Methods for Quantifying the Incertainty of Production Forecasts a Comparative Study

7/25/2019 Methods for Quantifying the Incertainty of Production Forecasts a Comparative Study

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Methods for quantifying the uncertainty of production forecasts

a comparative study

F.J.T. Floris,Delft Geoscience Research Centre1, P.O. Box 5028, 2600 GA Delft, The NetherlandsM.D. Bush,BPAmoco, Chertsey Road, Sunbury-on-Thames, Middlesex, TW17 7LN, UK

M. Cuypers,ELF Exploration UK, 30 Buckingham Gate, London SW1E 6NN, UK

F. Roggero,Institut Franais du Ptrole, 2 Avenue President Angot, Helioparc, 64000 Pau, France

A-R. Syversveen,Norwegian Computing Centre, P.O. Box 114 Blindern, Gaustadalleen 23, 0314 Oslo,

Norway

Abstract

This paper presents a comparison study in which several partners have applied methods to quantify uncertainty on

production forecasts for reservoir models conditioned to both static and dynamic well data. A synthetic case study

was set up, based on a real field case. All partners received well porosity/permeability data and historic production

data. Noise was added to both data types. A geological description was given to guide the parameterization of the

reservoir model. Partners were asked to condition their reservoir models to these data and estimate the probabilitydistribution of total field production at the end of the forecast period. The various approaches taken by the partners

were categorized. Results showed that for a significant number of approaches the truth case was outside the predicted

range. The choice of parameterization and initial reservoir models gave largest influence on the prediction range,

whereas the choice of reservoir simulator introduced a bias in the predicted range.

Introduction

Traditionally reservoir development decisions are based on a production forecast from a single history

matched reservoir model. To assess risk, some runs are made to check the sensitivity of the forecast.

However, formal quantification of risk requires full sampling of the forecast probability density function

for the quantity to be forecast.Several papers have appeared for the generation of full pdfs in the geosciences. The reduction in

uncertainty of hydrocarbon pore volume due to structural and porosity / permeability uncertainty is

studied in Berteig et al. (1988). Methods for quantifying the pdf of hydrocarbon volumes in place due to

top structure uncertainty are given in Abrahamsen et al. (1992), Samson et al. (1996) and Floris &

Peersmann (1998). Uncertainty in production from fields without history matching is described in Lia et

al. (1997). Landa & Horne (1997) investigate the reduction in uncertainty on reservoir description for a

synthetic case where saturation maps from 4D seismic have been included as history data.

In this paper, we focus on production forecast uncertainty quantification (PUNQ) methods. During the

execution of the EC sponsored PUNQ project (Bos, 2000) a number of new methods have been

developed which are published in previous papers. The aim of this paper is to report on an integrated

case study to which all methods were applied.

Work flow for uncertainty quantification

The general workflow contains a number of standard components. Instead of organizing this paper in

terms of the separate work flows, we choose to first categorize these workflow components as building

blocks and then build the work flows from them. The general outline of the work flows is given in Figure

1.

1The Delft Geoscience Research Centre is a collaboration between Delft University of Technology and the

Netherlands Organisation for Applied Scientific Research TNO


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Methods for quantifying uncertainty on production forecasts Page 2

Parameterization

In the Bayesian inversion approach, the key is to condition reservoir models to all available data. This

conditioning is done through parameters present in the reservoir model. In this study we have

concentrated on the spatial distribution of porosity and permeability. Their spatial distribution can be

parameterized in various ways. Below explanation is provided of the ways of parameterization used in

this study.

Grid blocks

The most general approach is to consider all grid block values as independent parameters. In this model

the porosity, and directional permeabilities for all 1761 active grid blocks are used as parameters,

resulting in about 5000 parameters. Through such an approach no preconceived idea about the geology is

incorporated in the reservoir model. All knowledge must be inferred from the well and production data

only. The main problems with this approach are the large number of parameters and the lack of spatial

continuity present in the resulting reservoir models.

RegionsThe use of homogeneous regions is a way to reduce the number of parameters. Through a proper

selection of regions, some geological or reservoir engineering concepts can be incorporated in the

reservoir model. Regions can either be used to follow geological layers or genetic units within layers, or

can be used to characterize draining areas of wells. With regions, less parameters are needed to

characterize the reservoir model, but preconceived ideas about the characteristics of regions may be

inappropriate. The assumption of homogeneity within a region may not be justifiable and lead to abrupt

changes between the boundaries of regions. The region approach has however been the standard

approach in reservoir history matching.

Pilot points

With the advent of geostatistics, a new class of spatial parameterization approaches has emerged. Theyare known as pilot point or master point approaches. The developments for characterizing aquifers in a

hydrogeological context started by de Marsily et al. (1984) with application to well interference testing.

They used only kriging solutions to the transmissivity fields. The approach was extended to the use of

Gaussian Random Fields by Ramaraoet al. (1995) and Gomez-Hernandezet al. (1996). Floris (1996)

describes the application to hydrocarbon reservoir characterizing involving multi-phase production data.

A number of pilot points are used to build smooth spatially correlated corrections to porosity /

permeability fields. The method produces in continuously varying heterogeneous reservoir models,

controlled by a limited number of pilot points. The pilot points are traditionally pre-fixed, but a method

for selecting a limited number of pilot points based on a combination of geological uncertainty and

sensitivity to production data has also been developed (Cuyperset al. (1998)).

Global parametersA final class of parameters are those that cannot be linked to a particular spatial location, called global or

underlying parameters. Examples are stochastic parameters such as mean values, standard deviation or

correlation lengths, or object parameters such as channel width and height. Also for property fields,

which are built from a weighted sum of basis functions, the weighting coefficients generally are not

localized and thus are global parameters. In the adaptive chain approach by Holden (1998) or the gradual

deformation method by Hu (2000), the coefficients in the linear combination of property fields used to

generating a new field are a form of global parameters.


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Objective function

In order to measure the extent to which a reservoir model is conditioned to the available history data, a

measure must be defined to quantify the mismatch between the simulated response of the reservoir model

and the history data. This measure is called the objective function.

Least Squares norm

Traditionally, the goal of history matching is to define a reservoir model that optimally reproduces the

observed production history of a field. The mismatch between the simulated production data and the

observed history data can be quantified using a Sum of Squares norm. For each quantity, e.g. Bottom

Hole Pressure (BHP), Gas Oil Ratio (GOR), Water Cut (WCT), and for each time step,tk, the difference

of the simulated value,osim

, and the observed value,oobs

, can be calculated. Squaring the numbers and

summing them up results in the SoSnorm. The number of data available for each quantity may differ, for

example BHP will be measured more frequently than GOR or WCT when using permanent pressure

gauges. To avoid that the more frequently measured observations overshadow the significance of other

observations, theSoSvalue can be divided by the total number of values available,

SoS(oobs

,p) =

k ijk

ksim

ij

kobs

ij

ijk

tjpiw

ptotow

nnn

2);()(111

(1)

Subscriptsi and j run over the wells and production data types and kruns over the report times and nw,np,

ntare the respective number of samples. Symbolp denotes the parameters, denotes the model +

measurement error andw denotes extra weighting factors.

When the difference between simulated and historic response are normalized by the measurement plus

modeling error, a normalized SoSnorm results. If all weightsw equal unity, then a normalized SoSof 1

implies that on average the simulated data is within the error band around the historic data. Variation of

the weightw can be used to assign more significance to particular production data for example on

reservoir engineering grounds.

Likelihood function

The definition of a Bayesian likelihood function relies more on the specification of a model for the

uncertainty associated with the observed production. When the measurement plus modeling error are

assumed to be independently Gaussian distributed, the likelihood function f(oobs

|p) follows formally as,

f(oobs

|p) =c

2

);()(

2

1exp

ijk

ksim

ij

kobs

ij

i j k

ptoto

(2)

wherec is a normalization constant. This likelihood function expresses the likelihood that the historic

data can be explained by the reservoir model for which the likelihood holds. If it is low, the historic

data are unlikely to come from the reservoir model. If it is high, then the reservoir model response fits the

historic data well. Note that the Bayesian formalism does not allow subjective weighting of the

mismatches as done in theSoSobjective function (both weighting with the number of samples and the

extra weights unless this is formally brought in as a model assumption in the production uncertainty

model.

Posterior distribution

It is generally believed that conditioning reservoir models on the least squares or likelihood function

alone is mathematically ill-posed. Several solutions to the optimization problem may occur. Moreover,

even for a single solution production data is often insensitive to some parameters (e.g. permeability in


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unswept areas) or only sensitive to a combination of parameters (e.g. harmonic mean of permeability

data). Consequently, individual parameter values can be changed quite dramatically without deterioration

of the history match. Note that parameter values, which are insensitive during the history matching

period, may become sensitive during the forecasting period. Through the Bayesian prior function, the ill-

posed mathematics of the optimization process can be resolved / reduced. The general formula for the

Bayesian posterior distribution is given by

)()|()|( pfpofcopf (3)

wheref(o|p) is the likelihood function and f(p) is the prior function. When both the prior distributions on

the parameters and the production uncertainty are assumed to be Gaussian, including the prior results in

an extra Sum of Squares term (here denoted in vector notation),

pp

T

pi j k ijk

ksim

ij

kobs

ij

pCp

ptoto

copf

1

2

);()(

2

1

exp)|(

(4)

where pis the vector of expectations ofp and Cpis the covariance matrix.

Optimization algorithms

In order to do the conditioning to production data, most approaches employ an optimization technique.

One technique is to adjust the parameters manually and inspect the improvement on the match visually.

However, here we have quantified the mismatch between observation and simulation data in terms of the

objective function, allowing the use of optimization algorithms.

Gradient optimizationThe most powerful tool for optimizing smooth functions is gradient optimization. Many variations on the

theme of gradient optimization exist. The steepest descent technique follows the direction opposite to the

gradient vector. The step size must be determined by, for example, a line search or a trust region

approach. Conjugate gradients employ only mutually orthogonalized gradient directions. Levenberg-

Marquardt switches from steepest descent to a Gauss-Newton approach, which also employs the second

derivative when the optimum is approached. The dog-leg method by Powell (1972) uses another

combination of the steepest descent direction and the direction proposed by the Gauss-Newton method to

further improve the optimization.

In general for numerical simulators, the gradient expressions are not present in closed form. Several

methods exist to obtain the gradient information. The most straightforward approach is to calculate the

sensitivity coefficients for each of the parameters by finite differences. Each iteration of the optimization

process now requiresN+1 simulation runs, whereNis the number of parameters. This is a very time-

consuming process if many parameters are present in the reservoir model. Using the Broyden (1965)

technique, the gradient information can also be updated using only the evaluation of the optimization

function for each new set of parameters. Thus, each iteration in the process requires only one extra

simulation run. The gradient only needs to be initialized, requiringNextra up front simulation runs. In

recent years some reservoir simulators calculate the gradient directly based on the finite difference

equations present within the reservoir simulator. For Simulator 3, a reservoir simulator used in this study

(see Table 2), the extra time needed to calculated the gradient to a single parameter is equivalent to

approximately 30 % of a traditional simulation run.


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Genetic Algorithms

The main problem with the gradient optimization technique is the danger of getting trapped in local

minima. Two techniques that perform global optimization are simulated annealing (e.g. Hegstad et al,

1994) and genetic algorithms (Goldberg, 1989). Tests have shown that for our case the SA and GA

methods required approximately the same number of simulation runs to arrive at an optimal solution. In

this work we have pursued the Genetic Algorithms, because they are easily parallelised and they can

cope with multiple optima.

In the Genetic Algorithm a population of random samples is evolved for a number of generations. During

the evolution of the population, rules inspired by genetics are used to select the best fitting reservoir

models. At the end of the optimization a population results which may consist of a number of optimal

clusters.

Table 1. Naming convention in Uncertainty Quantification methods.

ML2

Maximum Likelihood value

MAP2 Maximum A Posteriori value

ML+ Maximum Likelihood value + local characterization of the likelihood function

MAP+ Maximum A Posteriori value + local characterization of the posterior distribution

multi-ML/multi-MAP multiple ML or MAP values using different initial models

multi-ML+ /

multi-MAP+

multiple ML or MAP values using different initial models + local characterization of

the objective function around each peak

Oliver-prod Oliver approach using only samples from the production data in the objective function

Oliver-full Oliver approach using samples from both the prior and the production data in the

objective function

Posterior sampling Statistical sampling from the full posterior distribution

Uncertainty quantification

Having defined the parameters to be used in the reservoir model, and having a way to minimize the

objective function, results in a single conditioned reservoir model. This model is called the ML

(Maximum Likelihood) solution when only sum of squares or the likelihood function is used in the

objective function or the MAP (maximum a posteriori) solution if the prior term is included. Production

for the conditioned model can be forecast from the ML or MAP parameter values. The next step is to

consider the approaches used in quantifying forecast uncertainty. Table 1 summarizes the naming

convention.

ML+/MAP+

Having obtained the ML/MAP solution, it is possible to locally characterize the objective functionaround this solution and transfer this information into forecast uncertainty. Such an approach is called a

ML+/MAP+ approach. An example is local linearization of the posterior. The Scenario Test Method

(STM) is another such approach (Roggero, 1997). In the STM, a search for extreme high and low

forecasts is invoked, starting from the ML/MAP solution. As a constraint in the search, the objective

function value may not drop below a threshold value. The lower the threshold value is set, the more

extreme forecasts may be produced. This method is believed to result in a characterization of the pdf

envelope around a single ML/MAP solution. A third approach is the use of a Genetic Algorithm. As

2These methods give single estimates and thus do not quantify uncertainty. They are included for naming convention

only.


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detailed before, the GA results in a population of reservoir models. Near every optimum a set of

individuals may be present, which contain information about the posterior locally around that optimum.

Multi-ML/Multi-MAP

Another class of approaches start from multiple initial reservoir models. This approach is termed multi-

ML or multi-MAP depending on which objective function is optimized. If the objective function is truly

multi-modal, then different conditioned models are expected, which will result in different forecasts.

During the PUNQ project, a GA optimization code used by BPAmoco has been adapted, such that

starting from a single initial population the final population may be centered around a number of distinct

optima. Thus, with the resulting GA an initial random population may zoom into a number of distinct

optima.

Multi-ML+/multi-MAP+

These approaches are a combination of the previous two classes, where multiple ML/MAP solutions are

found and local characterization is performed.

Posterior sampling

The above multi-ML+/multi-MAP+ approaches only give a local characterization of the objective

function around a number of optima. When the space between the optima carries a significant

probability, this approach gives a restricted view of the possible reservoir models, and consequently may

result in an underestimation of the range of uncertainty as well as a severe bias in the forecasted mean.

Only through sampling of the complete posterior distribution can the full uncertainty be quantified. A

statistically correct way of sampling from the posterior distribution is obtained by using the class of

Markov-Chain Monte-Carlo techniques (Hegstad & Omre, 1997). A straightforward MCMC technique is

one where each time reservoir models are proposed from the prior model. After reservoir simulation, the

likelihood for the reservoir model is determined. This likelihood is used as a weighting factor for the

reservoir model forecast. The drawback of this technique is that many prior samples may be neededbefore a reservoir model with a reasonable likelihood occurs, each requiring a time-consuming reservoir

simulation run.

In order to improve this, a sequence of linked reservoir models can be generated, converging to samples

from the posterior distribution. Each new model is generated by making limited alterations to the current

model, creating a Markov-Chain. Still many samples may be needed in order to generate a useful number

of posterior samples. Further improvement can be gained by letting the proposed new model depend on a

set of previous models. To implement this, the GA approach can be used to select parent models and

generate child realizations from them. This leads to a new version of MCMC termed Adaptive Genetic

MCMC.

Oliver

In the approach suggested by Oliveret al. (1996), the advantages of the previous approaches are merged.The approach aims at sampling from the complete posterior distribution, but using an optimization

technique to reduce the number of reservoir simulation runs needed. In the approach a sample is drawn

from the prior reservoir model. Concurrently, a sample is also drawn from the production data. This

production sampling is done because the production data contains observation errors and the reservoir

model contains modeling error. Pairs of prior reservoir samples and production samples are subsequently

history matched. The matching criterion is formed by both the mismatch between the production sample

and the simulated production data and the deviation of the reservoir model from the sampled prior

reservoir model used as starting point for the optimization. The latter term regularizes the ill-posed

character of the optimization problem. It can be proved that this approach indeed leads to a correct

sampling of the posterior distribution for Gaussian models for the reservoir geology and linear models for


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the fluid flow. Oliveret al. (1996) do show that for a well test model, which is non-linear, the approach

still compares well with results from MCMC sampling.

Application of Oliver's approach to full-field reservoir engineering problems can be found in (Zhan Wu,

1998) and Floris & Bos (1998). For such multi-phase problems there is no proof that the approach

correctly samples from the posterior distribution.

PUNQ-S3 truth case description

The PUNQ-S3 case has been taken from a reservoir engineering study on a real field performed by one of

the industrial partners in the PUNQ project. It was qualified as a small-size industrial reservoir

engineering model. The model contains 19x28x5 grid blocks, of which 1761 blocks are active. A top

structure map of the field is shown in Figure 2. The field is bounded to the east and south by a fault, and

by a strong aquifer to the north and west. A small gas cap is located in the center of the dome shaped

structure. The field initially had six production wells located around the gas oil contact. Due to the strong

aquifer, there are no injection wells. The geometry of the field has been modeled using corner-point

geometry.The porosity/permeability fields were regenerated in order to have more control over the underlying

geological / geostatistical model. The corresponding geological description is given in Appendix A. A

geostatistical model based on Gaussian Random Fields has been used to generate the porosity /

permeability fields. The fields were generated independently for each of the five layers. Geostatistical

parameters, such as means and variograms were chosen to be as consistent as possible with the

geological model. Collocated co-simulation was used to correlate the porosity and permeability fields

statistically within each layer. To generate the fields, the Fortran program SGCOSIM from Stanford

University was used. It is part of an extension to the GSLIB software library (Deutsch & Journel, 1992).

Figure 3 shows the resulting fields for permeability in each of the five layers. The porosity fields have

similar characteristics.

The reservoir engineering model was completed with the PVT and aquifer data from the originalreservoir model and with power law relative permeability functions. There is no capillary pressure in the

model. The production scheduling resembles the history in the original model, i.e. a first year of extended

well testing, followed by a three year shut-in period, before field production commences. The well testing

year consists of four three-monthly production periods, each having its own production rate. During field

production, two weeks of each year are used for each well to do a shut-in test to collect shut-in pressure

data. A fixed fluid production constraint is imposed on the wells. After falling below a limiting bottom

hole pressure, they will switch to BHP-constraint. Using this completed reservoir engineering model, a

synthetic history is generated using the Simulator 1. The total simulation period is 16.5 years. Pressure,

water-cut and gas-oil ratio curves have been generated for each of the wells. Figure 4 shows a typical set

of production curves. The total oil recovery after the simulation period is 3.87 106

Sm3. The complete

data set suitable for Simulator 1 is publicly available on the internet at www.nitg.tno.nl/punq.

Gaussian noise was added to the well porosities / permeabilities and to the synthetic production data. Thestandard deviation on poro / perm values was set to 15 %. For the production data the Gaussian noise was

correlated in time to mimic the more systematic character of errors in such data. The noise level on the

shut-in pressures was 3 times smaller than on the flowing pressure, respectively 1 bar and 3 bar, to reflect

the more accurate shut-in pressures. The noise level on the GOR was set at 10 % before gas breakthrough

and 25 % after gas breakthrough, reflecting the difference between the solution and the free gas situation.

Similarly, Gaussian noise of 2 % before and 5 % after water breakthrough was used for the WCT.

Each of the partners in the project was given the noisy well porosities / permeabilities and synthetic

production history of the first 8 years (Note that this history includes 1 year of well testing, 3 years of

field shut-in, and covers 4 years of actual field production). The synthetic production data consisted of

the BHP, WCT and GOR for each of the six wells. Within the history period, two wells show gas

breakthrough and one well shows the onset of water breakthrough. All partners were asked to forecast the


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total oil production after 16.5 years including uncertainty estimates, using the Bayesian formalism. Note

that none of the partners were given the exact porosity / permeability grids, only the geological

description. Each of the partners used his own workflow to infer these grids. In a second stage, five

incremental wells were defined. For the truth case, the extra wells resulted in an incremental recovery of

1.46 106

Sm3. The partners were again asked to forecast the incremental recovery as a probability

distribution function.

Table 2. Approaches used by the different partners.

Partner +

Approach

Parameter

Domains

Independent

Parameter(s)

Spatial

technique

Uncertainty

Quantification

Optimization Reservoir

Simulator3

TNO-1 Homogeneous layers PORO, Kv, Kh,

uncorrelated

Piecewise

Constant

Oliver-full Dog-leg +

Broyden gradients

Simulator 1

TNO-2 Homogeneous

drainage area regions

PORO, Kv, Kh,

uncorrelated

Piecewise

Constant


Broyden gradients

Simulator 1

TNO-3 Homogeneous flow

path regions

PORO Piecewise

Constant


Broyden gradients

Simulator 1

Amoco-

Iso

Fixed pilot points

Isotropic variogram

PORO, Kv, Kh,

correlated statist.

GRF multi-ML+, start from

random prior models

GA Simulator 1

Amoco-

Aniso

Fixed pilot points

Anisotropic variogram

PORO, Kv, Kh,

correlated statist.

GRF multi-ML+, start from

random prior models

GA Simulator 1

Elf Pilot points selected PORO GRF multi-ML, start from

sampled prior models

GN-SD hybrid +

finite diff.

gradients

Simulator 1

NCC-GA Global parameters

acting on whole grid

PORO, Kv, Kh,

correlated

GRF multi-ML GA Simulator 2

NCC-AG

MCMC

Global parameters

acting on whole grid

PORO, Kv, Kh,

correlated statist.

GRF Posterior sampling GA Simulator 2

IFP-STM Fixed pilot points PORO Kriging Scenario Test Method

(ML+)

Gauss-Newton +

simulator

gradients

Simulator 3

IFP-Oliver Fixed pilot points PORO Kriging Oliver-Prod, start from

kriged solution

Gauss-Newton +

simulator gradients

Simulator 3

NCC-

Oliver

Fixed pilot points PORO, Kv, Kh,

correlated statist.

GRF Oliver-full Gauss-Newton +

simulator gradients

Simulator 3

Results

General setup

In order to be able to compare results from the different approaches, a fair measure for assessing forecast

uncertainty is required. This measure is determined by the objective function, but is also influenced by

the assumptions in the model. In the Bayesian context, the objective function has a prior geological

component and a production SoS or likelihood component. With the different parameterizations used bythe partners, prior pdfs cannot be made equivalent, so it was decided that a weak prior should be used.

The production component was preset so that partners used the same objective function. The objective

function used in most approaches is given by equation (1). Only for the NCC-MCMC case was the

likelihood function (4) used. The production data comprised BHP, GOR and WCT data from all six

wells. For the standard deviations, ijk, the noise levels applied to the synthetic case were used. The

weights,wijk, were set to 4 just prior and after breakthrough in the wells, with gas or water breakthrough

to give more weight to the occurrence of the breakthrough. The wells for which no water breakthrough

was observed received a zero WCT data point, again with a weight of 4, to penalize reservoir models that

3

The simulators used are numbered as follows 1=Eclipse, 2=More, 3=Athos.


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showed early breakthrough. WCT and GOR data after breakthrough and all pressure data were given a

weight of 1.

We note that it is hard to isolate the influence of the model assumptions, for example to differentiate

between a Gaussian Random Field model or a zone model for spatial character of the porosity /

permeability field, or to use a deterministic versus a stochastic poro/perm relationship versus independent

porosity and permeability. This hampers the comparison of the uncertainty ranges for forecasts from

various approaches.

Base Case

Table 2 shows which selections have been made in the workflow for the quantification of production

forecast uncertainty. Figure 5 shows the cumulative pdf curves generated by the various approaches used

by the partners. Figure 6 displays a summary in terms of the low-median-high ranges forecasted. The

results are classified into three groups, the TNO curves, the IFP/NCC-Oliver curves and the

Amoco/Elf/other-NCC curves.

TNO groupThe TNO curves are generated using piecewise homogeneous regions. The digits labeling the different

curves correspond to the number of layers (first digit), number of regions within each layer (middle digit

if it occurs) and parameter types (last digit), i.e. , or ,khand kv, respectively. The TNO curves show

larger uncertainty ranges than the other curves. Although their shapes are the similar, they show a mutual

shift along the horizontal axis. Initially the wider range was attributed to the use of production sampling

in the Oliver approach. Testing this hypothesis showed that dropping the production sampling does not

significantly affect the results. Other partners' results confirmed this. Further testing led to the final

conclusion that the use of homogeneous regions is not justified in this case. The homogeneous region

models do not lead to satisfactory history matches and result in large spreads in production forecast.

Parameterizations based on heterogeneous models, such as the pilot point approaches, are more

appropriate.

Amoco / Elf / NCC-GA,-MCMC group

The Amoco-Anisotropic curve and the Elf curve are practically identical. The main difference between

the approaches that generated the curves is the use of a different optimizer. Apparently, it is not critical

in this case to use global optimization or to consider multiple clusters. This result suggests that the

posterior surface may look like a plateau and not a landscape of individually identifiable peaks. The

Amoco-Isotropic curve shows a narrower uncertainty range than the Anisotropic curve because the

variogram used for the anisotropic case has a longer range in the NW-SE direction that accounts for the

presence of elongated channels. With the longer range, less variability is present in the reservoir porosity

/ permeability fields.

The NCC-MCMC curve has a smaller range; two reasons are forwarded to explain this. Firstly, in the

MCMC, the reservoir porosity/ permeability fields are linear combinations of Gaussian random fields,thus retaining Gaussianity. This extra constraint reduces the uncertainty ranges produced in the reservoir

models. Porosity/permeability fields generated with the pilot-point approaches do not satisfy Gaussian

constraints. Secondly, after detailed analysis of the results of various approaches (Omre et al., 1999), it

was noted that the optimization step used in most approaches leads to generation of extreme values of

parameters (many parameters end up at values prescribed at geological/physical limits). These extreme

values do result in good history matches, but give wider spreads in the forecasts. Since MCMC results

generally show much smoother optimized porosity/permeability fields their production forecast range is

smaller. NCC have used the posterior distribution from equation (4) for the dynamic data conditioning in

the NCC-GA and NCC-MCMC result. Since this criterion is weaker than the least squares norm from

equation (1), because of the lack of normalization, one would expect wider ranges for these NCC curves.


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Since the ranges are smaller we may conclude that the exact form of the objective function does not

significantly influence the result.

IFP / NCC-Oliver group

The two IFP curves and the NCC-Oliver curve have median values which are shifted to lower ranges.

This systematic shift is attributed to the use of another reservoir simulator for these approaches, i.e.

Simulator 3 versus Simulator 1. Note that there is no apparent shift between the approaches that use

Simulator 2 or Simulator 1. In the incremental study we will see that this shift vanishes when incremental

recovery is considered. The uncertainty range in the IFP-Oliver curve is very small. This can be

attributed to the fact that no prior sampling was used in the IFP-Oliver approach, and that the production

sampling does not contribute very much to the uncertainty range. In the NCC-Oliver approaches prior

sampling is used, resulting in an increase in the forecasted range. The STM curve has the largest range.

This is remarkable, because STM is believed to quantify the envelope of the forecast pdf around a single

posterior peak. Again this result is acceptable if the posterior surface looks like a large plateau.

Comparison to the truth caseFigure 6 shows that five out of the eleven estimated uncertainty ranges do not include the truth case

value, thus confirming the general experience that uncertainty is often underestimated. Note that the

TNO-2 approach with its large uncertainty range still doesnt include the truth case, and that NCC-

MCMC with its small uncertainty range does include the truth case. Again this indicates the importance

of using properly history-matched models and appropriate models for heterogeneity.

Incremental case

All partners were asked to generate production forecasts for the conditioned reservoir models, including

five incremental wells. The partners were asked to quantify the incremental recovery, the difference

between total oil production of the base case and the case with the incremental wells included. Again, the

result should be a cumulative distribution function.Figure 7 shows the cdf curves of incremental recovery for the incremental wells case. These curves can

be split into the TNO curve and all other curves. Again, we see that the homogeneous region

parameterization leads to a severe bias in the results. All other curves tend to be in agreement. Note

especially that the IFP curve now lines up with the other curves. The shift between Simulator 3 and

Simulators 1 and 2 is negated because it occurs both in the base case and in the case including the

incremental wells. In calculating the incremental recovery, the difference between the two results is

calculated, so canceling the shift.

Comparison to the truth case

Most approaches underestimate the incremental recovery of the truth case. The truth case value lies on

the high side of five of the curves. The NCC curves all have ranges which do not cover the truth case

value. The fact that the truth case value lies in the upside of the curves can be explained in hindsight. Thewell locations were chosen in such a way that the recovery from the truth case permeability field would

be optimized. For an ensemble of permeability fields, these well locations are not optimal and will

produce at a lower average rate.

Discussion

Before drawing conclusions from the results, some discussion is needed regarding the comparison of the

results from the various partners. In order to make an objective comparison between the various

uncertainty quantification approaches possible, the prior and likelihood measures were prescribed.

However, especially the results from the NCC approach make it clear that the uncertainty ranges are not

only influenced by the information used to define the objective function, but also on the underlying

assumptions made and techniques used in the reservoir modeling. These can be seen as an implicit form


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of prior information. In this study the important assumptions are related to the fluid flow model and the

spatial distribution of porosity and permeability. Making a particular choice for the fluid flow simulator

and for modeling the spatial distributions will affect the uncertainty quantification in an implicit way.

This makes comparison of uncertainty ranges somewhat ambiguous. In particular, it is not formally

possible to compare all results with a single true uncertainty range. Each approach has its own

underlying set of assumptions, resulting in its own uncertainty range. Therefore, we can only make

observations and not judgements in terms of the correctness of estimates.

Conclusions

From the PUNQ-S3 integrated case study we conclude the following

Using porosity and permeability multipliers for homogeneous layers or regions resulted in large

uncertainty ranges and a significant bias in the production forecast due to the poor quality of history

matches obtained with these models. Heterogeneous models parameterized by pilot points gave more

consistent results.

Production sampling (as introduced by Oliver et al. (1996)) did not significantly contribute to theforecast uncertainty range. Omitting the sampling of several prior geological models led to too

narrow ranges of uncertainty around the forecasts.

The use of different reservoir simulators to model the same reservoir led to a systematic difference in

production forecast for one of the (three) simulators. Fortunately, when considering incremental

recovery, the difference was negated.

Approaches which include an optimization step, showed a tendency to predict larger uncertainty

ranges. This is because the conditioned porosity/permeability fields contain significantly more

extreme high and/or low values.

Results suggest that the multi-dimensional posterior distribution looks more like a large plateau, than

many isolated peaks.

Comparison with the cumulative production forecast of the truth case showed that five out of theeleven predicted forecast ranges did not include the truth case value.

All approaches underestimated the incremental recovery of the truth case. The latter is probably

because the locations of the incremental wells were optimally designed for the truth case.

We realize that these conclusions may be specific to the current test case. For confirmation, the above

case study should in principle be repeated for many truth cases.

Acknowledgements

The European Commission is acknowledged for partly funding this project in the Joule-III Non-

Nuclear Energy Programme. We thank all other partners in the PUNQ project for the lively

discussions and valuable contribution to this study. We thank Elf for supplying a field case which

acted as a valuable integration tool.

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Website

http://www.nitg.tno.nl/punq


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Appendix A. Geological description.

This appendix gives a geological description of the heterogeneities that occur based on knowledge of the

regional geology (such as paleoslope, paleo water depth, gross environments of deposition, size and

shape of sedimentary bodies, structural trends and style) which is normally known from adjacent fields

and wildcat wells. In the geological interpretation, the layer thicknesses, which are all in the order of 5

meters, played an important role.

Sediments were deposited in a deltaic coastal plain environment. Layers 1, 3, and 5 consist of fluvial

channel fills encased in floodplain mudstone. Layer 2 represents marine or lagoonal clay with some distal

mouthbar deposits. Layer 4 represents a mouthbar or lagoonal delta encased in lagoonal clays.

Layers 1, 3, and 5 have linear streaks of highly porous sandstone (phi > 20 %), with an azimuth

somewhere between 110 and 170 degrees (SE). These sandstone streaks of about 800 m width are

embedded in a low porosity shale matrix (phi < 5 %). The width and the spacing of the streaks vary

somewhat between the layers. A summary is given in Table 3. In layer 2 marine or lagoonal shales occur , in which distal mouthbar or distal lagoonal delta

occur. They translate into a low-porous (phi < 5%), shaly sediment, with some irregular patches of

somewhat higher porosity (phi > 5%).

Layer 4 contains mouthbars or lagoonal deltas within lagoonal clays, so a flow unit is expected

which consists of an intermediate porosity region (phi ~ 15%) with an approximate lobate shape

embedded in a low-porosity matrix (phi < 5%). The lobate shape is usually expressed as an ellipse (ratio

of the axes= 3:2) with the longest axis perpendicular to the paleocurrent (which is between 110 and 170

degrees SE).

Table 3. Expected sedimentary facies with estimates for width and spacing for major flow units for each

layer.

Layer Facies W Spacing

1 Channel Fill 800 m 2-5 km

2 Lagoonal Shale


4 Mouthbar 500-5000 m 10 km



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List of tables

Table 1. Classification of Uncertainty Quantification methods.

Table 2. Approaches used by the different partners.

Table 3. Expected sedimentary facies with estimates for width and spacing for major flow units for each

layer.

List of Figures

Figure 1. General work flow used for the quantification of forecast uncertainty.

Figure 2. Top Structure map of the PUNQ-S3 case. The field contains both oil and gas. Black dots

indicate six initial wells located around the gas-oil contact. White dots indicate additional wells added ina later phase.

Figure 3. Horizontal permeability fields in the five layers for the synthetic PUNQ-S3 case.

Figure 4. Typical dynamic production data for a well showing bottom hole pressure (BHP), oil

production rate (OPR), gas-oil ratio (GOR) and water cut (WCT). After a 1 year extended production test

and 3 years field shut-in, field production starts at a fixed oil rate. History data runs until year 8 and is

forecasted until year 16.5. This well shows the start of water production. Two other wells show gas

breakthrough.

Figure 5. Cumulative distribution functions for the total oil production forecasted up to 16.5 years. Thecurves can be split up into three classes, the TNO curves, the IFP / NCC-Oliver curves and the Amoco /

Elf / other-NCC curves.

Figure 6. Summary of low - median high ranges of cumulative prouction at 16.5 years for all

approaches. The ranges were generated by ordering the samples and taking the 10 %, 50 % and 90 %

model.

Figure 7. Cumulative distribution functions for incremental recovery at 16.5 years using 5 more wells.

The curves can be split up into two classes, the TNO curve and the other curves.


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Figure 1. General work flow used for the quantification of forecast uncertainty.

Prior sampling Production

sampling

Reservoir model Production data

Parameterization

+ prior pdfs

History data

+ error data

Reservoir

simulation

History matched

reservoir model

Forecasting

Objective function

Calculation

Updating of

reservoir model


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4000

PUNQS2 XY plane 1 TOPS step 0

2000

0

Tops at step 0, 0 days

2339 2347 2355 2363 2371 2379 2387 2395 2403 2411

1600800 2400 3200

Figure 2. Top Structure map of the PUNQ-S3 case. The field contains both oil and gas. Black dots

indicate six initial wells located around the gas-oil contact. White dots indicate additional wells added in

a later phase.


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Figure 3. Horizontal permeability fields in the five layers for the synthetic PUNQ-S3 case.


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Figure 4. Typical dynamic production data for a well showing bottom hole pressure (BHP), oil

production rate (OPR), gas-oil ratio (GOR) and water cut (WCT). After a 1 year extendedproduction test and 3 years field shut-in, field production starts at a fixed oil rate. History data

runs until year 8 and is forecasted until year 16.5. This well shows the start of water

production. Two other wells show gas breakthrough.


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Figure 5. Cumulative distribution functions for the total oil production forecasted up to 16.5 years. The

curves can be split up into three classes, the TNO curves, the Amoco / Elf / other-NCC curves and theIFP / NCC-Oliver curves.

Cumulative Oil Production after 16.5 years (in million Sm3)

3 3.2 3.4 3.6 3.8 4 4.2

0

0.2

0.4

0.6

0.8

1

Cumulativedistributionfunction

TNO-1: 5x3 parsTNO-2: (5x6)x3 pars

TNO-3: (5x14)x1 pars

Amoco Isotropic

Amoco Anisotropic

Elf

NCC-GA

NCC-MCMC

IFP-STM

IFP-Oliver

NCC-Oliver

truth case


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Figure 6. Summary of low - median high ranges of cumulative production at 16.5 years for all

approaches. The ranges were generated by ordering the samples and taking the 10 %, 50 % and 90 %

model.


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Figure 7. Cumulative distribution functions for incremental recovery at 16.5 years using 5 more wells.

The curves can be split up into two classes, the TNO curve and the other curves.

0

Incremental Oil Production after 16.5 years (in million Sm3)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.2

0.4

0.6

0.8

1

Cumu

lativedistributionfunction

TNO-2: (5x6)x3 pars

Amoco IsotropicAmoco Anisotropic

Elf

NCC-GA

NCC-MCMC

IFP-STM

NCC-Oliver

truth case

Documents

Methods for Quantifying the Incertainty of Production Forecasts a Comparative Study