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Characterization of permeability fields between horizontal wells using a hybrid of cross -hole imagery and repeat interference test

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Characterization of permeability fields between horizontal wells using a hybrid of cross -hole imagery and repeat interference test

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Content C h a r a c t e r iz a t io n o f P e r m e a b il it y F ie l d s B e t w e e n H o r iz o n t a l W e l l s U s i n g A H y b r id o f C r o s s -H o l e Im a g e r y a n d R e pe a t In t e r f e r e n c e t e s t by Jamal Ali Alkhonifer A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Petroleum Engineering) August 2000 Copyright © 2000 Jamal Alkhonifer Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3093411 UMI UMI Microform 3093411 Copyright 2003 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES, CALIFORNIA 90007 This dissertation, written by ...............................Jam a1 _ A l i _ A lk h o n i f e r .............................. under the direction of h is. Dissertation Committee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillment of re quirements for the degree of DOCTOR OF PHILOSOPHY Dean of Graduate Studies Date ....July..lQ.»...2.Q.Q.Q DISSERTATION COMMITTEE Chairperson Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Dedication To the m em ory of m y beloved father, Ali. To m y m other, Sarah, whose unconditional and ceaseless love endow ed me w ith incentives to attain this work, and to m y dear wife, Sarah, w hose patience and devotion are forever heartfelt, and to my charm ing daughter, Eman, for bringing the greatest joy to m y life. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Jamal Alkhonifer Committee Chair: Iraj Ershaghi C h a r a c t e r iz a t io n o f P er m e a b il it y Fie l d s Be t w e e n H o r iz o n t a l W ells U s in g A H y b r id o f C ro ss-H o l e Im a g e r y a n d R e p e a t In t e r f e r e n c e test Success of m any displacem ent processes depends on correct estim ation of perm eability distribution. Among these one could nam e w ater flooding, gas injection, and SAGD (Steam assisted gravity Drainage). Traditional form ation evaluation techniques rely prim arily on shallow investigative tools and measurements. Such data are then correlated using geostatistical-modeling techniques. There is a need for perm eability m apping on a broader scale than that offered by local sam pling methods. Using the capabilities offered by horizontal holes, new approaches for characterization of perm eability fields can be devised. This study presents new concepts on the use of innovative detection and m onitory procedures and the interpretation aspects of such data to m ap perm eability fields betw een tw o horizontal or for that m atter vertical holes. The study presents a new form of m onitoring pressure interference data betw een tw o parallel horizontal wells. In pursuit of yet a better reservoir description, this concept is further investigated to include cross hole tom ography data. Spatial correlation length, as an integral part of reservoir characterization, and its influence on both the system effective perm eability and pressure responses is also investigated. Pressure transient variance is used as a stand-alone tool to identify perm eability correlation length. A m ethodology to utilize such a tool is presented in this study. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The proposed conceptual approach for m apping perm eability fields is based on integrating the responses at m ultiple isolated probing points along a horizontal well path, and m apping perm eability profiles from the application of a hybrid of deterministic and stochastic models. From this work, it is observed that perm eability distribution honoring a given PDF and semi-variograms alone is not conclusive b u t m ust be further constrained using data from other techniques such as repeat interference testing. It is also found that variations in pressure, in the observation well, due to perm eability contrasts, are captured only during the transient p art of the test. Inclusion of abundantly sam pled data, like traveltim e tom ography, is show n to enhance the estim ation of the perm eability field. This is done through techniques like ordinary cokriging, w hich not only recognize the trend of the perm eability distribution but also pick up details sm eared by other methods. Discrete w avelet transforms, DWT, is utilized in the appraisal of the system effective perm eability, keff It is observed that, w ith the same PDF of perm eability, reservoirs w ith long correlation lengths, exhibit large values of ke f } w hereas reservoirs w ith short correlation are show n to exhibit low keff This finding was instrum ental in quantifying the system correlation length through a type curve. Finally, the need to characterize and m ap fractures as flow units is a critical issue in some reservoirs. As part of this study a m athematical m odel is developed to serve as the base for extending the previous w ork to include naturally fractured reservoirs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgments I w ould like to express m y sincerest appreciations and indebtedness to Professor Iraj Ershaghi, m y academic advisor and committee chairman, for his guidance, diligence and boundless encouragem ent. His dedication and perpetual interest in research have instilled high standards into m y m ind for acquiring knowledge. I w ould like to extend m y profound gratitude to m y committee members, Dr. Bruce Davis and Professor Charles Sammis for serving in my dissertation committee, reviewing this m anuscript and m aking constructive recommendations. Spatial thanks to Professor Jim Moore and Professor Katherine Shing for their com m itm ent to serve on m y guidance committee. I w ould like to thank Saudi Aramco for providing m e the opportunity to pursuit m y graduate career. Portions of this study w ere supported by US DOE-Tideland. University of Southern California. iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Contents Dedication ii Acknowledgements iii List Of Tables vii List Of Figures viii Abstract xvi Chapter 1 1.1 Introduction 1 1.2 Objective And Organization 13 Chapter 2 A New Approach Of Permeability Characterization Using Interference Analysis Of Parallel Horizontal Wells 15 2.1 Introduction 15 2.2 Conceptual Test Design 16 2.3 Proof Of Concept 25 2.3.1 Base-1 Case 25 2.3.2 Base-2 Case 28 2.4 Incorporation Of Pressure Response Data 33 2.5 Results And Discussion 37 2.6 Determining Impervious Shale Distribution 37 2.7 Uniqueness Of Solution 40 2.8 Multiple Pulsing Points 41 2.9 Summary And Conclusion 42 Chapter 3 Permeability Cokriging With Cross-well Traveltime Tomography 43 3.1 Introduction 43 3.2 Cross-Well Traveltime Tomography 45 3.2.1 Traveltime Tomography Inversion 48 3.3 Mathematical Formulation 50 3.4 Comparison Between Cross-Well Tomography And Well Logging 52 3.5 Permeability Cokriging With Cross-Hole Tomography Data 53 3.5.1 Generation Of Synthetic Tomograms 54 3.5.2 Generation Of Primary Or Permeability Data 55 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 Chapter 5 Chapter 6 3.6 Pressure Interference Testing 61 3.7 Semi-Variograms And Cross Semi-Variograms 61 3.8 Estimation Of Primary Data For Entire Grid System 69 3.8.1 Ordinary Kriging 70 3.8.1.1 Results And Discussion 72 3.8.2 Kriging With An External Drift 74 3.8.2.1 Results And Discussion 75 3.8.3 Cokriging 77 3.8.3.1 Collocated Cokriging 79 3.8.3.2 Standardized Ordinary Cokriging 80 3.8.3.3 Results And Discussion 80 3.8.4 Ordinary Or Traditional Cokriging 82 3.8.4.1 Results And Discussion 82 3.8.5 Conclusion And Summary Of Applied Methods 84 Up-Scaling Permeability Maps 90 4.1 Introduction 90 4.2 Up-scaling 91 4.2.1 Cardwell And Parson Method 91 4.2.2 Up-scaling Using Discrete Wavelets Transforms 92 4.3 Mathematical Formulation 93 4.3.1 Up-scale 2-D Permeability 93 4.4 Results And Analysis 96 4.5 Effective Permeability Calculations 101 Estimation Of Semi-Variograms From Effective Permeability And Pressure Data 106 5.1 Importance Of Calculating The Correct Semi- Variogram 106 5.2 Proof Of Concept 107 5.2.1 Procedure For Proofing The Concept 107 5.2.2 Results And Discussions 108 5.3 Methodology 111 5.3.1 Procedure 111 5.4 A Case Study 118 5.5 Conclusion 124 Interference Testing Of Horizontal Wells In Naturally Fractured Reservoirs 125 6.1 Introduction 125 6.2 Mathematical Model Derivation 127 6.2.1 Pss-Interporosity Model 132 6.3 Type Curve Analysis 133 6.3.1 Pof Vs. to Type Curves 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.4 Conclusion 137 Chapter 7 Conclusion 138 Chapter 8 Recommendations For Future Work 142 References 144 Appendix-A Equations Of Dimensionless Parameters 149 Appendix-B Simulation Of Pseudo-Steady-State Dual Porosity Model 150 v i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Tables Table-1.1: Table-2.1: Table-2.2: Table-3.1: Table-3.2: Table-3.3: Table-3.4: Table-4.1: Table-4.2: Table 5.1: Table-5.2: Table-5.3: Table-5.4: Table-5.5: Table-5.6: Classification of modem environments of sand depositions Courtesy LeBlanc (1977) [3] 3 Reservoir Model Properties 25 Descriptive Statistics for base-2 case 32 Ranges of values of sonic compressional wave velocity and transit time for common rock matrix materials 48 Summary of the descriptive statistics for the tomographic data 55 Summary of the descriptive statistics for the permeability data 56 A Summary of descriptive statistics for all of the estimation methods in comparison to actual data set 85 Daubechies-4 and Daubechies-6 wavelet functions’ coefficients 94 Comparison table for CP and DWT methods performance 100 A layout of the procedure followed to generate the a 2 a p type curve 115 Horizontal effective permeability at X = 0.00 for five realizations 116 Horizontal effective permeability at X = 0.125 for five Realizations 116 Horizontal effective permeability at X = 0.25 for five realizations 116 Horizontal effective permeability at X = 0.50 for five realizations 116 Horizontal effective permeability at A , = 0.75 for five realizations 117 vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table-5.7: Table-5.8: Table-5.9: Table-B.l: Horizontal effective permeability at A , = 1.0 for five realizations Horizontal effective permeability at A , = 1.5 for five realizations Horizontal effective permeability at A . = 2.0 for five realizations Numerical simulator parameters for PSS dual porosity Model Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures Figure-1.1: Figure-1.2A: Figure-1.2B: Figure-1.3: Figure-1.4: Figure-1.5 A: Figure- 1.5B: Figure-1.6: Figure-1.7: Figure-1.8: Figure-1.9: Figure-1.10: Figure-2.1: Figure-2.2: Schematic representation for Reservoir Management Approach Courtesy Thakur (1990) [1] A normal fault showing possible layout for reservoir heterogeneity A reverse fault showing possible layout for reservoir heterogeneity A schematic of different completion methods to intercept vertical fractures (modified from Aguilera (1995) [6]) A schematic of high permeability bands juxtaposing low ones modeled in Chapter-2 A schematic for a drainage area for a vertical well A schematic for a drainage area for a horizontal well A schematic of the compressional wave velocity, Vp Tomograms Courtesy Mendoza and Cruz [19] A schematic of the shear wave velocity, Vs tomograms Courtesy Mendoza and Cruz [19] A schematic of the Poisson ratio map Courtesy Mendoza and Cruz [19] A schematic of the Density of the formation Courtesy Mendoza and Cruz [19] A schematic of the porosity map Courtesy Mendoza and Cruz [19] Example of Horizontal Well Pairs in Reservoir Exploitation Segments A, B, C, and D are not in communication with each other except through the formation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure-2.3A: Pressure distribution is different in each segment 18 Figure-2.3B: Pressure History at the Observation Well 19 Figure-2.4: Permeability anomalies created by channel sands and shale breakers 19 Figure-2.5: All curves join in to form one, representing the entire system 20 Figure-2.6: Horizontal Well Model 21 Figure-2.7: Type curves for Xd, Yd < 1.0 used in conjunction with Equation-2.5 and Equation-2.6 24 Figure-2.8: Type curves for Xd, Yd >1.0 used in conjunction with Equations 2.5 and Equation-2.6 24 Figure-2.9: First scenario of permeability distribution 26 Figure-2.10: Selected samples of the first scenario no conditioning data honored 27 Figure-2.11: Best possible realization to our true model (1st scenario) 29 Figure-2.12: Univariate distribution of the permeability field (2n d scenario) 30 Figure-2.13: Second scenario model semi-variogram 30 Figure-2.14: Base-2 case permeability field 32 Figure-2.15: A sample of the semi-variogram of the generated realizations compared with base-2 case. Geometric anisotropy has been reproduced in all of the realizations 34 Figure-2.16: Plot of the correlation index, Cl, for 20 runs. Run-18 has the Highest Cl value 35 Figure-2.17: Base-2 case as well as a sample of the many realizations generated. It is obvious that realization-18 bears the most resemblance to base-2 as predicted by Cl plot 36 Figure-2.18A: Interference tests for segment-A of the observation well compared with the base case 38 X Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure-2.18B: Interference tests for segment-B of the observation well compared with the base case 38 Figure-2.18C: Interference tests for segment-C of the observation well compared with the base case 39 Figure-2.18D: Interference tests for segment-D of the observation well compared with the base case 39 Figure-2.19: Pressure and pressure derivative response for selected realizations at the active well 40 Figure-2.20: Inclusion of multiple source points within the active well 41 Figure-3.1: Schematic diagram illustrating the scale of probing of cores, well logs, and pressure transient and seismic 44 Figure-3.2A: Cross-well tomography geometry for conventional wells 49 Figure-3.2B: Proposed Cross-well tomography geometry for horizontal wells 49 Figure-3.3: Schematic illustration of ray paths through a cell slowness model 51 Figures-3.4: a, b, c, d, and e: A carton illustration for the comparison between coss-well tomography and well logging 53 Figure-3.5: Gray level maps of Vp, compressional wave velocity. Runs-1, 2, 3, and 4 57 Figure-3.6: Gray level maps of Vp, compressional wave velocity. Runs-5, 6, 7, and 8 58 Figure-3.7: Semi-variograms for compressional wave velocity Runs-1, 2, 3, and 4 59 Figure-3.8: Semi-variograms for compressional wave velocity Runs-5, 6, 7, and 8 60 Figure-3.9: Gray level map of true permeability field 61 Figure-3.10: The three most commonly used variogram transitional models 64 X I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure-3.11: Experimental semi-variogram for permeability data set Figure-3.12: Model semi-variogram for primary variable permeability data set Figure-3.13: Experimental semi-variograms for secondary variable, Vp data set Figure-14: Experimental and Model semi-variograms for Vp data set Figure-3.15: Experimental cross-variogram primary and secondary data sets Figure-3.16: Experimental and Model cross-variograms Figure-3.17: Kriging finds the value of the point to be kriged so that it best fits the variogram model Figure-3.18A: True permeability field better continuity in the north-south direction Figure-3.18B: Kriged permeability field. Acted like a low pass filter smoothing out the data to the average value Figure-3.19: Cross validation of true vs. kriged permeability values Correlation coefficient = 0.25 Figure-3.20: Kriging with an external drift, using well and tomographic data Figure-3.21: Permeability kriged with an external drift. The drift improved the estimates compared to Ordinary Kriging Figure-3.22: Cross validation of true permeability vs. permeability kriged with and external drift; Correlation coefficient = 0.41 Figure-3.23: Cokriging using well and tomographic data. Note the cross- variogram model in addition to variogram models Figure-3.24: Permeability cokriged with tomographic data using Standardized cokriging with one unbiased condition, equation-3.21 Figure-3.25: Cross validation of true vs. Standardized Cokriged permeability values; Correlation coefficient = 0.50 65 66 66 67 67 68 70 73 73 74 75 76 76 77 81 81 xii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure-3.26: Permeability cokriged with tomographic data using Ordinary Cokriging with two unbiased conditions 83 Figure-3.27: Cross validation of true vs. Ordinary Cokriged Permeability values; Correlation coefficient = 0.88 84 Figure-3.28: A q-q plot of kriged permeability field 86 Figure-3.29: A q-q plot of kriged permeability field w/ external drift 86 Figure-3.30: A q-q plot of standardized cokriged permeability field 87 Figure-3.31: A q-q plot of ordinary cokriged permeability field 87 Figure-3.32: A p-p plot of standardized cokriged permeability field 88 Figure-3.33: A p-p plot of ordinary cokriged permeability field 88 Figure-4.1: The original fine scale permeability field (40 x 40) above and (20 x 20) up-scaled grids using the two methods 97 Figure-4.2: The original fine scale permeability field (40 x 40) above and (10 x 10) up-scaled grids using the two methods 98 Figure-4.3: The original fine scale permeability field (40 x 40) above and (5 x 5) up-scaled grids using the two methods 99 Figure-4.4: The original fine scale permeability field (64 x 64) 101 Figure-4.5 A: First resolution level up-scaled permeability field 32 x 32 102 Figure-4.5B: Second resolution level up-scaled permeability field 16x16 102 Figure-4.5C: Third resolution level up-scaled permeability field 8 x 8 103 Figure-4.5D: Fourth resolution level up-scaled permeability field 4 x 4 103 Figure-4.5E: Fifth resolution level up-scaled permeability field 2 x 2 104 Figure-4.5F: Sixth and final resolution level up-scaled permeability field l x l 104 Figure-5.1: A schematic of the multiple probing points in an observation horizontal well and the active well for an interference test 106 xiii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure-5.2A: Calculated semi-variogram for the first realizations at different elapsed times 109 Figure-5.2B: Calculated semi-variograms for the second realization at the same elapsed times 110 Figure-5.3: A sample of realizations at different correlation lengths 114 Figure-5.4: Gray scale maps at each level of resolution using DWT 118 Figure-5.5: Effective permeability type curve calculated using DWT 119 Figure-5.6A: A sample of simulated interference pressure tests at A , = 0.0 119 Figure-5.6B: A sample of simulated interference pressure tests at A = 0.125 120 Figure-5.6C: A sample of simulated interference pressure tests at A = 0.25 120 Figure- 5.6D: A sample of simulated interference pressure tests at A = 0.5 121 Figure- 5.6E: A sample of simulated interference pressure tests at A = 0.75 121 Figure- 5.6F: A sample of simulated interference pressure tests at A = 1.0 122 Figure- 5.6G: A sample of simulated interference pressure tests at A = 1.5 122 Figure- 5.6H: A sample of simulated interference pressure tests at A = 2.0 123 Figure-5.7: Type curve for a a p versus time 123 Figure-6.1: Horizontal well model schematic 128 Figure-6.2: Variation law in the Laplace domain Courtesy Aguilera and Ng [36] 131 Figure-6.3: Interference testing of horizontal wells in NFR XD= 0,YD= 4 135 Figure-6.4: Interference testing of horizontal wells in NFR XD= 3,YD= 2 135 Figure-6.5: Interference testing of horizontal wells in NFR XD = 3,YD = 5 136 Figure-6.6: Interference testing of horizontal wells in NFR XD= 5,Yd = 2 136 xiv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure-6.7: Figure-B.l: Figure-B.2: Interference testing of horizontal wells in NFR XD= 5,YD= 5 137 Stehfest Algorithm solution for dual porosity in conventional wells 159 Simulator solution for dual porosity in conventional wells 160 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract Success of m any displacem ent processes depends on correct estim ation of perm eability distribution. Among these one could nam e w ater flooding, gas injection, and SAGD (Steam assisted gravity Drainage). Traditional formation evaluation techniques rely prim arily on shallow investigative tools and m easurem ents. Such data are then correlated using geostatistical-modeling techniques. There is a need for perm eability m apping on a broader scale than that offered by local sam pling m ethods. Using the capabilities offered by horizontal holes, new approaches for characterization of perm eability fields can be devised. This study presents new concepts on the use of innovative detection and m onitory procedures and the interpretation aspects of such data to m ap perm eability fields betw een two horizontal or for that m atter vertical holes. The study presents a new form of m onitoring pressure interference data betw een tw o parallel horizontal wells. In pursuit of yet a better reservoir description, this concept is further investigated to include cross hole tom ography data. Spatial correlation length, as an integral part of reservoir characterization, and its influence on both the system effective perm eability and pressure responses is also investigated. xvi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Pressure transient variance is used as a stand-alone tool to identify perm eability correlation length. A m ethodology to utilize such a tool is presented in this study The proposed conceptual approach for m apping perm eability fields is based on integrating the responses at m ultiple isolated probing points along a horizontal well path, and m apping perm eability profiles from the application of a hybrid of deterministic and stochastic models. From this work, it is observed that perm eability distribution honoring a given PDF and semi-variograms alone is not conclusive b u t m ust be further constrained using data from other techniques such as repeat interference testing. It is also found that variations in pressure, in the observation well, due to perm eability contrasts, are captured only during the transient part of the test. Inclusion of abundantly sam pled data, like traveltim e tom ography, is show n to enhance the estim ation of the perm eability field. This is done through techniques like ordinary cokriging, which not only recognize the trend of the perm eability distribution but also pick up details sm eared by other m ethods. xvii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Discrete wavelet transforms, DWT, is utilized in the appraisal of the system effective permeability, ke fr It is observed that, w ith the same PDF of perm eability, reservoirs w ith long correlation lengths, exhibit large values of ke ff w hereas reservoirs w ith short correlation are show n to exhibit low ke fr This finding w as instrum ental in quantifying the system correlation length through a type curve. Finally, the need to characterize and m ap fractures as flow units is a critical issue in some reservoirs. As part of this study a m athem atical m odel is developed to serve as the base for extending the previous w ork to include naturally fractured reservoirs. xviii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter-1 1.1 Introduction The objective of reservoir management is to optimize recovery and profitability from oil and gas fields. There are many factors that can have a direct impact on recovery optimization. Among these are reservoir characteristics and completion methods. To accurately describe and characterize a given reservoir, a good understanding of the geological reasons for reservoir heterogeneity is needed. Depositional processes or post-depositional events can both affect reservoir rock heterogeneity. Reservoir heterogeneity can be approximately delineated using pressure transient tests, well logs, cores, production tests, and by other geostatistical techniques. In this study, as a measure of reservoir heterogeneity, permeability field as an intrinsic property will be scrutinized. Reservoir management Among the different definitions for reservoir management, Thakur (1990) [1] has defined it as the wise exploitation of the different means available to maximize benefits from a reservoir. According to Thakur, proper implementation of reservoir management is a key 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. factor in optimizing the recovery from any field. Fig.-l.l shows all the functional groups that can contribute to the success of reservoir management. Wiggin and Startzman (1990) [2] described reservoir management as "that set of operations and decisions by which a reservoir is identified, measured, produced, developed, monitored, and evaluated from its discovery through depletion and final abandonment". To identify, develop, and monitor a particular reservoir, one needs, as a subset of an integrated study, to develop a sound geological model for that reservoir. To do that, it is imperative that we research and understand the reservoir geological characteristics as well as the causes under which these characteristics are formed. Reservoir Engineering Geology Production Engineering Geophysics / Design & Construction y Engineering Gas and Chemical Engineering Reservoir Management Research & Service Labs Drilling Production Operations Figure-1.1 Schematic representation for Reservoir Management Approach Courtesy Thakur (1990) [1] 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Causes of reservoir heterogeneity Reservoir rocks are created under complex geological processes. These processes often involve many sequences of depositional cycles under various sedimentary environments. Such variations in deposition like deltaic and deep marine lead to reservoir heterogeneity. Recognition of the environment of deposition is a pivotal step in geologic description of a reservoir because the distribution and continuity of the reservoir and non-reservoir rock are usually related to depositional conditions. Table-1.1 shows the classification of modem environments of sand depositions [3]. Table-1.1 Classification of modern environments of sand depositions Courtesy LeBlanc (1977) [3] Continental Alluvial (fluvial) - Alluvial fan - Braided stream - Meandering stream (includes flood basins between meander belts) - Aeolian (can occur at various positions within continental and transitional environments) Transitional (Shallow Marine) Deltaic Birdfoot-lobate (fluvial dominated) Cuspate-arcuate (wave and current dominated) Estuarine (with strong tidal influence) Coastal Interdeltaic Barrier island (includes barrier islands, lagoons behind barriers, tidal channels, and tidal deltas) Chenier plain (includes mud flats and cheniers) Transgressive marine Marine Deep marine Outer shelf Inner shelf Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Of equal importance to depositional environments are the post- depositional processes. Faulting, folding, fracturing, subsidence, and diagenesis (compaction, cementation, solution, and fracturing) could all contribute to the complexity of the reservoir rock. Fig.-1.2A and Fig.-1.2B depict normal and reverse faulting systems respectively. Whereby the vertical bedding sequence of the formation has been altered along the fault line causing vertical permeability heterogeneity. These same systems can juxtapose layers that possess dissimilar permeability distributions giving rise to lateral permeability variations as well. A normal fault Figure-1.2A: A normal fault showing possible layout for reservoir heterogeneity A reverse fault Figure- 1.2B: A reverse fault showing possible layout for reservoir heterogeneity 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The presence of natural fractures is another scenario in which low and high permeability bands are juxtaposing one another; see Fig.- 1.3 and Fig.-1.4. In Chapter-2 we will model and examine such contrasting permeability distributions in more details. Fig.-1.3 schematically illustrates the importance of completing wells as vertical, deviated, or horizontal drainholes to ensure better interception of highly permeable areas; this brings us to the next section. CR O SS SECTION Matrix Fractures Figure-1.3 A schematic of different completion methods to intercept vertical fractures (modified from Aguilera (1995) [6]) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Horizontal Wells Matrix Fractures Figure-1.4 A schematic of high permeability bands juxtaposing low ones modeled in Chapter-2 Completion methods Drilling and subsequently completing wells as conventional (vertical) or horizontal drainholes can have a significant impact on the way a reservoir is produced and managed. Fig.-1.5A is a schematic representation of a drainage area for a vertical well where h, re V / and rw denote formation thickness, reservoir radius drained by vertical wells, and wellbore radius respectively. Fig.l.5B, on the other hand, depicts a drainage area of a horizontal well where h, re h , and L designate formation thickness, reservoir radius drained by horizontal wells, and horizontal wellbore length respectively. A vertical well drains a cylindrical volume, whereas a horizontal hole drains an ellipsoid, a three-dimensional ellipse. In most cases L > h (Fig.-1.5B), therefore, horizontal wells are expected to drain a larger reservoir volume than do vertical wells. Moreover, horizontal wells 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. possess larger contact area with the formation, and hence, greater radius of influence than vertical wells. As a result, information obtained from horizontal well testing, like transmissibility and storativity, will be more representative of lateral variations in formation properties than the ones obtained from vertical well testing. ev WWW WWW \ W \ WWWWWWW \-w \ 2rw Figure-1.5A A schematic for a drainage area for a vertical well Figure-1.5B A schematic for a drainage area for a horizontal well 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Drilling horizontal wells to increase the productivity has been proven to be successful in several fields. Furthermore, horizontal wells are effective in tight formations, thin formations, reservoirs with gas an d /o r water coning problems, and formations that are naturally fractured. Because most natural fractures of commercial importance are either vertical or semi-vertical in nature, the objective of horizontal well drilling would be to drill a horizontal drainhole normal to the principal direction of the natural fractures so that as many as possible of these fractures are intercepted. Under such circumstances conventional wells do not possess the same probability of success as directional or horizontal wells for exploiting naturally fractured reservoirs. Fig.-1.3 illustrates this concept. Pressure transient tests Use of pressure transient tests to help in the characterization of the different aspects of reservoir heterogeneity has been the subject of investigation by several authors in many publications [7], [8], [9], and [22]. In general, these types of tests can be classified as single well and multiple well tests. DST (Drill stem test), Repeat Formation, Drawdown, Buildup, Step Rate, and Falloff tests all fall under the single well test classification. Interference and pulse tests, on the other hand, involve more than a single well and hence fall under the multiple well test category. This study introduces a new concept in 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. pressure interference testing; whereby responses from a series of interference tests, between two horizontal wells, are modeled to map the permeability distribution between the two wells. Furthermore, investigation results are presented as how these tests can be integrated with other methods to enhance reservoir description. Horizontal hole pairs potential The concept of horizontal hole pairs has great potential applications for reservoir imaging. One useful application is to map and characterize permeability profiles in areal and vertical segments of a given reservoir using pressure data, as will be demonstrated in Chapter-2. Information obtained, however, can be used in designing SAGD (Steam Assisted Gravity Drainage) and other recovery processes in reservoirs where pairs of horizontal wells are planned for recovery optimization. A concept such as cross-hole tomography is another potential application of horizontal hole pairs. Conventionally, this application has been applied for imaging a 2-D region between vertical boreholes. A good implementation of cross-well tomography in vertical boreholes for the purpose of reservoir characterization problems was demonstrated by Mendoza and Cruz [19]. They carried out their experiment in an oil producing field of the West Permian Basin, Texas, U.S.A. Fig.-1.6 shows the compressional wave velocity, Vp tomograms whereas Fig.-1.7 depicts the shear wave velocity, Vs tomograms. From these tomograms they were able to delineate the imaged vicinity 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. between the two wells with 23 ft resolution. They computed Poisson ratio, a , Fig.-1.8, density of the formation, p , Fig.-1.9, and porosity, (p, Fig.-l.10. Similar application of cross-well tomography using horizontal hole pairs will be addressed in Chapter-3. The obvious advantage of horizontal wells over their conventional counterparts is the larger reservoir coverage they possess. Vp Tomograms 6.46 km/sec 5.05 km/sec Figure-1.6 A schematic o f the compressional wave velocity, Vp Tomograms Courtesy Mendoza and Cruz [19] 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Vs Tomograms 3.66 km/sec 2.75 km/sec m S a r Figure-1.7 A schematic of the shear wave velocity, Vs tomograms Courtesy Mendoza and Cruz [19] Poisson ratio map Figure-1.8 A schematic of the Poisson ratio map Courtesy Mendoza and Cruz [19] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Density map 2.70 qm/cm ^ / Figure-1.9 A schematic of the Density of the formation Courtesy Mendoza and Cruz [19] Porosity map ----.....................X r ^ „ i i A'9fV<>.A w 1 v • - ' . j h , ■. / y > -^> — ^■■v. -. W '- : . 1 / / 1 -i’ * y. ( f j k A t ' \* w r f { 4 y * v . f ^ •"/* * , 1 . A i - ■< , 1.W-V ~ V y ' > , / v ■ v ■ ■ ■ ■ ' ( % / . < * > ■ i vfc y - A < • > 4- * 6 / > 4 , ? / A A / -**.41 f r * * / ' ^ / A ^ > V 'Y - , . < s r - , > 4 / - 'tV> ^ ^ ^ - tV»* * * fe u .-* , * < * ^ V ^ V , "A. io .14>: 13.28: Figure-1.10 A schematic of the porosity map Courtesy Mendoza and Cruz [19] 1.2 Objectives and Organization Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The purpose of this study is to investigate the potential applications of horizontal well pairs in the characterization of permeability fields. This will be achieved using a hybrid of cross-hole imagery, repeat interference testing and other analytical and geostatistical techniques. This includes the estimation of semi- variograms in the lateral or inter-well direction and the development of a mathematical model for interference testing of horizontal wells in the presence of natural fractures. In Chapter-2 we will present a new approach in interference analysis of parallel horizontal wells for permeability characterization. Chapter-3 will focus on the improvement of estimated permeability fields using soft data obtained from cross-well travel time tomography. This will be accomplished using cokriging techniques. Up-scaling permeability maps for purposes such as calculating the effective permeability will be presented in Chapter-4. Two up-scaling methods will be scrutinized, namely, Cardwell and Parson method (CP method) and discrete wavelet transform method (DWT method). In Chapter-5 the importance of calculating the correct semi-variogram will be stressed; in light of this realization a systematic method to estimate the inter-well semi-variogram based on horizontal effective permeability and pressure data will be addressed. In Chapter-6 a mathematical model will be presented for interference testing of horizontal wells in 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. naturally fractured reservoirs. Along with the mathematical model derivations, all the pertinent equations and type curves will be presented in this Chapter. The conclusions of this study will be given in Chapter-7. Chapter-8 includes recommendations for future research. Appendix-A will contain the various dimensionless parameters that appeared in previous Chapters. Appendix-B includes the numerical simulation of dual porosity models used to cross validate our derived analytical model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter-2 A New Approach Of Permeability Characterization Using Interference Analysis Of Parallel Horizontal Wells 2.1 Introduction The presence of permeability anomalies such as channel sands and shale barriers in an otherwise uniform geological facies has implications on the accuracy of building representative geologic and flow models. Mapping permeability fields and characterizing heterogeneity in areal and vertical segments of a given reservoir have been major areas of research in the petroleum industry. Concepts such as cross-hole tomography [19], VSP [20] (Vertical Seismic Profile), and Seismic while Drilling have been proposed and tested for mapping lithological changes in strata and identification of flow boundaries. Estimation of reservoir permeability structure, however, has been primarily an inferred approach using porosity correlation or lumped estimates from pressure transient tests. Well test methods have long been useful tools to quantify reservoir properties such as storativity and transmissivity. A particular class of well tests including multiple wells is interference 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. testing providing the opportunity for recognition of permeability variations. This particular advantage can be greatly enhanced in horizontal well interference testing because of larger coverage of reservoir permeability structure. In this Chapter a new form of interference testing between two horizontal wells is proposed. Using conceptual testing procedures such as the one proposed in this study, one can further enhance estimation of permeability fields. 2.2 Conceptual Test Design Consider a segment of a reservoir between two horizontal wells. Examples of such settings include vertical segments considered for SAGD [12], and placement of horizontal wells in flat or dipping reservoirs for improved invasion and sweep efficiency, Fig-2.1. Figure-2.1 Example of Horizontal Well Pairs in Reservoir Exploitation 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The additional cost of drilling such wells may not be justified if the portion of the reservoir between the two horizontal wells includes high permeability channels that can lead to poor sweep efficiency. To test the concept proposed in this study, first a simple permeability structure is modeled where responses created from a point source in one horizontal well is monitored at several distinct points within the second horizontal well, Fig.-2.2. Active Horizontal Well. Entire Well is active B C D A I Observation Horizontal Well Figure-2.2 Segments A, B, C, and D are not in communication with each other except through the formation Numerical experimentations show significant variations in pressure response at various observation points, Fig.-2.3A and Fig.-2.3B.Such response variations are attributed to the particular permeability structure affecting the influence area between the source and response 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. points, Fig.-2.4 shows examples of scenarios in which such a response can occur. It is important to note that the variation in pressure due to permeability contrast is captured only during the transient part of the pressure response in the observation well. After that, the whole system will act as one. This behavior is depicted on Fig.-2.5. Similar results were found by Lasseter et al [13] except that their work focused on the estimation of vertical distribution profile of permeability on a layer basis. Observation Well Segments. Pressure vs Time For Base case-2 1 0 0 1: Segment-A 2: Segment-B 3: Segment-C 4: Segment-D Q . 10 1 0 0 1 Time, hrs Figure-2.3A Pressure distribution is different in each segment 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2020 q = 1000 b/D Q_ O riginal reservoir pressure « 1 9 6 0 1930 10 100 1 Time, hrs Figure-2.3B Pressure History at the Observation Well Channel Sand Shale Breakers Figure-2.4 Permeability anomalies created by channel sands and shale breakers Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 E 0 1 E 1 1 E 2 Time, hrs 1E3 Figure-2.5 All curves join in to form one, representing the entire system To quantitatively represent the model, pressure response function for a horizontal well (see Fig.-2.6) in an infinite reservoir is examined. The fundamental dimensionless pressure drop, PD equation for a horizontal well in an infinite reservoir is [33] and [34]: d’^d’Z d,Zw D ,Ld,tD) = ( - Y 2 '] / > 1 + X n / l 4 7 z > J er} s , 2 V t d j + erf V l- X A ii Jj J 1 + 2]T exp(- n27t2L2 Dr D )cos {nnZD )cos{n7c{Zw D )\ ■ d r D B = 1 J (2.1) where the various terms in this equation are defined in Appendix-A. Equation-2.1 can be used to estimate both wellbore dimensionless 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. pressure drop if evaluated at small values of (XD ,YD ) and can be used to calculate PD for an observation well located at a distance of (XD ,YD ) a way from the active well. Horizontal well — L/2 h Figure-2.6 Horizontal Well Model Active well: We use pressure drawdown analysis techniques [14] to calculate the permeability in the vicinity of the active well. To ensure that pseudo-radial flow develops at late times and according to the following equation [5] the active well was simulated for a period that exceeds tk le where _ I230l}</){ict ^ late 7 (2.2 ) K x This was performed to ensure the accuracy of our analysis for calculating the permeability in the vicinity of the pulsing well. 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Assuming there is no skin factor, pressure response during pseudo- radial flow period is given by [15]: P i~ P w f= H T T T log ** 2 ~ 2'023 M H w-J / (2.3) Equation-2.3 indicates that plotting Pwf versus t on semi-log coordinates will exhibit a semi-log straight line of slope m where The equivalent horizontal permeability is obtained from Equation-2.4 above. Observation well: We used applicable [16] horizontal interference test techniques to compute the permeability in the vicinity of each segment in the observation well. Malekzadeh and Taib [16] proposed several equations each associated with its corresponding type curves to compute the transmissibility and hence the permeability. We chose to work with the following PD equation: m = 162.6 qfiB (2.4) 'match V D Jmatch 'match match (2.5) 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For the sake of completeness Equation-2.6 below is used to compute the storativity that can also serve as a check to our model storativity. 0.03724<zH (PD L „ f dP I dt J match Upon obtaining a match of the plot of the pressure derivative of the observation well segments versus time with Fig.-2.7 or Fig.-2.8, depending on the distance from the observation well to the pulsing well, both Equation-2.5 and Equation-2.6 can be used to obtain the transmissibility and storativity. 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10.0 (XD.YD) Q Q _ 0 > > 0 Q 0 c o t o 0 Q _ 3 4 .0 0 C 0 CO 1 2 -0 b 0.0 0.0 0.1 1.0 tD/rDA 2 Figure-2.7 Type curves for X p, Y& < 1.0 used in conjunction with equations 2.5 and 2.6 0.20 - a C L > |S 0.15 - (XD,YD) v _ < D a Q > v - C O C O 0 > 0.10 ’ 0 - c o C O 0 0 0.05 c C D £ b 0.00 0.0 0.1 1.0 tD/rDA2 Figure-2.8 Type curves for XD , YD >1.0 used in conjunction with equations 2.5 and 2.6 Courtesy M alekzadeh and Taib [16] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.3 Proof of Concept To examine the proposed conceptual testing procedure, a series of numerical experimentations were designed and modeled. A model was selected to examine various permeability heterogeneity distributions; this model was represented by a 1600-grid system. Table-2.1 shows the reservoir properties used for the cases reported in this Chapter. Table-2.1 Reservoir Model Properties Number of Grids 40 x 40 x 1 Grid dimensions 50’ x 50’ x 40’ Porosity 20% C t 6.0 x 10'6 psi'1 Initial Saturation Single phase Production location, a horizontal well (6,6,1) extending parallel to x-axis to (35,6,1) Observation location a horizontal well (6,35,1) extending parallel to x-axis to (35,35,1) 2.3.1 Base-1 Case: This system was assumed to contain a simple distribution where permeability values consisted of four band values: 1.0, 50.0,100.0, and 200.0 md respectively. These values were distributed in a certain pattern as shown in Fig.-2.9. Let us call this image as base-1. Our goal 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. here is to reproduce this realization using the proposed testing concept otherwise known as the inverse problem. From the basic permeability distribution for this case, a number of realizations was made honoring all the statistical attributes of the base case data except the spatial permeability distribution. Fig.-2.10 depicts examples of some of the permeability maps. All of these realizations were generated with no permeability conditioning data. Looking at these images, it is easy to see how different they are from base-1. G my Level o f Pem eability 1 50 100 200 Figure-2.9 First scenario of permeability distribution 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 10 20 30 40 0 10 20 30 40 Gray Level of P erm eab ility 1 ____5 0 1 0 0 2 0 0 Figure-2.10 Selected samples of the first scenario no conditioning data honored Next, we took base-1 and subjected it to our interference test series. This is equivalent to the responses we get from the field or Po b s . We then calculated the permeability values in both the active and observation wells within the influence areas. These values served as our permeability conditioning data at those locations. New realizations were generated using the calculated conditioning data. We now take each generated realization and subject it to simulated interference test. The best realization of the permeability field will be the one that gives rise to the closest possible pressure distribution in the segments of the observation well. This was achieved utilizing the following Equation: Gray Level of P erm eab ility Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. y ] {^obs,! P°b s )' {Pcal,i P ca l ) (2.7) cat J where r is the product momentum correlation coefficient. The realization shown in Fig.-2.11 resulted in the highest value for r and thus was selected as a permeability field best describing Case 1. 2.3.2 Base-2 Case: In base-1 case, the spatial distribution of permeability was limited to four values. Such conditions are relaxed in this scenario and additional noise was introduced to our system to generate a more realistic permeability distribution. 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 10 20 30 40 G ray L e \ e lo f Pem e a b iliy 1 50 100 200 Figure-2.11 Best possible realization to our true model (first scenario) Fig.-2.12 shows the univariate distribution of the permeability. The principal feature of this distribution is its bi-modal characteristic where the second peak is log-normally distributed with a variance, 0 ln k 2 = 1.52, and a mean, (l,n k = 3.75. Fig.-2.13 shows the model directional semi-variogram selected for this study. The model shows that the range changes with direction, while the sill remains constant. This will introduce a geometric anisotropy into our system. As a consequence, our permeability field will possess a larger continuity in the North-South direction than in the East-West trend. 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Base-2 case univariate distribution 25 20 fr15 10 L L 5 0 200 300 400 1 0 0 0 Permeability values, md Figure-2.12 Univariate distribution of the permeability field (second scenario) Model directonal semi-variogram with geometric anisotropy. 9 000 E-W 720 0 ( 3 5400 N-S U i o ■= 360 0 The sill value is the same, however, the range changes with direction. An indication of geometric anisotropy in the system. 1800 • 2 5 0 500 750 1000 0 Separation vector, h, ft Figure-2.13 Second scenario model semi-variogram Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This model was intentionally chosen as such, to test the robustness of our algorithm to reproduce such a feature. The univariate and spatial distributions were then coupled together to generate our permeability field depicted in Fig.-2.14. This will be our base-2 case or true permeability field. Its principle features are the high permeability regions located at the North-West and at the center as well as the low permeability region located at South-West of the field. The previous image was then perturbed by changing the seed value for the simulated annealing [17] algorithm. We then obtained different realizations that possess the same statistical descriptive attributes as our base-2 realization, however, the spatial permeability distributions were different. Table-2.2 shows the univariate statistics for all of our generated realizations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure-2.14 Base-2 case permeability Held Table-2.2 Descriptive Statistics for base-2 case Mean 100.02 Standard Error 2.23 Median 84.07 Standard Deviation 89.28 Variance 7,970.77 Kurtosis 5.19 Skewness 1.86 Range 557.07 Minimum 2.93 Maximum 560 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The next step was to conduct simulated interference tests. We first started with base-2 permeability distribution as the initial reservoir condition. The permeability values that were obtained from these tests were used as conditioning data in the generation of the subsequent simulated annealing runs. 2.4 Incorporation of the Pressure Response Data Pressure distribution in a reservoir during production is often a very sensitive indicator of the presence and strength of barriers and faults and the magnitude and distribution of permeability in the producing zone. Likewise, pressure changes at observation wells, in response to the pulse at the active well, are influenced by the presence of the barriers or faults and the permeability field within the influence region between the pulse and the probing points. At this point we have many realizations that honor the descriptive statistics as well as the semi-variogram of our initial permeability field. All of these realizations are equally probable to represent our field. Fig.-2.15 shows samples of the semi-variograms of the generated realizations in comparison with base-2 case. To some degree all of them depict some resemblance to base-2 case. Semi-variograms by 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. themselves are inadequate criteria if not accompanied by other data. To narrow down our choices, we take each realization, and subject it to the simulated interference test. The best realization of the permeability field will be the one that gives rise to the closest possible pressure distribution to base-2. D irectional S em i-variogram for realization-1 Directional Sem i-variogram for realization-10 9900 7920 . 5940 1980 run-1 0 750 1000 500 0 250 9900 7920 „ 5940 3960 1980 run-10 0 0 250 500 750 1000 Separation vector, h, ft S eparation vector, h, ft Directional Sem i-variogram for reali2ation-17 D irectional Sem i-variogram for realization-18 9900 7920 E-W i 5940 3960 1980 run-17 750 1000 500 0 250 9900 7920 5940 1960 run-18 0 250 500 750 1000 S eparation vector, h, ft S eparation vector, h, ft Figure-2.15 A sample of the semi-variogram of the generated realizations compared with base-2 case Geometric anisotropy has been reproduced in all of the realizations 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Equation-2.7 was used to calculate r, the product momentum correlation coefficient, for each pressure response in each segment of the horizontal well for all realizations. In other words, for the example under discussion, a given realization will end up with five different values for r. One value is for the active well, and four values will each represent one segment of the four present in the observation well. Following that, a new correlation index, Cl, was defined as the weighted linear combination of all of the segments. The higher the correlation index the closer the realization to base-2 case. Fig.-2.16 is a plot of the correlation index for 20 runs. Clearly, our best choice will be realization-18 because it has the maximum correlation index. Fig.-2.17 depicts base-2 case as well as a sample of the many realizations generated. It is obvious that realization-18 bears the most resemblance to base-2 case among all, as predicted from our correlation index plot. Correlation Index for all the runs 1.00000 0 .9 9 9 9 5 — 0.9 9 9 9 0 0.9 9 9 8 0 0.9 9 9 7 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Run number Figure-2.16 Plot of the correlation index, Cl, for 20 runs. Run-18 has the highest C l value. 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B ase-2 case gray level permeability map * 1 Run-1 gray level permeability map Run-17 gray level permeability map Run-1 8 gray level perme ability map I .5 2 .7 Figure-2.17 Base-2 case as well as a sample of the many realizations generated. It is obvious that realization-18 bears the most resemblance to base-2 as predicted by C l plot 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.5 Results And Discussion As previously shown, run-18 (in base-2 case) was selected to be the best probable realization to represent our true permeability field. Let us examine the results more closely. Figures-2.18A, B, C, and D depict the pressure curves generated from the interference tests at the observation well in segments A, B, C, and D respectively. Looking at these curves, we see that realization-18 distinguishes itself from the rest of them by giving the best pressure match-up to the base case. This distinction ceases to exist for the pressure curves generated in the active well, Fig.-2.19. One possible explanation is that the entire portion of the active horizontal well is on production. This, of course, subjects the active well to the same pressure drop everywhere. If, however, the active well were to be segmented as well, then a different pressure drop well develop in each partition, depending on how porous or how tight the formation is in the vicinity of each segment. As a result, different pressure curves would be generated for each simulated realization. 2.6 Determining Impervious Shale Distribution Thus far we considered the areal permeability distribution, in the X-Y plane. Following the same approach outlined in the previous two scenarios, this method can also be used to model vertical shale distribution, in the X-Z plane. Fig.-2.2 as well as all of the equations 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. would still apply, however, the pulsing well location would be at a different depth form the observation well. 100 Base: 1 ru n l8 : 2 runlO: 3 ru n l: 4 runl7: 5 ( / > Cl Q _ 0.1 0.01 1 10 0.1 Time, Days Figure-2.18A Interference tests for segment-A of the observation well compared with the base case. 100 Base: 1 runl8: 2 runlO: 3 runl: 4 runl7: 5 Q. 10 0.1 Time, Days Figure-2.18B Interference tests for segment-B of the observation well compared with the base case 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 Base: 1 runl8: 2 runlO: 3 ru n l: 4 runl 7: 5 C O Q_ C L CL 0.1 10 Time, Days Figure-2.18C Interference tests for segment-C of the observation well compared with the base case 100 Base: 1 runl8: 2 runlO: 3 runl: 4 runl7: 5 C O Q. CL 0.1 1 10 Time, Days Figure-2.18D Interference tests for segment-D of the observation well compared with the base case 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1E3 1E 2 d . 1E 1 1E O 0.01 0.1 1 10 0.001 Time, Days Figure-2.19 Pressure and pressure derivative response for selected realizations at the active well 2.7 Uniqueness of the solution Many possible realizations exist that honor the univariate statistics as well as the semi-variograms. However, these two criteria alone are not adequate to render the closest permeability distribution for the true field. For instance, Fig.-2.15 shows a sample of realizations that fulfill the aforementioned criteria but are different in the spatial arrangement for permeability. As a result, each one would give a different pressure response. This new dimension of pressure curve matching for the segments' interference tests will drastically narrow down our selection among the equally probable realizations. In addition, the more information we have such as production data and 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the geology of the reservoir the more we can improve and refine our selection among the equally probable realizations. 2.8 Multiple Pulsing Points Thus far, the well generating the pressure waves is taken to be the entire horizontal portion. Fig.-2.20, however, suggests that the active horizontal well can also be segmented, and pulsed one segment at a time, while recording the pressure changes in the partitions of the observation well. This will add more constrains to our final solution because of the generation of more pressure curves, hence, more permeability conditioning data along the active well. The final product of all of this is the reduction of the number of equally probable realizations and the improvement of the delineation of the permeability field. Active Horizontal W e i A selected segment is pulsed one at a time A T ^ / \ w \ x /-~ . \ / X X / V: : . ''l \ > X / ' A A x / y < -. / y \ s \ ^ y / . . y y / A / A f Observation Horizontal Well Figure-2.20 Inclusion of multiple source points within the active well 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.9 Summary and Conclusion We have proposed a new conceptual approach for mapping permeability fields between two horizontal wells. • The proposed method can in principle be applied in the active well using DST's generated pulses monitored at a response well while the active well is under drilling. • Variations in pressure due to permeability contrast are captured only during the transient part of the pressure response in the observation well. • Permeability maps honoring the univariate statistics as well as the semi-variogram are not conclusive without the pressure curve matching from the interference test. • The procedure has application in designing SAGD and other recovery processes in dipping reservoirs where horizontal wells are planned for improved recovery. • Inclusion of multiple source points within the active well, Fig.-2.20, can further improve the delineation of the permeability fields. In the next Chapter, permeability maps are going to be further enhanced by the inclusion of other soft data, namely, traveltime tomography information. This will be achieved utilizing means such as cokriging. 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter-3 Permeability Cokriging with Cross-Well Traveltime Tomography 3.1 Introduction Describing the heterogeneity of a reservoir is a common task that faces both geological and reservoir engineering disciplines. Knowledge of the geology can be utilized to construct a reservoir model giving the most likely hydrodynamics response, however, such a model may result in a poor prediction of inter-well reservoir properties, i.e. porosity and permeability. An objective of reservoir engineering may require only a model of heterogeneity that gives a desired response from reservoir simulations. This response is often a result of inversion models giving a non-unique solution satisfying a given history matching for a field. Incorporating as much knowledge of the geology and well data as possible into a model, one can greatly reduce the usual gaps caused by the uncertainty of the large and small-scale heterogeneities. Information [18] describing the subsurface permeability field is usually obtained 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. using small-scale or micro tools like cores (inches) and well logs (inches to several feet). The large-scale or macro tools include pressure transient testing and seismic data, Fig.-3.1. The micro-probing tools have their limitations as permeability predictors; they are inherently limited to their scale of measurement. Furthermore, the scarcity of the cores and log data makes it even more difficult to describe permeability fields with the level of resolution often sought. The macro tools (hundreds of feet), specifically the conventional surface seismic, can provide extensive large-scale images of subsurface reservoirs, however, fine scale permeability values are not directly obtainable from such data. Pressure Transients and Seismic Core Samples Well Logs Figure-3.1 Schematic diagram illustrating the scale of probing of cores, well logs, and pressure transient and seismic. 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Cross-well travel time tomography provides a middle solution between the fine-scaled and large-scaled probing tools. An experiment [19] carried out in an oil-producing field of the West Permian Basin, Texas, U.S.A. (with 15 - 20 feet resolution) provides an idea of the potential usefulness of this tomographic method. Another field tomography project [20] in the Midway Sunset field, California, with a resolution of approximately 10 feet places even more confidence on this method. 3.2 Cross-Well Traveltime Tomography The prefix tomo is Greek for slice and therefore implies a 2-D reconstruction. Tomography is a term used to refer to a particular process whereby an image is produced for some physical properties of an object under study. The technique known as seismic traveltime tomography, Fig.-3.2A and Fig.-3.2B, can be used to obtain a host of petrophysical properties of the area of interest. Such petrophysical properties would include primary or compressional wave velocity, Vp , profiles, secondary or sheer wave velocity profiles, Vs , Poisson's ratio, a , density distribution, p , and also porosity, cp. Following is a brief description of all of the aforementioned properties from cross-well tomography prospective. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. P-waves and S-waves: P-waves push (compress) and pull (dilate) rocks in the direction the wave is traveling. S-waves, on the other hand, shake the particles at right angles to their direction of travel. The propagation of P-waves involves changing the volume and shape of the intervening material, while S-waves change only the shape. Because solids, liquids, and gases all resist being compressed and will elastically spring back once the force is removed, P-waves can travel through all types of matter. On the other hand, because S-waves change only the shape of the media through which they travel and because fluids (gases and liquids) do not resist changes in shape, fluids will not transmit S-waves. In water, for example, Vp = 2.4 mile/sec, while Vs = 0 mile/sec. However, in any solid media P-waves travel approximately 1.7 times that of S-waves. Vp / Vs ratio is a parameter that is directly obtained from these two quantities and is used as a qualitative indicator of lithology, type of fluid, porosity, and pressure. Poisson's ratio: Poisson's ratio, o , is defined as the rate of change of transverse contraction to longitudinal extension when a material is stretched. This elastic constant depends on the rigidity of the media. Mathematically, a is defined as, 1/2 — a = r \ 2 Y. \ vp j (3.1) 1- r \2 Y, vVp; 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. From the above relation, it is clear to see that a = 0.5 when Vs = 0, in the case of liquids which can not stand shear stress. In general, though, the range for a goes from 0.05 for extremely hard rocks, as in granite, to approximately 0.45 for unconsolidated materials. Besides its normal usage, this parameter, a , can be a qualitative indicator for fluid saturation. This was demonstrated [19] in the study performed on the Central Basin Platform in the Permian Basin of west Texas, USA. The lithology in that field usually ranges in the low values of Poisson's ratio. After a water injection program took place, the same lithology gave high values of a , confirming the presence of fluid saturation. D ensity: Density distribution, p , can be obtained for the tomographed area once the compressional velocity profile is known. Lindseth's linear regression equation is one method to generate a density contrast of a medium. Lindseth's linear regression equation is „ _ V p ~ 3 4 6 0 P ~ 0.308 V p ( 3 ' 2 ) Porosity: Porosity is another outcome that can be related to our cross-well travel time tomography. Willie's time average equation is one way to do that and is, 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. At-At f n - 1 — ma At -A t ( 3 * 3 ) LXl f ma where At is the formation travel time and can be computed using Vp profile, Atm a is the matrix travel time, and Atf is the fluid travel time. Ranges of values [21] of compressional wave velocity and transit time for common rock matrix materials are listed in Table-3.1. Table-3.1 Ranges of values of sonic compressional wave velocity and transit time for common rock matrix materials. Common rock matrix materials (V p)m a ft/sec A tm a fisec/ft Sandstones 18,000-19,500 55.5-51.0 Limestones 21,000-23,000 47.6-43.5 Dolomites 23,000 43.5 Anhydrite 20,000 50.0 Salt 15,000 66.7 Water 5,300 188.68 3.2.1 Traveltime Tomography Inversion A typical problem of a 2-dimensional inversion is to infer the compressional wave slowness (reciprocal of velocity) distribution of a giving medium. It is most convenient to develop inversion and tomography equations in terms of wave slowness models, because the pertinent formulas are linear in slowness. Given a set of observed first- arrival traveltimes between sources and receivers of known locations 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the slowness distribution is determined. Conventionally, cross-well tomography has been applied for imaging a 2-D region between vertical bore holes [19] and [20], Fig.-3.2A. In this work, however, we are proposing the same application using horizontal bore holes, Fig.- 3.2B. The obvious advantage of horizontal wells over their conventional counterparts is the larger reservoir coverage they possess. Well A Well B • Seismic wave transmitter i Seismic wave detector Area of interest Figure-3.2A Cross-well tomography geometry for conventional wells Well A Area of interest Well B • Seismic wave transmitter # Seismic wave detector Figure-3.2B Proposed Cross-well tomography geometry for horizontal wells 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3 Mathematical Formulation Suppose we have a set of observed traveltimes, tv t2 , ......... , tm from m source-receiver pairs in a medium of slowness s(x). Let P, be the ray path connecting the ith source-receiver pair. We can write js(x )d lp ‘ =f., 1= 1,2, ,J (3.4) P , given a block model of slowness, let be the length of ith ray-path through thejth cell \d l (3.5) P t ncell j For a model with n cells equation (3.4) can be written as Z L S-=t., i = 1, ,m y j 1 (3.6) 7 = 1 Note that for any given i, the length of the ray-path is non zero only if it intersects a given cell in the slowness model. Fig.-3.3 illustrates ray- path segmentation for a 2-D cell model. 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Source * S , S 2 S 3 S 4 S 5 S 6 S 7 / s 8 hs s 9 ®10 y ^ s n S i2 ®13 / s 14 S ,5 S 16 o Receiver Figure-3.3 Schematic illustration of ray paths through a cell slowness model Equation-3.6 can be written in matrix notation by defining the column vectors s and t and the matrix M as follows V ( t ^ ( lu hi • L ' s2 ^ 2 ^ 2 1 12 2 • h n s = ■ , t = , M = ■ KS n J ) J m \ 1m l Imn j equation (3.6) then becomes 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ms = t (3.7) There are several methods to solve equation (3.7) for the slowness model. The most widely used among them is the SIRT (Simultaneous Reconstruction Technique) and ART (Algebraic Reconstruction Technique) algorithms. 3.4 Comparison Between Cross-Well Tomography And Well Logging Reference [20] compares the two methods utilizing Fig.-3.4. The true geology is depicted in Fig.-3.4a. Suppose that well logging tools detect a low porosity layer in the interval 800-900 feet in well-A and well-B logging tools show that there is a low porosity layer in the interval 700- 800 feet. Because of the fact that well logging tools in general probe only from several inches to several feet into the formation, a geologist or an engineer will have to interpolate the logs. Such interpolation is depicted on Fig.-3.4c. Cross-well tomography, Fig.-3.4d, on the other hand, uses down hole seismic wave transmitters that are powerful enough to probe the area of interest by sending seismic energy. These seismic activities are recorded by many detectors that possess very high signal-to-noise ratio. Fig.-3.4e is the final tomography interpretation. These cartoons can qualitatively 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. demonstrate how cross-well tomography can capture the geology structure away from the wells that would have been otherwise missed using conventional logging tools. Well A Well B Well A Well B & - 500' - 600' - 700' - 800 - 900' -1000' , - y r ( ( " I 1 1) ) i i * _' > ((40) t r (a) True geology (b) Well logging Well A Well B k (c) Well loggiing interpretation Well A Well B Well A Well B - 500' - 600' - 700' - 800 - 900' - 1000 ' True geology (d) well tomography Well A Well B (e) Well tomography Interpretation Figures-3.4a, b, c, d, and e: A carton illustration for the comparison between cross-well tomography and well logging 3.5 Permeability Cokriging With Cross-Hole Tomography Data Sections 3.5.1 and 3.5.2 will describe in details how we generated synthetic tomograms (secondary or soft data) and permeability data (primary). 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.5.1 Generation of synthetic tomograms The following steps were followed to generate the synthetic compressional wave velocity, V normally obtained from cross-hole tomography: • Generate the bulk density, p b, using any distribution desired (in our case we selected a normal distribution with p = 2.64 and a = 0.038) • Use the Lindseth's linear regression equation that relates compressional wave velocity, Vp , to bulk density as follows, _ Up-3460 P b ~ 0.308Vp (3 -8 ) • Select an appropriate semi-variogram to distribute Vp data on a 64 X 64 X 1 grid system. • Generate multiple realizations and select one to perform the study, Fig.-3.5 and Fig.-3.6. The corresponding semi-variograms for these realizations are shown on Fig.-3.7 and Fig.-3.8. Run-2 in Fig.-3.5 was selected to represent our true field case. The corresponding semi-variogram of run-2 is depicted on Fig.-3.7. Note that with the above-mentioned procedure how non-unique our realizations are, even though they all follow the same model semi- 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. variogram as well as the same normal distribution and same descriptive statistics. A summary of the descriptive statistics for the tomographic data is presented on table-3.2. Table-3.2 summary of the descriptive statistics for the tomographic data __________________________________________________ Mean 18772.0 F Standard Deviation 1186.2 o Variance 2 1407151.1 < T Minimum 16848.9 M aximum 20927.9 3.5.2 Generation of the primary or permeability data Using our same bulk density distribution, the following relation, equation (3.9) was utilized to create a porosity map for our true field. « Pma P b *P ~ (3.9) P m a - P f Where in this case we chose p m a = 2.9 g/cc (in the case of Dolomite and Anhydrite) and p f = 0.85 g/cc in the case of oil. From the porosity we were able to generate our corresponding true field primary data image (permeability field). This was done utilizing the following: 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. k m = 250 (P 3 A (3.10) wi J or J c]n = 100 (P 2.25 A V c wi y (3.11) In real life, however, different porosity-permeability transformations would produce different results. Consequently, one can capitalize on this difference to come up with a unique correlation between compressional wave velocity data and permeability for the reservoir at hand. In our case, the final permeability filed is depicted as Run-2 in Fig.-3.9. A summary of the descriptive statistics for this permeability field is presented in table-3.3. Table-3.3 A summary of the descriptive statistics for the permeability data M ean V 48.07 Standard Deviation a 38.06 Variance a 2 1449.12 Minimum 6.25 M aximum 142.59 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Gray level Vp for run-2 <fray fewrf Vp for run-4 Gray level Vp for run-3 Figure-3.5 Gray level maps of VP , compressional wave velocity. R uns-1,2, 3, and 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure-3.6 Gray level maps of VP, compressional wave velocity. Runs-5,6,7, and 8 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Run-1 Run-2 Distance Distance Run-4 Run-3 Distance Distance Figure-3.7 Semi-variograms for compressional wave velocity R uns-1,2 ,3 , and 4 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Run-5 Run-6 Distance Distance Run-7 Run-8 Distance Distance Figure-3.8 Semi-variograms for compressional wave velocity R uns-5,6,7 , and 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Gray lavai of true permeability map Figure-3.9 Gray level map of true permeability Held 3.6 Pressure Interference Testing Our series of interference testing was performed on the permeability field depicted on Fig.-3.9, the corresponding permeability calculated along the horizontal wells were treated as hard or primary data that were subsequently honored at those locations. This study utilized guidelines by Alkhonifer and Ershaghi [22] when performing this task. 3.7 Semi-variograms and cross-semi-variograms A variogram, or a semi-variogram, is defined as half the average squared difference between paired data values from the same attribute, equation (3.12). I N{h) (3.12) 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where N (h) is the number of pairs and xt is the value at the start or tail of the pair i and yt is the corresponding end or head value. Similarly, a cross-semi -variogram is defined as half of the average squared difference of h-increments relative to two different attributes, equation (3.13). I n{ h ) 7 x y ^ = 2 N ( K ) ^ (3-13) where again N (h) is the number of pairs and x, is the value of attribute x at the tail of pair i and x, is the corresponding head value. The locations of the two values x, and x, are separated by vector h with specified direction(s) and distance tolerance, (y, - y, ) is the corresponding h-increment of the other attribute y. There are several variogram models developed for fitting sample variograms. Three standard variogram models are commonly used in the literature. [23] and [24]. 1. Exponential Variogram Model. This is a transitional model which is defined by an effective range a and a sill s: y(h)= 1 - exp( - — ) (3.13) a 2. Gaussian Variogram Model. This is also a transitional model that is often used to model extremely continuous processes. This model is defined by an effective range a and a sill s: 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3. Linear Variogram Model. This is not a transitional model since it does not reach a sill, but increases linearly with the separation distance h. Its standard equation takes the form: y{h)=\h | (3.15) 4. Spherical Variogram Model. Perhaps the most commonly used variogram model. It is a transitional model with a range a and a sill s, its normalized equation (normalized to the variance) takes the following form: y{h)=\ h 1.5— -0.5 if h< a V \a otherwise (3.16) Figure-3.10 depicts the three most commonly used variogram transition models. 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. exponantial model spherical model Gaussian model range Figure-3.10 The three most commonly used variogram transitional models The semi-variograms as well as the cross-semi-variograms used throughout these studies were the same. This was done to ensure proper comparison of all the final estimated permeability fields using the aforementioned methods. Figure-3.11 shows the experimental semi-variogram for the permeability field. The data set was obtained from the interference test series performed previously. The model semi-variogram for the same data is depicted in Fig.-3.12. Fig.-3.13 and Fig.-3.14 display the experimental and the model semi-variograms for the tomographic data respectively. The cross-semi-variogram for the two data sets is shown in Fig.-3.15. We notice that the shape looks more like a covariogram than it is a semi-variogram. This is due to the nature of the 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. relationship between the primary and the secondary variables being inversely proportional. In other words, porous areas are expected to transmit P-waves at a slower rate than does tighter matrix. The fitted or model cross-semi-variogram is shown in Fig.-3.16. Moreover, it is often possible to construct the directional cross-semi-variogram for data sets that are abundantly sampled, like in the case of tomographic operations. Fig.-3.16 shows that there is more continuity in the northern-southern direction than in the western-eastern trend. This is inferred from the determined range values, a, for the two directions. 6 00. 4 0 0 -1 & O s > 200.-I 100._ I lo o . 150. 0. 50. 200. 2 50. Distance Figure-3.11 Experimental semi-variogram for permeability data set 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. eoo._ 5 0 0 . 200.-I experimental variogram model variogram 1 0 0 .^ 100 . 130. 200 . SO. 2 5 0 . 3 0 0 . 0. Distance Figure-3.12 Model semi-variogram for primary variable permeability data set & o ‘S 1700000. _f 1600000.J 1500000. J 1400000 1300000. i 1200000.. 1100000-4 I000000.J 900000-4 600000-4 7 0 0 0 0 0 - i 600000.J 500000. J 4 O O O O O .4 300000-4 200000-4 100000. J 0 1 1 SO. 1 1 I 1 100. I I I I 1 50. 200 . 2 5 0 . 3 0 0 . Distance Figure-3.13 Experimental semi-variograms for secondary variable, Vp data set 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. kb o 1 7 0 0 0 0 0 . 4 1 600000. J 1 50 0 0 0 0 .4 1400000.-I 1 3 00000.4 1200000.J 1100000-4 1 000000. J 9 0 0 0 0 0 . 4 800000. J 7 0 0 0 0 0 .4 eooooo.J 500000. i 4 0 0 0 0 0 . 4 300000. J 200000 100000.4 o .i V experimental variogram model variogram 0. I - 5 0 . 100. 150. 2 0 0 . 2 5 0. 300. Distance Figure-14 Experimental and Model semi-variograms for VP data set \ k b o E\W ^ -34827. _ 0. 5 0 . 1 0 0 . 150. 200 . 2 5 0 . 300. Distance Figure-3.15 Experimental cross-variogram primary and secondary data sets 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. experimental variogram model variogram & o ■ g > \W 1 0 0 . 150. ZOO. 2 5 0 . 3 0 0 . 5 0 . 0 Distance Figure-3.16 Experimental and Model cross-variograms Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.8 Estimation of the primary data in the entire grid system To estimate the primary data at all the nodes of our grid system we used several geostatistical techniques that can be categorized into two groups: I. Methods utilizing only the sample values of the primary data we are trying to estimate. This class would include and is not limited to methods like: A. Polygon. B. Triangulation. C. Inverse distance methods. D. Ordinary Kriging. These point estimation methods are all linear and theoretically unbiased (i.e. mean residual or error, mR = 0). But what distinguishes ordinary kriging among them all is its aim of minimizing the error variance, a 2 R. As such, ordinary kriging is the only estimation method that would be considered in this category. II. Methods that not only consider the primary data of interest, but also one or more secondary variables. These secondary variables are usually spatially cross-correlated with the primary variable and thus contain some useful information about the 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. primary variable that we are trying to estimate. These methods would include and are not limited to: A. Kriging with an external drift B. Cokriging: • Collocated cokriging • Standardized ordinary cokriging • Ordinary or traditional cokriging . 3.8.1 Ordinary kriging This method is often associated with the acronym B.L.U.E. for "best linear unbiased estimator". Ordinary kriging is linear because its estimates are weighted linear combinations of the available data; it is unbiased since it tries to have mR, the mean residual or error, equal to zero; it is best because it aims at minimizing a 2 R, the variance of the errors. Fig.-3.17 depicts how schematically kriging works, [25]. Figure-3.17 Kriging finds the value of the point to be kriged so that it best fits the variogram model Model Variogram 102.3 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Mathematically, we solve the following System of equations, where, (c '-'1 1 c 12 Ct. n fvO f c \ ' “'1 0 r 22 C2 „ i w 2 c„ r n2 c nn i Cn 0 1 1 . . 1 O y y )Jt2 V (n + l)* (n + l) {n + \) x \ (n + 1) jc1 C • W — D (3.18) C : is the covariance matrix calculated from the covariance function. W : is kriging weights matrix. D : is the covariance matrix for the estimated point. (i : is the Lagrange parameter to constrain the solution of the kriging system. To solve for the ordinary kriging weights, W, we multiply the above equation by C" 1 , the inverse of the covariance matrix. If, for instance, we are estimating permeability data, then the estimated permeability value at a given location is calculated as, 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. n ^ estimated, i (3.19) 1 = 1 3.8.1.1 Results and Discussion The true permeability field is depicted on Fig.-3.18A. The final kriged permeability map is shown on Fig.-3.18B. Even though some of the principle features or the highlights of the original map were captured or recognized by this method, at least near or around the tested segments of the horizontal well, it is obvious that it did not do a satisfactory job in the estimation of the permeability field in undersampled locations. It acted like a low pass filter smoothing out the data. The correlation coefficient between the true and the kriged permeability data is 0.25. Fig.-3.19 shows a scatter plot of kriged values vs. true ones. We notice, that there is a big cluster of the estimated values exiting near or around the mean of the data set. That is the best kriging can do, in a case of undersampled data sets' situations such as ours. To derive a more informed permeability estimate in the areas where kriging assigned a near constant value equal to the mean of the data, additional information is needed to constrain the model. One can take advantage of other types of secondary data in the area. This leads us to methods like kriging with an external drift and cokriging with all of its forms. 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Gray level of true permeability mat Figure-3.18A True permeability field better continuity in the north-south direction Gray level of kriged permeability map Figure-3.18B Kriged permeability field. Acted like a low pass filter smoothing out the data to the average value Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. True perm eability values, md Figure-3.19 Cross validation of true vs. kriged permeability values. Correlation coefficient = 0.25 3.8.2 Kriging with an external drift Kriging with an external drift is a simple yet efficient algorithm for incorporating a secondary variable in the estimation of the primary variable. The drift or trend component is usually modeled as smoothly varying deterministic function whose unknown parameters are fitted from the secondary data. The function that defines the trend should be specified by the physics of the problem. The trend is usually modeled as a low order (<2) polynomial. Fig.3.20 schematically illustrates how kriging with an external drift works, [25]. 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Well Variogram model Kriging with External Drift K EDM ap Tomograms Figure-3.20 Kriging with an external drift, using well and tomographic data 3.8.2.1 Results and Discussion The true permeability field is depicted on Fig.-3.18A. The final result for this method is shown on Fig.-3.21. It is evident that more of the principle features or the highlights of the original map were captured or recognized by this method in comparison to the ordinary kriging estimates. This method better estimated areas that carried the least information, i.e. a way from the segments of the horizontal wells, where the hard data are. In ordinary kriging, however, such areas were more or less given a constant value that is equal to the mean of the hard data set. Fig.-3.22 illustrates this point where the scatter points are not as concentrated around the mean value as in Fig.-3.19. Instead an obvious departure of the scatter points from the average value towards the 45° lines took place in this approach. The correlation coefficient between the true and the kriged permeability values is 0.41 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. that is better than what ordinary kriging did. Not withstanding all of the above, this approach still acted as a low pass filter smoothing out data. Kriged with external drift perm, map Figure-3.21 Permeability kriged with an external d rift The drift improved the estimates compared to ordinary kriging ' ‘ . t f t* • V • • A *. * . • V . *• V *\ ••A* •• -• • , * < * • v . ' H J M V ; ... » i & y R * I H R K v S i v *N^v r I 3l| 57 S3 109 135 True permeability values, md Figure-3.22 Cross validation of true permeability vs. permeability kriged with and external drift; Correlation coefficient = 0.41 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.8.3 Cokriging Cokriging uses data defined on different attributes. In our situation, the permeability values are estimated from a combination of permeability samples and related tomograms data values. Fig.-3.23 schematically illustrates how cokriging works, [25]. Well data Variogram model Cokriged map Cross- variogram model Cokriging Variogram model Tomograms Figure-3.23 Cokriging using well and tomographic data. Note the cross- variogram model in addition to variogram models Kriging requires the determination of only one semi-variogram for the data at hand. In cokriging, however, we need to determine at least three semi-variograms. One describes the spatial variability of the primary data, Figures (3.11 and 3.12); the second describes the spatial variability of the secondary data, Figures (3.13 and 3.14); while the third being a cross-variogram describes the cross spatial variability 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. between the two data sets, Figures (3.15 and 3.16). The usefulness of the secondary data is often enhanced by the fact that primary variable of interest is undersampled. For example, in our situation all available hard data, permeability data, are known at only few locations where the probing points of the interference test series exist. Our secondary variable, however, covers more or less the entire region of interest. In areas where the only information we have is the secondary data, the cross-correlated information contained in Figures (3.15 and 3.16) plays a pivotal role in the cokriging estimates. The mathematical development of the cokriging system is identical to that of the ordinary kriging system except that the former involves a much bigger covariance matrix containing both the primary as well as the secondary variable information. The equivalent of equation-3.19 in cokriging is, where the first summation is for the primary data and the second summation is for the secondary data set. In our case, e s tim a te * ' ■ final cokriged permeability value at a given node. n m k estim ated (3.20) : cokriging weight for a given primary data point or permeability. 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. u. : cokriging weight for a given secondary data point or compressional wave velocity. k, : permeability data point involved in the final estimate. VP ) . : compressional wave velocity data point involved in the final estimate. As mentioned earlier, there are several ways to combine given data sets of different attributes into a cokriging system. Following is a closer look at each method. 3.8.3.1 Collocated cokriging This is a reduced method of cokriging that consists of retaining only the collocated secondary data. One obvious advantage of this class of cokriging is that it demands less, in terms of data and the subsequent joint modeling of the covariance matrix. Hence, less computational burdens. Here, we only need to construct the covariance matrix of the collocated data of the different attributes rather than the entire data set of the secondary variable. We will skip this type and focus on the full method. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.8.3.2 Standardized ordinary cokriging This approach consists of creating new secondary variables with the same mean as the primary variables. Then all the cokriging weights are constrained to some to one, i.e. n m 1 > , + Z “ , = 1 (3.21) /=1 7=1 From the above equation, we note that both the cokriging weights for the primary variable as well as the cokriging weights for the secondary variable directly contribute to the final estimate of the permeability values being estimated. If it is desired, however, to determine the final estimates of the permeability values using only cokriging weights for the primary variable then the second summation in the above equation should be set to zero. This is going to be tackled in ordinary or traditional cokriging. 3.8.3.3 Results and Discussion Fig.-3.24 is the result of this type of cokriging. It is obvious that the resulting map identifies the general trend of the original permeability field. It successfully estimates the general shape of the true data, i.e. the better continuity along the North-South direction. However, there is an obvious smoothing effects on the final estimates. 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Standardized cokriged permeability map * * * 48 Ji ■ * * » a * * *» "fjjff; , ___' w l l T y 1 ' Figure-3.24 Permeability cokriged with tomographic data using Standardized Cokriging with one unbiased condition, equation-3.21 True permeability values, md Figure-3.25 Cross validation of true vs. Standardized Cokriged permeability values; Correlation coefficient = 0.50 Fig.-3.25 shows the scatter plot of the true verses the estimated data points. A correlation coefficient of 0.50. A fat cloud around the mean 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. value of the data exists. This cloud is not as concentrated as in the ordinary kriging estimates that can be regarded as an improvement in our estimates to the true permeability map. 3.8.4 Ordinary or Traditional Cokriging In this approach the sum of the cokriging weights of the primary variable is set to one, where as the sum of the cokriging weights to any other variable is set to zero. These two conditions are: n m S w i = 1 a n d Z u i = 0 (3.22) ;=1 j = 1 The last constrain tends to limit the influence of the secondary variable on the final cokriged values to a certain extent. It can also cause some cokriged values to be negative. Consider the case where only two secondary data values are found equidistant from the point of estimation and from all primary data. Since they are equidistant, they must be weighted equally, and since the nonbiased condition for the secondary data requires the weights to sum to zero, one weight must be negative, the other positive. Although mathematically correct, this solution is difficult to be related to physical processes with such weighting scheme. On the other hand, having cokriging weights, for the secondary variable, to sum to zero will ensure that all of the 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. estimated primary data are weighted combination of only adjacent primary data. 3.8.4.1 Results and Discussion Fig.-3.26 is the result of this type of cokriging. It is obvious that the resulting map is by far the best estimate for the true permeability field. It not only identifies the general trend and shape of the original permeability map, but it also successfully picked up some of the small details that other methods smoothed out. Ordinary Cokrigod Parmaability mat Figure-3.26 Permeability cokriged with tomographic data using Ordinary Cokriging with two unbiased conditions Fig.-3.27 shows the scatter plot of the true verses the estimated data points. The correlation coefficient is 0.88. The cloud about the 45- degree line is the best we have so far. This cloud is not as concentrated 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. as in the previous estimates that can be regarded as an improvement in our determined true permeability map. True permeability values, md Figure-3.27 Cross validation of true vs. Ordinary Cokriged Permeability values; Correlation coefficient = 0.88 3.8.5 Conclusion and Summary of applied methods Table-3.4 summarizes the results of the estimation methods in comparison to the actual data set. Comparing its statistical outcomes to the original data set, traditional cokriging consistently proved to be the best technique considered so far. Other commonly used visual comparison tools like q-q and p-p plots are also shown. A q-q plot is a graph on which the quintiles form two distributions are plotted versus one another. A q-q plot of two identical distributions will plot as 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. straight line x = y. A p-p plot, on the other hand, compares the cumulative probabilities of two data distributions, e.g. campers an original data distribution to a distribution of simulated points. These plots compared each estimation method with the original data set. This comparison gave consistent results with table-3.4. Table-3.4 Summary of descriptive statistics for all of the estimation methods in comparison to actual data set. M ethod Statistics Original Data set Ordinary Kriging Kriging with an external drift Standardized Cokriging Traditional Cokriging Mean P 48.07 34.55 34.74 25.14 37.6 Standard Deviation o 38.06 9.02 9.25 19.1 30.4 Variance o2 1449.1 81.4 85.61 366.0 923.01 Minimum Value 6.25 11.2 10.1 0.14 0.10 Maximum Value 142.6 72.6 72.6 113.9 125.7 Correlation Coefficient P 1.0 0.21 0.41 0.50 0.88 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Q-Q Plot of Krtged Permeability Field 12 0._ 3- t E 8 0 . _ 4 40._ 40. 80. 120 . T m e Permeability Figure-3.28 A q-q plot of kriged permeability field q-q Plot o f Krtged Perm, w/ Ext. Drift Q eo._ 40._ 40. 0. 80. 1 20 . T rue Permeability Figure-3.29 A q-q plot of kriged permeability field w/ external drift Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. q-q Plot o f Stand. Cokriged Map 120._ id 40. 60. 120. T rue Permeability Figure-3.30 A q-q plot of standardized cokriged permeability field q-q Plot o f Ordinary Cokriged Map 120 ._ L o id V E f c Q _ 1 g > o o > . l a c ‘ E O 4-0 . eo. 0 . 120 . T rue Permeability Figure-3.31 A q-q plot of ordinary cokriged permeability field Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. P-P Plot o f Standardized Cokriged Uj T J ,2 0 _ .0 0 . .00 .20 .40 .B O ,B0 T rue Permeability Figure-3.32 A p-p plot of standardized cokriged permeability Held P-P Plot o f Ordinary C okriged Uap ,80_ « 4) ^ .B O I L .0 0 . .20 40 .00 .60 .8 0 T rue Permeability Figure-3.33 A p-p plot of ordinary cokriged permeability field 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In the next Chapter, up-scaling of permeability maps will be undertaken. This work will enable us to investigate the permeability distribution at different levels of resolution. In addition, system effective permeabilities will be generated and analyzed. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter-4 Up-Scaling Permeability Maps 4.1 Introduction As it is obvious from Chapter-2 and Chapter-3 work, many simulation runs ought to be performed to come up with the best possible realization that bears the most resemblance to the true permeability field. Consequently, we have to scale-up our stochastically generated permeability fields from fine to practical field simulation scales. Otherwise, simulations on blocks (that are in the order of millions in real fields) will be prohibitively expensive. Moreover, effective permeability calculations can be done using up- scaling techniques as will be demonstrated in the next Chapter. Scale-up of permeability has been addressed in several studies. Cardwell and Parsons [26] reached to the conclusion that effective permeability of heterogeneous blocks lie between the harmonic average as a lower bound and the arithmetic average as an upper limit. M. A. Malik et al [27] devised a method based on the Cardwell and Parsons (CP method). 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.2 Up-Scaling In this Chapter, two up-scaling methods will be scrutinized and applied to some of the generated realizations in the previous and next Chapters. 4.2.1 Cardwell and Parson Method As mentioned in the introduction this method suggests that the effective permeability of heterogeneous blocks lie between the harmonic average as a lower bound and the arithmetic average as an upper limit. The geometric average fulfills this condition. Reference [27] proceeded with the following equation, \ U N k g a = F h V 1 = 1 (4.1) where, k ca- is the geometric average permeability. k j : is the permeability of block z . N : is the number of fine-scale blocks within the course block. The way they proceeded is as follows: 1. Row s First, the fine-scale cells in each row are treated as layers in series and an effective permeability is computed as a harmonic mean for each row. The rows are now treated as layers in parallel to calculate the 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. effective permeability as an arithmetic average of all the harmonic means we already obtained. 2. Columns: Next, we switch to the columns where they are treated as layers in parallel and an effective permeability is calculated as an arithmetic average for each column. The columns are then treated as layers in series and an effective permeability is calculated as a harmonic mean of the already computed arithmetic averages. 3. Final step Now, a geometric average of the results of steps 1 and 2 above is taken according to equation-4.1 to calculate the scale-up permeability. [27] as well as [28] both agree on the fact that the reliability of the CP method is limited to log- normal distributions only. 4.2.2 Up-scaling using Discrete Wavelets Transforms Discrete wavelets transforms or DWT are being used in this study to up-scale permeability realizations. Also, a comparison between DWT and CP method in up-scaling is presented. Both case-2 and run-18 from Chapter-2 are used in this comparison. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2-D wavelets transforms are straightforward extensions of the 1- D version. At each level of 2-D data decompression we obtain an average estimate of the previous fine scale. In 2-D we obtain three components of data details that represent variations of data along the x-axis, variations of data along the y-axis, and variations of data along the diagonal direction. 4.3 Mathematical Formulation Panda et al [31] describes the wavelet transforms as "Mathematically, the wavelet transform is a method that projects a function f(x) on various vector sub-spaces that represent different scales. By defining a suitable scaling function cp{x) the projection of f(x) onto the sub-spaces can be viewed as successive approximation of f(x) at varying resolutions since it acts as a low-pass filter or as a smoothing function.". One can think of <p(x) as the averaging window commonly used in statistical analysis. Chu et al [32] presents a good mathematical derivation for the implementation of discrete wavelet transform. 4.3.1 Up-scale 2-D Permeability One can arrange the block permeability as a matrix, where ktJ represents permeability and the subscript i and j are indices for rows and columns respectively. 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. K° = '* 1 ,1 k k 1.2 1,3 ' k 2.2 • k ^ K\,n *,M . • K j K n (4.2) . ■ k m,n ) The superscript 0 indicates the initial state or the discrete approximation at scale 0. We need to convolute the permeability values along one direction with the low-pass filter first, followed by a convolution using the same filter on the permeability along another direction. For our work, we selected a standard Daubechies filter, D4 . Tabel-4.1 shows the coefficients for D, and D,. Table-4.1 Daubechies-4 and Daubechies-6 wavelet functions’ coefficients 4 2 •i & Closed form Open form 8 significant digits a ' Daubechies-4 Daubechies-6 D-4 D-6 h, (l+ V 3)/(4V 2) (1+VlO +^5 + 2VlO )/(16V2) 0.48296291 0.33267055 1 * 2 (3+V 3)/(4V 2) (5+ 710+ 3^ 5 + 2VlO )/(16V 2) 0.83651630 0.80689150 1 * 3 (3-V 3)/(4V 2) (10-2 V1 0 +2V5 + 2V 10 )/(16V 2 ) 0.22414386 0.45987750 1 * 4 (1-V 3)/(4V 2) (10-2 V lO -2^/5 + 2V10 )/(16V 2 ) 0.12940952 -0.1350110 1 * 5 - (5 + 7 1 0 -3 ^ 5 + 2V lO )/(16V 2} - -0.0854412 1 * 4 - (l+V lO -x/5 + 2V 10 )/(16-n /2 ) - 0.03522629 Equation-4.3 shows the general make-up for a four-coefficient low-pass filter matrix. 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. H l,LI2 ~ (K h2 h3 K 0 0 . 0 °1 0 0 K h2 h3 h4 0 . . . 0 0 0 0 0 0 hi h2 h3 hA . . 0 0 J 1 ? , K 0 0 0 0 . hx h2 y (4.3) Notice that the column dimension is always twice that of the row in matrix H. In multi-resolution analysis, a given matrix X with dimensions (L x n) at scale level m is written as Xm L n . Using this notation, we can decompose the matrix permeability in equation-4.2 at scale level 0 along the j direction as, {Kl LnJ=nL n,LKl„ (4.4) Equation-4.4 reduced the number of the permeability grids along the j direction by a factor of 2. The final permeability matrix at scale level 1 is determined by decomposing the resulting intermediate permeability matrix (K1 ^ n ) along the i direction in a similar manner, {K{n ,nn) = { H nnj K ' Lnj } (4.5) Combining equation-4.4 and equation-4.5 a recursive procedure can be established to compute equivalent permeability values at an arbitrary courser scale m from information at a finer scale m-1 as follows, jy-m TT j y m - \ ( t _t V L /2 ,n /2 L /2 ,L L,n V 1 n / 2 , n ) (4.6) 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Notice that after each level of decomposition the number of the permeability grids is reduced by a factor of 4. 4.4 Results and Analysis DWT has two filters associated with it, a low and a high pass filter. It is the high pass filter that enables us to pick up all the details. Even though we keep getting smoother and smoother data each time we acquire a courser grid, the details are hardly lost in the process. Moreover, the differences are always accounted for between one fine scale and the next coarser grid. This is what makes it possible to reconstruct the original fine-scale permeability field from any level of a coarse scale field. Fig.-4.1 depicts the original fine scale permeability field, 40 x 40 grid and the 20 x 20 up-scaled maps for the CP and the DWT methods. Fig.-4.2 and Fig.- 4.3 also show the up-scaled version from 40 x 40 to 10 x 10 and 5 x 5 grids respectively for the CP and the DWT methods. If we examine any level of up-scaling using the two methods, we will observe consistent better results using the DWT method. Unlike the CP method, DWT method does not suffer from the smearing or the sm oothing effects on the up-scaled m ap resolutions. M oreover, it captures and recognizes sudden permeability changes, i.e. porous and tight formations that are juxtaposed. This makes DWT more suitable in 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. up-scaling heterogeneous permeability fields, like in naturally fractured reservoirs or faulted reservoirs or any combination of the aforementioned. Base-2 case permeability map 40 x 40 grids Jt________Sl CP Up-scaled map 20 x 20 resolution DWT Up-scaled map 20 x 20 resolution Figure-4.1 Original fine scale k-field (40 x 40) above and (20 x 20) up-scaled grids using the two methods 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Base-2 case permeability map 40 x 40 grids CP Up-scaled map 10 x 10 resolution DWT Up-scaled map 10 x 10 resolution I ■ H I Figure-4.2 The original fine scale permeability Held (40 x 40) above and (10 x 10) up- scaled grids using the two methods 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Base-2 case permeability map 40 x 40 grids CP Up-scaled map 5 x 5 resolution DWT Up-scaled map 5 x 5 resolution Figure-4.3 The original fine scale permeability field (40 x 40) above and (5 x 5) up-scaled grids using the two methods 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Tabele-4.2 shows the standard deviations as well as the coefficient of variations for the original permeability field and for each level of resolution. It is obvious that smoothing effects are more pronounced in the CP method. Looking at the 5 x 5 resolution level, we can see that the coefficient of variation is greater in the DWT method than in the CP one. This indicates that the sharp spikes of the permeability field have not been lost by the averaging effects. This is what makes the DWT very efficient in picking up sharp discontinuities such as faults in the macro-scale level, whereas, keeping the heterogeneous identity at the micro-scale level. Table-4.2 Comparison table for CP and DWT methods performance Resolution level CP method DWT method 40 x 40 original map STDV = 89.25 CV =0.91 STDV = 89.25 CV = 0.91 20x20 STDV = 75.2 CV = 0.78 STDV = 77.8 CV = 0.88 10 x 10 STDV = 54.6 CV = 0.57 STDV =58.79 CV =0.68 5 x 5 STDV = 26.3 CV = 0.45 STDV = 48.6 CV = 0.486 STDY: the standard deviation. CV: the coefficient of variation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.5 Effective permeability calculations DWT method can be harnessed to compute the effective permeability for a given field. For this to be possible, though, the field grid system must be of 2n x 2n dimensions, where n is a positive integer. will be obtained after n levels of decompositions using DWT. Fig.- 4.4 is a realization of 26 x 26 dimensions. We notice that n = 6. In order for us to compute the effective permeability for such a realization, 6 levels of decompositions must be performed. Fig.-4.5A through Fig.4.5F each shows a different resolution level on a gray scale map for the same permeability field. Fig.-4.5F is our desired ke fr. Lamda - 0.5, Reatfzation-3 Figure-4.4 The original fine scale permeability field (64 x 64) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. eabfttt^jl 32x32 i . 32.3 Figure-4.5A First resolution level up-scaled permeability field 32 x 32 Up-scaled permeability 16 x 16 . 32.7 Figure-4.5B Second resolution level up-scaled permeability field 16 x 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Up-scaled permeability 8 x 8 Figure-4.5C Third resolution level up-scaled permeability field 8 x 8 Figure-4.5D Fourth resolution level up-scaled permeability field 4 x 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Up-scaled permeability 2 x 2 Figure-4.5E Fifth resolution level up-scaled permeability field 2 x 2 Up-scaled permeability 1 x 1 ygUktsi* 143. 129. 115. 101 . 67.7 7 3 9 60.1 46.3 32.6 16.6 5.0 Figure-4.5F Sixth and final resolution level up-scaled permeability field l x l Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter-5 will explore the potentiality of Discreet Wavelet Transforms, as a better technique for up-scaling permeability fields, to calculate system effective permeabilities. From such calculations, appraisal of spatial correlation length will be presented. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter-5 Estimation of Semi-variograms from Effective Permeability and Pressure Data 5.1 Importance of Calculating the Correct Semi- variogram The estimation of the semi-variogram in the lateral or inter-well direction is pivotal when performing reservoir characterization studies using stochastic modeling. This Chapter presents a systematic method to estimate the inter-well semi-variogram based on horizontal effective permeability and pressure data. The pressure data are measured at multiple probing points along an observation horizontal well in an interference test, Fig.-5.1. Entire W ell is A ctive Probing points along the observation well Figure-5.1 Schematic representation for the multiple probing points in the observation horizontal well and the active well for an interference test 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Several studies in the literature stressed the importance of estimating the horizontal semi-variogram to achieve a good reservoir description. Lucia and Fogg [29] stated that the principle difficulty in reservoir characterization is estimating the spatial distribution of petrophysical properties between vertical wellbores. Many other investigators made such observations. 5.2 Proof of Concept To test the procedure proposed in this study, first several runs were conducted to determine how sensitive and discernible semi- variograms are with respect to pressure data. 5.2.1 Procedure for Proofing the Concept For a given reservoir structure with a specific PDF of permeability, two realizations with two different correlation lengths were generated. These two realizations were created with the same seed in the simulating annealing algorithm. This was done to ensure unbaisedness in our results. We used X = 300 ft and 800 ft for the first and the second realization respectively. Using Fig.-5.1 schematic, pressure data were obtained from the observation well at multiple points along the well. Semi-variograms were then computed and plotted for each realization at different times. 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.2.2 Results and Discussions Fig.-5.2A depicts the calculated semi-variograms for the first realization at different elapsed times. Like wise, Fig.-5.2B shows the calculated semi-variograms for the second realization at the same elapsed times. The effect of X on the measured pressure data is very pronounced in both cases. This establishes the dependence of measured pressure data on the correlation length. Subsequent sections will examine and present this realization and quantify it. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T O O i i > M 200 J 100 J Distance, ft I I ► Distance, ft 1 1 > 1 Distance, ft 1520. _ 1 10 00.- Distance, ft I 1 Distance, ft I I > 1000. _ Distance, ft Figure-5.2A Calculated semi-variogram for the first realizations at different elapsed times 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Semi-variogram Semi-variogram Semi-variogram Distance, ft 500. J 1 1 0 . 400. Distance, ft o . 200. 400. Distance, ft 1000. _ Distance, ft i 1 > I in 0. 200. 400. 600. eoo 1000. Distance, ft Figure-5.2B Calculated semi-variograms for the second realization at the same elapsed times 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.3 Methodology The methodology adopted here is similar to what was done in reference [30]. The difference is that they used performance data to determine the correlation length. This work, however, focuses on pressure data. Specifically, pressure data obtained from interference testing for horizontal wells using multiple probing points. 5.3.1 Procedure 1. Assume that the reservoir heterogeneity structure could be represented using several correlation lengths. Correlation length, X, is defined in terms of the reservoir dimensions, i.e., X = 0.50 corresponds to correlation continuity up to one half the reservoir length. For this study we used correlation lengths of, A , = 0.00, X2 = 0.125, X3 = 0.25, X4 = 0.50, X5 = 0.75, X6 = 1.0, X7 = 1.5, Xs = 2.0. 2. Using simulating annealing algorithm, several realizations are generated for each correlation length. The system probability density function, PDF, is the same throughout this study. Fig.-5.3 shows a sample of those realizations at different correlation lengths. 3. For each realization the effective permeability, ke f f , is computed using discrete wavelet transforms, already discussed in Chapter-4. i l l Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure-5.4 portrays gray scale maps at each level of resolution. The final l x l dimension is our desired ke ff for that particular realization. 4. A type curve plot of k€ vs. A is generated. Fig.-5.5 represents this type curve. This graph could also give us some insight of the uncertainty associated with this method. 5. Pressure response is simulated for each correlation length to each realization and the following steps are followed to generate the second type curve of cr2 a p versus time: I. Obtain the time versus pressure readings for all the probing points along the observation horizontal well from the simulated pressure interference test. II. Compute Ap at each time step for all probing points. Fig.-5.6A through Fig.-5.6H portray a sample of the simulated interference pressure tests at A ,, A 2 / A 3 , A 4 , A 5 , A 6 / and \ respectively. III. For each time step, calculate the variance, cr^p, to all the Ap computed at the previous step. IV. Repeat steps I. - III. for all realizations at the particular A . V. Calculate the expected a2 a p values for all the realizations. VI. Generate a plot of a2 a p versus time for that particular A . VII. Repeat steps I. - VI. for all correlation lengths. 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. VIII. Calculate the expected o2 a p values for all the realizations. This expected value will give us the final type curve for this particular A ,. Fig.-5.7 is the type curve for cr2 a p versus time. 6. Using the first type curve of ke ff vs. X and/or the second cr2 A p type curve, a reasonable value of X can be calculated for a given permeability field with the same PDF. 7. From the PDF and its descriptive statistics, i.e., variance, the sill of the variogram is known. This will complete the picture as far as the nature of the semi-variogram we are dealing with. Table-5.1 illustrates these previous steps, where n is the number of time steps and j is the number of probing points, Pb, such that pn j is the pressure reading at time step n at probing point j. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure-5.3 A sam ple o f realizations at different correlation lengths 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 5.1 A layout of the procedure followed to generate the 0 * A p type curve Time P b p t l P b pt 2 ■ P b pt j P b p t l A p Pb pt 2 A p Pb pt j Ap o^A p t, Pi, P« Pn Ap„ A P,2 AP,i O^tAp,,.. APlj) t2 P2 , P2 2 P 2 1 A p2 , A P2 2 AP2 | ^ I AP 1 ' A P2 jl P„, P „2 P n i A P„, A P n 2 AP n i ^..(APm- APn |) At this point we need to calculate the expected a2 A p values for all the realizations. This expected value will give us the final type curve for this particular X . 5.4 A Case Study Following the procedure outlined in section 5.3.1, eight correlation lengths were assumed to represent the permeability heterogeneity. \ = 0.0, X 2 = 0.125, X 3 = 0.25, X 4 = 0.5, X 5 = 0.75, X 6 = 1.0, X 7 = 1.5, and X 8 = 2.0 The system's effective permeabilities were calculated for each realization using DWT. Fig.-5.4 depicts gray scale maps, to one of the realizations, for each level of resolution as up-scaling is performed. The final resolution of 1 x 1 represents ke f f to be used in the effective permeability type curve. Table-5.2 through Table-5.8 were generated and tabulated using DWT. 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table-5.2 Horizontal effective permeability at X = 0.00 for five realizations Realization # Keff Realization-1 42.47 Realization-2 42.42 Realization-3 41.78 Realization-4 41.00 Realization-5 42.10 Table-5.3 Horizontal effective permeability at X = 0.125 for five realizations Realization # Keff Realization-1 41.2966 Realization-2 41.5876 Realization-3 42.2019 Realization-4 41.694 Realization-5 40.4873 Table-5.4 Horizontal effective permeability at X = 0.25 for five realizations Realization # Keff Realization-1 41.7393 Realization-2 43.3846 Realization-3 42.8895 Realization-4 43.3294 Realization-5 43.7546 Table-5.5 Horizontal effective permeability at X = 0.50 for five realizations Realization # Keff Realization-1 46.6523 Realization-2 41.4833 Realization-3 42.84.78 Realization-4 46.113 Realization-5 44.8603 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table-5.6 Horizontal effective permeability at X = 0.75 for five realizations Realization # Keff Realization-1 42.603 Realization-2 45.2718 Realization-3 43.4745 Realization-4 48.278 Realization-5 47.6317 Table-5.7 Horizontal effective permeability at X = 1.0 for five realizations Realization # K eff Realization-1 44.1211 Realization-2 44.0342 Realization-3 44.341 Realization-4 50.2164 Realization-5 49.1054 Table-5.8 Horizontal effective permeability at X = 1.5 for five realizations Realization # K eff Realization-1 42.6924 Realization-2 39.8706 Realization-3 41.0863 Realization-4 42.2913 Realization-5 49.5694 Table-5.9 Horizontal effective permeability at X = 2.0 for five realizations Realization # Keff Realization-1 50.5085 Realization-2 46.9467 Realization-3 46.7881 Realization-4 47.9152 Realization-5 48.7982 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Lamda = 0.5, Realization Orginal permeability map 64 x 6 4 Up-scaled permeability 32 x 32 Up-scaled permeability Id x 16 Up-scaled permeability 8 x 8 Up-scaled permeability 4 x 4 Up-scaled permeability 2 x 2 Up-scaled permeability 1 x 1 Figure-5.4 Gray scale maps at each level of resolution using DWT Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Effective horizontal permeability obtained by DWT, keff 55.0 te 51.0 -§ 47.0 43.0 l j 39.0 35.0 0.75 0.25 0.50 1.00 1.25 0.00 1.50 1.75 2.00 Correlation length, lamda Figure-5.5 Effective permeability type curve calculated using DWT Interference test for realization-2 Lamda = 0.0 1 0 0 0.01 10 1 100 times, hrs Figure-5.6A: A sample of the simulated interference pressure tests at X = 0.0 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Interference test for realization-4 Lamda = 0.125 1 0 0 - T 1 0 - f c : Q . r a 0 ) Q 0.01 1 0 0 10 times, hrs Figure-5.6B A sample of the sim ulated interference pressure tests a t X = 0.125 Interference test for realization-4 Lamda = 0.25 1 0 0 10 1 0.1 0.01 10 100 times, hrs Figure-5.6C A sample of the sim ulated interference pressure tests at X = 0.25 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Interference test for realization-5 Lamda = 0.5 100 0. .2 " a i Q 0.01 100 10 1 times, hrs Figure- 5.6D A sample of the simulated interference pressure tests at X = 0.5 Interference test for realization-1 Lamda = 0.75 1 0 0 ... 10 1 0.1 0.01 1 0 0 10 1 times, hrs Figure- 5.6E A sample of the simulated interference pressure tests at X = 0.75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Interference test for realization-5 Lamda = 1.0 100 Q . < D O 0.01 10 100 1 times, hrs Figure- 5.6F A sample of the simulated interference pressure tests at X = 1.0 Interference test for realization-3 Lamda = 1.5 100 . ► * £7 ;.....:.................... ^ ... 10 1 0.1 0 .01 10 1 0 0 1 times, hrs Figure- 5.6G A sample of the simulated interference pressure tests at X = 1.5 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Interference test for realization-4 Lamda = 2.0 100 C L r o " a i D 10 100 times, hrs Figure- 5.6H A sample of the simulated interference pressure tests at X = 2.0 Delta P variance vs. time, all realizations 200 Lamda No. Correl. length 0 . 0 0 0 0.125 0,250 0.500 0.750 1 . 0 0 0 1.500 2.000 160 -120 20 30 50 60 40 0 10 tim e, hrs Figure-5.7 Type curve for O2 ^ versus time 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.5 Conclusion A systematic approach was devised for estimating spatial correlation length in the horizontal plane between the wells. This was done for correlated permeability fields having the same k PDF. Two type curve plots were generated. Fig.-5.5 depicts the ke ff vs. X type curve whereas Fig.-5.7 displays the o \p vs. X plot. From the first, it was observed that reservoirs with long correlation lengths, X , exhibit large values of ke ff but reservoirs with shorter continuity were shown to exhibit lower R values. Similar observations were found from the a \p vs. X plot, where reservoirs with long X gave large values of g\p whereas formations with short X produced low estimates. The influence of X on both the system ke ff and pressure response was easily noticeable. The previous findings were instrumental in quantifying the system X through the generated type curves. Up until this point, characterization of permeability fields was investigated using techniques that are numerically and geostatistically based. The next Chapter, however, will shift momentum towards developing analytical mathematical models that can be informative and contributive to the field of reservoir characterization. 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter-6 Interference Testing of Horizontal Wells in Naturally Fractured Reservoirs 6.1 Introduction The previous Chapters dealt with the characterization of permeability fields that are more or less homogeneous from a macro scale prospective. The need to characterize reservoirs that are not only heterogeneous in the micro-scale but also so in the macro-scale arises so often in real life. Naturally Fractured reservoirs are among the category of what we refer to as heterogeneous systems. In naturally fractured reservoirs that are considered for gas injection or pressure maintenance in general, a good characterization of the permeability field plays a pivotal role in the success and execution of such plans. In this Chapter a mathematical analytical model is developed for interference testing of horizontal wells in naturally fractured reservoirs. This model will help us determine the permeability values in the influence area of the segments of the horizontal wells much so like it did in Chapters two and three. The difference here, of course, is the natural fractures that are present. 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Horizontal wells possess larger contact area with the formation, and hence, greater radius of influence than vertical wells. As a result, information obtained from horizontal well testing, like transmissibility and storativity, will be more representative of the formation than the ones obtained from vertical well testing. A lot of work has been done in the area of single well testing for horizontal wells. F. Daviau, G. Bourdarot et al [33], and M. D. Clonts & Ramey [34], explored this area and obtained solutions to this problem based on the instantaneous source functions together with the Newman product method. This work was further extended by R. Des & AJ. Rosa [35], Aguilera & Ng. [36], and Stewart & Heriot [37], to incorporate dual porosity reservoirs. Multi-well testing in horizontal wells, on the other hand, received little attention compared to its conventional counterpart. D. Malek Zadeh and D. Tlab [38] developed a model for interference testing of horizontal wells in homogeneous reservoirs. Our work deals with the problem of interference testing of horizontal wells in naturally fractured reservoirs, or dual porosity models. Many authors have considered extending the solution of homogeneous reservoirs to naturally fractured reservoirs using what is known as the variation law. This transformation is achieved through 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the Laplace domain. F. Davlau and G. Bourdet et al suggested this approach and R. de S. Carvalho & Rosa tried it. In doing so, a lot of conditions and limitations had to be imposed on the results to a degree where the solution became of little practicality. To circumvent this problem, we used a very neat approach devised by Aguilera. This method exempts us from dealing with the Laplace space, and hence, with the potential inaccuracies that may arise from numerical inversions. This work provides the dimensionless pressure and dimensionless pressure derivative type curves along with the appropriate equations. Permeability, storativity, and all the different properties associated with naturally fractured reservoirs will be determined utilizing these equations. The model will be simulated using a numerical simulator to cross check its validity. 6.2 Mathematical Model Derivation Following is an outline of how we get the pressure distribution for a horizontal well which is infinite in the x and y directions and bounded by no flow boundaries top and bottom, in the z direction, see Fig.-6.1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. z A Horizontal well h O x Z y Figure-6.1 Horizontal well model schematic The solution presented here is similar to the one presented by Clonts and Ramey [34] using the instantaneous source functions together with the Neuman product method. Using green's function theory, we can best describe our model as the intersection for the following 1-D models: • x-direction: An infinite slab source in an infinite reservoir. • y-direction: An infinite plane source in an infinite reservoir. • z-direction: An infinite plane source in an infinite slab reservoir. Mathematically; the instantaneous source functions in the x, y, and z directions respectively are, (6.1) (6.2) cos n n :—^ h (6.3) 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The expression for the pressure drop caused by the production from a continuous source is: 1 , D AP= fqL{t)-S{X,Y,Z,t-T)dT (6.4) < P ct o where, qL (t) : flow per source length = q /L S(X, Y, Z, t - t): is the product of the three instantaneous source functions. Now let us introduce our dimensionless parameters as follows: I x k M W ) PD= y — (6.5) k t tD = - ------ 77— T f (6-6) qll k t l° ~ / 2 f X D (L/2) ( 6 ,? ) Y Y° (L/2) ( 6 ' 8 ) D = ~i (6-9) h L lkz y LD~ 2 h 4 f (6-,0) Substitution of equation-6.1 through equation-6.3 and equations-6.5 through equation-6.10 into equation-6.4 will yield our 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. solution for the distribution of the pressure in a horizontal well bounded by a no-flow boundaries top and bottom and infinite along the x and y directions. J Z ' d PD{XD,YD,ZD,tD)= ^~ \f{ X D,rDy f{YD,TDy f{ZD,tD)dtD where, f ( X D,rD)= (6.11) fl + X0 > ) ( M 1 — X n erf j . i d l2 V ^ J + erf D l2 V ^ » Jj / ( I D > T D)=- 7 = eX P f _ Y 2 ^ Lp K 4 td J f ( Z DtTD)- 1 + 2 ^ exp(- n 2 7r2 l}D TD)cos{nnZw D )cos(n^ZD)| n = 1 Note that the above solution assumes that k = k and k can be * y % different. Therefore, transmissibilities obtained are areal average in the x and y directions. MalekZadeh et al [38] considered interference testing in horizontal wells. Their work considers interference testing in horizontal wells in homogeneous reservoirs only. In this work, however, we are going to consider the case where we have heterogeneous reservoirs; i.e. naturally fractured reservoirs. Many authors [34] and [36] have considered extending the solution for homogeneous reservoirs to NFR by the variation law which is done in the Laplace space. Figure-6.1 shows this procedure. 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Step 1. Solution of Homogeneous problem in real time. Step 7. Solution o f double porosity in real time. Step 2. Step 6. Solution of Homogeneous Take the inverse Laplace problem in Laplce space. 5 Step 3. Multiply by S -fc Step 4. Replace s by sf(s) E H> Step 5. Devide by s. Figure-6.2 Variation law in the Laplace domain Courtesy A guilera and N g [36] This method stipulates that the boundary conditions of step #1 must not be a function of time. In other words, Dohamolt boundary type conditions can not be applied here. But the real difficulty in this procedure lies in step #2, where finding the Laplace transform of some solutions either does not exist or is extremely difficult to evaluate. F. Davlau et al [33] has suggested this approach and Carvalho and Rosa [35] have tried it, and in doing so, a lot of conditions had to be imposed on the results to a degree where the solutions became of little practicality. Moreover, the Laplace transform integral had to be broken into intervals and some of those intervals were estimated numerically, i.e., lDl g(s)= Je ‘ f (t)dt' where, tD l and tD s are dimensionless long time tDs and short times respectively. 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It is possible to achieve the same goal using a very neat approach devised by Aguilera [36]. He showed that it only requires substitution of the storage term in real time homogeneous reservoir solution by: ^=C0+{l-O))-f(t,l) (6.12) where the function f(t,z) depends on whether psss or unsteady state (transient) interporosity model is assumed. The following two equations are for pseudo-steady state interporosity model and for transient interporosity model respectively. - f - 1 1 -e T ( 6.i3) f (t, t ) = y /t/r • tanh Vr/7 (6.i4) where t is the time and T is the time constant approximately equal to the transition time reaching the state of the total system and equal to 1 / A. This method also eliminates the need of using the Stehfest algorithm to numerically invert to the real time from the Laplace space. 6.2.1 Psss-interporosity Model For this case, £ D will become: €d =CO + (l-C o)-(l-QX p(-tDA)) (6.15) Equation-6.15 will be used in conjunction with equation-6.11 to arrive to equation-6.16, which will describe the pressure distribution in the reservoir for this model. 132 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Z d PD{XD,YD,ZD,tD4 D) = - ~ Jf { X D,TD)-f{YD,TD)- f ( Z D,TD)dTD (6.16) where again, f ( X D,tD)= f ( . erf V V l + X, + erf j K 2^Jtd l - X , W J J f{YD,TD) = - ^ e x p f y 2 ' \ n *D V 4 T D J f ( Z D, t D )=[ 1 + 2 J e x p ( - n 27t2l}D TD )cos{nnZw D )cos(nxZD) V « = t 6.3 Type Curve Analysis Type curve analysis consists of finding a type curve that matches the actual response of the well and the reservoir during the test period. In this work, we consider a Pof vs. to type curve. This type curve can be used for the test data obtained from the field or from a simulator in the case of designing an interference test. Choosing which family set of type curves for analysis is dictated by the distance between the pulsing and observation wells. 6.3.1 PD f vs. tj, type curves Given some test data for P vs. t in an observation well, we plot them on the same size log-log scale as that of the type curve. With sliding the field data plot only vertically and horizontally on the 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. appropriate type curve, we then determine the right one. From this curve, we already established two important parameters, storage ratio, co, and the interporosity flow parameter, X, The remaining four important quantities, t, tD , P, and PD are determined from the match point. We then apply the following equations. now, using the value of co and equation-6.18 the (coC)f term can be found from: The interporosity flow parameter, X , will help us determine the shape factor, a, from the following equation: where km can be obtained from core analysis. Following are the type curves generated for the above equations for different values of Ul.2qjuB (PDf\ M (6.17) Having estimated kf , we then proceed to find {{(pC)m + {cpC)f ) from: 0.0010548 kf (t ) (6.18) (6.19) (6.20) 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PDf PDf storativity ratios, inter-flow parameters, and different distances from the active well to the observation well. 1E + 0 1 1 E + 0 0 - omega = l 1: omega =0.01 2: omega =0.001 3: omega =0.0001 1E-01 1E-3 1E-2 1 E - 1 1 E 0 1E1 1 E 2 1E3 1E4 1E5 tD ZD = ZwD = 0.5: LD = 10; XD = 0; YD = 4 Figure-6.3 Interference testing for horizontal wells in NFR XD = 0, YD = 4 1E+01 1 E + 0 0 • • 1: omega =0.01 2: omega =0.001 3: omega = 0.0001 omega = 1 IE-01 1E-3 1E-2 1E-1 1 E 0 1E1 1 E 2 1E3 1E4 1E5 tD ZD = ZwD = 0.5; LD = 10; XD = 3; YD = 2 Figure-6.4 Interference testing for horizontal wells in NFR XD = 3 , YD = 2 135 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1E+01 p 1 E + 0 0 • 1: omega =0.01 2: omega = 0.001 3: omega = 0.0001 omega = 1 m m i 1E-1 M tH f - 1 E 2 1 E - 0 1 1E1 1E-3 1E4 1E5 tD ZD = ZwD = 0.5; LD = 10; XD = 3; Y D = 5 Figure-6.5 Interference testing for horizontal wells in NFR XD = 3 , YD = 5 1E+01 P 1E+00 1: omega = 0.01 2; omega = 0.001 3: omega = 0.0001 4; la m ia = 0.01 6: lamda = 0.0001 - h - h + H —J- 1E4 - s— f i m i l - 1 E 0 1 E 2 1 E - 0 1 1E-2 IE-1 1E1 1E-3 1E5 tD Z D = ZwD - 0.5; LD = 10; X D = 5; YD = 2 Figure-6.6 Interference testing for horizontal wells in NFR XD = 5 , YD = 2 136 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1E+01 S-IE+OO 1: omega =0.01 2: omega = 0.001 3: omega = 0.0001 omega = 1 1E-01 1 E 0 1E1 1 E 2 1E-2 1E -1 1E3 1E4 1E5 1E-3 tD ZD = ZwD = 0.5; LD = 10; XD = 5; YD = 5 Figure-6.7 Interference testing for horizontal wells in NFR XD = 5, YD = 5 6.4 Conclusion An approach to extend solutions of homogeneous reservoirs to NFR was utilized. This method eliminated the need of using Stehfest algorithm to convert from Laplace space to real time. Two families of type curve plots were generated, namely, PD f vs. tD and PD f' vs. tD . From these, we can estimate storativity ratio, interflow parameter, permeability and porosity of the fractures. This mathematical model can serve as the base for extending the work that was done in the previous Chapters to include naturally fractured reservoirs. 137 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter-7 Conclusion This study w as conducted to investigate the potential applications of horizontal well pairs in characterizing perm eability fields. In this respect, the study included perm eability characterization using a new approach in repeat interference testing analysis for parallel horizontal wells. This concept w as further exam ined and subsequently enhanced to include soft data, namely, traveltim e tom ography data, w hich w ere cokriged w ith perm eability inform ation to yield m ore representative and detailed perm eability fields. This w ork required num erous sim ulation runs to come up w ith the best results. Consequently, up-scaling techniques w ere undertaken and examined. Discrete w avelet transform s, DWT, w as concluded to be our choice for the task of up-scaling fine perm eability m aps to their respective coarser grids. In addition, effective perm eability calculations were perform ed using DWT to ultim ately aid us in the estim ation of semi-variograms in the lateral or inter-well direction. Thus far, characterization of perm eability fields w as conducted through the utilization of num erical and geostatistical techniques. How ever, well test analysis from the point of view of conceptual m athem atical models 138 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. will always be informative and contributive to the field of reservoir characterization. As such, a m athem atical analytical m odel w as developed for interference testing of horizontal wells in naturally fractured reservoirs. A new conceptual approach for m apping perm eability fields betw een tw o horizontal wells has been proposed. From this work, it was established that perm eability distribution honoring the univariate statistics as well as the semi-variograms are not conclusive w ithout the pressure curve m atching from the repeat interference test. It w as also found that variations in pressure, in the observation well, due to perm eability contrasts are captured only during the transient part of the pressure response. This m ethod is suitable for im plem entation w hile the active well is under drilling through DST's generated pulses m onitored at a response well. Inclusion of m ultiple source points w ithin the active well can further im prove the delineation of the perm eability fields. Finally, the procedure has application in designing SAGD and other recovery processes in dipping reservoirs w here horizontal wells are planned for im proved recovery. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. W hen combined w ith other secondary data, like traviltim e tom ography information, our procedure outlined in Chapter-2 proved to be rew arding in the final estim ation of perm eability fields. A m ong the different techniques studied to include the secondary variables, ordinary or traditional cokriging gave the m ost prom ising outcomes. It not only identified the general trend and shape of the original perm eability distribution, but it also successfully picked up some of the small details that other m ethods sm oothed out. In up-scaling perm eability fields, tw o techniques w ere scrutinized, CP m ethod and DWT m ethod. Unlike the CP approach, DWT does not suffer from sm earing or sm oothing effects. U pon exam ination of any level of up-scaling using the tw o m ethods, w e observed consistent better results using DWT. It captured and recognized sudden perm eability changes, i.e. porous and tight formations that are juxtaposed. This m akes it m ore suitable in up-scaling heterogeneous perm eability fields, like in naturally fractured reservoirs or faulted reservoirs or any com bination of the aforementioned. DWT w as also utilized in the calculations of the effective perm eability betw een the wells. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Pressure variance analysis derived from interference tests betw een horizontal hole pairs was show n to be a potential tool to identify the perm eability correlation length. Two type curves w ere generated for this purpose. Effective perm eability type curve, calculated using DWT, and pressure variance type curve. These two are independent of each other and hence can serve to cross validate one another w hen estim ation of correlation length is desired. Finally, a m athem atical m odel w as developed for interference testing of horizontal wells in naturally fractured reservoirs. A n approach devised by Aguilera [36] to extend existing solution of hom ogeneous reservoirs to NFR w as utilized. This approach elim inated the need of Stehfest num erical inversion from Laplace space to real time. O ur m odel provided the dimensionless pressure and dimensionless pressure derivative type curves along w ith the appropriate equations. Permeability, storativity, and all the different properties associated w ith naturally fractured reservoirs were determ ined utilizing these equations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter-8 Recommendations for Future Work 1. A two-dim ensional stochastic m odeling was done throughout this study. Extending it to a three-dim ensional problem w ould be m ore representative and realistic in the delineation of spatial heterogeneity and its influence on both the pressure data and system 's effective permeability. 2. Inclusion of m ultiple source points w ithin the active well, m ay further constrain the non-uniqueness inherent nature associated w ith solving stochastic models. 3. Knowledge of perm eability anisotropy effects is helpful in developing a particular field for w ater injection m aintenance or for any other plan. Pressure interference data proposed in Chapter-2 has the potential of determ ining this im portant param eter. 4. Inclusion of m ore soft data, like production or injection history, besides tom ographic inform ation in the cokriging of the perm eability field will prove to be fruitful. 142 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5. Spherical variogram m odel has been used throughout this study. The effect of different models for representing spatial heterogeneity can be examined. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reference List [1] G. C. Thakur, SPE, Chevron U.S.A. Inc.: "Reservoir Management: A Synergistic Approach" SPE 20138, Presented At The Permian Oil and Gas Recovery Conference held in Midland, TX, March 8-9,1990. [2] M. L. Wiggins, and R. A. Startzman, SPE: "An Approach to Reservoir Management" SPE 20747, Presented At The Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held in New Orleans, LA, September 23-26,1990. [3] LeBlanc, R. J. Sr.:, " Distribution and Continuity of Sandstone Reservoirs-Part 1" J. Pet. Tech. July 1977. [4] R. M. Sneider, C. N. Tinker, and L. D. Meckel: "Deltaic Environment Reservoir types and their characterizations" J. Pet. Tech. November 1978. [5] S. D. Joshi, " Horizontal Well Technology " PennWellPublishing Company, Tulsa, Oklahoma, USA (1991). [6] R. Aguilera, "Naturally Fractured Reservoirs" Second Edition, 1995. [7] T. D. Streltsova,: Well Testing in Heterogeneous Formations, Exxon Monographs, John Wiley & Sons (1988) [8] M. M. Kamal, "Interference and Pulse Testing - A Review" JPT, December 1983. [9] M. M. Kamal, D. G. Freyder, and M. A. Murray, ARCO E&P Technology, SPE "Use of Transient Testing in Reservoir Management" SPE-28008, Prepared for Presentation at the University of Tulsa Centennial Petroleum Engineering Symposium held in Tulsa, Oklahoma, U.S.A., 29-31 August 1994. [10] Ten-when, and Inderwiesen, P.L., Texaco Inc.: " Reservoir Characterization With Crosswell Tomography: A Case Study in the Midway Sunset Field, California" SPE 22336, presented at the SPE International Meeting on Petroleum Engineering held in Beijing, China, 24-27 March, 1992. 144 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [11] Ahmed, Shabbir, Schlumberger:" Applications of VSP Inversion in Reservoir Characterization" SPE 25634, presented at the SPE Middle East Oil Technical Conference and Exhibition held in Bahrain 3-6 April 1993. [12] Siu, A.L.,* and Nghiem, L.X.,* Computer Modelling Group, and Gittins, S.D., Nzekwu, B.I.,* and Redford, D.A.,* Alberta Oil Sands Technology and Research Authority , ’ ‘ 'SPE Members:" Modeling Steam-Assisted Gravity Drainage Process in the UTF Pilot Project" SPE 22895, presented at the 66th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held in Dallas, TX, October 6-9,1991. [13] T. Lasseter, M. Karakas, J. Schweitzer: "Interpreting an RFT- Measured Pulse Test with a Three-Dimensional Simulator" SPE 14878, SPE Formation Evaluation, March 1988. [14] S. D. Joshi, " Horizontal Well Technology " PennWellPublishing Company, Tulsa, Oklahoma, USA (1991). [15] Goode, P. A. and Thambynayagam, R. K. M.: " Pressure Drawdown and Buildup Analysis for Horizontal Wells in Anisotropic Media" SPE Formation Evaluation, December 1987. [16] D. Malekzadeh, D. Taib: " Interference Testing of Horizontal Wells" SPE 22733, presented at the SPE Annual Technical Conference and Exhibition held in Dallas, Texas, U. S. A., 6-9 October 1991. [17] Deutsch, C. and Joumel, A.: GSLIB: Geostatistical Software Library and User's Guide, Oxford University Press, New York, (1992). [18] Al-Afaleg, N.: Characterization Of Permeability Fields And Their Correlation Structure From Pressure Transients, Ph.D. Dissertation, University Of Southern California, 1996. [19] J.A. Mendoza-Amuchastegui And L.C. Ramirez-Cruz: "Reconstruction Of Petrophysical Images Using Cross-Well Traveltime Topography" SPE 28674, Presented At The SPE International Petroleum Conference And Exhibition Of Mexico Held In Veracruz, Mexico, 10-13 October 1994 Instituto Mexicano Del Petroleo 145 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [20] T-W. Lo, P.L. Inderwiesen, D.R. Melton, And D.L. Howlett, Texaco Inc., And A.V. Lewis And M.F. Ponek, Texaco E&P Inc.; Nd. Livingston, Unocal Corp.; And W.B. Hatcher, Santa Fe Energy: "The Benefit And Reliability Of Using Cross-Well Tomography For Reservoir Characterization" SPE 22757, Presented At The 66th SPE Annual Technical Conference And Exhibition Of Society Of Petroleum Engineers Held In Dallas, Texas, 6-9 October 1991 [21] Schlumberger, Log Interpretation Principles/Applications. 1989. [22] Alkhonifer J., Ershaghi, I., University Of Southern California.: "Detection Of Channel Sands And Vertical Shale Continuity Using A New Approach In Interference Analysis Of Parallel Horizontal Wells" SPE 53928, Presented At The 1999 SPE Latin American And Caribbean Petroleum Engineering Conference Held In Caracas, Venezuela, 21-23 April 1999. [23] Deutsch, C. And Joumel, A.: GSLIB: Geostatistical Software Library And User 'S Guide, Oxford University Press, New York, (1992). [24] Isaaks, E.H. And Srivastava, R.M.: An Introduction To Applied Geostatistics, Oxford University Press, New York, (1989). [25] Jeffrey M. Yarns and Richard L. Chambers, Editors: Stochastic Modeling And Geostatistics, Principles, Methods, and Case Studies. AAPG Computer Application In Geology, No. 3. Tulsa, Oklahoma, USA. 1994. [26] Cardwell and Parsons: "Average Permeabilities of Heterogeneous Oil Sands" Trans, AIME, 160 (34) (1945). [27] M. A. Malik, L. W. Lake The University of Tulsa at Austin.: SPE 38310 " A Practical Approach to Scaling-Up Permeability and Relative Permeabilities in Heterogeneous Permeable Media" Presented At The 1997 SPE Western Regional Meeting held In Long Beach, California, USA 25-27 June 1997. [28] Abbaszadeh, M.: SPE 36179 "Evaluation of Permeability Upscaling Techniques and a New Algorithm for Interblock Transmissibilities" Presented at the 7th ADIPEC, Abu Dhabi, UAE, October 13-16 1996. 146 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [29] Lucia F. Jerry, SPE U. of Texas, and Fogg Graham E., SPE, University Of California.: "Geologicl/Stochastic Mapping of Heterogeneity in a Carbonate Reservoir" SPE 19597, Presented At The Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held in San Antonio, TX, October 8-11,1989. [30] Mohammed Al-Qahtani Y.: Characterization Of Spatially Correlated Permeability Fields Using Performance Data, Ph.D. Dissertation, University Of Southern California, 1996. [31] M. N. Panda, C. Mosher, and A. K. Chopra, ARCO Exploration and Production Technology: SPE 36516 " Application of Wavelets Transforms to Reservoir Data Analysis and Scaling" Presented At The 71s t Annual Technical Conference and Exhibition Held In Denver, Colorado, USA 06-09 October 1996. [32] Lifu Chu, R. A. Schatzinger, and M. K. Tham: SPE 36517 " Application of Wavelet Analysis to Upscaling of Rock Properties" Presented At The 71s t Annual Technical Conference and Exhibition Held In Denver, Colorado, USA 06-09 October 1996. [33] F. Davlau, G. Mouronval, G. Bourdarot, and P. Curutchet: "Pressure Analysis for Horizontal Wells". Paper SPE 14251. [34] Gringarten, A.C. and Ramey, H.J. Jr.: "The Use of Source and Green's Functions in Solving Unsteady-flow Problems in Reservoirs," SPE # 15116 SPEJ (Oct. 1973) 285-296, Trans., AIME, 255. [35] R. de S. Carvalho and A.J. Rosa, Petrbras: "Transient Pressure Behavior for Horizontal Wells in Naturally Fractured Reservoir". Paper SPE 18302 presented at the 63rd Annual Technical and Exhibition of SPE held in Houston, TX., Oct. 2-5, 1985. [36] Aguilera, R. and Ng, M.C. : "Transient Pressure Analysis of Horizontal Wells in Anisotropic Naturally Fractured Reservoirs". Paper SPE 19002 presented at SPE Joint Mountain Regional/Low Permeability Reservoirs Symposium and Exhibition, Denver, Colorado, March 6-8,1989. [37] Du, Kui-Fu, Heriot-Watt U. Stewart, Georgert, Edinburgh Petroleum Services Ltd.: "Transient Pressure Response of 147 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Horizontal Wells in Layered and Naturally Fractured Reservoirs With Dual-Porosity Behavior" Paper SPE 24682 presented at the 67th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held in Washington, DC, October 4-7, 1992. [38] D. Malekzadeh and D. Taib, U. of Oklahoma: " Interference Testing of Horizontal Wells". Paper SPE 22733 presented at the 68th Annual Technical and Exhibition of SPE held in Dallas, TX., Oct. 6-9,1991. [39] Stehfest, H. "Numerical Inversion of Laplace Transforms," Communications of the ACM (Jan. 1970), 13, No. 1,47-49. (Algorithm 368 with correction (Oct. 1970), 13, No. 10). [40] EI-Banbi, Ahmed H., Wattenbarger, R.A., and Maggard, J.B ./'Proper Use of Numerical Simulation for Pressure Transient Analysis and Pattern Displacements," paper presented at the 13th EGPC Conference and Exhibition, Cairo, Egypt, Oct. 2 1-24, 1996. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix-A A.l Dimensionless parameters The various dimensionless param eters that appeared in the previous equations are defined in field units as follows: , k vh(AP) PD = 7.08*10 — - (A-l) r D = D ZD (A-2) 0.0002637k t tD = --------;-------dr (A-3) ° m c t{L/2)2 x X D = ---------- (A-4) D (L/2) Yd = ---------- (A-5) D (L/2) z ° - - h i i (A-6) z»„ = % (A-7) h Ld = E (A-8) (2 h ) \ k v 149 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix-B B.l Objective This appendix shows how w e used conventional num erical reservoir sim ulator to m odel dual porosity reservoirs w ithout the need to m odify the program m ing code of the existing simulator. We followed the same approach as did reference [40]. However, w e dealt w ith m odeling horizontal wells instead. The final product w as utilized to cross validate our analytical m odel developed in Chapter-6. B.2 Pseudo-Steady-State Dual Porosity Models In PSS dual porosity, two layers model the reservoir: • One layer is to m odel the connected fracture system. • The other layer is to m odel the m atrix system. Data input "tricks" were used to give us the required storage and interporosity flow param eters (co and X). Table-B.l lists the input properties that w e used to m odel a PSS dual porosity behavior of a reservoir w ith know n properties, cp, k, h, co, a n d A. Notice that the formation perm eability, k, is the bulk fracture 150 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. permeability. The properties w ith subscripts 1 and 2 refer to the fracture and matrix layers respectively, used in the num erical model. The permeabilities, kh and kv refer to the horizontal and vertical perm eabilities in the num erical m odel respectively. L refers to the horizontal wellbore length. T ab le-B .l Num erical sim ulator param eters for PSS dual porosity m odel Layer h < P k h k v 1 Fracture Layer hi (px = {(pCO h )/h j k„ = k h / h t kv = A k h (h ,+ h 2 )/2 L 2 2 M atrix Layer h 2 (p2 ={(p{l-CO)h)/\ k,^ = zero kv = A kh (h ,+ h 2 )/2 L 2 B.3 Derivation of the model Analytical model relations: [<PC), l(p C ), + {<pC), J j2 ^ ~ j u (B.2) k f flow equation (PSS) assumption): 151 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. w here C j is the flow per unit volum e from the matrix to the fracture and is * i] = q / A h , therefore, qB Ah n Numerical Simulation Relations: Transmissibility betw een tw o grid blocks: k,k T = 2A r v2 (h2k1 + h1 k2) Flow equation: qB = — T{P2 - P,) (B.4) (B.5) (B.6) B.4 Procedure We sim ulated the PSS dual porosity m odel w ith a tw o layer model. Let layerl be used for the fracture system and layer 2 be used for the m atrix system. We then solved the storage equations to obtain the porosity that we input in the num erical model, We also equated the flow 152 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. from layer 2 to layer 1 with that of the analytical model to derive the permeability that we input in the simulator (numerical) model. Storage Relations Volume of fracture system = A h t Volume of m atrix system = A h2 where, A = total area betw een m atrix and fracture layers, hj = fracture layer thickness h2 = m atrix layer thickness Volume of fracture _ Ah2 _ /i, V = f total volume A(h] + h2) h{ + h2 V = hl /Z j + h2 Notice that Vf + Vm — 1 and since C 0 = 7------- r j 1 7 - (B.7) {<pVc,)f +{<pVc,)m for the sim ulation model, w e can use co = (q>2 h1) /[((p1 ht) + (cp2 h2)]. We also have the total porosity relation from the analytical model: (q> h) = [( (pfhf) + (cpmhm )] and the equivalent relation in the sim ulation 153 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. m odel is: ((ph) = (< plhl) + {(p2h2) (B.8) Solving equation-B.7 and equation-B.8 for ( < p i h ] ) + ( q> 2 h.2 ) and equating (<p i h \ ) + ( $ 2 ^ 2 ) - (< P i h i) / co, we, therefore, get: = ((ph) (B.9) 0) Equation-B.9 can be used to calculate < p j in the following equation: h (px = (p(Q — (B.10) K Substituting equation-B.10 into equation-B.8 (^> 2 ^- 2 ) — cp h(l-co) or (p2 = (p{ 1 - 0 ) ) ^ - (B.ll) K So we can select h i and h 2 and determine (p 7 and (P2 from equation-B.10 and equation-B.ll. Vertical Flow relations: Analytical model: qB = (a k m/ju )(Pm - Pj)Ah and the sim ulation model: qB = (kr/n )T(P2 - Pt) 154 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Writing the analytical model in terms of layer pressures and equating the tw o equations and realizing tat kr = 1 for single phase flow, w e get (ak2 /ju )(P2 - PJAh = (1/ju )T(P2 - P 3 ) simplifying, T = ak^Ah Realizing that the transmissibility is calculated w ith the harm onic average The permeability, k, is the bulk fracture permeability (formation permeability), k; £2 and refer to the vertical permeabilities in the fracture and matrix layers respectively. Putting a relation in terms of numerical model parameters: as: T = 2A [(ktk2 )/( h,k2 + h^)]. Equating and simplifying, (B.12) 2 Solving X relation for a , a = (X/L )(k/km). A k a (B.13) Substituting equation-B.13 into equaion-B.12 A k 2k, — — h = ------------------------ L k2 hlk1 + h2kx (B.14) 155 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For simplicity, we assume the two vertical permeabilities, kj and & 2 are the same and equal to kv (B.15) Solving equation-B.15 for kv , u _Akh(hl + h2) K., — * (B.16) H orizontal Flow The horizontal flow will have to obey the relation: khhf = kh where, kh = horizontal fracture perm eability hf = fracture layer perm eability k = formation bulk perm eability h = form ation thickness putting this equation in our num erical m odel terminology: khh1= kh Solving for the horizontal perm eability of the fracture layer The horizontal perm eability in the m atrix layer w as assum ed to be zero k ~ k - h K (B.17) 156 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. since the analytical m odel assumes that the flow occurs only in the fracture system tow ards the well. B.5 Validation of the procedure Example: In order to cross validate the previous procedure and be able to im plem ent it. A num erical example was em ployed for this purpose. This example is for a well-known and established solution of PSS dual porosity in conventional wells. The num erical solution by Stehfest [39] is em ployed to check our results obtained by the model. B.5.1 PSS Dual Porosity Example for conventional wells This is an example of a well producing at constant rate from an infinite reservoir. Neither wellbore storage nor skin w as considered in this example. The values of the storativity ratio, m, and interporosity flow coefficient, X , are: 0.01 and 1.0 E-5 respectively. We used the following reservoir and fluid properties in the num erical sim ulation model: k = 100 m d. h = 10 ft. (p - 0.20 ju = l c p 157 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ct = l.E-5 psi'1 The input data for the two layer numerical model were calculated according to Table-B.l. These values are: Laver-1 (fracture layer) h, = 10 ft. <p1 = 0.002 kh = 100 md kv = 1.6 md Laver-2 (matrix layer) h2 = 10 ft. (p2 = 0.198 kh = zero md kv = 1.6 md The dimensionless variables used to compare the numerical simulation solution with the analytical solution are defined by: PD = 7.08*10-3 QMBo 0.0002637^ t D ~ , 2 158 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Comparison between the solutions calculated by our model and the analytical model is shown in Fig.-B.l and Fig.-B.2. The figures show excellent match for all flow regimes. So, now we can use this model to obtain responses for interference tests in NFR of horizontal wells. 0.01 i : i .......• > ..........................:... .......................i..........................;...;........................... i....U ..U .:.:. 1E0 1E1 1E2 1E3 1E4 1E5 1E6 1E7 tD Figure-B.l Stehfest Algorithm Solution for Dual Porosity in Conventional Wells 159 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Q g C L Q 0.01 1 E 2 1E3 1 E 0 1E1 1E4 1E5 1 E 6 1E7 tD Figure-B.2 Sim ulator Solution for D ual Porosity in C onventional W ells 160 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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Asset Metadata

Creator Alkhonifer, Jamal Ali (author)

Core Title Characterization of permeability fields between horizontal wells using a hybrid of cross -hole imagery and repeat interference test

School Graduate School

Degree Doctor of Philosophy

Degree Program Petroleum Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, petroleum,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Ershaghi, Iraj (committee chair), Davis, Bruce, W. (committee member), Sammis, Charles G. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-238651

Unique identifier UC11339036

Identifier 3093411.pdf (filename),usctheses-c16-238651 (legacy record id)

Legacy Identifier 3093411.pdf

Dmrecord 238651

Document Type Dissertation

Rights Alkhonifer, Jamal Ali

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA