APPLICATION OF EXPERIMENTALDESIGN AND RESPONSE SURFACE

ANALYSIS TO GAS CAPBLOWDOWN

A REPORT SUBMITTED TO THE DEPARTMENT OFPETROLEUM ENGINEERING

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THEDEGREE OF MASTER OF SCIENCE

ByAmit KumarAugust 2002

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I certify that I have read this report and that in my opinion it is fully adequate, in scopeand in quality, as partial fulfillment of the degree of Master of Science in PetroleumEngineering.

_____________________________Dr. Khalid Aziz

(Principal Advisor)

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Acknowledgement

I wish to express my sincere thanks to Dr. Khalid Aziz for his guidance, patience, andencouragement. His insights were invaluable during the research. I would like to expressmy gratitude to SUPRI-B for funding this research.

I also wish to thank BHP Petroleum for providing the field data for the Skua field. I amgrateful to Peter Behrenbruch for his encouragement. I also received some useful pointersand suggestions from Dr. Christopher D. White, LSU and Dr. Adwaita Chawathe,ChevronTexaco for which I am thankful.

I must mention the help I received from my friends and colleagues at Stanford. I wouldespecially like to thank Rakesh Kumar, Ashish Dabral, and Sunderrajan Krishnan.

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Table of Contents

ACKNOWLEDGEMENT iii

LIST OF FIGURES v

LIST OF TABLES vi

ABSTRACT vii

1. INTRODUCTION 1

1.1 HISTORICAL DEVELOPMENT OF EXPERIMENTAL DESIGN 11.2 ORGANIZATION OF REPORT 3

2. EXPERIMENTAL DESIGN AND RESPONSE SURFACES 4

2.1 TERMS AND DEFINITION 42.1.1 DEFINITIONS 4

2.2 TYPES OF DESIGN 62.2.1 Classical Experimental Designs 6

2.2.3.1 Fitting A Model 92.2.3.2 Significance Testing for the Fitted Model and Individual Coefficients 11

2.3 SOME POPULAR DESIGNS 132.3.1 Plackett-Burman Design Error! Bookmark not defined.2.3.2 D-optimal Design 142.3.3 Box-Behnken Design 15

3. GAS CAP BLOWDOWN 16

3.1 EARLY AND DELAYED BLOWDOWN 163.2 IMPORTANT FACTORS IN GAS CAP BLOWDOWN 173.3 SIGNIFICANCE OF GAS CAP BLOWDOWN 183.4 SIMULATION MODEL 19

4. APPLICATION OF EXPERIMENTAL DESIGN 21

4.1 SETUP OF THE REGRESSION 214.2 COMPARISON OF DESIGNS 224.3 EFFECTS OF FACTORS 24

4.3.1 Main Effects 254.3.2 Interaction Effects 264.3.3 Nature of Residuals 31

4.4 CONCLUSIONS AND FUTURE WORK 35

NOMENCLATURE 37

REFERENCES 40

APPENDIX A. GENERATING A HALF-FRACTION DESIGN 42

APPENDIX B. ALGORITHM FOR CONSTRUCTING D-OPTIMAL DESIGNS 45

APPENDIX C. EXAMPLE ECLIPSE DATA FILE 47

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List of Figures

Figure 1. Pareto Chart of the Standardized Effects ..................................................................................... 24Figure 2. Main Effects for the Full Factorial Case ...................................................................................... 25Figure 3. Main Effects for Box-Behnken Design .......................................................................................... 26Figure 4. Main Effects for D-optimal Design with 49 Runs ......................................................................... 27Figure 5. Main Effects for D-optimal Design with 100 Runs ....................................................................... 27Figure 6. Interaction Effects for Full Factorial Design ............................................................................... 28Figure 7. Contour Plot of Response Surface for Horizontal Permeability and Relative Gas Cap Size........ 30Figure 8. Contour Plot of Response Surface for Horizontal Permeability and Gas Rate ............................ 30Figure 9. Contour Plot of Response Surface for horizontal permeability and Relative Water Zone Size .... 31Figure 10. Contour Plot of Response Surface for Horizontal Permeability and Horizontal Location of Well...................................................................................................................................................................... 32Figure 11. Contour Plot of Response Surface for Gas Rate and Relative Water Zone Size ......................... 32Figure 12. Contour Plot of Response Surface for Gas Rate and Horizontal Location of Well .................... 33Figure 13. Contour Plot of Response Surface for Relative Gas Cap Size and Horizontal Location of Well 33Figure 14. Normal Probability Plot of the Residuals (Full Factorial Design)............................................. 34Figure 15. Standardized Residuals versus the Fitted Values (Full Factorial Design) ................................. 34Figure 16. Standardized Residuals versus Observation Order (Full Factorial Design) .............................. 35Figure 17. Analysis of a Half-Fraction of the full 24 design; a 25-1 Fractional Factorial Design ................ 43

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List of Tables

Table 1 – Reservoir Properties, Dimensions and Initial Conditions ............................................................ 19Table 2 – Parameters Varied in Simulation Study....................................................................................... 20Table 3 – Factors and Their Levels ............................................................................................................. 21Table 4 – Comparison of Regressions ......................................................................................................... 23Table 5 -- Coefficients and P-Values for Regression Terms......................................................................... 29

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Abstract

Numerical Reservoir Simulation is often too expensive to allow exhaustive investigationof sensitivities to multiple parameters. Experimental Design and the associated techniqueof response surface analysis make available the tools to reduce the number of simulationsrequired. Moreover, the methodology provides a statistical framework to estimate theimpact of various parameters and their interactions. It accords not only accurate estimateof response, but also quantification of errors.

In the present work, second order polynomial response surface models were used toapproximate the relationship between recovery and six reservoir and productionparameters in case of Gas Cap Blowdown with the objective to identify the keyparameters that govern the recovery from Gas Cap Blowdown and to quantify theirimpact. High gas rate in presence of high horizontal permeability, a small gas cap, largewater support, and central location of the well were found to be the conditions for highrecovery. The effects of many factors were found to depend on other factors; the modelalso captured these interaction effects.

Different experimental designs were employed and compared with the exhaustive case. Itwas found that D-optimal design with the same number of runs as Box-Behnken designprovided a better fit to the observed reservoir behavior. There was only a slight increasein the goodness of fit using a D-optimal design with double the number of runs.Comparisons with the exhaustive case confirmed that all the designs used in this studygive consistent results about statistically significant effects.

Uncertainty analysis on the reservoir performance can be made using the regressionsdeveloped. Also, optimization of response can be performed using the regression modelas an inexpensive proxy for the flow simulator.

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Chapter 1

Introduction

Experimental Design and the associated technique of Response Surface Modeling can bea powerful tool to make effective use of reservoir simulation for sensitivity andoptimization studies.

In order to optimize data acquisition and field development, reservoir modeling teamsrequire knowledge of sensitivities to key parameters and uncertainty in the outcome. Thisis generally addressed by running a large number of simulations spanning the range ofuncertainty in geologic, engineering, and even financial parameters. Even though thevalues of model parameters remain uncertain, for a given set of parameters, the outcomeof reservoir simulation is deterministic and reasonably accurate. The complexity inreservoir simulation software and the high running time of complex simulation modelspose a limit on the number of simulations actually performed.

Experimental Design is used to select a moderate number of simulation runs and analyzethem to estimate the sensitivity of reservoir behavior to various factors. Moreover,accurate polynomial models called ‘Response Surfaces’ can be fitted to the simulatorresponse and can be used as a computationally efficient proxy for the reservoir simulatorin performing uncertainty analysis, parameter estimation, and optimization.

1.1 Historical Development of Experimental Design

Experimental Design has been widely used in agriculture, process control, qualitycontrol, and optimization; primarily, to select a set of experiments and analyze them toascertain the effect of variables, interaction of variables, and estimation of errors.Unfortunately, this technique has not been as widely used in the petroleum industry as itmight have been. A few applications in the petroleum area are cited:

• Chu(1990), to optimize the choice of the completion interval in steamflooding;• Damlseth et al. (1991), to a North Sea field development study;• Narayanan et al. (1999), to generate psuedofunctions;

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• Dejean and Blanc (1999), to build a simplified model for a process and estimate theuncertainties on the response predictions in combinations with experimental designand Monte Carlo simulations;

• White et al. (2000), for parameter estimation and uncertainty analysis;• Wang (2001), for simulation of turbidite reservoir using outcrop data;• Friedmann et al. (2001), to assess uncertainty in recovery predictions in channelized

reservoirs for both primary and waterflood processes; and• Manceau et al. (2001), to quantify the impact of important reservoir uncertainties on

cumulative oil production and optimize future field development.

Chu (1990) used two-level factorial design (see section 2.2.1) to predict steamfloodperformance. He studied the effect of various rock and fluid properties as well as designand operating variables. However, Chu did not include uncertainty assessment of theresponse. Damlseth et al. (1991) performed a simulation study based on modelsconstructed using an optimal design. The resulting response surface was used to identifyand assess important factors affecting the recovery as well as the interaction amongfactors of interest. Uncertainty assessment was done using Monte Carlo simulations onthe response surface obtained.

Narayanan et al. (1999) applied the response surface approach to waterflooding studiesfor models based on an outcrop. Response surface methodology was used to generatepseudofunctions for different reservoir descriptions without expensive fine gridsimulations. As the response surface technique was found to be simpler and cheaper thanfine grid simulations, it allowed easy generation of pseudofunctions for differentscenarios. The study also quantified the effects of different parameters of the reservoirmodel on the recovery responses and upscaled properties.

Dejean and Blanc (1999) used the same set of tools for a synthetic field but they studied alarge number of factors using a cheaper two-level design and retain only those parametersfor further analysis using three-level designs that were seen to have the most significanteffect on the response. In a similar work, Wang (2001) used the statistical tool ofprincipal component analysis to reduce a large number of geological parameters to amore manageable set and performed “designed” simulation on turbidite reservoirs usingoutcrop data.

White et al. (2000) employed the same techniques for estimating parameters andassociated uncertainty assessment. They also examined the effects of varyinggeostatistical parameters and compared geostatistical and quasi-deterministic models ofgeologic variability. They used the response surface models to test differences betweenscenarios, assess sensitivities to factors, and estimate the effects of measurements onresponse uncertainty. They also used Bayesian statistics to calculate maximum-likelihoodestimates of factors conditional to sets of responses.

Friedmann et al. (2001) performed response surface analysis on simulation runsgenerated using experimental design to obtain a simplified (polynomial) analog to thesimulation model. Moreover, they used neural networks trained on the simulation models

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generated using experimental design to construct recovery type curves. The limitation ofthe type curve approach is that they are suitable only in immature projects wheresufficient data is not available to build reliable simulation models. Moreover, since thetype curves are based on synthetic geologic models, their use is limited to cases where thereservoir is similar to the synthetic model.

Manceau et al. (2001) combined experimental design methodology with gradualdeformation methods and optimization techniques into a “Joint Modeling Method”. Inthis method, the production response is modeled with a “mean model” and a “variancemodel”. The mean model, after experimental design methodology, describes theproduction response as a function of the deterministic parameters like petrophysicalproperties and production parameters. The variance model describes the dispersion of theproduction response due to the non-continuous stochastic parameters like geostatisticalrealizations. Together these models provided the framework for quantifying the riskassociated with both deterministic and stochastic uncertainties.

A persistent problem with reservoir simulation studies is the large number of simulationsthat must be performed in order to estimate the impact of various parameters on fluidflow. Experimental design make it possible to select with statistical means only a smallsubset of the total number of runs otherwise required and response surface techniquefacilitates the analysis of the effect of the parameters along with their interacts with otherparameters. The general strategy employed in this work is applicable to all types ofsimulation studies that involve estimation of impact of a large number of parameters.

1.2 Organization of Report

The present chapter introduces the topic and outlines previous applications ofExperimental Design in petroleum engineering. Chapter 2 explains the methodology ofexperimental design and response surface analysis and Chapter 3 explains the problem ofgas cap blowdown to which these techniques are applied. Chapter 4 details the procedureadopted in this application and presents the results obtained. Chapter 4 also contains theconclusions that are drawn from the present study and suggests some of the future workthat might be done in this area. Appendix A illustrates some of the basic concepts behindthe generation of economical designs through the example of half-fraction designs.Appendix B outlines the use of algorithms to generate optimal designs. Appendix Ccontains an eclipse data file with all factors at their base values. Terms used in equationsare defined just after they are used, but for reference, they are also provided in theNomenclature Section.

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Chapter 2

Experimental Design and Response Surfaces

Myers and Montgomery (1995) describe Design of Experiments (DOE) as a method toselect experiments to maximize the information gained from each experiment and toevaluate statistically the significance of different factors.

An experimental design study aiming to generate response surfaces requires identificationof the various factors that cause changes in the response and predicting the variousfactors these variations in a mathematical form. The choice of the mathematical form (themodel) and selection of the set of experimental conditions (the design) both influenceeach other. For example, a quadratic model requires a design with at least three levels.One particular design can be used to generate different models. At the same time,different models may be used to estimate the parameters of a particular model.

The design provides a set of input values for the factors that are used to run theexperiments or simulations (numerical experiments). The results obtained from theexperiments are fitted to the model, using standard regression techniques. Statistical toolsallow the effects of the factors to be ascertained as well as the significance of theseeffects. The model fitted to the data can also be used as a cheap proxy for the actualprocess being studied by the experiments in order to perform optimization, risk analysis,etc.

2.1 Terms and Definition

It is convenient to use the standard terminology of Experimental Design in order tofollow the later discussion. Hence important terms are defined below based on Box et al.(1978).

2.1.1 Definitions

Designs are lists of different experimental conditions (or combination of factors) at whichexperiments (or simulations) are performed. With two factors, a complete 3-level designrequires 32 runs. In matrix notation, each row of the design matrix indicates a run,whereas each column contains the settings of each factor. e.g. a design with two factors Aand B and 2 runs may be given as:Run Factor A Factor B1 +1 -12 0 +1

Conventionally, in matrix notation, the same is expressed as follows:

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+1 -10 +1

Factors are the input variables. They could be the control variables in an experiment orstochastic. They are systematically varied in the simulation study or an experiment toassess their effects.

Levels. The values that a factor takes constitute its “levels”. A two-level factor, forexample, will take a high value and a low value. Sometimes the number of levels eachfactor takes identifies a design. e.g. in “a two level design”, all factors will have a highand a low value. The levels are customarily denoted by “+” or “-“ referring to the higherand the lower values respectively. In a 3-level design the levels are denoted by “+1”, “0”,“-1”, in reference to the high, the base, and the low setting of the values of the factorsrespectively.

Responses are the system outputs. We might consider oil in place, recovery efficiency,breakthrough time, initial rate, etc. as responses.

Main Effect. The main effect of a factor can be given as the difference between theaverage response at higher settings of the factor and the average response at lowersettings of the factor.

Interaction Effect. If the effect of a factor depends on the effect of another factor, the twofactors are said to interact. For example, the performance of dual multilateral wells in areservoir with anisotropy in horizontal permeability is affected by well orientation muchmore severely compared to a multilateral of quad configuration (Rivera et al., 2002).Hence, orientation and configuration of multilateral wells interact in their effect onreservoir performance.

Confounding. When the factors in a design are so arranged that some effects areindistinguishable from other effects, these effects are said to be “confounded”. Onlycertain confounding patterns are considered to generate designs in order that the effectsof interest do not confound with each other. The terms to be confounded with the meanterm in the model determine the entire confounding pattern by implication. (Appendix Apresents this concept with an example.)

Resolution. A design of resolution R is one where no l-way interactions are confoundedwith any other interaction of order less than R-l. (By convention, resolution is expressedin Roman.) A design of resolution R=III(3) does not confound main effects with oneanother but does confound main effect with two-factor interactions; whereas, A design ofresolution R=IV(4) does not confound main effects with two-factor interactions but doesconfound two-factor interactions with other two-factor interactions. In the same vein, thehigher resolution R=V (5), does not confound main effects and two-factor interactionswith each other, but does confound two-factor interactions with three-factor interactions.

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2.2 Types of Design

Broadly, there are two kinds of designs in use, namely:1. Classical Experimental Designs2. Optimal Experimental Designs

Response surfaces are the widely used tools to analyze the results of the experiments (orresponse) that may have been performed using an experimental design.

2.2.1 Classical Experimental Designs

Factorial designs are the simplest, requiring Lk experiments, where L is the number oflevels and k is the number of factors. Since the factors are set at either their maximum orminimum value in an two-level factorial experiment, these designs cannot go beyondestimating first-order effects and interactions. Hence, they are valid only over a limitedrange of values, or have low accuracy. Scaling the factors and response can help alleviateboth limitations in some cases.

Quadratic effects can be estimated only with a three-level or higher design. A three-levelfactorial, requiring 3k experiments, may be employed, where k is the number of factors.In this case, a factor will be set to its maximum, center, or minimum value. As thenumber of factors grow, a full factorial design raises the required number of experimentsto unacceptably high values.

To reduce the high cost of these designs for large number of factors, partial factorialdesigns are formulated that select only a subset of the full-factorial design. The accuracyof higher-order interactions is traded for lower number of experiments. Higher-orderinteractions that are considered to be insignificant are mixed together by a process knownas confounding (i.e. confusing) so that these effects can be no longer separated.

A notation, such as 2k-pR can describe fractional factorials, where k is the number of

factors, p is the fraction of the factorial, and R is the characteristic resolution of thedesign. Let us take an example of 2(11-7) design of resolution III (three). This means thatthe 2-level design accommodates overall k = 11 factors (the first number in parentheses);however, p = 7 of those factors (the second number in parentheses) were generated fromthe interactions of a 2(11-7), or, full 24 factorial design. As a result, the design does notgive full resolution; that is, there are certain interaction effects that are confounded with(identical to) other effects. Box-Behnken designs (Section 2.3.2) are a popular exampleof partial three-level designs. Generation of a half-fraction design is discussed inAppendix A.

Central composite designs are another family of classical experimental designs. Thedesign is computed by adding axial points to a smaller inner design that may be factorial

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or fractional factorial. Clearly, this approach combines the advantages of a two-leveldesign with those of a three-level design.

However, these designs also suffer from some shortcomings. Reusing points is difficult ifadditional points are added to a central composite design. Hence, if some experiments areperformed, the results can not be easily added to the existing design. Also, with largenumber of factors, large factor ranges are required in order to maintain desirablenumerical properties.

2.2.2 Optimal Experimental Designs

Experimental designs may be defined with respect to an optimality condition. Theseoptimal designs are quite flexible and also allow incorporating new data easily. Forexample, with a D-optimal design it is easy to add a few more results to an alreadyexisting data set (‘augmentation’).

However, there are some limitations of optimal designs as well. The criterion foroptimality of design may not be clearly related to the phenomenon being modeled. Also,there are different types of optimality conditions and it may not be entirely clear which ofthese abstract conditions are more suitable to the process being modeled. Moreover,computations of the optimality depend upon the response model selected. So a design thatis optimal for a quadratic model need not be optimal for linear model.

2.2.3 Response Surface Models

Response surface models are functions that are empirically fit to observed data fromresults of experiments or simulations. Usually, the data being fit were obtained at factorvalues specified by an experimental design and the model being fit is a polynomial inthose factors. If a response y in the process (or system) being modeled depends on inputvariables kξξξ ,...,, 21 the model that describes the process may be written as (Myers and

Montgomery, 1995):

εξξξ += ),...,,( 21 kfy (2.1)

Where the true response function f is unknown and can be very complicated in realapplications. ε is the error term that represents sources of variability not accounted for inf including both pure error and error due to lack of fit. ε may include measurement erroron the response, other sources of variability inherent in the process, the effect of variablesnot included in the model, etc. (Wang, 2001).

The expected value of y, denoted by η is given as (Myers and Montgomery, 1995):

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)()],...,,([)( 21 εξξξη EfEyE k +== (2.2)

Further, assuming ε to have a normal distribution with a zero mean (and variance 2σ ):

),...,,( 21 kf ξξξη = (2.3)

The variables kξξξ ,...,, 21 are the natural variables that are expressed in their natural units

of measurement. They are, however, converted (“coded”) to [-1,1] with the mean value atzero, before performing regression. In terms of the coded variables, the response functionis written as (Myers and Montgomery, 1995):

),...,,( 21 kxxxf=η (2.4)

A very simple first-order model, main effects model, in two variables x1 and x2 is:

22110 xx βββη ++= (2.5)

The constant 0β is an estimate of the mean of y over the experimental domain. It

corresponds to the value of y when all coded variables are at 0. The coefficient 1β is an

estimate of the gradient of y with respect to x1. Similarly, the coefficient 2β is an estimateof the gradient of y with respect to x2.

A more complicated first-order model, shown below, includes the two-factor interactionterm such that:

211222110 xxxx ββββη +++= (2.6)

The interaction term accounts for the variation in y due to x1 depending on the value ofx2, or the variation in y due to x2 depending on the value of x1. However, the first-ordermodels are likely to be useful only in a relatively small region of the independent variablespace. Also, they can not account for curvature in f. Hence, for most modeling studies,second-order models are employed. The second-order model in general form is given as(Myers and Montgomery, 1995):

jiji

ij

K

jjjjj

K

jjo xxxx ∑∑∑∑

<==

+++= ββββη2

2

1

(2.7)

It includes a constant, linear effects, single-term quadratic effects, and two-terminteractions (from left to right).

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Selection of the model is a subjective decision. It should be representative of theunderlying process. For example, a very complicated process is unlikely to beapproximated by the main-effects model but the general second-order model might beappropriate.

Once the process is approximated by a simple analytical method such as above that fitswell the true response surface, the identification of parameters that are actually influentialon the response as well as their possible interactions can also be obtained by a processcalled “screening”. Plackett-Burman designs (Section 2.3.1) are a good example of ascreening design.

2.2.3.1 Fitting A Model

In a general matrix notation, the model may be expressed as (Dejean and Blanc, 1999):

eXy += β (2.8)

Where,

X = (n*p) matrix called model matrix, or regression matrix. It depends both on theregression model and on the design of the experimentsy = (n*1) vector of observations of the responseβ = (p*1) vector of the coefficients (or parameters) of the model.n = the number of experiments, andp = the number of terms in the model (including the constant).The true coefficients β remain unknown and have to be estimated. The estimate of aresponse at all observation points may be given as:

βXy = (2.9)

Where,

y = n-vector of estimated responses, based on observations y

X = the regression matrix,

β = the vector of model coefficients obtained from a least-square fit to the observations.

Under the assumption that the error term is normally distributed with a zero mean andvariance of 2σ , only the mean of the response has to be modeled.

XbXEyE === )()( βη (2.10)

Where,

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b= an unbiased estimate of β .

Further, if we assume a function L, representing the “loss” resulting from an incorrectestimate of response, the loss function may be written as (Dejean and Blanc, 1999):

))'.(('.1

2 ββ XyXyeeeLn

ii −−===∑

=

(2.11)

The estimate of β such that ‘L’ in the above form is minimized is known as its least-square estimate. The least-square method can be used to fit a polynomial to observed data(Montgomery and Peck, 1982). The name is appropriate for this particular form of theloss function that penalizes overestimation as severely as an underestimation. This is themost commonly used estimate, but if another function models the loss incurred from anerror in estimate better than the square form for a particular problem, it can be used as theloss function. b is given as (Dejean and Blanc, 1999):

yXXXb ')'( 1−= (2.12)

Prime (′) indicates transpose of a matrix unless noted otherwise. b is a random variablewith the following properties:

β=)(bE (2.13)

)'()( 2 XXbCov σ= (2.14)

The covariance of b is directly related to the quality of the fit of the model and dependson the regression matrix X and the error variance. An estimate of error variance is givenby (Dejean and Blanc, 1999):

pn

en

ii

−=∑

=1

2

σ̂ (2.15)

The summation term in the above equation, which is the sum of squares of error terms, issometimes denoted as SSE. All other things being equal, a model with minimum errorvariance is desirable since it explains better the variability of the response. Details ofprocedures of calculating b are available in numerous texts (including Montgomery andPeck, 1982).

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2.2.3.2 Significance Testing for the Fitted Model and Individual Coefficients

The process of statistical testing involves advancing a hypothesis (called a “nullhypothesis) and an alternate hypothesis that is expected to be true if the null hypothesis isfalse, given a dataset. The alternate hypothesis may not be proved to be true but it may bepossible to quantify its plausibility by means of a probability (P-value). The nullhypothesis is tested against the alternate hypothesis by means of an appropriate statistic(called a “test”). The distribution of the test statistic, which is a real-valued function ofthe data, in the event of the null hypothesis being true is known as “null distribution”.Tables for several of null distributions (t-distribution, F-distribution, chi-squaredistribution, etc.) are widely available and can be readily computed. The value of the teststatistic from the data is compared with the null distribution in order to quantify theplausibility of the either hypothesis.

For the purpose of ascertaining the predictive power of the model in the design range offactors, significance testing is necessary. The test for significance of regression is a test todetermine if a linear relationship exists between the response variable and a subset of theregressor variables (or, factors). The appropriate hypotheses are (Myers andMontgomery, 1995):

0...: 1210 ==== −pH βββ (2.16)

joneleastatforH j ,0:1 ≠β (2.17)

The test statistic for Ho is an F-test such that:

)(

)1(

pnSS

pSS

MS

MSF

E

R

E

Ro −

−== (2.18)

Where,

SSR = the sum of squares due to the model,

SSE = the sum of squares due to the error,

MSR = the mean square of the regressors, and

MSE = the mean square error.

The total sum of squares, SST = SSR + SSE.

If ),1( pnpFFo −−> α , hypothesis H0 is rejected. This implies that at least one of thevariables contributes significantly to the model.

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The coefficient of multiple determination, R2, is defined as (Myers and Montgomery,1995)

T

E

T

R

SS

SS

SS

SSR −== 12 (2.19)

R2 is a measure of the amount of reduction in the variability of y obtained by using theregressor variables in the model. However, a large value of R2 does not necessarily implythat the regression model is good (Myers and Montgomery, 1995). Since R2 alwaysincreases as terms are added to the model, it is sometimes preferable to use an R2 statisticadjusted for number of terms, R2

adj , defined as (Myers and Montgomery, 1995)

)1(1

1)1(

)(1 22 R

pn

n

nSS

pnSSR

T

Eadj −

−−−=

−−

−= (2.20)

Adjusted R2 statistic does not always increase as variables are added to the model. If theadjusted R2 statistic differs significantly from the (original) R2 statistic, it may be a strongindication that unnecessary terms were added. Both R2 and adjusted R2 estimate thequality of fit, but a good fit does not guarantee good predictive value of the fitted model(Dejean and Blanc, 1999).

Sometimes it is necessary to test the significance of individual regression coefficient.When a variable is added to the regression model, the sum of squares for the regressionincreases while the error sum of squares decreases. Significance testing for individualcoefficient helps in making the decision whether the increase in regression sum ofsquares is sufficient to justify the use of an additional variable. Also, adding unimportantvariable may increase the mean square error thus making the model less useful. Theappropriate hypotheses for any coefficient jβ are (Myers and Montgomery, 1995):

0: =joH β (2.21)

0:1 ≠jH β (2.22)

The test statistic for this hypothesis is (Myers and Montgomery, 1995):

jjE

j

CMS

bt

.0 = (2.23)

Where,

b = the least square estimate of β ,

MSE = the mean square error, and

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Cjj = the diagonal element of the covariance matrix, given by (X’X)-1.

The hypothesis H0 is rejected if 1,2/0 || −−> kntt α . If the hypothesis Ho is not rejected, then

it is indicated that the term related to jβ can be deleted from the model.

The concept of P-value is also used to determine the statistical significance of the effectsin the model. The P-value quantifies the strength of evidence against the null hypothesisand in favor of the alternative. P-value may be defined as the probability that a variatewould assume a value greater than or equal to the observed value strictly by chance. α-value is defined as the number 0 ≤ α ≤ 1, such that P(z ≥ zobserved) ≤ α is consideredsignificant, where ‘P’ denotes P-value. A commonly used α−value is 0.05. Given thedegrees of freedom in the regression and the value of a test statistic, like F-test or t-test,P-value can be calculated or looked up from standard tables. If the P-value of an effect isless than α−value, hypothesis H0 is rejected, thus implying that at least one of thevariables is statistically significant. It may be noted however, that a high P-value does notnecessarily support the null hypothesis.

2.3 Some Popular Designs

Some designs are applied frequently because they offer certain advantages over the lesspopular ones. Three of them are discussed in the following account.

2.3.1 Plackett-Burman Design

In order to screen a large number of factors to identify those that may be important adesirable design would be one that allows one to test the largest number of factor maineffects with the least number of observations. One way to design such experiments is toconfound all interactions with "new" main effects. Such designs are also sometimescalled saturated designs, because all information in those designs is used to estimate theparameters, leaving no degrees of freedom to estimate the error. Since the added factorsare created by equating (see Appendix A), the "new" factors with the interactions of a fullfactorial design, these designs always will have 2k runs (e.g., 4, 8, 16, 32, and so on).Plackett and Burman (1946) showed how full factorial design can be fractionalized in adifferent manner, to yield saturated designs where the number of runs is a multiple of 4,rather than a power of 2. These two-level fractional designs can be used for k=N-1variables in N runs, where N is a multiple of 4. These highly fractionalized designs toscreen the maximum number of (main) effects in the least number of experimental runsare known as Plackett-Burman designs and are widely used for “screening” a largenumber of possible factors and retain only those that have higher main effect (Freidmannet al., 2001).

14

2.3.2 D-optimal Design

The D-optimal design procedures provide various options to select from a list of valid(candidate) points (i.e., combinations of factor settings) those points that will extract themaximum amount of information from the experimental region, given the respectivemodel that is expected to be fitted to the data. Details of the algorithms for generating theoptimal designs are provided in Appendix B.

When the factor level settings for two factors in an experiment are uncorrelated, that is,when they are varied independently of each other, then they are said to be orthogonal toeach other. Two column vectors X1 and X2 in the design matrix are orthogonal if X1'*X2=0. The more redundant the vectors (columns) of the design matrix, the closer to zero isthe determinant of the correlation matrix for those vectors; the more independent thecolumns, the larger is the determinant of that matrix. Thus, finding a design matrix thatmaximizes the determinant D of this matrix means finding a design where the factoreffects are maximally independent of each other. This criterion for selecting a design iscalled the D-optimality criterion (Poland et al., 2001).

A D-optimal design seeks to minimize the average size of the variance matrix ofparameter estimates by minimizing its “average eigenvalue”. In a least squares analysis,the variance matrix of the vector β , given a design matrix X, is proportional to (X'X)-1.The determinant of (X'X)-1 equals the product of its eigenvalues. Thus, the search for D-optimal designs aims to minimize |(X'X)-1|, where the vertical lines (|…|) indicate thedeterminant. As a practice, however, maximizing |(X'X)| is preferred to minimizing|(X'X)-1| since it avoids computation of the inverse of a maxtrix. (See Kuhfeld, 1997.)

A number of standard measures have been proposed to summarize the efficiency of adesign. D-efficiency is such a measure related to the D-optimality criterion. It is definedas follows (Narayanan, 1998):

D-efficiency = 100 * (|X'X|1/p/n)

Where,

p = the number of factor effects in the design (columns in X),n = the number of requested runs, or, the number of points in the optimal design.

It may be noted that since the p-th root of the determinant is the geometric mean of theproduct of its eigenvalues, it is a measure of goodness of design. This measure can beinterpreted as the relative number of runs (in percent) that would be required by anorthogonal design to achieve the same value of the determinant |X'X|. However, anorthogonal design may not be possible in many cases, that is, it is only a theoretical"yard-stick." Therefore, this measure can be used as a relative indicator of efficiency, tocompare other designs of the same size, and constructed from the same set of candidatepoints (Kuhfeld, 1997).

15

2.3.3 Box-Behnken Design

The equivalents of Plackett-Burman designs in the case of 3k-p designs are the so-calledBox-Behnken designs (Box and Behnken, 1960; also Box and Draper, 1987). Thesedesigns do not have simple design generators and have complex confounding ofinteraction. However, the designs are economical and therefore particularly useful whenit is expensive to perform the necessary experimental runs.

Box-Behnken designs were used in this study. This design has several advantages relativeto the alternatives. The Box-Behnken design reduces the number of required experimentsby confounding higher-order interactions. This reduction becomes more significant as thefactors increases in number. Box-Behnken designs do not require many moreexperiments than two-level. Also, unlike D-optimal designs, Box-Behnken designs do notrequire or depend on prior specification of the model. Further, unlike first-order (e.g.,two-level factorial) designs, Box-Behnken designs allow estimation of quadratic termsand do not yield a constant sensitivity of the response to the factor. Moreover, includingthe center point in the design reduces the estimation error for the most likely responses.Two-level designs do not include experiments at the design center-point and thus may beinaccurate in the most likely factor ranges. However, as the Box-Behnken design doesnot use the extreme values for all the factors simultaneously, the accuracy at the extremessuffers.

16

Chapter 3

Gas Cap Blowdown

Development of an oil field usually aims to maximize ultimate recovery as well as tominimize capital expenditure and operating expenditure. This may be achieved using atraditional development plan, or if need be, a less conventional approach.

It is the general practice to make use of all available natural reservoir drive mechanisms –gas in solution, primary gas cap, aquifer and compaction. This allows the intrinsic energyof the reservoir to drain out the maximum oil. However, sometimes one natural drive maybe so dominant that the benefits of the others may not be necessary. Specifically, a strongaquifer may obviate the need of a small gas cap for the purpose of conserving reservoirenergy and “blowing down” the primary gas cap may maximize the ultimate recoveryfrom a field.

3.1 Early and Delayed Blowdown

A regulatory requirement often imposed on blowdown is that the primary gas cap maynot be produced till the pool is in the final stages of depletion (more than, say, 90% of theultimate recovery). This constitutes the “Delayed Blowdown”. If the gas from the gas capis produced earlier than that stage, it is called “Early Blowdown” (Kuppe et al., 1998).

It was found that both forms of blowdowns may result in a comparable recovery ofhydrocarbons (Kuppe et al., 1998), but an early blowdown approach can conserve capitaland reduce the operating life of the reservoir by many years. It may be noted that in thisparticular case, a water fence was created by water injection between the gas cap and theoil leg in order to allow concurrent production of oil and gas still maintaining aseparation between the two.

The advantages of the early blowdown can be attributed to several factors. As thevoidage replacement is maintained, the argument that the gas cap is required as a sourceof energy no longer holds. A high recovery in the early blowdown is obtained bydepleting the gas cap before water influx as the superior gas mobility can outrun thewaterfront. The reduction in gas recovery due to liquid invasion is lesser in case of earlyblowdown because of less available time for the water influx. Hence, higher gassaturations lead to higher gas mobilities and better hydrocarbon recovery.

In the present study, only early blowdown is studied since it was found that only fewresults are available on this form of blowdown even though it has been appliedsuccessfully in some cases (Behrenbruch and Mason, 1993).

17

3.2 Important Factors in Gas Cap Blowdown

Two kinds of considerations are important when considering the relative importance ofdifferent drive mechanisms – Reservoir Energy and Fluid Displacement. To compare therelative benefits of water drive and gas cap drive, we can, therefore, discuss in theseterms. The following is based largely on Behrenbruch and Mason (1993):

Reservoir Energy. Intrinsic energy of different substances can account for their ability tocompensate reservoir voidage. The compression-volume product maintains the reservoirpressure. Considering typical values, gas has much lesser instantaneous expansioncapacity than water (Behrenbruch and Mason, 1993).

However, the closer proximity of the gas cap to the oil leg compared to the aquifer resultsin more effectiveness of the gas cap initially. The aquifer due to its larger distance fromthe oil leg is slow to respond to the pressure depletion in the oil leg.

Fluid Displacement. Aquifer strength has to be sufficiently high in terms of both size andconnectivity in order to sweep the oil at high pressure, but it is usually unknown in thereal cases.

Availability and use of make up gas and cost of re-injection also needs to be considered.Barring regulatory requirement of injection of gas, the value of the gas and the cost ofinjection have to be weighed against the value of additional oil recovery (if any).

Well placement is another factor. Due to updip movement of fluid contacts, it may bedesirable to place wells near the crest of the reservoir initially. Horizontal wells areknown to greatly enhance and accelerate the recovery of attic oil in some cases (Vo et el,1996).

The size of the gas cap is important because of its expansive energy and possiblycommercial value. As the aquifer sweeps the oil into the gas cap, pore space in the gascap zone is saturated with oil, which may be reduced again when water invasion starts.Part of the oil originally in the oil zone may be spread over the initial gas zone and be lostas residual oil saturation after displacement by water. A smaller volume of abandoned oiland a shorter operating life can balance this potential loss.

For a strong water drive reservoir, the gas cap blowdown from the crestally located wellscauses the aquifer to push up the oil column and re-saturate most of the original gas capvolume. This oil is potentially unrecoverable. The volume of oil loss due to re-saturationof the gas cap can be given in stock tank barrels (stb) as:

)1()1(,,,

,wc

go

o

oi

o

go

wc

gi

o

gog S

S

B

mNB

B

S

S

GB

B

SV

−=

−=φ (3.1)

18

Assuming that an alternative production policy will leave behind an abandonment oilcolumn of pore volume (φVab), the amount of abandoned oil in stb will be:

abo

wcab B

SV

,

)1( −φ (3.2)

The abandoned oil volume in case of gas cap blowdown will be, similarly,

o

orwab B

SVφ (3.3)

For gas cap blowdown to be favorable, the oil losses in case of gas cap blowdown mustbe less than those in case of an alternative strategy, i.e.

abo

wcab

o

orwab

wc

go

o

oi

B

SV

B

SV

S

S

B

mNB

,

, )1(

)1(

−<+

−φφ (3.4)

It may be noted that for the same width of abandonment column of oil, it will tend to becloser to the original gas-oil contact in case of strong water drive and closer to the oil-water contact in case of a weak water drive. Water injection may be needed in order tomake this feasible if the aquifer support is not enough.

3.3 Significance of Gas Cap Blowdown

The significance of gas cap blowdown lies in its ability to provide an alternative reservoirmanagement strategy in certain cases. It has been successfully applied in many cases andcan be extremely attractive with proper reservoir monitoring and control. Some casestudies can be seen in Lee (1993), Behrenbruch and Mason (1993), and Starzer et al.(1995)

A pitfall is the uncertainty in the drive mechanisms particularly in the aquifer size and thegas cap size. If the predicted values of the drive mechanism are not accurate,recompletion of wells may have to be resorted to in order to drain from a shrinking oilrim, or, unplanned fluid injection may be another expensive solution.

Gas coning and cusping can be severe in these cases due to greater gas mobility.Constrained oil rate and reconsidering completion intervals are ways to check this.However, these are not desirable solutions since lower initial completions will forcerecompletions later as the fluid contacts move up.

19

3.4 Simulation Model

The simulation model used in the present study is a regular grid, shoe-box model withconstant permeability (though vertical anisotropy is considered), and uniform porosity.Three phase flow is considered along with a Carter-Tracy analytic aquifer support at thebottom. Table 1 shows the important parameters for the base case of the model. Theparameters are based on the Skua field in East Timor Sea (Behrenbruch, 2000), but thereservoir model is synthetic. Six factors each with three levels were considered, as shownin Table 2. The factors were assumed to be independent. Using three levels instead of twoensures that curvature in the relation between a factor and the response can be captured.All the cases have a gas cap, an oil zone, and a water zone connected to a Carter-Tracyaquifer. The well is always completed just above the gas-oil contact. As seen in Table 2,each factor is varied over a possible range of values. The details of implementation canbe seen in Appendix C containing a sample Eclipse input data file.

Table 1 – Reservoir Properties,Dimensions and Initial Conditions

Property Value

φ , % 21

wiS , % 20

xk , md 50

yk , md 50

zk , md 0.5

xN 49

yN 9

zN 27

x∆ , m 55.56

y∆ , m 97.96

z∆ , m 2.8

Datum depth, m 2286.5

ip at datum depth,bars

228.6

Depth to top ofreservoir, m

2258.5

Reservoir Thickness,m

75.6

20

The full set of runs in this case requires 36 (=729) simulations. We manage to reduce thisrequirement drastically with the help of experimental design, as explained in the Chapter4. Moreover, more information can be extracted from the reduced number of runs, than isusually done by changing one factor at a time from its base value.

Table 2 – Parameters Varied in Simulation Study

HorizontalPermeability (md)

AquiferInfluxCoefficient

GasRate(105

sm3/day)

Oil-WaterContact(m)

Gas-OilContact(m)

Well Location inGrid

5 0.1 3 2297.7 2261.3 (1,1)50 1 6 2314.5 2278.1 (13,5)

500 10 12 2331.3 2294.9 (25,9)

21

Chapter 4

Application of Experimental Design

The simulation runs were designed using Box-Behnken and D-optimal designs, andanalyzed using response surface methods. The regressions so obtained were analyzed forrelative importance of the factors, their main effects and interaction effects. Both designsgive comparable results and are discussed in detail in the following sections.

4.1 Setup of the Regression

Six factors were chosen for their expected impact on oil recovery using gas capblowdown. The factors are denoted by KH (horizontal permeability), AQ (Aquiferstrength coefficient for the Carter-Tracy aquifer attached to the bottom blocks), GR(specified Gas Rate for the well), WI (the ratio of volume of the oil zone to that of thewater zone), M (the ratio of volume of the free gas cap to that of the oil zone), and LH(Horizontal Location of the well, measured along the diagonal from the top right cornerto the center of the model, from 0 to 1). A vertical anisotropy of 0.01 is assumed for thepurpose of this study. M and WI are varied by varying the fluid contacts. M is referred toas “relative gas cap size”, and WI as “relative water zone size”.It should be noted that inhereafter KH would refer to natural logarithm of horizontal permeability and AQ to thatof the aquifer influx coefficient, since their levels vary logarithmically, rather thanlinearly. The factors have the values at different levels as shown in Table 3. The responsevariable is the cumulative oil production after 8 years divided by the initial pore volumeof oil in 10-3 sm3/rm3. It represents the oil recovery, and is denoted by R.

Table 3 – Factors and Their Levels

KH AQ GR WI M LH

Level 1 1.61 -2.39 3.00 1.00 0.08 0.00

Level 2 3.91 0.00 6.00 1.86 0.54 0.50

Level 3 6.21 2.30 12.00 13.00 1.00 1.00

22

The coefficients of each term is calculated by first converting them using the followingrelation:

2

2min,max,

min,max,

ii

iii

ix ξξ

ξξξ

−

+−

=

Where,

x = coded value

ξ= raw “uncoded” value of the natural variable

The subscripts max and min refer to respectively the maximum and the minimum valuesof the factors. The subscript i refers to the factor number. This coding is done to eliminateany spurious statistical results due to different measurement scales for the factors. Forexample, the gas rate is to the order of 106 sm3/day whereas the relative gas cap size is ofthe order 0.1 to 1, and so the regression with uncoded values would give very highcoefficients for the gas rate compared to the relative gas cap size. (It is noted that theregressors are all rendered dimensionless by the coding scheme.)

Four different cases were considered to compare Box-Behnken design and D-optimaldesign. For 6 factors, the Box-Behnken design requires 49 runs including one run with allthe factors at their base (or center) value. This design is compared to a D-optimal designwith the same 49 number of runs. The candidates for the D-optimal design were all theruns possible with 3 levels and 6 factors, i.e. 36 (=729) runs. The design was optimal withrespect to the full quadratic response surface model. The same method was employed togenerate another D-optimal design with 100 runs. To compare these designs, the fullfactorial case consisting of all 36 cases was also considered.

The response surface model in all the cases was the full quadratic model, which can beconstructed from Equation 2.7 with the number of factors, K=6. Hence, the coefficientsto be estimated from the regression were 28 in number -- 1 constant, 6 for the linearterms, 6 for the square terms and 15 two-factor interaction terms.

Simulations were performed using a commercial flow simulator (ECLIPSE) for thesefour cases. Values for factors were input to the simulator as the four design casesprescribe and the responses so obtained as output were recorded. For each case,regression in the six factors is performed with respect to the response (R) using astandard statistical software (MINITAB) to obtain the response surface models.

4.2 Comparison of Designs

Regressions were significant for all the cases since the P-values were found to be 0.00. Itestablishes that the models fitted to the data account for the variability in the process

23

irrespective of the design chosen in this study. Table 4 shows the P-values for differentterms (linear, quadratic, and interaction) for the four cases discussed in Section 4.1. Thelinear terms are found to be significant in all designs. Only the Box-Behnken design failsto find the interaction terms significant for α = 0.05. The square terms are significantonly according to the full factorial design for the same α -value. (The concept of α -value is explained in Section 2.2.3.2.)

Table 4 – Comparison of Regressions

P-values for Regression Terms Goodness of Fit

Linear Square Interaction R-Sq Adjusted R-Sq

Box-Behnken 0.000 0.109 0.075 0.884 0.735

D-optimal(49runs)

0.000 0.247 0.002 0.922 0.822

D-optimal(100runs)

0.000 0.169 0.000 0.896 0.856

Full Factorial 0.000 0.000 0.000 0.868 0.863

For the same number of runs, D-optimal design shows much better R2 value than theBox-Behnken design. Since a D-optimal design is optimal in the sense of decreasedvariance of the parameter estimate (see Section 2.2.3.1 and Section 2.3.2), it is notsurprising that it shows a better fit. It may be recalled that R2 indicates the percentage ofvariability in the process explained by the fitted model. The other two cases have highersample size available for regression and thus may be expected to show a better fit.However, they suffer from the presence of outliers (Fig. 8 for the full factorial), whichprevents them from achieving a better fit than the cases with lower number of runs. Infact, as Table 4 indicates, the full factorial case has the smallest value of R2. It may beconcluded that the goodness of fit is high in all the cases, but more available data doesnot necessarily lead to better fit.

The R2adj values show the goodness of fit for the regressions adjusted for number of

terms. It is found to be lowest for the Box-Behnken design. Since the full factorial designis the exhaustive case, it is not surprising that it has the highest value of adjusted R2. TheD-optimal design with 49 runs shows a better fit than the Box-Behnken design eventhough it requires the same number of runs. The D-optimal design with 100 runs showsslight improvement in fit than the D-optimal design with 49 runs.

The difference between R2adj and R2 is the highest for the Box-Behnken case. It then

decreases with increasing number of runs. This difference indicates presence ofunnecessary terms in the model (Myers and Montgomery, 1995). Clearly, the Box-

24

Behnken case failed to attach significance to some terms that were found to be necessaryby the cases with a larger dataset.

4.3 Effects of Factors

A Pareto chart (Friedmann et al., 1999) is a bar chart that graphically ranks the effects offactors on the response so that the most important ones can be identified. It is generallyused with screening designs since it can be constructed from 2-level designs and thusrequires relatively small number of runs (Friedmann et al., 1999). Since the 3-level fullfactorial case is a superset of 2-level full factorial case, the Pareto chart could easily becomplied from existing results. The Pareto chart, in Fig. 1 is used to compare the relativemagnitude and the statistical significance of both main and interaction effects.

Figure 1. Pareto Chart of the Standardized Effects

As seen in Fig. 1, by far the most important factor is the main effect of KH. It is followedby the interaction effect of KH and M, which has only a slightly higher significance thanthe main effect of M. The interaction of KH and GR, the main effect of WI, theinteraction of KH and WI, and the main effect of GR are in order of decreasingsignificance, but are almost equal. The main effect of LH is slightly more important thanthe interaction effect of KH and LH, but less than the main effect of GR. The effects thatare below the dotted vertical line (α = 0.05) in Fig. 1 are insignificant. (The concept ofα -value is explained in Section 2.2.3.2.) Even though the Pareto chart is informative inranking the impact of effects, as mentioned earlier, it is based only on 2-levelinformation. It also ranks some three-factor interactions as significant, but the quadraticmodel can not capture these effects. Hence even in the full factorial case, the model

25

imposes limitations on the accuracy of the regression relative to the true responsefunction.

4.3.1 Main Effects

Figs. 2, 3, 4, and 5 show the main effects of the factors for the full factorial design, theBox-Behnken design, the D-optimal design with 49 runs, and the D-optimal design with100 runs respectively. The main effects plots are used to compare the relative strength ofthe effects across factors. In these figures, the means at each level of a factor are plottedand connected with a line. Center points and factorial points are represented by differentsymbols. A reference line at the grand mean of the response data is also shown.

All designs predict similar behavior for KH, though they differ in their estimation of themain effects of other factors. It is instructive to look at the P-values of individual factorsfrom Table 5. The factors whose estimates by different designs differ also have high P-values, or in other words, low significance. For example, AQ has very high P-values andits estimation differs in case of every design. On the other hand, KH or GR have low P-values in all designs and their estimates are very similar. It can be thus concluded that theimportant main effects are captured well by all designs, even though the full factorialcase is more accurate. The significance of the effects is also consistent with theinformation in the Pareto chart of Fig. 1.

Figure 2. Main Effects for the Full Factorial Case

As KH increases, so does recovery. This is to be expected since a high horizontalpermeability causes low near-wellbore pressure drop (drawdown), thus reducing gasconing and aiding recovery. AQ is not very significant since the aquifer is very strongeven at the lowest value of aquifer influx coefficient. From Fig. 2, it is observed that

26

increased specified gas rate increases oil recovery R. thus producing the gas at higher rateleads to faster blowdown of gas, thus more oil can be recovered. All designs show thatsmaller M leads to higher R.

Figure 3. Main Effects for Box-Behnken Design

4.3.2 Interaction Effects

Fig. 6 shows the plots of two-factor interaction as obtained from the full factorial case.An interactions plot is a plot of means for each level of a factor with the level of a secondfactor held constant. These plots are used to compare the relative strength of the effectsacross factors. However, the interpretation, as also for the main effects, is meaningfulonly if the interaction effects are statistically significant.

27

Figure 4. Main Effects for D-optimal Design with 49 Runs

Figure 5. Main Effects for D-optimal Design with 100 Runs

Most significant interaction effects are those of KH with M, GR, WI, and LH as seenfrom Table 5. The contours of the response surfaces for these pairs of factors are shownin Figs. 7, 8, 9, and 10 respectively. The other factors are held constant at their basevalues in these figures.

Higher values of R occur at high permeability (KH) and low M. Since the gas cap issmall it does not contribute significantly to the recovery compared to the aquifer, and athigher permeability producing the gas helps to recover more oil.

28

Figure 6. Interaction Effects for Full Factorial Design

Table 5 -- Coefficients and P-Values for Regression Terms

Box-Behnken D-optimal(49Runs)

D-optimal(100Runs)

Full Factorial

Term Coefficient

P-value

Coefficient

P-value Coefficient P-value Coefficient

P-value

Constant

34.84 0.548 13.84 0.723 47.23 0.102 52.84 0

KH 60.08 0 39.68 0 43.34 0 47.65 0AQ 0 1 -2.79 0.427 3.49 0.182 0.58 0.559GR 12.25 0.082 6.86 0.065 8.56 0.002 10.35 0WI -7.61 0.181 -8.56 0.02 -9.21 0.001 -9.09 0M -15.01 0.048 -22.63 0 -25.36 0 -20.31 0LH 4.53 0.512 1.34 0.704 5.95 0.024 5.11 0KH*KH 16.05 0.169 23.05 0.026 20.42 0.006 20.31 0AQ*AQ 4.14 0.716 -0.9 0.929 -0.19 0.978 0.16 0.921GR*GR -6.36 0.624 -6.42 0.568 -1.5 0.85 -7.69 0WI*WI 16.92 0.715 13.3 0.747 -8.07 0.779 -5.46 0.427M*M -8.22 0.472 12.11 0.229 -6.6 0.345 -6.96 0LH*LH -9.02 0.431 -11.02 0.273 -5.11 0.47 -8.76 0KH*AQ 0 1 -1.97 0.6 3.93 0.157 0.58 0.615KH*GR 32.37 0.001 7.02 0.071 9.69 0.001 14.4 0KH*WI -5.06 0.394 -8.34 0.032 -10.43 0 -10.27 0KH*M -29.94 0.004 -25.12 0 -29.34 0 -29.21 0KH*LH 5.95 0.521 1.28 0.733 5.44 0.052 5.68 0AQ*GR 0 1 1.04 0.778 2.25 0.415 0.56 0.619AQ*WI 0 1 -2.53 0.482 1.88 0.477 0.59 0.571AQ*M 0 1 -1.94 0.609 -1.88 0.496 -0.53 0.648AQ*LH 0 1 7.2 0.064 2.89 0.298 0.53 0.648GR*WI -4.9 0.509 -3.05 0.393 -0.64 0.808 -2.86 0.005GR*M 5.12 0.568 0.06 0.988 -1.55 0.575 0.97 0.394GR*LH 3.86 0.549 0.5 0.894 5.38 0.053 2.73 0.016WI*M -10.42 0.178 2.1 0.56 -0.93 0.724 -0.61 0.558WI*LH -0.29 0.969 -0.99 0.782 2.7 0.312 -0.12 0.908M*LH 0.56 0.951 -4.39 0.245 -4.82 0.085 -2.4 0.038

30

Figure 7. Contour Plot of Response Surface for Horizontal Permeability andRelative Gas Cap Size

Figure 8. Contour Plot of Response Surface for Horizontal Permeability and GasRate

It can be seen from Figs. 7-10 that higher R-values occur at higher horizontalpermeability consistent with the main effect of horizontal permeability. Similarly, thetendency of the other factors is to affect the recovery as manifest in their main effects.

31

For example, Fig. 8 shows that higher GR and higher horizontal permeability lead tohigher recovery, as is to be expected from their main effects (discussed in Section 4.3.1).Lower WI (Fig. 12) indicates bigger water zone size compared to the oil leg. As morewater is available to provide pressure support especially in the face of depleting gas zone,more oil recovery is obtained. A centrally located well provides more recovery at higherhorizontal permeabilities as seen in Fig. 10. Since this is a homogeneous reservoirwithout dip, the result is reasonable.

Interaction effects of GR with WI and LH are also significant. It is apparent that theeffect of specified gas rate on recovery is highly dependent on the location of the well(Fig. 11), as well as the size of water zone relative to that of the oil zone (Fig. 12). Highgas rate is more beneficial in case of large water zone, since a larger water zone is able toprovide more pressure support to aid recovery. Also, producing the well at higher gas rateyields more recovery in case of a centrally located well. From Fig. 13 it is indicated thatwith a smaller gas cap size, a centrally located well can accord better recovery.

Figure 9. Contour Plot of Response Surface for horizontal permeability and RelativeWater Zone Size

4.3.3 Nature of Residuals

Residuals are assumed to be normally distributed and independent. We need to verify thatthis indeed is the case for the regression to be considered useful. In Figs. 14-16, theresiduals in the plots are standardized by dividing the residual by its variance.Standardization eliminates the effect of location of data points in the predictor space(Myers and Montgomery, 1995).

32

In Fig. 14, the normal probability plot of residuals for the full factorial case is almoststraight showing that it is close to a normal distribution in the central portion of the data.Hence, the transformation of variables as a remedial measure is not necessary (Myers andMontgomery, 1995).

Figure 10. Contour Plot of Response Surface for Horizontal Permeability andHorizontal Location of Well

Figure 11. Contour Plot of Response Surface for Gas Rate and Relative Water ZoneSize

33

Figure 12. Contour Plot of Response Surface for Gas Rate and Horizontal Locationof Well

Figure 13. Contour Plot of Response Surface for Relative Gas Cap Size andHorizontal Location of Well

34

Figure 14. Normal Probability Plot of the Residuals (Full Factorial Design)

Figure 15. Standardized Residuals versus the Fitted Values (Full Factorial Design)

Fig. 15 contains the plot of residuals versus the fitted value for the full factorial case.There is a clear indication of a systematic error. Since the residuals are not scatteredrandomly, the variance of the original observation is not constant for all values of theresponse (Myers and Montgomery, 1995). Presence of outliers can also be seen whichindicates that the fitted model is not a very good approximation to the true response

35

surface in some regions of regressor space. This indicates the need to employ higherorder polynomials or explore alternative regression tools like neural networks. Thetraining data for the neural network may come from runs designed using experimentaldesign. Inferring the cause behind the artifact errors observed could be part of futurework.

Fig. 16 shows the residuals versus the observation order for the full factorial case. Theobservations were not arranged at random – the pattern of arrangement of values of thefactors is reflected in the plot. It can be concluded that some factors have strong influenceon the response. In fact, a closer inspection of the data indicated that large excursions ofresiduals from zero occur when the horizontal permeability is at the high level. Given ahigh horizontal permeability, the excursions are positive when the specified gas rate forthe well is at the high level and are negative when it is at the low level. It may besuggested that factors with insignificant main effect be removed and the factors withlarge influence on the response possibly transformed.

Figure 16. Standardized Residuals versus Observation Order (Full Factorial Design)

4.4 Conclusions and Future Work

It was found that experimental designs are very useful, not only to reduce the number ofruns required in a simulation study, but also to maximize information gained from thestudy. The important factors can be ascertained and ranked. Their main effects andinteractions can be estimated accurately. The designs compared provided consistentresults and comparison with the exhaustive case confirmed their usefulness.

In the specific case of gas cap blowdown, the process was well captured by the designsemployed. High gas rate in presence of high horizontal permeability, a small gas cap,

36

large water support, and central location of the well were found to be the conditions forhigh recovery. The regression obtained is strictly valid for this case in the ranges ofvalues of the factors chosen, but its application may be tested in more general cases. Therelative ranking of the factors and their interactions is intuitively satisfying and lendsweight to future applications of this tool for gas cap blowdown studies.

In this study only one response was considered, but the same methodology can be easilyextended to include more responses, thus providing an even more useful approximationto the flow simulator.

Choosing an optimum time for gas cap blowdown is a challenging task. A future studycan address this issue. During gas cap blowdown, the oil rises into the original gas capzone as gas is produced from the reservoir and hence the hysteresis of relativepermeability curve becomes important. The impact of this hysteresis is an importantaspect of this problem.

An immediate application could be choosing appropriate probability distributions for thefactors used in this study and use the regressions obtained to perform uncertainty analysison the response, using Monte Carlo Simulation (Wang, 2001). The important aspect ofthis approach is choice of physically meaningful distributions for the factors in order toachieve practical applicability. The regression obtained through different designs in thisstudy can be used for an optimization study. Validity of the regression should also bechecked on a real field case.

37

Nomenclature

AQ = regression factor corresponding to aquifer activity coefficient

b = unbiased estimate of β

B = volumetric factor, rb/stb

C = covariance matrix

E = expected value

f = true response function

F0 = F-test statistic for the null hypothesis

Fα = F-test statistic for a given α-value

G = gas in place (standard conditions), scf

GR = regression factor corresponding to gas rate specified for the well

H0 = null hypothesis

H1 = alternative to the null hypothesis

KH = regression factor corresponding to natural logarithm of horizontal permeability

LH = regression factor corresponding to horizontal location of the well

m = initial relative reservoir gas cap volume (compared to reservoir oil volume), ratio

M = regression factor corresponding to m

MS = mean square value

n = number of simulation runs ( or experiments)

N = oil in place (standard conditions), stb

p = number of (independent) parameters or factors

38

P = P-value

R = ratio of cumulative oil production in 8 years to initial oil pore-volume, 10-4 sm3/rm3

R2 = coefficient of multiple determination

R2adj = R2 adjusted for number of terms in the model

S = reservoir saturation by fluid (oil , gas or water), fraction

S = average saturation, fraction

SS = Sum of squares

t0 = t-test statistic for the null hypothesis

V = volume

W = water (or aquifer) volume

WI = regression factor corresponding to oil volume compared to water volume, ratio

X = regression matrix, or, model matrix

y = response (of the process)

Greek

α = cut-off value for significance testing

β = vector of size equal to the number of runs, containing the coefficients of the model

ε = error

ξ = natural variables in natural units of measurement

η = expected value of response

2σ = variance of error

σ̂ = estimate of variance of error

φ = porosity

39

abφ = pore space of abandonment oil rim

Subscripts

E = Due to error (SS or MS)

i = index

g = gas

gi = gas initial , scf

j = index

k = index

o = oil

oi = oil, initial, stb

or = oil, residual, fraction

org = oil residual after displacement by gas, fraction

orw = oil , residual – after displacement by water, fraction

o,g = oil in gas cap

o,ab = oil at abandonment

w = water

max = maximum value

min = minimum value

T = Total

Superscripts

-1 = inverse of a matrix

` = transpose of a matrix

40

References

Azancot, M., Ward, R., Gibson, S., “The Role of Subsea Systems as an Integral Partof Occidental's North Sea Development Strategy”, paper 17622, SPE InternationalMeeting on Petroleum Engineering, Tianjin, China, November 1-4, 1988.

Behrenbruch, P. and Mason, L.T., paper SPE 25353, “Optimal Oilfield Developmentof Fields With a Small Gas Cap and Strong Aquifer”, SPE Asia-Pacific Oil andGas Conference & Exhibition, Singapore, February 1993

Behrenbruch, P., BHP Petroleum, 2000, Personal Communication

Box, G.E.P. and Behnken, D.W.,: “Some New Three Level Designs for the Study ofQuantitative Variables”, Technometrics, November 1960.

Box, G.E.P., Hunter, W.G., and Hunter, J.S., “Statistics for Experimenters”, JohnWiley & Sons, 1978.

Box, G.E.P. and Draper, N.R., “Empirical Model-Building and Response Surface”,John Wiley & Sons, 1987.

Chu, C., “Optimal Choice of Completion Intervals for Injectors and Producers inSteamfloods,” paper SPE 25787, International Thermal Operations Symposium,Bakersfield, California, February, 1993.

Cook, R.D. and Nachtsheim, C.J., “ A Comparison of Algorithms for ConstructingExact D-Optimal Designs”, Technometrics, Vol. 22, No. 3, August 1980.

Damlseth, E., Hage, A., and Volden, R., “Maximum Information at Minimum Cost:A North Sea Development Study With Experimental Design”, JPT, December1992.

Dejean, J.P. and Blanc, G., “Managing Uncertainties on Production Predictions UsingIntegrated Statistical Methods”, paper SPE 56696, SPE Annual TechnologyConference and Exhibition, Houston, Texas, October 1999.

Friedmann, F., Chawathe, A., and Larue, D., “Assessing Uncertainty in ChannelizedReservoirs Using Experimental Designs”, paper SPE 71622, SPE AnnualTechnical Conference and Exhibition, Louisiana, New Orleans, September–October 2001.

Galil, Z., and Kiefer, J., “Time- And Space-Saving Computer Methods, Related toMitchell's DETMAX, for Finding D-Optimum Designs”, Technometrics, Vol. 22,No. 3, August 1980.

Kuppe, F.C., Chugh, S., and Kyles, J.D., “Gas Cap Blowdown of the Virginia HillsBelloy Reservoir”, presented at 49th Annual Technical Meeting of the PetroleumSociety, Clagary, Alberta, Canada, June 8-10, 1996.

41

Lee, B., “Samarang K5/7 Reservoir Simulation Study”, paper SPE 25351, SPE AsiaPacific Oil and Gas Conference and Exhibition, Singapore, 8-10 February 1993.

Myers, R.H. and Montgomery, D.C., “Response Surface Methodology”, John Wiley& Sons, 1995.

Manceau, E., Mezghani, M., Zabalza-Mezghani, I., and Roggero, F., “Combination ofExperimental Design and Joint Modeling Methods for Quantifying the RiskAssociated With Deterministic and Stochastic Uncertainties – An Integrated TestStudy”, SPE 71620, SPE Annual Technical Conference and Exhibition, NewOrleans, Louisiana, September-October 2001.

Montgomery, D. C., and Peck, E. A., “Introduction to Linear Regression Analysis”.John Wiley and Sons, 1982.

Narayanan, K., White, C.D., Lake, L.W., and Willis, B.J., “Response SurfaceMethods for Upscaling Heterogeneous Geologic Models”, paper SPE 51923,Reservoir Simulation Symposium, Houston, Texas, February 1999.

Narayanan, K., “Applications for Response Surfaces in Reservoir Engineering”, MSThesis, University of Texas at Austin, 1999.

Plackett, R.L. and Burman, J.P., “The Design of Optimum MultifactorialExperiments”, Biometrika, Vol. 33, Issue 4, June 1946.

Poland, J., Mitterer, A., Knodler, K., and Zell, A., “Genetic Algorithms Can Improvethe Construction of D-Optimal Experimental Designs”, N. Mastorakis (Ed.),Advances In Fuzzy Systems and Evolutionary Computation (Proceedings ofWSES EC 2001), pp. 227-231.

Rivera, N., Kumar, A., Kumar, A., and Jalali, Y., “Application of Multilateral Wellsin Solution Gas-Drive Reservoirs”, paper SPE 74377, SPE InternationalPetroleum Conference and Exhibition, Villahermosa, February 2002.

Starzer, M.R., Tenzer, J.R., Larson, J.W., Bunch, B.C., Boehm, M.C., “BlowdownOptimization for the East Coalinga Extension Field, Coalinga Nose Unit, FresnoCounty, California”, paper 29667, Western Regional Meeting, Bakersfield,California, March 1995.

Vo, D.T., Marsh, E.L., Sienkiewicz, L.J., and Mueller, M.D., “Gulf of MexicoHorizontal Well Improves Attic Oil Recovery in Active Water Drive Reservoir”,paper SPE 35437, SPE/DOE Tenth Symposium on Improved Oil Recovery,Tulsa, Oklahoma, April 1996.

Wang, F., “Designed Simulation for Turbidite Reservoirs Using The Bell Canyon 3DData Set”, MS Thesis, University of Texas at Austin, 2001.

White, C.D., Willis, B.J., Narayanan, K., and Dutton, S.P., “Identifying Controls onReservoir Behavior Using Designed Simulations”, paper SPE 62971, SPE AnnualTechnical Conference and Exhibition, Dallas, Texas, October 2000.

42

Appendix A

Generating a Half-Fraction Design

Generation of a half-fraction design is considered in order to get a feel for how fractionaldesigns are generated in general, and also to have a better understanding of the reasonsthat make them economical as well as informative. The description in this section isbased on Box et al. (1978).

The number of runs required by a full 2k factorial design increases geometrically as k isincreased. It is observed that when k is not small, the desired information can often beobtained by performing only a fraction of the full factorial design. In other words, theretends to be a redundancy in the in terms of an excess number of interactions that can beestimated and sometimes in an excess number of variables that can be studied. Fractionalfactorial designs exploit this redundancy. An example shown below illustrates whateffects can be estimated using only a half-fraction of a 25 factorial design, thus reducingthe task to performing only 24 experiments.

Let there be five variables denoted by 1 through 5, each set at a high and a low valuedenoted by ‘+’ and ‘-‘. The procedure for the half-fraction design is as follows:

1. The full 24 fractional design is written for 1,2,3, and 4.2. The column of signs for the 1234 interaction is written, and these are used to

define the levels of variables 5. This is denoted as 5=1234, and is said to be thegenerator for the design.

The interactions are obtained by multiplying the columns under each variable. Hence 12interaction is obtained by multiplying the columns under 1 and 2. Similarly, for higherfactor interactions.

Figure 17 presents the analysis of generation of the half-fraction design, 25-1 from the full24 design.

43

1 2 3 4 5=1234- - - - ++ - - - -- + - - -+ + - - +- - + - -+ - + - +- + + - ++ + + - -- - - + -+ - - + +- + - + ++ + - + -- - + + ++ - + + -- + + + -+ + + + +

Variables

Figure 17. Analysis of a Half-Fraction of the full 24 design; a 25-1 FractionalFactorial Design

To estimate the effect of two-way interaction 12, the signs of 1 and 2 are multiplied toobtain the signs for 12. When multiplying, ‘+’ is treated as ‘+1’ and ‘-‘ as ‘-1’. Thiscolumn is multiplied to the column containing the measurements for the correspondingruns, and the sum of the values in the resulting column when divided by 8 gives theestimate of the 12 interaction effect. (The divisor is 8 because the 12 interaction effect isthe difference between two averages of eight results.) Of course, this estimate is limitedby the approximation inherent in this design, namely neglecting three-factor and higherorder interactions.

With this reduced fraction design, 16 runs can be performed and 16 quantities estimated--the mean, the 5 main effects, and the 10 two-way interactions. But since the full fractiondesign calls for 32 runs, the remaining 16 effects, constituting of 10 three-wayinteractions, the 5 four factor interactions, and the 1 five-factor interactions, are notestimated. An attempt to measure the value of 123 interaction by multiplying the signs incolumns 1,2, and 3 provides (written row-wise to save space)

123= -++-+--+-++-+--+

This is found to be identical to the 45 interaction. Hence,

45= -++-+--+-++-+--+

Thus, 123=45. This means that the three-way interaction 123 is indistinguishable fromthe two-way interaction 45 in this design, or in other words, these two effects areconfounded. Equivalently, 123 and 45 are said to be “aliases”. In fact, on inspection itmay be found that this design lumps together two-factor interactions with three-factor

44

interactions but does not confound main effect with two-way interactions. The methodsto arrive at the confounding pattern in the design other than the tedious method ofinspection are described in detail in standard texts (Box, et al., 1978).

The half-fraction design is the simplest fractional design but the same basic principleshold for construction of other fractional designs, e.g. 2k-p or 3k-p.

45

Appendix B

Algorithms for Constructing D-Optimal Designs

Most algorithms for generating D-optimal designs are heuristics that sequentiallygenerate a good set of design points from a set of candidate points, the final setrepresenting the optimal design. The set of candidate points consists of all the factor levelcombinations that may potentially be included in the design. It is not necessary that theset of candidate points comprise of the full factorial design; for larger problems, afractional-factorial design may be a good set of candidate points. The algorithm forgenerating an optimal design searches the candidate points for a set of N design pointsthat is optimal according to a given efficiency criterion. N has to be specified by the user,but it must be greater than the number of parameters (Kuhfeld, 1997).

Dykstra (1971) presented a sequential search algorithm that starts with an empty designand adds candidate points so as to maximize the chosen efficiency criterion at each step.Since it starts with an empty set, it always finds the same design for a given problem.Moreover, even though it is fast, it is not very reliable in finding a globally optimaldesign (Kuhfeld, 1997).

Cook and Nachstein (1980) compared the relative performance characteristics of severalalgorithms. The Wynn-Mitchell algorithm (Mitchell and Miller, 1970 and Wynn, 1972)and the Van Schalkwyn algorithm (Van Schalkwyn, 1971) were found to be the fastestmethods to generate designs of acceptable efficiency, whereas the Fedorov algorithm(Federov, 1969, 1972) produced the most efficient designs. Mitchell (1974) generalizedthe Wynn-Mitchell procedure to the DETMAX algorithm. Cook and Nachstein (1980)also proposed a modified Fedorov algorithm that was found to be almost as efficient asthe Fedorov algorithm but nearly twice as fast.

If a candidate xi is defined by a point ui in the input space, such that i = 1, 2,…,n, the ncandidate points are defined by the d-dimensional Euclidean space and a regressionmodel. For a choice of p < n candidates, let the set of candidate points be

pp njj }...1{),...,( 1 ∈=ξ , we write p=||ξ and define the design matrix ')...(

1 pjj xxX =ξ ,

thus denoting the matrix composed of the chosen candidates, namely j1, j2,… ,jp.. Thebraces define the input space. It may be noted that the notation is after Poland et al.(2001).

46

The Federov algorithm starts with an N-point nonsingular design, say, )1(ξ . During the i-

th iteration, a point x(i) is removed from )1( −iX and a point x is added to it, so that theresulting increase in the determinant of the design matrix )(iX is maximal (Cook andNaschtein, 1980). This method requires that all candidate point and design point pairs bechecked before making the swap. This makes it slower than DETMAX.

The DETMAX algorithm starts with a random design of size p (the actual number ofcandidates). The value of p changes during the process as additions and deletions to thedesign are made depending on whether they improve the design (based on the criterion tobe maximized). The algorithm stops after there is no further improvement.

In a recent development, genetic algorithms were found to improve the construction of D-optimal experimental designs (Poland et al., 2001). They can provide a way to obtaingood practical designs that are good with respect to more than one optimality criterion.

The commercial statistical softwares usually implement more than one algorithm. Theuser can specify one algorithm to initialize the design, that can be further improved byapplying another method.

47

Appendix C

Example ECLIPSE Data File

RUNSPECTITLEskua (regular grids, non-dipping )

DIMENS -- dimensions of the model-- NX NY NZ

49 9 27 /

-- specify the phases presentOILGASWATERDISGAS

METRIC -- unit specification

START -- starting date for simulation run1 'JAN' 2001 /

-- some other sizes and dimensions necessary for memory allocation:EQLDIMS -- equilibration table size (not needed yet)

1 100 10 1 20 /TABDIMS -- size of saturation and pvt tables

1 1 40 40 /WELLDIMS -- max numb of WELLS/CONN per WELL/GROUPS/WELLperGROUP

2 5 1 2 /NSTACK -- usually 10

25 /AQUDIMS--mxnaqn mxnaqc niftbl nriftb nanaqu ncamax0 0 1 36 1 441 /

UNIFOUTGRID

DXV49*55.56 /DYV9*97.96 /

48

DZ11907*2.8 /

PERMX11907*50 /

PERMY11907*50 /

PERMZ11907*50 /

MULTIPLY

PERMZ 0.01 //

PORO11907*0.21 /

BOX1 49 1 9 1 1 /

TOPS441*2258.5 /

ENDBOX

-- request init and grid file, necessary for post processing of the simulation--INIT--GRIDFILE-- 2 /

NOGGF

PROPS == pvt and relperm tables =============================

--pvt dataGRAVITY42.0 1.03 0.72 /

-- SK.PROPS11 data:-- PVT Properties from Sk-3 PVT Report-- From Table 16 Diff Vapn at 205 degF Field Adjusted-- Converted to Metric

49

-- Live Oil Properties (With Dissolved Gas)

PVTO

28.60 39.97 1.114 0.607 /53.01 76.86 1.176 0.500 /80.40 116.36 1.244 0.421 /108.40 154.22 1.315 0.370 /138.93 190.90 1.390 0.332 /173.48 228.89 1.48 0.300

231.78 1.479 0.301245.09 1.474 0.303 /

/

-- Dead Gas Properties (No Vapourized Oil)

PVDG76.86 0.01513 0.01420116.36 0.00977 0.01560154.22 0.00730 0.01710190.90 0.00590 0.01865228.89 0.00496 0.02056245.09 0.00465 0.02150

/

-- Water PVT Properties (from HP Fluids PAC)

-- Assumptions : 9 scf/bbl gas, salinity (NaCl) 90,000 ppm-- Pressure FVF Compress Viscosity Viscosibility

PVTW227.586 1.04 43.5E-06 0.35 0.0 /

-- Rock Properties (from HP Fluids PAC)

-- Pressure Compressibility

ROCK227.586 50.75E-06 /

--satn function tablesSWFN---- Sw krw Pcw

50

--0.15 0.00 0.00.80 0.21 0.01.0 1.0 0.0 /

SOF3

-- So krow krogcw--

0.00 0.00 0.000.20 0.00 0.000.85 0.85 0.85 /

-------------------------------------------SGFN

-- Sg krg Pcg--

0.0 0.0 0.00.15 0.0 0.00.85 0.90 0.0 /

SOLUTION

EQUIL -- DATA FOR INITIALISING FLUIDS TO POTENTIAL EQUILIBRIUM-- DATUM DATUM OWC OWC GOC GOC RSVD RVVD SOLN-- DEPTH PRESS DEPTH PCOW DEPTH PCOG TABLE TABLE METH

2286.5 228.6 2314.5 02278.1 0 1 0 0/

RSVD -- variation of initial rs with depth-- depth rs

2286.5 178.12500 178.1

/

--aquifer; carter-tracy non-dip, irregular skuaAQUCT--No depth prs perm poro compress. Rin thick angle PVT infTab salt

1 2286.5 1* 600. 0.11 4.E-5 3.E3 340. 1* 1 1 /

AQUANCON--aquifer no I1 I2 J1 J2 K1 K2 ,connecting face ,influx coeff, influx coeff multiplier1 1 49 1 9 27 27 'K+' 1 /

51

/

SUMMARYEXCEL

WGOR/WBHP/WOPR/WGPR/FPRFOPTFGPTFWCTFAQRFOPVFGPVFOEFORMFFORMWFORMY

SCHEDULE

--RPTRST--'BASIC=4'--/WELSPECS == WELL SPECIFICATION DATA================================-- WELL GROUP LOCATION BHP PI-- NAME NAME I J DEPTH DEFN

'PROD1' 'DUMMY' 13 5 1* 'GAS' 2* 'STOP' /

/

COMPDAT -- COMPLETION SPECIFICATION DATA-- WELL LOCATION OPEN/ SAT CONN WELL KH S D AXIS-- NAME I J K1 K2 SHUT TAB FACT DIAM

'PROD1' 13 5

52

7 7 'OPEN' 0 1* 0.216 3* //

WCONPROD -- PRODUCTION WELL CONTROLS-- WELL OPEN/ CTRL OIL WATER GAS LIQU RES BHP-- NAME SHUT MODE RATE RATE RATE RATE RATE

'PROD1' 'OPEN' 'GRAT' 2* 600000 2* 40 /

/

-- timesteps can be refined by entering multiple TSTEP keywordsTSTEP -- and run it for 8 yrs

8*365./

ENDENDEND