Shameem Siddiqui Malgorzata Ziaja

QUANTIFICATION OF UNCERTAINTIES ASSOCIATED WITH RESERVOIR PERFORMANCE SIMULATION

By

Andrew Oghena, B.E, M.S.

A DISSERTATION

IN

PETROLEUM ENGINEERING

Submitted to the Graduate Faculty

of Texas Tech University in Partial Fulfillment of the Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

IN

PETROLEUM ENGINEERING

Approved

Lloyd R. Heinze Chairperson of the Committee

Shameem Siddiqui

Malgorzata Ziaja

Accepted

John Borrelli Dean of the Graduate School

May, 2007

EXECUTIVE SUMMARY

For this research, numerical simulation was utilized to build black oil and

compositional reservoir simulation models. The modeled reservoirs were used to quantify

uncertainty associated with reservoir performance by utilizing Black Oil Conditioning

technique. This method is performed after history matching. It involves perturbing the

history matched model to generate few realizations and simultaneously modeling each

realization using both black oil and composition simulation and, thereafter, condition the

black oil output with the compositional simulation results. This approach allows the use

of few simulation models to quantify simulated reservoir performance uncertainty.

The main source of uncertainty focused within this research is the uncertainty

associated with reservoir description. The reservoir description parameter of interest is

permeability. Ratio of vertical to horizontal permeability distribution of the history

matched black oil model was perturbed slightly to generate the few realizations.

It is well known, that black oil simulation model is limited in terms of its capacity

to provide detailed compositional information and, therefore, exhibit less fluid behavior

capacity. As a result, to more accurately account for the influence of reservoir description

and fluid behavior on simulated reservoir performance, this research provides a method

of conditioning black oil results by compositional simulation output and also proposes

two algorithms for estimating confidence interval during uncertainty assessment. The

assumption behind this technique is that all reservoirs have some element of

compositionality in their reservoir fluid.

In conclusion, this research also recommend sufficient history period whereby

observed field historical data can be utilized for acceptable reservoir history matching.

ii

ACKNOWLEDGEMENT

This research was conducted at Texas Tech University under the supervision of

Dr. Akanni Lawal and Dr. Lloyd Heinze. I like to express many thanks to Dr. Heinze for

his technical advice and direction in pursuing this study. In particular, I am grateful for

his patience during several lengthy deliberations to attend to the demands of this project

despite his tight schedule. I am also indebted to Dr. Lawal for initiating this research and

for his reservoir fluid phase behavior contribution. My gratitude also goes to the other

members of my committee, Dr. Shameem Siddiqui and Dr. Malgorzata Ziaja and to Ms.

Joan Blackmon for editing this dissertation report. I highly appreciate the moral support

from my parents, Ms. Unwana Ebiwok, my entire family members, and the financial

support from the Petroleum Engineering Department. Above all, I give special gratitude

to Almighty God for giving me the opportunity to undertake this study.

Andrew Oghena, Texas Tech University, May 2007

iii

TABLE OF CONTENTS

ACKNOWLEDGEMENT ii

ABSTRACT v

LIST OF TABLES vi

LIST OF FIGURES vii

NOMENCLATURE ix

CHAPTER I. INTRODUCTION AND STATEMENT OF PROBLEMS

1.1 Statement of Research Project 1 1.2 Fundamentals of Reservoir Performance Simulation 4 1.3 Sources of Uncertainties in Reservoir Performance Simulation 6 1.3.1 Geological Uncertainty 6 1.3.2 Upscaling Uncertainty 8 1.3.3 Model Uncertainty 12 1.3.3.1 Truncation, Stability and Round-off errors 17 1.3.4 Reservoir Description Uncertainty 18 1.4 Black Oil and Compositional Simulations 19 1.4.1 Black Oil Simulation 21 1.4.2 Compositional Simulation 23 1.5 Problem Definition 28 1.6 The economic significance of uncertainty quantification 30

II METHODS FOR UNCERTAINTY ESTIMATION 2.1 Literature Review 32

2.2 Definition of Simulation Input Parameter Associated Uncertainty 37 2.3 Model Parameterization 40 2.3.1 Grid Blocks 40 2.3.2 Regions 41 2.3.3 Pilot Points 41 2.4 Objective Function 42 2.4.1 Least Square 42 2.4.2 Likelihood Function 43 2.4.3 Posterior Distribution 44 2.5 Model Optimization Process 45 2.5.1 Gradient Optimization 45 2.5.2 Non-Gradient Technique 46 2.5.3 Root Mean Square Match Analysis 46

2.6 Uncertainty Quantification Algorithms 48 2.6.1 Linear Uncertainty Analysis 48 2.6.2 Probability Uncertainty Quantification 49 2.6.2.1 Input Parameter Probability Distribution Function 52 2.6.2.2 Output Parameter Probability Distribution Function 56 2.6.3 Quantification of Uncertainty Using Bayesian Approach 56 2.7 Uncertainty Forecasting 58


iv

III. RESERVOIR PERFORMANCE SIMULATION 3.1 Reservoir Heterogeneity 61 3.1.1 Heterogeneity Scale Effect 61 3.1.2 Heterogeneity Measurement 63 3.2 Reservoir Simulator 66

3.3 Reservoir Model Description 71 3.4 First Case Simulation: Natural Depletion Model 73 3.5 Second Case Simulation: Water-Alternate-Gas Model 79 IV. RESERVOIR PERFORMANCE ANALYSIS 4.1 History Matching and Optimization 83 4.2 Research Methodology 83 4.3 Observed History Data Duration 86 4.3.1 Well Testing Interpretation 93 4.4 Ultimate Recovery Uncertainty: Natural Depletion 95 4.4.1 Positive and Negative Confidence Interval Algorithms 97 4.5 Ultimate Recovery Uncertainty: Water-Alternate-Gas 102 4.6 Justification of the applied Uncertainty Quantification Method 104 4.7 Relating Research Findings to Existing Literature 105 V. CONCLUSIONS AND RECOMMENDATIONS 5.1 Conclusions 108 5.2 Recommendations 109

REFERENCES APPENDICES

A. RESERVOIR FLUID PRESSURE-VOLUME-TEMPERATURE PROPERTIES

B. SIMULATION MODEL DATA FILE

C. DATA FOR OBSERVED HISTORY DURATION

D. PLOTS OF BLACK OIL AND COMPOSITIONAL SIMULATION GENERATED DATA

E. DATA FOR OPTIMIZATION OF BLACK OIL WITH COMPOSITIONAL

SIMULATION


v

ABSTRACT

This research presents a method to quantify uncertainty associated with reservoir

performance prediction after history match by conditioning black oil with compositional

simulation. Two test cases were investigated.

In the first test case, a black oil history matched model of a natural depleted

volatile oil reservoir was used to predict reservoir performance. The same reservoir was

simulated with compositional model and the model used to forecast reservoir

performance. The difference between black oil and compositional models predicted

cumulative oil production were evaluated using an objective function algorithm. To

minimize the objective function, the black oil and compositional simulation reservoir

descriptions were equally perturbed to generate few multiple realizations. These new

realizations were used to predict oil recovery and their forecast optimized. Non-linear

analysis of the optimization results was used to quantify the range of uncertainty

associated with the predicted cumulative oil production. Similarly, a second test case was

studied whereby, the same volatile reservoir was produced under water-alternate-gas

injection scheme. As in the first test case, it is shown how optimization followed by non-

linear analysis of both the black oil and compositional simulation predictions can be used

to assess uncertainty in reservoir performance forecast.

It is well known that the disadvantage of the black oil is its inability to simulate

comprehensive reservoir fluid compositional data. To eliminate this limitation in

reservoir performance prediction, this research presents a technique that is based on

conditioning black oil output with compositional simulation in order to better account for

fluid phase behavior and reservoir description influence on reservoir performance.


Quantification of Uncertainties Associated with Reservoir Performance Simulation

vi

LIST OF TABLES

1.1 2 Parameters Equation of State Critical Compressibility Factor 27

2.1 Three types of distribution: Normal, Lognormal and Exponential 53

3.1 Influence of Heterogeneity Scale 62

3.2 Equations Solved by a Reservoir Simulator 68

3.3 Common Data Required for Reservoir Simulation 69

3.4 Common Reservoir Simulator Grid Dimensions 70

3.5 Reservoir Layer Data 74

3.6 Reservoir Model Data 75

3.7 Data for Relative Permeability and Capillary Pressure 76

3.8 Compositional Fluid Description 77

3.9 Peng-Robinson Fluid Characterization 77

3.10 Injection Gas Composition 80

4.1 Base Case Reservoir Description and Simulation Output 87

4.2 1% Reservoir Description Perturbation 87



4.5 Transient Pressure Interpretation 93

4.6 Conditioning of Black Oil Simulator with Compositional 95

4.7 Perturbed KV/KH and Corresponding Simulator Cumulative Oil Production

97

4.8 Black Oil Conditioning 106



vii

LIST OF FIGURES 1.1 Uncertainty Sources 2

1.2 Individual Uncertainties and Composite Uncertainty 3

1.3 History Matching Flow Chart 5

1.4 Upscaling Process 11

1.5 Finite Difference 13

1.6 3-Dimensional Discretized Model 14

1.7 Explicit Approximation 15

1.8 Implicit Formulation 15

1.9 Black Oil and Composition Simulation Processes 22

2.1 Root Mean Square Match Analysis 47

2.2 Cumulative Distribution Function 51

2.3 Cumulative Distribution Function Statistical Properties 51

2.4 Normal Probability Distribution 53

2.5 Discrete Histogram plot used to generate probability distribution function 55

2.6 Continuous probability distribution function 55

2.7 Relationship between input parameters and model result uncertainty 56

3.1 Dykstra-Parsons Coefficient Method 65

3.2 Grid Block 69

3.3 Reservoir Model Schematic 74

3.4 Reservoir Model Cross-section Schematic 80

3.5 Second Case Reservoir Model Schematic 81



viii

3.6 Oil saturation at time zero 81

3.7 Oil Saturation after 12 Years 82

4.1 Black Oil Conditioning Flow Chart 85

4.2 Two Months Observed History Data Matching 89

4.3 Six Months Observed History Data Matching 89

4.4 Twelve Months Observed History Data Matching 90

4.5 Eighteen Months Observed History Data Matching 90

4.6 Twenty Four Months History Data Matching 91

4.7 Forty Eight Months History Data Matching 91

4.8 Reservoir Performance Prediction 1 92

4.9 Reservoir Performance Prediction 2 92

4.10 Observed History Data Log-Log Plot 94

4.11 History Matched Model Log-Log Plot 94

4.12 Black Oil Simulator Forecast after Conditioning 96

4.13 Positive Confidence Interval Algorithm 98

4.14 Negative Confidence Interval Algorithm 98

4.15 Cumulative Oil Production Uncertainty Quantification 100

4.16 Water-cut Uncertainty Quantification 101

4.17 Uncertainty Forecast for WAG Scheme 103

4.18 Conventional Linear Analysis of Uncertainty 107



ix

NOMENCLATURE

Symbol Definition

a Attraction Term of EOS

A Dimensionless attraction term

b Van der Waals co-volume

B Dimensionless van der Waals co-volume

c Compressibility

C Carbon component

D Dimensional

f Fugacity

k Permeability

n Number of variables

L Liquid mole fraction

m Fugacity coefficient

P Pressure

Q Flow rate

R Universal Gas Constant

t Time

T Absolute Temperature

V Vapour mole fraction

x x-direction

y y-direction

z z-direction

Z Gas deviation/compressibility Factor

AIM Adaptive Implicit

BHP Bottom hole pressure

cdf Cumulative distribution function

GOR Gas-oil-ratio

IMPLICIT Fully implicit



x

IMPES Implicit Pressure Explicit Saturation

pdf Probability distribution function

RDM Reservoir description model

RMS Root mean square

WCT Water-Cut

Bo Oil formation volume factor

BO Black Oil

Cp Covariance matrix

COP Cumulative oil production

FPR Field pressure

FGOR Field gas-oil-ratio

FWCT Field water-cut

HC Hydrocarbon

MCMC Markov-Chain Monte-Carlo

MSCF Thousand standard cubic feet

mD Milli Darcy

PDE Partial Differential Equation

PVT Pressure-Volume-Temperature

STB Stock tank barrel

TOP Total oil production

WAG Water-Alternate-Gas

Rs Solution-gas-oil ratio

Sw Water Saturation

SoS Sum of square

V Molar volume

xi Oil mole fraction

yi Gas mole fraction

N Number of components

EoS Equation of State



xi

Subscript

i, j Component Identification

1, 2 Component Index

c Critical Property

l Liquid Phase

v Vapor Phase

m Mixture

w van der Waals Representation

Greek Letter

α LLS EOS Parameter

αij Binary Interaction Parameter of α

β LLS EOS Parameter

βij Binary Interaction Parameter β

μ Viscosity

φ Porosity

ρ Density

ω Acentric Factor

Ωw van der Waals Constant Parameter

Ω Dimensionless EoS Parameter,

σ Variance



- 1 -

CHAPTER 1

INTRODUCTION AND STATEMENT OF PROBLEMS

1.1 Statement of Research Project

The objective of this study is to quantify the uncertainties associated with

reservoir performance simulation. The term reservoir performance is defined as; oil and

gas production rates, gas-oil ratio, water-oil ratio, and cumulative oil production. This

research is focused on quantifying uncertainty associated with future cumulative oil

production prediction from black oil reservoir simulation model.

To achieve this research objective, black oil and compositional simulation models

were constructed for the same volatile oil reservoir and these model reservoir descriptions

were perturbed to generate few multiple realizations. The dynamic outputs of these new

realizations were matched to determine a range of possible outcomes. The range between

the smallest and largest cumulative oil production values quantify the uncertainty

associated with the reservoir simulation performance prediction.

Perturbation process is employed to generate multiple realizations. The parameter

adjustment process is performed on a single simulation model obtained after history

matching. In reservoir simulation studies where more than one model matched observed

history data, the aforementioned approach was carried out using more than one of the

matched models.

In conclusion, the problem statement to be answered is; how to develop a method

that can account for uncertainty resulting from both internal and external factors (figure



- 2 -

1.1) which translate into composite uncertainty associated with reservoir performance

prediction?

Figure 1.1: Uncertainty Sources

The uncertainties associated with individual reservoir characteristics such as:

hydrocarbon originally in place, aquifer size, sand continuity, shale continuity,

permeability, upscaling, mathematical model, and external factors (e.g. pump lifetime),

all add up to give a resultant total uncertainty associated with reservoir performance

prediction27, 28, 31, 48, 59, 76, 123, 144. This is simply put as the uncertainty in reservoir input

parameters lead to uncertainty in reservoir performance forecast (figure 1.2).

Uncertainty Sources in Reservoir

Performance Prediction

External Factors

Internal Factors

Flow Boundary

Condition (Geology

parameter)

Mathematic Model

Reservoir Characterization

Data Quality

Upscaling Technique

Field Management

Strategy

Surface Facilities



- 3 -

Figure 1.2: Individual uncertainties and Composite uncertainty

During reservoir description process, reservoir engineers assign values to

reservoir model parameters using incomplete data such as data which was measured from

a small portion of the reservoir to describe the entire reservoir26, 39, 47, 48, 68, 76, 83, 84, 94, 110,

120,155, 167. The incomplete data limit reservoir simulation model capacity to mimic actual

reservoir accurately leading to error in the model output.

Therefore, to address this problem, reservoir engineers carry out uncertainty

evaluation during reservoir simulation study in order to quantify the reservoir simulation

model in ability to mimic the actual reservoir (mismatch). Quantification of reservoir

simulation mismatch enables the assessment of the uncertainty associated with the

reservoir model performance prediction.

Management decision on field development is taken only when the associated

uncertainties with both the individual reservoir model parameter and the simulation

Uncertainty Quantification in

Reservoir Performance Simulation

Individual input Parameters

Uncertainties

Simulation result: Production Variables

& Reserves Uncertainties (Composite Uncertainty)

Sum



- 4 -

production forecast is well understood and quantified. If not a decision to obtain

additional reservoir data measurement is taken so as to better understand the reservoir.

1.2 Fundamentals of Reservoir Performance Simulation

The goal of reservoir performance simulation is to build a reservoir model that is

capable of predicting the actual reservoir performance (water cut, reservoir pressure, and

gas-oil-ratio, etc.) for different production scenarios by minimizing associated

uncertainties/errors in reservoir simulation. Minimization of the simulator errors is

achieved by performing reservoir history matching. History match process involves

comparing the simulator dynamic output with observed field production data8, 10, 20, 22, 32,

109, 121, 132, 133, 142,151, 166. When an acceptable match is obtained, the simulator is then used

to predict the reservoir future production performance.

The petroleum industry conventional approach to minimize the difference

between observed history data and simulation model result is to vary the model input

parameters until a match with the history data is achieved. This optimization process is

conducted using least square objective function algorithm. On the other hand, a more

recent approach involves constructing multiple reservoir simulation models and conduct

history matching of simulated and observed data. When a match is obtained, the matched

model(s) is used to forecast future reservoir performance and to quantify associated

uncertainty (figure 1.3). The major problem with this multiple realization technique is an

increase in the computation cost. While this technique was developed to minimize the

non-uniqueness of traditional history matching since a match with a single simulation



- 5 -

model may have resulted from compensation errors of the various interacting

parameters/factors89, 144. The fact is that more than one model can reproduce the real

reservoir observed history data.

Figure 1.3: History Matching Flow Chart

Observed history data measurement

Reservoir model

Parameterization and Prior pdf definition

History data

Reservoir model simulation

Objective function

Mismatch

History matched model

Reservoir prediction Uncertainty forecast

NO

YES



- 6 -

1.3 Sources of Uncertainties in Reservoir Simulation

For all data that are used in reservoir modeling there exists a certain degree of

uncertainty associated with each of the data. Figure 1.1 gives several sources of

uncertainties associated with reservoir performance simulation. A brief explanation of

some of these sources of uncertainties is discussed to highlight this research significance.

1.3.1 Geological Uncertainty

Uncertainties arising from geological data include errors in geological structure

exact locations, reservoir and aquifer sizes, reservoir continuity, fault position,

petrofacies determination, and insufficient knowledge of the depositional environment. A

number of techniques are available for the quantification of uncertainties. One of the

widely used techniques is to quantify the uncertainty in geological model with a

geostatistical tool.

Geostatistics involve synthesizing geological data using statistical properties such

as a variogram9, 20, 22, 47, 48, 68, 83, 90, 94, 97, 110, 121, 132, 138, 154 . This process enables the

geologists to generate multiple realizations of the geological models (Stochastic) which

allows quantification and minimization of uncertainties associated with geological

information.

The problem with geostatical modeling is that it is computationally difficult to

condition the model with dynamic data. Also, it is difficult to utilize traditional history

match process to condition the geostatistical models7, 9. Recently, a number of new

methods have been developed to condition geostatitical models to dynamic data.



- 7 -

Examples of these techniques are Simulated Annealing method59, 108, 109, 138. and Genetic

algorithm59, 108, 109, 138. These two methods are limited in practical application for large

fields modeling because of computational costs that result from numerous grid cells.

The aforementioned two stochastic techniques which are used to condition

geostatistical model using dynamic data involve the construction of multiple realizations

of the geological model. These independent but equi-probable realization models are

judged as a good model or not by using criteria such as Markov Chain Monte Carlo

simulation to either accept or reject a realization model. This process is also heavily

computational. The generation of different realizations results in discontinuity which can

thwart the effective conditioning of the initial model with dynamic data9.

Another method used to condition the geostatical model to production data is the

pilot point technique22, 59. The pilot point method is carried out by selecting certain points

in the reservoir and perturbing their values. The change resulting from the perturbation is

propagated by Kriging to the remaining parts of the reservoir. This method provides an

approximate solution to the history match inverse problem.

Gradual deformation is another technique that could be applied to reduce

geological model uncertainty133. The use of gradual deformation in geostatistical

modeling is an effective inversion algorithm that constrains the initial model to dynamic

data through the generation of gradual match improvement at the same time it honors the

statistical characterizations. In this method a new single realization is generated by linear

combination of two initial models. This is a continuous way to minimize the initial model



- 8 -

uncertainty instead of generating independent models as in the case of stochastic

methods.

The limitation of the gradual deformation method is that each new model is

controlled by a set of deterministic parameters. This limits the accuracy of gradual

deformation technique because a good history match of the geological model with

observed history data may result from compensating the errors associated with the model

input data because the observed data can be matched with more than one set of model

input data (inverse problem).

Furthermore, in mature fields, geological modeling the integration of well test

pressure data, production data and geological description can decrease uncertainty in the

geological model9, 84, 155.

1.3.2 Upscaling Uncertainty

After seismic survey, the geologist builds the small-scale geologic model (static

model). As a result of the fine scale level it is computationally expensive to investigate

reservoir flow behavior using the geologic model. Consequently, the geological model is

upscaled into a coarse scale model generally called a reservoir simulation model by

reservoir engineers in order to evaluate the reservoir flow behavior. During upscaling

reservoir properties (permeability, porosity and relative permeability) are upscaled so as

to reduce the number of grid blocks composing the simulation modell6, 11, 14, 19, 35, 36, 44, 54,

71, 81, 82, 116, 122, 135, 156, 159, 162, 168.



- 9 -

Upscaling could be defined as the process of representing small-scale features in a

reservoir simulation model. Upscaling is the process that explains how reservoir

properties at different scales are integrated into a model so that the simulation model

mimics the real reservoir behavior. Nearly all petroleum reservoirs are heterogeneous at

all scales ranging from microscopic to gigascopic, and they are mostly anisotropic with

spatial variation of rock and fluid properties. These variations in rock-fluid parameters

control reservoir fluid flow and reservoir performance.

To predict a reservoir performance having well spacing of 1km, reservoir

thickness of 100m and smallest heterogeneity scale of 1mm and to describe the reservoir

heterogeneity down to 1mm in a 3-D model it requires 1017 grid points cells to represent

all the reservoir properties. This number of grid cells is quite high for current computer

capacity to handle and human mind to comprehend or interpret.

In the petroleum industry two types of reservoir models are constructed: fine grid

and coarse grid models. The fine grid model is employed to geologically characterize a

reservoir although in most modeling the model areal resolution is still coarse due to

computational costs for the finer grids. On the other hand, the coarse grid model is used

to evaluate reservoir performance prediction.

At the moment, due to computer limitation most fine grid models are constructed

to contain between 107-108 grid cells while coarse grids are in the range of 105-106 grid

cells. It is obvious that both the fine and coarse grid models differ in their level of

resolution and a means of transforming the fine grid to coarse grid model is needed.

Furthermore, to investigate the uncertainty associated with reservoir performance



- 10 -

forecast, the uncertainty in each reservoir parameters needs to be evaluated. Evaluating

uncertainty of fine grid model individual reservoir parameters involves thousands of fine

grid simulation runs and these high-quantity runs are limited by current available

computer capability and high computational costs. As a result, upscaled/coarse grid

model of the fine-scale model is required.

Therefore, the upscaling technique is needed to transform the fine grid into coarse

grid model (see figure 1.4). A number of upscaling techniques are available in the

industry. The different approaches can be classified according to two broad methods54:

(1) the type of parameter to be upscaled, and (2) the method of computing the parameter.

These methods have various degrees of limitations in the ability to translate a fine-scale

model into a coarse-scale model.

Irrespective of the particular upscaling technique employed to generate the coarse

model, utmost care should be taken to ensure that the upscaled model input parameter(s)

is equivalent to the fine scale model parameter. For example, accurate upscaling of

residual oil saturation and initial water saturation are vital because these two parameters

determine the amount of oil that can be recovered from the reservoir. Some parameters

such as porosity and saturation are accurately upscaled using simple volume averaging

techniques. While absolute and relative permeability have varied upscaling algorithms.

As a result, significant amount of uncertainties exist when permeabilities are upscaled

from fine scale into coarse grid.



- 11 -

Recognizing the fact that upscaling uncertainty exists, there is a need for proper

quantification of upscaling uncertainty, which is important for better reservoir

performance prediction.

Figure 1.4: Upscaling Process

The following is a general gridding guidelines and gridding rules of thumb

• Choose the minimum number of grid blocks to solve the problem

• Pore volume considerations

With the exception of aquifers, no single grid block should have more than 20%

of the total pore volume of the system.

• Pressure drop considerations

No more than 10 to 20% of the total pressure drop in the simulation grid should

be between two adjacent grid blocks.

• Relative grid block sizes

Grid block dimensions should not change by more than a factor of 3 between

blocks. - For example, the size of a grid block should not be larger than three

times, or smaller than one third the size of its neighbors.

Upscaling of fine scale model into a coarse scale model is conducted in both black

oil and compositional simulation. Detailed literature on upscaling of black oil model can

be found in reference 54. While less work has been done on upscaling of compositional

Fine Scale Coarse Scale

Reduce Grid number

Upscale Reservoir Properties



- 12 -

simulation, most of the work done so far on compositional upscaling14, 41, 51, 71, 92, 101, 122,

126,147, 162 has been the adjustment of K-value flash calculation in order to account for

using coarse grid to represent fine scale. This K-value adjustment resulted in Alpha factor

method which serves as modifiers. The modifiers are introduced into numerical

simulation flow equation to relate fluid composition flowing out of the grid to the fluid

composition within the grid.

1.3.3 Model Uncertainty

The mathematical model used in numerical reservoir simulation is derived by

integration of three fundamental equations which are, conservation of mass, Darcy’s

equation and equation of state4, 13, 30, 46, 98, 112. The resulting mathematical model for three-

dimensional, single-phase flow equation is:

Equation 1 is solved analytically for p( x,y,z,t).

( ) 1−−−−−−∂∂

=+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂ φβ

μβ

μβ

μβ

tQ

zpck

zypck

yxpck

x

( ) ( )

( ) ( )

( ) ( ) 0,,,,0,0,,

0,,,,0,,0,

0,,,,0,,,0

.)0,,,(

=∂∂

=∂∂

=∂∂

=∂∂

=∂∂

=∂∂

=

tLyxyptyx

yp

tzLxyptzx

yp

tzyLxptzy

xp

pzyxp

z

y

x

i



- 13 -

This mathematical model, equation 1, is a non-linear partial differential equation

(PDE) which can not be easily solved analytically. As a result, the PDE is converted to a

numerical model using Taylor series approximation57. The numerical model is derived by

replacing the partial derivatives in the PDE with finite differences (Equation 2) evaluated

at specific values of x, y, z, and t as outlined below also see depicted in fig 1.5

----------------------------------------- 2

Figure 1.5: Finite Difference

With this approximation the differential equation is transformed into an algebraic

equation that can be easily solved using matrix. The resulting finite difference

formulation is given in equation 3 and the numerical model can be represented in three

directions as shown in figure 1.6.

tpp

tp

xpp

xp

nn

tt

ii

xx

n

i

Δ−

=∂∂

Δ−

=∂∂

+

=

+

=

1

1



- 14 -

DHpFpCpBpGpEpAp kjikjikjikjikjikijikji =++++++ +++−−− 1,,,1,,,1,,1,,,,,,1 ----------3

Figure 1.6: 3-Dimensional Discretized Model

The resulting numerical equation includes pressure terms evaluated at two

different points in time. These times are the initial time, t = t0 and at a selected future time

called time step, t = t1. Knowing the pressure at the initial time, we have to solve the

numerical equation for pressure at the given time step. At subsequent time steps, pressure

will be calculated at multiple points in a three dimensional model.

In the process of deriving the numerical model by approximating the PDE, a

number of ways can be used to form the finite-differences112. If the space derivative is

transformed by central differences and time derivative by forward difference, we have

explicit approximation of the PDE (see figure 1.7).



- 15 -

Figure 1.7: Explicit Approximation

On the other hand, if the space derivative is transformed by central difference and

a backward difference for the time derivative, implicit approximation is obtained (figure

1.8).

Figure 1.8: Implicit Formulation



- 16 -

The resulting numerical equation coefficient is solved using n × n matrix. For one

dimensional, single phase numerical model, a tridiagonal matrix is formed. In a four-cell

system, the matrix is depicted as in equation 4:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

DDDD

PPPP

BACBA

CBACB

---------------------------------------------------- 4

This matrix representation consists of three non-zero diagonals and is easily

solved. The computation time needed to obtain pressure solution (pn+1) for the implicit

approximation is more than that for the explicit method. The problem complexity

increases with increased number of dimensions and for multiple phases present. For two

phases, the fluid flow equation applies to each flowing phase individually such that at

each time step there are two unknowns to be solved, po ,and Sw in each grid block

equations 5 & 6.

Oil: ( )( )wooo

o

oro St

Qx

pckkx

−∂∂

=+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂ 1φβ

μβ

------------------------ 5

Water: ( ) ( )www

co

w

wrw St

Qx

ppckkx

φβμβ

∂∂

=+⎟⎟⎠

⎞⎜⎜⎝

⎛∂−∂

∂∂

------------------- 6



- 17 -

1.3.3.1 Truncation, Stability and Round-off Errors in Numerical Model There are three errors which are consequences of PDE discretization. These errors

are truncation, round-off, and stability errors57, 112. The truncation error results from the

substitution of the partial derivatives in the differential equation with approximate finite

differences. When solving the numerical model, if numerical solution convergences then

the numerical solution will approach the exact solution as the change in space, Δx, and

change in time, Δt, approaches zero12, 38, 72, 73, 105, 111, 113, 114, 115, 116, 117, 118. As a result, it is

an approximate solution which does not mimic the actual reservoir exactly.

Round-off error, on the other hand, is a result of using a computer to solve the

numerical model because the computer can not represent real numbers accurately.

Stability error consists of the approximation method used in transforming the

PDE into a numerical model and the PDE itself. Instability in numerical solution can be

defined as a feedback process whereby one error leads to another error (truncation or

round-off errors, respectively). As the error increases, the rate of error growth increases

so that the error growth gets so large that the solution is lost.

Stable numerical solution = Error growth rate is constant

Unstable numerical solution = Error growth rate is exponential

Apart from the algorithms used to generate the numerical model, model

uncertainties can also arise from the type of reservoir simulator used. The simulator used

can either be mass balance or streamline, finite difference or finite element. The inherent

uncertainty results from the inadequacy to completely translate the continuous mass

balance and flow equations into discrete approximates and the use of a computer to solve



- 18 -

the equation. As a result, the numerical model that is used in reservoir simulation

contains uncertainty which need to be quantified.

1.3.4 Reservoir Description Uncertainty

Reservoir characterization process involves the determination of reservoir rock

and fluid properties that will accurately represent the actual reservoir. It is likely that area

of reservoir simulation with greatest uncertainty because the actual reservoir description

can not be achieved even at the end of the field life. This research is aimed at minimizing

the uncertainty associated with reservoir description during reservoir simulation.

Reservoir properties used in characterizing a reservoir are oil rock and fluid

properties26, 39, 47, 68, 83, 94, 110, 120, 167. The reservoir rock parameters are porosity and

permeability. These rock properties are obtained by several methods such as core and

well log analysis. Apart from instrumental and measurement errors associated with the

rock parameters derived from the aforementioned methods, the vital source of error is

that these measurements represent a very small area of the reservoir compared with the

entire reservoir to describe in reservoir simulation model.

Using the parameter obtained from a relatively small portion of the reservoir to

describe the whole reservoir will result in high uncertainty because we are not certain of

the individual parameter continuity (or remaining constant) from point of measurement to

other reservoir locations. For example the continuity of permeability from the

measurement point ‘A’ to the point ‘B’ in the reservoir.



- 19 -

One way of reducing the discussed uncertainty is to generate a representative

statistical distribution function from the range of measured rock data and function is used

to populate the entire reservoir grid cells. The statistical distribution function for

permeability is derived based on the understanding of natural distribution and the trend

exhibited by the measured permeability data. The common statistical distribution trend

exhibited by permeability is a log-normal function33, 49, 69.

The statistical algorithm used to generate the reservoir description model is an

approximation. Therefore, the generated reservoir description is an approximation of the

real reservoir. The algorithm used to generate the reservoir description model is

conditioned to production data so as to obtain a match and the calibrated model is used to

predict reservoir performance8, 154, 155.

1.4 Black Oil and Compositional Simulation

The selection of an appropriate reservoir simulation model to be used during any

given reservoir simulation will depend on the reservoir at hand and the available data12, 41,

38, 78, 105, 106, 157, 162. This is because the simulation model will only be useful if it is capable

of simulating the actual reservoir and the fluid phases38, 41, 78, 105. Two commonly used

finite difference simulations for modeling hydrocarbon reservoir processes are black oil

and compositional simulations41.

The main dissimilarity between black oil and compositional simulation is the data

included in the fluid properties section. In black oil simulation, the fluid property section

is defined by the PVT table which includes formation volume factors and solution gas-oil



- 20 -

ratio versus pressure101. On the other hand, for compositional simulation in addition to

what is contained in black oil model, the compositional PVT table includes composition

(oil and gas mole fractions, xi and yi) as single-valued versus pressure obtained from

equation of state flash calculation51, 58, 92, 126. The compositional model better accounts for

fluid phase behavior when compared to black oil model, particularly in volatile oil and

miscible gas injection modeling18, 41, 42, 65, 79, 101, 147,153, 162.

It is well known that the black oil PVT table can be converted to compositional

PVT table18, 65, 79, 87. Also, it is understood that the fundamental reservoir simulation mass

balance equation is applied to both black oil and compositional simulation. But the

formulation code for solving the numerical model continuity/mass balance equation is

different in some simulators65, 78, 106, 140, 150, 153. For the simulator used in this research,

ECLIPSE, the formulation coded for black oil is fully Implicit (IMPLICIT) and Adaptive

Implicit (AIM) for compositional simulation.

The formulation mode is the solution procedure used to solve reservoir simulation

mass balance equation. There are three types of solution procedures, IMPLICIT, IMPES

(Implicit Pressure Explicit Saturation) and AIM13, 30, 98, 112. The IMPLICIT option is

totally stable, generally allowing for large timesteps and used for difficult (high

throughput ratio) reservoir problems such as water coning. It is robust and efficient for

black oil reservoir runs while its efficiency is limited by numerous components when

used for compositional runs. IMPES is potentially unstable, faster than IMPLICIT and

less sustainable to dispersion problems, it is commonly used for easy problems (cells

where the solution is changing slightly) and small timesteps simulation studies such as



- 21 -

history matching. AIM formulation mode is between the fully Implicit and the IMPES

solution methods and it has the advantages of both the IMPLICIT and IMPES and neglect

their disadvantages. The AIM formulation permits grid cells in difficult regions to be

solved with the IMPLICIT method, while cells in the easy regions are solved by the

IMPES method. With ECLIPSE compositional simulator AIM is the default mode.

Consequently, the reservoir problem investigated in this research is how to

simultaneously use both black oil and compositional simulation to better improve

reservoir description and fluid phase behavior so that the simulated model will be able to

mimic the actual reservoir response and thereby better quantify the uncertainty associated

with reservoir model performance prediction.

1.4.1 Black Oil Simulation

When the hydrocarbon fluid phases are distinct such that there is negligible mass

transfer between the liquid and gaseous phases a black oil simulation is applied to

simulate the reservoir process. With black oil simulation there is no need to separate the

hydrocarbon fluid into individual components for reservoir characterization. The fluids in

black oil runs are oil, water and gas.

In black oil modeling, reservoir fluid Pressure-Volume-Temperature (PVT)

properties are generated as a function of saturation pressure. This is because the black oil

model is used for reservoir simulation under the assumption that reservoir fluid properties

are strong functions of pressure. Therefore PVT pressure cell experiment and PVT

correlations are commonly used to obtain reservoir fluid PVT properties. In pressure cell



- 22 -

experiments, PVT properties are derived using either Constant volume depletion (CVD)

or constant composition expansion (CCE) PVT experiments. Black oil model quality can

be improved by using finer pressure intervals during a PVT experiment. On the other

hand, PVT correlations are derived and used in a given oil province with similar oil

characteristics.

In a given reservoir simulation and for every timestep the outlined stages in figure

1.9 occur depending on whether the black oil or the compositional simulation is in use.

BOS fluids: oil, gas, and water CS. fluids: HC components and water

Figure 1.9: Black Oil vs. Compositional Simulation Processes

Black Oil Simulation(BOS)

Compositional Simulation(CS)

Flow equation solution for each cell subject to material

balance.

Flow equation solution for each cell subject to material

balance

PVT data lookup fromsupplied tables

Iterative solution of cubic equation of state for eachcomponent in each cell

Iterative flash of componentmixture to equilibrium

conditions for each cell



- 23 -

1.4.2 Compositional Simulation (CS)

Where significant mass transfer exists between hydrocarbon liquid and gas phases

the appropriate way of modeling the reservoir process is to use a compositional

simulator38, 41, 98, 101, 153, 162. Compositional model is generally used during reservoir

simulation when the oil formation volume factor is greater than two101, 153. In a

compositional model we utilize more than two hydrocarbon components. The fluids in

CS runs are the hydrocarbon components (C1, C2, … Cn) and brine water. It is used for

volatile oil reservoir, gas condensate reservoir, gas injection, solution-gas, and gas-cap

drive reservoir simulation studies. It is vital to use a compositional model when the

reservoir pressure decline is significant and fluid properties vary from one location in the

reservoir to another location.

Reservoir processes with compositional effect are commonly encountered in

volatile oil and gas condensate reservoirs and gas injection recovery mechanisms

(enhanced oil recovery)92, 141, 152, 153. In CS, the hydrocarbon fluids are described using

hydrocarbon components. The number of components for flash calculation varies from

four to ten depending on the simulation process objective and end use of the hydrocarbon

fluid. In CS, reservoir fluid properties are function of pressure and composition; as a

result a continuous equation or function is required to describe the fluid.

A cubic Equation of State, EoS, preferably four parameters EoS, is used to

characterize the hydrocarbon fluids. The EoS generates the phase fugacities and Z-factors

which are used to determine inter-phase equilibrium and fluid densities. Some of the

cubic equations of state available in the literature are:



- 24 -

Original EoS

Van dal Waal VDW

Two Parameters EoS:

Peng-Robinson PR

Redlich-Kwong, RK

Soave-Redlich-Kwong SRK

Zudkevitch-Joffe-Redlich-Kwong ZJ

Three Parameters EoS:

Clausius

Schmidt-Wenzel

Four Parameters EoS:

Lawal-Lake-Silberberg LLS

Himpan-Danes-Gaena

Each of these respective cubic EoS can be written in the generalized form such as

equation 7:

( )( ) ( )

( )[ ]1

11

70

2210

212

21211

212

012

23

++−=

+−−+−=

−−+=

−−−−−−−−−−−−−−−−−−−−−−−−−=+++

BBmmABEBmmBmmmmAE

BmmEWith

EZEZEZ



- 25 -

The difference in the aforementioned EoS is the m1 and m2 fugacity coefficients.

The fugacity coefficients are obtained by equation 8:

( )( ) ( ) ( ) ( )

( )( )

jk

kjjkjk

n

jjj

n

j

n

kjkkj

jjiji

iiiii

Where

AAA

BxB

AxxA

xAWhere

ZBB

BmZBmZ

BB

ABmmABZpxf

δ

δ 2/1

1

1 1

1

2

21

1

8

1ln2

ln/ln

−=

=

=

=Σ

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

−+⎟⎟⎠

⎞⎜⎜⎝

⎛++

⎟⎠⎞

⎜⎝⎛ −Σ

−+−−=

∑

∑∑

∑

=

= =

is the binary interaction coefficient between hydrocarbon components and

between hydrocarbon and non-hydrocarbon components.

Equations 7 and 8 are the mixing rules used in all the available EoS while their

difference is the manner in which EoS A and B parameters are calculated. The A and B

parameters are given by equation 9 and 10 below:

( )

( )

( )

( )jTand

jTwhere

TP

jTB

TP

jTA

b

a

rj

rjbj

rj

rjaj

,

,

10,

9, 2

Ω

Ω

−−−−−−−−−−−−−−−−−−−−−−−−−Ω=

−−−−−−−−−−−−−−−−−−−−−−−−−Ω=

are functions of the reduced temperature, T and acentric factor, w



- 26 -

For example using PR EoS51, 58 we have equation 11.

( ) ( )( )[ ]( )

0,

126992.054226.137464.01, 22/12

bb

rjjjaa

jT

TwwjTo

Ω=Ω

−−++Ω=Ω---------11

In each gridblock phase-equilibrium calculation is performed at the end of each

timestep. The cubic equation is solved to determine the Z-factor and fugacity. Three

density solutions are obtained with the smallest root for liquid and largest root for gas

phase. The fugacities in the liquid and gas phases must be equal (see equation 12) in

order to obtain a system in thermodynamic equilibrium which is vital for the CS process.

( )iii

iViL

xpTff

ff

,,=

=------------------------------------------------------------12

Selected EoS is used to obtain liquid and vapor phases fugacities. And the process

of obtaining liquid and vapor fugacities is commonly referred to as flash vaporization

calculation51, 92, 126, 153.

During the simulation process, equilibrium constants simply referred to as K-

values (Equation 13) for each component is calculated at each timestep to define the

inter-phase equilibria. Each component mole fractions (compositions) in the liquid and

gaseous phases are defined by the equations 14 and 15.



- 27 -

( )

( ) 1511

1411

13

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−+

=

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−+

=

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−=

VKzK

y

VKz

x

xy

K

i

iii

i

ii

i

ii

The summation of K-ratio and calculated liquid and vapor density is used by the

simulation to calculate condensate/liquid droplet in condensate reservoir simulation as

depicted in equation 16.

( ) 161 −−−−−−−−−−−−−−−−−−−−−−−−−⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

V

Lii KKDropletl

l

The advantage of the four-parameter EoS over the two-parameter EoS is that the

two-parameter EoS do not predict liquid properties such as density very well. For

example, the critical compressibility predicted by the following EoS two-parameter EoS

are given in table 1.1 whereas for hydrocarbons Zc is less than 0.29

Table 1.1: Two-Parameters EoS Critical Compressibility Factor

EoS Zc Peng-Robinson 0.307 Redlich-Kwong 0.333 Van der Waals 0.375

However, the two-parameter EoS can be tuned as proposed by Peneloux et. al. to

improve the hydrocarbon liquid property predicted. This tuning is achieved by a process

referred to as the volume shift.



- 28 -

1.5 Problem Definition

It is a known fact that reservoir performance prediction obtained from reservoir

simulation models can not be exact. This is generally accepted industry-wide and

reported by numerous authors3, 15, 21, 27, 28, 59, 63, 75, 77, 85, 89, 93, 125, 134, 144, 103 . There is and

there will be always an associated uncertainty with future production forecast. On the

encouraging side, active research is on-going to address the issue of uncertainty

quantification.

When the future production performance of a reservoir is to be forecasted,

reservoir model of the real reservoir is built and the model is conditioned with observed

data. Once a match is achieved, the model is used to predict future reservoir performance.

The problem with this single-model conventional history-matching is that more than one

combination of the input parameters can match the historical data. This means that

history matching is a non-unique problem28, 59, 89, 144. As a result, the future prediction

obtained with a single matched model contains significant uncertainty, which need, to be

quantified.

From the aforementioned, it is obvious that to reduce the uncertainty associated

with future reservoir performance prediction, more than one model should be constructed

to match with observed history data before carrying out reservoir performance prediction

and quantification of associated uncertainty. All the constructed reservoir models are

history matched such that every model that matches the history data is subsequently used

for reservoir performance prediction and uncertainty quantification. The use of multiple



- 29 -

models is limited by computational cost and hence based on current computational

ability. It is not economical for large reservoir simulation.

Various schools of thought exist over the best way to quantify uncertainties

associated with reservoir performance prediction59, 89, 93, 144. Some believe that generating

a single reservoir description model that is a condition with production data is sufficient

to quantify the uncertainty associated with simulated reservoir performance while others

argue that the most feasible and practical method is to quantify uncertainty to generate

multiple realizations of the reservoir and condition the models with available data as a

better approach to quantify the uncertainty in the simulated reservoir performance. The

first approach is referred to as the deterministic technique while the latter is called the

stochastic reservoir modeling.

In this research, Black Oil Conditioning (BOC) technique is proposed that is

capable of better quantifying uncertainty in reservoir performance simulation. This

method is performed after history matching and it involves simultaneously modeling the

same reservoir using both black oil and composition simulation and, thereafter, condition

the black oil output with the compositional simulation result. The main source of

uncertainty focused on in this research is the uncertainty associated with reservoir

description. The reservoir description parameter of interest is the permeability. The ratio

of vertical to horizontal permeability distribution of both the black oil and composition

simulation was equally perturbed slightly to generate multiple reservoir realizations.

Thereafter, the simulated black oil and compositional model results are minimized using

an objective function algorithm.



- 30 -

The rationale behind the BOC method comes from the fact that the black oil

model is limited in terms of its capacity to provide detailed compositional information

and thus has less ability to describe fluid behavior. Therefore, to better account for the

influence of reservoir description and fluid behavior on reservoir performance this

research provides a method of conditioning the black oil results by the compositional

simulation output. The assumption behind this technique is that all reservoirs have some

element of compositionality in their reservoir fluid.

It is worth stressing that the ability of compositional simulation to describe

reservoir fluid in greater detail than black oil model is because in a black oil simulator,

the PVT properties are function of pressure only and they are derived at given pressure

intervals. On the other hand, in compositional simulation reservoir fluid properties

(density, viscosity, etc) are function of pressure and composition. As a result, a

continuous equation is used to model the fluid behavior. Consequently, black oil model

output is not equal to compositional simulation results.

1.6 Economic Significant of Reservoir Uncertainty Quantification

During the life of a reservoir, the pre-reservoir and post-reservoir performance

evaluations are generally not equal. This is due to inadequate quantification of

uncertainties associated with the reservoir model input parameters and the resulting

composite uncertainty associated with the pre-reservoir performance prediction.

The decision to develop a reservoir is based on the prediction of production

performance following history-matching process. Likewise, in some instances the



- 31 -

decision to obtain additional reservoir measurement data is taken when the uncertainty of

the forecast is great.

This necessity is the reason for accurate quantification of uncertainty associated

with reservoir performance forecast so that projected recovery will be accurately

estimated for economic decisions. These vital reasons underlined the economic

importance of increasing interest to properly quantify the uncertainties associated with

reservoir performance simulation.



- 32 -

CHAPTER II

METHODS FOR UNCERTAINTY QUANTIFICATION

2.1 Literature Review

Reservoir engineers believe in the existence of uncertainty associated with

reservoir simulation prediction following history matching. However, the techniques for

uncertainty quantification have been an area of active debate and increasing research

activities. It has led to a number of studies focusing on statistical and non-statistical

uncertainty quantification methods.

The uncertainties in reservoir performance simulation can be divided into two: (1)

the uncertainty associated with the model individual input parameters, and (2) the

composite/total uncertainty associated with the reservoir simulation output such as

cumulative oil production. The composite uncertainty is a consequence of the uncertainty

associated with the input parameters and the numerical model.

A number of methods have been reported for quantifying uncertainty associated

with input parameters as well as the resulting total uncertainty in the reservoir simulation

output3, 15, 21, 27, 28, 31, 39, 59, 60, 61, 63, 77, 85, 89, 93, 97, 100, 103, 107, 125, 134, 144, 148. The standard

principle common to all the techniques is to reduce uncertainty in the input parameter by

conditioning the model with observed history data (i.e. field measured oil, gas and water

production rates, gas-oil-ratio and reservoir pressure). This principle is a sound approach

because the historical data are direct responses of the actual reservoir that responds

according to the actual parameters. It is these actual reservoir parameters that history

matching tries to estimate.



- 33 -

The process of constraining reservoir model with historical data is referred to as

history matching8, 10, 15, 20, 22, 24,25, 27, 32, 34, 37, 62, 67, 86, 89, 96, 108, 109, 121, 131, 132, 133, 137, 138, 142, 144,

146, 151, 160, 164, 165, 166, 168. History matching involves the determination of a set of reservoir

parameters that will make the model output as close as possible to the observed history

data. There are two areas of interest in history matching. Firstly, the different approaches

for constructing reservoir models for history matching and secondly the varied methods

for generating an appropriate misfit algorithm to calculate the difference between the

model data and the historical data.

The first report on history matching was by Kruger in 1961142. Kruger

acknowledges the need for simulation calculated pressure to be equivalent to the actual

field pressure and introduced an approximate adjustment factor for each grid. This idea

was modified by Jacquard142 with an electric analyzer that was used to model analog

reservoirs. He reported an agreement between electric resistance-capacity network and a

reservoir model. With this method it is theoretically feasible to determine spatial

variation of reservoir properties. Jacquard and Jain142 reported an approach based on

steepest descent method with a two-dimensional model and successfully applied to

theoretical reservoirs. They142 suggested that dividing reservoirs into zones and having a

longer period of historical data would yield better agreement during history matching.

Dupuy142 extended the work of Jacquard and Jain using theoretical reservoir. From his

investigation, he reported that in reservoir history matching, whereby, one has the

knowledge that parameter perturbation was not possible, using the least-square error

criteria is not enough standard for misfit matching. He suggested the use of some fudge



- 34 -

factor in addition with the least-square error criterion. Jahns142 used the principle of

Jacquard and Jacquard and Jain to determine the effect of perturbation in one zone on all

the other remaining zones and used convolution techniques to estimate the total

individual zones effects as well as generating the values of the reservoir parameters that

is used in subsequent simulation run. He concluded that using regression analysis will

improve the misfit matching. He also noted that both storage factor (фch) and

transmissibility (kh/μ) need adjustment during history matching.

Coats et al. 39 based their work on the aforementioned techniques. They suggested

a random selection of the reservoir parameters values for simulation runs and application

of regression analysis on the simulation results. Coat et al. 39 bounded the resulting

regression analysis solution using linear programming. This procedure yielded good

results in some cases where in some cases it gives extreme values such as negative

storativity and transmissibility. Slater and Durrer142, and Thomas et al. 151 based their

work using the same principles (Gauss-Newton and Gradient methods) to propose a

balanced error-weighted approach that systematically minimizes the misfit between

simulation data and actual field data in order to achieve a reasonable history match.

The aforementioned pioneer investigators acknowledged the fact that building

reservoir model to match historical data will be achieved by parameter modifications.

Also, these earlier investigators made use of non-linear regression methods that are based

on determination of sensitivity coefficients which are the partial derivatives of reservoir

dynamic variables such as reservoir pressure as function of reservoir parameters such as

permeability. The parameter is perturbed at each simulation run in order to evaluate the



- 35 -

sensitivity of the reservoir variable to the parameter that was perturbed. At each time a

reservoir parameter (typically permeability and porosity) is perturbed the full simulation

run is performed. This approach is time intensive and limits the regression method

efficiency.

In the petroleum industry literature two different methods have been recognized

to generate reservoir models for uncertainty quantification. These approaches are

deterministic15, 24, 34, 89, 121, 142, 151, and stochastic48, 68, 78, 86, 89, 94, 133, 142, 144 methods.

Stochastic techniques involves the generation of multiple reservoir model realizations

that will be conditioned by historical data. The advantage of this approach is that it aimed

at minimizing the non-uniqueness of history matching121, 144. By non-unique it implies

that during history matching more than one combination of model input parameters can

match the history data. The disadvantage of building multiple realizations for uncertainty

quantification is that each simulation run can be very expensive especially for modeling

large reservoirs. Therefore, stochastic application is limited by computational cost,

although, it is viewed by many as the most feasible way to quantify uncertainty

associated with reservoir performance prediction59, 63, 89, 92, 99, 100, 121, 144. The use of

streamline simulation could help to reduce the problem of computational time144.

Streamline simulation application is not applicable to all reservoirs, especially for

reservoirs that are highly heterogeneous.

On the other hand, the deterministic approach involves the use of a single

reservoir model for uncertainty quantification. This approach is fast and easy to use but it

is a less accurate method of quantifying uncertainty89. The following uncertainty



- 36 -

quantification methods59: Linear Uncertainty Analysis, Perturbation Methods, and the

Scenario Test Method, are faster and easier to use. These methods quantify the reservoir

uncertainty associated with the performance prediction by using a single reservoir

description model (RDM). The deterministic approach of using one RDM to quantify

uncertainty underestimates the associated uncertainty because it does not recognize the

fact that other RDMs could honor the available data since when one model matches the

observed data, it may have resulted from a compensating error103, 144. This is why more

than one RDM should be used in the quantification of uncertainty since it is well known

that the process of history matching is non-unique144.

The Bayesian technique has been widely used to assess uncertainty in reservoir

parameter15, 37, 59, 61, 62, 102, 132,161, 167, 168. Bayesian method provides a link between a prior

distribution function and posterior probability distribution function through a likelihood

function assuming a continuous probability distribution. A prior distribution function

assesses the uncertainty in a simulator input parameter while the posterior probability

distribution function can be used to quantify the uncertainty associated with individual

input parameter as well as the reservoir model output variable after the model as been

conditioned with observed history data. The use of percentiles of the posterior probability

distribution function (P10, P50, P90 or lower case, most likely, upper case) 31 to evaluate

uncertainty is limited due to computation time involved during the history match. Also

the method of deriving the models is heuristic such that there is no assurance that the

models are equivalent to the probability distribution function.



- 37 -

During uncertainty quantification, a sensitivity study is used to determine the

input parameter contributing the highest influence on the composite uncertainty89, 142, 151.

This is carried out by calculating the rate of change of the model output variable to a

given input parameter. The input parameter with highest gradient is considered to have

the greatest impact on the reservoir simulation model output. This process is referred to

as gradient technique10, 22, 24, 25, 89.

2.2 Definition of Simulation Input Parameter Associated Uncertainty

A reservoir simulation input parameter is the parameter that is entered into a

numerical simulation model so that the model will mimic the actual reservoir behavior.

This input parameter is not usually known with 100% certainty, therefore, it contains

some degree of associated uncertainty. The degree of uncertainty associated with the

input parameter need to be quantified so that the uncertainty associated with the

simulation result can also be determined.

One method that is frequently used to quantify uncertainty in reservoir parameter

is probability distribution function33, 37, 49, 59, 61. Probability distribution describes the

chance of obtaining a parameter value. In reality, the distribution itself may never be

known. In practice an experimental probability distribution is determined, thereafter, we

look for a theoretical distribution that would have produced such a sampling distribution

– curve fitting. For a continuous distribution a probability density f(x) is assigned to each

x such that the probability of a value lying between x + dx is given by f(x)dx. Therefore

the probability of x lying in between y and z is given by equation 17:



- 38 -

( ) ( ) 17Pr −−−−−−−−−−−−−−−−−−−−−=≤≤ ∫z

y

dxxfzxyobabilty

Cumulative probability is when the probability of x is equal to or smaller than a

given value x0 as given by equation 18:

( ) ( ) 18Pr0

0 −−−−−−−−−−−−−−−−−−−−−−=≤ ∫∞−

x

dxxfxxobabilty

After the uncertainty associated with each individual input parameter is

determined by probability density function, they are treated individually (converted into a

cumulative probability distribution) and transformed into a composite uncertainty

(cumulative probability distribution) associated with the simulation model result -

Markov Chain Morte Carlo method.

The limitation of any uncertainty quantification method is how the uncertainty

associated with the input parameter is determined. This effect is higher with external

factors where the practical experience of the engineer is vital in the definition of feasible

parameter range.

One other area of concern is the interaction between the individual input

parameters. The dependence between the model parameter needs to be determined and

quantified. In some cases the relationship between these parameters are nonlinear which

complicate the use of sensitivity analysis to quantify the influence of individual

parameter uncertainty on the composite output uncertainty27. To reduce parameter



- 39 -

interaction effect non-interaction and linear relationship can be assumed with certain

interval that can give reasonable degree of accuracy.

Some of the model input parameters are obtained by experimental measurement

(direct method) while majority of the input data are derived from indirect measurement

such as established correlations27 due to high cost of the direct methods. For the

parameters obtained from linear correlations, their standard deviation from the actual

value can be determined using linear regression analysis that will permit uncertainty

quantification with coefficient of variation.

If the input parameter was measured by experimental procedures, the coefficient

of variation can be reduced by repeated measurements (stochastic approach) using the

same core. Repeated measurements with the same core can be used to reduce the

uncertainties in some parameters such as porosity, capillary pressure and absolute

permeability while it is not possible for relative permeability core measurement due to

hysteresis and the possibility of a change in wetting conditions27. Uncertainties associated

with relative permeability measurements can be quantified by assessing the random

errors in oil and water rates and differential pressure. The uncertainty is highest at the end

point for water relative permeability.

The uncertainties associated with PVT parameters such as formation volume

factor, viscosity, solution gas-oil ratio, and fluid density can be reduced by using an

appropriate PVT cell experimentation procedure and a robust cubic equation of state for

density and fugacity calculations.



- 40 -

2.3 Model Parameterization

During reservoir simulation process the reservoir model is constructed such that it

is conditioned to all available data. The conditioning is achieved by using the model

parameters such as porosity and permeability. The spatial distribution of porosity and

permeability can be parameterized using a number of methods59. These methods include

the use of grid blocks, regions and pilot points.

2.3.1 GRID BLOCKS

Grid blocks are the building blocks of a reservoir model. They can be depicted in

one, two and three dimensions. And they can be in radial, rectangular, and unstructured

shapes grid blocks. These grid blocks are assigned with property values. In general

reservoir grid block values are referred to as parameters. Usually, porosity and directional

permeability are assigned to each active grid block of a reservoir model. The advantage

of grid block parameterization approach is that the model is free of predetermined

knowledge of the reservoir geology. While the drawback is that each grid block is

defined by more than one parameter resulting in numerous parameters to be handled and,

secondly, there is discontinuity in the reservoir parameters from one grid block to the

next block.



- 41 -

2.3.2 Regions

One common method for reducing the number of parameters is by utilizing

homogeneous regions. Region zonation involves the assumption that a given zone of the

reservoir has uniform parameter that is different to the other zone. In most cases regions

can be described as layers. While in some cases layers are divided into genetic units that

represent regions. Under this approach the reservoir is divided into smaller zones in

which the parameters are assumed to have uniform values37, 137.The primary advantage of

the regions approach is the use of fewer parameters to model the reservoir. On the other

hand, the assumption of homogeneous zone results in a boundary between two zones

which is characterized with abrupt changes from one zone to the next. Further, the

predetermined notion of the homogenous nature of each region may be wrong.

2.3.3 Pilot Points

Pilot point method involves the use of prefixed point or master point to construct

smooth variation in porosity and permeability fields throughout the reservoir. This

approach enables continuous variation in heterogeneous reservoirs parameters from one

point to another. These pilot points are few numbers of defined points. The pilot point

approach relies on geostatistical techniques to define the spatial variation in reservoir

parameters starting from the predefined points.



- 42 -

2.4 Objective Function

During reservoir history match the reservoir model is conditioned to the observed

history data34, 37, 59, 86, 89, 151, 166. In order to measure the extent of the conditioning, a

mismatch between the reservoir model output and the history data is quantified. The

mismatch quantification is referred to as objective function. Three types of objective

function algorithms are commonly used to measure mismatch between simulated

reservoir response and observed history data. These algorithms are; Least Square,

Likelihood Function, and Posterior Distribution.

2.4.1 Least Square

The mismatch between reservoir simulated data and observed history data can be

quantified by using sum of squares algorithm. This is achieved by calculating the

difference for each data (e.g. BHP, WCT, GOR) at each time step and squaring the

obtained value before summing them up.

A simplified sum of square algorithm, SoS, is given by equation 19:

19)(1

2 −−−−−−−−−−−−−−−−−−−−−−−−−−−= ∑=

n

i

iobs

isim ffZ

Where i

simf = Reservoir simulated data i

obsf = Observed history data Z = Measurement of the mismatch (SoS)

Robust SoS algorithms are shown in equation 20 and 21:



- 43 -

2012

1

−−−−−−−−−−−−−−−−−−−−−−−⎟⎟⎠

⎞⎜⎜⎝

⎛ −= ∑

=

n

i i

iobs

isim

iff

wn

Zσ

21112

−−−−−−−−−−−−−−−−−−−−⎟⎟⎠

⎞⎜⎜⎝

⎛ −= ∑∑

n

j ij

ijobs

ijsim

ij

n

i

ffw

nnZ

σ

Where n = Number of measurement taken for each variable σ = Reservoir model plus measurement error w = Weighting factor

For the robust SoS algorithm the total number of measurements taken is included

because it is common to have more measured data of one variable such as BHP compare

to another variable. To eliminate the effect of having one variable measured data higher

than the other the algorithm is divided by the number of measurements taken. Also, the

measurements plus the modeling error, w, is used to normalize the SoS. This is a fudge

factor accounting for unbiasedness in measurement such that when it is taken as one it

means that the reservoir simulated data is within the error limit of the historical data.

2.4.2 Likelihood Function

The likelihood function is a measure of how well the simulated data match the

observed history data. If it is assumed that the model plus measurement errors are

independently Gaussian distributed, Bayesian likelihood function can be expressed as

given by equation 22. When the likelihood function is high it means that the model

simulated data match the observed history data.



- 44 -

2

21

)/( ∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

−

i i

obsi

simi dd

cepofσ

--------------------------------------------22

Where

c = Normalization constant

p = Parameter

2.4.3 Posterior Distribution

It is a known fact that history matching is a non-unique process144. As a result,

single solutions such as using least square method and likelihood function will likely

result in inaccurate mismatch estimation. This is due to the fact that during reservoir

history matching, some parameters (e.g. unswept zones permeability) may be insensitive

to the mismatch quantification process, whereas, it is sensitive to the forecasted result. To

reduce this effect, Bayesian posterior probability distribution function is employed by

linking the a prior distribution to the posterior distribution through a likelihood function.

Bayesian posterior distribution is general represented by equation 23:

(p|o) = cf(o|p)f(p)--------------------------------------------------- 23 Where f(p|o) = Bayesian posterior probability distribution f(p) = Bayesian a priori probability distribution f(o|p) = Likelihood function

To apply equation 23, let assume that the reservoir parameter a prior distribution

is Gaussian. Also, if the uncertainty associated with observed data is Gaussian, then the

Bayesian posterior probability distribution is given by equation 24:



- 45 -

( ) ( ) ( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−−+⎟⎟

⎠

⎞⎜⎜⎝

⎛ −= −−

∑ pppi i

obsi

simi vpCvp

ddceopf 1

2

21

/σ

------------24

Where pC = Covariance matrix

pv = Parameter expectation vector

2.5 Model Optimization Process

In reservoir simulation, the model output for example, well pressure is modified

so that the difference between the model result and the actual field data is minimize. The

minimization process is referred to as optimization process and a number of optimization

techniques have been used to achieve the minimization process. One of these techniques

is to manually adjust the model input parameters to achieve a reduction of the mismatch.

A better approach is the use of optimization algorithms which is made possible as a result

of the objective function algorithms. Optimization of the objective function is performed

by using either gradient method or non-gradient techniques.

2.5.1 Gradient Optimization

Gradient optimization has been widely used to optimize objective functions.

Different gradient optimization techniques exist. They include the Steepest descent,

Conjugate gradient, Gauss-Newton, and Dog-leg techniques. Each of these algorithms

can be employed to optimize the objective function. The gradient method involves

calculation of the objective function gradient (i.e. gradient of the solution e.g., well

pressure) with respect to model input parameter10, 24, 25, 32, 89. The limitation of the



- 46 -

gradient optimization technique is the possibility of having been trapped in local minima.

The gradient method is a non-linear optimization algorithm that relies on a single model

for perturbation. It is fast and easy method but of less accuracy.

2.5.2 Non-Gradient Technique

In order to overcome the problem of having been trapped in local minima, global

or non-gradient optimization techniques have been introduced to minimize the objective

function. Examples of the non-gradient techniques are, simulated annealing and genetic

algorithms. These non-gradient methods do not calculate the gradient of the objective

function47, 48, 59, 68, 89, 108, 138. Simulated annealing method optimizes the objective function

by construction large number of model realizations for the optimization process.

Meanwhile, for the genetic algorithm approach, a number of realizations (child

realizations obtained from parent realizations) are generated such that genetic techniques

are used to determine the best matched model which are usually more than one. Non-

gradient methods are computational expensive and of less application in large fields

simulations.

2.5.3 Root Mean Square Match Analysis

Root mean square analysis is used to assess error size after an initial guess is used

and at subsequent iterations. In most reservoir simulation optimization process, a

threshold value that is less than two (see fig. 2.1) is an indication of acceptable match

between the simulation output and the actual field data. The threshold value is obtained



- 47 -

from root means square, RMS, match analysis. RMS provides an average value of the

difference between simulated and observed history data, it is an overall measurement of a

history match. It is defined by equation 25:

WhereNOFRMS 252

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−×

=

N = Total number of observation

OF = Objective Function

RMS sensitivity can be defined by calculating the partial derivatives of the RMS with

respect to individual input parameter. The sensitivity will explain how the RMS will vary

with respect to the perturb parameter and therefore it can be used to determined the most

sensitive parameter during the history matching.

Figure 2.1: Root Mean Square Match Analysis according to Bos, 2002

6

5

4

3

2

1

0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Fo r e c a s t i ng rms

History Matching, rms

Worst forecas

t

Best match

No simplerelationshi

p

Forecasting with calibrated models carries inherent uncertainty! This needs to be quantified.



- 48 -

2.6 Uncertainty Quantification Algorithms

Some of the commonly used uncertainty quantification algorithms in the

petroleum industry are linear uncertainty analysis, probability techniques and Bayesian

methods, and Markov-Chain Monte-Carlo technique.

2.6.1 Linear Uncertainty Analysis

Uncertainty associated with reservoir parameters that can be quantified using

interval mathematics or linear uncertainty analysis include measurement errors and any

parameter that can be generated with more than one technique89. Linear uncertainty

analysis is often used when it is difficult to assign a probability value for the uncertainty

associated with the input or output parameter.

With linear uncertainty analysis, uncertainty associated with an input parameter is

estimated by determining the input parameter range which, in turn, is used to determine

the confidence interval of the model output parameter. In this analysis each parameter has

an upper and lower limit interval that is used to quantify uncertainty. For example, when

porosity values lie between 15% and 35% then the range of possible porosity value

is %1025 ± . In another example, let the most likely value of parameter y be y0 this will

translate to a most likely value x0 for x if we assumed that model error is negligible. In

addition, let the confidence interval on parameter y be y0 yΔ± , applying linear

uncertainty analysis the corresponding confidence interval xΔ on x is given by equation

26:



- 49 -

)(

26)(

xfwhere

yxfx

′

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−Δ′=Δ

is the derivative of the output with respect to the input parameter value.

The application of interval mathematics is limited to uncertainty quantification

where probability method can not be applied. The disadvantage of the interval

mathematics is that the associated total uncertainties are described by one interval.

2.6.2 Probability Uncertainty Quantification

In the petroleum industry probability techniques have widely used to quantify

uncertainty especially if probability distribution of the uncertain parameter is known.

Probability can be defined simply as the likelihood of an event to occur. In a sample of a

large number the probability of having an event is the ratio of the number of times the

event will occur to the total number of samples. This principle is employed in uncertainty

analysis.

For uncertainty quantification, uncertainty associated with a parameter is

quantified using the probability assigned to the parameter. For example, let the

probability that the porosity of a reservoir is 25% be P, it means that in a large number of

core sample analysis, the number of times a porosity value of 25% is obtained with

respect to the total samples is equal to the fraction P.

Using probability to quantify uncertainty in reservoir simulation involve two

stages. The first stage is to describe the uncertainty associated with the model input



- 50 -

parameter with probability distribution. The second stage is to estimate the uncertainty

associated with the model output parameter and with probability distribution. Probability

distribution is generated based on the assumption that a variable is considered to occur

over a certain range. The values of the variable are represented by frequency of

occurrence. A frequency distribution (histogram) derived from N total samples is

transformed into a probability distribution by dividing each frequency with N, thereafter,

a theoretical probability distribution function that can be used to represent the distribution

is plotted using histogram.

For a discrete variable, v, (i.e. v only assume integer values), a probability of p(v)

is associated to each value of v such that the summation of all probabilities will equal

one. If the distribution is continuous, each v will have a probability density f(v) so that

the probability of finding a value that lie between x and x + dx is f(v)dv.

There are two types of commonly used distribution functions49, 59, 90: cumulative

distribution function, CDF, and probability distribution function, PDF. See equations 27

and 28, and figures 2.2 and 2.3 for examples of CDF and PDF, respectively.

( ) ∫∞−

=≤=0

)(0

v

dvvfvvPCDF --------------------------- 27

Equation 27 is a CDF that represents the probability of v 0v≤

dvvfyvxPPDFy

x∫=≤≤= )()( ----------------------- 28

Equation 28 is an example of a PDF representing the probability of v lying between x and

y.



- 51 -

F(v) = probability of V≤ v

Figure 2.2: Cumulative Distribution Function

Some common statistical properties of a CDF are:

Median = F(0.5)

Upper Quartile = F(0.75)

Interquartile range = F(0.5) – F(0.75)

Figure 2.3: Cumulative Distribution Function Statistical Properties

1.0

0.0

F(x)

0.75

0.50

0.25

X Medianx

F(v)

v

1

0



- 52 -

2.6.2.1 First stage: input parameter probability distribution function

The first step is performed by using statistical methods to determine probability

distribution of the input parameter. One of the many statistical methods involves the use

of histogram plot to determine the probability distribution. There are five commonly used

models of distributions which are mostly used to represent frequency distribution

depicted by histogram plot. These models are: Uniform, Normal, Lognormal, Gamma,

and Exponential distributions (see table 2.1).

With the aid of statistical technique, such as a histogram (figures 2.5 and 2.6), the

probability distribution is then used to generate a probability density function, PDF. The

PDF is a useful quantification tool employed by reservoir engineers to quantify

uncertainty associated with reservoir simulation input parameter.

Normal distribution should occur when the parameter value is due to the

summation of more than one independent cause and the representative curve is

symmetrical about the mean value (figure 2.4). Its equation is a function of mean and

variance. On the other hand, most reservoir parameters do not follow normal distribution

patterns. Rather, their logarithm is normally distributed. Table 2.1 depicts the three

common probability distribution functions and their statistical properties.



- 53 -

Figure 2.4: Normal Probability Distribution

Table 2.1: Three types of distribution: Normal, Lognormal and exponential

This method of uncertainty quantification deals with the uncertainty associated with

individual parameters of the probability distributions instead of the distribution. The

method work best when the uncertain parameters are assumed independent. The mean

and variance of the individual parameters are calculated. The weighted average values of

Normal Lognormal Exponential Probability density function f(x)

( )2

2

2 2exp

21

σμ

πσ

−−

x ( )2

2

2logexp

21

σμ

πσ−

−x

x

xλλ −exp

Statistical properties 2σ

μ

=

=

Variancemean

( )

( ) ( )22

2/2

2exp1exp

expσμσ

σμ

+

+

−=

=

Variance

mean

2

1var

1

λ

λ=

=

iance

mean



- 54 -

all possibilities in the parameter probability distribution give rise to the mean. While the

variance represents the weighted average of the squares of differences from the mean.

The mean and variance can be calculated by using three values of a given parameter as

follows:

1. When there is a ten percent chance of occurrence of a smaller value of

the parameter to exist – Lower Case value

2. The Most-Likely value

3. When there is a ten percent chance of occurrence of a higher value to

exist – Upper Case value

When the mean and variance of the data of interest is calculated, for example,

reservoir oil production performance, an estimation of upside potential (10%), 50%, and

downside risk (90%) is determined.



- 55 -

0

100

200

300

400

500

600

0 5 10 1 1 10 2 2 10 2 2 10 2 3 10 2

Gamma Ray Reading, API Units

Figure 2.5: Discrete Histogram plot used to generate PDF

Figure 2.6: Continuous PDF, F(x) = df/dx

f(x)

x0



- 56 -

2.6.2.2 Second stage: output parameter probability distribution function

During reservoir simulation runs the numerical model calculates an output result

based on the input parameters. As a result, uncertainty is propagated through the model

calculation process. The uncertainty associated with the simulation output parameter can

be characterized or represented using probability distribution function that depends on the

input parameters probability distribution function, see figure 2.7.

Plus equal Input parameter 1 Input parameter 2 Model Output uncertainty distribution uncertainty distribution uncertainty Dis.

Figure 2.7: Relationship between input parameters & model result uncertainty

2.6.3 Quantification of Uncertainty using Bayesian Approach

Bayes’ theorem provides a statistical means to obtain posterior probability density

function (PDF) from a priori PDF and a likelihood function15, 37, 59, 61, 62, 132. The a priori

PDF quantify uncertainty associated with model input parameter, while the likelihood

function accounts for the probability that the observed production data would be obtained

irrespective of reservoir description model. On the other hand, the posterior PDF quantify

uncertainty in the parameter when model result has been matched with production data.

Bayesian algorithm for quantifying parameter uncertainty can be represented by equation

29;



- 57 -

f(p|o) = cf(o|p)f(p)--------------------------------------------------- 29 Where f(p|o) = Bayesian posterior probability distribution f(p) = Bayesian a priori probability distribution f(o|p) = Likelihood function

The major difficulty of achieving a good history match model is the problem of

non-uniqueness. The main cause of this problem is the inability to properly estimate the

actual reservoir spatial varying parameters such as porosity and permeability which is

what history matching is trying to achieve. One way of reducing the non-uniqueness is by

constraining the parameter space into smaller units in order to obtain a priori statistical

information about the uncertain parameter. These smaller zones are assumed to be

homogeneous, hence, the parameter space can be reduced into a fewer dimensional space.

In Bayesian uncertainty quantification technique the a priori statistical

information of the unknown reservoir parameter is included in the objective function. The

statistical information such as variance is applied to check that the estimation does not

deviate from the parameter assumed mean value. Therefore, this statistical term, variance,

acts to constrain the parameter space to a small extent that is centered on the parameter a

priori estimation.

The a priori information of the unknown reservoir parameter is obtained from a

given location in the reservoir as a point measurement. This point measurement can be

derived from well testing interpretation, core analysis, or log evaluation.



- 58 -

2.7 Uncertainty Forecasting

After achieving a reservoir model that matched the observed history data by

minimizing the objective function. The matched model is used to forecast future reservoir

performance prediction such as cumulative oil production. The next step is to forecast the

uncertainty associated with the predicted future reservoir performance. In most cases, the

technique used to obtain the matched model is employed to quantify the uncertainty

associated with the future performance prediction. For example, when the objective

function (least square, likelihood function or posterior) is used to obtain a matched model

which is subsequently used for future production prediction then the uncertainty

associated with the prediction can be quantified by perturbing the objective function

around the optimal model. Some of commonly used methods are linearization of the

posterior, genetic algorithm, gradient optimization, and scenario test method.

The scenario approach involved using a single matched model to estimate high

and low predictions around the optimal model. This process involved locally

characterizing the objective function about the optimal model. Also, if more than one

matched model was obtained during the optimization process it follows that multiple

uncertainties will be quantified. Gradient optimization approach involves slight

perturbation of the objective function so as to quantify range of associated uncertainty.

On the other hand, the genetic algorithm involves generating child realizations from

parent realizations and using genetic principles to select more than one best fit model to

forecast associated uncertainty.



- 59 -

The disadvantage of quantifying reservoir prediction associated uncertainty by

locally characterizing the objective function about the optimal reservoir model is that it

undermines the fact that other possible models can exist. Consequently, this method can

lead to underestimation of the actual uncertainty range. One method of preventing this

problem is the use of Markov-Chain Monte-Carlo (MCMC) technique to assess

uncertainties associated with reservoir performance prediction. The Monte-Carlo

technique is carried out by generating new reservoir model from prior model and

estimating the likelihood of the new model. The calculated likelihood is then used as a

weight-factor for subsequent models. The disadvantage of MCMC technique is it may

involve generating a large number of reservoir models before obtaining an acceptable

likelihood value. This is time demanding and computational cost. Another method,

although equally computational expensive, is the use of geostatistical technique to

generate multiple initial reservoir realizations that are condition with history data. After

history matching the range of uncertainty is determined by using all the matched models

to forecast future reservoir performance.

This research is aimed at forecasting uncertainty associated with predicted

reservoir future performance following history matching by constraining black oil

simulation model with compositional model. It is carried out by performing the usual

history matching procedure of objective function optimization to obtain a history match

model with a black oil simulator. Thereafter, the matched model is run simultaneously

using both black oil and compositional simulators. The output of the two simulation

models are optimized using a least square objective function algorithm. The mismatch



- 60 -

between black oil and compositional simulation results is minimized by manually

adjusting their reservoir parameters equally. This process of minimizing the misfit is

employed to determine minimum and maximum deviations between the two models,

which is then used to account for the range of uncertainty that is associated with the

reservoir performance prediction.



- 61 -

CHAPTER III

RESERVOIR PERFORMANCE SIMULATION

3.1 Reservoir Heterogeneity

A vital factor that influence oil and gas reservoir performance is reservoir

heterogeneity. Reservoir heterogeneity occurs at various scales and these scales range

from micros to giga.

3.1.1 Heterogeneity Scale Effect

The giga and mega scales are the largest heterogeneity scales. Reservoir structures

exhibiting this scale size are large sealing faults that control both vertical and horizontal

sweep efficiency, resulting in compartmentalization (flow unit) of reservoirs.

After the mega scale, the next largest scale of heterogeneity is the macro scale.

This scale characterizes the permeability zonation within a genetic unit. The macro scale

heterogeneity influences reservoir sweep efficiency as well as reservoir continuity. It

extends laterally over several feet. Heterogeneity at this scale is likely to have a large

effect on reservoir pressure behavior in the near-well zone.

On the other hand, micro scale, which follows the mega scale on heterogeneity

scale sizes, involves variation between different pore sizes. Reservoir features that exhibit

this scale size have a large impact on residual oil saturation.

It can be inferred from the aforementioned different scales of reservoir

heterogeneity that productivity index may be largely dependent on the prevailing

reservoir heterogeneity. Table 3.1, shown below, outlines the different types of reservoir



- 62 -

heterogeneity and their effect on reservoir continuity, reservoir sweep efficiency and oil

recovery.

Table 3.1: Influence of Heterogeneity Scale

Heterogeneity Influence Reservoir

Heterogeneity

Type

Reservoir

Continuity

Sweep Efficiency

Horizontal Vertical

ROS in

swept zones

Sealing fault

Semi-sealing fault

Non-sealing fault

S

M

M

S

S

S

S

S

Genetic unit

boundaries

Permeability

zonation in

genetic units

S

S

M

S

S

M

Shale in genetic

units

Cross-bedding

M

M

S

M

M

S

Pore types

Texture types

S

S

Open fractures

Tight fractures

S

M

S S

S

M: moderate effect

S: strong effect

ROS: residual oil saturation

Source: Weber et al., SPE paper 19582



- 63 -

3.1.2 Heterogeneity Measurement

According to Lake and Jensen, 1989, reservoir heterogeneity can be defined as

“Heterogeneity is the quality of the medium which causes the flood front, the boundary

between the displacing and displaced fluids, to spread as the displacement proceeds. For

a homogeneous medium, rate of spreading is zero. As the degree of heterogeneity

increases, the amount of spreading increases’. In addition, Lake and Jensen, classified

reservoir heterogeneity measurements into three types:

1. Static Measurement using Correlation (a). This involves measurement of

reservoir heterogeneity in which reservoir rock samples are taken as

independent data belonging to a given population and the spatial relationship

between the samples is neglected. The methods that utilizes this approach are:

a. Dykstra-Parsons coefficient

b. Lorenz coefficient

c. Coefficient of variation

2. Static Measurement using Correlation (b). This category is similar to the first

except that determination of heterogeneity is a function of measured rock

samples and qualitative evaluation of spatial correlation. The correlation

between one well to another enables the estimation of the interwell zone

reservoir properties. Examples of methods belonging to this category are:

a. Capillary pressure curve

b. Polasek and Hutchinson’s heterogeneity factor



- 64 -

3. Dynamic Measurement of Heterogeneity. The dynamic method involves

estimation of reservoir heterogeneity from the flow of fluid. This implies that

the well has to be producing before this measurement can be obtained. Some

of the methods that utilizes the dynamic approach are:

a. Dispersivities (Autocorrelation)

b. Channeling factor (e.g. Koval)

The Dykstra-Parsons coefficient method is widely used to assess reservoir

heterogeneity. Dykstra and Parsons, 1950, calculated the Dykstra-Parsons coefficient by

using minipermeameter measurements. The Dykstra-Parsons coefficient is an indicator of

permeability variations. It involves measurement of permeability at half-foot intervals of

core samples to calculate permeability and assigned probability values to the permeability

data before ranking the permeability in decreasing magnitude. Thereafter, a log-normal

plot of the permeability and assigned probability is made. The plot best fit straight line is

used to estimate 84th percentile permeability, K0.84 and the median permeability, K0.50.

The Dykstra-Parsons coefficient method is depicted in figure 3.1 and is calculated with

equation:

50.0

84.050.0

KKK

CDP−

= ---------------------------------------------- 30



- 65 -

Figure 3.1: Dykstra-Parsons Coefficient Method

In this study, reservoir permeability is taken as the most vital reservoir property

that control flow of fluid and reservoir heterogeneity is assessed by considering

permeability distribution because permeability variation is a good indicator of reservoir

heterogeneity. As a result, during the reservoir simulation process a base case reservoir

simulation model is perturbed by modifying the model permeability to generate multiple

realizations. These multiple realizations reservoir performance predictions depict the

influence of permeability variation or reservoir heterogeneity on oil and gas recovery.

The advantage of proper estimation of reservoir heterogeneity (permeability) is that

realistic measurement of heterogeneity reduces history matching time during reservoir

simulation.



- 66 -

3.2 Reservoir Simulator

A reservoir simulator is a mathematical model of a system that is simply an

equation which relates the behavior of the system, expressed in terms of observable

variables, to some parameters which describe the system. These equations are frequently

described as physical laws. Examples of mathematical models applied to petroleum

reservoirs are material balance equation and decline curve analysis. These models are

very useful in conducting analytical reservoir performance evaluation but because of the

simplifying assumptions, they are of less use for detailed reservoir description purposes.

As a result, a more detailed mathematical model is constructed by subdividing the

reservoir into small volume elements, referred to as grid, and applying the laws of mass

conservation and fluid flow to each grid. By letting the elements tend to zero volume, the

equations for movement of fluid in a porous medium can be constructed. The resulting

equations are non linear differential equations which are almost always too difficult to

solve analytically. As a result, approximations are made in order to solve the equations at

discrete points in space and time and it is this discretization which leads to the

requirement to solve large linear matrix systems. The discretized partial differential

equation is referred to as numerical model, which is easier to solve.

A simulator or numerical model can be described as a series of numerical

operations whose results represent the reservoir behavior. A simulator can be referred to

as a tool for integrating all of the factors that influence reservoir production and it is



- 67 -

basically solutions to conservation equations that represent physical laws. According to

Lake, 1989, the equations that comprise a simulator can be divided into 2 groups:

1. Conservation of

a. Mass

b. Energy

2. Empirical laws

a. Darcy

b. Capillary pressure

c. Phase behavior

d. Fick

e. Reaction rates

Table 3.2 depicts the equations solved by a typical simulator and table 3.3 and 3.4

show some common data and grid dimensions required for reservoir simulation study,

respectively, while figure 3.2 shows a schematic of simulation grid block.

It is not technically possible to have a single simulator that can represent all

possible cases of flow. As a result, Lake, 1989, classified simulators as follows:

1. Dimensionality (1-D, 2-D and 3-D)

2. Numerical algorithm

a. Finite difference

b. IMPES

c. Implicit

d. Direct solvers



- 68 -

3. Vectorization

4. Physical properties

a. Single-phase (gas or oil)

b. Black oil

c. Compositional

d. Thermal

Table 3.2: Equations Solved by a Reservoir Simulator, Lake 1989



- 69 -

Table 3.3: Common Data Required for Reservoir Simulation

Property Sources Permeability Pressure Transient Testing, Core Analyses, Correlations Porosity Core Analyses, Well Log Data Structure, Thickness Geologic Maps, Core Analyses, Well Log Data Relative Permeability and Capillary Pressure Laboratory Core Flow Tests Saturations Well Log Data, Core Analyses, Pressure Cores, Log-Inject-

Log, Single-Well Tracer Tests PVT Data Laboratory Analyses of Reservoir Fluid (Formation Volume Factors, Samples, Correlations Gas Solubility, Viscosity, Density)

Figure 3.2: Grid Block

Conservation law is applied on the grid block as follows:

1. Rate in – Rate Out = Rate of Accumulation

2. For each reservoir fluid component (oil, gas and water)

3. In each grid block

In

Out

y

In

Ouz

I Ou x

Grid block



- 70 -

Table 3.4: Common Reservoir Simulator Grid Dimensions



- 71 -

3.3 Reservoir Model Description

The first step in reservoir simulation study is to construct the best possible

reservoir description using all available geologic and engineering data. An accurate

reservoir description is essential to the success of the reservoir simulation study. The

degree of detail or complexity of the reservoir description is a function of the problem

under investigation. Nevertheless, good understanding of the reservoir controls on

production performance is required regardless of the complexity of the simulation

method adopted.

A reservoir description model is used to quantify uncertainty associated with

predicted production variables. The uncertainty assessment accuracy is dependent on

reservoir model validity27, 93. As a result, the model should capture the key uncertainties

associated with the reservoir description model so that acceptable uncertainties in the

production variables can be quantified. Once the reservoir description model has been

constructed the remaining task is primarily to solve a set of differential equations with

respect to saturation and pressure in time and space to calculate reservoir performance.

In this study, reservoir performance simulation of the Society of Petroleum

Engineers (SPE) fifth comparative solution project is investigated79 by using a petroleum

industry standard reservoir simulator – ECLIPSE. The original SPE project involved

simulating a synthetic volatile oil reservoir with black oil and compositional simulators

with different simulator providers. However, this research is focused on using the

ECLIPSE compositional simulator to condition the black oil model so as to quantify the

range of uncertainty associated with the black oil model performance prediction.



- 72 -

The fifth comparative solution reservoir description model is a synthetic reservoir

consisting of three-dimensional, three-phase flow in heterogeneous, single-porosity

reservoir. Capillary forces, gravity, and viscosity are defined by Darcy’s law in terms of

relative permeability. And flow is considered isothermal. The black oil model PVT table

consists of gas-oil capillary pressure versus gas saturation. While solution gas oil ratio,

Rs, oil formation volume factor, Bo, and oil viscosity are defined as a function of oil

pressure (see Appendix A and B for further details). On the other hand, the

compositional model uses a two-parameter Peng Robinson six-components EoS to

characterize hydrocarbon fluid and utilizes equation of state fugacity derived K-values

(Appendix A and B). The K-values were generated internally by the ECLIPSE 300

simulator as the original reservoir fluid expands during natural depletion and WAG

injection scenario, respectively. In both black oil and compositional simulators (i.e.

ECLIPSE 00 and ECLIPSE300) IMPLICIT formulation code is applied.



- 73 -

3.4 First Case Reservoir Simulation

The first test case model objective is to simulate a volatile oil reservoir with black

oil model and, thereafter, condition the model results with compositional simulation

output so as to quantify the uncertainty associated with the reservoir predicted production

performance. The synthetic reservoir consisting of three layers was modeled with 7×7×3

Cartesian grids (figure 3.3). Numerical dispersion problems resulting from the coarseness

of the grid is ignored. A single production well that produced at a maximum oil rate of

12000 STB/D is located in grid block i=7, j=7 and k=3. The well shut-in criteria were

minimum BHP of 1000 psi and a limiting WOR and GOR of 5 STB/STB and 10

MSCF/STB, respectively. The simulation model input data are given in tables 3.5 – 3.7

and it was run for ten years without pressure support. Similarly, a compositional model of

the same volatile oil reservoir description model was constructed in which hydrocarbon

fluids were describe with six components Peng-Robinson characterization, Table 3.8 and

3.9 gives the detailed equation of state parameters used for the compositional simulation

model. The percentage of each six components composing the reservoir oil is given in

Table 3.8. From Table 3.8 it is obvious that the reservoir oil is very light. Appendix B

outlines the ECLIPSE input data file for both black oil and compositional simulation

models.



- 74 -

Figure 3.3: Reservoir model schematic

Table 3.5: Reservoir Layer Data

Layer Thickness

(feet)

Porosity

(fraction)

Horizontal

Perm. (mD)

Vertical Perm.

(mD)

1 20.0 0.3 500.0 50.0

2 30.0 0.3 50.0 50.0

3 50.0 0.3 25.0 25.0

Layer Initial

So

Initial

Sw

Initial

Poil (psia)

Elevation

(feet)

1 0.8 0.2 3984.3 8335

2 0.8 0.2 3990.3 8360

3 0.8 0.2 4000.0 8400

Production Well



- 75 -

Table 3.6: Reservoir Model Data

Grid Dimension Areally:7 x 7 in 3 layers

Water Density 62.4 lb/cuft

Oil Density 38.53 lb/cuft

Gas Density 68.64 lb/cuft

Water Compressibility 3.3 x 10-6 psi-1

Rock Compressibility 5.0 x 10-6 psi-1

Water Formation Volume Factor 1.00 RB/STB

Water Viscosity 0.70 cp

Reservoir Temperature 160 oF

Separator Conditions (Flash Temperature

and Pressure)

60 oF

14.7 psia

Reservoir Oil Saturation Pressure 2302.3 psia

Oil Formation Volume Factor (above

bubble point pressure)

-21.85 x 10-6 RB/STB/PSI

Reference Depth 8400.0 ft

Initial Pressure at Reference Depth 4000.0 psia

Initial Water Saturation 0.20

Initial Oil Saturation 0.80

Areal Grid Block Dimensions 500 ft x 500 ft

Reservoir Dip 0

Trapped Gas, Corresponding to Initial Gas

Saturation

20%

Wellbore Radius 0.25 ft

Well KH 10000.0 md/ft

Well Location; Grid Cell Center Production well: I = 7, J = 7

(Completed in Layer 3)

WAG Injection well: I = 1,

J=1 (Completed in Layer 1)



- 76 -

Table 3.7: Data for Relative Permeability and Capillary Pressure

Sw Pcow Krw Krow

0.2000 45.0 0.0 1.0000

0.2899 19.03 0.0022 0.6769

0.3778 10.07 0.0180 0.4153

0.4667 4.90 0.0607 0.2178

0.5556 1.80 0.1438 0.0835

0.6444 0.50 0.2809 0.0123

0.7000 0.05 0.4089 0.0

0.7333 0.01 0.4855 0.0

0.8222 0.0 0.7709 0.0

0.9111 0.0 1.0000 0.0

1.000 0.0 1.0000 0.0

Liq. Sat. Pcgo Krlig Krg

0.2000 30.000 0.0 1.0000

0.2889 8.000 0.0 0.5600

0.3500 4.000 0.0 0.3900

0.3778 3.000 0.0110 0.3500

0.4667 0.800 0.0370 0.2000

0.5556 0.030 0.0878 0.1000

0.6444 0.001 0.1715 0.0500

0.7333 0.001 0.2963 0.0300

0.8222 0.0 0.4705 0.0100

0.9111 0.0 0.7023 0.0010

0.9500 0.0 0.8800 0.0

1.000 0.0 1.0000 0.0

Residual oil to gas flood = 0.15



- 77 -

Critical gas saturation = 0.05

Table 3.8: Compositional Fluid Description

Reservoir Fluid Composition (Mole Fraction)

C1 0.50

C3 0.03

C6 0.07

C10 0.20

C15 0.15

C20 0.05

Table 3.9: Peng-Robinson Fluid Characterization

Component PC (psia) TC (OR) MW Accentric

Factor

Critical

Z

C1 667.8 343.0 16.040 0.0130 0.290

C3 616.3 666.7 44.100 0.1524 0.277

C6 436.9 913.4 86.180 0.3007 0.264

C10 304.0 1111.8 142.290 0.4885 0.257

C15 200.0 1270.0 206.000 0.6500 0.245

C20 162.0 1380.0 282.000 0.8500 0.235

All components have equal omega A & B

0777961.0

4572355.0

=Ω

=ΩOB

OA

Peng-Robinson parameters A and B, for each component are given by

equation 31 and 32, respectively:



- 78 -

( )[ ]

49.0,.....01666.0164423.048503.1379642.049.0,.........26992.054226.137464.0

32

31/11

32

2

22

⟩+−+=

⟨−+=

−−−−−−−−−−−−−−−−−−−−−−−−−−−−Ω=

−−−−−−−−−−−−−−−−−+⎟⎟⎠

⎞⎜⎜⎝

⎛Ω=

wwwwkwwwk

WhereTT

PPB

TTKTT

PPA

CC

OB

CCC

OA

With the exception of the component below all binary interaction

coefficients are zero.

Interaction between

C1 and C15 = 0.05

C1 and C20 = 0.05

C3 and C15 = 0.005

C3 and C20 = 0.005

Peng-Robinson EoS was used to determine fluid densities at separator conditions.



- 79 -

3.5 Second Case Reservoir Simulation

The synthetic black oil reservoir of section 3.4 was modified by adding one WAG

injection well that is located at grid block i=1, j=1 and k=1 see Figure 3.4 and 3.5. The

reservoir is produced under natural drive mechanism at 12000 STB/D for two years

which allows the average reservoir pressure to decline rapidly below the initial saturation

pressure. The reservoir oil initial saturation pressure is 2300 psia. The WAG injection

scheme starts after two years of natural production raising the reservoir average pressure

from the natural depletion state to minimum miscibility pressure condition. The reservoir

oil minimum miscibility pressure is in the range of 3000 to 3200 psia. WAG injection

was one year cycle of alternating water injection followed by an enriched methane

solvent at maximum injection BHP of 10,000 psia, water rate of 12,000 STB/D and gas

rate of 12,000 MSCF/D. Table 3.10 depicts the injectant solvent composition. The

synthetic reservoir was simulated with a black oil simulator (ECLIPSE100) and was run

for 12 years. Figures 3.6 and 3.7 depict the reservoir oil saturation at the beginning and

end of the simulation period. As in Section 3.4, the black oil model result was condition

with compositional simulation model of the same reservoir description to determine

uncertainty in reservoir performance prediction.



- 80 -

Table 3.10: Injection Gas Composition

Injection Gas/Solvent Composition (Mole Fraction)

C1 0.77

C3 0.20

C6 0.03

C10 0.00

C15 0.00

C20 0.00

Figure 3.4: Reservoir Model Cross-section Schematic

Layer 1

Layer 2

Layer 3

WAG injection

0.3

0.3

0.3

20

30

50

500

50

200

500

50

200

50

50

25

ф H Kx Ky Kz

Oil production



- 81 -

Figure 3.5: Second Case Reservoir Model Schematic

Figure 3.6: Oil Saturation at Time Zero

Production Well

InjectionWell



- 82 -

Figure 3.7: Oil Saturation after 12 Years



- 83 -

CHAPTER IV

RESERVOIR PERFORMANCE ANALYSIS

4.1 History Matching and Optimization

In history matching, simulated model output is conditioned to observe history data

by modifying the model parameter so that the simulated data matches the history data. In

this research, the reservoir model that was described in Chapter 3 is taken as the history

matched model. Consequently, Chapter three models for the two scenarios considered are

assumed as the single best model that can reproduce the actual reservoir observed history

data. Then, the next step is to use the matched model for future production forecast. And,

after making the prediction, assessment of uncertainty associated with the forecast is

performed (see figure 4.1).

4.2 Research Methodology

This research proposes a method to quantify uncertainty associated with reservoir

performance simulation by performing the following steps:

1. Obtain a history match black oil model.

2. Construct a compositional simulation model of the matched model.

3. Perturb slightly the black oil and compositional reservoir description

parameters that control the reservoir output (e.g. permeability).

4. Minimize the difference between the two models output by using a statistical

sum of square objective function algorithm. The optimization process is used

to determine lowest and highest deviations of the two models output.



- 84 -

5. The lowest and highest deviations quantify the range of uncertainty

associated with predicted reservoir future performance.

In case of multiple history match models steps 1 through 5 are performed on each

model or a selection of three of the matched models.

For this study reservoir performance simulation of SPE fifth comparative solution

project is evaluated79. In this SPE project a volatile oil reservoir was simulated with both

black oil and compositional simulators. This research is focused on taking the SPE

synthetic reservoir project a step ahead by conditioning the black oil model results with

compositional simulation model output in order to assess uncertainty in the reservoir

performance simulation.

In addition, IMPLICIT formulation code was used in both black oil and

compositional cases (although IMPES and AID formulations are commonly applied in

compositional model). This approach is to reduce the difference between the two models

to mainly how the fluid phase behavior is treated.



- 85 -

Figure 4.1: Black Oil Conditioning Flow Chart

History Matched Model

Black Oil Simulator Compositional Simulator

Black Oil Output Compositional Output

Objective Function Optimization

Uncertainty Quantification



- 86 -

4.3 Observed History Data Duration

The volatile reservoir described in Section 3.4 was used to investigate the

duration of observed history data that is sufficient for a good history match. It should be

noted that duration of observed history will vary from one reservoir to another. The

variation is a function of reservoir rock and fluid properties, reservoir drive mechanism,

type of production scheme and number and location of wells in the reservoir. In this

investigation, a single producing well located at one corner of the reservoir, which is

perforated in one layer out of three layers that are hydrodynamically connected, is study.

The reservoir is a multiphase flow in heterogeneous single-porosity medium. This

investigation was performed by simulating the base case reservoir (which is assumed as

the real reservoir) for 2, 6, 12, 18, 24, and 48 months (see Tables 4.1, 4.2, 4.3 and 4.4 and

Appendix C). Thereafter, the reservoir description (permeability) was varied from 1%,

10%, 20%, 30%, 75% and 90% of the initial value and run for the same number of

months as in the base case model. The simulated data, BHP, GOR, WCT, and TOP of

both the base case and perturb models were matched as depicted in Figures 4.2 thru 4.9.

From Figures 4.2 to 4.9, it is concluded that for the reservoir under investigation,

observed historical data of 18 months are sufficient for a good history match if the model

is 75% and above close to the actual reservoir. (If the model is between 50 – 70% of the

actual reservoir more than 18 months data is required) This means that a good reservoir

simulation model of the real reservoir will be obtained after 18 months of producing the

actual reservoir (i.e. having 18 months plus of observed historical data for history



- 87 -

matching). Consequently, future reservoir predictions that will be obtained from

calibrated history matched simulation model are reliable for field development.

Table 4.1: Base Case Reservoir Description and Simulation Output

Base Case 6 Months TIME FGOR FPR FWCT

(DAYS) (MSCF/STB) (PSIA)

0 0 3993.75 0 1 0.5728 3981.823 2.28E-06 4 0.5728 3946.034 3.74E-06

13 0.5728 3838.561 5.84E-06 30 0.5728 3635.205 9.35E-06 60 0.5728 3274.432 1.52E-05 90 0.5728 2902.931 2.08E-05

120 0.5728 2529.859 2.60E-05 150 0.527151 2286.751 3.51E-05 180 0.512511 2240.178 4.02E-05

Permx Permy PermZ Layer1 500 500 50 Layer2 50 50 50 Layer3 200 200 25

Table 4.2: 1% Reservoir Description Perturbation

1% TIME FGOR FPR FWCT (DAYS)

(MSCF/STB) (PSIA)

0 0 3993.75 0 1 0.5728 3992.667 1.61E-06 4 0.5728 3989.562 4.67E-06

13 0.5728 3980.926 8.98E-06 30 0.5728 3965.781 1.27E-05 60 0.5728 3940.67 1.55E-05 90 0.5728 3916.537 1.71E-05

120 0.5728 3893.097 1.82E-05 150 0.5728 3870.162 1.89E-05 180 0.5728 3847.622 1.95E-05

Permx Permy PermZ Layer1 5 5 0.5 Layer2 0.5 0.5 0.5 Layer3 2 2 0.05



- 88 -



(MSCF/STB) (PSIA)

0 0 3993.75 01 0.5728 3981.825 4.55E-064 0.5728 3946.014 8.36E-06

13 0.5728 3838.521 1.19E-0530 0.5728 3635.045 1.55E-0560 0.5728 3273.753 2.08E-0590 0.5728 2902.881 2.58E-05

120 0.52391 2577.909 3.73E-05150 0.51046 2366.895 4.65E-05180 0.520919 2273.761 5.14E-05

Permx Permy PermZ Layer1 150 150 15Layer2 15 15 15Layer3 60 60 7.5



(MSCF/STB) (PSIA)

0 0 3993.75 01 0.5728 3981.823 2.44E-064 0.5728 3946.025 4.01E-06

13 0.5728 3838.552 6.16E-0630 0.5728 3635.196 9.67E-0660 0.5728 3274.408 1.55E-0590 0.5728 2902.921 2.10E-05

120 0.5728 2529.849 2.62E-05150 0.523139 2287.649 3.60E-05180 0.509729 2240.616 4.15E-05

Permx Permy PermZ Layer1 450 450 45Layer2 45 45 45Layer3 180 180 22.5



- 89 -

WATER-CUT MATCHING AFTER 2 MONTHS

0

0.000005

0.00001

0.000015

0.00002

0.000025

0.00003

0 1 4 13 30 60

TIME, Days

WA

TER

-CU

T

BASE CASE1%10%20%30%75%90%

Figure 4.2: Two Months Observed History Data Matching

SIX MONTHS WATER-CUT MATCH

0

0.00001

0.00002

0.00003

0.00004

0.00005

0.00006

0 20 40 60 80 100 120 140 160 180 200

TIME, DAYS

WA

TER

-CU

T

BASE CASE1%10%20%30%75%90%

Figure 4.3: Six Months Observed History Data Matching



- 90 -

12 MONTHS WATER-CUT MATCH

0

0.00001

0.00002

0.00003

0.00004

0.00005

0.00006

0.00007

0.00008

0 50 100 150 200 250 300 350 400

TIME, DAYS

WA

TER

-CU

T

BASE CASE1%10%20%30%75%90%

Figure 4.4: Twelve Months Observed History Data Matching


0

0.00002

0.00004

0.00006

0.00008

0.0001

0.00012

0 100 200 300 400 500 600

TIME, DAYS

WA

TER

-CU

T BASE CASE1%75%90%

Figure 4.5: Eighteen Months Observed History Data Matching



- 91 -

24 MONTHS DATA WATER-CUT MATCH

0

0.00002

0.00004

0.00006

0.00008

0.0001

0.00012

0.00014

0.00016

0 100 200 300 400 500 600 700 800

TIME, DAYS

WA

TER

-CU

T BASE CASE1%75%90%

Figure 4.6: Twenty Four Months History Data Matching


0

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003

0.00035

0 200 400 600 800 1000 1200 1400 1600

TIME, DAYS

WA

TER

-CU

T

BASE CASE90%75%

Figure 4.7: Forty Eight Months History Data Matching



- 92 -

10 YEARS PREDICTION WITH 18 MONTHS 75% MATCHED MODEL

0

2000000

4000000

6000000

8000000

10000000

12000000

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, DAYS

CU

M. O

IL P

RO

DU

CTI

ON

, STB

OBSERVED OPTSIMULATED OPT

Figure 4.8: Reservoir Performance Prediction 1

10 YEARS PREDICTION WITH 24 MONTHS 75% HISTORY MATCHED MODEL

0

2000000

4000000

6000000

8000000

10000000

12000000

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, DAYS

CU

M. O

IL P

RO

DU

CTI

ON

, STB

OBSERVED DATASIMULATED DATA

Figure 4.9: Reservoir Performance Prediction 2



- 93 -

4.3.1 Well Testing Interpretation

Once a history matched model is achieved such as the reservoir simulated model

that matched twenty four months observed history data, the next step is to validate the

history matched model using transient pressure analysis. A log-log derivative plot

analysis of pressure changes with respect to superposition time is a proven standard

technique for reservoir behavior interpretation.

As a result, log-log derivative analysis using Eclipse WellTest interpretation

software was used to analysis section 4.1.1 observed history pressure data and the

matched simulation model generated pressure data. The interpretation of each transient

pressure response gave reservoir parameters depicted in table 4.5.

Table 4.5: Transient Pressure Interpretation

Reservoir Parameter History Data Simulated Data Difference

Initial Pressure 3981.82 3981.82 - Skin Factor -7.1034 -7.1033 0.0001 Permeability 10.0579 10.0579 -

From Table 4.5, a validation conclusion is made that the simulation model used to

match twenty four months observed history data is an acceptable representative model of

the real reservoir. See Figures 4.10 and 4.11 for the log-log pressure match of observed

history data and simulated pressure response, respectively.



- 94 -

Figure 4.10: Observed History Data Log-Log Plot

Figure 4.11: History Matched Model Log-Log Plot



- 95 -

4.4 Ultimate Recovery Uncertainty: Natural Depletion

Steps one through step five described in Section 4.2 were used on the

black oil and compositional simulation models of Section 3.4 to forecast the range

of uncertainty associated with the synthetic reservoir ultimate recovery. The

permeability KV/KH ratio of the reservoir third layer was perturbed manually (up

to 100 percent of initial value) in both black oil and compositional simulators as

given in Table 4.6. The perturbed models were used to forecast ten years

production. The difference between black oil generated cumulative oil production

and that of compositional simulator were optimized using sum of square objective

functions given by equation 33. After the optimizations process the lowest and

highest objective function values were selected to define the range of uncertainty

associated with the reservoir performance prediction see Figure 4.12 and data in

Appendix E.

( ) 33... 2 −−−−−−−−−−−−−−−−−−−−−−−−−−= ∑ BOComFO

Table 4.6: Conditioning of Black Oil Simulator with Compositional

HM: history match

COP: cumulative

Oil production.

Optimized COP KV/KH

Confidence Interval

3.56373E+12 0.25 100% 2.20751E+12 0.225 2.28327E+12 0.2 2.37008E+12 0.175 2.34844E+12 0.15 3.60626E+12 0.125 HM 2.52426E+12 0.1 2.55908E+12 0.075 2.67512E+12 0.05 3.3686E+12 0.025

3.03698E+12 0.005 -100%



- 96 -

100% CONFIDENCE INTERVAL: CONDITIONING

7000000

7500000

8000000

8500000

9000000

9500000

10000000

10500000

11000000

11500000

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, DAYS

CUM

. OIL

PRO

D., S

TB

HMKV/KH - 9/40KV/KH - 1/40

Figure 4.12: Black Oil Simulator Forecast after Conditioning



- 97 -

4.4.1 Positive and Negative Confidence Interval Algorithms

The assumed history matched black oil model KV/KH permeability ratios of the

third layer were perturbed until a ratio of 1 was obtained. For each perturbation ratio, the

model was used to forecast future oil recovery and the difference in cumulative oil

production between the model and the history matched model were calculated (Table

4.7). Thereafter, plots of the difference in cumulative oil production vs. KV/KH ratio

were made. The plots were used to derive positive and negative algorithms that could be

used to estimate corresponding cumulative oil production for the reservoir at any given

KV/KH perturbation ratio. Figures 4.13 and 4.14 depict these algorithms.

Table 4.7: Perturbed kv/kh and Corresponding Simulator COP KV ∆KV COP ∆COP KV/KH

1 -24 11471837 -738440 0.005 5 -20 10827802 -94405 0.025

10 -15 10591709 141688 0.05 15 -10 10498428 234969 0.075 20 -5 10452052 281345 0.1

HM: 25 0 10733397 0 HM: 125 50 25 10670168 63229 0.25

100 75 10560330 173067 0.5 150 125 10456699 276698 0.75 175 150 10433452 299945 0.875 200 175 10496487 236910 1

COP: Cumulative Oil Production



- 98 -

POSITIVE CONFIDENCE INTERVAL

y = -2E+06x3 + 3E+06x2 - 825189x + 128930R2 = 0.9963

0

50000

100000

150000

200000

250000

300000

350000

0 0.2 0.4 0.6 0.8 1 1.2

KV/KH

Cha

nge

in C

um. O

il Pr

od. S

TB

Figure 4.13: Positive Confidence Interval Algorithm

NEGATIVE CONFIDENCE INTERVAL

y = 3E+09x3 - 7E+08x2 + 5E+07x - 955130R2 = 0.9968

-800000

-600000

-400000

-200000

0

200000

400000

0 0.02 0.04 0.06 0.08 0.1 0.12

KV/KH

Chan

ge in

Cum

. Oil

Prod

., ST

B

Figure 4.14: Negative Confidence Interval Algorithm



- 99 -

In all the previous perturbation, the KV/KH adjustment was carried out using the

third layer. Perturbation was performed only on the third layer after cross-section

examination of the reservoir which revealed that the layer will have significant influence

on recovery. To validate this point, KV/KH of the reservoir first and second layers were

perturbed in addition to the third layer and each new realization was used to make

prediction. The total oil recovery and field water cut data (Figures 4.15 and 4.16,

Appendix C) were plotted to define the range of associated uncertainty. From Figure

4.15, the range of associated uncertainty with ten years cumulative oil production is from

10.1 MMSTB to 10.75 MMSTB and this is equivalent to only when third layer KV/KH

was perturbed.



- 100 -

BLACK OIL SIMULATION: CUM. OIL PROD. UNCERTAINTY RANGE

00.250.5

0.751

1.251.5

1.752

2.252.5

2.753

3.253.5

3.754

4.254.5

4.755

5.255.5

5.756

6.256.5

6.757

7.257.5

7.758

8.258.5

8.759

9.259.5

9.7510

10.2510.5

10.7511

0 250 500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000 3250 3500 3750 4000

Mill

ions

TIME, DAYS

CU

MM

. OIL

PR

OD

., ST

B BASE CASE3RD LY - 1/41ST LY -1/53RD LY - 1/23RD LY - 1, 1ST LY - 1/23RD LY - 11ST LY -1, 3RD LY - 1/23RD LY -1, 1ST LY -1

Figure 4.15: Cumulative Oil Production Uncertainty Quantification



- 101 -

BLACK OIL SIMULATION: WCT UNCERTAINTY RANGE

0.00E+00

5.00E-05

1.00E-04

1.50E-04

2.00E-04

2.50E-04

3.00E-04

3.50E-04

4.00E-04

1 90243 396 546 699 850

1003115

4130

7146

0161

1176

4191

5206

8222

1237

1252

4267

5282

8297

9313

2328

5343

6358

9

TIME, DAYS

WC

T, S

TB/S

TB

BASE CASE3RD LY - 1/41ST LY - 1/53RD LY -1/23RD LY - 1, 1ST LY - 1/23RD LY -11ST LY -1, 3RD LY -1/23RD LY -1, 1ST LY -13RD LY - 3/4

Figure 4.16: Water-Cut Uncertainty Quantification



102

4.5 Ultimate Recovery Uncertainty: Water-Alternate-Gas

The synthetic reservoir producing under WAG scheme of Section 3.5 was allowed

to run for thirteen years. And steps 1 to 5 of Section 4.2 were applied to investigate the

ultimate oil recovery uncertainty. Three new realizations were generated (Appendix C)

which were used to forecast the range of uncertainty associated with the reservoir

performance prediction. These new realizations that quantify the uncertainty range are

high, low and most likely case models as given in Figure 3.17 and Appendix C. From

Figure 4.17 the range of uncertainty associated with predicted total oil recovery is

between 24.65 and 24.68 MMSTB and the three cases recovery are:

High Case: 24,681,318 STB

Most Likely 24,663,478 STB

Low Case: 24,655,026 STB



103

UNCERTAINTY FORECAST

88.5

99.510

10.511

11.512

12.513

13.514

14.515

15.516

16.517

17.518

18.519

19.520

20.521

21.522

22.523

23.524

24.525

1 913 1153 1430 1547 1777 1804 1879 2200 2345 2594 2920 3033 3312 3603 3735 4033 4252 4444

Mill

ions

TIME, DAYS

TOTA

L O

IL P

RO

D.,

STB

UPSIDE CASEDOWNSIDEMOST LIKELY

Figure 4.17: Uncertainty Forecast for WAG Scheme



104

4.6 Justification of the Applied Uncertainty quantification method

It is a known fact that black oil is limited by its inability to generate

comprehensive compositional data. Also, it is well understood that black oil PVT table

consisting of Bo, Rs versus pressure can be used to simulate equivalent compositional

model values of mole fractions x and y (fluid composition) and saturated oil and gas

phase molar densities versus pressure. In addition, simulation mass balance equation is

the same for both black oil and compositional models the only difference between these

models is compositional derived equation of state PVT, which is more detailed than black

oil PVT, which is simpler. Furthermore, in black oil simulation, a simple check of the

total mole fraction is used to determine phase appearance or disappearance while for

compositional simulation Newton-Raphson flash calculation is performed to determine

liquid and vapor (L and V) mole fractions. Therefore, it can simply be said that

compositional simulation is more detailed and more precise than black oil model when

describing reservoir fluid phase behavior. Consequently, compositional simulation model

result can be used to condition black oil model output and the conditioning transformed

into quantification of uncertainty in reservoir performance prediction. This technique of

black oil conditioning is proposed in this research.



105

4.7 Relating Research Approach to Conventional Method

ECLIPSE simulator SIMOPT package is widely used in the industry to quantify

uncertainty associated with reservoir performance prediction. This package was used to

investigate Section 3.4 synthetic reservoir model uncertainty range. The simulation

optimization process was carried out by using only the single assumed history matched

black oil model of Section 3.4 Thereafter, the model permeability distribution was

perturbed slightly so as to quantify uncertainty associated with the reservoir performance

prediction.

Furthermore, linear uncertainty quantification method proposed by Lepine et al.89

was also used to assess the reservoir uncertainty by considering 100% confidence

interval. The resulting uncertainty quantification is given in Table 20 and Figure 4.18. In

the conventional method, KV/KH value corresponding to %100± confidence interval is

used only in the black oil model to forecast production and assessment of uncertainty

associated with the prediction. While the black oil conditioning technique proposed in

this research, objective function (Equation 33) optimization of few multiple realizations

between %100± were used to select the models with minimum and maximum objective

function values. Thereafter, the selected two models were used to forecast oil recovery as

well as to assess uncertainty associated with the reservoir performance prediction. See

Table 4.8 for the objective function optimization results.



106

Table 4.8: Black Oil Conditioning

Optimized COP KV/KH

Confidence Interval

3.56373E+12 0.25 100% 2.20751E+12 0.225 Conditioning2.28327E+12 0.2 2.37008E+12 0.175 2.34844E+12 0.15 3.60626E+12 0.125 HM 2.52426E+12 0.1 2.55908E+12 0.075 2.67512E+12 0.05 3.3686E+12 0.025 Conditioning

3.03698E+12 0.005 -100%

Comparison of Figure 4.18 obtained by conventional uncertainty quantification

method with Figure 4.12 derived from black oil conditioning method proposed in this

study revealed that the proposed technique for assessing uncertainty gives better

quantification of uncertainty associated with reservoir performance prediction.



107

100% CONFIDENCE INTERVAL: CONVENTIONAL

7000000

7500000

8000000

8500000

9000000

9500000

10000000

10500000

11000000

11500000

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, DAYS

CU

M. O

IL P

ROD

., ST

B

HMKV/KH - 1/4KV/KH - 1/200

Figure 4.18: Conventional Linear Analysis of Uncertainty



108

CHAPTER V

CONCLUSIONS AND RECOMMENDATIONS

5.1 Conclusions

The results of this research have shown that uncertainties associated with

reservoir performance simulation are better quantified when reservoir description and

reservoir fluid phase behavior are adequately represented. The following conclusions are

made:

Black oil conditioning technique can be utilized to quantify uncertainty associated

with simulated reservoir performance by generating few reservoir realizations

from a history matched model. This is a cost effective approach to assess reservoir

performance uncertainty.

Two analytical equations are presented for calculating negative and positive

confidence intervals, which can be used to assess oil recovery with varying

reservoir permeability. These equations are functions of reservoir heterogeneity.

18 months history period is sufficient for observed historical data to be utilized

for acceptable history matching if the simulated model is able to mimic the actual

reservoir up to 75% and above.

24 months plus history period is sufficient for acceptable history match if the

simulated reservoir model mimic the real reservoir less than 75%.

It should be noted that the results presented in this research are quite exact (close to ideal

conditions). This is due to the fact all the analysis was carried out utilizing synthetic

reservoir models.



109

5.2 Recommendations

This study is not exhaustive. There are areas requiring further investigation. These

are detailed as follows:

In this research, synthetic reservoir model was used as the assumed history

matched model and for the assessment of uncertainty associated with reservoir

performance prediction. It is suggested that a real reservoir should be used to

perform both the history matching and quantification of uncertainty associated

with the reservoir performance prediction.

Peng-Robinson cubic equation of state was used in the compositional

simulator. Peng-Robinson fails to properly account for hydrocarbon liquid

behavior. As a result, a robust cubic equation of state such as Lawal-Lake-

Silberberg four parameter equation of state should be investigated.

Additional computational cost resulting from simultaneously using

compositional and black oil simulator in the prediction stage after history

matching was not taken into consideration. This should be considered in order

to account for the cost implication of black oil conditioning technique when

compared to conventional uncertainty quantification methods.



110

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114 Peaceman, D.W., Interpretation of Wellblock Pressures in Numerical Reservoir Simulation: Part 3 – Off-Center and Multiple Wells within a Wellblock, SPE 16976, Annual Technical Conference and Exhibition (Sept., 1987).



122

115 Peaceman, D.W., A New Method for Representing Multiple Wells with Arbitrary Rates in Numerical Reservoir Simulation, SPE 29120, SPE Reservoir Engineering (Nov., 1995).

116 Peaceman, D.W., Effective Transmissibilities of a Gridblock by Upscaling - Comparison of a Direct Methods with Renormalization, paper SPE 36722, SPE Journal (Sept., 1997).

117 Peaceman, D.W., A New Method for Calculating Well Indexes for Multiple Wellblocks with Arbitrary Rates in Numerical Reservoir Simulation, SPE 79687, SPE Reservoir Simulation Symposium (Feb., 2003).

118 Peaceman, D.W. and Rachford, H.H., Numerical Calculation of Multidimensional Miscible Displacement, SPE 471, Annual Fall Meeting (Oct., 1960).

119 Pedrosa, O.A. and Aziz, K., Use of a Hybrid Grid in Reservoir Simulation, paper SPE 13507, SPE Reservoir Engineering (Nov., 1986).

120 Phan, V. and Horne, R.N., Determining Depth-Dependent Reservoir Properties using Integrated Data Analysis, paper SPE 56423, presented at Annual Technical Conference and Exhibition (Oct., 1999).

121 Portella, R.C.M. and Prais F., Use of Automatic History Matching and Geostatistical Simulation to improve Production Forecast, paper SPE 53976, presented at SPE Latin America and Caribbean Petroleum Engineering Conference (April, 1999).

122 Pedersen, C. and Thibeau, S., Smørbukk Field: Fluid Modeling and Upscaling Issues to Simulate the Gas Cycling Process in Lower Tilje Formation, paper 83959, presented at Offshore Europe 2003, Aberdeen (Sept. 2003).

123 Philippe, L., Swaby, P.A., Rowbotham, P.S., Dubrule, O. and Haas, A., From Seismic to Reservoir Properties with Geostatistical Inversion, SPE 57476, SPE Reservoir Evaluation and Engineering (Aug., 1999).



123

124 Quandalle, P., Eighth SPE Comparative Solution Project: Gridding Techniques in Reservoir Simulation, paper SPE 25263, presented at SPE Symposium on Reservoir Simulation, New Orleans (March 1993).

125 Qvreberg, O., Damsleth, E. and Haldorsen, Putting Error Bars on Reservoir Engineering Forecasts, paper SPE 20512, Journal of petroleum technology (June, 1992).

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127 Ramey, H.J., Practical Use of Modern Well Test Analysis, paper SPE 5878, presented at 46th Annual California Regional Meeting of the Society of Petroleum Engineers of AIME (April, 1976).

128 Ramey, H.J., Pressure Transient Testing, Journal of Petroleum Technology (July 1982).

129 Ren, W., Mclennan, J.A., Cunha, L.B. and Deutsch, C.V., An Exact Downscaling Methodology in Presence of Heterogeneity: Application to the Athabasca Oil Sands, paper SPE 97874, presented at SPE International Thermal Operations and Heavy Oil Symposium, Calgary (Nov. 2005).

130 Renard, Ph. And Marsily, G., Calculating Equivalent Permeability: A Review, Advances in Water Resources, Vol. 20, Nos 5-6, pp253-278, (1997).

131 Romero, C.E., Carter, J.N., Zimmerman, R.W. and Gringarten, A.C., Improved Reservoir Characterization through Evolutionary Computation, paper SPE 62942, presented at Annual Technical Conference and Exhibition (Oct., 2000).

132 Roggero, F., Direct Selection of Stochastic Model Realizations Constrained to Historical Data, SPE 38731, Annual Technical Conference and Exhibition (Oct., 1997).

133 Roggero, F. and Hu, L.Y., Gradual Deformation of Continuous Geostatistical Models for History Matching, paper SPE 49004, presented at SPE Annual Technical Conference and Exhibition (Sept., 1998).



124

134 Rotondi, M., Nicotra, G., Godi, A., Contento, F.M., Blunt, M.J. and Christie, M.A., Hydrocarbon Production Forecast and Uncertainty Quantification: A Field Application, SPE 102135, presented at SPE Annual Technical Conference and Exhibition (Sept., 2006).

135 Sablok, R. and Aziz, K., Upscaling and Discretization Errors in Reservoir Simulation, paper SPE 93372, presented at SPE Reservoir Simulation Symposium, Houston (Feb. 2005).

136 Saleri, N.O., Reservoir Performance Forecasting: Accelerated by Parallel Planning, SPE 25151, Journal of Petroleum Technology (July, 1993).

137 Schulze-Riegert, R.W., Axmann, J.K., Haase, O., Rian, D.T. and You, Y.L., Optimization Methods for History Matching of Complex Reservoirs, paper SPE 66393, presented at SPE Reservoir Simulation Symposium (Feb., 2001).

138 Sen, M.K., Datta-Gupta, A., Stoffa, P.L., Lake, L.W. and Pope, G.A., Stochastic Reservoir Modeling using Simulated Annealing and Genetic Algorithms, paper SPE 24754, presented at Annual Technical Conference and Exhibition (Oct., 1994).

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142 Slater, G.E. and Durrer, E.J., Adjustment of Reservoir Simulation Models to Match Field Performance, SPE 2983, Annual Fall Meeting (Oct., 1970).

143 Stern D., Practical Aspects of Scaleup of Simulation Models, paper SPE 89032, Distinguished Author Series (Sept. 2005).



125

144 Subbey, S., Christie, M. and Sambridge, M., A Strategy for Rapid Quantification of Uncertainty in Reservoir Performance Prediction, paper SPE 79678, presented at SPE reservoir simulation symposium (Feb., 2003).

145 Suzuki, K and Hewett, T.A., Sequential Scale-Up of Relative Permeabilities, paper SPE 59450, presented at SPE Asia Pacific Conference, Japan (April 2000).

146 Tan, T.B. and Kalogerakis, N., A Fully Implicit Three-Dimensional Three-Phase Simulator with Automatic History-Matching Capability, SPE 21205, SPE Symposium on Reservoir Simulation (Feb., 1991).

147 Tang, D.E. and Zick, A.A., A New Limited Compositional Reservoir Simulator, SPE 25255, Symposium on Reservoir Simulation (Feb., 1993).

148 Tavassoli, Z., Carter, J.N. and King, P.R., Errors in History Matching, SPE 86883, Journal of Petroleum Technology (Sept., 2004).

149 Thibeau, S., Smorbukk field: Impact of Small Scale Heterogeneity on Gas Cycling Performance, paper SPE 75229, presented at SPE/DOE Improved Oil Recovery Symposium (April, 2002).

150 Thiele, M.R., Batycky, R.P. and Blunt, M.J., A Streamline-Based 3D Field-Scale Compositional Reservoir Simulator, SPE 38889, Annual Technical Conference and Exhibition (Oct., 1997).

151 Thomas, L.K, Hellums, L.J. and Reheis, G.M., A Nonlinear Automatic History Matching Technique for Reservoir Simulation Models, paper SPE 3475, Society of Petroleum Engineers Journal (Dec., 1972).

152 Todd, M.R., O’Dell, P.M., and Hirasaki, G.J., Methods for Increased Accuracy in Numerical Reservoir Simulators, SPE 3516, Annual Fall Meeting (Oct., 1971).



126

153 Todd, M.R. and Longstaff, W.J., The Development, Testing and Application of a Numerical Simulator for Predicting Miscible Flood Performance, paper SPE 3484, Journal of Petroleum Technology (July, 1972).

154 Tran, T.T., Wen, X. and Behrens, R.A., Efficient Conditioning of 3D Fine-Scale Reservoir Model to Multiphase Production Data Using Streamline-Based Coarse-Scale Inversion and Geostatistical Downscaling, paper SPE 74708, SPE Journal (Dec. 2001).

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156 Wang, K., Sepehrnoori, K. and Killough, J.E., Ultrafine-Scale Validation of Upscaling Techniques, paper SPE 95774, presented at SPE Annual Technical Conference and Exhibition, Dallas (Oct. 2005).

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158 Weisenborn, A.J., and Schulte, A.M., Compositional Integrated Sub- Surface-Surface Modeling, paper SPE 65158, presented at the SPE European Petroleum Conference (Oct., 2000).

159 Wen, X.-H., Durlofsky, L.J. and Chen, Y., Efficient Three-Dimensional Implementation of Local-Global Upscaling for Reservoir Simulation, paper SPE 92965, presented SPE Reservoir Simulation Symposium, Houston (Feb. 2005).

160 Williams, M.A., Keating, J.F. and Barghouty, M.F., The Stratigraphic Method: A Structured Approach to History-Matching Complex Simulation Models, paper SPE 38014, presented at SPE Reservoir Simulation Symposium (June, 1997).

161 Wills, H.A., Graves, R.M. and Miskimins, J., Don’t Be Fooled by Bayes, paper SPE 90717, presented at Annual Technical Conference and Exhibition (Sept., 2004).



127

162 Young, L.C. and Hemanth-Kumar, K., Compositional Reservoir Simulation on Microcomputers, SPE Petroleum Computer Conference (June, 1989).

163 Wu, X.H. and Parashkevov, R.R., Effect of Grid Deviation on Flow Solutions, paper SPE 92868, presented at SPE Reservoir Simulation Symposium (Jan., 2005).

164 Wu, Z., Reynolds, A.C. and Oliver, D.S., Conditioning Geostatistical Models to Two-Phase Production Data, paper SPE 49003, presented at Annual Technical Conference and Exhibition (1998).

165 Xue, G. and Datta-Gupta, A., Structure Preserving Inversion: An Efficient Approach to Conditioning Stochastic Reservoir Models to Dynamic Data, SPE 38727, Annual Technical Conference and Exhibition (Oct., 1997).

166 Yang, P.H. and Watson, A.T., Automatic History Matching with Variable- Metric Methods, paper SPE 16977, presented at Annual Technical Conference and Exhibition (Sept., 1987).

167 Yang. P.H. and Watson, A.T., A Bayesian Methodology for Estimating Relative Permeability Curve, SPE 18531, Journal of Petroleum Technology (1991).

168 Zhang, P., Pickup, G.E. and Christie, M.A., A New Upscaling Approach for Highly Heterogeneous Reservoirs, paper SPE 93339, presented at SPE Reservoir Simulation Symposium, Houston (Feb. 2005).

169 Zhang, F., Skjervhelm, J.A., Reynolds, A.C. and Oliver, D.S., Automatic History Matching in a Bayesian Framework, Example Applications, paper SPE 84461, presented at Annual Technical Conference and Exhibition (Oct., 2003).



128

APPENDIX A

RESERVOIR PRESSURE-VOLUME-TEMPERATURE (PVT) PROPERTIES

Constant Composition Expansion Derived Pressure-Volume Relations @ 160oF

Pressure, psia

Relative Volume

Liquid Saturation

4800.0 0.9613 1.00004500.0 0.9649 1.00004000.0 0.9715 1.00003500.0 0.9788 1.00003000.0 0.9869 1.00002500.0 0.9960 1.00002302.0 1.0000 1.00002000.0 1.0668 0.90771800.0 1.1262 0.84281500.0 1.2508 0.73751200.0 1.4473 0.62031000.0 1.6509 0.5344500.0 2.9317 0.288314.7 164.088 0.000014.7 @ 60oF 77.5103 0.0100



129

Differential Vaporization of Oil @ 160oF

A. Oil relative volume = oil barrels at specified pressure and temperature per residual oil

barrel at 60oF

B. Gas formation volume factor = gas surface volume at 14.7 psia and 60oF per one

reservoir barrel of gas at given pressure and temperature

C. Solution gas/oil ratio = volume in gas in SCF at given pressure and temperature per

barrel at 14.7 psia and 60oF

Pressure, psia

Oil Relative Volume

Gas Density, G/CC

Oil Density, G/CC

Oil Viscosity, CP

Gas Viscosity, CP GOR

Comp. Factor, Z

4800.0 1.2506 0.1115 0.5628 0.272 0.0170 572.8 0.8663 4500.0 1.2554 0.1115 0.5607 0.265 0.0170 572.8 0.8663 4000.0 1.2639 0.1115 0.5569 0.253 0.0170 572.8 0.8663 3500.0 1.2734 0.1115 0.5527 0.240 0.0170 572.8 0.8663 3000.0 1.2839 0.1115 0.5482 0.227 0.0170 572.8 0.8663 2500.0 1.2958 0.1115 0.5432 0.214 0.0170 572.8 0.8663 2302.3 1.3010 0.1115 0.541 0.208 0.0170 572.8 0.8663 2000.0 1.2600 0.0955 0.549 0.224 0.0159 479.0 0.8712 1800.0 1.2350 0.0851 0.5541 0.234 0.0153 421.5 0.8764 1500.0 1.1997 0.0698 0.5617 0.249 0.0145 341.4 0.8872 1200.0 1.1677 0.0549 0.569 0.264 0.0138 267.7 0.9016 1000.0 1.1478 0.0452 0.5738 0.274 0.0134 222.6 0.9131 500.0 1.1017 0.0222 0.5853 0.295 0.0127 117.6 0.9490 14.7 1.0348 0.0011 0.5966 0.310 0.0107 0 0.9947 14.7 1.0000 0.0011 0.6174 0.414 0.0107 0 0.9947

GOR: Solution Gas-Oil Ratio



130

Constant Composition Expansion: Solvent Gas Pressure-Volume Relations @ 160oF

A. Relative volume = volume per volume of the original charge @ 4800 psia and

160oF

B. Gas formation volume factor = volume of gas at 14.7 psia and 60oF relative to

1 reservoir barrel of gas at specified pressure and temp.

C. Volatile oil in solvent gas = oil in stock tank barrels per MSCF of gas at 160oF

Pressure, psia

Gas Relative Volume

Gas Formation Volume Factor

Gas Density, G/CC

Gas Molecular Weight

Gas Viscosity, CP

Comp. Factor, Z

Volatile Oil in Solvent Gas

4800.0 1.0000 1.7191 0.3072 23.76 0.038 0.8943 0.04500.0 1.0343 1.6620 0.2970 23.76 0.037 0.8672 0.04000.0 1.1053 1.5551 0.2779 23.76 0.034 0.8238 0.03500.0 1.2021 1.4298 0.2555 23.76 0.031 0.7839 0.03000.0 1.3420 1.2809 0.2289 23.76 0.027 0.7501 0.02500.0 1.5612 1.1007 0.1967 23.76 0.023 0.7272 0.02302.3 1.6850 1.0201 0.1823 23.76 0.022 0.7228 0.02000.0 1.9412 0.8853 0.1582 23.76 0.019 0.7233 0.01800.0 2.1756 0.7901 0.1412 23.76 0.018 0.7296 0.01500.0 2.6812 0.6413 0.1146 23.76 0.016 0.7493 0.01200.0 3.4969 0.4913 0.0878 23.76 0.014 0.7818 0.01000.0 4.3477 0.3951 0.0706 23.76 0.013 0.8100 0.0500.0 9.6364 0.1785 0.0319 23.76 0.012 0.8977 0.014.7 363.9816 0.00448 0.0008 23.76 0.011 0.9969 0.014.7 @ 60oF 304.5530 0.00600 0.0010 23.76 0.010 0.9945 0.0



131

PVT table for 4 component solvent – Repressurization data

Pressure, psia

Oil Relative Volume, RB/STB

Gas Formation Volume Factor, RB/MCF

Solution Gas, MCF/STB

Oil Viscosity, CP

Gas Viscosity, CP

Solvent Viscosity, CP

14.7 1.0348 211.416 0.0000 0.3100 0.0107 0.011 500.0 1.1017 5.9242 0.1176 0.2950 0.0127 0.012 1000.0 1.1478 2.8506 0.2226 0.2740 0.0134 0.013 1200.0 1.1677 2.3441 0.2677 0.2640 0.0138 0.014 1500.0 1.1997 1.8457 0.3414 0.2490 0.0145 0.016 1800.0 1.2350 1.5202 0.4215 0.2340 0.0153 0.018 2000.0 1.2600 1.3602 0.4790 0.2240 0.0159 0.019 2302.3 1.3010 1.1751 0.5728 0.2080 0.0170 0.0 2500.0 1.3278 1.1025 0.6341 0.2000 0.0177 0.023 3000.0 1.3956 0.9852 0.7893 0.1870 0.0195 0.027 3500.0 1.4634 0.9116 0.9444 0.1750 0.0214 0.031 4000.0 1.5312 0.8621 1.0995 0.1670 0.0232 0.034 4500.0 1.5991 0.8224 1.2547 0.1590 0.0250 0.037 4800.0 1.6398 0.8032 1.3478 0.1550 0.0261 0.038



132

APPENDIX B

SIMULATION MODEL DATA FILE

The ECLIPSE input data file outlined below for both black oil and compositional model

are the initial models which were assumed as the history matched model.

BLACK OIL MODEL INPUT FILE -- "Fifth Comparative Solution Project: -- Evaluation of Miscible Flood Simulators" -- J.E. Killough, C.A. Kossack -- The 5th SPE Symposium on Reservoir Simulation, -- San Antonio, TX, February 1-4, 1987 -- Case 1B: -- 1. 4-component, solvent model -- 2. Production for 2 years: -- (1). Oil rate = 12000 STB/D, -- (2). Min production BHP = 1000 PSIA -- 3. WAG injection starts at the end of year 2 with 1-year cycle: -- (1). Gas rate = 12000 MSCF/D -- (2). Water rate = 12000 STB/D -- (3). Max injection BHP = 10000 PSIA NOECHO RUNSPEC ------------------------------------------------------------------- TITLE Fifth Comparative Solution Project - Case 1B DIMENS -- NX NY NZ 7 7 3 / OIL WATER



133

GAS DISGAS FIELD SOLVENT MISCIBLE 1 20 NONE / TABDIMS 1 1 40 40 / EQLDIMS 1 20 / WELLDIMS 3 3 1 3 / START 1 JAN 1987 / NSTACK 50 / TRACERS -- NOTRAC NWTRAC NGTRAC NETRAC DIFF 0 0 1 0 DIFF / UNIFOUT UNIFIN GRID ------------------------------------------------------------------- INIT GRIDFILE 0 1 / DXV 7*500 / DYV 7*500 /



134

DZ 49*20 49*30 49*50 / TOPS 49*8325 / PORO 147*0.3 / PERMX 49*500 49*50 49*200 / PERMY 49*500 49*50 49*200 / PERMZ 49*50 49*50 49*25 / RPTGRID / PROPS ------------------------------------------------------------------- STONE SWFN -- SW KRW PCOW 0.2 0 45.0 0.2899 0.0022 19.03 0.3778 0.0180 10.07 0.4667 0.0607 4.90 0.5556 0.1438 1.8 0.6444 0.2809 0.5 0.7000 0.4089 0.05 0.7333 0.4855 0.01 0.8222 0.7709 0.0 0.9111 1.0000 0.0 1.00 1.0000 0.0 / SGFN -- SG KRG PCOG 0.00 0.000 0.0 0.05 0.000 0.0



135

0.0889 0.001 0.0 0.1778 0.010 0.0 0.2667 0.030 0.001 0.3556 0.05 0.001 0.4444 0.10 0.03 0.5333 0.20 0.8 0.6222 0.35 3.0 0.65 0.39 4.0 0.7111 0.56 8.0 0.80 1.0 30.0 / SOF3 -- SO KROW KROG 0.00 0.0 0.0 0.0889 0.0 0.0 0.1500 0.0 0.0 0.1778 0.0 0.0110 0.2667 0.0 0.0370 0.3 0.0 0.0560 0.3556 0.0123 0.0878 0.4444 0.0835 0.1715 0.5333 0.2178 0.2963 0.6222 0.4153 0.4705 0.7111 0.6769 0.7023 0.80 1.0 1.0 / SOF2 -- SO KROW 0.00 0.0 0.0889 0.0 0.1500 0.0 0.1778 0.0 0.2667 0.0 0.3 0.0 0.3556 0.0123 0.4444 0.0835 0.5333 0.2178 0.6222 0.4153 0.7111 0.6769 0.80 1.0 /



136

-- Gas/solvent saturation functions SSFN -- KRG* KRS* 0.0 0.0 0.0 1.0 1.0 1.0 / PVTW -- PREF BW CW VISW CVISW 4000 1.0 3.3D-6 0.7 0 / ROCK -- PREF CR 4000 5.0D-6 / DENSITY -- OIL WATER GAS 38.53 62.40 0.06864 / SDENSITY -- SOLVENT 0.06243 / -- Todd-Longstaff mixing parameter TLMIXPAR 0.7 / -- Miscibility function table MISC 0.0 0.0 0.1 0.3 1.0 1.0 / -- Miscible residual oil saturation tables --SORWMIS -- 0.0 0.05 -- 1.0 0.05 / -- Reservoir dry gas PVT data PVDG -- PG BG VISG 14.7 211.4160 0.0107 500.0 5.9242 0.0127



137

1000.0 2.8506 0.0134 1200.0 2.3441 0.0138 1500.0 1.8457 0.0145 1800.0 1.5202 0.0153 2000.0 1.3602 0.0159 2302.3 1.1751 0.0170 2500.0 1.1025 0.0177 3000.0 0.9652 0.0195 3500.0 0.9116 0.0214 4000.0 0.8621 0.0232 4500.0 0.8224 0.0250 4800.0 0.8032 0.0261 / -- Solvent PVT data PVDS -- PS BS VISS 14.7 223.2140 0.011 500.0 5.6022 0.012 1000.0 2.5310 0.013 1200.0 2.0354 0.014 1500.0 1.5593 0.016 1800.0 1.2657 0.018 2000.0 1.1296 0.019 2302.3 0.9803 0.022 2500.0 0.9085 0.023 3000.0 0.7807 0.027 3500.0 0.6994 0.031 4000.0 0.6430 0.034 4500.0 0.6017 0.037 4800.0 0.5817 0.038 / -- Reservoir live oil PVT data PVTO -- RS PO BO VISO 0.0000 14.7 1.0348 0.310 / 0.1176 500.0 1.1017 0.295 / 0.2226 1000.0 1.1478 0.274 / 0.2677 1200.0 1.1677 0.264 / 0.3414 1500.0 1.1997 0.249 / 0.4215 1800.0 1.2350 0.234 / 0.4790 2000.0 1.2600 0.224 / 0.5728 2302.3 1.3010 0.208



138

3302.3 1.2792 0.235 4302.3 1.2573 0.260 / 0.6341 2500.0 1.3278 0.200 / 0.7893 3000.0 1.3956 0.187 / 0.9444 3500.0 1.4634 0.175 / 1.0995 4000.0 1.5312 0.167 / 1.2547 4500.0 1.5991 0.159 / 1.3478 4800.0 1.6398 0.155 5500.0 1.6245 0.168 / / -- Define tracer associated with reservoir gas TRACER -- NAME PHASE TG GAS / / RPTPROPS / SOLUTION ------------------------------------------------------------------- EQUIL -- DATUM DATUM OWC OWC GOC GOC RSVD RVVD SOLN INIT -- DEPTH PRESS DEPTH PCOW DEPTH PCOG TABLE TABLE METH METH 8400 4000 9000 0 7000 0 1 1* 0 / RSVD -- DEPTH RS 8200 0.5728 8500 0.5728 / -- Tracer associated with free gas TBLKFTG 147*0 / -- Tracer associated with dissolved gas TBLKSTG 147*1 / RPTSOL RESTART=2 /



139

SUMMARY ------------------------------------------------------------------- -- Field vectors FOPR FOPT FWPR FWPT FNPR FNPT FGPR FGPT FWIR FWIT FNIR FNIT FTPRTG FTPTTG FGOR FWCT FPR -- Well vectors WBHP PROD INJW INJG / WWIR INJW / WNIR INJG / WWIT INJW / WNIT INJG / -- Simulator performance vectors



140

PERFORMANCE SEPARATE SCHEDULE ------------------------------------------------------------------- RPTRST BASIC=2 / DRSDT 0 / TUNING 2* 2*0.001 / / 2* 50 1* 2*16 / WELSPECS -- WELL GRUP LOCATION BHP PI 3* XFLOW -- NAME NAME I J DEPTH DEFN PROD G 7 7 8400 OIL 3* NO / / COMPDAT -- WELL -LOCATION- OPEN/ SAT CONN WELL -- NAME I J K1 K2 SHUT TAB FACT DIAM PROD 7 7 3 3 OPEN 1* 1* 0.5 / / WCONPROD -- WELL OPEN/ CNTL OIL WATER GAS LIQU RES BHP -- NAME SHUT MODE RATE RATE RATE RATE RATE PROD OPEN ORAT 12000 1* 1* 1* 1* 1000 / / WECON -- GRUP MIN MIN MAX MAX MAX WORK END -- NAME ORAT GRAT WCT GOR WGR OVER RUN? PROD 1* 1* 0.8333 10.0 1* WELL YES / / -- Production for 2 years TSTEP 2*365 /



141

-- Start WAG cycles -------------------------------------------------------- -- Define WAG injection wells WELSPECS -- WELL GRUP LOCATION BHP PI 3* XFLOW -- NAME NAME I J DEPTH DEFN INJG G 1 1 8335 GAS 3* NO / INJW G 1 1 8335 WAT 3* NO / / -- Complete WAG injection wells COMPDAT -- WELL -LOCATION- OPEN/ SAT CONN WELL -- NAME I J K1 K2 SHUT TAB FACT DIAM INJG 1 1 1 1 OPEN 1* 1* 0.5 / INJW 1 1 1 1 OPEN 1* 1* 0.5 / / -- Define constraints for WAG injection wells WCONINJE -- WELL INJ OPEN/ CNTL SURF RESV BHP -- NAME TYPE SHUT MODE RATE RATE LIM INJW WAT OPEN RATE 12000 1* 10000 / INJG GAS OPEN RATE 12000 1* 10000 / / -- Set solvent faction for gas injector WSOLVENT -- WELL SOLVENT -- NAME CONC INJG 1.0 / / -- Set WAG cycle periods to 1 year WCYCLE -- WELL ON OFF STARTUP MAX CNTL -- NAME TIME TIME TIME TSTEP TSTEP? INJW 365 365 1* 10 YES / INJG 365 365 1* 10 YES / / -- Start with the water injector open and the gas injector shut. WELOPEN



142

INJW OPEN / INJG SHUT / / -- Advance to the start of the first gas injection period and -- open the gas injector. It will start cycling. TSTEP 365 / WELOPEN INJG OPEN / / -- Advance to 20 years TSTEP 17*365 / END ------------------------------------------------------------------- COMPOSITIONAL SIMULATION INPUT FILE -- "Fifth Comparative Solution Project: -- Evaluation of Miscible Flood Simulators" -- J.E. Killough, C.A. Kossack -- The 5th SPE Symposium on Reservoir Simulation, -- San Antonio, TX, February 1-4, 1987 -- Case 1A: -- 1. 6-component, full compositional model -- 2. Production for 2 years: -- (1). Oil rate = 12000 STB/D, -- (2). Min production BHP = 1000 PSIA -- 3. WAG injection starts at the end of year 2 with 1-year cycle: -- (1). Gas rate = 12000 MSCF/D -- (2). Water rate = 12000 STB/D -- (3). Max injection BHP = 10000 PSIA NOECHO RUNSPEC -------------------------------------------------------------------



143

TITLE Fifth Comparative Solution Project - Case 1A DIMENS -- NX NY NZ 7 7 3 / FIELD OIL WATER GAS COMPS 6 / IMPLICIT TABDIMS 1 1 40 40 / EQLDIMS 1 20 / WELLDIMS 3 3 1 3 / START 1 JAN 1987 / UNIFOUT UNIFIN GRID ------------------------------------------------------------------- INIT GRIDFILE 0 1 / DXV 7*500 / DYV



144

7*500 / DZV 20 30 50 / TOPS 49*8325 / PORO 147*0.3 / PERMX 49*500 49*50 49*200 / PERMY 49*500 49*50 49*200 / PERMZ 49*50 49*50 49*25 / RPTGRID / PROPS ------------------------------------------------------------------- NCOMPS 6 / -- Peng-Robinson EOS EOS PR / -- Peng-Robinson correction PRCORR -- Reservoir temperature RTEMP 160 / -- Standard temperature and pressure in Deg F and PSIA STCOND 60 14.7 / -- Component names



145

CNAMES C1 C3 C6 C10 C15 C20 / -- Critical temperatures Deg R TCRIT 343.0 665.7 913.4 1111.8 1270.0 1380.0 / -- Critical pressures PSIA PCRIT 667.8 616.3 436.9 304.0 200.0 162.0 / -- Critical Z-factors ZCRIT 0.290 0.277 0.264 0.257 0.245 0.235 / -- Molecular Weights MW 16.04 44.10 86.18 149.29 206.00 282.00 / -- Acentric factors ACF 0.013 0.1524 0.3007 0.4885 0.6500 0.8500 / -- Binary Interaction Coefficients BIC 0.0 0.0 0.0 0.0 0.0 0.0 0.05 0.005 0.0 0.0 0.05 0.005 0.0 0.0 0.0 / STONE SWFN -- SW KRW PCOW 0.2 0 45.0 0.2899 0.0022 19.03 0.3778 0.0180 10.07 0.4667 0.0607 4.90 0.5556 0.1438 1.8 0.6444 0.2809 0.5 0.7000 0.4089 0.05 0.7333 0.4855 0.01 0.8222 0.7709 0.0



146

0.9111 1.0000 0.0 1.00 1.0000 0.0 / SGFN -- SG KRG PCOG 0.00 0.000 0.0 0.05 0.000 0.0 0.0889 0.001 0.0 0.1778 0.010 0.0 0.2667 0.030 0.001 0.3556 0.05 0.001 0.4444 0.10 0.03 0.5333 0.20 0.8 0.6222 0.35 3.0 0.65 0.39 4.0 0.7111 0.56 8.0 0.80 1.0 30.0 / SOF3 -- SO KROW KROG 0.00 0.0 0.0 0.0889 0.0 0.0 0.1500 0.0 0.0 0.1778 0.0 0.0110 0.2667 0.0 0.0370 0.3 0.0 0.0560 0.3556 0.0123 0.0878 0.4444 0.0835 0.1715 0.5333 0.2178 0.2963 0.6222 0.4153 0.4705 0.7111 0.6769 0.7023 0.80 1.0 1.0 / -- Total composition vs. depth ZMFVD -- DEPTH C1 C3 C6 C10 C15 C20 1000.0 0.5 0.03 0.07 0.2 0.15 0.05 10000.0 0.5 0.03 0.07 0.2 0.15 0.05 / -- Surface densities: only the water value is used



147

DENSITY 1* 62.4 1* / ROCK -- PREF CR 4000 5.0E-6 / PVTW -- PREF BW CW VISW CVISW 4000 1.0 3.3E-6 0.70 0.0 / RPTPROPS / SOLUTION ------------------------------------------------------------------- EQUIL -- DATUM DATUM OWC OWC GOC GOC RSVD RVVD SOLN INIT -- DEPTH PRESS DEPTH PCOW DEPTH PCOG TABLE TABLE METH METH 8400 4000 9000 0 7000 0 1* 1* 0 1 / RPTRST BASIC=2 SOIL SGAS SWAT VOIL VGAS PCOG PCOW PSAT / RPTSOL / SUMMARY ------------------------------------------------------------------- -- Field vectors FOPR FOPT FWPR FWPT FGPR FGPT FWIR FWIT



148

FGIR FGIT FGOR FWCT FPR -- Well vectors WBHP P IWAG / WWIR IWAG / WGIR IWAG / -- Simulator performance vectors PERFORMANCE RUNSUM SCHEDULE ------------------------------------------------------------------- RPTRST BASIC=2 SOIL SGAS SWAT VOIL VGAS PCOG PCOW PSAT / -- Controls for AIM AIMCON 6* -1 / RPTPRINT 0 1 0 1 1 1 0 1 0 0 / -- 1-stage separator conditions SEPCOND -- SEP GRUP STAGE TEMP PRESS -- NAME NAME # SEP G 1 60 14.7 / / -- Define production well WELSPECS



149

-- WELL GRUP LOCATION BHP PI -- NAME NAME I J DEPTH DEFN P G 7 7 8400 OIL / / -- Complete production well COMPDAT -- WELL -LOCATION- OPEN/ SAT CONN WELL -- NAME I J K1 K2 SHUT TAB FACT DIAM P 7 7 3 3 OPEN 1* 1* 0.5 / / -- Associate separator with wells WSEPCOND -- WELL SEP -- NAME NAME P SEP / / -- Define production constraints WCONPROD -- WELL OPEN/ CNTL OIL WATER GAS LIQU RES BHP -- NAME SHUT MODE RATE RATE RATE RATE RATE P OPEN ORAT 12000 1* 1* 1* 1* 1000 / / -- Economic limits: max WOR=5 (WCT=0.8333) and GOR=10 WECON -- GRUP MIN MIN MAX MAX MAX WORK END -- NAME ORAT GRAT WCT GOR WGR OVER RUN? P 1* 1* 0.8333 10 1* WELL Y / / -- Production for 2 years TSTEP 2*365 / -- Start WAG cycles -------------------------------------------------------- -- Define WAG injection well WELSPECS -- WELL GRUP LOCATION BHP PI -- NAME NAME I J DEPTH DEFN IWAG G 1 1 8335 GAS /



150

/ -- Complete WAG injection well COMPDAT -- WELL -LOCATION- OPEN/ SAT CONN WELL -- NAME I J K1 K2 SHUT TAB FACT DIAM IWAG 1 1 1 1 OPEN 1* 1* 0.5 / / -- Define injection gas (solvent) stream WELLSTRE -- STREAM ---------- FRACTION ----------- -- NAME C1 C3 C6 C10 C15 C20 SOLVENT 0.77 0.20 0.03 0.0 0.0 0.0 / / -- Define gas (solvent) injection target WCONINJE -- WELL INJ OPEN/ CNTL SURF RESV BHP -- NAME TYPE SHUT MODE RATE RATE LIM IWAG GAS OPEN RATE 12000 1* 10000 / / -- Define injected gas (solvent) type WINJGAS -- WELL FLUID STREAM -- NAME TYPE NAME IWAG STREAM SOLVENT / / -- Define water injection target WELTARG -- WELL CNTL CNTL -- NAME MODE VALUE IWAG WRAT 12000 / / -- Define WAG well injection scenarios WELLWAG -- WELL WAG FIRST INJ 2ND INJ -- NAME TYPE FLUID PERIOD FLUID PERIOD IWAG T W 365 G 365 / /



151

-- Advance to 20 years TSTEP 18*365 / END



152

APPENDIX C

DATA FOR OBSERVED HISTORY DURATION

SIX MONTHS HISTORY PERIOD SIMULATION

Base Case 6 Months

FWCT TIME (DAYS)

FGOR (MSCF/STB)

FPR (PSIA)

0 0 3993.75 0

1 0.5728 3981.823 2.28E-06

4 0.5728 3946.034 3.74E-06

13 0.5728 3838.561 5.84E-06

30 0.5728 3635.205 9.35E-06

60 0.5728 3274.432 1.52E-05

90 0.5728 2902.931 2.08E-05

120 0.5728 2529.859 2.60E-05

150 0.527151 2286.751 3.51E-05

180 0.512511 2240.178 4.02E-05 Perm x Perm y Perm z

Layer 1 500 500 50 Layer 2 50 50 50 Layer 3 200 200 25



153

1% PERMEABILTY VARIATION FWCT TIME

(DAYS)FGOR (MSCF/STB)

FPR (PSIA)

0 0 3993.75 0

1 0.5728 3992.667 1.61E-06

4 0.5728 3989.562 4.67E-06

13 0.5728 3980.926 8.98E-06

30 0.5728 3965.781 1.27E-05

60 0.5728 3940.67 1.55E-05

90 0.5728 3916.537 1.71E-05

120 0.5728 3893.097 1.82E-05

150 0.5728 3870.162 1.89E-05


Layer 1 5 5 0.5 Layer 2 0.5 0.5 0.5 Layer 3 2 2 0.05



154

10% PERMEABILITY VARIATION

FWCT TIME (DAYS)

FGOR (MSCF/STB)

FPR (PSIA)

0 0 3993.75 0

1 0.5728 3983.67 6.23E-06

4 0.5728 3956.762 1.23E-05

13 0.5728 3884.277 1.71E-05

30 0.5728 3758.267 2.00E-05

60 0.5728 3552.992 2.24E-05

90 0.5728 3360.842 2.42E-05

120 0.5728 3179.709 2.58E-05

150 0.569315 3010.405 2.75E-05


Layer 1 50 50 5 Layer 2 5 5 5 Layer 3 20 20 2.5



155


FWCT

TIME (DAYS)

FGOR (MSCF/STB)

FPR (PSIA)

0 0 3993.75 0

1 0.5728 3981.815 5.51E-06

4 0.5728 3945.993 1.07E-05

13 0.5728 3838.374 1.54E-05

30 0.5728 3634.753 1.94E-05

60 0.5728 3273.146 2.45E-05

90 0.563856 2951.856 2.86E-05

120 0.536526 2704.234 3.40E-05

150 0.521017 2509.644 4.06E-05





156


FWCT TIME (DAYS)

FGOR (MSCF/STB)

FPR (PSIA)

0 0 3993.75 0

1 0.5728 3981.825 4.55E-06

4 0.5728 3946.014 8.36E-06

13 0.5728 3838.521 1.19E-05

30 0.5728 3635.045 1.55E-05

60 0.5728 3273.753 2.08E-05

90 0.5728 2902.881 2.58E-05

120 0.52391 2577.909 3.73E-05

150 0.51046 2366.895 4.65E-05





157


FWCT

TIME (DAYS)

FGOR (MSCF/STB)

FPR (PSIA)

0 0 3993.75 0

1 0.5728 3981.824 2.73E-06

4 0.5728 3946.024 4.54E-06

13 0.5728 3838.55 6.79E-06

30 0.5728 3635.193 1.03E-05

60 0.5728 3274.37 1.60E-05

90 0.5728 2902.92 2.15E-05

120 0.5728 2529.847 2.67E-05

150 0.516892 2289.838 3.76E-05





158


FWCT TIME (DAYS)

FGOR (MSCF/STB)

FPR (PSIA)

0 0 3993.75 0

1 0.5728 3981.823 2.44E-06

4 0.5728 3946.025 4.01E-06

13 0.5728 3838.552 6.16E-06

30 0.5728 3635.196 9.67E-06

60 0.5728 3274.408 1.55E-05

90 0.5728 2902.921 2.10E-05

120 0.5728 2529.849 2.62E-05

150 0.523139 2287.649 3.60E-05





159

12 MONTHS HISTORY PERIOD SIMULATION

Base Case 12 Months

FWCT TIME (DAYS)

FGOR (MSCF/STB)

FPR (PSIA)

0 0 3993.75 0

1 0.5728 3981.823 2.28E-06

4 0.5728 3946.034 3.74E-06

13 0.5728 3838.561 5.84E-06

30 0.5728 3635.205 9.35E-06

60 0.5728 3274.432 1.52E-05

90 0.5728 2902.931 2.08E-05

120 0.5728 2529.859 2.60E-05

150 0.527151 2286.751 3.51E-05

180 0.512511 2240.178 4.02E-05

210 0.503134 2197.59 4.47E-05

240 0.49678 2157.168 4.87E-05

270 0.497577 2118.387 5.19E-05

300 0.500208 2081.004 5.53E-05

330 0.507403 2044.646 5.94E-05





160

1%

FWCT TIME (DAYS)

FGOR (MSCF/STB)

FPR (PSIA)

0 0 3993.75 0

1 0.5728 3992.667 1.61E-06

4 0.5728 3989.562 4.67E-06

13 0.5728 3980.926 8.98E-06

30 0.5728 3965.781 1.27E-05

60 0.5728 3940.67 1.55E-05

90 0.5728 3916.537 1.71E-05

120 0.5728 3893.097 1.82E-05

150 0.5728 3870.162 1.89E-05

180 0.5728 3847.622 1.95E-05

210 0.5728 3825.404 2.00E-05

240 0.5728 3803.458 2.04E-05

270 0.5728 3781.727 2.07E-05

300 0.5728 3760.198 2.10E-05

330 0.5728 3738.862 2.13E-05





161

10%

FWCT TIME (DAYS)

FGOR (MSCF/STB)

FPR (PSIA)

0 0 3993.75 0

1 0.5728 3983.67 6.23E-06

4 0.5728 3956.762 1.23E-05

13 0.5728 3884.277 1.71E-05

30 0.5728 3758.267 2.00E-05

60 0.5728 3552.992 2.24E-05

90 0.5728 3360.842 2.42E-05

120 0.5728 3179.709 2.58E-05

150 0.569315 3010.405 2.75E-05

180 0.554003 2863.75 3.00E-05

210 0.539348 2735.072 3.29E-05

240 0.52601 2621.798 3.61E-05

270 0.519964 2522.127 3.97E-05

300 0.515686 2440.792 4.29E-05

330 0.512769 2373.479 4.60E-05





162

20%

FWCT TIME (DAYS)

FGOR (MSCF/STB)

FPR (PSIA)

0 0 3993.75 0

1 0.5728 3981.815 5.51E-06

4 0.5728 3945.993 1.07E-05

13 0.5728 3838.374 1.54E-05

30 0.5728 3634.753 1.94E-05

60 0.5728 3273.146 2.45E-05

90 0.563856 2951.856 2.86E-05

120 0.536526 2704.234 3.40E-05

150 0.521017 2509.644 4.06E-05

180 0.515848 2371.951 4.65E-05

210 0.523763 2291.692 5.03E-05

240 0.526275 2263.709 5.28E-05

270 0.526073 2245.489 5.45E-05

300 0.524902 2228.419 5.58E-05

330 0.523292 2212.152 5.69E-05





163

30%

FWCT TIME (DAYS)

FGOR (MSCF/STB)

FPR (PSIA)

0 0 3993.75 0

1 0.5728 3981.825 4.55E-06

4 0.5728 3946.014 8.36E-06

13 0.5728 3838.521 1.19E-05

30 0.5728 3635.045 1.55E-05

60 0.5728 3273.753 2.08E-05

90 0.5728 2902.881 2.58E-05

120 0.52391 2577.909 3.73E-05

150 0.51046 2366.895 4.65E-05

180 0.520919 2273.761 5.14E-05

210 0.523066 2244.897 5.43E-05

240 0.522283 2219.799 5.63E-05

270 0.520327 2196.383 5.78E-05

300 0.518105 2174.385 5.90E-05

330 0.516341 2153.621 6.02E-05





164

75%

FWCT TIME (DAYS)

FGOR (MSCF/STB)

FPR (PSIA)

0 0 3993.75 0

1 0.5728 3981.824 2.73E-06

4 0.5728 3946.024 4.54E-06

13 0.5728 3838.55 6.79E-06

30 0.5728 3635.193 1.03E-05

60 0.5728 3274.37 1.60E-05

90 0.5728 2902.92 2.15E-05

120 0.5728 2529.847 2.67E-05

150 0.516892 2289.838 3.76E-05

180 0.504346 2241.437 4.42E-05

210 0.501011 2198.997 4.92E-05

240 0.505036 2158.233 5.32E-05

270 0.508089 2119.061 5.72E-05

300 0.513796 2081.11 6.15E-05

330 0.52436 2044.014 6.66E-05





165

90%

FWCT TIME (DAYS)

FGOR (MSCF/STB)

FPR (PSIA)

0 0 3993.75 0

1 0.5728 3981.823 2.44E-06

4 0.5728 3946.025 4.01E-06

13 0.5728 3838.552 6.16E-06

30 0.5728 3635.196 9.67E-06

60 0.5728 3274.408 1.55E-05

90 0.5728 2902.921 2.10E-05

120 0.5728 2529.849 2.62E-05

150 0.523139 2287.649 3.60E-05

180 0.509729 2240.616 4.15E-05

210 0.500368 2198.168 4.64E-05

240 0.49943 2157.619 5.02E-05

270 0.501218 2118.703 5.36E-05

300 0.505299 2081.102 5.74E-05

330 0.513904 2044.483 6.18E-05





166


Base Case 18 Months TIME FGOR FPR FWCT (DAYS)

(MSCF/STB) (PSIA)

0 0 3993.75 01 0.5728 3981.823 2.28E-064 0.5728 3946.034 3.74E-06

13 0.5728 3838.561 5.84E-0630 0.5728 3635.205 9.35E-0660 0.5728 3274.432 1.52E-0590 0.5728 2902.931 2.08E-05

120 0.5728 2529.859 2.60E-05150 0.527151 2286.751 3.51E-05180 0.512511 2240.178 4.02E-05210 0.503134 2197.59 4.47E-05240 0.49678 2157.168 4.87E-05270 0.497577 2118.387 5.19E-05300 0.500208 2081.004 5.53E-05330 0.507403 2044.646 5.94E-05360 0.521464 2008.809 6.44E-05390 0.542494 1972.788 7.06E-05420 0.573459 1935.676 7.85E-05450 0.613992 1898.323 8.80E-05480 0.65278 1863.844 9.32E-05510 0.699453 1831.338 9.78E-05540 0.752294 1800.404 0.000103



167


(MSCF/STB) (PSIA)

0 0 3993.75 01 0.5728 3992.667 1.61E-064 0.5728 3989.562 4.67E-06

13 0.5728 3980.926 8.98E-0630 0.5728 3965.781 1.27E-0560 0.5728 3940.67 1.55E-0590 0.5728 3916.537 1.71E-05

120 0.5728 3893.097 1.82E-05150 0.5728 3870.162 1.89E-05180 0.5728 3847.622 1.95E-05210 0.5728 3825.404 2.00E-05240 0.5728 3803.458 2.04E-05270 0.5728 3781.727 2.07E-05300 0.5728 3760.198 2.10E-05330 0.5728 3738.862 2.13E-05360 0.5728 3717.704 2.15E-05390 0.5728 3696.712 2.17E-05420 0.5728 3675.878 2.20E-05450 0.5728 3655.194 2.22E-05480 0.5728 3634.655 2.24E-05510 0.5728 3614.246 2.26E-05540 0.5728 3593.975 2.27E-05



168


(MSCF/STB) (PSIA)

0 0 3993.75 0.00E+001 0.5728 3981.824 2.73E-064 0.5728 3946.024 4.54E-06

13 0.5728 3838.55 6.79E-0630 0.5728 3635.193 1.03E-0560 0.5728 3274.37 1.60E-0590 0.5728 2902.92 2.15E-05

120 0.5728 2529.847 2.67E-05150 0.516892 2289.838 3.76E-05180 0.504346 2241.437 4.42E-05210 0.501011 2198.997 4.92E-05240 0.505036 2158.233 5.32E-05270 0.508089 2119.061 5.72E-05300 0.513796 2081.11 6.15E-05330 0.52436 2044.014 6.66E-05360 0.537602 2008.33 7.17E-05390 0.547054 1975.515 7.51E-05420 0.561646 1944.467 7.88E-05450 0.580482 1915.301 8.29E-05480 0.603495 1888.213 8.73E-05510 0.635291 1862.69 9.17E-05540 0.676047 1838.208 9.54E-05



169


(MSCF/STB) (PSIA)

0 0 3993.75 01 0.5728 3981.823 2.44E-064 0.5728 3946.025 4.01E-06

13 0.5728 3838.552 6.16E-0630 0.5728 3635.196 9.67E-0660 0.5728 3274.408 1.55E-0590 0.5728 2902.921 2.10E-05

120 0.5728 2529.849 2.62E-05150 0.523139 2287.649 3.60E-05180 0.509729 2240.616 4.15E-05210 0.500368 2198.168 4.64E-05240 0.49943 2157.619 5.02E-05270 0.501218 2118.703 5.36E-05300 0.505299 2081.102 5.74E-05330 0.513904 2044.483 6.18E-05360 0.528194 2008.354 6.71E-05390 0.549253 1972.143 7.36E-05420 0.578708 1935.317 8.12E-05450 0.601094 1901.543 8.64E-05480 0.632212 1870.326 9.15E-05510 0.677521 1840.711 9.57E-05540 0.725026 1812.442 0.0001



170


Base Case 24 Months TIME FGOR FPR FWCT (DAYS)

(MSCF/STB) (PSIA)

0 0 3993.75 01 0.5728 3981.823 2.28E-064 0.5728 3946.034 3.74E-06

13 0.5728 3838.561 5.84E-0630 0.5728 3635.205 9.35E-0660 0.5728 3274.432 1.52E-0590 0.5728 2902.931 2.08E-05

120 0.5728 2529.859 2.60E-05150 0.527151 2286.751 3.51E-05180 0.512511 2240.178 4.02E-05210 0.503134 2197.59 4.47E-05240 0.49678 2157.168 4.87E-05270 0.497577 2118.387 5.19E-05300 0.500208 2081.004 5.53E-05330 0.507403 2044.646 5.94E-05360 0.521464 2008.809 6.44E-05390 0.542494 1972.788 7.06E-05420 0.573459 1935.676 7.85E-05450 0.613992 1898.323 8.80E-05480 0.65278 1863.844 9.32E-05510 0.699453 1831.338 9.78E-05540 0.752294 1800.404 0.000103570 0.809489 1770.578 0.000108600 0.86998 1741.353 0.000114630 0.955911 1713.052 0.000121660 1.052185 1685.696 0.000129690 1.158652 1659.097 0.000137720 1.272151 1633.254 0.000146



171


(MSCF/STB) (PSIA)

0 0 3993.75 0 1 0.5728 3992.667 1.61E-06 4 0.5728 3989.562 4.67E-06

13 0.5728 3980.926 8.98E-06 30 0.5728 3965.781 1.27E-05 60 0.5728 3940.67 1.55E-05 90 0.5728 3916.537 1.71E-05

120 0.5728 3893.097 1.82E-05 150 0.5728 3870.162 1.89E-05 180 0.5728 3847.622 1.95E-05 210 0.5728 3825.404 2.00E-05 240 0.5728 3803.458 2.04E-05 270 0.5728 3781.727 2.07E-05 300 0.5728 3760.198 2.10E-05 330 0.5728 3738.862 2.13E-05 360 0.5728 3717.704 2.15E-05 390 0.5728 3696.712 2.17E-05 420 0.5728 3675.878 2.20E-05 450 0.5728 3655.194 2.22E-05 480 0.5728 3634.655 2.24E-05 510 0.5728 3614.246 2.26E-05 540 0.5728 3593.975 2.27E-05 570 0.5728 3573.839 2.29E-05 600 0.5728 3553.835 2.31E-05 630 0.5728 3533.944 2.33E-05 660 0.5728 3514.18 2.34E-05 690 0.5728 3494.544 2.36E-05 720 0.5728 3475.035 2.38E-05



172


(MSCF/STB) (PSIA)

0 0 3993.75 01 0.5728 3981.824 2.73E-064 0.5728 3946.024 4.54E-06

13 0.5728 3838.55 6.79E-0630 0.5728 3635.193 1.03E-0560 0.5728 3274.37 1.60E-0590 0.5728 2902.92 2.15E-05

120 0.5728 2529.847 2.67E-05150 0.516892 2289.838 3.76E-05180 0.504346 2241.437 4.42E-05210 0.501011 2198.997 4.92E-05240 0.505036 2158.233 5.32E-05270 0.508089 2119.061 5.72E-05300 0.513796 2081.11 6.15E-05330 0.52436 2044.014 6.66E-05360 0.537602 2008.33 7.17E-05390 0.547054 1975.515 7.51E-05420 0.561646 1944.467 7.88E-05450 0.580482 1915.301 8.29E-05480 0.603495 1888.213 8.73E-05510 0.635291 1862.69 9.17E-05540 0.676047 1838.208 9.54E-05570 0.717177 1814.641 9.91E-05600 0.760561 1791.826 0.000103630 0.805666 1769.518 0.000107660 0.852345 1747.493 0.000111690 0.911639 1725.97 0.000116720 0.980449 1705.046 0.000122



173


(MSCF/STB) (PSIA)

0 0 3993.75 01 0.5728 3981.823 2.44E-064 0.5728 3946.025 4.01E-06

13 0.5728 3838.552 6.16E-0630 0.5728 3635.196 9.67E-0660 0.5728 3274.408 1.55E-0590 0.5728 2902.921 2.10E-05

120 0.5728 2529.849 2.62E-05150 0.523139 2287.649 3.60E-05180 0.509729 2240.616 4.15E-05210 0.500368 2198.168 4.64E-05240 0.49943 2157.619 5.02E-05270 0.501218 2118.703 5.36E-05300 0.505299 2081.102 5.74E-05330 0.513904 2044.483 6.18E-05360 0.528194 2008.354 6.71E-05390 0.549253 1972.143 7.36E-05420 0.578708 1935.317 8.12E-05450 0.601094 1901.543 8.64E-05480 0.632212 1870.326 9.15E-05510 0.677521 1840.711 9.57E-05540 0.725026 1812.442 0.0001570 0.776274 1785.233 0.000105600 0.830215 1758.625 0.00011630 0.891767 1732.56 0.000115660 0.971848 1707.341 0.000122690 1.061685 1682.823 0.000129720 1.159643 1658.905 0.000137



174


BASE CASE 48 MONTHS TIME FGOR FOPT FPR FWCT (DAYS)

(MSCF/STB) (STB) (PSIA)

0 0 0 3993.75 0 1 0.5728 12000 3981.823 2.28E-06 4 0.5728 48000 3946.034 3.74E-06

13 0.5728 156000 3838.561 5.84E-06 30 0.5728 360000 3635.205 9.35E-06 60 0.5728 720000 3274.432 1.52E-05 90 0.5728 1080000 2902.931 2.08E-05

120 0.5728 1440000 2529.859 2.60E-05 150 0.527151 1800000 2286.751 3.51E-05 180 0.512511 2160000 2240.178 4.02E-05 210 0.503134 2520000 2197.59 4.47E-05 240 0.49678 2880000 2157.168 4.87E-05 270 0.497577 3240000 2118.387 5.19E-05 300 0.500208 3600000 2081.004 5.53E-05 330 0.507403 3960000 2044.646 5.94E-05 360 0.521464 4320000 2008.809 6.44E-05 390 0.542494 4680000 1972.788 7.06E-05 420 0.573459 5040000 1935.676 7.85E-05 450 0.613992 5397939 1898.323 8.80E-05 480 0.65278 5725190 1863.844 9.32E-05 510 0.699453 6028294 1831.338 9.78E-05 540 0.752294 6309348 1800.404 0.000103 570 0.809489 6569979 1770.578 0.000108 600 0.86998 6811580 1741.353 0.000114 630 0.955911 7033723 1713.052 0.000121 660 1.052185 7237839 1685.696 0.000129 690 1.158652 7425737 1659.097 0.000137 720 1.272151 7598492 1633.254 0.000146 750 1.409122 7756160 1608.069 0.000157 780 1.552855 7900667 1583.45 0.000167 810 1.709555 8033581 1559.202 0.000178 840 1.870063 8155667 1535.438 0.000189 870 2.025405 8267844 1512.25 0.000201 900 2.182132 8370816 1489.531 0.000212 930 2.336698 8465260 1466.915 0.000225 960 2.480039 8552065 1445.03 0.000236 990 2.611721 8632169 1424.056 0.000248



175

75% TIME FGOR FOPT FPR FWCT (DAYS)


0 0 0 3993.75 0 1 0.5728 12000 3981.824 2.73E-06 4 0.5728 48000 3946.024 4.54E-06

13 0.5728 156000 3838.55 6.79E-06 30 0.5728 360000 3635.193 1.03E-05 60 0.5728 720000 3274.37 1.60E-05 90 0.5728 1080000 2902.92 2.15E-05

120 0.5728 1440000 2529.847 2.67E-05 150 0.516892 1800000 2289.838 3.76E-05 180 0.504346 2160000 2241.437 4.42E-05 210 0.501011 2520000 2198.997 4.92E-05 240 0.505036 2880000 2158.233 5.32E-05 270 0.508089 3240000 2119.061 5.72E-05 300 0.513796 3600000 2081.11 6.15E-05 330 0.52436 3960000 2044.014 6.66E-05 360 0.537602 4311689 2008.33 7.17E-05 390 0.547054 4637873 1975.515 7.51E-05 420 0.561646 4941943 1944.467 7.88E-05 450 0.580482 5225343 1915.301 8.29E-05 480 0.603495 5489564 1888.213 8.73E-05 510 0.635291 5736646 1862.69 9.17E-05 540 0.676047 5969369 1838.208 9.54E-05 570 0.717177 6188958 1814.641 9.91E-05 600 0.760561 6396205 1791.826 0.000103 630 0.805666 6591848 1769.518 0.000107 660 0.852345 6776521 1747.493 0.000111 690 0.911639 6950159 1725.97 0.000116 720 0.980449 7112907 1705.046 0.000122 750 1.055887 7265631 1684.6 0.000128 780 1.137442 7409039 1664.566 0.000134 810 1.223556 7543607 1644.942 0.000141 840 1.315051 7669682 1625.727 0.000148 870 1.425184 7787125 1606.856 0.000157 900 1.534848 7897082 1588.296 0.000165 930 1.652501 8000286 1569.945 0.000173 960 1.77508 8097076 1551.814 0.000181 990 1.89721 8187826 1533.977 0.00019

1020 2.015661 8272940 1516.464 0.000198 1050 2.132898 8352765 1499.251 0.000207



176

90% TIME FGOR FOPT FPR FWCT (DAYS)


0 0 0 3993.75 0 1 0.5728 12000 3981.823 2.44E-06 4 0.5728 48000 3946.025 4.01E-06

13 0.5728 156000 3838.552 6.16E-06 30 0.5728 360000 3635.196 9.67E-06 60 0.5728 720000 3274.408 1.55E-05 90 0.5728 1080000 2902.921 2.10E-05

120 0.5728 1440000 2529.849 2.62E-05 150 0.523139 1800000 2287.649 3.60E-05 180 0.509729 2160000 2240.616 4.15E-05 210 0.500368 2520000 2198.168 4.64E-05 240 0.49943 2880000 2157.619 5.02E-05 270 0.501218 3240000 2118.703 5.36E-05 300 0.505299 3600000 2081.102 5.74E-05 330 0.513904 3960000 2044.483 6.18E-05 360 0.528194 4320000 2008.354 6.71E-05 390 0.549253 4680000 1972.143 7.36E-05 420 0.578708 5035902 1935.317 8.12E-05 450 0.601094 5361485 1901.543 8.64E-05 480 0.632212 5661975 1870.326 9.15E-05 510 0.677521 5942008 1840.711 9.57E-05 540 0.725026 6203454 1812.442 0.0001 570 0.776274 6447573 1785.233 0.000105 600 0.830215 6675513 1758.625 0.00011 630 0.891767 6887893 1732.56 0.000115 660 0.971848 7084609 1707.341 0.000122 690 1.061685 7266901 1682.823 0.000129 720 1.159643 7435969 1658.905 0.000137 750 1.262953 7592636 1635.592 0.000145 780 1.384624 7736993 1612.814 0.000154 810 1.513797 7870175 1590.526 0.000164 840 1.653069 7993651 1568.552 0.000174 870 1.798974 8108008 1546.913 0.000184 900 1.942987 8213904 1525.717 0.000194 930 2.080459 8312047 1504.997 0.000204 960 2.224727 8402833 1484.544 0.000215 990 2.363855 8486791 1464.181 0.000226

1020 2.49303 8564601 1444.492 0.000236 1050 2.612129 8636968 1425.541 0.000247



177

MATCHED MODEL PREDICTION

75% - 12 MONTHS 90% - 12 MONTHS FGOR FPR FWCT FGOR FPR FWCT

0 9E-08 2.03E-13 0 0 2.56E-140 9.025E-05 6.4E-13 0 7.225E-05 7.29E-140 0.000121 9.03E-13 0 8.281E-05 1.024E-130 0.00012996 9.02E-13 0 8.464E-05 1.024E-130 0.00374544 6.4E-13 0 0.0005476 9E-140 0.00013689 4.9E-13 0 1E-04 4E-140 0.00013689 4.9E-13 0 0.0001 4E-14

0.000105 9.52586496 6.25E-12 1.60888E-05 0.8055063 8.1E-136.67E-05 1.58533281 1.6E-11 7.74041E-06 0.191844 1.69E-124.51E-06 1.97824225 2.03E-11 7.64982E-06 0.3341996 2.89E-126.82E-05 1.13465104 2.03E-11 7.02356E-06 0.2035814 2.25E-120.000111 0.45441081 2.81E-11 1.32565E-05 0.100109 2.89E-120.000185 0.01127844 3.84E-11 2.59172E-05 0.0095844 4.41E-120.000288 0.39891856 5.18E-11 4.22609E-05 0.0264713 5.76E-12

0.00026 0.22877089 5.33E-11 4.52988E-05 0.206843 7.29E-12 0.001088 15.32183028 2.39E-10 0.000165236 1.8791263 2.84633E-11



178


0 0 2.56E-14 0 9E-08 2.025E-130 7.225E-05 7.29E-14 0 9.025E-05 6.4E-130 8.281E-05 1.024E-13 0 0.000121 9.025E-130 8.464E-05 1.024E-13 0 0.00012996 9.025E-130 0.0005476 9E-14 0 0.00374544 6.4E-130 1E-04 4E-14 0 0.00013689 4.9E-130 0.0001 4E-14 0 0.00013689 4.9E-13

1.60888E-05 0.8055063 8.1E-13 0.000105239 9.52586496 6.25E-127.74041E-06 0.191844 1.69E-12 6.66692E-05 1.58533281 1.6E-117.64982E-06 0.3341996 2.89E-12 4.50687E-06 1.97824225 2.025E-117.02356E-06 0.2035814 2.25E-12 6.81589E-05 1.13465104 2.025E-111.32565E-05 0.100109 2.89E-12 0.000110503 0.45441081 2.809E-112.59172E-05 0.0095844 4.41E-12 0.000184654 0.01127844 3.844E-114.22609E-05 0.0264713 5.76E-12 0.000287543 0.39891856 5.184E-114.52988E-05 0.206843 7.29E-12 0.000260434 0.22877089 5.329E-114.56815E-05 0.4165412 9E-12 2.07935E-05 7.43707441 2.025E-112.75541E-05 0.1282356 7.29E-12 0.000139539 77.28519744 9E-140.00016636 10.365824 2.56E-12 0.001122934 288.2321108 2.601E-11

0.000423082 42.022806 2.89E-12 0.002429093 593.892026 3.481E-110.000481 87.845631 4.41E-12 0.004116726 982.9604448 3.721E-11

0.00074355 144.90863 7.0756E-

12 0.005813646 1429.157538 5.42138E-11

0.002052464 287.56679 6.16889E-

11 0.014730441 3394.286221 4.11261E-10



179


0 9E-08 1.9807E-13 0 0 2.431E-140 9.025E-05 6.41234E-13 0 7.225E-05 7.413E-140 0.000121 9.05355E-13 0 8.281E-05 1.027E-130 0.00012996 8.794E-13 0 8.464E-05 9.875E-140 0.00374544 6.87108E-13 0 0.00054756 7.689E-140 0.00013689 5.30953E-13 0 1E-04 5.899E-140 0.00013689 4.07216E-13 0 0.0001 4.471E-14

0.000105239 9.52586496 6.45189E-12 1.60888E-05 0.80550625 7.494E-136.66692E-05 1.58533281 1.6322E-11 7.74041E-06 0.191844 1.712E-124.50687E-06 1.97824225 2.05262E-11 7.64982E-06 0.33419961 2.843E-126.81589E-05 1.13465104 2.00594E-11 7.02356E-06 0.20358144 1.984E-120.000110503 0.45441081 2.749E-11 1.32565E-05 0.10010896 2.788E-120.000184654 0.01127844 3.85549E-11 2.59172E-05 0.00958441 4.119E-120.000287543 0.39891856 5.10853E-11 4.22609E-05 0.02647129 5.691E-120.000260434 0.22877089 5.24272E-11 4.52988E-05 0.20684304 6.882E-122.07935E-05 7.43707441 2.02474E-11 4.56815E-05 0.41654116 8.516E-120.000139539 77.28519744 1.2673E-13 2.75541E-05 0.12823561 7.749E-120.001122934 288.2321108 2.62678E-11 0.00016636 10.36582416 2.584E-120.002429093 593.892026 3.523E-11 0.000423082 42.02280625 3.149E-120.004116726 982.9604448 3.78341E-11 0.000481 87.84563076 4.49E-120.005813646 1429.157538 5.47357E-11 0.00074355 144.9086288 7.077E-120.008521446 1941.600845 7.96633E-11 0.001103203 214.7895425 1.060E-110.011972384 2547.594392 1.12283E-10 0.001581194 298.3323473 1.540E-110.022573527 3188.420449 1.86583E-10 0.004114469 380.5932774 3.138E-110.039935766 3818.84449 2.98167E-10 0.006454051 468.501696 4.673E-11

0.06101563 4471.931256 4.301E-10 0.009402541 562.9325664 6.55936E-

110.085089998 5154.148698 5.8648E-10 0.012658028 657.9994523 8.972E-11 0.243839192 24516.82635 2.10488E-09 0.03736595 2870.715675 3.202E-10



180

48 MONTHS MATCHED MODEL 10 YEARS PREDICTION OBSERVED 10 YEARS HISTORY

TIME FGOR FOPT FWCT TIME FGOR FOPT FWCT (DAYS)

(MCF/ STB) (STB)

(DAYS)

(MCF/STB) (STB)

0 0 0 0 0 0 0 01 0.57 12000 2.44E-06 1 0.57 12000 2.28E-064 0.57 48000 4.01E-06 4 0.57 48000 3.74E-06

13 0.57 156000 6.16E-06 13 0.57 156000 5.84E-0640 0.57 480000 1.17E-05 40 0.57 480000 1.14E-05

121 0.57 1452000 2.64E-05 121 0.57 1452000 2.62E-05365 0.51 4380000 6.69E-05 365 0.52 4380000 6.50E-05

524.84 0.68 5866501 9.65E-05 521.77 0.70 5952404 9.89E-05730 1.02 7102293 0.00013 730 1.12 7229839 0.00014

912.5 1.59 7821784 0.00017 912.5 1.79 7935202 0.000191095 2.26 8300417 0.00023 1095 2.45 8393343 0.00025

1277.5 2.75 8634867 0.00027 1277.5 2.82 8715890 0.000291460 2.91 8887812 0.0003 1460 2.87 8961569 0.000311825 2.73 9254713 0.00032 1825 2.59 9295078 0.000312190 2.39 9495055 0.00031 2190 2.24 9534457 0.00032555 2.04 9677956 0.00029 2555 1.86 9715776 0.000292920 1.69 9823144 0.00027 2920 1.48 9859771 0.000263285 1.35 9942660 0.00025 3285 1.14 9978785 0.000243650 1.05 1E+07 0.00023 3650 0.84 1E+07 0.00021



181

APPENDIX D

PLOTS OF BLACK OIL AND COMPOSITIONAL SIMULATION GENERATED DATA



182

A COMPARISON OF BLACK OIL AND COMPOSITIONAL SIMULATION GOR

0

0.5

1

1.5

2

2.5

3

3.5

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, days

GOR, MSCF/STB

COM - GORBO - GOR

COMPARISON OF BLACK OIL AND COMPOSITIONAL SIMULATION OIL PRODUCTION

0

2000000

4000000

6000000

8000000

10000000

12000000

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, days

Oil Produtcion, stb

COM - Oil prod.BO - Oil prod.



183

BLACK OIL AND COMPOSITIONAL SIMULATION COMPARISON - FPR

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, days

Field Pressure, psia

COM - FPRBO - FPR

BLACK OIL AND COMPOSITIONAL SIMULATION COMPARISON - OPR

0

2000

4000

6000

8000

10000

12000

14000

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, days

Oil Prod. Rate, STB/ day

COM - OPRBO - OPR



184

WATER-CUT MATCHING AFTER 2 MONTHS

0

0.000005

0.00001

0.000015

0.00002

0.000025

0.00003

0 1 4 13 30 60

TIME, Days

WA

TER

-CU

T

BASE CASE1%10%20%30%75%90%

BLACK OIL AND COMPOSITIONAL SIMULATION COMPARISON - WPR

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, days

Water Prod. Rate, STB/day

COM - WPRBO - WPR



185

PRESSURE MATCH

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 10 20 30 40 50 60 70

TIME, DAYS

PRES

SUR

E, P

SIA BASE CASE

1%10%20%30%75%90%

10 YEARS PREDICTION WITH 2 MONTHS HISTORY DATA

0

0.5

1

1.5

2

2.5

3

3.5

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, DAYS

GO

R, S

CF/

STB




186


0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, DAYS

PRES

SUR

E, P

SIA



0

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003

0.00035

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, DAYS

WA

TER

-CU

T




187

24 MONTHS GOR MATCH

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 100 200 300 400 500 600 700 800

TIME, DAYS

GO

R, S

CF/

STB

BASE CASE1%75%90%

24 MONTHS DATA PRESSURE MATCH

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 100 200 300 400 500 600 700 800

TIME, DAYS

PRES

SUR

E, P

SIA

BASE CASE1%75%90%



188

24 MONTHS DATA WATER-CUT MATCH

0

0.00002

0.00004

0.00006

0.00008

0.0001

0.00012

0.00014

0.00016

0 100 200 300 400 500 600 700 800

TIME, DAYS

WA

TER

-CU

T BASE CASE1%75%90%


0

0.5

1

1.5

2

2.5

3

3.5

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, DAYS

GO

R, S

CF/

STB




189


0

2000000

4000000

6000000

8000000

10000000

12000000

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, DAYS

CU

M. O

IL P

RO

DU

CTI

ON

, STB



0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, DAYS

PRES

SUR

E, P

SIA




190

48 MONTHS GOR MATCH

0

0.5

1

1.5

2

2.5

3

3.5

0 200 400 600 800 1000 1200 1400 1600

TIME, DAYS

GO

R, S

CF/

STB

BASE CASE75%90%


0

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003

0.00035

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, DAYS

WA

TER

-CU

T




191

48 MONTHS CUM. OIL PROD. MATCH

0

1000000

2000000

3000000

4000000

5000000

6000000

7000000

8000000

9000000

10000000

0 200 400 600 800 1000 1200 1400 1600

TIME, DAYS

CU

M. O

IL P

RO

DU

CTI

ON

, STB

BASE CASE90%75%

48 MONTHS PRESSURE MATCH

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 200 400 600 800 1000 1200 1400 1600

TIME, DAYS

PRES

SUR

E, P

SIA

BASE CASE75%90%



192

SIX MONTHS GOR MATCH

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 20 40 60 80 100 120 140 160 180 200

TIME, DAYS

GO

R, S

CF/

STB

BASE CASE1%10%20%30%75%90%


0

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003

0.00035

0 200 400 600 800 1000 1200 1400 1600

TIME, DAYS

WA

TER

-CU

T

BASE CASE90%75%



193

SIX MONTHS PRESSURE MATCH

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 20 40 60 80 100 120 140 160 180 200

TIME, DAYS

PRES

SUR

E, P

SIA BASE CASE

1%10%20%30%75%90%

SIX MONTHS WATER-CUT MATCH

0

0.00001

0.00002

0.00003

0.00004

0.00005

0.00006

0 20 40 60 80 100 120 140 160 180 200

TIME, DAYS

WA

TER

-CU

T

BASE CASE1%10%20%30%75%90%



194

12 MONTHS GOR MATCH

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 50 100 150 200 250 300 350 400

TIME, DAYS

GO

R, S

CF/

STB

BASE CASE1%10%30%20%75%90%


0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 50 100 150 200 250 300 350 400

TIME, DAYS

PRES

SUR

E, P

SIA BASE CASE

1%10%20%30%75%90%



195

18 MONTHS GOR MATCH

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 100 200 300 400 500 600

TIME, DAYS

GO

R, S

CF/

STB

BASE CASE1%75%90%


0

0.00001

0.00002

0.00003

0.00004

0.00005

0.00006

0.00007

0.00008

0 50 100 150 200 250 300 350 400

TIME, DAYS

WA

TER

-CU

T

BASE CASE1%10%20%30%75%90%



196


0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 100 200 300 400 500 600

TIME, DAYS

PRES

SUR

E, P

SIA

BASE CASE1%75%90%


0

0.00002

0.00004

0.00006

0.00008

0.0001

0.00012

0 100 200 300 400 500 600

TIME, DAYS

WA

TER

-CU

T BASE CASE1%75%90%



197

75% MATCH MODEL GOR PREDICTION

0

0.5

1

1.5

2

2.5

3

3.5

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, DAYS

GO

R, S

CF/

STB

OBSERVED GORSIMULATED GOR

75% MATCHED MODEL CUM. OIL PROD. PREDICTION

0

2000000

4000000

6000000

8000000

10000000

12000000

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, DAYS

CU

M. O

IL P

RO

DU

CTI

ON

, STB




198

75% MATCHED MODEL PRESSURE PREDICTION

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, DAYS

PRES

SUR

E, P

SIA

OBSERVED PRESSURESIMULATED PRESSURE

75% MATCHED MODEL WATER-CUT PREDICTION

0

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003

0.00035

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, DAYS

WA

TER

-CU

T

OBSERVED FWCTSIMULATED FWCT



199

50% MATCHED MODEL 10 YEARS GOR PREDICTION

0

0.5

1

1.5

2

2.5

3

3.5

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, DAYS

GO

R, S

CF/

STB

OBSERVED GORSIMULATED GOR

50% MATCHED MODEL 10 YEARS CUM. OIL PROD. PREDICTION

0

2000000

4000000

6000000

8000000

10000000

12000000

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, DAYS

CU

M. O

IL P

RO

DU

CTI

ON

, STB




200

50% MATCHED MODEL 10 YEARS PRESSURE PREDICTION

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, DAYS

PRES

SUR

E, P

SIA

OBSERVED PRESSURESIMULATED PRESSURE

50% MATCHED MODEL 10 YEARS WATER-CUT PREDICTION

0

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003

0.00035

0 500 1000 1500 2000 2500 3000 3500 4000

TIME, DAYS

WA

TER

-CU

T

OBSERVED WATER-CUTSIMULATED WATER-CUT



201

APPENDIX E

OPTIMIZATION OF BLACK OIL WITH COMPOSITION History matched model optimization

BLACK OIL COMPOSITIONAL TIME FGOR FOPT FWCT FGOR FOPT FWCT

(days)

(MSCF/STB) (STB)

(MSCF/STB) (STB)

OPTIMIZATION

1 0.556 12000 2.18E-06 0.5728 12000 2.28E-06 03 0.556 36000 3.31E-06 0.5728 48000 3.74E-06 1.4E+08

15 0.556 180000 6.22E-06 0.5728 156000 5.84E-06 5.7E+0831 0.556 372000 9.58E-06 0.5728 372000 9.56E-06 059 0.556 708000 1.51E-05 0.5728 708000 1.50E-05 090 0.556 1080000 2.05E-05 0.5728 1080000 2.08E-05 0

120 0.556 1440000 2.51E-05 0.5728 1440000 2.60E-05 0151 0.501 1812000 3.78E-05 0.5264 1812000 3.53E-05 0181 0.500 2172000 3.88E-05 0.5123 2172000 4.04E-05 0212 0.493 2544000 4.26E-05 0.5026 2544000 4.50E-05 0243 0.498 2916000 4.65E-05 0.4966 2916000 4.90E-05 0273 0.485 3276000 4.86E-05 0.4975 3276000 5.22E-05 0304 0.482 3648000 5.12E-05 0.5009 3648000 5.58E-05 0334 0.487 4008000 5.46E-05 0.5089 4008000 6.00E-05 0365 0.497 4380000 5.89E-05 0.5246 4380000 6.54E-05 0396 0.518 4752000 6.46E-05 0.5477 4752000 7.20E-05 0424 0.538 5088000 7.05E-05 0.5781 5088000 7.96E-05 0455 0.567 5460000 7.78E-05 0.6188 5452088 8.90E-05 625944485 0.607 5812363 8.70E-05 0.6608 5775205 9.40E-05 1.3E+09516 0.654 6145273 9.15E-05 0.7096 6083869 9.88E-05 3.7E+09546 0.700 6443400 9.61E-05 0.7632 6360856 0.0001 6.1E+09577 0.756 6728185 0.0001 0.8231 6625599 0.00011 1.0E+10608 0.815 6991648 0.00011 0.8909 6869974 0.00012 1.4E+10638 0.877 7227884 0.00011 0.9782 7087364 0.00012 1.9E+10669 0.989 7449742 0.00012 1.0819 7293336 0.00013 2.4E+10699 1.104 7646486 0.00013 1.1915 7476696 0.00014 2.8E+10730 1.232 7831889 0.00014 1.3122 7650216 0.00015 3.3E+10761 1.372 8000091 0.00015 1.4599 7807899 0.00016 3.6E+10789 1.509 8140210 0.00016 1.5973 7939549 0.00017 4.0E+10



202

820 1.700 8280622 0.00017 1.7611 8073172 0.00018 4.3E+10850 1.862 8405024 0.00018 1.9201 8191968 0.00019 4.5E+10881 2.021 8522470 0.0002 2.0768 8304521 0.0002 4.7E+10911 2.190 8626256 0.00021 2.2358 8404433 0.00022 4.9E+10942 2.351 8724112 0.00022 2.3917 8498897 0.00023 5.0E+10973 2.500 8813764 0.00023 2.5352 8585603 0.00024 5.2E+10

1003 2.627 8893849 0.00024 2.6612 8663162 0.00025 5.3E+101034 2.741 8970367 0.00026 2.7739 8737345 0.00026 5.4E+101064 2.832 9039271 0.00026 2.8658 8804204 0.00027 5.5E+101095 2.904 9105694 0.00027 2.9413 8868679 0.00028 5.6E+101126 2.957 9167901 0.00028 2.9986 8929100 0.00029 5.7E+101154 2.992 9221008 0.00029 3.0379 8980702 0.00029 5.7E+101185 3.016 9276423 0.00029 3.0693 9034556 0.0003 5.8E+101215 3.018 9327314 0.0003 3.0881 9083914 0.0003 5.9E+101246 3.013 9377228 0.0003 3.0943 9132283 0.00031 6.0E+101276 3.001 9423251 0.0003 3.0911 9176857 0.00031 6.0E+101307 2.982 9468566 0.0003 3.0802 9220740 0.00031 6.1E+101338 2.957 9511813 0.00031 3.0639 9262621 0.00032 6.2E+101368 2.930 9551873 0.00031 3.0427 9301420 0.00032 6.2E+101399 2.898 9591489 0.00031 3.0151 9339805 0.00032 6.3E+101429 2.865 9628268 0.00031 2.9854 9375459 0.00032 6.3E+101460 2.830 9664710 0.00031 2.9522 9410807 0.00032 6.4E+101491 2.794 9699681 0.00031 2.9172 9444729 0.00032 6.5E+101519 2.761 9730134 0.00031 2.8839 9474251 0.00032 6.5E+101550 2.723 9762541 0.00031 2.8463 9505631 0.00032 6.6E+101580 2.686 9792744 0.00031 2.8083 9534849 0.00032 6.6E+101611 2.648 9822783 0.00031 2.7682 9563885 0.00032 6.7E+101641 2.611 9850811 0.00031 2.7287 9590965 0.00032 6.7E+101672 2.573 9878714 0.00031 2.6876 9617918 0.00032 6.8E+101703 2.537 9905606 0.00031 2.6466 9643896 0.00032 6.8E+101733 2.501 9930731 0.00031 2.607 9668169 0.00032 6.8E+101764 2.464 9955782 0.00031 2.5665 9692370 0.00031 6.9E+101794 2.429 9979205 0.00031 2.5281 9715004 0.00031 6.9E+101825 2.392 1.00E+07 0.0003 2.4887 9737593 0.00031 7.0E+101856 2.354 1.00E+07 0.0003 2.4494 9759421 0.00031 7.0E+101884 2.320 1.00E+07 0.0003 2.4141 9778543 0.00031 7.1E+101915 2.281 1.00E+07 0.0003 2.3748 9799021 0.00031 7.1E+101945 2.243 1.00E+07 0.0003 2.3366 9818224 0.00031 7.2E+101976 2.203 1.00E+07 0.0003 2.2967 9837443 0.00031 7.2E+102006 2.164 1.00E+07 0.0003 2.2578 9855486 0.0003 7.3E+102037 2.123 1.00E+07 0.00029 2.2173 9873567 0.0003 7.3E+102068 2.082 1.00E+07 0.00029 2.1765 9891113 0.0003 7.3E+10



203

2098 2.042 1.00E+07 0.00029 2.1369 9907616 0.0003 7.3E+102129 2.000 1.00E+07 0.00029 2.0958 9924183 0.0003 7.4E+102159 1.959 1.00E+07 0.00029 2.056 9939783 0.00029 7.4E+102190 1.917 1.00E+07 0.00028 2.0148 9955461 0.00029 7.5E+102221 1.874 1.00E+07 0.00028 1.9737 9970720 0.00029 7.5E+102249 1.836 1.00E+07 0.00028 1.9366 9984173 0.00029 7.6E+102280 1.794 1.00E+07 0.00028 1.8955 9998684 0.00028 7.6E+102310 1.753 1.00E+07 0.00027 1.8558 1.00E+07 0.00028 7.6E+102341 1.712 1.00E+07 0.00027 1.8148 1.00E+07 0.00028 7.7E+102371 1.672 1.00E+07 0.00027 1.7754 1.00E+07 0.00028 7.7E+102402 1.631 1.00E+07 0.00027 1.7349 1.00E+07 0.00027 7.7E+102433 1.590 1.00E+07 0.00026 1.6946 1.00E+07 0.00027 7.8E+102463 1.551 1.00E+07 0.00026 1.656 1.00E+07 0.00027 7.8E+102494 1.512 1.00E+07 0.00026 1.6163 1.00E+07 0.00027 7.8E+102524 1.473 1.00E+07 0.00026 1.5782 1.00E+07 0.00026 7.9E+102555 1.435 1.00E+07 0.00025 1.5393 1.00E+07 0.00026 7.9E+102586 1.396 1.00E+07 0.00025 1.5006 1.00E+07 0.00026 7.9E+102614 1.396 1.00E+07 0.00025 1.4663 1.00E+07 0.00026 8.0E+102645 1.325 1.00E+07 0.00025 1.4287 1.00E+07 0.00025 8.0E+102675 1.325 1.00E+07 0.00025 1.3928 1.00E+07 0.00025 8.1E+102706 1.289 1.00E+07 0.00024 1.3562 1.00E+07 0.00025 8.1E+102736 1.253 1.00E+07 0.00024 1.3212 1.00E+07 0.00025 8.2E+102767 1.218 1.00E+07 0.00024 1.2855 1.00E+07 0.00024 8.2E+102798 1.183 1.00E+07 0.00024 1.2504 1.00E+07 0.00024 8.3E+102828 1.148 1.00E+07 0.00023 1.2168 1.00E+07 0.00024 8.3E+102859 1.115 1.10E+07 0.00023 1.1827 1.00E+07 0.00024 8.3E+102889 1.082 1.10E+07 0.00023 1.15 1.00E+07 0.00023 8.4E+102920 1.049 1.10E+07 0.00022 1.1169 1.00E+07 0.00023 8.4E+102951 1.013 1.10E+07 0.00022 1.082 1.00E+07 0.00023 8.5E+102979 0.979 1.10E+07 0.00022 1.0501 1.00E+07 0.00022 8.5E+103010 0.950 1.10E+07 0.00021 1.0162 1.00E+07 0.00022 8.6E+103040 0.919 1.10E+07 0.00021 0.9847 1.00E+07 0.00022 8.6E+103071 0.889 1.10E+07 0.00021 0.9534 1.00E+07 0.00021 8.7E+103101 0.860 1.10E+07 0.00021 0.924 1.00E+07 0.00021 8.7E+103132 0.833 1.10E+07 0.0002 0.8948 1.00E+07 0.00021 8.8E+103163 0.806 1.10E+07 0.0002 0.8666 1.00E+07 0.0002 8.9E+103193 0.779 1.10E+07 0.0002 0.8402 1.00E+07 0.0002 8.9E+103224 0.754 1.10E+07 0.00019 0.8137 1.00E+07 0.0002 8.9E+103254 0.729 1.10E+07 0.00019 0.7887 1.00E+07 0.0002 8.9E+103285 0.705 1.10E+07 0.00019 0.7636 1.00E+07 0.00019 8.9E+103316 0.682 1.10E+07 0.00019 0.7391 1.00E+07 0.00019 9.0E+103344 0.658 1.10E+07 0.00018 0.7175 1.00E+07 0.00019 9.1E+10



204

3375 0.638 1.10E+07 0.00018 0.6943 1.00E+07 0.00019 9.1E+103405 0.616 1.10E+07 0.00018 0.6724 1.00E+07 0.00018 9.1E+103436 0.595 1.10E+07 0.00018 0.6503 1.00E+07 0.00018 9.2E+103466 0.574 1.10E+07 0.00017 0.6294 1.00E+07 0.00018 9.2E+103497 0.554 1.10E+07 0.00017 0.6083 1.00E+07 0.00018 9.3E+103528 0.533 1.10E+07 0.00017 0.5877 1.00E+07 0.00018 9.3E+103558 0.514 1.10E+07 0.00017 0.5682 1.00E+07 0.00017 9.3E+103589 0.495 1.10E+07 0.00017 0.5485 1.00E+07 0.00017 9.4E+103619 0.476 1.10E+07 0.00016 0.5298 1.00E+07 0.00017 9.4E+10

3.6E+12



205

Model with Minimum Sum of Square Optimization


(days)

(MSCF/STB) (STB)

(MSCF/STB) (STB)

OPTIMIZATION

1 0.5728 12000 2.25E-06 0.5563 12000 2.15E-06 04 0.5728 48000 3.71E-06 0.5563 36000 3.28E-06 1.40E+08

13 0.5728 156000 5.82E-06 0.5563 180000 6.20E-06 5.80E+0831 0.5728 372000 9.54E-06 0.5563 372000 9.57E-06 059 0.5728 708000 1.50E-05 0.5563 708000 1.51E-05 090 0.5728 1080000 2.08E-05 0.5563 1080000 2.05E-05 0

120 0.5728 1440000 2.61E-05 0.5563 1440000 2.52E-05 0151 0.5268 1812000 3.53E-05 0.501 1812000 3.77E-05 0181 0.512 2172000 4.03E-05 0.4998 2172000 3.86E-05 0212 0.5018 2544000 4.47E-05 0.4896 2544000 4.19E-05 0243 0.493 2916000 4.89E-05 0.479 2916000 4.53E-05 0273 0.4925 3276000 5.20E-05 0.4751 3276000 4.82E-05 0304 0.4957 3648000 5.56E-05 0.4689 3648000 5.06E-05 0334 0.5063 4008000 6.02E-05 0.4815 4008000 5.47E-05 0365 0.5249 4380000 6.61E-05 0.5016 4380000 5.99E-05 0396 0.5502 4752000 7.31E-05 0.5226 4752000 6.60E-05 0424 0.5835 5088000 8.13E-05 0.5427 5088000 7.22E-05 0455 0.6277 5450626 9.13E-05 0.5811 5460000 8.08E-05 8.80E+07485 0.6734 5773233 9.63E-05 0.634 5808086 9.02E-05 1.20E+09516 0.7267 6080598 0.0001 0.6929 6134943 9.56E-05 3.00E+09546 0.7847 6355700 0.00011 0.7575 6425013 0.000102 4.80E+09577 0.8487 6618018 0.00011 0.8227 6700446 0.000108 6.80E+09608 0.9277 6859092 0.00012 0.8891 6954282 0.000114 9.10E+09638 1.0195 7073357 0.00013 0.9711 7179874 0.000121 1.10E+10669 1.1286 7276312 0.00014 1.042 7396197 0.000128 1.40E+10699 1.2437 7456546 0.00015 1.1842 7588170 0.000139 1.70E+10730 1.3784 7626154 0.00016 1.3262 7767642 0.00015 2.00E+10761 1.5273 7780729 0.00017 1.505 7929155 0.000163 2.20E+10789 1.6738 7909759 0.00018 1.6926 8061434 0.000176 2.30E+10820 1.8432 8040423 0.00019 1.851 8194654 0.000188 2.40E+10850 2.004 8156403 0.0002 1.9896 8313125 0.000199 2.50E+10881 2.1662 8266042 0.00022 2.123 8425876 0.00021 2.60E+10911 2.3273 8363207 0.00023 2.2859 8525794 0.000223 2.60E+10942 2.4832 8455022 0.00024 2.4351 8620321 0.000236 2.70E+10



206

973 2.6261 8539358 0.00026 2.581 8707107 0.000249 2.80E+101003 2.7449 8614897 0.00027 2.6941 8784907 0.00026 2.90E+101034 2.8438 8687348 0.00028 2.7877 8859502 0.00027 3.00E+101064 2.9196 8752862 0.00029 2.8598 8926958 0.000279 3.00E+101095 2.9794 8816230 0.0003 2.9145 8992209 0.000287 3.10E+101126 3.0229 8875761 0.0003 2.9515 9053524 0.000294 3.20E+101154 3.0507 8926709 0.00031 2.9722 9106024 0.000299 3.20E+101185 3.0681 8980022 0.00031 2.9764 9161035 0.000304 3.30E+101215 3.0723 9029021 0.00032 2.9678 9211672 0.000307 3.30E+101246 3.066 9077156 0.00032 2.9518 9261453 0.00031 3.40E+101276 3.0524 9121605 0.00032 2.9306 9307435 0.000313 3.50E+101307 3.0327 9165441 0.00033 2.9038 9352786 0.000315 3.50E+101338 3.0078 9207350 0.00033 2.8728 9396137 0.000316 3.60E+101368 2.979 9246233 0.00033 2.8401 9436344 0.000318 3.60E+101399 2.9459 9284747 0.00033 2.8047 9476146 0.000319 3.70E+101429 2.912 9320554 0.00033 2.7697 9513124 0.000319 3.70E+101460 2.8761 9356070 0.00033 2.7331 9549788 0.00032 3.80E+101491 2.8389 9390170 0.00033 2.696 9584996 0.00032 3.80E+101519 2.8042 9419854 0.00033 2.6627 9615665 0.00032 3.80E+101550 2.7652 9451421 0.00033 2.6256 9648309 0.000321 3.90E+101580 2.7269 9480822 0.00033 2.593 9678707 0.000321 3.90E+101611 2.6878 9510040 0.00033 2.5593 9708913 0.000321 4.00E+101641 2.6509 9537277 0.00033 2.5285 9737059 0.000321 4.00E+101672 2.6137 9564367 0.00033 2.4967 9765045 0.000322 4.00E+101703 2.578 9590449 0.00033 2.4645 9791990 0.000322 4.10E+101733 2.5439 9614791 0.00033 2.4328 9817142 0.000321 4.10E+101764 2.5085 9639033 0.00033 2.3991 9842201 0.000321 4.10E+101794 2.4738 9661686 0.00033 2.3656 9865627 0.000321 4.20E+101825 2.4373 9684278 0.00033 2.3299 9889004 0.00032 4.20E+101856 2.4002 9706100 0.00033 2.2933 9911598 0.000319 4.20E+101884 2.3661 9725211 0.00032 2.2596 9931397 0.000318 4.30E+101915 2.3277 9745678 0.00032 2.2221 9952612 0.000317 4.30E+101945 2.29 9764869 0.00032 2.1851 9972516 0.000316 4.30E+101976 2.2505 9784078 0.00032 2.146 9992450 0.000314 4.30E+102006 2.212 9802112 0.00032 2.1075 1.00E+07 0.000313 4.40E+102037 2.172 9820187 0.00032 2.0668 1.00E+07 0.000311 4.40E+102068 2.1319 9837728 0.00032 2.0257 1.00E+07 0.000309 4.40E+102098 2.0929 9854226 0.00031 1.9853 1.00E+07 0.000307 4.50E+102129 2.0524 9870791 0.00031 1.9431 1.00E+07 0.000305 4.50E+102159 2.0129 9886390 0.00031 1.902 1.00E+07 0.000302 4.50E+102190 1.9719 9902073 0.00031 1.8595 1.00E+07 0.0003 4.50E+102221 1.9307 9917339 0.0003 1.817 1.00E+07 0.000298 4.60E+10



207

2249 1.8935 9930802 0.0003 1.7787 1.00E+07 0.000295 4.60E+102280 1.8521 9945330 0.0003 1.7365 1.00E+07 0.000293 4.60E+102310 1.8122 9959049 0.0003 1.6959 1.00E+07 0.00029 4.70E+102341 1.771 9972881 0.00029 1.6544 1.00E+07 0.000287 4.70E+102371 1.7315 9985957 0.00029 1.6146 1.00E+07 0.000285 4.70E+102402 1.6909 9999152 0.00029 1.574 1.00E+07 0.000282 4.80E+102433 1.6505 1.00E+07 0.00029 1.534 1.00E+07 0.000279 4.80E+102463 1.6117 1.00E+07 0.00028 1.4956 1.00E+07 0.000277 4.80E+102494 1.5721 1.00E+07 0.00028 1.4565 1.00E+07 0.000274 4.90E+102524 1.5345 1.00E+07 0.00028 1.4192 1.00E+07 0.000271 4.90E+102555 1.4961 1.00E+07 0.00028 1.381 1.00E+07 0.000268 4.90E+102586 1.4582 1.00E+07 0.00027 1.3434 1.00E+07 0.000266 4.90E+102614 1.4243 1.00E+07 0.00027 1.3432 1.00E+07 0.000266 5.00E+102645 1.3872 1.00E+07 0.00027 1.2733 1.00E+07 0.00026 5.00E+102675 1.3518 1.00E+07 0.00027 1.2732 1.00E+07 0.00026 5.10E+102706 1.3157 1.00E+07 0.00026 1.2383 1.00E+07 0.000258 5.10E+102736 1.2813 1.00E+07 0.00026 1.2029 1.00E+07 0.000255 5.10E+102767 1.2461 1.00E+07 0.00026 1.1691 1.00E+07 0.000252 5.20E+102798 1.2115 1.00E+07 0.00025 1.1346 1.00E+07 0.000249 5.20E+102828 1.1785 1.00E+07 0.00025 1.1008 1.00E+07 0.000247 5.20E+102859 1.1449 1.00E+07 0.00025 1.0688 1.00E+07 0.000244 5.30E+102889 1.1129 1.00E+07 0.00025 1.033 1.00E+07 0.00024 5.30E+102920 1.0773 1.00E+07 0.00024 0.9996 1.00E+07 0.000237 5.40E+102951 1.0424 1.00E+07 0.00024 0.9667 1.00E+07 0.000233 5.40E+102979 1.012 1.00E+07 0.00024 0.9352 1.00E+07 0.00023 5.40E+103010 0.98 1.00E+07 0.00023 0.9079 1.00E+07 0.000227 5.50E+103040 0.9501 1.00E+07 0.00023 0.8786 1.00E+07 0.000224 5.50E+103071 0.9204 1.00E+07 0.00023 0.8512 1.00E+07 0.000221 5.60E+103101 0.8926 1.00E+07 0.00022 0.8238 1.00E+07 0.000218 5.60E+103132 0.8648 1.00E+07 0.00022 0.798 1.00E+07 0.000215 5.60E+103163 0.8379 1.00E+07 0.00022 0.7721 1.00E+07 0.000212 5.70E+103193 0.8126 1.00E+07 0.00021 0.7469 1.10E+07 0.000209 5.70E+103224 0.7873 1.00E+07 0.00021 0.7231 1.10E+07 0.000206 5.70E+103254 0.7634 1.00E+07 0.00021 0.6991 1.10E+07 0.000204 5.80E+103285 0.7393 1.00E+07 0.00021 0.6764 1.10E+07 0.000201 5.80E+103316 0.7157 1.00E+07 0.0002 0.6535 1.10E+07 0.000198 5.80E+103344 0.695 1.00E+07 0.0002 0.6312 1.10E+07 0.000196 5.90E+103375 0.6726 1.00E+07 0.0002 0.6114 1.10E+07 0.000194 5.90E+103405 0.6514 1.00E+07 0.0002 0.59 1.10E+07 0.000191 5.90E+103436 0.63 1.00E+07 0.00019 0.5697 1.10E+07 0.000189 6.00E+103466 0.6097 1.00E+07 0.00019 0.5492 1.10E+07 0.000186 6.00E+103497 0.5892 1.00E+07 0.00019 0.5299 1.10E+07 0.000184 6.00E+10



208

3528 0.5692 1.00E+07 0.00019 0.5103 1.10E+07 0.000182 6.10E+103558 0.5502 1.00E+07 0.00019 0.4912 1.10E+07 0.000179 6.10E+103589 0.531 1.00E+07 0.00018 0.4731 1.10E+07 0.000177 6.10E+103619 0.5128 1.00E+07 0.00018 0.4548 1.10E+07 0.000175 6.20E+103650 0.4944 1.00E+07 0.00018 0.4376 1.10E+07 0.000173 6.20E+10

2.20E+12



209

Model with Maximum Sum of Square Optimization


(days)

(MSCF/ STB) (STB)

(MSCF/STB) (STB)

OPTIMIZATION

1 0.5728 12000 2.48E-06 0.5563 12000 2.36E-06 04 0.5728 48000 3.95E-06 0.5563 36000 3.50E-06 1.4E+08

13 0.5728 156000 6.04E-06 0.5563 180000 6.41E-06 5.8E+0831 0.5728 372000 9.73E-06 0.5563 372000 9.74E-06 059 0.5728 708000 1.52E-05 0.5563 708000 1.52E-05 090 0.5728 1080000 2.09E-05 0.5563 1080000 2.05E-05 0

120 0.5728 1440000 2.60E-05 0.5563 1440000 2.51E-05 0151 0.5232 1812000 3.63E-05 0.5012 1812000 3.87E-05 0181 0.5129 2172000 4.19E-05 0.5009 2172000 4.01E-05 0212 0.5058 2544000 4.69E-05 0.4974 2544000 4.38E-05 0243 0.5058 2916000 5.01E-05 0.4911 2916000 4.65E-05 0273 0.5009 3276000 5.27E-05 0.4868 3276000 4.91E-05 0304 0.5015 3648000 5.60E-05 0.4831 3648000 5.18E-05 0334 0.5109 4008000 6.03E-05 0.4875 4008000 5.49E-05 0365 0.5255 4380000 6.54E-05 0.503 4380000 5.94E-05 0396 0.5438 4752000 7.13E-05 0.5198 4752000 6.46E-05 0424 0.5682 5088000 7.80E-05 0.5342 5088000 6.98E-05 0455 0.5983 5448720 8.56E-05 0.5582 5460000 7.71E-05 1.3E+08485 0.6242 5769722 9.02E-05 0.587 5808029 8.40E-05 1.5E+09516 0.6609 6078324 9.39E-05 0.6182 6139204 8.79E-05 3.7E+09546 0.6994 6357205 9.76E-05 0.6506 6437874 9.11E-05 6.5E+09577 0.7432 6625791 0.000102 0.6887 6725339 9.48E-05 9.9E+09608 0.7892 6876338 0.000106 0.7313 6993283 9.88E-05 1.37E+10638 0.836 7103068 0.00011 0.7757 7235674 0.000103 1.76E+10669 0.8922 7321312 0.000115 0.8249 7469275 0.000107 2.19E+10699 0.9563 7518011 0.000121 0.8861 7679892 0.000113 2.62E+10730 1.0236 7706895 0.000126 0.9555 7881569 0.000118 3.05E+10761 1.1068 7882230 0.000133 1.0271 8068628 0.000124 3.47E+10789 1.1879 8030416 0.000139 1.1231 8225692 0.000131 3.81E+10820 1.2907 8182401 0.000147 1.2207 8386590 0.000139 4.17E+10850 1.3999 8318581 0.000156 1.3472 8529615 0.000148 4.45E+10881 1.5063 8449589 0.000163 1.4476 8667431 0.000155 4.75E+10911 1.61 8567907 0.000171 1.5657 8790760 0.000163 4.97E+10942 1.7206 8681853 0.000178 1.6889 8909032 0.000172 5.16E+10973 1.8393 8787926 0.000187 1.804 9018984 0.00018 5.34E+10



210

1003 1.966 8883365 0.000196 1.9596 9117121 0.00019 5.46E+101034 2.0934 8974773 0.000205 2.0898 9210896 0.000199 5.58E+101064 2.2071 9057108 0.000213 2.204 9295243 0.000208 5.67E+101095 2.3123 9136331 0.000222 2.3093 9376304 0.000216 5.76E+101126 2.4063 9210326 0.000229 2.4022 9451949 0.000224 5.84E+101154 2.4823 9273298 0.000236 2.4766 9516318 0.00023 5.91E+101185 2.5559 9338774 0.000243 2.5433 9583241 0.000237 5.98E+101215 2.6178 9398537 0.000248 2.5999 9644285 0.000242 6.04E+101246 2.6728 9456781 0.000254 2.649 9703730 0.000248 6.10E+101276 2.7184 9510121 0.000259 2.6886 9758133 0.000252 6.15E+101307 2.7585 9562263 0.000264 2.7204 9811290 0.000257 6.20E+101338 2.7913 9611664 0.000268 2.747 9861613 0.000261 6.25E+101368 2.8174 9657090 0.000272 2.7685 9907842 0.000264 6.29E+101399 2.8393 9701674 0.000275 2.7888 9953155 0.000268 6.32E+101429 2.8568 9742772 0.000278 2.8018 9994877 0.000271 6.36E+101460 2.8687 9783210 0.000281 2.8102 1E+07 0.000273 6.38E+101491 2.876 9821759 0.000284 2.814 1E+07 0.000276 6.41E+101519 2.8792 9855133 0.000286 2.8143 1E+07 0.000278 6.43E+101550 2.8792 9890419 0.000288 2.8107 1E+07 0.000279 6.45E+101580 2.8757 9923096 0.000289 2.8039 1E+07 0.000281 6.46E+101611 2.8681 9955374 0.000291 2.7926 1E+07 0.000282 6.48E+101641 2.8574 9985286 0.000292 2.7781 1E+07 0.000283 6.50E+101672 2.843 1E+07 0.000292 2.76 1E+07 0.000283 6.52E+101703 2.8252 1E+07 0.000293 2.739 1E+07 0.000283 6.54E+101733 2.8051 1E+07 0.000293 2.7164 1E+07 0.000283 6.57E+101764 2.7819 1E+07 0.000293 2.691 1E+07 0.000283 6.59E+101794 2.7574 1E+07 0.000293 2.6646 1E+07 0.000282 6.61E+101825 2.7302 1E+07 0.000292 2.6358 1E+07 0.000282 6.64E+101856 2.7015 1E+07 0.000292 2.6057 1E+07 0.000281 6.66E+101884 2.6744 1E+07 0.000291 2.5773 1E+07 0.00028 6.69E+101915 2.6432 1E+07 0.00029 2.5451 1E+07 0.000279 6.71E+101945 2.612 1E+07 0.000289 2.5127 1E+07 0.000278 6.74E+101976 2.5788 1E+07 0.000287 2.4782 1.1E+07 0.000277 6.76E+102006 2.5459 1E+07 0.000286 2.4439 1.1E+07 0.000275 6.79E+102037 2.5112 1E+07 0.000285 2.4082 1.1E+07 0.000274 6.82E+102068 2.4757 1E+07 0.000283 2.3715 1.1E+07 0.000272 6.84E+102098 2.4408 1E+07 0.000281 2.3352 1.1E+07 0.00027 6.87E+102129 2.4042 1E+07 0.00028 2.2973 1.1E+07 0.000268 6.90E+102159 2.3682 1E+07 0.000278 2.2599 1.1E+07 0.000266 6.92E+102190 2.3306 1E+07 0.000276 2.2211 1.1E+07 0.000264 6.95E+102221 2.2925 1E+07 0.000274 2.1819 1.1E+07 0.000262 6.98E+102249 2.2578 1E+07 0.000272 2.1455 1.1E+07 0.00026 7.00E+10



211

2280 2.2192 1E+07 0.00027 2.1055 1.1E+07 0.000258 7.03E+102310 2.1816 1E+07 0.000267 2.0667 1.1E+07 0.000255 7.06E+102341 2.1426 1E+07 0.000265 2.0267 1.1E+07 0.000253 7.09E+102371 2.1048 1E+07 0.000263 1.9881 1.1E+07 0.000251 7.12E+102402 2.0656 1E+07 0.00026 1.9484 1.1E+07 0.000248 7.14E+102433 2.0264 1E+07 0.000258 1.9091 1.1E+07 0.000246 7.17E+102463 1.9886 1E+07 0.000256 1.8712 1.1E+07 0.000243 7.20E+102494 1.9496 1.1E+07 0.000253 1.8324 1.1E+07 0.000241 7.23E+102524 1.9122 1.1E+07 0.000251 1.7952 1.1E+07 0.000239 7.25E+102555 1.8738 1.1E+07 0.000248 1.7572 1.1E+07 0.000236 7.28E+102586 1.8357 1.1E+07 0.000246 1.7194 1.1E+07 0.000234 7.31E+102614 1.8015 1.1E+07 0.000243 1.6856 1.1E+07 0.000231 7.33E+102645 1.764 1.1E+07 0.000241 1.6485 1.1E+07 0.000229 7.36E+102675 1.728 1.1E+07 0.000239 1.6128 1.1E+07 0.000227 7.39E+102706 1.6911 1.1E+07 0.000236 1.5763 1.1E+07 0.000224 7.41E+102736 1.6557 1.1E+07 0.000234 1.5413 1.1E+07 0.000222 7.44E+102767 1.6194 1.1E+07 0.000231 1.5055 1.1E+07 0.000219 7.46E+102798 1.5836 1.1E+07 0.000229 1.47 1.1E+07 0.000217 7.49E+102828 1.5492 1.1E+07 0.000226 1.4698 1.1E+07 0.000217 7.53E+102859 1.5139 1.1E+07 0.000224 1.4359 1.1E+07 0.000214 7.56E+102889 1.4801 1.1E+07 0.000221 1.4014 1.1E+07 0.000212 7.60E+102920 1.4456 1.1E+07 0.000219 1.3685 1.1E+07 0.00021 7.64E+102951 1.4117 1.1E+07 0.000216 1.3349 1.1E+07 0.000207 7.67E+102979 1.3816 1.1E+07 0.000214 1.3018 1.1E+07 0.000205 7.70E+103010 1.3486 1.1E+07 0.000212 1.2722 1.1E+07 0.000203 7.74E+103040 1.3172 1.1E+07 0.00021 1.2398 1.1E+07 0.0002 7.77E+103071 1.2851 1.1E+07 0.000207 1.2088 1.1E+07 0.000198 7.80E+103101 1.2543 1.1E+07 0.000205 1.1772 1.1E+07 0.000196 7.84E+103132 1.223 1.1E+07 0.000203 1.1469 1.1E+07 0.000194 7.87E+103163 1.192 1.1E+07 0.0002 1.116 1.1E+07 0.000191 7.90E+103193 1.1624 1.1E+07 0.000198 1.0856 1.1E+07 0.000189 7.93E+103224 1.1321 1.1E+07 0.000196 1.0564 1.1E+07 0.000187 7.97E+103254 1.1032 1.1E+07 0.000194 1.0236 1.1E+07 0.000184 8.00E+103285 1.0711 1.1E+07 0.000191 0.9924 1.1E+07 0.000181 8.03E+103316 1.0389 1.1E+07 0.000188 0.9613 1.1E+07 0.000178 8.07E+103344 1.0108 1.1E+07 0.000185 0.9312 1.1E+07 0.000175 8.10E+103375 0.9807 1.1E+07 0.000182 0.9048 1.1E+07 0.000173 8.13E+103405 0.9524 1.1E+07 0.00018 0.8765 1.1E+07 0.00017 8.16E+103436 0.9241 1.1E+07 0.000177 0.8498 1.1E+07 0.000168 8.20E+103466 0.8974 1.1E+07 0.000175 0.823 1.1E+07 0.000165 8.23E+103497 0.8705 1.1E+07 0.000172 0.7977 1.1E+07 0.000163 8.26E+103528 0.8443 1.1E+07 0.00017 0.7722 1.1E+07 0.00016 8.29E+10



212

3558 0.8196 1.1E+07 0.000167 0.7474 1.1E+07 0.000158 8.33E+103589 0.7946 1.1E+07 0.000165 0.7239 1.1E+07 0.000156 8.36E+103619 0.771 1.1E+07 0.000163 0.7002 1.1E+07 0.000153 8.39E+103650 0.7472 1.1E+07 0.00016 0.6778 1.1E+07 0.000151 8.42E+10

3.37E+12


Shameem Siddiqui Malgorzata Ziaja

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Transcript of Shameem Siddiqui Malgorzata Ziaja