Copyright. Ankesh Anupam

Transcription

1 Copyright by Ankesh Anupam 2010

2 The Thesis Committee for Ankesh Anupam Certifies that this is the approved version of the following thesis: Hierarchical Modeling of Fractures for Naturally Fractured Reservoirs APPROVED BY SUPERVISING COMMITTEE: Supervisor: Sanjay Srinivasan, Supervisor Mrinal Sen

3 Hierarchical Modeling of Fractures for Naturally Fractured Reservoirs by Ankesh Anupam, B.Tech Thesis Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Master of Science in Engineering The University of Texas at Austin August 2010

4 Dedication This thesis is dedicated to my loving family.

5 Acknowledgements I would like to express my sincere gratitude to my supervisor and mentor, Dr. Sanjay Srinivasan. I consider myself extremely privileged to get an opportunity to work with him. I want to thank him for being so supportive over the last two years and sharing his immense knowledge with me. I learnt a lot under his guidance not only making me a technical sound engineer but also a better person. He will always be a source of inspiration for me to achieve better and bigger goals in my life. I would also like to thank to Dr. Mrinal Sen to be so patient and accommodating with my thesis and providing me with valuable feedbacks. I would also like to extend my earnest regards to my research group, especially Juliana Leung, Cesar Mantilla, Selin Erzybek, Brandon Henke and Harpreet Singh. Their useful discussion and ideas have helped me a lot during my research and made it a smooth sailing journey for last two years. I would like to convey my heartiest thankfulness to all my friends, who made these years the most memorable days of my life. I would especially like to thank Lokendra Jain, Ankur Gandhi and Sayantan Bhowmik to be my best pals and be there for me every moment I needed them. I would like to express my gratitude to all the faculty members for sharing their knowledge with me. I would also like to thank Jin Lee for her help with administrative support, Dr. Roger Terizen for his computer support and all other staff members at Petroleum and Geosystem Engineering Department for their support. Finally I would like to thank my parents to whom I owe almost everything in my life. I would also like to thank my wonderful brother Abhishek, who have provided me encouragement and support in all my endeavors. v

6 Abstract Hierarchical Modeling of Fractures for Naturally Fractured Reservoirs Ankesh Anupam, M.S.E The University of Texas at Austin, 2010 Supervisor: Sanjay Srinivasan Discrete Fracture Networks (DFN) models have long been used to represent heterogeneity associated with fracture networks but all previous approaches have been either in 2D (assuming vertical fractures) or for simple models within a small domain. Realistic representation of DFN on field scale models have been impossible due to two reasons - first because the representation of extremely large number of fractures requires significant computational capability and second, because of the inability to represent fractures on a simulation grid, due to extreme aspect ratio between fracture length and aperture. This thesis presents a hierarchal approach for fracture modeling and a novel random walker simulation to upscale the fracture permeability. The modeling approach entails developing effective flow characteristics of discrete fractures at micro and macrofracture scales without explicitly representing the fractures on a grid. Separate models were made for micro scale and macro scale fracture distribution with inputs from the seismic data and field observations. A random walker simulation is used that moves walkers along implicit fractures honoring the intersection characteristics of the fracture vi

7 network. The random walker simulation results are then calibrated against highresolution flow simulation for some simple fracture representations. The calibration enables us to get an equivalent permeability for a complex fracture network knowing the statistics of the random walkers. These permeabilities are then used as base matrix permeabilities for random walker simulation of flow characteristics of the macro fractures. These are again validated with the simulator to get equivalent upscaled permeability. Several superimposed realizations of micro and macrofracture networks enable us to capture the uncertainty in the network and corresponding uncertainty in permeability field. The advantage of this methodology is that the upscaling process is extremely fast and works on the actual fractures with realistic apertures and yields both the effective permeability of the network as well as the matrix-fracture transfer characteristics. vii

8 Table of Contents List of Tables... xiv List of Figures...xv Chapter 1: Introduction Overview Fracture and Fracture System Classification of Naturally Occurring Fractures Tectonic Fractures Fault Related Fracture Systems Fold Related Fracture Systems Regional Fractures Contractional Fractures Desiccation Fractures Syneresis Fractures Thermal Contractional Fractures Mineral Phase Change Fractures Factors Affecting Flow Behavior Through Fractures Fracture Morphology Open Fracture Deformed Fractures Vuggy Fractures Precipitation and Dissolution Dolomitization Fracture Sets and Fracture Spacing Fracture Modeling: Objectives and Challenges research objective Thesis outline...13 viii

9 Chapter 2: Literature Review Overview Models of fractured porous media Dual porosity model Discrete Fracture Network (DFN) models Fractal Models Geomechanical Fracture Models Upscaled Permeability of DFN Model Permeability Tensor for Fractured Rock (Oda s Method) Impermeable Matrix Permeable Matrix Limitations of Oda s Approach Upscaling of Fractured rock using Homogenization Network Simulation Random Walk Method...28 Chapter 3: Stochastic Modeling of Fracture Network in a Carbonate Reservoir Overview Stratigraphic Overview Units of Lower Cretaceous Unit CRT Unit CRT Unit CRT Units of upper cretaceous UNIT CRT Unit CRT Unit CRT Breccias units Breccia description Parameters for Fracture Modeling Fracture Set...33 ix

10 3.3.2-Fracture Azimuth and Dip Fracture Intensity Fracture Length Object Based Modeling for Fractures Model Creation Length Distribution Dip and Azimuth Distribution Spatial Distribution Moving Window Averages Distribution of subsequent fracture sets Work Flow for Fracture Simulation Conclusion...44 Chapter 4: Hierarchical Modeling of Flow in Fractures Limitations of Discrete Fracture Network Model A New Modeling Approach Hierarchical Modeling Continuous Space Modeling Random Walker Simulation Basic Formulation Equation of an ellipse Equation of a plane containing the ellipse Distance of a point from the fracture plane Test for the point if it lies in fracture Intersection of two ellipses in 3-D Step1: Equation of the plane containing the fractures and equation of fracture Step2: Points of intersection between the fracture and common intersecting line Step3: Points of intersection between fractures Remarks...58 x

11 Chapter 5: Microfracture Modeling Overview Simulation of Fracture Network Fracture Property Generation Centre of the Fracture Fracture Length Dip and Azimuth Aperture Fracture Set Generation Handling Edge Effects Check for intersection to the boundary Centers of replicated ellipse: Random walker simulation Percolation Analysis Movement of Random Walkers Results of Random Walker Simulation Permeability Upscaling and Uncertainty Quantification Permeability Anisotropy Percolation Threshold Validation of Results Calibration Procedure Reasons for Disagreements Permeability Model for the Field Conclusion...86 Chapter 6: Modeling Macrofractures by Integrating the Effective Properties of Microfractures Overview Macrofracture Modeling Fracture Property Generation Conditioning to fracture intensity map...88 xi

12 6.2.3 Generating Multiple Fracture Sets Results Of Macrofracture Simulation Random walker simulation Algorithm of Random Walker movement through Macrofractures...93 Movement of Walker initially placed in fracture...95 Movement of Walkers initially placed in Matrix...98 Walker Percolation Interpretation of Output Validation of Random Walker Simulation Generating an Upscaled Permeability Map Results and Discussion Conclusion Chapter 7: Modeling the Impact of Diagenesis Overview Geostatistical Analysis of Diagenesis Modeling Impact of Diagenesis on Effective Permeability Effect of Diagenesis on Microfractures Effect of Diagenesis on Macrofractures Estimation of Altered Permeability Combining Macro and Micro Fractures Conclusion Chapter 8: Conclusions and Recommendations for Future Work Conclusions Recommendations for Future Work Appendix A: FORTRAN Implementation of Object Based Modeling A.1 Overview A.2 Implementation of Object Based Fracture Modeling in a Gridded Framework xii

13 Appendix B: Implementation Microfracture Modeling and Percolation Analysis137 B.1 Code for Microfracture Modeling B.2 Sample Output of Microfracture Model B.3 Implementation of Percolation Analysis B.4 Sample Output from Percolation Analysis B.5 Detailed Table of Results of Percolation Analysis Appendix C: Implementation of Macrofracture Modeling and Random Walker Simulation C.1 Overview C.2 Code for Macrofracture Modeling C.3 Code for Random Walker Simulation on Continuous Domain Bibliography Vita xiii

14 List of Tables Table 3-1Mean and variance for dip and azimuth distribution Table 5-1 Anisotropic permeability values as a function of fracture intensity Table 5-2 Percolation Threshold in the three directions Table B-1 Showing Results from Percolation Analysis in Z Direction Table B-2 Permeabilities obtained from Percolation Analysis in Z Direction Table B-3 Showing Results from Percolation Analysis in X Direction Table B-4 Permeabilities obtained from Percolation Analysis in X Direction Table B-5 Showing Results from Percolation Analysis in Y Direction Table B-6 Permeabilities obtained from Percolation Analysis in Y Direction xiv

15 List of Figures Figure 1-1 Conjugate shear fractures in outcrop from Trinidad. (Adapted from S.Serra and D.B.Felio)... 4 Figure 1-2 Orthogonal regional fractures in Devonian Antrim shale, Michigan Basin (adapted from Nelson, 1985)... 5 Figure 3 Contractional chickenwire fracturing observed in core. (Adapted from Nelson, 1979, courtesy of AAPG)... 7 Figure 1-4 Orthogonal joints in sandstone showing multiple fracture sets. Cedar Mesa, Utah Figure 2-5 The Fractured rock divided in to Periodic Unit Cells (PUCs). The length of entire system is L (global) and the length of PUC is l (local) where l /L << Figure 3-1 The lithologic sequence observed in the reservoir. The Breccia zone is at the top followed by the upper and lower cretaceous Figure3-2 Intensity Map for Fracture Set 1 Figure3-3 Intensity Map for Fracture Set Figure3-4 Intensity Map for Fracture Set 3 Figure3-5 Intensity Map for Fracture Set Figure3-6 Intensity Map for Fracture Set 5 Figure3-7 Intensity Map for Fracture Set Figure 3-8 Generation of first fracture set. The fracture intensity map is input to the simulation. One slice through the fracture model is shown on the right Figure 3-9 Generation of second fracture set. The smoothened fracture map corresponding to the first fracture set is provided as conditioning information along with the intensity map for the second fracture set Figure 3-10 Generation of third fracture Set. The smoothened fracture map corresponding to the first and second fracture sets is provided as conditioning information along with the intensity map for the third fracture set. The simulated map on the right shows fractures belonging to the first, second and third sets Figure 3-11 Generation of fourth fracture set. The smoothened fracture map corresponding to the previous fracture sets is provided as conditioning information along with the intensity map for the fourth fracture set. One slice through the fracture model consisting of all fracture sets up to the fourth is shown on the right Figure 3-12 Generation of fifth fracture set. The smoothened fracture map corresponding to the previous fracture sets is provided as conditioning information along with the intensity map for the fifth fracture set Figure 3-13 Generation of last fracture Set. The smoothened map corresponding to all prior fracture sets is used as conditioning information along with the intensity map for the sixth fracture set Figure 3-14 Different realizations of fracture model with different weight given to the prior existing fractures. a) realization obtained corresponding to scaling factors of one assigned to the map of pre-existing fractures, b) realization when no weight given to prior existing fractures (i.e scaling factor of zero) xv

16 Figure 4-1 Ellipse with azimuth 90 0 and dip 0 0. The major axis which corresponds to length of fracture is 2a and the minor axis is 2b Figure 4-2 Rotated ellipse with its major axis rotated by an angle θ about the Z axis (perpendicular to the plane of ellipse, not shown in figure) in counter clockwise direction Figure 4-3 Fracture plan rotated along its major axis by dip angle φ. The ellipse has been rotated about its major axis by an angle φ Figure 4-4 Figure showing the plane in which the fracture lies. The ellipse represents a fracture with any arbitrary dip φ and azimuth θ Figure 5-1 Fractures cutting across the edges of the simulation block. The fractures that cut across the boundary of the domain are replicated on the other side. Replicated fractures are shown to be of same color Figure 5-2 Percolation of a random walker through a connected network of fracture. Random walkers are allowed to move only through the fractures and reach the other side only if a connecting path exists Figure 5-3 Shows Cumulative Distribution Function of permeability for fracture intensity of The plot shows the bracket of permeability uncertainty arising due different fracture patterns for fracture intensity of Figure 5-4 Shows Cumulative Distribution Function of permeability for fracture intensity of 0.5. The order of permeability variation given as ratio of maximum to minimum permeability is much lower compared to Figure Figure5-5 Plot shows maximum, minimum and average upscaled permeability corresponding to each fracture intensity value for flow in the z-direction. The range of permeability brackets the uncertainty in permeability due to uncertainty in fracture network Figure 5-6 Plot shows the maximum, minimum and average upscaled permeability versus fracture intensity for flow in the x- direction Figure 5-7 Plot shows the maximum, minimum and average upscaled permeability versus fracture intensity for flow in the y-direction Figure 5-8 Variation in average effective permeability in three directions as the fracture intensity is varied. The permeability in z-direction is higher than that compared in X and Y directions Figure 5-9 Plot showing the percolation threshold in three directions as a function of fracture intensity. Z direction has the lowest values indicating a much greater permeability in Z direction compared to other directions Figure 5-10 Comparison of upscaled permeability values calculated using the random walker to that obtained by applying flow based upscaling Figure 5-11 Map of maximum upscaled permeability in the z direction computed using the random walker simulation. It can be clearly observed that the microfracture permeability is high near the anticlines where the fracture intensity values are high Figure 5-12 Map of average permeability in the Z direction computed using the random walker simulation xvi

17 Figure 5-13Minimum permeability map in the z direction. The pink color down the anticlines shows the matrix permeability. These are areas with low fracture intensity values falling below maximum percolation threshold. Figure 5-11 through Figure 5-13 bracket the uncertainty in upscaled permeability in the z-direction Figure 5-14 Spatial variation in maximum permeability in x-direction. The pink regions show the areas below the minimum percolation threshold Figure 5-15 Spatial variation in average permeability in x direction. The size of regions with permeability equal to the matrix permeability has increased Figure 5-16 Spatial variation in minimum permeability in the x direction. Pink regions show the areas with fracture intensity below the minimum percolation threshold. Figure 5-14 through Figure 5-16 bracket the uncertainty in upscaled permeability in the x direction Figure 6-1 Realizations of the macrofracture model (a) shows the cross sectional view of the macrofracture model and is an approximate representation on a gridded system. (b) shows how the macro fractures after the horizons have been superimposed. This is also approximate because the ellipse has been approximated by a decagon to facilitate visualization using Petrel Figure 6-2 Shows transition of walker to a connecting fracture. The walker initially placed at point A inside fracture 1 has to move to the connecting fracture 2. Case (a), when both points of intersection are in the direction of positive pressure gradient and walker can move to any point on line BC. Case (b), when one of the points (D) of intersection is in direction of negative pressure gradient. The walker can move to any point on the line BC Figure 6-3 Calibrating the median travel time of random walkers to flow based upscaled permeability. There exists a negative correlation showing that the travel time increases as permeability of system decreases Figure 6-4 Final map of permeability in the z-direction. The map includes the effect of both micro and macro fractures. Areas in red are the regions where permeability has been enhanced significantly due to the presence of macrofractures Figure 6-5 Final map of permeability in the x-direction. Since the fractures show an isotropic behavior aerially, this map also represents permeability in Y-direction. It should be noted that permeabilities are much less in x-direction compared to z-direction shown in Figure Figure 6-6 Map of interporosity flow parameter λ calculated using random walker simulation. Red areas highlight the areas where flow takes mostly through fractures Figure 7-1 Probability map of diagenetic alteration. The map has been generated using indicator simulation. Red areas show regions of high probability of porosity alteration due to diagenetic alteration while blue shows regions least affected by diagenesis Figure 7-2 Reduction in permeability in z-direction of microfractures as a function of the probability of diagenesis. The higher the probability of diagenesis, greater the reduction in permeability xvii

18 Figure 7-3 Plot of Original Permeability verses Modified permeability when single or multiple macrofracture cuts through the grid. Different lines in the plot correspond to different values of probability of diagenesis xviii

19 Chapter 1: Introduction 1.1 OVERVIEW Characterizing flow through porous media is key towards forecasting reservoir flow performance. All reservoirs around the world can be broadly classified as either sandstone or carbonate reservoirs. The flow in a sandstone reservoir is predominantly through intergranular pores, and is governed mainly by Darcys law. Flow in a carbonate reservoir is mainly controlled by the complex system of fracture network. These fractures have a great influence on the effective permeability of the medium. In contrast to sandstone reservoirs, the matrix permeability is extremely low in carbonate reservoirs. Matrix stores hydrocarbon fluids while fractures act as the pathway for the hydrocarbons to flow to the well. Typically for carbonate reservoirs, matrix has high porosity but very low permeability while on the other hand fractures have low porosity but very high permeability. If fracture is present it essentially provides a resistance free conduit for fluid flow. Therefore there is a need for effective characterization of fractures if we want to get any handle on the production forecast of a carbonate reservoir. An important aspect of fracture modeling is to understand the distribution of these fractures and factors affecting their formation. 1.2 FRACTURE AND FRACTURE SYSTEM A fracture can be defined as a planer discontinuity formed as a result of shear or tensile failure of the rock (Ranalli and Gale, 1976). According to the geomechanical definition, a fracture can be defined as the plane of stress release formed when a rock fails under stress. The spatial extent of fractures can vary from millimeters to several 1

20 kilometers. The orientation and the degree of fracturing depend on the tectonic setting and stress conditions prevalent at that time. Since paleostress conditions guide the formation of fractures, these fractures are always formed in a swarm, cluster or group. A group of fractures that are genetically related in their origin are often referred to as a fracture set. Each fracture set has specific orientation depending on the stress conditions prevalent. The orientation of individual fracture may however vary within a very small range with respect to the mean orientation of the set. The subsurface stress conditions are also a function of time and vary with the tectonic setting over a geological time period. These changes in stress directions lead to formation of multiple fracture sets in reservoirs. The orientation of each set of fractures is representative of the paleostress condition associated with that set. For this reason sub surface fractures are often used to study the changes in stress regime over time. It is important to understand that though stresses are the predominant factors for fracturing, actual fracturing is controlled by various other factors such tensile and compressive strengths of rock, temperature, pore pressure etc. Temperature and rock type control the extent of fracturing. Temperature at a given depth is controlled by the geothermal gradient in that region. As the temperature increases, failure strength of rock decreases and its behavior starts to shift from brittle to ductile. Fracturing will be more efficient in brittle reservoir rocks, where the fractures are relatively extended and have large openings. These can be called macrofractures. Whereas in less brittle rocks, the fractures are of limited extent and have relatively small openings. These fractures can be called as microfractures or fissures. For modeling purpose it is very important to characterize both micro and macro fractures and understand their combined affect on the flow performance of reservoirs. 2

21 1.3 CLASSIFICATION OF NATURALLY OCCURRING FRACTURES Fractures can be classified into the following types based on their origin as suggested by Stearns and Friedman (1972) Tectonic Fractures Tectonic fractures are those whose origin can, on the basis of orientation, distribution, and morphology, be attributed to local tectonic events. Tectonic fractures are a result of shear failure of a rock as it is observed in most outcrops. These have specific orientation and are very closely related to the orientation of local faults and folds. The order of formation of fracture sets is also very important. The formation of later ones is affected by the previously existing fracture sets. Multiple tectonic events lead to formation of intersecting fracture sets that form an ideal three-dimensional connecting network for the fluid flow. Tectonic fractures can be further divided into two different categories Fault Related Fracture Systems Tectonic forces which lead to faulting also produces clusters of fractures in vicinity of the fault. These fracture sets are formed in three distinct preferential directions: (1) Parallel to the fault; (2) Conjugate to the fault i.e., having the same dip as the fault but in the opposite direction; (3) Direction bisecting the acute angle between the other two fracture sets. Several researchers have studied fault fracture relationship (Stearns, 1964; Norris, 1966; Tchalenko and Ambraseys, 1970; Freund, 1974) to determine the direction of principal stresses at the time of fracturing Fold Related Fracture Systems Orientation of fractures formed due to folding is very complex and relates to the stress and strain history of the rock during the initiation and growth of fold. Upon 3

22 investigation by different researchers (Nickelson and Hough, 1967; Ardnt, 1969; Burger and Thompson, 1970; McQuillon, 1974; Reik and Currie, 1974) it was concluded that folding can result in five distinct fracture sets, whose orientations are guided by dip and azimuth direction of the fold. Figure 1-1 Conjugate shear fractures in outcrop from Trinidad. (Adapted from S.Serra and D.B.Felio) Regional Fractures Regional fractures are those developed over large areas with relatively little change in orientation, no offset across fracture plane, and perpendicular to major bedding surface (Stearns, 1972). They can extend up to 100s of miles. They differ from tectonic fractures in that they are developed in simple geometry, having large spacing, crosscutting all local structures. The origin for these type of fractures is rather inconclusive, however many believe that these are formed as a result of vertical earth movements (Harper, 1966; Nelson, 1975). From the reservoir flow aspect these are macrofractures extending to field scale. 4

23 Figure 1-2 Orthogonal regional fractures in Devonian Antrim shale, Michigan Basin (adapted from Nelson, 1985) Contractional Fractures This is a collection of tension or extension fractures associated with a general bulk volume reduction throughout the rock. These may be formed as a result of desiccation, syneresis, thermal gradients, and mineral phase changes. Their impact on the flow behavior is not very clear. However they constitute an integral part of fracture network and provide permeable pathways for flow. The problem is that it is very difficult to predict their presence and their extent in subsurface. They can be of following types Desiccation Fractures Desiccation fractures are most commonly observed contractional fractures, and are formed as a result of shrinkage upon loss of water. These are generally developed in clay rich sediments (Netoff, 1971; Khale and Floyd, 1971) and are not important from the reservoir flow perspective. 5

24 Syneresis Fractures Syneresis refers to the chemical process of bulk volume reduction as a result of subsurface dewatering. These form a three-dimensional polygonal network of fractures isotropically distributed within the sediment (White, 1961; Picard, 1965; Donovan and Foster, 1972). This brings us again to the notion of microfractures which are small in extent, isotropically distributed, and form a complex fracture network. These are more important for hydrocarbon flow because they form a fracture system interconnected in three dimensions and have a large volume extent Thermal Contractional Fractures These fractures are formed by contraction of hot rock as it cools down. Its formation depends upon the depth of burial and thermal gradient across the material. They are not important in petroleum production Mineral Phase Change Fractures These are formed as a result of volume reduction due to mineral phase changes especially in clay and carbonates. Thirteen percent reduction in molar volume occurs when phase changes from calcite to dolomite. Similarly phase change from montmorillonite to illite leads to a volumetric change. These types of fractures are common in dolomite reservoirs where phase changes lead to widespread chickenwire fracturing. 6

25 Figure 3 Contractional chickenwire fracturing observed in core. (Adapted from Nelson, 1979, courtesy of AAPG) 1.4 FACTORS AFFECTING FLOW BEHAVIOR THROUGH FRACTURES The presence of fractures always impacts the permeability of the porous medium. However the effect of fractures on flow can vary immensely, ranging from having no practical impact to a multi-fold increase in permeability. The overall permeability is a function of permeability of an individual fracture and their intersection characteristics. It is very important to note that mere presence of conducting fractures doesn t impact reservoir performance until the fractures are able to conduct fluid to long distances by forming an intersecting network. Overall, the factors affecting the effective permeability of the medium can be grouped into two - the first governing the permeability of individual fractures, and the second governing the fracture network characteristics. However there is a significant overlap in this classification as will be seen in the ensuing discussions. 7

26 1.4.1 Fracture Morphology Fracture morphology refers to the structure of individual fractures. It can be observed from outcrops or inferred from well logs. The following classification of fracture morphology has been suggested (Roland, 1985) Open Fracture It refers to fractures completely open to flow and having maximum permeability. These are very rarely found in subsurface Deformed Fractures These are the fractures which were initially open but have subsequently altered by deformation. Two end members of this type of fractures are gouge filled fractures and slickensided fractures. Gouge is defined as finely abraded material occurring between the walls of fracture as a result of shear motion. Due to presence of these fine particles, there is a considerable reduction in permeability perpendicular to the fracture (Brock, 1973; Engelder, 1973; Jamison and Stearns, 1982). Slickenside refers to a polished or striated fracture side wall resulting from either pulverization or creation of glass from grain melting. These slickenside surfaces act as barriers for flow across the fractures. On the other hand, the permeability along the fracture is enhanced due to presence of smooth surfaces, which lowers the frictional resistance to flow. These fractures can be commonly observed in outcrops. The deformation of the fractures can be an important reason for expecting permeability anisotropy in the system. It should be noted that deformation can lead to permeability lower than the matrix permeability, in the direction perpendicular to fracture orientation. 8

27 Vuggy Fractures This refers to the vugs that develop adjacent to the fracture due to flow of reactive fluid through the fractures, dissolving some of the matrix. Though they don t have much impact on permeability, they considerably increase the fracture porosity Diagenesis Diagenesis is any chemical, physical, or biological change undergone by sediment after its initial deposition and during and after its lithification. These changes happen at relatively low temperatures and pressures and result in changes to the rock's original mineralogy and texture. Diagenesis in carbonate systems has profound effect on fracture permeabilities. There are various diagenetic processes which affect the reservoir quality. The most important diagenetic effects are listed below Precipitation and Dissolution Precipitation and dissolution take place when a reactive fluid flows through the porous media. The process is of special significance for carbonate reservoirs where the primary constitutive mineral calcite has considerable solubility in water. Dissolution produces vugs and cave features that provide additional space for hydrocarbon storage. Development of karst features are a result of this process. Precipitation on the other hand causes either reduction in aperture or complete closure of fractures. Partially or completely mineralized fractures are very commonly observed in cores and outcrops. The bearing of this on reservoir quality is immense. A prospective hydrocarbon play can be rendered totally unproductive by diagenetic effects. The degree of diagenesis usually varies from one part of the reservoir to another. However it is not a localized process and affects the permeability of a fracture and a fracture network as a whole. Modeling diagenesis is a big challenge in characterizing carbonate reservoirs. 9

28 Important diagenetic events in this category are calcite dissolution and precipitation, sulphate precipitation and alteration, karst and cave formation and aragonite dissolution Dolomitization It is a geochemical process where magnesium (Mg 2+ ) ions replace calcium (Ca 2+ ) ions in calcite to form dolomite. The volume of a dolomite crystal is less than that of the calcite crystal; therefore replacement increases the pore space. Dolomites are good reservoir rocks and are resistant to any further diagenesis Fracture Sets and Fracture Spacing Fracture sets as mentioned earlier are clusters of fractures with common tectonic origin. They have specific orientations with very little variation and are largely nonintersecting. Each fracture set is associated with a particular reservoir stress state. As the stress conditions change over geological time new fracture sets are formed. Formation of multiple fracture sets is necessary for formation of fracture networks, which can carry fluid to long distances. For tectonically active areas, for example near convergent or divergent plate boundaries where the changes in stress directions are more frequent multiple fracture sets can be expected. Fracture spacing is defined as the average distance between regularly spaced fractures measured perpendicular to a parallel set of fractures of a given orientation. Other terms used to quantify abundance of fractures in a reservoir are fracture intensity or fracture density. The smaller the fracture spacing for a fracture set, the greater is the connectivity of the fractures network. 10

29 Figure 1-4 Orthogonal joints in sandstone showing multiple fracture sets. Cedar Mesa, Utah. 1.5 FRACTURE MODELING: OBJECTIVES AND CHALLENGES Characterizing fracture network is crucial for understanding the flow behavior of carbonate reservoirs. The aim of the modeling is to effectively characterize the fracture permeability. This task has proven extremely difficult over the past years. The difficulty comes from the complexity of the network topology and the wide disparity in the length scales of fractures and matrix blocks. The sparse data available to model the fracture network also implies that it is not possible to come up with a single reliable model. The attempt is rather to bracket the uncertainty in permeability due to fractures. Usually very little information is available about the sub-surface fractures during field development. Direct evidence of fractures comes from FMI (Formation Micro Imager) logs and cores. However this information is very limited, as only those fractures that intersect the wellbores can be observed through logs and cores. Other indirect 11

30 information about fractures can be obtained from seismic data, production data, well testing etc. All these indirect evidences are at different length scales and can only provide qualitative information. Based on the available data, the best which can be done is to generate a suite of models each of which honors the available data, and is equally probable. Given the enormous number of fractures that can be present in a field, the modeling process is extremely demanding in terms of computational time. The modeling procedures employed so far have been either too simplistic or on a small grid scale. Little has been done so far in an attempt to rigorously model fractures at the field scale using multiple realizations. Finding effective permeability of fracture network entails characterization of intersections between the fractures and interactions between the fractures and the matrix. Flow based upscaling of fractured network using conventional simulators require discretization of fractures on a gridded system. The extreme aspect ratio between the fracture aperture and fracture length makes it impossible to represent fractures with realistic apertures in a gridded domain. Few other methods (Oda et al., 1985; Douglas and Arbogast) have also been proposed for upscaling fracture networks, but none of them is rigorous enough to allow upscaling without any simplifying assumptions. 1.6 RESEARCH OBJECTIVE The objective of my research is to effectively characterize the flow in a naturally fractured carbonate reservoir. It is important to point that the permeability of a fractured system is not only a function of the degree of fracturing but also on the intersection characteristics of the fractures. There can be enormous differences in permeability for the same amount of fracturing but corresponding to different fracture pattern characteristics. This leads to uncertainty in the fracture network due to scarcity of the available data, as 12

31 well as uncertainty due to the effective upscaled permeability. The primary research objective is to represent the total uncertainty in the permeability field at the field scale using an effective and computationally inexpensive method. 1.7 THESIS OUTLINE In Chapter 2, a literature review of some of the previous works done on modeling and upscaling fracture networks is presented. Chapter 3 gives a description of the initial fracture network model that formed the basis for further modeling presented in subsequent chapters. Chapter 4 introduces the new hierarchical approach for modeling fractures at micro and macro length scales. It explains the advantages of this approach and the basic equations used in this modeling approach. Chapter 5 deals with the simulation and upscaling of microfractures. It presents the results for upscaling microfractures and the associated uncertainty in permeability distribution. Permeability anisotropy has also been investigated in this chapter. Chapter 6 explains the simulation and upscaling of macrofracture network. It introduces a new random walker technique to upscale the macrofractures integrating the microfracture characteristics. It also presents the application of the modeling techniques to an actual field data to generate permeability distribution in each grid of the simulation model. Chapter 7 presents an approach to model the affect the diagenesis on the effective permeability of a fracture network. Chapter 8 discusses the conclusions drawn from the research work and also explores the avenues for future research work. 13

32 Chapter 2: Literature Review 2.1 OVERVIEW In this chapter relevant literature relating to modeling and calculation of effective properties of fracture network in porous media is reviewed. We begin with a discussion of various modeling approaches that have been used to model fractures. Next, a review of some of the popular methods for upscaling fracture networks is discussed. At the end, a brief introduction to random walkers is presented, because a random walker has been used for upscaling in this thesis work. 2.2 MODELS OF FRACTURED POROUS MEDIA Different modeling approaches have historically been used to model fractured media. There are at least four different classes of fracture models worth discussing Dual porosity model Dual porosity model was initially introduced by Barenblatt et al. (1960) and was formally extended by Warren and Root (1963) for porous media. The fractured rock is envisaged as consisting of two porous systems, the matrix which has high porosity but low permeability and fractures which have low porosity but high permeability. The flow of fluid takes place independently in both the porous media with some interchange of fluid taking place at the interface. It is a continuum model and has been solved (Liu and Chen(1990), DeSwaan(1976)) using different boundary conditions. The following transport equations can be written for flow through fractures and matrix. f P 1. f (. f øc f t K P ) = q t µ m P 1. m (. m ømct K P ) = q t µ (2-1) (2-2) 14

33 The following nomenclature have been used in equations 2-1 and 2-2 f, m = Superscripts representing fracture and matrix respectively K = Permeability P = Pressure c t = Total compressibility Ø = Porosity µ= Fluid viscosity q= Flow rate As is true for any continuum model, the properties of the matrix and fracture are averaged properties of the medium. This implies that the averaging volume should be greater than the REV (Representative Elementary Volume) of both the matrix and the fracture. Closemann (1975) and Abdassah and Ershaghi (1986) extended the double porosity to a triple porosity model for fractured carbonate reservoirs. Triple porosity models consider two different types of matrix properties. The second type of matrix is added to account for the storage and flow associated with vugs and cave features in carbonate reservoirs. The double porosity model is very simple and needs very few parameters to characterize the fractured systems. It gives a good understanding of the flow mechanism inside a fractured rock but is inherently too simple to capture the influence of complex fracture networks. 15

34 2.2.2 Discrete Fracture Network (DFN) models With the advent of fast computers with large memories, the focus shifted towards modeling fractures in rocks explicitly based on observations made in outcrops and accounting for geomechanics. The pioneering work was first done by Snow et al (1969), who modeled fractures as a bundle of infinitely extending capillary tubes. Since then many different DFN models have been developed. In two-dimensional models (Long et al. 1982; Smith and Schwartz, 1984; Robinson et al 1984; Mukhopadhyay and Sahimi, 1992), fractures are represented as lines distributed randomly in a plane. The centers of the line of fractures follow a Poisson distribution while the length and orientation of a fracture can be selected from any given distribution. For these models the degree in fracturing or fracture intensity (λ l ) is defined as the average frequency of fractures intersecting a scanline. Hestir and Long (1990) analyzed various two-dimensional models to determine fracture connectivity and relate the parameters of fracture networks to standard percolation network parameters. The use of two-dimensional networks to model fractures is restrictive because it is difficult to represent the actual fracture network with fractures exhibiting arbitrary dips realistically in a two-dimensional model. More complicated three-dimensional models were developed (Long et al. 1985, Charlaix et al. 1987, Pollard et al. 1976) to allow for realistic representation of fracture network. Fractures are represented as flat planes of finite dimensions and in most of the cases as circular discs. Billaux et al. (1989) used Poisson process to distribute the centers of the fractures and a log normal distribution for the radius of the fracture. The main disadvantage of these approaches is the significant increase in computational time, especially as the model size increases. To apply a continuum approach to such models an appropriate REV has to be found and that is non-trivial as demonstrated in Leung (2009). 16

35 It turns out that usually fracture networks encountered in nature are slightly above their percolation threshold (Sahimi et al.1993) and in this case the REV is so large that the averaging becomes meaningless. Clemo and Smith (1989) proposed a hierarchical modeling of fractures. The fractures that form the backbone of the network are represented individually and are called primary fractures. The rest of the fractures, that are less important from the standpoint of flow are grouped together into network blocks and are represented by their effective properties Fractal Models Fractals refer to objects that are self-similar at all length scales. Turcotte et al. (1986) was the first to point out the power law size distribution for fractures which is a typical characteristic of fractal objects. Many researchers have proposed synthetic fractal models of fracture networks. Barnsley (1988) proposed an iterative scheme to generate fractal fracture patterns. The initial simple shapes called propagators are perturbed iteratively using a set of numerical transformations, to generate finer scale fractures. After several iterations the pattern converges to a fractal pattern. Barton et al. (1987) constructed fracture model by selecting randomly from frequency distribution of fracture trace lengths, spacing, orientation and crossing obtained from analysis of actual fracture trace maps Geomechanical Fracture Models In these models the fracture network is generated by modeling deformation in rocks that lead to initiation and propagation of fractures. Thus, an attempt is made to replicate the actual process of fracture generation based on the stresses and the elastic properties of the rock. Fracture mechanics is based on the theory of fracture propagation 17

36 by Griffiths (1921), which deals with the stability of a single crack and its propensity to propagate. However the fracture propagation theory works well only for solids that are more or less homogenous. For natural rocks that are inherently heterogeneous it is difficult to obtain reasonable results using this theory. Even small heterogeneities in the rock strongly affect the process of fracture propagation. Therefore it is inappropriate to use average rock properties to generate fractures, as heterogeneities are important for this process and cannot be neglected. Sahimi and Goddard (1986) generated fracture model for heterogeneous rocks. The solid is modeled as a bond network where each bond is considered as a spring whose extension and failure is governed by linear elasticity equation given by Hook s law. To incorporate heterogeneity the spring constants for the bond were made to vary and were picked up from a probability density function. This work was further extended by Arcangelis et al. (1989) to include the effect of torsional and bending forces for the bonds. 2.3 UPSCALED PERMEABILITY OF DFN MODEL There have been a few different approaches used to find upscaled permeability of Discrete Fracture Network (DFN) models. Some of the most important ones are discussed below Permeability Tensor for Fractured Rock (Oda s Method) Oda et al. (1985) introduced a new upscaling approach for fractured porous medium. A symmetric permeability tensor is found which depends only on the distribution of geometrical properties of the fractures (aperture, size and orientation).this approach is widely used for upscaling for its simplicity and therefore is reviewed in detail here. 18

37 If the fractured rock can be assumed to be homogenous, anisotropic porous medium, the flow of fluid obeys the Darcy s law in which the superficial flow velocity ν is related to the flow gradient / xi by a proportionality coefficient k ij, commonly known as permeability tensor. The permeability tensor is independent of the fluid property and flow gradient and is an intrinsic property of the porous medium. Darcy law is given as v i g = k µ ij x i (2-3) where g is the gravitational acceleration and µ is the kinematic viscosity of the fluid. Long et al (1982) suggested that if sufficient number of fractures is present, the rock mass behaves more like a porous medium and hence Darcy law can be applied Impermeable Matrix Consider a rock mass with volume V as the flow region. It is homogenously cut by m (V) fractures whose centers occur at random in the volume. The apparent flow velocity is defined as 1 1 v = vdv = v dv ( f ) ( f ) i i i V V V ( c ) V (2-4) Here v is the local flow velocity in the fractures and is the volume ( c ) i associated with the fractures. Due to absence of any reliable information about the shape of the fractures, the fractures are assumed as circular discs with radius r and aperture t. The orientation of the fractures is given by unit normal vector n, which covers the entire solid angle Ω. 19

38 ( ) Now a joint probability density function (PDF) f (, rt, ) n is introduced such that f n, rt, dωdrdt gives the fraction of total number of fractures having their normal unit vectors oriented inside around n, with radii ranging from r to r dr and apertures ranging from t to t dt. The PDF satisfies the following relation 00Ω ( ) f n, rt, dω dr dt = 1 (2-5) Let N be the number of fractures with their radii less than r, apertures less than t and unit normal vectors oriented inside a part of total solid angle ΔΩ. It can be expressed as t r ( V) N m f n, rt, d dr dt 00ΔΩ ( ) = Ω (2-6) Let dn and dv be the number and volume of (n, r, t) fractures i.e the fractures with unit normal vector between n and n + d and the diameters and apertures ranging from r to r + dr and from t to t + dt. dn ( ) ( V ) = m f n, r, t dω dr dt (2-7) 2 πrt π 2 ( V ) V m f n ( ) d = dn = r t, r, t dω dr dt (2-8) 4 4 Now let the flow region be subjected to two constant pressure boundaries ø 1 and ø 2. Assuming a linear pressure gradient between the boundaries the gradient is given as 1 2 J = p (2-9) L where L is the distance between the two boundaries and p is the unit vector pointing in the direction of J. It must be noted that the assumption for constant pressure gradient is only acceptable if there is sufficient number of fractures present in the flow 20

39 region, making the system almost homogenous. Let J (f) be the component J projected in the direction of (n, r, t) fracture. ( f ) = ( δ nn ) J J (2-10) ij i j i δ ij is the Kronecker delta and n i and J i are components of n and J projected on orthogonal reference axes x i (i = 1,2,3). J (f) is the projection of total pressure gradient J in the direction of (n, r, t) fracture. Now assuming flow through fracture as flow between infinitely extending parallel plates. The mean flow velocity is given as g 2 ( f ) g 2 v = i λ t J λ t ( δij nn i j) i μ = μ J (2-11) Here λ is a dimensionless constant chosen depending on the fracture roughness and morphology. Typical values range from 0 λ 1/12. Using equation 2-8 and 2-11, equation 2-4 can be modified as 1 vi = vi dv V ( f ) ( f ) ( c V ) ( V ) g π m 3 2 [ tr ( δij nn i j ) (, rt, ) d drdt] i μ 4 f n Ω J V 00Ω = λ (2-12) Since orientations of all the fractures in the flow domain are known, the integration can be performed over all the fractures to calculate the average flow velocity. A comparison between the equation 2-12 and equation 2-3 (Darcy s law), equivalent permeability tensor k ij can be found as k = λ( P δ P ) (2-13) ( f ) ij kk ij kk 21

40 Where ( V ) π m 3 2 ij = i j 4V 00Ω (,, ) P t r nn E n r t dωdr dt kk (2-14) P = P + P + P (2-15) Permeable Matrix In rocks where matrix also contributes to the flow, equation 2-5 can be modified to include the matrix contribution. Since the permeable matrix behaves like an ideal porous medium an additional permeability tensor is added to the permeability tensor arising due to fractures. The final expression is given as 1 ( ) ( ) 1 v v d v d ( ) ( ) f f m m i = i V + i V V ( c) V ( m V V ) g ( f ) ( m) = ( k ij + k ij ) J i (2-16) µ Limitations of Oda s Approach One of the major limitations of Oda s approach is that it has been formulated on the basic assumption that the block is fully divided by fractures such that there is always flow across the domain. It always gives non-zero permeability regardless of whether the fractures become disconnected and the flow region is completely impermeable. To correct this shortcoming a threshold value of a ij was introduced in equation 2-13, below which the flow region becomes impermeable. If the permeability of system falls below the threshold value it is considered as having zero permeability. The modified equation is given as k = λ( P δ P ) - a (2-17) ( f ) ij kk ij kk ij 22

41 However, determination of the threshold value a ij is specific to the fracture network under investigation, and no generalized value can be assumed for threshold constant. As a result Oda s approach fails for fracture networks very near to percolation threshold, which is the case found in natural fractured rocks as pointed out by Sahimi et al Even when there are a large number of fractures, the upscaled permeability must depend on the number of connected paths that are formed across the fractured medium. There is no consideration about the intersection characteristics of the fractures in Oda s approach suggesting that the spatial location of the fractures and their connectivity do not matter. Such an assumption is counter-intuitive. Despite these drawbacks, Oda s approach remains popular in practice due to its simplicity Upscaling of Fractured rock using Homogenization The approach of homogenization for upscaling fractured systems was first introduced by Douglas and Arbogast (1990). This work was extended further by Zanganeh and Salimi (2007) where they used homogenization to obtain macroscopic multiphase flow model equations based on equations at the microscopic level. The basic methodology for the process of homogenization is as follows. The first step is to sub divide the fractured reservoir into periodic unit cells (PUC) as shown in Figure 2-5. Each PUC consists of matrix surrounded by fractures and represents the fractured medium at local scale. Two characteristic length scales are defined; L is the global or macroscopic length scale whereas l is the microscopic or local length scale. An important requisite for method of homogenization is the separation of the characteristic length scales. It is required that /L 1, where is the scaling ratio. 23

42 L PUC l Figure 2-5 The Fractured rock divided in to Periodic Unit Cells (PUCs). The length of entire system is L (global) and the length of PUC is l (local) where l /L << 1 The following nomenclature is used in the ensuing discussion f, m = Subscripts denoting fractures and matrix respectively * = Superscripts denoting the property evaluated/measured at local scale = Porosity, = Concentration of component j in phase α Ω = Saturation of phase α = Denotes the entire medium on global scale = Denotes the domain of unit cell = Unit normal vector pointing outward from matrix to fracture Ω m = Boundary of matrix fracture interface k = Scaling Ratio = Flow velocity vector = Permeability 24

43 The second step is to derive the transport equations at local scale for both matrix and fracture. It is assumed that the differential of flux in the transport equation can be split into a large scale (global) contributionδ, and a small scale (local) contributionδ. The mass conservation equation for fracture at local scale (inω ) can be written as t ϕ C S = Δ. C u Δ. C u * * * f α j, f α f b α j, f α f s α j, f α f α α α (2-18) For the matrix the local and global properties are the same i.e. and. This implies that the volume support for local scale is larger than the REV for matrix and thus there is no need for splitting the flux term in to local and global contributions. The mass conservation equation for matrix at local scale (inω ) can be written as t α ϕ C S = Δ. C u m α j, m αm α j, f αm α (2-19) The continuity equation for the mass flux at matrix fracture interface can be written as: C u. n= C u. n (2-20) * αjf, αf αjm, αm The averaged form of transport equation for flow in fracture and matrix can be written as: * * * < ϕ fcα j, fsα f >= <Δ b. Cα j, fuα f > <Δ s. Cα j, fuα f > t α α α < ϕ mcα j, msαm >= < Δ. Cα j, fuαm > t (2-21) (2-22) α α 25

44 The spatial average of any parameter over a domain D can be done as shown in equation 2-23 where denotes the spatial co-ordinates of any point in domain D. 1 < x >= xdrd D (2-23) D Divergence theorem can be used further to simplify the volume integral into surface integral. It is given as 1 1 Δ. xdrd = D D xdσ D (2-24) D D In equation 2-23 σ D denotes the surface co-ordinates of domain D. All these lead to the final averaged equation over the unit cell: < ϕ C S >+< ϕ C S > = <Δ. C u > t (2-25) * * f α j, f α f m α j, m αm b α j, f α f α t α α The solution to Equation 2-25 is still complicated but can be solved using different simplifying assumptions. As for the most common oil-water system, ignoring capillary pressure in fractures and assuming a capillary-gravity equilibrium at all times leads to λwmλom ϕ S = P t λ + λ m wm c, m wm om (2-26) where λ wm and λ om are the relative mobilities of water and oil phase and P c,m is the capillary pressure in the matrix. The changes in the phase saturations in the matrix add to the flux flowing through the fracture. The overall flux flowing through the medium can be used to find the effective permeability of the combined system. 26

45 2.3.3 Network Simulation Network simulations have long been used to estimate effective permeability, conductivity and diffusivity of porous medium (Nicholson and Petropoulos, 1977; Dullion, 1975; Koplik et al., 1984; Seeburger and Nur, 1984; Doyen, 1988, Bryant et al. 1993). These works vary in the amount of detail included in the network, but the essential idea behind all of them is the same. The same models can be used for calculating the effective permeability of fractured rock. The flow problem is established by relating the pressure drop to the flow rates at each node. Standard transport equations like Hagen- Poiseuille, Forchheimer (1901), Darcy, Brinkman(1947) etc. are used to relate flow rate and pressure depending on the type of porous medium and the fluids we intend to model. Mass balance equation can be written for each intersection node as: Q = ij 0 j (2-27) Where is the flowrate between node j and node i. The summation is performed over all the nodes connecting to node i for which the equation is being written. Similar equations can be written for all the intersection nodes in the network. Substituting the transport equation, a set of simultaneous equations is obtained in terms of nodal pressure. These equations can be solved to get pressure field at all intersecting node. Usually the imposed condition for such network problems is the flow rate. Equivalent pressure drop can be used to back calculate the effective permeability of the network. The major drawback of this method is that it assumes no flux interchange between matrix and fractures. The pressure field is solved assuming steady state flux across the block and no flow through the matrix. Nevertheless this approach is useful for rocks with negligible matrix permeability or where the matrix contribution is known to be very small. 27

46 2.4 RANDOM WALK METHOD A random walk is a mathematical formalization of the trajectory of a particle that takes successive random steps. Random walk analyses have been applied in computer science, physics, ecology, economics, psychology and a number of other fields as a fundamental model for random processes in time. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock etc. can all be modeled as random walks. Various types of random walks are of interest. Often, random walks are assumed to be Markov processes (memory less processes), but other, more complicated walks like non-markovian elephant walks are also of interest. Some random walks are on graphs, others on a line, in the plane, or in higher dimensions. Random walks also vary with regard to the time parameter. Often, the trajectory of a walker at discrete time intervals is computed. However, in some cases, the walker makes discrete moves at random times. Several properties of random walks, including dispersal distributions, first-passage times and encounter rates, have been extensively studied (Havlin and Ben-Avraham, 1987; Haus and Kehr, 1987). Random walks have been long been used to determine equivalent transport properties like diffusivity and conductivity (Huges 1994) for disordered systems. Some of the modeling work using random walk has been done by Bunde et al. (1985), Sahimi and Stauffer (1991), Zheng and Chiew (1989), Kim and Torquato (1991). A popular random walk model is that of a random walk on a regular grid, where at each step the walk jumps to another site according to some probability distribution. The probability distribution is determined by a kernel or potential function that determines the transition probability for the walker to move to the next location. The kernel function depends on the physical process we intend to replicate. For example for modeling diffusion, the kernel function is 28

47 determined by Fick s law where the diffusivity constant and the concentration difference between the two locations determine the transition probability In this thesis, random walks have been used to find the effective permeability of fractured systems. These random walks have been done on a continuous space where both the transition time and the position of walker are continuous. The kernel function determining each transition is governed by Darcy law, where the effective permeability of each path and the pressure field determine the transition probability. This is discussed in detail in later chapters. 29

48 Chapter 3: Stochastic Modeling of Fracture Network in a Carbonate Reservoir 3.1 OVERVIEW This chapter presents an object-based approach for modeling fracture networks in a major oil and gas producing carbonate reservoir. The field has more than a hundred wells and is currently under production. The ultimate goal is to characterize flow through the fracture network for future performance prediction of the reservoir. 3.2 STRATIGRAPHIC OVERVIEW The field consists of three major anticlinal structures. The regional stratigraphy consists of the brecciated formation at the top, followed by the cretaceous sequence divided into three distinct zones Cretaceous Superior, Cretaceous Middle and Cretaceous Inferior. These are characterized predominantly by limestone and dolomite facies that exhibit a high degree of natural fractures. The flow in the reservoir is mainly through these fractures, with matrix contribution being minimal. The breccia is the major producer in this sequence. The following grouping of facies, consistent with the stratigraphic sequence described earlier (and shown in Figure 3-1) was made. 1. Units of breccias of upper cretaceous (Brecha-1, Brecha-2) 2. Units of Upper cretaceous: CRT_4, CRT_5 and CRT_6 3. Units of Lower Cretaceous CRT_1, CRT-2 and CRT-3 30

49 BRECCIA Upper Cretaceous Lower Cretaceous Figure 3-1 The lithologic sequence observed in the reservoir. The Breccia zone is at the top followed by the upper and lower cretaceous Units of Lower Cretaceous The lower cretaceous is subdivided into three geologic petrophysical units from base to top: CRT_1, CRT-2 and CRT Unit CRT-1 This unit overlies the Jurassic and Triassic sequences and is composed of mudstone to wackestone, with bioclasts and intraclasts mainly dolomitized. The entire unit is highly dolomitized and has abundant vuggy porosity. 31

50 Unit CRT-2 This unit has sediments of mudstone-wackestone of dolomitized clasts with abundant vuggy porosity and microfractures. The layer limit is not well defined, mainly due to dolomitization processes. The presence of vugs and caverns formed as a result of process of dissolution increase the porosity of a rock Unit CRT-3 This unit comprises of thin layers of mudstone, light gray dolomitize interspersed with bentonite and vuggy porosity Units of upper cretaceous Upper cretaceous is subdivided into three geologic-petrologic units from base to top CRT-4, CRT-5 and CRT UNIT CRT-4 The unit is represented by dolomitized limestones with intervening 10 to 20 cm layers of mudstone with clay. There is also brecciated limestone and dolomitized limestone compacted and well cemented. To the base of this unit are limestones containing both cemented and partially open fractures Unit CRT-5 This unit comprises of wackestone of bioclasts and dolomites interspersed with shale. To the top of this unit, dolomite is the dominant facies. Stratification thickness varies from 10 to 40 cm Unit CRT-6 The rocks of this unit are dolomitized limestone with layers of mudstone pyrite disseminated. The upper contact of the unit CRT-6 is an erosional truncation. 32

51 3.2.3 Breccias units It is possible to recognize two units of breccia: one corresponding to the microbreccia and the expulsion material in the upper part of the breccia, named here as Brecha-2. Underlying Brecha-2 is Brecha-1 which merges into the Cretaceous formation. This is the highest producing formation of the field and therefore much of the effort has gone to study and characterize this formation Breccia description Breccia contains fragments of limestone and dolomitize of different sizes. The original composition of the fragments is variable with mudstone, wackestone, packstone and grainstone. The dominant size of the fragments is between few millimeters to 10 cm (gravel and bigger). The presence of vugs and fractures are common in the majority part of the breccia zone. 3.3 PARAMETERS FOR FRACTURE MODELING As discussed earlier all the lithology units of the field are fractured and have gone through extensive diagenesis. DFN (Discrete Fracture Network) modeling approach has been adopted to model the fractures. The model generates discrete fractures on the field scale based on input parameters. The key input data used for the existent fracture model are listed below Fracture Set A total of six different fracture sets have been modeled, each having a different mean orientation. The decision about the number of fracture sets was based on the structural geologist s interpretation that was in turn based on the observations from cores and logs. Core and log (FMI) analysis from a number of wells show distinct preferential 33

52 directions for fractures. These preferential directions were grouped in to six different fracture sets. It is also important to specify the order of formation of fracture sets that directly relates to the order of occurrence of tectonic events, which led to formation of these fracture sets Fracture Azimuth and Dip Fracture azimuth is the angle subtended by the fracture to the geographical north direction. Fractures exhibit a scatter around the mean direction that is assumed to be characterized by a Gaussian distribution. The mean and variance of the Gaussian distribution for each fracture set were inferred from cores and FMI logs. Fracture dip is defined as the angle between the fracture axis and the horizontal plane. Dip of the fracture set is interpreted from the dip of the formations and is nearly constant for all the fracture sets. The dip is also assumed to follow a normal distribution, with mean and variance inferred from log and core data available from the wells. The mean and variance for dip and azimuth distributions for the six fracture sets, used as input is shown in Table 3-1 below Table 3-1Mean and variance for dip and azimuth distribution Summary Mean Variance Azimuth Dip Azimuth Dip Set Set Set Set Set Set

53 3.3.3 Fracture Intensity Fracture intensity is defined as fracture area per unit bulk volume. It is a measure of the degree of fracturing and controls the spatial distribution of fractures. Maps for fracture intensity were obtained by structural interpretation and were provided as inputs for this study. Thus in total there are six fracture intensity maps for each fracture set that are at a coarser resolution. Since the resolution of seismic data is very coarse, it is used as a soft data for controlling the degree of fracturing in different parts of the reservoir. Fracture intensity maps for all the six fracture sets as used for fracture modeling are shown below in Figure3-2 through Figure 3-7 Figure3-2 Intensity Map for Fracture Set 1 Figure3-3 Intensity Map for Fracture Set 2 Figure3-4 Intensity Map for Fracture Set 3 Figure3-5 Intensity Map for Fracture Set 4 35

54 Figure3-6 Intensity Map for Fracture Set 5 Figure3-7 Intensity Map for Fracture Set Fracture Length For fracture length distribution a power law distribution is assumed. Power law distribution for fracture intensity has been verified from many different outcrop studies (Odeling, 1997; Laubach and Lake, 1997). The power law can be expressed as N - n = ml (3-1) where L is the length of fracture, N is the cumulative number of fractures with length greater than L, n is the power law exponent, and m is the proportionality constant. The power law distribution implies that the number of long fractures is much less compared to short fractures and that decrease is exponential. For using a power law we need to specify the power law exponent and either the cumulative number of fracture or the minimum length of fracture as an input. An exponent of 3.1 was used for all the six fracture sets and a minimum size of two grid length has been specified. The power law exponent was based on a detailed statistical study of the fractures observed in cores and logs along several wells. 36

55 3.4 OBJECT BASED MODELING OF FRACTURES In object based modeling approach the essential features of the system are treated as objects that are placed in the domain stochastically. The shapes of the objects differ depending on the feature we want to model. For fracture modeling the fractures are modeled as planar objects. The fractures are dropped in the given domain and are distributed satisfying given input statistics. In this study, fractures are represented as elliptical objects that are distributed satisfying the statistics described above. A suite of fracture models are generated that can be used to characterize the uncertainty in flow. The various steps involved in the modeling process are discussed below Model Creation Fractures are modelled on a grid that is of size 145x215x66. We model fracture sets, one at a time following a chronological order. Fracture lengths, dip, azimuth and fracture intensity maps are given as input Length Distribution It is desired that the length of fractures follow the power law model with power law coefficient of 3.1. The minimum length of fracture has been specified as the dimension of two grid blocks whereas the maximum fracture length can be given as an input. Based on the maximum and minimum fracture length, CDF (Cumulative Distribution Function) for fracture length distribution following a power law statistics is computed. A random pick from this CDF allows enforcing a power law distribution for fracture length. 37

56 3.4.3 Dip and Azimuth Distribution The dip and the azimuth were assumed to follow a normal distribution. The dip and azimuth of each fracture in a set was sampled from a normal distribution with the mean and variance of the given fracture set as specified in Table Spatial Distribution The fracture intensity maps were used as soft data for spatially distributing fractures. We first start with generation of fractures for the first fracture set. To use the intensity data as soft input we compute the simulated fracture intensity within moving windows. This procedure is described next Moving Window Averages The centre of the fracture is selected randomly from the domain. But the fracture is generated only when the fracture intensity calculated using a window average at all grid locations intersected by the fracture is less than the fracture intensity data. The condition implies that at every grid location intersected by the fracture, the average fracture intensity for the window accounting for all fractures intersecting that window must be less than the fracture intensity provided as input to the simulation. If there are already sufficient numbers of fractures within the window satisfying the fracture intensity map, that centre location of the fracture is rejected. A new random point for the centre of fracture is selected and the same test is performed again. The fracture length for the newly selected centre is the same as for the discarded centre location to ensure that the power law distribution for length is not violated. The moving window can be taken of different window sizes. The window size can be interpreted in terms of the influence area of a previously existing fracture. Greater the 38

57 window size, greater is the volume of influence of the fracture that prevents other fractures to be simulated within the neighbourhood of the existing fracture Distribution of subsequent fracture sets The process outlined above is for the generation of the first fracture set. For generation of the fracture sets 2 through 6, the same procedure is followed for the length, dip and azimuth distribution. The length distribution follows the power law with exponent of 3.1. The dip and azimuth also follow a Gaussian distribution with mean and variance of the distribution being different as enumerated in Table 3-1. However the spatial distribution of fractures in any set has to be consistent with the fractures in the previous sets. The generation of new fractures are affected by the presence of pre existing fractures. Pre existing fractures inhibit the formation of new fractures in their vicinity. To take this in to account, the generation of next fracture set is conditioned to both the fracture intensity map for that set, as well as the fracture density of the pre existing fractures. The fracture distribution corresponding to the previous set is smoothened using a window average in order to generate a smoothened fracture map. The size of the window depends upon our interpretation of the region of influence of pre existing fractures. The same is done for generation of all further fracture sets. So in affect the generation of last fracture set is conditioned to its fracture intensity map and all the prior existing fractures. Two scaling factors are also introduced to allow different degree of conditioning to the fracture intensity maps and prior existing fractures. These scaling factors can take any value between zero and one. Scaling factor of one implies that the fracture will be generated only when the moving window average condition is satisfied for all the grid blocks intersected by the fracture. It turns out that this condition is very stringent and 39

58 generates fractures which are almost non-intersecting. To relax this condition scaling factor less than one can be used which will generate fracture even if only a fraction (equal to scaling factor given as input) of total number of grids intersected by the fracture satisfy the moving window condition. The two scaling factors determine the degree of adherence to fracture intensity maps and prior existing fracture map and is an input to the model. Its affect is shown in Figure 3-14 below. 3.5 WORK FLOW FOR FRACTURE SIMULATION The workflow for sequentially simulating the six fracture sets is described pictorially in Figure 3-8 through Figure 3-13 below. The figures show one slice through a 3 D simulation model. The areal intensity is honored in the average over several realizations of the fracture model. Intensity Map (Set 1) Fracture Set 1 Figure 3-8 Generation of first fracture set. The fracture intensity map is input to the simulation. One slice through the fracture model is shown on the right. 40

59 Smoothened Fracture Map Intensity Map (Set 2) Fracture Set 2 Figure 3-9 Generation of second fracture set. The smoothened fracture map corresponding to the first fracture set is provided as conditioning information along with the intensity map for the second fracture set. Smoothened Fracture Map Intensity Map (Set 3) Fracture Set 3 Figure 3-10 Generation of third fracture Set. The smoothened fracture map corresponding to the first and second fracture sets is provided as conditioning information along with the intensity map for the third fracture set. The simulated map on the right shows fractures belonging to the first, second and third sets. 41

60 Smoothened Fracture Map Intensity Map (Set 4) Fracture Set 4 Figure 3-11 Generation of fourth fracture set. The smoothened fracture map corresponding to the previous fracture sets is provided as conditioning information along with the intensity map for the fourth fracture set. One slice through the fracture model consisting of all fracture sets up to the fourth is shown on the right. Smoothened Fracture Map Intensity Map (Set 5) Fracture Set 5 Figure 3-12 Generation of fifth fracture set. The smoothened fracture map corresponding to the previous fracture sets is provided as conditioning information along with the intensity map for the fifth fracture set. 42

61 Smoothened Fracture Map Intensity Map (Set 6) Final Fracture Model Figure 3-13 Generation of last fracture Set. The smoothened map corresponding to all prior fracture sets is used as conditioning information along with the intensity map for the sixth fracture set. The process of sequentially simulating the fracture sets replicates the geological sequence of formation of fractures. It can be noticed that the fractures hardly intersect each other. To make a model with more fracture intersections, the size of the averaging windows and the weight assigned to the smoothened fracture maps were adjusted. Figure 3-14 below show two different realizations obtained by assigning different weights to prior existing fractures. The number of intersections is much greater in Figure 3-14(b), where no weight is assigned to prior existing fractures. This process can be repeated to generate multiple realizations of the fracture model. The working code for the object based simulation of fracture networks is given in Appendix A. 43

62 (a) (b) Figure 3-14 Different realizations of fracture model with different weight given to the prior existing fractures. a) realization obtained corresponding to scaling factors of one assigned to the map of pre-existing fractures, b) realization when no weight given to prior existing fractures (i.e scaling factor of zero). 3.8 CONCLUSION This modeling process can rapidly generate multiple realizations of fractures. The fractures represented here are field scale fractures, which extend over multiple grid blocks. But as we will discuss in the next chapter, this modeling approach on a gridded system has serious limitations with realistic representation of the fractures that typically have apertures that are considerably smaller than the grid dimensions. It is not possible to find the effective permeability of individual grid blocks on which we need to perform flow simulations. Due to these limitations, we will resort to a hierarchical modeling approach on an un-gridded domain. This new approach is introduced in the next chapter. 44

63 Chapter 4: Hierarchical Modeling of Flow in Fractures In the last chapter object based modeling was used to model the discrete fracture network (DFN). The algorithm is capable of generating multiple realizations of fracture network at field scale. However, the basic motive of modeling, which is to generate permeability values for individual grid blocks in the simulation domain and quantifying uncertainty in effective permeability, cannot be fulfilled using this modeling approach. Some of the limitations of the object-based modeling approach are discussed below. 4.1 LIMITATIONS OF DISCRETE FRACTURE NETWORK MODEL In object based modeling, fractures are generated as elliptical objects. These objects fill up the domain until they satisfy the target proportion of fractures (fracture intensity). Every realization of the generated model consists of fractures distributed over the entire field. When fractures are generated using object based modeling approaches, the minimum size of the geocellular model is itself too coarse to represent small-scale fractures that can fall within the modeling cell. Yet, the distribution of these small-scale fractures controls the effective permeability of the grid block. Only by characterizing these small-scale micro fractures, quantification of uncertainty in effective permeability can be accomplished. The aperture of the fractures represented using an object-based approach is controlled by the resolution of the geocellular grid. The apertures of fractures are generally of the order of millimeters, whereas lengths can be of the order of miles. Due to this extreme aspect ratio between fracture aperture and fracture length, it is impossible to represent both on any practical grid scale. Representation of fractures on gridded domain results in unrealistically large fracture apertures. This produces fictitious intersections between the fractures resulting in flawed results for any type of connectivity analysis. 45

64 An alternative would be to model flow through the fractured media using a continuous domain that is not discretized into grid blocks. The continuous coordinates of fractures are used to move particles through the media. This approach is elaborated in the next section. 4.2 A NEW MODELING APPROACH In order to address the limitations of the previous object-based modeling approach, a unique hierarchical flow-modeling scheme is implemented. The main aspects of this scheme are described below Hierarchical Modeling In order to take into account the effect of small scale fractures, a hierarchical modeling approach is adopted. This consists of developing separate fracture models at two different length scales - the microfracture model and the macrofracture model. Microfractures are small-scale fractures lying within a grid block. These fractures typically fall below the resolution of seismic, and thus cannot be observed through seismic data and yet they govern the effective permeability of individual grid blocks. On the other hand, macro-fractures are field scale fractures extending through multiple grid blocks. They affect the connectivity over long length scales. As will be seen shortly, modeling fractures at different length scales simplifies the modeling process. Generating fracture models at each scale is extremely fast and less memory demanding. At the end, different realizations of micro and macro-fractures can be superimposed on each other to quantify total uncertainty in fracture distribution Continuous Space Modeling As stated earlier, there exists an extreme aspect ratio between fracture length and fracture aperture. For example, for a fracture of length one kilometer and aperture one 46

65 millimeter, the aspect ratio is In order to represent this fracture on a gridded system, the grid dimension cannot exceed one millimeter. In such a case, representation of a single fracture will require 10 6 grid blocks. This clearly shows that it is practically impossible to represent fracture networks on a gridded domain. Therefore to have a realistic representation of fracture aperture, all further modeling is done on a spatially continuous domain. Fractures are represented as elliptical discs in 3-D space, with their thickness equal to actual fracture aperture. Since fractures are no longer being represented on a grid, any length and aperture can be taken. For each fracture, the properties required for complete characterization of the fracture are the centre of the ellipse (x,y,z), dip, azimuth and length. These properties can be made to follow any distribution. This is a very powerful paradigm in terms of fracture representation. Now the entire modeling can be done even without explicitly representing the fractures Random Walker Simulation To find the upscaled permeability of the discrete fracture network, a new and unique random walker simulation is used. This simulation works in a continuous space domain. Random walkers are moved from one side to the other side of the domain. Each step of the walker movement is determined stochastically using preset transition rules. The statistics generated by the random walker simulation is an indicator of system connectivity, and the ease with which the fluid can flow through the system. These statistics correlate directly to equivalent permeability of fractured system. The strength of the technique is that it can be used to upscale the fracture network on a continuous space domain and is extremely fast compared to any other upscaling technique. Such a upscaling is not possible using a usual flow simulator which works on a gridded system. 47

66 4.3 BASIC FORMULATION For all the subsequent modeling processes, fractures will be represented as elliptical objects in 3-D space with thickness equal to fracture aperture. Their thickness is extremely small compared to other dimensions of the ellipse and can be neglected for all practical considerations. In such a case the fracture can be considered as an elliptical plane, and can be directly dealt with using the analytical equation of an ellipse. The ultimate objective is to model the movement of random walkers through the media containing fractures. However, instead of moving the walkers along a discrete grid, the attempt will be to model the movement of walkers along continuous spatial coordinates. For this it is necessary to evaluate at each step of the particle movement if the particle is inside a fracture and the extent of the particle s movement through a fracture. The geometric equations below are useful to find the extent of fractures and model their intersection with each other. Derivation of these equations is discussed in the rest of this chapter Equation of an ellipse The aim is to find the equation of ellipse with any given arbitrary azimuth and dip. Azimuth is defined as the angle between the major axis of the ellipse and the geographical north, whereas dip is the angle made by the plane of the ellipse to the horizontal plane. The general equation of the ellipse can be written as: x a y + =1 b (4-1) The equation 4-1 represents an ellipse with major axis a, minor axis b (a > b) and centre at the origin. It is shown in Figure 4- below. 48

67 y 2b (0,0) 2a x Figure 4-1 Ellipse with azimuth 90 0 and dip 0 0. The major axis which corresponds to length of fracture is 2a and the minor axis is 2b Equation 4-1 depicts a fracture with azimuth 90 0 and dip 0. Next the ellipse is rotated about Z-axis to an arbitrary azimuth angle. This is shown in the Figure 4-2 given below. North Y y X Azimuth (90-Azimuth)=θ x East Figure 4-2 Rotated ellipse with its major axis rotated by an angle θ about the Z axis (perpendicular to the plane of ellipse, not shown in figure) in counter clockwise direction. The equation of the ellipse in rotated co-ordinates is given by X a Y b = (4-2) 49

68 Lowercase x and y show the original co-ordinates whereas uppercase X and Y show the transformed co-ordinates. The relation between original and transformed co-ordinates is given as X = x Cos θ + y Sinθ Y = x Sin θ + y Cosθ (4-3) Equation of the ellipse can now be written in terms of original co-ordinate system as 2 2 ( xcosθ + y Sinθ) (- x Sinθ+ ycosθ) + = 1 (4-4) 2 2 a b Equation 4-4 represents an ellipse with its centre at origin and rotated by an angle θ about the Z-axis in counter clockwise direction. Now the ellipse is rotated about its major axis to have a dip φ. This is shown in Figure 4-3 below Dip(φ) Azimuth(θ) North Plane Z =0 Figure 4-3 Fracture plan rotated along its major axis by dip angle φ. The ellipse has been rotated about its major axis by an angle φ. The ellipse which was initially in 2-D after rotation lies in 3-D. The co-ordinates after the azimuth and dip rotations are given as below: 50

69 X Z = X Cosϕ = X Sinϕ (4-5) Y = Y Where X, Y and Z are the coordinates after the azimuth rotation, and X, Y and Z are the coordinates after the dip rotation. Using equation 4-5 the equation of the rotated ellipse can be written in 3-D as: X acos = φ Y b (4-6) Z = X Tan ϕ Equation 4-6 in terms of original x, y, z co-ordinates and shifting the centre to any arbitrary point ( xc, yc, zc) is given as (( ) ) ( ) ( ) 2 2 x- xc Cosθ + ( y- ycsinθ - x- xc Sinθ+ ( y- yc) Cosθ + = 1 (4-7) a Cos ϕ b (( ) ) z = zc x xc Cosθ + (y yc)sinθ Tanφ (4-8) Equations 4-7 and 4-8 satisfactorily define the coordinate of any point lying within the rotated ellipse in 3-D Equation of a plane containing the ellipse Equation of a plane in 3-D is defined by the normal vector to the plane. General equation of plane is given as n. X = 0 ( Xc) (4-9) 51

70 Where, n Normal vector to the plane Any point on the plane ( Xc, Yc, Zc ) Consider an ellipse with an arbitrary dip φ and azimuth (90- θ).this is shown in Figure 4-4Figure 4-3. Equation of the plane containing this ellipse in terms of its dip and azimuth is given by: ( ) ( ) X Xc Sinϕ. Cos θ + ( Y Yc) Sinϕ. Sin θ + Z Zc Cos ϕ = 0 (4-10) Figure 4-4 Figure showing the plane in which the fracture lies. The ellipse represents a fracture with any arbitrary dip φ and azimuth θ Distance of a point from the fracture plane For any point to lie inside the fracture its distance from the plane (containing the fracture) must be less than the aperture of the fracture. If the distance is greater than the aperture, the point lies in the matrix. Let ( X1, Y1, Z1) be the co-ordinates of the point. Distance from plane of fracture is given by Equation

71 Distance = ( ) ϕ θ ( ) ϕ θ ( ϕcosθ) ( ϕ Sinθ) ( Cos ) X1 Xc Sin. Cos + Y1 Yc Sin. Sin + ( Z1 Zc)Cosϕ ( ( Sin. + Sin. + φ )) = 1 ( ). ( ). ( ) = X1 Xc Sin ϕ Cosθ + Y1 Yc Sin ϕ Sinθ + Z1 Zc Cos ϕ (4-11) In the above equation the distance can be both negative and positive depending on which side of the plane the point lies. For comparison to fracture aperture, absolute value of the distance is taken Test for the point if it lies in fracture Even if the distance of the point from the fracture plane is less than the aperture, the point may not lie in the fracture. It depends on the location of the fracture on the plane. To check if the point lies inside the fracture, the equation of the ellipse is used. The following two steps are used to check if the point lies inside fracture. (1) Find a point on fracture plane such that the line joining the test point and the point on the plane is normal to the plane. Let its co-ordinates be( X1, Y1, Z1 ). X 1 = X 1 Distance* Sin ϕ. Cosθ Y 1 = Y 1 Distance * Sin ϕ. Sinθ (4-12) Z 1 = Z 1 Distance * Cos ϕ (2) For the point ( X1, Y1, Z1 ) using the equation of ellipse. on the plane, find if it lies in the fracture or not by If (( 1- ) + ( 1- ) ) -( 1- ) + ( 1- ) ( ) 2 2 X Xc Cosθ Y Yc Sinθ X Xc Sinθ Y Yc Cosθ + -1< 0 (4-13) acosϕ b 53

72 Then the point lies inside the fracture. Using Equation 4-12 and Equation 4-13 it can be tested if any given point lies inside the fracture or not. It must be noted that the distance used in the Equation 4-12 should be used with its proper sign (-/+) as obtained from Equation Intersection of two ellipses in 3-D Another complication that arises when modelling the movement of particles through a continuous system of coordinates are possible fracture intersections that might lie in the path of the particle. Consider two ellipses in 3-D with the following attributes Dip Azimuth Centre Major Axis Minor Axis Ellipse 1 Ø1 θ1 ( X1,Y1,Z1 ) a1 b1 ( ) Ellipse 2 Ø2 θ 2 X2, Y2, Z2 a2 b2 The intersection of two ellipses in 3-D will be a line with both of its ends lying on anyone of the ellipses. It is difficult to solve the two quadratic equations simultaneously, so we proceed through a step by step process. The aim is to find the end points of the common line, joining the points of intersection of the two ellipses Step1: Equation of the plane containing the fractures and equation of fracture The equations of the planes containing the ellipses can be written as ( ) θ θ ( ) X -X1 SinØ1. Cos 1 + ( Y -Y1) SinØ1. Sin 1+ Z -Z 1 CosØ 1 = Plane 1 ( ) θ θ ( ) X -X 2 SinØ2. Cos 2 + ( Y -Y 2) SinØ2. Sin 2+ Z -Z 2 CosØ 2 = 0 - Plane 2 (4-14) These two equations when solved simultaneously yields the line formed by the intersection of these two planes. It is evident that the common line of intersection must lie on both the planes and therefore any point on the line must satisfy the equation of both 54

73 planes. However this line may or may not intersect the ellipses which lie on either of these planes. Furthermore even if it intersects both the ellipses there may not be any common line segment lying inside both the ellipses. The equation of the two ellipses can be written as. (( X-X1)Cosθ1 + ( Y-Y1) Sinθ1) (-( X-X1) Sinθ1 +(Y-Y1) Cosθ1) a1 Cos Ø =1 b (( X-X2)Cosθ2 + ( Y-Y2) Sinθ2) (-( X-X2) Sinθ2 +(Y-Y2) Cosθ2) a2 Cos Ø =1 b (4-15) (4-16) The intersection between two fractures has to satisfy Equations 4-15 and 4-16 (equations of the ellipses) as well as Equation 4-14(equation of the plane) jointly. In order to accomplish this, the line of intersection between two fracture planes is solved first using Equation Subsequently, the intersection of that line with the two fracture ellipses is calculated Step2: Points of intersection between the fracture and common intersecting line First we solve the equation of the line with ellipse 1. The equation of line expressed in Equation 4-14 has been re-written, as shown in Equation 4-17, to simplify the ensuing discussion. ax 1 + a2y + a3z = k1 b1x + b2y + b3z = k2 (4-17) 55

74 Where, ( θ ) ( ) ( θ ) ( ) a 1 = cos 1 * sin Ø1 b 1 = cos 2 * sin Ø2 ( ) θ ( ) a 2 = sin ( θ1)* sin Ø1 b 2 = sin ( 2)* sin Ø2 ( ) ( ) a 3 = cos Ø1 b 3 = cos Ø2 k 1 = xc1* a 1 + yc1* a 2 + zc1* a3 k2 = xc2* b1 + yc2* b2 + zc2* b3 Eliminating z from Equation 4-17 we get a relationship between x and y co-ordinates given as y = px + r (4-18) ab 1 3 kb 1 3 b1 k 2 Where p = a3 and r = a3 a2 b3 a2 b3 b 2 b 2 a3 a3 Putting Equation 4-18 in Equation 4-15 of the ellipse we get a quadratic expression in x as 2 Ax + Bx + C = 0 (4-19) Where, 2 ( θ θ ) ( θ θ ) ( ) A = p Sin 1+ Cos 1 b1 + p Cos 1 Sin 1 a1 Cos Ø1 ( θ θ ) ( θ θ θ ) B = 2 [ Cos 1+ p Sin 1 r Sin 1 X1 Cos 1 Y1 Sin 1 b1 ( p Cos θ1 Sin θ1 ) ( r Cos θ1 X1 Sin θ1 Y1 Cos θ1 ) ( a1 Cos Ø1) + + C = ( r Sinθ1 X1 Cosθ1 Y1 Sinθ1) b ( r Cos θ1 X1 Sin θ1 Y1 Cos θ1) ( a1cosø1 ) ( a1 b1 Cos Ø1) ] 56

75 The discriminant of the quadratic expression in Equation 4-19 is given as 2 ( 4 ) Discriminant = B AC (4-20) If the discriminant is positive then we have real roots for the quadratic equation. In other words the line intersects the ellipse only if the discriminant is greater than zero.in this case, the roots (points of intersection) are given by Roots ( x1, x2 ) = B + B 2 A 2 / 4 AC (4-21) Equation 4-21 gives co-ordinates of point of intersection of the plane with the first ellipse. Following the same equations the points of intersection of plane with second ellipse can be found. Thus in total four points of intersection are obtained, two for each ellipse (conditioned to the fact that all the four points of intersection are real). These four points specify the limits of the intersection of the ellipse with the line formed by the intersection of the planes on which these ellipses lies Step3: Points of intersection between fractures It s clear that if any one of the ellipses doesn t intersect the common line; there can be no common line segment between the fractures. On the other hand even though real points of intersection between the planes containing the fracture and each individual fracture are obtained, it doesn t ensure the existence of a common line segment. If such a line segment exists its end points must lie inside one of the ellipse and on the boundary of the other ellipse. This can be verified as 1. For the first two points of intersection with first ellipse, check whether they lie inside second ellipse 2. For the second two points, check whether these points lie inside first ellipse. 57

76 These checks can be performed easily using Equation 4-13 for each of the four points of intersection. The result of this check can be either no common point when none of the points of intersection lies inside the other ellipse or two common points. The line joining the two common points is the line segment lying in both the ellipses and is the line of intersection of two fractures. 4.4 REMARKS The equations discussed above will be used for the random walker simulation discussed next, that will be used to calculate effective properties of the fracture media. Details of the procedure, to model the displacement of random walkers through microfractures are discussed in detail in the next chapter. 58

77 Chapter 5: Microfracture Modeling 5.1 OVERVIEW Microfractures can be defined as small scale, sub seismic fractures extending within a grid block. Nevertheless it is important to model these microfractures, because of their influence on the effective permeability of each grid block. Modeling microfracture is the key to proper characterization of anisotropy and uncertainty in the permeability value for the grid block. Each grid block has an associated fracture intensity value that is an indicator of the degree of fracturing in that region. In this work, it is assumed that the given fracture intensity maps are representative of both the degree of micro and macrofracture intensity in a region. In the case of sub-grid microfractures, the fracture intensity value is assumed representative of the number of micro-fractures within the block. This assumption is justified because higher density of macro-fractures implies increased tectonic activity in the region and that in turn results in more intense micro-fracturing. Using this assumption, fracture intensity value for an individual grid can be used for modeling microfractures. Once the microfracture model has been generated, the effective permeability of the microfracture distribution is calculated using the random walker. This effective permeability is subsequently specified as the matrix permeability for computing the effective permeability of the macrofractures. Different micro fracture models are developed for each grid block depending on their fracture intensity. For each grid (with a given intensity) block multiple realizations of microfracture model are generated. These multiple realizations represent the uncertainty in fracture distribution for that grid. The uncertainty in equivalent permeability of the grid can be quantified by performing random walker simulation on all microfracture realizations with the same 59

78 fracture intensity. The uncertainty quantification is presented in terms of maximum, minimum and average permeability values corresponding to a specific fracture intensity. The process is repeated for all the values of fracture intensity. At the end of it, a bracket for permeability variation is obtained for each grid block. First I will discuss in detail the simulation of microfractures and then move on to random walker simulation used for obtaining effective properties of the microfracture network. 5.2 SIMULATION OF FRACTURE NETWORK As stated earlier each fracture is being represented using the equation of an ellipse. The fracture intensity of the grid is taken as an input. Microfracture simulation comprises the following steps Fracture Property Generation Centre of the Fracture A single flow simulation grid block is divided into 100x100x100 grid blocks in order to represent the microfractures. The size of the domain is big enough to contain sufficient number of microfractures to have a statistically stable distribution of fractures. The centre of fracture (Xc,Yc,Zc) is generated randomly within the domain implying that there is no spatial preference for micro fractures within the simulation grid block. However the abundance of microfractures definitely depends on its vicinity to macro fracture which is given by fracture intensity of that grid block Fracture Length The length of the fracture is the length of the major axis of the ellipse. The ratio of major to minor axis can replicate fractures with different shapes. 60

79 The length distribution follows a power law distribution with an exponent of 3.1. This exponent value is consistent with the value used for the object based simulations. The maximum length of the microfracture will be half the length of the domain size. This ensures that all the fracture generated lie within the domain (grid) Dip and Azimuth Both of these parameters follow a normal distribution with mean and variance specified in Table Aperture The aperture is picked from a uniform distribution with a linear dependence on length of the fracture. The following relation is used for the fracture aperture. Lfrac Aperture = A + A A * p * 1 ( ) min max min ( + Lfrac ) (5-1) Where, A min = Minimum Aperture A max = Maximum Aperture L frac = Length of Fracture p = Random number (0, 1) Equation 5-1 ensures that the aperture generated for the fracture is always between the maximum and minimum aperture values. The maximum and minimum values are based on some of the observed fracture apertures in cores. The maximum value is around 2-3mm while minimum value is 1μm. In order to avoid strict proportionality between aperture and length, a random number has been inserted in to the equation. 61

80 5.2.2 Fracture Set Generation Fractures/ellipses can be generated stochastically to fill up the 3-D domain using the properties described above. At the sub-seismic resolution, it is assumed that generation of new microfractures is independent of prior existing microfractures. And therefore the generation of a new fracture set is independent of any prior existing fracture sets. This allows microfractures to be able to cut across each other without any restrictions and form an ideal 3-D fracture network for fluid flow. When the desired fracture intensity is achieved, the generation of ellipses is stopped. The fracture intensity of the domain is given by: Surface Area of the Fractures Fracture Intensity = BulkVolume π ab( Area of anellipse) = Volume of Domain, (5-2) Where, a = Major axis of the ellipse b = Minor axis of the ellipse At the end of the simulation we have microfractures that reflect the properties for the fracture set. The actual number of fractures generated depends upon the fracture intensity specified as an input Handling Edge Effects In practice, we will never have a standalone volume/domain in which all the fractures lie completely inside the system. Therefore in order to make a small domain representative of a large extending infinitely system, periodic boundary condition is imposed. Whenever a fracture cuts any of the boundaries /surface, it is replicated exactly on the opposite side. This simulates both the fractures going out of the domain and also 62

81 the fractures cutting into the domain from outside. Using periodic boundary condition also ensures the conservation of fracture area lying inside the domain and hence the fracture intensity of the domain. Figure 5- schematically shows this implementation to handle edge effects. Figure 5-1 Fractures cutting across the edges of the simulation block. The fractures that cut across the boundary of the domain are replicated on the other side. Replicated fractures are shown to be of same color. First the fracture is checked to see if it cuts any of the boundaries. If not there is no need for replicated ellipse. However if there is an intersection, there can be three different cases 1. If the fracture cuts only one boundary, there will only be one replicated ellipse. 2. If the fracture intersects 2 boundaries; there will be three replicated ellipses. 3. If the fracture cuts all the 3 boundaries, there will be seven replicated ellipses 63

82 Check for intersection to the boundary ( Xc Yc Zc ) ( ) Let,, be the centre of the ellipse and Xm, Ym, Zm be the dimensions of the domain. Using the equation of ellipse the maximum and minimum coordinates in all the three directions can be found as ( ) Max / Min x coordinate of ellipse Xmax / Xmin = (Xc+/- b Sin θ + a Cos θ) Cosϕ ( ) Max / Min y coordinate of ellipse Ymax / Ymin = (Yc+/- a Sin θ + b Cos θ ) ( ) Max / Min z coordinate of ellipse Zmax / Zmin (5-3) = (Zc+/- b Sin θ + a Cos θ) Sinϕ For each ellipse, the maximum and minimum extents in each co-ordinate direction can be determined to find if it intersects the domain. It must be noted that for the replicated ellipse at least one of the centres will be outside the domain Centers of replicated ellipse: Let (X, Y, Z ) be the centre of replicated ellipse and (X, Y, Z) be the centre of original ellipse. Xm, Ym, Zm are the dimensions of the domain in X, Y and Z directions respectively. Case 1: When the ellipse cuts one of the domain boundaries If it cuts only one of the boundaries (for example in X direction), then there will be only one replicated ellipse. The centre of the replicated ellipse will be Or (5-4) 64 ( 0) X c = Xc + Xm Y c = Yc Z c = Zc if Xmin < ( ) X c = Xc Xm Y c = Yc Z c = Zc if Xmax > Xm

83 Similarly the centres of the replicated ellipses can be found if it intersects the domain in Y and Z directions. Case 2: When the ellipse cuts two domain boundaries If it cuts two boundaries simultaneously (for example in X and Y direction), then there will be three replicated ellipse. The case occurs when ellipse cuts one of the edges of the domain. The centre of the three replicated ellipse in this case are given below: ( ) X c = Xc Xm Y c = Yc Z c = Zc if Xmax > Xm Or (5-5) X c = Xc + Xm Y c = Yc Z c = Zc if Xmin < 0 ( ) ( ) X c = Xc Y c = Yc Ym Z c = Zc if Ymax > Ym Or (5-6) X c = Xc Y c = Yc + Ym Z c = Zc if Ymin < 0 ( ) X c = ( Xc-Xm ) or ( Xc + Xm ) Y c = ( Yc-Ym ) or ( Yc +Ym ) Z c = Zc (5-7) In exactly the same way the centres of the replicated ellipse can be determined if it cuts (X&Z) and (Y&Z) directions respectively. Case 3: When the ellipse cuts all the three domain boundaries If it cuts all the three boundaries (X, Y & Z) simultaneously, then there will be seven replicated ellipse. This case occurs when ellipse cuts one of the corners of the domain. The centre of the seven replicated ellipse in this case are given below 65

84 ( ) X c = Xc Xm Y c = Yc Z c = Zc if Xmax > Xm Or (5-8) X c = Xc + Xm Y c = Yc Z c = Zc if Xmin < 0 ( ) ( ) X c = Xc Y c = Yc Ym Z c = Zc if Ymax > Ym Or (5-9) X c = Xc Y c = Yc + Ym Z c = Zc if Ymin < 0 ( ) ( ) X c = Xc Y c = Yc Z c = Zc Zm if Zmax > Zm Or (5-10) X c = Xc Y c = Yc Z c = Zc + Zm if Zmin < 0 ( ) X c = ( Xc-Xm ) or ( Xc + Xm ) Y c = ( Yc-Ym ) or ( Yc +Ym ) Z c = Zc (5-11) X c = ( Xc-Xm ) or ( Xc + Xm ) Y c = Yc Z c = Zc+Zm) or ( Zc-Zm) (5-12) X c =Xc Y c = ( Yc-Ym ) or ( Yc +Ym ) Z c = Zc+Zm) or ( Zc-Zm ) (5-13) ( ) ( ) ( ) ( ) Z c = (Zc + Zm ) or ( Zc Zm) X c = Xc Xm or Xc + Xm Y c = Yc Ym or Yc + Ym (5-14) The process stated above is used to implicitly generate multiple realizations of the micro fracture models which are then investigated through movement of random walkers to analyze the connectivity and find the equivalent permeability of the system. 66

85 The code and output for the simulation of microfractures is listed in Appendix B. It is to be emphasized that because all these realizations are on continuous (un-gridded) spatial coordinates and therefore cannot be visualized using regular pixel based plots. 5.3 RANDOM WALKER SIMULATION The next task is to upscale the micro fracture models and calculate the upscaled permeability value for each flow simulation grid block. For each value of fracture intensity, multiple realizations of micro fracture model are considered in order to obtain a distribution of upscaled permeability and quantifying the uncertainty in the effective permeability value for that grid block. In order to calculate this effective permeability, percolation analysis is performed on each micro fracture model to assess if the microfracture network in the block forms a connected pathway across the block. The effective permeability of the block is calculated proportional to the arrival time characteristic of the walkers for those blocks that exhibit percolation. This procedure is described in the next section Percolation Analysis Percolation analysis tests the connectivity of the fracture network. A fracture network is said to be percolating if there exists at least one connected path through the fractures from one side of the domain to the other. This analysis is very relevant because the upscaled permeability is largely a function of fracture connectivity. Even if the domain has large number of fractures but they don t form a connected path across the block, then there is hardly any effect of fractures on the upscaled permeability of the system. Furthermore, larger the number of fracture connected paths, higher is the equivalent/upscaled permeability of the system. 67

86 Percolation analysis is performed by moving random walkers from one side of the domain to the other side with walkers moving only through the fractures. The only constraint put on the movement of the walker is that it must move towards the other side of the domain, following the imposed pressure gradient across the system. Thus if the walker has no further path to move forward through the fractures it just stops. Only those walkers which find a connecting path along the fractures are able to reach the other side of the domain. The assumption for this analysis is that the matrix is non-conducting and fluid flow is only through the fractures. This assumption applies very well for a carbonate reservoir where fluid flow is only through the fractures and matrix has a very low permeability. This is shown schematically in Figure 5-2 below. Figure 5-2 Percolation of a random walker through a connected network of fracture. Random walkers are allowed to move only through the fractures and reach the other side only if a connecting path exists. To perform percolation analysis, a computer code has been developed in FORTRAN. The code takes the number of random walkers and the fracture model as 68

87 input and produces the percolation result as output. The source code and sample output is listed in Appendix B. The detailed algorithm used for random walker movement is discussed below Movement of Random Walkers 1. The fracture parameters are read from the input file. The number of fractures depends on the fracture intensity. 2. All intersection points among the fractures are calculated including all the replicated ellipses. All the fractures intersecting the inlet face of the block are found first. If the walker has to percolate then it needs to starts from one of the fractures that intersect the face. 3. At the start the random walkers are placed in one of the fractures on the inlet face of the domain. The starting fracture is selected randomly among all the fractures intersecting that face. 4. All possible paths for the next movement of the random walker are analyzed. The only constraint is that the movement should be in the direction of the pressure gradient across the block. There may be multiple options for the next movement of the walker. The number of paths depends on the number of fractures intersecting the fracture currently occupied by the walker. 5. Among the available paths one of the paths has to be selected for further movement of the walker. For the case of microfractures we are interested in knowing the percolation characteristic of the system. Therefore in this case all the possible paths of movement which allow the movement in the direction of the potential gradient are equally probable. And for that reason the next path 69

88 for movement is selected randomly among the available paths. Once the path is decided the walker is advanced to the next fracture. 6. The same process as in steps 4 & 5 are followed again to find the possible paths for further movement of walker to the next fracture. At each step it is tested if the fracture on which the walker resides intersects the other side of the domain or not. 7. If the walker reaches a fracture that intersects the other face of the domain, the walker percolates. On the other hand if the walker is somewhere in the middle and has no possible path for moving further, it cannot percolate. 8. The above steps (1-7) are repeated for all subsequent walkers. The number of walkers should be large enough to explore all possible connected paths through the fracture network to get a statistically stable result. 9. At the end of the simulation, the number of percolating walkers is compiled. For non-percolating systems this number is zero and the effective permeability due to microfractures is assigned to be zero md. For percolating system the number is non-zero and higher the number, the greater is the number of connected paths in the system and consequently, higher the effective permeability. 70

89 5.4 RESULTS OF RANDOM WALKER SIMULATION Below are the results of percolation analysis performed on different microfracture models for different fracture intensities. A detailed table of results is presented in Appendix B Permeability Upscaling and Uncertainty Quantification The aim of percolation analysis is to find the upscaled permeability of microfractures for each grid block based on its fracture intensity. It is assumed that there is a linear correspondence between the upscaled permeability and the fraction of walkers percolating, as both of them are the indicators of connectivity of the system. This linear correspondence is used to express the effective permeability as a weighted average of matrix and fracture permeability, where the weights are the fraction of walkers percolating as shown in Equation 5-15 = ( )* ( ) Upscaled Permeability Fraction of percolating Walkers Kfrac + Fraction of non percolating walkers * Kmat (5-15) Where, Kmat = Average Matrix Permeability Kfrac = Average Fracture Permeability For the present case we use the microfracture fracture permeability as millidarcy and matrix permeability as 0.1 millidarcy, which are typical for carbonate fracture-matrix systems. Here the fracture permeability is considered to be constant, which may not be true in reality. The permeability of individual microfractures is a function of fracture aperture and roughness which may also vary along the length of the fracture. 71

90 There can be many different distributions of fracture network having the same fracture intensity. This is represented by multiple realizations of fracture network for the same fracture intensity. When random walker simulation is performed on all these realizations, the output is not a single permeability but a range of permeability values, each one corresponding to a particular realization. The CDF (cumulative distribution function) represents uncertainty in effective permeability corresponding to the given fracture intensity. Figure 5-3 and Figure 5-4 show the CDF of effective permeability distribution for fracture intensities of 0.15 and 0.5. The same process is repeated for each value of fracture intensity to give an uncertainty envelope for all fracture intensities. 1 CDF of Effective Permeability Distribution for Fracture Intensity of CDF Permeability(md) Figure 5-3 Shows Cumulative Distribution Function of permeability for fracture intensity of The plot shows the bracket of permeability uncertainty arising due different fracture patterns for fracture intensity of

91 1 CDF of Effective Permeability Distribution for Fracture Intensity of CDF Permeability(md) Figure 5-4 Shows Cumulative Distribution Function of permeability for fracture intensity of 0.5. The order of permeability variation given as ratio of maximum to minimum permeability is much lower compared to Figure 5-3. The percolation study can be performed in all three directions on the same fracture network. The same network can be percolating in one direction but not in other directions. This may lead to permeability anisotropy in the system. Therefore to completely characterize the network it is important to study percolation in all the three directions. Percolation study has been performed on microfractures in all the three directions to quantify the permeability anisotropy of the system. Figure5-5 through Figure 5-7 shows the variation in maximum, minimum and average effective permeability in the three directions X, Y and Z with change in fracture intensity. 73

92 Permeability vs Fracture Intensity (Z-Dirn) Permeabilty (md) Avg Perm Maxm. Perm Minm. Perm Fracture Intensity Figure5-5 Plot shows maximum, minimum and average upscaled permeability corresponding to each fracture intensity value for flow in the z-direction. The range of permeability brackets the uncertainty in permeability due to uncertainty in fracture network Permeability vs Fracture Intensity (X-Dirn) Permeabilty (md) Avg Perm Max. Perm Min Perm Fracture Intensity Figure 5-6 Plot shows the maximum, minimum and average upscaled permeability versus fracture intensity for flow in the x- direction. 74

93 Permeability vs Fracture Intensity (Y-axis) Permeabilty (md) Avg Perm Max Perm Min Perm Fracture Intensity Figure 5-7 Plot shows the maximum, minimum and average upscaled permeability versus fracture intensity for flow in the y-direction. For each value of fracture intensity a maximum, minimum and average permeability is calculated in the three directions as shown in Figure5-5 through Figure 5-7. The minimum, maximum and average permeability value corresponding to a fracture intensity are computed over several realizations constrained to that value of fracture intensity. The maximum and minimum values come from the realization that has the maximum and minimum numbers of percolating walkers. While the average permeability, is the permeability corresponding to the average number of walkers percolating computed over all the realizations. Another important point to note is that the variation in permeabilities is higher at lower fracture intensities and all the three permeabilities merge to a single point at higher intensities and remain constant with further increase in fracture intensity. As the number 75

94 of fractures increase beyond a limit in a percolating medium, the effective permeability is limited by the maximum permeability of individual fracture and remains constant. The values obtained above can be used to populate microfracture permeability in all the grids. Based on the fracture intensity in each grid block, the permeability tensor due to microfractures can be assigned Permeability Anisotropy On comparing the upscaled permeability obtained in the three directions, anisotropy in the permeability values can be observed. Figure 5-8 and Table 5-2 show the variation in average permeability in the three directions with fracture intensity. Permeabilty (md) Average Permeability vs Fracture Intensity (All Directions) Z Dirn Y Dirn X Dirn Fracture Intensity Figure 5-8 Variation in average effective permeability in three directions as the fracture intensity is varied. The permeability in z-direction is higher than that compared in X and Y directions. 76

95 Table 5-2 Anisotropic permeability values as a function of fracture intensity. Fracture Intensity Kx(md) Ky(md) Kz(md) The z-direction permeability is much higher compared to that in the x and y directions. The x and y direction permeabilities exhibit very similar values of permeabilities showing that medium is isotropic in areal sense but is anisotropic vertically. In the present case, we expect the x and y direction permeabilities to be equivalent because the azimuth distribution of the fracture sets span 360 o. However in the z- direction, the mean dip direction for all the six fracture sets is around Thus, there is a directional preference in the dip distribution of the fractures and consequently, the permeability obtained in the z-direction is quite different from that in the x and y directions. This results in the permeability in z direction to be much higher than that in the x and y directions. To confirm this observation, the random walker simulations were run with the dip angle close to 0 0. Hardly any connectivity was observed in the z 77

96 direction in this case implying that vertical anisotropy is due to preferential dip orientation of fractures Percolation Threshold The percolation analysis helps define percolation thresholds in all three directions. We define the minimum and the maximum percolation threshold in terms of the fracture intensity. Minimum percolation threshold is the maximum fracture intensity below which there is no connectivity in the system. This is very important because if the fracture intensity in a given reservoir is less than the minimum threshold then fractures do not influence the flow in the reservoir. In this case there is no need for characterizing and modeling the fractures. In a similar fashion the maximum percolation threshold is defined as the maximum fracture intensity above which there is always a percolating network in the system. In other words if the fracture intensity is above the maximum threshold value, the flow in the grid block is entirely through and the fractures have to be studied and modeled in detail. There is a small range of values of fracture intensities between the maximum and minimum percolation threshold that signify the transition of the system from a nonpercolating to a percolating system. If fracture intensity lies in this range, the impact of microfractures on flow cannot be assigned with certainty. The thresholds can be different in different directions. Based on the analysis Table 5-3 and Figure 5-9 show the threshold values obtained in the three directions. Since the z-direction exhibits the maximum connectivity the thresholds are lower in that direction and are almost equal in the x and y directions. It is important to note that these 78

97 thresholds are strongly dependent on the orientation and number of fracture sets present. If the orientations or the distribution of fracture sets change, different values of threshold intensities could be found. However the analysis performed is general and can be done on any fracture network to find the critical thresholds. Table 5-3 Percolation Threshold in the three directions Threshold/Direction Z Dirn X Dirn Y Dirn Minimum Percolation Threshold Maximum Percolation Threshold Fraction of Realizations showing Percolation Percentage Percolation vs Fracture Intensity (All Directions) Y dirn Z dirn X dirn Minimum Percolation Threshold Maximum Percolation Threshold Fracture Intensity Figure 5-9 Plot showing the percolation threshold in three directions as a function of fracture intensity. Z direction has the lowest values indicating a much greater permeability in Z direction compared to other directions 79

98 5.5 VALIDATION OF RESULTS A validation study was attempted to verify the accuracy of the results obtained by percolation analysis. The best approach is to compare the effective permeabilities obtained from the percolation analysis to the actual flow based upscaled permeability on fine scale models. For this task, the commercial simulator (CMG) has been used. Different sets of fracture models with different fracture intensities were created. These models were created on a smaller domain exhibiting a simple fracture network. The models were intentionally kept small and simple to keep the simulator run time manageable. The basic steps for the calibration process are listed below Calibration Procedure 1. Fracture models are generated in a 20x20x20 gridded domain using ellipsim. Ellipsim code first generates the centres of the ellipse/fractures and then populates the grids with 1 s if the fracture lies in it. Otherwise the grids have a value At the end of simulation two different output files are generated. The first one is the fracture simulation file containing the fracture information as Boolean objects on a gridded basis. This served as permeability input for the simulator (CMG). The second file contains the fracture properties (centre, dip, azimuth and length). This will be input to the random walker code to perform percolation analysis. 3. The grid blocks in the fine scale model marked as fractures were given a permeability of 10000md whereas the matrix was given 0.1md. This is done to ensure that the flow takes only through the fractures and the matrix is nonconducting. This file is then loaded as the permeability field for the simulation. 4. Flow was simulated maintaining the opposite faces as constant pressure boundaries. Once steady state is established the equivalent permeability of the system can be calculated using the flow rate calculated by the simulator. 80

99 5. The second file containing the fracture properties is run using the random walker code to find the fraction of total number of walkers that percolate. In order to approximate the fractures as represented in the simulator, the aperture of the fracture was taken to be of the size of 1 grid block (unrealistically large). 6. The above steps 1-5 were performed for multiple realizations from ellipsim for different fracture intensities. The attempt is to find a correlation between the permeability obtained from flow simulation and the fraction of random walker percolating. Figure 5-10 shows the results obtained from this calibration study. The agreement between the two process for computing upscaled permeability is quite satisfactory. The plot clearly shows a linear correlation between the fraction of walkers percolating and the upscaled permeability of the system. Permeability obtained from random walker simulation (md) Comparision of Permeability obtained from simulator and Random Walker Simulation y = x R² = Upscaled Permeability obtained from Flow simulation Figure 5-10 Comparison of upscaled permeability values calculated using the random walker to that obtained by applying flow based upscaling. 81

100 5.5.2 Reasons for Disagreements It can be observed that the correlation developed in Figure 5-10 is good but not a perfect match. The following reasons might be reasons for this disagreement: 1. The flow in the simulator is only along the x-y-z direction, not necessarily along the fracture which may be in any oblique direction in which the actual fracture extends. A random walker is free to move in any direction the fracture extends whereas in a simulator there is no flow between diagonally placed grids. 2. The fractures taken as input for simulator are not the exact representation of the fractures in the random walker code. The basic reason being that the fracture in the simulator has a minimum thickness of one grid, which may not be realistic. When we represent a continuous feature in a grid system, we always lose some accuracy. 5.6 PERMEABILITY MODEL FOR THE FIELD The effective permeability corresponding to the microfracture can be assigned to individual grid blocks based on the fracture intensity in that block to obtain permeability maps. The upscaled permeability values in each direction are shown in these maps Figure 5-11 through Figure There are permeability maps corresponding to the maximum, minimum and average permeability values in each grid block corresponding to the fracture intensity in that block. 82

101 Figure 5-11 Map of maximum upscaled permeability in the z direction computed using the random walker simulation. It can be clearly observed that the microfracture permeability is high near the anticlines where the fracture intensity values are high Figure 5-12 Map of average permeability in the Z direction computed using the random walker simulation. 83

102 Figure 5-13Minimum permeability map in the z direction. The pink color down the anticlines shows the matrix permeability. These are areas with low fracture intensity values falling below maximum percolation threshold. Figure 5-11 through Figure 5-13 bracket the uncertainty in upscaled permeability in the z-direction. Figure 5-14 Spatial variation in maximum permeability in x-direction. The pink regions show the areas below the minimum percolation threshold 84

103 Figure 5-15 Spatial variation in average permeability in x direction. The size of regions with permeability equal to the matrix permeability has increased. Figure 5-16 Spatial variation in minimum permeability in the x direction. Pink regions show the areas with fracture intensity below the minimum percolation threshold. Figure 5-14 through Figure 5-16 bracket the uncertainty in upscaled permeability in the x direction. 85

104 5.7 CONCLUSION In this chapter we dealt with modeling the microfractures. These fractures control the permeability of the grid block. Percolation analysis was used to find the upscaled permeability of microfractures. Important findings of this analysis are the permeability anisotropy and the existence of percolation thresholds. Based on this analysis the uncertainty in the permeability field can be quantified. These fractures are at scales smaller than the grid block in a flow simulation model. To capture the effect of the complete fracture distribution, the upscaled permeability corresponding to the microfractures have to be combined with large-scale fractures that span across several simulation grid blocks. In the next chapter we will deal with the modeling of macrofractures. The upscaled permeability due to both micro and macrofracture will be computed. 86

105 Chapter 6: Modeling Macrofractures by Integrating the Effective Properties of Microfractures 6.1 OVERVIEW Macrofractures can be defined as fractures that span more than one grid block. These are large-scale features that can sometimes be seen through seismic interpretation. They provide a resistance-free conduit for fluid flow over long distances. They can act as high permeability streaks adversely affecting waterflood and EOR processes or cause drilling problems for wells. In the previous chapter, microfracture models were developed and the upscaled permeability corresponding to a specific intensity of microfractures in a grid block was calculated. These calculated permeabilities do not take into account the influence of large-scale macrofractures. The subject of discussion in this chapter is the modeling of macrofractures and its integration with the microfracture model in order to come up with parameters for a flow simulation model such as effective permeability, matrix-fracture transfer coefficient etc. 6.2 MACROFRACTURE MODELING The aim is to simulate macro fractures at field scale using fracture intensity as conditioning data. The distribution of macro fractures for each fracture set should comply with the fracture intensity map for a given fracture set. The other parameters that are input are power law distribution for the fracture length and normal distribution for the fracture dip and azimuth. Ellipses in 3-D with a finite realistic aperture are used to represent macrofractures. The principal difference now is that, unlike microfractures, these fractures are long and extend through multiple grid blocks. Macrofracture modeling requires that the fracture intensity maps available on a gridded basis, be used as conditioning data for generating macro fractures on a 87

106 continuous space basis (since the fracture apertures cannot be represented on a grid). The following steps were performed for generating the macrofracture model Fracture Property Generation The center, length, dip and azimuth of the fracture are obtained by sampling from appropriate probability distributions. Dip and azimuth follow a normal distribution with mean and variance specified for each fracture set while length follows a power law distribution with exponent 3.1. The centers of the fracture are generated randomly inside the domain (145x215x66). This domain size is the same over which the fracture intensity data is available Conditioning to fracture intensity map Once the fracture properties of fracture are generated, all the grid blocks that are intersected by the fracture are identified. This can be calculated by using the equation of the ellipse in 3-D. If we decide to keep this fracture, the fracture intensities of all grid blocks intersected by this fracture are updated by one. This is consistent with the basic definition of fracture intensity that is fracture area per unit volume. In order to check whether this fracture should be kept or not, the simulated fracture intensity is checked against the fracture intensity map. This is done in the following way. 1. Two different macrofracture models are generated. The first one is generated on the gridded system by calculating the intersection of ellipses with individual grid blocks and the second having the properties of the ellipse in a continuous space domain. The first model is used for conditioning the macrofracture model to the fracture intensity map, as both are on a gridded system. While the second is the actual 88

107 macrofracture model used for performing random walk simulations in a continuous space framework. 2. The first fracture model is initialized with zero as all grid locations. As the fractures are simulated, the grid locations intersected by fractures are changed to ones 3. For generating each new fracture the sum of fracture intensity for all grid blocks intersected by the fracture is computed. This sum is computed over both the gridded fracture map and the conditioning fracture intensity map and is a function of the length, dip and azimuth of the fracture. 4. If the simulated fracture intensity is less than the prescribed fracture intensity there is scope for generation of more fractures. In such a case, the simulated fracture is retained. On the other end if simulated intensity is more, the fracture is rejected, and a new centre for the fracture is generated. This process of conditioning the fracture model ensures the use of fracture intensity map as soft data. Since the grids over which the sum is computed is arbitrary (depending on fracture length, dip and azimuth) the fracture intensity map is not honored strictly. The adherence is based on grouping of grid blocks rather than individual grid nodes. This makes it possible to use fracture intensity map as the soft data Generating Multiple Fracture Sets The conditioning process is continued till the fracture intensity condition for the given fracture set is satisfied. As one fracture set is completed the process moves on to generation of the next fracture set. The first fracture model generated from the prior fracture sets is smoothed using a window averaging process and subsequently the smoothened map is used for conditioning the next fracture set. This ensures that the new 89

108 fracture set will be conditioned to all the prior existing fracture sets. One realization of the macro fracture model is generated when all the fracture sets are generated. This method can be used to generate multiple realizations of macro fracture models, with each realization honoring the fracture intensity data. Again it is to be emphasized that while the conditioning is checked on a gridded basis, the final model in terms of fracture centers and intersections are used for the random walk simulation. 6.3 RESULTS OF MACROFRACTURE SIMULATION At the end of simulation two output files are generated. The first one is the implicit fracture model that contains all the properties of the ellipses (macro fractures) such as fracture center, aperture, orientation, etc. This will be used for subsequent analysis of the macro fracture network. The second file contains the explicit fracture model generated on a gridded system. This model is useful for visualization purpose and is only an approximate representation of the actual fracture network. It is approximate because the aperture of the fracture represented in these models is much larger than the actual aperture of the fracture. It is therefore useful only for visualization of the fracture network. These models are loaded for reservoir intervals bound by the actual horizons. A commercial software (Petrel) is used for the visualization and for this reason, each fracture is approximated using 10 points equally separated along a decagon. Figure 6- shows two different realizations of the macrofracture model. The code for macrofracture simulation is given in Appendix C 90

109 (a) (b) 91

110 (a) (b) Figure 6-1 Realizations of the macrofracture model (a) shows the cross sectional view of the macrofracture model and is an approximate representation on a gridded system. (b) shows how the macro fractures after the horizons have been superimposed. This is also approximate because the ellipse has been approximated by a decagon to facilitate visualization using Petrel. 92

111 6.4 RANDOM WALKER SIMULATION Complete characterization of the fracture network requires integration of both the models microfracture and macrofracture models. These models are not independent. They represent fractures at different length scales and are conditioned to the same fracture intensity data. The reason for adopting this hierarchical modeling approach is to reduce the computation time and to be able to develop model for the whole field. The ultimate objective of the random walker simulation is to find the upscaled permeability of fracture network integrating both the microfracture and the macrofracture models. Explicit representation of fractures at both scales is not possible given the wide range of length and aperture distributions. Therefore to incorporate the effect of microfractures, the upscaled microfracture permeability is used as matrix permeability when simulating the movement of particles through the macrofracture. With the introduction of finite permeability for matrix, the random walkers can now move through both fractures and matrix. This is unlike the random walker simulation used for upscaling microfractures where walkers could move only through the fractures. The time taken by the random walker to reach the other end of the domain is calculated. The path of each walker is an emulation of the actual flow streamline and the time taken is a measure of the tortuosity of that streamline. Therefore it is expected that the average of the time taken by the walkers should directly correlate with the upscaled permeability of the system. The detailed algorithm used for random walker simulation is discussed below Algorithm of Random Walker movement through Macrofractures The random walkers are moved from one side of the domain to the other side and the time taken by each walker is calculated. The following step-by-step procedure is implemented: 93

112 1. The macro-simulation parameters (center of ellipsoids, aperture, length, orientation etc) are input to the random walker simulation. 2. Using these input intersections between the fractures are calculated. These calculations are done using the analytical equation of ellipse. They give the coordinates of all the points of interaction between the fractures. 3. Since it is also desired to compare the results to flow based upscaling, the system used here is simplified. For the present case constant matrix and fracture permeability has been used. The matrix and fracture permeability has been taken as 0.1md and 10000md. The domain used for flow simulation is 20x20x20. This flow simulation is only for validating the results of the random walk simulation. 4. Initially the walker is placed on the starting face of the domain. An arbitrary point on the starting face is selected randomly to start the movement of the walker. A uniform pressure field is assumed between the starting and the ending face between which the walker has to travel. 5. There are two options for the starting point. The walker can either lie in a fracture or in the matrix. If it lies in a fracture, all the fracture intersections at the location of the starting point are found. It is extremely rare to find walker lying in multiple fractures simultaneously at the starting point. However if such a case arises, one of the fractures in which the walker lies is selected randomly as the starting fracture. 6. If the walker lies in the matrix, distances to the nearest fractures are calculated. The number of nearest fractures for which the distance is stored is an input parameter and can be varied depending on the fracture intensity. The distances calculated is the distance joining the present position of the walker to the nearest point on the fracture. The co-ordinates of that closest point on the fractures are 94

113 also calculated. For this calculation only those fractures are considered which allow the particle to move in the direction of positive pressure/potential gradient. Movement of Walker initially placed in fracture 7. If the walker is initially placed in a fracture, the walker can move either to the connecting fracture or to the matrix. The number of possible paths for further movement of walker is (n+1), where n is the number of fractures intersecting the initial fracture and allowing the walker to move in direction of positive pressure gradient. One extra path is for the possible movement of the walker from fracture to the matrix. In case there are no available paths through fractures the walker is forced to move to the matrix. 8. For movement of the walker, a cumulative distribution function (CDF) is calculated based on the number of possible paths. The probability to take a particular path is made proportional to the harmonic mean of the permeability of the initial point (fracture) and the destination point (either a fracture or matrix). The probability (P i ) to take i th path out of total (n+1) path is given as P = i k i n+ 1 k j j= 1 2k js, k je, k j = (Effective permeability along a particular path) k + k js, je, (6-1) (6-2) Where k j,s = Permeability of the fracture the walker is initially placed k j, e = Permeability of the destination point (matrix/fracture) along the j th path. 95

114 9. The path for the movement is determined by randomly sampling the cdf. If the movement into the matrix is selected, the particle is moved to any arbitrary point within the same grid block in the direction of positive pressure gradient. Since the matrix has much lower permeability compared to the fracture, the probability for fracture-to-matrix transition is very low. 10. However, if the walker chooses to move to a connecting fracture, it can move to any point on the line of intersection of two fractures. In this case two possible positions of random walker are considered. This is schematically shown in Figure 6-2 below. The walker initially is placed at point A inside fracture 1 and has to move to the connecting fracture 2. 96

115 C 1 A B 2 Direction of Maximum Pressure Gradient (a) A 2 C 1 D B Direction of Maximum Pressure Gradient (b) Figure 6-2 Shows transition of walker to a connecting fracture. The walker initially placed at point A inside fracture 1 has to move to the connecting fracture 2. Case (a), when both points of intersection are in the direction of positive pressure gradient and walker can move to any point on line BC. Case (b), when one of the points (D) of intersection is in direction of negative pressure gradient. The walker can move to any point on the line BC In the first case (Figure 6-2) both the points of intersection (B & C) lie in the direction of positive pressure gradient and therefore the two farthest paths of transition for walker are AB and AC respectively. Any point on line BC can be considered as final position of the walker. However if one of the point of intersection lies in the direction of negative pressure gradient, the walker can move to only a segment of the line of 97

116 intersection. This is shown in Figure 6-2, where the point D lies in the direction of negative pressure gradient with respect to initial position of walker at point A. The walker can only move to the segment BC of the line of intersection CD. The position of point B is found by interpolating between the points of intersection such that the pressure gradient is the minimum. Any point on the segment BC can be taken as final position of walker. 11. The time calculated for this movement is based on the pressure gradient, the distance and the harmonic mean of permeability between the starting and the ending point. Movement of Walkers initially placed in Matrix 12. If the walker is initially placed in the matrix, it can either continue moving through the matrix or move back to a nearby fracture. As stated in step 6 the distances to nearest fractures are already known. The maximum movement in a single step has been restricted to unit grid length. Thus only those fractures lying within a unit grid distance to the present walker position is considered for movement. If there are no such fractures the walker will continue to move through the matrix. 13. For the movement of walker there will be (n+ 1) paths, where n is the number of nearby fractures with distance less than unity. The cdf is constructed based on the number of possible paths in the same manner as stated in step 8. For the first n paths, the harmonic mean of fracture and matrix permeability is used while for the last path the harmonic mean of matrix permeability of the two adjacent grids is used. 98

117 14. The path for the movement is determined by randomly sampling from the cdf. If the walker chooses to move to the fracture it is moved to the nearest point on the chosen fracture. For this path the time of transition is calculated based on the distance, pressure gradient and permeability along the path. 15. On the other hand if the walker chooses to move through the matrix it is moved through the matrix to an arbitrary point in the adjacent grid, in the direction of positive pressure gradient. It should be noted that the walker has much higher probability of moving back in to the fracture if there are any nearby fractures. Walker Percolation 16. Depending on the current position of the walker (either in fracture or matrix), the path for further movement is selected based on the steps discussed before. The movement of the walker is continued till it reaches the other end of the domain. For each movement of the walker the transition time is determined depending on the distance, the pressure gradient in that direction and the permeability of the path. 17. The total time taken by the walker is the sum of the time taken for each step. The total time taken by the walker is an indicator of the equivalent resistance of the porous medium along the travel path of the walker. 18. A large number of walkers are allowed to move through the porous media and the time taken for each walker to percolate is calculated. The number of walkers must be large enough to obtain statistically stable results for travel time distribution. This happens when most possible paths of percolation are explored by the walkers. 99

118 6.4.2 Interpretation of Output The random walker simulation gives the time taken by the random walkers to move from one side of the domain to the other moving through both the fractures and the matrix. The main simulation output is the travel time for a large number of random walkers. The time taken for each walker is a characteristic of the path traversed that is in turn a characteristic of the porous medium. Since there are several random walkers, the statistical properties of the travel time distribution is examined in order to arrive at the effective properties of the fractured medium. Different statistical parameters were derived to characterize the travel time distribution. It was found that the median of the travel time correlates best to the upscaled permeability (as discussed in the validation section discussed next). The number of matrix to fracture transitions for each walker is also recorded. This parameter must also correlate to the commonly used interporosity flow parameter (λ) of dual porosity model, which is a measure of matrix to fracture transitions. The code for the random walker simulation and sample output is given in Appendix C 6.5 VALIDATION OF RANDOM WALKER SIMULATION An attempt was made to verify the results of random walker simulation with those from flow based upscaling. For this purpose, random walker simulation is run after making some simplifications and the median of the travel time distribution is compared to the upscaled permeability. The steps are listed below. 1. Macrofracture models were generated on a 20x20x20 domain size. Simulation was performed to generate multiple models of macrofractures filling different proportions of domain. This is done to cover models with different degree of connectivity and different upscaled permeability. 100

119 2. Two files are generated for each model. The first contains the properties of the macrofractures (dip, azimuth, length and centre of the fracture) on which the random walker simulation is carried out. The second one has the fractures explicitly simulated on a gridded basis. This file is used as permeability input to the simulator for flow based upscaling. The grid blocks simulated as fractures are assigned a permeability of 10000md while the rest of the grid blocks are given a permeability of 0.1md. 3. Flow simulation was run on the generated models to find the flow based upscaled permeability of the system. Also the file containing the fracture properties is run using the random walker code to find the travel time of the walkers. In order to approximate the fractures as represented in the simulator, the aperture of the fracture was taken to be of the size of 1 grid block (unrealistically large). 4. The median of the travel time distribution is calculated and is plotted against the upscaled permeability. Steps 1-4 are performed for all the fracture models to obtain a reliable correlation. Figure 6-3 shows a scatter plot between the median travel time and the upscaled permeability. The median of the travel time shows a negative correlation with effective permeability of the medium. Increase in median travel time indicates an increase in average resistance offered by the medium and hence a decrease in effective permeability of the medium. The straight line fit is quite satisfactory except for very low values of permeabilities. A perfect correlation cannot be expected, for the simple reason that the fractures on the gridded system used for flow based upscaling is only an approximate representation of the actual fracture network. 101

120 40000 Median of Travel time Vs Upscaled Permeability Median of the travel time y = x R² = Upscaled Permeability(md) Figure 6-3 Calibrating the median travel time of random walkers to flow based upscaled permeability. There exists a negative correlation showing that the travel time increases as permeability of system decreases. 6.6 GENERATING AN UPSCALED PERMEABILITY MAP Random walker simulation of macrofractures is intended to couple the effect of macro and micro fractures on the effective permeability of the system. The desired end results of this procedure are field scale permeability models, combining both the macro and micro fracture models. Many such permeability models can be generated by superimposing different realizations of micro and macro fracture models and that quantifies the uncertainty in the permeability field arising due to the uncertainty in the fracture distribution. The following steps are followed to generate a field scale permeability model and assess the uncertainty in effective permeability. 102

121 1. Any one of the simulation grid blocks is selected at random. Based on the fracture intensity in that block, microfracture permeability can be allocated based on the calibrations presented in the previous chapter. For each value of fracture intensity there is a range of microfracture permeability (Section 5.4). Thus any value of microfracture permeability can be chosen and the associated variations represent the uncertainty. Microfracture permeability is assigned as matrix permeability for the random walker simulation. 2. One of the realizations of macrofractures is chosen. The permeability of the macrofractures is determined using the Hagen-Poiseuille s equation of flow between parallel plates. 2 W 2 Fracture Permeability = length 12 (6-3) where, W = Aperture of the fracture The aperture is taken from a uniform distribution bounded by maximum and minimum aperture value. This assumption is only to mimic reasonable variations in aperture. In the next chapter, the variation in aperture will be related to diagenesis. The values for maximum and minimum aperture are based on observations from cores and log data. ( Lfrac Lmin ) W = Amin + p* ( Amax Amin )* ( L L ) max min (6-4) ( ) ( 3 ) Where, A = Minimum Aperture 1mm min A = Maximum Aperture mm maz L frac Lmax = Lengthof the fracture = Maximum fracturelength 103

122 Lmin = Minimum fracturelength p = Random Number from Uniform Distribution (0,1) 3. Each macrofracture extends over multiple grid blocks. Therefore for the chosen grid block it is important to know the macrofractures that cut through the block. In case no macrofracture intersects a block, the permeability of the grid remains unaffected and is equal to the microfracture permeability. However if there is one or multiple macrofractures passing through the grid, the effective grid block permeability has to be computed accounting for the macrofracture(s). 4. To find the macrofractures intersecting a grid block, analytical equations of the ellipsoid are used. 5. However if there are one or more macrofractures passing through the grid, random walker simulation is performed. To perform simulation the grid block is re-scaled to 20x20x20 domain. This scaling is done to directly apply the correlation obtained in calibration study which is obtained for the same domain size. The centers and size of the macrofractures are adjusted accordingly. Median of the travel time distribution is used to find the upscaled permeability. 6. Steps one through six is are repeated for all the grids in the field. At the end we get an upscaled permeability of each grid block 6.7 RESULTS AND DISCUSSION Random walker simulation was performed over the entire simulation domain to generate permeability maps. Since the field exhibits vertical anisotropy, simulation was performed in X and Z to get the complete permeability tensor. The permeability in the Y direction is the same as that in the X direction. Another important output generated is the fracture-to-matrix transfer coefficient (interporosity flow parameter λ) for each grid 104

123 block. The entire dataset was loaded on to Petrel and the selected reservoir horizons were superimposed. Figures 6-4 and 6-5 are some snapshots of the spatial variation in the z-directional permeability (K Z ) and x-directional permeability K X. Figure 6-4 Final map of permeability in the z-direction. The map includes the effect of both micro and macro fractures. Areas in red are the regions where permeability has been enhanced significantly due to the presence of macrofractures. 105

124 Figure 6-5 Final map of permeability in the x-direction. Since the fractures show an isotropic behavior aerially, this map also represents permeability in Y- direction. It should be noted that permeabilities are much less in x-direction compared to z-direction shown in Figure 6-4 Figure 6-4 and Figure 6-5 shows the final permeability map obtained in Z and X directions. The base permeability is determined by the microfracture permeability. The red areas show the zones of very high permeability. The permeability in these regions has been enhanced enormously due to the macrofractures intersecting the grid blocks. These red areas are high permeability streaks and provide a path of least resistance for the fluid to travel through large distances. It should be noted that both Figure 6-4 and Figure 6-5 correspond to one realization of the macrofractures. Different realizations of macrofractures will result in a 106

125 suite of effective permeability models that represent the uncertainty in upscaled permeability. Figure 6-6 Map of interporosity flow parameter λ calculated using random walker simulation. Red areas highlight the areas where flow takes mostly through fractures. Figure 6-6 and shows the matrix to fracture transfer coefficient for the entire field. These are calculated based on average number of matrix to fracture transitions a walker undergoes before it percolates. The areas in red show the grids where the matrix to transfer coefficient is very nearly equal to one and the flow takes mostly through the fractures. These are the same grids where the permeability is very high due to presence of macrofractures. On the other hand the pink areas show no matrix to fracture flow, mainly due to absence of macrofractures in those areas. 107

126 6.7 CONCLUSION In this chapter we described an approach to model and upscale macrofractures. The matrix permeability for the flow simulation grid blocks were identified to the anisotropic permeability corresponding to the microfractures. The random walker simulation presented in this chapter therefore integrates the effect of both micro and macro fractures. The technique presented in this chapter is several times faster than the flow based upscaling technique and can account for fractures exhibiting a wide range of apertures. The movement of random walkers is tracked on a continuous spatial domain and consequently the fractures (and their small apertures) do not have to be explicitly simulated on a grid. The next chapter presents an approach to model the variation in fracture aperture due to diagenesis and the resultant impact on upscaled permeability. 108

127 Chapter 7: Modeling the Impact of Diagenesis 7.1 OVERVIEW Diagenesis can be defined as any chemical or physical change in the sediment after its initial deposition. Diagenesis significantly affects the quality of the reservoir. Its effect is more pronounced in carbonate reservoirs. It plays a major role in determining reservoir connectivity and therefore it is crucial to model diagenesis for proper assessment of reservoir flow behavior. The major diagenetic processes are precipitation and dissolution, which are associated with reactive fluid transport. Flow of reactive fluid through existing fractures can cause either precipitation or dissolution of rock material. Fresh water flowing through the fractures is capable of dissolving carbonate (CaCO 3 ) rock material. When fluid flowing in the fracture is fully saturated, the dissolved carbonates can get precipitated, if there is any perturbation in mineral concentration, temperature etc. There are many different factors which affect the equilibrium concentration of Ca 2+ in solution, leading to either dissolution or precipitation. Such processes can change the morphology of the fractures, increase or decrease the aperture or even close the fracture. Other than Calcite, Gypsum (CaSO 4 ) and Aragonite are other calcium minerals prone to dissolution and precipitation. Some other diagenetic processes like dolomatization, karsts and cave formations etc. are also important in carbonate reservoirs. In this chapter, the focus of discussion will be on the methodology adopted for modeling effect of diagenesis on fracture network permeability. The aim is to quantify the effect of diagenesis in terms of change in effective permeability of the system. 109

128 7.2 GEOSTATISTICAL ANALYSIS OF DIAGENESIS The objective is to use the well information in order to construct spatial models for the diagenetic processes. Since there is considerable ambiguity in the interpretation of logs, in order to derive variations in rock connectivity caused by diagenesis, the models have to provide a realistic depiction of uncertainty. The geostatistical analysis of sonic log data was performed. Sonic log travel time is a measure of porosity of the rock. This log is often used to distinguish between the primary and secondary porosity of the rock. These logs are recorded along the well and are available for most of the wells of the field. Travel time histograms were constructed using the data available along the wells. Indicator variable was defined using a threshold cut off for sonic travel time. This cut off was selected based on a cluster analysis of the travel time versus gamma ray cross plot. Mineral alterations and precipitation of clays due to diagenesis affect gamma ray measurements. Details of the cluster analysis procedure can be found in Srinivasan and Sen, Indicator variograms were constructed and indicator simulation was performed to generate probabilistic diagenetic maps for the field. The model generated by indicator simulation gives the probability of diagenetic alteration at each location. Figure 7- shows the results of the indicator simulation. 110

129 Figure 7-1 Probability map of diagenetic alteration. The map has been generated using indicator simulation. Red areas show regions of high probability of porosity alteration due to diagenetic alteration while blue shows regions least affected by diagenesis The regions with high probability of diagenetic alteration (red areas) are the ones where the effect of diagenesis is maximum and considerable alteration in permeability has taken place. On the other hand the areas with low probability (blue areas) have their original permeability preserved. 7.2 MODELING IMPACT OF DIAGENESIS ON EFFECTIVE PERMEABILITY Fracture network permeabilities computed till now (chapter 5 and chapter 6) assume fracture apertures uniformly sampled within a range. It is expected that diagenesis affects equally micro and the macro fractures. The effect of diagenesis is modelled separately for both scales of fracture models and then combined together to calculate the altered permeability of the system. 111