UTILIZATION OF ARTIFICIAL NEURAL NETWORKS IN THE ... · Figure 4.15: Thickness prediction errors with and without the output functional links: region’s area/region’s permeability

The Pennsylvania State University

The Graduate School

Department of Energy and Geo-Environmental Engineering

UTILIZATION OF ARTIFICIAL NEURAL NETWORKS IN THE

OPTIMIZATION OF HISTORY MATCHING

A Thesis in

Petroleum and Natural Gas Engineering

by

Asha Ramgulam

2006 Asha Ramgulam

Submitted in Partial Fulfillment of the Requirements

for the Degree of

Master of Science

May 2006

I grant The Pennsylvania State University the nonexclusive right to use this work for the University's own purposes and to make single copies of the work available to the public on a not-for-profit basis if copies are not otherwise available.

Asha Ramgulam

The thesis of Asha Ramgulam was reviewed and approved* by the following:

Turgay Ertekin Professor of Petroleum and Natural Gas Engineering George E. TrimbleChair in Earth and Mineral Sciences Graduate Program Chair in Petroleum and Natural Gas Engineering Thesis Co-Advisor

Peter B. Flemings Professor of Geosciences Thesis Co-Advisor

Robert W. Watson Associate Professor of Petroleum and Natural Gas Engineering and Geo-

Environmental Engineering

Zuleima T. Karpyn Assistant Professor of Petroleum and Natural Gas Engineering

*Signatures are on file in the Graduate School

iii

ABSTRACT

Artificial neural networks are becoming increasingly popular in the oil and gas

industry. In the past, studies have been done on the use of artificial neural networks in

reservoir characterization, field development and formation damage prediction, to name a

few. The aim of this study is to provide guidelines to successfully train and develop an

artificial neural network (ANN) that will predict reservoir properties that can give an

improved history match when input into the reservoir simulation model. An ANN was

developed to improve the history match with a ‘small’ number of simulation runs for a

reservoir that produced oil, gas and water for ten years. Due to a lack of specific

protocols for this type of study, the trial and error process was utilized to establish

guidelines and suggestions.

The neural network was developed by using an inverse solution method to

formulate the training and testing data. Normalization of the data simplified the neural

network, improved its effectiveness and enhanced its performance. The feed-forward

network with back-propagation and the hyperbolic tangent sigmoid function (tansig) in

the hidden layers of the network proved to be most effective in the training/learning

process.

Results indicated that functional links and eigenvalues of various matrices were

effective in the training/learning process. The functional links and eigenvalues provided

the network with the necessary connections that linked the inputs to the required outputs.

iv

It was necessary to input production differences between the historical and simulated

performances at specific times to successfully train the network and predict realistic

property values for the reservoir. Data structure and production time intervals influenced

the training time as well as the accuracy of the predictions. If time intervals were too

short, training times were longer, memorization occurred, and the network could not

accurately predict the reservoir’s properties. Most of the effective functional links that

were successful in the training/learning process included relationships between

permeability and other factors such as porosity, areas of the regions in the reservoir and

the distances from the producer to the boundaries of the reservoir.

The M4.1 reservoir in the Tahoe Field located in the Gulf of Mexico was used as

a case study to illustrate the use of artificial neural networks in decreasing the amount of

numerical reservoir simulations required to obtain an improved history match. The

effective parameters, obtained from network development, were applied to data from the

M4.1 reservoir simulations to determine which functional links and architecture would be

most effective in training the network. It was observed that some of the functional links

and network structures that were effective in network development were also effective in

the ANN developed for the M4.1 reservoir while some were not.

v

TABLE OF CONTENTS

LIST OF FIGURES .....................................................................................................vii

LIST OF TABLES.......................................................................................................xii

NOMENCLATURE ....................................................................................................xv

ACKNOWLEDGEMENTS.........................................................................................xvii

Chapter 1 INTRODUCTION AND PROBLEM STATEMENT ...............................1 1.1 Introduction.....................................................................................................1 1.2 History matching ............................................................................................2 1.3 Artificial neural networks ...............................................................................4 1.4 Artificial neural networks and history matching: Problem statement ............7

Chapter 2 RESERVOIR SIMULATION MODEL USED TO FORMULATE TRAINING AND TESTING DATA....................................................................11

2.1 Simple multi-phase reservoir modeled with one region and one producer ....11 2.2 Multi-phase reservoir modeled with four regions and one producer..............13 2.3 Multi-phase reservoir modeled with four regions and three producers. .........13

Chapter 3 DATA FORMULATION AND TRAINING STRATEGY.......................17

3.1 Training and testing data ................................................................................17 3.2 Feed-forward network with back-propagation ...............................................19 3.3 Error definitions..............................................................................................23

Chapter 4 CASE STUDIES AND NETWORK DEVELOPMENT...........................24

4.1 Criteria used to determine the structure of the artificial neural network........24 4.2 Homogeneous reservoir with one region and one producing well .................25 4.3 Reservoir with four regions and one producing well......................................31 4.4 Reservoir with four regions and three producing wells..................................49

Chapter 5 UTILIZATION OF AN ARTIFICIAL NEURAL NETWORK FOR PREDICTION OF M4.1 RESERVOIR PROPERTIES AND THE RESULTING HISTORY MATCH ......................................................................58

5.1 Utilization of ANN in predicting the properties for a single-region homogeneous property reservoir ...................................................................59 5.1.1 Training and testing the ANN with data generated from the single-

region M4.1 reservoir simulations .........................................................60

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5.1.2 Prediction of properties for the M4.1 reservoir ....................................65 5.2 Utilization of the ANN in predicting the properties of the reservoir that is

delineated into four regions ...........................................................................66 5.2.1 Training and testing of the ANN with data generated from the

delineated Tahoe M4.1 reservoir simulations ........................................70 5.2.2 Predicting the properties in each region of the M4.1 reservoir ............73

Chapter 6 DISCUSSION AND CONCLUSIONS......................................................81

BIBLIOGRAPHY........................................................................................................91

Appendix A RESERVOIR ROCK AND FLUID PROPERTIES USED TO FORMULATE TRAINING AND TESTING DATA USING LANDMARK’S DTVIP RESERVOIR SIMULATOR .......................................88

A.1 Initialization Data ..........................................................................................88 A.2 Rock property data.........................................................................................89 A.3 Black oil PVT data.........................................................................................94

Appendix B TRAINING AND TESTING SAMPLE DATA................................95

Appendix C SAMPLE MATLAB PROGRAMS USED IN DESIGNING, TRAINING AND TESTING THE ARTIFICIAL NEURAL NETWORK .........100

C.1 Extraction of data from historical production and simulated runs.................100 C.2 Training the network......................................................................................106 C.3 Output of training errors ................................................................................107 C.4 Output of testing errors ..................................................................................109 C.5 Prediction of reservoir’s properties................................................................112

Appendix D RESERVOIR ROCK AND FLUID PROPERTIES USED TO BUILD THE TAHOE M4.1 RESERVOIR SIMULATION MODEL.................116

Appendix E INSERT: “TAHOE FIELD CASE STUDY – UNDERSTANDING RESERVOIR COMPARTMENTALIZATION IN A CHANNEL-LEVEE SYSTEM”.............................................................................................................118

vii

LIST OF FIGURES

Figure 1.1: Three-layer neural network. Inputs are in the first layer and the outputs are in the last layer of the network. The middle layer provides the connecting link between the input and output layers............................................5

Figure 1.2: The simulation model is discretized into a grid and the simulations are run. The properties for k, ø and h are varied and simulations done that give different production profiles for each combination of properties.................8

Figure 1.3: Differences between the numerical simulations and the historical field performance at specific times are input to the network for training. The varied properties (k, ø and h) for each simulation run are used as the output data in the training process. An ANN is constructed so that the desired outputs can be obtained............................................................................................................9

Figure 1.4: A new data set that consists of small disparities between the numerical model and the historical field performance is input to the expert system developed. The output from this expert system is a combination of k, ø and h that are representative of the reservoir.....................................................9

Figure 1.5: The ANN predicted properties (k, ø and h) obtained from the expert system are input to the simulation model. This numerical simulation should give an improved history match. ..........................................................................10

Figure 2.1: The 16x16 grid block system used to formulate the training data for network development. The grid shows the initial fluid distribution in the reservoir. This is a multi-phase reservoir with one homogeneous property region and one producing well. ............................................................................12

Figure 2.2: The 16x16 grid block system used to formulate the training data for network development. The grid shows the initial fluid distribution for the multi-phase reservoir with four homogeneous regions and one producing well. ......................................................................................................................14

Figure 2.3: Transmissibility in different regions of the reservoir. The Channel, Region 2, has the least permeability and transmissibility in the reservoir as a result of the limited communication across the channel.......................................15

Figure 2.4: The 16x16 grid block system used to formulate the training data for network development. The grid shows the initial fluid distribution for the multi-phase reservoir with four homogeneous regions and three producing wells. .....................................................................................................................16

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Figure 4.1: Average training errors for each property using different data structures and different numbers of neurons in the hidden layers of a 4-layer neural network. .....................................................................................................27

Figure 4.2: Comparison of 2, 3 and 4 hidden layers on network performance for a single-region, homogeneous property reservoir with and without pressure differences as inputs to the network. The network that had 3 hidden layers gave the lowest prediction errors for permeability, porosity and thickness. ........28

Figure 4.3: Structure of the back-propagation network (BPN) used in the case study of a single-region, square homogeneous property reservoir with one producer. The delta values for each case are at specific times, as a result, there are multiple delta values for each case. .......................................................29

Figure 4.4: Reservoir property prediction errors using two properties as input data while the third property was predicted..........................................................34

Figure 4.5: Reservoir property prediction errors using one property as input data while the other two properties were predicted......................................................34

Figure 4.6: Average training errors with and without time as an input in the training data using a 4-layer neural network and 25 neurons in the hidden layers. Time used as an input decreased the average training errors....................35

Figure 4.7: Reservoir property prediction errors in each region using time, ∆qo, ∆qg, ∆qw, and ∆p as inputs and k, ø and h as outputs. The prediction errors were higher for all three properties.......................................................................37

Figure 4.8: Average training errors with and without the input functional links: time(yrs) and ∆qo∗ time(yrs). The training errors for permeability, porosity and thickness were decreased with the use of these input functional links. .........38

Figure 4.9: Average testing errors with and without the input functional links: time(yrs) and ∆qo∗ time(yrs). The testing errors for permeability, porosity and thickness were decreased with the use of these input functional links. .........39

Figure 4.10: Average permeability prediction error was decreased from 88% to 52% with the combined use of the output functional links kh and k/ø.................40

Figure 4.11: Average porosity prediction error was decreased from 8% to 4% with the combined use of the output functional links kh and k/ø. ........................40

Figure 4.12: Average thickness prediction error was decreased from 33% to 20% with the combined use of the output functional links kh and k/ø. ........................41

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Figure 4.13: Permeability prediction errors with and without the output functional links: region’s area/region’s permeability and closest and furthest distance from the well to the boundaries/region’s permeability. Both functional links decreased the prediction errors. ............................................................................42

Figure 4.14: Porosity prediction errors with and without the output functional links: region’s area/region’s permeability and closest and furthest distance from the well to the boundaries/region’s permeability.........................................43

Figure 4.15: Thickness prediction errors with and without the output functional links: region’s area/region’s permeability and closest and furthest distance from the well to the boundaries/region’s permeability.........................................43

Figure 4.16: Prediction errors of k, ø and h using 2, 3, 4 and 5 hidden layers in the network with varying output functional links. Inset: Average prediction errors of the properties k, ø and h. The 4-hidden layer network gave the lowest average prediction errors. ..........................................................................44

Figure 4.17: Structure of the back-propagation network (BPN) used in the case study of a square homogeneous reservoir with four regions and one producer. The network predicts k, ø and h for each region. .................................................47

Figure 4.18: Actual and predicted permeability values utilizing the final network. The use of the eigenvalue as an output functional link further decreased the prediction errors....................................................................................................48

Figure 4.19: Actual and predicted porosity values utilizing the final network. The use of the eigenvalue as an output functional link further decreased the prediction errors....................................................................................................48

Figure 4.20: Actual and predicted thickness values utilizing the final network. The use of the eigenvalue as an output functional link further decreased the prediction errors....................................................................................................49

Figure 4.21: Average prediction errors of permeability, porosity and thickness using different numbers of hidden layers and varying functional links in the network. The network with six hidden layers gave the lowest prediction errors for k, ø and h...............................................................................................50

Figure 4.22: Comparison of the base case permeability with the ANN-predicted permeability values using the output functional link permeability thickness.......52

Figure 4.23: Comparison of the base case porosity with the ANN-predicted porosity using the output functional link permeability thickness. ........................53

x

Figure 4.24: Comparison of the base case thickness with the ANN-predicted thickness using the output functional link permeability thickness. ......................53

Figure 4.25: Comparison between the historical profile and the simulated profile obtained from scenario 1. The graphs show large deviations from the historical profile, the largest being observed in the water production..................55

Figure 4.26: Comparison between the historical profile and the simulated profile obtained from scenario 2. The graphs show large deviations from the historical profile, the largest being observed in the water production..................56

Figure 4.27: Comparison between the historical profile and the simulated profile obtained from the ANN-predicted properties. There is a good history match for the oil and gas production as well as the pressure profile. The history match for water production is improved but it is not as close a match as the oil and gas production profiles. ............................................................................57

Figure 5.1: Initial fluid distribution and the location of the producing wells in the M4.1 reservoir shown on a curvilinear grid..........................................................59

Figure 5.2: Architecture of the neural network used to predict the properties of the single-region M4.1 reservoir...........................................................................64

Figure 5.3: Training errors for k, ø and h for 38 simulated cases. The average training error for each property was 15%, 3% and 7% respectively, with an overall average training error of 8%.....................................................................64

Figure 5.4: Testing errors for k, ø and h for 10 simulated cases. The average testing error for each property was 11%, 3% and 8% respectively with an overall average testing error of 7%.......................................................................65

Figure 5.5: Simulated production profiles for the single-region M4.1 reservoir using the ANN-predicted properties compared to the historical production data........................................................................................................................66

Figure 5.6: The four delineated regions in the M4.1 reservoir on a curvilinear grid: Region North of Fault A, the Channel, the East Levee and the West Levee respectively. The regions were delineated according to compartmentalization and varying permeability in each region. The Channel, Region 2, was simulated as having the least transmissibility and permeability due to the restricted communication between the East and West Levees. ...........67

Figure 5.7: Architecture of the neural network used to predict the properties of the single-region M4.1 reservoir...........................................................................70

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Figure 5.8: Training errors for k, ø and h in each region of the reservoir. The overall average training error was 35%. ...............................................................71

Figure 5.9: Testing errors for k, ø and h in each region of the reservoir. The average testing error was 20%..............................................................................72

Figure 5.10: Production profiles for the M4.1 reservoir using ANN-predicted properties for each region in the numerical simulation model compared to the historical production data. ....................................................................................74

Figure 5.11: Training and testing errors for the GOC and the OWC obtained from the ANN. The average training errors for the GOC and OWC were 0.72% and 0.53% respectively. The average testing errors for the GOC and the OWC were 0.92% and 0.85% respectively. .........................................................79

Figure 5.12: Improved history match to water production with ANN-predicted GOC at 10,100 ft and OWC at 10,300 ft. .............................................................80

Figure A.1: Relative permeability to oil and water in a two-phase system. ...............92

Figure A.2: Relative permeability to oil and gas in a two-phase system....................92

Figure A.3: Capillary pressure curve in the oil-water system. ...................................93

Figure A.4: Capillary pressure curve in the gas-oil system. .......................................93

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LIST OF TABLES

Table 3.1: Transfer functions that can be used in the middle layers of the network during the training/ learning process [Hagan et al, 1996]. ...................................21

Table 3.2: Inputs to the network that were used to predict the reservoir’s properties. The inputs consist of time, and the differences between the base case field performance and the numerical simulations. Acceptable differences were randomly generated between small negative and positive values. ..............22

Table 4.1: Twenty-two values for permeability, porosity and thickness were randomly generated within the ranges shown. The production and pressure profiles obtained by inputting these values into the simulation model were used to formulate the training and testing data. ....................................................26

Table 4.2: Inputs and outputs, at specific times, used in the training of the neural network for the single-region homogeneous property reservoir...........................26

Table 4.3: Comparison of ANN-predicted reservoir properties using a narrow data range and a wide data range in the formulation of the training data. Smaller prediction errors were observed with the training data set that was formulated using the wide data range and a larger number of simulations. .........31

Table 4.4: Randomly generated properties for 35 cases used in the reservoir simulations for generating training data for the reservoir with four regions. .......32

Table 4.5: Two reservoir properties and the differences between field performance and the base case simulation were used as inputs. One reservoir property was predicted. The output functional links were used as forcing functions in training the network..........................................................................33

Table 4.6: One reservoir property and the differences between field performance and the base case simulation were used as inputs. Two reservoir properties were predicted. The output functional links were used as forcing functions in training the network..............................................................................................33

Table 4.7: Inputs used to train the network. These inputs are at specific times for each case ...............................................................................................................36

Table 4.8: Outputs from the network..........................................................................36

Table 4.9: Inputs used in training the network for the reservoir with four homogeneous regions and three producing wells. ................................................51

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Table 4.10: Outputs used in training the network for the reservoir with four homogeneous regions and three producing wells. Permeability thickness product is the functional link used as a forcing function within the network.......51

Table 4.11: Comparison of the base case properties and ANN-predicted properties in each region of the reservoir. Two additional scenarios for reservoir properties were also simulated to compare the deviations from the historical field performance. .................................................................................55

Table 5.1: Randomly generated properties used in the 38 simulation runs for formulating the training data for the single-region reservoir................................61

Table 5.2: Randomly generated properties used in the 10 simulation runs for formulating the testing data for the single-region reservoir. ................................61

Table 5.3: Inputs and outputs used in training the network for the single-region homogeneous property reservoir. .........................................................................62

Table 5.4: Final inputs and outputs used in training the network for the single-region homogeneous property reservoir. ..............................................................63

Table 5.5: Randomly generated properties used in the 38 simulation runs for formulating the training data for the reservoir with four regions. ........................68

Table 5.6: Randomly generated properties used in the 10 simulation runs for formulating the testing data for the reservoir with four regions. ..........................69

Table 5.7: Predicted properties for each region of the M4.1 reservoir using two training data sets. The training data was formulated using a narrow data range with 38 simulations and a wide data range with 53 simulations. The ANN-predicted properties using both networks were similar. .......................................75

Table 5.8: Randomly generated values for the GOCs and OWCs used in 38 simulations for formulating the training data for the reservoir with four regions...................................................................................................................77

Table 5.9: Randomly generated values for GOCs and OWCs used in 10 simulations for formulating the testing data for the reservoir with four regions...................................................................................................................78

Table 5.10: Inputs and outputs to the network used in the prediction of the properties k, ø, h, GOC and OWC........................................................................78

Table A.1: Rock and fluid data used in the initialization of the artificial neural network (ANN) model. .........................................................................................88

xiv

Table A.2: End point input data used to generate relative permeability curves based on Corey’s Model for the water-oil system. ...............................................90

Table A.3: End point input data used to generate relative permeability curves based on Corey’s Model for the gas-oil system....................................................90

Table A.4: Two-phase relative permeability and capillary pressure data derived from end points obtained from laboratory data. ...................................................91

Table A.5: Black oil PVT data derived from fluid analysis in the M4.1 reservoir.....94

Table A.6: Black oil PVT data used in the reservoir simulator to design the ANN model. ...................................................................................................................94

Table D.1: Rock and fluid data used in the initialization of the M4.1 simulation model. ...................................................................................................................116

Table D.2: Properties of the fluid components used in the compositional model for the M4.1 reservoir in the Tahoe Field.............................................................117

xv

NOMENCLATURE

µo Oil viscosity, cp

µg Gas viscosity, cp

mse Mean square error performance function

msereg Mean square error with regularization performance function

Bg Gas formation volume factor, RB/MSCF

h Thickness, ft

k Permeability, md

krg Gas-phase relative permeability

krnw Relative permeability of the non-wetting phase

krog Oil-phase relative permeability in the presence of gas

krow Oil-phase relative permeability in the presence of water

krw Water-phase relative permeability

ø Porosity

Rv Solution oil-gas ratio, RB/STB

Bo Oil formation volume factor, RB/STB

Siw Irreducible saturation of the wetting phase

Sw Saturation of the wetting phase

Swn Normalized wetting phase saturation

t Time

psat Saturation pressure, psia

Rs Solution gas-oil ratio, SCF/STB

xvi

traingdx Network training function that updates weight and bias values according

to gradient descent momentum and an adaptive learning rate.

trainscg Network training function that updates weight and bias values according

to the scaled conjugate gradient method.

∆qo Differences between the simulated oil production and the historical oil

water production at specific times, STB/D

∆p Differences between the simulated pressure and the historical

pressure at specific times, psia

∆qg Differences between the simulated gas production and the historical gas

production at specific times, SCF/D

∆qw Differences between the simulated water production and the historical

production at specific times, STB/D

xvii

ACKNOWLEDGEMENTS

I would like to take this opportunity to thank some of the people who have been

an essential part of my studies at Penn State. First, I would like to thank Dr. Turgay

Ertekin for his invaluable guidance, inspiration and support in the development of this

work and throughout my studies at Penn State. My gratitude also goes out to Dr. Peter

Flemings for his support, guidance and enthusiasm that encouraged me to excel in my

endeavors. He has been a strong motivating force and an integral part of my studies in the

Petroleum Geosystems Initiative.

Thanks to my PNGE professors at Penn State; they have all contributed to my

successful development as a petroleum engineer. Special thanks go to Dr. Robert Watson

and Dr. Zuleima Karpyn who were always willing to provide their insight and direction.

In addition, I would like to thank Heather Nelson, Tom Canich, Damian Futrick and the

EGEE staff who assisted with many administrative and computer issues.

Joe Razzano and Chekwube Enunwa, my team members, were great to work with

and with whom many ideas were shared and developed along the way. Special thanks to

them for enriching my experience. I would also like to thank my family who has been a

constant source of support throughout this journey.

xviii

This study would not have been possible without the sponsors of the Petroleum

Geosystems Initiative. Thanks go to Shell Oil Company, Chevron Corporation, Kent and

Helen Newsham, Doug and Ellen Heller and the Geosystems Enrichment Fund.

Chapter 1

INTRODUCTION AND PROBLEM STATEMENT

This chapter provides general information on history matching and artificial

neural networks (ANN). A brief description is given to illustrate how artificial neural

networks are utilized to optimize the history matching process. The subsequent chapters

describe the procedures, programs and results of the neural network modeling.

1.1 Introduction

The utilization of artificial neural networks is becoming increasingly popular in

the oil and gas industry. This study provides some guidelines on the use of artificial

neural networks for history matching. A “simple” reservoir simulation model was

designed to simulate the depletion of fluids in a hydrocarbon reservoir. The model design

was influenced by data from the M4.1 reservoir in the Tahoe field located in the Gulf of

Mexico. The reservoir’s characterization was based on work done by Enunwa et al.

(Appendix E). Landmark’s DTVIP reservoir simulation model was used to simulate a

“small” number of simulation runs using estimated reservoir properties such as

permeability, porosity and thickness. Initially, the model was a simple black oil model for

a single-region, homogeneous property reservoir with one producer. The degree of

complexity of the model was then increased by adding regions and two more producing

wells. Network development consisted of training and testing to ensure that the artificial

2

neural network had been successfully trained and validated. The predictions were then

carried out with a novel data set.

1.2 History matching

The main objective of history matching is to test, validate and improve a reservoir

simulation model. Reservoir modeling is one of the most viable and reliable ways to

predict production performance and to understand reservoir flow mechanisms [Schiozer,

1999]. It is considered to be one of the most powerful predictive tools available to a

reservoir engineer [Ertekin et al, 2001]. A reservoir model is constructed by using

geological and geophysical data from specific wellbores that penetrate a hydrocarbon

reservoir. These data are in the form of well tests, seismic surveys, well log data, etc.

However, a large majority of the reservoir properties remains unknown and there are

many uncertainties. Consequently, the available data need to be adjusted for a simulation

model to accurately predict reservoir performance. Once a reservoir model is constructed

based on available data, it must be tuned with known reservoir behavior, which is known

as history matching. History matching is used to adjust the proposed model until

simulation results match the observed data [Schiozer, 1999].

History matching can be either manual or automatic. Automatic history matching

uses computer logic to adjust reservoir engineering data rather than direct engineering

judgment. However, manual history matching is more widely used as it utilizes the

engineering judgment of experienced professionals on the field. History matching

3

attempts to determine parameters such as porosity, permeability and thickness by

matching the properties of each gridblock in a simulator model that result in simulated

well pressures and production data that match as closely as possible to those measured

during production [Dye et al, 1986]. An initial model that is developed using available

data does not generally match historical performance. Normally, there will be deviations

from the historical performance and, as a result, adjustments to the reservoir simulation

model must be made so that there is an improved match. Once the historical production

data are matched, it indicates that the model approximates the actual reservoir behavior.

The simulation model can then be used to more accurately predict the behavior of the

reservoir as the hydrocarbons are being produced.

The final history-matched model is not unique. There may be several history-

matched models that can provide equally acceptable matches to past reservoir

performance but may yield significantly different future predictions [Ertekin et al, 2001].

However, a history-matched model gives a better understanding of the reservoir and

clarifies uncertainties such as aquifer support, paths of fluid migration, communication

barriers and depletion mechanisms. In addition, areas of by-passed reserves and unusual

operating conditions can be more easily identified. Once the historical production data

are matched, a much greater confidence can be placed in the predictions made with the

model [Ertekin et al, 2001].

4

1.3 Artificial neural networks

Artificial neural networks are information processing systems that are a rough

approximation and simplified simulation of the biological neuron network system. The

first practical application of artificial neural networks came in the late 1950s when Frank

Rosenblatt and his colleagues demonstrated their ability to perform pattern recognition

[Hagan et al, 1996]. However, interest in neural networks dwindled due to its limitations

as well as the lack of new ideas and powerful computers [Hagan et al, 1996]. With some

of these hurdles overcome in the 1980s, and with the development of the back-

propagation algorithm for training multilayer perceptron networks, there was a renewed

interest in the field. Since then, artificial neural networks have been improved and

applied in aerospace, automotive, defense, transportation, telecommunications,

electronics, entertainment, manufacturing, financial, medical and the oil and gas industry

to name a few.

Artificial neural networks have the ability to recognize complex patterns quickly

with a high degree of accuracy, it makes no assumptions about the nature and distribution

of the data and they are not biased in their analysis. In addition, artificial neural networks

have non-linear tools and as such are good at predicting non-linear behaviors. Neural

networks form a broad category of computer algorithms that solve several types of

problems, including pattern classification, functions approximation, pattern completion,

pattern association, filtering, optimization and automatic control [Mohaghegh, 2000].

5

Two primary elements make up neural networks: processing elements, which

process information, and interconnections or links between the processing elements. The

structure of the neural network is defined by the interconnection architecture between the

processing elements. Information processing within an artificial neural network occurs in

the processing elements that are called neurons which are grouped into layers. Signals are

passed between neurons over the connecting links as shown in Figure 1.1.

In this example, there are three inputs into the network, therefore, there are three

input neurons in the first layer; there are three outputs from the network, consequently,

there will be three output neurons in the output layer. The optimum number of middle

layers and the number of neurons in these layers are determined by trial and error during

Figure 1.1: Three-layer neural network. Inputs are in the first layer and the outputs are in the last layer of the network. The middle layer provides the connecting link between theinput and output layers.

Input Layer(3 neurons)

Middle Layer Output Layer (3 neurons)

Inputs Outputs

6

the training/learning process. Multi-layer networks are generally, more powerful than

single-layer networks [Hagan et al, 1996]. Each connecting link has an associated weight,

which, in a typical neural network, multiplies the signal being transmitted. Each neuron

applies an activation function (usually non-linear) to its net input to determine its output

signal [Mohaghegh, 2000]. The middle layers in the network are used to develop an

internal representation of the relationship between the variables and are chosen to satisfy

a specific problem that the neurons are attempting to solve. Through this process, which

is known as training, the output neurons are taught to give the correct answer depending

on the stimulus presented to the input neurons.

Neural networks can be classified into two major categories on the basis of

training methods: supervised and unsupervised. Unsupervised neural networks are mainly

clustering and classification algorithms in which no feedback is provided to the network.

Unsupervised networks only require inputs which are classified into groups and clusters.

During a supervised training process, both input and output are presented to the network

to permit learning on a feedback basis [Mohaghegh, 2000]. The training set is used to

develop the desired network which can then predict the parameters that are required. A

trained network consists of a set of weights that give the desired output for the

corresponding inputs. The training process determines the magnitude of the weights that

produces the desired output for the corresponding inputs. The resulting network can then

be used for predictions.

7

Artificial neural networks in the oil and gas industry are based on supervised

training algorithms that have the potential for solving many of the challenging and

complex problems in the oil and gas industry [Mohaghegh, 2000]. The use of artificial

neural networks in the oil and gas industry has been steadily increasing over the years.

Previously, some of the studies done on the applications of neural networks have been in

reservoir characterization, field development, two-phase flow in pipes, and identification

of well test interpretation models, completion analysis, formation damage prediction,

permeability prediction and fractured reservoirs.

1.4 Artificial neural networks and history matching: Problem statement

History matching requires a large number of simulation runs because there are

many uncertainties in reservoir characterization. The procedure of history matching is

costly, time-consuming and difficult. Time is an important factor and as such tools are

required that can make the best use of time and engineering judgment. This work

explores the possibilities of using artificial neural networks to minimize the amount of

time and simulation runs required in obtaining a history match for a reservoir.

This study provides a link between history matching using a reservoir simulation

model and an artificial neural network. The reservoir simulator is used to provide the

necessary training and testing data that are used to train and validate the network. The

resulting network is then employed to make predictions with a data set for which it has

not been trained. The intent of this thesis is to establish guidelines and suggestions for the

8

construction of artificial neural networks that can assist in optimizing the history

matching process by predicting the properties of the reservoir that can lead to improving

the history match with fewer simulation runs.

Initially, a numerical simulation model is defined using a grid system and a base

case is set up that represents the historical production of the field. A “small” number of

simulations are carried out by varying the history matching parameters such as

permeability, porosity and thickness that results in different production histories for each

case (Figure 1.2). The differences in fluid production between each case and the base

case at specific times are used to formulate the training data. An inverse solution is used

to set up the training of the network that uses these differences as input parameters. The

varied properties of the reservoir, i.e. permeability, porosity and thickness that were used

to formulate the production for each case are used as the target data for training the

network.

time

qo

time

qg

time

qw

time

p

time

qo

time

qo

time

qo

timetime

qo

time

qg

time

qgqg

time

qw

timetimetime

qw

time

p

time

pp

Figure 1.2: The simulation model is discretized into a grid and the simulations are run. The properties for k, ø and h are varied and simulations done that give differentproduction profiles for each combination of properties.

9

An artificial neural work can then be constructed that will train the expert system

in mapping the disparities between field performance and the results generated by

numerical simulations (Figure 1.3).

The expert system that has been trained and tested can then be used to predict a

combination of the history matching parameters that will generate acceptable prescribed

disparities between the numerical model and the field performance (Figure 1.4).

…….Training…….TrainingTraining

Figure 1.3: Differences between the numerical simulations and the historical field performance at specific times are input to the network for training. The varied properties(k, ø and h) for each simulation run are used as the output data in the training process. An ANN is constructed so that the desired outputs can be obtained.

Reservoir Parameters

Prediction Reservoir Parameters

Prediction

Figure 1.4: A new data set that consists of small disparities between the numerical model and the historical field performance is input to the expert system developed. The output from this expert system is a combination of k, ø and h that are representative of thereservoir.

10

The history matching parameters obtained from the expert system are then input

into the simulation model and compared with actual historical production data to ensure

that the guidance suggested by the system is accurate (Figure 1.5). If the artificial neural

network has been successful there should be an improved history match.

time

qo

time

qg

time

qw

time

p

time

qo

time

qo

time

qo

time

qo

time

qg

time

qg

time

qg

time

qg

time

qw

time

qw

time

qw

time

p

time

p

time

p

time

p

Figure 1.5: The ANN predicted properties (k, ø and h) obtained from the expert system are input to the simulation model. This numerical simulation should give an improved history match.

Chapter 2

RESERVOIR SIMULATION MODEL USED TO FORMULATE TRAINING AND TESTING DATA

This chapter describes the design of the reservoir simulation model from which

the training and testing data were generated. A multi-phase black oil model was used to

simulate the depletion of hydrocarbons in the reservoir. Figure 2.1 illustrates the initial

fluid distribution of the square shaped reservoir that is discretized into a 16x16 grid block

system on a rectangular grid. The parameters such as rock and fluid properties used to

build the reservoir simulation model are described in Appendix A. The training and

testing data were formulated from production data for each phase by varying the

permeability, porosity and thickness in the reservoir simulation model.

2.1 Simple multi-phase reservoir modeled with one region and one producer

Initially, the reservoir was considered to be a single-region, homogeneous

property reservoir with one producing well (Figure 2.1). A “small” number of simulation

runs were done using Landmark’s DTVIP reservoir simulator. Randomly generated

values for permeability, porosity and thickness were generated within estimated ranges

and used to simulate depletion of the reservoir for each case. The range of values chosen

was estimated from core data obtained from the M4.1 reservoir in the Tahoe field located

in the Gulf of Mexico (Appendix E). The average properties that were used as the base

case (historical production) for permeability, porosity and thickness were estimated at 80

12

md, 0.3 and 150 ft respectively. The differences between the varied numerical

simulations and the base case field performance at different times were then used to

formulate the training and testing data (Appendix B).

2DVIEW Study[ANN_Sim] Case[Base] Time[01-JAN-1994, 0 days] TimeStep[ 0] Azimuth[ 0.00] Inclination[ 0.00]

4

TERNARY DIAGRAM

SG0.00

SG0.90

SW0.10

SW1.00

SO0.00

SO0.90

Producing Well

0miles

.5

Figure 2.1: The 16x16 grid block system used to formulate the training data for network development. The grid shows the initial fluid distribution in the reservoir. This is a multi-phase reservoir with one homogeneous property region and one producing well.

13

2.2 Multi-phase reservoir modeled with four regions and one producer

The reservoir model was made more complex by dividing the field into regions

similar to that described by Enunwa et al (Appendix E). A different base case was then

used with each region having its own average properties. Thirty-five randomly generated

cases (Table 4.4) were used to formulate the training and testing data. Pressure data in the

M4.1 reservoir indicated that there is limited communication across a channel that is

located between the East and West Levee (Appendix E). This limited communication was

simulated by using lower permeability in the Channel, Region 2 (Figure 2.2 & 2.3).

Figure 2.3 shows the transmissibility across each of the regions due to the varied

permeability.

2.3 Multi-phase reservoir modeled with four regions and three producers.

Two additional wells were then added to the reservoir simulation model

(Figure 2.4). Simulations were done for the same 35 cases (Table 4.4) and the same base

case with permeability, porosity and thickness of 80 md, 0.3 and 150 ft. respectively.

14

Figure 2.2: The 16x16 grid block system used to formulate the training data for network development. The grid shows the initial fluid distribution for the multi-phase reservoir with four homogeneous regions and one producing well.

15

2DVIEW Study[Comparisons] Case[Base] Time[01-JAN-1994, 0 days] TimeStep[ 0] Azimuth[ 0.00] Inclination[ 0.00]

4

Y-DIRECTION TRANSMISSIBILITY[TY] (RB-CP/DAY-PSI)

7.206.405.604.804.003.202.401.600.80 8.000.00

Producing Well

0 .5miles

Figure 2.3: Transmissibility in different regions of the reservoir. The Channel, Region 2, has the least permeability and transmissibility in the reservoir as a result of the limited communication across the channel.

16

Figure 2.4: The 16x16 grid block system used to formulate the training data for network development. The grid shows the initial fluid distribution for the multi-phase reservoir with four homogeneous regions and three producing wells.

Chapter 3

DATA FORMULATION AND TRAINING STRATEGY

This chapter describes the procedure for formulating the training and testing data

that were used in the development of the artificial neural network (ANN). In addition, it

explains the significance of normalizing the data and the use of functional links within

the network. The methodology employed in the use of the feed-forward network with

back-propagation is also described.

3.1 Training and testing data

The production profiles obtained from the numerical simulations were divided

into training and testing data sets. The training data consisted of the differences between

the base field performance and the numerical simulations every 90 days during

production for the period 1994 to 2004 (Appendix B). These differences in production

and pressure (∆qo, ∆qg, ∆qw, ∆p) were input to the network and the varied properties for

each simulation (k, ø and h) were used as the output target data. The network was

constructed such that when trained, the results obtained were close to the target data.

The testing data were extracted from the training data and they were not used in

the training data set. The differences between the historical field performance and the

simulated cases (∆qo, ∆qg, ∆qw, ∆p) with known properties (k, ø and h) were input to the

18

network and the properties for each region were predicted. The ANN-predicted properties

were compared to the known values of k, ø and h that gave the differences (∆qo, ∆qg,

∆qw, ∆p) that were used as the input. These test data were used to verify and validate the

network performance for prediction.

The neural network was developed with an initial small number of input

parameters. The input data consisted of time, ∆qo, ∆qg, ∆qw, ∆p and associated functional

links that were dependent on the case study. Initially, time was in increments of 30 days

and the delta values were the differences between the base case performance and the

numerical simulations at corresponding times (Appendix B). As the network became

more complex and training times became longer, the amount of data was decreased and

data were input every 90 days. The input data were pre-processed by normalizing and

utilizing functional links.

Normalizing the data standardizes the numerical range of the input data and

enhances the fairness of training by preventing an input with large values from swamping

out another input that is equally important but with smaller values [Al-Fattah, 1994].

Functional links were also used to enhance the network’s performance. These are

mathematical functions of the input or target data which amplify subtle differences in the

data and provide the advantage of additional input or output parameters to or from the

network, which can provide more connections to the middle layer and correspondingly

improve predictive capabilities [Doraisamy, 1998].

19

Functional links influence the network’s training/learning process so there is a

better understanding between the inputs and outputs. One example a functional link used

was the reciprocal of time. In this instance, the functional link did not provide an

improvement to the neural network, illustrating that some functional links may have no

effect on the results. Effective functional links were determined by trial and error. An

example of an effective functional link used in the target data was permeability thickness

product, kh (Section 4.3). The target output of the neural network was the permeability,

porosity, thickness, and the functional link kh. In this case, the use of a functional link

amplified small changes that improved the training and testing errors and gave improved

prediction results.

3.2 Feed-forward network with back-propagation

The feed-forward network with back-propagation was utilized in the training

process. This network was chosen because it is the most well-known and widely used

feed-forward network used in engineering applications [Doraisamy, 1998]. In addition, it

is easy to implement, trains faster than other types of networks, and solves many types of

problems correctly [Centilmen, 1999]. There was a large data set to be trained and as

such, the speed of training was an important factor in training the network presented in

this thesis.

The feed-forward network with back-propagation operates in two steps. First, the

input data are presented to the input layer and the transfer functions process the

20

information through the layers until the network’s response is generated at the output

layer. The optimum transfer function in each layer is determined by trial and error.

Transfer functions are listed in Table 3.1.

Second, the network’s response is compared to the desired output and if it does

not agree, an error is generated. The error signals are then transmitted back from the

output layer to each node in the intermediate layers. Each intermediate layer receives a

portion of the total error signal, based roughly on the relative contribution the unit made

to the original output [Ali, 1994]. Based on the error signals received, connection weights

between layer neurons are then updated until the network parameters have been

determined that gives a pre-defined performance goal. Using this method, the network

learns to reproduce outputs by learning patterns contained within the data. Once the

network is trained, it can then make predictions from a novel set of inputs.

The resulting ANN developed from training can be used to predict the required

outputs of the reservoir’s properties. Predicting the properties of the reservoir was

achieved by using time, ∆qo, ∆qg, ∆qw, and ∆p as inputs to the network at specific times

(Table 3.2). These inputs were randomly generated between small positive and negative

values that were the allowable deviations from the historical production at each time.

Using these inputs to the network, it was then possible to predict the output properties of

k, ø and h. The ANN-predicted properties were then input into the simulator to give an

improved history match for the reservoir.

21

Table 3.1: Transfer functions that can be used in the middle layers of the network duringthe training/ learning process [Hagan et al, 1996].

Hard Limit a = 0, n < 0 a = 1, n ≥ 0 hardlim

Symmetric Hard Limit

a = -1, n < 0 a = +1, n ≥ 0 hardlims

Linear a = n purelin

Saturating Linear

a = 0, n < 0 a = n, 0 ≤ n ≤ 1

a = 1, n > 1satlin

Symmetric Saturating

Linear

a = -1, n < -1 a = n, -1 ≤ n ≤ 1

a =1, n > 1satlins

Log-Sigmoid logsig

Hyperbolic Tangent Sigmoid

tansig

Positive Linear a = 0, n < 0 a = n, 0 ≤ n poslin

Competitive a = 1, neuron with max n a = 0, all other neurons compet

Name Input/Output Relation Icon MATLAB Function

nea −+=

11

nn

nn

eeeea −

−

+−

=

C

22

Table 3.2: Inputs to the network that were used to predict the reservoir’s properties. The inputs consist of time, and the differences between the base case field performance and the numerical simulations. Acceptable differences were randomly generated between small negative and positive values.

1 9.00E-06 -9.61E-06 4.52E-07 -9.70E-06

90 -5.38E-06 3.63E-06 7.60E-06 5.36E-06

180 2.14E-06 -2.41E-06 -6.54E-06 9.42E-06

270 -2.80E-07 6.64E-06 9.59E-06 9.80E-06

360 7.83E-06 5.63E-08 -4.57E-06 5.78E-06

450 5.24E-06 4.19E-06 -4.95E-06 -1.23E-06

540 -8.71E-07 -1.42E-06 7.51E-06 -3.38E-08

630 -9.63E-06 -3.91E-06 4.75E-06 -5.72E-06

720 6.43E-06 -6.21E-06 -7.27E-06 2.87E-06

810 -1.11E-06 -6.13E-06 -9.76E-06 -3.60E-06

900 2.31E-06 3.64E-06 7.88E-06 9.20E-06

990 5.84E-06 -3.94E-06 -6.02E-06 4.53E-06

1080 8.44E-06 8.33E-07 -4.03E-06 -1.76E-06

1170 4.76E-06 -6.98E-06 3.23E-06 4.89E-06

1260 -6.47E-06 3.96E-06 -4.31E-06 -4.64E-06

1350 -1.89E-06 -2.43E-06 -6.16E-07 -1.20E-06

1440 8.71E-06 7.20E-06 -8.70E-06 8.67E-06

1530 8.34E-06 7.07E-06 9.77E-06 3.67E-06

1620 -1.79E-06 1.87E-06 1.66E-06 -5.75E-06

1710 7.87E-06 -6.90E-08 -1.53E-06 6.78E-06

1800 -8.84E-06 8.00E-06 3.10E-07 2.58E-06

1890 -2.94E-06 6.43E-06 -3.32E-06 -7.32E-06

1980 6.26E-06 2.90E-06 -1.34E-06 -5.86E-06

2070 -9.80E-06 6.36E-06 -5.48E-06 2.14E-06

2160 -7.22E-06 3.20E-06 1.60E-06 2.60E-06

2250 -5.94E-06 -3.16E-06 5.21E-06 -2.59E-06

2340 -6.03E-06 -4.21E-06 5.96E-07 1.50E-06

2430 2.08E-06 -3.18E-06 2.81E-06 -9.72E-07

2520 -4.56E-06 6.82E-07 -5.82E-06 -9.12E-06

2610 -6.02E-06 4.54E-06 -2.40E-06 -9.46E-06

2700 -9.69E-06 -3.81E-06 5.67E-06 -3.75E-06

2790 4.94E-06 6.77E-06 3.62E-06 -9.74E-06

2880 -1.10E-06 1.36E-06 -7.78E-07 -2.32E-06

2970 8.64E-06 -2.59E-06 1.36E-06 3.66E-06

3060 -6.80E-07 4.05E-06 5.88E-06 -8.14E-06

3150 -1.63E-06 9.31E-07 -8.82E-06 -9.29E-06

3240 6.92E-06 -1.10E-06 2.06E-06 2.25E-06

3330 5.03E-07 3.89E-06 -8.99E-06 2.17E-06

3420 -5.95E-06 2.43E-06 -1.69E-06 -9.68E-06

3510 3.44E-06 5.90E-06 -3.90E-06 -9.67E-06

3600 6.76E-06 9.14E-06 7.49E-06 -6.20E-06

∆p (psia)Time (days) ∆qo (STB/D) ∆qg (MSCF/D) ∆qw (STB/D)

23

3.3 Error definitions

Errors define the performance of the network being trained and as such

three types of errors were used throughout this study. The following gives a definition for

each type of error:

• Training errors – errors obtained when the trained network’s response is

compared to the desired output for each set of input and output data for a specific

case. The average training error is the average of all training errors for all cases.

• Testing errors – errors obtained when the network’s response to the inputs of an

untrained data set are compared to known output values for a specific case. The

average testing error is the average of all testing errors for all cases.

• Prediction errors – errors obtained when predicted property values are compared

to the known property values. For example, the properties (k, ø and h) for the base

case are compared to the ANN-predicted values for permeability, porosity and

thickness. The average prediction error is the average error for a specific property

throughout the reservoir.

Chapter 4

CASE STUDIES AND NETWORK DEVELOPMENT

This chapter describes the methodologies employed in the development of the

artificial neural network (ANN). The case studies are presented in increasing order of

complexity. The initial case study focuses on a square reservoir considered to be a single

homogeneous property region with one producer. Subsequent studies divide the reservoir

into four regions and add two producers to the field. The ANN models developed uses the

feed-forward network with back-propagation. The convergence criterion and the number

of epochs set for each study are adjusted such that the network does not over-fit the data

and prevents over-training that can cause the network to memorize rather than generalize.

4.1 Criteria used to determine the structure of the artificial neural network

The optimum size and structure of the ANN was determined by trial and error.

Some of the factors that were taken into consideration when determining the best

architecture for training the network were the following:

• Data structure – the structure of data inputs to and outputs from the

network

• Number of hidden layers – the number of middle layers in the network

that would optimize the training/learning process

25

• Number of neurons in each layer – the number of neurons within the

hidden layers of the network that would produce the desired characteristics

of the output layer

• Transfer function – a linear or non-linear function in each layer that is

chosen to satisfy some specification of the problem that the neuron is

attempting to solve

• Training algorithm – a training function that updates the weight and bias

values within the network

• Performance functions – a function that measures the network’s

performance

• Functional links – mathematical functions of inputs or outputs which

amplify subtle differences in the data

4.2 Homogeneous reservoir with one region and one producing well

The rock and fluid properties used to construct the numerical simulation model

for generating the training and testing data are described in Appendix A. Figure 2.1

shows the reservoir’s initial fluid distribution and the location of the producer for the base

case. The base case consisted of permeability, porosity and thickness values of 80 md, 0.3

and 150 ft respectively (Table 4.1). Twenty-two randomly generated values of

permeability, porosity and thickness within the ranges shown in Table 4.1 were varied in

the simulation model. The differences (∆qo, ∆qg, ∆qw, ∆p) between the field performance

for the base case and each of the twenty-two simulations at specific times were used as

26

the inputs for training the network while the values of k, ø and h were the target data

(Table 4.2).

Table 4.1: Twenty-two values for permeability, porosity and thickness were randomly generated within the ranges shown. The production and pressure profiles obtained by inputting these values into the simulation model were used to formulate the training and testing data.

BASE 80 0.3 150

1 70 0.34 2092 49 0.33 1283 58 0.25 774 52 0.23 2535 15 0.27 1936 33 0.34 1217 88 0.34 1518 63 0.31 2089 82 0.23 224

10 38 0.27 24811 19 0.27 11612 95 0.25 9413 71 0.26 18714 47 0.32 26215 72 0.29 20416 16 0.26 15217 93 0.25 28318 13 0.34 28819 40 0.33 14620 100 0.31 26621 100 0.35 30022 10 0.22 60

Run # Permeability Porosity Thickness

Range 10<k<100 0.22<Φ<0.35 50<h<300

Table 4.2: Inputs and outputs, at specific times, used in the training of the neural network for the single-region homogeneous property reservoir.

Inputs Description Outputs Description∆qo (STB/D) Oil production difference from base case k Permeability∆qg (MSCF/D) Gas production difference from base case Φ Porosity∆qw (STB/D) Water production difference from base case h Thickness∆p (psi) Pressure difference from base case - -

27

In order to determine whether the data structure that was input to the network

would have any effect on the network’s performance and prediction results, two data

structures were examined. The first data structure consisted of all time steps for a specific

case followed by the second case, third case etc., for the 22 cases. The second data

structure consisted of the first time step for each case followed by the second time step

for each case etc. as shown in Appendix B. It was determined that the second data

structure gave lower average training errors for all properties (Figure 4.1). The number of

neurons in the hidden layers of a 4-layer network structure was also modified to

determine its effects. It was observed that the network with the larger number of neurons,

25 in this case and the second data structure gave better results. If the number of neurons

was larger than 25, no improvements were observed.

0

1

2

3

4

5

6

7

8

9

10

Permeability Porosity Thickness

Reservoir Properties

Ave

rage

Tra

inin

g Er

ror (

%)

Data Structure 1, 10 neurons in hidden layers




Figure 4.1: Average training errors for each property using different data structures and different numbers of neurons in the hidden layers of a 4-layer neural network.

28

The architecture of the network, i.e. the number of hidden layers, as well as the

training function for each layer was examined to determine the best structure for training

this network. The inputs with and without pressure difference values that consist of 2, 3

and 4 hidden layers were examined to determine the optimum number of hidden layers

for training the network. It was found that a 5-layer network with three hidden layers and

the inclusion of pressure differences as inputs gave the lowest prediction errors

(Figure 4.2). The most effective transfer function in all layers except the last layer was

the hyperbolic tangent sigmoid function (tansig). Best results were obtained when the

linear function (purelin) was used in the final layer of the network.

0

0.5

1

1.5

2

2.5

3

1 2 3 4 5Number of Hidden Layers in Network

Pred

ictio

n Er

rors

(%)

Permeability ErrorsPorosity ErrorsThickness Errors

Figure 4.2: Comparison of 2, 3 and 4 hidden layers on network performance for a single-region, homogeneous property reservoir with and without pressure differences as inputs to the network. The network that had 3 hidden layers gave the lowest prediction errors for permeability, porosity and thickness.

29

The 5-layer neural network was successfully trained with production and pressure

data from 1994 to 2004 using the feed-forward network with back-propagation learning

algorithm. In this case, it was determined that the network should have 4, 20, 25, 25 and 3

neurons in each of the layers as shown in Figure 4.3.

The network was validated using the test data (Appendix B), after which

predictions for the permeability, porosity and thickness were obtained by inputting small

randomly generated values of ∆qo, ∆qg, ∆qw, and ∆p (Table 3.2), at specific times, to the

Figure 4.3: Structure of the back-propagation network (BPN) used in the case study of asingle-region, square homogeneous property reservoir with one producer. The delta values for each case are at specific times, as a result, there are multiple delta values for each case.

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

h

kφ

Input Layer (4 neurons) Middle Layers

20 - 25 - 25

Output Layer (3 neurons)

daystime

pq

qq

daystime

pq

qq

w

g

o

w

g

o

3600

90

=

∆∆

∆∆

=

∆∆

∆∆

M

M

M

30

trained network. The ANN-predicted values for permeability, porosity and thickness of

the reservoir were 82 md, 0.31, and 162 ft respectively, i.e. prediction errors of 2.5%,

3.3% and 8% respectively. The performance function used in training this network was

mean square error (mse) and the training function used was traingdx, which updates

weight and bias values according to gradient descent momentum and an adaptive learning

rate.

In many petroleum and natural gas producing fields, there is a large amount of

uncertainty associated with the properties that are input into the simulation model. It is

important to account for this uncertainty in the network so that it can accurately predict

the reservoir’s properties that can give a better history match. As a result, the range was

expanded (Table 4.3) and thirteen additional simulations were done to generate field

performance within a wider range of reservoir properties for permeability, porosity and

thickness. The network that used a narrow range of data and a lesser number of

simulations predicted the reservoir’s properties with good accuracy. The prediction errors

obtained for k, ø and h were 2.5%, 3.3% and 8% respectively (Table 4.3).

The data set using the wide range of properties proved to be more effective in

predicting the properties of the reservoir. The prediction errors for k, ø and h were

improved to 2.5%, 0% and 2.7% respectively. Therefore, as expected, if there is a greater

amount of uncertainty with the data, a larger number of simulations should be performed

that will make available a wider range of training data set that can more effectively train

the network. Providing the network with a larger amount of training data within a wide

31

data range improved the predictive capability and decreased the prediction errors giving

evermore improved results (Table 4.3).

4.3 Reservoir with four regions and one producing well

Figure 2.2 shows the four regions of the reservoir and the location of the

producing well. Table 4.4 shows the ranges within which the properties were randomly

generated and the absolute values of k, ø and h for each region. These randomly

generated properties were input to the simulation model to formulate the training data for

the network. The base case (Table 4.4) consisted of k, ø and h for each region and was

used to generate the historical production and pressure profile from the simulation model.

Table 4.3: Comparison of ANN-predicted reservoir properties using a narrow data range and a wide data range in the formulation of the training data. Smaller prediction errors were observed with the training data set that was formulated using the wide data range and a larger number of simulations.

k (md) ø h (ft)

k ø h

Narrow Data Range

Wide Data Range

- - -

Percentage Prediction Errors from ANN

Predicted Properties 82 0.3 146 2.5 0.0 2.7

2.5 3.3 8.0

- - -Ranges 10<k<300 0.15<ø<0.35 20<h<300

0.31 162

Ranges 10<k<100 0.22<ø<0.35 50<h<300

Base Properties 80

Predicted Properties 82

0.3 150

32

Table 4.4: Randomly generated properties for 35 cases used in the reservoir simulations for generating training data for the reservoir with four regions.

k1 ø1 h1 k2 ø2 h2 k3 ø3 h3 k4 ø4 h4

Ranges 40<k1<150 0.18<ø1<0.26 50<h1<150 10<k2<150 0.26<ø2<0.34 50<h2<150 40<k3<150 0.24<ø3<0.32 100<h3<200 30<k4<100 0.22<ø4<0.3 150<h4<3001 72.173 0.23184 71.128 21.538 0.32608 105.77 120.84 0.27032 101.44 80.068 0.26924 198.82

2 130.41 0.24901 92.111 20.108 0.28701 70.545 64.481 0.26699 148.58 35.666 0.2916 231.62

3 149.34 0.18247 132.95 19.61 0.27579 136.05 94.629 0.26517 141.64 89.211 0.221 195.98

4 47.46 0.20927 64.247 32.06 0.30081 75.735 40.773 0.31577 177.29 80.286 0.24407 250.91

5 147.04 0.18751 82.163 18.613 0.27679 77.764 74.781 0.28281 148.81 90.928 0.29721 201.86

6 141.51 0.20677 131.79 36.62 0.3064 78.817 121.23 0.29616 152.26 91.051 0.22817 297.09

7 145.46 0.18059 79.338 22.355 0.33834 123.48 126.94 0.31 178.2 83.309 0.26768 203.1

8 101.95 0.24524 57.995 28.791 0.29737 99.701 84.031 0.3189 159.12 76.862 0.25752 152.93

9 144.56 0.182 142.83 30.232 0.32417 55.467 83.988 0.31082 112.64 93.142 0.27739 271.91

10 75.005 0.21394 66.677 37.535 0.29464 69.129 67.686 0.27239 110.97 87.502 0.28871 297.42

11 69.249 0.1827 88.677 17.715 0.27929 122.46 50.11 0.29017 166.29 88.286 0.23485 196.78

12 46.719 0.23414 98.601 20.943 0.31077 147.37 106.24 0.27084 199.71 74.794 0.25699 206.84

13 76.604 0.20624 131.09 47.053 0.28937 120.45 131.57 0.30783 134.62 40.934 0.29226 262.2

14 72.167 0.22481 90.571 19.39 0.29452 130.15 72.283 0.28215 117.61 84.822 0.22177 216.42

15 84.273 0.21398 129.5 15.119 0.28415 70.625 91.209 0.30459 106.79 81.198 0.28083 164.98

16 43.752 0.19201 51.938 26.505 0.29457 127.27 45.008 0.27148 130.94 69.531 0.28367 298.89

17 71.622 0.18515 145.67 35.389 0.31524 137.27 49.429 0.31694 133.48 80.634 0.24511 230.44

18 64.158 0.22932 135.49 17.262 0.30098 130.74 96.446 0.24241 137.62 46.782 0.23887 170.18

19 102.25 0.23032 148.63 26.129 0.33931 125.66 118.7 0.3163 195.22 64.29 0.25956 169.7

20 135.71 0.19413 75.07 37.551 0.27473 140.46 43.898 0.29715 171.93 59.652 0.27482 279.6

21 68.049 0.21263 98.251 23.036 0.30472 92.1 147.75 0.29172 177.93 85.697 0.29557 256.24

22 105.1 0.23649 82.461 46.931 0.29896 115.84 94.719 0.27573 161.77 83.439 0.25514 299.74

23 97.496 0.21438 55.74 40.285 0.2918 50.13 117.08 0.25397 164.92 46.722 0.25979 272.36

24 54.206 0.19687 79.097 26.045 0.29943 148.2 128.34 0.30682 175.63 74.458 0.22789 267.34

25 40.415 0.22728 117.61 17.844 0.33751 91.231 96.834 0.31761 114.78 46.208 0.26436 264.54

26 75.84 0.24698 145.22 26.011 0.32616 84.76 95.297 0.2508 159.95 73.116 0.25486 285.75

27 79.684 0.25106 54.027 16.179 0.27027 116.82 123.02 0.26008 189.86 48.795 0.2345 165.47

28 149.34 0.25085 132.16 12.583 0.32738 94.368 66.192 0.3128 117.19 99.385 0.2323 181.76

29 143.4 0.25703 72.362 14.814 0.32316 133.99 143.74 0.29412 181.89 83.222 0.2947 277.66

30 87.755 0.24157 149.97 11.174 0.28759 102.95 49.749 0.28985 106.93 63.752 0.2309 261.44

31 142.91 0.20944 76.053 44.608 0.31908 75.714 99.731 0.28098 195.57 96.164 0.29988 210.76

32 94.652 0.18545 73.028 26.568 0.26476 65.63 139.38 0.24028 131.73 55.252 0.27525 195.76

33 146.82 0.18871 102.51 41.677 0.30469 105.78 134.75 0.25815 100.52 35.907 0.29688 221.13

34 145.34 0.25495 106.07 42.656 0.32262 146.13 109.46 0.31828 175.99 96.396 0.28204 171.71

35 119.28 0.22088 132.6 25.813 0.27126 145.13 68.86 0.3089 130.87 78.358 0.22006 224.15

BASE 75 0.22 100 20 0.3 125 75 0.28 165 55 0.26 250

33

Initially, the network was used to predict one of the reservoir properties, for

example permeability, while the other two properties (ø and h), ∆qo, ∆qg, ∆qw, and ∆p

were used as inputs (Table 4.5). This resulted in average prediction errors for

permeability, porosity and thickness of 36%, 7% and 16% respectively (Figure 4.4).

Second, one of the reservoir properties, for example porosity, in addition to ∆qo, ∆qg,

∆qw, and ∆p were used as inputs while the other two reservoir properties (k and h) were

predicted (Table 4.6). This improved the prediction errors for thickness but that of

permeability and porosity was not significantly affected (Figure 4.5).

Table 4.5: Two reservoir properties and the differences between field performance and the base case simulation were used as inputs. One reservoir property was predicted. The output functional links were used as forcing functions in training the network.

Inputs Outputs Inputs Outputs Inputs Outputs

h ø ∆qo ∆qg ∆qw ∆p

k associated functional

links

k ø ∆qo ∆qg ∆qw ∆p

h associated functional

links

k h

∆qo ∆qg ∆qw ∆p

ø associated functional

links

Table 4.6: One reservoir property and the differences between field performance and the base case simulation were used as inputs. Two reservoir properties were predicted. The output functional links were used as forcing functions in training the network.

Inputs Outputs Inputs Outputs Inputs Outputs

h ∆qo ∆qg ∆qw ∆p

k ø

associated functional

links

k ∆qo ∆qg ∆qw ∆p

h ø


links

ø ∆qo ∆qg ∆qw ∆p

k h


links

34

0

20

40

60

80

100

Region 1 Region 2 Region 3 Region 4

Regions

Pred

ictio

n Er

rors

(%)


Average k error = 36%Average ø error = 7%Average h error = 16%

Figure 4.4: Reservoir property prediction errors using two properties as input data while the third property was predicted.

0

20

40

60

80

100


Regions

Pred

ictio

n Er

rors

(%)



Figure 4.5: Reservoir property prediction errors using one property as input data while the other two properties were predicted.

35

The ANN structure developed in section 4.2 did not prove as effective a predictor

in this reservoir with four regions. Therefore, in an effort to have a better representation

of the data to the network, the time factor was considered as another input. Data are input

to the network at specific time intervals for each case as its inclusion would better

describe combinations of data at each time. Figure 4.6 shows the average training errors

obtained using a 4-layer network with and without time as an input factor. The inclusion

of time gives the network a better understanding of the relationship between the input and

output data as the reservoir was depleted.

0

1

2

3

4

5

6

7

8

9

10

Permeability Porosity Thickness

Reservoir Properties

Ave

rage

Tra

inin

g Er

rors

(%)

Errors without time as inputErrors with time as input

Figure 4.6: Average training errors with and without time as an input in the training data using a 4-layer neural network and 25 neurons in the hidden layers. Time used as an input decreased the average training errors.

36

The next step was to predict the permeability, porosity and thickness values in

each region of the reservoir (Table 4.8) when the inputs consisted of time, ∆qo, ∆qg, ∆qw,

and ∆p (Table 4.7). The structure of the input and output data for training the network is

shown in Appendix B.

Table 4.7: Inputs used to train the network. These inputs are at specific times for each case

Inputs Descriptiontime (days) Time at which production is recorded∆qo (STB/D) Oil production difference from base case∆qg (MSCF/D) Gas production difference from base case∆qw (STB/D) Water production difference from base case∆p (psi) Pressure difference from base case

Table 4.8: Outputs from the network. Outputs Description

k1 Permeability North of Fault A (Region 1)Φ1 Porosity North of Fault A (Region 1)h1 Thickness North of Fault A (Region 1)k2 Permeability in Channel (Region 2)Φ2 Porosity in Channel (Region 2)h2 Thickness in Channel (Region 2)k3 Permeability in East Levee (Region 3)Φ3 Porosity in East Levee (Region 3)h3 Thickness in East Levee (Region 3)k4 Permeability in West Levee (Region 4)Φ4 Porosity in West Levee (Region 4)h4 Thickness in West Levee (Region 4)

37

When all reservoir properties were predicted there was an increase in the

prediction errors of the network. The prediction errors of k, ø and h were 88%, 8% and

33% respectively (Figure 4.7). This illustrated that there was a lack of connectivity

between the inputs and the desired outputs within the network. It was also observed that,

as the number of inputs decreased, the prediction errors of the properties increased with

the exception of the porosity predictions that were not significantly affected.

0

20

40

60

80

100

120

140

160

180

200


Regions

Pred

ictio

n Er

rors

(%)



Figure 4.7: Reservoir property prediction errors in each region using time, ∆qo, ∆qg, ∆qw, and ∆p as inputs and k, ø and h as outputs. The prediction errors were higher for all three properties.

38

The large prediction errors shown in Figure 4.7 are not acceptable and

consequently, the data sets were re-examined and analyzed. It is observed that there were

large variations among the input values, for example values ranging from -1000 to 1000.

These large variations were decreased by reducing the magnitudes of the larger values by

dividing by a constant, for example 100. However, this magnitude reduction did not

improve the understanding of the network. The sensitivity of the network to various input

functional links was also tested. The sensitivities were small and it was found that time

variable in years and days as inputs decreased the training and testing errors. In addition,

the functional link ∆qo∗ time (yrs) also improved the results. Overall, these two input

functional links improved the average training and testing errors (Figure 4.8 & 4.9).

0

1

2

3

4

5

6

7

8

9

10

Region1 Region2 Region3 Region4Reservoir Regions

Ave

rage

Tra

inin

g Er

rors

(%)

Permeability Errors Without Input Functional Links

Permeability Errors With Input Functional Links

Porosity Errors Without Input Functional Links

Porosity Errors With Input Functional Links

Thickness Errors Without Input Functional Links

Thickness Errors With Input Functional Links

Figure 4.8: Average training errors with and without the input functional links: time(yrs) and ∆qo∗ time(yrs). The training errors for permeability, porosity and thickness were decreased with the use of these input functional links.

39

Using the output functional links, permeability thickness (kh) and

permeability/porosity (k/ø) in the target data showed improvements in the prediction

results (Figures 4.10, 4.11 & 4.12). This indicated that the network now had a better

understanding of the effects of permeability thickness and the relationship between the

permeability and porosity of the reservoir. The prediction errors for k, ø and h were

reduced from 88%, 8% and 33% to 52%, 4% and 20% respectively with the use of these

two output functional links.

0

1

2

3

4

5

6

7

8

9

10

Region1 Region2 Region3 Region4Reservoir Regions

Ave

rage

Tes

ting

Erro

rs (%

)

Permeability Errors Without Input Functional Links

Permeability Errors With Input Functional Links

Porosity Errors Without Input Functional Links

Porosity Errors With Input Functional Links

Thickness Errors Without Input Functional Links

Thickness Errors With Input Functional Links

Figure 4.9: Average testing errors with and without the input functional links: time(yrs)and ∆qo∗ time(yrs). The testing errors for permeability, porosity and thickness were decreased with the use of these input functional links.

40

0

20

40

60

80

100

120

140

160

180

200


Reservoir Regions

Pred

ictio

n Er

rors

(%)

Permeability Errors Without Output Functional Links

Permeability Errors With Output Functional Link, kh

Permeability Errors With Output Functional links, kh andk/phi

Average error without functional links = 88%Average error with functional link, kh = 57%Average error with functional links, kh and k/ø = 52%

Figure 4.10: Average permeability prediction error was decreased from 88% to 52% with the combined use of the output functional links kh and k/ø.

0

20

40

60

80

100


Reservoir Regions

Pred

ictio

n Er

rors

(%)

Porosity Errors Without Output Functional Links

Porosity Errors With Output Functional Link, kh

Porosity Errors With Output Functional links, kh and k/phi


Figure 4.11: Average porosity prediction error was decreased from 8% to 4% with the combined use of the output functional links kh and k/ø.

41

The prediction errors were still somewhat large. In order to train the network to

better understand the physical parameters of the reservoir, the distances from the

producer to the boundaries of the reservoir (D) and the regions’ areas (A) were also taken

into consideration. The distances from the producer to the boundaries divided by the

permeability in each region (D/k) and the area of each region divided by the permeability

in each region (A/k) were used as output functional links. This resulted in an overall

decrease in the property prediction errors. The inclusion of only the closest boundary did

not improve results; both boundaries (closest and furthest) must be included in the output

functional links.

0

20

40

60

80

100


Reservoir Regions

Pred

ictio

n Er

rors

(%)

Thickness Errors Without Output Functional Links

Thickness Errors With Output Functional Link, kh

Thickness Errors With Output Functional links, kh and k/phi


Figure 4.12: Average thickness prediction error was decreased from 33% to 20% with the combined use of the output functional links kh and k/ø.

42

The distances from the producer to the boundaries of the reservoir and the

regions’ areas are important factors in the development of the network and of course, in

the production of fluids from the reservoir. Figures 4.13, 4.14 and 4.15 show the effect of

adding these functional links as outputs from the neural network on the prediction results.

Both functional links, D/k and A/k, decreased the prediction errors of the properties. The

average prediction error was reduced from 25% to 13% with the use of A/k values as

output functional links. A combination of the functional links, A/k and D/k did not

improve results. Therefore, the functional link, A/k was the preferred functional link

chosen that resulted in lower average prediction errors of 13%.

0

20

40

60

80

100


Reservoir Regions

Pred

ictio

n Er

rors

(%)

Permeability Errors Without Boundaries and Areas as Functional Links

Permeability Errors With Boundaries/k as Output Functional Link

Permeability Errors With Areas/k as Output Functional Links

Average error without functional links = 52%Average error with boundaries/k as functional link = 14%Average error with areas/k as output functional link = 14%

Figure 4.13: Permeability prediction errors with and without the output functional links:region’s area/region’s permeability and closest and furthest distance from the well to the boundaries/region’s permeability. Both functional links decreased the prediction errors.

43

0

20

40

60

80

100


Reservoir Regions

Pred

ictio

n Er

rors

(%)

Porosity Errors Without Boundaries and Areas

Porosity Errors With Boundaries/k as Output Functional Link

Porosity Errors With Areas/k as Output Functional Links

Average error without functional links = 4%Average error with boundaries/k as functional links = 5%Average error with areas/k as functional links = 4%

Figure 4.14: Porosity prediction errors with and without the output functional links: region’s area/region’s permeability and closest and furthest distance from the well to theboundaries/region’s permeability.

0

20

40

60

80

100


Reservoir Regions

Pred

ictio

n Er

rors

(%)

Thickness Errors Without Output Functional Links

Thickness Errors With Boundaries/k as Output Functional Links

Thickness Errors With Areas/k as Output Functional Links

Average error without functional links = 20%Average error with boundaries/k as functional link = 29%Average error with areas/k as functional links = 22%

Figure 4.15: Thickness prediction errors with and without the output functional links: region’s area/region’s permeability and closest and furthest distance from the well to theboundaries/region’s permeability.

44

The optimum number of layers for the network was obtained by trying different

numbers of hidden layers and varying the output functional links. The results of two

scenarios with and without A/k show that better results were obtained with A/k

(Figure 4.16). It was found that, regardless of the functional links used, the optimum

number of hidden layers was constant and the least prediction errors for permeability,

porosity and thickness were obtained using 4 hidden layers in the network (Figure 4.16).

0

20

40

60

80

100

1 2 3 4 5 6

Number of Hidden Layers in Network

Pred

ictio

n Er

rors

(%)

Permeability errorsPorosity errorsThickness errorsBest permeability resultBest porosity resultBest thickness result

0

10

20

30

40

0 1 2 3 4 5 6Number of Hidden Layers

Ave

rage

Err

ors

(%)

Average Errors with Areas

Figure 4.16: Prediction errors of k, ø and h using 2, 3, 4 and 5 hidden layers in the network with varying output functional links. Inset: Average prediction errors of the properties k, ø and h. The 4-hidden layer network gave the lowest average prediction errors.

45

Other factors that were taken into consideration when designing the network was

the type of network training function and the network performance function. The training

functions tried were ‘traingdx’ and ‘trainscg’ while the network’s performance functions

tried were ‘mse’ and ‘msereg’. ‘mse’ measures the network’s performance according to

the mean squared errors while ‘msereg’ measures the network’s performance as the

weighted sum of two factors – the mean squared errors and the mean squared weight and

bias values. ‘Traingdx’ updates weights and bias values according to gradient descent

momentum while ‘trainscg’ updates weights and bias values according to a scaled

conjugate gradient method [Hagan et al, 1996]. The combination of ‘msereg’ training

function and ‘traingdx’ performance function decreased the errors for all three properties.

In an attempt to provide the network with additional information that can improve

the network’s understanding between the inputs and outputs, eigenvalues of the following

matrices were introduced as output functional links. In this way, the eigenvalues will

serve as a forcing function in the output. The eigenvalues were all tried as outputs since

the properties of the reservoir are unknowns.

Case 1:

111

11

φφ khk

222

22

φφ khk

333

33

φφ khk

444

44

φφ khk

λ1 , λ2 λ3 , λ4 λ5 , λ6 λ7 , λ8

Case 2:

111

11

hkhk φ

222

22

hkhk φ

333

33

hkhk φ

444

44

hkhk φ

λ1 , λ2 λ3 , λ4 λ5 , λ6 λ7 , λ8

46

Case 3:

111

11

hkhk φ

222

22

hkhk φ

333

33

hkhk φ

444

44

hkhk φ

λ1 , λ2 λ3 , λ4 λ5 , λ6 λ7 , λ8

Case 4:

411

3111

211

/DDhDkDhkk

φφ

411

3111

211

/DDhDkDhkk

φφ

λ1 , λ2, λ3 λ4, λ5 , λ6

411

3111

211

/DDhDkDhkk

φφ

411

3111

211

/DDhDkDhkk

φφ

λ7, λ8, λ9 λ10, λ11, λ12

where D1, D2, D3, and D4, are the distances from the center of the producing well to the

boundaries of the reservoir.

Case 5:

4

4444

3

3333

2

2222

1

1111

kAhk

kA

hk

kAhk

kAhk

φ

φ

φ

φ

λ1, λ2, λ3, λ4

where A1, A2, A3 and A4 are the areas of the respective regions.

47

The functional links that included the eigenvalues using the first four cases did

not improve the performance of the network. However, case 5, which used the first

eigenvalue of the 4x4 matrix did show some improvement in the prediction and testing

errors. The final structure, which predicted the permeability, porosity and thickness for

each region using 7 inputs and 37 outputs are shown Figure 4.17. Satisfactory results for

the properties were obtained using this network.

The average training and testing errors using this network were all less than 10%.

The average prediction errors for k, ø and h were now 9%, 2% and 20% respectively

(Figure 4.18, 4.19, & 4.20), a decrease of 5%, 2% and 2% respectively over the previous

case which used only the functional links, k*h, k/ø, and A/k. The most critical factors in

training this network was observed to be the functional links k*h and k/ø.

Figure 4.17: Structure of the back-propagation network (BPN) used in the case study of a square homogeneous reservoir with four regions and one producer. The network predicts k, ø and h for each region.

Input Layer (7 neurons) Middle Layers

50-100-100-50

Output Layer (37 neurons)

∆∆

∆∆

∗∆

∆∆

∆∆

∗∆

pqqq

tqyrst

dayst

pq

qq

tqyrst

dayst

w

g

o

o

w

g

o

o

)()3600(

)()90(

M

M

EigenvaluekAAAA

kAAAAkkhkhk

hh

kk

44321

14321

4411

4411

41

41

41

/,,,

/,,,//

→

→→

→→→

φφ

φφ

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

48

0

20

40

60

80

100

120

k1 k2 k3 k4

Region's Permeability

Perm

eabi

lity

(md)

0

20

40

60

80

100

Pred

ictio

n Er

ror (

%)

Base Permeability

Predicted Permeability

Errors with Functional Linksand Eigenvalue

Errors with Functional Links

Average Error with Eigenvalue = 9%Average Error without Eigenvalue = 14%

Figure 4.18: Actual and predicted permeability values utilizing the final network. The use of the eigenvalue as an output functional link further decreased the prediction errors.

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

ø1 ø2 ø3 ø4

Region's Porosity

Poro

sity

0

20

40

60

80

100

Pred

ictio

n Er

ror (

%)

Base Porosity

Predicted Porosity

Errors with Functional Linksand Eigenvalue

Errors with Functional Links


Figure 4.19: Actual and predicted porosity values utilizing the final network. The use of the eigenvalue as an output functional link further decreased the prediction errors.

49

4.4 Reservoir with four regions and three producing wells

Figure 2.4 shows the reservoir with four regions and three producing wells. The

training for the development of this network began with the last stage of section 4.3

which used 7 inputs and 37 outputs. However, this gave prediction errors of 84%, 14%

and 27% for permeability, porosity and thickness respectively. As a result of these large

errors, the data were treated as a new data set and a new network was developed. The

same data structure, training function and performance functions described in the

previous section proved to be effective.

0

50

100

150

200

250

300

h1 h2 h3 h4

Region's Thickness

Thic

knes

s (ft

)

0

20

40

60

80

100

Pred

ictio

n Er

ror (

%)

Base Thickness

Predicted Thickness

Errors with Functional Linksand EigenvalueErrors with Functional Links


Figure 4.20: Actual and predicted thickness values utilizing the final network. The use of the eigenvalue as an output functional link further decreased the prediction errors.

50

Various functional links with different numbers of hidden layers were examined

and tested to determine the optimum architecture and most effective functional links. The

most effective functional links were found to be permeability thickness product (kh) and

permeability/porosity (k/ø) with average property prediction errors as shown in

Figure 4.21. However, a combination of both links did not improve the results;

consequently kh was the preferred functional link used as it gave lower prediction errors.

The optimum number of hidden layers was determined to be six (Figure 4.21) regardless

of the output functional links used. The final inputs and outputs that effectively trained

the network are described in Tables 4.9 & 4.10.

0

20

40

60

80

100

120

1 2 3 4 5 6 7 8 9 10

Number of Hidden Layers in Network

Pred

ictio

n Er

rors

(%)

Permeability errorsPorosity errorsThickness errorsBest permeability resultBest porosity resultBest thickness result

Figure 4.21: Average prediction errors of permeability, porosity and thickness using different numbers of hidden layers and varying functional links in the network. The network with six hidden layers gave the lowest prediction errors for k, ø and h.

51

Table 4.9: Inputs used in training the network for the reservoir with four homogeneousregions and three producing wells.

Inputs Descriptiontime (days) Time at which production is recordedtime (yrs) Time (yrs) at which production is recorded

∆qo (STB/D) Oil production difference from base case∆qg (MSCF/D) Gas production difference from base case∆qw (STB/D) Water production difference from base case∆p (psi) Pressure difference from base case

∆qo*time(yrs) Oil production * time(yrs)

Table 4.10: Outputs used in training the network for the reservoir with fourhomogeneous regions and three producing wells. Permeability thickness product is the functional link used as a forcing function within the network.

Outputs Descriptionk1 Permeability North of Fault A (Region 1)Φ1 Porosity North of Fault A (Region 1)h1 Thickness North of Fault A (Region 1)k2 Permeability in Channel (Region 2)Φ2 Porosity in Channel (Region 2)h2 Thickness in Channel (Region 2)k3 Permeability in East Levee (Region 3)Φ3 Porosity in East Levee (Region 3)h3 Thickness in East Levee (Region 3)k4 Permeability in West Levee (Region 4)Φ4 Porosity in West Levee (Region 4)h4 Thickness in West Levee (Region 4)

k1h1 Permeability Thickness (Region 1)k2h2 Permeability Thickness (Region 2)k3h3 Permeability Thickness (Region 3)k4h4 Permeability Thickness (Region 4)

52

Many of the functional links that were used in the 1-well case were not effective

in the 3-well case. The distances from the producer to the boundaries of the reservoir and

the regions’ areas also did not prove to be an important consideration with the 3-well

case. This could indicate a lack of connectivity between the inputs and the outputs or that

another factor plays a more important role in the understanding of the network. This can

be further explored by adding output functional links. The ANN was trained using 7

inputs, 16 outputs and 6 hidden layers. The number of neurons in each of the hidden

layers was 50, 100, 150, 100, 75 and 50 respectively. The results obtained for the

permeability, porosity and thickness were compared with the base case as shown in

Figures 4.22, 4.23 and 4.24.

0

20

40

60

80

100

120

140

k1 k2 k3 k4Region's Permeability

Perm

eabi

lity

(md)

0

20

40

60

80

100

120

140

160

Pred

ictio

n Er

ror (

%)

Base Permeability

Predicted Permeability

Errors

Figure 4.22: Comparison of the base case permeability with the ANN-predicted permeability values using the output functional link permeability thickness.

53

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

ø1 ø2 ø3 ø4Region's Porosity

Poro

sity

(Fra

ctio

n)

0

20

40

60

80

100

Pred

ictio

n Er

ror (

%)

Base Porosity

Predicted Porosity

Errors

Figure 4.23: Comparison of the base case porosity with the ANN-predicted porosity using the output functional link permeability thickness.

50

100

150

200

250

300

h1 h2 h3 h4

Region's Thickness

Thic

knes

s (ft

)

0

20

40

60

80

100

Pred

ictio

n Er

ror (

%)

Base Thickness

Predicted Thickness

Errors

Figure 4.24: Comparison of the base case thickness with the ANN-predicted thickness using the output functional link permeability thickness.

54

The errors associated with the predicted values were large and seem to indicate

poor performance in the predicting capability of the network. However, these apparently

large prediction errors were due to relatively small magnitudes in the predicted values.

Table 4.11 shows the base case properties, the ANN-predicted values, as well as two

imposed scenarios for k, ø and h so that comparisons can be made. Figures 4.25 and 4.26

show the history match obtained with the input of properties from the two imposed

scenarios into the simulation model. These comparisons show large discrepancies

between the simulated profiles and historical profiles. It would be expected that a history

match would be improved with the ANN-predicted properties.

The input of the ANN-predicted properties to the simulation model illustrated that

there were improvements to the history match with respect to fluid production and the

pressure profile (Figure 4.27). However, even though there was an improved history

match to the water production, the match was not as good as that observed for the oil and

gas production and the pressure profile. This indicated that the network does not have a

good understanding of the relationship between the properties of the reservoir and the

water production. Others possible factors that may be examined to improve the network’s

understanding of water influx into the reservoir are the relative permeability curves and

the use of functional links that could link the water properties to the rock properties.

55

Table 4.11: Comparison of the base case properties and ANN-predicted properties in each region of the reservoir. Two additional scenarios for reservoir properties were also simulated to compare the deviations from the historical field performance.

k1 75 85 40 150Φ1 0.22 0.2 0.18 0.26h1 100 102 50 150k2 20 52 10 150Φ2 0.3 0.29 0.26 0.34h2 125 79 50 150k3 75 73 40 150Φ3 0.28 0.3 0.24 0.32h3 165 139 100 200k4 55 66 30 100Φ4 0.26 0.3 0.22 0.3h4 250 244 150 300

Base Case (History)

Predicted ImprovementsProperties Imposed

Scenario 1Imposed

Scenario 2

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5000

10000

15000

20000

OIL

PR

OD

UC

TIO

N R

ATE

(STB

/ D

AY)

Oil ProductionSimulated Worse Case Historical Base

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1000

2000

3000

4000

5000

AVE

RA

GE

PRES

SUR

E (W

T B

Y H

C P

V) (P

SIA

)

Average Field PressureSimulated Worse Case Historical Base

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20000

40000

60000

80000

100000

120000

140000

GA

S PR

OD

UC

TIO

N R

ATE

(MSC

F / D

AY)

Gas ProductionSimulated Worse Case Historical Base

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100

200

300

400

500

600

WA

TER

PR

OD

UC

TIO

N R

ATE

(STB

/ D

AY)

Water ProductionSimulated Worse Case Historical Base

Figure 4.25: Comparison between the historical profile and the simulated profile obtained from scenario 1. The graphs show large deviations from the historical profile, the largest being observed in the water production.

56

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5000

10000

15000

20000

OIL

PR

OD

UC

TIO

N R

ATE

(STB

/ D

AY)

Oil ProductionSimulated Best Case Historical Base

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2000

2500

3000

3500

4000

4500

5000

AVE

RA

GE

PRES

SUR

E (W

T B

Y H

C P

V) (P

SIA

)

Average Field PressureSimulated Best Case Historical Base

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20000

40000

60000

80000

100000

120000

140000

GA

S PR

OD

UC

TIO

N R

ATE

(MSC

F / D

AY)

Gas ProductionSimulated Best Case Historical Base

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500

1000

1500

2000

2500

WA

TER

PR

OD

UC

TIO

N R

ATE

(STB

/ D

AY)

Water ProductionSimulated Best Case Historical Base

Figure 4.26: Comparison between the historical profile and the simulated profile obtained from scenario 2. The graphs show large deviations from the historical profile, the largestbeing observed in the water production.

57

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5000

10000

15000

20000

OIL

PR

OD

UC

TIO

N R

ATE

(STB

/ D

AY)

Oil Production Simulated Historical Base

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2000

3000

4000

5000

AVE

RA

GE

PRES

SUR

E (W

T B

Y H

C P

V) (P

SIA

)

Average Field Pressure Simulated Historical Base

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20000

40000

60000

80000

100000

120000

140000

GA

S PR

OD

UC

TIO

N R

ATE

(MSC

F / D

AY)

Gas Production Simulated Historical Base

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100

200

300

400

500

600

WA

TER

PR

OD

UC

TIO

N R

ATE

(STB

/ D

AY)

Water Production Simulated Historical Base

Figure 4.27: Comparison between the historical profile and the simulated profile obtained from the ANN-predicted properties. There is a good history match for the oil and gas production as well as the pressure profile. The history match for water productionis improved but it is not as close a match as the oil and gas production profiles.

Chapter 5

UTILIZATION OF AN ARTIFICIAL NEURAL NETWORK FOR PREDICTION OF M4.1 RESERVOIR PROPERTIES AND THE RESULTING HISTORY

MATCH

The main reservoir in the Tahoe field, termed the M4.1, is a Late Miocene sand

located approximately 10,000 ft below sea level. The reservoir was formed by turbidite

flows that entered an unconfined slope setting and deposited a NW-SE trending channel-

levee system. This channel-levee system is draped over an anticlinal dome and cut by

normal faults. Faulting forms an updip trap in the M4.1 and plays an important role in the

compartmentalization of the reservoir (Appendix E).

The rock and fluid properties used to build the simulation model for the M4.1

reservoir are outlined in Appendix D and the reservoir is characterized in Appendix E.

The actual production history from three producers in the M4.1 reservoir was used as the

base case to formulate the training and testing data for the artificial neural network

(ANN). This chapter describes the development of an ANN used in training, testing and

predicting the permeability, porosity and thickness for the M4.1 reservoir in the Tahoe

field. The ANN-predicted properties were then input into the M4.1 reservoir simulation

model and compared to the actual production history.

59

5.1 Utilization of ANN in predicting the properties for a single-region homogeneous property reservoir

The M4.1 reservoir was initially considered to be a single-region, homogeneous

property reservoir with three producers and an average permeability, porosity and

thickness throughout. The initial fluid distribution in the reservoir simulation model and

the location of the producers are shown in Figure 5.1. The fluid distribution in Figure 5.1

is the result of case 36 shown in Table 5.1.

2DVIEW Study[Tahoe_Results] Case[NewRange] Time[01-JAN-1994, 0 days] TimeStep[ 0] Azimuth[ 0.00] Inclination[ 0.00]

7833_4S4S

A1S

ST2

STBP1

7 S

TERNARY DIAGRAM

SG0.00

SG0.90

SW0.10

SW1.00

SO0.00

SO0.90

A3ST

A1STBP1

4ST2

0 .5miles

Figure 5.1: Initial fluid distribution and the location of the producing wells in the M4.1 reservoir shown on a curvilinear grid.

60

5.1.1 Training and testing the ANN with data generated from the single-region M4.1 reservoir simulations

Thirty-eight combinations of randomly generated properties (Table 5.1) were

input to the reservoir simulation model to generate different production profiles. The

ranges within which the properties were randomly generated were estimated from field

data in the M4.1 reservoir (Table 5.1). As previously described, the training data

consisted of the differences between the historical production and the simulated

production profiles at specific times, using different combinations of properties

(Appendix B).

An additional ten simulations, with different property combinations, were run to

generate the production profiles from which the test data was formulated to test and

validate the network (Table 5.2). The differences between the historical production and

the ten simulated production profiles were used to generate the test data, which was in the

same format as the training data (Appendix B). The formulated test data was used as

inputs to the trained network that predicted the properties (k, ø and h) of the reservoir.

The predicted properties obtained from the trained ANN were then compared to the

properties that were input to the simulator to determine whether the network had been

successfully trained.

61

Table 5.1: Randomly generated properties used in the 38 simulation runs for formulatingthe training data for the single-region reservoir.

k ø hRanges 30<k<120 0.22<ø<0.31 80<h<200

1 73.66 0.274 119.292 62.67 0.274 135.713 88.20 0.236 151.664 50.15 0.267 142.055 82.84 0.261 127.656 97.60 0.259 164.997 94.52 0.274 146.038 72.91 0.280 117.449 87.98 0.274 145.71

10 66.93 0.267 136.0511 56.34 0.247 143.5512 62.17 0.268 163.1313 74.04 0.274 162.0914 62.17 0.256 138.6915 67.95 0.271 117.9716 46.20 0.260 152.2617 59.27 0.266 161.7118 56.16 0.253 143.5119 77.84 0.286 159.8020 69.20 0.260 166.7721 81.13 0.276 156.1322 82.55 0.267 164.9523 75.40 0.255 135.7924 70.76 0.258 167.5725 50.33 0.287 147.0426 67.57 0.270 168.9227 66.92 0.254 131.5428 81.88 0.281 131.3729 96.29 0.292 166.4830 53.11 0.262 155.3231 95.85 0.277 139.5232 78.96 0.241 116.5433 89.79 0.262 132.4934 98.46 0.294 149.9735 73.08 0.255 158.1936 56.25 0.265 160.0037 30.00 0.225 87.5038 112.50 0.305 200.00

Table 5.2: Randomly generated properties used in the 10 simulation runs for formulatingthe testing data for the single-region reservoir.

k ø hRanges 30<k<120 0.22<ø<0.31 80<h<200

1 71.25 0.248 140.002 83.06 0.266 168.353 67.42 0.264 163.644 74.48 0.276 124.815 70.19 0.268 140.846 102.60 0.268 138.767 76.48 0.249 119.948 69.94 0.263 133.869 75.24 0.275 137.41

10 92.27 0.267 154.39

62

The neural network inputs and outputs used in the training/learning process for

the single-region, homogeneous property M4.1 reservoir are shown in Table 5.3. Using

these network inputs and outputs and 3 hidden layers in the network, the average training

and testing errors were 12% and 7% respectively. The network’s performance function

and training function used in the training/learning process was ‘msereg’ and ‘traingdx’

respectively.

Some of the functional links that proved to be effective in network development

in Chapter 4 were not as effective in training the network developed for the M4.1

reservoir. The inclusion of ∆qo∗ time(yrs) as an input functional link that previously

proved to be effective in the training process (Section 4.3) did not improve the training or

testing results in the training of this network. In addition, eigenvalues, areas and the

distances to the boundaries also did not prove to be as effective. As a result, these

functional links were not used in the development of this network.

Table 5.3: Inputs and outputs used in training the network for the single-region homogeneous property reservoir.

Inputs Description Outputs Description

∆qo (STB/D) Oil production difference from historical data k Permeability

∆qg (MSCF/D) Gas production difference from historical data Φ Porosity

∆qw (STB/D) Water production difference from historical data h Thickness

63

Previously developed networks in Chapter 4 illustrated that time, kh and k/ø

proved to be effective functional links. In addition, since the gas/oil ratio was one of the

historical data that was available, this was also included as input data to the network

(Table 5.4). These additional network inputs (time and ∆gor) and output functional links

(kh and k/ø), shown in Table 5.4, resulted in improvements to the training errors. The

average training error was reduced to 8% but the average testing error remained at 7%

(Figures 5.3 and 5.4). The best results for the training and testing errors were obtained by

using the inputs and outputs as described in Table 5.4 and the network’s architecture is

described in Figure 5.2. This ANN, which gave acceptable results, was then used to

predict the permeability, porosity and thickness of the M4.1 reservoir.

Table 5.4: Final inputs and outputs used in training the network for the single-region homogeneous property reservoir.

Inputs Description Outputs Descriptiontime (days) Time at which production is recorded k Permeability∆qo (STB/D) Oil production difference from historical data Φ Porosity∆qg (MSCF/D) Gas production difference from historical data h Thickness∆qw (STB/D) Water production difference from historical data kh Functional Link, kh

∆GOR (SCF/STB) GOR difference from historical data k/ø Functional Link, k/ø

64

Figure 5.2: Architecture of the neural network used to predict the properties of thesingle-region M4.1 reservoir.

0 5 10 15 20 25 30 35 4012

14

16

18Training Errors vs Case Number

Per

m E

rror %

0 5 10 15 20 25 30 35 402.5

3

3.5

4

Phi

Erro

r %

0 5 10 15 20 25 30 35 406

7

8

9

Case Number

Thic

knes

s E

rror %

Figure 5.3: Training errors for k, ø and h for 38 simulated cases. The average training error for each property was 15%, 3% and 7% respectively, with an overall average training error of 8%.

Input Layer (5 neurons)

Middle Layers20 - 25 - 25

Output Layer(5 neurons)

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

∆∆

∆∆

∆∆

∆∆

gorq

qq

dayst

gorq

qq

dayst

w

g

o

w

g

o

)3600(

)90(

M

M

φ

φ

/kkhh

k

65

5.1.2 Prediction of properties for the M4.1 reservoir

The ANN predictions showed that the permeability, porosity and thickness of the

M4.1 reservoir have average values of 70 md, 0.266 and 147 ft respectively. These values

were input to the simulation model and the production profiles compared to the historical

data (Figure 5.5). The simulated gas and oil production matched the actual production

fairly closely but the water production did not. The properties considered here may not

have had a significant effect on the water production, which could have been due to other

factors in the simulator that need to be adjusted, for example the relative permeability to

1 2 3 4 5 6 7 8 9 100

20

40Testing Errors vs Test Number

Per

m E

rror %

1 2 3 4 5 6 7 8 9 100

5

10

Phi

Erro

r %

1 2 3 4 5 6 7 8 9 100

10

20

Test Number

Thic

knes

s E

rror %

Figure 5.4: Testing errors for k, ø and h for 10 simulated cases. The average testing error for each property was 11%, 3% and 8% respectively with an overall average testing errorof 7%.

66

water, the gas/oil contact (GOC) , the oil/water contact (OWC) or the presence of an

aquifer.

5.2 Utilization of the ANN in predicting the properties of the reservoir that is delineated into four regions

Due to the compartmentalization and varying connectivity between the

compartments in the M4.1 reservoir, it was delineated into four regions with varied

permeability, porosity and thickness within the ranges shown in Tables 5.5 and 5.6. The

regions delineated were the area North of Fault A, the Channel, the East Levee and the

West Levee respectively (Figure 5.6).

01/94 09/96 06/99 03/02 12/040

1000

2000

3000

4000

5000

6000

OIL

PR

OD

UC

TIO

N R

ATE

(STB

/ D

AY)

Simulated Oil Production Historical Oil Production

01/94 09/96 06/99 03/02 12/040

10000

20000

30000

40000

50000

(SC

F / S

TB)

Simulated GOR Historical GOR

01/94 09/96 06/99 03/02 12/040

10000

20000

30000

40000

50000

60000

70000

80000

GA

S PR

OD

UC

TIO

N R

ATE

(MSC

F / D

AY)

Simulated Gas Production Historical Gas Production

01/94 09/96 06/99 03/02 12/040

5000

10000

15000

20000

WA

TER

PR

OD

UC

TIO

N R

ATE

(STB

/ D

AY) Simulated Water Production Historical Water Production

Figure 5.5: Simulated production profiles for the single-region M4.1 reservoir using the ANN-predicted properties compared to the historical production data.

67

Each region in the reservoir used a different combination of the properties k, ø

and h (Tables 5.5 & 5.6) for the simulation runs that were used in formulating the training

and testing data. The differences between the simulated production and the historical

production, from the three producers, at specific times were used to generate the training

and testing data for input to the network.



22.5020.0017.5015.0012.5010.007.505.002.50 25.000.00

Region 1

Region 2

Region 4

Region 3

0 .5miles

Fault A



22.5020.0017.5015.0012.5010.007.505.002.50 25.000.00

Region 1

Region 2

Region 4

Region 3

0 .5miles



22.5020.0017.5015.0012.5010.007.505.002.50 25.000.00

Region 1

Region 2

Region 4

Region 3



22.5020.0017.5015.0012.5010.007.505.002.50 25.000.00

Region 1

Region 2

Region 4

Region 3



22.5020.0017.5015.0012.5010.007.505.002.50 25.000.00



22.5020.0017.5015.0012.5010.007.505.002.50 25.000.00



22.5020.0017.5015.0012.5010.007.505.002.50 25.000.00

Region 1

Region 2

Region 4

Region 3

0 .5miles

0 .5miles

0 .500 .5miles

Fault A

Figure 5.6: The four delineated regions in the M4.1 reservoir on a curvilinear grid: Region North of Fault A, the Channel, the East Levee and the West Levee respectively. The regions were delineated according to compartmentalization and varying permeability in each region. The Channel, Region 2, was simulated as having the least transmissibility and permeability due to the restricted communication between the East and West Levees.

68

Table 5.5: Randomly generated properties used in the 38 simulation runs for formulating the training data for the reservoir with four regions.

k1 ø1 h1 k2 ø2 h2 k3 ø3 h3 k4 ø4 h4

Ranges 40<k1<150 0.18<ø1<0.26 50<h1<150 10<k2<50 0.26<ø2<0.34 50<h2<150 40<k3<150 0.24<ø3<0.32 100<h3<200 30<k4<100 0.22<ø4<0.3 150<h4<3001 72.173 0.23184 71.128 21.538 0.32608 105.77 120.84 0.27032 101.44 80.068 0.26924 198.822 130.41 0.24901 92.111 20.108 0.28701 70.545 64.481 0.26699 148.58 35.666 0.2916 231.623 149.34 0.18247 132.95 19.61 0.27579 136.05 94.629 0.26517 141.64 89.211 0.221 195.984 47.46 0.20927 64.247 32.06 0.30081 75.735 40.773 0.31577 177.29 80.286 0.24407 250.915 147.04 0.18751 82.163 18.613 0.27679 77.764 74.781 0.28281 148.81 90.928 0.29721 201.866 141.51 0.20677 131.79 36.62 0.3064 78.817 121.23 0.29616 152.26 91.051 0.22817 297.097 145.46 0.18059 79.338 22.355 0.33834 123.48 126.94 0.31 178.2 83.309 0.26768 203.18 101.95 0.24524 57.995 28.791 0.29737 99.701 84.031 0.3189 159.12 76.862 0.25752 152.939 144.56 0.182 142.83 30.232 0.32417 55.467 83.988 0.31082 112.64 93.142 0.27739 271.91

10 75.005 0.21394 66.677 37.535 0.29464 69.129 67.686 0.27239 110.97 87.502 0.28871 297.4211 69.249 0.1827 88.677 17.715 0.27929 122.46 50.11 0.29017 166.29 88.286 0.23485 196.7812 46.719 0.23414 98.601 20.943 0.31077 147.37 106.24 0.27084 199.71 74.794 0.25699 206.8413 76.604 0.20624 131.09 47.053 0.28937 120.45 131.57 0.30783 134.62 40.934 0.29226 262.214 72.167 0.22481 90.571 19.39 0.29452 130.15 72.283 0.28215 117.61 84.822 0.22177 216.4215 84.273 0.21398 129.5 15.119 0.28415 70.625 91.209 0.30459 106.79 81.198 0.28083 164.9816 43.752 0.19201 51.938 26.505 0.29457 127.27 45.008 0.27148 130.94 69.531 0.28367 298.8917 71.622 0.18515 145.67 35.389 0.31524 137.27 49.429 0.31694 133.48 80.634 0.24511 230.4418 64.158 0.22932 135.49 17.262 0.30098 130.74 96.446 0.24241 137.62 46.782 0.23887 170.1819 102.25 0.23032 148.63 26.129 0.33931 125.66 118.7 0.3163 195.22 64.29 0.25956 169.720 135.71 0.19413 75.07 37.551 0.27473 140.46 43.898 0.29715 171.93 59.652 0.27482 279.621 68.049 0.21263 98.251 23.036 0.30472 92.1 147.75 0.29172 177.93 85.697 0.29557 256.2422 105.1 0.23649 82.461 46.931 0.29896 115.84 94.719 0.27573 161.77 83.439 0.25514 299.7423 97.496 0.21438 55.74 40.285 0.2918 50.13 117.08 0.25397 164.92 46.722 0.25979 272.3624 54.206 0.19687 79.097 26.045 0.29943 148.2 128.34 0.30682 175.63 74.458 0.22789 267.3425 40.415 0.22728 117.61 17.844 0.33751 91.231 96.834 0.31761 114.78 46.208 0.26436 264.5426 75.84 0.24698 145.22 26.011 0.32616 84.76 95.297 0.2508 159.95 73.116 0.25486 285.7527 79.684 0.25106 54.027 16.179 0.27027 116.82 123.02 0.26008 189.86 48.795 0.2345 165.4728 149.34 0.25085 132.16 12.583 0.32738 94.368 66.192 0.3128 117.19 99.385 0.2323 181.7629 143.4 0.25703 72.362 14.814 0.32316 133.99 143.74 0.29412 181.89 83.222 0.2947 277.6630 87.755 0.24157 149.97 11.174 0.28759 102.95 49.749 0.28985 106.93 63.752 0.2309 261.4431 142.91 0.20944 76.053 44.608 0.31908 75.714 99.731 0.28098 195.57 96.164 0.29988 210.7632 94.652 0.18545 73.028 26.568 0.26476 65.63 139.38 0.24028 131.73 55.252 0.27525 195.7633 146.82 0.18871 102.51 41.677 0.30469 105.78 134.75 0.25815 100.52 35.907 0.29688 221.1334 145.34 0.25495 106.07 42.656 0.32262 146.13 109.46 0.31828 175.99 96.396 0.28204 171.7135 119.28 0.22088 132.6 25.813 0.27126 145.13 68.86 0.3089 130.87 78.358 0.22006 224.1536 75.00 0.22 100.00 20.00 0.30 125.00 75.00 0.28 165.00 55.00 0.26 250.0037 40.00 0.18 50.00 10.00 0.26 50.00 40.00 0.24 100.00 30.00 0.22 150.0038 150.00 0.26 150.00 50.00 0.34 150.00 150.00 0.32 200.00 100.00 0.30 300.00

69

Table 5.6: Randomly generated properties used in the 10 simulation runs for formulating the testing data for the reservoir with four regions.

k1 ø1 h1 k2 ø2 h2 k3 ø3 h3 k4 ø4 h4

Ranges 40<k1<150 0.18<ø1<0.26 50<h1<150 10<k2<50 0.26<ø2<0.34 50<h2<150 40<k3<150 0.24<ø3<0.32 100<h3<200 30<k4<100 0.22<ø4<0.3 150<h4<300TEST1 100 0.2 110 25 0.28 100 85 0.25 150 75 0.26 200TEST2 144.51 0.21558 91.027 34.152 0.3277 100.28 56.596 0.29159 183.85 96.979 0.23092 298.25TEST3 65.425 0.22923 139.36 20.888 0.30201 120.95 116.77 0.30544 156.81 66.581 0.22094 237.42TEST4 106.75 0.24335 55.789 17.953 0.27621 92.889 81.621 0.29282 137.04 91.61 0.29151 213.52TEST5 93.458 0.25375 85.287 10.611 0.31377 80.462 134.6 0.26736 170.27 42.107 0.23593 227.33TEST6 138.04 0.23906 131.32 39.871 0.32705 68.965 133.9 0.26318 154.66 98.582 0.2439 200.09TEST7 123.83 0.1941 50.986 27.804 0.26157 69.343 105.29 0.2673 144.49 49.001 0.27292 214.94TEST8 90.211 0.21246 63.889 47.273 0.3145 118.22 94.621 0.28273 169.46 47.663 0.24275 183.89TEST9 42.035 0.25484 70.277 28.64 0.29036 80.276 138.97 0.29817 162.13 91.302 0.25754 236.97TEST10 130.35 0.25335 69.872 26.746 0.32654 104.17 130.38 0.26474 179.48 81.611 0.22518 264.05

70

5.2.1 Training and testing of the ANN with data generated from the delineated Tahoe M4.1 reservoir simulations

The developed network and functional links, that effectively trained the ANN

developed for the single-region reservoir, were used in the training process using data

formulated from the M4.1 simulations. In this case the predicted properties were

permeability, porosity, thickness and associated functional links in each region of the

reservoir (Figure 5.7).

Figure 5.7: Architecture of the neural network used to predict the properties of thesingle-region M4.1 reservoir.

Input Layer (5 neurons)

Middle Layers20 - 25 - 25

Output Layer(20 neurons)

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

4411

4411

41

41

41

// φφ

φφ

kkhkhk

hh

kk

→→

→→→

∆∆

∆∆

∆∆

∆∆

gorq

qq

dayst

gorq

qq

dayst

w

g

o

w

g

o

)3600(

)90(

M

M

71

The network was not trained as effectively as the single-region case as seen from

the training and testing errors in Figures 5.8 and 5.9 respectively. The average training

error increased from 8% to 35% and the average testing error increased from 7% to 20%.

This could be as a result of the network having a larger number of outputs to predict and

the connectivity is not as clearly defined as with the single-region reservoir.

0 5 10 15 20 25 30 35 4035

40

45Training Errors vs Case Number: North of Fault A

Avg

Per

m E

rror %

0 5 10 15 20 25 30 35 409

10

11

Avg

Phi

Erro

r %

0 5 10 15 20 25 30 35 4028

30

32

34

Case Number

Avg

h E

rror %

0 5 10 15 20 25 30 35 4025

30

35

40Training Errors vs Case Number: Channel

Avg

Per

m E

rror %

0 5 10 15 20 25 30 35 404.5

5

5.5

6

Avg

Phi

Erro

r %

0 5 10 15 20 25 30 35 4024

26

28

30

Case Number

Avg

h E

rror %

0 5 10 15 20 25 30 35 4030

35

40Training Errors vs Case Number: East Levee

Avg

Per

m E

rror %

0 5 10 15 20 25 30 35 406

6.5

7

7.5

Avg

Phi

Erro

r %

0 5 10 15 20 25 30 35 4012

13

14

15

Case Number

Avg

h E

rror %

0 5 10 15 20 25 30 35 4020

22

24

26Training Errors vs Case Number: West Levee

Avg

Per

m E

rror %

0 5 10 15 20 25 30 35 407

7.5

8

8.5

Avg

Phi

Erro

r %

0 5 10 15 20 25 30 35 4014

16

18

Case Number

Avg

h E

rror %

Figure 5.8: Training errors for k, ø and h in each region of the reservoir. The overallaverage training error was 35%.

72

As a result of the higher training and testing errors, as well as the added

uncertainty of the ranges utilized, fifteen additional simulations were done using

randomly generated properties within wider ranges as shown in Table 5.7. These

additional simulations were used to formulate a larger training data set. However, these

additional training data did not significantly change the predicted properties (Table 5.7)

of each region but it did increase the training times.

1 2 3 4 5 6 7 8 9 100

50

100

150Testing Errors: North of Fault A

Per

m E

rror %

1 2 3 4 5 6 7 8 9 100

10

20

Phi

Erro

r %

1 2 3 4 5 6 7 8 9 100

50

100

Case Number

h E

rror %

1 2 3 4 5 6 7 8 9 100

50

100

150Testing Errors: Channel

Per

m E

rror %

1 2 3 4 5 6 7 8 9 100

5

10

15

Phi

Erro

r %

1 2 3 4 5 6 7 8 9 100

20

40

60

Case Number

h E

rror %

1 2 3 4 5 6 7 8 9 100

50

100Testing Errors: East Levee

Per

m E

rror %

1 2 3 4 5 6 7 8 9 100

5

10

15

Phi

Erro

r %

1 2 3 4 5 6 7 8 9 100

10

20

30

Case Number

h E

rror %

1 2 3 4 5 6 7 8 9 100

50

100Testing Errors: West Levee

Per

m E

rror %

1 2 3 4 5 6 7 8 9 100

10

20

Phi

Erro

r %

1 2 3 4 5 6 7 8 9 100

10

20

30

Case Number

h E

rror %

Figure 5.9: Testing errors for k, ø and h in each region of the reservoir. The averagetesting error was 20%.

73

The same functional links, input and outputs (Table 5.4) used in section 5.1

worked just as effectively in training this network. Initially, three hidden layers were used

in the training/learning process. However, with the variation of the number of hidden

layers, it was observed that the use of one hidden layer gave ANN-predicted properties

that were similar to the network that used a larger number of hidden layers. As a result,

one hidden layer with 50 neurons was used to train this network.

5.2.2 Predicting the properties in each region of the M4.1 reservoir

The network developed above was used to predict the properties in each region of

the reservoir. Table 5.7 shows the inputs to the network, the outputs, and the ANN-

predicted properties in each region of the reservoir. The use of a larger number of

simulations with an expanded data range for formulating the training data did not

significantly change the predicted properties; the values were of the same order

(Table 5.7). The ANN-predicted values were then input to the simulator so as to compare

with the historical production of the reservoir.

The simulated production using the ANN-predicted properties using the narrow

data range and the wide data range gave the same result (Figure 5.10). The simulated oil

and gas production obtained from the ANN-predicted properties matched the actual

production data fairly well (Figure 5.9). However, as in the case of the single-region

reservoir, the match for water production was not as good. As explained previously, this

deviation from the historical water production could be due to differences in relative

74

permeability to water, the gas/oil contact (GOC), the oil/water contact (OWC), or the

presence of an aquifer. It could also indicate a lack of connectivity between the inputs

and outputs within the network. This connectivity could be improved with the use of

additional functional links, which can be explored in future studies.

01/94 09/96 06/99 03/02 12/040

1000

2000

3000

4000

5000

6000

OIL

PR

OD

UC

TIO

N R

ATE

(STB

/ D

AY)

QOP QOPH Simulated Oil Production Historical Oil Production

01/94 09/96 06/99 03/02 12/040

10000

20000

30000

40000

50000

(SC

F / S

TB)

GOR GORH Simulated GOR Historical GOR

01/94 09/96 06/99 03/02 12/040

10000

20000

30000

40000

50000

60000

70000

80000

GA

S PR

OD

UC

TIO

N R

ATE

(MSC

F / D

AY)

QGP QGPH Simulated Gas Production Historical Gas Production

01/94 09/96 06/99 03/02 12/040

5000

10000

15000

20000

WA

TER

PR

OD

UC

TIO

N R

ATE

(STB

/ D

AY)

QWP QWPH Simulated Water Production Historical Water Production

Figure 5.10: Production profiles for the M4.1 reservoir using ANN-predicted properties for each region in the numerical simulation model compared to the historical production data.

75

Table 5.7: Predicted properties for each region of the M4.1 reservoir using two training data sets. The training data wasformulated using a narrow data range with 38 simulations and a wide data range with 53 simulations. The ANN-predicted properties using both networks were similar.

k1 ø1 h1 k2 ø2 h2 k3 ø3 h3 k4 ø4 h4

40<k1<150 0.18<ø1<0.26 50<h1<150 10<k2<50 0.26<ø2<0.34 50<h2<150 40<k3<150 0.24<ø3<0.32 100<h3<200 30<k4<100 0.22<ø4<0.3 150<h4<300

20<k1<200 0.15<ø1<0.32 20<h1<200 5<k2<75 0.18<ø2<0.35 20<h2<200 20<k3<200 0.18<ø3<0.33 50<h3<250 20<k4<150 0.18<ø4<0.32 100<h4<300

164.0 70.0 0.26 215.0107.0 102.0 0.28103.0 0.21 97.0 31.0

95.0 0.22 98.0 26.0 0.28

Training data from 53 cases

and 10 test cases

Training data from 38 cases

and 10 test cases

time ∆qo ∆qg ∆qw ∆GOR

time ∆qo ∆qg ∆qw ∆GOR

Channel East Levee West Levee

149.0 70.0 0.260.30 105.0 92.0 227.0

Inputs

0.29

Wide Data Range

Narrow Data Range

Outputs: k, ø, h, kh and k/ø

Region North of Fault A

In order to test the effectiveness of training the network with a property that

presents uncertainty in the simulation model, a new training data set was generated. The

simulations were done using the same randomly generated properties as in Table 5.5 and

Table 5.6, as well as randomly generated values for the GOCs and OWCs (Table 5.8 and

Table 5.9). The GOC values were randomly generated between 9,900 ft and 10,200 ft

while the OWC values were randomly generated between 10,200 ft and 10,500 ft. The

differences between the historical production and the simulated production were used to

formulate the training and testing data as described in Section 5.1.1.

The network was successfully trained using one hidden layer with 50 neurons and

the inputs and outputs shown in Table 5.10. The training errors obtained for the GOC and

the OWC were 0.72% and 0.53% respectively while the testing errors obtained for the

GOC and OWC were 0.92% and 0.85% respectively (Figure 5.11). Small randomly

generated values of production differences (Section 3.2, Table 3.2) were used as inputs to

the network for the prediction of k, ø, h, GOC and OWC in the M4.1 reservoir. Using

these inputs, the ANN-predicted values for GOC and OWC from the trained network

were 10,100 ft and 10,300 ft respectively. Inputting the predicted values for k, ø, h, GOC

and OWC for each region into the simulator showed that there is an improved history

match for water production (Figure 5.12) while that of oil and gas remained the same as

shown in Figure 5.10.

77

Table 5.8: Randomly generated values for the GOCs and OWCs used in 38 simulations for formulating the training data for the reservoir with four regions.

GOC (ft) OWC (ft)Ranges 9,900<GOC<10,200 10,200<OWC<10,500

1 10185 103262 9969 104543 10082 103584 10046 102615 10167 104026 10129 104517 10037 102068 9906 104049 10146 10314

10 10033 1045011 10085 1035112 10138 1041313 10177 1032914 10121 1029115 9953 1025716 10022 1025817 10181 1040518 10175 1029119 10023 1036320 10168 1024521 9800 1020022 10006 1031423 10144 1045824 9903 1045625 9942 1037826 9961 1034927 9960 1047028 10081 1044629 9982 1020530 9960 1044531 9700 1010032 10124 1030333 9700 1000034 10180 1030235 10040 1036036 10200 1041037 9900 1020038 10200 10500

78

Table 5.9: Randomly generated values for GOCs and OWCs used in 10 simulations for formulating the testing data for the reservoir with four regions.

GOC (ft) OWC (ft)Ranges 9,900<GOC<10,200 10,200<OWC<10,500

1 9963 103252 9900 102253 10135 104624 10104 102055 10038 104306 10070 104917 10138 104978 9918 104379 10081 10332

10 9915 10349

Table 5.10: Inputs and outputs to the network used in the prediction of the properties k,ø, h, GOC and OWC.

Inputs Description Outputs Description

time (days) Time at which production is recorded k Permeability

∆qo (STB/D) Oil production difference from historical data Φ Porosity

∆qg (MSCF/D) Gas production difference from historical data h Thickness

∆qw (STB/D) Water production difference from historical data kh Functional Link, kh

∆GOR (SCF/STB) GOR difference from historical data k/ø Functional Link, k/ø

- - GOC Gas/Oil Contact

- - OWC Oil/Water Contact

79

1 2 3 4 5 6 7 8 9 100

1

2

3Testing Errors vs Case Number

GO

C T

estin

g E

rrors

%

1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

OW

C T

estin

g E

rror %

0 5 10 15 20 25 30 35 40

0.65

0.7

0.75

0.8Training Errors vs Case Number

GO

C T

rain

ing

Erro

rs %

0 5 10 15 20 25 30 35 400.5

0.52

0.54

0.56

0.58

OW

C T

rain

ing

Erro

rs %

Figure 5.11: Training and testing errors for the GOC and the OWC obtained from theANN. The average training errors for the GOC and OWC were 0.72% and 0.53% respectively. The average testing errors for the GOC and the OWC were 0.92% and 0.85% respectively.

80

When there is high uncertainty with respect to a specific property, this uncertainty

can be incorporated into the network by modifying the output data in the training data set

to include the varied properties (k, ø, h , GOC and OWC) and associated functional links.

The input data can be formulated from the production profiles as previously described. In

this way, reservoir characteristics, rock properties and fluid properties can be built into

the network so as to predict data that present uncertainty in the simulation model. This

study concentrates on the prediction of k, ø and h, but in future studies other factors such

as end point data for relative permeability, oil and gas exponents, saturations, density etc.

can be varied to formulate training and testing data, and the ANN used in predicting the

values that can give an improved history match.

01/94 09/96 06/99 03/02 12/040

5000

10000

15000

20000W

ATER

PRO

DUCT

ION

RATE

(STB

/ DA

Y)

QW P QW PH

GOC=10250; OWC=10410 GOC=10100; OWC=10300 Historical Water Production

GOC=10100 ft, OWC=10300 ft

GOC=10250 ft, OWC=10410 ft

01/94 09/96 06/99 03/02 12/040

5000

10000

15000

20000W

ATER

PRO

DUCT

ION

RATE

(STB

/ DA

Y)

QW P QW PH

GOC=10250; OWC=10410 GOC=10100; OWC=10300 Historical Water Production

GOC=10100 ft, OWC=10300 ft

GOC=10250 ft, OWC=10410 ft

GOC=10100 ft, OWC=10300 ft

GOC=10250 ft, OWC=10410 ft

Figure 5.12: Improved history match to water production with ANN-predicted GOC at 10,100 ft and OWC at 10,300 ft.

Chapter 6

DISCUSSION AND CONCLUSIONS

This study developed a methodology using artificial neural networks and an

inverse solution method to train and predict the permeability, porosity and thickness of a

reservoir that gave an improved starting point for a history match. This goal was achieved

by using an inverse method to train the network using production differences between

historical data and reservoir simulations. Each simulation had varied properties that were

randomly generated between realistic ranges for the specific reservoir under study. The

trained network was then tested and used to predict the reservoir’s properties.

Artificial neural networks (ANNs) can effectively be used to decrease the number

of simulations required to obtain a history match. A lack of established procedures for

network construction led to the use of trial and error in determining the optimum network

used in the training/learning process. The number of neurons in the layers was

determined ad hoc; except for the first and the last layers whose dimensions were

determined by the input and target data. This study provided some guidelines in the

development of ANNs for history matching. It can pave the way towards further

development of more universal networks that can predict many of the uncertainties in a

reservoir simulation model that can lead to improved history matching with a lesser

number of simulations.

82

The ANN was trained using a feed-forward network with back-propagation that

has proven to be successful in engineering applications. The network was built in stages

in which the degree of complexity was increased. At each stage of network development,

a new network with different inputs and functional links was developed.

Time was an important factor in training the network. Whenever time was

included, there were improved results. However, each case can have a large number of

time steps, resulting in a large number of inputs that can cause the network to take longer

training times. In addition, the network can become very complicated and a large number

of neurons had to be used in the hidden layers to successfully train the network. Using

data at shorter time intervals can also cause the problem of overtraining and

memorization and as such the performance goal must be carefully monitored. It was

determined that production taken every 90 days provided the network with sufficient data

for successful training.

The network was trained and developed using the production differences between

reservoir simulations and the historical production data at specific times, which proved to

be quite effective. Another approach that was taken in the development of the network

was to use the total difference in the areas under the curves, i.e. the historical versus

simulated curves, as inputs to the network. This did not take the time steps into

consideration. The observation made here is that the total difference between the curves

is either always positive or always negative. This did not allow for proper training when

83

there were no positive or negative values. As a result, the ANN-prediction results were

unrealistic.

Utilization of the ANN proved to be useful in predicting the properties, k, ø and h,

of the M4.1 reservoir that gave a good starting point for a history match. Training the

network was done using two data sets, one with a narrow range and one with a wide

range of properties to account for greater uncertainties. Both networks predicted similar

results for k, ø and h and they both gave almost identical history matches. In this case, the

uncertainty in the data was not extremely high and as such expanding the training data set

did not significantly improve the results. However, in reservoirs in which there is a high

degree of uncertainty, using a wide data range should prove to be more effective in

predicting the unknowns.

A complicated network yielded approximately the same results as a simple

network for the M4.1 reservoir with four regions and three producers. Therefore, a simple

ANN structure was used to train this network. The functional links kh, and k/ø proved to

be the most effective in the training process. The other functional links that were used in

developing the network did not improve the training and testing errors for the M4.1

reservoir. During the ANN development in Chapter 4, specific property values were

known and the network was tailored to give those specific values. However, in the

network development for the M4.1 reservoir, the properties were unknown and the

predictions using some of the specific functional links were not very different. The

predicted values were all of the same order and the history match obtained by inputting

84

these values into the simulator were similar. Even though some of the functional links did

not give any visible improvements in the predicted properties for the M4.1 reservoir, they

may prove to be useful in obtaining improved history matches for other reservoir studies

of this nature.

Some of the conclusions that this study has led to are as follows:

1) An increase in the amount of neurons in each layer of the network resulted in

increased training times but gave better results. However, there was a point at

which increasing the number of neurons can be ineffective, causing overtraining

and memorization.

2) Multi-layer networks proved to be effective in training the network. However, if

the number of hidden layers were too high the network became inefficient and

ineffective.

3) In network development, smaller networks gave larger training, testing and

prediction errors while larger networks performed better. However, when applied

to the M4.1 reservoir the simple network was just as effective as the more

complex networks.

4) Greater uncertainties in the data require a larger amount of training data to

successfully train the network.

5) Less data (every 90 days versus every 30 days) did not have a significant impact

on the outputs from the network. Production inputs to the network every 90 days

effectively trained the network while lessening the training times and the

complexity.

85

6) The most effective transfer function in the hidden layers was the hyperbolic

tangent sigmoid function (tansig) while the linear function (purelin) was most

effective in the last layer.

7) As the reservoir became more complex, there was difficulty in finding effective

functional links for training the network. Two of the most effective functional

links was found to be kh and k/ø. The areas of the regions and the distances from

the producers to the boundaries of the reservoir also helped the neural network to

understand the behavior of the reservoir.

8) Eigenvalues of various matrix constructions can be a valuable link to

mathematically characterize the data sets and feed the network with specific data

that can accurately characterize the reservoir.

9) Some of the functional links that proved to be effective in the ANN developed in

Chapter 4 did not prove to be as effective when applied to a M4.1 reservoir in the

Tahoe field. These functional links could prove to be effective in other history

match studies.

The application of ANNs in the prediction of the properties for the M4.1 reservoir

has proven to be promising. An ANN was developed that predicted properties that gave a

close match to the oil and gas production from three producers in the M4.1 reservoir.

Uncertainties in the simulation model can be presented to the network and appropriately

trained to give realistic values for reservoir properties that are unclear. The training data

should consist of specific ranges for the properties that we are attempting to predict and

should be appropriate for the field under study. If field data are not available then the

86

ANN can be developed to encompass a wider range of properties that is realistic for the

region being studied.

The work done in this study has been on a field level, future work can consider

the differences in production and pressures at individual wells which can be used as

additional inputs to the network. This will provide further links within the network that

should improve the network’s prediction accuracy. The use of additional functional links

that will improve the connectivity between the inputs and the outputs can also be

explored. This study focused on the prediction of k, ø and h but it has been illustrated that

artificial neural networks can be adapted to effectively predict other properties of the

reservoir with a reduced number of simulations.

BIBLIOGRAPHY

Ali, J.K., Neural Networks: A New Tool for the Petroleum Industry? Society of Petroleum Engineers, Paper No. 27561, 1994.

Al-Fattah, S.M., Startzman, R.A., Predicting Natural Gas Production Using Artificial Neural Network. Society of Petroleum Engineers, Paper No. 68593, 2001.

Centilmen, A., Applications of Neural-networks in Multi-well Field Development. The Pennsylvania State University, Department of Mineral Engineering, 1999.

Doraisamy, H. Methods of Neuro-Simulation for Field Development – A Thesis in Petroleum and Natural Gas Engineering. The Pennsylvania State University, Department of Mineral Engineering, 1998.

Dye, L.W., Horne, R.N., Aziz, K. A New Method for Automated History Matching of Reservoir Simulators. Society of Petroleum Engineers, Paper No. 15137, 1986.

Ertekin, T., Abou-Kassam, J. H., King, G. Basic Applied Reservoir Simulation. Henry L. Doherty Memorial Fund of AIME, Society of Petroleum Engineers, Richardson, Texas, 2001.

Hagan, M.T., Demuth, H.B., Beale, M. Neural Network Design, PWS Publishing Company, 1996.

Mohaghegh, S. Virtual-Intelligence Applications in Petroleum Engineering, Parts 1-3. Society of Petroleum Engineers, Paper Nos. 58046, 61926 & 62415, 2000.

Schiozer, D.J., Use of Reservoir Simulation, Parallel Computing and Optimization Techniques to Accelerate History Matching and Reservoir Management Decisions, Society of Petroleum Engineers, Paper No. 53979, 1999.

Appendix A

RESERVOIR ROCK AND FLUID PROPERTIES USED TO FORMULATE TRAINING AND TESTING DATA USING LANDMARK’S DTVIP RESERVOIR

SIMULATOR

This appendix describes the parameters used in the reservoir simulation model in

the formulation of the training and testing data. The varying factors for all cases were the

permeability, porosity and thickness of the respective regions as described in Chapter 4.

A.1 Initialization Data

Table A.1: Rock and fluid data used in the initialization of the artificial neural network (ANN) model.

Type of Fluid Model Black Oil

Stock Tank Water Density 1.027 g/cc

Water Salinity 75,000 ppm

Water Formation Volume Factor 1.028

Water Viscosity 0.3615 cp

Water Compressibility 2.52 x 10-6 psi-1

Rock Compressibility 3.2 x 10-6 psi-1

Reservoir Pressure 4921 psia

Reservoir Temperature 203 oF

Gas Oil Contact 9,675 ft.

Oil Water Contact 9,925 ft.

Saturation Pressure 4880 psia

89

A.2 Rock property data

The rock property data were based on Corey’s permeability correlations. The

assumptions made here are that the water is the wetting phase, gas is the non-wetting

phase, and oil is the intermediate wetting phase in a three-phase system. End points were

used in the formulation of the tables according to Corey’s model which is a probability

model based on channel flow considerations. Corey’s model uses two sets of two-phase

relative permeability data to model the three-phase relative permeabilities. The following

equations govern the use of the Corey’s model [Error! Not a valid link.].

krw represents the relative permeability of the wetting phase in a three-phase system

obtained from measured two-phase data and is given by Eq. A.4.

where Swn is the normalized wetting-phase saturation and is given by Eq. A.5.

krg represents the relative permeability of the non-wetting phase. The relative

permeability of the non-wetting phase is given by Eq. A.6.

),( wrw Sfk = (Eq. A.1)

)( grg Sfk = (Eq. A.2)

).,( gwro SSfk = (Eq. A.3)

4wnrw Sk = (Eq. A.4)

wirr

iwwwn S

SSS

−−

=1

(Eq. A.5)

)1()1( 22wnwnrnw SSk −−= (Eq. A.6)

90

The end point saturations shown in Table A.2 and A.3 were used to model the

relative permeability of the fluids in this reservoir. The resulting relative permeability is

given in Table A.4. Figures A.1, A.2, A.3 and A.4 illustrate the curves obtained. The

capillary pressure curves were modeled according to core data and capillary pressure

laboratory measurements from the Tahoe Field. The capillary pressure exponent was

determined by trial and error until a match with experimental data was obtained.

Table A.2: End point input data used to generate relative permeability curves based on Corey’s Model for the water-oil system.

Connate water saturation 0.1 Oil relative permeability at connate water saturation 0.85 Residual oil saturation to water 0.02 Water relative permeability at residual oil saturation 0.9 Oil curve exponent 3 Water curve exponent 2.5 Entry Capillary Pressure 0.9 psia Maximum Capillary Pressure 32 psia Capillary Pressure Exponent 2.7

Table A.3: End point input data used to generate relative permeability curves based on Corey’s Model for the gas-oil system.

Critical gas saturation 0 Residual oil saturation to gas 0.05 Gas rel. permeability at connate water sat 0.87 Oil curve exponent 4 Gas curve exponent 1.5

91

Table A.4: Two-phase relative permeability and capillary pressure data derived from end points obtained from laboratory data.

Water-Oil Tables Gas-Oil Tables

Sw Krw Krow Pcow Sg Krg Krog Pcog 0.1 0 0.85 32 - - - -

0.15 0.0007 0.7132 11.27 0 0 0.85 0.9 0.2 0.0039 0.5919 5.63 0.05 0.0114 0.7305 0.91

0.25 0.0108 0.4852 3.46 0.1 0.0322 0.6216 0.92 0.3 0.0222 0.3922 2.44 0.15 0.0592 0.5231 0.93

0.34 0.035 0.327 1.98 0.2 0.0911 0.4347 0.95 0.39 0.0561 0.2562 1.63 0.25 0.1274 0.3558 0.97 0.44 0.0835 0.1964 1.41 0.3 0.1674 0.2863 1 0.49 0.1177 0.1467 1.26 0.35 0.211 0.2256 1.04 0.54 0.1591 0.1063 1.17 0.4 0.2578 0.1733 1.08 0.59 0.2082 0.074 1.1 0.45 0.3076 0.1291 1.15 0.64 0.2655 0.049 1.05 0.5 0.3603 0.0925 1.24 0.69 0.3313 0.0304 1.01 0.55 0.4156 0.0629 1.37 0.74 0.406 0.0172 0.98 0.6 0.4736 0.0399 1.58 0.78 0.4724 0.01 0.96 0.65 0.534 0.0228 1.9 0.83 0.5641 0.0042 0.94 0.7 0.5968 0.0111 2.44 0.88 0.6657 0.0012 0.93 0.75 0.6618 0.004 3.46 0.93 0.7776 0.0002 0.91 0.8 0.7291 0.0007 5.63 0.98 0.9 0 0.9 0.85 0.7985 0 11.27

1 1 0 0 0.9 0.87 0 32

92

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Water Saturation

Rel

ativ

e Pe

rmea

bilit

y to

Oil

in th

e Pr

esen

ce o

f Wat

er

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rel

ativ

e Pe

rmea

bilit

y to

Wat

er

krw

krow

krwkrow

Figure A.1: Relative permeability to oil and water in a two-phase system.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Gas Saturation

Rel

ativ

e Pe

rmea

bilit

y to

Oil

in th

e Pr

esen

ce o

f Gas

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rel

ativ

e Pe

rmea

bilit

y to

Gas

krgkrog

krgkrog

Figure A.2: Relative permeability to oil and gas in a two-phase system.

93

0

5

10

15

20

25

30

35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Water Saturation

Pres

sure

(psi

a)

Figure A.3: Capillary pressure curve in the oil-water system.

0

5

10

15

20

25

30

35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Gas Saturation

Pres

sure

(psi

a)

Figure A.4: Capillary pressure curve in the gas-oil system.

94

A.3 Black oil PVT data

Table A.5: Black oil PVT data derived from fluid analysis in the M4.1 reservoir.

API Gravity of Stock Tank Oil 33.5 Molecular Weight of Residual Oil 241 Gas Oil Ratio 974 SCF/STB Gas Gravity 0.6 Reservoir Temperature 203 oF Separator Temperature 60 oF Separator Pressure 14.65 psia Maximum Pressure Entry 5000 psia

Table A.6: Black oil PVT data used in the reservoir simulator to design the ANN model.

psat (psia)

RS (SCF/STB)

Bo (RB/STB)

Bg (RB/MSCF) GR µo(cp) µg (cp)

5000.00 1004.90 1.5194 0.6727 0.6 0.3964 0.0239 4974.39 998.71 1.5164 0.6751 0.6 0.3978 0.0238 4948.79 992.52 1.5133 0.6775 0.6 0.3991 0.0238 4923.18 986.34 1.5103 0.6799 0.6 0.4005 0.0237 4897.57 980.17 1.5073 0.6823 0.6 0.4019 0.0236 4871.96 974.00 1.5042 0.6848 0.6 0.4034 0.0236 4386.23 858.30 1.4481 0.7378 0.6 0.4328 0.0224 3900.50 745.20 1.3941 0.8079 0.6 0.468 0.0212 3414.77 634.94 1.3424 0.9027 0.6 0.5107 0.0199 2929.04 527.84 1.2932 1.0365 0.6 0.564 0.0187 2443.31 424.32 1.2468 1.2332 0.6 0.6323 0.0176 1957.58 324.93 1.2032 1.5449 0.6 0.7232 0.0165 1471.84 230.50 1.1629 2.0817 0.6 0.8506 0.0156 986.11 142.32 1.1264 3.1736 0.6 1.0412 0.0148 500.38 62.88 1.0946 6.4297 0.6 1.3512 0.0142 14.65 0.90 1.0706 226.9946 0.6 1.8512 0.0138

Appendix B

TRAINING AND TESTING SAMPLE DATA

time (days) ∆qo ∆qg ∆qw ∆p time

(yrs) ∆qo * time

(yrs) Sample Input Data

Case 1 90 -74 95 -0.001 0 0.00274 -0.20274 Case 2 90 -35 42 1.809 1 0.00274 -0.09589 Case 3 90 82 -101 1.822 9 0.00274 0.22466 Case 4 90 -2 2 -0.104 2 0.00274 -0.00548 Case 5 90 -95 120 1.825 7 0.00274 -0.26027 Case 6 90 -132 177 -5.074 -7 0.00274 -0.36164 Case 7 90 -37 46 1.823 10 0.00274 -0.10137 Case 8 90 160 -200 1.827 18 0.00274 0.43836 Case 9 90 -169 218 -2.415 -6 0.00274 -0.46301

Case 10 90 -212 278 -5.030 -11 0.00274 -0.58082 Case 11 90 45 -56 1.822 11 0.00274 0.12329 Case 12 90 -11 14 1.821 8 0.00274 -0.03014 Case 13 90 -50 65 -1.321 -3 0.00274 -0.13699 Case 14 90 20 -23 1.821 7 0.00274 0.05480 Case 15 90 43 -53 1.829 11 0.00274 0.11781 Case 16 90 -173 229 -5.157 -10 0.00274 -0.47397 Case 17 90 -59 76 1.820 4 0.00274 -0.16164 Case 18 90 204 -256 1.820 11 0.00274 0.55890 Case 19 90 103 -128 1.826 13 0.00274 0.28219 Case 20 90 -106 140 -3.143 -4 0.00274 -0.29041 Case 21 90 -117 148 -0.675 -2 0.00274 -0.32055 Case 22 90 -162 216 -5.300 -8 0.00274 -0.44384 Case 23 90 -29 41 -2.388 -5 0.00274 -0.07945 Case 24 90 -36 48 -1.983 1 0.00274 -0.09863 Case 25 90 -18 25 -1.590 -3 0.00274 -0.04932 Case 26 90 -117 155 -3.822 -7 0.00274 -0.32055 Case 27 90 231 -290 1.820 16 0.00274 0.63288 Case 28 90 73 -88 1.830 10 0.00274 0.20000 Case 29 90 -182 236 -2.969 -4 0.00274 -0.49863 Case 30 90 5 -5 -1.355 -3 0.00274 0.01370 Case 31 90 -130 164 1.826 6 0.00274 -0.35616 Case 32 90 36 -47 1.818 7 0.00274 0.09863 Case 33 90 -20 23 1.815 2 0.00274 -0.05480 Case 34 90 -5 8 1.831 14 0.00274 -0.01370 Case 35 90 15 -17 1.819 5 0.00274 0.04110

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

96

Case 1 3600 -67.68 -502.4 19.553 -24.44 9.863 -667.5 Case 2 3600 140.41 -202.9 238.17 -50 9.863 1384.8 Case 3 3600 510.11 -136.1 241.77 210.3 9.863 5031.2 Case 4 3600 63.44 178.48 -38.7 112.87 9.863 625.7 Case 5 3600 268.26 -708 241.67 67.435 9.863 2645.9 Case 6 3600 -759.3 1681.8 -765 140.43 9.863 -7489.2 Case 7 3600 313.79 -1144 241.58 -8 9.863 3094.9 Case 8 3600 967.47 -308.3 242.76 234.3 9.863 9542.2 Case 9 3600 -1.547 189.04 -453.5 45.435 9.863 -15.3

Case 10 3600 -1010 1939.1 -867.4 121.43 9.863 -9958.2 Case 11 3600 459.33 -325 241.78 163.87 9.863 4530.3 Case 12 3600 316.25 -1200 241.34 -36 9.863 3119.2 Case 13 3600 158.48 -8.304 -422.8 -122.4 9.863 1563.1 Case 14 3600 360.85 -144.6 240.34 183.3 9.863 3559.1 Case 15 3600 919.46 -638 242.71 164.3 9.863 9068.7 Case 16 3600 -992.1 1807.5 -854.7 51 9.863 -9785.5 Case 17 3600 234.17 -1077 238.27 -10 9.863 2309.6 Case 18 3600 923.27 -346.6 242.63 141.3 9.863 9106.2 Case 19 3600 882.92 -1723 242.68 -53 9.863 8708.3 Case 20 3600 -68.26 231.13 -617.8 -61.44 9.863 -673.3 Case 21 3600 -85.46 -584.4 -200.2 -78.44 9.863 -842.9 Case 22 3600 -864.4 1402.1 -783.6 53 9.863 -8525.8 Case 23 3600 174.19 1272.9 -442.2 148.87 9.863 1718.0 Case 24 3600 176.18 37.957 -348.9 43.435 9.863 1737.6 Case 25 3600 125.36 494.04 -413.9 -17 9.863 1236.5 Case 26 3600 -328.3 677.22 -659.3 -4 9.863 -3238.3 Case 27 3600 953.58 187.74 242.65 248.74 9.863 9405.1 Case 28 3600 745.02 -677 242.46 145.87 9.863 7348.1 Case 29 3600 -199.4 -425.4 -606.1 -155.9 9.863 -1966.9 Case 30 3600 168.95 690.57 -358 101.43 9.863 1666.4 Case 31 3600 169.97 -1256 241.23 -31.44 9.863 1676.4 Case 32 3600 394.86 422.61 241.89 215.3 9.863 3894.5 Case 33 3600 195.01 -213.6 240.48 -20 9.863 1923.4 Case 34 3600 868.22 -2181 242.72 -75.44 9.863 8563.2 Case 35 3600 325.04 -669.7 239.47 74.435 9.863 3205.9

97

k1 ø1 h1 k2 ø2 h2 k3 ø3 h3 k4 ø4 h4 k1h1 k2h2 k3h3 k4h4 Sample Target Data

72.173 0.2318 71.128 21.538 0.3261 105.77 120.84 0.2703 101.44 80.068 0.2692 198.82 5133.5 2278.1 12258.0 15919.1 130.41 0.249 92.111 20.108 0.287 70.545 64.481 0.267 148.58 35.666 0.2916 231.62 12012.2 1418.5 9580.6 8261.0 149.34 0.1825 132.95 19.61 0.2758 136.05 94.629 0.2652 141.64 89.211 0.221 195.98 19854.8 2667.9 13403.3 17483.6 47.46 0.2093 64.247 32.06 0.3008 75.735 40.773 0.3158 177.29 80.286 0.2441 250.91 3049.2 2428.1 7228.6 20144.6

147.04 0.1875 82.163 18.613 0.2768 77.764 74.781 0.2828 148.81 90.928 0.2972 201.86 12081.2 1447.4 11128.2 18354.7 141.51 0.2068 131.79 36.62 0.3064 78.817 121.23 0.2962 152.26 91.051 0.2282 297.09 18649.6 2886.3 18458.5 27050.3 145.46 0.1806 79.338 22.355 0.3383 123.48 126.94 0.31 178.2 83.309 0.2677 203.1 11540.5 2760.4 22620.7 16920.1 101.95 0.2452 57.995 28.791 0.2974 99.701 84.031 0.3189 159.12 76.862 0.2575 152.93 5912.6 2870.5 13371.0 11754.5 144.56 0.182 142.83 30.232 0.3242 55.467 83.988 0.3108 112.64 93.142 0.2774 271.91 20647.5 1676.9 9460.4 25326.2 75.005 0.2139 66.677 37.535 0.2946 69.129 67.686 0.2724 110.97 87.502 0.2887 297.42 5001.1 2594.8 7511.1 26024.8 69.249 0.1827 88.677 17.715 0.2793 122.46 50.11 0.2902 166.29 88.286 0.2349 196.78 6140.8 2169.4 8332.8 17372.9 46.719 0.2341 98.601 20.943 0.3108 147.37 106.24 0.2708 199.71 74.794 0.257 206.84 4606.5 3086.4 21217.2 15470.4 76.604 0.2062 131.09 47.053 0.2894 120.45 131.57 0.3078 134.62 40.934 0.2923 262.2 10042.0 5667.5 17712.0 10732.9 72.167 0.2248 90.571 19.39 0.2945 130.15 72.283 0.2822 117.61 84.822 0.2218 216.42 6536.2 2523.6 8501.2 18357.2 84.273 0.214 129.5 15.119 0.2842 70.625 91.209 0.3046 106.79 81.198 0.2808 164.98 10913.4 1067.8 9740.2 13396.0 43.752 0.192 51.938 26.505 0.2946 127.27 45.008 0.2715 130.94 69.531 0.2837 298.89 2272.4 3373.3 5893.3 20782.1 71.622 0.1852 145.67 35.389 0.3152 137.27 49.429 0.3169 133.48 80.634 0.2451 230.44 10433.2 4857.8 6597.8 18581.3 64.158 0.2293 135.49 17.262 0.301 130.74 96.446 0.2424 137.62 46.782 0.2389 170.18 8692.8 2256.8 13272.9 7961.4 102.25 0.2303 148.63 26.129 0.3393 125.66 118.7 0.3163 195.22 64.29 0.2596 169.7 15197.4 3283.4 23172.6 10910.0 135.71 0.1941 75.07 37.551 0.2747 140.46 43.898 0.2972 171.93 59.652 0.2748 279.6 10187.7 5274.4 7547.4 16678.7 68.049 0.2126 98.251 23.036 0.3047 92.1 147.75 0.2917 177.93 85.697 0.2956 256.24 6685.9 2121.6 26289.2 21959.0 105.1 0.2365 82.461 46.931 0.299 115.84 94.719 0.2757 161.77 83.439 0.2551 299.74 8666.7 5436.5 15322.7 25010.0

97.496 0.2144 55.74 40.285 0.2918 50.13 117.08 0.254 164.92 46.722 0.2598 272.36 5434.4 2019.5 19308.8 12725.2 54.206 0.1969 79.097 26.045 0.2994 148.2 128.34 0.3068 175.63 74.458 0.2279 267.34 4287.5 3859.9 22540.4 19905.6 40.415 0.2273 117.61 17.844 0.3375 91.231 96.834 0.3176 114.78 46.208 0.2644 264.54 4753.2 1627.9 11114.6 12223.9 75.84 0.247 145.22 26.011 0.3262 84.76 95.297 0.2508 159.95 73.116 0.2549 285.75 11013.5 2204.7 15242.8 20892.9

79.684 0.2511 54.027 16.179 0.2703 116.82 123.02 0.2601 189.86 48.795 0.2345 165.47 4305.1 1890.0 23356.6 8074.1 149.34 0.2509 132.16 12.583 0.3274 94.368 66.192 0.3128 117.19 99.385 0.2323 181.76 19736.8 1187.4 7757.0 18064.2 143.4 0.257 72.362 14.814 0.3232 133.99 143.74 0.2941 181.89 83.222 0.2947 277.66 10376.7 1984.9 26144.9 23107.4

87.755 0.2416 149.97 11.174 0.2876 102.95 49.749 0.2899 106.93 63.752 0.2309 261.44 13160.6 1150.4 5319.7 16667.3 142.91 0.2094 76.053 44.608 0.3191 75.714 99.731 0.281 195.57 96.164 0.2999 210.76 10868.7 3377.5 19504.4 20267.5 94.652 0.1855 73.028 26.568 0.2648 65.63 139.38 0.2403 131.73 55.252 0.2753 195.76 6912.2 1743.7 18360.5 10816.1

98

146.82 0.1887 102.51 41.677 0.3047 105.78 134.75 0.2582 100.52 35.907 0.2969 221.13 15050.5 4408.6 13545.1 7940.1 145.34 0.255 106.07 42.656 0.3226 146.13 109.46 0.3183 175.99 96.396 0.282 171.71 15416.2 6233.3 19263.9 16552.2 119.28 0.2209 132.6 25.813 0.2713 145.13 68.86 0.3089 130.87 78.358 0.2201 224.15 15816.5 3746.2 9011.7 17563.9

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

72.173 0.2318 71.128 21.538 0.3261 105.77 120.84 0.2703 101.44 80.068 0.2692 198.82 5133.5 2278.1 12258.0 15919.1 130.41 0.249 92.111 20.108 0.287 70.545 64.481 0.267 148.58 35.666 0.2916 231.62 12012.2 1418.5 9580.6 8261.0 149.34 0.1825 132.95 19.61 0.2758 136.05 94.629 0.2652 141.64 89.211 0.221 195.98 19854.8 2667.9 13403.3 17483.6 47.46 0.2093 64.247 32.06 0.3008 75.735 40.773 0.3158 177.29 80.286 0.2441 250.91 3049.2 2428.1 7228.6 20144.6

147.04 0.1875 82.163 18.613 0.2768 77.764 74.781 0.2828 148.81 90.928 0.2972 201.86 12081.2 1447.4 11128.2 18354.7 141.51 0.2068 131.79 36.62 0.3064 78.817 121.23 0.2962 152.26 91.051 0.2282 297.09 18649.6 2886.3 18458.5 27050.3 145.46 0.1806 79.338 22.355 0.3383 123.48 126.94 0.31 178.2 83.309 0.2677 203.1 11540.5 2760.4 22620.7 16920.1 101.95 0.2452 57.995 28.791 0.2974 99.701 84.031 0.3189 159.12 76.862 0.2575 152.93 5912.6 2870.5 13371.0 11754.5 144.56 0.182 142.83 30.232 0.3242 55.467 83.988 0.3108 112.64 93.142 0.2774 271.91 20647.5 1676.9 9460.4 25326.2 75.005 0.2139 66.677 37.535 0.2946 69.129 67.686 0.2724 110.97 87.502 0.2887 297.42 5001.1 2594.8 7511.1 26024.8 69.249 0.1827 88.677 17.715 0.2793 122.46 50.11 0.2902 166.29 88.286 0.2349 196.78 6140.8 2169.4 8332.8 17372.9 46.719 0.2341 98.601 20.943 0.3108 147.37 106.24 0.2708 199.71 74.794 0.257 206.84 4606.5 3086.4 21217.2 15470.4 76.604 0.2062 131.09 47.053 0.2894 120.45 131.57 0.3078 134.62 40.934 0.2923 262.2 10042.0 5667.5 17712.0 10732.9 72.167 0.2248 90.571 19.39 0.2945 130.15 72.283 0.2822 117.61 84.822 0.2218 216.42 6536.2 2523.6 8501.2 18357.2 84.273 0.214 129.5 15.119 0.2842 70.625 91.209 0.3046 106.79 81.198 0.2808 164.98 10913.4 1067.8 9740.2 13396.0 43.752 0.192 51.938 26.505 0.2946 127.27 45.008 0.2715 130.94 69.531 0.2837 298.89 2272.4 3373.3 5893.3 20782.1 71.622 0.1852 145.67 35.389 0.3152 137.27 49.429 0.3169 133.48 80.634 0.2451 230.44 10433.2 4857.8 6597.8 18581.3 64.158 0.2293 135.49 17.262 0.301 130.74 96.446 0.2424 137.62 46.782 0.2389 170.18 8692.8 2256.8 13272.9 7961.4 102.25 0.2303 148.63 26.129 0.3393 125.66 118.7 0.3163 195.22 64.29 0.2596 169.7 15197.4 3283.4 23172.6 10910.0 135.71 0.1941 75.07 37.551 0.2747 140.46 43.898 0.2972 171.93 59.652 0.2748 279.6 10187.7 5274.4 7547.4 16678.7 68.049 0.2126 98.251 23.036 0.3047 92.1 147.75 0.2917 177.93 85.697 0.2956 256.24 6685.9 2121.6 26289.2 21959.0 105.1 0.2365 82.461 46.931 0.299 115.84 94.719 0.2757 161.77 83.439 0.2551 299.74 8666.7 5436.5 15322.7 25010.0

97.496 0.2144 55.74 40.285 0.2918 50.13 117.08 0.254 164.92 46.722 0.2598 272.36 5434.4 2019.5 19308.8 12725.2 54.206 0.1969 79.097 26.045 0.2994 148.2 128.34 0.3068 175.63 74.458 0.2279 267.34 4287.5 3859.9 22540.4 19905.6 40.415 0.2273 117.61 17.844 0.3375 91.231 96.834 0.3176 114.78 46.208 0.2644 264.54 4753.2 1627.9 11114.6 12223.9 75.84 0.247 145.22 26.011 0.3262 84.76 95.297 0.2508 159.95 73.116 0.2549 285.75 11013.5 2204.7 15242.8 20892.9

79.684 0.2511 54.027 16.179 0.2703 116.82 123.02 0.2601 189.86 48.795 0.2345 165.47 4305.1 1890.0 23356.6 8074.1

99

149.34 0.2509 132.16 12.583 0.3274 94.368 66.192 0.3128 117.19 99.385 0.2323 181.76 19736.8 1187.4 7757.0 18064.2 143.4 0.257 72.362 14.814 0.3232 133.99 143.74 0.2941 181.89 83.222 0.2947 277.66 10376.7 1984.9 26144.9 23107.4

87.755 0.2416 149.97 11.174 0.2876 102.95 49.749 0.2899 106.93 63.752 0.2309 261.44 13160.6 1150.4 5319.7 16667.3 142.91 0.2094 76.053 44.608 0.3191 75.714 99.731 0.281 195.57 96.164 0.2999 210.76 10868.7 3377.5 19504.4 20267.5 94.652 0.1855 73.028 26.568 0.2648 65.63 139.38 0.2403 131.73 55.252 0.2753 195.76 6912.2 1743.7 18360.5 10816.1 146.82 0.1887 102.51 41.677 0.3047 105.78 134.75 0.2582 100.52 35.907 0.2969 221.13 15050.5 4408.6 13545.1 7940.1 145.34 0.255 106.07 42.656 0.3226 146.13 109.46 0.3183 175.99 96.396 0.282 171.71 15416.2 6233.3 19263.9 16552.2 119.28 0.2209 132.6 25.813 0.2713 145.13 68.86 0.3089 130.87 78.358 0.2201 224.15 15816.5 3746.2 9011.7 17563.9

Appendix C

SAMPLE MATLAB PROGRAMS USED IN DESIGNING, TRAINING AND TESTING THE ARTIFICIAL NEURAL NETWORK

C.1 Extraction of data from historical production and simulated runs

% Asha Ramgulam % Pennsylvania State University % MS, Petroleum and Natural Gas Engineering, Spring 2006, % This program interpolates using time to calculate ∆qo, ∆qg, ∆qw and ∆GOR at the same times as the historical data. It formulates the training data for specific times, i.e. every 90 days for 10 years.

clear all;

format compact;

% Loads all the data

load history.mat; %File with historical production from the Tahoe Field

load GORH.txt; % File with historical GOR from the Tahoe Field

load properties.txt; % File with the properties for each region for all

cases

%Files with different production outputs generated from the Tahoe

simulations

load run1.txt;

load run2.txt;

load run3.txt;

load run4.txt;

load run5.txt;

101

load run6.txt;

load run7.txt;

load run8.txt;

load run9.txt;

load run10.txt;

load run11.txt;

load run12.txt;

load run13.txt;

load run14.txt;

load run15.txt;

load run16.txt;

load run17.txt;

load run18.txt;

load run19.txt;

load run20.txt;

load run21.txt;

load run22.txt;

load run23.txt;

load run24.txt;

load run25.txt;

load run26.txt;

load run27.txt;

load run28.txt;

load run29.txt;

load run30.txt;

load run31.txt;

load run32.txt;

load run33.txt;

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load run34.txt;

load run35.txt;

load run36.txt;

load run37.txt;

load run38.txt;

% Input of data into an array

run1=run1;

run2=run2;

run3=run3;

run4=run4;

run5=run5;

run6=run6;

run7=run7;

run8=run8;

run9=run9;

run10=run10;

run11=run11;

run12=run12;

run13=run13;

run14=run14;

run15=run15;

run16=run16;

run17=run17;

run18=run18;

run19=run19;

run20=run20;

run21=run21;

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run22=run22;

run23=run23;

run24=run24;

run25=run25;

run26=run26;

run27=run27;

run28=run28;

run29=run29;

run30=run30;

run31=run31;

run32=run32;

run33=run33;

run34=run34;

run35=run35;

run36=run36;

run37=run37;

run38=run38;

% File with the different cases run for Tahoe and ANN. This contains % different permeability, porosity and thickness for each of the four % regions properties2=properties;

% Calculation of the values for the functional links k*h and k/ø.

[row,col]=size(properties2); % row = No. of cases

for k=1:row

properties2(k,13)=properties(k,1)*properties(k,3);




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properties2(k,17)=properties(k,1)/properties(k,2);




end

save properties2.mat

% Interpolates to obtain the oil, gas and water flowrates, and the GOR at the same times as the historical data, i.e. every 90 days for 10 years j=1;

for time2=90:90:3770

for a=1:row

run_numbera(j,1)=time2;

run_numbera(j,2)=interp1(runa(:,2),runa(:,3),time2);




end

j=j+1;

end

% Calculates the difference between the historical data and the simulated data at the same times to create the delta values for b=1:row

deltab(:,1)=history(:,1);

deltab(:,2)=history(:,2)-run_numberb(:,2);


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deltab(:,5)=history(:,5)-run_numberb(:,5); end

[r,c]=size(delta1);

for k=1:row

for i=1:r

deltak(i,6:25)=properties2(k,1:20);

end

end

save delta.mat

% Creates the input and target files. % The input file consists of time, ∆qo, ∆qg, ∆qw and ∆gor % The target file consists of the properties permeability, porosity and % thickness in each region as well as the functional links kh and k/ø InputAll=[];

TargetAll=[];

for i=1:row

InputAll=[InputAll;deltai(:,1:5)];

TargetAll=[TargetAll;deltai(:,6:25)];

end

save InputAll.mat

save TargetAll.mat

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C.2 Training the network

% Outputs consist of the properties and the functional links kh and k/ø % 4 regions with different ranges of data for each region are used % Reservoir contains 3 producers % Region 1: Region North of Fault A % Region 2: Channel % Region 3: East Levee % Region 4: West Levee clear all;

format compact;

% Loads the input and target data

load InputAll.mat;

load TargetAll.mat;

%Transposes the input and target data

InputAll_T=InputAll';

TargetAll_T=TargetAll';

%Normalization of the input and target data

[n_InputAll_T,mini,maxi,n_TargetAll_T,mint,maxt] =

premnmx(InputAll_T,TargetAll_T);

% Inputs are time, ∆qo, ∆qg, ∆qw and ∆gor % Training the network using one hidden layer with 50 neurons PR=[ones(5,1)*-1 ones(5,1)];

net=newff(PR,[50,20],'tansig','purelin','traingdx','learngdm','msereg

');

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% Network parameters net.trainparam.goal=0.0001;

net.trainparam.epochs=75000;

net.trainparam.show=10;

net=train(net,n_InputAll_T,n_TargetAll_T);

weight = net.IW 1,1;

bias = net.b 1;

save Training1.mat

C.3 Output of training errors

clear all;

load Training1.mat % File with all variables from the training program

[ri,ci]=size(InputAll); % Size of the input file with training data

rr=ri/row; % Gives the number of time steps for each case

% Calculates the training errors, post normalizes the data and gives the output for each case for cases=1:row

for time=1:rr

error2cases(:,time)= postmnmx(sim(net,n_InputAll_T(:,(time-

1)*row+cases)),mint,maxt);

error2cases(:,time)=error2cases(:,time)-(TargetAll_T(:,(time-

1)*row+cases));

error2cases(:,time)=abs(error2cases(:,time));

error2cases(:,time)=error2cases(:,time)./(TargetAll_T(:,(time-

1)*row+cases))*100;

end

end

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% Calculates the average property error for each region

for i=1:row

avgerror_k1(i)=mean(error2i(1,:));




avgerror_phi1(i)=mean(error2i(2,:));




avgerror_h1(i)=mean(error2i(3,:));




end

% Graphical output of errors for Region 1

figure();

subplot(3,1,1);plot([1:row],avgerror_k1(:),'bo-')

title('Training Errors vs Case Number: North of Fault A')

ylabel('Avg Perm Error %')

grid on;

subplot(3,1,2);plot([1:row],avgerror_phi1(:),'bo-')

ylabel('Avg Phi Error %')

grid on;

subplot(3,1,3);plot([1:row],avgerror_h1(:),'bo-')

xlabel('Case Number')

ylabel('Avg h Error %')

grid on;

109

fprintf(1,' %g\n');

fprintf(1,'Permeability Error North of Fault A =

%g\n',mean(avgerror_k1(:)))

fprintf(1,'Porosity Error North of Fault A =

%g\n',mean(avgerror_phi1(:)))

fprintf(1,'Thickness Error North of Fault A =

%g\n',mean(avgerror_h1(:)))

C.4 Output of testing errors

% Contains data for 10 test cases

clear all;

format compact;

load Training1.mat; % File with all variables from the training program

load TestAll.mat; % File with the formulated input test data

load Testproperties.mat % File containing the properties for the 10

test cases

Testproperties2_T=Testproperties2'; %File with the properties and

functional links in the target data

[No_Rows,No_Col]=size(Testproperties2); % No_Rows = No. of cases

TestAll_T=TestAll';

110

% Normalization of input data using the pre-defined values of mini and % maxi from the training file

[n_TestAll_T] = tramnmx(TestAll_T,mini,maxi);

for cases=1:No_Rows

error5cases(:,:= postmnmx(sim(net,n_TestAll_T(:,cases)),mint,maxt);

error3cases(:,:)=error5cases(:,:)-(Testproperties2_T(:,cases));

error3cases(:,:)=error3cases(:,:)./(Testproperties2_T(:,cases))*100

error3cases(:,:)=abs(error3cases(:,:));

end

% Calculates the errors for each property in each region

for i=1:No_Rows

avg_testerror_k1(i)=mean(error3i(1,:));

avg_testerror_phi1(i)=mean(error3i(2,:));

avg_testerror_h1(i)=mean(error3i(3,:));










end

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% Output of Errors for each case and each property for Region1

figure();

subplot(3,1,1);plot([1:No_Rows],avg_testerror_k1(:),'bo-')

title('Testing Errors: North of Fault A')

ylabel('Perm Error %')

grid on;

subplot(3,1,2);plot([1:No_Rows],avg_testerror_phi1(:),'bo-')

ylabel('Phi Error %')

grid on;

subplot(3,1,3);plot([1:No_Rows],avg_testerror_h1(:),'bo-')

xlabel('Case Number')

ylabel('h Error %')

grid on;

fprintf(1,' NORTH OF FAULT A: %g\n');

fprintf(1,'Permeability error = %g\n',mean(avg_testerror_k1(:)))

fprintf(1,'Porosity error = %g\n',mean(avg_testerror_phi1(:)))

fprintf(1,'Thickness error = %g\n',mean(avg_testerror_h1(:)))

112

C.5 Prediction of reservoir’s properties

% This is the prediction file that uses DataTest1 file with small delta % values to predict the properties (k, ø and h) of the field % Inputs consist of time, very small ∆qo, ∆qg, ∆qw and ∆GOR at each time step for one case. % Output of k, phi and h for each region of the field

clear all;

format compact;

load Training1.mat;

load delta.mat;

[No_Rows,c]=size(delta1); % No_Rows = No. of time steps for 10 years

taken every 90 days

i=1;

for time=90:90:3770

DataTest1(i,1)=time;

i=i+1;

end

% Random generation of very small delta values for ∆qo, ∆qg, ∆qw and ∆GOR as inputs for each time step that will generate the properties of the field dt1 = -0.00001; dt2 =0.00001;

del_qo = dt1 + (dt2-dt1)* rand(No_Rows,1);

DataTest1(:,2)=del_qo;

dt3 = -0.00001; dt4 = 0.00001;

del_qg = dt3 + (dt4-dt3)* rand(No_Rows,1);

DataTest1(:,3)=del_qg;

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dt5 = -0.00001; dt6 =0.00001;

del_qw = dt5 + (dt6-dt5)* rand(No_Rows,1);

DataTest1(:,4)=del_qw;

dt6 = -0.00001; dt7 =0.00001;

del_GOR = dt6 + (dt7-dt6)* rand(No_Rows,1);

DataTest1(:,5)=del_GOR;

i=i+1;

%Transposes the input data %Input data consists of very small variations of the base case DataTest1_T=DataTest1';

% Normalization of the input data using the pre-defined values of mini and maxi determined from training the network [n_DataTest1_T] = tramnmx(DataTest1_T,mini,maxi);

% Output of results using the trained network

case_test=1;

for time=1:No_Rows

resultcase_test(:,time)=postmnmx(sim(net,n_DataTest1_T(:,(case_test-

1)*No_Rows+time)),mint,maxt);

end

% Calculation and output of the average predicted properties k,ø and h in each region of the field avg_k1=sum(resultcase_test(1,:))./No_Rows;

avg_phi1=sum(resultcase_test(2,:))./No_Rows;

avg_h1=sum(resultcase_test(3,:))./No_Rows;

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fprintf(1,' ---------- %g\n');

fprintf(1,'Permeability North of Fault A = %g\n',avg_k1);

fprintf(1,'Porosity North of Fault A = %g\n',avg_phi1);

fprintf(1,'Thickness North of Fault A = %g\n',avg_h1);

avg_k2=sum(resultcase_test(4,:))./No_Rows;



fprintf(1,' %g\n');

fprintf(1,'Permeability in Channel = %g\n',avg_k2);

fprintf(1,'Porosity in Channel = %g\n',avg_phi2);

fprintf(1,'Thickness in Channel A = %g\n',avg_h2);




fprintf(1,' %g\n');

fprintf(1,'Permeability in East Levee = %g\n',avg_k3);

fprintf(1,'Porosity in East Levee = %g\n',avg_phi3);

fprintf(1,'Thickness in East Levee = %g\n',avg_h3);




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fprintf(1,' %g\n');

fprintf(1,'Permeability in West Levee = %g\n',avg_k4);

fprintf(1,'Porosity in West Levee = %g\n',avg_phi4);

fprintf(1,'Thickness in West Levee = %g\n',avg_h4);

Appendix D

RESERVOIR ROCK AND FLUID PROPERTIES USED TO BUILD THE TAHOE M4.1 RESERVOIR SIMULATION MODEL

This appendix describes the parameters used in the Tahoe M4.1 reservoir

simulation model. There were many similarities between the Tahoe M4.1 reservoir model

and the ANN model that was initially used to develop the network. The rock properties,

relative permeability, and capillary pressure data were the same as the initial ANN model

described in Appendix A. However, the structure (Appendix E), the gas/oil and the

oil/water contacts were different as described in Table D.1. In addition, the M4.1

reservoir simulation model was not square, a curvilinear grid was used (Figure 5.1), and a

compositional model was used to simulate the fluid behavior. The compositional model

uses the Peng Robinson equation of state to model the fluid that consisted of five

components as described in Table D.2.

Table D.1: Rock and fluid data used in the initialization of the M4.1 simulation model.

Type of Fluid Model CompositionalStock Tank Water Density 1.027 g/cc

Water Salinity 75,000 ppmWater Formation Volume Factor 1.028

Water Viscosity 0.3615 cpWater Compressibility 2.52 x 10-6 psi-1Rock Compressibility 3.2 x 10-6 psi-1

Reservoir Pressure 4921 psiaReservoir Temperature 203 oF

Gas Oil Contact 10,250 ft.Oil Water Contact 10,410 ft.Saturation Pressure 4880 psia

117

Table D.2: Properties of the fluid components used in the compositional model for the M4.1 reservoir in the Tahoe Field.

Component Initial Mole Fractions

Molecular Weight

Critical Temperature

(oF)

Critical Pressure

(psia)

Critical Z-factor

Acentric Factor

Omega A (Ωai)

Omega A (Ωbi)

Parachor

P1 0.9 16.28 -116.75 668 0.2889 0.0138 0.455676 0.0777779 70.6 P2 0.0673 38.12 152.22 658.08 0.2887 0.1277 0.4713395 0.0796221 133.9 P3 0.020866 93.6 485.03 438.42 0.2673 0.2899 0.4641667 0.0786007 303.1 P4 0.011478 171.84 744.29 288.01 0.2448 0.4873 0.4572355 0.0777961 542.5 P5 0.000356 355.97 1135.58 153.99 0.2019 0.9865 0.4572355 0.0777961 1101

Appendix E

INSERT: “TAHOE FIELD CASE STUDY – UNDERSTANDING RESERVOIR COMPARTMENTALIZATION IN A CHANNEL-LEVEE SYSTEM”

by

Enunwa, Chekwube; Razzano III, Joseph; Ramgulam, Asha; Flemings, Peter B.; Ertekin, Turgay and Karpyn, Zuleima, Pennsylvania State University, 2005.

Refer to packet insert.

Documents

UTILIZATION OF ARTIFICIAL NEURAL NETWORKS IN THE ... · Figure 4.15: Thickness prediction errors with and without the output functional links: region’s area/region’s permeability