Comput GeosciDOI 10.1007/s10596-015-9491-x

ORIGINAL PAPER

Updating joint uncertainty in trend and depositionalscenario for reservoir exploration and early appraisal

Celine Scheidt1 · Pejman Tahmasebi1 ·Marco Pontiggia2 ·Andrea Da Pra2 ·Jef Caers1

Received: 3 July 2014 / Accepted: 21 April 2015© Springer International Publishing Switzerland 2015

Abstract Computationally efficient updating of reservoirmodels with new production data has received considerableattention recently. In this paper however, we focus on thechallenges of updating reservoir models prior to produc-tion, in particular when new exploration wells are drilled.At this stage, uncertainty in the depositional model is highlyimpactful in terms of risk and decision making. Mathemati-cally, such uncertainty is often decomposed into uncertaintyof lithological trends in facies proportions which is typi-cally informed by seismic data, and sub-seismic variabilityoften modeled geostatistically by means of training images.While uncertainty in the training image has received consid-erable attention, uncertainty in the trend/facies proportionreceives little to no consideration. In many practical appli-cations, with either poor geophysical data or little wellinformation, the trend is often as uncertain as the trainingimage, yet is often fixed, leading to unrealistic uncertaintymodels. The problem is addressed through a hierarchicalmodel of probability. Total model uncertainty is divided intofirst uncertainty in the training image, then uncertainty inthe trend given the uncertain training image. Our method-ology relies on an efficient Bayesian updating of thesemodel parameters (trend and training image) by modelingforward-simulated well facies profiles in low-dimensionalmetric space. We apply this methodology to a real field casestudy involving wells drilled sequentially in the subsurface,

� Celine Scheidtscheidtc@stanford.edu

1 Energy Resources Engineering Department, StanfordUniversity, Stanford, CA, USA

2 ENI, Milan, Italy

where as more data becomes available, uncertainty in bothtraining image and trend require updating to improve char-acterization of the facies.

Keywords Multiple-point geostatistics · Uncertainty ·Bayesian · Reservoir model updating · Distance-basedmodeling

1 Introduction

Reservoir model updating with production data has recentlyreceived considerable attention both in literature and indus-trial practice. In particular, the use of ensemble models ina data assimilation framework has been developed with theaim of reducing uncertainty on reservoir forecasting [9] (see[1] for a review). Less attention has been given to updatingmodels at the early exploration and appraisal phase of devel-opment of a reservoir, when no production data is available.Nevertheless, at early exploration and appraisal phase, thesame modeling questions arise, i.e., how to update multi-ple models when new wells are drilled, thereby possiblyreducing uncertainty on key reservoir parameters. At thisstage of reservoir development, little information on thereservoir is known; perhaps only a few wells are drilled,with available seismic surveys often of varying quality. Asa consequence, a significant degree of uncertainty in thecalculation of reserves and/or recovery factors is present.Next to structural uncertainty, the most critical uncertaintyin green fields lies in the proportion and trend of pay faciesand depositional scenarios, possibly with associated priorbeliefs. Given these large uncertainties, rebuilding modelsentirely with newly obtained data is often deemed imprac-tical, and procedures for updating prior beliefs on modelingparameters are rarely if ever employed in the industry today.

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This paper addresses the question of updating uncertaintyon pay facies proportion, trend and depositional scenariowhen a new well is drilled and new well-log data becomesavailable. New well information does not only provide localinformation of reservoir heterogeneity near its location, butit may also reveal significant global insight into variationsin trend and depositional scenario (for example reveal addi-tional lithologies). In the worst case, it may reveal thatcurrent beliefs are inconsistent with the newly acquired dataor change the understanding of the nature of the deposi-tional system. In such cases, there would be no alternativebut to construct an entire new set of models. In this paper, weaddress the situation where the prior set of models is consis-tent with the new data, and hence, the new well informationallows the reduction of uncertainty on facies proportions,trend, and beliefs in depositional scenario. The set of priormodels should subsequently be updated to account for thereduced uncertainty in the input parameters and the newdata.

Few studies have been performed for updating the priorprobabilities given new static well data. Of note is thework of Caumon et al. [4] and Maharaja et al. [13], whichfocused on updating the uncertainty in the global net-to-gross (defined as proportion of pay facies) when a new wellhas been drilled. The prior distribution of the net-to-grosswas updated using a Bayesian framework. They proposedto randomize both the geological scenario and the priornet-to-gross value using a bootstrap procedure.

Most studies to update probabilities of training imagesuse production data, which is not available in early appraisal[10, 16]. In this paper, the proposed methodology relies onthe Bayesian modeling techniques developed in [16], but isadapted for static well data, Park et al. [16] developed anapproach for updating probabilities associated with deposi-tional scenarios (training images in their application) whennew production data becomes available within a Bayesianframework. The methodology required the generation of aset of prior models and their forward flow simulation toupdate the probabilities of the training images, without anyexplicit history matching. The method presented in [16] willbe extended in several aspects. First, more than one uncer-tain component is considered in this paper, and second,the methodology is extended to continuous parameters. Theapproach was additionally adapted to deal with static welldata instead of dynamic production data. Finally, an auto-mated procedure was developed to estimate the parametersrequired by the methodology.

The approach was applied to a real turbidite reservoirwhere only a single well was drilled and a second well hasbeen planned and was eventually drilled in October 2013.The main uncertainties for this field lie in the trend (width ofthe main channel belt containing the pay facies) and in thedepositional scenario. First, a description of the field and its

available data are presented; then, a detailed description ofthe methodology is illustrated with two synthetic well data.The methodology is then applied to the actual newly drilledwell and corroborated by comparison with rejection sam-pling. Finally, a Monte Carlo study is presented to validatethe obtained updated probabilities.

2 Reservoir case study description

The field under investigation is a turbidite reservoir inearly stage of development. The dimensions of the field are8.5 × 13.5 × 0.08 km at its largest point, discretized into170 × 275 × 55 grid cells. Only one well (w1) has beendrilled and a second well (w2) was scheduled to be drilled.Both well locations are shown in Fig. 1. At the time ofthe initial modeling phase of the reservoir, well w2 was notdrilled yet; hence, it was not used. A 3D seismic surveywas performed, but is of relatively poor quality and resolu-tion. Therefore, in this initial modeling phase, one needs toaccount for uncertainty in the facies architecture, locations,and proportions, represented by depositional scenarios andtrends in the reservoir model.

2.1 Modeling of uncertainty in depositional scenario

Four facies have been identified from well w1 and the seis-mic data. They consist of background shales, thin bed sands,bedded sands, and massive sands. Due to the low quality ofthe seismic, the location and the proportion of each faciesare highly uncertain. In addition, considerable uncertainty ispresent in geological heterogeneity, architecture, dimension

Fig. 1 A single layer of the field, showing one realization of thefacies, and the locations of the only well drilled (w1) and the upcomingwell (w2)

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Fig. 2 Three differentdepositional scenariosrepresented as TIs

of geobodies, and facies proportions. Given the little amountof data available, different depositional hypotheses havebeen made for this study. The facies are modeled usinga multi-point statistics (MPS) algorithm, where the depo-sitional scenarios are represented as training images (TI).Even though uncertainty in depositional scenario can bedescribed by various parameters that enter into the creationof a training image (e.g., geobody size, proportion, etc.,we refer to [20] for an example), training images providea natural way for geologists to conceptualize interpreta-tion of depositional scenarios (major differences in the jointparameters) [14]. For this study three TIs (shown in Fig. 2)represent depositional scenario uncertainty, which were pro-vided by geological experts of the operating company.The statement of such possible scenarios, in any practical

setting, will always be a subjective decision. All three train-ing images have the four facies mentioned above and allrepresent a levee channel complex, which was defined as themost probable depositional scenario. The training imagesdiffer in the spatial arrangement and have major differencesin proportion of the channels and levee. As an example, onlyone channel and lower percentage of sand can be observedfor ti2, compared to ti1. In addition, some levees that arepresent in ti1 and ti2 are not present in ti3.

It can be seen in Fig. 3 that both wells (w1 and w2)

are located close to each other. Only a sub-region of thereservoir around the two wells is thus considered in thisstudy, for convenience. The number of grid cells in the sub-region illustrated in Fig. 3 (right) is 120 × 140 × 55, whichcorresponds to approximately 6 × 7 × 0.08 km.

Fig. 3 Use of a sub-grid aroundthe wells

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2.2 Modeling of uncertainty in horizontal trendand proportions

Most commonly, trends and proportions in the reservoir arederived from seismic data and are represented as a set ofprobability maps per facies. Example of the combined prob-ability map for the pay facies, provided by the operatingcompany, is shown in Fig. 4. In Fig. 4, a main channel beltcontaining the levee channel complex can clearly be seen;however, its width is not clearly defined, due to the poorresolution of the seismic. In particular, one difficulty is thatprobability maps require the target proportions to be definedfor each of the facies, which in itself is uncertain [13]. Infor-mation on target proportions in the early appraisal stageis very limited and incomplete, due to lack of data (onlyone well in this case). As a consequence, probability mapscalibrated with deterministic proportions, such as those dis-played in Fig. 4 may not properly reflect uncertainty [22,23]. For that reason, in this paper, we use auxiliary variables[5] instead of probability maps. An auxiliary variable is acontinuous property that reflects some property of the train-ing image in a certain neighborhood (support). An auxiliaryvariable can for example represent the facies proportions,object size, and orientation, at a given location [5]. An auxil-iary variable must be defined for both the training image andthe simulation grid. Compared to using probability maps,the advantage of using auxiliary variables, in addition to itssimplicity, is that only one (3D) map needs to be defined,regardless of the number of facies in the training image.

Pay - Facies

Fig. 4 Probability map for the pay facies derived from seismic dataprovided by the operating company. The width of the main channelbelt is highly uncertain

In the case study investigated, an auxiliary variable indicat-ing where the levee channel complex should preferably belocated is used as shown in Fig. 5. Figure 5 shows one ofthe training images (top left) and its corresponding auxil-iary variable (top right) as well as two examples of auxiliaryvariable for the simulation grid (bottom). A high value (red)of the auxiliary variable indicates the presence of leveesand channels, whereas a low value (blue) for the auxiliaryvariable implies the presence of shale at the given location.

As mentioned earlier, a significant uncertainty is presentin the width of the belt that contains the levee channel com-plex. A simple, low-dimensional parameterization is usedin this paper to account for uncertainty in trend. By usinga single parameter w, a set of auxiliary variables (both forthe TI and the simulation grid) with varying widths can begenerated (Fig. 5, bottom). The uncertainty on w is takenuniformly distributed between 3.5 and 7 km (the prior distri-bution of w) and defines the width of the auxiliary variablealong the x-axis. Note that the auxiliary variables shownin Fig. 5 (bottom) vary in depth with respect to the ver-tical trend observed in the seismic. The auxiliary variableguides the MPS simulation to place channels in regionsinside the belt. The prior uncertainty in the auxiliary vari-able leads to a prior uncertainty in the facies proportions: anarrower belt will contain fewer channels and levees (andmore shale) than a wider belt. This procedure is formulatedin a Bayesian context, where uncertainty on facies propor-tion becomes an output of the workflow and is a functionof the width of the auxiliary variable, the well data, and thetraining image proportions. In this manner, our work differsfrom [13] and [23], where uncertainty in facies proportionsare determined ahead of the modeling workflow, then variedwhile generating MPS realizations.

2.3 Creation of a set of initial/prior models

Having defined the training images and a set of auxil-iary variables, a series of reservoir models are generated.Any MPS algorithm can be used for that purpose; herewe use multi-scale cross-correlation-based simulation (MS-CCSIM) [24, 25]. MS-CCSIM is a pattern-based tech-nique that uses cross-correlations on an overlapping regionbetween a previously simulated pattern and the pattern tobe simulated to ensure spatial continuity of geological fea-tures. Boolean methods [8, 27, 28] could be alternativelyemployed if conditioning to wells is not too time-consumingand proper sampling methods [12, 15] for such conditioningare employed to avoid artifacts.

Note that the reservoir models are only conditioned to thewell w1, since at the time of the modeling study, the secondwell had not yet been drilled. One hundred reservoir modelswere generated for each TI with varying width w sampledfrom a uniform prior distribution. Examples of models are

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Fig. 5 Example of trainingimage (TI) and its correspondingauxiliary variable (AXTI) (top);example of auxiliary variablesfor the simulation grid withdifferent width w (bottom)

shown in Fig. 6, where models generated with a small w

tend to concentrate channels/levees in the center whereasmodels generated with a large w place such bodies moredispersed from the channel axis.

The next section describes in detail the proposed method-ology to update the prior probabilities of the depositionalscenario (TI) and trend after a new vertical well is drilledand facies profiles along the well-bore become available.Updating the prior beliefs recognizes that some of the trendvalues or TIs that were thought possible after drilling thefirst well may now be highly unlikely given the informa-tion from the new well. The method is first illustrated usingsynthetic well data, then applied to the actual well drilled.

3 Methodology

The notations used throughout this paper are the following:

• TI Random variable representing the training imagechoice, with discrete outcomes tik , k = 1,. . . , K with K

representing the number of training images.• W Random variable representing the trend (described

by the belt width), with outcomes w, w ∈ [3.5, 7].• P Random variable representing the proportion of pay

facies in the model with outcomes p.

• D Observable response, namely the uncertain responseprior to drilling the well. D has outcomes d, which inthis case represent the facies profile along a single wellpath.

• dobs Outcome of D; in this application, the observeddata at the newly drilled well.

• fTI,W |d (tik ,w|dobs) Joint probability density of TI andW given data dobs.

• fW |TI,d (w|tik,dobs) Probability density of W , given tikand dobs.

A Bayesian formulation is used to update the probabili-ties of the uncertain parameters. In this context, the termprior uncertainty refers to the probability distributions ofthe uncertain parameters that express the beliefs and uncer-tainties in the model parameters (in this case TI and trend)before drilling the new well w2. In the presence of uncertainmodel parameters TI and W , the posterior distribution ofmodels f (m|dobs) can be written as follows:

f (m|dobs) =∫tik∈{1,2,3}

∫w∈[3.5,7]

f (m|tik ,w,dobs)fTI,W |D

(tik ,w|dobs) dtikdw (1)

In this paper, we focus on the study of the latter term,fTI,W |D (tik ,w|dobs), which corresponds to updating theprior uncertainty of parameters (TI and W in this example)

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Fig. 6 Example of realizationsfor different training images andchannel belt widths w

given new observed well data. The first term of Eq. 1 isobtained by generating models conditioned to dobs (in thiscase conditional simulation, including w2), using a MPSalgorithm.

We will rely on the following sequential decompositionof fTI,W |D (tik ,w|dobs):fTI,W |D (tik ,w|dobs) = fW |TI,D (w|tik,dobs)

×P (TI = tik|D = dobs) (2)

which divides the task of determining fTI,W |D (tik ,w|dobs)into determining first, the updated probability of the dis-crete variable P (TI = tik|D = dobs) given the observeddata and then, for each outcome tik of the discrete vari-able, the densities of the continuous variable (trend in thiscase) given the observed data: fW |TI,D (w|tik,dobs). Next,

we will present a direct estimation method of both termsin Eq. 2 using a combination of distance-based modeling[16, 18, 19] and kernel smoothing [21], thereby avoidingany complex Markov chain Monte Carlo (McMC) meth-ods. A synthetic well is used for illustration purposes (welldata is shown in Fig. 7) in this section. The facies profile atthe new well location consists of mostly pay facies (29 %of thin beds, 27 % of bedded sand, and 5 % of massivesand).

3.1 Modeling of the probability of the training imagegiven the data

First, the probability of the TI given the data P(TI =tik|D = dobs) is evaluated using the direct estimation

Fig. 7 Low-dimensional representation of the well facies dobs extracted at the new well location for the prior models, for each TI (left). Theobserved synthetic well data is illustrated and its location in space is shown by the black cross; corresponding probability density f (dr|tik) (right)

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Table 1 Updated probabilities for each training image, given thesynthetic well data

ti1 ti2 ti3

P(TI = tik |dobs) 0.2 0 0.8

methodology presented in [16]. For completeness, a briefreview of this work is presented, which will then allowcontrasting with what is presented here.

To estimate P (TI = tik|D = dobs), Park et al. [16] useBayes’ rule as follows:

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

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

(3)

Where f (dobs|tik) represents the density of the observeddata given the training image tik. Only the probability den-sities f (dobs|tik) need to be estimated since the priorsP(TI = tik) are specified. The density f (dobs|tik) is esti-mated directly from a set of prior models each with responsed (flow response in their application). However, since dcan be a high-dimensional variable, a low-dimensional rep-resentation dr of the model responses d is constructed. Ametric space is created where Park et al. define the dis-tance as the difference in production response betweenany two models from the prior. The observed data dobscan be represented in this metric space as well (denoteddrobs), as its distance to any other models can be evalu-ated. Multi-dimensional scaling (MDS, [3]) is used to createan equivalent low-dimensional (dimension typically up to10) Euclidean space on which f

(drobs|tik

)can be estimated

directly, without any need for McMC sampling. The under-lying assumption of the method is that f (dobs|tik) can beapproximated by the density f

(drobs|tik

)of points/models

for a given TI at the location of drobs in the low-dimensional

MDS space. Park et al. employed an adaptive kernel den-sity estimation method where the bandwidth is determinedby clustering to obtain an approximation off

(drobs|tik

). The

use of the kernel density estimate explains the need for alow-dimensional MDS space. It was shown in [16] that ker-nel smoothing can be successfully applied in dimensions upto 10.

In this paper, a similar procedure is followed to obtainP (TI = tik|D = dobs). One major difference is the purposeof the modeling study, which is to update the trend and TIgiven the facies profile measured at the newly drilled well,as opposed to the use of production data in [16]. Hence, thedefinition of a new distance is necessary, in order to projectthe well facies profiles in low-dimensional space. As inall distance-based modeling approaches, the distance needsto be tailored to the response of interest. When evaluatingP (TI = tik|D = dobs), uncertainty in training image is ofinterest; therefore, the distance should be designed to distin-guish between different patterns at the newly drilled well. Toevaluate a distance between prior models and observed welldata, well facies must be extracted from the prior modelsat the well location (w2). A multi-point histogram (MPH)approach is used (see for example, [2, 7, 11]), where his-tograms of patterns found at the new well location for eachreservoir model and the observed well are computed. Thehistogram represents the frequency distribution of the pat-terns that appears in the well. A J-S divergence distance isused to evaluate the differences in pattern distribution and toproject the models in a low-dimensional space [26]. Basedon the pair-wise MPH distance, MDS can be applied to rep-resent in Euclidean space the differences in patterns foundat the wells. Figure 7 (left) shows, for each training image,a 2D projection of the prior models (colored dots) and thedata (black cross) in MDS space, based on their pattern sim-ilarity. The first and second dimensions of the MDS spacehave a contribution of 60 and 9 % of the total variance,respectively.

Fig. 8 Low-dimensionalrepresentation of the priormodels, based on the proportionof pay facies. Points are coloredaccording to the value of w

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Fig. 9 Prior set of models injoint space (Dr, TI, W). Pointsare colored according to thevalue of w

Having defined a low-dimensional representation of D(3D in this application), which is denoted Dr, we employkernel density estimation to directly estimate f

(drobs|tik

). In

this work, an automated procedure to define the bandwidthhas been implemented and is described in the Appendix.Figure 7 (right) shows an illustration of the densitiesf (dr|tik) estimated for each training image. Note that forillustration purposes, the densities are evaluated in the spacespanned by dr, but to estimate f

(drobs|tik

), the densities

only need to be evaluated at the location of the new data,i.e., at drobs.

The probabilities of each TI given the observed well dataare then obtained by assuming f (dobs|tik) ∼= f

(drobs|tik

)and using Eq. 3. Table 1 shows the resulting probabilities.It can be observed that ti2 has a very low probability andcould be rejected from subsequent modeling. In addition, ti3has a much higher probability than ti1 when such a well isobserved.

Now that the second term in Eq. 2 is determined, thefirst term, namely, fW |TI,D (w|tik,dobs), requires estima-tion. This term consists of estimating for each trainingimage the values of the trend that are most likely plausiblewith the observed well.

3.2 Modeling of the probability of the trend giventhe training image and the observed data

The method of Park et al. [16] presented in the previoussection is limited to only one type of uncertainty (the TI)which has discrete outcomes. Now, a joint probability distri-bution must be evaluated since uncertainty is present in bothtrend and TI. In addition, the parameter W which definesthe trend is continuous; hence, a joint probability of mixedparameters (continuous and discrete) must be calculated.

As before, the probability density of the trend giventhe data and the TI fW |TI,D (w|tik,dobs) can be esti-mated using a low-dimensional representation of D. ForfW |TI,D (w|tik,dobs), the low-dimensional projection of dmust be designed to distinguish between different val-ues of the trend. Models generated with a small w willmost likely show no pay facies at the new well loca-tion. As the width increases, more pay facies will beobserved. To distinguish the values of w, a suitable low-dimensional representation for realizations of D is theproportion of each facies in the well extracted from themodels. Since three different types of pay facies arepresent in the models, d can be represented in the 3D

Fig. 10 Probability densities: fW |TI,D (w|tik,dr ) (left) and fW |TI,D(w|tik,dr

obs

)(right)

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Fig. 11 Probability density ofthe training image and the beltwidth given the new well data:fTI,W |D (tik, w|dobs) (left);probability density of thetraining image and theproportion of pay facies giventhe new well data:fTI,P |D (tik, p|dobs) (right)

space of the proportions of pay facies. The resultinglow-dimensional representation of d for all three TIs is pro-vided in Fig. 8, where points are colored according to thevalue of the trend. Note that in this particular case, the defi-nition of a metric and application of MDS were not needed,as d can be projected in a 3D space and does not needadditional compression.

Figure 8 shows that large values of w (red points, widebelt) tend to be grouped on the right side of the graph, closeto the observed data (black cross) whereas small values of w

tend to be located further on the left side. As a consequence,it is expected that the probability density of W given thenew data is high for large values of w and then graduallydecreases as w decreases. A mathematical evaluation is pre-sented next, where the probability density is estimated foreach TI.

The main difficulty in estimating the density comparedto the previous section lies in the fact that the trend variableis a continuous variable, contrary to the case of a discreteTI variable. Hence, a probability density must be calculatedinstead of a probability. We address this point by employ-ing an additional dimension to the space Dr representing thevalues of w. An example of such a space is illustrated inFig. 9. Note that for illustrative purposes, only one dimen-sion is used to represent the low-dimensional representationof the data, Dr in Fig. 9, even though a 3D space was usedin this example.

fW |TI,D (w|tik,dobs) is approximated by fW |TI,D(w|tik ,drobs) which is estimated in the space represented in Fig. 9.Since the TI is a discrete parameter and its value is assumedfixed to tik , the density fW |TI,D

(w|tik,dr

obs

)is estimated

independently for each TI. Again, kernel density estimationis used to evaluate fW |TI,D (w|tik,dr) and then the valuesat dr = drobs are taken. Kernel smoothing is applied jointlyto both dr and w; hence a bandwidth for the trend mustbe evaluated as well. Details on how to compute the band-width automatically are provided in the Appendix. Figure 10shows the probability densities fW |TI,D (w|tik,dr) (left) andfW |TI,D

(w|tik,drobs

)(right).

At this point, both terms of Eq. 2 have been estimated.Remaining is the multiplication of those terms, to obtain

the final probability density fTI,W |D (tik, w|dobs), which ispresented in Fig. 11 (left).

Since the total proportion p of pay facies in the mod-els and the auxiliary variable width are highly dependent oneach other, one can determine the updated joint probabilitydensity of the TI and the proportion fTI,P |D (tik, p|dobs) inthe exact same way as for the width. The only difference isthat the density is estimated in the joint space (Dr, TI, P)

instead of (Dr, TI, W). The updated joint probability densityfTI,P |D (tik, p|dobs) is shown in Fig. 11 (right).

Given the observed well data shown in Fig. 7, the pro-posed approach shows that ti2 is not likely to occur. Thedepositional setting represented in ti2 could thus be removedfrom the study. In addition, only large values for w are pos-sible, which indicates a wide belt containing the pay facies.Not surprisingly, the updated proportion of pay facies p inthe model is quite high, with values varying between 40 and75 %. This confirms what was expected, as the well containsmostly sand facies.

3.3 Addition of a third well

In this section, a synthetic example is provided where athird well w3 is sequentially drilled. At this point, the facies

Fig. 12 Drilling of a third well w3. Location and synthetic well faciesprofile

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Fig. 13 Low-dimensional representation of the well facies profile extracted from the prior set of models and the real observed well data (blackcross) for w3. MPH-based distance (left) and proportions of pay facies (right)

profiles of wells w1 and w2 are known, as well as theupdated probabilities obtained after drilling well w2. Thenext step is to update the set of existing models to reflectthe updated joint probability and, if necessary, create newmodels conditioned to all available well data (w1 and w2

in this example). In this case, none of the initial mod-els are conditioned to w2; hence, a new set of 300 faciesmodels conditioned to w1 and w2 is generated with param-eter values respecting the updated distributions of both thebelt width and the training image (shown in Fig. 11). Inparticular, 60 models are created for ti1 and 240 for ti3, cor-responding to the updated probability of 0.2, 0, and 0.8 forti1, ti2, and ti3, respectively. Due to the indication of a largechannel belt after drilling of well w2, w3 is placed at a loca-tion further away from the assumed center of the channelcompared to the two preceding wells. The well location andits profile are shown in Fig. 12. Note the presence of onlyone sand facies (bedded sands) in the well and a proportionof pay facies of about 0.4.

With this new well data in Fig. 12, the proceduredescribed above is applied again using the new set of mod-els. The low-dimensional spaces obtained for the trainingimage and the trend are presented in Fig. 13. It is clear

from Fig. 13 (left) that ti1 is incompatible with the data,as all models constructed using ti1 are far from the datain this low-dimensional representation. The updated prob-ability densities for the width of the channel belt and theproportion of pay facies are shown in Fig. 14. Note thatsince ti1 and ti2 were successively rejected from the study,only the probability density of the width for ti3 remains non-zero. We observe that the probability density function of W

is narrower than in Fig. 11; channel belts below 5.7 km areno longer possible. However, in comparing Figs. 11 and 14,the uncertainty in the proportion of pay facies for ti3 has notchanged significantly by drilling well w3.

Even though the example uses three training images,the procedure can be applied similarly to more than threetraining images, as long as the MPS algorithm can createa sufficient number of realizations to perform the analy-sis. In addition, in the case where multiple wells are drilledat the same time, the application of the workflow remainsthe same. The difference will be in the calculation of thedistance: the distances between models at each well loca-tion need to be aggregated into a single distance whichis subsequently used to project all the model responses inlow-dimensional space.

Fig. 14 Updated joint probability density function for w3: fTI,W |D (ti3, w|dobs) (left) and fTI,P |D (ti3, p|dobs) (right)

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Fig. 15 Low-dimensional representation of the well facies profile extracted from the prior set of models and the real observed well data (blackcross) for w2: MPH-based distance (left) and proportions of pay facies (right)

The method was presented using two synthetic wellsdrilled sequentially as the observed data. In the next section,the methodology is applied using the real observed datafrom well w2, which was drilled in late 2013.

4 Results

In the actual case study the operating company reportedthat well w2 encountered only shale; no reservoir pay facieswere found. In this section, we apply the above estima-tion procedure to update all probabilities and densities giventhis specific well data. The updated probabilities are thencompared with rejection sampling. Finally, a resamplingprocedure is applied to validate the proposed method.

4.1 Application to the real observed data

The proposed approach is applied to the real observed wellprofile (100 % shale). First, the probability of the TI giventhe observed well data dobs is computed and then the prob-ability densities of the trend for a given TI and dobs areestimated. The definitions of the distances remain the samefor each expression of the probability; the only change isthe location of the observed data in MDS space. Illustra-tions of the MDS spaces including the observed data forboth the training image (left) and the trend (right) are dis-played in Fig. 15. In both maps, the location of the new welldata is right at the edge of the cloud of points, which is notsurprising given that it traversed 100 % shale.

The updated probabilities of each TI given the new welldata are shown in Table 2. It can be concluded that the newwell is not very informative on the training image, althoughti3 shows a slightly higher probability.

The updated joint probabilities of the TI and the trend(and proportion) given the observed well data are displayedin Fig. 16. Given that the newly drilled well did not recordproducing sands, the updated probabilities suggest that

narrow belts (small values of w) are more likely to occurthan wide belts. Note that it is possible to obtain a 100 %shale well for a wide belt for ti1 and ti2. This observationhighlights one main advantage of the procedure; it accountsfor the possibility that a shale well can be obtained evenwith a wide belt, due to channel sinuosity architecture orsimple bad luck. Interestingly too, the updated probabilitiesof the proportion of pay facies show that a larger propor-tion for ti3 is possible, compared to ti1 and ti2. The largerproportion of pay facies for ti3 can be explained by the factthat in general ti3 contains more channel levees than ti1 andti2, but they are of smaller size, thus increasing the possi-bility of missing a sand body. Finally, the new well doesnot provide significant information on the type of deposi-tional scenario. ti3 is shown to be slightly more probablethan ti1 and ti2. These results emphasize that working withthe fixed probability maps shown in Fig. 4 does not prop-erly account for the uncertainty in the channel belt width. Inother words, the channel belt width in the probability mapsin Fig. 4 was fixed at about 6.5 km, which, according to thejoint probability shown in Fig. 16 is very unlikely. In thenext section, rejection sampling is performed to determinethe “true” joint probability density of the trend (and propor-tion) and the TI, which is then used to validate the proposedapproach.

4.2 Real observed data: comparison with rejectionsampler

In most situations, rejection sampling is not possiblebecause it requires considerable CPU time. Here, since thewell contains only shale observations, it is relatively easy to

Table 2 Updated probabilities for each training image, given theobserved well data

ti1 ti2 ti3

P(TI = tik |dobs) 0.33 0.30 0.37

Comput Geosci

Fig. 16 Updated joint probability density function for w2: fTI,W |D (tik, w|dobs) (left) and fTI,P |D (tik, p|dobs) (right)

obtain models that contain only shale at the well location bya simple random sampling. This would evidently not be thecase for variable facies profiles at the well, such as the syn-thetic well data used above. Rejection sampling is appliedas follows:

1. Draw randomly a sample tik from the prior.2. Draw randomly a w from the prior.3. Generate a single model m with that tik and w.4. Extract the well data from the model at the well loca-

tion.5. If the well is in all shale, keep the model, otherwise

reject it.

Rejection sampling was applied until 850 models presenting100 % shale at the well location w2 were obtained. The rela-tive frequency histogram of the uncertain parameters TI andW are presented in Fig. 17 (left). In Fig. 17 (right), the ker-nel smoothed densities obtained by the proposed approachesare displayed again for comparison. Both methods providesimilar distributions. Figure 18 confirms the validity of theupdated joint probability of the TI and proportion of pay

facies P . In particular, the density of P for ti3 is much widerthan for ti1 and ti2.

4.3 Validation using a resampling procedure

As mentioned earlier, rejection sampling is much more dif-ficult to apply for non-shale wells. In order to confirm thevalidity of the obtained joint probability density, the follow-ing resampling procedure is applied. The idea underlyingthis section is based on the total probability formula:

fTI,W (tik ,w) =∫

dobs

fTI,W |D (tik ,w|dobs) f (dobs)ddobs (4)

One can see in Eq. 4 that if a randomization is performed ondobs, the integration of the conditional probabilities shouldaverage out to the prior probabilities. As a consequence,one way to validate the proposed approach is to do a ran-domization of dobs (observed well facies profile), evaluatethe conditional density fTI,W |D (tik ,w|dobs) given that dobsusing the proposed approach and integrate it over many dobs

Fig. 17 Joint probability density distribution for (TI, W ) obtained by rejection sampling (left) and the proposed methodology (right)

Comput Geosci

Fig. 18 Joint probability density distribution for (TI, P ) obtained by rejection sampling (left) and the proposed methodology (right)

(right-hand side in Eq. 4). The validity of the procedureto estimate the joint probabilityfTI,W |D (tik ,w|dobs) is thenverified if the procedure retrieves the prior joint density(left-hand side in Eq. 4). Note that the resampling pro-cedure evaluates the quality of the obtained distributionsto reproduce the correct posterior, on average. It doesnot specifically validate the results for the particular dataobserved.

The procedure is the following:

1. Draw a tik and aw from their prior distribution (uniformfor both variables).

2. Generate a single modelm with the tik and w.3. Extract well data at the well location and take it as

observed well data dobs.4. Evaluate fTI,W |D (tik ,w|dobs) (or fTI,P |D (tik ,p|dobs))

using the methodology presented above.5. Sample repeatedly from the resulting distribution

fTI,W |D (tik ,w|dobs) (or fTI,P |D (tik ,p|dobs)).

Fig. 19 Resampling procedure: prior relative frequency histogram of the uncertainty parameters (left) and relative frequency histogram resultingfrom the resampling procedure (right) for W (top) and P (bottom)

Comput Geosci

6. Repeat the procedure many times.

The densities fTI,W |D (tik ,w|dobs) and fTI,P |D (tik ,p|dobs)are evaluated using the same set of 300 prior models (100per TI, with a channel belt width varying uniformly) as forthe above studies. The procedure was repeated 1500 times,with different dobs. For each iteration of the procedure, 1000samples were drawn from the conditional density distribu-tions (step 5). Figure 19 displays the left-hand side (LHS)and right-hand side (RHS) of Eq. 4 for the width in theauxiliary variable (top) and the proportion of pay facies(bottom).

One can observe that distributions close to the prior areretrieved. A two-sample Kolmogorov-Smirnov (K-S) testwas applied on samples from RHS and LHS and confirmsthe null hypothesis: the samples from the RHS and LHSare drawn from the same distribution. This shows the valid-ity, on average, of the evaluation of fTI,W |D (tik ,w|dobs) andfTI,P |D (tik ,p|dobs). Because of the approximate nature ofthis procedure, one cannot expect to obtain a perfect matchbetween the RHS and LHS of Eq. 4.

This randomization procedure confirms that the probabil-ity density of uncertain parameters can be approximated bythe density of points in the reduced joint space (Dr, TI, W)

or (Dr, TI, P). In addition, it confirms as well that the band-width estimation is robust and leads to reasonable densityestimations.

5 Conclusions

This paper proposes a methodology to jointly update uncer-tainty in both depositional scenario and trend when newwell data becomes available. The methodology is designedfor fields in early development, with little available data(only a few wells, no production), hence considerable uncer-tainty. The method was applied successfully to a real fieldwhere data was obtained through well log measurements ata newly drilled well. Uncertainty in trend (represented byuncertainty in the width of the belt containing the pay facies)was modeled using auxiliary variables instead of probabil-ity maps. By varying the width of the auxiliary variable,uncertainty both in the belt width and in the proportion ofpay facies was accounted for. A major advantage of thisapproach is that target proportions do not need to be spec-ified a priori, they are outputs of the proposed modelingprocedure.

Updating the prior probabilities of the uncertain param-eters to account for new data is a crucial part of a suc-cessful modeling effort and leads to better decision making.The prior probabilities of uncertain parameters (in thiscase, depositional scenario and trend) were jointly updatedusing a combination of distance-based modeling and kernel

smoothing. The proposed method extends the idea of [16]in several aspects. It has been adapted to handle multipleuncertain parameters, as well as a “mixture” of continuousand discrete parameters. Even though only two uncertainparameters were used in the application, the methodologycan be applied to more uncertain parameters. In addition,the proposed workflow is applicable to any type of data(well profile, seismic data, GPR, well production, etc.), aslong as a relatively low-dimensional representation of thedata can be obtained, via the definition of a distance. Forexample, in [20], the data was a 3D seismic cube and a6D space was constructed. Finally, a procedure to estimateautomatically the bandwidth in the kernel smoothing wasdeveloped as the bandwidth choice may influence signifi-cantly the density estimates. A validation of the approachwas provided through a resampling technique, which con-firms the robustness of the proposed automated bandwidthcalculation.

The approach was applied to real field data, where thenew well was drilled entirely into shale. The prior proba-bilities of uncertain parameters (depositional scenario andtrend) were updated given this new data. The joint probabil-ity of the TI and the width of the channel belt as well as thejoint probability of the TI and the proportion of pay facieswere evaluated using a rejection sampling procedure. Com-parison of the joint probabilities with rejection samplingfurther validates the proposed approach.

The proposed approach is particularly applicable at earlystages of reservoir exploration and development when largeuncertainties in interpretation or trend of the reservoir mod-els are present. At much later stages, with many more wellsdrilled including availability of production data, much of thestated uncertainty will be resolved and hence other uncer-tainties will become more prevalent. It remains to be inves-tigated how this approach extends to the late productionstages.

Acknowledgments We appreciate the donation of this dataset byENI.

Appendix A: Kernel smoothing and estimationof the kernel bandwidth

The proposed workflow requires the estimation ofthe probability density functions f

(drobs|tik

)and

fW |TI,D(w|tik,drobs

)using a kernel smoothing approach [6,

21]. The kernel density estimate is formulated as follows:

fH(x) = 1

n

n∑i=1

KH(x − xi)

where

Comput Geosci

• x = (x1, x2, ..., xn)T and xi = (xi1, xi2, ..., xid)T , i =

1, ..., n are d vectors, d being the dimension of thespace.

• H is the d × d bandwidth matrix which is symmetricand positive definite.

• K is the kernel function which is a symmetric multi-variate density: KH(x) = |H|−1/2 K(H−1/2x).

The choice of the kernel function K is not as crucial to theaccuracy of kernel density estimators as the bandwidth H[29]. As a consequence, the standard multivariate normalkernel function is used: K(x) = (2π)−d/2 exp(−1/2xT x).In the following sections, details on how to compute thebandwidth for the Gaussian kernel are presented. A descrip-tion of the procedure is given first for the evaluation off

(drobs|tik

)(defined in Section 3.1) and second for the

evaluation of fW |TI,D(w|tik,dr

obs

)or fP |TI,D

(p|tik,dr

obs

)(Section 3.2).

A.1. Estimation of the kernel bandwidthwhen estimating f

(drobs|tik

)

Section 3 shows that the evaluation of P(TI = tik|D = dobs)only requires the evaluation of f

(drobs|tik

). As a conse-

quence, it makes sense to estimate the bandwidth basedon the points dr in MDS space in the vicinity of the low-dimensional representation of the data drobs. Points locatedfar from drobs will not impact the density f

(drobs|tik

); thus,

the choice of the bandwidth for those points has less impor-tance. As a consequence, only a neighborhood around theobserved data in reduced MDS space is considered to eval-uate the bandwidth Hd . A k-medoid clustering technique isused, where the observed data drobs is fixed as a medoid. Thenumber of clusters is defined using the silhouette index [17].The points belonging to the cluster containing the observeddata drobs are subsequently used to estimate the bandwidth.Silverman’s rule of thumb [21] is then used to estimate thebandwidth Hd for each dimension in the MDS space. Thebandwidth is defined as a diagonal matrix with diagonal ele-

ments Hii defined as√Hii =

(4

dMDS+2

) 1dMDS+4

n−1

dMDS+4 σi ,

where σi is the standard deviation of the ith dimensionof the points dr in the cluster containing drobs and dMDS

is the dimension of the MDS space. This bandwidth wasdemonstrated to be good estimation for the Gaussian ker-nel and Gaussian distributions. Even though the distributionof points in MDS is not necessarily Gaussian, tests haveshown that it is a valid estimation. The density f

(drobs|tik

)is then obtained in the following manner: f

(dr

obs |tik) =

1Ntik

tik∑j=1

KHd

(dr

obs − drj

), with Ntik representing the num-

ber of prior models for tik .Next, details on how to evaluate the density for the trend,

given the training image and the data are presented

A.2. Estimation of the kernel bandwidthwhen estimating fW |TI,D

(w|tik,drobs

)

or fP |TI,D(p|tik,drobs

)

The calculation of fW |TI,D (w|tik,dobs) differs from the cal-culation of f

(drobs|tik

)in two aspects. First, the distribution

of the variable W must be estimated, which is a probabilitydensity function (pdf) since W is a continuous variable. Inthe previous section, the density was only computed at a sin-gle value: drobs. Second, the distribution of W is conditionalto the additional variable Dr; hence, the joint space (Dr, W)must be considered.

The second point shows the need of the definition of ajoint bandwidth, one for dr and one for w The evaluation ofthe bandwidth for dr, denoted Hd , is evaluated in the exactsame manner that was described in the previous section.This is because, when evaluating fW |TI,D

(w|tik,drobs

), only

models in a close neighborhood of drobs are of interest. Sil-verman’s rule of thumb is applied to the points belongingto the cluster containing drobs. For the trend, it also makessense to use values of w that correspond to points dr that areclose to drobs, as only those values will be considered whenevaluating the density fW |TI,D (w|tik,dobs). The bandwidthfor the trend, denotedHw, is thus evaluated by applying Sil-verman’s rule of thumb to the values of w corresponding topoints dr in the cluster containing drobs. The kernel smooth-ing is subsequently applied in the joint space (Dr, W), usingthe bandwidth

Hd,w =(Hd 00 Hw

)

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