SPE-170636-MS

Integration of Principal-Component-Analysis and Streamline Informationfor the History Matching of Channelized Reservoirs

C. Chen, Shell International Exploration and Production; G. Gao, Shell Global Solutions US Inc.; J. Honorio, MIT;

P. Gelderblom, Shell Global Solutions International; E. Jimenez, Qatar Shell GTL Limited; T. Jaakkola, MIT

Copyright 2014, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in Amsterdam, The Netherlands, 2729 October 2014.

This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contentsof the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflectany position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the writtenconsent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations maynot be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract

Although Principal Component Analysis (PCA) has been widely applied to effectively reduce the numberof parameters characterizing a reservoir, its disadvantages are well recognized by researchers. First, PCAmay distort the probability distribution function (PDF) of the original model, especially for non-Gaussianproperties such as facies indicator or permeability field of a fluvial reservoir. Second, it smears theboundaries between different facies. Therefore, the models reconstructed by PCA are generally unac-ceptable for geologists.

A workflow is proposed to seamlessly integrate Cumulative-Distribution-Function-based PCA (CDF-PCA) and streamline information for assisted-HM on a two-facies channelized reservoir. The CDF-PCAis developed to reconstruct reservoir models using only a few hundred of principal components. It inheritsthe advantage of PCA to capture the main features or trends of spatial correlations among properties, andmore importantly, it can properly correct the smoothing effect of PCA. Integer variables such as faciesindicators are regenerated by truncating their corresponding PCA results with thresholds that honor thefraction of each facies at first, and then real variables such as permeability and porosity are regeneratedby mapping their corresponding PCA results to new values according to the CDF curves of differentproperties in different facies. Therefore, the models reconstructed by CDF-PCA preserve both geological(facies fraction) and geostatistical (non-Gaussian distribution with multi-peaks) characteristics of theiroriginal or prior models. Our preliminary results indicate that the history-matched model using theCDF-PCA alone may not satisfy the requirement of geologists, e.g., some channels may becomedisconnected during history-matching. Therefore, we propose a method of combining CDF-PCA togetherwith streamline information. Because velocity of the tracer in the streamline provides connectivityinformation between injectors and producers, it enhances channel connectivity without over-correction oncell-based permeability during the process of history matching.

The CDF-PCA method is applied to a real-field case with three facies to quantify the quality of themodels reconstructed. The history matching workflow is applied to a synthetic case. Our results show thatthe geological facies, reservoir properties, and production forecasts of models reconstructed with CDF-PCA are well consistent with those of the original models. The integrated HM workflow of CDF-PCA

with streamline information generates reservoir models that honor production history with minimalcompromise of geological realism.

IntroductionBoth object-based and MPS-based (Multi-Point Statistics) models generate relatively more geologicallyrealistic channel bodies compared to conventional two-point geostatistics-based techniques. However,conditioning such models to production data and correctly sampling the posterior probability distributionare challenging problems. One of the major challenges is that the number of parameters to be tuned duringthe process of history-matching (HM) is too large to be effectively handled by available HM workflows,especially when adjoint gradient is unavailable. Another challenge is that the models obtained after HMgenerally violate or distort the geological and geostatistical characteristics (e.g., channelized reservoir) ofthe original or prior models.

A geological model which is used to describe the depositional environment and to predict futureproduction is highly relevant to field development planning. To best predict the future reservoir perfor-mance and reduce the reservoir uncertainty, the geological models need to be conditioned to any types ofavailable data, e.g. 3D seismic data, well log data, production data, time-lapse seismic data, core analysisdata, etc. There are many tools or algorithms to generate geological models subject to geostatisticaldescription; however, it is not easy to adjust geological models to match data and retain geologicalconsistency, especially when the geological model contains channels, lobes or such non-Gaussian objectbased models.

In only a few cases, e.g. aeolian system, wave-dominated delta system or one facies problem, thereservoir model parameters (e.g. permeability and porosity) can be assumed random Gaussian fields thatcan be characterized by variogram. A Gaussian field can be modeled with two-point statistics [14], inwhich a variogram is used to quantify how properties are correlated with each other spatially. Manymethods have been developed to quantify uncertainties of reservoirs model parameters and productionforecasts through conditioning to production data, such as Ensemble Kalman filter [1] method and therandomized maximum likelihood (RML) method [16]. Both methods are formulated on the basis ofBayesian framework under the assumption of prior Gaussian reservoir models. However, in most cases,the prior reservoir model is non-Gaussian, e.g. a fluvial system, where the uncertain parameters are highlyrelevant to structures and patterns that cannot be characterized by simple two-point correlations. Fordifferent lithology/facies, rock properties could be sampled from very different probabilistic distributionsso that the overall probability distribution of rock properties is non-Gaussian. To model non-Gaussiangeology, object-based modeling [11] and Multi-Point-Statistics modeling [19] techniques are able toprovide clear shape and boundary of geological bodies and some detailed geological/channelized features.The channelized geostatistical descriptions (e.g. lithology distribution, channel width, wavelength, ori-entation, etc.) are usually input to a geological modeling software to generate unconditional realizationsby sampling with random seeds. By running simulations of these realizations of reservoir models, one canquantify the uncertainty of production forecasts. However, conditioning these non-Gaussian models tovarious production data and properly sampling their posterior probability distribution are still verychallenging tasks.

Recently, KarhunenLoeve (K-L) expansion[6] has been introduced to approximate geological modelswith an infinite linear combination of orthogonal functions. As a linear K-L approximation, PrincipalComponent Analysis (PCA) has been applied to effectively reduce the number of parameters character-izing a reservoir and preserve major characteristics of two-point geostatistics. PCA has been applied inmany fields, such as face recognition, history matching [15, 21], seismic interpretation [18] etc. Mo-hamed[15] and Chen [4] used PCA for model reparameterization and history matching the syntheticBrugge field. PCA decouples the selection of the basis functions and the estimation of coefficients. Inanother words, the basis function represents the parameter/model uncertainty in a library of the given prior

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realizations[15, 21] and a given set of coefficients represent a realization of reservoir model, e.g.stochastic spatial distribution of properties. The coefficients can be calibrated with the observed dynamicdata. By applying PCA, a reservoir model can be reconstructed by a linear combination of eigenvectorsobtained with a given set of prior reservoir models using only a few principal components. In fact, PCAcan be regarded as a smoothing operator or tool that removes high frequency noise from the original datasets. The fewer the number of principal components, the smoother the reconstructed parameters. The mostoutstanding advantage of PCA is its capability of capturing the strong spatial correlations amongparameters, with only a few principal components. However, its smoothing effect also raises some issuesbecause some subtle but important statistical details (e.g., cumulative density or probability densityfunction) of prior models are no longer preserved. For the channelized facies problem, PCA transformsdiscrete facies indicators to real numbers, smears boundaries between different facies, and distorts theprobability distributions of reservoir parameters, especially when the prior models present multi-peaknon-Gaussian PDF. Sarma et al. [17] applied a Kernel-PCA method, which is a nonlinear form of K-Lexpansion, to address the limitations associated with the linear K-L expansion. The results of Kernel-PCAare better than the results of PCA. However, as shown by Sarma et al. [17], the Kernel-PCA also generatesunclear boundaries between facies.

M. Khaninezhad[12, 13] applied the K-SVD method to capture the channel features for the fluvialmodels, and compared their K-SVD results with results of PCA for several synthetic cases. Their resultsindicate that by sparsifying the prior models to learn a geologic dictionary and using sparse reconstructiontechniques, it is feasible to represent and estimate complex non-Gaussian geologic features from limitedflow data. However, computational cost for applying K-SVD to a large reservoir model is quite expensive.

One of our major objectives is to reconstruct channelized geological and reservoir models with muchfewer number of uncertain parameters so that we can condition these models to production data throughautomatic/assisted history matching using model-based derivative-free optimization algorithms. Thegeological and reservoir models obtained by HM have to capture major flow dynamics and are alsoconstrained by histogram of lithology distribution and correlations between reservoir properties of theoriginal model. The following four important questions must be addressed when regenerating geologicalrealizations using new parameters.

1. Are new parameters able to regenerate both static models (e.g. facies) and dynamic models (e.g.permeability field)?

2. Are the models regenerated by new parameters still geologically realistic and relevant? In anotherwords, does the new geological and reservoir models satisfy geologists requirement?

3. Are the realizations generated from sampling the probability distribution of these new uncertainparameters (generally they are Gaussian and much easier to sample) are equivalent to (but notbiased from) the realizations generated from sampling the prior probability distribution?

4. Is it feasible to match history production data by tuning these new parameters, .e.g., usingderivative-free optimization methods, especially when adjoint gradient is unavailable?

PCA provides possibility to capture some main features of non-Gaussian models, but loses some minorfeatures. As a model reconstructed with PCA tends to filter out low frequency information and thestatistics after reconstruction deviate from the original statistics, a straightforward idea is to use normalscore transformation[5] to obtain a non-Gaussian field. Several researchers have applied normal scoretransform and/or truncated Gaussian to history matching problems with non-Gaussian prior models (e.g.,channelized models). Zhao et al.[22] applied the truncated Plurigaussian models to generate facies modelsand the Ensemble Kalman Filter to update the associated permeability field. Zhao et al.[22] consideredcorrecting water saturation, which tends to be deviated from its original distribution at each assimilationtime step, using normal score transform; however, their results indicate that such a correction does notyield significant improvement over standard EnKF. Zhou et al.[2325] applied normal score transform on

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each element of the state vector (e.g. pressure and saturation fields) before EnKF updating step andback-transform after EnKF updating step.

Inspired by the normal score transform applied in these prior researches, we proposed a new procedure(CDF-PCA) that can reproduce exactly the same CDF of each original individual realization. Integervariables such as facies indicators are regenerated by truncating their corresponding PCA results withthresholds that honor the fraction of each facies at first, and real variables such as permeability andporosity are regenerated by mapping their corresponding PCA results to new values according to the CDFcurves of different properties in different facies. Therefore, the models reconstructed by CDF-PCApreserve both geological (facies fraction) and geostatistical (non-Gaussian distribution with multi-peaks)characteristics of their original or prior models. If parameters defining the CDF curves are stronglycorrelated to those new uncertain parameters (or PCA coefficients), analytical response-surface-modelsusing Radial-Basis-Function (RBF) [27] are constructed to represent those CDF transform mappingfunctions. If they are weakly correlated to PCA coefficients, the facies fraction and the statistics of rockproperties are treated as independent uncertain variables which can be quantified by prior information andto be tuned during history matching. By seamlessly integrating the CDF-based mapping functions withPCA, the new method (also called CDF-PCA) inherits the advantage of PCA to capture the main featuresor trends of spatial correlations among properties, and preserves the PDF of all geological and reservoirproperties by properly correcting the smoothing effect of the traditional PCA. For discrete properties (e.g.,facies indicators), the CDF-mapping of CDF-PCA is similar to the truncated Gaussian method, and theirroots can be traced back to the theory of inverse cumulative density function [3]. Truncated Gaussian isapplied on a Gaussian random field. In contrast, the CDF-PCA is able to capture connectivity uncertaintyin the given training realizations; therefore the reconstructed field with CDF-PCA is not limited to aGaussian field.

Our preliminary results indicate that the history-matched model using the CDF-PCA alone may notsatisfy the requirement of geologist, e.g., some channels regenerated by the CDF-PCA during history-matching may become broken or disconnected. To alleviate such kind of negative impacts and thereforeto improve the image quality, we propose a method of combining CDF-PCA together with streamlineinformation which is relevant to flow behavior. We split the history matching process to two steps. Firstly,we use a streamline simulator to generate the velocity map based on a model generated from PCA beforeperforming CDF-mapping. Although this model is not perfect, it is sufficient for streamline simulator togenerate the velocity map. Because velocity of the tracer [20] in the streamline provides connectivityinformation between injectors and producers, it will enhance channel connectivity without over-correctionon cell-based permeability during the process of history matching. Secondly, we use velocity map asmultipliers to adjust facies generated from PCA and then apply CDF-mapping. In the second step, faciesin high velocity area gain higher weight to be sand and vise versa.

We also demonstrate the feasibility of conditioning to production data by tuning the PCA-coefficientswith the simultaneous perturbation and multivariate interpolation (SPMI) optimization algorithm [4, 8,and 29] that does not require analytical or adjoint derivatives of the objective function with respect toparameters to be tuned during the process of HM. The SPMI algorithm is a derivative-free optimizationtool and has been proven robust in In-situ Upgrading Process (IUP) production optimization problems [8,29]. In each iteration, the SPMI will generate perturbation and searching points simultaneously, and aquadratic model will be constructed by interpolating perturbation points and searching points iteration byiteration, using both the value of the objective function and the available derivatives evaluated at eachpoint. SPMI is a massively parallelized model-based optimization method with or without derivatives.

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MethodologyPCA with CDF mapping to reconstruct facies and properties

The mean reservoir model can be estimated from Ne unconditional samples m(j) (for j 1, 2,. . ., Ne)

by,

where m(j) is a Nm dimensional vector and Nm Ne. Using PCA, we can approximate a reservoirmodel m by y,

(1)

where is an N Nm matrix that is composed of the first N orthogonal basis vectors (correspondingto the first N largest Eigen-values of Cm, the covariance matrix of the original model), and (calledPCA-coefficients vector) is an N dimensional random vector with mean zero and identity covariancematrix. PCA maps m from a high (Nm) dimensional domain to in a much lower (N) dimensional domain.However, when applying PCA to discrete facies indicators, they become real numbers, boundariesbetween different facies become unclear, and the probability distributions of reservoir parameters aredistorted.

Because the PCA model in Eq. 1 is derived from minimizing the mean-square-error (MSE) of faciesindicators and reservoir properties between the original prior model and the reconstructed model.Therefore, it is reasonable to assume that the orders for most data in the original model are preserved inthe reconstructed PCA model. If the value of a property (e.g., porosity) in the i-th gridblock is greater thanthe value of the same property in the j-th gridblock for the original model, i,prior j,prior, then it is most

probable that the same order holds for the PCA mode, i.e., i,PCA j,PCA. Based on this observation, we

propose a CDF mapping method to remapping real valued facies indicators obtained with PCA to discretefacies indicators that honor the fraction of each facies of the prior model, and then remapping reservoirproperties (e.g., porosity and permeability) in each gridblock from their original PCA values to new valuessuch that the new model preserves the prior PDF or CDF of these properties.

Schematically, the CDF mapping is shown in Figure 1. The dark blue curve in Figure 1(a) shows a

Figure 1CDF mapping. The left subfigure (a) shows a cumulative density function (CDF) for a prior realization of a 2-facies model. The rightsubfigure (b) shows a CDF for a realization reconstructed with PCA. By taking the same CDF values, the y1 and y2 in the right subfigure can bemapped into, respectively, 0 and 1 in the left subfigure.

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cumulative density function (CDF) for a realization of a 2-facies Egg model. As it has only two integervalues, 0 or 1, the CDF curve is a stair-like curve where no point appears between 0 and 1. However, theCDF curve of the model reconstructed by PCA becomes a continuous curve, and the approximated valuesof facies indicators in all gridblocks become real values ranging from 0.5 to 1.5, see the dark blue curveshown in Figure 1(b). A natural way to obtain integer type facies indicators is to simply truncate the PCAresults either to 1 (when y0.5) or to 0 (otherwise). However, the fraction for each facies (or the CDFcurve) of the truncated facies indicators are not guaranteed the same as that of the original model, becausethe threshold for the truncation is fixed to 0.5. In contrast, when CDF mapping is applied, we alwaysguarantee that the fraction for each truncated facies indicators are exactly the same as that of the originalmodel. In Figure 1(b), we take one variable (y1) and follow the path in red color to find its percentile value(F(y1)). Then we map F(y1) to the left subfigure and find its corresponding new value of 0 (m1, prior 0)with the same percentile value on the prior CDF curve. For y2, we follow the green lines and map to theleft subfigure to find its corresponding value of 1. After we repeat the procedure for all gridblocks (or yis),we obtain a new facies model which has the same CDF curves as the prior facies model. In fact, theCDF-mapping for integer variables (facies) can be regarded as a truncation operator defined as: f (y;) 0 for y and f (y ; ) 1 for y , where is the threshold that is predetermined by the prior model.For the example shown in Figure 1, 0.495. The switching point is referred to as the fraction of thefirst facies, which corresponds to the threshold. For example, the switching point in left subfigure ofFigure 1 is 0.61.

The CDF-mapping procedure discussed above can also be applied for real variables (e.g. permeability,porosity, Net-to-Gross values, etc.). For each realization, we include both facies indicators and rockproperties in the original model mprior and the PCA approximate model y. We use two steps to reconstructthe realization of facies and rock properties. First, we use CDF-mapping to reconstruct facies indicators.Second, known the facies indicator for each gridblock, we calculate the CDF curves of the rock propertiesfor each individual facies from prior realizations and then apply the CDF-mapping procedure for rockproperties, i.e., remapping the value of a rock property in a gridblock (the corresponding element in thevector y) to a new value according to the CDF curve of that property for the facies that has been alreadyregenerated in the same gridblock.

Figures 2 to 4 show results obtained by applying CDF-mapping for both facies and permeability fieldof the Egg model. Figure 2 shows the first step to identify facies. An unconditional realization generatedfrom the prior Egg model is shown in Figure 2a. For the purpose of comparison, we also show thecorresponding model obtained with PCA (Figure 2b) and the corresponding model obtained withCDF-PCA (Figure 2c). We first obtain the basis matrix from the covariance matrix of the prior modelthat is estimated from a set of training realizations (1000 realizations for this example). Note that the

Figure 2The first step to identify facies, Egg example. The fourth subfigure shows the blue curve lies upon the black curve after applyingCDF-mapping. (a) Original facies; (b) Reconstructed facies with PCA; (c) Reconstructed facies with CDF-PCA; (d) Comparison of cumulative densityfunction between (a)-(c) facies models.

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training realizations can be generated by any geological modeling method, e.g., object based modeling,multipoint statistics, etc. A model y (Figure 2b) corresponding to the prior realization of mprior shown inFigure 2(a) is then reconstructed with PCA (Eq. 1) by solving the corresponding principal componentsfrom . We should note that the mprior shown in Figure 2(a) is not in the training

set. For the given mprior, the fraction of facies 0 is known (0.42), see the black curve with stars shown inFigure 2(d). Therefore, the threshold ( 0.49) for facies is determined. For a realization that is to begenerated during history matching procedure, the threshold or switching point can be treated either as afunction of the coefficient or as an uncertain parameter to be tuned. Using the PCA model (y) and thethreshold value ( 0.49), the 2-facies model mCDF-PCA is reconstructed (as shown in Figure 2c). TheFigure 2d shows the corresponding CDF curves for the prior model (black curves with stars), the PCAmodel (the green curve), and the CDF-PCA model (the blue curve), respectively. Obviously, the CDFcurve of the CDF-PCA model is exactly the same as that of the prior model, i.e., the blue curve is identicalto the black curve with stars. However, the CDF curve of the PCA model (the green curve) deviates fromthe prior CDF curve significantly.

Figures 3(a) and (b) show the second step to obtain permeability for facies 0 and facies 1 usingCDF-mapping. In both subfigures, the black curve with stars, the blue curve, and the green curvecorrespond to CDF curves of the normalized permeability for the original/prior realization, the CDF-PCArealization, and the PCA realization, respectively. The fact that the black and the blue curves are identicalclearly validates that the CDF-mapping preserves the probability distribution of the original model.

Figure 3The second step to reconstruct permeability, Egg example. For different facies, normalized permeability is adjusted based on priorhistogram information.

Figure 4Comparison of normalized permeability, Egg example. (a) Original permeability; (b) Reconstructed permeability with PCA; (c)Reconstructed permeability with CDF-PCA; (d) Comparison of cumulative density function (CDF) curves for three permeability models.

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Figures 4(a), (b), and (c) illustrate the permea-bility fields of the prior model, the model recon-structed by PCA, and the model reconstructed byCDF-PCA, respectively. The noticeable differencebetween Figure 4(a) and Figure 4(b) clearly indi-cates that PCA may distort the permeability fieldsignificantly. In contrast, after CDF-mapping, thenew model reconstructed with CDF-PCA shown inFigure 4(c) is quite similar to the original or priormodel shown in Figure 4(a). The difference betweenthe CDF curve of PCA (the green curve) and theCDF curve of the prior (the black curve with stars)is also noticeable, as shown in Figure 4(d). Theoriginal CDF curve of permeability (in black withstars) clearly indicates that the prior distribution isnon-Gaussian, and the original PDF curve has two peaks: one peak corresponds to the shale facies withvery low permeability with the normalized k being about 0.1 in Figure 4(d); whereas the other peakcorresponds to the sand facies with quite large permeability with the normalized k being about 1.0 inFigure 4(d). In contrast, the CDF curve of permeability from the traditional PCA shown as the green curvein Figure 4(d) looks like a normal (Gaussian) distribution, and its PDF curve has only one peak.

Determination of switching point and parameters defining CDF curves for random realizationsWhen performing CDF-mapping, we need to specify the CDF curves for reservoir properties or

switching point for facies indicators. Typically, different realizations have different CDF curves ordifferent switching points. For a model that is reconstructed corresponding to a prior realization generatedfrom geostatistical software, the switching points for facies and CDF curves for real valued reservoirproperties in each facies can be determined from the prior realization. However, when a new realizationis generated with CDF-PCA but it does not correspond to any of those prior realizations, the CDF curvesor switching points used for the CDF-mapping for this new realization are not readily available. One cantreat the switching point and the statistic parameters that define the CDF curves of rock properties asindependent random/uncertain variables. If we assume that rock properties in the same facies followGaussian distribution, then the statistic parameters to define the CDF curves of rock properties couldinclude the mean value (CDF,i,j) and the standard deviation (CDF,i,j) of the i-th property for the j-th facies.

The mean value and the standard deviation of a specific property in a specific facies may vary for differentrealization. Based on those prior realizations, we can quantify the uncertainty of these new uncertainparameters, i.e., to determine their mean values ( and ) and their covariance matrix. In sucha situation, the parameters to be adjusted during assisted history matching include PCA coefficients, theswitching point and the statistics parameters to define the CDF curve of each rock property for each facies.For a 2-facies problem, the total number of parameters is nc 1 4 nr, where nc is the number of PCAcoefficients, 1 is for the switching point, and nr is the number of rock properties. 4 comes from twoparameters (mean value and standard deviation) for each type of rock property for two facies.

In a more general case, one can assume that CDF parameters are correlated to , but with correlationcoefficient not equal to 1. They depend on in some degree, but not completely determined by . Forexample, as shown in Figure 5, a response surface with RBF is built between switching points and PCAcoefficients for Egg example and 1000 QA/QC switching points were used to check the quality of theresponse surface. If the quality of response surface meets satisfactory, i.e. the points are close to thediagonal straight line in this QA/QC plot, there are strong correlation between switching points and thePCA coefficients. One can use the predicted switching point as mean value and use history matching

Figure 5QA/QC for response surface between switching points andPCA coefficients, Egg example.

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procedure to tune the noise for the switching point. In this case, a Support-Vector-Regression [26, 28]could be applied to capture both their dependence on and their stochastic features, e.g., using the RBFkernel. Research on this respect is beyond the scope of this paper.

Conditioning reservoir models to hard data with CDF-PCAHard data here is referred to data obtained from direct measurements (e.g., well log data), typically

point data obtained from wells. The term hard data is used to emphasize the fact that the modeling methodshould exactly reproduce these measured values at their corresponding locations. However, trainingrealizations can be generated by object-based models (e.g., channelized models) which may or may nothave been conditioned to hard data, e.g., when new hard data acquired from a new wells that are drilledlater. These realizations can be easily reconstructed by PCA and conditioned to hard data by minimizingthe objective function defined in Eq. 2.

(2)

where mhard is a column vector of dimension Nhard, the number of hard data, and Chard is the Nhard Nhard covariance matrix for measurement errors of hard data. The matrix Chard becomes diagonal when allthe measurement errors of hard data are independent to each other. One should pre-calculate the index ofthe hard data in the vector m. Then y is a sub-vector consisting of values with the same index as mhardin y. ( 0)

T ( 0) is a regularization term and is the regularization coefficient, where 0 denotesthe PCA coefficients of an unconditional realization that is generated without conditioning to new harddata and denotes the new coefficients of the conditional realization by conditioning to new hard data. Weuse 1 when exists. In case of no measurement error associated with hard data, does not

exist, and it is equivalent to set Chard to an identity matrix while setting to a very small positive value,

e.g., 10-8 in our implementation. We can solve from p H 0,

(3)

where is a sub-matrix of that corresponds to the indices of new hard data. After solving the new

coefficients , we can follow the CDF-PCA workflow discussed above to generate the conditionalrealization. Comparison between results generated from unconditional and conditional realizations areshown in Figure 6. In this example, the training realizations (or images) are generated by multi-point-statistics approach. An unconditional realization is regenerated with PCA and CDF-PCA, as shown inFigure 6(a) and (b), respectively. For the unconditional realization shown in Figure 6(b), the faciesindicators at the four corners of (0,0), (60,0), (0,60) and (60,60) are 0, 0, 0 and 1, respectively. Afterdrilling 4 new wells that are located on the four corners of the reservoir domain, hard data (facies

Figure 6Realization generated with (a) PCA without hard data, (b) CDF-PCA without hard data, (c) PCA conditioned to hard data, (d) CDF-PCAconditioned to hard data. Blue is facies 0 and yellow is facies 1. Hard data in the four corner gridblock are known. The red open circle representsfacies 0 and the red solid dot represents facies 1.

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indicators) in the four corner gridblocks are known. The two open red circles at the tow top cornersindicate that the two wells are drilled through facies 0 and the two solid red dots at the two bottom cornersindicate that the other two wells are drilled through facies 1. Core samples reveal the true facies indicatorsare 1, 1, 0 and 0. Obviously, the unconditional realization does not match the hard data. The conditionalrealization after conditioning to hard data using Eq. 3 is shown in Figure 6(c) with PCA and theconditional CDF-PCA realization is shown in Figure 6(d). The conditional realization yields correct faciesat all four corners. Because no reservoir simulation is required for conditioning to hard data, it is quiteconvenient to determine with Eq. 3 even if the number of PCA components is huge. Therefore, theCDF-PCA provides a very efficient way of generating conditional realizations by conditioning to newhard data without losing information from the original unconditional realizations.

Conditioning reservoir models to production data with CDF-PCAA reservoir model usually consists of millions of gridblocks and there are a few unknown properties

(permeability, porosity, net-to-gross ratio, initial water saturation, etc.) at each gridblock. Due to reservoirheterogeneity, history matching is typically an ill-conditioned problem, i.e. the number of unknownparameters is much larger than the number of available observed data. History matching is usually posedas a mathematical problem of generating posterior samples by conditioning to production data within aBayesian framework. When CDF-PCA is applied for reparameterization, it has been proved that the newuncertain parameter vector is Gaussian with zero mean and identical covariance matrix. Therefore, thelogarithmic-PPDF (posterior-probability-density-function) of after conditioning to measurement datadobs is given by the Bayes rule:

(4)

where dobs is an Nd dimensional observation data vector and CD represents the Nd Nd covariancematrix of measurement errors for dobs; and g*() g(m()) is the corresponding predicted data vectorobtained by running the reservoir simulation with reservoir model m(). During the process of historymatching, we adjust the parameter vector to minimize the objective function O() for the maximum aposteriori (MAP) estimate.

Adjustment of PCA-models using streamline informationTo improve the quality of the history matched model in terms of connectivity for object based models,

we account for the flow behavior information. Flow relevant information should be considered to assistcorrecting the incorrect facies indicator before any types of static-information-based correction. In anotherword, the idea is to update reservoir properties within stream tubes before applying CDF-mappingcorrection.

The interstitial velocity indicates the speed of a particle moving from one grid to another grid.Therefore, high velocity area implies that the particle moves in a high permeability zone and vise versa.For example, if a gridblock in a prior model was identified as facies shale with low permeability but thevelocity in this gridblock (e.g., by running streamline simulation) is much higher than the velocities inother nearby gridblocks with shale facies, it is a good indicator that this grodblock should be connectedto a channel nearby, and therefore we should adjust the facies on this gridblock from shale facies to sandfacies with high permeability, because it is more probable to observe a relatively high velocity in agridblock when the gridblock connects to a channel instead of depositing randomly on a shale background.

Figure 7 illustrates an example of adjusting PCA-models using the streamline (or velocity) informa-tion. Starting from the top left figure (the facies model obtained with PCA), there are two paths in Figure7: the top path shows the example of using streamline information to highlight channels and thenperforming CDF-mapping approach; the lower path shows the example of directly using CDF-mappingto obtain the facies model. As shown in the upper left subfigure (the original PCA model) and the bottomright subfigure (the CDF-PCA model) in Figure 7, the channel branch near the top tends to be broken from

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the results generated from PCA. However, the velocity map (see the left of the two pictures shown in theupper middle subfigure in Figure 7) suggests that the top channel branch should be connected. The rightof the two pictures shown in the upper middle subfigure in Figure 7 is obtained after correcting the originalPCA results (the top left). The connectivity of the upper channel branch is improved by correcting theoriginal PCA-model using the velocity (or streamline) information. By gradually increasing the weightingfactor of velocity information, the upper channel branch will become connected again.

Case StudiesQuantifying quality of reconstructed models for a field caseThe CDF-PCA method is applied to a real field example to quantify the quality of the reconstructedmodels and to prove that the method is capable for a large scale and complex channelized model. Thereservoir model has 0.2 million gridblocks, and is composed of sand, levee and shale facies. 1000unconditional realizations were generated with object-based prior geological model, and they were usedto form the basis matrix of (see Eq. 1) for both PCA and CDF-PCA. Two measures were used toquantify the quality of reconstructed data sets, including the fraction of incorrect facies (Ef) and themean-square-error (MSE or the 2 norm) of permeability and porosity between the images of the originaland the reconstructed data sets (CDF-PCA). As 2 norm is not a good choice for evaluating the differenceof facies between two models, we use Ef for facies instead. The fraction of incorrect facies, Ef, is actuallythe normalized 0 norm, i.e., calculated by normalizing the number of gridblocks with incorrect facies bythe total number of gridblock.

In this example, a set of basis matrix with different number of principal components (N 25, 50,100, 200, 400, and 800) was tested to show the impact of N on the quality of the reconstructed model.Table 1 summarizes quality analysis for models reconstructed with different methods for the real fieldcase. The results listed on the second and the third rows in Table 1 clearly show that the CDF-PCA canimprove the quality of the reconstructed facies model significantly. The PCA solution minimizes the totalMSEs for facies, permeability, and the porosity between the original and the reconstructed data set. WhenN is small (e.g., less than 200), the MSEs of CDF-PCA for permeability (the fourth row) and porosity(the sixth row) become larger than those of PCA (the fifth row and the seventh row), because the CDFcorrection makes the new solution deviates from the PCA solution that minimizes the total MSEs.

Figure 7An example of combining streamline information with CDF-PCA.

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However, when N is large enough (e.g., larger than 200), the increment in MSEs of CDF-PCA forpermeability and porosity becomes negligible. Therefore, the gain from improving the quality of thereconstructed facies model, and more importantly, from preservation of the non-Gaussian characteristicsof the CDF-PCA weights much more than the lose due to the negligible increment in MSEs for thepermeability and the porosity, see Figure 8(a).

Figure 8(a) illustrates the relative comparison for quality of facies (in black), permeability (in red), andporosity (in blue) models reconstructed by CDF-PCA method and traditional PCAtruncation method.When compared to the traditional truncation method (PCAtruncation) where the threshold for truncationis fixed to 0.5, the CDF-PCA method can reduce the fraction of incorrect facies by 22~42%, see the backcurve with open circles in Figure 8(a). Similarly, the negative relative improvement shown by the red (forpermeability) and blue (for porosity) curves indicates that the MSEs of CDF-PCA for permeability andporosity are larger than those of the PCA, when N is less than 200. Plots shown in Figure 8(b) indicatethat the measures of model error between the prior model and the reconstructed model with CDF-PCA(fraction of incorrect facies or MSEs for permeability and porosity) decrease as the number of principalcomponents increases. When N 200, the fraction of incorrect facies is less than 0.1, and the MSEs forpermeability and porosity are less than 0.15.

In the following example, only 50 principal components are used to reconstruct the facies, permeabilityand porosity models of the real field case. Figure 9 shows images of facies distribution in one layer of theprior realization (the top left), the PCA realization (the top middle), and the CDF-PCA realization (the topright). Obviously, the facies model reconstructed with PCA (the top middle subfigure) is unacceptable forgeologists, because there is no clear boundary between three facies. It is very difficult to identify the leveefacies from the PCA facies image. Even though we have reduced the number of uncertain parameters from600,000 to 50 with a reduction factor of 12,000, the facies model reconstructed by the CDF-PCA is quitesimilar to the original model. As we expected, the CDF-PCA preserves fractions (or PDF) of all threefacies, e.g., by comparing the bar plots shown on the bottom right (generated from the realization

Table 1Quality analysis of models reconstructed with different methods for the real field case

Number of principal components 25 50 100 200 400 800

Fraction for incorrect facies (CDF-PCA) 0.1824 0.1654 0.1391 0.0972 0.0519 0.0007

Fraction for incorrect facies (PCATruncation) 0.2332 0.2197 0.1905 0.1352 0.0745 0.0012

MSE for permeability(CDF-PCA) 0.2455 0.2249 0.1911 0.1451 0.1014 0.0345

MSE for permeability (PCA) 0.2075 0.1914 0.1714 0.1384 0.1051 0.0494

MSE for porosity(CDF-PCA) 0.2275 0.2095 0.1810 0.1409 0.1028 0.0408

MSE for porosity(PCA) 0.1925 0.1790 0.1612 0.1312 0.1010 0.0485

Figure 8(a) Relative improvement in quality, and (b) impact of the number of PCA coefficients on quality of facies, permeability, and porositymodels reconstructed by CDF-PCA method.

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reconstructed with CDF-PCA) with those shown on the bottom left (generated from the prior realization).In contrast, except for those inactive cells (as indicated by the left-most blue bar on the bottom middlesubfigure), the PDF curves for the realization reconstructed with PCA behaves more like a Gaussianinstead of three distinct bars.

Figure 10 shows images of facies (subfigures on the top row), permeability (subfigures on the secondrow), and porosity (subfigures on the bottom row) in a vertical cross-section plane. These images aregenerated form a prior realization (subfigures on the left-most column), reconstructed with PCA (sub-figures on the middle column) and CDF-PCA (subfigures on the right-most column), respectively.Compared to the prior model, the porosity and permeability fields reconstructed by PCA are muchsmoother and they are unacceptable for geologists and reservoir engineers. In contrast, after CDFcorrection, the porosity and permeability fields reconstructed by CDF-PCA are quite similar to theiroriginal counterparts. When more principal components are added, the difference between the modelsreconstructed by CDF-PCA and their original prior models will become negligible; see plots shown inFigure 8(b).

For the prior realization, some levee facies are within the channel facies (see the thin light green leveein the red channel on the top left subfigure in Figure 9). However, such subtle features disappear in thereconstructed facies model with CDF-PCA shown on the top right subfigure in Figure 9. You may alsonotice that more levee facies (with indicator of 1) locate between the shale facies (with indicator of 0) andthe channel facies (with indicator of 2) in both the prior model and the model reconstructed withCDF-PCA. According to the geological or depositional process, levee facies behaves much like antransitional facies between channel facies and shale facies, and we also follow such a natural order ofdepositional process to order these facies indicators, i.e., 0 for shale, 1 for levee, and 2 for channel.However, in a more general case, such kind of natural order among facies may not exist, especially whenmore facies co-exist. The CDF-PCA method presented in this paper fails to reproduce satisfactory faciesimages for such a case. Because of this limitation, in the following sections of this paper, we will befocused on reconstructing models with only two facies. Inspired by these observations, we haveinvestigated an alternative method of CDF-based pluri-PCA to overcome the limitations of the CDF-PCA,where all facies are reconstructed according to the rock-type-rules and the order of facies indicators will

Figure 9Layer images of facies (the first row) and their corresponding PDF (the second row) for the prior model (the left column), and the modelsreconstructed by PCA (the middle column) and CDF-PCA (the right column) with 50 principal components.

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not impact the results of the reconstructed models. Details of the CDF-based pluri-PCA and how toautomatically determine the rock-type-rules based on prior realizations are beyond the scope of this paper,and it will be discussed in more details in another paper to be published later.

Quantifying uncertainty of production forecasts for the Egg modelIn this section, we use the Egg model to demonstrate the similarity of production uncertainty quantifi-cation between the prior models and the models reconstructed with CDF-PCA method. Egg model is ameandering channel system and the channels are populated with object-based modeling approach and thefacies 0 (sand) is modeled with channel fraction of 0.42. The facies 1 (shale) is the background lithology.The orientation, amplitude, wavelength, width and thickness of channels are sampled with triangledistribution and their mean values are listed in Table 2. The minimum (or maximum) value of the triangledistribution is defined as the min_factor (or max_factor) multiplied by the mean value. The porosity andpermeability in the sand body are modeled with Sequential Gaussian Simulation. The correlation lengthsalong major (or long) principal axis, the minor (or short) principal axis, and the vertical axis are,respectively, 173 m, 93.5 m and 3.2 m. The azimuth is -102.0 degree. Egg model contains 60 60 10gridblocks. The grid size is x y z 4.572m. There are 6 producers and 4 injectors in thereservoir. Note that the models have been conditioned to hard data of facies and well log data ofpermeability and porosity already. The standard deviation of permeability and porosity are estimated fromwell log data directly. Only water and oil phases are present in the system.

Figure 10Vertical cross-sectional images of facies (the first row), permeability (the second row), and porosity (the third row) for the prior model(the left column), and the models reconstructed by PCA (the middle column) and CDF-PCA (the right column) with 50 principal components.

Table 2Parameters for Egg model

Sand fraction 0.42 Thickness mean 3 m

Orientation mean 53 Sand permeability mean 2,163 md

Amplitude mean 125 m Sand permeability std 1,320 md

Wavelength mean 847 m Shale permeability mean 2 md

Width mean 84.3 m Shale permeability std 0.5 md

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Figure 11 shows one realization of permeability field with the well locations in the reservoir. The redcolor indicates the sand deposition in channel with high permeability and the light blue color indicates thematrix with low permeability. The gray wells are injectors and the golden wells are producers.

In this case, we treat the shale fraction and the statistic parameters that define the CDF curves of rockproperties as uncertain parameters to be solved in addition to PCA coefficients. Figure 12 compares thefirst layer for one realization of the original training model (the left subfigure) with the model recon-structed with PCA (the middle subfigure) and the model reconstructed with CDF-PCA (the rightsubfigure). In this example, 100 principal coefficients were used to reconstruct the model for both PCAand CDF-PCA. After reducing the number of uncertain parameters from 108,000 to 100 with a reductionfactor of 10,800, the facies model reconstructed with the CDF-PCA is almost identical to the trainingmodel. Figure 13 shows the PCA and CDF-PCA images reconstructed with a group of random coefficientssampled from Gaussian distribution with mean zero and variance 1. The CDF-PCA image shown in theright subfigure in Figure 13 is randomly sampled and it still captures the main features (meandering

Figure 11Initial Egg model with 10 wells.

Figure 12Comparison of the first layers for facies models for Egg model.

Figure 13Reconstructed facies with random coefficients for Egg model.

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channels) of the training image shown in Figure 12, although locations of channels in both figures are notexactly the same.

For the purpose of quality analysis and quality control (QA/QC), we have generated two ensembles of1000 realizations with object based modeling approach. The ensemble 1 is used as training realizationsand the ensemble 2 is used as QA/QC realizations. We compute PCA coefficients for both ensembles, andthen reconstruct their corresponding realizations using CDF-PCA.

Figure 14 shows the comparison of probability density function of different production forecasts datafor the Egg model, including cumulative oil (a) and cumulative water (b), all are evaluated at the end ofthe 8 years production period. There are six curves in each subfigure. The black and blue curves,respectively, represent the production forecasting results obtained from ensemble 1 of the trainingrealizations and the ensemble 2 of the QA/QC realizations. The green and purple curves are, respectively,the production forecasting results obtained from ensemble 1 and 2 realizations that are reconstructed withCDF-PCA using 100 principal coefficients. The red curve corresponds to the pdf of production forecastsobtained from 1000 realizations that are generated with CDF-PCA by sampling the 100 random principalcoefficients. For the purpose of QA/QC, we also use 200 coefficients to reconstruct the ensemble 2(QA/QC realizations) and show its probability density function in light blue color. As shown in Figure 14,the black curves are quite close to the blue curves, which clearly indicate that the quality of using 1000training realizations is reasonably satisfactory. The light blue curve is fairly close to the purple curve,which indicates that using 100 principal coefficients are sufficient for this case. Results listed in Table 3are P10-P50-P90 values of cumulative oil and cumulative water for these six different scenarios. Allcurves in different colors shown in the same subfigure in Figure 14 are very close to each other, and the

Figure 14Comparison of probability density function of production data for Egg model. Black: Training, ensemble 1; Green: CDF-PCA, ensemble1; Red: CDF-PCA, random; Blue: QA/QC, ensemble 2; Light blue: CDF-PCA, ensemble 2, 200 components; Purple: CDF-PCA, ensemble 2, 100components.

Table 3Comparison of P10-50-90 values of cumulative oil and cumulative water

Cumulative oil (STB) Cumulative water (STB)

P10 P50 P90 P10 P50 P90

Ensemble-1: training 7.381e5 8.281e5 9.586e5 1.358e6 1.485e6 1.576e6

Ensemble-1: CDF-PCA-100 7.45E05 8.38E05 9.67E05 1.35E06 1.48E06 1.57E06

Ensemble-2: QA/QC 7.349e5 8.302e5 9.563e5 1.357e6 1.483e6 1.578e6

Ensemble-2: CDF-PCA-100 7.403e5 8.391e5 9.698e5 1.343e6 1.474e6 1.573e6

Ensemble-2: CDF-PCA-200 7.325e5 8.365e5 9.623e5 1.351e6 1.477e6 1.581e6

Random: CDF-PCA-100 7.130e5 8.184e5 9.473e5 1.366e6 1.496e6 1.600e6

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P10-50-90 values for different scenarios listed in thesame column in Table 3 are also very close to eachother, which further validates that the uncertaintyquantifications based on realizations of geologicaland reservoir models reconstructed/regenerated bythe CDF-PCA using 100 principal coefficients arealmost the same as good as using their originalrealizations. In summary, CDF-PCA not only hon-ors the uncertainty characteristics of the originalstatic models, but also honors the uncertainty char-acteristics of their production forecasts.

Assisted history matching by integration ofCDF-PCA with streamline information for theEgg modelIn this history matching example, an additional re-alization, which is not within the training data set,was generated as the true model. This true model was used to generate true measurement data ofwater injection rate in all water injectors and water production rates in all producers by running reservoirsimulation under the liquid rate constraints of 200 BBL/day for all producers and maximum bottom-holepressure constraints of 400 bars for all water injectors. The production period is 8 years. The observed datais generated by adding synthetic measurement noise to the true measurement data. The standarddeviation of measurement error is 5 BBL/day for the rate measurement. The parameters to be tunedinclude PCA coefficients, the switching point of facies and the statistical parameters that define the CDFcurves of permeability and porosity in each facies. The parallelized optimization tool SPMI is applied tominimize the objective function defined by Eq. 4.

Figure 15 shows the normalized objective function versus iterations. As shown by the red curve inFigure 15, the normalized objective function converges to 14.5 after 25 iterations if only CDF-PCA isapplied. The blue curve in Figure 15 indicates that the normalized objective function can converge to asmaller value of 9.38 after 23 iterations by integration of CDF-PCA with streamline correction. Even atthe first iteration where the initial prior model was used, streamline correction reduces the normalizedobjective function from 74 to 64, because the initial models after performing CDF-mapping with andwithout streamline correction are different. Figure 7 shows how streamline information can be used toimprove channel connectivity in facies model. Visual inspection tells us that the facies model obtainedwith combining streamline information and CDF-mapping is better than the model obtained withCDF-mapping only. Although the channels in black circles in both subfigures are disconnected, the onewith streamline information correction shows better connectivity after CDF-mapping.

In this example, we use 500 PCA components to reconstruct geological realization. Different perme-ability realizations in the first and the tenth layers of the Egg model are illustrated in Figures 16 and 17.The top left subfigure is the true model; the top middle one is the prior realization generated by objectbased modeling method; the top right one is the model reconstructed with CDF-PCA corresponding to theprior realization; the bottom left is the best model obtained after history matched using the CDF-PCAapproach; and the bottom right one is the best matched model using the hybrid approach of CDF-PCA andstreamline information. Because streamline enhances channel connectivity, the history matched modelwith streamline correction (the bottom right) looks more similar to the prior model than the one (thebottom left) obtained with CDF-PCA alone.

Figure 15Normalized objective function VS iterations.

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Figure 16Comparison of the 1st layer of models. Upper left: True; Upper middle: Initial generated by an object-based geological modeling tool;Upper right: Initial by CDF-PCA; Lower left: Best model by CDF-PCA; Lower right: Best model by CDF-PCAstreamline.

Figure 17Comparison of the 10th layer of models. Upper left: True; Upper middle: Initial by Initial generated by an object-based geologicalmodeling tool; Upper right: Initial by CDF-PCA; Lower left: Best model by CDF-PCA; Lower right: Best model by CDF-PCAstreamline.

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From Figures 16 and 17, the following two evidences are observed by comparison between the priormodel (top middle) and the best matched model (bottom right) that is obtained by the hybrid approach ofCDF-PCA with streamline information:

1. The injector myinj3 and the producer myprod1 are most probably connected through a channelin the upper layers of the reservoir;

2. The injector myinj1 and the producer myprod1 are most probably connected through a channelin the lower layers of the reservoir.

This information could help geologist to improve the geologic model, especially if justified by thegeologic concept and potentially with other soft information. Figure 18 compares cross-sections of the fivemodels. These models show that the channel connectivity in the vertical direction between layers is alsogeologically reasonable.

Plots shown in Figures 19 and 20 include the observed data (green) of water production rate in eachproducer (Figures 19) and water injection rate in each injector (Figures 20), and those predicted with theinitial (red) model, the HM matched model with CDF-PCA alone (black) and the HM matched model withCDF-PCA plus streamline correction (blue). Overall, the matching quality is improved significantly,although we still see some gaps in some producers and injectors. Except for the producer myprod3 (seethe bottom left subfigure in Figure 19), the HM matched model obtained from the hybrid approach ofCDF-PCA with streamline correction yields predictions (blue curves) that match the observed data (greencurves) much better than those of applying CDF-PCA alone. Because the streamline correction enhancesthe connectivity between producers and injectors, it is most probable that the thin channel in the blackcircle shown on the bottom right subfigure in Figure 17 connects the producer myprod3 and the injectormyinj4, and therefore results in earlier water breakthrough in myprod3. In our current implementation,the weighting factor for streamline correction is fixed. We believe that introducing the weighting factoras a new uncertain parameter to be tuned during the process of HM would further improve theperformance.

Figure 18Comparison of the outcrop of models. Upper left: True; Upper middle: Initial by Initial generated by an object-based geological modelingtool; Upper right: Initial by CDF-PCA; Lower left: Best model by CDF-PCA; Lower right: Best model by CDF-PCAstreamline.

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Discussion and ConclusionsWe propose a new procedure that reproduces exactly the same CDF curves of facies and reservoirproperties in each facies of each original individual realization. By seamlessly integrating the CDF-basedmapping functions with PCA, the new method (also called CDF-PCA) has the following advantages:

It inherits the advantage of capturing correct spatial correlations of PCA, and preserves geosta-tistics of all physical properties by properly correcting the smoothing effect of PCA.

It can be used for reconstructing the training realizations of non-Gaussian fields, e.g., thosegenerated with multi-point-statistics modeling or object based modeling tools.

It provides a convenient way of conditioning non-Gaussian model (e.g. channelized model) to harddata with the analytical solution.

Figure 19Production water rate matches for Egg model. Red: Prior model; Blue: Best match by CDF-PCAstreamline; Black: Best match byCDF-PCA only; Green: measurement data.

Figure 20Injection water rate matches for Egg model. Red: Prior model; Blue: Best match by CDF-PCAstreamline; Black: Best match byCDF-PCA only; Green: measurement data.

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It not only honors the uncertainty characteristics of the original static models, but also honors theuncertainty characteristics of their production forecasts.

The CDF-PCA method is applied to a field case to quantify the quality of the reconstructed three-faciesmodels. When natural order among facies exists, we show that the quality of the reconstructed model isquite satisfactory. To overcome the limitations of the CDF-PCA that requires the existence of naturalordering among facies, an alternative method of CDF-based pluri-PCA has been investigated, where allfacies are reconstructed according to the rock-type-rules and the order of facies indicators will not impactthe results of the reconstructed models. Details of the CDF-based pluri-PCA and how to automaticallydetermine the rock-type-rules based on prior realizations are beyond the scope of this paper, and it willbe discussed in more details in another paper to be published later.

We also apply the CDF-PCA method to a synthetic case. Our results show that the geological facies,reservoir properties, and production forecasts of models reconstructed with CDF-PCA are well consistentwith those of the original models. The integrated HM workflow of CDF-PCA with streamline informationgenerates reservoir models that honor production history with minimal compromise of geological realism.

Usually, it is feasible to sample the prior uncertain model (or its PDF) by generating enoughunconditional realizations and then to quantify the uncertainty of production forecasting by runningreservoir simulation with these unconditional realizations. To sample the posterior uncertain model (or itsPDF), one needs to perform history matching using each unconditional realization as the initial guess, e.g.,when applying the randomized maximum likelihood (RML) method. The high dimension of unknownsmakes it quite expensive to perform HM for such a large number of unconditional realizations, especiallywhen the adjoint-gradient is not available. The CDF-PCA approach can effectively reduce the number ofuncertain parameters without compromising the uncertainty quantification characteristics of the originalor prior model, which makes it feasible generating posterior realizations by history matching productiondata. Furthermore, the Gaussian features for the reduced set of uncertain parameters lends itself morenaturally to both EnKF and RML methods that are formulated by assuming Gaussian prior distribution.The CDF-PCA plus streamline correction HM workflow can be easily integrated with EnKF and RMLmethods to further quantify uncertainty of reservoir properties and production forecasting by conditioningto production data.

AcknowledgementThe authors would like to thank Shell International Exploration and Production Inc. for permission topublish this paper. We also want to thank the following colleagues for their suggestions: Faruk O. Alpak,Douglas Leyden, Matthew Wolinsky, Jim Jennings, Gosia Kaleta and Jeroen Vink. We also want to thankMr. Hai Vo for providing MPS training realizations in the section of Conditioning reconstructed modelto hard data.

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Integration of Principal-Component-Analysis and Streamline Information for the History Matching ...IntroductionMethodologyPCA with CDF mapping to reconstruct facies and propertiesDetermination of switching point and parameters defining CDF curves for random realizationsConditioning reservoir models to hard data with CDF-PCAConditioning reservoir models to production data with CDF-PCAAdjustment of PCA-models using streamline information

Case StudiesQuantifying quality of reconstructed models for a field caseQuantifying uncertainty of production forecasts for the Egg modelAssisted history matching by integration of CDF-PCA with streamline information for the Egg model

Discussion and Conclusions

AcknowledgementReferences