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
Copyright 2006, Andrew Oghena
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
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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
Andrew Oghena, Texas Tech University, May 2007
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.
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
4.3 30% Reservoir Description Perturbation 88
4.4 90% Reservoir Description Perturbation 88
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
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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
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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
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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
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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
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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
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
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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
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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
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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
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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
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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.
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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
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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.
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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
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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.
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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
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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
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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
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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).
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
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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
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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.
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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:
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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.
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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( )
( ) 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.
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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.
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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.
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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.
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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.
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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:
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( ) ( ) 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
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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.
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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.
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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.
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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:
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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.
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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:
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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( ) ( ) ( )⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−−+⎟⎟
⎠
⎞⎜⎜⎝
⎛ −= −−
∑ 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
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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
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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.
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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:
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)(
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
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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.
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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
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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.
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
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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.
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
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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;
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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.
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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.
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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
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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.
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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Table 3.4: Common Reservoir Simulator Grid Dimensions
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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.
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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.
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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.
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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
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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)
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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
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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:
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( )[ ]
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.
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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.
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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
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Figure 3.5: Second Case Reservoir Model Schematic
Figure 3.6: Oil Saturation at Time Zero
Production Well
InjectionWell
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Figure 3.7: Oil Saturation after 12 Years
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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.
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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.
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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
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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
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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
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Table 4.3: 30% Reservoir Description Perturbation
30% TIME FGOR FPR FWCT (DAYS)
(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
Table 4.4: 90% Reservoir Description Perturbation
90% TIME FGOR FPR FWCT (DAYS)
(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
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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
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
18 MONTHS WATER-CUT MATCH
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
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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
48 MONTHS WATER-CUT MATCH
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
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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
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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.
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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Figure 4.10: Observed History Data Log-Log Plot
Figure 4.11: History Matched Model Log-Log Plot
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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%
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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
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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
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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
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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.
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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
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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
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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
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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
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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.
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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.
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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.
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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
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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.
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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.
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83 Lamy, P., Swaby, P.A., Rowbotham, P.S., Dubrule, O. and Haas, A., From Seismic to Reservoir Properties with Geostatistical Inversion, SPE 57476, Annual Technical Conference and Exhibition (1998).
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89 Lepine, O.J., Bissell, R.C., Aanonsen, S.I., Pallister, I.C. and Barker, J.W., Uncertainty Analysis in Predictive Reservoir Simulation Using Gradient Information, SPE 57594, Journal of Petroleum Technology (Sept., 1999).
90 Lerche, I., Geological Risk and Uncertainty in Oil Exploration, Academic Press, California (1997).
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95 Mahani, H. and Muggeridge, A.H., Improved Coarse Grid Generation Using Vorticity, paper SPE 94319, presented at SPE Europec/EAGE Annual Conference, Spain (June 2005).
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96 Makhlouf, E.M., Chen, W.H., Wasserman, M.L. and Seinfeld, J.H., A General History Matching Algorithm for Three-Phase, Three-Dimensional Petroleum Reservoirs, paper SPE 20383, SPE Advanced Technology Series.
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98 Mattax, C.C. and Dalton, R.L.: Reservoir Simulation, SPE Monograph 13, Richardson (1990).
99 Millar, D., New Workflows Reduce Forecast Cycle Time, Refine Uncertainty, Journal of Petroleum Technology (July, 2006).
100 Mohaghegh, S.D., Quantifying Uncertainties Associated with Reservoir Simulation Studies Using Surrogate Reservoir Models, SPE 102492, presented at SPE Annual Technical Conference and Exhibition (Sept., 2006).
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102 Newendorp, D.P. and Campbell, J.M., Bayesian Analysis: A Method for Updating Risk Estimates, SPE 3463, Journal of Petroleum Technology (Feb., 1972).
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105 Odeh, A.S., Reservoir Simulation…What is it? SPE 2790, Journal of Petroleum Technology.
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107 Okano, H., Pickup, G.E., Christie, M.A., Subbey, S. and Monfared, H., Quantification of Uncertainty in Relative Permeability for Coarse-Scale Reservoir Simulation, SPE 94140, SPE Europe/EAGE Annual Conference (June, 2005).
108 Ouenes, A., Bhagavan, S., Bunge, P.H. and Travis, B.J., Application of Simulated Annealing and other Global Optimization methods to Reservoir Description: Myths and Realities, paper SPE 28415, presented at Annual Technical Conference and Exhibition (Sept., 1994).
109 Ouenes, A., Brefort, B., Meunier, G. And Dupere, S., A New Algorithm for Automatic History Matching: Application of Simulated Annealing Method.
110 Painter, S., Paterson, L. and Boult, P., Improved Technique for Stochastic Interpolation of Reservoir Properties, SPE 30599, Journal of Petroleum Technology (March, 1997).
111 Peaceman, D.W., Interpretation of Well-Block Pressure in Numerical Reservoir Simulation, SPE 6893, Annual Fall Technical Conference and Exhibition (Oct., 1977).
112 Peaceman, D.W.: Fundamentals of Numerical Reservoir Simulation, Elsevier Scientific Publishing Company, Amsterdam (1977).
113 Peaceman, D.W., Interpretation of Well-Block Pressures in Numerical Reservoir Simulation with Nonsquare Grid Blocks and Anisotropic Permeability, SPE 10528, Journal of Petroleum Technology (June, 1983)
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).
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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).
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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).
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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).
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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
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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
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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
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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
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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
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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 /
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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
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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 /
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-- 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
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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
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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 /
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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
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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 /
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-- 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
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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 -------------------------------------------------------------------
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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
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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
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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
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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
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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
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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
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-- 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 /
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/ -- 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 / /
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-- Advance to 20 years TSTEP 18*365 / END
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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
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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
180 0.5728 3847.622 1.95E-05 Perm x Perm y Perm z
Layer 1 5 5 0.5 Layer 2 0.5 0.5 0.5 Layer 3 2 2 0.05
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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
180 0.554003 2863.75 3.00E-05 Perm x Perm y Perm z
Layer 1 50 50 5 Layer 2 5 5 5 Layer 3 20 20 2.5
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20% PERMEABILITY VARIATION
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 Perm x Perm y Perm z
Layer 1 100 100 10 Layer 2 10 10 10 Layer 3 40 40 5
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30% PERMEABILITY VARIATION
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 Perm x Perm y Perm z
Layer 1 150 150 15 Layer 2 15 15 15 Layer 3 60 60 7.5
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75% PERMEABILITY VARIATION
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 Perm x Perm y Perm z
Layer 1 375 375 37.5 Layer 2 37.5 37.5 37.5 Layer 3 150 150 18.75
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90% PERMEABILITY VARIATION
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 Perm x Perm y Perm z
Layer 1 450 450 45 Layer 2 45 45 45 Layer 3 180 180 22.5
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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
360 0.521464 2008.809 6.44E-05 Perm x Perm y Perm z
Layer 1 500 500 50 Layer 2 50 50 50 Layer 3 200 200 25
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
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
360 0.5728 3717.704 2.15E-05 Perm x Perm y Perm z
Layer 1 5 5 0.5 Layer 2 0.5 0.5 0.5 Layer 3 2 2 0.05
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
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
360 0.519712 2322.975 4.84E-05 Perm x Perm y Perm z
Layer 1 50 50 5 Layer 2 5 5 5 Layer 3 20 20 2.5
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
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
360 0.521479 2196.605 5.79E-05 Perm x Perm y Perm z
Layer 1 100 100 10 Layer 2 10 10 10 Layer 3 40 40 5
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
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
360 0.51521 2133.93 6.14E-05 Perm x Perm y Perm z
Layer 1 150 150 15 Layer 2 15 15 15 Layer 3 60 60 7.5
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
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
360 0.537602 2008.33 7.17E-05 Perm x Perm y Perm z
Layer 1 375 375 37.5 Layer 2 37.5 37.5 37.5 Layer 3 150 150 18.75
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
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
360 0.528194 2008.354 6.71E-05 Perm x Perm y Perm z
Layer 1 450 450 45 Layer 2 45 45 45 Layer 3 180 180 22.5
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
166
18 MONTHS HISTORY PERIOD SIMULATION
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
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
167
1% TIME FGOR FPR FWCT (DAYS)
(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
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
168
75% TIME FGOR FPR FWCT (DAYS)
(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
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
169
90% TIME FGOR FPR FWCT (DAYS)
(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
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
170
24 MONTHS HISTORY PERIOD SIMULATION
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
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
171
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-05 210 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
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
172
75% TIME FGOR FPR FWCT (DAYS)
(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
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
173
90% TIME FGOR FPR FWCT (DAYS)
(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
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
174
48 MONTHS HISTORY PERIOD SIMULATION
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
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
175
75% TIME FGOR FOPT FPR FWCT (DAYS)
(MSCF/STB) (STB) (PSIA)
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
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
176
90% TIME FGOR FOPT FPR FWCT (DAYS)
(MSCF/STB) (STB) (PSIA)
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
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
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
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90% - 18 MONTHS 75% - 18 MONTHS FGOR FPR FWCT FGOR FPR FWCT
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
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75% - 24 MONTHS 90% - 24 MONTHS FGOR FPR FWCT FGOR FPR FWCT
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
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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
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181
APPENDIX D
PLOTS OF BLACK OIL AND COMPOSITIONAL SIMULATION GENERATED DATA
Andrew Oghena, Texas Tech University, May 2007
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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.
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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
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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
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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
OBSERVED DATASIMULATED DATA
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186
10 YEARS PREDICTION WITH 2 MONTHS HISTORY DATA
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 DATASIMULATED DATA
10 YEARS PREDICTION WITH 2 MONTHS HISTORY DATA
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 DATASIMULATED DATA
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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%
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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%
10 YEARS PREDICTION WITH 24 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
OBSERVED DATASIMULATED DATA
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
189
10 YEARS PREDICTION WITH 24 MONTHS HISTORY DATA
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
10 YEARS PREDICTION WITH 24 MONTHS HISTORY DATA
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 DATASIMULATED DATA
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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%
10 YEARS PREDICTION WITH 24 MONTHS HISTORY DATA
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 DATASIMULATED DATA
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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%
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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%
48 MONTHS WATER-CUT MATCH
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%
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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%
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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%
12 MONTHS PRESSURE MATCH
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%
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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%
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%
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
196
18 MONTHS PRESSURE MATCH
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%
18 MONTHS WATER-CUT MATCH
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%
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
OBSERVED OPTSIMULATED OPT
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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
OBSERVED OPTSIMULATED OPT
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
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
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
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Quantification of Uncertainties Associated with Reservoir Performance Simulation
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
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
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
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
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
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
205
Model with Minimum Sum of Square Optimization
BLACK OIL COMPOSITIONAL TIME FGOR FOPT FWCT FGOR FOPT FWCT
(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
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
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
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
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
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
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
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
209
Model with Maximum Sum of Square Optimization
BLACK OIL COMPOSITIONAL TIME FGOR FOPT FWCT FGOR FOPT FWCT
(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
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
Andrew Oghena, Texas Tech University, May 2007
Quantification of Uncertainties Associated with Reservoir Performance Simulation
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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
Andrew Oghena, Texas Tech University, May 2007