00066350_prediction of oil production with confidence intervals

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This paper was prepared for presentation at the SPE Reservoir Simulation Symposium heldin Houston, Texas, 1114 February 2001.

This paper was selected for presentation by an SPE Program Committee following reviewof information contained in an abstract submitted by the author(s). Contents of the paper,

as presented, have not been reviewed by the Society of Petroleum Engineers and aresubject to correction by the author(s). The material, as presented, does not necessarilyreflect any position of the Society of Petroleum Engineers, its officers, or members. Papers

presented at SPE meetings are subject to publication review by Editorial Committees of theSociety of Petroleum Engineers. Electronic reproduction, distribution, or storage of any partof this paper for commercial purposes without the written consent of the Society of

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AbstractWe present a prediction methodology for reservoir oilproduction rates which assesses uncertainty and yields

confidence intervals associated with its prediction. Themethodology combines new developments in the traditionalareas of upscaling and history matching with a new theory

for numerical solution errors and with Bayesian inference.We present recent results of coworkers and ourselves.

IntroductionA remarkable development in upscaling

1, 2allows reduction

in computational work by factors of more than 10,000

compared to simulations using detailed geological models,while preserving good fidelity to the oil cut curves generatedfrom solutions of the highly detailed geologies. In common

engineering practice, the detailed geology models are tooexpensive for routine simulation. This is especially the caseif an ensemble of realizations of the reservoir is to be

explored. The ensemble allows consideration of distinctgeological scenarios, an issue of greater importance in manycases than errors associated with upscaling of detailed

geology to obtain a coarse grid solution.Upscaling allows rapid solutions and is a key to good

history matching. We formulate history matching

probabilistically to allow quantitative estimates of predictionuncertainty3, 4. A probability model is constructed fornumerical solution errors. It links the history match to

prediction with confidence intervals.The error analysis establishes the accuracy of fit to be

demanded by the history match. It defines a Bayesian

posterior probability for the unknown geology. Thus historymatching defines a revised ensemble of geologies, with

revised probabilities or weights. Prediction is based on theforward solution, averaged with these weights. Confidenceintervals are also defined by the probability weights for the

ensemble together with error probabilities for the forward

solution.Results of the prediction methodology will be described,

based on simulated geologies and simulated reservoir flowproduction rates. Efficient scaleup allows a sizable numberof geologies to be considered. The Bayesian framework

incorporates prior knowledge (for example fromgeostatistics or seismic data) into the prediction. We showthat a history match to past production rates improvesprediction significantly.

The plan of this paper is to pick one fine grid reservoirfrom an ensemble and regard its solution as a stand in forproduction data. Other reservoirs in the ensemble are

evaluated on the basis of the quality of their match to thisdata. They are upscaled, simulated on a coarse grid, and the

upscaled solution is compared to production history fromthe data. Probability of mismatch between the coarse gridsolution and the data weights each realization in a balanced

manner according to (a) its prior probability and (b) thequality of its match to data. We thus define a posteriorprobability on the ensemble, which is used for prediction.

Uncertainty in the prediction has two sources: uncertainty

in the geology, or history match, as discussed above, anduncertainty in the forward simulation, also conducted oncoarse grids. The total uncertainty receives contributionsfrom these two sources, and its analysis leads to confidenceintervals for prediction.

The intended application of this prediction methodology isto guide reservoir development choices. For this purpose,simulation of an ensemble of reservoir scenarios is

important to explore unknown geological possiblities.Statistical methods are important to assess the ensemble ofoutcomes.

The methods are intended for use by reservoir managersand engineers. For this purpose, the methods will need to beaugmented by inclusion of factors omitted from the present

study. The significance of our methods is their ability topredict the risk, or uncertainty associated with productionrate forecasts, and not just the production rates themselves.

The latter feature of this method, which is not standard, isvery useful for evaluation of decision alternatives.

Stochastic History MatchingProblem Formulation. Stochastic history matching isbased on an ensemble of geological realizations. To

simplify this study, we fix the geologic model aside from the

SPE 66350

Prediction of Oil Production With Confidence IntervalsJames Glimm, SPE, SUNY at Stony Brook and Brookhaven National Laboratory; Shuling Hou, SPE, Los Alamos NationalLaboratory; Yoon-ha Lee, SUNY at Stony Brook; David Sharp, SPE, Los Alamos National Laboratory; and Kenny Ye,SUNY at Stony Brook.


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JAMES GLIMM SPE 663502

permeability field, which is taken to be a random variable

simple form of the Darcy and Buckley-Leverett equations5

0== pKv ; 0=v , .(1)

( ) 0=+

sfv

t

s, ....(2)

where is a relative mobility, Kthe absolutepermeability, v velocity, p pressure, s the water

saturation and f the fractional flow flux. We consider

these equations in a two dimensional (reservoir cross

section) geometry, 10 x , 10 z in dimensionless

units. Assume no flow across the boundaries 1,0=z and

a constant pressure drop across the boundaries 1,0=x .The absolute permeability Kis spatially variable, with anassumed log normal distribution. We characterize the

covariance ( )Kln by correlation lengths 50/1=zl and

=lx ( )0180604020 .,.,.,.,. . Thus, ( )Kln is actually aGaussian mixture, and is not Gaussian. This distribution for

Kis called the prior distribution. Each realization is aspecific choice ofK.

We consider an ensemble defined by 500 realizations of

K, 100 for each of the five correlation lengths, selectedaccording to the above Gaussian distribution. Each K isspecified on a 100 x 100 grid (the fine grid). K and the

fractional flow functions f are then upscaled to grids at the

levels 5 x 5, 10 x 10, and 20 x 20. Each of the coarse gridupscaled reservoirs is also solved, in all cases for up to 1.4

pore volumes of injected fluid (1.4 PVI).We select one of the geologies,

0iK , as representing the

exact but unknown reservoir. We observe the oil cut

0if generated by the fine grid solution for times

00 tt (PVI). This data represents past, historical data,

and using it, we seek to predict production for 10 ttt =1.4 PVI, i.e. into the future. The solution is (a) historymatching, to select a revised ensemble of geologies, which

reflect agreement with history data, and (b) forwardsimulation, averaged over the revised (posterior) ensemble,to predict the future production.

The Bayesian Framework. In the Bayesian framework,

the prediction problem is solved by assigning a probability,or likelihood to any degree of mismatch between the coarse

grid oil cut ( )tcj and the observed history

( ) 00,0 tttfO i = , where ( )tfi0 is the oil cut for the

reservoir0i

K computed on the fine grid (and the fine grid is

conceptually considered to be exact). The probability or

likelihood of the observation given the geology Kis

denoted ( )KOp | . A mismatch could arise due to

measurement errors, or as we consider here, due to use of acoarse grid in a simulation analysis.

According to Bayes theorem, the posterior probability for

the geology defined by the permeability realizationKis

=

dKKpKOp

KpKOpOKp

)()(

)()()( , .(3)

where ( )Kp is the prior probability for the realization K.The prior probability is defined, for example, by methods of

geostatistics 6, 7, 8, 9, 10, and in the present context it is definedby the above mixture of Gaussians with specified correlationlengths.

In the absence of errors, there would be no mismatch, and

we could accept geology jK as a history match only if

( ) ( )tftc ij 0 . This is of course unrealistic, as errors do

occur. Since jKOp | assumes 0ij KK = is exact, the

mismatch is assumed to be due to an error in determining

jc . We write jjj cfe = as the error. Measurement

errors also contribute to the mismatch likelihood, but for

simplicity we concentrate on scale up and numericalsolution errors only. Thus, jKOp | is the probability of

the error je .

A Probability Model for Solution Errors. To characterizethe errors statistically, we introduce the mean

( ) ( ) ==N

j jte

Nte

1

1, (4)

and the sample covariance

( ) ( ) ( )( ) ( ) ( )( ) = =

N

j jj

s sesetete

N

tsC1

1

1, , ...(5)

The precision matrixs is the operator inverse to sC . We

make the approximation 0=e . Consider the expectation

dtdstetsseee st

o

s )(),()(),(1

= , .(6)

In Gaussian statistics, ),(21 ees is proportional to the

log probability of the error e .We also introduce the discrepancy

( ) ( ) ( )tctftd jiij = , .(7)as the difference between the fine grid solution of one

reservoir and the coarse grid solution of another. For

ji = , we have iii ed = is an error. Bona fide

discrepancies, ijd ( )ji are typically systematically

larger than errors jjj de = . This fact allows us to

differentiate between errors (good matches to history) anddiscrepancies (poor match to history). See Fig. 1.

The discrepancies play a critical role in evaluation of the

Bayes likelihood )( KOp . This likelihood assumes that

the geology Khas been chosen exactly. Hence, under thisassumption, the discrepancy is an error, and the negative

probability of the likelihood is ),( dds defined by (6).Thus,


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SPE 66350 PREDICTION OF OIL PRODUCTION WITH CONFIDENCE INTERVALS 3

),(2

1)(ln ddKOp s , .(8)

where0ji

dd= if jKK= .

Fig. 1 Typical errors (lower, solid curves) and discrepancies(upper, dashed curves), plotted vs. PVI. The two families ofcurves are clearly distinguishable.

Model Reduction. The precision matrixs and its inverse,

the sample coveriancesC , are defined nonparametrically in

terms of data. The data is a study of errors resulting from

upscaling of a fine grid geology specification. According to

well-established principles of statistics, the models or

sC for solution errors should be limited in complexity bythe amount of data used to define it. For this reason, we

project the errors e to a p dimensional space, ee pp = ,

and equivalently, we truncates and sC to obtain

pp projected precision and covariance matrices

p

s

p

s

p = and ps

p

s

p CC = . The general

principles governing data smoothing state thatp should be

neither too small, to avoid bias, nor too large, to avoiddispersion, also known as overfitting of data. Conceptually,

errors are averaged into pbins along the PVI axis, although

better results can be obtained through higher order methods

using finite element spaces4, 12

.

Prediction

Window Based Prediction. We show a simple method of

prediction, based on a window to accept or reject candidatereservoirs in the ensemble as being acceptable orunacceptable matches to history data. Window based

prediction assumes a large number of fine grid solutions,and thus is not practical, but it serves to introduce the ideas.If the window is narrow, window based prediction is alsomore accurate, and thus by comparison to other predictionmethods, it indicates the amount of information lost in

various practical, but approximate prediction methods. Weintroduce a tolerance, or window size, and accept into a

reduced, or history matched ensemble0i

R those realizations

for which

00,)()( 0 tssfsf ij , ..(9)

Here, 0t is the present time, and the tolerance is chosen

subjectively, to make the history matched ensemble small,

but not too small. For a small window size , this methodrequires a large starting ensemble. Oil cut curves for the

Fig. 2 Top: Oil cut curves for the complete ensemble.Bottom: Oil cut curves for the reduced ensemble. The reduced

ensemble is defined by (9) with 08.0= and 8.0)( 00 =tfi .

unconstrained and constrained ensembles are shown in Fig.2. To realize window based prediction within a Bayesian

framework, we define a likelihood function )( KOp to be

1 if the realization gives an oil cut lying within the window

and 0 otherwise.

A Metric for Prediction. We evaluate prediction by the

percent reduction in error, relative to the base case of prior

prediction. For any prediction )(sp of the future oil cut,

we define the prediction error as

dssfspPE it

ti )()( 01

00= , ...(10)

The choice


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)(1

)(1

sfN

spj

N

jprior == , ..(11)

defines the fine grid prior prediction; similarly we define the

coarse grid prior prediction with jc replacing jf in (11).

The prior prediction erroroiprior

PE , is defined by (10) with

(11) substituting for )(sp . We thus define the per cent

error reduction,

( ) 1001,1

1

0

0

=

=

=

o

o

iprior

N

i

i

N

i

PE

PEER , .(12)

To assess the window based prediction scheme, we define

the restricted ensemble0i

R as in (9), and

)(1

)(0

0

sfN

sp jiRji

= , .(13)

where 0iN is the number of elements of 0iR . The resultingerror reduction is presented in Table 1, first averaging over

100 choices of 0i of a specified correlation length, and then

averaging these averages over all five correlation lengths.

The three right columns in Table 1 specify initial times

0t for which the fine grid oil cut satisfies

4.0,6.0,8.0)( 00 =tfi .

Table 1. Window Based Prediction Error Reduction(per cent)

Present Oil Cut

0.8 0.6 0.5 0.4

0.2 53 59 63 62

0.4 33 43 42 43

0.6 37 46 49 51

0.8 42 53 58 62

1.0 43 55 61 65

Mean 42 51 55 57

Bayesian Prediction. We use Bayes formula (3) with thelikelihood (8) defined by the nonparametric error model (4),

(5), (6). We define p using piecewise linear elements and

a value of 16=p . We consider 0t , the present time, to

correspond to an oil cut of 0.6. Bayes formula defines aweight, or probability, jw for each realization jK . The

posterior prediction )(sp is then

)()(1

scwspjj

N

j == , ..(14)

We use (11) with jf replaced by jc to define the prior

prediction and (10), (12) to define per cent error reduction.

The results are presented in Table 2 for three levels of scaleup (5 x 5, 10 x 10, 20 x 20). Table 2a gives error reductionpercentages for a present oil cut of 0.8. Table 2b assumes a

present oil cut of 0.6. Results for a present oil cut of 0.4(not presented here) show continued improvement.

Table 2a. Bayesian Prediction Error Reduction(per cent). Present Oil Cut = 0.8

Level of Scale Up

5 x 5 10 x 10 20 x 20

0.2 21 40 45

0.4 11 09 14

0.6 10 08 07

0.8 11 11 081.0 13 22 15

Mean 13 18 18

Table 2b. Bayesian Prediction Error Reduction(per cent). Present Oil Cut = 0.6

Level of Scale Up

5 x 5 10 x 10 20 x 20

0.2 29 50 55

0.4 18 25 22

0.6 23 28 33

0.8 21 28 27

1.0 23 33 45

Mean 23 32 36

In Fig. 3 we show the histogram for posterior predicted error

prior predicted error for 10 x 10 scale up and a present oilcut of 0.6. The skewing of this distribution to negative

Fig. 3 Histogram of posterior predicted error prior

predicted error.

values represents the improvement (and sometimes a verysignificant improvement) of the posterior over the prior.

Confidence Intervals. A confidence interval is a range ofpossible outcomes likely to include the true outcome. For

example, a 5%-95% confidence interval is an interval ofoutcomes that will include the true outcome with 90%probability, but with a 5% chance that the outcome will lieoutside the interval on either the low or the high side. To

determine confidence intervals, we need to convolve twosources of uncertainty or error: the choice of the correct

geology and the accuracy of the forward simulation, for


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SPE 66350 PREDICTION OF OIL PRODUCTION WITH CONFIDENCE INTERVALS 5

times tin the future, 10 ttt , using a coarse grid withknown error statistics. To assign probabilities to outcomes,we need to reassess our metric for prediction. We consider

the total future oil production ,)(0

1

0

dssfi

t

t compared to

the predicted production .)(10

dsspt

t

Thus, we remove the

absolute values in (10).

In place of the mean prediction )(sp we need to

examine the probability density function )(pdf for thecorresponding random variable

( )dssescdssf iit

tjjj

t

t +=0

0)()()( , ..(15)

Here the random variables are ja)( , governed by the

Bayes posterior distribution, and jeb)( , governed by both

Bayes posterior and by the model for error statistics. Here

we introduce an approximation, and evaluate je in the priordistribution, i.e. to be independent of the observed oil cut

0if in the past, and independent of the random variable jc

We make this approximation because error statistics areexpensive to generate, and in practice their customization to

specific reservoirs may not be feasible.To find confidence intervals for (15), we sort values by

magnitude, and exclude the top and bottom 5% in

probability. The outer boundaries

0ic retained in this sort

are the confidence intervals. The result still depends on the

choice of exact geology 0i . To facilitate comparison of

confidence bounds for distinct 0i , we non-dimensionalize

the confidence intervals to obtain confidence intervals forrelative error

dssp

dsspc

t

t

t

ti

)(

)(

1

0

1

00

, (16)

The confidence intervals (16) have a weaker dependence on

0i than do the

0ic . We average them. For comparison, we

report the mean (with respect to average over 0i ) of the

non-dimensionalized prediction error

( )

dssf

dsspsf

i

t

t

i

t

t

)(

)()(

0

1

0

0

1

0, .(17)

The mean refers to a systematic bias in the predictionmethods. We also report the standard deviation for the

prediction, defined relative to both the Bayes posterior and

the 0i summations. See Table 3. The standard deviation

contains information similar to that of the confidence

intervals. For Gaussian errors, the 5%-95% confidence

intervals are equal to 1.645 , i.e., a multiple 1.645 ofthe standard deviation. Here we consider 10 x 10 scale-up,

p (binning dimension) = 16, present oil cut = 0.6.

Table 3. 5%-95% Confidence Intervals forPrediction of Future Production, as a Per Cent ofFuture Production. Comparison to PredictionStandard Deviation and Mean as Per Cent ofFuture Production.

ConfidenceIntervals

StandardDeviation

Mean

0.2 [-25,9] 6 10.4 [-27,12] 11 -4

0.6 [28,13] 20 -8

0.8 [-30,17] 28 -14

1.0 [-29,14] 25 -15

Mean [-28,13] 18 -8

DiscussionThe prediction error reduction is consistent with earlier

results3, 4 based on a smaller sample size (50 vs. 500

realizations). The error reduction shows understandabletrends. As the present time is moved later, moreinformation is available to constrain the predictions, and the

reduction of prediction error increases. (The error

decreases). As the problem is solved more accurately (withless scale up), the error reduction also increases. The

window based predictions using only fine grid solutions areconsistently more accurate than the Bayesian posteriorpredictions using upscaled (approximate) solutions. The

best predictions, and largest error reductions are obtainedwith the smallest correlation lengths, i.e. for reservoirs notdominated by narrow conduction bands. Such reservoirs

show less variability for production and are thus easier topredict. We note a systematic tendency (bias) for theupscaled solutions to overpredict production, especially for

the highly layered reservoirs (with long correlation length).

Conclusions A systematic method is presented for assessing uncertaintyand combining incomplete information. Scientifically basedprobabilities allow improved management of risk.

The benefit of this technology is a rational, objective, andsystematic method to combine all available partialknowledge, to formulate to make predictions, based on this

knowledge, and to assess uncertainty. Key ingredients ofthis technology are:1. Upscaling for rapid solutions2. Error estimated to generate probabilities and

likelihoods

3. Bayesian statistical inference

With these methods, we achieve a reduction of predictionerror by 30% in comparison to prior prediction, for a sampleproblem. We achieve 5%-95% relative percentage

confidence intervals of [-28%, +13%].Further studies are needed to assess various methods to

describe uncertainty in prediction. We will also investigate

trade offs between increased computational work (ensemblesize, degree of scale up) and the narrowing of confidenceintervals. Finally, it will be of interest to find the inherentlimits, set by lack of knowledge, for accuracy of prediction.

Nomenclatures

C = sample covariance)(sci = coarse grid oil cut for geology i


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)(sdij = discrepancy

)(sei = error

ER = error reduction (per cent)

)(sfi = fine grid oil cut for geology i

zx

ll , = correlation lengths

iK = absolute permeability of geology

i

N = number of realizations

O = observation (of oil cut history)

)(Kp = Bayes prior

)( KOp = Bayes likelihood function

)( OKp = Bayes posterior

)(sp = predicted oil cut

PE = prediction error PVI = pore volumes injected fluid

R = restricted ensemble of reservoirsts, = times (inPVI)

0t = present time

1t = final time )4.1( PVI=

jw = posterior probability of

realization jK

= variances = sample precision operator

p = projection onto p dimensional

binning subspace

Subscripts

ji, = indices for sample geologies

0i = geology taken as exact

p = dimension of data subspace used

for binning or error modelreduction

AcknowledgmentsThe work of James Glimm is supported in part by the NSFGrant DMS-9732876, the Army Research Office GrantDAAG559810313, the Department of Energy Grants DE-FG02-98ER25363 and DE-FG02-90ER25084, and Los

Alamos National Laboratory under Contract#C738100182X. Shuling Hou is supported by theDepartment of Energy under Contract W-7405-ENG-36.

Yoon-ha Lee is supported by the Department of Energygrant DE-FG02-90ER25084. David Sharp is supported bythe Department of Energy under Contract W-7405-ENG-36.

Kenny Ye is supported in part by Los Alamos NationalLaboratory under Contract #C738100182X.

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