spe-119139-pa

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1/8March 2010 SPE Journal 31

Comparison of Stochastic SamplingAlgorithms for Uncertainty Quantification

Linah Mohamed,SPE, Mike Christie,SPE, and Vasily Demyanov,SPE,

Institute of Petroleum Engineering, Heriot-Watt University

Copyright 2010 Society of Petroleum Engineers

This paper (SPE 119139) was accepted for presentation at the SPE Reservoir SimulationSymposium, The Woodlands, Texas, USA, 24 February 2009, and revised for publication.

Original manuscript received for review 3 November 2008. Revised manuscript received forreview 13 March 2009. Paper peer approved 19 March 2009.

Summary

History matching and uncertainty quantification are two importantresearch topics in reservoir simulation currently. In the Bayesianapproach, we start with prior information about a reservoir (e.g.,from analog outcrop data) and update our reservoir models withobservations (e.g., from production data or time-lapse seismic).The goal of this activity is often to generate multiple models thatmatch the history and use the models to quantify uncertainties inpredictions of reservoir performance. A critical aspect of generat-ing multiple history-matched models is the sampling algorithmused to generate the models. Algorithms that have been studiedinclude gradient methods, genetic algorithms, and the ensembleKalman filter (EnKF).

This paper investigates the efficiency of three stochastic sampling

algorithms: Hamiltonian Monte Carlo (HMC) algorithm, ParticleSwarm Optimization (PSO) algorithm, and the NeighbourhoodAlgorithm (NA). HMC is a Markov chain Monte Carlo (MCMC)technique that uses Hamiltonian dynamics to achieve larger jumpsthan are possible with other MCMC techniques. PSO is a swarmintelligence algorithm that uses similar dynamics to HMC to guidethe search but incorporates acceleration and damping parametersto provide rapid convergence to possible multiple minima. NA is asampling technique that uses the properties of Voronoi cells in highdimensions to achieve multiple history-matched models.

The algorithms are compared by generating multiple history-matched reservoir models and comparing the Bayesian credibleintervals (p10p50p90) produced by each algorithm. We showthat all the algorithms are able to find equivalent match qualitiesfor this example but that some algorithms are able to find good

fitting models quickly, whereas others are able to find a morediverse set of models in parameter space. The effects of the differ-ent sampling of model parameter space are compared in terms ofthe p10p50p90 uncertainty envelopes in forecast oil rate.

These results show that algorithms based on Hamiltoniandynamics and swarm intelligence concepts have the potential to beeffective tools in uncertainty quantification in the oil industry.

Introduction

History matching and uncertainty quantification are two importantresearch topics in reservoir simulation currently. Automated andassisted history-matching concepts have been developed overmany years [e.g., Oliver et al. (2008)]. In recent years, researchhas been quantifying uncertainty by generation of multiple his-

tory-matched reservoir models rather than just seeking the besthistory-matched model. A practical reason for using multiplehistory-matched models is that a single model, even if it is thebest history-matched model, may not provide a good prediction(Tavassoli et al. 2004).

Most uncertainty quantification studies use a Bayesian approach,where we start with prior information about a reservoir expressed asprobabilities of unknown input parameters to our model. These priorprobabilities can come from a number of sources (e.g., from analog

outcrop data) or previous experience with an analog reservoir froma similar depositional environment. The prior probabilities are thenupdated using Bayes rule, which provides a statistically consistentway of combining data from multiple sources. The data that are usedto update the prior probabilities are observations about the reservoir(e.g., production data or time-lapse seismic data).

By generating multiple models that match history and are con-sistent with known prior data, we are able to estimate uncertaintiesin predictions of reservoir performance. A critical aspect of uncer-tainty quantification, therefore, is the sampling algorithm used togenerate multiple history-matched reservoir models.

There are a number of algorithms that have been used in thepetroleum literature to generate history-matched models, and thesealgorithms fall into two principal types: data assimilation methods

and calibration methods (Christie et al. 2005). Data assimilationmethods calibrate a number of estimates of model parameterssequentially to points in a time series of observed data. In calibra-tion methods, on the other hand, a complete run of the simulationis carried out and the match quality to the historical productiondata is used to move the estimates of the model parameters towarda better solution.

The main data-assimilation method used for history matchingin the oil industry is EnKF. Evensen (2007) describes the theoryof EnKF and shows a number of applications of EnKF to historymatching field examples. Liu and Oliver (2005) showed that EnKFcompared favorably with gradient-based methods when applied tohistory match a truncated Gaussian geostatistical model of faciesdistributions. The number of applications of EnKF is growingrapidly, with several papers presented at the 2009 SPE Reservoir

Simulation Symposium.Gradient methods are highly efficient and have been widely

used in history-matching problems. These methods require thecalculation of the derivative of the objective function with respectto the reservoir model parameters as either gradients or sensitivitycoefficients. Techniques available include steepest descent, Gauss-Newton, and Levenberg-Marquardt, which can be found in somemodern commercial history-matching software, for example SimOpt(2005). Early work by Bissell et al. (1994) history matched two realcase studies using gradient methods and found that the results werecomparable with hand matches. Lepine et al. (1999) estimated uncer-tainty of production forecasts using linear perturbation analysis. Arange of possible future production profiles and the confidence inter-vals for the future production performance were derived to quantifythe uncertainty. The main problem of using gradient-based methodsis that they can easily get trapped in local minima (Erbas 2006).

Recently, stochastic methods have gained some popularity fortheir relatively simple implementations. They are equipped withvarious heuristics for randomizing the search and, hence, explor-ing the global space and preventing entrapment in local minima.Examples of stochastic methods include genetic algorithms (GAs)and evolutionary strategies. GAs have been used widely in historymatching and are available in a variety of forms, including binarycoded GAs and real-coded GAs. Romero et al. (2000) applied amodified GA to a realistic synthetic reservoir model and studiedthe main issues of the algorithm formulation. Yu et al. (2008) usedgenetic programming to construct proxies for reservoir simulators.

Bayes Theorem and Uncertainty Quantification.Bayes theo-

rem is the formal rule by which probabilities are updated givennew data (Jaynes 2003). Bayes theorem is:

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2/832 March 2010 SPE Journal

p m Op O m p m

p O( | )

( | ) ( )

( )= . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

p(m|O) is the posterior probability (i.e., the probability of themodel given the data).p(O|m) is the likelihood term (i.e., the prob-ability of the data assuming that the model is true).p(m) is the priorprobability and can be given as a sum of independent probabilitiesfor model parameters or as some more complex combination ofmodel inputs. The normalizing constantp(O) is sometimes referredto as the evidence; if this term is small, it suggests that the modeldoes not fit the data well.

Sampling algorithms often work using the negative log ofthe likelihoodusually called the misfit M. Assuming that themeasurement errors are Gaussian, independent, and identicallydistributed, the misfit can be computed using the conventionalleast-squares formula,

Mq q

t

t

T

=( )

=

obs sim 2

21 2

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

where Tis the number of observations, q is the rate, superscriptsobs and sim refer to observed and simulated, and 2is the varianceof the observed data. Other statistical models for observationalnoise give rise to different expressions for the misfit.

Sampling Algorithms and

Uncertainty Quantification

The purpose of this paper is to investigate the efficiency of threestochastic sampling algorithms for generating history-matchedreservoir models. The algorithms are: HMC, PSO, and NA. HMCis an MCMC technique that uses Hamiltonian dynamics to achievelarger jumps than are possible with other MCMC techniques. PSOis a swarm intelligence algorithm that uses similar dynamics toHMC to guide the search but incorporates acceleration and damp-ing parameters to provide rapid convergence to possible multipleminima. NA is a sampling technique that uses the properties of Vor-onoi cells in high dimensions to achieve multiple history-matchedmodels. NA has been used in a number of reservoir history-match-ing studies (Subbey et al. 2003, 2004; Rotondi et al. 2006; Erbasand Christie 2007). A new, more efficient PSO technique called

flexi-PSO has been developed recently by Kathrada (2009) andapplied to history matching of synthetic reservoir models.

The Neighbourhood Algorithm. NA is a stochastic samplingalgorithm that was developed originally by Sambridge (1999a)for solving geophysical inverse problems. It is a derivative-freemethod that aims at finding an ensemble of acceptable modelsrather than seeking for a single solution. The key approximationin NA is that the misfit surface is constant in the Voronoi cellsurrounding a sample point in parameter space. Quantifying theuncertainty using NA involves two phases: a search phase, in whichwe generate an ensemble of acceptable solutions of the inverseproblem; and an appraisal phase, in which NA-Bayes (NAB) (Sam-bridge 1999b) computes the posterior probability on the basis ofthe misfits of the sampled models and the Voronoi approximationof the misfit surface.

The search phase can be summarized as follows: Step 1: Initialize NA with a population of an initial set of ninit

models randomly generated in the search space by a quasirandomnumber generator; for each model, the forward problem is solvedand the corresponding misfit valueis obtained.

Step 2: Determine the nr models having the lowest misfitvalues among the previously generated models ns.

Step 3: Generate a total of ns new models using a Gibbssampler in the nrVoronoi cells previously selected.

Step 4: NA returns to Step 2, and the process is repeated untilit reaches the user-defined number of iterations.

Thus, a total of N = ninit+nsnumber of iterations models isgenerated by the algorithm. The ratio ns/nr controls the behavior

of the algorithm: The lowest value of ns/nr=1 aims to explore thespace and find multiple regions of good fitting models; as the value

of ns/nr is increased, the algorithm tends to improve the matchesobtained at the expense of finding multiple clusters of good-fittingmodels. A general guideline is to start with a value of ns/nr=2 toobtain a balance between exploration and exploitation.

Because the sampling density of the misfits obtained is notrelated to the posterior probability density, a separate calculationhas to be carried out to compute probabilities of the models. Thiscalculation assumes that the misfit is constant in each Voronoi celland calculates the probability as the exponential of the negativemisfit times the volume of the Voronoi cell. The resulting ensembleof models with their posterior probabilities can then be used toestimate the p10p50p90 uncertainty envelopes.

PSO.PSO is a population-based stochastic optimization algorithmdeveloped by Kennedy and Eberhart (1995). It is known as a swarmintelligence algorithm because it was originally inspired by simula-tions of the social behavior of a flock of birds. Many studies haveshown PSO to be a very effective optimization algorithm. PSOis easy to implement and computationally efficient and has beenapplied successfully to solve a wide variety of optimization prob-lems (Kennedy and Eberhart 1995; Eberhart and Shi 2001).

The main computational flow is described in the following steps: Step 1: Initialize the PSO algorithm with a population of

particles of ninitmodels at locations randomly generated in param-eter space. Each particle is also assigned a random velocity. Foreach model, the forward problem is solved and the relevant misfitvalue M is obtained.

Step 2: At each iteration, evaluate the fitness of individualparticles.

Step 3: For each particle, update the position and value ofpbest, the best solution the particle has seen. If current fitness valueof one particle is better than its pbestvalue, then its pbestvalue andthe corresponding position are replaced by the current fitness valueand position, respectively.

Step 4: Find the current global best fitness value and the cor-responding best position gbestacross the whole populationspbestandupdate if appropriate.

Step 5: Update velocity for each particle using Eq. 3. Theupdated velocity is determined by the previous iterations velocityand by the distances from its current position to bothpbestand gbest.

vwv c p x c g

i

k

i

k

i i

k

+

= + ( )+

1

1 1 2 2rand randbest besstk

i

kx( ), . . . . . (3)

where:vi

kis the velocity of particle iat iteration k;xi

kis the position of particle iat iteration k;c1is a weighting factor, termed the cognition component, which

represents the acceleration constant which changes the velocity ofthe particle towardspbesti;

c2 is a weighting factor, termed the social component, whichrepresents the acceleration constant which changes the velocity ofthe particle towards gkbest;

rand1and rand2are two random vectors with each componentcorresponding to a uniform random number between 0 and 1;

pbesti is thepbestof particle i;gkbestis the global best of the entire swarm at iteration k;and w is an inertial weight that influences the convergence of

the algorithm.Initially, the values of the velocity vectors are randomly gener-

ated with v v vi

k= [ ]0 max max, . If one component in a particle violatesthe velocity limit vmax, its velocity will be set back to the limit.

Step 6: Update the position for one particle by using

x x vi

k

i

k

i

k+ += +1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

Step 7: Repeat Steps 2 to 6 until the maximum number ofiterations is reached or another stopping criterion is attained.

Because PSO was developed as an optimization tool, wehave to extend the algorithm to quantify uncertainty in reservoir

modeling. We choose to extend the algorithm using the sameconcepts as the NA by running the NAB code, which computes

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the posterior probability under the assumption that the misfitsurface is constant in each Voronoi cell surrounding a particle(Sambridge 1999b).

Hamiltonian Monte Carlo.HMC is an MCMC method that wasintroduced by Duane et al. (1987) for sampling from distributions.HMC aims to fix some of the problems that MCMC algorithms cansuffer from [e.g., the Metropolis algorithm (Metropolis et al. 1953)can suffer from slow exploration of the probability distribution ifthe step size is too small or can suffer from excessive rejection ofproposed locations if the step size is too high].

The idea of HMC is to regard each set of parameter variables

as a point in parameter space and introduce a set of auxiliarymomentum variables u. We define a potential energy as the nega-tive logarithm of the posterior probability U(x) =logp(m|O), anda kinetic energy as K(u) =uTu/2. The Hamiltonian or total energyis thenH(x,u) =U(x) +K(u). Our extended probability distribution,assuming the momenta are sampled from a normal distributionwith mean zero and variance 1, is

p x u e e e p m O N uH x u U x K u

, | ; ,,( ) = ( ) ( ) ( ) ( ) ( ) 0 1 . . . . . . . (5)

So, if we can sample from the extended distribution, we canobtain samples from the posterior probability distribution of inter-est by discarding the information on the momentum variables.

We generate new samples by solving Hamiltons equations of

motion:

uH x u

x=

( )

,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

and

xH x u

u= ( )

,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

Hamiltonian dynamics is time-reversible and preserves volume instate space and total energy. This means that, if we simulate thedynamics exactly, we will leave the extended density invariant(Bishop 2006). In practice, we simulate the Hamiltonian dynamicsusing the leapfrog algorithm

u tt

u tt

tU x t

x

x t

+

=

( )( )

+

2 2

( tt x t tu t t

) ( ) .= + +

2

. . . . . . . . . . . . . . . . . . . . . . (8)

Although the leapfrog algorithm simulation is time-reversible andvolume-preserving, it preserves energy only to second order in the

timestep size t. The effect of this error can be eliminated by applyinga Metropolis accept/reject step at the end of the trajectory.

The steps for the HMC algorithm are as follows: Step 1: Initialize the HMC algorithm with a population of an

initial set of ninitmodels randomly generated in the search spaceby a quasirandom number generator; for each model, the forwardproblem is solved and the relevant misfit valueM is obtained. Thepopulation of models with their misfits will be used to approximatethe surface and obtain the gradients (see the next section).

Step 2: Generate a new momentum vector ufrom the Gauss-ian distributionp u K u( ) ( )( )exp . This step is considered asthe stochastic part of the algorithm, which ensures that the whole

phase space is explored. Step 3: Starting from the current state, performLleapfrog steps

with a step size t, resulting in the new state (x(tL),u(tL)). Step 4: Employ the Metropolis rule, make the next

sample x u x tL u tLk k+ +( ) = ( ) ( )( )1 1, , , with probabilitymin exp1, , , ( ) ( )( )( )( ) ( )( )H x tL u tL H x uk k , where k is theiteration number.

Computing the Gradients. Implementing HMC requires us tocompute the gradient of the negative logarithm of the posteriordistribution, which is the sum of the misfit and the negative log ofthe prior. There are two approaches that can be followed to com-pute this term: If gradients of the solution with respect to uncertainvariables are available (e.g., from an adjoint code), then this termcan be computed directly; alternatively, it can be computed lessaccurately from a proxy.

In the application presented here, we use a general regressionneural network (GRNN) (Specht 1991; Nadaraya 1964; Watson1964) with Gaussian kernels to approximate the misfit surfaceand then compute the gradient from the approximated surface. Acritical step in the regression is that of computing the kernel-widthparameter. In our application, we used a fraction of the averagedistance between points in each dimension. More sophisticatedmethods are available [e.g., Kanevski and Maignan (2004)]. It isimportant to note that there are some restrictions on updating thesurface to ensure consistency with the MCMC assumptions.

Field ApplicationThe Teal South Reservoir

Teal South is a reservoir located in the Gulf of Mexico, approxi-mately 144 km southwest of Morgan City, Louisiana, as shown inFig. 1.The 4,500-ft-deep sand is bounded on three sides by faultsand closed by dip to the north, and is shown in Fig. 2.Fluids areproduced from a single horizontal well through solution-gas drive,aquifer drive, and compaction drive. Production started in late1996, and data are available in the form of monthly oil, water, andgas measurements, as well as two pressure measurements of 3,096psi initially and 2,485 psi after 570 days of production (Christieet al. 2002).

~ 144 km southwest of Morgan City, Louisiana

Fig. 1The Teal South field location.Fig. 2The Teal South 4,500-ft-deep sand structure map andthe 11115 simulation grid.

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Teal South Model Uncertain Parameters.We used a relativelycoarse simulation model with a grid size of 11115. We set up themodel with eight uncertain parameters: horizontal permeabilitiesof all five layers, a single value for kv/kh, rock compressibility, andaquifer strength. We chose uniform priors in the logarithms of thevariables, as shown in Table 1.

Fig. 3shows the observed production rates for oil and water as afunction of time. The oil rate peaked after 80 days of production andthen declined rapidly. Water production started after the oil rate peakedand stayed steady for the majority of the time. The first 181 days ofproduction data were used in the history match, and the remaining3 years is used as prediction data to measure the predictive quality ofthe history matches. The dashed black line shows the data used forthe history matching (six measurements out of 41 measurements). Thesimulator production was controlled to match total liquid rate, and

history matching was carried out by matching the field oil rate.

A least-squares misfit was used to measure how well a specificset of reservoir model parameters fit the observed data. The stan-dard deviation of the oil production measurement errors was set to100 STB/D. Use of a least-squares misfit is based on the assump-tion that the measurement errors are Gaussian and independent.

Algorithm Setup Specifications

For all the methods, we started from same initial population composedof 30 models generated randomly in parameter space. Fig. 4 showsthe two different 2D projections of the initial models. In Fig. 4a, weshow the models plotted against the scaled values of two layer perme-ability multipliers (P1 and P2), and, in Fig. 4b, the models are plottedagainst the scaled aquifer strength and the scaled rock compressibilitymultipliers. The points are color-coded according to the misfit.

We ran 45 iterations for both NA and PSO algorithms. ForNA, the parameters we chose were: ns/nr=30/15. For PSO, theparameters we chose were: c1= c2 = 2 with a linear decrease inthe inertial weight wfrom 0.9 to 0.4 (Eberhart and Shi 2007). Thetotal number of reservoir model simulations was 1,380.

We ran a single HMC chain of 1,350 reservoir model simula-tions starting from a random one of the 30 initial points used forNA and PSO. We started the HMC sampling using gradients esti-mated from the GRNN with kernel-width adjustment factor equalto 0.4. The leapfrog step sizes were chosen to be =0.02 i,wherei= (Cii), and C is the covariance matrix. At each HMCiteration, the number of leapfrog steps taken was randomly drawn

from a uniform distribution from 10 to 25.

2,500

2,000

1,500

1,000

500

0

1,000

800

600

400

200

0

0 200 400 600 800 1,000 1,200 1,400

Time, days

OilRate,

STB/D

WaterRate,

STB/D

0 200 400 600 800 1,000 1,200 1,400

Time, days

Fig. 3Production history (oil rate and water rate) for Teal South reservoir.

)b()a(

4.06.08.010.0

4.06.08.010.0

1.0

0.8

0.5

0.2

0.0

1.0

0.8

0.5

0.2

0.00.0 0.3 0.5 0.8 1.00.0 0.3 0.5 0.8 1.0

Scaled Aquifer StrengthScaled P1

ScaledP

2

ScaledRockCompressibility

Fig. 4Two different 2D projections of initial population of 30 randomly generated models in 3D parameter space.

TABLE 1PARAMETERIZATIONFOR THE TEAL SOUTH MODEL

Parameter Units Prior range

kh (for each of the five layers) md 10{1...3}

kv/kh 10{2...1}

Aquifer strength million STB 10{79}

Rock compressibility psi1 10

{4.0963.699}

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Results and Discussion

We will first compare the performance of the two sampling algo-rithms NA and PSO. Both NA and PSO generate multiple history-matched models. Uncertainty quantification is then carried outusing a separate code that converts the posterior probability densityat each sampled location to a posterior probability (equal to densitytimes volume of Voronoi cell surrounding a point).

Fig. 5a shows the best history match obtained by NA andPSO. There is little to choose between the two history-matchedmodels. Fig. 5b shows the optimal values for the five horizontalpermeabilities; the best-fitting parameters found by NA and PSOare different (although the differences are not large). Two othersets of parameter values providing almost equally good matchesare shown in Fig. 6.

Fig. 7shows the progress of the mean generational minimummisfit of NA and PSO. To generate this figure, we ran five runs ofNA and PSO. NA and PSO used identical sets of points for eachrun, with a new set of random starting conditions generated foreach run. We then plotted the mean generational minimum misfitalong with the standard deviation around each point. We can seethat NA and PSO reach the same misfit but that, on average, PSOreduces the misfit in each generation more rapidly than NA.

The sampling history of NA is shown in Fig. 8,and that of PSO isshown in Fig. 9.Each figure consists of eight plots, showing the evolu-tion of the parameters sampled as a function of the number of sampled

points. The points are color-coded according to the misfit, showingthe concentration of sampling at low misfit values as the algorithm

sampling evolves. Note that the white (light) points have misfit 10 orabove and include many models that do not match at all well.

The parameter values for the good history-matched models canbe seen by looking at the range of the black (dark) points. Bothalgorithms appear to concentrate sampling for the permeability

)b()(a

NA maximum likelihood model

PSO maximum likelihood model

Observed data used for HM period

Observed data for prediction

2500

2000

1500

1000

500

0

0 200 400 600 800 1000 1200

Time, days

FOPR

180

160

140

120

100

80

60

40

20

0kh1 kh2 kh3 kh4 kh5

NA1

PSO1

Fig. 5Comparison of the best history matches (a) and the corresponding permeability estimates (b) obtained from NA and PSO.

)b()(a

180

160

140

120

100

80

60

40

20

0

kh1 kh2 kh3 kh4 kh5

NA2

PSO2

180

160

140

120

100

80

60

40

20

0

kh1 kh2 kh3 kh4 kh5

NA3

PSO3

Fig. 6Two alternative sets of parameter values providing almost equivalent match qualities to the maximum likelihood model.

NA

PSO6.0

5.5

5.0

4.5

0 10 20 30 40

GenerationalMinimum

Generation

Fig. 7Evolution of the mean generational minimum misfit forNA and PSO.

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values in similar zones, although NA appears to be holding ontotwo possible minima for log(kh2) (upper right plot), whereas PSOhas homed in on the higher value. The sampling for rock com-pressibility, aquifer strength, and log(kv/kh) evolves differently forboth algorithms. Nonetheless, the best matches obtained are verycomparable in quality, as shown in Fig. 5.

Fig. 10 shows the sampling history for HMC. Note that, inHMC, the algorithm is not continually trying to improve the degree

of match; rather, it is constantly sampling models that are accept-able history matches. Most of the models generated have misfits

0 500 1000

0 500 1000

P1 P2

P3 P4

P5 log(kv/kh)

Rock compressibility Aquifer strength

1.00.80.60.40.20.0

1.00.80.60.40.20.0

1.00.80.60.4

0.20.0

1.00.80.60.40.20.0

ScaledParame

terValue

Misfit Evaluation Number

Fig. 9Sampling history of PSO for each of the eight unknownparameters.

0 500 1000

0 500 1000

P1 P2

P3 P4


1.00.80.60.40.20.0

1.00.80.60.40.20.0

1.00.80.60.4

0.20.0

1.00.80.60.40.20.0

ScaledParame

terValue


P5 log(kv/kh)

Fig. 8Sampling history of NA for each of the eight unknownparameters.

0 200 400 600

P1 P2

P3 P4


1.00.80.60.40.20.0

1.00.80.60.40.20.0

1.00.80.60.40.2

0.0

1.00.80.60.40.20.0

ScaledParam

eterValue


0 200 400 600

4.0

6.0

8.0

10.0

P5 log(kv/kh)

Fig. 10Sampling history of HMC for each of the eight un-known parameters.

of eight or belowcorresponding to an average deviation fromobserved values of 1.5 standard deviations or below.

Uncertainty QuantificationComparison of NA, PSO, andHMC.HMC is designed so that the models generated sample cor-rectly from the posterior distribution (within sampling error), whichmeans that p10p50p90 predictions can be generated from anappropriate sum of the generated models. Neither PSO nor NA has

this property, so a separate assessment has to be made to determinethe probability of each of the sampled models. We used the NABcode (Sambridge 1999b) to determine these probabilities for PSOand NA. NAB works by running a Gibbs sampler on the approximatemisfit surface generated by assuming that the misfit is constant overeach of the Voronoi cells surrounding a sampled point.

2,500

2,000

1,500

1,000

500

0

0 200 400 600 800 1,000 1,200

P90P50P10TruthEnd of history

Time, days

FieldOilProductionRa

te,

STB/D

Fig. 11Uncertainty intervals generated by NA.

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Figs. 11 and 12 show the Bayesian credible intervals (p10p50p90) for oil rate after history matching to the first 181 daysof production. Both NA and PSO have produced similar ranges.The equivalent plot for HMC is shown in Fig. 13.

Fig. 14 shows the cumulative density function of the oil rateat two times. Fig. 14a is the CDF for the oil rate after 181 daysat the end of the history matching period. The uncertainty plots

for all three algorithms are very similar. Fig. 14b shows the CDFafter 1,187 days. In this case, there is a greater difference betweenthe algorithms, although they are still similar. HMC provides aslighter wider p10p90 range after the end of the history-matchingperiod than NA and PSO. The solid black vertical lines show theobserved value at the end of history match time (left) and at theforecasted time (right).

Conclusions

This paper investigated the efficiency of three stochastic samplingalgorithms: HMC, PSO, and NA. We compared the algorithms bygenerating multiple history-matched reservoir models for the TealSouth reservoir with eight unknown parameters.

Specific conclusions drawn are as follows:

NA, PSO, and HMC are able to find equivalent match qualitiesfor this example.

PSO is able to obtain a good history match in fewer iterationsthan NA for this example, and this behavior is robust to changingthe initial random starting conditions.

PSO tends to concentrate sampling more in the low-misfit regionsthan NA; for each algorithm, this behavior is likely to depend onthe algorithm parameter setting though, and so general conclu-sions are hard to draw.

NA and PSO need a separate calculation to go from sampledmodels to forecasts of uncertainty.

HMC is able to generate samples from the posterior in one step. NA, PSO, and HMC are all able to produce equivalent forecasts

of uncertainty.These results show that algorithms based on Hamiltonian

dynamics and swarm intelligence concepts have the potential to beeffective tools in uncertainty quantification in the oil industry.

Acknowledgments

We would like to thank the sponsors of Phase II of the UncertaintyQuantification Project at Heriot-Watt for their support: BG, BP,Chevron, ConocoPhillips, Eni, StatoilHydro, JOGMEC, Shell,TransformSW, and the UK DTI (BERR). We also thank Schlum-

berger-GeoQuest for the use of the Eclipse reservoir simulatorsoftware.

2,500

2,000

1,500

1,000

500

0

0 200 400 600 800 1,000 1,200


Time, days

FieldOilProductionRate,

STB/D

Fig. 12 Uncertainty intervals generated by PSO.

2,500

2,000

1,500

1,000

500

0

0 200 400 600 800 1,000 1,200


Time, days

FieldO

ilProductionRate,

STB/D

Fig. 13Uncertainty intervals generated by HMC.

)b()a(

1.0

0.8

0.6

0.4

0.2

0.0

1.0

0.8

0.6

0.4

0.2

0.0

CumulativeDensityFunction

CumulativeDensityFunction

Field Oil Production Rate, day 181

NAPSOHMCObserved

NAPSOHMCObserved

Field Oil Production Rate, day 1,187

1,100 1,200 1,300 1,400 1,500 100 150 200 250

Fig. 14 Cumulative distributions from NA, PSO, and HMC at (a) and after (b) the end of history matching.

7/26/2019 SPE-119139-PA


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Linah Mohamed is a PhD student in petroleum engineering atHeriot-Watt University, UK. She holds an MS degree in computerscience from the University of Khartoum, Sudan. Mohamedsprincipal research interests are history matching and uncer-tainty quantification using stochastic sampling techniquesbased on Hamiltonian dynamics, swarm intelligence concepts,and machine learning.Mike Christieis professor of reservoir engi-neering at Heriot-Watt University in Edinburgh, where he runs aresearch group on uncertainty quantification in reservoir simula-tion. Before joining Heriot-Watt, he spent 18 years working for BP,holding positions in research and technology development inthe UK and USA. He holds a bachelors degree in mathematicsand a PhD degree in plasma physics, both from the Universityof London. Christie has been an SPE Distinguished Lecturer andwas awarded the SPE Ferguson medal in 1990. Vasily Demyanovcarries out research in spatial statistics and machine learn-ing applications to model uncertainty of subsurface reser-voir predictions. He holds a PhD in physics and mathematicsfrom the Russian Academy of Sciences and has worked witha wide range of spatial statistics and artificial neural networksapplications in environmental and geosciences fields. He haspublished several papers in the leading geosciences, environ-mental, statistics, and computer science editions, including anaward winning paper in Pedometrics. Demyanov is a coauthor

of the chapters on geostatistics and artificial neural networksin Advanced Mapping of Environmental Data: Geostatistics,Machine Learning and Bayesian Maximum Entropy. Advancedkernel learning methods, stochastic optimisation algorithms, andBayesian statistics are among his current research interests.

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