A Streamlines Based Multivariate Regression Model to Quantify the Impact of Reservoir Heterogeneity on Ultimate Recovery. Abdullah Al-Najem, MS

Transcription

1 A Streamlines Based Multivariate Regression Model to Quantify the Impact of Reservoir Heterogeneity on Ultimate Recovery by Abdullah Al-Najem, MS A Dissertation In Petroleum Engineering Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Approved Dr. Mohamed Y. Soliman Chair of Committee Dr. Habib Menouar Dr. Shameem Siddiqui Dr. Ravi Vadapalli Dr. Dominick J. Casadonte, Jr. Interim Dean of the Graduate School December, 2013

2 Copyright 2013, Abdullah Al-Najem

3 DEDICATION I dedicate this work to My parents, Tarfah Abdullah and Ali Al-Najem, for their love and support throughout my entire life, My adorable wife, Amani, and my kids, Lujain, Safanh and Ali, for their love, endless patience, and understanding, My brothers and sisters for their unconditional love and relentless support and to my father-in law and mother-in law for their support and prayers.

4 ACKNOWLEDGMENTS My first gratitude goes to the Almighty God for granting me the strength, courage and perseverance to complete this work. I feel extremely privileged to have the opportunity to pursue a doctoral degree at Texas Tech University, as well as to be involved in such an exciting research topic. I am greatly indebted to my supervisor, Dr. Mohamed Y. Soliman, for taking me into his research group and for constantly supporting me all these years. I especially am grateful for all of his guidance, advice, and encouragement during the course of this study. I would like to thank the members of my advisory committee Dr. Habib Menouar, Dr. Shameem Siddiqui and Dr. Ravi Vadapalli, for their careful reviews and useful comments and suggestions. I also would like to extend my sincere thanks to Dr. Md. Rakibul Sarker for his help in using FORTRAN. Dr. Md. Rakibul Sarker was my lighthouse during the stormy and darkest nights, and always helped me find my way back. Acknowledgements also are due to Dr. Benaissa Chidmi for his help in developing the multivariate regression models using Matlab and the very useful discussions that followed. I also wish to thank the people at the Department of Petroleum Engineering for their support and contributions to my academic achievements. Further, I would like to thank my sponsor company, Saudi Aramco, for giving me the opportunity to pursue my doctoral degree. Last, but not least, I would like to acknowledge my dear friends, new and old, who have been always there for me. ii

5 TABLE OF CONTENTS Acknowledgments... II Abstract... VIII List of Tables... X List of Figures... XII Chapter 1: Reservoir Heterogeneity Introduction Definition of Heterogeneity Heterogeneity Classifiers Static Measures Dykstra-Parsons Coefficient V DP Lorenz coefficient, L C Coefficient of Variation, C V Dynamic Measures Koval Factor Dispersion Simulation Based Dynamic Measures...9 Chapter 2: Static Estimators Inconsistencies in Predicting Hydrocarbon Ultimate Recovery Generation of the Permeability (K) Fields Permeability Distributions Statistical Analysis Description of the Streamline Simulation Model Simulation Runs Necessary to Conduct the Heterogeneity Study Different Models Dynamic Interactions with Streamlines FORTRAN Code for Static Classifiers Calculations FORTRAN Code Results of the Static Heterogeneity Indices Lorenz Coefficient - L C Dykstra-Parsons Coefficient - V DP Coefficient of Variations (Cv)...35 iii

6 2.8 Problem Statement...38 Chapter 3: Streamline Simulation as an Alternative Summary Theory and Background about Streamline Simulation Literature Review of the Streamlines Technology Historical Perspective Development of Streamline Simulators Black Oil Simulation Compositional Simulation Thermal Simulators Extensions and Improvements of Streamline Simulators Improving Streamline Mathematical Calculations Flow Based Grids and Complex Cell Geometries Petroleum Engineering Applications of Streamline Simulation History Matching...71 I. Delineation of Drainage Zones...72 II. Direct Methods...75 III. Optimization Methods...76 IV. Data Integration Methods Reservoir Management...83 I. Sweep Efficiency...85 II. Rate Optimization...88 III. Well Placement...92 IV. Enhanced Oil Recovery Upscaling, Ranking, and Characterizing Fine-Grid Geological Models...97 I. Upscaling...98 II. Uncertainty Quantification and Ranking of Geological Models...99 III. Characterization of Reservoir Heterogeneity iv

7 Chapter 4: Streamlines Mathematical Formulations Introduction Governing IMPES Equations: Pressure Equation Governing IMPES Equations Material Balance Equation Coordinate Transform Gravity Tracing Streamlines Timestepping in Streamlines Simulations Chapter 5: Development of the Streamline Based Dynamic Measure - SL_TOF Index Mining of Streamlines Information FORTRAN Code for Extracting PETREL Dynamic Production Data TOF_Hete Streamline Derived Index Case-1: Low Average Permeability Model AvgK15md Case-2: Intermediate Average Permeability Model AvgK75md Case-3: High Average Permeability Model AvgK150md Case-4: Low Mobility Ratio Model Mu Case-5: Intermediate Mobility Ratio Model Mu Case-6: High Mobility Ratio Model Mu Case-7: Producers Liquid Rate Model LiquidRate1500 bbls Case-8: Producers Liquid Rate Model LiquidRate2000 bbls Comparison of TOF_Hete and the Static Estimators; L C and V DP Chapter 6: The Streamlines based Multivariate Regression Models Introduction Theory and Background of the Multivariate Regression Estimation of the β using the Least Squares principle Description of the Streamlines based Multivariate Regression Models Matlab Code Sensitivity Cases of Streamlines based Multivariate Regression Models v

8 6.4.1 Low Permeability Case Avgk15md A. Separate Regression Models Base Case B. Pooled Standard Deviations Multivariate Regression Model C. Pooled Models Cumulative Water Injection as a Variable Intermediate Permeability Case Avgk75md A. Separate Regression Models Base Case B. Pooled Standard Deviations Multivariate Regression Model C. Pooled Models Cumulative Water Injection as a Variable High Permeability Case Avgk150md A. Separate Regression Models Base Case B. Pooled Standard Deviations Multivariate Regression Model C. Pooled Models Cumulative Water Injection as a Variable Low Mobility Ratio Mu A. Separate Regression Models Base Case B. Pooled Standard Deviations Multivariate Regression Model C. Pooled Models Cumulative Water Injection as a Variable Intermediate Mobility Ratio Mu A. Separate Regression Models Base Case B. Pooled Standard Deviations Multivariate Regression Model C. Pooled Models Cumulative Water Injection as a Variable Intermediate Mobility Ratio Mu A. Separate Regression Models Base Case B. Pooled Standard Deviations Multivariate Regression Model C. Pooled Models Cumulative Water Injection as a Variable Intermediate Mobility Ratio LiquidRate A. Separate Regression Models Base Case B. Pooled Standard Deviations Multivariate Regression Model C. Pooled Models Cumulative Water Injection as a Variable Intermediate Mobility Ratio LiquidRate A. Separate Regression Models Base Case B. Pooled Standard Deviations Multivariate Regression Model vi

9 C. Pooled Models - Cumulative Water Injection as a Variable Combining the Different Heterogeneity Sensitivities Scenarios Chapter 7: Validation of the Multivariate Regression Model Results Introduction Average Permeability Cases Mobility Ratio Cases Producer s Constraint Cases Chapter Conclusion Recommendations References Appendices Appendix-A: Histogram Plots for the Cases Avgk75md and Avgk150md Appendix-B: An Example of a Bivariate Linear Regression Solution in Matrix Form vii

10 ABSTRACT Reservoir heterogeneity plays an important role in reservoir performance, especially under secondary and tertiary displacements. Many different estimators have been proposed to characterize reservoir heterogeneity. Most of these so-called heterogeneity classifiers concentrate on variations in permeability because permeability directly impacts flow and varies significantly within reservoirs compared to other reservoir properties. The heterogeneity classifiers typically fall into two groups: static and dynamic. The two static parameters most often used to quantify reservoir heterogeneity are the Dykstra-Parsons coefficient (V DP ) and the Lorenz coefficient (L C ). The static heterogeneity classifiers however, are inadequate to account for the complex and nonlinear interrelationships among permeability, pressure, and changing fluid saturation within porous media. Hence, a number of dynamic heterogeneity classifiers that implicitly include fluid distribution in flow path and its connected structures have been proposed. These classifiers include fast simulation, permeability connectivity estimates, and streamline simulation. Among these, streamline simulation has gained wide attention during the last two decades. Computation of streamlines is fast and effective and yields direct relationships between time-of-flight (TOF) and dynamic data such as production and tracer breakthrough curves. In this study, we developed multivariate regression models using streamline information such as TOF and streamline density to assist practicing engineers to better quantify the impact of reservoir heterogeneity on ultimate hydrocarbon recovery and also to easily predict fluid saturation changes during primary or enhanced recovery periods. viii

11 For this purpose, we made extensive streamline simulation runs covering a wide range of permeability distributions, mobility ratios, and producers' constraints using a popular streamline simulator. These numerical experimentations showed the presence of strong interrelationships between streamline simulations (as the independent variables), namely TOF and streamline density, and fluid saturation changes and oil recovery factors (as the dependent variables). Additionally, most of the plotted data points of predicted oil recovery values and simulated values fall close to the ideal 45 line, indicating a high degree of predictive accuracy of the models. Other existing classifiers, though yielding reasonable accuracy, were not nearly as accurate as the new criterion. ix

12 LIST OF TABLES 2.1 Statistical Information of one realization for the 8 standard deviations Streamlines attributes outputted as a result of including ALLOC under the SCHEDULE section Streamlines attributes outputted as a result of including TOF under the SCHEDULE section Results of calculating the area under the curve and the TOF_Index Avgk15md Results of calculating the area under the curve and the TOF_Index Avgk75md Results of calculating the area under the curve and the TOF_Index Avgk150md Results of calculating the area under the curve and the TOF_Index Mu Results of calculating the area under the curve and the TOF_Index Mu Results of calculating the area under the curve and the TOF_Index Mu Results of calculating the area under the curve and the TOF_Index LiquidRate1500bbls Results of calculating the area under the curve and the TOF_Index LiquidRate2000bbls Summary of the Equations, R 2, adjusted R 2 for the Separate Models AvgK15md Summary of the Equations, R 2, adjusted R 2 for the Separate Models AvgK75md Summary of the Equations, R 2, adjusted R 2 for the Separate Models AvgK150md Summary of the Equations, R 2, adjusted R 2 for the Separate Models Mu Summary of the Equations, R 2, adjusted R 2 for the Separate Models x

13 Mu Summary of the Equations, R 2, adjusted R 2 for the Separate Models Mu Summary of the Equations, R 2, adjusted R 2 for the Separate Models LiquidRate1500bbls Summary of the Equations, R 2, adjusted R 2 for the Separate Models LiquidRate2000bbls xi

14 LIST OF FIGURES 1.1 Dykstra-Parsons coefficient plot of typical petroleum reservoirs Lorenz coefficient plot illustrating greater concavity toward upper corner Generating permeability fields to conduct the heterogeneity study Histogram of st.dev0.25 Kavg15md Histogram of st.dev0.50 Kavg15md Histogram of st.dev1.0 Kavg15md Histogram of st.dev1.25 Kavg15md Histogram of st.dev1.50 Kavg15md Histogram of st.dev2.00 Kavg15md Histogram of st.dev2.50 Kavg15md Histogram of st.dev3.00 Kavg15md D and 3D schematics of the streamline simulation model Summary of the simulation runs needed to conduct the heterogeneity study Snapshot of permeability distribution of the different models (8 standard deviations) Snapshot of the streamlines interactions with the increase of degree of heterogeneity Flowchart of the heterogeneity classifiers calculations FORTRAN code Lorenz Plot of the Model Avg15md Lorenz Plot of the Model Avg75md Lorenz Plot of the Model Avg150md Lorenz plot of 15 realizations of the model with st.dev0.5 and AvgK=150md...26 xii

15 2.19 Differences of cumulative oil production of the model with st.dev0.5 and AvgK=150md Lorenz plot of 15 realizations of the model with st.dev2 and AvgK=150md Differences of cumulative oil production of the model with st.dev2 and AvgK=150md Differences of cumulative water production of the model with st.dev2 and AvgK=150md Cumulative oil production vs. Lc of the three different average permeability models (15md, 75md and 150md) Total liquid vs. Lc of the three different average permeability models (15md, 75md and 150md) Cumulative water production vs. Lc of the three average permeability models (15md, 75md and 150md) Cumulative oil production vs. Vdp of the three average permeability models (15md, 75md and 150md) Total liquid vs. V DP of the three average permeability models (15md, 75md and 150md) Cumulative water production vs. V DP of the three average permeability models (15md, 75md and 150md) Cumulative oil production vs. C V of the three average permeability models (15md, 75md and 150md) Total liquids vs. C V of the three average permeability models (15md, 75md and 150md)...37 xiii

16 2.31 Cumulative water production vs. C V of the three average permeability models (15md, 75md and 150md) Streamline tracing and velocity vector mapping in a saturation grid Generalized flow diagram for streamline simulation Scaling of CPU time: streamlines vs. finite difference Streamlines capturing sweep/drainage areas associated with injectors and producers Streamlines information aid in re-balancing rates in different patterns Timeline that summarizes the development and enhancement of streamlines simulators Summary of the main streamlines technology application s categories Operator splitting applied to streamlines to calculate gravity D gridblock with known interstitial velocity field and origin Velocity gradient across the gridblock The correct face that the streamline is leaving requires the minimum time Pressure and total velocity resulting from performing step 1 and Flowchart of timestepping in streamlines simulators Inclusion of streamlines based parameters in the FRONTSIM data deck Flowchart of the FORTRAN code used to read and re-arrange the PETREL extracted data TOF profile of 8 standard deviations of the Avg15md case TOF profile of 8 standard deviations of the Avg75md case TOF profile of 8 standard deviations of the Avg150md case xiv

17 5.6 TOF profile of 8 standard deviations of the mobility ratio=2 case TOF profile of 8 standard deviations of the mobility ratio=5 case TOF profile of 8 standard deviations of the mobility ratio=10 case TOF profile of 8 standard deviations of the liquidrate=1500 case TOF profile of 8 standard deviations of the liquidrate=2000 case Recovery factor comparison of the different cases vs. Lorenz coefficient Recovery factor comparison of different cases vs. Dykstra-Parsons coefficient Recovery factor comparison of the different cases vs. TOF_Hete index Regression Residuals Comparison of simulated and predicted cumulative liquid production rates Homo_LowK model Comparison of simulated and predicted cumulative liquid production rates Stde0.25_LowK model Comparison of simulated and predicted cumulative liquid production rates Stde0.50_LowK model Comparison of simulated and predicted cumulative liquid production rates Stde1.0_LowK model Comparison of simulated and predicted cumulative liquid production rates Stde1.25_LowK model Comparison of simulated and predicted cumulative liquid production rates Stde1.50_LowK model Comparison of simulated and predicted cumulative liquid production rates Stde2.0_LowK model xv

18 6.9 Comparison of simulated and predicted cumulative liquid production rates Stde2.5_LowK model Comparison of simulated and predicted cumulative liquid production rates Stde3.0_LowK model Comparison of simulated and predicted cumulative liquid production rates LowK - pooled models Comparison of simulated and predicted cumulative liquid production rates LowK pooled models including cumulative injection rates Comparison of simulated and predicted cumulative liquid production rates Homo_MedK model Comparison of simulated and predicted cumulative liquid production rates Stde0.25_MedK model Comparison of simulated and predicted cumulative liquid production rates Stde0.50_MedK model Comparison of simulated and predicted cumulative liquid production rates Stde1.0_MedK model Comparison of simulated and predicted cumulative liquid production rates Stde1.25_MedK model Comparison of simulated and predicted cumulative liquid production rates Stde1.50_MedK model Comparison of simulated and predicted cumulative liquid production rates Stde2.0_MedK model xvi

19 6.20 Comparison of simulated and predicted cumulative liquid production rates Stde2.5_MedK model Comparison of simulated and predicted cumulative liquid production rates Stde3.0_MedK model Comparison of simulated and predicted cumulative liquid production rates MedK - pooled models Comparison of simulated and predicted cumulative liquid production rates MedK pooled models including cumulative injection rates Comparison of simulated and predicted cumulative liquid production rates Homo_HiK model Comparison of simulated and predicted cumulative liquid production rates Stde0.25_HiK model Comparison of simulated and predicted cumulative liquid production rates Stde0.50_HiK model Comparison of simulated and predicted cumulative liquid production rates Stde1.0_HiK model Comparison of simulated and predicted cumulative liquid production rates Stde1.25_HiK model Comparison of simulated and predicted cumulative liquid production rates Stde1.50_HiK model Comparison of simulated and predicted cumulative liquid production rates Stde2.0_HiK model xvii

20 6.31 Comparison of simulated and predicted cumulative liquid production rates Stde2.5_HiK model Comparison of simulated and predicted cumulative liquid production rates Stde3.0_HiK model Comparison of simulated and predicted cumulative liquid production rates HiK pooled models Comparison of simulated and predicted cumulative liquid production rates HiK pooled models including cumulative injection rates Comparison of simulated and predicted cumulative liquid production rates Homo_Mu2 model Comparison of simulated and predicted cumulative liquid production rates Stde0.25_Mu2 model Comparison of simulated and predicted cumulative liquid production rates Stde0.50_Mu2 model Comparison of simulated and predicted cumulative liquid production rates Stde1.0_Mu2 model Comparison of simulated and predicted cumulative liquid production rates Stde1.25_Mu2 model Comparison of simulated and predicted cumulative liquid production rates Stde1.50_Mu2 model Comparison of simulated and predicted cumulative liquid production rates Stde2.0_Mu2 model xviii

21 6.42 Comparison of simulated and predicted cumulative liquid production rates Stde2.5_Mu2 model Comparison of simulated and predicted cumulative liquid production rates Stde3.0_Mu2 model Comparison of simulated and predicted cumulative liquid production rates Mu2 - pooled models Comparison of simulated and predicted cumulative liquid production rates Mu2 pooled models including cumulative injection rates Comparison of simulated and predicted cumulative liquid production rates Homo_Mu5 model Comparison of simulated and predicted cumulative liquid production rates Stde0.25_Mu5 model Comparison of simulated and predicted cumulative liquid production rates Stde0.50_Mu5 model Comparison of simulated and predicted cumulative liquid production rates Stde1.0_Mu5 model Comparison of simulated and predicted cumulative liquid production rates Stde1.25_Mu5 model Comparison of simulated and predicted cumulative liquid production rates Stde1.50_Mu5 model Comparison of simulated and predicted cumulative liquid production rates Stde2.0_Mu5 model xix

22 6.53 Comparison of simulated and predicted cumulative liquid production rates Stde2.5_Mu5 model Comparison of simulated and predicted cumulative liquid production rates Stde3.0_Mu5 model Comparison of simulated and predicted cumulative liquid production rates Mu5 - pooled models Comparison of simulated and predicted cumulative liquid production rates Mu5 pooled models including cumulative injection rates Comparison of simulated and predicted cumulative liquid production rates Homo_Mu10 model Comparison of simulated and predicted cumulative liquid production rates Stde0.25_Mu10 model Comparison of simulated and predicted cumulative liquid production rates Stde0.50_Mu10 model Comparison of simulated and predicted cumulative liquid production rates Stde1.0_Mu10 model Comparison of simulated and predicted cumulative liquid production rates Stde1.25_Mu10 model Comparison of simulated and predicted cumulative liquid production rates Stde1.50_Mu10 model Comparison of simulated and predicted cumulative liquid production rates Stde2.0_Mu10 model xx

23 6.64 Comparison of simulated and predicted cumulative liquid production rates Stde2.5_Mu10 model Comparison of simulated and predicted cumulative liquid production rates Stde3.0_Mu10 model Comparison of simulated and predicted cumulative liquid production rates Mu10 pooled models Comparison of simulated and predicted cumulative liquid production rates Mu10 pooled models including cumulative injection rates Comparison of simulated and predicted cumulative liquid production rates Homo_LiquidRate1500 model Comparison of simulated and predicted cumulative liquid production rates Stde0.25_LiquidRate1500 model Comparison of simulated and predicted cumulative liquid production rates Stde0.50_LiquidRate1500 model Comparison of simulated and predicted cumulative liquid production rates Stde1.0_LiquidRate1500 model Comparison of simulated and predicted cumulative liquid production rates Stde1.25_LiquidRate1500 model Comparison of simulated and predicted cumulative liquid production rates Stde1.50_LiquidRate1500 model Comparison of simulated and predicted cumulative liquid production rates Stde2.0_LiquidRate1500 model xxi

24 6.75 Comparison of simulated and predicted cumulative liquid production rates Stde2.5_LiquidRate1500 model Comparison of simulated and predicted cumulative liquid production rates Stde3.0_LiquidRate1500 model Comparison of simulated and predicted cumulative liquid production rates LiquidRate pooled models Comparison of simulated and predicted cumulative liquid production rates LiquidRate1500 pooled models including cumulative injection rates Comparison of simulated and predicted cumulative liquid production rates Homo_LiquidRate1500 model Comparison of simulated and predicted cumulative liquid production rates Stde0.25_LiquidRate1500 model Comparison of simulated and predicted cumulative liquid production rates Stde0.50_LiquidRate1500 model Comparison of simulated and predicted cumulative liquid production rates Stde1.0_LiquidRate1500 model Comparison of simulated and predicted cumulative liquid production rates Stde1.25_LiquidRate1500 model Comparison of simulated and predicted cumulative liquid production rates Stde1.50_LiquidRate1500 model Comparison of simulated and predicted cumulative liquid production rates Stde2.0_LiquidRate1500 model xxii

25 6.86 Comparison of simulated and predicted cumulative liquid production rates Stde2.5_LiquidRate1500 model Comparison of simulated and predicted cumulative liquid production rates Stde3.0_LiquidRate1500 model Comparison of simulated and predicted cumulative liquid production rates LiquidRate pooled models Comparison of simulated and predicted cumulative liquid production rates LiquidRate1500 pooled models including cumulative injection rates Comparison of simulated and predicted cumulative liquid production rates all cases pooled models including water injection rates Comparison of the low, med and high permeability cases simulated cumulative oil production Comparison of the low, med and high permeability cases predicted cumulative oil production Comparison of the low, med and high permeability cases simulated oil recovery factor Comparison of the low, med and high permeability cases predicted oil recovery factor Comparison of the low, med and high mobility ratio cases simulated cumulative oil production Comparison of the low, med and high mobility ratio cases predicted cumulative oil production xxiii

26 7.7 Comparison of the low, med and high mobility ratio cases simulated oil recovery factor Comparison of the low, med and high mobility ratio cases predicted oil recovery factor Comparison of the liquid rate cases simulated cumulative oil production Comparison of the three producing rates constraint with time Comparison of the liquid rate cases predicted cumulative oil production Comparison of the liquid rate cases simulated recovery factor Comparison of the liquid rate case predicted recovery factor xxiv

27 CHAPTER 1 RESERVOIR HETEROGENEITY 1.1 Introduction Heterogeneity, the spatial variation in reservoir properties, is a ubiquitous feature in all naturally-occurring porous media and one of the most important factors governing fluid flow (Lund et al., 1995). Reservoir properties include (but are not limited to) permeability, porosity, thickness, saturation, faults and fractures, rock facies, and rock characteristics (Ahmed, 2006). Reservoir heterogeneity plays an important role in reservoir performance, especially under secondary and tertiary displacements where thief zones or recirculation of injected water become detrimental (Shook and Mitchell, 2009). On the other hand, primary recovery almost is not affected by variations of properties within the reservoir, and a properly averaged, homogenoeus permeability will be sufficient to reveal the reservoir performance, as was proved by Walsh and Lake (2007) in heterogeneous media with crossflow. As such, the accurate prediction of the reservoir properties as a function of spatial location is essential in order to achieve a proper reservoir description and be able to predict ultimate hydrocarbon recovery. 1.2 Definition of Heterogeneity In terms of dynamic responses, heterogeneity is defined as the dispersity of displacement front of flooding process (Lake and Jensen, 1989). Statically, heterogeneity is described as the complexity of flow path and contrast of permeability. The most pronounced form of heterogeneity involves permeability. 1

28 Most heterogeneity measures concentrate on the level of permeability variation in a reservoir, however, variations in other properties, such as porosity, cation exchange capacity, and amounts of clay minerals, often being ignored. This practice usually is justified by two arguments according to Jensen et al. (1997). First, permeability is a property which, by its very nature, has a direct impact upon flow. Consequently, it must be included in any measure of heterogeneity. Second, permeability variations typically are much larger than variations of other properties (i.e., vary by more than an order of magnitude between different strata) (Ahmed, 2006). Hence, changes in permeability can dominate easily the influence of variations in other properties. 1.3 Heterogeneity Classifiers Whatever the reservoir properties involved, heterogeneity measures can be classified into two groups according to Lake and Jensen (1989): static measures and dynamic measures. Static measures describe the distribution in permeability (K) and porosity ( ) of a given model. On the other hand, dynamic measures implicitly include the distribution in flow path length and connected structure of flow paths (Shook and Mitchell, 2009) Static Measures Miller and Lents (1947), Dykstra and Parsons (1950), Schmalz and Rahme (1950) were the first researchers to develop static measures to help quantify the impact of the degree of heterogeneity and predict the ultimate hydrocarbon recovery. Here, we consider three heterogeneity measures: the Dykstra-Parsons coefficient, the Lorenz coefficient, and the coefficient of variation. These measures are used frequently because of the relatively low 2

29 cost to obtain estimates and give some idea about heterogeneity effects in models. All three use permeability data, but porosity data may be included as in the case of the Lorenz coefficient. In addition, these static measures assume that permeability is a scalar quantity Dykstra-Parsons Coefficient, V DP A measure of permeability variability that is widely used in the petroleum industry is the Dykstra-Parsons (1950) coefficient of variation, or V DP. V DP is a statistical measure of non-uniformity of a set of data (Tiab, 2012). In general, it is well recognized that the permeability data is log-normally distributed due to the nature of the geological processes that created permeability, which tend to leave permeabilities distributed around the geometric means (Ahmed, 2006). Dykstra and Parsons recognized this feature and introduced the permeability variation that characterizes a particular distribution (Houseworth, 1991). The common practice in the industry is to apply V DP to the permeability; nevertheless, it can be extended to treat other rock properties. V DP is a dimensionless number that ranges from 0 to 1. A homogenous reservoir has a coefficient of permeability variation that approaches 0, whereas an extremely heterogeneous reservoir has a coefficient of permeability variation that approaches 1. Figure 1.1 shows theoretical log-normal permeability distributions and their 3

30 corresponding V DP. Petroleum reservoirs typically have V DP between 0.5 and 0.9. Figure 1.1: Dykstra-Parsons coefficient plot of typical petroleum reservoirs (Carlson, 2003) Calculation of the V DP involves plotting the frequency distribution of the permeability data on a log-normal probability graph. This is done by arranging the permeability values in descending order and then calculating the percent of the samples with permeabilities greater than or equal to that permeability value. Note that in order to avoid values of 0 or 100%, which are not present on the probability scale, the percent greater than or equal to the value is normalized by N+1, where N is the number of samples (Houseworth, 1991). The data are plotted on a log-normal probability graph. Normally, such a plot gives a straight line, at least when the central 4

31 portion is used. The mid-point of the permeability distribution (%>=50) is the median permeability where K 50 is the permeability value at %>=50, which is the log mean permeability, and K 84.1 is the permeability value at %>=84.1, which is one standard deviation from the mean. Hence, V DP, is defined as: (1.1) The Dykstra-Parsons method is commonly referred to as a permeability ordering technique. It should be noted that if all the permeabilities are equal, the numerator (or Equation 1.1) would be 0, and the V DP also would be 0. This would be the case for a completely homogeneous system (Ahmed, 2006) Lorenz coefficient, L C Another measure of heterogeneity widely used in the petroleum industry is the Lorenz coefficient (L C ). Miller and Lents (1947) introduced the concept of layer flow capacity (kh) and storage (φh) and used them to predict the performance of gas cycling operations. Stiles (1949) and Schmalz and Rahme (1950) built on that and used the cumulative of the flow capacity ( kh) and cumulative thickness ( h) to construct a diagram they identified as a flow capacity diagram. In his study, Stiles (1949) applied this capacity distribution diagram to estimate the oil recovery for a field under waterflooding. Schmalz and Rahme (1950), on the other hand, used the flow capacity diagram 5

32 to compute a single parameter to describe the variations in reservoir properties (i.e., degree if heterogeneity) within a pay zone. Obviously, both studies did not account for the variations in porosity since they only used ( kh) and ( h) (Shook and Mitchell, 2009). The L C of variation is obtained by plotting a graph of cumulative kh versus cumulative φh known as a flow capacity plot. The plot can be used to describe the reservoir heterogeneity quantitatively by calculating the L C (Ahmed, 2006). Figure 1.2: Lorenz Coefficient plot illustrating greater concavity toward upper corner (Ahmed, 2006) The L C ranges between 0 and 1 for a completely homogeneous system and completely heterogeneous system, respectively. Figure 1.2 shows an illustration of the L C plot, which is known as the flow capacity distribution a plot of the normalized Σkh versus Σφh. In the case of a homogenous system where all permeabilities are equal, the plot will give a straight line. However, as the degree of heterogeneity 6

33 increases (as in the case of heterogeneous systems), the curve starts to exhibit greater concavity toward the upper left corner. The L C is defined by the Equation 1.2: (1.2) Even though the L C accounted for porosity, unlike the Dykstra- Parsons coefficient, it has some disadvantages. First, L C is not a unique measure of reservoir heterogeneity because several permeability distributions can have the same value of L C (Martin, 1996). Additionally, like the V DP, the permeability ordering technique of zonation ignores the physical location of the rock and basically would assume that all the values of equal permeability are in communication with each other (Porges, 2006) Coefficient of Variation, C V The coefficient of variation Cv is gaining more attention as a measure of heterogeneity and increasingly has been applied in geological and engineering studies (Jensen et al. 1997, Siddiqui et al. 2006). It is a dimensionless measure of sample variability or dispersion, and it is given by the Equation 1.3: (1.3) where E(k) is the expectation of the permeability k and Var(k) is the variance. 7

34 The unique feature about Cv is that, in the case of data from different populations, the Cv stays relatively constant because the mean and standard deviation often change together. In other words, larger variations in Cv of two different samples would indicate a dramatic difference in the population associated with both samples (Jensen et al., 1997, Martin, 1996) Dynamic Measures It is very obvious that the static estimators fail to describe the permeability connectivity within the reservoir, which, as stated above, certainly would impact recovery. So, to better measure heterogeneity and help in ranking geologic models, a number of dynamic measures have been proposed (Shook and Mitchell, 2009). These measures should be superior to static measures since they most directly characterize flow. Jensen and Lake (1989) classified dynamic measures based on end member displacement. A displacement that has an uneven front is called channeling, and its progress is to be characterized by a Koval factor. If it has an even front, it is dispersive and is characterized by a dispersion coefficient Koval Factor Koval factor is used in miscible flooding to incorporate empirically the effect of heterogeneity on viscous fingering (Koval, 1963). It is defined as reciprocal of the dimensionless breakthrough time in a unit-mobility ratio displacement, as shown by Equation 1.4: (1.4) 8

35 In this Equation, t D is the volume of displacing fluid injected divided by the pore volume of the medium. For a homogenous medium, t D =1 and H k =1. There is no upper limit on H k obviously; large values of H k are detrimental to recovery Dispersion If a miscible displacement proceeds through a 1D homogeneous permeable medium, its concentration at any position x and time t is given by Equation 1.5: [ ] (1.5) The most important parameter in this expression is K l, the longitudinal coefficient. K l has been found to be proportional to v according to Equation 1.6: (1.6) where α l is the longitudinal dispersivity, an intrinsic property of the medium, and v is the mean interstitial velocity of the displacement. Clearly, the degree of spreading depends on the dispersivity; the larger α l, the more spreading occurs Simulation-based Dynamic Measures Another classification of dynamic measures was introduced by Shook and Mitchell (2009). Shook and Mitchell classified dynamic measures as fast simulation, permeability-connectivity estimates, and streamline simulation. Ballin et al. (1992) proposed using a combination of a Fast simulator (FS), which can be a tracer simulation or a simplified flow mode, and a 9

36 comprehensive flow simulator (CS), which can be a full physics finite difference simulator. The method relies on identifying an appropriate set of FS response parameters that are rank-preserving approximations of the CS model. Deutsch and Srinivasan (1996) also used the same idea of FS models to rank alternative Earth models. Even though these two studies showed some success in ranking models, their disadvantage is that they require the model to be upscaled, which might result in unrealistic approximations. Hird and Dubrule (1998) and Ballin et al. (2002) introduced the concept of Resistivity Index (RI) to rank models by estimating the reservoir connectivity. This technique tries to identify least resistive path between an arbitrary grid block and a producer. 10

37 CHAPTER 2 STATIC ESTIMATOR S INCONSISTENCIES IN PREDICTING HYDROCARBON ULTIMATE RECOVERY 2.1 Generation of the Permeability (K) Fields To illustrate the shortcomings of the static measures, multiple realizations of permeability fields were generated with some very wide variations in permeability values. As mentioned earlier, it generally is recognized that the permeability data are log-normally distributed. That is, the geologic processes that create permeability in reservoir rocks appear to leave permeabilities distributed around the geometric mean (Jensen et al., 1997). To create permeability fields that honor such a feature, the LOGNORM.INV function in Microsoft Excel was used. This function is the inverse of the cumulative log-normal distribution function of x for a supplied probability. The different arguments needed to use this function in Excel are probability, mean, and standard deviation, as shown by Figure 2.1. Figure 2.1: Generating permeability fields to conduct the heterogeneity study 11

38 Because I needed to run this for an N number of grids, which is equivalent to the simulation model size (X*Y) and because I also needed to ensure that I could generate multiple realizations for each average permeability and standard deviation, I replaced the probability within this function with the RAND() function in Excel. The RAND() function returns an evenly distributed random real number greater than or equal to 0 and less than 1. A new real, random number is returned every time the worksheet is calculated. For this study, 15 realizations with three different average permeabilties (15 md, 75 md, and 150 md) and 8 different standard deviations were generated. The standard deviation in this function basically is the ln(x) since the intent is to generate a log-normally distrusted function. For example, for the 10x10 grid size alone, the number of multiple realizations of the generated permeability fields is 360 (3*8*15). 2.2 Permeability Distributions Statistical Analysis In order to make sure the randomly generated permeability fields are lognormally distributed, the histogram of each realization was examined. In the following section, the histograms of the different realizations for the 15 md average permeability are presented, as shown in Figures 2.2 to 2.9. Overall, the generated multiple realizations honored the log-normally distrusted feature of permeability distribution that normally is found in petroleum reservoirs. The histograms of the other average permeabilities (i.e., Kavg=75 md and 150 md) exhibits similar trends and are presented in Appendix A. 12

39 Frequency Frequency Texas Tech University, Abdullah Al-Najem, December Histogram of St.dev Kavg=15md Realization#1 Realization#2 Realization#3 Realization#4 Realization#5 Realization#6 Realization#7 Realization#8 Realization#9 Realization#10 Realization#11 Realization#12 Realization#13 Realization#14 Realization# Bins Figure 2.2: Histogram of st.dev0.25 Kavg15md Histogram of St.dev0.5 Model - Kavg=15md Realization#1 Realization#2 Realization#3 Realization#4 Realization#5 Realization#6 Realization#7 Realization#8 Realization#9 Realization#10 Realization#11 Realization#12 Realization#13 Realization#14 Realization# Bins Figure 2.3: Histogram of st.dev0.5 Model - Avg15md 13

40 Frequency Frequency Texas Tech University, Abdullah Al-Najem, December Histogram of St.dev1.0 Model - Kavg=15md Realization#1 Realization#2 Realization#3 Realization#4 Realization#5 Realization#6 Realization#7 Realization#8 Realization#9 Realization#10 Realization#11 Realization#12 Realization#13 Realization#14 Realization# Bins Figure 2.4: Histogram of st.dev1.0 Model - Avg15md Histogram of St.dev1.25 Model - Kavg=15md Realization#1 Realization#2 Realization#3 Realization#4 Realization#5 Realization#6 Realization#7 Realization#8 Realization#9 Realization#10 Realization#11 Realization#12 Realization#13 Realization#14 Realization# Bins Figure 2.5: Histogram of st.dev 1.25 Model Avg 15 md 14

41 Frequency Frequency Texas Tech University, Abdullah Al-Najem, December Histogram of St.dev1.50 Model - Kavg=15md Realization#1 Realization#2 Realization#3 Realization#4 Realization#5 Realization#6 Realization#7 Realization#8 Realization#9 Realization#10 Realization#11 Realization#12 Realization#13 Realization#14 Realization# Bins Figure 2.6: Histogram of st.dev1.5 Model - Avg15md Histogram of St.dev2.0 Model - Kavg=15md Realization#1 Realization#2 Realization#3 Realization#4 Realization#5 Realization#6 Realization#7 Realization#8 Realization#9 Realization#10 Realization#11 Realization#12 Realization#13 Realization#14 Realization# Bins Figure 2.7: Histogram of st.dev2.0 Model - Avg15md 15

42 Frequency Frequency Texas Tech University, Abdullah Al-Najem, December Histogram of St.dev2.5 Model - Kavg=15md Realization#1 Realization#2 Realization#3 Realization#4 Realization#5 Realization#6 Realization#7 Realization#8 Realization#9 Realization#10 Realization#11 Realization#12 Realization#13 Realization#14 Realization# Bins Figure 2.8: Histogram of st.dev2.5 Model - Avg15md Histogram of St.dev3.0 Model - Kavg=15md Realization#1 Realization#2 Realization#3 Realization#4 Realization#5 Realization#6 Realization#7 Realization#8 Realization#9 Realization#10 Realization#11 Realization#12 Realization#13 Realization#14 Realization# Bins Figure 2.9: Histogram of st.dev3.0 Model - Avg15md 16

43 Table 2.1 presents statistical information of one of the realizations for the eight standard deviations. It is very clear from the variances, for instance, that the reservoirs include more variations and become more heterogeneous as the standard deviation increases. Such a conclusion is very important to carry this heterogeneity study in order to make sure a wide range of variations is examined. Table 2.1: Statistical Information of one realization for the 8 standard deviations St.dev. Mean Variance Median Maximum Minimum Lower Quartile Upper Quartile E E E Description of the Streamline Simulation Model A ¼ of five spot pattern streamline model was built using FRONTSIM, the Schlumberger streamline simulator. The base case 3D simulation model is 660 ft by 660 ft by 490 ft with a Cartesian grid of 10x10x13 grid blocks in the x, y, and z directions, respectively. The sizes of each grid block in both x and y directions were uniform and were kept constant at 66 ft; however, the size of the layers in the z direction varied, as shown in Figure In this model, two phases were considered, and both the injector and the producer were completed in the first 10 layers. In addition, the injection and 17

44 production rates were maintained constant at 1000 STB/D. Porosity is 24% and constant, and the vertical permeability is maintained at 10% of horizontal permeabilities (i.e., K x and K y ). The simulation was run for 6000 days (>16 years). Figure 2.10: 2D and 3D Schematics of the Streamline Simulation Model 2.4 Simulation Runs Necessary to Conduct the Heterogeneity Study Figure 2.11 is a flowchart that summarizes the different parts proposed to carry out broadly the heterogeneity impact on hydrocarbon ultimate recovery, develop a dynamic streamline-based index, and develop a streamlines-based multivariate regression model. As can be seen from Figure 2.11, there are three major sensitivity scenarios. These are grid size, mobility ratio, and producer well control. As for the grid size effect, the base case is the 10x10 under which three different average permeability models (i.e., 15 md, 75 md, and 150 md) were generated. Under each one of the average permeabilities, there were 8 different standard deviations; under each standard deviation, there were 15 realizations. The average permeabilities of these different systems were of the ratios of 1 to 2 and 1 to

45 The second sensitivity scenario was the mobility ratio. In this case, beside the base mobility ratio, which is 1.6, another three mobility ratios were examined. Those were 2, 5, and 10. Similar to the grid size case, under each case of the mobility ratio, 8 different standard deviations were investigated. The last sensitivity scenario examined was the producer well control. In this case, beside the base case producer rate of 1000 bbls/d, two different rate constraints were tested. Again, under each of these producer constraint cases, 8 different standard deviations were examined. Based on that, the total number of FRONTSIM models needed was 408 simulation runs. The average simulation runtime of an individual case is approximately 10 minutes. Thus, the total time required to carry out the 408 simulations cases was 4080 minutes, or 68 hours. Figure 2.11: Summary of the simulation runs needed to conduct the heterogeneity study 19

46 2.5 Different Models Dynamic Interactions with Streamlines Streamline models can easily help to visualize how flood fronts evolve and interact with reservoir heterogeneity. Additionally, the geometry and density of the streamlines reflect the impact of geology on fluid paths, providing better resolution in regions of faster flow, as in the case of naturally fractured reservoirs (Datta-Guptta, 2007). Figure 2.12 shows the permeability distribution of one of the models and how it changed as the permeability variations (i.e., standard deviation) increased. The scale in Figure 2.12 was altered to go from 1 md to 500 md, and the gray or black grids are those with very high permeability values. Also, Figure 2.13 shows a snapshot of the streamline interactions with the increase of degree of heterogeneity. A very interesting observation here especially with the high standard deviations models (i.e., high permeability values) is that the streamlines interact with the changes in permeability and tend to break and turn to find the least resistant flow paths. While in the case with little variation in permeability, the streamlines are more-or-less straight lines connecting the injector to the producer. Such information is invaluable because streamlines information, such as TOF or streamlines density, can be used to develop a more accurate model to predict the impact of degree of heterogeneity on the hydrocarbon ultimate recovery. 20

47 Figure 2.12: Snapshot of permeability distribution of the different models (8 standard deviations) Figure 2.13: Snapshot of the streamlines interactions with the increase of degree of heterogeneity 21

48 2.6 FORTRAN Code for Static Classifiers Calculations In order to present the limitations and inconsistencies of the static classifiers, a FORTRAN code was developed to help efficiently and rapidly calculate the Lorenz Coefficient (Lc), the Dykstra-Parsons (Vdp), and coefficient of variations (Cv) of each of the realizations. As stated earlier, having 480 permeability distributions makes it almost impossible to carry out all the tedious calculations involved in calculating these static classifiers. Briefly, the FORTRAN code began by asking for the way of calculating the histogram for the permeability field provided. The histogram within the code can be calculated in terms of bin size or bin number. These histograms are needed as part of the statistical analysis of the multiple realizations. Once done with the histogram, the code sorted the permeability data and then generated the frequency table. Then, it calculated the average permeability (Kavg), geometric mean (Kg-mean), and standard deviation (st.dev) of the realization provided. Once these were calculated, the Cv was calculated. Next, the code called upon a subroutine from the FORTRAN library to help calculate the V DP, which is needed to help find the probability of each permeability, fit a line, and calculate the R 2. These are the procedures needed to calculate the Dykstra-Parsons variable, or V DP. The next estimator that was calculated was the Lorenz coefficient (L C ). The code calculates the Kh and normalized cumulative of Kh and φh. Once that was done, the area under the curve was calculated. Once the area under the curve was known, the code used this information to calculate the L C. 22

49 All of these calculations were outputted and sorted in a file called Results. Once this was completed for one realization, the next realization started, followed by the next, until the last realization specified at start was reached. Then, the code will call gnuplot, a command-line program that can generate two- and threedimensional plots of functions, data, and data fits, to plot the histograms and once this task is completed the FORTRAN code will then END. Figure 2.14 summarizes the different steps of the static classifiers calculations FORTRAN code. Figure 2.14: Flowchart of the heterogeneity classifiers calculations FORTRAN code 2.7 FORTRAN Code Results of the Static Heterogeneity Indices Lorenz Coefficient: L C As mentioned above, the severity of deviation from a straight line in the Lorenz plots indicates the degree of heterogeneity. Figures 2.15, 2.16, and 2.17 show the Lorenz plots of the three different average permeabilities 23

50 Normalized K_h Texas Tech University, Abdullah Al-Najem, December 2013 generated (i.e., 15 md, 75 md, and 150 md). The green line represents the homogeneous system, a completely uniform system where all permeabilities are equal. Because the Lorenz ranges between 0 for a homogeneous system and 1 for a heterogeneous system, the multiple realizations generated cover a wide range of heterogeneity variations. This supported the previously discussed histograms of the different models (i.e., Section 2.2, Figures 4 11) that have a great coverage of permeability variations to help get a better idea about the sensitivity of different factors, such as mobility ratios and production rates on the ultimate hydrocarbon recovery. 1 Lorenz Plot - AvgK=15 md HOMO stdev0.25r1 stdev0.5r1 st.dev1r1 st.dev1.25r1 st.dev=1.5r1 st.dev2r1 st.dev2.5r1 st.dev3r Normalized Phi_h Figure 2.15: Lorenz Plot of the Model Avg15md 24

51 Normalized K_h Normalized K_h Texas Tech University, Abdullah Al-Najem, December 2013 Lorenz Plot - AvgK=75 md HOMO stdev0.25r1 stdev0.5r1 st.dev1r1 st.dev1.25r1 st.dev=1.5r1 st.dev2r1 st.dev2.5r1 st.dev3r Normalized Phi_h Figure 2.16: Lorenz Plot of the Model Avg75md 1 Lorenz Plot - AvgK=150 md HOMO stdev0.25r1 stdev0.5r1 st.dev1r1 st.dev1.25r1 st.dev=1.5r1 st.dev2r1 st.dev2.5r1 st.dev3r Normalized Phi_h Figure 2.17: Lorenz Plot of the Model Avg150md 25

52 As mentioned earlier, Lorenz calculations involve many assumptions. Figure 2.18 shows the Lorenz plot of the 15 realizations generated with an average permeability of 150 md and a standard deviation of 0.5. As can be seen here, the 15 realizations gave different Lorenz plots, which accordingly resulted in producing different cumulative oil productions from each realization, as illustrated in Figure Figure 2.18: Lorenz plot of the 15 realizations of the model with st.dev0.5 and AvgK=150md 26

53 Cum. Oil Production, STB Millions Cumulative Oil Production - AvgK=150 md -15 Realizations (σ=0.5) Time, days R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 Figure 2.19: Differences of cumulative oil production of the model with st.dev0.5 and AvgK=150md Such impact gets worse as the degree of permeability variations increases (i.e., in the case of high standard deviations). Examining the same model but with a higher standard deviation revealed a huge difference in the Lorenz coefficient between the different 15 realizations, as can be seen in Figure Additionally, huge differences exist in breakthrough times and ultimate hydrocarbon recovery, as depicted in Figure For instance, one model had a breakthrough time of about 600 days, while another model had a breakthrough time of about 1600 days a difference of 1000 days, as illustrated in Figure

54 Cum. Production, STB Millions Texas Tech University, Abdullah Al-Najem, December 2013 Figure 2.20: Lorenz plot of the 15 realizations of the model with st.dev2 and AvgK=150md 2.5 Cumulative Oil Production - AvgK=150 md -15 Realizations (σ=2) R1 R3 R5 R7 R9 R11 R2 R4 R6 R8 R10 R R13 R14 R Time, days Figure 1.21: Differences of cumulative oil production of the model with st.dev2 and AvgK=150md 28

55 Cum. Production, STB Millions R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 Cumulative Water Production - AvgK=150 md -15 Realizations (σ=2) Time, days Figure 2.22: Differences of cumulative water production of the model with st.dev2 and AvgK=150md Figure 2.23 shows the cumulative oil production versus the Lorenz coefficients for all 120 realizations at 15 md, 75md, and 150md. A very exciting observation is that the lower average permeability produced higher oil rates compared to the other two higher average permeability models. Additionally, the impact of permeability variations (i.e., increase in standard deviations) was remarkably clear and strong. For example, all 15 realizations of low Lorenz coefficients in all three average permeability models almost produced the same amount of cumulative oil. However, more than 25% of variation in cumulative oil production from each realization was observed in the case of high Lorenz coefficients (i.e., L C of ). 29

56 Figure 2.23: Cumulative oil production vs. L C of the three different average permeability Models (15md, 75md and 150md) Figure 2.24 shows the cumulative liquid production versus the Lorenz coefficient. Here, as expected, the total liquid production of the high permeability models showed higher cumulative productions. Additionally, by examining the cumulative water productions from all the realizations as depicted in Figure 2.25, the increase of permeability variations resulted in water moving faster toward the producer in some areas, leaving some oil behind and eventually resulting in lower ultimate hydrocarbon recovery. 30

57 Figure 2.24: Total liquid vs. L C of the three different average permeability models (15 md, 75 md, and 150 md) Figure 2.25: Cumulative water production vs. L C of the three average permeability models (15 md, 75 md, and 150 md) 31

58 Another interesting observation from these plots is that the models tend to segregate into different classes as a result of the wide permeability variations. Four different classes with different ranges of Lorenz coefficients can be identified from these plots of cumulative liquid, oil, and water productions versus Lorenz coefficients. These are nearly homogeneous (L C < 0.2), slightly heterogeneous (0.2 < L C < 0.3), intermediate heterogeneous (0.4 < L C < 0.7), and highly heterogeneous (L C > 0.7) Dykstra-Parsons Coefficient: V DP The observations found with the Lorenz coefficient are not unique and are true for other static estimators, such as the Dykstra-Parsons (V DP ) and coefficient of variations (C v ) static estimators. As with the Lorenz plots, Figure 2.26 shows the cumulative oil production of the three different models versus V DP. Because of the fewer permeability variations within the lower average permeability model (i.e., K avg =15 md) compared to the higher models, more oil was produced. In addition, the prediction is good for the cases of lower V DP, but it gets very inconsistent as the heterogeneity degree increases. As with the L C, the results tended to segregate into the same four different classes mentioned above. 32

59 Figure 2.26: Cumulative oil production vs. V DP of the three average permeability models (15 md, 75 md, and 150 md) Figure 2.27 shows the cumulative liquid production versus the V DP. The heterogeneity variations resulted in the oil following longer paths to reach the producer, giving the chance for water to reach the producer faster. This resulted in producing higher water volumes from the models with higher permeability variations compared to the lower one, as can be seen in Figure

60 Figure 2.27: Total liquid vs. V DP of the three average permeability models (15 md, 75 md, and 150 md) Figure 2.28: Cumulative water production vs. V DP of the three average permeability models (15 md, 75 md, and 150 md) 34

61 Coefficient of Variations (Cv) As described above, C V is a dimensionless measure of sample variability or dispersion. It is a good measure to consider in this study because of its unique feature: Large changes in C V between two samples indicate a dramatic difference in the populations associated with those samples (Jensen et al., 1997). This is very clear as represented by classes I and II of Figure 2.29, where more scatter is happening compared to the L C and V DP (Figures 2.23 and 2.26). In addition, the results tend to spread more horizontally. This is because C V has no limit. For instance, the generated multiple realizations variations ranged from C V =0.25 to C V =9. Figure 2.29 shows the cumulative oil production versus C V. As with the other two static estimators, higher oil productions were observed for the lower permeability models. In addition, the models tend to segregate into the same classes identified previously. 35

62 Figure 2.29: Cumulative oil production vs. C V of the three average permeability models (15 md, 75 md, and 150 md) Figure 2.30 and 2.31 show the cumulative liquid and cumulative water productions versus C V. As with the other static estimators, more liquid productions are occurring for the higher average permeabilities models but with higher percentage of water compared to oil resulting in some bypassed oil. 36

63 Figure 2.30: Total liquids vs. C V of the three average permeability models (15 md, 75 md, and 150 md) Figure 2.31: Cumulative water production vs. C V of the three average permeability models (15md, 75md, and 150 md) 37

64 Based on these results, it is fair to state that static classifiers are inconsistent, failed to predict accurately the hydrocarbon ultimate recovery, and are inadequate to account for the complex and nonlinear interrelationships among permeability, pressure, and changing fluid saturation within porous media. Hence, a dynamic estimator that implicitly includes fluid distribution in flow path and its connected structures is needed to more accurately predict the impact of heterogeneity on ultimate recovery. 2.8 Problem Statement The full description of geological systems requires knowledge of the unknown spatial distribution of various static parameters and their impact on reservoir heterogeneity and hydrocarbon ultimate recovery. Generally, this knowledge tends to be unattainable due to the localized nature of sources of information, such as cores and well tests. Various static and dynamic estimators have been employed in an attempt to quantify heterogeneity. The Dykstra-Parson (V DP ) and Lorenz (L C ) coefficients are the two most widely used static parameters for the quantification of reservoir heterogeneity. The static heterogeneity classifiers are inadequate to account for the complex and nonlinear interrelationships among permeability, pressure, and changing fluid saturation within porous media. On the other hand, the Koval factor and dispersion are limited to miscible displacements (i.e. Channeling and Dispersive). Introducing some simulation-based dynamic measures, such as fast simulation, permeability connectivity estimators, and streamlines simulation, helped in this regard. Even though the first two classes of the simulation-based 38

65 dynamic measures showed some success in assessing heterogeneity, the first class which is the fast simulation require some scaleup; the permeability connectivity estimator is limited to the depletion studies because it cannot account for the additional dynamics resulting from injection. On the other hand, streamline simulation has gained wide attention during the last two decades. Computation of streamlines is fast, has minimal numerical dispersion, and yields direct relationships between time-of-flight (TOF) and dynamic data, such as production and tracer breakthrough curves. Therefore, there must be a good relationship between the streamlinesderived variables and the heterogeneity degree within the reservoir and its impact on hydrocarbon ultimate recovery. As such, multivariate regression models using streamline information, such as TOF and streamlines-density will be developed to assist practicing engineers to better quantify impact of reservoir heterogeneity on ultimate recovery and also to predict saturation changes during primary or enhanced recovery periods. 39

66 CHAPTER 3 STREAMLINE SIMULATION AS AN ALTERNATIVE 3.1 Summary Streamline and streamtube methods have been used in fluid flow computations for many years. Early applications for hydrocarbon reservoir simulation were first reported by Fay and Pratts in the 1950s. Streamline-based flow simulation has made significant advances in the last 15 years. Today s simulators are fully three-dimensional and fully compressible, and they account for gravity and complex well controls. Most recent advances also allow for compositional and thermal displacements. Streamline simulation has undergone several phases within its short stretch in the petroleum industry. Initially, the main focus was on the speed advantage and less on fluid flow physics. Next, the focus shifted to extend its applicability to more complex issues, such as compositional and thermal simulations, which require the inclusion of more physics and potentially reduce the advantage of computational time. Recently, the focus again has shifted towards the application of streamline technologies to areas where it can complement finite difference simulation, such as revealing important information about drainage areas, flood optimization and improvement of sweep efficiency, quantifying uncertainties, etc. 3.2 Theory and Background about Streamline Simulation Streamlines are integrated curves that are locally tangential to a defined velocity field at a given instant in time (Datta-Gupta, 2007 & Thiele et al,. 2010) as illustrated in Figure 3.1. Modeling fluid flow and transport using streamlines dates 40

67 back to the study of well patterns and total recovery by Muskat and Wyckoff in A comprehensive historical overview of earlier streamlines work can be found in Batycky (1997), Datta-Gupta and King (1998), Thiele (2001), Moreno et al. (2004), and Datta-Gupta (2007). Figure 3.1: Streamline tracing and velocity vector mapping in a saturation grid (Ibrahim et al. 2007) Streamline simulation is an implicit pressure explicit saturation (IMPES) type of reservoir simulation that solves the pressure Equation implicitly and then solves the saturation/conservation Equations explicitly. Thus, streamline simulators operate on the principle of de-coupling the pressure Equation from the saturation Equation. This simplification allows a heterogeneous 3-D domain to be decomposed into a number of 1-D streamlines where all fluid calculations are carried out. 41

68 Figure 3.2: Generalized flow diagram for streamline simulation (Gerritsen M., 2008) Figure 3.2 is a generalized flow diagram for a streamline simulation. The 3- D grid is first initialized starting with input data such as grid geometry, rock and fluid properties, well locations, injection rates, and boundary conditions. Then, using finite difference (FD) approximations, the pressure distribution is derived to generate instantaneous velocity vectors perpendicular to the computed pressure contours. The velocity field is then used to trace the streamlines. The specified wells (i.e., source or sink) and boundary conditions govern the initiation and termination of all traced streamlines, which, once established, make up the second grid system needed to help solve the fluid flow Equation along the streamlines. The second grid system works under a local timestep controlled by fluid movement computations, making it a time-variant grid. The solution obtained from the second grid then is mapped back onto the original 3-D Cartesian grid to account for fluid phase distribution and saturations. For subsequent cycles, the pressure is solved again, and the streamlines are redrawn. The process continues until the end of the 42

69 simulation. This dual grid approach distinguishes streamline simulators from conventional finite difference simulators. This simulation process involves many different mathematical calculations on both grid systems in order to solve the pressure and transport Equations. The following references thoroughly explain the mathematical formulations: Batycky et al. 1997, Ingebrigsten et al. 1999, Doi and Suzuki 2000, Lolomari et al. 2000, Gautier et al. 2001, Jessen and Orr Jr. 2002, Di Donato and Blunt, 2003, Moreno et al. 2004, Gerristen et al. 2005, Mallison et al. 2006, Cheng et al There are several advantages to employing streamline simulations for modeling fluid flow. One of the main advantages that attracted researchers to streamline technology in the first place was its computational speed. Due to their 1- D nature, transport calculations are not constrained by grid instability, allowing for larger timestamps and minimizing numerical diffusion. Batycky (1997) and Osako and Datta-Guptta (2007) reported that the streamline technique exhibits a nearlinear scaling of the CPU time and was faster than FD simulations by factors of 1 to 3 orders of magnitude, especially for large models (i.e., > 10,000 grid blocks), as can be seen in Figure

70 Figure 3.3: Scaling of CPU time: streamlines vs. finite difference (Osako & Datta-Gupta, 2007) Another advantage is the visualization potential offered by streamline simulations. For example, visualizing the source-sink relationships based on streamline density can help obtain a quantitative flow indicator. This visualization is extremely useful in optimizing waterfloods/gas floods because the benefits of injection can easily be quantified over time, as can be seen in Figure 3.4. Figure 3.4: Streamlines capturing sweep/drainage areas associated with injectors and producers (Marco, 2001) Additionally, this powerful aspect of streamline visualization, with the help of the quantitative flow indicators, can assist in identifying fluid loss to wells outside a pattern and then balancing different patterns using well allocation factors (WAF) between injectors and producers. Figure 3.5 shows a simple case of using 44

71 streamline information to help re-balance rates. To optimize injection efficiency of each injector, the rates are changed from one time to another until even distribution of streamlines associated with injectors or producers are obtained. Figure 3.5: Streamline information aids in re-balancing rates in different patterns (Marco, 2001) These are just some of the advantages of streamline methods. It is highly recommend that the reader refers to AlNajem et al. (2012) for attaining a better idea about the different advantages of streamline technology and also the wide range of petroleum engineering applications that symbolize the relevance and validity of streamline simulation in addressing reservoir engineering concerns. Like FD simulation, streamline simulation has its limitations. Two important ones are mapping between coordinates and modeling of fluid flow complex physics. Streamline simulation contains two separate grids: an underlying physical grid where the pressures and the velocities are calculated and the streamline time-of-flight (TOF) grid where the fluid transportation is calculated. Streamlines are re-generated at each pressure update. This means that the saturations from the old set of streamlines must be mapped back to a new set of streamlines. Streamlines transport saturations rather than conserved volumes, which are defined only implicitly in the TOF coordinate. Because of the re-sampling of 45

72 the implicit volume from TOF coordinates to physical coordinates, potential mass balance errors and, to some extent, numerical dispersion may be introduced. Another major limitation of streamline simulation results from its main advantage defined above: computational speed. When dealing with complex physics like high compressibility, capillary effects, and phase behavior, the computational speed decreases. This is due to the need for more frequent resampling of streamlines, which means more frequent solving of the pressure Equation. Streamline-based flow simulation has made significant advances in the last 15 years. Today s simulators are fully 3-D and fully compressible, and they account for gravity, fracture flow, and non-uniform conditions as well as complex well controls. Most recent advances also allow for compositional and thermal displacements. 3.3 Literature Review of Streamlines Technology In this section, we present a comprehensive review of the evolution and advancement of streamline simulation technology. This section offers a general overview of most of the material available in the literature on the subject. This work includes the review of more than 200 technical papers and gives a chronological advancement of streamline simulation technology from 1996 to present. Firstly, three major areas are identified. These are the development of streamline simulators, enhancements to current streamline simulators, and applications of streamline simulators. In view of the fact that this state-of-the-art technology has been employed for a wide range of applications, this literature 46

73 review defines four major application areas that symbolize the relevance and validity of streamline simulation in addressing reservoir engineering concerns. These are reservoir characterization, reservoir management, history matching, and upscaling and ranking of geological models Historical Perspective A thorough review of the early development of streamline simulation technology is available in Batycky (1997) and Datta-Guptta (2007). In this section, the current state of streamline simulation and their applications is placed in a historical perspective. The main advantages of streamline early methods that attracted early researches like Fay and Pratts (1951) and Higgins and Leighton (1962) were freedom from numerical dispersion and computational speed associated with a slowly varying velocity field. These methods employed a 1-D analytical solution to fluid flow (such as the Buckley-Leverett solution), which were mapped along each streamline. Using 1-D analytical solutions is very attractive in certain engineering applications, such as the comparison of a standard FD simulation with dispersion-free solutions, ranking and screening of reservoir models, and understanding the effects of phase behavior and heterogeneity in compositional simulation. Notable work that employs analytical streamline includes Thiele et al. (1996), Peddibhotla et al. (1996), and Jessen and Orr (2002). A very important drawback of using analytical solutions is that they cannot be applied to flow simulation of real fields since they cannot account 47

74 properly for gravity, changing well conditions (such as infill wells), and nonuniform initial conditions. To account for these effects, numerical solutions along streamlines are required, combined with operator-splitting techniques (Batycky et al., 1997) Development of Streamline Simulators Tremendous developments have been made in geostatistical methods to help develop multi-million 3-D fine scale static models that are conditioned to geophysical data. This has introduced outstanding challenges to any reservoir simulation method to predict accurately and realistically the flow performance of any recovery processes. Those challenges can be resolved by either advances in computer hardware or by improvement to the simulation method (Batycky, 1997). Although impressive, current computing technology limits researchers from simulating such multi-million-cell models on practical time-scales. Recent developments in streamline simulations offer significant potential to meet some of these challenges (Datta-Guptta, 2007). Using standard computer resources, streamline simulation can model field scale displacements of large models both faster and more accurately than FD methods (Batycky, 1997). The following is a literature review of the work in which fully functional streamline simulators has been developed. To follow the traditional categories of flow simulators, the literature in this section was divided into three groups, namely black oil simulation, compositional simulation, and thermal simulation. The review within this 48

75 section suggests that today s streamline simulators are fully 3-D, fully compressible, and account for pressure-dependent properties such as gravity and capillarity and complex mechanisms such as dual media, dissolution, geochemical reactions and non-isothermal flow Black Oil Simulation Batycky (1997) and Batycky et al. (1997) were the first to develop a 3-D, two-phase streamline simulator to model field scale phenomena such as heterogeneity, gravity, changing well conditions, and non-uniform initial conditions using numerical solutions along streamlines. Tracer, waterfloods, and first-contact miscible (FCM) displacement comparisons showed speed-up factors up to three orders of magnitude over FD methods. As a matter of fact, due to the reduced numerical diffusion, FCM and tracer displacements showed more accurate results. Moreover, for a million gridblocks waterflood displacement solved on a standard workstation, the streamline method agreed very well with the FD method, which took two CPU days to run, even though it was upscaled 16 times. Jang et al. (2002) and Jang et al. (2006) introduced an advection dispersion ratio into the streamline simulator to model dispersive transport flow effectively. This approach was applied to transport in an artificially fractured sample, and results showed that tracer breakthrough curve matched the the experiment. Additionally, the authors tested the allocation of various advection-dispersion ratios 49

76 to different streamlines, and results showed that a representative advection-dispersion (AD) ratio is applicable when the AD ratio is large. On the other hand, various AD ratios should be allocated to each streamline when AD ratio is small. As an attempt to take advantage of the recently developed operator splitting algorithms, Rodriguez et al. (2003) developed a full 3-D streamline simulator for two-phase incompressible flow including capillary pressure but used the operator splitting technique to separate capillary and gravity terms from the convective terms. This treatment was faster than using FD, but it made the streamline method slower than the cases where capillary forces are not considered. By modifying the pressure and saturation Equations, Berenblyum et al. (2003) modified a streamline simulator to include capillary and gravity effects. The new simulator was applied to several simulation cases. The results showed that capillary forces stabilize the displacement front and increase oil production even for strong viscous-dominated cases. Di Donato (2003) developed a streamline-based dual porosity simulator to model waterflooding in fractured reservoirs. The results and run times of the streamline-based approach and FD simulator give similar answers when using the same transfer function, but they were at least one-to-two orders of magnitude faster for models with 100,000 to 1,000,000 grid blocks. Similarly, Moreno et al. (2004) developed a 50

77 3-D, incompressible streamline reservoir simulator for dual-porosity reservoirs that accounts for capillary, gravity, and transfer functions. The simulator was used to model several water-oil displacement test cases, and comparisons with a commercial FD dual-porosity model were favorable, considerably reducing CPU time and numerical dispersion compared to traditional FD. The authors believed that results could be significantly improved when using a finer FD grid for the calculation of the velocity field. By improving the discretatizaion along the streamlines and employing an adaptive-implcit (AIM) formulation, Thiele et al. (2004) extended the work by Di Donato (2003) to complex geological grids with multiple relative permeability regions. Using this approach, numerical tests showed more channeling along high permeability fracture regions, giving earlier breakthrough times as compared FD simulators. However, the run time was nine times faster than FD. Kozlova et al. (2006) developed a mathematical model for a three-phase compressible dual porosity streamline simulator. The solution algorithm used sequential approach, and saturation Equations are solved along each streamline by a standard upwind finite difference method. The streamline-based approach showed good agreement with the FD simulator but with an order of magnitude improvement in simulation time. Cheng et al. (2006) generalized streamline models to compressible flow by introducing an effective density along 51

78 streamlines, incorporating a density-dependent source term in the streamline saturation Equation, and using this information to incorporate cross-streamline effects via pressure updates and remapping of saturations. The approach was tested on a highly heterogeneous carbonate reservoir in West Texas with over 30 years of production history. The proposed formulation resulted in a significant improvement in performance prediction compared to commercial streamline simulators. Obi and Blunt (2006) modeled CO 2 sequestration into a one million grid model of the North Sea aquifer using a streamline-based simulation that incorporates advection, dissolution, and porosity and permeability changes. The results showed that the advective transport of CO 2 was dominated by high permeability channels resulting in low sequestration efficiencies. Qi et al. (2007) modified the work by Obi and Blunt (2006) by incorporating trapping and permeability hysteresis correlation in order to improve CO 2 storage in North Sea aquifers. The simulation results suggested that a combination of brine and CO 2 injection can lead to optimal storage efficiency by first maintaining stable displacement and also inject approximately 25% of the mass of CO 2 stored during chase water injection. The authors believed that this approach would enable at least 90% of the CO 2 to be trapped. 52

79 In cases where dual porosity cannot be employed, a Darcy- Stokes compound single porosity model is needed to make Darcy-flow region, which obeys Darcy s law, and a free-flow region, which satisfies a Navier-Stokes flow. Peng et al. (2007) developed a twophase, slightly compressible streamline-based Darcy-Stokes model and successfully applied it to the prediction of flow behaviors in the Tahe reservoir having meter-sized cavities. The streamline based Darcy-Stokes model resolved the convergence issue seen in other methods and greatly reduced the computational time. Beraldo et al. (2007) extended a 3-D incompressible streamline simulator to model oil composition variations by including an API tracking option. Using SPE 10th comparative Solutions Project, numerical results showed that API tracking used within streamline simulators reduced run time while giving similar results as the FD method. Using the compressible formulations proposed by Cheng et al. (2006) and Osako and Datta-Gupta (2007), Beraldo et al. (2008) extended his previous work to compressible cases to model accurately oil-quality variation. Instead of time-of-flight, the authors used cumulative streamtube volume as the distance coordinate for transport solution along streamlines. The numerical tests showed that this implementation produced results similar to those of FD at different levels of compressibility, and it was considerably faster than available commercial FD and streamline simulators. 53

80 To improve the accuracy and efficiency of solute transport modeling using streamline simulation, Jeong et al. (2008) developed a 2-D streamline simulation model using a semi-analytical solution considering advection, dispersion, sorption, and decay and complex boundary conditions such as the Dirichlet boundary, the first-order degraded boundary, and the Cauchy boundary. The differences of plume spreads between SL models using semi-analytical solution and SL using numerical solution were so clear. CPU comparisons showed that the developed model is times faster. Thiele et al. (2008) extended a streamline simulator to model field-scale polymer flooding. The SL was tested using 1-D, 2-D, and 3-D displacement cases, and it compared favorably with the FD simulator but ran ten times faster. AlSofi et al. (2009) extended a streamline simulator to handle polymer flooding with Newtonian and non-newtonian behavior. The results of a 2-D heterogeneous model for unconditionally stable flooding showed that injection requirements doubled, and a shear-thinning fluid decreased recovery by 2% compared to a Newtonian fluid. To determine the impact of volumetric strain variations on average reservoir pressure and production rate profiles, Rodríguez de la Torre (2010) developed a two-way explicit approach for coupling streamline simulator and geomechanics. The performance of the approach was problem-dependent; the more complex the models are, 54

81 the larger the geomechanical response is. The author believed that powerful keywords in streamline simulators are needed to account for changes in porosity and permeability with pressure in order to measure geomechanical response after each timestep rather than at the end of the simulation. To investigate the effects of CO 2 and aqueous-phase densities on the migration extent for a long time scale after injection, Takasawa et al. (2010) developed a streamline-based 3-D model incorporating dissolution, geochemical reaction, porosity, and permeability change due to precipitated solids from the reaction and gravity segregation in open-boundary aquifers. The model was used successfully to simulate 20 years of CO 2 injection and 200 years of fluid migration in 3-D homogeneous and heterogeneous aquifers of different pressures and temperatures Compositional Simulation The first 3-D, two-phase compositional streamline simulator was developed by Thiele et al. (1997), where he used a 1-D compositional FD simulator to move components numerically along streamlines and then map the 1-D solutions back onto an underlying Cartesian grid to obtain a full 3-D compositional solution. This approach was tested on a 3-D heterogeneous model with 518,400 gridblocks and 36 wells for a condensing-vaporizing gas drive with four components. The run time was three days, which is the same 55

82 amount of time the FD simulator took to simulate the upscaled 28,800 gridblock version of the problem. The streamline method described by Thiele et al. (1997) assumes the pressure to be constant throughout the global timestep, which would result in inaccuracies when extended to three phases. So, Crane et al. (2000) developed a 3-D, three-phase compositional streamline reservoir simulator that uses 1-D fully implicit solutions of FD simulations to solve for pressure and fluid composition together along each streamline and account for most of the physical effects that depend on the changes in pressure. This SL simulator was tested extensively, and comparison against FD compositional simulators showed that it is significantly faster and requires less memory since it uses larger timesteps and because it did not need to invert very large matrix systems associated with a fully implicit, large 3-D multicomponent compositional simulator. Most compositional streamline calculations performed to-date have used numerical simulations for solutions along streamlines. Analytical solutions were limited to only a small number of components. A breakthrough algorithm was developed by Jessen et al. (2000) to help obtain analytical 1-D solutions with an arbitrary number of components. This breakthrough helped Jessen and Orr (2002) to employ these analytical 1-D solutions in a streamline simulator to evaluate displacement performance of miscible gas injection processes 56

83 in a heterogeneous porous media. The 1-D two-phase solution allows any number of components to be present in the injected gas as well as in the reservoir fluid. The comparison of 2-D and 3-D cases of a commercial FD compositional simulator, streamlines combined with a dispersion-free analytical 1-D solution and streamlines combined with a dispersed numerical 1-D solution showed this approach to produce more accurate recovery free from numerical dispersion in addition to increased speed factors of about two and three orders of magnitude. Jessen and Orr (2004) presented an approach for implementing gravity effects in compositional streamline simulators. The method was demonstrated to conserve mass. Its application added only marginally to the overall CPU requirement. The new approach was demonstrated to be in excellent agreement with commercial FD simulators for prediction of flows in 2-D vertical and multi-well 3-D geometries. The analytical solution described above is only applicable to constant initial and injection conditions. For displacements effected by gravity, where components move in directions not aligned with the streamlines (like in walter-alternating-gas (WAG) processes), the assumption cannot be expected to provide accurate results. In order to accurately and rapidly model miscible WAG processes that involve considerable compositional exchange, Yan et al. (2004) developed a 3- D, three-phase compositional streamline simulator for the WAG 57

84 processes. The comparison of 2D and 3D numerical tests with a commercial FD compositional simulator showed good agreement, while the SL was much faster with increased speed factors increasing with the simulation model size. Osako (2006) extended two- and three-phase black oil and compositional streamline formulations to compressible flow using relative density for total fluids along streamlines, a density dependent source-term in streamline saturation/composition conservation Equation and cross-streamline effects via pressure updates and remapping of saturations. The results on 20 years of waterflooding in the Goldsmith San Andres Unit (GSAU) showed that this approach closely follows results from an FD simulator. In addition, usage of an optimal coarsening method for selection of the number of segments along streamline for 1D solution for compositional cases resulted in a significant reduction of computational time while maintaining close agreement with FD results. Along the same line of speeding up the compositional streamline simulator and, due to some asymptotic properties of the gas-liquid flow in particular, to a high difference between phase mobilities Oladyshkin et al. (2007) developed an asymptotic streamline compositional simulator that works on a total splitting between the thermodynamics and hydrodynamics (HT splitting), leading to an analytical or semi-analytical solution for the 58

85 multicomponent flow problem along the streamlines. The numerical tests on a nine-component mixture gas condensate flow showed that a commercial FD compositional simulator required two days, whereas the asymptotic simulator took less than two minutes (ratio 1:1500) Thermal Simulators To minimize the grid orientation effects associated with unfavorable mobility ratios of hot water displacement, Pasarai and Arihara (2005 & 2007) developed a thermal streamline simulator for hot waterflooding that incorporates effects of compressibility, nonisothermal flow, and physical diffusions of gravity, capillary, and conduction. To examine the accuracy and efficiency of the simulator, they investigated two modeling approaches for solving mass and heat transport Equations. In comparison with a commercial thermal simulator, the first proposed approach was much faster and relatively grid insensitive, but lower accuracy was observed. On the other hand, the results of the second approach, including full physics, were more accurate but less efficient. Zhu et al. (2010) extended a streamline simulator to include thermal effects of temperature-dependent viscosity and thermal compressibility for two-phase hot waterflooding. This thermal streamline simulator was tested on 2D heterogeneous quarter of fivespot problems, and it produced accurate results at a much lower computational cost than a commercial FD thermal simulator. By 59

86 implementing pressure updating along streamlines and Glowinskischeme operator splitting, Zhu et al. (2011) extended the work by Zhu et al. (2010) to steamflooding to account for large volume change, strong gravity override, and high couple of energy and mass transport. The authors demonstrated the viability of this work using several 2D cases, and results showed accurate solutions compared to finite volume thermal simulators but with less computational overheads Extensions and Improvements of Streamline Simulators Many researchers have applied streamline methods to bridge the gap between fine-scale geologic models and flow simulation in unfavorable mobility ratios displacements. As a result, a common belief was established that streamline simulation mostly is effective for large models with convection-dominated flow, where the recovery mechanism is dominated by heterogeneity effects instead of complex physics processes such as compressibility, capillary imbibition, gravity segregation, and compositional effects (Datta-Gupta, 2007). As explained in the preceding section, experience in industry does show that streamline technology has evolved considerably with the inclusion of such complex physics processes and also has gained better recognition by the industry realizing the great benefits and wide applicability of streamline models in addressing such complex mechanisms. This section is divided into two parts. The first part highlights ongoing research to overcome limitations of streamline technology and to improve the accuracy and efficiency of the mathematical calculations along streamlines. 60

87 The second part explores the tremendous efforts that have been accomplished in regards to employing streamline method to help with highly distorted grids while, at the same time, overcoming the challenge of precise tracing of streamlines. The literature within this section has observed great work in terms of improving the accuracy and efficiency of the streamline method, which includes (but is not limited to) better mappings to and from streamlines, optimized timestep selection and streamline coverage, and irregular time-offlight based saturation Improving Streamline Mathematical Calculations Ingebrigtsen et al. (1999) implemented two different approaches to handle pressure and saturation couplings for calculations of three-phase compressible flow using streamlines. The first method was a sequential IMPES-type method where additional steps were included for reducing mass discrepancy error. The second method was an implicit method where both pressure and saturation were solved simultaneously along the streamlines. The numerical tests showed that both methods compared well with results of the FD simulator, but the implicit method was significantly slower. Guevara-Jordan and Rodriguez-Hernandez (2001) combined the singular valued decomposition (SVD) algorithm and the classical method of images to determine the streamline distribution in a sectional homogeneous, one-phase and incompressible reservoir with an arbitrary number of permeability regions and shapes. The numerical 61

88 examples showed that the new method has substantial reduction in CPU times in comparison with the finite and boundary element techniques. Osako et al. (2003) proposed an approach for timestep selection during streamline simulations including the aligned and transverse flux terms into the streamline formulations, a grid based corrector algorithm to update the saturation to account for the transverse flux, and a discrete CFL (Courant-Fredrich-Levy) formulation to ensure numerical stability. The authors tested this approach using homogeneous and heterogeneous quarter of five-spot patterns at various mobility ratios. The results demonstrated the impact of the transverse flux correction on the accuracy of the solution and on the appropriate choice of timesteps across a range of mobility ratios. Mallison et al. (2004) extended the compositional streamline simulation to improve mappings to and from streamlines that are necessary to obtain reliable predictions of gas injection processes. The authors tested this improvement using a simple model for miscible flooding based on incompressible Darcy flow. Results indicated that the mappings offer improved resolution and reduced mass balance errors relative to the standard mappings. The mappings also required fewer streamlines to achieve a desired level of accuracy. Matringe and Gerritsen (2004) used the mixed hybrid finite element method (MHFEM) for solving the pressure field to improve 62

89 streamline tracing results over the traditional FD method. The authors also presented two new methods to improve streamline tracing within grid cells. The first is an adaptive mesh refinement-inspired tracing method that provides more accurate tracing where needed. The second is a higher order variation of Pollock's (1988) tracing that uses a second order interpolation of the velocity field. Both methods proposed are flux conservative and efficient. Jimenez et al. (2005) investigated transverse spatial errors resulting from the number or placement of streamlines by providing an analytic proof and a numeric demonstration of the order of spatial convergence of the mass balance discretization error. The authors also provided a simple and exact means of calculating the time-of-flight for arbitrary corner point cells, or unstructured grids, in 2D or 3D for either compressible or incompressible flow. Using this new time-offlight formulation, a series of cross-sectional finite difference simulations were analyzed to identify grid orientation errors in the numerical calculation of flux and spatial error. Matringe et al. (2005) developed a fully adaptive streamline method that optimizes streamline coverage to promote local streamline adaptation. To determine what streamline density is needed in different regions, an indicator based on Krigging mapping developed by Mallison et al. (2004) was used. The numerical tests showed that 63

90 this approach was able to coarsen or refine the streamline grid efficiently according to the error indicator. To reduce the complexity of numerical efficiency and avoid grid distortion and mapping, Wang et al. (2006) integrated a twophase, multi-component compositional streamline method with Eulerian-Lagrangian formulation to provide a mass-conservative framework through solving the pressure and transport Equations on the fixed mesh. The authors presented numerical experiments to compare with the widely used upwind method for compositional flow. The results indicated that the Eulerian-Lagrangian formulation generated stable and accurate solutions that resolve moving steep fronts and are physically reasonable, even when using a timestep two orders of magnitude higher than that used by the upwind method. Likewise, Kippe et al. (2007) proposed a modification to the standard mapping algorithm of streamlines to improve the mass-conservation properties of the method and provide high-accuracy production curves using few streamlines. This new method was verified using the Model 2 from the 10th SPE Comparative Solution Project. They generated highly accurate production curves compared to the reference solutions in less than 10 minutes using one processor on a standard desktop computer. Andrianov and Bratvedt (2007) accelerated the saturation transport using adaptive mesh refinement (AMR) along streamlines. 64

91 The authors studied the performance of the resulting multi-resolution scheme on a synthetic and real reservoir models. It was shown that the use of the AMR technique provided up to a five-fold acceleration. Osako and Datta-Gupta (2007) built on the work of Cheng et al. (2006) and used the concept of effective density to redefine the bistream functions and decouple the 3D conservation Equations to 1D transport Equations for overall compositions along streamlines. The authors also proposed an optimal gridding strategy for efficient solution of 1D compositional Equations along streamlines. This technique was benchmarked using a series of numerical experiments in a 2D five-spot pattern and a 3D heterogeneous case with gravity effects. The results highlighted the importance of rigorous treatment of compressibility effects. Podgornova et al. (2007) proposed an explicit analytic approach to PVT calculations in the framework of compositional streamline simulators to reduce the time required for the flash calculations. The authors compared the technique developed with an FD commercial compositional simulator. It was shown that the combination of high-contrast geology and analytical thermodynamic PVT approach led to a significant increase of CPU efficiency. By employing local boundary layer (LBL) refinement construction, which allows examining each cell face in isolation from other cell faces and other cells, Jimenez (2008) proposed a method that 65

92 produces consistent representation of streamlines and velocities near faults and non-neighbor connections. Comparisons of this method and the standard Pollock algorithm on field scale models with complex fault juxtapositions and several non-neighbor connections revealed that LBL construction clearly depicted the slippage in streamline trajectories to account for flux discontinuities at the fault surface, while the standard Pollock algorithm either terminates the trajectories or results in incorrect tracing across the fault surface. Nourozieh et al. (2008) developed a method to move streamline-based saturation forward using irregular time-of-flight space instead of the common method of transforming the irregular time-of-flight space onto a regular space, which results in the averaging of saturations and the introduction of numerical dispersion. A comparison of the two techniques using a synthetic waterflood model showed that this method reduced the smearing of the front, gave more accurate breakthrough times, and minimized numerical dispersion. By assuming thermodynamic equilibrium along streamlines, Lazaro-Vallejo et al. (2011) extended a three-component, two-phase streamline simulator to include two new phase behavior algorithms. The authors applied this method to the Stage 2 CO2CRC Otway project, and similar residual trapping results were obtained as an FD simulator but with a reduction of CPU time by 80%. 66

93 Flow-Based Grids and Complex Cell Geometries The first attempt to use streamlines to generate grids was demonstrated by Verma and Aziz (1996). They used arbitrary selected streamlines and y-axis parallel lines at equal intervals as gridblock boundaries. Using the natural property of streamlines in providing a concentration of gridlines in high velocity regions, Edwards et al. (1998) developed a grid generation technique based on streamlines and equi-potentials in order to optimize nodal placement. The comparison of this technique with the conventional Cartesian grids showed that this method improved accuracy, gave close results similar to the fine Cartesian grids, and drastically reduced the number of gridblocks used. Prévost et al. (2001) showed how to extend streamlines method to curvilinear corner point geometry (CPG) grids and unstructured triangular grids while employing a consistent full tensor finite volume scheme in each case. The authors tested this technique using a quarter of a five-spot model with non-orthogonal and triangular grids and for another case of a curved-boundary reservoir. The results demonstrated the benefits of the new streamline tracing algorithms and showed that the resulting velocity field is divergence-free in the physical space for each case. Building on that work, Prévost et al. (2003) introduced a local post-processing for numerically calculated fluxes, leading to a consistent and flux-continuous piecewise constant representation for 67

94 the velocity. A comparison of a homogeneous cubic domain with 27,000 structured grid streamlines simulation (Batycky et al., 1997), and 1,728 control volumes for unstructured simulations showed excellent agreement, indicating the accuracy of unstructured streamline simulation. Along the same line, Matringe et al. (2005) proposed a unified formulation for high-order streamline tracing on unstructured quadrilateral and triangular grids based on the use of the stream function. The authors tested this new streamline algorithm using the mixed finite element method (MFEM) on an isotropic and homogenous quarter of a five-spot and on a heterogeneous system with fully unstructured grids. In both cases, the results showed that the high order discretization and tracing method produced better, smoother physical streamlines, which are less sensitive to the simulation grid. This work was extended by Matringe (2006) to 3D grids. In this extension, the velocity field is interpolated from the multipoint flux approximation (MPFA) sub fluxes using mixed finite element velocity shape functions, and the streamlines are then integrated to arbitrary accuracy. Two main advantages were noticed using this method. First, the streamlines were more accurate; second, direct interpolation of velocity from the MPFA sub fluxes avoided the expensive flux postprocessing. 68

95 To identify high flow regions and to select appropriate points on streamlines as grid nodes, Gherabati et al. (2008) introduced a new method of grid coarsening that encompassed the tracing of streamlines from the reservoir boundary to the well and monitored the velocity trend along streamlines. The generated grid pattern was finer in high flow regions and could adapt itself successfully based on the type of geological features presented. The pressure was the main criterion used for comparing the results, and it was found that the response of the model with coarse grids was more consistent with a fine model. As a continuation of previous work, Matringe et al. (2008) proposed a streamline tracing method for the reconstruction of the velocity field with high-order accuracy from fluxes provided by MPFA discretization schemes on general triangular or quadrilateral grids. This reconstruction relies on the correspondence between MPFA fluxes and the degrees of freedom of a mixed finite-element method (MFEM) based on the first-order Brezzi-Douglas-Marini space. The numerical experiments showed that in the presence of misaligned anisotropic permeability or grid distortion, higher-order velocity reconstruction based on BDM1 recovers more accurate streamlines and TOF than existing methods with no additional computational cost. Zhang et al. (2011) developed a streamline tracing algorithm for unstructured polygonal cells using cell face fluxes and minimum velocity variance (MVV). The authors evaluated eight different 69

96 velocity interpolation schemes used in the literature using single cell models and a 2D quarter five-spot case. The authors believed that the MVV with quadrilateral refinement is the only method that satisfies local conservation, has an analytical solution, uses only cell boundary flux information, and generates satisfactory streamline tracing results for the test cases. Figure 3.6 presents a timeline that summarizes the development and enhancement of streamline simulators during the time from 1996 until present and also how this technology has progressed and evolved over time. Figure 3.6: Timeline that summarizes the development and enhancement of streamline simulators Petroleum Engineering Applications of Streamline Simulation State-of-the-art streamlines technology has been employed for a wide range of applications. In this section, we defined three major application areas that symbolize the relevance and validity of streamline simulation in addressing reservoir engineering concerns. These are history matching; 70

97 reservoir management; and upscaling, ranking, and characterization of finegrid geological models. Under each main category, different subareas have been identified in order to better represent the specific application of streamline simulation. Figure 3.7 summarizes the main categories and also the subcategories under them. Figure 3.7: Summary of the main streamline technology applications categories The following sections highlight the work that has been done under each category and its sub-categories History Matching History matching is an inverse problem that is known to be highly nonlinear and ill-posed, producing a set of non-unique solutions (Subbey et al., 2003, Maucec et al., 2007, Stephen et al., 2007, Kazemi and Stephen, 2010). Obtaining a reliable history match typically requires numerous flow simulations. One major benefit of streamline simulation is that, due to its efficiency, many more flow simulations 71

98 can be evaluated in a given time compared to standard FD simulators. As a consequence, many authors have employed streamline simulation in history matching algorithms that require hundreds or thousands of flow simulations (for example, Subbey et al., 2003). Beyond the speed of the flow simulation, there are many additional benefits that streamline simulations provide to history matching, many of which have been described already in other reviews: for example, Datta- Gupta (2007), Stenerud and Lie (2007) and Thiele et al. (2010). In this section, many of the diverse applications of streamline simulation to history matching are reviewed, ranging from the simple use of streamline-delineated drainage zones in traditional history matching practices to sophisticated use of streamline information for data integration. I. Delineation of Drainage Zones One significant benefit of streamline simulation applied to history matching is that the streamlines identify instantaneous flow directions and drainage regions of the producing wells, which are often very complex and varying throughout the lifetime of a reservoir. When a producing well does not match its historical data at a particular time, the streamlines can associate the mismatch with the petrophysical properties in a particular drainage zone. Several authors have employed this association in various history matching methods, which are described below. 72

99 Emanuel and Milliken (1998) described a history matching methodology that uses flow paths described by 3D streamlines to assist in changing reservoir parameters. Two examples involving more than 60 wells were described, which showed its power and utility. In both cases, very satisfactory well-by-well history matches were obtained after just a few iterations. Using dynamic streamline-derived drainage zones, Hoffman and Caers (2003) proposed a regional probability perturbation method (RPM) for modifying geostatistical realizations in a geologically consistent manner. The methodology is an extension of the global perturbation method (GPM), and it relies on a region-wise perturbation of the probability model. Tests using a two-facies (sand and shale), fractured reservoir model with three fault blocks showed that the RPM converged faster and showed better matches for pressure and water-cuts than GPM. To facilitate the choice of regions-of-interest to history match, Maschio and Schiozer (2004) used streamline simulation to determine flow paths among producers and injectors or between an aquifer and a producer. They combined this with another two methods namely, the global history match and the regions-ofinterest history match. The proposed approach was applied to an offshore field with 22 years of production history. The results 73

100 showed the use of the regions of main flow path determined by streamlines provided better results. Devegowda et al. (2007) examined two techniques for covariance localization using streamline-specific information to mitigate small sample sizes and non-linear dynamics in non- Gaussian settings problems associated with the use of Ensemble Kalman Filter (EnKF). The first approach utilizes streamline trajectories to delineate regions that will the have largest influence on solution. The second approach uses streamline-based sensitivities, which will be discussed below. The proposed methods of covariance localization are shown to exhibit superior performance vis-à-vis the traditional form of the EnKF demonstrated with synthetic examples and real field cases. Kazemi and Stephen (2010) investigated a history matching technique by focusing on sub-volumes of the parameter space and used streamlines to help choose where changes are required. The streamline-guided approach gave a 70% improvement in history matching from the base model and around a 40% reduction of misfit in prediction. In a more sophisticated example, Watanabe and Datta-Guptta (2011) defined streamline zones per fluid phase order to improve the traditional distancebased localizations used in EnKF. The discontinuities in water and gas phase streamlines were used to define flow relevant covariance 74

101 localization for water-cut and GOR data assimilation during EnKF updating. The authors compared both methods using two cases, and results showed that phase streamlines-based localization outperforms traditional distance based localization in terms of capturing low permeability barriers affecting gas flow and high permeability streaks contributing to aquifer supports. II. Direct Methods Streamline-based history matching techniques can go beyond the ability to delineate drainage zones. One set of examples is described in this subsection as direct methods. Direct methods were developed first by Wang and Kovscek (2000) and employ the fractional flow curves at the producing wells and relate them to changes effective permeability (and/or porosity) values for history matching. This approach was benchmarked with 2D incompressible cases where the effect of gravity was not important and illustrated the computational efficiency, generality, and robustness of the proposed procedure. The approach was extended by Agarwal and Blunt (2001) and Agarwal and Blunt (2003) who developed a history matching method for water cut data using a commercial, compressible three-phase streamline simulator with gravity. Data from a North Sea field was used to test this full-physics technique, which gave a reasonable history match and a good prediction of future 75

102 performance. Lately, Batycky et al. (2007) applied a similar method to a large waterflood. de Hann et al. (2001) and Kretz et al. (2004) proposed similar approaches for history matching of time-lapse seismic. de Hann et al. (2001) used time lapse 3D seismic saturation contours to determine the velocity of the flow between saturation contour pairs. By applying Darcy s Law to this velocity field, a realistic permeability map could be inverted. The authors tested this technique on five randomly created, cases and it allowed identifying different zones of high and low permeabilities. Kretz et al. (2004) modified a given set of flow properties (e.g., porosity, permeability, cell volume, etc.) for history matching observed fluid fronts obtained from 4D seismic. The approach was tested using several 2D cases. The matching of fluid fronts needed few iterative steps leading to a more active use of 4D seismic in reservoir management. III. Optimization Methods Gradient-based optimization methods require the calculation of sensitivity coefficients. By drawing a correspondence between streamline modeling and ray tracing in seismology, Vasco et al. (1998) developed a method for analytically calculating the sensitivity coefficients of the time-offlight (TOF) with respect to the gridblock petrophysical properties. 76

103 The TOF can then be related to the breakthrough of injected fluids or tracer. The fact that the sensitivity coefficients are calculated analytically greatly enhances the efficiency of the optimization procedure. Because the sensitivity calculations involve evaluation of 1D integrals along streamlines, it scales very well with respect to model size (Qassab et al., 2003, Cheng et al., 2006). This breakthrough development has led to a significant body of research and applications on history matching, some of which are described below. He et al. (2001) developed a generalized travel time inversion method for production data integration that minimizes a travel time shift derived through maximizing a cross-correlation between observed and computed production responses at each well. The travel time sensitivities were computed analytically using a single streamline simulation that accounts for gravity and changing field conditions, and the minimization is relatively insensitive to the initial model. Starting with well logs and seismic data, the Goldsmith San Andres Unit (GSAU) in the West Texas field example water-cut history was integrated in less than two hours using a simple PC. Gautier et al. (2001) extended the previous work of Vasco et al. (1998) to compute sensitivity coefficients of pressure, saturation, and production data in two-phase flow and non-unit 77

104 mobility ratios for varying boundary conditions. The results showed that the objective function gradient was non-monotonic when sensitivity coefficients for saturation only were considered. In contrast, when only pressure is considered, the objective function becomes smooth and converges fast. Qassab et al. (2003) applied the streamline-based production data integration method proposed by He et al. (2001) to a giant Saudi Arabian carbonate reservoir to condition a multimillion-cell geologic model in order to match 30 years of historical production responses. For this field, production data integration was carried out in less than six hours on a PC, and the geologic model derived was found to be consistent with field observations. Vasco et al. (2003) developed a trajectory-based streamline approach to invert time-lapse seismic amplitude and travel time to image reservoir flow properties, such as permeability, by computing and matching TOF. The application to actual time-lapse data from 40 years of historic production of the Bay Marchand field in the Gulf of Mexico indicated that the method was robust, even in the presence of noise. By defining a generalized travel time inversion (GTTI), Cheng et al. (2004) proposed an approach to history match FD models using streamline trajectories, TOF, and parameter 78

105 sensitivities in an inversion algorithm to update the reservoir model during FD simulation. The approach was applied to a synthetic example as well as two field examples with multimillion cell geologic models and with 30 years of production history. The entire history matching of the field example took nine hours on a PC. Al-Harbi (2004) extended a streamline model to treat fracture and matrix as separate continua connected through a transfer function. In addition, the author computed reservoir properties versus production response sensitivities in naturally fractured reservoirs (NFR) using the GTTI. These derived sensitivities were applied in conjunction with a dual porosity FD simulator to help incorporate the compressibility effects and the complex physics. Al-Huthali and Datta-Gupta (2004) and Al- Huthali and Datta-Gupta (2004) developed a streamline-based technique to describe fluid transport in NFR through a dual-media approach. This technique was tested on five- and nine-spot patterns, and results showed close agreement with recovery and saturation profiles with a marked reduction in numerical dispersion and grid orientation effects. Using the double-continua formulation developed by Al-Huthali and Datta-Gupta (2004) and flow paths produced by a streamline simulator, Lino (2006) developed a simulation model for tracer response analysis in NFR. The model 79

106 was validated using heterogeneous permeability distributions and a multiple injection/production scheme including mega-fractures. The saturation fronts and tracer responses were less smeared than those obtained from a commercial FD simulator and had improved depiction of channeling flow. Vargas-Guzmán et al. (2009) used streamline-based GTTI to condition efficiently and rapidly a one million cells prior geological model with measured dynamic data and to interpret spatial distributions of high permeability zones. The permeability modifications statistical analysis revealed that marcopores and fractures are the main causes of high permeability zones. Additionally, the existence of localized high permeability fracture zones was associated with specific rocks and high curvature regions. Stenerud et al. (2007) combined multiscale streamline methods with the GTTI method for history matching highresolution geologic models. The approach helped to derive a fast quasi-linear method, demanding a few iterations, and enforce basis functions to be updated adaptively only in areas with relatively large sensitivities to production response. This approach was tested using a highly detailed 3D geomodel consisting of more than one million simulation cells and 69 producers, and history matching of 80

107 a seven-year period was accomplished in less than 40 minutes on an ordinary PC. Using streamline based water-cut (WC) and GOR sensitivities, Oyerinde et al. (2007) developed a three-phase flow history matching method that involves simultaneous GTTI inversion of WC, GOR, and flowing BHP data. This approach was examined using a highly faulted, West African reservoir with over 30 years of production history. The results of WC and GOR match illustrated the effectiveness of this method. Devegowda et al. (2007) used streamline-based sensitivities for covariance localization applied to EnKF. Efendiev et al. (2008) used streamline-based sensitivities generated through a single streamline simulation to provide analytical approximations to a two-stage Markov Chain Monte Carlo (MCMC) approach used for the sampling of a permeability field. The numerical tests showed a several-fold increase in the acceptance rate and significant reduction in computational time required for MCMC sampling. To depict the movement and trapping of CO 2 in the aquifer, Rey et al. (2010) developed a streamline tracing algorithm using component fluxes generated by an FD compositional simulator. The authors also used a compositional streamline simulator to determine the sensitivity of the time-lapse seismic attributes; 81

108 specifically, they interpreted saturation differences to changes in reservoir properties such as permeability and porosity. These sensitivities were then used in an inverse modeling algorithm to calibrate the geologic model to time-lapse seismic data. IV. Data Integration Methods A separate approach, which is distinctive from the optimization methods described above, fall under the category of data integration methods. These methods use the same streamlinespecific information used for the direct methods, but instead of applying the history matching modifications directly to the grid block properties, the information is incorporated into the geomodeling process. As such, these methods are termed geologically-consistent. The first example of this approach is described by Caers et al. (2002), who adapted the method of Wang and Kovscek (2000) to modify the geological model while at the same time maintaining the spatial correlation structure of the grid block properties. Similar approaches were proposed by Le Ravalec-Dupin and Fenwick (2002) and Caers (2003). These authors developed a procedure that combines a streamline-based history matching method and a geostatistical parameterization technique to estimate the distributions of porosity and permeability in heterogeneous petroleum reservoirs. A special feature of the suggested 82

109 methodology is that a single fluid flow simulation can be run for a group of reservoir model perturbations. This property makes the entire history match process computationally efficient. Gross et al. (2004) and Fenwick et al. (2005) employed a combination of geostatistical tools to relate production data to petrophysical properties at multiple scales. The methodology was applied for history matching a giant Saudi Arabian carbonate oil reservoir with over 500 producers and with over 50 years of historical data. The permeability corrections provided by the streamlines helped achieving noticeable history matching improvements. Gross (2006) developed a statistical technique to establish consistency across field data such as facies type, permeability, porosity, and net-to-gross by obeying inter-dependence revealed using streamline simulators. The technique was tested using a 10,000 grid block, 2D model with 16 producers and 9 injectors. The results showed satisfactory match of water-cut and pressure without destroying geological consistency Reservoir Management Managing a water and miscible flooding requires an understanding of how injectors displace fluid to producers (Grinestaff, 1999). A key element to a successful flood is good sweep efficiency, which can be significantly impacted by reservoir heterogeneity 83

110 resulting in areas of water cycling and poor sweep as flood matures (Izgec et al., 2010).Traditionally, most flood management has been restricted to static allocations or sensitivity studies centered on finite difference simulation (Thiele and Batycky, 2003, Batycky et al., 2005, Ghori et al., 2007, Izgec et al., 2010). Static allocations, which use reservoir properties and neighboring well distances, have been used by engineers to quantify injector to producer relationship. However, such approach did not permit the evaluation of changes in injection profiles, water influx and cross flow over time (Grinestaff, 1999). Likewise, detailed finite difference simulation can provide some answers related to injectors/producers connectivity but computational times and data extraction makes well level decisions on a daily basis too expensive (Grinestaff, 1999). As an alternative, streamline simulation is an ideal tool to help optimize large floods because it provides efficient and fast means to capture detailed fluid movements resulting from conformance control (vertical and/or areal by mechanical means), infill drilling, well conversion and pattern realignment (Grinestaff, 1999, Thiele and Batycky, 2003, Izgec et al., 2010). Thus, total flow rates and phase rates between well pairs can be calculated. This important information can help engineers identify areas of extreme cycling, patterns with poor sweep, or local voidage imbalances (Thiele and Batycky, 2003, Batycky et al., 2005, Singh et al., 2010). 84

111 Following is a literature review of the work in which streamlines have been applied to optimize flood management and hence allow engineers to focus on understanding reservoir and well behavior rather than reacting to it (Grinestaff, 1999). In this section, four areas have been identified namely sweep efficiency, rate optimization, well placement and enhanced oil recovery. The review period has observed intense efforts by researchers especially during the last three years to fully and widely implement streamlines technology in addressing reservoir management issues. A very interesting observation resulting from exploiting streamlines in reservoir management is the introduction of several streamline based reservoir engineering indicators as a result of the powerful aspects of streamlines visualization. Some of those are time-of-flight to the producer (TTP), Predominant Streamline Fields (PSF), Weak Streamline Fields (WSF), Strength Index of Streamline Field (SISF), depletion capacity maps, flow-based well allocation factors, streamline-based flood efficiency maps and variations of TOF distribution maps. I. Sweep Efficiency Chakravarty et al. (2000) conducted a study of the high permeability, high recovery (over 70%) Saladin (Barrow Group) reservoir (offshore Western Australia) using 3D Streamline Simulation to investigate any remaining potential prior to 85

112 abandonment. A very good history match was achieved in a short time and results indicated two distinct areas of significant bypassed oil and drilling in one of recommended target areas has yielded significant recovery of bypassed oil. Grinestaff and Caffrey (2000) used a high-resolution 3D streamline model to study regional and individual wells performance of the Northwest Fault Block (NWFB) area of Prudhoe Bay, Alaska. Streamline modeling provided accurate results of vertical and area wide sweep efficiencies and water cycling. Injection was hence reduced by 30-40%, resulting in increased production and cost savings. By examining the time-of-flight to the producer (TTP), Sharif and MacDonald (2001) used streamline simulation to evaluate flow behavior and production performance of 105 scenarios of a 4.2 million blocks geologic model. By observing evolution of drained volume as TTP increases, the results showed that wells drained most of swept volume for TTP<1,000, after which increase in drained volume is not significant. Additionally, between TTP of 1,000 and 10,000, presence of pockets of additional drained volume was identified. Thiele and Batycky (2003) described an automated approach to optimize injection and production well efficiencies in a waterflood using streamline-based flow simulation. By knowing 86

113 injection efficiencies or well allocation factors (WAF) across the field for each injector, water can be reallocated from lowefficiency to high efficiency wells. Liu (2010) introduced Predominant Streamline Fields (PSF) and Weak Streamline Fields (WSF) by statistically analyzing the Strength Index of Streamline Field (SISF), which is a function of both allocation factor and swept volume of specific well pair at a given time step. The results of testing this on the Gudong field, China, indicated that PSF behavior was consistent with a lot of other history data, which helped EOR and infill drilling decisions in this field. Using a streamline tracing algorithm, Al-Zawawi et al. (2011) enhanced a FD simulator, to generate streamlines. The authors presented two cases in which number of injectors was reduced without impacting overall injection efficiency in the first case while injection efficiency was improved by optimizing well location in the second case. To assess the impact of scale precipitation deep in the reservoir on scale management economics, Akuanyionwu and Wahid (2011) used a commercial streamline simulator to model insitu scale precipitation in the Green field (North Sea) and tracked scaling dynamic concentration ions present at the high-risk producer area. Because streamlines showed a drop in scaling ion 87

114 concentration (sulfate) in some areas as the water moves toward the producers, frequencies of estimated scale squeeze treatment were highly reduced and thus, eliminated the need of building a sulfate reducing plant. II. Rate Optimization Grinestaff (1999) demonstrated the ability of streamline simulation, dynamic injection pattern allocations to proactively manage waterflood patterns through time. For a major field simulated using this modeling tool, over 15% of total flood throughput was removed by lowering injection rates leading to a 75% reduction in total fluid rate with no loss of oil production. Guimarães et al. (2005) developed an automated optimization methodology combining three tools: traditional simulation, streamline simulation and quality map. By applying efficiency of injector well (Ef) function obtained from streamline simulation, water injection and allocation of new wells were more reliably controlled. The proposed methodology was applied to an offshore field with water injection and results showed that efficient control of fluid flow allowed the best production strategy. Thiele and Batycky (2006) employed streamline information to develop the concept of Injector Efficiency (IE). They developed a heuristic optimization procedure which rebalances injection volumes from inefficient to efficient injectors, 88

115 thus improving oil production. A similar procedure was proposed by Ghori et al. (2007) who utilized streamline simulation to optimize and set injection targets for 160 water injectors supporting 520 oil producers in a Saudi Arabian field. Using predefined independent segments established from streamline patterns, rates in efficient injectors were increased and rates of inefficient injectors were decreased. This proposed method increased the injectors' efficiencies by 10% and overall efficiency of each segment by 5%-7%. Rodriguez et al. (2008) used streamlines technology in conjunction with IPR target per segment, reservoir pressure distribution and maximum water injection plant facilities to effectively manage water injection in a giant field. The new approach improved injection efficiency for all segments simultaneously in a single run. To give a better measure of an injector s true effectiveness, Batycky et al. (2008) developed a streamline based flood surveillance method using flow-based well allocation factors (WAF). By comparing results of this approach against full historymatched flow simulation models of several field cases, the authors confirmed that only first order flow effects such as well locations, historical rates, major geological features are required to build such model. 89

116 Using a penalized misfit function for optimization, Taware et al. (2010) proposed a streamline based approach for computing sensitivity of injection and production rates in smart wells, given multiple geologic models, to maximize oil recovery and NPV. The results of 2D and 3D cases showed that as the acceleration component was increased, the proposed arrival time optimization approached the NPV optimization. Alhuthali et al. (2010) computed the sensitivity of arrival times with respect to well rates using streamlines. The approach was tested on the Brugge field and a super-giant Middle Eastern field containing vertical and horizontal wells and completed with ICVS. The results showed increased cumulative oil production while substantially decreasing associated water production. Izgec et al. (2010) and Izgec et al. (2011) used streamline simulations to derive Hydrocarbon F-Φ Curve and Hydrocarbon Lorenz Coefficient (LC-HC) and used them for optimum waterflood management. The authors tested this approach using both synthetic data and field data. The speed, flexibility to start optimizing at any arbitrary time regardless of history in addition to ability to handle large problems illustrated the power of this method. Bostan et al. (2011) utilized streamline simulations to optimize well allocation factor of an undersaturated oil field in Iran 90

117 with an area of 10 km 2. Based on streamlines information, using same amount of water, injection efficiency was improved by about 7% in 3 years. Su et al. (2011) used two streamline optimization algorithms to optimize 60 years waterflooding of the Burgan Field in Kuwait. The first optimizes injection efficiency based on total production rates and the second is utilizes remaining oil ranking, in which streamline bundles with more remaining oil get more injection. Both methods were very effective and oil gains were about the same for the first 25 years. However, the authors believed that the second method would result in higher ultimate recovery based on the comparison between some wells that were not completely shut-in when used remaining oil ranking algorithm. By equalizing average time-of-flight amongst producing wells using analytically computed weighting factors of injection and production rates, Park and Datta-Gupta (2011) proposed a waterflood rate optimization workflow using streamline-based flood efficiency maps. The 60,000-grid block Brugge field model and a CO 2 pilot project in the Goldsmith San Andrea Unit (GSAU) were tested. The results not only showed optimized flood rates but confirmed the practicality of evaluating injection efficiency using the variations of TOF distribution maps. 91

118 III. Well Placement To select new wells best locations, understand drainage volume and speed up process, Dehdari et al. (2008) employed two tools: compositional simulation using different property maps, and streamlines simulation for optimizing one of Iranian oil fields. This methodology helped in defining several scenarios to maximize oil production and the streamline technique improved recovery by 22 MM bbls compared to the conventional methods. Singh et al. (2010) used a streamline model with Singular Value Decomposition technique (SVD) to estimate flood front movement in the mature Assam oil field. The authors demonstrated the calculation of breakthrough time; streamline distribution and areal sweep efficiencies, sensitivity of solution to number of boundary points and number of image wells for four cases. Such information resulted in identifying fresh areas for infill drilling. A comparison of this technique with FD technique showed a remarkable similarity. Sayyafzadeh et al. (2010) used streamline density information to understand fluid movement within the reservoir and to develop a best practice for future developments such as infill drilling or converting dead producers into injectors. Following this approach, the simulation results showed that the ultimate oil 92

119 recovery increased by 70%, which compares favorably with the EOR processes by requiring less investment and operating cost. Sehbi et al. (2011) optimized number of hydraulic fracture stages in tight gas reservoirs by examining the evolution of gas streamlines derived drainage volumes as a function of number of fracture stages. The application to the Cotton Valley formation showed that fractures in the middle have considerably less streamline density compared to outer fractures in an indication of the shielding of the inner fractures by the outer fractures. Ten hydraulic fracture stages with a fracture half-length of 500 ft was found to be the optimal completion strategy for the horizontal well studied. Kang et al. (2011) optimized well placement in a naturally fractured tight gas reservoirs by examining the evolution of the gas streamlines derived well drainage volumes. The application to a tight gas field in the Rocky Mountain region showed that presence of natural fractures tends to enhance well drainage volumes and accelerate production depending upon distribution and orientation of fractures. Additionally, defining undrained volumes helped introducing a "depletion capacity" map for optimization of infill locations. 93

120 IV. Enhanced Oil Recovery Maure et al. (2001) utilized streamline simulation to infer inter-well connectivity and allocation factors in Victor Claro reservoir layers C2-D2, La Ventana field, northern Cuyo Basin in the Mendoza Province of Argentina to understand the microbial enhanced oil recovery (MEOR) mechanisms and improve biotreatments. The results showed that the original oil production decline rates improved and the two combined mechanisms helped explain the changes. Seto et al. (2003) developed a streamline compositional simulator to model enhanced condensate recovery by gas injection. The authors compared streamline and FD results for 2D and 3D cases and the results showed that numerical dispersion associated with compositional FD simulation resulted in optimistic recovery predictions while compositional streamline simulation reduced these effects, thus producing more realistic predictions. In addition, speed-up factors of two to three orders of magnitude were observed. Li (2004) extended UTCOMP streamline simulation (Strator) to model four-phase flow for a nine-component injection of the combination of Prudhoe Bay Gas (PBG) and Natural Gas Liquid (NGL) into a reservoir containing a 12-component viscous oil (Shrader). Results indicated that Strator was many 94

121 times faster than FD simulator. In addition, the viscous fingering was clearly observed during streamline simulation while displacement front was very blurry during FD simulation due to high numerical dispersions. Bhambri and Mohanty (2005) and Bhambri and Mohanty (2005) developed a four-phase streamline module that works with an existing FD simulator to study gas or WAG injections in medium viscosity reservoirs. The authors tested this module using 2D and 3D homogenous quarter of five-spot models with vertical wells. Results showed that four phases exist near gas-oil displacement front and sweep efficiency increases with WAG ratio. Bhambri (2007) developed black oil and compositional streamline modules with parallel option to study large scale waterflooding, gas injections, and WAG processes in low temperature reservoirs. The comparison with FD revealed close agreement with a speed up factor of 12 for a 10,000 grid cell 2D waterflood simulation. However, water saturation maps showed some instabilities and oscillations for very big time steps (100 days) and its resolution is affected by number of streamlines launched. By using a data reduction algorithm to reduce number of calls to the Rieman solver, Juanes and Lie (2005) and Juanes and 95

122 Lie (2008) developed a framework that employ streamline method, a front tracking algorithm and the analytical Riemann solver to efficiently simulate first-contact miscible (FCM) gas injection processes. The Authors tested this framework using the 10th SPE comparative solution project and demonstrated that WAG scheme recovery efficiency was higher than waterflooding or gas injection alone. To delineate the advantages and limitations of streamline simulations for high mobility ratio waterfloods, Osako et al. (2009) used streamline simulation to predict the performance of mechanistic and field-scale heavy oil waterflood models for oilwater viscosity ratio range of 10 to Besides showing similar results as FD, streamline showed less grid orientation effects and computational advantage for large-scale field model. However, for viscosity ratios of ten or higher, SL generated instabilities and the results were sensitive to pressure update. In the domain of polymer floods, in addition to the work of Thiele et al. (2008), and AlSofi et al. (2009), Clemens et al. (2010) used streamline based well pattern to efficiently manage polymer injection projects. Based upon comparison of the cumulative oil produced as a function of polymer injected,, the authors introduced a new metric: the polymer injection efficiency as a function of time for each pattern. The simulation results from the Romanian field 96

123 tested showed an anticipated recovery increase of 5.3% of OOIP compared to the base waterflood case. Nielsen et al. (2010) developed a two-phase microbial enhanced oil recovery (MEOR) compositional streamline simulator with gravity effects to study oil displacement by water containing bacteria, substrate, and surfactant. This streamline simulator was found to produce similar results as FD simulator for 1D and 2D displacements with increased and faster oil recovery Upscaling, Ranking, and Characterizing Fine-Grid Geological Models Recent enhancements in software, hardware, integration expertise, data collection and interpretation have improved the industry's ability to build detailed 3D large multi-million cell geologic models of the subsurface (Idrobo et al., 2000). Due to their computational time and costs, upscaling is needed to efficiently and reasonably translate detailed grids to coarser grids without compromising the anticipated reservoir performance (Chawathé and Taggart, 2001). Equally, there exists a need to rank a large number of generated reservoir model realizations to quantify the potential sources of errors or uncertainties. For both tasks, streamlines method is especially appealing. The streamline time-of-flight provides a powerful means of visualization. By comparing the time-of-flight from the fine scale detailed geologic model with that from an upscaled model, the validity of upscaling techniques are examined in addition to 97

124 level of upscaling (Datta-Gupta, 2007). Additionally, a unique feature of streamlines method is its ability to efficiently compute the sensitivity of the production data to reservoir parameters such as porosity and permeability providing a great potential for considering a large number of plausible reservoir descriptions (Efendiev et al., 2008). Following is a literature review of the work in which streamlines has been applied for bridging the gap between fine scale detailed geological models and flow simulation models through providing a judicious upscaling technique and appropriate ranking criteria to select a subset of equi-probable reservoir realizations. I. Upscaling Portella and Hewett (1998) developed an upscaling method that generates a 2D or 3D coarse grid, based on streamtubes. It also calculates upscaled relative permeability curves, or pseudofunctions using semi-analytical streamtube method. The results from streamtube coarse grid were in very good agreement with the fine grid solution. By defining impermeable media based on increased tortuosity of streamline flow path, Doi and Suzuki (2000) applied a streamline method for upscaling vertical permeability of a Middle Eastern heterogeneous limestone reservoir. The final effective permeability showed that the streamline method was effective as it 98

125 was able to match the simulated formation pressure with the historical data. Chawathe and Taggart (2001) used 3D streamline simulation for vertically and areally upscaling a 0.6 million-cell sector of a gas condensate field on the West coast of Africa in order to better understand flux distributions. A high-permeability streak seen in the static geological model was clearly identified in addition to discovery of approximately three more relatively prolific layers. By combining the methods of minimizing variation and flux, manual judgment and streamline simulation, Nair and Al- Maraghi (2006) applied a hybrid upscaling approach to the 157- layer geological model of a carbonate reservoir in Kuwait to generate a vertically upscaled model with 42 layers. The streamline simulation results showed that flow paths are directly related to the overall heterogeneity and water arrival time was closely related to the presence (or removal) of high permeability pathways. The streamline model results were comparable with the fine-scale model. II. Uncertainty Quantification and Ranking of Geological Models To rigorously accounts for interaction between flow field and underlying heterogeneity, Idrobo et al. (2000) proposed a 99

126 connectivity criterion based on streamline time-of-flight and used it to rank geostatistical realizations. For the test example on North Robertson Unit field, West Texas, 50 permeability realizations were ranked based on volumetric sweep efficiency and the percent recovery and the swept volumes at different times were clearly consistent with the ranking. By accounting for gravity, fluid contacts, changing streamlines, fractional flow effects, choice of well locations, rates, boundary conditions, and patterns, Gilman et al. (2002) described a practical process for dynamically ranking various geologic realizations for a mature field with 60 years of historical production using uniform well spacing and streamline simulation. The streamline simulation gave a 50% variation in future oil recovery using a uniformly spaced, 40-acre 5 spot pattern. Kovscek and Wang (2005) employed a streamline simulator as a proxy for uncertainty quantification of CO2 sequestration. Streamline simulation allowed for a selection of a representative subset of equi-probable reservoir models that encompass uncertainty with respect to true reservoir geology. In a similar approach, Acosta (2010) used a streamline simulator to rank 20 equi-probable geostatistical realizations for a model with 1.8 million active cells in the Ceuta-Tomoporo field in West 100

127 Venezuela by finding the least cumulative errors of the historical and simulated oil productions and the water-cuts. Scheidt and Caers (2009) used streamline simulation as a distance measure of dissimilarity between reservoir models. Using clustering and distance measures, their approach identifies a small subset of reservoir models for standard flow simulation. The uncertainty in future production was captured accurately when simulating fewer than 10% of the total number of reservoir models. Odai and Ogbe (2011) utilized average breakthrough time (ABT) derived from a 3D streamline simulator for ranking equiprobable realizations. Tests indicated that high ABT very reasonably correlated with cumulative waterflood recovery. III. Characterization of Reservoir Heterogeneity To quantify the uncertainties of small-scale geological features, Hastings et al. (2001) proposed a semi-analytical streamline method based on a new solution for the Buckley- Leverett flow in heterogeneous media with varying relative permeabilities. The method assumed that larger scale heterogeneities (faults, channel bodies, etc.) define the streamlines geometry but small scale properties (e.g., cross-bedding) governed the flow along them which was validated using a fluvial reservoir cross-section. The ability to model contrasts in relative permeability helped accurately reproduces the main features of the 101

128 production curve with smoother profile compared to a commercial streamline simulator. Pranter et al. (2005) used streamline simulations to assess the variability of petrophysical properties on fluid flow of the dolomite outcrops model of the Mississippian Madison formation at Sheep Canyon, Wyoming. Results indicated that as the component of the long-range periodicity increases, a corresponding increase in breakthrough time and sweep efficiency occurs. Additionally, heterogeneity caused by lateral petrophysical cyclicity should be incorporated for hole effect magnitudes that are greater than 10% of the petrophysical variance. To quantify intermediate-scale heterogeneity of fluvial reservoirs, Pranter et al. (2007) conducted single phase, 2D and 3D streamline simulations on the fluvial point-bar outcrop model, Williams Fork Formation, Colorado to compare relative effects of lithology, grain size, and petrophysical heterogeneity on breakthrough time (BTT), reservoir volume swept at BTT, and sweep efficiency. For 2D models, results showed that lithologic heterogeneity has a greater effect on recovery efficiency than petrophysical heterogeneities. However, the 3D models showed that lithologic distribution has a more significant effect on BTT and sweep efficiency compared to perophysical distribution if well patterns oriented perpendicular to paleoflow direction. 102

129 Using streamline TOF and volumetric flow rate information, Shook and Mitchell (2009) developed a new method for estimating heterogeneity in Earth models. This method was tested using 450 models that were constructed using a wide range of Dykstra-Parsons coefficient and correlation lengths, and two different well patterns. This study showed that the Lorenz coefficient as determined from dynamic data is the single best measure of heterogeneity. 103

130 4.1 Introduction CHAPTER 4 STREAMLINES MATHEMATICAL FORMULATIONS What we have just explained involves a lot of mathematical calculations on both grid systems in order to solve pressure and transport Equations. Here is a list of some of references where the mathematical formulations have been thoroughly explained and covered. These include but not limited to (Batycky et al., 1997, Ingebrigsten et al. 1999, Doi and Suzuki 2000, Lolomari et al. 2000, Gautier et al. 2001, Jessen and Orr Jr. 2002, Di Donato and Blunt, 2003, Moreno et al. 2004, Gerristen et al. 2005, Mallison et al. 2006, Cheng et al. 2006). 4.2 Governing IMPES Equations: Pressure Equation The streamline method is an implicit pressure and explicit saturation (IMPES) formulation with the pressure field solved for implicitly and the oil/gas/water saturations solved for explicitly. The governing Equation for pressure, Ρ, for multi-phase, incompressible flow without capillary and dispersion effects is given by Equation 4.1: where ( ) (4.1) (4.2) and where D is the depth below datum, g is gravitational acceleration constant, k is the permeability tensor, krj is the relative permeability, μ j is viscosity, and ρj is the phase density of phase j. 104

131 The only unknown in Equations 4.1 and 4.2 is the pressure (P). Once P is defined, total velocity u t, is derived from the 3D solution to the pressure Equation and the application of Darcy s law. (4.3) 4.3 Governing IMPES Equations Material Balance Equation The governing saturation Equation for the IMPES method for multi-phase incompressible flow is given by Equation 4.4: (4.4) Where is known from Equation 4.3 and the phase fractional flow is given by: (4.5) Equations 4.1 and 4.4 form the IMPES set of nonlinear Equations to be used in the streamline simulator. In the streamline method, the 3D Equation is being transformed into a multiple 1D Equations that are solved along the streamlines. On the other hand, in conventional FD, Equation 4.4 is solved in its full 3D form with the previously calculated pressure field. 4.4 Coordinate Transform The gridblock faces containing injectors are used to launch the streamlines. As the streamlines are traced from injectors to producers, the time-of-flight (travel time) along the streamline is determined, which is defined as: 105

132 (4.6) Taking derivative of both sides (4.7) Rearranging we get, (4.8) Substituting in Equation 4.4 gives (4.9) Equation 4.9 is the governing pseudo 1D phase material balance Equation transformed along a streamline coordinate. It is pseudo 1D since the gravity term is typically not aligned along the direction of a streamline. 4.5 Gravity Because the gravity term is not aligned along a streamline direction, Equation 4.9 is split into two parts using what is called operator splitting as shown in Figure

133 Figure 4.1: Operator splitting applied to streamlines to calculate gravity (modified from Thiele, 2005) Operator splitting solves the conservation Equation by breaking it up into two parts, in which the solution of the first part becomes the initial conditions for the next. While the order of the solution in mathematically immaterial, streamline based simulation solves the convective step first followed by the gravity step. Thus, a convective step along streamlines is taken as shown by Equation 4.10: = 0 (4.10) Then, a gravity step is taken along gravity lines as shown by Equation 4.11: (4.11) For simplicity, the z-coordinate direction is assumed to be aligned with the gravity lines. These are 1D Equations that are solved using standard finitedifference numerical techniques. 107

134 4.6 Tracing Streamlines Tracing streamlines from injectors to producers is based on the analytical description of a streamline path within a gridblock as outlined by Pollock (1988). The underlying assumption is that the velocity field in each coordinate direction varies linearly and is independent of the velocities in the other directions. Pollock's method is attractive because it is analytical and consistent with the governing material balance Equation (Thiele, 2001). Considering a 2D gridblock as shown in Figure 4.2, for which we know the interstitial velocity field and have defined a local coordinate system and origin. The velocity in the x direction as shown in Figure 4.3, is defined by Equation 4.12: ( - (4.12) Where = velocity gradient across the gridblock and is given by Equation 4.13: = (4.13) Figure 4.2: 2D gridblock with known interstitial velocity field and origin (Thiele, 2001) 108

135 Figure 4.3: Velocity gradient across the gridblock (Thiele, 2005) Knowing that = / one can integrate Equation 4.12 to determine the time it takes to exit from each possible face. = [ ] (4.14) = [ ] (4.15) and = [ ] (4.16) The correct face that the streamline will exit is the one that requires the smallest value of Δt e as shown in Figure

136 Figure 4.4: The correct face that the streamline is leaving requires the minimum time (Thiele, 2001) Knowing the minimum time, then the exact exit location of the streamline is determined by re-writing Equations 4.14 through 4.16 as: [ ( ) ] (4.17) [ ( ) ] (4.18) [ ( ) ] (4.19) 110

137 Once the exit point location (x e, y e, z e ) is known, it becomes the inlet position of the neighboring block and so the streamline can be traced from block to block. 4.7 Timestepping in Streamlines Simulations In field-scale displacements, the streamline paths change with time because of the changing mobility field and/or changing boundary conditions. Thus, the velocity field is updated periodically according to these changes by solving for the pressure. In section 3.2 of chapter 3, we briefly touched on how the streamlines simulator works. Here, we will give more detailes about the timestepping in streamlines simulation and more specifically about the calculations of pressure, saturations along streamlines, how that saturation is mapped back to the underlying grids and also how gravity is accounted for. To move the 3D solution forward in time globally by Δt, the following algorithm is usually used in streamline simulators as presented in Figure 4.6: 1. Solve for pressure using Equation 4.1 at the start of new timestep. 2. Use Darcy s law to define the total velocity at gridblock faces of the 3D Cartesian grid system as shown in Figure

138 Figure 4.5: Pressure and total velocity resulting from performing step 1 and 2 (Lolomari et al., 2000) 3. Next, the streamlines are trace from injectors to producers. For each streamlines drawn, the current saturation of each gridblock that the streamline passes through is recorded. This helps generate a profile of saturation vs. τ for each streamlines line. Then, move the simulation forward by Δt and solve Eqn. 16 to find the new saturation. This involves several timesteps within the numerical solvers because Δt >> Δt1D. Once the new saturation profile along each streamlines is generated, the average properties of all streamlines within each gridblock are mapped back to the underlying 3D grid. This helps determining the saturation distribution at t n If Gj 0, first, trace the saturation distribution calculated in the convective step (3) as a function of z along a gravity line. Then, move the simulation forward by Δt to calculate the new saturations using Equation Once the new saturation profiles (accounting for gravity) are generated, the average properties of all gravity lines within each gridblock are mapped back to the underlying 3D grid. This information is used to decide on the final saturation distribution at t n Return to step

139 Figure 4.6: Flowchart of timestepping in streamlines simulators (modified from Ibrahim et al., 2007) 113

140 CHAPTER 5 DEVELOPMENT OF THE STREAMLINE BASED DYNAMIC MEASURE (SL_TOF INDEX) 5.1 Mining of Streamlines Information In all the 480 simulation cases that were ran, the SCHEDULE section within the FRONTSIM data deck was altered to include several streamlines related parameters such as ALLOC, FLOWS, FLUXDENS, TOF and TOFSENS as shown in Figure 5.1. A full description of each of these parameters will be given below. Figure 5.1: Inclusion of streamlines based parameters in the FRONTSIM data deck First, ALLOC gives streamline scalar attributes for allocation data. Table 5.1 summarizes the attributes that are outputted as part of including this keyword under the SCHEDULE section of the simulation model. Table 5.1: Streamlines attributes outputted as a result of including ALLOC under the SCHEDULE section ALLOC: Gives streamline scalar attributes for visualization of allocation data. Attributes are PFRACRES PFRACOIL PFRACWAT PFRACGAS IFRACRES IFRACOIL IFRACWAT IFRACGAS Reservoir fluid production fraction per streamline start point. Oil production fraction per streamline start point. Water production fraction per streamline start point. Gas production fraction per streamline start point. Reservoir fluid injection fraction per streamline end point. Oil injection fraction per streamline end point. Water injection fraction per streamline end point. Gas injection fraction per streamline end point. 114