Comput GeosciDOI 10.1007/s10596-013-9343-5

ORIGINAL PAPER

History matching and uncertainty quantification of faciesmodels with multiple geological interpretations

Hyucksoo Park · Céline Scheidt · Darryl Fenwick ·Alexandre Boucher · Jef Caers

Received: 27 May 2012 / Accepted: 21 January 2013© Springer Science+Business Media Dordrecht 2013

Abstract Uncertainty quantification is currently one ofthe leading challenges in the geosciences, in particularin reservoir modeling. A wealth of subsurface dataas well as expert knowledge are available to quantifyuncertainty and state predictions on reservoir perfor-mance or reserves. The geosciences component withinthis larger modeling framework is partially an interpre-tive science. Geologists and geophysicists interpret datato postulate on the nature of the depositional environ-ment, for example on the type of fracture system, thenature of faulting, and the type of rock physics model.Often, several alternative scenarios or interpretationsare offered, including some associated belief quantifiedwith probabilities. In the context of facies modeling,this could result in various interpretations of faciesarchitecture, associations, geometries, and the way theyare distributed in space. A quantitative approach tospecify this uncertainty is to provide a set of alternative3D training images from which several geostatisticalmodels can be generated. In this paper, we considerquantifying uncertainty on facies models in the earlydevelopment stage of a reservoir when there is still

H. Park · C. Scheidt · J. Caers (B)Energy Resources Engineering Department,Stanford University, Stanford, CA, USAe-mail: jcaers@stanford.edu

D. FenwickStreamsim Technologies, San Francisco, CA, USA

A. BoucherAdvanced Resources and Risk Technologies,Sunnyvale, CA, USA

considerable uncertainty on the nature of the spatialdistribution of the facies. At this stage, productiondata are available to further constrain uncertainty. Wedevelop a workflow that consists of two steps: (1) deter-mining which training images are no longer consistentwith production data and should be rejected and (2)to history match with a given fixed training image. Weillustrate our ideas and methodology on a test casederived from a real field case of predicting flow ina newly planned well in a turbidite reservoir off theAfrican West coast.

Keywords Popper–Bayes theorem · Inversemodeling · Training image · Geological scenarios ·History matching

1 Introduction

Realistically assessing reservoir uncertainty is one ofthe important challenges in reservoir engineering. Oneof the most critical stages for such assessment is the ap-praisal to early production stage: with few wells drilledand limited information at hand, modeling uncertaintyabout reservoir heterogeneity is often critical in thedesign of new platforms or planning of new wells. Atthis stage, one typically has well logs from a few wells,3D seismic data, well-testing data, and early productionhistory. Next to the reservoir structure, the facies archi-tecture and its spatial distribution are important factorsto consider. Due to the limited amount of information,there may still be considerable uncertainty on the na-ture of the facies depositional models. Often, at thisearly development stage, the type of depositional sys-tem (deltaic, fluvial, turbidite) may be resolved, but the

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nature of the facies architectural elements and the fa-cies associations and relationships in 3D remain highlyuncertain. Several plausible scenarios may be offeredas interpretation based on well-log and seismic data. Inthis paper, we consider the situation where a limited set(ten or fewer) of such interpretations on the style ofspatial continuity is provided. The problem we addressis on how to provide an uncertainty statement about(1) the 3D spatial distribution of multiple facies and,derived thereof, (2) the uncertainty in the productionforecast of newly planned wells.

The formulation of the problem can be posed ei-ther as a sampling problem or a stochastic optimiza-tion problem (see [29] for a review). In the samplingapproaches [28, 30, 44], one would first formulate aposterior distribution of the reservoir properties basedon prior information of these properties (geostatisticsis often used) and a likelihood distribution specifyinghow well the data should be matched. Then, Markovchain Monte Carlo (McMC) is used to sample thatposterior distribution (see [16, 19, 27] for recent ex-amples in flow). While this approach has appeal froman uncertainty quantification point of view, one runsinto considerable problems practically, in particularwhen the modeling of forward responses (flow simu-lation) is CPU demanding. The optimization approachconsists of building geologically realistic models thatalso match the production history. The problem isposed as a stochastic optimization problem and solvedusing various such techniques (see [2, 5, 31] for re-cent example benchmark comparisons). The reservoirproperties are parameterized, and these parametersare estimated using a least-square or maximum likeli-hood framework. Some methods require single-modelperturbations, meaning one needs to restart the opti-mization to get another history-matched model; othersuse ensemble methods (e.g., EnKf [17], and its manyvariations; see [1] for a review) and aim to update theensemble to improve the history match. The differencebetween the sampling and optimization approach oftenlies in the purpose: sampling methods aim for realis-tic uncertainty quantification, while history-matchingmethods aim for efficient reservoir model construc-tion but may not reflect realistic uncertainty. If oneaccepts the notion that uncertainty quantification isessentially a quantification of lack of understanding,then such lack of understanding should be dependenton how one formulates the problem (e.g., using theBayesian theory) and not on the particular history-matching technique used to solve it [44]. For example,what many benchmark comparisons essentially show ishow uncertainty quantification varies depending on thehistory-matching technique.

In this paper, we consider an additional complexity,namely that part of the prior model has a high levelof discrete uncertainty, often also termed scenario un-certainty. For example, one may have several differentvariogram models, or several different Boolean modeldefinitions, scenarios of fracture hierarchies, or severalhypotheses on the layering structure or fault interpreta-tion [6]. Such uncertainty in the prior model is a directresult of the uncertainty in the interpretations made,which is the norm in real field applications. Often, ge-ological scenario uncertainty has a considerable impacton flow response uncertainty (see [18] for a compre-hensive analysis for facies models). In this paper, wewill model the uncertainty in geological interpretationof facies through 3D training images and MPS [39, 40],one of a number of possible ways to handle geologicalscenarios. This approach provides the ability for geol-ogists to inject geological realism (depositional modelinterpretation, facies associations, dimensions, geome-tries, and architectures) lacking in other covariance-based facies modeling approaches such as truncatedGaussian [3, 26].

This paper presents a methodology whereby theproblem is decomposed into two parts, which are (1)the modeling of probabilities of each scenario (andpossible rejection of scenarios) and (2) sampling withineach scenario. In the “rejection” part, without any his-tory matching, we will model the probability of eachgeological scenario given the production data and at-tempt to reject geological scenarios with low proba-bility. In the “sampling” part, we will use an existingtechnique for generating facies distribution constrainedto production data using a single given training image,namely the probability perturbation method, althoughany other techniques could be used. We will evaluatethis methodology using a realistic reservoir case. Wewill compare the uncertainty quantification resultingfrom applying this methodology with the rejection sam-pler, which is the only exact sampler in the nonlinear,non-Gaussian case (McMC samplers only converge inthe limit). In the conclusions, we will discuss how thisapproach is general and could be applied to other typeof inverse problems, other than history matching.

2 Reservoir case study description

2.1 Reservoir context

The deepwater turbidite offshore West Coast Africa(termed WCA) reservoir considered in this paper islocated in a slope–valley system and has been used as

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Fig. 1 Structural model withfaults of the WCA reservoir,location of injector, producerwells, and planned welllocations

New well plannedP3

P1P4

P2

a case study in a previous work on quantifying uncer-tainty and history matching (see [22, 37]). The locationof the WCA reservoir is offshore in 1,600 ft of water andis 4,600 ft below sea level, resulting in large drilling costsand risk. The WCA reservoir is a slope–valley systemdivided into four structural blocks by a number of faultswith different fluid contacts: CC, CD, CE, and CW (seeFig. 1). Although the faults reduce the transmissibilitybetween CW, CC, and CE, the three segments are stillin communication with one another. However, CD isnot in communication with the others. An aquifer existsin the east of the reservoir, and a fault separates it fromthe rest of the reservoir. The reservoir structure is fairlywell known. The water–oil contact is located around5,440 ft and varies for the different segments.

In this paper, we do not directly use the productiondata of the real field; instead, we create what could betermed a virtual case, namely by an existing reservoirmodel as truth and then generate production data fromit by forward simulation. We will treat the result of thisforward run as field data. Although evidently idealized,the advantage of doing so is that (1) the data can bepublished without issues of disclosing sensitive infor-mation and (2) we can control certain elements of thehistory-matching problem that are inconsequential to

the studied methodology (such as erroneous data andother issues in model construction that are not studiedhere).

2.2 Geosciences

In this field, three different geological scenarios arerepresented by three training images (TIs, Fig. 2). EachTI has 78 × 59 × 116 grid cells. These scenarios wereprovided by the operating company and have beencreated by geological experts. The first geological sce-nario (TI1) has sand facies with low channel thickness,low width/thickness ratio, and low channel sinuosity.The second geological scenario (TI2) has sand facieswith high channel thickness, low width/thickness ratio,and low channel sinuosity. The third geological sce-nario (TI3) has sand facies with low channel thickness,low width/thickness ratio, and high channel sinuosityincluding levee. Figure 3 shows an example of threeprior models created by the geostatistical algorithmsnesim [39], an algorithm available in the public domain[33]. The three prior models are generated based onthe three different geological scenarios (TIs). As iscommon with these techniques, the reproduction of the

Fig. 2 Geologicalinterpretation of faciesgeometry, architecture, andassociation in terms of threetraining images

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Fig. 3 Two example priormodels for each trainingimage. Models are 3D, butonly 2D sections are shown

TI1 TI2 TI3

channel geometry is not perfect. Each TI has threedifferent sand facies, and the fourth facies representsthe shale or nonpay part of the reservoir. The amountof sand facies is about 48 % of the gross volume. Thehighest quality facies (channel facies for all TIs, shownin red in Fig. 2) has a constant permeability of 2,000 mdand a constant porosity of 0.3 and occupies about 28 %of the gross volume. The second quality facies (channelfacies in TI1 and TI2, levee in TI3 shown in yellow inFig. 2) has a permeability of 400 md and a porosity of0.2 and occupies about 10 % of the gross volume. The

last sand facies (channel facies in TI1 and TI2, leveein TI3 shown in sky blue in Fig. 2) has a permeabilityof 100 md and a porosity of 0.15 and occupies about10 % of the gross volume. The fourth facies (bluebackground in all TIs in Fig. 2) is non-pay (representedby zero porosity) and occupies about 52 % of the grossvolume. In this case, the prior probabilities of each TIare initially given as 50, 25, and 25 %, respectively,by the expert geologist. Facies are well characterizedat the well locations used as so-called hard data forall geostatistical models. The geostatistical technique

Fig. 4 Historic water ratedata

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employed allows for including a facies probability mapderived from seismic and rock physics modeling (asdone in WCA, see [22]) but was not included in thiscase study.

2.3 Flow model and production data

The geostatistical simulation was directly mapped tothe stratigraphic grid, and no upscaling was performed.The flow model has approximately 100,000 active cells,the exact number varying for each simulation. Thereservoir has four injection wells and four produc-tion wells (Fig. 1). Well positions are taken from theactual field. Injection/production of water/oil during2,000 days is performed with a fixed liquid rate of2,000 stb/day. Viscosities for oil and water are 15 and1 cP, respectively. Densities for oil and water are 57and 63 lbm/ft3, respectively. To simulate the reservoirmodels, streamline simulation is performed directly onthe geostatistical model. The production data consist ofwater rate per each production well over 2,000 days (seeFig. 4). Note that matching the water rate is equivalentto matching the oil rate in this case, since the total liquidrates of the wells are fixed in the simulation. Althoughwe do not match pressures in this particular case, themethod can incorporate pressures at the producingwells in the objective function with no difference inthe approach given below. Many of the elements ofthis case are also present in actual field problems: large3D models, uncertain geological interpretation, multi-ple facies categories representing different orders ofmagnitude in petrophysical properties, complex faciesarchitecture, and time-consuming flow models.

3 Methodology

3.1 Decomposition of the problem

In this paper, we use the sampling approach to gener-ate models, since we wish to ensure a realistic quan-tification of uncertainty. We also rely on a Bayesianapproach, meaning, we consider the geological modelinformation as prior and the production data as a meansto determine a posterior by simple application of theBayes’ rule. An important component of our approachis not to start immediately with creating models thatmatch data, as there is a risk that the uncertainty repre-sented by such ensemble is understated. Instead, we usethe idea proposed by Popper [32] and later discussedwith a geoscientific context by Tarantola [45], namely

of attempting to reject models instead of accept/includemodels (through history matching). In this particularcase, we will first consider rejecting concepts/scenarios,such as training images, then only after rejecting, pro-ceed with sampling actual reservoir models that matchthe data up to a specified precision. Such a screeningidea has previously been considered in the contextof structural modeling [13, 41, 42]. This idea can besimply derived by application of the following rule ofconditioning:

P (M, TI|D = dobs) =K∑

k=1

P (M| TI = tik, D = dobs)

× P(TI = tik |D = dobs ). (1)

In other words, we attempt to determine the jointuncertainty in the facies model M (a vector of dis-crete variables) and training image variable TI (hav-ing discrete outcomes tik, k = 1, . . ., K) given thatthe data are decomposed into two parts. The firstpart, P (TI = tik |D = dobs ), consists of determining theprobability of the training image from production datadobs alone. The second part, P (M| TI = tik, D = dobs),essentially calls for sampling using a fixed/given train-ing image. Consider first a proposal on how to modeland use the probability P (TI = tik |D = dobs ). Often, inmost practical cases with considerable uncertainty, theterm P (TI = tik |D = dobs ) in Eq. 1 is the most critical:it requires investigating whether geological interpreta-tion is in fact consistent with data. Secondly, any modelsM generated depend first and foremost on the set ofretained training images.

3.2 Rejection based on metric space modeling

To establish P (TI = tik |D = dobs ), we want to avoidany history matching using optimization techniques(inverse modeling) as this risks artificially reducinguncertainty. Instead, we would like to apply the rejec-tion principle and estimate this probability by forwardmodeling only. Then, this probability can be used toreject training image models as well as determine thenumber of models to sample from each training image.To estimate P (TI = tik |D = dobs ), we use Bayes’ ruleas follows:

P (TI = tik |D = dobs ) = f (dobs| tik) P (TI = tik)K∑

k=1f (dobs| tik) P (TI = tik)

.

(2)

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Since the priors P (TI = tik) are specified by the geo-scientist, one is left with estimating the likelihoodsf (d| tik), which we will perform without relying on clas-sical Gaussian or least-square assumptions. The mostimportant remaining problem in achieving this is thepossible high dimension of the response vector d. Inour reservoir case with four production wells and 25time steps, the dimension of the forward response d is100. However, in order to determine f (d| tik), we areinterested in modeling the variation of d for each giventraining image, not the response d itself. The variationof d for a given training image can be analyzed througha set of scoping runs (per TI) and metric space modeling[36, 37].

To illustrate the method and its generality, considera simple example, using covariances instead of train-ing images. Two sets of covariance-based models aregenerated, but with different anisotropy direction (45◦and 135◦) in the covariance function (see Fig. 5). Awater-cut response from water injection in the lowerleft corner and production from a well in the upper rightcorner is generated for 100 models of each covariancemodel. The water data (dobs) are shown in Fig. 5 withred dots.

The distance between any two models is defined asthe difference between these water-cut responses as afunction of time:

d(mi, m j

) =Np∑

k=1

√(gik − g jk

)T (gik − g jk

)with

gik = g (mi)k (3)

where Np denotes the number of wells and gik is thewater-cut response for model mi and well k.

Using this distance, we create the 2D multidimen-sional scaling (MDS) plot shown in Fig. 6 (top). MDSallows visualizing all the prior models by their mutualdifference of response, as well as the water-cut data(dobs), represented as a black cross in Fig. 6. The densityof points from each training image around the water-cut data indicates how likely each geological scenariois for the given data. For example, in the case wherethe water-cut data correspond to the red cross in Fig. 6,then only models having an anisotropy of 135◦ (purplepoints) are present around the water-cut data. Con-sequently, models having an anisotropy of 45◦ can berejected from the matching process, because they arenot consistent with the data. What is also clear in Fig. 6is the heteroscedasticity of the density of the responsevariables: the left tail has less dense points than theright tail. Hence, we cannot rely on commonly usedlikelihood models relying on Gaussian assumptions of-ten with a fixed error variance.

Instead, we estimate the density of points aroundthe water-cut data using an adaptive kernel densityestimation. Kernel smoothing is applied in the low-dimensional space created by MDS (by retaining onlythe most important eigen components), which results ina density map per geological scenario (Fig. 6, bottom).An adaptive kernel smoothing is used to account for thechange in variance (heteroscedasticity) of the responsevariables as is evident in Fig. 6. The value of densityobtained at the location corresponding to the water-cutdata is defined as the likelihood function f (dobs| tik) for

45 degrees 135 degrees

Wat

er-c

ut

Time, days

Fig. 5 Example used to illustrate the estimation of P(Scenario|Data): (left) two realizations with different anisotropy and (right) watercut data and 200 forward model responses

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Fig. 6 (Top) MDS plot ofwater rate data and modelresponse (bottom) kernelsmoothing of likelihoodfunctions

eige

n-co

mpo

nent

2

eigen-component 1

for 45 degrees for 135 degrees

each covariance model. Table 1 shows the estimation ofthe probabilities (Eq. 2) for a set of alternative water-cut data dobs (five crosses in Fig. 6). To compare theaccuracy of these estimations, we perform rejectionsampling (requiring 4,000 flow simulations; for a fullexplanation of rejection sampling, see Section 4.3),resulting in the probabilities in the right column ofTable 1. The method produces probabilities similar torejection sampling.

To summarize, we use the following procedures for-mulated in metric spaces and applied them for eachtraining image in the case study:

1. Generate N geostatistical facies models for thegiven training image tik.

Table 1 Probability of covariance scenario given production dataobtained using (1) method and (2) rejections sampler

P (45◦|di) Rejection sampler

d1 (blue) 0.74 0.76d2 (black) 0.46 0.49d3 (red) 0.01 0d4 (green) 0.23 0.19d5 (gray) 0.66 0.67

2. Run the flow simulator on each of the N models toget N responses di.

3. Calculate the distance (Eq. 3) between any tworesponses and between the responses and the datadobs. Create a distance matrix.

4. Apply MDS on the distance matrix.5. Project the responses in a low-dimensional space

(dim = M << N).6. Model f (d| tik) using adaptive kernel density esti-

mation in this low-dimensional space.7. Calculate the likelihood value f (dobs| tik).8. Calculate using Eq. 2 P (TI = tik |D = dobs ).

Two conditions are necessary to render estimatingf (d| tik) in step 6 above applicable in real field cases.The dimension M should be low (5D–10D) to makeadaptive kernel smoothing practical for given N. Thedimension M is case specific and depends on the vari-ation of the forward model responses, which itselfdepends on the variation of the geostatistical modelsgenerated for a given training image and how that vari-ation is translated into response variation, which itselfis dependent on the type of flow physics and boundaryconditions (wells). For the flow modeling cases studieswe have investigated with MDS (see [4, 9, 36, 37], for

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example), we find that in most cases, a dimension ofM = 10 or lower is sufficient to represent such varia-tion, at least for cases of early production (few wells)with large uncertainty.

A common confusion with MDS is that it does notaim for dimension reduction of models as done forexample in PCA or KPCA [35]; MDS simply aimsat approximating the non-Euclidean distance betweenmodel responses di with a Euclidean distance, i.e., itfinds in what low-dimensional space such approxima-tion works. The second condition is that the number ofscoping models/runs N is a fraction of the total num-ber of flow simulations needed for the entire history-matching methodology. A bootstrap technique [38] canbe used to determine the number N of simulations pergeological scenario.

The probability obtained from Eq. 2 allows com-puting the number of history-matched models per ge-ological scenario that should be generated by sampling.To generate a total of Nhm models matching the data,the number of models per scenario is determined byP (TI = tik |D = dobs ) × Nhm. If this number is less than1, then the scenario is rejected.

3.3 Sampling facies models

Once decided on the style of spatial continuity, a num-ber of techniques are available to generate an ensemble

of models to represent P (M = m| TI = tik, D = dobs).We rely on the probability perturbation method (PPM;[7, 10, 23]) which has been proven practical in numer-ous published real field application of facies as well asother discrete reservoir properties (see for example thefield cases published in [11, 12, 21, 22, 25, 34]). In thispaper, we will use a particular form of PPM termedregional PPM [20]. In regional PPM, the field is decom-posed into regions, either statically based on geologi-cal consideration or dynamically based on the stream-line geometry which delineates drainage regions of theproducing wells. The advantage is that perturbationis varied per region, resulting in faster convergence.Although there is no theoretical proof, PPM appears tobe more of a sampling technique with properties similarto the rejection sampler as compared to a stochasticoptimization technique such as gradual deformation(see [8] for a quantitative comparison).

4 Application to case study

4.1 Kernel density estimation

Based on the distance matrix calculated from runningflow simulation on the set of 180 prior models (60 foreach training image, see Fig. 7), an MDS plot in 2D iscreated (see Fig. 8). The observed response is repre-sented as a black cross. To represent more than 99 %

Fig. 7 Comparing priormodel responses (water rateover time in days) with fieldproduction data

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eigen-component 1

eige

n-co

mpo

nent

2

1|obsf ( ) ( ) ( )tid2|obsf tid 3|obsf tid

Fig. 8 (Top) MDS plot of water rate data for WCA and model response (bottom) kernel smoothing of likelihood functions for eachtraining image

of the variance, we need ten dimensions as given bythe eigenvalue structure obtained from MDS. Kernelsmoothing will be done in those ten dimensions, butplots will only be shown in two dimensions. The MDSplot clearly demonstrates the effect of varying the train-ing image, as each realization response is color-codedwith the training image being used. Responses from TI3

appear to be the densest among all the TIs around theobserved data location on the MDS map. TI1 appearsto be more distant from the plotted response. Figure 8shows the kernel density estimation results for eachTI plotted in 2D. The cross represents the location ofgiven production data. As expected, TI3 has the highestdensity value at the given observed data location onthe MDS map. We apply Bayes’ rule (Eq. 2) with priorvalues

P (TI = ti1) = 0.50, P (TI = ti2) = 0.25, and

P (TI = ti3) = 0.25 to get

P (TI = ti1 |D = dobs ) = 0.01,

P (TI = ti2 |D = dobs ) = 0.39, and

P (TI = ti3 |D = dobs ) = 0.60.

Based on this result, we reject TI1 and retain only TI2

and TI3 for history matching.

4.2 Regional PPM

We now proceed with sampling models for TI2 and TI3

constrained to production data using regional PPM. Us-ing the probability defined above, 29 posterior (history-matched) models were generated, 12 from TI2 and 17from TI3. Figure 9 shows the streamline geometry ofone flow simulation and one example of the regiondefinition. Note that the regions are defined accordingto the streamlines and thus they change as the modelsare updated. A model is accepted in regional PPM if theobjective function (mismatch in water cut) is lower thana defined threshold t. Figure 10 shows a few posteriormodels, and quality of the match to production datacan be assessed in Fig. 11 (left). The nature of theregional PPM algorithm is such that, although regions

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Fig. 9 (Left) Streamlinesimulation and (right)example of region geometryfor one iteration in regionalPPM

Streamline geometry at final time step Example of region geometry

are used for model perturbation, they do not affect thereproduction of channel geometries or create artifactsat the region boundaries [20]. The total computationaleffort is 508 (PPM) + 180 (scoping) = 688 runs or ∼24flow simulations per model.

4.3 Comparison with rejection sampler

Although requiring considerable CPU, for the sake ofthis case study and for validating the proposed method-ology, we apply the following implementation of therejection sampler on both TI variable and facies model:

1. Draw randomly a TI from the prior.2. Generate a single geo-model m with that TI.

3. Run the flow model simulator to obtain a responsed = g(m).

4. Accept the model if RMSE (dobs, g (m)) < t, t beingthe same threshold as defined in PPM (correspond-ing to a uniform likelihood).

RMSE is a root mean square difference between fieldand model response. A large number of tries, namely7,236 flow simulations, are needed to obtain the sameamount of models as PPM. The outcome of this rejec-tion sampler is twofold: (1) a set of 29 history-matchedmodels and (2) updated TI probabilities. For those, weobtain:

P (TI = ti1 |D = dobs ) = 0.03 (1 model)P (TI = ti2 |D = dobs ) = 0.34 (10 models)P (TI = ti3 |D = dobs ) = 0.63 (18 models)

Fig. 10 A few posteriormodels, only for TI2 and TI3,notice the absence of anyregion artifacts. Models are3D, but only 2D sections areshown

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Rejection samplerMethodology

Fig. 11 Comparing history match quality for methodology and rejection sampler

The updated TI probability obtained by rejection sam-pling is very similar to the one obtained by kernelsmoothing in metric space.

4.4 Forecasting a newly planned well

In order to compare the prediction performance of theproposed methodology, production is forecast for anadditional year, based on the rejection sampler andproposed methodology. Water rate predictions in thenewly placed production well for both methodologiesare shown in Fig. 12. The P10, P50, and P90 quantilesshow good agreement with the rejection sampler.

1000

Rejection

P90600

800PPM

400

P50200

Wat

er R

ate,

stb

/day

P102000 2100 2200 23000

Time, days

Fig. 12 P10-P50-P90 quantiles of water at the prediction year forrejection sampling and PPM

5 Discussion and conclusions

This paper proposes a method consisting of two parts toquantify uncertainty of facies with multiple geologicalinterpretations. The first component, namely modelingthe probability of each interpretation (whether usingtraining images or other techniques) given the pro-duction data, is the more critical contribution of thepaper. There is a very practical reason for modelingP (TI = tik |D = dobs ) and that reason can be found inthe very nature of reservoir modeling. Geoscientists of-ten do not consider in great detail or in any quantitativefashion the production history when interpreting thestyle of reservoir heterogeneity of facies. Instead, theyfocus on generating geological understanding and inter-pretations based on analog outcrops, principles of de-position, genesis, and process, using core, well-log, andseismic data for interpretations. Because of the uncer-tainty in such interpretations, some may be inconsistentwith production history; there is certainly no guaranteethat they will be consistent. The procedure proposedhere establishes a way to detect this inconsistency andrejects outright low or zero probability interpretationsbefore proceeding with any history matching.

We believe the proposed methodology has a largepractical appeal in cases with considerable uncertaintyin the geological model description and when criticaldecisions require a realistic quantification of financialrisk. Several aspects of this statement can be motivatedas follows.

Geosciences In planning new wells, there is little hopeof having any prediction power if the models are not

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geologically realistic or do not represent other datasuch as well logs and seismic. The reservoir geosciencesare an important and integral part of establishing suchprediction capability and on its own require substan-tial modeling effort and expertise. The methodologywe propose allows including state-of-the-art geologicaland geophysical modeling components into the history-matching workflow. We directly use the models gen-erated by a geostatistical techniques constrained towell-log and seismic data, layering, zonation, or stratig-raphy. There is no need for parameterizing a highlycomplex 3D model (and its uncertainty). While suchparameterization would have appeal from an optimiza-tion point of view, it may (1) lose information built intocomplex 3D geological models, (2) require continuityin the objective function if gradients are used, and(3) be problem/case specific and therefore difficult toimplement from a software engineering point of view(see below).

Computational Modern paradigms in the computa-tional geosciences such as cluster computing and GPUcomputing have changed the way problems can besolved [14, 15, 43, 46]. While running 600+ flow sim-ulations would have appeared impractical 10 years ago,parallelization makes such effort possible within a day.Interestingly enough, even the rejection sampler ap-pears to become less out of reach than just one decadeago.

Software engineering While many worthy ideas andmethods have been proposed in history matching, veryfew make it to actual usable and extendable softwareimplementations whether open source or commercial.The software challenge is inherently linked to all com-putational geosciences, and in a reservoir context, thischallenge lies in merging the geosciences world with theflow modeling world. One particular implementation ofPPM within the open source software SGeMS has beenpublished [24]. Implementations of MDS are availablein commercial software and underway in open-sourcecodes. Reliance on direct perturbation of geomodels(PPM) and scoping/screening in metric space (MDS)make the software challenge achievable in the currentstate-of-the-art software packages.

Extensions Whether the approach works for laterstage reservoir development (more wells, more pro-duction, multiple flow processes, and multiple type ofproduction data) remains to be investigated. At suchstage, there should be considerable insight into the typeof geological heterogeneity as well as well control of thespatial distribution of facies through extensive logging.Nevertheless, it should be understood that the most

consequential investments are not made at that stageand that history matching has the largest impact whenuncertainty is considerable. The methodology extendsto other cases than flow where large and discrete uncer-tainty in the prior exists. Current research is focused onapplication in geophysical inversion (seismic and EM)as well as fracture modeling.

Acknowledgment We appreciate the donation of this datasetby Chevron.

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