The Pennsylvania State University. The Graduate School. Department of Energy and Mineral Engineering

Transcription

1 The Pennsylvania State University The Graduate School Department of Energy and Mineral Engineering DEVELOPMENT OF AN ARTIFICIAL NEURAL NETWORK MODEL FOR DESIGNING WATER FLOODING PROJECTS IN THREE-PHASE RESERVOIRS A Thesis in Energy and Mineral Engineering by I-AN LAI 2016 I-An Lai Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science August 2016

2 The thesis of I-An Lai was reviewed and approved* by the following: ii Turgay Ertekin Professor of Petroleum and Natural Gas Engineering Thesis Advisor Hamid Emami-Meybodi Assistant Professor of Petroleum and Natural Gas Engineering John Yilin Wang Associate Professor of Petroleum and Natural Gas Engineering Turgay Ertekin Professor of Petroleum and Natural Gas Engineering Head of the Department of Department or Graduate Program *Signatures are on file in the Graduate School

3 iii ABSTRACT Water flooding is a predominant secondary recovery method used in conventional petroleum reservoirs. The performance of a water flooding project will be impeded if a free gas phase arises in the reservoir. The main purpose of this study is to apply artificial neural network technology to solve problems related to water flooding projects in three-phase reservoirs. An artificial neural network (ANN) is a computational tool developed based on biological neural systems. The computation of an artificial neural network is similar to the way the human brain can predict the results based on previous knowledge gained through experiencing various situations. This study presents three artificial neural network proxy models that are applied to predict production profiles, project design parameters, and reservoir properties. Users can save time by using these ANN models instead of using numerical simulations and thus can achieve more desirable recovery targets of water flooding projects. In this thesis, various scenarios are created by the combination of reservoir properties, project design parameters, PVT parameters, and rock and fluid parameters. Then, a commercial reservoir simulator is utilized to generate the production profiles of oil, water, and gas in water flooding projects. An artificial neural network is developed with the ascertained properties (reservoir properties, project design parameters, PVT parameters, and rock and fluid parameters) as inputs with production rates and time intervals as outputs. In the final stage, graphical user interfaces are created to facilitate users access to these ANN models. Users can input the parameters and the results are displayed in the graphical user interface. The selected ranges of input data are also displayed at the end through a graphical user interface in order to enhance its usability.

4 iv TABLE OF CONTENTS List of Figures... v List of Tables... vi Acknowledgements... vii Chapter 1 Introduction... 1 Chapter 2 Problem Statement... 3 Chapter 3 Literature Review Water Flooding Water Flooding with Initial Gas Saturation Artificial Neural Networks Artificial Neural Network History ANN Structure Transfer Function Training and Learning Function Applications of Artificial Intelligence in the Oil and Gas Industry Pressure-Volume-Temperature Property Properties of Natural Gas z-factor Gas Compressibility Gas Viscosity Gas Formation Volume Factor Properties of Reservoir Oil Solution Gas-Oil Ratio (GOR) Oil Formation Volume Factor (FVF) Oil Viscosity Oil Compressibility Properties of Reservoir Water Water Formation Volume Factor (FVF) Water Viscosity Water Compressibility Relative Permeability and Capillary Pressure Chapter 4 Reservoir Modeling and Data Generation Reservoir Description Well Pattern and Grid Block Sensitivity Analysis Generation of Reservoir Properties and Project Design Parameters Water Injection Time Generation of PVT Properties Generation of Rock and Fluid Properties Data Collection... 43

5 v Chapter 5 Artificial Neural Network Development Introduction Training of ANN Forward ANN for Predicting Production Profiles Inverse ANN for Predicting Project Design Parameters Inverse ANN for Predicting Reservoir Properties Graphical User Interface Graphical User Interface for Forward ANN Graphical User Interface for Inverse ANN for Predicting Project Design Parameters Graphical User Interface for Inverse ANN for Predicting Reservoir Properties Chapter 6 Results and Discussion Result of Forward Artificial Neural Network Model Error Analysis Results of the Inverse ANN models Result of Inverse ANN Model for Predicting Project Design Parameters Result of Inverse ANN Model for Predicting Reservoir Properties Accuracy Analysis Accuracy Analysis of the Inverse ANN for Predicting Project Design Parameters Accuracy Analysis of the Inverse ANN for Predicting Reservoir Properties Chapter 7 Case Study The Elk Basin Madison Field Review of Field Data Monte Carlo Simulation Results and Discussion The Probabilistic Analysis for Oil Recovery The Probabilistic Analysis for Water Breakthrough Time Chapter 8 Conclusion and Future Work Summary Conclusion Recommendations References Appendix A Parameters for sensitivity analysis Appendix B Forward ANN Inputs Appendix C Forward ANN Prediction

6 vi LIST OF FIGURES Figure (3-1): Mobility ratio vs. breakthrough areal sweep efficiency [6]... 8 Figure (3-2): Saturation profile during a water flood in three reservoir system [6]... 9 Figure (3-3): Neuron structure [8] Figure (3-4): Multiple-Input Neuron [8] Figure (3-5): Multiple layers of an ANN model [8] Figure (3-6): Structure of the feed forward back-propagation network Figure (3-7): Structure of the cascade back-propagation network Figure (4-1): eservoir pattern of the water flooding projects Figure (4-2): Comparison of a five-spot pattern and a 1/8 section of five-spot pattern Figure (4-3): Sensitivity analysis of a grid block system Figure (4-4): Grid block system of this study Figure (4-5): Method of choosing the starting time of water flooding Figure (4-6): Estimation of GOR for three randomly selected cases Figure (4-7): Estimation of oil FVF for three randomly selected cases Figure (4-8): Estimation of z-factor for three randomly selected cases Figure (4-9): Estimation of oil viscosity for three randomly selected cases Figure (4-10): Estimation of gas viscosity for three randomly selected cases Figure (4-11): Estimation of oil compressibility for three randomly selected cases Figure (4-12): Estimation of water-oil relative permeability curves for three randomly selected cases Figure (4-13): Estimation of capillary pressure in water-oil interface for three randomly selected cases Figure (4-14): Estimation of gas-oil relative permeability curves for three randomly selected cases... 42

7 vii Figure (4-15): Estimation of capillary pressure in gas-oil interface for three randomly selected cases Figure (4-16): Diagram of data collection of oil rate points and time intervals Figure (4-17): Diagram of data collection of water rate points Figure (4-18): Diagram of data collection of gas rate points Figure (5-1): Performance plot of the Forward ANN model Figure (5-2): Structure of the Forward ANN model Figure (5-3): Structure of the inverse ANN model for predicting project design parameters Figure (5-4): Structure of the inverse ANN model for predicting reservoir properties Figure (5-5): GUI of the forward ANN model Figure (5-6): GUI for the inverse ANN for predicting project design parameters Figure (5-7): GUI for the inverse ANN for predicting reservoir properties Figure (6-1): Performance of the forward ANN model Figure (6-2): Error for forward ANN prediction Figure (6-3): Comparison of oil production profiles generated by ANN & simulator in Case Figure (6-4): Comparison of water production profiles generated by ANN & simulator in Case Figure (6-5): Comparison of gas production profile generated by ANN & simulator in Case Figure (6-6): Comparison of oil production profile generated by ANN & simulator in Case Figure (6-7): Comparison of water production profile generated by ANN & simulator in Case Figure (6-8): Comparison of gas production profile generated by ANN & simulator in Case Figure (6-9): Comparison of oil production profile generated by ANN & simulator in Case

8 Figure (6-10): Comparison of water production profile generated by ANN & simulator in Case Figure (6-11): Comparison of gas production profile generated by ANN & simulator in Case Figure (6-12): Error distribution in the prediction of time interval Figure (6-13): Error distribution in the prediction of oil rate Figure (6-14): Error distribution in the prediction of water rate Figure (6-15): Error distribution in the prediction of gas rate Figure (6-16): Test Case 27 with the highest error in oil rate prediction Figure (6-17): Test case 31 with the highest error in oil rate prediction Figure (6-18): Test Case 17 with the maximum error in water rate prediction Figure (6-19): Test Case 14 with the minimum error in water rate prediction Figure (6-20): Test Case 7 with the minimum error in gas rate prediction Figure (6-21): Test Case 5 with the minimum error in gas rate prediction Figure (6-22): Error distribution in the prediction of oil rate points Figure (6-23): Error distribution in the prediction of water rate points Figure (6-24): Error distribution in the prediction of gas rate points Figure (6-25): Diagram of data collection of oil rate points Figure (6-26): Diagram of data collection of gas rate points Figure (6-27): Average error in the prediction of project design parameters Figure (6-28): Comparison of pattern area predicted by ANN and simulator Figure (6-29): Comparison of water injecting rate predicted by ANN and simulator Figure (6-30): Comparison of bottom-hole pressure of production well predicted by ANN and simulator Figure (6-31): Comparison of bottom-hole pressure of injection well predicted by ANN and simulator Figure (6-32): Comparison of water injection time predicted by ANN and simulator viii

9 ix Figure (6-33): Average error in the prediction of reservoir properties Figure (6-34): Comparison of porosity predicted by ANN and simulator Figure (6-35): Comparison of permeability predicted by ANN and simulator Figure (6-36): Comparison of thickness predicted by ANN and simulator Figure (6-37): Accuracy analysis scheme Figure (6-38): Comparison the cumulative oil production, the result of accuracy analysis of the inverse network for predicting project design parameters Figure (6-39): Comparison the cumulative water production, the result of accuracy analysis of the inverse network for predicting project design parameters Figure (6-40): Comparison the cumulative gas production, the result of accuracy analysis of the inverse network for predicting project design parameters Figure (6-41): Comparison the cumulative oil production, the result of accuracy analysis of the inverse network for predicting reservoir properties Figure (6-42): Comparison the cumulative water production, the result of accuracy analysis of the inverse network for predicting reservoir properties Figure (6-43): Comparison the cumulative gas production, the result of accuracy analysis of the inverse network for predicting reservoir properties Figure (7-1): Elk Basin Madison structure and location of wells [38] Figure (7-2): Monte Carlo simulation of oil recovery Figure (7-3): Monte Carlo simulation of water breakthrough time

10 x LIST OF TABLES Table (3-1): Training functions in MATLAB NN Toolbox Table (3-2): Training functions in MATLAB NN Toolbox Table (3-3): Learning functions in MATLAB NN Toolbox Table (3-4): Constants of the Dranchuk and Abou-Kassem EOS Table (4-1): Selected ranges of reservoir properties covered in the study Table (4-2): Selected ranges of project design parameters covered in the study Table (4-3): Selected ranges of PVT parameters covered in the study Table (4-4): Selected ranges of rock and fluid parameters covered in the study Table (5-1): Inputs and outputs of the forward ANN model Table (5-2): Inputs and outputs of the inverse ANN model for predicting project design parameters Table (5-3): Inputs and outputs of the inverse ANN model for predicting reservoir properties Table (6-1): The hidden layers, neurons, and transfer functions of the forward ANN Table (6-2): Input data of the sample testing cases Table (6-3): Comparison of error in different scenarios Table (6-4): The hidden layers, neurons, and transfer functions of the inverse ANN for predicting project design parameters Table (6-5): The hidden layers, neurons, and transfer functions of the inverse ANN for predicting reservoir properties Table (7-1): Summary of Elk Basin Madison zone characteristics [38] Table (7-2): Studied parameters and the selected ranges of each parameter of the Elk Basin Madison field Table (7-3): Inputs of Case 4542 in the Monte Carlo simulation Table (7-4): The comparison of the projection design parameters of the inverse network and of the forward network

11 xi ACKNOWLEDGEMENTS I would like to thank my advisor Dr. Turgay Ertekin for being my master research advisor and academic mentor during my study in Pennsylvania State University. His continuous, illuminated, patient and erudite guidance is essential in my two years knowledge growth. I truly appreciate his contribution at every step of my work. I would like to thank my committee members Dr. Hamid Emami-Meybodi and Dr. John Yilin Wang. I would also like to thank doctoral students Quin Sun and Jiang Zhang for their advice and opinion during my master research. I would like to thank my friends Ting-An Chen, Chen-Fu Tsai and Jason Jan for their encouragement and help when I am having a hard time. Finally, I would like to show my deepest gratitude to my family. This thesis is dedicated to my father.

12 1 Chapter 1 Introduction Using natural recovery mechanisms such as liquid and formation expansion most of the time is not efficient for the recovery of oil, as natural recovery methods leave behind a significant amount of original oil in place. After the primary recovery stage, oil production goes to the next phase, secondary recovery. Secondary recovery involves application of gas and water injection to displace and expel oil to surface. There are billions of barrels of oil produced by water flooding recovery across the world [1]. Water flooding is used extensively as a secondary oil recovery method for three major reasons. First, water is readily available and can be injected easily as an injection fluid. Second, water can flow quickly into oil reservoirs and expel light and medium density of oil efficiently. Third, using the water flooding technique can lower capital investment and operating costs. Design of a water flooding project becomes more difficult if free gas is present in the reservoir. Because of its high compressibility, oil saturation changes during water flooding. Thus, the amount of oil recovered is more challenging to predict. In order to improve the efficiency of oil recovery modeling, the focus of this study is to predict production profiles of reservoir fluids in water flooding projects in three-phase systems via artificial neural network based proxy models. An artificial neural network (ANN) is a mathematical model which is inspired by biological neural networks. With the interconnection of artificial neurons and layers, an ANN can efficiently solve non-linear problems. ANNs are widely used in the academic research and engineering industries.

13 2 In this study, three artificial neural network proxy models are developed to be employed for solving problems related to modeling water flooding projects. The first ANN model is a forward network which is designed to predict the production profiles of oil, water, and gas and the water breakthrough time. The second ANN model is an inverse looking network that is specifically developed to predict project design parameters with given reservoir properties and production data. The third ANN model is also an inverse network which is developed to predict the reservoir skeleton properties. In association with the networks, graphical user interfaces are created in order to provide users with an accessible platform for utilizing these three artificial neural networks. Chapter 2 elaborates on the problem analyzed in this study, while Chapter 3, divided into three parts, reviews previous studies related to the relevant parameters and instruments for oil recovery modeling. The first part of Chapter 3 provides the background of water flooding recovery, and the second part explains the development, structure, and functions of an artificial neural network model. The third part reviews correlations applied to estimate PVT properties, and rock and fluid properties. Chapter 4 discusses the water flooding model used in this study and the generation of the studied properties within the ranges selected. Additionally, the method of processing production data for ANN training is explained. Chapter 5 discusses the development of three ANN models (one forward ANN and two inverse ANNs), while Chapter 6 presents and analyzes the performance of these three ANN models. Chapter 7 discusses a case study (based on the Elk Basin Madison Field) with the application of a forward network and a probabilistic analysis of the results. Finally, Chapter 8 highlights the conclusions drawn from the developed ANN models in water flooding projects and suggests some relevant future work.

14 3 Chapter 2 Problem Statement Using water flooding schemes is one of the most common secondary oil recovery methods utilized in petroleum and natural gas industry. Before water flooding is employed, primary production is achieved by depending on naturally present reservoir energy such as reservoir rock and fluid expansion, aquifer expansion and gravitational forces [2]. Primary recovery depletes this naturally occurring reservoir pressure during production. Water flooding can be applied hereon to increase the recovery rate of the reservoir by maintaining the reservoir pressure. This mechanism of water flooding becomes much more complex if the reservoir fluids contain free gas. Typically, numerical simulations of water flooding are relatively costly in terms of time. In this study, artificial neural network (ANN) technology is utilized to reduce the time necessary for simulating water flooding in petroleum reservoirs. ANN modeling has become popular in the petroleum and natural gas industry due to its capability of solving non-linear complex problems and because it has proven to be a powerful approach in predicting reliable results. This study focuses on developing an ANN model for predicting the following parameters: Oil production rate Water production rate Gas production rate Durations of different segments of water flooding projects Before the development of the ANN model, the first step is to prepare a training set which consists of different combinations of the modeling parameters. These parameters include reservoir properties, water flooding project design properties, pressure-volume-temperature (PVT) properties, and rock and fluid properties. The reservoir properties include thickness, depth,

15 4 permeability, and porosity. The project design parameters include pattern area, bottom-hole pressure of the injection well, bottom-hole pressure of the production well, water injection rate and injection time. The PVT parameters include initial pressure, bubble point pressure, gas density, oil density, temperature, z-factor, gas compressibility, gas viscosity, gas formation volume factor, solution gas-oil ratio, oil formation volume factor, oil viscosity, oil compressibility, water formation volume factor, water viscosity, and water compressibility. The rock and fluid properties include the initial oil saturation, initial water saturation, initial gas saturation, residual oil saturation, residual water saturation, residual gas saturation, relative permeability, and capillary pressure characteristics at the a water-oil interface and oil-gas interface. By combining these studied parameters, various scenarios are generated and simulated by a commercial reservoir simulator. The next step is to collect the data from production profiles of oil, water and gas for the training of the artificial neural network. There are two discontinuities of production profiles; hence, data pretreatment is an essential approach to decompose the production profiles and extract the data. The studied parameters are entered into the ANN model as inputs and the production rates of oil, water, and gas and time intervals are entered into the ANN model as outputs during the training process. Once the ANN proxy model is developed, it should provide satisfying predictions of oil, water and gas production profiles and time intervals within seconds. The following steps outline the workflow of the development of ANN models in this study: 1. Build an ideal reservoir model by utilizing a commercial Black-Oil simulator 1 and verifying the results. 2. Create different reservoir scenarios by combining various reservoir properties, project design parameters, PVT parameters, and rock and fluid parameters. 1 CMG: Computer Modeling Group

16 5 3. Extract production data from the outcomes of the Black-Oil simulation runs. 4. Load the various reservoir properties, project design parameters, PVT parameters, and rock and fluid parameters as input data and then load production data as outputs to train an artificial neural network. 5. Identify an ANN model by comparing the performance of ANN models in different structures. 6. Apply the designed ANN model in a real water flooding case and conduct a probabilistic analysis of the performance of the network. Finally, a graphical user interface (GUI) is constructed for the ANN model by applying the MATLAB GUIDE TOOL in the MATLAB 2 work platform. The GUI is a user-friendly and efficient approach for users to access the performance of the artificial neural network models. 2 MATLAB: a tool for numerical computation by The MathWorks, Inc

17 6 Chapter 3 Literature Review In this chapter, the protocol of water flooding is introduced and discussed. This chapter also describes the development of the artificial neural network (ANN) technology and reviews ANN structures as well as training functions, transfer functions, learning functions, and performance functions. At the end of this chapter, the correlations for PVT properties and reservoir fluid and rock properties are presented. 3.1 Water Flooding Water flooding is typically a secondary recovery method of hydrocarbon production. In the water flooding process, water is injected into a reservoir in order to increase the reservoir pressure and push oil into the production well. Water flooding can increase the recovery factor and prolong the production period of a reservoir. The first water flooding occurred accidentally in Pithole City, Pennsylvania in At that time, water flooding was discovered as a way to maintain reservoir pressure and allow wells to have a longer production period [3]. The first systematic operation of five-spot pattern water flooding was conducted at Bradford oil field in Pennsylvania in 1924 [4]. Within ten years, water flooding became a common procedure in enhanced oil recovery. In the early 1950 s, the technique of water flooding was widely recognized as a useful secondary recovery method. Water flooding is dominant among fluid injection methods because of three main reasons. First, a water source is typically easy to acquire. The water source can be the produced water such

18 7 as saline water or brine from oil reservoirs. Oceans, rivers, streams, and aquifers are also common water sources for water flooding [1]. Second, water is more mobile than other fluids and it can displace light and medium density oil more efficiently than other fluids. Water injection can also generate high oil recovery in the nearby production wells because the reservoir pressure is increased. Third, injecting water is more economically favorable because the capital investment and costs of operation are lower than injecting other fluids. The water flooding method can be discussed on a micro-level (at the scale of rock pores) and a macro-level (at the scale of entire reservoirs). On a micro level, the efficiency of water-oil displacement depends on reservoir fluid properties and the characterization of reservoir rock. Before conducting a water flooding project, the cores of a reservoir need to be analyzed in a laboratory. Rock properties such as porosity, permeability, and wettability are measured in order to understand water flooding performance. Also, the reservoir fluid properties are essential factors for water flooding. For example, oil viscosity and oil density are two crucial factors determining the efficacy of the water-oil displacement. The oil viscosity is related to the mobility which is defined as the effective permeability of the rock, divided by the fluid viscosity [5]. The definition of mobility of fluid is described as λ (3.1) where λ is mobility, k is permeability and µ is viscosity. In a uniform rock sample, a fluid with higher viscosity has lower mobility and vice versa since the permeability is a property of the rock skeleton. As for oil density, a crude oil with high API has low viscosity so the mobility of this crude oil is expected to be high. Also, the water-oil mobility ratio (M) is a major factor in deciding the efficiency of water-oil displacement. The definition of the mobility ratio is as follows: (3.2)

19 8 When the water-oil mobility ratio is increasing, the water-oil displacement efficiency will decrease as shown in Figure (3-1). In other words, the water-oil mobility ratio has a negative correlation with the water oil displacement efficiency. Figure (3-1): Mobility ratio VS. Breakthrough areal sweep efficiency [5] At the macro scale of an entire reservoir during water flooding recovery, the effect of the reservoir geology, gravitational force, and geometry can determine how successful a water flooding will perform. Also, water-oil displacements in different directions (horizontal and vertical displacement) are important considerations. Then, the appropriate pattern applied to water flooding work will determine the production performance. The following equation [1] shows the efficiency of oil recovery in water flooding: (3.3) where E RWF = Overall water flood recovery efficiency, fraction E D = Displacement efficiency within the volume swept by water, fraction E A = Areal sweep efficiency, fraction E I = Vertical sweep efficiency, fraction

20 Water Flooding with Initial Gas Saturation Some reservoirs in water flooding project contain initial gas saturation. There are a number of studies about the effect of the initial gas saturation on water flooding recovery [6-9]. The presence of a gas phase was found to lower the residual oil saturation during water flooding recovery [6]. Figure (3-2) illustrates a saturation profile of the water flooding in a three-phase reservoir. Injection water displaces both oil and gas and the oil bank is caused by the water injection. In the displacement process, the oil bank expels the free gas and leaves the trapped gas. If the pressure is high enough, the trapped gas will be dissolved in the oil bank since injecting water into a reservoir causes repressurization of the reservoir. When the trapped gas is dissolved in the oil bank, the oil saturation will be the same as the initial saturation in the oil bank [3]. Whether trapped gas will be dissolved in the oil bank or not depends on the pressure of a reservoir. With the presence of a gas phase, the prediction of oil production will be more complicated because the oil saturation will be affected. Figure (3-2): Saturation profile during a water flood in three reservoir system [5]

21 Artificial Neural Networks Artificial Neural Network History The concept of an Artificial Neural Network (ANN) was inspired by the biological neural system of the human brain. In the 1940s, Warren McCulloch and Walter Pitts developed the basic arithmetic and logical functions of an artificial neural network system. However, the development of ANN was not achieved in that time because of the inefficiency of computing machines. In 1957, Frank Rosenblatt invented the perception network which was applied to pattern recognition. Pattern recognition can be implemented by a perception neural network with the weight vector of various patterns. The perception network was the first practical neural network [10]. However, there are some constraints in perception networks, which can only solve limited problems. By, in the 1980s, the development of neural networks had made a considerable progress because of new ideas concerning neural network design and due to the increased computational power of computers. The most significant breakthrough was the discovery of back propagation algorithms by David Rumelhart and James MacClelland, which can be applied in multi-layered neural networks. After the 1980s, the application of the neural network has been common in different fields of study as an essential computational tool [10] ANN Structure An artificial neural network is a mathematical model simulating a biological neural system. Like a human neural system, an artificial neural network can accept input signals, transform the inputs into output signals, and pass the output signals to the next neural system. Figure (3-3) is a schematic diagram of a biological neuron structure.

22 11 Figure (3-3): Neuron structure [10] A single neuron contains three parts which are dendrites, the cell body, and the axon. Dendrites are tree-like nerves, and the responsibility of dendrites is to receive input signals and transport them to the cell body. Dendrites also transform the inputs signals while transporting them to the cell body. The cell body sums up, thresholds, and transforms the input signals and conveys these signals to the axon. The axon, the longest nerve fiber, conveys the signals processed in the cell body to the next neuron from the end of the axon. An Artificial Neural Network (ANN), composed of artificial neurons, is a mathematical model which mimics this biological neural system in order to solve complicated problems. There are three sets of calculations in each artificial neuron: multiplication, summation, and activation [11]. Figure (3-4) illustrates how a neuron works with multiple-input information.

23 12 Figure (3-4): Multiple-input neuron [10] It can be illustrated in the following mathematical form: ( ( (3.4) where a = output W = weight of each input p = input b = bias f = the transfer function. Each input value is multiplied by individual weights when it enters an artificial neuron. Then, the summation function is used in the middle of the artificial neuron in order to sum all weighted input values and biases. At the end of the artificial neuron calculation, the sum of the weighted inputs and biases passes through activation functions, which are also known as transfer functions. The mathematical relationships in an artificial neuron are quite straightforward, but the interconnecting neurons in an artificial neural network are capable of solving highly sophisticated problems.

24 13 As for layers in an artificial neural network, either a single layer or a multi-layer structure can be employed depending on the complexity of the problems under consideration. In each layer, there are the weight matrix, the sum of inputs, and the transfer functions. By interconnecting these layers, a multi-layers network is developed. Figure (3-5) shows an ANN model with multiple layers. Figure (3-5): Multiple layers of an ANN model [10] ANN models can be classified into many different types by the structure of the neurons and the layers interconnectivity. The two most common types of ANN models are Feed forward back-propagation and Cascade back-propagation. The feed forward back-propagation network contains a back-propagation learning algorithm, with an input layer, hidden layers, and an output layer. In the feed forward network, the working flow is executed from the neurons that receive inputs in the first layer through the hidden layers to the output layer continuously. In other words, the forward network is a straight computation in one direction without back-loops. Figure (3-6) shows the neurons and the layers interconnecting structure of a feed forward back propagation. However, the transportation of

25 14 input data in the Cascade network is different from that in the feed forward network. In cascade back-propagation, input data are connected to neurons in all layers. The information flows in the cascade network not only move in one direction from input to output but also flow in the opposite direction. Figure (3-7) illustrates the neurons and the layers interconnecting structure of a Cascade back-propagation. The back-propagation algorithm means errors can be sent back to the network. The learning process of an ANN model can amend the weights and biases of the input data toward the direction, and as a result error performance drop sharply via the general principle of the Least Mean Squared error [12]. Figure (3-6): Structure of the feed forward back-propagation network Figure (3-7): Structure of the cascade back-propagation network

26 Transfer Function An artificial neural network can be represented by its transfer functions. A transfer function is responsible for transferring the summation of the input data from the previous layer and creating outputs for the next layer [13]. Transfer functions can be categorized into three main types of functions which are step functions, linear functions and non-linear functions. Various transfer functions can be chosen in solving different problems. This study uses three common transfer functions, which are Hyperbolic Tangent Sigmoid, Log-Sigmoid, and Linear function. These three transfer functions are found to generate better results than ones generated by other transfer functions. Table (3-1) shows the formulas of transfer functions in the MATLAB NN tools. Tangent Sigmoid (tansig) and Log-Sigmoid (logsig) are non-linear functions. The non-linear functions are utilized in hidden layers of an artificial neural network in order to normalize the inputs of each neuron. With the normalized inputs, neurons can conduct computation more accurately and the training of the network can be implemented more efficiently. The Tangent Sigmoid function is often used for pattern recognition problems. The results from the Tangent Sigmoid function are compressed between -1 and 1. For the Log-Sigmoid function, input data can be any value between plus and minus infinity and outputs are shifted into the range from 0 to 1. The Log-Sigmoid transfer functions are usually applied in multilayer networks which are trained in back-propagation algorithms [13]. On the other hand, the linear transfer function (purelin) is typically utilized in the output layer of ANN models in order to present the result of the network. Due to the complexity of the problem in this study, the non- linear transfer functions are used in hidden layers, whereas in the output layers, the linear transfer function is employed.

27 16 Table (3-1): Transfer Functions in MATLAB NN toolbox [13] Training and Learning Function Training functions are used to adjust the weights of inputs in each neuron in hidden layers. Based on training processes, networks are categorized as unsupervised and supervised training networks. The unsupervised network has inputs and weights. As for supervised training networks, not only inputs and weights but also outputs exist in the network. In the supervised network, the weights are modified iteratively until errors meet the error tolerance of the output data and the provided target [14].

28 17 The scaled conjugate gradient function (tainscg) is applied as a training algorithm in this study. With low consumption of time in line-search and small memory requirement, the scaled conjugate gradient function improves the efficiency of the training process because the convergence rate is faster than in other back propagation training algorithms such as standard back-propagation [15]. By comparing the performances of the networks with different training functions listed in Table (3-2), the network with tainscg function can provide better prediction. As for choosing learning algorithms, the gradient descent with momentum weight and bias learning function (learngdm) is selected for this study because it can generate better prediction than other learning function list in Table (3-3). This function calculates the momentum constant along with the changes in inputs, errors, learning rate, and weights. Table (3-2) and Table (3-3) list the training functions and learning functions of the MATLAB NN Toolbox 3 [13]. Table (3-2): Training functions in MATLAB NN Toolbox [13] Training Functions trainb Batch training with weight and bias learning rules trainbfg BFGS quasi-newton backpropagation trainfgc BFGS quasi-newton backpropagation for use with NN model reference adaptive controller trainbr Bayesian regularization trainbuwb Batch unsupervised weight/bias training trainc Cyclical order incremental update traincgb Powell-Beale conjugate gradient backpropagation traincgf Fletcher-Powell conjugate gradient backpropagation traincgp Polak-Ribiere conjugate gradient backpropagation traingd Gradient descent backpropagation traingda Gradient descent with adaptive learning rule backpropagation traingdm Gradient descent with momentum backpropagation traingdx Gradient descent with momentum and adaptive learning rule backpropagation trainlm Levenberg-Marquardt backpropagation trainoss One step secant backpropagation trainr Random order incremental training with learning functions trainrp Resilient backpropagation (Rprop) trains Sequential order incremental training with learning functions trainscg Scaled conjugate gradient backpropagation 3 MATLAB NN Toolbox: an artificial neural network tool in MATLAB platform developed by MathWorks, Inc.

29 18 Table (3-3): Learning functions in MATLAB NN Toolbox [13] Learning Functions learncon Conscience bias learning function learngd Gradient descent weight/bias learning function learngdm Gradient descent with momentum weight/bias learning function learnh Hebb weight learning function learnhd Hebb with decay weight learning rule learnis Instar weight learning function learnk Kohonen weight learning function learnlv1 LVQ1 weight learning function learnlv2 LVQ2 weight learning function learnos Outstar weight learning function learnp Perceptron weight and bias learning function learnpn Normalized perceptron weight and bias learning function learnsom Self-organizing map weight learning function learnsomb Batch self-organizing map weight learning function learnwh Widrow-Hoff weight and bias learning rule Applications of Artificial Intelligence in the Oil and Gas Industry The applications of artificial neural networks are common in many fields of studies such as engineering, science, and business. With the ability of learning, data processing, pattern recognition, and data optimization, an artificial neural network is a prevalent tool in data analysis. In the petroleum and natural gas industry, the artificial neural network technique is widely used. The applications of artificial neural network models include lithology identification, well logs interpretation, well testing, reservoir simulation, reservoir characterization, reservoir stimulation and enhanced oil recovery. At the Pennsylvania State University, there are many of studies utilizing the artificial neural network successfully in the petroleum and natural gas engineering field of study [16-19].

30 Pressure-Volume-Temperature Property Pressure-Volume-Temperature (PVT) properties are essential for the production of a reservoir. Normally, petroleum engineers prefer to conduct laboratory analyses of the reservoir fluids to understand the fluids physical properties. However, in absence of PVT analysis, some correlations have to be employed to estimate the physical properties of reservoir fluids. In this section, the correlations for physical properties of reservoir fluids are reviewed Properties of Natural Gas z-factor Z-factor is one of the most important properties of natural gas. The Dranchuk and Abou- Kassem [20] developed an equation of state (EOS) with eleven constants. This EOS is described as follow: ( ( ( ( (3.5) where ( (3.6) ( ( (3.7) (3.8) ( ( (3.9) ( ( ( ( (3.10) The constants for the Dranchuk and Abou-Kassem EOS are listed in Table (3-4).

31 20 Table (3-4): Constants of the Dranchuk and Abou-Kassem EOS A 1 = A 2 = A 3 = A 4 = A 5 = A 6 = A 7 = A 8 = A 9 = A 10 = A 11 = In the Dranchuk and Abou-Kassem EOS, pseudo pressure and temperature are applied. Based on 264 different gas samples, Sutton created a correlation [21] for pseudo pressure (P r ) and pseudo temperature (T r ). Sutton s correlation is applied in this study and is expressed in equations below: (3.11) (3.12) (3.13) (3.14) where P pc = Pseudo- critical pressure, psi T pc = Pseudo- critical temperature, R = gas gravity, (Air=1.0) P r = Pseudo- reduce or reduced pressure, psi T r = Pseudo- reduce or reduced temperature, R Gas Compressibility The definition of the gas compressibility in an isothermal condition is ( ) (3.15) The ρ is gas density which is defined as

32 21 (3.16) Combining the above two equations, the gas compressibility equation can be derived as ( ) (3.17) An explicit relation for gas compressibility is developed by utilizing the Dranchuk and Abu-Kassem correlation. First, pseudo- reduced compressibility, c r, should be defined as follows: (3.18) Then, the c r can be derived in the equation below: ( ( ) ( ) ) (3.19) The derivative term is defined by applying Dranchuk and Abu-Kassem EOS. ( ) ( ( ( ( ( ) (3.20) and ( ( ) [ ( ] ( (3.21) Once the pseudo- reduced compressibility value, c r, is calculated, the gas compressibility value, c g, can be obtained from the following equation: (3.22) Gas Viscosity Lee, Gonzales, and Eakin [22] developed a semi-empirical gas viscosity correlation, as shown below. This viscosity correlation gives an accurate estimation of gas viscosity as long as the z factor of the gas is correct. ( ( (3.23)

33 22 where (3.24) ( ( (3.25) (3.26) (3.27) where M is the molecular weight Gas Formation Volume Factor The definition of the gas formation volume factor is the ratio of the volume of gas at the reservoir temperature and pressure (V R ) to the volume at the standard or surface temperature and pressure (V sc ). (3.28) This definition can be derived when standard pressure is psi, and standard temperature is R. (3.29) where RB/SCF = reservoir barrels per standard cubic feet T = temperature R P = pressure, psi z = z-factor

34 Properties of Reservoir Oil The correlations for oil properties are more complicated than they are for gas because oils are made up of many different components. The correlations for oil properties will be different when pressure is higher or lower than the bubble point pressure. In this study, the initial pressure is set lower than the bubble point pressure in order to establish that reservoirs are in three-phase conditions at the beginning of the production. Hence, the following discussion of the correlations of oil properties focuses on the scenario in which the reservoir pressure is below the bubble point pressure Solution Gas-Oil Ratio (GOR) In order to estimate the solution GOR value (R so ), the Standing s correlation [23, 24] is applied in this study. This correlation has two different parts because of the pressure of the reservoir and the bubble point pressure. Since the water flooding projects in this study are all in three phase reservoir systems, the correlation used for the pressure lower than the bubble point pressure is manipulated. When, ( (3.30) where (3.31) T = temperature, F

35 Oil Formation Volume Factor (FVF) The oil formation volume factor (FVF) is defined as the volume of oil at stock-tank conditions divided by the volume of oil at elevated pressure and the temperature in the reservoir. Standing [23, 24] also developed a correlation for the oil formation volume factor, Bo, for Black- Oil reservoir. The following equation is the correlation of B o for the condition that the pressure is lower than the bubble point pressure. When (3.32) ( (3.33) ( (3.34) where = oil specific gravity T = temperature, F Oil Viscosity The oil viscosity is a fundamental fluid property in a water flooding project because oil viscosity is relative to oil mobility which is one important factor in determining whether the water flooding is effective or not. The Egbogah [25] correlation, which is modified from the Beggs and Robinson [26] correlation, is used in computing dead oil viscosity. When, [ ( ] ( (3.35) where = dead oil viscosity, cp

36 25 T = temperature, F In addition, the Beggs and Robinson [26] correlation for the live oil viscosity with solution gas is showed as follows: (3.36) where ( ( Oil Compressibility The oil compressibility correlation applied in this study was developed by Villena-Lanzi [27]. This correlation is for the isothermal oil compressibility coefficient, c o, for Black-Oil reservoirs at a pressure lower than the bubble point pressure. When, ( ( ( ( ( ( (3.37) where Solution GOR at the bubble point pressure, scf/ STB P = pressure, psi P b = bubble point pressure, psi T = temperature, F = oil garvity

37 Properties of Reservoir Water The correlations for water physical properties are comparatively more straightforward than the correlations for oil and gas because the composition of water is related to dissolved solids. Also, the influence of the pressure and temperature variation in water properties is relatively smaller than it for gas or oil properties [28] Water Formation Volume Factor (FVF) The McCain s [29, 30] correlation for the water formation volume factor, B w, is used in this study. The correlation is described as below. ( ( ) (3.38) where (3.39) T = temperature, F P = pressure, psi (3.40) Water Viscosity The correlation used for water viscosity was developed by McCain [29, 30]. This correlation is for the formation water viscosity, µ w1, at the reservoir temperature and the atmosphere pressure. The McCain s correlation is described as follows: (3.41)

38 27 where (3.42) (3.43) T = temperature, F S = salinity, % by weight solids Once the formation water viscosity at atmosphere pressure is determined, the formation water viscosity at reservoir pressure, µ w can be estimated by McCain s correlation. (3.44) Water Compressibility Water compressibility is influenced by the presence of free gas in the reservoir. McCain [29, 30] developed a correlation for formation water compressibility at the condition that the reservoir pressure is below the bubble point pressure: ( ) ( ) (3.45) 3.4 Relative Permeability and Capillary Pressure In different reservoirs lithology, the relative permeability of fluids and the capillary pressure of the fluid interface are different. Relative permeability and capillary pressure are essential indicators of rock type in a reservoir. The definition of relative permeability is the ratio of the effective permeability for a particular fluid to reference permeability for the rock. Relative permeability and capillary pressure data are obtained from laboratory analysis. However, in the

39 28 lack of real data from analysis, the relative permeability and capillary pressure of a particular rock type must be estimated via using correlations. Corey [31, 32] correlation for relative permeability and capillary pressure are used in the estimation of water-oil relative permeability, gas-oil relative permeability, and capillary pressure in water-oil and gas-oil interfaces. Also, the Stone correlation is applied in calculating the three-phase relative permeability in this study [33]. The modification of Corey's formulation for relative permeability and capillary pressure by Zahoor and Derahman [34] are described as follows: (( ( (3.46) (( ( (3.47) ( ( (3.48) (3.49) (( ( (3.50) (( ( (3.51) ( ( (3.52) (3.53) K rw = Water relative permeability K rwro = Water relative permeability at residual oil saturation K row = Oil relative permeability of water-oil system K roiw = Oil relative permeability at irreducible water saturation K rog = Oil relative permeability of gas-oil system K rg = Gas relative permeability K rgro = Gas relative permeability at residual oil saturation S w = Water saturation

40 29 S wirr = Irreducible water saturation S orw = Residual oil saturation of water-oil system S L = Liquid Saturation S org = Residual oil saturation of gas-oil system S gc = Critical gas saturation P dw = Capillary pressure at irreducible water saturation P dl = Capillary pressure at critical liquid saturation P cow = Capillary pressure of water-oil system P cog = Capillary pressure of gas-oil system n w = Corey water exponent n ow = Corey oil exponent of water-oil system n g = Corey gas exponent n og = Corey oil exponent of gas-oil system = Characteristic constant

41 30 Chapter 4 Reservoir Modeling and Data Generation In this chapter, scenarios of water flooding project in this study will be described. Also, a grid block system of reservoirs will be defined via sensitivity analysis. Then, the generation of the reservoir properties, project designs parameters, PVT parameters, and rock, and fluid properties will be discussed. Lastly, the methods for data collection of the results from reservoir simulator will be presented. 4.1 Reservoir Description The reservoir model used in this study is a two-dimensional single-permeability, singleporosity Cartesian Black-Oil reservoir. The reservoir simulator used in modeling reservoirs in this study is CMG IMEX 4 Black-Oil simulator. Figure (4-1) shows a 2D reservoir pattern utilized in this study. In order to generate various scenarios of the reservoir model, four major reservoir system properties (reservoir parameters, project design parameters, PVT parameters, and rock and fluid parameters) are studied. MATLAB can create these parameters randomly via the function rand with the specified maximum and minimum values. In addition, MATLAB is also used in combining the parameters and developing various reservoir models with certain assumptions. The main assumptions of the reservoir condition of the water flooding projects are listed as follows: 1. Homogeneous and isotropic reservoir. 4 CMG IMEX: Computer Modeling Group, Implicit-Explicit Black-Oil simulator

42 31 2. Uniform grid distribution. 3. Single layer reservoir. 4. Three phase system. 5. Constant reservoir temperature. 6. Production period is 20,000 days. 7. Gravitational effects are neglected. 8. Reservoir pattern is one-eighth section of five-spot pattern. 9. Skin factor is neglected. 10. All natural gases are sweet gas. 11. The injected fluid is water. 12. All cases are produced in primary recovery and injected water after primary recovery Figure (4-1): Reservoir pattern of the water flooding projects

43 Well Pattern and Grid Block Sensitivity Analysis This study models a one-eighth section of a five-spot water-flooding pattern with one production well and one injection well, located within a three-phase reservoir. With the simplified reservoir pattern, the simulator consumes less computational time, and the production profiles of different scenarios can be generated and observed more efficiently. Figure (4-2) illustrates the comparison of a five-spot pattern and the one-eighth section of five-spot pattern which is used in this study. Figure (4-2): Comparison of a five-spot pattern and a one-eighth section of five-spot pattern In order to provide successful simulations of reservoir models, the first step is to determine the grid system of a reservoir. It was known that the finer grid size is, the more accurate the production rate is in the reservoir simulation. However, the smaller size of the grid means the more blocks in a system. As the number of grid blocks is increased, the simulation takes longer computational times. Thus, the grid sensitivity analysis is conducted in order to find the minimum number of grid blocks without decreasing the accuracy of the runs. A representative reservoir was built with reservoir properties, project design, PVT data, and rock and fluid properties listed in Appendix A. The number of grid blocks tried is 1 1, 5 5, 13 13, 22 22, 31 31, and with one layer. Figure (4-3) shows oil production rate comparison in different grid block systems. The oil production profile shows that there are no noticeable deviations in

44 Oil Rate, bbl/day grid blocks and grid blocks After a grid sensitivity analysis for the oil field is 33 conducted, a grid system of is considered appropriate to run all of the simulations. Figure (4-4) depicts the grid block system and reservoir pattern in this study. Oil Rate VS. Time Time, day 41X41 of grid 31X31 of grid 22X22 of grid 13X13 of grid 5X5 of grid 1X1 of grid Figure (4-3): Sensitivity analysis of a grid block system Figure (4-4): Grid block system of this study (blue color)

45 Generation of Reservoir Properties and Project Design Parameters The first step in developing an artificial neural network is to generate trusted databases with the selected ranges. The selected rock properties and project design parameters are specified within the studied ranges in the following table (4-1) and table (4-2) [2, 5-7]. The reservoir properties are selected randomly within the given ranges by MATLAB in order to have realistically fair cases. The reservoir properties involved in this study are thickness, permeability, porosity, residual oil saturation, residual water saturation, residual gas saturation, oil saturation, water saturation, gas saturation, initial pressure, and depth. Also, the project design parameters include pattern area, water injection rate, the bottom-hole pressure of the production well, bottom-hole pressure of the injection well, and the time of water injection. Table (4-1): Selected ranges of reservoir properties covered in the study Reservoir parameters Minimum Maximum Unit Thickness ft Permeability md Porosity Depth ft Table (4-2): Selected ranges of project design parameters covered in the study Project Design Minimum Maximum Unit Parameters Pattern area Acre Water injection rate Barrel BHP of Production Well Psi Injection Time 0.4 Time (Oil Rate=2.5bbl/d) 0.7 Time (Oil Rate=2.5bbl/d) Day BHP of Injection Well 0.7 Depth 1.06 Depth Psi

46 Water Injection Time In this study, all scenarios begin with primary recovery and water injected in a certain time period. In primary recovery, a reservoir undergoes production till oil rate is 2.5 barrels per day. The time that the oil production rate equals to 2.5 barrels per day is recorded and the starting time of water flooding is a portion of the time. The water injection time is determined randomly from 0.4 to 0.7 of the time that the oil production rate reaches 2.5 barrels per day. Figure (4-5) illustrates the method that determines the water injection time. The time interval between two black lines is 0.4 to 0.7 of the time at which the oil rate is down to 2.5 barrels per day. In this time interval, a time will be selected randomly by MATLAB as the water injection time. t total q oil = 2.5 bbl/d Figure (4-5): Method of choosing the starting time of water flooding Generation of PVT Properties In simulating of the reservoir fluids, the physical properties of reservoir oil, water, and gas are varie in different scenarios. Those physical properties of reservoir fluids can be estimated via the correlations that are discussed in Chapter 3. With these correlations, simulating the PVT

47 Pressure, psi 36 data of a system is much easier and more efficient via computer programming. In order to generate a PVT database, several PVT parameters such as initial pressure, bubble point pressure, temperature, density of oil and density of gas are studied. Table (4-3) lists the selected PVT parameters and the studied range of each parameter. Figures (4-6) through (4-11) show the estimations of PVT properties including GOR, oil FVF, z-factor, oil viscosity, gas viscosity, and oil compressibility for three randomly selected scenarios. Table (4-3): Selected ranges of PVT parameters covered in the study PVT parameters Minimum Maximum Unit Initial Pressure Psi Bubble Point Pressure 1.1 Pi 1.4 Pi Psi Temperature F Density of oil API Density of gas SG Estimatation of GOR Case 1 Case 2 Case GOR, scf/stb Figure (4-6): Estimation of GOR for three randomly selected cases

48 z factor Pressure, psi 37 Estimatation of oil FVF Case 1 Case 2 Case Bo, rb/stb Figure (4-7): Estimation of oil FVF for three randomly selected cases Estimation for z factor pressure, psi Case 1 Case 2 Case 3 Figure (4-8): Estimation of z-factor for three randomly selected cases

49 Pressure, psi Pressure, psi Estimation of oil viscosity Case 1 Case 2 Case oil viscosity, cp Figure (4-9): Estimation of oil viscosity for three randomly selected cases 6000 Estimation of gas viscosity Case 1 Case 2 Case gas visocisty, cp Figure (4-10): Estimation of gas viscosity for three randomly selected cases

50 Pressure, psi Estimatation of oil compressibility oil compressibility, 1/psi Case 1 Case 2 Case 3 Figure (4-11): Estimation of oil compressibility for three randomly selected cases Generation of Rock and Fluid Properties In this study, the lithology of a reservoir is viewed as a changeable property in different system. With diverse rock types, a three-phase reservoir has various relative permeability and capillary pressure in water-oil interface and gas-oil interface. In Chapter 3, the Corey s correlations and Stone correlations are employed in simulating the relative permeability and capillary pressure of a system. The following Table (4-4) shows the selected rock and fluid parameters used to model relative permeability and the capillary pressure simulation and the studied range of each parameter. Figures (4-12) through (4-15) illustrate the example of estimations of relative permeability and capillary pressure in water-oil system and gas-oil system for three randomly selected scenarios. The PVT parameters of those three random selected case are listed in the Appendix B.

51 40 Table (4-4): Selected ranges of rock and fluid parameters covered in the study Rock and Fluid Parameters Minimum Maximum Water relative permeability at residual oil saturation (K rwro ) Oil relative permeability at irreducible water saturation (K roiw ) Corey water exponent (n w ) 2 4 Corey oil exponent of water-oil system (n ow ) 2 4 Gas relative permeability at residual oil saturation (K rgro ) Corey gas exponent (n g ) 2 4 Corey oil exponent of gas-oil system (n og ) 2 4 Capillary pressure at irreducible water saturation (P dw ) 3 5 Capillary pressure at critical liquid saturation (P dl ) 1 3 Characteristic constant (λ) -4-2 Residual oil saturation (S or ) Irreducible water saturation (S wirr ) Residual Gas saturation (S gr ) Oil saturation (S o ) Swirr-Sgr Water saturation (S w ) Swirr 1-So-Sgr Gas saturation (S g ) 1-So-Sw 1-So-Sw

52 Capillary Pressure, psi Relative Permeability 41 Water-Oil Relative Permeability Curves krw(case1) krow(case1) krw(case2) krow(case2) krw(case3) krow(case3) Water Saturation Figure (4-12): Estimation of water-oil relative permeability curves for three randomly selected cases 8.8 Capillary Pressure in Water-Oil Interface Case 1 Case 2 Case Water Saturation Figure (4-13): Estimation of capillary pressure in water-oil interface for three randomly selected cases

53 Capillary Pressure Relative Permeability 42 Gas-Oil Relative Permeability Curves krg(case1) krog(case1) krg(case2) krog(case2) krg(case3) krog(case3) Gas Saturation Figure (4-14): Estimation of gas-oil relative permeability curves for three randomly selected cases 4 Capillary Pressure in Gas-Oil Interface Case 1 Case 2 Case Oil Saturation Figure (4-15): Estimation of capillary pressure in gas-oil interface for three randomly selected cases

54 Data Collection In order to cover the selected range of reservoir properties, project design parameters, PVT parameters, and rock and fluid properties, 1,427 different scenarios are created with a different properties combination. The number of the scenarios is chosen by comparing the performance of the developed ANN models with different number of scenarios and some impractical cases are excluded. MATLAB is utilized to generate different reservoir properties, project design parameters, PVT parameters, and rock and fluid properties combination within the selected ranges. With the different scenarios, a commercial simulator, CMG-IMEX, is employed for running all of the different water flooding cases and for estimating the production profiles of oil, water and gas. The production profiles of oil, water and gas are the key performance indicators of the simulation. In the production profiles, the oil production trends have two discontinuities so the data pretreatment must be conducted. The production profiles can be replicated by dissecting the production profiles into three time intervals and selecting production rate points. The first time interval is from the beginning of oil production to the time at which lowest oil production rate occurs. During this time interval, the oil production rate is declining and reaches the local minimum. The second time interval is from the time of minimum oil production rate to the water breakthrough time at which the peak oil rate occurs. The oil rate is ascending during the second time interval. The last time interval is from the water breakthrough time to the end of the production. During the last time interval, the oil production rate is decreasing because the effect of water injection is fading. With the three time intervals, the production rates are collected by dividing each time interval into five parts. Five production rate points are collected in each time interval and the number of production rate points of a production profile is fifteen. In addition, the data collection of water and gas production profiles using the same time intervals of the oil production profile in each scenario and the

55 44 method of selecting production rate points is the same. The following diagrams, Figures (4-16), (4-17), and (4-18) show the method of selecting production rate points schematically in this study. For an oil production profile, the collected data are Oil Rates 1 through 15 and Time Intervals 1, 2, and 3. For a water production profile, the collected data are Water Rates 1 through 15. Lastly, for a gas production profile, the collected data are Gas Rates 1 through 15. Hence, with 48 data sets which contain 15 oil rate points, 15 water rate points, 15 gas rate points and three time intervals, the production profiles of a three-phase reservoir can be reproduced. There are two reasons why fifteen production rates are chosen in a production profile. First, the artificial neural network with fifteen data points of a production profile can perform much well than ones with more data points of a production profile. The other reason is that extracting fifteen data points of a production profile can capture the trend of the production profiles far well than extracting less data points of a production profile. Figure (4-16): Diagram of data collection of oil rate points and time intervals

56 45 Figure (4-17): Diagram of data collection of water rate points Figure (4-18): Diagram of data collection of gas rate points

57 46 Chapter 5 Artificial Neural Network Development 5.1 Introduction Within the data brackets selected reservoir properties, project design parameters, PVT parameters, and rock and fluid parameters, combinations are created for 1,427 different scenarios. The oil, water, and gas production profiles are generated by the reservoir simulator, CMG-IMEX. The reservoir properties, project design parameters, PVT parameters and rock and fluid parameters are extracted in one file as input data for ANN training. In addition, the oil, water, gas production rate and time intervals are extracted in one file as output data. Furthermore, some functional links are applied as outputs. Then, the MATLAB Neural Network toolbox is used to develop ANN models with these input data and output data. A forward ANN model and two inverse ANN models are developed in this study. In order to achieve the suitable structure of ANN, hidden layers, neurons, training and transfer algorithms are chosen via trial and error. The database used for developing network is listed in Appendix B and the forward artificial neural network predictions are listed in Appendix C. At the end, a graphical user interface (GUI) is built via MATLAB Guide Toolbox 5 with the ANN models to access the performance of the ANN more efficiently. 5 MATLAB Guide Toolbox: a graphical user interface tool in MATLAB platform developed by MathWorks, Inc.

58 Training of ANN Preparation and processing of input and output data should be done before training an artificial neural. The combination of different reservoir properties, project design parameters, PVT parameters, and rock and fluid parameters are entered into the ANN network as the input data; the oil, water, gas production rate points and time intervals are entered into the ANN model as the output data. Then, the inputs and outputs had to be processed before using input data and output data in an expert system. In general, input data and output data had to be normalized between -1 and 1 in order to make sure the learning process is successful. After normalization of the data, inputs and outputs need to be divided into three sets namely training, validation, and testing data sets. The training data sets are used in computing the gradient and updating the weights and biases of the network to satisfy the output of the simulator [9]. The validation data sets are used to validate the network by comparing the output of the simulator to the output of the validation data sets and adjusting the weights of the input until the output of the network meets the output of the simulator. When the output of the network satisfies the output of the validation data set, the testing data sets will test the network by computing the error between the output of the simulator and the output of the testing data set. The performances of those three data sets are plotted in the performance plot, Figure (5-1).

59 48 Figure (5-1): performance plot of the forward model The performance plot shows that the training, validation, and testing errors follow a declining pattern in the initial stage of the training process. However, if the validation error is rising while the training error and testing error are decreasing, the network reaches the point of network memorization also known as overfitting. When the network meets the memorization, the network cannot be trained after the memorization. The training of the network is stopped because the overfitting occurs. Even though the error of training data set is dropping to a minuscule value after the overfitting, when new testing data is introduced to the network, the error of new testing

60 49 data is significant. Also, a regression plot is an important indicator for the training process. The regression line, which can be accessed during the training process, shows the behavior of the ANN training. The ANN training is completed and can be terminated when the regression constant is close to 1. The structure of hidden layers, neurons in each hidden layer, and training, transfer, and learning algorithms are crucial for finding a network with the best output. In order to attain the best performance of the ANN model and the minimum difference between the prediction of the ANN model and the results of the simulator, a large number of training processes with various structures of an ANN model are conducted via a trial and error method to find the most suitable architecture, number of hidden layers, number of neurons in each layer, transfer function, learning function and training function. 5.3 Forward ANN for Predicting Production Profiles The forward ANN is designed for the prediction of oil, water and gas production profiles in diverse reservoir scenarios. The studied reservoir properties, project design parameters, PVT parameters, and rock and fluid parameters are inputs of the forward ANN model. The oil, water, gas rates and three-time intervals generated from the simulator are outputs of the forward ANN model. Also, some functional links are created in order to reduce the differences between the prediction of the forward network and the results of the simulator. The selection of these functional links is based on a mathematical relationship between the input data and output data. By providing more active nodes of the forward ANN model, the functional links can improve the outcome of the network. The functional links are given as outputs in this network. There are 14 functional links in the forward artificial neural network. Table (5-1) lists the input, output parameters of the forward ANN model.

61 50 Table (5-1): Inputs and outputs of the forward ANN model INPUT Reservoir Properties Thickness Depth Permeability Porosity Project Design Parameters PVT Parameters Pattern Area Bottom hole Pressure of Production well Bottom hole Pressure of Injection well Water injection rate Injection Time Bubble Point Pressure Initial Pressure Temperature Density of oil Density of gas Rock and Fluid Parameters K rwro, K roiw, K rgro, S o, S w, S g, S or, S wirr, S gr, n w, n ow, n g, n og, P dw, P dl, λ OUTPUT Oil Rate Oil Rate 1, Oil Rate 2, Oil Rate 3, Oil Rate 4, Oil Rate 5, Oil Rate 6, Oil Rate 7, Oil Rate 8, Oil Rate 9, Oil Rate 10, Oil Rate 11, Oil Rate 12, Oil Rate 13, Oil Rate 14, Oil Rate 15 Water Rate Water Rate 1, Water Rate 2, Water Rate 3, Water Rate 4, Water Rate 5, Water Rate 6, Water Rate 7, Water Rate 8, Water Rate 9, Water Rate 10, Water Rate 11, Water Rate 12, Water Rate 13, Water Rate 14, Water Rate 15 Gas Rate Gas Rate 1, Gas Rate 2, Gas Rate 3, Gas Rate 4, Gas Rate 5, Gas Rate 6, Gas Rate 7, Gas Rate 8, Gas Rate 9, Gas Rate 10, Gas Rate 11, Gas Rate 12, Gas Rate 13, Gas Rate 14, Gas Rate 15 Time Intervals Time Interval 1, Time Interval 2, Time Interval 3 Functional Links [Time Interval 3/Time Interval 1] [Time Interval 3/(Time Interval 1+Time Interval 2)] [(Time Interval 3+ Time Interval 2)^2] [10 Time Interval 1 ] [10 Time Interval 2 ] [Oil Rate 5/ Time Interval 1] [Oil Rate 5 Oil Rate 6] [Oil Rate 5 Porosity Permeability] [Gas Rate 5 Gas Saturation] [Gas Rate 6 Gas Saturation] [Gas Rate 5 Time Section 1] [Time Interval 1 Time Interval 2/ Time Interval 3] [log(gas Rate 5)] [log(gas Rate 6)] The number of data sets is an essential element for developing a reliable network. If data sets are not enough for the training, results of an ANN could become unreliable. If too many data sets are utilized in an expert system, over-fitting and memorization may occur during the training process. At first, 500 cases are employed in training the forward ANN model. However, the

62 51 network performs a large variation between the output of ANN model and the output of the simulator. With further examination, the ANN model with 500 data sets failed to capture the trends of the production profiles and the time intervals of water flooding projects. Therefore, more cases are added to the training of the forward network. In addition, some non-practical cases are eliminated in order to improve the accuracy of the database. Finally, the total number of data sets is 1,427 for developing the forward ANN model. The data sets are separated randomly into 95% for training data sets, 2.5% for validation data sets and 2.5% for testing data sets via a MATLAB function, divideran. Also, the input data and output data are in logarithmic form and scaled between -1 and 1 to ensure a better training. Because some inputs and outputs in the data are large or close to zero, the logarithmic input and output data can assure that the ANN model does not generate negative predictions, which are impractical in production rate and time intervals. When the number, the division and transformation of the data sets are done, the next step is to select training algorithm and transfer function of the ANN model. The back propagation scaled conjugate gradient training function (trianscg) is applied in this study because it was found to generate better results comparing to other transfer functions in the MATLAB ANN toolbox. Also, transfer functions implemented in the study are the Log-sigmoid transfer function (logsig) and the Tangent Sigmoid transfer function (tansig) since they were found to generate better outcomes than the other transfer functions. As for the learning function, the gradient descent with momentum weight and bias (learngdm) function is selected. In addition, the mean sum of squares of the network error with regularization (msereg) function is used as the performance function. As for the architecture of a network, the number of hidden layers and neurons for each hidden layer determine the time in calculation and the final accomplishment of a network. A MATLAB script was designed as a tool for finding a better structure of an ANN model via trial and error iteration automatically. In the beginning, the script creates a given number of structures with a certain amount of hidden layers and neurons located in each layer randomly. Transfer

63 52 functions of each hidden layer are also selected randomly in different structures. Then, the MATLAT runs these given networks, and then records and compares the performance of those ANN models. The best structure is achieved with the minimum difference between the prediction of the ANN model and the result of the simulator. In this study, the architecture of the forward ANN model with three hidden layers, 34, 58, and 92 neurons for each hidden layer correspondingly provides the best prediction of production profiles. The transfer function of each hidden layer is tansig. Figure (5-2) shows the forward network providing satisfactory results. Figure (5-2): Structure of the Forward ANN model 5.4 Inverse ANN for Predicting Project Design Parameters The inverse ANN model is developed to predict the project design parameters of water flooding projects that can be utilized to satisfy a desired water flooding recovery over a specified

64 53 production period of 54 years (20,000 days). The reservoir properties, PVT parameters, rock and fluid parameters, oil, water, and gas production profiles over the production period are the inputs for the inverse network. The outputs of the inverse network are pattern area, bottom-hole pressure of production well, bottom-hole pressure of injection well, water injection rate, and timing of water flooding. Table (5-2) lists the inputs and outputs of the inverse ANN model for predicting project design parameters. Table (5-2): Inputs and outputs of the inverse ANN model for predicting project design parameters INPUTS Reservoir Properties Thickness Depth Permeability Porosity PVT Parameters Bubble Point Pressure Initial Pressure Temperature Density of oil Density of gas Rock Properties Parameters K rwro, K roiw, K rgro, S o, S w, S g, S or, S wirr, S gr, n w, n ow, n g, n og, P dw, P dl, λ Oil Rate Oil Rate 1, Oil Rate 2, Oil Rate 3, Oil Rate 4, Oil Rate 5, Oil Rate 6, Oil Rate 7, Oil Rate 8, Oil Rate 9, Oil Rate 10, Oil Rate 11, Oil Rate 12, Oil Rate 13, Oil Rate 14, Oil Rate 15 Water Rate Water Rate 1, Water Rate 2, Water Rate 3, Water Rate 4, Water Rate 5, Water Rate6, Water Rate 7, Water Rate 8, Water Rate 9, Water Rate 10, Water Rate 11, Water Rate 12, Water Rate 13, Gas Rate Water Rate 14, Water Rate 15 Gas Rate 1, Gas Rate 2, Gas Rate 3, Gas Rate 4, Gas Rate 5, Gas Rate 6, Gas Rate 7, Gas Rate 8, Gas Rate 9, Gas Rate 10, Gas Rate 11, Gas Rate 12, Gas Rate 13, Gas Rate 14, Gas Rate 15 Time Intervals Time Interval 1, Time Interval 2, Time Interval 3 OUTPUTS Project Design Parameters Bottom-hole Pressure of Production Well Bottom-hole Pressure of Injection Well Water Injection Rate Injection Time Pattern Area Functional Links log (Injection Time) log(depth) Initial Pressure 0.5 Injection Time 10 log(bhp of Injection Well)

65 54 The number of datasets used in this inverse ANN model is 1,427 sets, which are including 1,355 training, 36 validation, and 36 testing sets. In order to achieve the best performance, many training processes have been studied via trial and error to choose the best structure, number of hidden layers, number of neurons in each layer, transfer function, learning function, and training function. The feed forward back-propagation structure with two hidden layers provides acceptable results. The first hidden layer has 72 neurons and a logsig transfer function, whereas the second hidden layer has 88 neurons and a tansig transfer function. Moreover, input layer has 73 neurons and output layer has 5 neurons. In addition, the training algorithm of this network is trainscg. The structure of designed network is shown in Figure (5-3). Figure (5-3): Structure of the inverse ANN model for predicting project design parameters

66 Inverse ANN for Predicting Reservoir Properties In this ANN model, the reservoir skeleton properties are estimated for given project design parameters, PVT parameters, rock and fluid parameters, oil rate, water rate, gas rate, and time intervals of the production profiles. The outputs of this inverse ANN model are porosity, permeability, and thickness. Table (5-3) lists the inputs and outputs of the inverse ANN model. Table (5-3): Inputs and outputs of the inverse ANN model for predicting reservoir properties INPUTS Fluid Properties Parameters Bubble Point Pressure Initial Pressure Temperature Density of oil Density of gas Rock and Fluid Properties Parameters K rwro, K roiw, K rgro, So, Sw, Sg, Sor, Swirr, Sgr, nw, now, ng, nog, P dw, P dl, λ Reservoir Property Oil Rate Water Rate Depth Oil Rate 1, Oil Rate 2, Oil Rate 3, Oil Rate 4, Oil Rate 5, Oil Rate 6, Oil Rate 7, Oil Rate 8, Oil Rate 9, Oil Rate 10, Oil Rate 11, Oil Rate 12, Oil Rate 13, Oil Rate 14, Oil Rate 15 Water Rate 1, Water Rate 2, Water Rate 3, Water Rate 4, Water Rate 5, Water Rate6, Water Rate 7, Water Rate 8, Water Rate 9, Water Rate 10, Water Rate 11, Water Rate 12, Water Rate 13, Water Rate 14, Water Rate 15 Gas Rate Gas Rate 1, Gas Rate 2, Gas Rate 3, Gas Rate 4, Gas Rate 5, Gas Rate 6, Gas Rate 7, Gas Rate 8, Gas Rate 9, Gas Rate 10, Gas Rate 11, Gas Rate 12, Gas Rate 13, Gas Rate 14, Gas Rate 15 Time Intervals Time Interval 1, Time Interval 2, Time Interval 3 Project Design Parameters Bottom-hole Pressure of Production Well Bottom-hole Pressure of Injection Well Water Injection Rate Depth Injection Time Pattern Area OUTPUTS Reservoir Properties Thickness Permeability Porosity Functional Links Porosity Thickness Permeability Thickness

67 56 The number of datasets used in this inverse ANN model is 1427 sets which are the same data sets used in the previous ANN model employed for estimating project design parameters. The training, validation and testing data sets are randomly assigned in percentages of 95, 2.5 and 2.5 respectively. In order to find the architecture of the network with the best performance, many training processes have been studied via trial and error in order to choose the most effective structure, number of hidden layers, number of neurons in each layer, transfer function, learning function and training function. The feed forward back-propagation structure with four hidden layers provides acceptable results. The first hidden layer has 20 neurons and uses a tansig transfer function; the second hidden layer has 90 neurons and a logsig transfer function; the third hidden layer has 52 neurons with a logsig transfer function; the fourth hidden layer has 66 neurons with a logsig transfer function Moreover, the input layer has 75 neurons and the output layer has 3 neurons. In addition, the training algorithm, trainscg, is utilized in this inverse ANN model. Figure (5-4) shows the structure of the designed network.

68 57 Figure (5-4): Structure of the inverse ANN model for predicting reservoir properties 5.6 Graphical User Interface A graphical user interface (GUI) is developed to enhance efficiency and user-friendliness, especially for reservoir engineers. Three GUIs are created for the forward ANN, for the inverse ANN for predicting project design parameters, and for the inverse ANN for predicting reservoir properties respectively Graphical User Interface for Forward ANN A graphical user interface (GUI) is designed for the forward ANN model. There are four parts of the input parameters which are reservoir properties and project design parameters, PVT

69 58 parameters, and rock and fluid parameters. In this graphical user interface, the predictions of production rates and time sections are demonstrated as three plots which are oil, water, and gas production profiles. A user can choose values in the input parameter blanks. Once the required input data are designated, the ANN predictions of fluid production profiles are predicted via pushing the Start button. Figure (5-5) shows the graphical user interface of this forward ANN model. Figure (5-5): GUI of the forward ANN model Graphical User Interface for Inverse ANN for Predicting Project Design Parameters A GUI is created for the inverse ANN predictor for project design parameters. The GUI contains four parts of input data, which are reservoir properties, PVT parameters, rock and fluid parameters, and production profile data. Once these required inputs are selected, users can acquire the prediction of inverse ANN model including pattern area, water injection rate, bottom-hole pressure of production well, bottom-hole pressure of injection well, and water injection time in

70 the columns at the lower side of the GUI. Figure (5-6) shows the GUI for the inverse ANN for predicting project design parameters. 59 Figure (5-6): GUI for the inverse ANN for predicting project design parameters Graphical User Interface for Inverse ANN for Predicting Reservoir Properties Once the inverse ANN model for predicting the reservoir properties is developed, a graphical user interface (GUI) is designed for users to acquire the prediction of the network. Four groups of parameters including project design parameters, PVT parameters, rock and fluid parameters and production profiles of oil, water, and gas should be specified in order to access the outcomes of the inverse ANN model. Once the inputs are specified, the prediction of porosity, permeability, and thickness are displayed immediately in the lower side of the GUI. Figure (5-7) shows the graphical user interface for the inverse ANN for predicting reservoir properties.

71 Figure (5-7): GUI for the inverse ANN for predicting reservoir properties 60

72 61 Chapter 6 Results and Discussion This chapter presents an in-depth analysis of the performance of three artificial neural network models developed in the study. The results of each artificial neural network will be discussed and compared with the results generated by the reservoir simulator graphically. Moreover, the error between the prediction and the output of simulator is an essential consideration of the performance of an artificial neural network model. After the training process of an ANN model, an ANN model can predict the output values with the input of the testing data sets. The difference of the ANN prediction and the output of the testing data sets generated by the commercial simulator can be described in the following mathematical form: ( (6.1) In addition, average error of the total prediction is also a crucial performance indicator of an expert system. The following equation describes calculation of the average error. ( (6.2) N is the number of testing data sets. The further discussion of the error and the average error of ANN models prediction will be presented in this chapter. 6.1 Result of Forward Artificial Neural Network Model After the comparison with ANN models of water flooding projects with different numbers of datasets and the ANN with or without functional links, the three hidden layer feed

73 62 forward back-propagation structure can provide satisfactory results. The distribution of the neurons in three hidden layers is 34, 58, 92 neurons, and the sum of the neurons is 184. The input layer contains 30 neurons which are reservoir properties, project design parameters, rock, and fluid parameters. The output layer has 62 neurons of which the first three neurons represent the time intervals, each 15 neurons for oil, water, gas production rate and 14 functional links. Table (6-1) illustrates the neurons and transfer functions in each hidden layers, input, and output of the forward ANN model. Table (6-1): The hidden layers, neurons, and transfer functions of the forward ANN Layer Neurons Transfer Function Input (Reservoir, Project Design, Rock, and Fluid) 30 Hidden layer 1 34 Tangent Sigmoid Hidden layer 2 58 Tangent Sigmoid Hidden layer 3 92 Tangent Sigmoid Output (Time Intervals & Production Rate) 48 Pure Lin Output (Functional Links) 15 The performance of this forward ANN training is displayed in Figure (6-1) as shown below. Figure (6-1) shows that the best performance occurred at epoch 4,179 which means the validation error reaches the minimum value in the iteration. After epoch 4,179, the network is over-training because the validation error is increasing. The early stopping method is utilized in achieving the best performance of this network. The best performance is the goal point showed in the performance plot.

74 63 Figure (6-1): Performance of the forward ANN model. The error between the output of the simulator and the predicted value from the forward network of the testing sets is showed in Figure (6-2) below. The average error of the forward ANN model prediction is 13.7%, which is less than the desired tolerance (15%), so the forward network has the capability to predict production profile of oil, water, and gas accurately.

75 Error, % 64 Forward ANN prediction Testing Sets Figure (6-2): Error for forward ANN prediction Figure (6-2) shows that the best prediction occurs in Test Case 5 and the worst prediction is in Test Case 16. Test Case 24 has a moderate error level in ANN prediction. The reservoir properties, project design parameters, PVT parameters, and rock and fluid parameters of these three cases are listed in Table (6-2).

76 65 Table (6-2): Input data of the sample testing cases Inputs Case 16 Case 24 Case 5 Pattern area (acre) Thickness (ft) Permeability (md) Porosity Residual oil saturation Residual water saturation Residual Gas saturation Oil saturation Water saturation Gas saturation Initial Pressure (psi) Depth (ft) Water injection rate (bbl) Injection Time (day) BHP of Production Well (psi) BHP of Injection Well (psi) Bubble Point Pressure (psi) Temperature (F) Density of oil (API) Density of gas (SG) K rwro K roiw S wirr S orw n w n ow K rgro S gc S org n g n og P dw P dl lambda

77 66 Figure (6-3), Figure (6-4), and Figure (6-5) show production profiles of the Test Case 16 with a maximum error of 26%. In the Test Case 16, the error in time interval prediction is 32%; the error in oil rate prediction is 29.5%; the error in water rate prediction is 12.8%; the error in gas rate prediction is 35.7%. As shown in these figures, the ANN predictions of the production profiles of oil, water, and gas are shifted slightly because the error in the prediction of time intervals is 32%. However, the predictions in production rates of oil, water, and gas have smaller ranges of error, so the forward network can still capture the trend of production profiles. Figure (6-3): Comparison of oil production profiles generated by ANN and simulator in Case 16

78 67 Figure (6-4): Comparison of water production profiles generated by ANN and simulator in Case 16 Figure (6-5): Comparison of gas production profiles generated by ANN and simulator in Case 16

79 Oil Rate SC(bbl/day) 68 Figures (6-6), (6-7), and (6-8) show that Test Case 24 resulted in a median error of 12.2%. In the Test Case 24, the error in time interval prediction is 6.9%; the error in oil rate prediction is 9.7%; the error in water rate prediction is 10%; the error in gas rate prediction is 17.2%. The ANN prediction for Case 24 clearly can match the production profiles generated by the simulator better than in the previous case (Case 16) Oil Rate v.s.time Plot CMG ANN Time(days) x 10 4 Figure (6-6): Comparison of oil production profiles generated by ANN and simulator in Case 24

80 Gas Rate SC(ft 3 /day) Water Rate SC(bbl/day) Water Rate v.s.time Plot CMG ANN Time(days) x 10 4 Figure (6-7): Comparison of water production profiles generated by ANN and simulator in Case 24 6 x 105 Gas Rate v.s.time Plot 5 CMG ANN Time(days) x 10 4 Figure (6-8): Comparison of gas production profiles generated by ANN and simulator in Case 24 The best result for the ANN prediction occurs in the Test Case 5 with an error of 4.9%. The error in the prediction of time interval is 3.5%; the error in oil rate prediction is 8%; the error

81 Water Rate SC(bbl/day) Oil Rate SC(bbl/day) 70 in the water rate prediction is 1.4 %; the error in the gas rate prediction is 5.7%. Figures (6-9), (6-10), and (6-11) show the comparison of the production profiles of oil, water and gas generated by the simulator and by the ANN Oil Rate v.s.time Plot CMG ANN Time(days) x 10 4 Figure (6-9): Comparison of oil production profiles generated by ANN and simulator in Case Water Rate v.s.time Plot CMG ANN Time(days) x 10 4 Figure (6-10): Comparison of water production profiles generated by ANN and simulator in Case 5

82 Gas Rate SC(ft 3 /day) x 104 Gas Rate v.s.time Plot 10 CMG ANN Time(days) x 10 4 Figure (6-11): Comparison of gas production profiles generated by ANN and simulator in Case Error Analysis The mean error of the forward ANN prediction is 13.7% and the mean errors in the prediction of time intervals, oil rates, water rates, and gas rates are 17.8%, 15.6%, 6% and 18% respectively. Figures (6-12), (6-13), (6-14), and (6-15) show the error distribution of the 36 testing sets in the prediction of time intervals, oil rate, water rate, and gas rate respectively. Figure (6-12): Error distribution in the prediction of time interval

83 72 In Figure (6-12), Test Case 15 has the highest error of 44% and the lowest error occurs at Test Case 14 of 2.2%. The equations of error calculation are showed in the equation (6.1) and (6.2). The time interval prediction is essential for the network to match the production profiles because production profiles which always have two discontinuities in all scenarios are dissected into three parts by three time intervals. The production profiles generated by the forward ANN model would shift if the error in time interval prediction is high. In order to provide a better prediction, some functional links which are related to time intervals are added in the forward ANN model. Figure (6-13): Error distribution in the prediction of oil rate In Figure (6-13), the highest error of 37% in oil rate prediction occurs in the Test Case 27. Figure (6-16) shows the oil production profile of Test Case 27 which has the highest error in oil rate prediction. According to Figure (6-16), the highest error is located in the Oil Rate 6. Figure (6-17) shows the Test Case 31 with the lowest error of oil rate production which is 6%. The highest error in oil rate prediction of Test Case 31 occurs in the Oil Rate 5.

84 73 Figure (6-14): Error distribution in the prediction of water rate From Figure (6-14), the highest error in water rate prediction occurs in Test Case 17 and the minimum error is in Test Case 14. Figure (6-18) and Figure (6-19) show water production profiles of Test Case 17 with a 24.6% error and Test Case 14 with a 1.2% error. Figure (6-15): Error distribution in the prediction of gas rate Figure (6-15) shows the maximum error of gas rate prediction occurs in Test Case 5 and the minimum occurs in Test Case 7. Figure (6-20) and Figure (6-21) plot the gas production profiles of Test Case 7 with a 42.2% error and Test Case 5 with a 5.7% error. In Figure (6-20), high errors are located in Gas Rate 1, Gas Rate 5, and Gas Rate 6. The high error in the Test Case 5 occurs in the Gas Rate 1 as shown in Figure (6-21).

85 Oil Rate SC(bbl/day) Oil Rate SC(bbl/day) Oil Rate v.s.time Plot CMG ANN Time(days) x 10 4 Figure (6-16): Test Case 27 with the maximum error in oil rate prediction Oil Rate v.s.time Plot CMG ANN Time(days) x 10 4 Figure (6-17): Test case 31 with the minimum error in water rate prediction

86 Water Rate SC(bbl/day) Water Rate SC(bbl/day) Water Rate v.s.time Plot CMG ANN Time(days) x 10 4 Figure (6-18): Test Case 17 with the maximum error in water rate prediction Water Rate v.s.time Plot CMG ANN Time(days) x 10 4 Figure (6-19): Test Case 14 with the minimum error in water rate prediction

87 Gas Rate SC(ft 3 /day) Gas Rate SC(ft 3 /day) x 105 Gas Rate v.s.time Plot 3 CMG ANN Time(days) x 10 4 Figure (6-20): Test Case 7 with the maximum error in gas rate prediction 12 x 104 Gas Rate v.s.time Plot 10 CMG ANN Time(days) x 10 4 Figure (6-21): Test Case 5 with the minimum error in gas rate prediction

88 Error, % Error, % 77 Also, if the error distributions are presented in plots of mean error versus each production rate point number, the performance of the ANN model can be interpreted and discussed more thoroughly. Figures (6-22), (6-23) and (6-24) illustrate the error distribution of production rate point numbers of oil, water, and gas. 35 Error in Oil Rate Prediction Oil Rate Point Number Figure (6-22): Error distribution in the prediction of oil rate point numbers Error in Water Rate Prediction Water Rate Point Number Figure (6-23): Error distribution in the prediction of water rate point numbers

89 Error, % Error in Gas Rate Prediction Gas Rate Point Number Figure (6-24): Error distribution in the prediction of gas rate point numbers In the error distribution in the prediction for oil rate point numbers, Figure (6-22), the high error occurs in the Oil Rate 5 and the Oil Rate 6. In comparison with Figure (6-25), the Oil Rate 5 is the local minimum of the oil production rate. Because of the low value of the Oil Rate 5, the error of the forward ANN model prediction is very high. However, the Oil Rate 5 takes a small proportion of the cumulative oil production. Also, the error of the Oil Rate 14 and the Oil Rate 15 are relatively high. Figure (6-25) shows that in the end stage of the water flooding projects, the oil rates are small and stable and take small proportion of the cumulative oil recovery production. Also, Figure (6-24) indicates that the highest error in gas rate prediction occurs in the Gas Rate 6. However, this high error is not an issue, because the gas rate will be negligible at the Gas Rate 6, because in Figure (6-26), which illustrates the method of collecting gas rate data, gas production rates become very limited for the Gas Rate 5 through 15 since water injection takes place in the reservoir.

90 79 Therefore, absolute errors cannot evaluate the accuracy of the networks since they sometimes lead to wrong conclusions. For instance, the range of oil rates is between 1 to 200 barrels per day for this study. If the neural network forecasts a value of 4 for 1 barrel per day, that represents 300% of absolute error and is considered as a poor performance. Conversely, if the network predicts a value of 203 for the 200 barrels per day, that represents only 1.5% of absolute error and is interpreted as a relatively good performance. The difference of absolute errors between two cases is 298.5% even though both cases have exactly same deviation of 3 barrels per day. If the error of the Oil Rate 5, Oil Rate 14 and Oil Rate 15 can be neglected, the error in oil rate prediction will be lower. The error in the prediction of oil rate will decrease from 15.6% to 14% if the Oil Rate 5, Oil Rate 14 and Oil Rate 15 are excluded in the mean error calculation. As for the error in the gas rate prediction, the error will decline from 18% to 16.2% if Gas Rate 5 and Gas Rate 6 are neglected in the mean error calculation. Hence, the total mean error of the forward ANN model will be 13% if some production rate points are excluding in the mean error calculation. Figure (6-25): Diagram of data collection of oil rate points

91 80 Figure (6-26): Diagram of data collection of gas rate points Also, if converting the prediction of production profiles into cumulative production, the error of the forward ANN prediction will become lower. The errors in prediction of cumulative oil, water, and gas production are 9%, 5.5%, and 16.4% correspondingly. The mean error in the cumulative production prediction is 10.3% and the average error in the prediction (including time section) is 11.6%. Also, Table (6-3) lists the error comparison in three different errors which are the absolute error, the absolute error excluding some production rate points, and the error in cumulative production. If the performance of the forward network is interpreted in the cumulative production, the difference between the network prediction and the output of the simulator is lowest.

92 81 Table (6-3): Comparison of error in different scenarios Absolute Error Error excluding some Production Rate Points Error in Cumulative Production Error in Time 17.8% 17.8% 17.8% Interval prediction Error in Oil Rate 15.6% 14.2% 9.0% Prediction Error in Water 6.0% 6.0% 5.5% Rate Prediction Error in Gas Rate 18.0% 16.8% 16.4% Prediction Mean Error 13.7% 13.0% 11.6% 6.2 Results of the Inverse ANN models Result of Inverse ANN Model for Predicting Project Design Parameters By using reservoir characteristics, PVT parameters, rock and fluid parameters, oil, water, and gas recovery and time interval of water flooding project, this inverse model conducts the prediction of project design parameters including the area of water flooding pattern, bottom-hole pressure (BHP) of production well, bottom-hole pressure of injection well, water injection time, and water injecting rate. A feed forward back-propagation network with two hidden layers is found to provide a satisfactory prediction. The distribution of the neurons in two hidden layers is 72, and 88 neurons and the summary of neurons is 160. The transfer function used for the first hidden layer is logsig and for the second hidden layer is tansig. The input layer contains 73 neurons including reservoir properties, PVT parameters, rock and fluid parameters, oil, water, gas rates and time interval of the production profile. The output layer has 9 neurons, which have five project design parameters and 4 functional links. The structure of the inverse ANN structure is presented in Table (6-4).

93 Table (6-4): The hidden layers, neurons, and transfer functions of the inverse ANN for predicting project design parameters Layer Neurons Transfer Function Input (Reservoir, Rock, and Fluid Parameters, 73 Time Intervals and Production Rate) Hidden layer 1 72 Log- Sigmoid Hidden layer 2 88 Tangent Sigmoid Output (Project Design Parameters) 5 Pure Lin Output (Functional Links) 8 The results are illustrated graphically by comparing the ANN prediction of parameters and the actual data generated by the simulator. The average error in the project design parameter prediction is described below in Figure (6-27). 82 Average Error Bottom-hole Pressure of Injection Well Injection Time Bottom-hole Pressure of Produciton Well Water Injection Rate Pattern Area Error % Figure (6-27): Average error in the prediction of project design parameters Figures (6-28), (6-29), (6-30), and (6-31) present the comparison of the result of inverse network prediction and the result generated by the simulator. The prediction of the area of water flooding pattern has 6.2% as the average error, as shown in Figure (6-28). Figure (6-29) shows the result of the result of the water injection rate. The average error in water injection rate prediction is 2.5%. Figures (6-30) and (6-31) show that the average error of the bottom-hole

94 Water Injection Rate (bbl/d) pressure of the production well and the bottom-hole pressure of injection well are 10.3% and 8.9%, respectively. Lastly, Figure (6-32) shows the average error in water injection time is 6.8%. 83 Figure (6-28): Comparison of pattern area predicted by ANN and simulator Water Injection Rate CMG v.s. ANN results CMG ANN Testing Sets Figure (6-29): Comparison of water injecting rate predicted by ANN and simulator