Abstract The conventional paradigm for predicting futurereservoir performance from existing production datainvolves the construction of reservoir models that match thehistorical data through iterative history matching. This isgenerally an expensive and difficult task and often resultsin models that do not accurately assess the uncertainty ofthe forecast. We propose an alternative re-formulation ofthe problem, in which the role of the reservoir model isreconsidered. Instead of using the model to match the his-torical production, and then forecasting, the model is usedin combination with Monte Carlo sampling to establisha statistical relationship between the historical and fore-cast variables. The estimated relationship is then used inconjunction with the actual production data to produce astatistical forecast. This allows quantifying posterior uncer-tainty on the forecast variable without explicit inversion orhistory matching. The main rationale behind this is thatthe reservoir model is highly complex and even so, stillremains a simplified representation of the actual subsurface.As statistical relationships can generally only be constructedin low dimensions, compression and dimension reductionof the reservoir models themselves would result in furtheroversimplification. Conversely, production data and forecast
2 Energy Resources Engineering Department,Stanford University, Stanford, CA, USA
3 Geological Sciences Department, Stanford University,Stanford, CA, USA
variables are time series data, which are simpler and muchmore applicable for dimension reduction techniques. Wepresent a dimension reduction approach based on func-tional data analysis (FDA), and mixed principal componentanalysis (mixed PCA), followed by canonical correlationanalysis (CCA) to maximize the linear correlation betweenthe forecast and production variables. Using these trans-formed variables, it is then possible to apply linear Gaussianregression and estimate the statistical relationship betweenthe forecast and historical variables. This relationship isused in combination with the actual observed historical datato estimate the posterior distribution of the forecast vari-able. Sampling from this posterior and reconstructing thecorresponding forecast time series, allows assessing uncer-tainty on the forecast. This workflow will be demonstratedon a case based on a Libyan reservoir and compared withtraditional history matching.
Forecasting future reservoir performance from existing pro-duction data requires the integration of many disciplines.While a large variety of methods are available [13, 18],the practical application of these methods remains lim-ited and often practice resorts to manual adjustment [31].In reality, considerable complexity presents itself. First,reservoir models need to be constructed consisting of struc-tural, lithological, and petrophysical components. Thesecomponents may be interrelated to some degree. Severaluncertainties exist, such as geological scenario uncertainty(discrete uncertainty); uncertainty on spatial distribution of
lithologies, porosity, and permeability (spatial uncertainty);and uncertainty in the structural framework in terms ofnumber of faults (categorical uncertainty), fault hierarchy(scenario uncertainty), and fault throw (continuous param-eter uncertainty). In addition, uncertainties may exist thataffect fluid behavior, such as relative permeability, PVTproperties or phase contacts, and initial and boundary con-ditions. Whether using a sampling approach to addressthe history-matching problem or an optimization approach,some forms of iteration must be done, in order to perturball of these components to achieve multiple history-matchedmodels that will hopefully represent realistic uncertainty.In this sense, sampling methods are preferred as they canincorporate prior uncertainty and hence, generate posteriorsolutions for uncertainty quantification, which are requiredfor decision-making purposes. The amount of parametersinvolved however may become very large, including theissue of dealing with spatial uncertainty. For the latter, sev-eral dimension reduction techniques have been proposed[14, 20].
To date, there is not one single method that can addressall these complexities. In other words, account properly forprior model uncertainty on all components of the model(fluid, rock, structure, petrophysics, and boundary) andsample models based on a Bayesian formulation of the pos-terior, matching all dynamic data, such as four-dimensional(4D) seismic, well test, production data, etc. Most publi-cations address part of the problem, either by focusing onpetrophysical properties only but not structure or by focus-ing on a fixed spatial model while perturbing simple engi-neering parameters. Generating usable and comprehensivesoftware also remains elusive.
In this paper, we make a first small step towards refor-mulating and perhaps, rethinking and re-evaluating thepractice of history matching in terms of real field appli-cations involving full complexity, going beyond the usualsimplified academic problems treated in many publications.Our approach builds on the previously published predictionfocus analysis (PFA; see [11, 24, 25]). In this approach,it is advocated to not generate multiple models that match
dynamic data, and then only run a forecast model on thehistory-matched models. Instead, the role of reservoir mod-els is reconsidered; rather, models are generated in order toestablish a direct statistical relationship between data vari-ables and forecast/prediction variables, then using this esti-mated relationship and the actual production data to producea statistical forecast. In this regard, we quantify posterioruncertainty on the forecast without history matching indi-vidual models. The rationale here is that reservoir modelingis complex and that reservoir models are extremely highdimensional and despite this, still remains a simplified rep-resentation of the actual subsurface geological and fluidcomplexity. Any sparse representation or dimension reduc-tion method would further simplify an already simplifiedreality. On the other hand, production data and forecast vari-ables are simple time-series on which statistical dimensionreduction techniques as well multivariate modeling can bereadily applied, without much loss of information. In thispaper, we apply this idea on a real field case and showthat roughly the same forecast can be obtained as with thetraditional approach of generating multiple history-matchedmodels. The structure of the paper is as follows. First, wereview the prediction-focused analysis in broad terms aswell as provide specific details. Next, we present a bootstrapmethod that allows evaluating the confidence in the method.We then apply the method to two versions of the same realfield case, one involving structural uncertainty and one not.We provide appropriate limitations and discussion in con-cluding this paper and identify scenarios in which traditionalinversion is still required.
2 Prediction-focused analysis
2.1 General overview
In prediction-focused analysis, one considers not just datavariables and model variables but also the intended purposein terms of prediction variables. Figure 1 illustrates the dif-ference in approach between traditional model building by
Fig. 1 Two views on addressingthe forecasting issue. Thetraditional framework (left)applies causal analysis to matchthe models to the data, then usethose matched models forforecasting. The proposedmethodology (right) usesevidential analysis by which themodel is used to construct astatistical relationship betweenthe data and forecast
Comput Geosci (2017) 21:315–333 317
history matching and then forecasting and an approach thatincludes the forecast in an integrated fashion. In generalterms, we represent time-varying historical data variables(such as watercut, oil rate, pressure, etc.) as vector d. Thereservoir model is represented by m. The latter consists ofthe spatial model, the structural model, the fluid model, andall associated parameters, hence, is very high dimensional.The (future) forecast (such as cumulative oil, water cut, andvolume) is represented by h. Clearly, the dimension of m ismuch larger than that of h and d.
dim (m) � dim (h), dim (d) (1)
In addition, we observe actual field production data (or anyother data), which we term dobs.
The idea of a prediction focus analysis is straightforward:using a stated prior on m, namely f (m), one generates, byMonte Carlo (or other methods, such as quasi Monte Carlo),a set of N prior models: {m1m2. . .mN }. These models arethen evaluated through the data forward model:
d = Gd(m) (2)
and the forecast forward model:
h = Gh(m) (3)
These are forward functions are deterministic functions thatgenerates the data and forecast variables for a given reser-voir model m. These functions are determinsitric (assumedexact); observation error will be treated later. In practice,these functions are generally reservoir simulators that gen-erate the expected historical and future production rates fora given reservoir model. Applying them to each prior modelresults in the pairs of data:{{d1, h1}, {d2, h2}, . . . , {dN , hN }
}(4)
Next, a statistical dimension reduction method is employed(e.g., MDS, PCA, kernel principal component analysis(KPCA), functional principal component analysis (FPCA),and canonical functional component analysis (CFCA), seelater) to generate reduced dimension vectors, d∗ and h∗or interms of the sample:
{{d∗1, h
∗1}, {d∗
2, h∗2}, . . . , {d∗
N , h∗N }}
where dim (d∗) << dim (d); dim (h∗) << dim (h)
(5)
This joint sample is used to construct a multivariate distribu-tion, assuming this is now possible because of the reduceddimensions (see next section on the specifics): f
(d∗, h∗).
The observed data is reduced in dimension using the samedimension reduction method, obtaining d∗
obs . The reduced
observations are then used to condition the multivariatedistribution as f
(d∗, h∗|d∗
obs
)which can be used to obtain:
f(h∗|d∗
obs
) =∫
d
f(h∗|d∗
obs
)f(d∗) dd∗ (6)
After backtransformation, this results in the desired poste-rior uncertainty on the forecast f (h|dobs). Note that theonly requirement for the dimension reduction technique isfor it to bijective namely having a uniqueness property:
if h∗ =rh (h);d∗ =rd (d) then h=r−1h
(h∗);d=r−1
d
(d∗)
(7)
with rh and rd the multivariate functions representingthe respective mapping from higher to lower dimensions.The bijectivity allows moving from one lower to higherdimensional space and vice versa without encountering anill-posed inverse problem. For example, multidimensionalscaling and KPCA are not bijective as they require the solu-tion of what is termed a pre-image problem (see [22]),while PCA, NLPCA, and FPCA are bijective operations (seeSatija and Caers [24]).
2.2 Specifics
Figure 2 provides an overview of the specific componentsthat are presented in this section. Preferably, the multivari-ate distribution in Eq. 6 should be multivariate Gaussian. Insuch case, the conditional distribution can be obtained usingGaussian process regression, equivalent to simple krigingthe forecast from the data [28]. A multivariate Gaussian canbe reasonably assumed when the relationship between d∗and h∗is linear:
d∗ = G · h∗ (8)
and when the marginal distributions are Gaussian. Gaussianmarginal distributions can always be obtained by means ofhistogram transformations, but obtaining a linear relation-ship is more challenging. To that extent, Satija and Caers[24] recognize that output of flow simulators represent sys-tematic, physical “signals,” varying in time. For example,cumulative curves obtained from multiple reservoir modelshave similar functional behavior (they start at zero, breakthrough, and then increase) and they are not purely stochas-tically varying time series. As a result, they propose toreduce dimension of such signals by means of a statisticaldimension reduction method that capitalizes on such sys-tematic behavior, namely FPCA. To that extent, consider thebasis expansion of the forecast variables as follows:
h (t) ∼=L∑
i=1
kξ,iξ i (t) (9)
318 Comput Geosci (2017) 21:315–333
Fig. 2 General overview of the proposed methodology. The workflowuses Monte Carlo sampling and forward modeling to produce a setof prior historical and forecast-response curves. FPCA and CCA areused to reduce the dimension of the responses and maximize a linearcorrelation between the two variables. Performing Gaussian process
regression and sampling from the resulting posterior distribution yieldsa set of updated forecasts conditioned to the observed historical data.Undoing the CCA/FPCA transformations produces the posterior fore-casts as a set of time series responses from which updated quantilescan be computed
Using a spline basis has the advantage of computational easeof evaluation as well as establishing derivatives. The choiceof the number of basis is a modeling choice and will need tobe tuned for each case, usually using cross-validation [20].We will use such spline basis throughout the paper. PCA onthe coefficients of this linear combination characterizes thefunctional variations in the time series data and is referredto as FCA:
h (t) ∼=K∑i=1
h fi φ h,i (t) (10)
Hence, FPCA represents a time series as a lin-ear combination of K orthonormal eigen-functions{φ h,1 (t) , φh,2 (t) · · · φh,K (t)
}with coefficients h f . Note
that FPCA, like PCA is bijective. A similar decompositioncan be achieved for the data variables:
d (t) ∼=K∑i=1
d fi φd,i (t) (11)
Applying FPCA to the set of N prior samples one obtains:
{{d f
1 , h f1 }, {d f
2 , h f2 }, . . . , {d f
N , h fN }
}(12)
Given the non-linear nature of the forecast and data responsemodel, Eqs. 2 and 3, it is not necessarily guaranteed that oneobserves, after FPCA, a linear relationship between the pair-wise components of d f and h f . This is usually attributed tothe presence of cross-correlations among the functional datavariables. Therefore, a linearizing operation is performedby means of canonical correlation analysis (CCA), whichis a more general form of partial least squares [29]. CCArelies on a linear transformation of both d f and h f suchthat the components in such transformation are maximallycorrelated, or in terms of notation:
Fig. 3 Situations where CFCA could yield an improper estimate of the posterior. A non-linear relationship between the data and forecast (left),inconsistent prior (middle), and insufficient models that match the data in the prior (right)
thereby maximizing the correlations between pairwisecomponents of hci and dci while constraining all theintercomponent correlations between hci and hcj �=i , betweendci and d
cj �=i , and between h
ci and d
cj �=i to 0.
If a linear correlation is observed between dc and hc,then a linear model G is regressed and a Gaussian likelihoodmodel formulated as follows:
L(hc
) = exp
(−1
2
(Ghc − dcobs
)TC−1dc
(Ghc − dcobs
))(16)
Cdc refers to the data covariance matrix of the canoni-cal components, which is estimated from the data errorcovariance in the original time domain using a Monte Carloapproach proposed in [12]. Since prior and likelihood are
multivariate Gaussian, the posterior is also Gaussian andthe posterior mean and covariance can be readily estimatedusing classical methods [28] as shown in Eqs. 17 and 18.
h̃ = h̄cprior+ChGT(GCHG
T +Cdc +CT
)−1(dcobs−Gh̄cprior
)
(17)
C̃h = Ch − ChGT
(GChG
T + Cdc + CT
)−1GCh (18)
CT is the covariance of the error that arises due to the linearfitting in Eq. 8, which can be readily estimated empiricallyfrom the residuals in Eq. 8. Once the posterior mean andcovariance is estimated, the posterior distribution is thensampled and the obtained samples are backtransformed into
Fig. 4 Location of theHameimat Trough in the SirteBasin where the N-97 field islocated. Image source: [2]
320 Comput Geosci (2017) 21:315–333
Table 1 Prior distribution ofuncertain reservoir parametersused in generation of bothcases of scenario 1
Parameter OWC μoil TF1 TF1 TF1 TF1
Distribution U [1061, 1076] N (0.3, 0.2) U [0.2, 0.8] U [0.2, 0.8] U [0.2, 0.8] U [0.2, 0.8]
Parameter Krw Kro Swir Sor nw noDistribution N (0.3, 0.2) N (0.7, 0.2) N (0.2, 0.2) N (0.2, 0.2) N (2.5, 0.2) N (2.0, 0.2)
The scoping set of 500 models were sampled from these distributions
actual signals, from which quantiles can be calculated (seeFig. 2).
2.3 Assumptions
There exist a number of underlying assumptions for thismethodology to work in actual practice. These pertain to theconsistency of the prior, the effectiveness of the dimensionreduction, and the existence of a linear relationship betweendata and forecast in the canonical space. We would like toemphasize these before presenting application cases.
2.3.1 Dimension reduction
The “compression” by means of FPCA must be significant.FPCA attempts to represent response variables from sim-ulators by means of few principal components. The lowerthe amount of components, the easier the Gaussian processregression. This compression will become more difficult asmore wells are involved; hence, the procedure is likely toapply at early development stages.
2.3.2 Consistency of the prior
The Bayesian formulation of the forecasting problemrequires the specification of a subjective belief on a hypoth-esis (e.g., the reservoir model). Evidence is then gathered(data) and then the probability of the evidence under thehypothesis is evaluated (the likelihood). The importance ofthis subjective prior is well known, and some authors inthe statistical world are working on methods of falsifica-tion [9, 10]. If the “data” falls outside “the prior” as stated,then the probability of the data under the prior hypothesis isvery small; consequently, the posterior is also very small forthe given hypothesis. It may be tempting to perform ad hocmodifications to the prior (such as multiplying permeabilitywith some value around a well based on the production data)with the purpose of ensuring that the prior range encom-passes the observed data. However, this has been refuted as“ad hoc” [5] and can lead to incorrect posteriors. Indeed,any ad hoc modification of the prior will only lead to pos-teriors to be again inconsistent with observation at a futuredate. Accordingly, the prior distribution should be selected
Fig. 5 Structural model andhorizontal permeability used forscenario 1. The location of theexisting producers are denotedby P1 ... P5, while the locationof the new well to be drilled isdenoted by PNEW. Theuncertain reservoir parametersand their prior distributions arelisted in Table 1
Comput Geosci (2017) 21:315–333 321
Fig. 6 Production data until day 3500 for each of the five existing producers for each prior model (gray). The production profiles were generatedby forward simulating the prior models using a streamline simulator. The observed production is shown in red
to be wide enough and/or properly sampled to allow mean-ingful statistical analysis (see Fig. 3). A number of scenariosmay still occur: (A) the data is not informative hence his-tory matching is needed to predict h (the original intent ofPFA [25] is to detect this); (B) the data may not be coveredby the prior, which may be due to an inconsistent (e.g., toonarrow) prior or the number of samples (N ) from the prioris insufficient; and (C) the data is covered by the prior butan insignificant amount of samples is available to estimatethe posterior of h. The latter may occur when the data lies inthe extremes of the prior. These situations must be assessedprior to performing PFA.
2.3.3 Linearity in canonical space
The relationship between the canonical components of dand h can be modeled by means of a linear regression. Anydeviation of this linear model will lead to increased modelerror, and hence, large wide uncertainty when this modelerror is included in the forecasting (as in [24]).
Any violations of these assumptions may lead to noreduction in posterior uncertainty or unreliable forecasts.These issues therefore beg for a quantitative assessment ofthe prediction power of the procedure, which is covered inthe next section. In instances where the predictive powerprovided by PFA is low, traditional history matching willstill be required.
2.4 Confidence vs uncertainty
An uncertainty statement for a forecast, can be as simpleas stating a posterior PDF of that forecast. Since all reser-voir modeling forecasts are obtained from limited amountof samples from that PDF, a question of confidence on thestated uncertainty is required. For example, if an uncertaintyinterval based on quantiles is provided (often in terms ofP10–P90), then the confidence on those quantiles needs tobe calculated. First, this is relevant for testing whether theposterior quantiles are different from the prior quantiles,
Fig. 7 The forecasted response for the new well from days 3500 to7500, generated by forward modeling each of the prior models (gray).The true (in reality unknown) forecast is shown in red
322 Comput Geosci (2017) 21:315–333
Fig. 8 Reconstructions of theoriginal time series using onlythe first four components ofFPCA along with the originalproduction data from producer 1for select prior models. Thequality of the reconstructionserves as a metric for selectingappropriate basis splines
as this would indicate predictivity of the data towards thatparticular forecast. Note that uncertainty and confidenceintervals need not be related. One may be very confident ona wide uncertainty and less confident on a narrow uncertainty.Since this depends on a number of factors (the prior uncertainty,the particular data and forecast, the flow model, etc.), aquantitative method is required to establish this relationship.
In this section, we develop a hypothesis test, which testswhether the data informs the forecast (prediction variablesh) using the above (Fig. 2) procedure. Based on the p valueof this hypothesis test, we then plot confidence vs uncer-tainty for several combinations of field data, to investigatewhat combination of historical data should be used for thatforecast. It is clear that the most optimal situation occurswhen we have small posterior uncertainty in combinationwith high confidence. A low confidence or large posterioruncertainty suggests that conventional inversion techniquesmay be required.
Fig. 9 Cumulative sum of the FPCA eigenvalues for the historicalresponses of producer 1. This indicates that over 99 % of the variabilityin P1’s responses can be captured by just the first 5 eigencomponentsof FPCA. This provides a metric for diagnosing the effectiveness ofthe dimension reduction
In general, the data dobs is informative when there is asignificant difference between the prior distribution f (h)
and the posterior f (h|dobs). Since both these pdfs are mul-tivariate and functional, it would be difficult to develop asimple measure. As a proxy, we propose using the firstfunctional component h f
1 containing the maximum vari-ability of H [16, 19, 21]. This would also be a necessarycondition for f (h) and f (h|dobs) to be different. The prob-lem of comparing pdfs is now simpler as only a comparisonof univariate distributions is needed. We therefore define atheoretical difference between prior and posterior functionalcomponent as
δ = Δ(f(h f1
), f
(h f1 |dobs
))(19)
� can be any cdf- or pdf-based difference such as L1-norm difference [7], the Kolmogorov-Smirnov difference[15, 26], Jensen-Shannon or Kullback-Liebler divergence
[17]. The prior distribution is estimated as f̂(h f1
)directly
from the N scoping runs. The posterior cumulative distri-
bution is estimated as f̂(h f1 |dobs
)from the M posterior
samples obtained from the Gaussian process regression andCCA backtransform. Thus, an empirical difference measureis estimated as
δ̂ = Δ(f̂(h f1
), f̂
(h f1 |dobs
))(20)
Fenwick [7] proposed a bootstrap-based hypothesis test totest for significant difference between two cdfs based onthe L1 norm. For the purpose of hypothesis testing, the
Table 2 Percentage of variance captured by the first 4 eigenvalues ofFPCA for each existing production well’s historical data
Well P1 P2 P3 P4 P5
Percentage of variance (%) 98.521 99.579 99.533 99.895 99.832
This suggests that the first 4 eigenvalues are sufficient to represent themajority of the variation in the prior models’ historical response
Comput Geosci (2017) 21:315–333 323
Fig. 10 Each of the prior models plotted in the reduced dimension(functional components) of data vs forecast. The correlation betweenthe first components (left) shows a weak correlation between dataand forecast ρ = 0.2915. Likewise, the cross-correlation between the
first component of the data and the second component of the forecastshows a similarly weak correlation ρ = −0.2577. Performing regres-sion and estimation using these components could result in inaccurateassessments of posterior forecast uncertainty
null hypothesis is that the prior and the posterior distribu-tions of the first functional component are not different. Forbootstrapping, B datasets are drawn from the existing Nscoping runs without replacement, resulting in a B bootstrapestimate of the difference
ˆ̂δb = ΔCDF
( ˆ̂Fb(h f1
),
ˆ̂Fb(h f1 |dobs
)), b=1, . . . ,B (21)
The number of times ˆ̂δb ≤ δ̂ is measure by how strong
the predictivity is coinciding with the number of times thenull hypothesis is rejected. The more the null hypothesis isrejected, the more confident we are on the informativenessof the data, hence a good measure is:
ω = 1
B
B∑b=1
i( ˆ̂δb ≤ δ̂
)i = 1 if ˆ̂
δb ≤ δ̂, 0 else (22)
In the case study section, we illustrate the use of thismeasure for various combinations of historical productiondata.
3 Case study
3.1 Case description
The WintersHall Concession C97-I in the N-97 field islocated in the Western Hameimat Trough of the Sirte Basinof north-central Libya (Fig. 4). The geological setting of theSirte Basin is described in detail by Ahlbrandt [2]. The pri-mary hydrocarbon source bed in the Sirte Basin has beenidentified by Ahlbrandt [2] as the Late Cretaceous SirteShale. The reservoir under consideration, the WintersHallConcession C97-I, is a fault-bounded horst block with theUpper Sarir Formation sandstone reservoir [3]. Complexinteractions of the dextral slip movements within the riftsystem have led to the compartmentalization of the reser-voir. Initial structural modeling attempts and interpretationfrom seismic data suggested the presence of up to fourfaults, each with unknown displacements and transmissibil-ity, due to uncertainty in the interpretation of the seismic
Fig. 11 The result of performing canonical correlation analysis onthe functional components of data and forecast to transform the mod-els into canonical space (hc vs dc). The correlation between the firstcanonical components is much stronger than its functional counterpart
(ρ = 0.8941). Furthermore, the cross-correlations between the firstdata canonical component and the second forecast canonical compo-nent is very low (ρ = 0.0297)
324 Comput Geosci (2017) 21:315–333
Fig. 12 Drawing from the posterior distribution yields posterior sam-ples in canonical space. Only the first two dimensions are shown here,but each sample is actually a point in 4D space. The reduction in vari-ance of the posterior samples in comparison with the prior modelsindicates that a reduction in forecast uncertainty is achieved
image. Based on this realistic settings and the reservoirmodel provided by the operator, we have generated a fewvirtual cases by which we hope to illustrate the validity ofthe direct forecasting approach.
3.2 Scenario 1: flow parameter uncertainty
In this first illustration, we consider the scenario wherethe structural geology and depositional environment areassumed to be well understood and the major sources ofuncertainty reside in the flow parameters. The reservoir isthus composed of five distinct compartments resulting infour uncertain fault transmissibilities. A large aquifer islocated in the lowest compartment, but the depth of this oilwater contact also remains uncertain. Other uncertain reser-voir parameters are relative permeabilities for the oil andwater phases and oil viscosity. The relative permeabilites
of the oil and water are modeled using the Corey Expres-sions [4], which requires three parameters for each phase(irreducible saturation, end point permeability, and Coreyexponent). The prior distributions of these parameters arelisted in Table 1. The structural model contains four faults asshown in Fig. 5 and is used for all realizations. Likewise, athree-facies training image containing sand channels is usedwith SNESIM [27] and Gaussian simulation to populate thegrid with the appropriate facies and reservoir parameters(porosity/permeability) (see Fig. 5).
We consider the situation where five producers and threeinjectors have already been drilled at the locations depictedin Fig. 5. The field has been in production for 3500 days,and production data is available for all five wells. A deci-sion needs to be made regarding the economic feasibilitydrilling of a 6th producer in the smallest reservoir compart-ment (denoted PNEW) and an additional injector (denotedINEW). Specifically, this decision will be made based onthe forecasted performance of this new well over the next4000 days. Therefore, we will seek to estimate the P10–P50–P90 forecasts of PNEW based on the first 10 years ofproduction data from the existing five producers.
3.2.1 Generating prior realizations of model, data,and forecast variables
A prior set of models is required to establish a statisticalrelationship between the data and forecast. In this case, aset of 500 prior reservoir models is generated by apply-ing Monte Carlo to the prior distributions in Table 1. Thenumber of models was selected to ensure that the prior dis-tributions were sufficiently sampled. The prior models wereforward modelled using a streamline simulator (3DSL) overall 7500 days to encompass both the 3500 days of produc-tion data, as well as the 4000 days of forecast required to
Fig. 13 By undoing the CCA and FPCA, the posterior forecast sam-ples are transformed from canonical space back into the time domain.The posterior P10, P50, and P90 forecasts are shown along withthe prior P10, P50, and P90 curves (left). The sampled forecasts are
shown with the original forecasts from the prior models as well asthe reference (right). This illustrates that CFCA does indeed provide areduction in the forecast uncertainty
Comput Geosci (2017) 21:315–333 325
Fig. 14 Prior and posterior P10,P50, and P90 forecasts when themeasurement error CD is 100(stb/day)2 (left) and 300(stb/day)2. As the measurementerror is increased, the posteriorforecasts exhibits smallerreduction in uncertaintycompared with the prior
make the decision regarding the new well. For illustration,one of the generated models is used as a reference case, andits production data used as the “observed” production data.The prior distribution of reservoir performance for each ofthe existing wells as well as the forecast for the new well isshown in Figs. 6 and 7.
3.2.2 Dimension reduction
The first requirement in establishing a relationship betweenthe data and forecast is low dimensionality of both variables.In this case, the data and forecast are time series responses,that while technically are infinite dimensional, have beendiscretized into vectors of lengths 150 and 170, respec-tively. Directly establishing a statistical correlation betweenvariables of such dimension is still infeasible, and thus aprojection into a lower dimension using FPCA is first per-formed. The critical step of FPCA is the selection of anappropriate basis functions. In this particular instance, 6th-order splines with 20 knots as basis functions are found toprovide reasonable approximations for both the data andforecast variables. The effectiveness of the projection isverified by performing FPCA, and then reconstructed theoriginal time series using the basis functions and harmonicscores seen in Fig. 8.The choice of splines is then iterativelyadjusted to minimize the average RMS between the origi-nal and reconstructed curves. It is also important to ensurethat oscillations caused by Runge’s phenomena [8] do notoccur over any of the models, when high-order splines areused. The cumulative sum of the FPCA eigenvalues is usedto ascertain the effectiveness of this compression. FromFig. 9, one observes that the compression is indeed signifi-cant, as 98.52 % of the variability in Producer 1’s response
is captured by the first 4 eigenvalues. Table 2 shows thepercentage of the variability represented by the first 4 eigen-
values for each of the existing producers(d fP1, d
fP2. . .d
fP5
)
in addition to the well to be drilled(hfPNEW
).
Since the production data is composed of multiple wells,the relationship between each of the existing wells andthe new well must be quantified. However, redundancybetween the data from each production well exists, as theyare obtained from the same underlying reservoir model. Forinstance, a shallow oil water contact in the aquifer wouldcause both P3 and P4 to experience early water break-through. Consequently, a second dimension reduction isapplied to the production data in the form of a mixed PCAon the matrix d f
Producers obtained by concatenating (d fP1,
d fP2 . . .d
fP5). In our example, 97.42 % of the variability of
the wells is captured by the first eight components of themixed PCA. This effectively reduces the 20-dimensionald fProducers into an 8-dimensional d f , thus each prior model
is represented by a single point in 12-dimensional space(eight components from historical d f and four componentsfrom the forecast h f ). This is illustrated in two dimensionscorresponding to the two largest eigenvalues in Fig. 10.
3.2.3 Canonical correlation of d f –hf and regression
Dimension reduction from an infinite dimensional timeseries into a 12-dimensional space enables the possibility ofperforming regression between data and forecast. However,this is contingent on a sufficiently strong linear correlationbetween the variables or else the resulting estimates fromregression will be inconclusive. In this example, the cor-relation between first functional components of data and
Table 3 Difference between P10 and P90 averaged over the forecast period with varying levels of measurement error.
Average P10–P90 stb/day 565.102 271.23 283.71 344.65 527.95
As the measurement error is increased to extreme values, the posterior forecast returns to that of the prior
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Fig. 15 P10, P50, and P90forecasts from direct forecastingand rejection sampling. Twoerror covariances wereillustrated: 50 (stb/day)2 (left)and 150 (stb/day)2 (right). Thenumber of models remainingafter rejection was 690 and 987in each case. A similar reductionin uncertainty was achieved bydirect forecasting usingsubstantially less computationaleffort
forecast is rather poor (ρ = 0.2915). This is due to thepresence of cross-correlations between the first and secondcomponents (ρ = −0.2577) (Fig. 10). To fully capitalizeon the full multivariate correlation between all componentof h and all components of d, a CCA is performed totransform the models into a canonical space (dc and hc).This subsequently increases the correlation between thefirst components from 0.2915 in the functional domain to0.8941 in canonical space, similarly the cross-correlationbetween the first and second components is reduced to0.0297 (Fig. 11). Now that a linear correlation has beenestablished in low dimensions, the corresponding linearrelationship G is obtained via Eq. 8. The application of lin-ear Gaussian regression (Eq. 16) to estimate the posterioron the forecast components requires that hc must be trans-formed using a normal score transform first to obtain hcGauss.Gaussian regression thus produces a multivariate normalposterior f (hcGauss|dobs) that is easily sampled to produceforecast components conditioned to dcobs(Fig. 12). To obtainthe corresponding forecasts as time series, the normalscore transform and canonical transform must be backtrans-formed. The resulting posterior samples now in functionalspace, are used in conjunction with the original eigenfunc-tions from FPCA to reconstruct h(t). One hundred posteriorforecasts were sampled and converted into time series asshown in Fig. 13 along with the P10–P50–P90 curvesof the posterior forecasts. The posterior quantiles exhibita significant reduction in uncertainty in comparison withthe prior.
3.2.4 Accounting for measurement error
In the previous example, the observed data was assumedto be error-free; however, this is often not the case inreal applications. This error is accounted for in the directforecasting workflow by the Cc
D term in Eqs. 17 and 18.
However, as the measurement error can only be estimatedin the original time domain (modeled as a zero mean Gaus-sian with diagonal covariance matrix CD), the proceduredescribed in Section 2.2 and Hermans et al. [12] must beapplied to obtain Cc
D It is evident from Eq. 18, that increas-ing magnitudes of measurement error will result in largerposterior uncertainty That is to say, the less reliable theobserved data, the less informative it is of the forecast Asan illustration varying levels of measurement noise (100and 300 (stb/day)2) were assumed, and posterior distribu-tions for each were computed (Fig. 14; Table 3). This alsoshows that increasing the measurement noise to extremevalues causes the posterior uncertainty to approaches theprior uncertainty. This is in accordance with Eq. 18, as anynew information, regardless of how reliable, cannot increaseuncertainty.
Fig. 16 Posterior forecast quantiles for the situation where the newwell is to be after 7000 days instead of 3600 days. The inclusion ofadditional informative data reduces the forecast uncertainty
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Fig. 17 Prior production data for P4 along with the true observed (left)and the subsequent models in canonical space using only P4 as the data(right). In this instance, the data does not inform the forecast due to thecompartment in which P4 is located having very little communication
with the compartment of the new well. This manifests as poor correla-tion in the canonical space. This means the posterior will provide littlereduction in forecast uncertainty in comparison with the prior
Fig. 18 Prior production data for P5 along with the true observed (left)and the subsequent models in canonical space using only P5 as the data(right). Water breakthrough has yet to occur in P5 for most of the priormodels, and as a result, the majority of the models are producing at the
production limits. This means that for a given production profile in P5,the forecasts for the new well could vary widely. This manifests in thecanonical plot as a vertical cluster of models around the observed data
Fig. 19 Due to the lack ofinformativty between P4 (left)and P5 (right), the posterioruncertainty does not showmarked reduction from the prioruncertainty
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Fig. 20 By using a combinatorial of historical data from a subset ofproducers, and running the bootstrap test for statistical significance, weobtain not only an updated forecast uncertainty but also a confidencein this uncertainty. Each point represents some combination of wells,its posterior forecast uncertainty (P10–P90 averaged over 4000 days offorecast)
3.2.5 Comparison with rejection sampling
To compare the reduction in uncertainty provided by directforecasting with an idealized case, where multiple history-matched models are available, 15,000 realizations weresampled from the prior distribution and forward simulated.
Fig. 22 The forecasted response for the new well from days 3500 to7500, generated by forward modeling each of the prior models (gray).The true (in reality unknown) forecast is shown in red. An injector hasbeen added in the same compartment as this producer
Rejection sampling is performed on the resulting responsesto identify the realizations that match the production datausing the same methodology as described in Satija and
Fig. 21 Production data until day 3500 for each of the five existing producers for each prior model (gray). The production profiles were generatedby forward simulating the prior models using a streamline simulator and variable well schedules. The observed production is also shown (red)
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Fig. 23 Reconstructions of thetime original time series usingonly the first five components ofFPCA along with the originalproduction data from producer 1for select prior models. A largernumber of knots is required inthis scenario due to theadditional complexities in therate
Caers [24]. The likelihood is defined for the data observed ineach realization in the rejection sampling set using Eq. 23.
CD is a diagonal matrix, as the noise is modeled as a zeromean Gaussian. Rejection sampling was performed for twoCD values, (σ 2 = 50 and 150 (stb/day)2) which resultedin 690 and 987 models, respectively. The P10–P50–P90curves for the corresponding forecasts of these matchedmodels were computed and shown in Fig. 15, along withthe computed P10–P50–P90 curves from direct forecasting.In both instances, direct forecasting has comparable uncer-tainty reduction with rejection sampling, while utilizingonly a fraction of the computational cost.
3.2.6 Updating forecasts with additional data
Assimilation of additional observed data remains a chal-lenge in convention history-matching problems. Techniquesbased on the Ensemble Kalman Filter [1], have beenproposed, but handling large systems with non-linearitiesand non-Gaussian geostatistical priors remains a topic ofresearch. Due to the complexity of the reservoir models, thehistory-matched models are often inconsistent with actual
future observations. This requires another round of costlyhistory matching and model updating in order to obtain anupdated forecast. Conversely, in PFA, the prior models justneed to be forward modeled for a longer simulation time toaccount for the new data. Consider the case where the newwell is to be drilled after 7000 days rather than 3500. Toobtain an updated forecast, Direct Forecasting is repeated,but with the original prior models forward simulated to11,000 days (7000-day production and 4000-day forecast).The resulting quantiles are shown in Fig. 16 and exhibit areduction in uncertainty compared with the previous 3500-day case. This should be expected as the incorporation ofadditional informative data should provide a reduction in theforecast uncertainty.
3.2.7 Confidence vs uncertainty
To understand if data is informative on the forecast, considerthe scenario in which we only use a single production well’sdata to forecast the new well. In Fig. 17, only data fromproducer 4 is used to make the forecast. As seen in Fig. 5,producer 4 is separated from the compartment in which thenew well is located by two faults and potentially has a lowlevel of communication. Performing CFCA shows that onlya 0.0341 correlation exists between hc and dc, which is
Fig. 24 The posterior P10, P50,and P90 forecasts are shownalong with the prior P10, P50,and P90 curves (left). Thesampled forecasts are shownwith the original forecasts fromthe prior models as well as thereference (right). Nomeasurement error was assumedhere
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Fig. 25 Prior and posterior P10,P50, and P90 forecasts when themeasurement error CD is100 stb/day (left) and300 stb/day
insufficient to reduce forecast uncertainty (Fig. 19). Thisrepresents a case where the data is not related to the fore-cast. Conversely, considering only producer 5 as the data,and examining the production profiles from the prior mod-els Fig. 18, one observes that most of the prior models havenot experienced water breakthrough after 3500 days. Con-sequently, they are all producing oil at the production limits.This manifests in the canonical domain as a cluster of mod-els that yield widely different forecasts. It follows that thequantiles obtained from CFCA using just P5, do not showa marked decrease in uncertainty when compared with theprior (Fig. 19).
To quantify the level of confidence on the updatedforecast quantiles, the bootstrap procedure outlined inSection 2.3 is applied. To assess both the reduction in uncer-tainty as well as the confidence we have in each updatedforecast, a combinatorial selection of wells was used toupdate the forecast (e.g., P1, P1–P2, etc.). The bootstrap wasperformed 500 times, and the average forecast uncertainty(mean difference between P10 and P90 over the duration ofthe forecast) is plotted against the computed ω of Eq. 22(Fig. 20). This shows that in general lower confidence cor-relates with larger posterior forecast uncertainty, and it canbe seen that using just P4 and P5 results in smaller uncer-tainty reduction and poor confidence. Conversely, usingall five wells produces both higher confidence and loweruncertainty. It should also be noted that some combina-tions of wells such as (P1, P3, and P4) do produce reduced
uncertainty, but with low confidence, and this must be takeninto account when decisions are made.
3.3 Scenario 2: flow parameter uncertaintywith variable well schedules
To illustrate a scenario where the data exhibits discontinu-ities and complexities commonly seen in real field cases,both the injector and producer schedules from the previ-ous case were modified to include shut-ins and pressurechanges (Fig. 21). Since the historical well schedule isalready known when constructing the prior models, it isincorporated directly into the forward model (gd in Eq. 2).An additional injector was also added to the same compart-ment as the producer to be forecasted, to induce additionalvariability in the forecasts (Fig. 22). Once again, 500 priormodels were sampled according to the prior parameterdistributions in Table 2.
3.3.1 Dimension reduction of discontinuous time series data
Due to the shut-ins and additional complexities in the histor-ical production data, the 6th-order 20-knot B-splines usedas basis functions in the previous scenario are insufficientto capture the variability among the prior models. However,this is resolved by increasing the number of knots to 30,as evident in the reconstructions shown in Fig. 23. Despitethis increase in number of knots, the FPCA and mixed PCA
Fig. 26 The two differentdepositional scenarios depictedby training images containingchannels (left) and ellipsoids(right)
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procedures outlined in Section 2.2 are still able to identifyredundancies in the data. For each producer, FPCA identi-fied that the first 5 eigenvalues were required to represent99 % of the variability (in contrast to 4 in the previous sce-nario), while the mixed PCA indicated that 12 total scores(vs 8 previously), are required to represent the entire set ofhistorical production rates.
3.3.2 Posterior forecasts
Applying CCA on the scores obtained by FPCA/mixedPCA resulted in correlations of 0.8821 in the first canoni-cal components of the data and forecast. Consequently, thereduction in uncertainty in the posterior forecasts (Fig. 24)is comparable with that of the previous scenario. Likewise,increasing the measurement error to 100 and 300 stb/daycauses the posterior uncertainty to revert towards the prioruncertainty (Fig. 25). This demonstrates that if the compres-sion provided by FPCA is significant, and CCA is success-ful in maximizing linear correlations, direct forecasting isable to provide reductions in uncertainty regardless of thenature of the historical and forecast data variables.
3.4 Scenario 3: with structural and depositionaluncertainty
In the third scenario, both structural and spatial uncertaintiesare considered. A combination of the structural uncer-tainty modeling approaches suggested [6, 30] is applied. As
Fig. 27 Field-level production data until day 3000 for each priormodel (gray). The production profiles were generated by forward sim-ulating the prior models using a streamline simulator. The observedproduction is shown in red
Fig. 28 The forecasted response for the new well from days 3000 to6000, generated by forward modeling each of the prior models (gray).The true (in reality unknown) forecast is shown in red
described in Bellman et al. [3], there could be between oneand four faults in the reservoir at the given locations. There-fore, the number of fault parameter (numFaults) is modeledas an integer random number. The throws or displacementsof each fault (Throwi) is uncertain and must abide by rulessuch as geologically older faults must have higher displace-ments than newer faults. This indicates a joint distributionbetween the Throwi and numFaults parameters:
Throwi+1 ≤ Throwi∀i ≤ numFaults
Throwi = 0∀i > numFaults
Fig. 29 Scatter plot of the first and second canonical componentsof the data and forecast variables. The observed data projected intocanonical space is indicated by the red line. The correlation in thisinstance is moderate (ρ = 0.65)
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Fig. 30 Comparison of posterior quartiles obtained from direct fore-casting and rejection sampling with prior quartiles of the predictedwater cut in the prediction
where Throw1 is the displacement in the oldest fault. Thestochastic model-generation algorithm detailed in Satija[23] is used to sample a value for each flow displacementparameter under these constraints. To account for the spatialuncertainty, two depositional scenarios are modeled usingtwo training images shown in Fig. 26. One TI contains sinu-soidal channels, while the other contains ellipsoidal lobes.Finally, two separate marginal distributions of net to grossare considered (50 and 70 %) and used as an input toSNESIM along with the TI to generate the resulting realiza-tions. In the scenario, one producer and one injector havealready been in place and producing for 3000 days, a deci-sion is required regarding drilling a new producer at a givenlocation in the highest fault block in the reservoir. Such adecision would be based on the new well-forecasted per-formance over the next 3000 days, such as if the operatorconsiders the well to be unprofitable if the water cut risesabove a certain threshold. To perform PFA, 500 modelsare sampled from the prior and simulated using 3DSL. Theresulting responses are shown in Figs. 27 and 28. The sameworkflow as the first scenario was applied, and resulting hc
vs dc plot is illustrated in Fig. 29. It should be noted that thecorrelation in this example is not as linear as the previouscases without structural uncertainty (0.61). As before, rejec-tion sampling consisting of 4700 models, resulting in 221history-matched models, was used and the posterior boundsin both cases show similar results as in Fig. 30.
4 Discussions and conclusions
Forecasting problems in the oil and gas industry are gen-erally formulated as iterative data inversion or history-
matching problems that are computationally expensive anddifficult. However, as the goal of forecasting is to informa decision, rather than obtaining the matched models, wepresent a reformulation of the forecasting problem wherethe role of the reservoir model is reconsidered. Instead ofattempting to use it to match the data, the model is insteadused to establish a statistical relationship between the his-torical data and forecast. This estimated relationship is thusused to obtain a statistical forecast based on actual observedproduction data. To establish such a relationship, a stronglinear correlation is required in a low dimension to renderthe statistical procedure feasible. In this paper, we use FPCAfollowed by CCA to achieve these requirements. The appli-cation of this methodology to three scenarios based on theWintersHall Concession C97-I reservoir in Libyan demon-strated that CFCA is able to provide updated estimatesof the forecast uncertainty that is comparable with rejec-tion sampling, but at a fraction of the computational cost.The confidence in these updated forecasts can be gaugedthrough a bootstrap test of statistical significance that wehave also presented in this paper. This level of confidencein the updated uncertainty should be taken into considera-tion when making decisions based off the forecast, as wellas identifying scenarios where traditional inversion (his-tory matching) is still required. While PFA was presentedin this paper using time series as the responses, an exten-sion of this research could be the application to other typesof responses common in the Earth Sciences such as satu-ration maps or time-lapse data. The reduction in requiredcomputational time and complexity of this methodology incomparison with history matching could have considerableimpact on how forecasting is done in the Earth Sciences.An additional appealing characteristic of this non-iterativeprocedure is that (1) it can be perfectly parallelizes mak-ing it extremely computationally efficient and (2) it can bedone “offline” outside of reservoir modeling and flow sim-ulation software thereby making it completely general froma software implementation point of view.
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