Comput Geosci (2011) 15:545–563DOI 10.1007/s10596-011-9223-9

ORIGINAL PAPER

A multi-resolution workflow to generate high-resolutionmodels constrained to dynamic data

Céline Scheidt · Jef Caers · Yuguang Chen ·Louis J. Durlofsky

Received: 19 May 2010 / Accepted: 4 January 2011 / Published online: 26 January 2011© Springer Science+Business Media B.V. 2011

Abstract Distance-based stochastic techniques haverecently emerged in the context of ensemble model-ing, in particular for history matching, model selectionand uncertainty quantification. Starting with an initialensemble of realizations, a distance between any twomodels is defined. This distance is defined such thatthe objective of the study is incorporated into the geo-logical modeling process, thereby potentially enhancingthe efficacy of the overall workflow. If the intent isto create new models that are constrained to dynamicdata (history matching), the calculation of the distancerequires flow simulation for each model in the initialensemble. This can be very time consuming, especiallyfor high-resolution models. In this paper, we present amulti-resolution framework for ensemble modeling. Adistance-based procedure is employed, with emphasison the rapid construction of multiple models that haveimproved dynamic data conditioning. Our intent is toconstruct new high-resolution models constrained todynamic data, while performing most of the flow sim-ulations only on upscaled models. An error modelingprocedure is introduced into the distance calculationsto account for potential errors in the upscaling. Basedon a few fine-scale flow simulations, the upscalingerror is estimated for each model using a clustering

C. Scheidt (B) · J. Caers · L. J. DurlofskyDepartment of Energy Resources Engineering,Stanford University, Stanford, CA 94305, USAe-mail: scheidtc@stanford.edu

Y. ChenChevron Energy Technology Company, San Ramon,CA 94583, USA

technique. We demonstrate the efficiency of themethod on two examples, one where the upscalingerror is small, and another where the upscaling error issignificant. Results show that the error modeling proce-dure can accurately capture the error in upscaling, andcan thus reproduce the fine-scale flow behavior fromcoarse-scale simulations with sufficient accuracy (interms of uncertainty predictions). As a consequence,an ensemble of high-resolution models, which are con-strained to dynamic data, can be obtained, but with aminimum of flow simulations at the fine scale.

Keywords History matching · Upscaling ·Error modeling · Distance-based techniques ·Uncertainty quantification · Kernel KL expansion

1 Introduction

Proper uncertainty quantification in reservoir perfor-mance requires, in an ideal situation, an ensembleof multiple high-resolution (sometimes referred to asfine-scale) reservoir models that are consistent withdynamic data (i.e., the models are history matched).High-resolution models often contain a level of detailthat is too fine for rapid subsurface flow simulation,which renders history matching or field optimizationsoverly time consuming. The problem is even morecomputationally challenging when an ensemble of high-resolution reservoir models is involved. To limit thetime required for history matching and optimization,two approaches are mainly applied in the literature:generating an ensemble of multiple low-resolution(sometimes referred to as coarse-scale) reservoir

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models, or using only a few high-resolution models.Neither of these approaches is fully satisfactory. In thefirst approach, upscaling techniques are often requiredto capture the impact of important fine-scale featuresin low-resolution models. However, the integrationof dynamic data into the low-resolution model maydestroy its consistency with the corresponding high-resolution model, which can result in a loss of geologicalrealism. The capability of such low-resolution modelsto predict future reservoir production and quantify itsuncertainty may, therefore, be called into question. Inthe latter approach, the construction and use of onlya few high-resolution models is often not adequate forproper uncertainty quantification.

The objective of this work is to present a workflowto rapidly generate multiple, high-resolution modelsthat are reasonably constrained to production data.This enables an efficient and realistic assessment ofuncertainty. If more precise history-matched modelsare required, the models constructed here can be usedas initial guesses in a history matching algorithm. Theproposed workflow uses distance-based ensemble mod-eling techniques, which have emerged in recent years[4, 20, 21]. As in other ensemble techniques (such asEnsemble Kalman Filter), distance-based techniquesfor dynamic data integration require a large initial en-semble of models and flow simulation on each memberof the ensemble. This may be computationally pro-hibitive for high-resolution models. In order to reducethe computational time requirement, a multi-resolutionframework is proposed. In this methodology, most ofthe flow simulations are performed on upscaled models,though the procedure still allows us to generate high-resolution models that are constrained (approximately)to available production data. In practice, some amountof error may be introduced in the upscaling step, espe-cially for complex flow systems. These upscaling errorsare accounted for in the workflow through a novel errormodeling procedure.

The novelty in this paper lies in the proposedworkflow, which combines various elements of previ-ous research in the areas of ensemble-level modeling[4, 20, 21], upscaling [8], ensemble history matchingwith distances [4, 15], and error modeling [10, 13, 14].None of these components in itself fully addresses thechallenges noted above, though the proposed workflowoffers a starting point for an overall solution. Wepropose a reformulation of distance-based ensemblemodeling techniques by introducing a multi-resolutionframework. This framework accounts for potentialerrors due to model coarsening by estimating upscalingerror and using it in the computation of the distancebetween models.

This paper proceeds as follows: we first review pre-vious research which is relevant to our workflow. Wethen describe the method developed in this paper, firstby presenting the essential elements of the distance-based technique employed to constrain models toproduction data, and then by introducing the multi-resolution workflow which includes an error modelingapproach. Finally, we apply the method to two differenttest cases and demonstrate its efficiency. We end thepaper with some concluding remarks.

1.1 Notation

The following notation will be used throughout thepaper:

– d: dimension of the mapping with Multi-Dimensional Scaling (MDS)

– L: total number of models (L is typically on theorder of 100)

– xi: a fine-scale reservoir model represented as avector

– xci : a coarse-scale reservoir model represented as a

vector– xi,d: location of model xi in MDS space– X = [x1,. . . , xL]T : the model matrix or the set of all

fine-scale models– e: upscaling error– g: forward simulation– dij: distance between fine-scale models xi and x j

– dcij: distance between coarse-scale models xc

i and xcj

– dc+eij : distance between coarse-scale models xc

i andxc

j taking into account upscaling errors– K: kernel matrix (L × L)

– ϕ: nonlinear function defining the transformationbetween the MDS space and the feature space

– �: matrix of models X mapped in feature space– y: a vector of independent random Gaussian

deviates

2 Distance modeling, upscaling and errormodeling—review

The workflow described below is based upon ap-proaches from three different areas: distance-basedmodeling, upscaling, and error modeling. We now dis-cuss relevant work on these three topics.

2.1 Distance-based modeling

In recent years, several papers have been publishedshowing the benefits of distance-based techniques

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applied to ensembles of models for uncertainty quan-tification, model selection and history matching [2, 4,15, 20, 21]. Scheidt and Caers [20] show how a dis-tance which is well correlated to the difference in flowresponse can be used to perform uncertainty quan-tification efficiently. In their examples, they employeda fast streamline simulation to evaluate the distancesbetween models. In the context of dynamic data inte-gration, Suzuki and Caers [23] employed a Hausdorffdistance to successfully perform history matching usingsearch algorithms, such as a neighborhood algorithm ortree-search algorithm. Park and Caers [15] presented adistance-based Ensemble Kalman Filter (EnKF) tech-nique, which was shown to improve upon the classicalEnKF technique. A metric space was introduced, whichis obtained from a distance (difference in tracer simu-lation results) between any two models. More recently,Caers et al. [4, 17] presented a post-image inverse prob-lem formulation and solution. In this approach, froman initial set of models, new models are constructedwhich are better constrained to production data thanthe initial set. The solution of the post-image problemis used as an illustrative method in the work presentedin this paper, and will be discussed in more detail later.Note that any other distance-based history matchingtechnique may be used in the proposed framework.

2.2 Upscaling techniques

Upscaling techniques are widely used to enable flowsimulations at reasonable computational cost. A varietyof upscaling techniques exist (e.g., see [8] for a recentreview), which can be classified as single-phase pa-rameter upscaling or multiphase parameter upscaling,depending on the quantities that are upscaled. The useof upscaled absolute permeability (or transmissibility)represents the most commonly applied coarsening tech-nique in practice. For cases with large upscaling ratios,the use of multiphase upscaling (e.g., upscaled rela-tive permeabilities) or nonuniform (flow-based) coarsegrids may be required to capture the transport of in-jected fluid and its effect on system mobility. Basedon the size of region over which the flow problem issolved in the upscaling calculations, upscaling meth-ods can also be classified as local, quasi-global, andglobal approaches, in order of increasing accuracy andcomputational cost. In addition, specialized upscalingin the vicinity of wells can often improve the accuracyof coarse models. More detailed discussion of a vari-ety of single-phase upscaling approaches can be foundin [6, 8].

The workflow developed in this paper is not specificto a particular upscaling technique, as our multi-

resolution framework can accommodate a wide vari-ety of upscaling procedures. As discussed earlier, theupscaling process in general may introduce bias andrandom errors depending on the complexity of the flowproblem and the specific upscaling method applied. Inthis work, we develop an error modeling procedure toappropriately account for these errors. Next, we reviewprevious work on modeling upscaling errors.

2.3 Error modeling

A number of previous researchers have introducedtechniques to estimate upscaling errors. In [10], a pro-cedure for variance correction was introduced, but biascorrection (i.e., a systematic shift in the curves) was notdiscussed. Omre and Lødøen [13] presented an errormodel which takes into account bias and changes in er-ror structure. The basic idea is to use coarse-scale fluidflow simulations to predict the results from the fine-scale fluid flow simulation. The correction is sufficientlyaccurate if the coarse-scale simulations capture themost important features of the fine-scale simulations.O’Sullivan and Christie [14] proposed an error modelwhich contains the covariance between fine and coarse-scale simulation results to account for the temporal cor-relation in the error. This error covariance is includedin the calculation of the misfit function. Both Omre andLødøen [13] and O’Sullivan and Christie [14] showedthat when the error is modeled properly, an ensem-ble of coarse-scale models can provide a reasonableapproximation to the fine-scale flow response. In ourwork, we develop an error expression similar to that inO’Sullivan and Christie, which accounts for the errorover the time of the flow simulation.

3 Overall methodology

History matching, or in more general terminology, pro-duction data integration, is an inverse problem thatshould be solved through Bayes’ rule if the aim is toquantify posterior uncertainty, i.e., uncertainty in thereservoir after history matching. According to Bayes’rule [24], this requires formulating a prior, which inour framework is done by generating a set of initialmodels that match any data available prior to inte-grating production data. In general, this prior includesgeological information constraining the style of geolog-ical heterogeneity (typically variogram, training imagesor Boolean models) as well as well-log, core and seis-mic data. Bayes’ rule requires formulating a likelihoodwhich captures the precision by which the data shouldbe matched. Knowing the prior and likelihood, the

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posterior can be constructed and samples can be drawnfrom it.

In order to implement the data integration proce-dure in practice, we must determine a proper parame-terization of the prior model such that it can easily beupdated into a posterior. In Bayes’ rule, the “prior” isnot a single model but an ensemble of models, in factan entire probability density function. Hence, we seeka stochastic parameterization for the prior, preferablyone that can handle a large variety of modeling ap-proaches (variogram-based, Boolean, multi-point, etc).Then, using optimization techniques, this parameteri-zation enables us to update the prior into the posterior.

In this paper, we employ a distance-based modelingtechnique that provides a stochastic parameterizationof an ensemble of reservoir models. The problem ofupdating the prior into the posterior is formulated asa post-image problem [4, 17], which is analogous to thepre-image problem described in the computer scienceliterature [22]. The distance forms the basis for theensemble parameterization and ensemble expansionformulation. In addition, the post-image solution em-ploys the classical Karhunen–Loeve expansion (KLE)and kernel-based techniques using a distance matrix,which allows new models to be generated and/or ad-justed to new data [19]. Note however that the post-image problem is not an optimization procedure that isdesigned to perfectly match available dynamic data, butrather a data integration method which will improvethe match. The next section provides a thorough reviewof this approach; a full account is published as a bookchapter in [4] and is also available in [17].

3.1 Constraining ensemble models to productiondata—a distance approach

The parameterization used in the proposed method isbased on the KLE. However, instead of a traditionalcovariance-based expansion, a distance-based KLE isformulated. We first review existing parameterizationsrelying on KLE (e.g., [18]) and then show the benefitsof a distance-based kernel KLE for constraining modelsto production data.

3.1.1 Standard KLE

The KLE formulation relies on the eigenvalue de-composition of the spatial covariance matrix C of themodels being simulated, namely C = W�CWT , withW = [w1,...,wN] and �C = [λ1,...,λN] the eigenvectorsand eigenvalues respectively of C and N the number ofgrid blocks in each model (all models have the same N).Note that if C is calculated as the ensemble covariance

matrix of L models, then only the first L eigenvaluesare nonzero. A correlated Gaussian variable x can thenbe simulated from a set of standard Gaussian deviatesy :x = W�

1/2C y (refer to [12]).

This parameterization assumes that the random fieldx is Gaussian, thus it is not appropriate for non-Gaussian models. In the latter case, the nonlinear formof the KLE is preferred, which introduces the notion ofkernels, and is denoted kernel KLE.

3.1.2 Kernel KLE and the pre-image problem

The concept of kernel KLE is similar to the stan-dard KLE described previously, except that a high-dimensional function ϕ : xi �→ ϕ (xi) ⇒ X �→ � is usedto first transform points from the initial space to a high-dimensional space called the feature space. The KLE isthen performed in the feature space:

ϕ(x) = �b with b = 1√L

VKy (1)

with:

– � representing the matrix of the initial ensemble ofL models, projected in the feature space

– VK representing the eigenvectors of the kernelmatrix K, which is defined as K = �� T

– y being a standard Gaussian vector.

With kernel KLE, the generation of a new randomGaussian vector y allows for the creation of new mod-els in the feature space [18]. Once a model is thusdefined in the feature space, we need to determinethe actual physical values corresponding to this model.This determination entails the solution of the so-calledpre-image problem. This consists of finding a modelrealization x by inverting the function ϕ. The solution tothe pre-image problem is accomplished by minimizingthe difference between the given feature expansion �band ϕ(x̂new) (a fixed-point optimization is applied in[18, 22]). The minimization of ‖ϕ(x̂new) − �b‖ for aGaussian kernel results in the new model x̂new being anonlinear combination of the existing prior models inthe ensemble:

x̂new =L∑

i=1

βopti xi with

βopti = bik′ (x̂new, xi

)

L∑i=1

bik′ (x̂new, xi) and

L∑

i=1

βopti = 1, (2)

Comput Geosci (2011) 15:545–563 549

where b = (b 1,. . . ,b L) is defined as in Eq. 1 and k′ isthe derivative of the kernel function k with respect toits argument. For more details refer to [22].

Note that the weights βopt depend on the mod-els xi, hence this constitutes a nonlinear combinationof models. In the context of history matching, theobjective function is then minimized using gradient-based techniques to optimize the vector of standarddeviates y.

The approach described above has the followinglimitations:

– The parameterization relies on kernel definitionsprovided in the literature [22] such as a polynomialkernel [18], which may not be informative withrespect to the objective of the parameterization.

– Solving the pre-image problem in high-dimensionalspace (the space of the models) is not trivialfor high-resolution models with a large numberof cells.

– To constrain any given model to production data,many evaluations of the forward simulation g (flowsimulation) are required, which may render themethod prohibitive in terms of CPU.

The next section is devoted to distance-based kernelKLE, which has been introduced by Caers et al. [3].We will see how distance-based kernel KLE is well-suited to our problem and can address the limitationsmentioned above.

3.1.3 Distance-based kernel KLE and the post-imageformulation

Application-tailored parameterization To generatenew models that integrate dynamic data, Caerset al. [3, 4] proposed making the parameterizationof the initial ensemble of models dependent on theproduction data itself. This is a typical though criticalstep in distance-based approaches which distinguishesthem from other existing ensemble approaches[9, 18].

The kernel (a measure of dissimilarity by itself) istailored for dynamic data integration by employing adistance that is dependent upon the production data.Flow simulations are used to compute the distance be-tween any two model realizations from the prior [4, 17].It is important to note that the distance between eachmodel and the production data (denoted data below)is also computed. This is equivalent to the traditional

objective function as applied in standard history match-ing procedures:

dij = d(g (xi) , g

(x j

)) ∀i, j and

di,data = d(g (xi) , data

) ∀i (3)

where g represents the forward model (flow simulator).Examples of expressions of distances are given in thetest cases below.

Note that unlike other distance-based methods, thepost-image problem requires the use of a distancewhich allows for the comparison of the forward modelto data. Therefore, distances such as the Hausdorff dis-tance cannot be used, as in uncertainty quantification.

It should be also noted that calculating the distancesis the most costly step in the distance-based KLE ap-proach. However, in the workflow, the large majorityof flow simulations are performed in this step; few ifany additional simulations are subsequently requiredusing this approach. We note that clustering techniquesor other procedures can be applied to further limit thenumber of flow simulations [20, 21].

The use of distance-based kernel KLE offersthree main advantages compared to traditional historymatching techniques:

– The use of the dissimilarity distance intrinsicallyintegrates flow information within the definition ofthe kernel and thus within the parameterizationEq. 1. In this way, the current procedure differsfrom other KPCA-based history matching methods,such as that of Sarma et al. [18], in which thedynamic response was not included in the kernel.

– In addition to accounting for the distance betweenmodel responses and production data (equivalentto the objective function used in traditional historymatching), the parameterization also accounts forthe distance between the different flow simula-tion models. This information, which is commonlyignored in standard history matching procedures,may result in improved efficiency for the currentapproach.

– The definition of a flow-based distance betweenmodels allows for a drastic reduction of the di-mensionality of the space in which the historymatching is performed. This dimensionality reduc-tion is based on the variability of the responsebetween flow simulation models, not on the vari-ability in the geological model properties. By usingMulti-Dimensional Scaling (MDS) with the dis-tances defined in Eq. 3, we can represent models

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in an extremely low-dimensional space (e.g., 2–5dimensions) instead of a high-dimensional space(the model space or perhaps a lower-dimensionalspace using PCA or KPCA on the model prop-erties). We now describe the Multi-DimensionalScaling for dimensionality reduction.

Dimensionality reduction: Multi-Dimensional Scaling(MDS) MDS is a statistical tool that creates a low-dimensional (dD) map of the object locations given aset of distances between objects (such as prior mod-els), where the distance is given in Eq. 3. The modellocations xd,i represent models xi in dD space and thelocation of the “truth data” xtrue

d represents the “true”(unknown) model in dD. Note that d is usually small(less than 10), reducing considerably the dimension ofthe problem. Figure 1 shows an application of MDS forthe second test case presented below. For more detailson MDS, refer to [4, 20].

By using the locations of the models (given by MDS)instead of the models themselves, the objective functioncan be expressed in a simpler and much less costlymanner. Specifically, we now seek to find new locationsxnew

d which minimize the distance to the location of the“truth” case xtrue

d . Given the model expansion in fea-ture space Eq. 1, the minimization of this distance canbe formulated as the following optimization problem[4, 17]:

yopt = arg minynew

d(xnew

d , xtrued

)with

ϕ(xnew

d

) = �1√L

VKynew, (4)

with:

– � representing the locations of the initial L modelsprojected in the feature space

– VK representing the eigenvectors of the distance-based kernel matrix K, which is defined asK = ��T

– ynew being a standard Gaussian vector– xnew

d and xtrued representing the locations of the new

and truth models respectively.

The problem defined in Eq. 4 is referred to in [4, 17] asthe post-image problem. Note that Eq. 4 does not de-pend on the transfer function g; thus it can be evaluatedefficiently, in contrast to other kernel KLE approaches[18]. The objective function is only dependent on thelocation of the “truth” in the MDS space, which isknown. Note also that Eq. 4 assumes the relationshipbetween the distances derived from the location of themodels and the distances obtained from flow responseto be linear/proportional, i.e.:

d(xd,i, xd, j

) ∝ d(g (xi) , g

(x j

)) ∀i, j,

hence d(xd,i, xtrue

d

) ∝ d(g (xi) , data

). (5)

Given a solution yopt for Eq. 4, a model expansionis generated in the feature space. To obtain multiplesolutions, we apply a global stochastic optimizationtechnique on y. A suitable technique is gradual defor-mation [11] since it searches in the space of Gaussianrandom variables. The creation of multiple y leads tomultiple sets of weights, hence multiple new models.

Solving the post-image problem Eq. 4 requires solv-ing a pre-image problem, i.e., finding optimal weights

Dissimilarity Distance Matrix D

1 2 3 ... data

1 0 d12 d13 ... d1data

2 d21 00 d23 ... d2data

3 d31 d32 0 ... d3data

... ... ... ... ... ...

data ddata1 ddata2 ddata3 ... 0

Model 1 Model 2

Model 3

2D projection of Metric Space

MDS

Single reservoir modelDefined by dissimilarity distance

“truth”data : data

d12

d13 d32

Fig. 1 Multi-dimensional scaling. Examples of a few models, distance matrix derived from the models and 2D space showing thelocations of the models

Comput Geosci (2011) 15:545–563 551

βopti to generate new model locations xd,i for each

y. This enables us to compute d(xnew

d , xtrued

). The

pre-image problem is solved in the low-dimensionalMDS space and thus converges rapidly. This is a keydifference with other kernel KLE frameworks, wherethe pre-image is solved in the space of models, and thusconvergence might be more challenging.

The final step is the reconstruction of the actual“image” xnew or model realization given the knowledgeof its location xnew

d . To obtain xnew, Scheidt et al. [19]employ an unconstrained optimization, which consistsof applying the weights β

opti defined by the pre-image

solution Eq. 2 to the model realizations themselves:

xnew =L∑

i=1β

opti xi.

Note that if Eq. 5 is not satisfied, then the solutionof Eq. 4 will not result in d

(xnew

d , xtrued

) = 0. The so-lution in this case, as with any nonlinear optimizationproblem, is to take the solution of Eq. 4 as an addi-tional initial model and then reapply the workflow. Thisprocess is iterated until a sufficiently accurate solutionis found.

It is important to emphasize that the solution of thepost-image problem is not designed to produce pre-cisely history-matched models (if Eq. 5 is not satisfied).The purpose is rather to provide rapid quantification ofuncertainty using models that are approximately con-strained to the dynamic data (an accurate assessment ofuncertainty does not require every model to preciselymatch the historical data). These models can also beused as input to a detailed history matching procedureif this is necessary.

There are several important points to note:

– The initial ensemble of models (the prior) needs tobe large enough (as in EnKF or any KLE method)for the parameterization to be effective.

– Solving the post-image problem requires perform-ing a flow simulation for each initial model.

– In theory, the solution of the post-image prob-lem does not require any new flow simulations. Inpractice, however, flow simulation will typically beperformed on newly constructed models.

– Strong nonlinearities in the flow problem may re-duce the accuracy of the history matching. General-ization of the method to handle such systems andchallenging geological models is currently underinvestigation [16].

– The new models generated are a nonlinearlyweighted sum of the existing models, since theweights are a function of the locations of the models(xd,i). Hence, the method is best suited for contin-uous models (Gaussian or non-Gaussian distribu-

tions). A generalization of the method to discrete(facies) models is proposed in [17].

– The weights are a function of the distance of theexisting models to data, as well as the distancebetween models.

3.1.4 Comparison of post-image solutionwith Ensemble Kalman Filter

While the goal of this paper is not to compare or test theefficiency of the post-image solution to the EnsembleKalman Filter method (EnKF), it is worth pointingout some differences in these two approaches. EnKFhas been shown to be capable of integrating historicaldata starting from an ensemble of reservoir models[9]. See [1] for a thorough review of EnKF. EnKFupdates models efficiently since it requires only oneforward simulation per model per update, as opposedto potentially hundreds for conventional methods. BothEnKF (at least the most common implementations) andthe post-image solution start with an initial ensemble ofmodels and perform as many flow simulations as thereare models. However, there are notable differences inthe manner in which the models are updated. EnKFupdates models from the initial set at each update stepusing successive linear combinations of models. Thecoefficients in the linear combination are defined fromthe error in prediction obtained by flow simulationsat each update step. The post-image solution uses theinitial simulations to parameterize the ensemble. Incontrast to EnKF, it constructs new models that in-tegrate dynamic data (after flow simulation has beenperformed up to the last update step) using a nonlinearcombination of initial models.

The ensemble techniques for data integration men-tioned above (distance-based KLE or EnKF), requireperforming a flow simulation for each initial model,which may be too expensive computationally for high-resolution models. To address this, we can define adistance which requires less computation to evaluate,as we now describe.

3.2 Adaptation of distance-based modelingto multi-resolution flow simulations

Figure 2 illustrates the key concepts of the workflowproposed in this paper. In the previous section, weshowed that the distance-based approach to ensem-ble modeling requires the evaluation of a forwardsimulation g for each model in the initial ensemble(Fig. 2a). For high-resolution models, evaluating g forevery model may be CPU prohibitive. In order toreduce the computation associated with calculation

552 Comput Geosci (2011) 15:545–563

0 200 400 600 800 10000

50

100

150

200

250

300

350

400

TIME (days)

Wat

er P

rodu

ctio

n at

P1

1 ..

data

Upscaling

Initial Ensemble

Coarse-scale

Simulation

WI*kx* ky*

1e

2e

Distance: cijd

Distance: ecijd +

Coarse & Fine scale Simulations

Post-Image problem

MDS & Clustering

(a) (b)

(d)(e)

(f)

(c)

Fig. 2 Illustration of the multi-resolution workflow: a constructan initial ensemble of models, b upscale to generate coarsemodels, c perform coarse-scale simulations—in red is the “truth”

production data, d create MDS map and perform clustering inthe MDS space, e evaluate upscaling errors from a few fine-scalesimulations, f generate new models using the proposed technique

of this distance, we propose the use of coarse-scalesimulations instead of fine-scale simulations to definethe distance. Thus, we need to generate coarse-scalemodels xc

i , i = 1, . . . , L from the fine-scale modelsxi, i = 1, . . . , L (Fig. 2b). Any upscaling technique canbe used; we refer to [6, 8] for discussion of differentupscaling techniques.

3.2.1 Accounting for upscaling errors

Upscaling will, in general, introduce errors in thecoarse-scale solution. The magnitude and type of errorsobserved depend on the complexity of the flow problemas well as on the specific upscaling method employed.As a consequence of these errors, flow responses forthe coarse-scale models will differ from those of thecorresponding high-resolution models. Here, we pro-pose to compute the potential error e introduced bythe upscaling procedure and to incorporate it in thecalculation of the distance. We thus approximate thefine-scale distance by running coarse-scale simulations

and including an estimate of the upscaling error. Thedistance is then defined as follows:

dc+eij = d

(g(xc

i

) + ei, g(xc

j

) + e j) ∀i, j

where ei = g(xi) − g(xc

i

). (6)

The key question here is how to evaluate the error inthe upscaling procedure without performing fine-scalesimulations for all models.

Our approach is based upon the intuitive assumptionthat models with similar flow responses at the coarse-scale possess similar upscaling errors. In other words,we assume that the upscaling error is not random buthas a structure that is correlated with the coarse-scaleflow response. As a consequence, if we group modelswith similar coarse-scale responses, we can estimatethe error from a prototype model in that group (e.g.,the model nearest the cluster center), and associate thesame error to other models in the cluster.

We employ a k-means clustering technique [20] togroup models with similar coarse-scale responses in

Comput Geosci (2011) 15:545–563 553

order to estimate the error e. Again, the procedure usesthe concept of distance. This distance is a function ofcoarse-scale flow response only and is defined as:

dcij = d

(g(xc

i

), g

(xc

j

)) ∀i, j. (7)

The clustering results in N groups of models withsimilar responses, where N is a fraction of the totalnumber of coarse-scale models. We select the N coarse-scale reservoir models that are closest to the centroidsof the clusters (the cluster centroids are shown assquares in Fig. 2d). The corresponding high-resolutionmodels are then simulated. Having evaluated the flowresponses for all N models, the upscaling error (Fig. 2e)for each selected model can be readily determined asek = g (xk) − g

(xc

k

), k = 1, . . . , N . Note that the error

is computed for each time step of the simulation, withT representing the total number of time steps. Thus, wehave ek = (

ek,1, . . . , ek,T)

.Since the clustering technique groups models with

similar flow responses [20], we consider that each clus-ter Ck has a single upscaling error ek. Each modelwithin a cluster therefore has the same upscaling error:ei = ek if i ∈ Ck, i = 1, . . . , L, k = 1, . . . , N. This ap-proach appears to suffice for the examples consideredbelow. An alternative approach would be to perform aninterpolation (inverse distance weighting) to estimatethe upscaling error for the models where no fine-scalesimulation is performed.

Note that the error is computed from only a few fine-scale simulations. If the estimation of the error is not

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554 Comput Geosci (2011) 15:545–563

sufficiently accurate, new fine-scale simulations can beiteratively added in order to obtain a more accurateestimation of e for each cluster.

3.2.2 Application to distance-based multi-resolutionmodeling with ensembles

Once we have a representation of the upscalingerror for each coarse-scale model (and thus of the dis-tance between fine-scale models), the distance-basedtechnique can be applied using the modified expres-sion for the distance (Eq. 6). New fine and coarse-scale models are constructed by solving the post-imageproblem using the weights defined in Eq. 2. To verifythe accuracy of the newly generated models (Fig. 2f),flow simulation is performed on the high-resolutionmodels. Given the new flow simulation on the high-resolution model, we can update the estimation of theerror by upscaling the model and comparing the fineand coarse-scale solutions. This procedure can be it-erated to ensure that the upscaling error is modeledcorrectly. We note finally that the multi-resolutionframework applied here could be used with otherdistance-based history matching techniques, such as thedistance-based Ensemble Kalman Filter [15].

It is important to note that one advantage of thismethod is that coarse and fine-scale models are gen-erated simultaneously and both are potentially con-strained to the production data. We are not, however,primarily interested in the coarse-scale models. Coarse-scale models are simply used as an intermediate stepto calculate the distances. Our objective is to generatehigh-resolution models that are constrained (as much aspossible) to production data, without performing manyfine-scale simulations.

The entire workflow, without considering potentialiterations, is presented in Fig. 2.

4 Test cases

In this section, two test cases involving water injectionare presented. The first example shows an applicationof the multi-resolution framework for a case where anadvanced upscaling technique is applied and accuratecoarse-scale models are obtained. The second exampleemploys a standard upscaling algorithm and exhibitssignificant upscaling errors. For each case, the perme-ability field (generated using a variogram-based algo-rithm) is the only property that varies between differentmodels.

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4.1 Case where minor upscaling errors are present

For the first test case, an initial ensemble containing300 models is considered. The fine-scale models are 2D,non-Gaussian, and contain 51 × 51 grid blocks (physicaldimensions are 1020 ft × 1020 ft). The models were gen-erated using sequential Gaussian simulation [7], witha histogram transform. They share the same histogramand variogram (with dimensionless correlation lengthsof lx = 0.7, ly = 0.5, and an orientation of 30◦) and arenot constrained to any well data. Some of the fine-scalerealizations are shown in Fig. 3a. We consider a five-spot pattern. The injection well is operated at a constantinjection rate and the producers are specified to havefixed bottom hole pressures. The water–oil end-pointmobility ratio (M) for this case is 2. Simulations are per-formed using a standard, commercial flow simulator.

Adaptive local-global upscaling [6] was applied tocoarsen the models to 17 × 17 grid blocks. This methoduses (coarse) global information in the specification ofthe local boundary conditions applied in the upscal-ing, which enables it to provide more accurate coarsemodels than a standard (local) upscaling procedure.Upscaled transmissibilities in the x and y directions

are generated. Some of the coarse-scale models (trans-missibility in the x direction) are shown in Fig. 3b.Even though we only employ a single-phase upscalingtechnique, the resulting coarse-scale models are of ahigh-level of accuracy and almost no error is observedbetween the fine and coarse-scale simulations (seequantiles, Fig. 3c). Figure 3c shows the estimated P10,P50, and P90 quantiles of the field water rate, for fine(solid line) and coarse (dashed line) simulations. Notethat in many cases, performing fine-scale simulations onall models would not be feasible, and thus the fine-scaleP10, P50, and P90 values would not be known.

The objective is to construct new, high-resolutionmodels that are constrained to the water injection andproduction rate at each well. The models could addi-tionally be constrained to oil production rate, and simi-lar results were obtained using either water productionrate or water and oil production rates. The distanceemployed is defined as follows Eq. 6:

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Fig. 7 “Truth” model,E-Type (mean) and varianceof the ten newly constructedmodels

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model i, well iw, and time t

The post-image solution described previously is thenapplied using the above distance, which is a function ofcoarse-scale simulation results only. Ten new fine-scalemodels are generated using the optimal set of weights,

generated by solving the post-image problem. To illus-trate the potential of the methodology, we performedfine-scale simulations on those ten models (Fig. 4),although this is not required in practice. Figure 4ashows the location of the models in the MDS map, thecross representing the “truth” case as mapped using thedistance above. Note that we show a 2D projection ofthe MDS map, although in reality this space is in 5D.Figure 4b shows that the water production rate curvesat each well for the new models (red curves) are closer

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Comput Geosci (2011) 15:545–563 557

to the reference production (green dots) than the waterproduction rate curves from the initial ensemble (graycurves).

Even though the new models do not match exactlythe reference production data, they are constrainedto dynamic data and can be used to perform furtherhistory matching or uncertainty quantification in futureproduction. If an improved match is desired, more iter-ations can be performed. The improvement in objectivefunction of the newly generated models is also shownin Fig. 5, where the histogram of the objective functionshows much smaller values for the new models (red)than the initial ones (blue). A different way to illustratethe efficiency of the method is to map the locations ofthe new models (blue “X”) in the MDS space (Fig. 5b).All the new models are located around the truth case(green “X”).

The method is able to create many different modelsthat are constrained to the same dynamic data. In or-

der to maintain uncertainty, the models should showsufficient variability. To illustrate this last point, weshow in Fig. 6 three of the newly constructed models.We observe that the images look very different; vari-ability is therefore preserved in the new models.

Figure 7 represents, respectively, the referencemodel (i.e., the high-resolution model that was usedto generate the truth production profile), the ensembleaverage (E-type), and ensemble variance derived fromthe ten new models. Low values of permeability areevident in the lower portion of the E-type image, nearwells P1 and P4 (which are located at the lower cornersof the model, see Fig. 3). Although permeability at P1(lower left) is relatively high in the reference realiza-tion, low values in the E-type model result because,in the reference model, well P1 does not produce anywater, and is thus associated with low permeabilityvalues. The same is true for well P4 (lower right). Onthe other hand, wells P2 and P3 do produce water, and

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2D projection of MDS space –use of dc 2D projection of MDS space –use of dc+e

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Fig. 9 a MDS space created from distances using the coarse-scale simulations. The fine-scale models are selected from kernelk-means in this space. b MDS space created from distances

using the coarse-scale simulations and error expressions. Thepost-image solution is applied using this space. c Water injec-tion/production profiles for each well

558 Comput Geosci (2011) 15:545–563

the high permeability values in the upper portion of thereference model that cause this are captured in the E-type image.

4.2 Case where significant error in upscalingis observed

The second test case involves a 3D model, with 100realizations of log-normally distributed permeabilityfields generated using sequential Gaussian simulation[7]. The models cover a volume of 550 ft × 550 ft ×125 ft and contain 55 × 55 × 25 fine-scale grid blocks.Model properties are not constrained to any well data.The permeability fields are characterized by dimension-less correlation lengths lx = ly = 0.5, lz = 0.1 and σ =1.73, where σ 2 is the variance of log k. Three of the fine-scale models are shown in Fig. 8a.

The coarse-scale models are generated using stan-dard local permeability upscaling. No multiphase para-meter upscaling or near-well upscaling is used in thiscase and the fine-scale models are uniformly coarsenedto 11 × 11 × 5 (Fig. 8b). We use Stanford’s GeneralPurpose Research Simulator GPRS [5] to simulate boththe fine and coarse-scale models. Five wells are con-sidered (as shown in Fig. 8a), with both the injectionand the production wells specified to operate at fixedbottom hole pressures. The water–oil end-point mo-bility ratio M is 50. In this example, because we areusing a simple upscaling procedure in conjunction witha relatively large coarsening factor and a high fluid mo-bility ratio, we expect the coarse-scale models to displaysome error relative to the fine-scale solutions. Figure 8cshows the estimated P10, P50, and P90 quantiles of thefield water rate for fine and coarse simulations. Cleardiscrepancies between the fine and coarse models are

“Truth” model

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(c)

Fig. 10 a MDS space created from distances using the coarse-scale simulations. The fine-scale models are selected from kernelk-means in this space. b MDS space created from distances

using the coarse-scale simulations and updated error expressions.The post-image solution is applied using this space. c Waterinjection/production at each well

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Fig. 11 Water injection and production rates at each well

Fig. 12 a Histogram of theobjective function (distanceto the reference) for theinitial ensemble of models(red) and the new set ofmodels (blue). b Locations ofthe new models in MDS space

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evident. Similar observations are obtained for otherquantities, such as oil production rate.

The objective is to construct new high-resolutionmodels that are constrained to water injection and pro-duction rates (as in the first case, oil rate could also beemployed). To take into account the errors in upscaling,the distance is defined as (from Eq. 6):

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)+ eiw

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)2

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(xiw

i,t

)defined as before

– eiwi,t the upscaling error evaluated for model i, well

iw, and time t.

In order to estimate the upscaling errors, a small set offine-scale simulations is required. We performed fivefine-scale simulations, chosen by means of k-meansclustering as described earlier. These models are rep-resented as squares in Fig. 9a. After the fine-scalesimulations are performed, the errors are evaluated,and a new MDS space is constructed using the distancedefined above, where the historical data can be mappedas well (the “truth” is located at the green “X” shownin Fig. 9b).

The post-image solution is applied to construct newmodels that are constrained to production data. Ahistogram transform is applied to ensure that the newmodels are characterized by the same histogram as theinitial models. Results are presented in Fig. 9c. In grayare the water injection and production curves for theinitial ensemble of models. The green dots representthe injection and production data that is being matched(the truth). The red curves represent the water injectionand production curves from a model generated usingthe post-image formulation accounting for upscalingerrors. It is evident that the estimation of the waterproduction for the injection well I1 and the productionwell P1 are not accurate. The next step therefore is

to iterate and obtain a new estimation of the error.The newly created fine-scale model is then upscaledusing the same technique as for the original models,and a flow simulation is performed on the coarse-scale.The upscaling error is subsequently updated using thisadditional information.

For this case, the additional information is adequateto obtain a new model with sufficient accuracy, asshown in Fig. 10. The update of the error using the newfine and coarse simulations generated from the firstiteration results in an improved description of the error,and a better result from the post-image.

We next generate nine more models using the same(updated) error model to obtain a total of ten high-resolution models constrained to the dynamic data.Figure 11 shows the injection and production ratesfor each well for the initial ensemble and the newlygenerated models (red).

Although the new high-resolution models do notmatch the historical data perfectly, they show clearimprovement compared to the initial ensemble. Thiswas achieved by performing a total of only seven fine-scale flow simulations (five for the clustering and twoin the iterations). This is also illustrated in Fig. 12,where the histogram of the objective function (distanceto the “truth”) is shown for the initial models (red)and the newly generated models (blue). Figure 12bshows the locations of the new models (blue X’s) inthe MDS space. We observe that the new models areconcentrated around the true model (green X).

Figure 13 shows some of the new models generatedby the workflow. Note that the models appear quite

Fig. 15 Locations of newwells (in red)

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Comput Geosci (2011) 15:545–563 561

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Fig. 16 Water injection and production rates for each well for the ten new (high-resolution) models

different, although they are constrained to dynamicdata. This is important for uncertainty quantification.Figure 14 shows the “true model,” i.e., the model whose

production data was chosen as a reference, the E-type(mean) and the variance of the ten new models. We ob-serve that the mean of the new models reproduces some

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562 Comput Geosci (2011) 15:545–563

of the features observed on the true model, especiallynear the wells (corners). In particular high permeabilityvalues are present at the bottom corner of the model(well P1), which is expected since well P1 has significantwater production.

Constructing models that integrate dynamic data isnot the final goal. Rather, we wish to use these modelsto provide predictions (and associated uncertainty) offuture reservoir performance. To measure the perfor-mance of the new models, we compare their predictionof water production at four new wells with predictionsusing the initial ensemble. Wells P5, P6, P7, and P8are added to the reservoir after 500 days of production(well locations are shown in Fig. 15), and wells P1,P2, P3, and P4 are shut-in at this time. Results forwater production rate for the new wells, in the ten newmodels, are shown in Fig. 16. These predictions arereasonable compared to the reference (green dots), andshow a degree of uncertainty. Note that in this example,the reference data is known because it is simulatedusing a reference model derived from the same prior.

Figure 17 shows the predictions obtained from thefive initial prior models that are the closest to thereference data (and are of similar distance to the ref-erence as the newly generated models). This provides areference for the uncertainty in the predictions. Com-parison of Figs. 16 and 17 shows that the new modelsgenerated from our workflow generally retain the ap-propriate degree of variation in the predictions. Thus,the proposed workflow does not artificially reduce theestimate of uncertainty, which is a desirable quality inhistory matching procedures. Note that the predictionsfor wells P5 and P7 for each set (new models and initialmodels) have the same behavior, over-prediction forwell P5 and under-prediction for well P7. There is somediscrepancy, however, in the predictions for well P8.

5 Concluding remarks

In this paper we presented a new multi-resolutionframework to constrain an ensemble of models to dy-namic data. Starting with an initial set of geologicalrealizations, we demonstrated how to construct newhigh-resolution models that are constrained to dynamicdata by performing flow simulations on the associatedupscaled models. The method employs distance-basedtechniques, where the distance between any two mod-els is defined from coarse-scale simulations. An errormodeling procedure is used in the calculation of thedistance to account for potential errors in the upscaling.Variability between new models is preserved, which isimportant for the quantification of uncertainty in future

production. The method is general in that it can handlesimulation errors through the updating of the distancefunction. In this study, these errors resulted from theuse of a simplified upscaling procedure, though er-rors due to the use of other proxy models could alsobe treated. The method can be applied not only tothe post-image problem, but also to other distance-based techniques, such as the distance-based EnsembleKalman Filter [15].

We presented two example cases that involved vary-ing degrees of upscaling error. In both cases, the useof distances based on coarse-scale simulations was suc-cessfully used to construct high-resolution models thatare constrained to dynamic data. In the 3D example,standard permeability upscaling was applied, and up-scaling errors were significant. For this case, perform-ing an additional fine-scale simulation for the initialhistory-matched model enabled us to update the errormodel. This was shown to lead to improved accuracy inthe history-matched models.

The multi-resolution framework should be testedusing alternative distance-based methods for the dataintegration. Future work should also include appli-cation of the method to cases with more complexgeology and flow systems (e.g., channelized systems,three-phase flow), in which the upscaling errors may bemore significant and less systematic.

Acknowledgements We thank Chevron Energy TechnologyCompany for funding this research. A portion of this work waspresented at the 12th European Conference on the Mathematicsof Oil Recovery (held in Oxford, UK, September 2010).

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