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Comput Geosci (2017) 21:1443–1458DOI 10.1007/s10596-017-9633-4
ORIGINAL PAPER
Production forecasting and uncertainty quantificationfor naturally fractured reservoirs using a new data-spaceinversion procedure
Wenyue Sun1 ·Mun-Hong Hui2 ·Louis J. Durlofsky1
Received: 30 September 2016 / Accepted: 26 February 2017 / Published online: 22 March 2017© Springer International Publishing Switzerland 2017
Abstract A new method for production forecasting anduncertainty quantification, applicable for realistic naturallyfractured reservoirs (NFRs) represented as general discrete-fracture-matrix (DFM) models, is developed and applied.The forecasting procedure extends a recently developeddata-space inversion (DSI) technique that generates produc-tion predictions using only prior-model simulation resultsand observed data. The method does not provide poste-rior (history-matched) geological models. Rather, the DSImethod treats production data as random variables. Theprior distribution is estimated from the flow simulationsperformed on prior geological models, and the posteriordata-variable distribution is sampled using a data-space ran-domized maximum likelihood method. The DSI treatmentrequires the parameterization of data variables to renderthem approximately multivariate Gaussian. The complexproduction data considered here (resulting from frequentwell shut-ins) is treated using a new reparameterization thatinvolves principal component analysis combined with his-togram transformation. The DSI method is first applied fortwo-dimensional DFM systems involving multiple fracturescenarios. In this case, comparison with a rejection sam-pling procedure is possible, and we show that the DSI resultsfor P10, P50, and P90 statistics are consistent with rejec-tion sampling results. The DSI method is then applied to a
� Wenyue [email protected]
1 Department of Energy Resources Engineering,Stanford University, Stanford, CA, 94305, USA
2 Chevron ETC, San Ramon, CA, 94583, USA
realistic NFR that has undergone 15 years of primary pro-duction and is under consideration for waterflooding. Toconstruct the DSI representation, around 400 prior DFMmodels, which correspond to different geologic conceptsand properties, are simulated. Two different reference ‘true’models, along with different data-assimilation durations, areconsidered to evaluate the performance of the DSI proce-dure. In all cases, the DSI predictions are shown to beconsistent with the forecasts from the ‘true’ model and toprovide reasonable quantification of forecast uncertainty.
Keywords Data-space inversion · Naturally fracturedreservoir · Discrete-fracture-matrix model · Uncertaintyquantification · History matching · Model inversion
1 Introduction
Naturally fractured reservoirs (NFRs), which account for asignificant portion of worldwide oil and gas resources, posesignificant challenges for reliable reservoir performanceforecasting due to their complex geology and high degreeof heterogeneity. Discrete-fracture-matrix (DFM) modelingmethods have been developed and applied to simulate flowin NFRs [7, 9, 12, 15, 16, 21], and models of this type arenow well established. Before these approaches can be usedfor practical reservoir management, however, they must firstbe capable of matching observed data and providing appro-priate uncertainty quantification. A number of existing dataassimilation, or history-matching, approaches (see [24, 25,31, 32] for comprehensive discussions) are based on theassumption of a fixed grid that is invariant between real-izations. Further, the grid is often assumed not to changebetween prior and posterior (i.e., history-matched) models.These assumptions may be problematic for DFM models, as
1444 Comput Geosci (2017) 21:1443–1458
the simulation grid can change substantially when fracturesare added, removed, or shifted. Thus, in order to use DFMmodels for practical decision making, it will be importantto devise data-assimilation procedures that are not impactedby variability in grid structure.
A method recently presented by Sun and Durlofsky [30],which is referred to as a data-space inversion (DSI) proce-dure, has some features that may render it applicable fordata assimilation with complex DFM models. DSI workssolely in the data space, with the goal of providing produc-tion forecasts and associated uncertainty. The method doesnot involve posterior (history-matched) geological models,so limitations related to gridding and model representationdo not arise. The basic idea in DSI is to view productiondata as a random vector with a prior distribution estimatedfrom flow results that are generated by simulating an ensem-ble of prior-model realizations. The determination of theconditional distribution given observed data is then posedwithin a Bayesian framework, and this distribution is sam-pled using a data-space randomized maximum likelihood(RML) method [18, 23, 26]. Sun and Durlofsky [30] appliedthe DSI procedure to a bimodal channelized system and to aGaussian permeability model. In both cases, they observedclose agreement between results from DSI and those froma strict rejection sampling procedure for key quantities suchas P10, P50, and P90 production rate predictions. The time-consuming step in DSI is the simulation of the prior models(in the cases considered in [30], 500 such simulations wereused), though these can all be run in parallel.
The DSI method shares some similarities with a numberof ensemble-based approaches, such as ensemble Kalmanfilter (EnKF) procedures [1, 5, 34] and ensemble smoother(ES) techniques [2, 6]. These approaches start from an ini-tial ensemble of prior models and associated simulationresults, which are assembled into so-called state vectors.For reservoir-forecasting problems, the state vectors includeboth static model parameters and dynamic production data,which are jointly updated during each iteration when assim-ilating observed data. Because EnKF and ES techniquesrequire the representation of multiple reservoir models onthe same (fixed) grid, these approaches are not directlyapplicable to the DFM models considered in this paper.
Previous studies have addressed the problem of dataassimilation for NFRs. These include the use of thegradual deformation method for history-matching a two-dimensional fractured model [10, 14]. With this treatment,only a single history-matched model is generated, and it isnot clear how to properly construct multiple posterior mod-els for uncertainty quantification. In other work, Cherpeauet al. [3] applied a Metropolis sampling algorithm [22]to generate multiple posterior models for a fractured sys-tem. In this case, however, only a few large-scale fractures,in contrast to the numerous fractures in NFR systems,
were considered. For the Tengiz field, a real NFR system,Dehghani et al. [4] and King et al. [17] applied experi-mental design procedures for uncertainty assessment. Thesestudies used small numbers of experiments, however, so theuncertainty quantification was somewhat heuristic.
Several approaches have also been presented to gener-ate forecasts without updating model parameters. Scheidtet al. [28] and Satija and Caers [27] proposed prediction-focused analysis (PFA) approaches that aim at building a sta-tistical relationship between simulated well data, obtainedfrom an ensemble of prior models, during the history-matching and forecasting periods. When data are observed,the corresponding forecasts are computed using the con-structed statistical relationship. Krishnamurti et al. [19]and Mallet et al. [20] investigated multi-model ensemble-forecasting approaches, in which the forecasts are obtainedas the weighted average of results generated from simu-lating multiple plausible physical models. The weights aredetermined by constraining the forecasts to match observeddata. Both the PFA and multi-model ensemble-forecastingapproaches have some shortcomings, however, whichinclude limitations on the amount of data that can be han-dled and limited treatment of measurement error (see [29]and [30] for further discussion of these issues). This maylimit the applicability of these methods for realistic produc-tion forecasting with DFM models.
In this paper, we extend the general DSI procedure andthen apply it for data assimilation in complex DFM models.Because DSI works in data space, it is already well-suitedfor this problem. However, as we will show later, DSIrequires the reparameterization of production data variables,and the approach developed for this by Sun and Durlof-sky [30] cannot be directly applied to some of the casesconsidered here. This is because our previous treatmentwas based on the existence of a few production stageswith discernible features or patterns (e.g., primary produc-tion at constant oil rate, followed by primary productionat a specified minimum bottom-hole pressure, followed bywater injection, etc). The stages that appear for the realis-tic system considered in this study are more complicated, inthat wells experience multiple shut-ins, and this reduces theapplicability of our previous treatment. We thus introduce ageneral histogram transform technique, in conjunction withprincipal component analysis (PCA), to reparameterize theproduction data. This new treatment is then incorporatedinto the DSI formulation. The generation of multiple con-ditional production forecasts is again accomplished using adata-space RML method. We emphasize that, even thoughthe simulation of realistic DFM models can be time con-suming, the DSI procedure enables the efficient generationof multiple forecasts once the prior models are simulated.Additional advantages and limitations of the DSI approachare discussed in [30].
Comput Geosci (2017) 21:1443–1458 1445
This paper proceeds as follows. In Section 2, we reviewthe general DSI framework introduced in [30]. Reparame-terization of data variables using principal component anal-ysis and histogram transformation is described in Section3. In Section 4, the DSI procedure is applied to synthetictwo-dimensional DFM systems involving realizations froma range of geological scenarios, and the DSI results are com-pared to those from a rejection sampling procedure. Then,in Section 5, we present DSI results for realistic NFR mod-els. We conclude with a summary and suggestions for futurework in Section 6.
2 Data-space inversion formulation
In this section, we provide a brief description of the DSIformulation introduced by Sun and Durlofsky [30]. Pleaserefer to that paper for a complete description of the basicprocedure.
Let m denote the vector containing reservoir modelparameters such as grid-block permeabilities and porosi-ties. The forward model relating model parameters to thedynamic production data is expressed as
dfull = g(m), (1)
where dfull is a column vector of dimension Nd that includesdynamic data of different types (e.g., oil and water pro-duction rates) from multiple wells at different time steps.Data contained in dfull are normally uncertain as the modelparameters in m are initially uncertain. We refer to the ele-ments in dfull as data variables. We consider dfull to includedata variables during both the history-matching and fore-
cast periods, i.e., dfull =[dT
hm, dTpred
]T
, where dhm and
dpred are column vectors containing data variables dur-ing the history-matching and forecast/prediction periods,respectively.
The observed (measured) data variables during thehistory-matching period are assembled into a column vec-tor, dobs, of dimension Nobs. Our goal is to predict thefuture quantities of interest, dpred, conditioned to observa-tions dobs. We emphasize that the goal of the DSI procedureis to generate estimates of dpred without calibrating modelparameters.
We begin by writing the conditional probability densityfunction (PDF) of dfull given dobs as
p(dfull|dobs) = p(dobs|dfull)p(dfull)
p(dobs)
∝ p(dobs|dfull)p(dfull), (2)
where p(dfull) is the prior PDF of dfull and p(dobs|dfull),referred to as the likelihood function of dfull, represents the
conditional PDF of observing dobs given dfull. The produc-tion forecasts, which comprise part of dfull, can be generatedby sampling the posterior distribution of dfull using Eq. 2once the likelihood function and prior PDF are defined.
Because dobs contains direct measurements of dhm, therelationship between dobs and dfull can be written as
dobs = Hdfull + ε, (3)
where H is an Nobs × Nd matrix, consisting of zeros andones, that simply extracts the elements in dfull correspond-ing to dhm, and ε denotes the measurement error vector. Themeasurement errors are assumed to be Gaussian distributedwith zero mean and covariance matrix CD. Therefore, thelikelihood function of dfull is Gaussian and can be writtenas follows:
p(dobs|dfull)
= p(ε = dobs − Hdfull)
∝ exp
(−1
2(Hdfull − dobs)
T C−1D (Hdfull − dobs)
). (4)
Sun and Durlofsky [30] provided analytical solutions forthe posterior distribution p(dfull|dobs) under the conditionthat the prior PDF of dfull is Gaussian. For cases when thisprior PDF is far from Gaussian, they showed that improvedDSI performance can be obtained by reparameterizing thedata variables in dfull into new variables that are nearlymultivariate Gaussian. If we let η denote the new (approxi-mately) multivariate Gaussian vector with prior mean ηpriorand covariance matrix Cη, the relationship between dfull andη can be written as
dfull = f(η). (5)
Thus, instead of directly sampling p(dfull|dobs) from Eq. 2,we first formulate and sample the conditional distributionof η given dobs. The ensemble of forecasts for dfull can thenbe generated by applying Eq. 5 to the corresponding condi-tional samples of η. Combining Eqs. 4 and 5, the conditionaldistribution of η can be expressed as
p(η|dobs)
∝ p(dobs|η)p(η)
∝ exp
(−1
2(H f(η) − dobs)
T C−1D (H f(η) − dobs)
−1
2(η − ηprior)
T C−1η (η − ηprior)
). (6)
In the following section, we describe the reparameter-ization of data variables and then present the data-spacerandomized maximum likelihood (RML) method to samplethe resulting posterior distribution.
1446 Comput Geosci (2017) 21:1443–1458
3 Reparameterization of data variables
As discussed in the Introduction, Sun and Durlofsky [30]introduced a pattern (or production-stage) based mappingtechnique suitable for handling time-series production datavariables that exhibit a few discernible patterns or features.This approach requires pre-specifying the production stages(and data patterns), and it may encounter difficulties whenthe production data display complex behavior with no obvi-ous patterns, as occurs with the data considered in this paper.In this section, we describe a new reparameterization pro-cedure that does not require any pre-specification of datapatterns and is thus applicable to any type of productiondata.
3.1 PCA representation of data variables
DSI entails first performing flow simulation for multiple(Nr) independently generated prior reservoir models. Wedesignate (dfull)1 to (dfull)Nr to be the data vectors obtainedby solving Eq. 1 for prior models m1,m2, . . . ,mNr . Wethen apply principal component analysis (PCA) to the sim-ulated data. As discussed in [30], this treatment eliminatesthe need to compute the inverse of Cη, and also reduces the
number of variables to be determined. To apply PCA, wefirst construct the centered matrix X ∈ R
Nd×Nr as
X = [ (dfull)1 − dprior (dfull)2 − dprior . . .
. . . (dfull)Nr − dprior]. (7)
We assume there is no modeling error associated with theforward model (Eq. 1). Therefore, (dfull)i , i = 1, 2, . . . , Nr
can be viewed as prior realizations of the random vectordfull, with the prior mean of dfull estimated as dprior =(1/Nr)
∑Nri=1(dfull)i .
We then perform singular value decomposition ofX/
√Nr − 1, which provides the eigenvalues and eigenvec-
tors of the covariance matrix XXT /(Nr−1). The square rootof the eigenvalues (ordered by descending value) form thediagonal elements of the matrix �, and the correspondingeigenvectors comprise the columns of the matrix U . Thus,new data realizations, which we refer to as dPCA, can begenerated through application of
dPCA = UΣξ + dprior = Φξ + dprior, (8)
where ξ is a vector containing uncorrelated variables withstandard normal distribution, and Φ is the basis matrix. The
Fig. 1 Example illustrating theimpact of the PCAparameterization of productiondata
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Comput Geosci (2017) 21:1443–1458 1447
leading columns in Φ are associated with the largest eigen-values of the covariance matrix XXT /(Nr − 1). In general,only the first Nl < Nr columns are retained, which meansthat Φ ∈ R
Nd×Nl and ξ ∈ RNl×1. See [30] for further details
on the construction of Φ.We now apply the PCA procedure described above to
simulated water production rate (WPR) data from a set of400 prior models. The data, shown in Fig. 1a, are normal-ized by the maximum WPR value among all wells at alltime steps. The prior geological models, which are intendedto represent a real naturally fractured reservoir, will bedescribed below. The data shown in Fig. 1a are reportedmonthly for 11 years, from around 2200 to 6200 days.Assuming for now that data vector dfull contains only theseWPR data, with Nd = 132, we have (dfull)i ∈ R
132×1
(i = 1, 2, . . . , Nr). Using the PCA parameterization (Eq. 8),we generate 400 new data realizations. These are shown asthe gray curves in Fig 1b. It is clear that the PCA realizationsdisplay unphysical behavior, particularly negative WPR val-ues. Figure 1c and d shows the histograms of the simulateddata and PCA data realizations at 6200 days. The histogramfrom the PCA realizations (Fig. 1d) is close to Gaussian andclearly quite different than the histogram from the simulateddata.
3.2 Histogram transformation
To improve the ability of the PCA parameterization to pre-serve the physical character of the production data variables,we introduce an additional histogram transformation step.The basic idea is to transform a PCA realization dPCA
(obtained using Eq. 8) to another realization d = hT(dPCA)
that has a specified target histogram for each variablein d. In this paper, the target histograms/distributions areobtained empirically from the ensemble of simulated data,as these data vectors represent the prior realizations of ran-dom vector dfull (Fig. 1c, for example, shows the targethistogram of WPR at 6200 days).
Because dPCA is a linear transformation of the multi-variate (standard) normally-distributed vector ξ (Eq. 8), thedistribution of dPCA is multivariate Gaussian with meandprior and covariance matrix ΦΦT . Thus, the initial his-togram/distribution of each variable in dPCA is Gaussian,with mean defined by the corresponding row of dprior
and variance defined by the diagonal term in ΦΦT . LetfI(d
j
PCA) and fT(dj
PCA) denote the initial and target cumu-lative distribution functions (CDFs) for row j in dPCA.Then, each element in the target realization d is obtained asd
j
PCA = f −1T fI(d
j
PCA), j = 1, . . . , Nd.Figure 2 shows the PCA data realizations before and after
this histogram transformation. It is clear that this treatmentsignificantly improves the ability of PCA to reproduce the
physical, non-negative character of the simulated WPR data(Fig. 1a). The red, blue, and green curves in Fig. 2a indicatethree particular PCA data realizations. The correspondingrealizations after histogram transformation are shown inFig. 2b. Although the red curve in Fig. 2a displays negativeWPR values, these are all non-negative after the histogramtransformation.
3.3 DSI formulation with reparameterization
Incorporating PCA and the histogram transform, we expressthe data vector dfull as a function of ξ as
dfull ≈ f(ξ) = hT(dPCA) = hT(Φξ + dprior). (9)
We take ξ to be the reparameterized vector representing dfull
in Eq. 5, that is η = ξ . Because ξ is multivariate standardnormal, Eq. 6 becomes
p(ξ |dobs) ∝ exp
(−1
2(H f(ξ) − dobs)
T C−1D (H f(ξ) − dobs) − 1
2ξT ξ
),
(10)
where f(ξ) is defined in Eq. 9. Equation 10 characterizesthe posterior distribution of new parameters ξ given dobs.Note that the (possibly ill-conditioned) (η−ηprior)
T C−1η (η−
ηprior) term in Eq. 6 no longer appears. Rather, after appli-
cation of PCA, this term now appears as ξT ξ .To sample this distribution, we apply the randomized
maximum likelihood (RML) method in the data space. Theobjective function is thus defined as
ξ rml = arg minξ
[(H f(ξ) − d∗
obs)T C−1
D (H f(ξ) − d∗obs)
+(ξ − ξ∗)T (ξ − ξ∗)]. (11)
Here, d∗obs are perturbed observations from the Gaussian dis-
tribution N[dobs, CD], and ξ∗ are sampled from the standardnormal distribution N[0, I ]. In this work, the quasi-Newtonline search algorithm within Matlab is used to find the mini-mum of the objective function in Eq. 11. This minimizationis generally accomplished in just a few seconds since theobjective function evaluations are very fast.
We have now completed the general description of theDSI procedure with our new reparameterization. Addi-tional algorithmic details are provided in [30]. The overallapproach applied here can be summarized as follows:
1. Perform flow simulation on an ensemble of prior modelrealizations, conditioned to prior geological informa-tion, to obtain simulated data (dfull)i , i = 1, 2, . . . , Nr.
2. Using PCA and histogram transformation, representdata variables dfull as a function of ξ (Eq. 9).
1448 Comput Geosci (2017) 21:1443–1458
3. Apply the RML method (in the data space) to samplethe conditional distribution of ξ given dobs (Eq. 10).Then, generate the corresponding forecasts throughapplication of Eq. 9.
It is important to reiterate that the only time-consumingstep in the entire procedure is performing the flow simu-lations for the prior models. These simulations can all berun in parallel, so the elapsed (wall-clock) time might cor-respond to only the time required for a few simulations if asufficient number of processors are available. If necessary,these runs can be accelerated by using upscaled DFM mod-els. The recent procedure presented in [15] could be appliedfor this purpose. The total computation time needed to gen-erate multiple forecasts with the data-space RML procedureis on the order of minutes.
4 Numerical results for synthetic models
In this section, the DSI procedure described above is appliedto synthetic two-dimensional DFM examples. The priormodels in these cases involve DFM realizations gener-ated from a range of fracture scenarios. The DSI resultsfor uncertainty quantification will be compared to thoseobtained from rejection sampling (RS).
4.1 Problem setup and prior-model predictions
The model covers an area of 2100 m × 2100 m and is ofthickness 100 m. An oil-water system is considered. Vis-cosities of oil and water at standard conditions are 1 cp and0.3 cp, respectively. The initial oil and water saturations are0.9 and 0.1. The oil and water matrix relative permeabilityfunctions
(kmro and km
rw
)are specified as
kmro =
(1 − Sor − Sw
0.9 − Sor
)2
, kmrw = 0.8
(Sw − 0.1
0.9 − Sor
)2
,
(12)
where Sw denotes water saturation and Sor is the resid-ual oil saturation (in this example Sor will be treated as anuncertain parameter). In the fractures, straight-line relativepermeability functions are specified for both phases, i.e.,kfro = 1 − Sw and k
frw = Sw. Capillary pressure effects are
ignored. The matrix and fracture porosities are assumed tobe constant at 0.2 and 0.3, respectively. One water injectionwell is placed at the center of the model, and four produc-tion wells are located near the corners, as shown in Fig. 3.The injector operates at a fixed bottom-hole pressure (BHP)of 3600 psi, and all producers operate at a fixed oil produc-tion rate (OPR) of 700 STB/day, subject to a minimum BHPof 2500 psi.
Figure 3 shows two model realizations corresponding todifferent fracture densities and orientations. Fractures aredisplayed as black or red lines and the background grayregion corresponds to matrix. The red fractures are the frac-tures intersected by wells. These are taken to be hard dataand are the same in all realizations. In each model, thereare two sets of black fractures, which are assumed to beorthogonal to each other.
Table 1 presents the six model parameters that are con-sidered to be uncertain. Fracture density denotes the averagefracture surface area per rock volume. The fracture den-sities for the models shown in Fig. 3a and b are 0.006and 0.01/m, respectively. Fracture orientation, in the x-yplane, represents the orientation for one set of fractures (theother set of fractures is oriented orthogonally). Matrix per-meability is uncertain but always nonzero, which enablessome flow through the matrix. The fracture permeabilitykf depends on the fracture aperture d via the relationshipkf = 1000d2 (the units of kf and d are md and mm, sofor d = 10 mm, kf = 105 md). Finally, residual oil satu-ration, which impacts the relative permeability curves (seeEq. 12), and rock compressibility, are also considered to beuncertain.
All parameters are assumed to be independently and uni-formly distributed a priori within the specified ranges shown
Fig. 2 Example illustratingimprovement in WPRrealizations when PCA is usedin conjunction with histogramtransformation
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Fig. 3 Fracture networks for two DFM realizations. Fractures areshown in black or red, and matrix is in gray. Injection (I1) andproduction (P1 to P4) wells are also displayed
in Table 1. Any set of these six model parameters definesa particular geological scenario. Within a scenario, an infi-nite number of realizations can be generated. In this work,we randomly select 100 geological scenarios (consistentwith the ranges in Table 1), and for each of these sce-narios we generate 100 fracture realizations. This resultsin a total of 10,000 prior-model realizations. It is impor-tant to emphasize that the fracture networks are differ-ent for all realizations, though realizations in the samescenario display the same average fracture density andorientation.
Figure 4 presents simulation results, for injector I1 andproducers P1 and P2, for 200 prior models (selected ran-domly from the set of 10,000). The simulation data at allwells are reported every 60 days over a total simulation timeframe of 7680 days. The variability in these prior simula-tion results is very large. For example, the predicted OPRplateau periods, shown in Fig. 4c and f, range from thestart to the end of simulation time frame for both P1 andP2. In addition, the impact of the (uncertain) fracture net-work on flow can be seen from the water production rate(WPR) results. When the injector and producer are con-nected by fractures, the injected water reaches the producerquickly. This phenomenon is evident in Fig. 4d, where wesee very early water breakthrough times (less than 100 days)for some realizations.
4.2 Rejection sampling procedure
Sun and Durlofsky [30] applied a rejection sampling (RS)procedure to provide reference results for uncertainty quan-tification. Though computationally expensive, RS can pro-vide a proper sampling of the posterior distribution of theproduction data. In this study, we apply a rejection samplingprocedure that differs slightly from that used in [30]. Themethod now proceeds as follows:
1. Generate model realization m consistent with priorgeological information.
2. Sample a variable p from a uniform distribution withinthe range [0,1].
3. Accept m as a posterior realization if p ≤ L(m)/SL,where L(m) is the likelihood function and SL is a valuethat is greater than or equal to the maximum of thelikelihood function.
The likelihood function L(m) is defined as
L(m) = c exp
(−1
2O(m)
), (13)
where c is a normalization constant. The quantity O(m)
is the weighted mismatch between observed and simulateddata, written here as
O(m) = 1
Nobs(g(m) − dobs)
T C−1D (g(m) − dobs). (14)
Because O(m) is nonnegative, we have L(m) ≤ c forany model m. Therefore, we set SL to be c in this study. Itis important to observe that the mismatch O(m) in Eq. 14is computed as the mean of the weighted data mismatch,rather than as the sum of the squared data mismatch. The lat-ter representation was used in [30], meaning that the factorof 1/Nobs in Eq. 14 did not appear in that work. The use ofthe 1/Nobs factor decreases the value of the mismatch func-tion O(m) and thus results in the acceptance of more priormodels within the RS procedure (in the absence of this scal-ing, few if any of the 10,000 prior model results would beaccepted).
For the purposes of strict uncertainty quantification, wedo not believe it is appropriate to introduce this 1/Nobs
Table 1 Uncertain modelparameters, ranges, and valuesfor ‘true’ models. Allparameters are assumed to beuniformly distributed a priori
Parameter Lower bound Upper bound ‘True’ model 1 ‘True’ model 2
Fracture density (×10−3/m) 6 10 8.8 6.6
Fracture orientation 25 65 65 62
Fracture aperture (mm) 8 12 8 9.4
Matrix permeability (mD) 0.3 0.5 0.43 0.34
Rock compressibility (×10−5/psi) 1 5 2.1 4.3
Residual oil saturation 0.1 0.3 0.19 0.16
1450 Comput Geosci (2017) 21:1443–1458
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Fig. 4 Simulated production results from prior models (two-dimensional synthetic case)
factor when we have zero model error and Gaussian mea-surement error (as we have here). Thus, we cannot claim thatthe resulting RS predictions provide the actual uncertaintyassessment. However, the comparison between the DSI andRS results will still be valid as along as the likelihood func-tions used for DSI and RS are consistent. This means wecan still use the RS results to assess the DSI procedure.To enable a valid comparison with RS, the DSI objectivefunction (Eq. 11) is now written as
ξ rml = arg minξ
[(ξ − ξ∗)T (ξ − ξ∗) + (H f(ξ) − d∗
obs)T
× (NobsCD)−1(H f(ξ ) − d∗obs)
], (15)
where we have (NobsCD)−1 instead of C−1D in the data mis-
match term. In addition to the use of Eq. 15 in place ofEq. 11, in the DSI RML procedure we must also sam-ple (perturbed) observations from the Gaussian distributionN[dobs, NobsCD] rather than from N[dobs, CD].
4.3 Results for two ‘true’ models
We now apply the DSI procedure for two different ‘true’synthetic two-dimensional models. ‘True’ model 1 corre-sponds to predictions that, visually, fall around the middleof the range of predictions from the prior models (Fig. 4),
and ‘true’ model 2 corresponds to predictions toward theedge of the ranges. In both cases, observed data are gener-ated by adding random noise to the true data (true data arethe predictions from the selected ‘true’ model). The noiseis sampled from the Gaussian distribution N[0, CD]. Thestandard deviations of the noise are set to be 30 psi forBHP data, 100 STB/day for WIR data, and 5% of the cor-responding true data, with a maximum of 20 STB/day forWPR and OPR data (as noted above, in the RML com-putations we perturb the observations with noise sampledfrom N[0, NobsCD] rather than N[0, CD]). The historicalperiod is taken to be the first 1200 days, with well dataobserved every 60 days at I1, P1, and P2, which results in 70observed data points. We focus on forecasts for productiondata for I1, P1, and P2. The total simulation time is 7680days and we take uniform 60-day time steps. We thus haveNd = 7 × 128 = 896, and (dfull) ∈ R
896×1.As discussed in Section 4.1, we generated a total of
10,000 prior realizations from a set of 100 geological sce-narios. For the RS procedure, we use 9800 of the 10,000prior models. The 200 models discarded correspond to thosefor the two geological scenarios from which the two ‘true’models were generated (which means that the true scenariois not represented in the set of prior realizations). For theDSI method, we use a total of Nr = 1000 prior models,which are randomly selected from the set of 9800 prior mod-els used with RS. See [30] for a discussion of the choice of
Comput Geosci (2017) 21:1443–1458 1451
Nr. As noted there, if too few models are accepted basedon the RML offset criterion, this suggests that more priormodels are needed. In this paper, we use the same ‘energy’criterion described in [30] to determine the number of prin-cipal components to retain in the Φ matrix in Eq. 8. In thiscase, a total of Nl = 9 principal components are required.
Forecasting results for ‘true’ model 1, in terms of P10,P50, and P90 statistics, from prior models, RS and DSI,are presented in Fig. 5. The red dots represent observeddata, and the dashed red curves denote the true data. Thegray regions indicate the P10–P90 range of forecasts fromprior models, which are obtained using the ensemble ofprior forecasts shown in Fig. 4. In the RS procedure, 268models (out of the 9800 simulated) are accepted. For DSI,a total of 100 forecasts are generated. The posterior P10,P50, and P90 statistics estimated from the RS and DSI fore-casts are shown as dashed blue and black curves in Fig. 5.The RS and DSI results are seen to be in generally closeagreement (though some differences are evident in the P90results in Fig. 5d), which demonstrates that DSI is able toprovide uncertainty quantification results that are consistentwith those from a much more expensive RS procedure. Wereiterate that, for the results in this paper, this consistency isdemonstrated for the likelihood function in Eq. 14, thoughwe could also apply DSI without the 1/Nobs scaling.
The DSI forecasts for all quantities in Fig. 5 displaynarrower uncertainty ranges than the prior forecasts. Uncer-tainty is reduced dramatically for, e.g., P1 oil and waterproduction rates. Also of interest is the fact that the true data(dashed red curves) consistently fall within the P10–P90range of the DSI forecasts.
We now consider ‘true’ model 2. Prior and posterior flowresults for this case are shown in Fig. 6. As noted earlier, theflow responses for ‘true’ model 2 are toward the edge of theprior-model predictions. In fact, the true data fall outside ofthe P10–P90 range of forecasts from the prior models for I1WIR (Fig. 6a) and for P2 BHP and OPR (Fig. 6e and f). Asa result, only 27 models out of the 9800 prior models con-sidered are accepted in the RS procedure (in contrast to 268for ‘true’ model 1). Again, a total of 100 DSI forecasts aregenerated. As shown in Fig. 6, despite the fact that relativelyfew data responses lie ‘near’ that of ‘true’ model 2, we seethat the DSI results still agree reasonably well with the RSresults (some differences are evident, however, for P2 OPRin Fig. 6f). It is noteworthy that the true data are alwaysencompassed within the P10–P90 range of DSI forecasts,even when they fall outside the P10–P90 range of prior fore-casts (as in Fig. 6a and f). As was the case for ‘true’ model1, the uncertainty reduction in the posterior predictions isagain significant.
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line defines the history-match period. Gray-shaded regions representthe P10 to P90 range of predictions from prior models. Black and bluecurves denote the P10, P50, and P90 results from DSI and RS
1452 Comput Geosci (2017) 21:1443–1458
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We reiterate that neither ‘true’ models 1 and 2, norany realizations from their corresponding scenarios, wereincluded in the set of prior models used with DSI. Thismeans that, at least for the cases presented above, the DSImethod is able to provide reasonable uncertainty quantifica-tion, even when the true fracture/matrix property scenario isnot contained in the prior ensemble. Although the systemsconsidered here are relatively simple, history-matching thistype of model can still be challenging for traditional model-inversion algorithms due to the difficulty of parameterizingfracture networks. The DSI method may indeed be attrac-tive for such systems if the primary goal is the productionforecast and associated uncertainty assessment, rather thanthe generation of a set of posterior models.
5 Numerical results for realistic field models
The results in Section 4 demonstrated the performanceof DSI for synthetic two-dimensional cases. We nowapply the method using an ensemble of three-dimensionalDFM models generated to characterize a real naturallyfractured carbonate oil reservoir. This example againinvolves uncertainty in the geological scenario, though onlysix different fracture-network realizations are considered
(our emphasis is now on uncertainty in other reservoirparameters, as described below). The realizations containaround 9000 discrete fractures, and the resulting DFM gridsinclude about 900,000 total cells, ∼600,000 of which corre-spond to matrix control volumes, and ∼300,000 to fracturecontrol volumes. The basic reservoir model is illustrated inFig. 7. Many of the fracture and matrix properties, discussedbelow, are similar to those applied in [11, 13]. There are 11
Fig. 7 One of the unstructured DFM grids generated for the natu-rally fractured reservoir of interest. Color indicates relative depth ofdiscrete-fracture cells. Underlying grid indicated by the gray lines
Comput Geosci (2017) 21:1443–1458 1453
active producers and one shut-in well that will be evaluatedfor use as an injector in a pilot waterflood.
We now describe how the DFM model properties areassigned. We use an experimental design involving fouruncertain parameters, shown in Table 2, to create an ensem-ble of ∼400 DFM models. In addition to the base-casegeologic concept, two other possibilities are considered: onewith additional fractures within the aquifer, and one withadditional fractures in the target pilot waterflood area. Twostochastic fracture networks are considered for each of thesethree concepts, giving rise to a total of six fracture real-izations. For fracture permeabilities, we specify values thatare scaled in accordance with fracture apertures (which arein turn correlated with fracture length) and are calibratedto well test data. Matrix properties such as porosity andirreducible water saturation vary spatially and are derivedfrom a well-calibrated, structured reservoir model of thefield. The base-case mean matrix and fracture permeabil-ity values differ by 5 orders of magnitude. The fracture andmatrix cells are assigned different relative permeabilities,capillary pressures, and rock compressibilities. We specifystraight-line relative permeabilities, zero capillary pressure,and higher rock compressibility for fractures. To reflectuncertainty in fracture permeability, fracture pore volume,and matrix permeability, global multipliers are applied forthe low and high cases (see Table 2 for multiplier values).Finally, uncertainty in the degree of water imbibition inthe matrix is captured through the use of three differentwater-oil capillary pressure curves.
The actual reservoir has undergone primary depletion for5400 days. A water-injection pilot, which entails injectionfor 60 days and monitoring for 800 days, is being consideredto evaluate the improved-oil-recovery potential for the field.The water is of viscosity slightly higher than that of thereservoir oil at existing temperature and pressure. Here wewill consider a (typical) scenario in which the history-matchperiod is shorter than the prediction period. We thus specifya 2400-day history-match period during which data are col-lected (except where otherwise indicated), and then considera prediction period that involves 3000 days of additionalprimary depletion followed by the 800-day pilot-evaluation
Table 2 Set of uncertain parameters considered for six differentstochastic DFM realizations. These various combinations define theensemble of prior DFM models for the realistic field case
Parameter Low Medium High
Fracture permeability × 0.2 × 1 × 10
Fracture pore volume × 0.7 × 1 × 0.7
Matrix permeability × 0.05 × 1 × 20
Matrix water imbibition level Low Mid High
period (for a total prediction period of 3800 days). Duringthe initial 5400-day period, the 11 producers are constrainedto operate at their individual historical oil production rates(these wells experience multiple shut-ins for operationalreasons). The observed data include static well pressures(SWPs) and water production rates (WPRs).
In the pilot-evaluation period, the shut-in well is used toinject water for 60 days (the time during which water is actu-ally injected is brief to minimize operational disruptions).This well operates at a target rate subject to a maximumbottom-hole pressure (BHP). During the 800-day evaluationperiod, the producers are specified to operate at prescribedtarget oil rates subject to minimum BHP constraints. Again,SWPs and WPRs are monitored at all producers during thisperiod.
Figure 8 shows simulation results for producers P1 andP3 (which are two wells that display relatively high WPRs),for 398 prior models. The SWP data are normalized by themaximum static well pressure observed for any well at anytime. The data plotted in blue correspond to 79 prior mod-els under the geologic concept ‘base + aquifer fractures,’and the data plotted in green correspond to 319 prior mod-els from the other two geologic concepts (defined above).Because wells are controlled by oil production rate for mostof the simulation period (as noted above), we focus on theforecasts for SWP and WPR. In the DSI procedure, thedata vectors (dfull)i , i = 1, 2, . . . , Nr, include SWP andWPR data at 11 wells (the shut-in water-injection well is notconsidered). The simulation data at all wells are reportedmonthly for the total simulation time frame of 6200 days (or17 years), resulting in Nd = 17 × 12 × 2 × 11 = 4488, and(dfull)i ∈ R
4488×1. Though the dimension of the data vectoris large, the total number of principal components retainedfor the basis matrix (Eq. 8) is only 13. This large amount ofreduction results from the strong cross-correlation betweenthe data variables in dfull.
Rather than use actual field data, we will evaluate theperformance of the DSI procedure for two different geo-statistical ‘true’ models that correspond to different waterproduction behavior. One of these ‘true’ models is selectedfrom the ‘green’ set of models and one is from the ‘blue’ setof models. The true data are then generated by performingflow simulation on the ‘true’ model. Observed data (whichare used in DSI) are obtained by adding random Gaussiannoise, sampled from N[0, CD], to the true data. The stan-dard deviation of measurement noise is now specified to be2% of the corresponding pressure/rate data in all cases. WellSWPs and WPRs are observed every 2 months at all wells.This results in 902 observed data points for a 2400-dayhistory-match period. It is important to emphasize that nei-ther of the ‘true’ models is included in the Nr prior modelsused in the DSI procedure. Due to the limited number (398)of prior models available, the rejection sampling procedure
1454 Comput Geosci (2017) 21:1443–1458
Fig. 8 Simulated productionresults from prior models forrealistic field case. Green andblue curves represent forecastscorresponding to differentgeological concepts
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described in Section 4.2 will not be applied here. Therefore,the objective function shown in Eq. 11, instead of that in Eq.15, is used to generate the DSI forecasts.
5.1 DSI results for ‘true’ model 3
‘True’ model 3 is selected from the ‘green’ set of models.These models generally lead to less water production thanmodels from the ‘blue’ set (as is evident in Fig. 8), though‘true’ model 3 was specifically chosen because it leads tonon-negligible WPRs. This case should therefore be some-what challenging for the DSI approach since many of theforecasts within the ‘green’ scenario are far from the trueresults. Figure 9a and b presents DSI results for well P1.The red points again correspond to the observed data (thesepoints lie to the left of the vertical dot-dash line at 2400 days,which indicates the history-match period), and the dashedred curves represent the forecast from the ‘true’ model. Thegray-shaded regions represent the P5 to P95 forecast rangefrom the prior models (note that in Section 4 we consideredP10 to P90 ranges). The solid black curves represent the P5,P50, and P95 results obtained using DSI. In Fig. 9b and c,only one curve (P95) is evident, since the P5 and P50 curvesin these figures correspond to zero water production.
For both SWP and WPR, the DSI results in Fig. 9a andb display narrower uncertainty ranges (P5–P95) compared
with those from the prior models. It is also apparent that thetrue forecasts (dashed red curves) consistently lie within theP5–P95 range of the DSI predictions. Because WPR dis-plays large variations due to the frequent well shut-ins, forimproved clarity, we present forecasts of cumulative waterproduction (CWP) in Fig. 9c. The cumulative distributionfunction (CDF) of CWP at the end of the forecasting period(6200 days) is shown in Fig. 9d. Substantial reduction inforecast uncertainty through use of the DSI procedure isevident in both plots.
Forecasts for well P3 are shown in Fig. 10. The SWPat P3 displays larger variations in the forecasting periodthan were observed for P1. Nonetheless, the DSI procedurestill narrows the range of uncertainty considerably com-pared with that from the prior models. In addition, the trueforecasts consistently lie within the P5–P95 DSI interval.CWP forecasts (and the degree of uncertainty reduction) forwell P3 (Fig. 10b and c) are generally similar to those forP1, and the true data are again captured within the P5–P95range. It is noteworthy that even though neither well experi-ences water breakthrough during the history-match period,forecast uncertainty in CWP (and WPR) is still reducedsubstantially after assimilating the observed data. Thisdemonstrates that the DSI approach captures correlationsbetween data of different types (SWP and WPR in this case)and at different wells.
Comput Geosci (2017) 21:1443–1458 1455
Fig. 9 Statistics of productionforecasts for well P1 from priormodels and DSI. Red points andred dashed curves indicateobserved data and predictionsfrom ‘true’ model 3. Verticaldot-dash line defineshistory-match period.Gray-shaded regions representthe P5 to P95 range ofpredictions from prior models.Black curves denote the P5, P50,and P95 results from DSI
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5.2 DSI results for ‘true’ model 4
We now present results for ‘true’ model 4. This model wasselected from the ‘blue’ set of models in Fig. 8, which cor-respond to higher WPRs. DSI results for ‘true’ model 4are shown in Fig. 11. The SWP forecasts for the two wells(Fig. 11a and d) are comparable to those for ‘true’ model3, even though the fluctuations are greater here. The CWPresults (Fig. 11b and e) are quite interesting in this case.For both wells, the P5–P95 uncertainty intervals are actu-ally expanded, rather than reduced as we might expect, afterapplication of the DSI procedure. However, the true data
now consistently fall within the P5–P95 range, while theylie slightly above the P95 prior from around 3000 days to5000 days in Fig. 11b. This lack of uncertainty reduction isalso evident in the CWP CDFs shown in Fig. 11c and f.
Water breakthrough does not occur over the 2400-day history-match period in Fig. 11. To demonstrate theuncertainty reduction that can be achieved once waterbreakthrough is observed, we now extend the history-matching period to 3000 days, at which time water is pro-duced in well P3. We now observe significant uncertaintyreduction relative to the prior forecasts for CWP in bothwells (Fig. 12b and e), with the true data again falling within
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Fig. 10 Statistics of production forecasts for well P3 from prior models and DSI for ‘true’ model 3. Lines, shadings, and colors have the samemeanings as in Fig. 9
1456 Comput Geosci (2017) 21:1443–1458
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Fig. 11 Statistics of production forecasts, for wells P1 and P3, from prior models and DSI for ‘true’ model 4. Lines, shadings, and colors havethe same meanings as in Fig. 9
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Fig. 12 Statistics of production forecasts, for wells P1 and P3, from prior models and DSI for ‘true’ model 4. History-match period here is 3000days. Lines, shadings, and colors have the same meanings as in Fig. 9
Comput Geosci (2017) 21:1443–1458 1457
the P5–P95 DSI interval. This high degree of uncertaintyreduction is also apparent in the CWP CDFs in Fig. 12c andf, as well as in the SWP forecasts, especially for well P3(Fig. 12d).
It is noteworthy that there are only 79 prior models inthe ‘blue’ set of models, from which ‘true’ model 4 wasselected. The other 319 prior models used by DSI pro-vide quite different forecasts (e.g., low WPRs, as seen inFig. 8). Nonetheless, the DSI procedure is still able to pro-vide reasonable uncertainty quantification results for thiscase. This again demonstrates that DSI can be appliedwith prior models that derive from different geologicalscenarios/concepts. This is in contrast to many existinghistory-matching approaches, such as the PCA-based meth-ods described in [32, 33], which require one scenario (i.e.,training image) at a time to be considered. It is also impor-tant to emphasize that no additional simulation runs wereperformed when more data were included in the inver-sion. Again, this is in contrast to most existing approaches,which require additional simulations when new data areintroduced.
6 Concluding remarks
In this paper, we extended the data-space inversion (DSI)procedure developed by Sun and Durlofsky [30] to enableits application to a realistic naturally fractured carbonatereservoir. In that case, due to the complex well controls (fre-quent shut-ins) employed, the production responses fromthe prior models do not display clear stages/patterns. Thismotivated the development of a new data reparameterizationtechnique that entails the use of PCA along with histogramtransformation. These treatments were implemented into adata-space RML procedure in order to generate multipleproduction forecasts.
We first applied the new DSI treatments to a synthetictwo-dimensional DFM system. This case involved realiza-tions drawn from a wide range of geological scenarioscharacterized by different fracture densities, orientationsand apertures, and different matrix properties. Two different‘true’ models were considered, and DSI predictions werecompared with rejection sampling results (using the meanof the weighted data mismatch) for both cases. Acceptableagreement between DSI and RS was achieved, suggestingthat DSI can be used for cases that include a variety ofgeological scenarios. We then applied the DSI procedureto a realistic three-dimensional naturally fractured system.Two different ‘true’ models, which corresponded to dif-ferent fracture scenarios, were again considered. In bothcases, we observed reasonable uncertainty quantificationresults using the DSI method, even when the forecastsfrom the ‘true’ model were outside the P5 to P95 range of
predictions from prior models. The impact of varyingamounts of historical production data on uncertainty reduc-tion was also demonstrated for one of the cases.
The DSI procedure has the advantage of requiring flowsimulations only for prior models, and these can all be runin parallel. Once these simulations are performed, the data-space RML procedure can be run in a matter of minutes.Another advantage of DSI is that additional simulation runsneed not be performed if new data become available. Inaddition, the impact of quantities such as measurement error(appearing in CD), which may also be used to approximatemodel error, can be efficiently studied. In this case, onlydata-space computations, not additional flow simulations,are required if different CD matrices are considered. Andfinally, DSI is well-suited for studies in which different real-izations are represented on different grids, as is typically thecase with DFM models (many traditional data-assimilationapproaches are limited in this regard).
The significant limitation with DSI, however, is the factthat it does not provide posterior (history-matched) geolog-ical models. Such models are of course required in someapplications. Other limitations of the DSI methodology arediscussed in [30]. These include the need for a sufficientlybroad set of prior-model scenarios, as were considered inSection 4 for the synthetic case. The DSI procedure in itscurrent form cannot be used for cases in which future welllocations or settings are not known when the DSI modelis constructed. This means that the method is not yet suit-able for use in optimizing well controls (e.g., time-varyingBHPs) or the placement of new wells. Finally, the proceduremay require the simulation of a large number of prior real-izations in problems involving complicated data responsesand/or a large amount of data.
In future work, we plan to extend DSI treatments toaddress several of these issues. For practical cases withmany wells and large data volumes, it may be beneficialto decompose the reservoir into multiple regions and tothen apply DSI region by region. It will also be useful toestablish linkages between DSI and traditional model-basedinversion methods. This may allow us to identify a smallnumber of realizations, and associated weightings, that pro-vide flow responses in agreement with those from DSI. Thiscould facilitate the application of DSI for optimization. Wealso plan to consider the use of DSI for the design of areservoir surveillance system (as in [8]) or to investigatethe value of information. For both problems, the impact onuncertainty reduction of different data types, and data mea-sured at various locations, can be quantified very efficientlyusing DSI.
Acknowledgments We thank Chevron ETC and the Stanford SmartFields Consortium for financial support.
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References
1. Aanonsen, S.I., Nævdal, G., Oliver, D.S., Reynolds, A.C., Valles,B.: The ensemble Kalman filter in reservoir engineering—areview. SPE J. 14(3), 393–412 (2009)
2. Chen, Y., Oliver, D.S.: Ensemble randomized maximum likeli-hood method as an iterative ensemble smoother. Math. Geosci.44(1), 1–26 (2012)
3. Cherpeau, N., Caumon, G., Caers, J., Levy, B.: Method forstochastic inverse modeling of fault geometry and connectivityusing flow data. Math. Geosci. 44(2), 147–168 (2012)
4. Dehghani, K., Fischer, D., Skalinski, M.: Application of inte-grated reservoir studies and techniques to estimate oil volumes andrecovery—–Tengiz field, Republic of Kazakhstan. SPE Reserv.Eval. Eng. 11(2), 362–378 (2008)
5. Evensen, G.: The ensemble Kalman filter: theoretical formulationand practical implementation. Ocean Dyn. 53(4), 343–367 (2003)
6. Evensen, G., van Leeuwen, P.J.: An ensemble Kalman smootherfor nonlinear dynamics. Mon. Weather Rev. 128(6), 1852–1867(2000)
7. Geiger, S., Matthai, S.K., Niessner, J., Helmig, R.: Black-oil simu-lations for three-component, three-phase flow in fractured porousmedia. SPE J. 14(2), 338–354 (2009)
8. He, J., Xie, J., Sarma, P., Wen, X.H., Chen, W.H., Kamath, J.:Model-based a priori evaluation of surveillance programs effec-tiveness using proxies. Paper SPE 173229 presented at the SPEReservoir Simulation Symposium, Houston, Texas, USA 23–25February (2015)
9. Hoteit, H., Firoozabadi, A.: Multicomponent fluid flow by discon-tinuous Galerkin and mixed methods in unfractured and fracturedmedia. Water Resour. Res. 41(11) (2015). doi:10.1029/2005WR004339
10. Hu, L., Jenni, S.: History matching of object-based stochasticreservoir models. SPE J. 10(3), 312–323 (2005)
11. Hui, M.H., Heidary-Fyrozjaee, M., Kamath, J.: Scaling gravity-drainage oil recovery from fractured reservoirs using 3D gravity-drainage scaling relationships. Paper SPE 172295-MS, presentedat the SPE Annual Caspian Technical Conference and Exhibition,Astana, Kazakhstan 12–14 November (2014)
12. Hui, M.H., Kamath, J., Narr, W., Gong, B., Fitzmorris, R.E.:Realistic modeling of fracture networks in a giant carbonatereservoir. Paper IPTC 11386-MS, presented at the InternationalPetroleum Technology Conference, Dubai, United Arab Emirates4–6 December (2007)
13. Hui, M.H., Mallison, B., Heidary-Fyrozjaee, M., Narr, W.: Theupscaling of discrete fracture models for faster, coarse-scale sim-ulations of IOR and EOR processes for fractured reservoirs.Paper SPE 166075-MS, presented at the SPE Annual Techni-cal Conference and Exhibition, New Orleans, Louisiana, USA 30September–2 October (2013)
14. Jenni, S., Hu, L., Basquet, R., De Marsily, G., Bourbiaux, B.:History matching of a stochastic model of field-scale fractures:Methodology and case study. Oil Gas Sci. Technol. 62(2), 265–276 (2007)
15. Karimi-Fard, M., Durlofsky, L.J.: A general gridding, discretiza-tion, and coarsening methodology for modeling flow in porousformations with discrete geological features. Adv. Water Resour.96, 354–372 (2016)
16. Karimi-Fard, M., Durlofsky, L.J., Aziz, K.: An efficient discretefracture model applicable for general purpose reservoir simulators.SPE J. 9(2), 227–236 (2004)
17. King, G.R., Jones, M., Tankersley, T., Flodin, E., Jenkins, S.,Zhumagulova, A., Eaton, W., Bateman, P., Laidlaw, C., Fitzmorris,R., Ma, X., Dagistanova, K.: Use of brown-field experimentaldesign methods for post-processing conventional history matchresults. Paper SPE 159341-MS, presented at the SPE Annual Tech-nical Conference and Exhibition, San Antonio, Texas, USA 8–10October (2012)
18. Kitanidis, P.K.: Parameter uncertainty in estimation of spatialfunctions: Bayesian analysis. Water Resour. Res. 22(4), 499–507(1986)
19. Krishnamurti, T.N., Kishtawal, C., Zhang, Z., LaRow, T.,Bachiochi, D., Williford, E., Gadgil, S., Surendran, S.: Multimodelensemble forecasts for weather and seasonal climate. J. Clim.13(23), 4196–4216 (2000)
20. Mallet, V., Stoltz, G., Mauricette, B.: Ozone ensemble forecastwith machine learning algorithms. J. Geophys. Res. 114(D5),148–227 (2009)
21. Matthai, S., Mezentsev, A., Belayneh, M.: Control-volume finite-element two-phase flow experiments with fractured rock repre-sented by unstructured 3D hybrid meshes. Paper SPE 93341-MS,presented at the SPE Reservoir Simulation Symposium, TheWoodlands, Texas, USA 31 January–2 February (2005)
22. Mosegaard, K., Tarantola, A.: Monte Carlo sampling of solutions toinverse problems. J. Geophys. Res. 100(B7), 12,431–12,447(1995)
23. Oliver, D.S.: On conditional simulation to inaccurate data. Math.Geosci. 28(6), 811–817 (1996)
24. Oliver, D.S., Chen, Y.: Recent progress on reservoir history match-ing: a review. Computat. Geosci. 15(1), 185–221 (2011)
25. Oliver, D.S., Reynolds, A.C., Liu, N.: Inverse theory for petroleumreservoir characterization and history matching. Cambridge Uni-versity Press, Cambridge (2008)
26. Reynolds, A.C., He, N., Oliver, D.S.: Reducing Uncertaintyin Geostatistical Description with Well-Testing Pressure Data.In: Reservoir Characterization–Recent Advances, pp. 149–162.American Association of Petroleum Geologists, Tulsa (1999)
27. Satija, A., Caers, J.: Direct forecasting of subsurface flow responsefrom non-linear dynamic data by linear least-squares in canonicalfunctional principal component space. Adv. Water Resour. 77, 69–81 (2015)
28. Scheidt, C., Renard, P., Caers, J.: Prediction-focused subsur-face modeling: investigating the need for accuracy in flow-basedinverse modeling. Math. Geosci. 47(2), 173–191 (2015)
29. Sun, W.: Data Driven History Matching for Reservoir ProductionForecasting. Master’s thesis, Stanford University (2014)
30. Sun, W., Durlofsky, L.J.: A new data-space inversion procedure forefficient uncertainty quantification in subsurface flow problems.Math Geosci. (2017). doi:10.1007/s11004-016-9672-8
31. Tarantola, A.: Inverse problem theory and methods for modelparameter estimation. SIAM (2005)
32. Vo, H.X., Durlofsky, L.J.: A new differentiable parameterizationbased on principal component analysis for the low-dimensionalrepresentation of complex geological models. Math. Geosci.46(7), 775–813 (2014)
33. Vo, H.X., Durlofsky, L.J.: Data assimilation and uncertaintyassessment for complex geological models using a new PCA-based parameterization. Computat. Geosci. 19(4), 747–767(2015)
34. Wen, X.H., Chen, W.H.: Real-time reservoir model updating usingensemble Kalman filter with confirming option. SPE J. 11(4),431–442 (2006)
