and the dif e.g. 10, we n we would re ncrease in the ertainty scenar ns, and a 1:5 ra thodology for is needed for

ned using the H he hypothesis ely counter-in e: This approa ) using EnOp ate a gradient, 11 that for a rees. Increasi n is minor at b ormulation. T n angle for in se plots an ens

ase. (Note the

thesis used in fferent ensemb need significa quire a ratio ensemble size rio we would

atio for the mo

the original (b higher ensem

HPG formula and achieve ntuitive. Thus

ach was follow pt gradients. S , for a sufficie perturbation ng the ratio w best. These re he results are creasing ratios semble of 100 different vertic n this paper a ble sizes of m antly higher ra of 1:10 with t e of model rea require lower odified formu

lue) and modif mble sizes of m tion, illustrate the desired co we test the M wed by Van E ince we are d ently small per size equal to will improve t esults are a ma based on the s with the HPG model realizat cal scales). and achieve model realizati atios (1:1100) the modified alizations. All r ratios: 1:800 ulation.

fied (red) form model realizatio

e that for sma onfidence inte MIG formulat Essen et al. (2 dealing with a rturbation size 0.01 we wou the gradient e arked improve highest uncer G formulation. tions is used. the desired 9 ions. We obse ) when using formulation. l these results

with the orig

using the high ons to satisfy ll ensemble si erval. For a tw ion to estimat 2009) albeit us two dimensio e, as illustrated uld at best nee

estimate, and ement in term rtainty case wh 11 . (b) The 95% erve the We are inal hest the izes wo-te a sing onal d in ed a the s of hile

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12 Fig req det suf con the Fig wo In sm 3D illu we sys act flu is a the is inj En g. 11: Illustrati quired even for

The effect o terministic ca fficiently sma nfidence inter e ratio increase g. 12: Hypothes ould require sig

summary, for mall ensembles

D Reservoir M

The ‘Egg M ustration of th ells (blue) and

stem in the fo tive. A detaile uid properties u assumed to be e injectors are simulated for ection rates as We use a com nOpt gradient on of the mea r small ensemb of perturbatio ase, that a dec all perturbation rval even for

es, although, i

sis testing res gnificantly sma

r this case usin s of model rea

Model: “Egg M

odel’, first int he permeabilit d four produc orm of discre ed description used for all en e incompressi rate-controlle a period of 3 s controls we mmercial fully of J with re an angle for in ble sizes comp

n size estima creasing pertu n size (  = 0 small ensemb it is significan

sults using the aller ratio comp

ng the MIG fo alizations. Model” troduced by v ty field of a si ction wells (r ete permeabili n of a standard nsemble mem ible. The botto ed between 1 3600 days or have of 40x8 y implicit blac espect to the ncreasing ens pared to Fig (10

ates from the urbation size 0.001) a ratio ble sizes of m ntly lower than

e traditional me pared to the HP ormulation giv van Essen (200 ingle model r ed) completed ity fields mod dized version mbers are given om hole press and 79.5 m3/d slightly less t = 320 control ck oil simulat controls u. A semble sizes a 0) for the highe

MIG formul results in an of 1:3 is suffi model realizati n the ratio req

ethodology for PG formulation

ves much bett

09), is a chan realization wit d in all the l deled with 60 of this Egg M n in Table 1. N sures of the pr day. The initia than 10 years ls, i.e. a 320 d tor (Eclipse, 2 An in-house and ratios. We est uncertainty lation is show increase in th ficient to satisf ons. We obse quired when us r different pert n. er angles for s nelized reserv th the location ayers. The m 0 × 60 × 7 = 2 Model is given No capillary p roducers are c al reservoir pr s. There are 4 dimensional pr 011) for the r simulator is u e observe a si case. wn in Fig. 12 he quality of fy our null hy erve that for a sing the HPG

turbation sizes

smaller ratios

voir with seve ns of the eight model represen 5.200 grid ce n in Jansen et pressures are in constrained be essure is at 40 0 control time roblem. eservoir simu used to calcu ignificant redu 2. We observ the gradient ypothesis and an increasing formulation. s () and ratios especially wh en vertical lay ht (mainly peri nts a channeli ells of which t al. (2013). T included and t etween 385 an 00 bars. Produ me steps of 90 ulations to esti

ulate the adjo

SPE 173236-uction in the r ve, ‘akin’ to estimated. Fo achieve the 9 perturbation s s. Lower val hen working w yers. Fig. 13 is ipheral) inject ized depositio 18.553 cells The reservoir the reservoir r nd 400 bar, wh uction of the f days, thus us mate the (robu oint gradient, -MS ratio the or a 95% size lues with s an tion onal are and rock hile field sing ust) see

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SP Kr can Fig det spa cov ‘re dif to con con sig pla dif fun Fig (ind ang wit ob E 173236-MS raaijevanger et n be found in g. 13: Reservoi Deterministi The methodo terministic ca ace. Randoml vered all featu elevant’ points fferent initial s 79.5 m3/day. ntrol points, a ntrol trajector gnificant scope ateau in the ob fferent parame nction value a g. 14: Illustratio dicative of the The Steep Re Fig. 15 disp

gle for an inc th different ra serve that we t al. (2007) fo Jansen et al. ( r model displa ic case ology present ase. When wo ly creating po ures of the obj s which captur strategies. Fig Thus we have as was done f ry. Fig. 14 als e for optimiza bjective funct eters are requ at iteration num

on of the optim steeper region

egion lays the resul creasing ensem andom seeds.

need an ensem

or details. Note (2013).

ying the positi

ted above is fi rking with a oints to evalu jective functio re the nature o g. 14 shows th e 35 points to for the Rosen so shows two ation, and a fl tion space. W uired to achiev mber 7 is an ar mization proces n of the curve)

lts for the poi mble size with

The left subpl mble size of a

e that the mod

on of the injec

first tested on 320-control p ate our metho on space, due

of the objectiv he optimization test the meth nbrock functio dashed regio atter region, i While using the ve the necessa rtifact of the p ss (blue curve) and a black el ints encapsula h a constant p lot of Fig. 15 approximately

del has been b

ctors (blue) and

a single reali problem it is, odology is al to the ‘emptin ve function sp n process for hodology, i.e., on, we now te ons, a steep re indicated with e same hypoth ary confidenc plotting algori ) using adjoint lipse (indicativ ated by the re perturbation s illustrates the y 150 to satisfy benchmarked f d producers (re ization of the of course, im so risky since ness’ of a 320 pace we perfo 35 iterations f , instead of us est the gradien egion, indicat h the black ell hesis, we inve ce intervals. N ithm. t gradients. Th ve of the flatter ed ellipse in F ize, = 0.01. e mean angle o

y the null hyp

for the differe

ed) Egg Model a mpossible to vi e we will not -dimensional rm an adjoint for an initial s sing a large nu nt quality for ed with a red ipse, which is estigate the tw Note: the appa

e curve is divi r region of the Fig. 14. It dep The results a of the differen othesis. We a ent simulators and thus we ar isualize the o t be sure that space. Thus, i t- based optim strategy of con umber of rand r 35 poins alo d ellipse, in w s indicative of wo regions sep arent decrease

ided into two p curve)

picts the decre are the mean

nt points in th also observe th , details of wh re dealing wit bjective funct t our points h in order to obt mization from t nstant rates eq domly distribu ong a pre-defi which there ex f a peak, ridge parately, beca e in the object

parts, a red elli

ease in the m of 5 experime he red ellipse. hat increasing 13 hich th a tion have tain two qual uted ined xists e or ause tive ipse mean ents We the

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14 ens est gra num 400 res Th Re [eq gra for des We neg stra equ ens cov Fig (14 the Sm reg fla ach cor hyp her a s equ fur bet imp not gre bec dir semble size le timate a high-adient estimat merical round 0 and 300 sam sults we have he results pres eynolds (2013 quations (5) a adients and the r different per sired confiden e observe that gative impact aight gradient uation (11). W semble sizes variance vecto g. 15: (a) illustr 4), while (b) dis The Flatter R The general e straight grad moothing of th gions the mod

tter region is t hieve the 95% rrelation lengt pothesis at mo re were obtain smoothed grad uation (11). F rther single sm tter quality gr proves with a t observe a si eater than 90 cause it impli rection. eads to higher -quality gradi te decreases, w d-off errors. E mples respect used a time-c sented in Fig 3) [equations and (6)]. Irres e modified an rturbation size nce intervals w t smoothing o t on the gradi t obtained fro We also note th and inferior or, i.e. equatio

rates the perfo splays the effec

Region trend in the re dient which g he gradients, dified formula the sensitivity % confidence ths varying fr ost of the poin ned through an dient given by Fig. 16 shows moothing, i.e. radients for al an increase in ignificant imp degrees, i.e. ies that even

r-quality gradi ent. We also while for sma E.g., with  = tively to satisf correlated cov g. 15(a) are o (17) and (18 spective of th nd original form es and correla will then be di of the gradien ent quality fo om equation ( hat a double s results for lar on (13), achiev ormance of the ct of using diffe esults observe gives good es on the other ation of Do an y of the gradie interval for rom 8 to 12 c nts. It is virtu n exhaustive t y equation (15 s the quality o equation (13) ll ensemble si ensemble siz provement in g within the f for smaller e ients. Thus fo tested (result ller perturbati 0.1 (i.e. 10 t fy the null hy variance matri obtained with 8)]. This perf he gradient fo mulation grad ation lengths ( ifferent. Fig. 1 nt, i.e. pre-mul or this set of p 14), i.e. equa smoothing of rger ensemble ves better resu

e two different erent versions

ed for the stee timates for th hand, achiev nd Reynolds (2 ent estimate qu this region w control time s ally impossib trial and error

) achieves a b of the differen ), which is aki zes. Note: irre ze. We also ob gradient quali first quadrant ensemble size or a 320-dimen ts not shown) ion sizes we a times larger), ypothesis and ix with a corr h equation (14 forms better t ormulation us dients converg (results not sh 15(b) shows th ltiplication of points and a  ation (15), per the straight g e sizes. Note ults than those

formulations of the ‘smooth eper region is he steeper reg es much-high 2013) perform quality to the c we observe th steps in conjun le to know a-approach. Sim better angle co nt smoothed g kin to a double espective of t bserve that fo ity. Also note with respect s we are able

nsional proble that for incr also obtain inf

and  =0.000 achieve the d relation length 4) and the m than using eq ed, an increa ge for larger en hown), althoug he effect of sm f the gradient  = 0.01. We rforms better radient, i.e. eq : pre-multipli e obtained wit used to estima hed’ gradient a also observed gion does not her-quality gra ms best. A maj choice of the c hat we need s nction with an priori the corr milar to the re ompared to di gradient formu e smoothing o the different g or this region w e that, irrespec to the adjoin e to approxim em with appro easing perturb ferior gradien 01, (i.e. 100 ti desired confid h equal to 20 modified formu quation (14) w sing ensembl nsemble sizes gh the ensemb moothing on th by the covari e also observe than using th quation (16), g ication of the h equation (11

ate the ‘straigh and their relativ

d for the flatte achieve good adients in this ajor difference correlation len smaller pertur n ensemble si rect correlatio esults for the s rectly estimat ulations. Whe of equation (14 gradient estim with ensemble ctive of ensem nt gradient. T mate, in gener oximately 150 bation sizes t nt estimates, m imes smaller) dence interval which was ar mulation sugge with the orig le size leads s. This is trend

ble size neede he different gr iance matrix, e that single s he smoothed g gives better re e gradient giv 1). ht gradient’ g g ve behavior. er region. How d gradients in s flatter regio e in gradient e ngth used. Alt rbation sizes, ize equal to 3 on length, and steeper region ting a smoothe en using equa 4), i.e. equatio mates used, the

le sizes higher mble size, the This is particu ral, a roughly SPE 173236-0 samples we he quality of most likely du ), we would n . To obtain th rbitrarily chos ested by Do ginal formulat to higher-qua d is also obser ed to achieve radient estima has a margin smoothing of gradient given esults for sma ven by the cro

given by equa wever, estimat n the flatter p on. Note, in b estimation for though we do  = 0.001, 300 to satisfy d the lengths u n, we observe t ed gradient us ation (11), the on (16), achie e gradient qua r than 300 we angles are ne ularly import ‘correct’ up--MS can the e to need hese sen. and tion ality rved the ates. ally f the n by aller oss-tion ting part. both the not and the used that sing en a eves ality e do ever tant, -hill

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SP Fig req ens the det mo gen opt thi 25 two Fig Fig usi elli E 173236-MS g. 16: Comparis Robust case The main pu quired to estim semble of 100 e different pe

tails. The hyp odel. The cont nerate the poi timization usi s control traje iterations due o regions usin g. 17: Six rando . g. 18: Illustratio ing adjoint gra pse (indicative

son of the effec

urpose of this mate a high-0 equi-probab rmeability fie pothesis tests trols and the f ints to test ou ing the adjoint ectory. The re e to the comp ng the same re omly chosen re on of the expe adients. The cu e of the flatter r ct of smoothin study is to qu quality gradie ble geological

elds and the d are performe fluid model a ur methodolog t method as de esult of the rob putational com easoning as ex ealizations disp ected objective urve is divided region of the c g and different

uantify the qua ent for a rea

models, six o different direc ed to estimate s well as the gy, the same a

escribed in Va bust optimiza mplexity invol xplained above

playing the unc

e function valu d into two parts curve).

t gradient estim

ality of the ‘ro alistic reservo of which are d ctions and or e a ‘robust’ e other properti approach is u an Essen et al ation is display lved. Again, f e. certainty in the ue over 100 rea ts, a red ellipse

mates for the f

obust’ ensemb ir test case. T displayed in F rientations of ensemble grad ies are exactly used as describ l. (2009), and

yed in Fig. 18 for the analys

e geological m alizations with e (indicative of latter region (b ble gradient an Thus, to test Fig. 17. The u the channels dient using al y the same as bed above. Th then assess th 8. Note that th sis, we divide

odels; see Jan

the robust op f the steeper r black ellipse) o nd determine our methodo uncertainty is s, see Jansen ll hundred rea

for the determ hat is, we firs he quality of th he optimizatio e the optimiza nsen et al. (201 ptimization pro region of the c of Fig. (14). the optimal r ology we use captured throu et al. (2013) alizations of ministic case. st perform rob he gradient al on was limited ation process i 3) for details. ocess (blue cu curve) and a bl 15 atio e an ugh for this To bust ong d to into rve) lack

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16 enc the all rob equ bet of 13 of in tha adv obj thi com qua Fig mo Fo for gra wh We usi for sat The Steep Re Following th capsulated by e original and

the five poin bust formulati uations (5) an tween 80 and 5 experiment ,15 and 16), w the gradient l that the smoo at using a rela vantage of us jective functio s region. Note mpared to usi ality gradients g. 19: Illustrate odel realization r the Rosenb rmulations. Fo adient quality here, similar to e also observ ing the modif rmulations, al tisfies the hyp

egion he analysis pr y the red ellips modified form nts we are abl ion [based on nd (6)] we nev 90 degrees fo ts with differe we observe mu leads to better othed version atively larger sing the mod on) is from wi e : when usin ing smoothed s for both grad

s the differenc n using the HPG

rock function or this case we for the two o the Rosenbr e, as expected fied formulat lthough the M othesis at all t rovided for th se as shown in mulation usin le to satisfy o n equations (1 ver satisfy ou or all the poin ent random se uch better ang r estimates of of equation ( r perturbation dified formula ithin this regio ng non-smooth gradients. No dient formulat ce in gradient q G ratio formula n we observed e observe a si ratio formula rock results, w d, that as the ion. The resu MIG formulati the points, eve

he determinis n Fig. 18. We ng the computa our null hypot

9) and (20)]. ur hypothesis a nts when using eeds.) If we u gles as shown the ensemble (14), i.e. equa size ( = 1) ation with th on so achievin hed gradients, ote: while the tions with the

quality betwee ation. d significant milar trend in tions. Fig. 20 we always ach ratio increas ults indicate t ion still perfo en for the 1:1

stic case we f have five poi ationally attra thesis, i.e. the However if w at any of the g equation (14 use the‘smooth in Fig. 19, th e gradient, and ation (15), ach ) for this regi he 1:1 ratio b ng good angle , i.e. equation e 1:1 ratio sati HPG formula en the original differences in n the results. F 0(a) depicts th hieve significa es the gradien that the modi orms better. I ratio. first consider ints (control s active 1:1 ratio e angle is less we use the 1:1 points. With 4). (As for the hed’ versions hough the hypo

d we observe hieves better r ion leads to th because the m es at acceptabl n (14), the gra isfies our hyp ation as depict and modified f n gradient qu Fig. 20 consist he results obta antly better gr nt quality imp ified formulat Irrespective o the steeper p strategies) whi o have been c than 10 degr 1 ratio with th the original f e deterministic of the gradie othesis is still a similar tren results than eq he best result maximum con le computation adient quality othesis, using ted in Fig. (19 formulations fo uality when u ts of two plots ained when u radient estima proves. Fig. 2 tion is less s f the ratio fo part of the op ich lie in this compared. We rees, when us he original fo formulation w c case, all resu ent as given b l not satisfied. nd as for the d quation (11). ts. These resu ntribution (lar nal costs is m is inferior (re g higher ratios 9). or varying rati using the HPG s which show using the orig ates with the M

20(b) depicts sensitive to th ormulation the SPE 173236-ptimization cu region for wh e observe that sing the modif

rmulation [ba we observe ang

ults are the m by equations ( . Thus smooth deterministic c We also obse ults highlight rgest increase more important

esults not show s leads to high os of controls G and MIG r the differenc ginal formulati MIG formulati the results w he different r e modified fo -MS urve hich t for fied ased gles mean (11, hing case erve the e in t for wn) her-per atio e in ion, ion. when atio orm,

(19)

SP Fig hig for [ba qua ‘sm for var cor Fig mo Sim fla Fig mo sat pro allo for sam wit com we E 173236-MS g. 20: (a) Differ ghlights the gra

The Flatter R The 1:1 ratio r the steeper p ased equations ality as shown moothed’ grad rmulations. No ries between rrelation lengt g. 21: Illustrate odel realization milar to the re tter region. F g. 22 (a) depi odified formul tisfied our hyp oblem dimens

owable error rmulation, we mples per geo thin the optim mputationally e suggest the u

rence in gradie adient quality w

Region o with the mod

part of the op s (5) and (6)] n in Fig. 21. W dient, given b ote: similar to 8 and 17. It th which has a s the differenc n using the HPG esults shown a ig. 22 consist

icts the result lation. We ob pothesis for on sion, the differ

margin of 2 would satisfy ological realiz mization proce y challenging a

use of the mod

ent quality for when using the

dified formula ptimization cu , the gradient We also obser by equations o the determin requires a su a strong impac ce in gradient q G formulation.

above for the ts of two plots ts obtained w bserve that the

nly a few poin rence between 20 degrees th y our hypothe zation to estim ess we would and therefore, dified formula

the two ratio f e modified grad ation [based on urve. However quality is not rve, slightly d (11, 13,15 an nistic case we ubstantial num ct on the gradi quality betwee steeper region s which show when using the

e same trend i nts for this rob n 10 and 20 de en, with the esis at most of mate a ‘good’

need to perfo , since the inc ation with a 1:

formulations (H dient formulati

n equations (1 r, for the flatt t as high. An different to ou nd 16), give use a smaller mber of simul ient quality. en the original n, the differen w the differenc e original for in the results bust case. We egrees is marg MIG ratio f f the points fo ’ gradient. Th orm 5000 rese crease in obje 1 ratio. HPG and MIG) ion. Note the d

19) and (20)] l ter part, thoug

increase in th ur previous ob very similar r perturbation lations to per and modified f nt ratio formul ce in gradient rmulation and exists, althou e could argue t ginal. Thus if formulation an or a ratio of 1 his would imp ervoir simulat ctive function

for the origina ifferent vertica leads to very h gh still better he ratios signi bservations tha results for bo size ( = 0.01 rform an exte formulations fo ations show a quality for th d 22 (b) depic ugh on the fla that for the ro we were to ha nd both the o :50, i.e. we w ply that, to est ions since the n value for the

al gradient for al scales

high quality g r than the orig ificantly impro at the differen oth the modif 1) and a corre ensive search or varying rati a similar trend he different ra cts the results atter part of th obust case and ave a hypothe original or m would need to timate a grad e ensemble siz e flatter region mulation while gradient estima ginal formulat

oves the grad nt versions of

fied and orig elation length t

for the ‘corr

os of controls

d for the point atio formulatio s when using he curve we h d, considering esis which had modified grad apply 50 con dient at one po ze is 100. Thi n is less than 17 e (b) ates tion ient f the inal that ect’ per ts in ons. the have our d an ient ntrol oint is is 1%

Figure

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