NUMERICAL INVERSE INTERPRETATION OF PNEUMATIC TESTS IN UNSATURATED FRACTURED TUFFS AT THE APACHE LEAP RESEARCH SITE by Velimir Valentinov Vesselinov Copyright C) Velimir Valentinov Vesselinov A Dissertation Submitted to the Faculty of the DEPARTMENT OF HYDROLOGY AND WATER RESOURCES In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY WITH A MAJOR IN HYDROLOGY In the Graduate College THE UNIVERSITY OF ARIZONA 200 0 2 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE As members of the Final Examination Committee, we certify that we have read the dissertation prepared by entitled Velimir Valentinov Vesselinov Numerical inverse interpretation of pneumatic tests in unsaturated fractured tuffs at the Apache Leap Research Site and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy A Teum d Peter J Wierenp. Arthur W. Warrick Donald E. Date Myers Date Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copy of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requireme t 1 Dissertation Director Shlomo P. Neuman Date 3 STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the copyright holder. SIGNED: 4 ACKNOWLEDGEMENTS This dissertation would have been impossible without the support of many. First and foremost, my advisor Dr. Shlomo P. Neuman. I am indebted for his guidance, patience, and support. I acknowledge his invaluable contributions to the accomplished research. He was always available to answer my questions and discuss everything from science to politics. He was the best mentor I have ever had and helped me a lot to develop myself as a researcher and person. I am thankful to everybody who contributed to the research of Apache Leap Research Site through the years, especially to Amado Guzman, Walter A. Illman, Dick Thompson and Guoliang Chen who conducted most of pneumatic tests I have analyzed. I am also thankful to the support and guidance of our Project Manager, Mr. Thomas J. Nicholson, United States Nuclear Regulatory Commission. I would also like to thank Drs. Thomas Maddock III, Donald E. Myers, Arthur W. Warrick, and Peter J. Wierenga for serving on my dissertation committee, and Dr. Jesus Carrera for the beneficial comments and discussions on my research. A substantial part of my research would have been impossible without the support of High-Speed Computing Committee and Computer Operations Group at the Center for Computing and Information Technology, The University of Arizona. I run my inverse model in parallel using the full capabilities of our supercomputer, which extended significantly my analysis. Part of the simulation and inverse modeling work was conducted during a summer internship with the Geoanalysis Group at Los Alamos National Laboratory. I am grateful to Dr. George A. Zyvoloski for his help in the implementation of FEHM, and to Dr. Carl W. Gable for his assistance in the use of X3D. Finally, I am thankful for the continued support of my family and friends. This research was financially supported by the United States Nuclear Regulatory Commission under contracts NRC-04-95-038 and NRC-04-97-056. 5 To Rumi and Neda. 6 TABLE OF CONTENTS LIST OF FIGURES 8 LIST OF TABLES 14 ABSTRACT 16 CHAPTER 1. INTRODUCTION CHAPTER 2. APACHE LEAP RESEARCH SITE 2.1. Site description 2.2. Previous investigations 2.2.1. Laboratory and field investigations 2.2.2. Single-hole pneumatic tests 2.2.3. Cross-hole pneumatic tests 2.3. Geostatistical analysis of spatial variability 2.3.1. Air permeability 2.3.2. Fracture density, matrix porosity, water content and van Genuchten a 2.3.3. Conditional simulations 2.4. Conceptualization of hydrogeologic conditions during pneumatic tests 18 CHAPTER 3. NUMERICAL INVERSE METHODOLOGY 3.1. Background 3.1.1. Governing equations 3.1.2. Inverse problem 3.2. Stochastic numerical inverse model 3.2.1. Pilot point formulation 3.2.2. Maximum likelihood approach 3.2.3. Optimization 3.2.4. Linearized analysis of estimation errors 3.2.5. Calculation of sensitivity matrix 3.2.6. Model identification criteria 3.3. Numerical inverse model for ALRS 3.3.1. Computational domain 3.3.2. Boundary and initial conditions 3.3.3. Borehole effects 3.3.4. Computational grid 26 26 31 31 33 39 41 41 54 54 68 71 71 71 74 78 79 81 83 88 91 92 93 93 93 95 96 7 TABLE OF CONTENTS—Continued 3.3.5. Program FEHM 3.3.6. Parameterization 3.3.7. Program PEST 3.3.8. Parallel processing 3.3.9. Code interaction 3.4. Linearization of governing equations 105 108 109 112 113 113 INTERPRETATION OF SINGLE-HOLE TESTS . . . . 115 4.1. Single-hole test JG0921 118 4.2. Single-hole test JGC0609 125 4.3. Single-hole test JHB0612 132 4.4. Single-hole test JJA0616 138 4.5. Discussion 145 CHAPTER 4. CHAPTER 5. INTERPRETATION OF CROSS-HOLE TESTS 146 5.1. Qualitative analysis 147 5.2. Inverse analysis of cross-hole tests 158 5.2.1. Uniform medium 158 5.2.2. Non-uniform medium 176 5.3. Discussion 210 5.3.1. Comparison between numerical and analytical analyses . 210 5.3.2. Comparison between present stochastic inverse model and previous applications of pilot-point method 211 5.3.3. Comparison between different models 5.3.4. Scale effects 5.3.5. Comparison between spatial patterns of permeability obtained by various methods 5.3.6. Comparison between permeability and porosity estimates . 5.3.7. Borehole effects CHAPTER 6. REFERENCES CONCLUSIONS 212 212 218 222 222 229 236 8 LIST OF FIGURES FIGURE 2.1. Location map of Apache Leap Research Site [after Guzman et al., 1996]) 27 Plan view of tested boreholes and plastic cover FIGURE 2.2. 29 FIGURE 2.3. Three-dimensional perspective of boreholes at the site 30 FIGURE 2.4. Three-dimensional perspective showing center locations of 3-m test intervals employed by Rasmussen et al. [1990] 32 FIGURE 2.5. Lower hemisphere Schmidt equal-area projection of fractures identified by Rasmussen et al. [1990]. Contours indicate number of fractures per unit area of projection circle. 34 FIGURE 2.6. Three-dimensional perspective showing center locations of neutron probe measurements performed by Thompson et al. [1995] 35 Scatter plot of air permeability versus fracture density [data FIGURE 2.7. from Rasmussen et al., 1990] 37 FIGURE 2.8. Three-dimensional perspective showing center locations of 1-m single-hole test intervals employed by Guzman et al. [1996] 38 FIGURE 2.9. Omni-directional sample and model variograms for various parameters at a minimum separation distance of 3 m 43 FIGURE 2.10. Kriged estimates and kriging variances of log io k at y = 7 m using a power model (left), and an exponential model with second order drift (right) 44 FIGURE 2.11. Variation of air permeability with depth and support scale (data from Guzman et al. [1996] and Rasmussen et al. [1990]) 46 FIGURE 2.12. Histograms of log io k 1-m and 3-m scale data (data from Guzman et al. [1996] and Rasmussen et al. [1990]) 47 FIGURE 2.13. Histograms of log io k at various scales along borehole Y2 (data from Guzman et al. [1996]) 49 FIGURE 2.14. Omni-directional sample variograms for log lo k with various supports and a power variogram model fitted to 1-m data 50 FIGURE 2.15. Three-dimensional perspective showing center locations of singlehole test intervals for combined set of 1-m (circles) and 3-m (squares) 51 scale measurements FIGURE 2.16. Kriged log lo k estimates obtained using 1-m scale data from boreholes X2, Y2, Y3, Z2, V2 and W2A (left) and same together with 3-m scale data from boreholes Xl, X3, Yl, Z1 and Z3 (right) 53 FIGURE 2.17. Kriged estimates of various parameters at y = 7 m 57 FIGURE 2.18. Three-dimensional representation of kriged log lo k [m 2 ]. 58 FIGURE 2.19. Kriged log io k [m 2 ] along various y-z planes 59 60 FIGURE 2.20. Kriged log io k [m 2 ] along various x-z planes 9 LIST OF FIGURES—Continued FIGURE 2.21. Kriged log io k [m 2 ] along various x-y planes FIGURE 2.22. Three-dimensional representation of kriged fracture 61 density {coun ts /m]. 62 2.23. Three-dimensional representation of kriged matrix porosity 63 log io 0,, [m 3 /m 3 ] FIGURE 2.24. Three-dimensional representation of kriged water content 8 [m 3 /m 3 ] 64 FIGURE 2.25. Three-dimensional representation of kriged van Genuchten's log io a [k P al 65 FIGURE 2.26. Three-dimensional representation of estimated air-filled matrix porosity log io çb,,,, [m 3 /m 3 ] 66 FIGURE 2.27. Random conditional realizations of various parameters at y = 7 m 67 FIGURE Computational domain and explicitly simulated boreholes. . . Side views of computational grid for injection borehole Y2. . . Vertical cross-sections through computational grid for injection borehole Y2 FIGURE 3.4. Three-dimensional representation of computational along injection borehole Example of log io k distribution throughout computational doFIGURE 3.5. main. Example of log lo 0 distribution throughout computational doFIGURE 3.6. main. FIGURE 3.1. FIGURE 3.2. FIGURE 3.3. 94 97 98 100 102 103 4.1. Pressure data from first injection step of test JG0921 interpreted by various inverse models 121 FIGURE 4.2. Relative change of borehole storage effects due to variation of air compressibility pressure versus relative time for all injection steps 122 and recovery of test JG0921 FIGURE 4.3. Relative pressure versus relative time for all injection steps and recovery of test JG0921 123 FIGURE 4.4. Pressure data from for all injection steps and recovery of test 124 JG0921 interpreted by inverse model. Pressure data from first injection step of test JGC0609 interFIGURE 4.5. 127 preted by various inverse models FIGURE 4.6. Relative change of borehole storage effects due to variation of air compressibility pressure versus relative time for all injection steps 128 and recovery of test JGC0609. Relative pressure versus relative time for all injection steps and FIGURE 4.7. 129 recovery of test JGC0609 FIGURE 10 LIST OF FIGURES—Continued Pressure data from for all injection steps and recovery of test 4.8. JGC0609 interpreted by inverse model. 130 Pressure data from first injection step of test JHB0612 interFIGURE 4.9. preted by various inverse models 134 FIGURE 4.10. Relative change of borehole storage effects due to variation of air compressibility pressure versus relative time for all injection steps and recovery of test J11B0612. 135 FIGURE 4.11. Relative pressure versus relative time for all injection steps and recovery of test JHB0612 136 FIGURE 4.12. Pressure data from for all injection steps and recovery of test JHB0612 interpreted by inverse model. 137 FIGURE 4.13. Pressure data from first injection step of test JJA0616 inter preted by various inverse models 141 FIGURE 4.14. Relative change of borehole storage effects due to variation of air compressibility pressure versus relative time for all injection steps 142 and recovery of test JJA0616 Relative pressure versus relative time for all injection steps and FIGURE 4.15. 143 recovery of test JJA0616. FIGURE 4.16. Pressure data from for all injection steps and recovery of test 144 JJA0616 interpreted by inverse model FIGURE FIGURE 5.1. Location of monitoring intervals and packers along tested bore- 148 holes. Log-log plots of observed pressure buildups (dots; kP a) versus 5.2. 150 time (days) and match points (circles) during test PP4. kP Log-log plots of observed pressure buildups (dots; a) versus FIGURE 5.3. 151 time (days) and match points (circles) during test PP5. Log-log plots of observed pressure buildups (dots; kP a) versus FIGURE 5.4. 152 time (days) and match points (circles) during test PP6. Log-log plots of observed pressure buildups (dots; kP a) versus FIGURE 5.5. 153 time (days) and match points (circles) during test PP7. Log-log plots of observed pressure buildups (dots; kP a) versus FIGURE 5.6. 154 time (days) and match points (circles) during test PP8. Measured variations in barometric pressure and pressure reFIGURE 5.7. 155 sponses in some of intervals during test PP4. test PP4. 156 temperature measurements collected during Some air FIGURE 5.8. Separate matches between observed (small dots) and simulated FIGURE 5.9. (thick curves) responses for test PP4 assuming uniform medium. . . . 160 FIGURE 5.10. Separate matches between observed (small dots) and simulated (thick curves) responses for test PP5 assuming uniform medium. . . . 161 FIGURE 11 LIST OF FIGURES—Continued FIGURE 5.11. Separate matches between observed (small dots) and simulated (thick curves) responses for test PP6 assuming uniform medium. . . . 162 FIGURE 5.12. Separate matches between observed (small dots) and simulated (thick curves) responses for test PP7 assuming uniform medium. . . 163 FIGURE 5.13. Separate matches between observed (small dots) and simulated (thick curves) responses for test PP8 assuming uniform medium. . . . 164 FIGURE 5.14. Separate matches between observed (small dots) and simulated (thick curves) responses for test PP4 assuming uniform medium and adjusting borehole storage parameters along injection and observation intervals. 171 FIGURE 5.15. Estimates of pneumatic properties for test PP4 obtained with and without borehole effects. 172 FIGURE 5.16. Estimates of pneumatic properties for test PP4 obtained for different initial barometric pressure. 174 FIGURE 5.17. Analytically versus numerically derived pneumatic properties for test PP4. 175 FIGURE 5.18. Three-dimensional representation of 32 pilot points. 178 FIGURE 5.19. Matches between observed (small dots) and simulated (thick curves) responses obtained by simultaneous inversion of PP4 data with 32 pilot points; log io q is assumed to be uniform 180 FIGURE 5.20. Three-dimensional representation of kriged log io k estimated by simultaneous inversion of PP4 data with 32 pilot points; log io is assumed to be uniform 181 FIGURE 5.21. Matches between observed (small dots) and simulated (thick curves) responses obtained by simultaneous inversion of PP4 data with 32 pilot points 182 FIGURE 5.22. Histogram of residuals between observed and simulated responses 184 FIGURE 5.23. Two-dimensional representation of covariance matrix of estimation errors at the pilot points 187 FIGURE 5.24. Two-dimensional representation of correlation matrix of estimation errors at the pilot points 188 FIGURE 5.25. Absolute components of the estimation covariance matrix versus distance between the pilot points 189 FIGURE 5.26. Absolute components of the estimation correlation matrix ver189 sus distance between the pilot points FIGURE 5.27. Two-dimensional representation of eigenvectors of covariance 191 matrix of estimation errors at the pilot points. 12 LIST OF FIGURES—Continued FIGURE 5.28. Maximum absolute eigenvector components associated with pa- rameter estimates FIGURE 5.29. Three-dimensional representation of kriged log o k estimated by simultaneous inversion of PP4 data with 32 pilot points. FIGURE 5.30. Three-dimensional representation of kriged log o q5 estimated by simultaneous inversion of PP4 data with 32 pilot points. FIGURE 5.31. Sample variograms of kriged log o k and log o q5 estimates obtained by simultaneous inversion of PP4 data with 32 pilot points. . . FIGURE 5.32. Forward simulation of test PP5 applying parameter fields estimated by simultaneous inversion of PP4 data with 32 pilot points. . . FIGURE 5.33. Forward simulation of test PP6 applying parameter fields estimated by simultaneous inversion of PP4 data with 32 pilot points. . . FIGURE 5.34. Forward simulation of test PP7 applying parameter fields estimated by simultaneous inversion of PP4 data with 32 pilot points. . . FIGURE 5.35. Matches between observed (small dots) and simulated (thick curves) responses obtained by simultaneous inversion of PP4 data with 64 pilot points FIGURE 5.36. Three-dimensional representation of 72 pilot points. FIGURE 5.37. Matches between observed (small dots) and simulated (thick curves) responses for test PP4 obtained by simultaneous inversion of PP4, PP5 and PP6 data. FIGURE 5.38. Matches between observed (small dots) and simulated (thick curves) responses for test PP5 obtained by simultaneous inversion of PP4, PP5 and PP6 data. FIGURE 5.39. Matches between observed (small dots) and simulated (thick curves) responses for test PP6 obtained by simultaneous inversion of PP4, PP5 and PP6 data. FIGURE 5.40. Three-dimensional representation of kriged log o k estimated by simultaneous inversion of PP4, PP5 and PP6 data. FIGURE 5.41. Three-dimensional representation of kriged log o 0 estimated by simultaneous inversion of PP4, PP5 and PP6 data. FIGURE 5.42. Histograms of air permeability data obtained using various approaches FIGURE 5.43. Histograms of air-filled porosity data obtained using various approaches FIGURE 5.44. Air permeability along boreholes estimated from single-hole tests and by simultaneous inversion of PP4, PP5 and PP6 data FIGURE 5.45. Sample variograms of log o k estimated from single-hole data and through stochastic inversion of cross-hole data 192 193 194 195 197 198 199 200 202 205 206 207 208 209 216 217 220 221 13 LIST OF FIGURES—Continued FIGURE 5.46. Air permeability versus air-filled porosity estimated by stochastic inversion of PP4 data at pilot points (open big circles) and within the computational domain 223 FIGURE 5.47. Simulated responses with (solid curve) and without (dashed curve) borehole effects for cross-hole test PP4 in log-log plots 225 FIGURE 5.48. Simulated responses with (solid curve) and without (dashed curve) borehole effects for cross-hole test PP4 in semi-log plots 226 FIGURE 5.49. Simulated air pressure [MPa] in computational region at the end of cross-hole test PP4 with borehole effect 227 FIGURE 5.50. Simulated air pressure [MPa] in computational region at the end of cross-hole test PP4 with borehole effect 228 14 LIST OF TABLES 2.1. TABLE 2.2. TABLE 2.3. Borehole coordinates and length. 28 Summary statistics of the geostatistically analyzed data. . . . 46 Summary statistics for log io k [772, 2 ] from single-hole tests at various scales along borehole Y2 48 TABLE 2.4. Results from Kolmogorov-Smirnov test of Gaussianity 56 TABLE Single-hole pneumatic tests analyzed by inverse modeling [after TABLE 4.1. Guzman et al., 1996] 116 Single-hole test data analyzed by inverse modeling [after GuzTABLE 4.2. man et al., 1996] 116 119 TABLE 4.3. Parameter estimates for test JG0921. Eigenanalysis of covariance matrix of estimation errors obtained TABLE 4.4. from interpretation of the first step of test JG0921 120 Eigenanalysis of covariance matrix of estimation errors obtained TABLE 4.5. from interpretation of all the injection steps and recovery of test JG0921. 120 126 Parameter estimates for test JGC0609. TABLE 4.6. Eigenanalysis of covariance matrix of estimation errors obtained TABLE 4.7. 131 from interpretation of the first step of test JGC0609 Eigenanalysis of covariance matrix of estimation errors obtained TABLE 4.8. from interpretation of all the injection steps and recovery of test JGC0609. 131 133 TABLE 4.9. Parameter estimates for test JHB0612. TABLE 4.10. Eigenanalysis of covariance matrix of estimation errors obtained from interpretation of all the injection steps and recovery of test JHB0612. 133 TABLE 4.11. Parameter estimates for test JJA0616 140 TABLE 4.12. Eigenanalysis of covariance matrix of estimation errors obtained 140 from interpretation of the first step of test JJA0616. TABLE 4.13. Eigenanalysis of covariance matrix of estimation errors obtained from interpretation of all the injection steps and recovery of test JJA0616. 140 Point-to-point cross-hole tests interpreted by numerical inverse TABLE 5.1. model Parameters identified for cross-hole test PP4 treating the medium TABLE 5.2. as spatially uniform Parameters identified for cross-hole test PP5 treating the medium TABLE 5.3. as spatially uniform Parameters identified for cross-hole test PP6 treating the medium TABLE 5.4. as spatially uniform Parameters identified for cross-hole test PP7 treating the medium TABLE 5.5. as spatially uniform 147 165 166 167 168 15 LIST OF TABLES—Continued Parameters identified for cross-hole test PP8 treating the medium as spatially uniform 169 Summary statistics for log io k [n1, 2 ] identified for cross-hole tests TABLE 5.7. 170 treating the medium as spatially uniform Summary statistics for log io 0 [m 3 /m 3 ] identified for cross-hole TABLE 5.8. 170 tests treating the medium as spatially uniform TABLE 5.9. Parameters identified for cross-hole test PP4 treating the medium as spatially uniform and adjusting borehole storage at injection and ob171 servation intervals TABLE 5.10. Parameters identified at the pilot points by numerical inversion of data from cross-hole test PP4 using 32 pilot points. 185 TABLE 5.11. Parameters identified at the pilot points by numerical inversion of data from cross-hole tests PP4, PP5 and PP6 using 72 pilot points. 204 TABLE 5.12. Comparison of different numerical inverse models using model identification criteria (numbers in brackets show model ranking). . . . 213 Summary statistics for log io k [772. 2 ] identified using different data TABLE 5.13. 214 and methods TABLE 5.14. Summary statistics for log io q5 [m 3 /m 3 ] identified using different 214 data and methods TABLE 5.6. 16 ABSTRACT A three-dimensional stochastic numerical inverse model has been developed for characterizing the properties of unsaturated fractured medium through analysis of singleand cross-hole pneumatic tests. Over 270 single-hole [Guzman et al., 1996] and 44 cross-hole pneumatic tests [Illman et al., 1998; Inman, 1999] were conducted in 16 shallow vertical and slanted boreholes in unsaturated fractured tuffs at the Apache Leap Research Site (ALRS), Arizona. The single-hole tests were interpreted through steady-state [Guzman et al., 1996] and transient [Illman and Neuman, 2000b] analytical methods. The cross-hole tests were interpreted by analytical type-curves [Illman and Neuman, 2000a]. I describe a geostatistical analysis of the steady-state single-hole data, and numerical inversion of transient single-hole and cross-hole data. The geostatistical analysis of single-hole steady-state data yields information about the spatial structure of air permeabilities on a nominal scale of 1 m. The numerical inverse analysis of transient pneumatic test data is based on the assumption of isothermal single-phase airflow through a locally isotropic, uniform or non-uniform continuum. The stochastic inverse model is based on the geostatistical pilot point method of parameterization [de Marsily, 1978], coupled with a maximum likelihood definition of the inverse problem [Carrera and Neuman, 1986a]. The model combines a finite-volume flow simulator, FEHM [Zyvoloski et al., 1997], an automatic mesh generator, X3D [Trease et al., 1996], a parallelized version of an automatic parameter estimator, PEST [Doherty et al., 1994], and a geostatistical code, GSTAT [Pebesma and Wesseling, 1998]. The model accounts directly for the ability of all borehole intervals to store and conduct air through the system; solves the airflow equations in their original nonlinear form accounting for the dependence of air compressibility on absolute air pressure; can, in principle, account for atmospheric pressure fluctuations at the soil surface; provides kriged estimates of spatial variations in air permeability 17 and air-filled porosity throughout the tested fractured rock volume; and is applied simultaneously to pressure data from multiple borehole intervals as well as to multiple cross-hole tests. The latter amounts to three-dimensional stochastic imaging, or pneumatic tomography, of the rock as proposed by Neuman [1987] in connection with cross-hole hydraulic tests in fractured crystalline rocks near Oracle, Arizona. The model is run in parallel on a supercomputer using 32 processors. Numerical in- version of single-hole pneumatic tests allows interpreting multiple injection-step and recovery data simultaneously, and yields information about air permeability, air-filled porosity, and dimensionless borehole storage coefficient. Some of this cannot be accomplished with type-curves [Inman and Neuman, 2000b]. Air permeability values obtained by my inverse method agree well with those obtained by steady-state and type-curve analyses. Both stochastic inverse analysis of cross-hole data and geostatistical analysis of single-hole data, yield similar geometric mean and similar spatial pattern of air permeability. However, I observe a scale effect in both air permeabil- ity and air-filled porosity when I analyze cross-hole pressure records from individual monitoring intervals one by one, while treating the medium as being uniform; both pneumatic parameters have a geometric mean that is larger, and a variance that is smaller, than those obtained by simultaneous stochastic analysis of multiple pressure records. Overall, my analysis suggests that (a) pneumatic pressure behavior of unsaturated fractured tuffs at the ALRS can be interpreted by treating the rock as a continuum on scales ranging from meters to tens of meters; (b) this continuum is representative primarily of interconnected fractures; (c) its pneumatic properties nevertheless correlate poorly with fracture density; and (d) air permeability and air-filled porosity exhibit multiscale random variations in space. 18 Chapter 1 INTRODUCTION An important part of current environmental problems is related to flow and transport in the vadose zone. Still, there is no well-established methodology for the characterization of unsaturated, especially fractured medium properties. Among major difficulties are those associated with the complexity of the governing processes and the pronounced non-uniformity and scale effects of medium properties. A potentially advantageous approach is to conduct gaseous pressure and tracer field tests which overcome many of the limitations associated with hydraulic flow and tracer tests [Guzman and Neuman, 1996]. Experience with gaseous field tests in fractured rocks is limited and much of it has been accumulated at the Apache Leap Research Site (ALRS) near Superior, Arizona [Rasmussen et al., 1990; Guzman et al., 1994, 1996; Guzman and Neuman, 1996; Inman et al., 1998]. Similar studies have also been performed at Los Alamos, New Mexico [Cronk et al., 1990], the Yucca Mountain site, Nevada [LeCain, 1996, 1998; Wang et al., 1998; Huang et al., 1999] and at Box Canyon, Idaho [Benito et al., 1998, 1999]. The objective of my dissertation is to characterize unsaturated fractured rocks at the ALRS through analysis and interpretation of previously conducted pneumatic (air-injection) tests. To accomplish this objective, I have developed a novel geostatistical three-dimensional stochastic numerical inverse model. The major questions that I address are: Is it plausible to represent the fractured medium at the site as an equivalent continuum, either deterministic or stochastic? Is it possible to simulate the pneumatic tests, and identify equivalent pneumatic properties of the medium, by means of a numerical inverse model? Is it possible to resolve the spatial variability of these properties in detail across a site? Is it feasible to obtain high-resolution images of subsurface medium heterogeneity by interpreting several 19 pneumatic tests simultaneously (which would amount to "pneumatic tomography" or "high-resolution stochastic imaging" of the subsurface [Neuman, 1987])? How accurately can the non-uniformity of the medium be estimated? Is there a scale effect when rock properties are identified on a fine and a coarse scale? What is the influence of open borehole intervals on pressure propagation and storage in the rock? Can a numerical inverse model complement, or be an alternative, to the standard analytical type-curve method of field test analysis? Flow in fractured rocks occurs predominantly through open fractures which tend to be more permeable than the surrounding rock matrix. There are two basic approaches to simulating flow in fractured rocks: discrete and equivalent continua. In the discrete approach, flow takes place through a network of intersected planes whose properties are prescribed either deterministically [e.g. Long et al., 1982; Dershowitz, 1984; Wang and Narasimhan, 1985] or stochastically [e.g. Chiles and de Marsily, 1993]. In practice, it is difficult to define in detail the geometry of a fracture network: size, shape, orientation, density, apertures and roughness of individual fractures. Even when some such data are available, it is still difficult to correctly represent flow within each fracture, along its intersections, and between the fractures and surrounding matrix blocks. There is a limited number of actual applications of this approach [Dyerstorp and Andersson, 1989; Cacas et al., 1990; Dershowitz et al., 1991]. As discussed in Chapter 2, at the ALRS, it has not been possible to determine the pneumatic prop- erties of individual fractures. The bulk pneumatic properties do not correlate with fracture occurrence in boreholes at the site. This supports an earlier conclusion by Neuman [1987, 1988b] that the hydraulic properties of fracture rocks must be deter- mined directly by means of hydraulic field tests, rather than surmised indirectly from fracture geometry. Similar results have been also obtained at other fractured rock sites worldwide such as Oracle, Arizona [Jones et al., 1985]; Chalk River, Canada [Raven et al., 1985]; and Stripa, Sweden [Gale et al., 1987]. To address spatial variability, he 20 proposed to consider the fractured rock as an equivalent stochastic continuum with properties that form spatially correlated (regionalized) random fields, which is valid if flow on the scale of interest is not dominated be a small number of individual fractures. If flow is dominated by a few major fractures or fracture zones, one can adopt a mixed discrete-continuum approach [cf. Andrews et al., 1986; Carrera et al., 1990a; Carrera and Martinez-Landa, 2000] which treats them as thin "aquifers" embedded with in a less permeable rock continuum. Equivalent approaches can be categorized by the number of continua they consider. Fractures and porous blocks can be simulated individually (as I do) or simultaneously as dual or multiple continua. Most common are the double-porosity model of Barenblatt et al. [1960] and Warren and Root [1963], and double-porosity/double-permeability model of Duguid and Lee [1977]. A major task of our field of study is to understand and predict the fluid flow and contaminant transport through natural hydrogeological systems. This requires the definition of a conceptual model and corresponding model parameters, which represent adequately hydrogeological processes of interest. For a given model, the identification of model parameters and associated uncertainties is called an inverse problem. Solution of the inverse problem is preferable to be obtained through analysis of data representing system behavior under controlled conditions; quite often, however, the analyzed data represent arbitrary, uncontrolled fluctuations of state variables. The inverse problem is often not well posed, i.e. the solution is non-unique or unsta- ble with respect to the model parameters, due to both model errors and insufficient quality and quantity of observational data. Numerical inverse methods have been used widely in hydrogeological research and, more recently, application. Milestone papers include those of Neuman [1973], Neuman and Yakowitz [1979], Neuman et al. [1980], Neuman [1980], Carrera [1984] and Carrera and Neuman [1986a, b, c]. Reviews and summaries can be found in the publications of Yeh [1986], Kool et al. [1987], Carrera [1988], Ginn and Cushman [1990], Carrera et al. [1993a], Sun [1994], McLaughlin and Townley [1996] and Carrera 21 et al. [1997]. In order to address the spatial non-uniformity of medium properties, inverse techniques have been coupled with geostatistics so that medium properties can be treated as spatially correlated (regionalized) random fields. This allows to capture in one parsimonious way heterogeneity at a relatively high resolution with only a small number of unknown parameters. de Marsily [1978] proposed the pilot-point method (discussed in Chapter 3) and Clifton and Neuman [1982] developed method which penalizes departures of inverse estimates from prior kriging estimates. Later developments of geostatistical inverse methodology were based on both analytical [Rubin and Dagan, 1987] and numerical [Kitanidis and Vomvoris, 1983; Hoeksema and Kitanidis, 1984; Sun and Yeh, 1992; Yeh et al., 1996; Yeh and Zhang, 1996; Zhang and Yeh, 1997] co-kriging techniques. G6mez-Herncindez et al. [1997] and Capilla et al. [1997] developed a sequential self-calibrated method based on "master locations", which is somewhat similar to an extension of the pilot-point method proposed by LaVenue and Pickens [1992]. Important comparisons between various geostatistical inverse methodologies have been performed by Rubin and Dagan [1987], Keidser and Rosbjerg [1991] and Zimmerman et al. [1998]. However, applications have been few in number and virtually all were two-dimensional [e.g. Clifton and Neuman, 1982; de Marsily et al., 1984; Certes and de Marsily, 1991; LaVenue et al., 1995; Carrera et al., 1997; Zimmerman et al., 1998]. Some of the applications were related to Performance Assessment of the Waste Isolation Pilot Project (WIPP) near Carlsbad, New Mexico, conducted by the U.S. Department of Energy. There have been very few three-dimensional numerical inverse simulations of transient flow, and these have not considered the heterogeneity of medium porosity; uniform values of porosity were either estimated by an inverse model or defined a priori. Carrera et al. [1990b] analyzed a cross-hole test conducted at the Chalk River site, Canada. Data from multiple observation points were inverted simultaneously to es- timate uniform transmissivities of four major fracture zones and the permeability of the surrounding fractured medium. Carrera et al. [1993b] and Sanchez-Vila et al. 22 [1993] developed a mixed discrete-continuum model for the El Cabril site (for storage of low and intermediate level radioactive waste in Spain). The model was calibrated against a 5-year record of natural head fluctuations, and it was successfully tested against a 1-month pumping test. Huang et al. [1999] interpreted a series of pneumatic cross-hole tests at Yucca Mountain by means of an inverse model that considers porosity as known and uniform, and used manual trial-and-error to estimate permeability within a series of zones. Martinez-Landa et al. [2000] designed an inverse model for the Grimsel test site for the Full Engineered Barrier Experiment (FEBEX). Uniform permeabilities of major fractures and the surrounding matrix were estimated through simultaneous analysis of five cross-hole tests. Sauer et al. [2000] interpreted a series of cross-hole tests in a system of aquifers and aquitards having uniform anisotropic properties. In all these studies, medium permeability was not regarded as a stochastic random field. Only G6mez-Herncindez et al. [2000] used a geostatistical inverse model [Gómez-Hernandez et al., 1997] to interpret three-dimensional saturated flow in a synthetic fractured medium. Interpretation of welltest data has been traditionally performed by means of typecurves derived from analytical solutions [e.g. Gringarten, 1982; Neuman, 1988a, 1998; Batu, 1998] which allow simple, fast and reliable means of assessing pneumatic prop- erties of the medium. Such analytical models, however, are restricted to simplified representations of the hydrogeological environment. An alternative is to apply numerical inverse methods, which, in general, are complex, difficult and time consuming. Though computationally intensive and prone to discretization and round-off errors, numerical methods are much more flexible in terms of their ability to simulate realistic hydrogeological conditions. There is a limited number of interpretations of welltests by numerical inverse models. Lebbe et al. [1992], Lebbe and de Breuck [1995, 1997], Hvilshoj et al. [1999] and Lebbe and van Meir [2000] applied numerical inverse mod- els to interpret well tests conducted in various hydrogeological conditions assuming axi-symmetric radial flow in isotropic or anisotropic, uniform or locally uniform multi- 23 layer (aquifer/aquitard) medium (see also Sauer et al. [2000] discussed in the previous paragraph). Lebbe and van Meir [2000] also demonstrated the importance of simul- taneous interpretation of multiple tests for the accurate characterization of medium properties. Cardenas et al. [1999] analyzed by a numerical inverse model "seal performance" air-injection tests in packed-off intervals at the WIPP site near Carlsbad, New Mexico. Numerical inverse models have been also successfully applied to ana- lyze various types of laboratory tests [e.g. Finsterle and Pruess, 1995; Finsterle and Persoff,, 1997; Finsterle and Faybishenko, 1999]. Simultaneous interpretation of a series of welltest by means of a stochastic inverse model which accounts for medium heterogeneity with a high resolution can be defined as "tomography" or "stochastic imaging." It has been recognized in the literature [e.g. Neuman, 1973; Carrera and Neuman, 1986a] that as much information as possible should be collected and simultaneously interpreted to obtain accurate characterization of medium properties. The idea of "high-resolution stochastic imaging," however, was originally proposed over a decade ago by Neuman [1987] in connection with interpretation of hydraulic cross-hole tests in saturated fractured crystalline rocks at the Oracle site near Tucson, Arizona [Hsieh et al., 1985]. Later, the concept has been explored by Tosaka et al. [1993], Bohling [1993], Gottlieb and Dietrich [1995], Masumoto et al. [1995, 1996, 1998], Butler et al. [1999] and Liu and Yeh [1999] (see also Gottlieb [1992]). Tosaka et al. [1993] analyzed a series of transient cross-hole tests in two- and three-dimensional synthetic media. While porosity was assumed a known constant, the inverse model estimated uniform permeability at the grid blocks of a computational region through simultaneous interpretation of the test data. This procedure identified properly the spatial pattern but not the magnitude of the permeability field. Gottlieb and Dietrich [1995] did a similar two-dimensional analysis considering transient flow between a source and a sink, and obtained similar results. Masumoto et al. [1998] extended the analysis of Tosaka et al. [1993] to synthetic multi-rate injection and recovery tests. For two-dimensional synthetic cases, Liu and 24 Yeh [1999] studied the influence of injection rate, distance between test boreholes and lengths of injection and observation intervals on inverse estimates depending on stochastic properties of the medium. They used a numerical inverse model based on iterative co-kriging of pressure and permeability data, while requiring that the spatial correlation of permeability field known a priori [Yeh et al., 1996]. There were also studies on tomographic inversion of tracer test data [e.g. Kunstmann et al., 1997; Vasco arid Datta-Gupta, 1999]. Tomographic imaging has been initially developed and is widely applied in physics, geophysics and medicine [e.g. Jackson, 1996; Ramm, 1997], where in contrast to hy- drogeology, the tomographic data represent steady-state measurements. Steady-state data, however, do not allow identifying the storage properties of a medium. Timevariation in the measurements complicates the inversion, but provides additional information and, therefore, in principle should improve the quality of inverse estimates. There have been a few applications of geophysical tomographic methods in hy- drogeological research. At the Oracle site, Ramirez [1986] demonstrated that tomographic images of electromagnetic resistivity correlate with kriging images of hydraulic conductivity obtained from single-hole tests [Jones et al., 1985; Neuman et al., 1985]. This is the first study to compare geophysical tomographic images with measured and kriged permeabilities. In Chapter 2, I provide a description of the Apache Leap Research Site and field and laboratory studies conducted there previously, with special emphasis on singlehole and cross-hole pneumatic tests. I discuss the statistical and geostatistical analyses of various ALRS rock properties and define the hydrogeologic conditions that prevail during pneumatic tests at the site. In Chapter 3, I develop a stochastic numerical inverse model for the simulation and interpretation of both single-hole and cross-hole transient pneumatic tests at the ALRS. Chapter 4 presents and discusses corresponding results from single-hole tests, and Chapter 5 does so for the cross-hole 25 tests. The dissertation ends with a comprehensive list of findings and conclusions in Chapter 6. 26 Chapter 2 APACHE LEAP RESEARCH SITE 2.1 Site description The Apache Leap Research Site (ALRS) is located in central Arizona near Superior approximately 160 km north of Tucson (Figure 2.1). Regional geology is characterized by a dacite zoned ash-flow tuff sheet overlying carbonate rocks. The volcanic sheet covers an area of 1,000 km 2 and varies considerably in thickness about an average of 300 m [Peterson, 1961]. This dissertation concern field studies conducted in the upper part of the tuff sheet, within a layer of slightly welded fractured tuffs. The site is located within the Pinal Mountains at an elevation of 1,200 m above sea level. The average barometric pressure at the site is approximately 87 kPa. Climate is temperate and dry, with a mean annual air temperature of about 30°C and a mean annual precipitation of less than 500 mm. Most of the precipitation occurs during the summer (from mid-July to late-September) and the winter (from mid-November to late-March). Infiltration conditions are not favorable in the summer, when not only air temperatures and evapotranspiration demand are high but also precipitation events have high intensity and short duration. However, in the winter due to lower air temperatures and evapotranspiration demand as well as longer duration and lower intensity precipitation events, the conditions for infiltration and groundwater recharge are much more favorable. The regional water table lies at a variable depth of more than 600 m. Except for a relatively thin perched zone of saturation at a depth of approximately 150 m, the rock above the water table is unsaturated. The site is similar in many respects to Yucca Mountain in southern Nevada where a candidate high-level radioactive waste repository site is still being characterized by the U.S. Department of Energy. 27 FIGURE 1996]). 2.1. Location map of Apache Leap Research Site [after Guzman et al., 28 Borehole 2.1. Borehole coordinates and length. Top Bottom Length x [m] y [m] z [rr ] x [m] y [m] z [m] [m] X1 X2 X3 Y1 Y2 Y3 Z1 Z2 Z3 V1 V2 V3 W1 W2 W2A W3 10.324 20.44 30.427 10.204 20.036 30.068 19.52 9.8 0. 1.248 4.238 7.238 5.24 4.85 5.24 5.24 TABLE 10.038 10.031 10.037 5.084 5.2 5.347 0. 0.032 0. 6.844 6.844 6.844 11.46 22.43 21.46 31.46 0.02 -0.02 0.04 -0.03 -0.31 -0.27 -0.33 -0.2 0. 0. 0. 0. 0. 0. -0.03 0. -2.715 -2.619 -2.552 -1.951 -1.842 -1.681 31.568 32.074 31.975 1.248 4.238 7.238 5.24 3.815 5.24 5.24 10.038 10.031 10.037 5.084 5.2 5.347 0. 0.032 0. 6.844 6.844 6.844 0.252 -3.931 0.148 -3.846 -13.02 -23.07 -32.94 -12.18 -22.18 -32.02 -12.38 -22.47 -31.98 -33.53 -28.25 -30.48 -11.24 -23.08 -21.34 -35.34 18.44 32.61 46.64 17.19 30.94 44.90 17.04 31.50 45.22 33.53 28.25 30.48 15.85 35.05 30.14 49.93 The site includes 22 vertical and inclined (at about 45 0 ) boreholes. In my study, I discuss data from 16 of these boreholes-X1, X2, X3, Yl, Y2, Y3, Z1, Z2, Z3, V1, V2, V3, Wl, W2, W2A and W3. The remaining boreholes are located outside the pneumatic test area. Borehole locations (Table 2.1) are defined in a local Cartesian coordinate system (x, y, z) with origin at the lower lip of the casing in borehole Z3, and a vertical z-axis pointing downward. A plan view and a three-dimensional perspective of the boreholes are shown in Figures 2.2 and 2.3, respectively. Due to a drilling error, boreholes V2 and W2 intersect at depth. All boreholes are open, except for the upper 1.8 m of each borehole which is cased. The boreholes span a rock volume of approximately 35 x 36 x 35 (44,100) m 3 . A total of 270-m oriented core was retrieved from the boreholes and stored at the University of Arizona Core Storage Facility. Shortly after the completion of drilling, a surface area of 47 x 27.5 (1,293) m 2 that includes all boreholes was covered with a thick plastic sheet (Figure 2.2) to minimize infiltration and evaporation. 29 - • W3 - 30 - - - W2 1111W2A - - 20 - - - - W1 0 - - V1 V2 V3 41 411 . - -- X2 • X1 0 • X3 eY1•y2•Y3 - - 0 _ • Z3•Z2 • Z1 - - Plastic cover - - -10 _ - i 1 1 0 10 x[m] FIGURE 2.2. Plan view of tested 20 30 boreholes and plastic cover. 40 30 0 -10 N -20 B -30 -40 FIGURE 2.3. Three-dimensional perspective of boreholes at the site. 31 2.2 Previous investigations Early work related to my area of study at the ALRS included the drilling of 16 boreholes and the conduct of various laboratory and field experiments. Detailed summary of these studies can be found in the Ph.D. dissertation of Illman [1999]. Here I concentrate mainly on laboratory and field investigations conducted by Rasmussen et al. [1990] and Thompson et al. [1995], single-hole pneumatic tests conducted by Rasmussen et al. [1990, 1993], Guzman et al. [1994, 1996] and Guzman and Neuman [1996], and cross-hole pneumatic tests conducted by Illman et al. [1998] and Inman [1999]. 2.2.1 Laboratory and field investigations Rasmussen et al. [1990] conducted a series of laboratory and field determinations of rock properties over 105 3-m borehole intervals in 9 of the boreholes—X1, X2, X3, Yl, Y2, Y3, Z1, Z2 and Z3 (Figure 2.4). Laboratory measurements on core segments from the 3-m borehole intervals included various matrix properties. For the purposes of my study the most relevant are the "oven-dry" matrix air permeability k [m 2 ], the effective matrix porosity O m [m 3 /m 3 ] and the parameter a [kPct -1 ] of the van Genuchten [1980] retention model (other parameters of this retention model are considered to be constant: the dimensionless parameter n = 1.6 and the residual water content 0, = 0). The measured values of k, On, and a lie in the following respective ranges: 3.8 x 10' 6 - 1.0 x 10' m 2 , 0.14 - 0.28 m 3 /m 3 and 0.0102 - 0.0642 kPa -1 . Rasmussen et al. [1990] defined the location and geometry of fractures intersected by the above nine boreholes through examination of oriented cores. They observed a total of 224 fractures. Using these data, I plotted fracture orientations on a Schmidt equal-area plot as shown in Figure 2.5 (another version of a similar plot by Yeh et al. [1988] has the strikes of all fractures rotated by 180 ° degrees). Though the fractures 32 o -10 N -20 2 -30 -40 2.4. Three-dimensional perspective showing center locations of 3-m test intervals employed by Rasmussen et al. [1990]. FIGURE 33 exhibit a wide range of inclinations and trends, most are near vertical, strike northsouth and dip steeply to the east. Fracture density for the 3-m intervals ranges from 0 to 4.3 counts I m. Thompson et al. [1995] determined the water content 19 [m 3 /m 3 ] at 159 locations along all boreholes, except W2A and W3, using neutron probes (Figure 2.6). The measurements were repeated eight times between May 1990 and April 1991. The results show that water content is nearly constant over time. Measured values vary from 0.071 to 0.124 m 3 /m 3 and are consistently lower than the matrix porosity ck n, (see above). The support scale of neutron probe measurements is inversely proportional to the local water content [cf. Jury et al., 1991]. 2.2.2 Single-hole pneumatic tests Over the 3-m borehole intervals defined in the previous subsection (Figure 2.4), Rasmussen et al. [1990, 1993] conducted a series of single-hole pneumatic tests by injecting air at a constant mass rate between two inflated packers while monitoring pressure within the injection interval. The injection flow rate varied from 6 x 10 -8 to 2 x 10 -5 kg/s (from 0.003 to 0.98 l I min under standard conditions) and the steady-state pres- sure buildup in the injection interval ranged from 86.1 to 151 kPa. Pressure is said to have reached stable values within minutes in most test intervals. Air permeability was calculated using steady state formula derived by Dachler [1936] and adapted to isothermal airflow by Rasmussen et al. [1990, 1993]. The air permeability values defined from 87 successful tests vary from 4.2 x 10 -17 to 2.4 x 10- 12 m2. The sample locations span a rock volume of approximately 32 x 10 x 28 (8,960) m 3 . Figure 2.7 shows a scatter plot of air permeability versus fracture density for the 3-m borehole intervals. There clearly is no correlation between fracture density and air permeability. Similar results have been obtained at other fractured rock sites worldwide such as Oracle, Arizona [Jones et al., 1985]; Chalk River, Canada [Raven 34 \ E Xi A >OE + X3 o 11 x 12 -0- 13 -0- Z1 +72 11 73 FIGURE 2.5. Lower hemisphere Schmidt equal-area projection of fractures identified by Rasmussen et al. [1990]. Contours indicate number of fractures per unit area of projection circle. 35 o -10 N -20 3 -30 -40 FIGURE 2.6. Three-dimensional perspective showing center locations of neutron probe measurements performed by Thompson et al. [1995]. 36 et al., 1985]; and Stripa, Sweden [Gale et al., 1987]. Guzman et al. [1994, 1996] and Guzman and Neuman [1996] conducted a much larger number of single-hole pneumatic injection tests of considerably longer duration over borehole intervals with different lengths. Their tests were performed under highly controlled conditions, subject to strict quality assurance, within 6 of the boreholesX2, Y2, Y3, Z2, V2 and W2A. A total of 184 borehole segments were tested by setting the packers 1 m apart (Figure 2.8); additional tests were performed in segments of lengths 0.5, 2.0 and 3.0 m in borehole Y2, and 2.0 m in borehole X2, bringing the total number of tests to more than 270. The sample locations span a rock volume of approximately 31 x 20 x 28 (17,360) m 3 . The tests were conducted by maintaining a constant injection rate until air pressure became relatively stable and remained so for some time. The injection rate was then incremented by a constant value and the procedure repeated. Two or more such incremental steps were performed in each borehole segment while recording the air injection rate, pressure, temperature and relative humidity during injection and, in most cases, recovery. The injection flow rate varied 9.8 x 10 -8 to 4.4 x 10 -4 kgls (from 0.0049 to 22.0 //min under standard conditions) and the steady-state pressure buildup in the injection interval ranged from 0.49 to 273 kPa. The time required for pressure in the injection interval to stabilize typically ranged from 30 to 60 min, increased with flow rate, and might have at times exceeded 24 h, suggesting that steady-state permeability values published in the literature for this [Rasmussen et al., 1990] and other sites, based on much shorter air injection tests, may not be entirely valid. Guzman and Neuman [1996] have shown that the measured pressure responses are affected by two-phase flow and inertia effects. For each relatively stable period of injection rate and pressure, air permeability was estimated by treating the rock around each test interval as an infinite three-dimensional uniform, isotropic continuum within which air flows as a single phase under steady state, in a pressure field exhibiting prolate spheroidal symmetry. For each test interval, I have defined the midrange value for the logio- 37 CS" • • • • • • • • • • • • • • • • • • 1-C \ I I— I LIJ • • CO '71- 1— Lli , • • • • • • 111111 1— SD • •• • • • 1111111 1 • • • • 11111111 1 • ai,_ [w] Ameewied ID OMB • •• • • IMO se • • • 1111111 1' 1.0 iii i_ 71'1 C.0 1— LI, ,_ ,- N- w r- 38 0 -10 N -20 2 -30 -40 FIGURE 2.8. Three-dimensional perspective showing center locations of 1-m singlehole test intervals employed by Guzman et al. [1996]. transformed air permeability estimates. In terms of these midrange values, local-scale air permeabilities vary by orders of magnitude, from 7.5 x 10' to 8.7 x 10- 14 m2, across the site. Illman et al. [1998] and Illman and Neuman [20001)] have interpreted data from the first step of 40 single-hole pneumatic tests by means of analytically derived typecurves, based on the assumption of single-phase airflow through a uniform, isotropic porous continuum. They considered the case where injection takes place at a point, and accounted for the effects of storage and skin in the injection interval. Their anal- 39 ysis was performed by means of type-curves based on previously published analytical solutions for liquid flow under a spherical and radial flow regime, including cases where a single horizontal or vertical fracture intersects the injection interval. To render them applicable to air, the authors recast the original analytical solutions in terms of pseudo-pressure [Al-Hussaing et al., 1996; Raghavan, 1993] and developed expressions and type-curves in terms of pseudo-pressure derivatives. Although the governing airflow equation was linearized in terms of either pressure, p, or pressure-squared, p 2 , (see Section 3.4) their results demonstrated that the two approaches produce similar results. They also concluded that during single-hole tests, skin effects are negligible but storage effects are significant. Using the same methodology, Illman and Neuman [2000b] also analyzed independently the data from all injection steps and recovery of a single test. This analysis confirmed the earlier findings of Guzman and Neuman [1996] and Vesselinov and Neuman [2000] that, although two-phase and inertia effects are pronounced for some of the tests, they do not impact significantly the estimation of air permeability. The type-curve analysis did not allow reliable identification of air-filled porosity or dimensionless borehole storage. Nevertheless, air permeability values obtained by type-curve and steady-state analyses agreed very well. Some of these results will be discussed in Chapter 4. 2.2.3 Cross hole pneumatic tests - Illman et al. [1998] and Illman [1999] designed and conducted a total of 44 cross-hole pneumatic tests, incorporating 16 of the boreholes at the site. The tests span a volume of fractured rock larger than that previously subjected to single-hole testing. In each cross-hole test, air was injected at a constant mass flow rate into a relatively short borehole interval while monitoring air pressure and temperature in the injection and observation intervals and barometric pressure, air temperature and relative humidity at the surface. Pressure measurements are relative, representing changes in absolute 40 air pressure with time. The injection flow rate varied from 1 x 10 -5 to 2 x 10 -3 kgls (from 0.5 to 100 //min under standard conditions) and the pressure buildup in the injection interval reached a maximum of 140 kPa. A more detailed description of some of the tests and results is given in Chapter 5. Assuming single-phase airflow through a uniform, isotropic porous continuum, Illman et al. [1998] analyzed the cross-hole tests by means of analytically derived type-curves. These type-curves were a modified version of those developed by Hsieh and Neuman [1985] for the interpretation of cross-hole hydraulic tests in saturated fractured crystalline rocks at their Oracle site near Tucson [Hsieh et al., 1985]. The type-curves allow the injection and observation intervals to be represented by lines having arbitrary lengths and spatial orientations. Illman et al. [1998] modified them for single-phase airflow in the presence of storage and skin in observation intervals. Since both pressure, p, and pressure-squared, p 2 , based methods for linearization of the governing airflow equation (see Section 3.4) gave similar results for single- hole tests, Illman et al. [1998] adopted the simpler p-based type-curves for their interpretation of cross-hole tests. They considered the case where injection takes place at a point, and observation along a line. To date, only one of these tests (labeled PP4) was fully analyzed in this manner [Inman et al., 1998; Illman, 1999; Inman and Neuman, 2000a]. The most up-to-date results are those given by Illman and Neuman [2000a]. They are discussed in Chapter 5. 41 2.3 Geostatistical analysis of spatial variability Core and single-hole measurements, conducted over short segments of a borehole, provide information only about a small volume of rock in the immediate vicinity of each measurement interval. Available data from the ALRS indicate that rock properties, measured on such small scales, vary erratically in space in a manner which renders the rock randomly heterogeneous. A major question is how to describe this spatial and directional dependence of medium properties in untested portions of the rock. The analyses of Guzman et al. [1994, 1996], Chen et al. [1997] and Illman et al. [1998] suggest that it is possible to interpolate some of the core and single-hole test measurements at the ALRS between boreholes by geostatistical methods, which view the corresponding variables as spatially correlated random fields. This is especially true for air permeability k, matrix porosity O m , fracture density, water content 0, and the van Genuchten water retention parameter a, for each of which there are enough measurements to constitute a workable geostatistical sample. Standard geostatistical analysis provides best (minimum variance) linear unbiased estimates of how each such quantity varies in three-dimensional space, together with information about the uncertainty of these estimates. A geostatistical analysis of the above site variables was originally conducted by Chen et al. [1997]. I have repeated, slightly modified, and extended their interpreta- tions. 2.3.1 Air permeability An omni-directional sample variogram y(h) of log io air permeability k [m 2 ] data as - a function of separation distance h [in], obtained from steady state interpretation of 1-m scale single-hole pneumatic injection tests, is shown in Figure 2.9. The shape of the sample variogram implies statistical non-homogeneous random log io k field with 42 homogeneous spatial increments (also called in the geostatistical literature "intrinsic" random field of "second order"). It was analyzed by Chen et al. [1997] using three structural models: • power variogram -y(h) = 0.2720 4 5 ; • exponential variogram -y(h) = 0.495 [1 — exp(-1.26h)] of residuals about a firstorder polynomial trend given by —16.45 0.0489x + 0.0561y — 0.182z; and • exponential variogram ry(h) = 0.581 [1 — exp(-1.67h)] of residuals about a second-order polynomial trend given by —16.76 H- 0.0566x H- 0.0463y — 0.233z 0.00126x 2 0.00154y 2 — 0.00718z 2 0.0000501xy 0.00530xz 0.00720yz. To select the best among these three models, Chen et al. [1997] employed the Maximum Likelihood Cross Validation approach of Samper and Neuman [1989a, b], coupled with the generalized least squares drift removal approach of Neuman and Ja- cobson [1984]. The analysis of Chen et al. [1997] showed that whereas the exponential variogram model with a second-order drift fits the data most closely, all the model discriminating criteria (AIC, BIC, O m and dm ; these criteria are also discussed in Chapter 3, page 92) consistently rank the power model as best. The reason is that the power model is most parsimonious with only two parameters, while the exponential variogram model with second-order drift is least parsimonious with twelve parameters. As is shown in Figure 2.10, both models yield very similar kriged estimates of log lo k, but rather different measures of the associated estimation uncertainty. The geostatistical analysis of Chen et al. [1997] was based on 1-m scale air permeability data [Guzman and Neuman, 1996] in six boreholes—X2, Y2, Y3, Z2, V2 and W2. I have augmented the 1-m data with air permeabilities obtained from 3-m test intervals [Rasmussen et al., 1990] in four additional boreholes Xl, X3, Yl, Z1 and Z3. 43 k) gio k [rn 2 ] 2.0 1-m support data 0.003 Power model 1.5 o Second-order residuals Spherical model • 0.002 e as 0.001 0.0 Fracture density [counts/m] , 0 [M 3/M 3] • 1.5 • 0.8 I. • - - 0 1.0 E El sr 0, 0 . 0.5 - - -o- - - Data • -O. -•• , ,o 1 . /c) ...... .. .. œ > 0.2 ,o 'S. e .. -e- - - Original data o Second-order residuals Spherical model - - Exponential model 0.0 0.02 FIGURE 2.9. Omni-directional sample and model variograms for various parameters at a minimum separation distance of 3 m. 44 -14A -14.9 -15 4 -15.9 -16A Estimation variance 0 I 0.48 0.38 028 0.18 -15 20 25 FIGURE 2.10. Kriged estimates and kriging variances of log io k at y = 7 m using a power model (left), and an exponential model with second order drift (right). 45 To check whether this augmentation is justified, I compared 1-m and 3-m scale single-hole air permeability data from boreholes X2, Y2, Y3 and Z2. Figure 2.11 shows how these vary with depth in each borehole. The figure also shows 0.5-m, 2-m and 3-m scale data obtained by Guzman et al. [1996] in borehole Y2. The figure demonstrates that as the support scale increases, the amplitude and frequency of spatial variations in air permeability decrease. Nevertheless, the 3-m data captures the overall trend of non-uniformity observed by 1-m data. One also notes that 3-m scale permeabilities obtained by Rasmussen et al. [1990] are consistently lower than those obtained by Guzman et al. [1996]; I attribute this systematic difference to the relatively short duration of tests conducted by Rasmussen et al. [1990]. I have included in Figure 2.11 information about the location of fractures intersected by the boreholes. One notes that correlation between measured local permeabilities and fracture locations/densities is poor. Histograms obtained from all the 1-m and 3-m scale data are presented in Figure 2.12. They suggest that population distributions associated with the two data sets are not significantly different. Summary statistics of the data samples are listed in Table 2.2. Recall that not all data come from the same boreholes. A qualitative comparison of the underlying population distributions can be performed by means of the non-parametric Kolmogorov-Smirnov test [cf. Hollander and Wolfe, 1973]. The Kolmogorov-Smirnov Z value is defined as the largest absolute difference between the sample distributions. The computed Z is compared to a critical value depending on sample sizes. If Z is greater than the critical value, the null hypothesis that the population distributions are similar is rejected. In practice, computer programs such as SPSS [1997] directly compute the two-tailed asymptotic significance level of the probability that the population distributions are not substantially different from each other. The significance level in our case is 0.2 and therefore the null hypothesis cannot be rejected at the typically accepted significance level of 0.05. A similar conclusion is reached using the parametric two-sample T test. 46 o Y3 10 — 20 30— 1-m scale ao - - - 3-m scale (1990) 8- I 1E-16 1E-14 1E-15 1E-13 1E-18 I I I 11111 I 1E-17 I 1 I 11111J 1E-16 Air permeability [mq I 1 1 11111 Fracture I 1 1 1 11111 1E-15 I 1 I I 1111 1E-14 1E-13 k (m2] "k2 kt 10 20 30 1 1 1 1 1111 1 1 I I I I 111 1 1 1 1 1 I 11 permeability with depth and support scale (data from Guzman et al. [1996] and Rasmussen et al. [1990]). FIGURE 2.11. Variation of air TABLE 2.2. Summary statistics of the geostatistically analyzed data. Sample size Parameter k [m 2 log io 184 - 1-m data 87 data - 3-m 227 and 3-m data - 1-m 105 Fracture density [count s/m] 105 log io O m [m 3 /m, 3 ] Minimum Maximum Mean Variance CV -17.03 -16.38 -17.13 0.0 -13.06 -11.62 -11.62 4.33 -15.25 -15.11 -15.22 0.767 0.763 1.037 0.870 0.695 -0.05 -0.0687 -0.0572 1.09 -0.845 0.071 -1.99 -0.561 0.124 -1.19 -0.760 0.0986 -1.68 2.94 x 10 -3 1.21 x 10 -4 2.09 x 10 -2 -0.0713 0.112 -0.0861 ] 0 [m 3 /m, 3 ] log io a [kPa -1 ] 159 105 47 O Cn1 C•I "or ar dir .dor Air .or .0" .0" 099999999119999 ii imi 00119911999999-99 07 Air, Jr , .ordrir .de Fir Z.", .0" MI1111111111111111111111111111111111111111111 ArZ .44;.a.Z.de Ar.e.e.or 4B 1 RITIIJIB IBUTE IMPRIBRINIB IJIJI 11101119 111 , 1111 o O CN1 6O 6o Aouenbeal a/0q 9 6 o 48 TABLE 2.3. Summary statistics for log io k [m 2 ] from single-hole tests at various scales along borehole Y2. Minimum Maximum Mean Variance Support scale Sample size CV 0.5 m 1m 2m 3m 54 28 14 9 —16.31 —16.19 —16.15 —15.80 -13.12 —13.32 —13.55 —13.61 —15.10 14.99 —14.95 —14.78 — 0.424 0.584 0.498 0.378 —0.0431 —0.0501 —0.0472 —0.0416 Histograms of the log io -transformed air permeability data at various (0.5, 1, 2 and 3 Tri) scales along borehole Y2 [Guzman et al., 1996] are presented in Figure 2.13 (these data have already been presented in different format in Figure 2.11). Summary statistics of the data are listed in Table 2.3. The histograms suggest that the underlying population distributions are not significantly different from each other; nevertheless, as support scale increases, the range of the data decreases, and the sample histograms becomes less skewed and closer to log-normal. The four distributions are compared qualitatively using the Kruskal-Wallis H test [cf. Hollander and Wolfe, 1973], which is a non-parametric alternative to the one-way analysis of variance (ANOVA) test. The H statistic is calculated on the basis of sums of ranks for combined samples. It is approximately chi-square (x 2 ) distributed with degrees of freedom equal to the total number of samples minus one. The null hypothesis that the population distributions do not differ in mean rank is rejected if the critical value of X 2 for the desired significance level is less than the computed H value. SPSS [1997] directly computes the corresponding significance value, which in our case is equal to 0.62. Therefore, the underlying population distributions do not differ statistically and do not indicate a statistically significant scale effect. Finally, I compare in Figure 2.14 the omni-directional sample variograms of 1-m, 3-m and combined 1-m and 3-m log io k data. Though the sample variograms differ somewhat from each other at large separation distances, they are otherwise quite close. Attempts on my part to represent the 3-m data by a variogram model that views them as a sample from a statistically homogeneous random field with a linear 49 MOM dor 1 111111111111111111 111111111111111111111 IIIIIIIIIIIIIIIIIlIIII11 1111111111111i1111 111111111111111111 /ZZYYZZZZ/ZZZZZ/ZZZ/ZZ ZZ FZZZZZLIZZ ‘„..\\.\:\ N.'"\ N.N.\.\\.\\X\N"\\X\NNN.N. 111111111111111 11111111 11111111111111111111111111111111111111 bWFJW 1=11111111111111111111111 "ZZZ/ZZZZZZZZZZZZ erZZZZZZ Z LA 0 LO 0 in CICIINNr 6 6 6 6 6 6 Ao u en bail angel au 0 r 6 00 6 6 50 );.*..;°' 2.0 1.5 - - -e- - - 1-m support data - --o----- 3-m support data ..... fr /... o 1-m and 3-m support data/: ,/ /.: Power model .1 0.5 0.0 FIGURE 5 10 15 Distance [m] 20 2.14. Omni-directional sample variograms for log io 25 k with various supports and a power variogram model fitted to 1-m data. or quadratic spatial drift were not successful. This supports the earlier variogram analysis of 1-m log io k data by Chen et al. [1997]. My analysis justifies the addition of the 3-m scale air permeability data of Ras- mussen et al. [1990] to the 1-m data of Guzman et al. [1996]. Locations of the combined set of 227 air permeability measurements are depicted in Figure 2.15; log io permeability [m 2 ] values range from —17.13 to —11.62 with mean, variance and coefficient of variation equal to —15.2, 8.7 x 10 -1 and —6.1 x 10 -2 , respectively (Table 2.2). The sample locations span a rock volume of approximately 32 x 20 x 32 (20,480) m 3 . I kriged these data using the power variogram model obtained by Chen et al. [1997] for the 1-m scale data. 51 0 -10 -20 N 2 -30 -40 2.15. Three-dimensional perspective showing center locations of single-hole test intervals for combined set of 1-m (circles) and 3-m (squares) scale measurements. FIGURE 52 Figure 2.16 compares kriged images of log io k I have generated along four vertical sections at y = 0, 5, 7 and 10 m using 1-m data (left column) and the combined set of 1-m and 3-m data (right column). Boreholes providing information to produce the respective kriged maps are included in the figure. The two sets of kriged images differ substantially from each other. This is most pronounced at y = 0 m, which passes through the Z-series of boreholes: here the inclusion of data from boreholes Z1 and Z3 has caused estimated permeability in the upper right corner of the section to be much higher than it is without these data. The effect extends to all four cross-sections, which exhibit elevated permeabilities near the upper-right corner. Along sections at y = 5 m and 7 m, which pass close to the Y and V series, respectively, the addition of data from Yl affects the shape and size of a prominent high-permeability zone which extends through Y2 (see corresponding peak in Figure 2.15). The addition of data from X3 to the set reveals corresponding high- and low-permeability zones in section y = 10 m, which correlate well with similar zones intersected by Y3 in section y = 5 M. 53 0 ly= O ml Z2 0 -5 -5 -10 -10 î_ 15 -15 Z3 Z2 Z1 -20 -25 -25 -30 -30 x [m] Y2 0 Y3 -5 -10 I -15 -20 -25 -30 x [m] X2 0 -5 -5 -10 -10 7 -15 -20 -20 -25 -25 -30 -30 x [m] x [m] FIGURE 2.16. Kriged log lo k estimates obtained using 1-m scale data from boreholes X2, Y2, Y3, Z2, V2 and W2A (left) and same together with 3-m scale data from boreholes Xl, X3, Yl, Z1 and Z3 (right). 54 2.3.2 Fracture density, matrix porosity, water content and van Genuchten a Chen et al. [1997] performed geostatistical analyses of fracture density [counts I m], log lo matrix porosity Om, [m 3 /m 3 ], water content 8 [m 3 /m 3 ] and log io van Genuchten a [kPal data. Summary statistics of the data are listed in Table 2.2 on page 46. Figure 2.9 (page 43) shows corresponding omni-directional sample variograms and models. The latter for each rock parameter are • fracture density [counts/m], exponential model with variance 0.69 and an inte- gral scale 2.5 m; • 1og 10 (q5,, [m 3 /m 3 ]), exponential model with variance 0.00315 and an integral scale 3 m; • 0 [m 3 /m 3 ], spherical model (sill 0.561 and correlation scale 6.29 m) of residuals about a second-order polynomial trend defined by 8.3145 + 0.2234x — 0.05905y+0.010397z-0.0049492x 2 +0.00077045y 2 +0.0028264z 2 -0.001986xy+ 0.0039864xz — 0.0080923yz; and • log io (a [k P a 1), exponential model with variance 0.0223 and integral scale 2.25 - rn. 2.3.3 Conditional simulations Figure 2.17 shows kriged images of log io k (utilizing the combined set of 1-m and 3-m data), fracture density, log io O m , 0 and log io a in a vertical plane corresponding to y = 7.0 m around which many of the available data are clustered. There clearly is no correlation between log lo air permeability and fracture density along this plane (nor anywhere else in the domain of investigation). Matrix porosity is consistently 55 higher than water content throughout the tested rock mass, reflecting the fact that the medium is not fully saturated. A three-dimensional representation of kriged log io k based on the combined set of data is shown in Figure 2.18. Figures 2.19 through 2.21 show corresponding sections in the y-z, z-x and x-y planes, respectively. Similar three-dimensional representations of kriged fracture density, matrix porosity log io Om , water content 0 and van Genuchten's log io a are presented in Figures 2.22 — 2.25, respectively. To estimate the pore space available for airflow, I subtracted the kriged estimates for water content (Figure 2.24) from those of matrix porosity (Figure 2.23) to obtain a three-dimensional representation of log io air-filled matrix porosity, q5,„, as shown in Figure 2.26. The three-dimensional images of kriging estimates in Figures 2.18 —2.26 represent a rock volume of approximately 45 x 33 x 33 (49,005) m 3 , encompassing the locations of all the analyzed data sets. Kriged estimates of hydrogeologic variables are smooth relative to the corresponding random fields. To generate less smooth and more realistic images that honor the available data, I used GCOSIM3D, a sequential Gaussian conditional simulation code developed for three-dimensional data by Gómez-Herndndez and Cassiraga [1994]. The code is applied separately to log io k (conditioned on the combined set of 1-m and 3- m data), fracture density, log io çbm , 19 and log io a data on the assumption that each of these variables is Gaussian. The normality test is performed using a one-sample Kolmogorov-Smirnov goodness-of-fit test [cf. Hollander and Wolfe, 1973], where the Kolmogorov-Smirnov D value, similarly to Z value defined above, is determined by the largest absolute difference between the sample and hypothesized theoretical distributions. Again, the computed D is compared to a critical value depending on the sample size. If D is greater than the critical value, the null hypothesis that the population distribution is Gaussian should be rejected. Computer programs such as SPSS 56 TABLE 2.4. Results from Kolmogorov-Smirnov test of Gaussianity. Parameter log io k [m 2 ] 1-m data — 3-m data — 1-m and 3-m data — Fracture density [counts I m] log lo O m [m 3 /m 3 ] 0 [m 3 /m 3 ] residuals log io a [kPar-1 ] Most extreme absolute difference Kolmogorov- Smirnov Z Asymptotic significance 0.046 0.135 0.065 0.194 0.114 0.075 0.076 0.652 1.263 0.978 1.984 1.173 0.947 0.778 0.789 0.082 0.294 0.001 0.127 0.331 0.581 [1997] directly computes the two-tailed significance level of the probability that the population distribution is not substantially different from the theoretical distribution. Calculated significance levels for all data samples are given in Table 2.4. They show that all data except fracture density pass the test of Gaussianity at a significance level of 0.05. Figure 2.27 shows conditionally simulated images of parameters in a vertical plane corresponding to y = 7 m. These images are clearly much less smooth than are their kriged counterparts in Figure 2.17. The results demonstrate that all the above data are amenable to continuum geo- statistical analysis and exhibit distinct spatial correlation structures. This suggests that each data set can be viewed as a sample from a random field, or stochastic continuum. This is so despite the fact that the rock is fractured and therefore me- chanically discontinuous. The conclusion is supported strongly by similar findings in many other fractured rock sites including crystalline rocks at Oracle, Arizona [Jones et al., 1985; Neuman and Depner, 1988]; Stripa, Sweden [SKB, 1993; Neuman, 1998]; and Fanay-Augeres, France [Cacas et al., 1990; Neuman, 1998]. It justifies the ap- plication of continuum flow and transport theories and models to fractured porous tuffs on scales of a meter or more, as proposed over a decade ago in the context of crystalline rocks by Neuman [1987] and affirmed more recently by Tsang et al. [1996]. 57 log io k [m 2 ] 0-0.695 -0.72 -0.745 -0.77 -0.795 -0.82 o 10 x [m] X 10 [m] 115 20 e [m 3 /m 3 } Fracture density [counts/m 0.3 N -20 FIGURE 2.17. Kriged estimates of various parameters at y = 7 m. 25 58 30 -10 FIGURE 2.18. Three-dimensional representation of kriged log io k [m 2 ]. -12.5 -13 -13.5 -14 -14.5 -15 -15.5 -16 -16.5 -17 59 x = 12 m x = -3 m x = 27 m -10 -20 -30 -30 (.1 X = X = 30m x= 15m M -10 -10 -20 '4 -20 -30 -30 1 y Cm] x=3m x= 18m -10 -20 -30 n 10 0 y x = 6 m -10 x = 21 m ' "§: -20 N -20 -30 10 2 Y Irn1 x=9m x = 24 m -10 X -20 FIGURE 2.19. Kriged log lo k [m 2 ] along various y-z planes. 60 a -10 " -20 -30 -30 -20 0 0 -10 -10 -10 " -20 " -20 -20 .30 -30 -30 T. [m] 0 .10 -10 " .20 " -20 -30 -30 T, FIGURE 2.20. Kriged log lo k [m 2 ] along various x-z planes. 61 z = -30 m [011 z = -18m 20 20 10 -E- 10 7. 0 20 r 10 0 20 [ mj FIGURE 2.21. Kriged log io k [m 2 ] along various x-y planes. 62 30 1.85 1.6 1.35 1.1 0.85 0.6 0.35 0.1 - FIGURE 2.22. [counts lm]. 10 Three-dimensional representation of kriged fracture density 63 -10 FIGURE 2.23. Three-dimensional representation of kriged matrix porosity log io O m [m 3 1 n13]. 64 -10 FIGURE 2.24. Three-dimensional representation of kriged water content 0 [m3/m3]. 65 -1.4 -1.5 -1.6 -1.7 -1.8 -10 FIGURE 2.25. Three-dimensional representation of [kPa]. kriged van Genuchten's log io a 66 -10 FIGURE 2.26. Three-dimensional representation of estimated air-filled matrix poros- ity log lo O ma [m3/m3]. 67 FIGURE 2.27. Random conditional realizations of various parameters at y = 7 m. 68 2.4 Conceptualization of hydrogeologic conditions during pneumatic tests The conceptual hydrogeological model of airflow at the site during pneumatic tests was originally defined by Guzman et al. [1996] and Guzman and Neuman [1996]. Here, I follow their findings and conclusions. The conceptualization is consistent with pneumatic-test interpretations by Rasmussen et al. [1990], Illman et al. [1998], Inman [1999], Illman and Neuman [2000b] and Inman and Neuman [2000a]. Pneumatic tests at the ALRS were conducted in a fractured unsaturated rock. To prevent evaporation and infiltration, a plastic cover was placed on the ground surface. Due to the high capillary retention properties of the porous (matrix) blocks of rock (see Section 2.2.1), water is drawn from fractures into the matrix, leaving the fractures primarily air-filled, and making it difficult for air to flow through matrix blocks. As a result, airflow during pneumatic tests is considered to occur primarily through fractures, most of which appear to contain little water. Moreover, since matrix blocks appear to be virtually saturated with water, air storage effects in the matrix blocks are not expected to play a role during pneumatic tests. While twophase flow of air and water may be taking place, it appears valid to disregard the movement of water and treat air as a single gas phase. This is so despite findings by Guzman and Neuman [1996] that, during single-hole tests, (1) pressure in the injection interval typically rises to a peak prior to stabilizing at a constant value, and (2) air permeability increases systematically with applied pressure. These observations are most probably due to a two-phase flow effect whereby water in the rock is displaced by air during injection. The air entry pressure of matrix can be estimated approximately [van Genuchten, 1980] as the inverse of van Genuchten's a. This yields air entry values that range from 15.6 to 98 kPa. These compare with air pressures in injection intervals during pneumatic tests, which however die rapidly with distance from these intervals. Therefore, one can expect two-phase flow effects to be important at most 69 in the close vicinity of the injection interval. Guzman and Neuman [1996] conducted numerical simulations demonstrating that the permeability determined from pneumatic tests under unsaturated conditions is generally lower than intrinsic permeability. Nevertheless pneumatic permeability approaches intrinsic permeability as the applied pneumatic pressure increases. It follows that air permeabilities and porosities derived from pneumatic test data should reflect closely the intrinsic properties of the surrounding fractures, which are relevant to both unsaturated and saturated conditions. It has also been demonstrated that: • Air permeabilities determined in situ from steady-state single-hole test data are much higher than those determined on core samples of rock matrix in the laboratory [Rasmussen et al., 1990], suggesting that the in situ permeabilities represent the properties of fractures at the site [Guzman and Neuman, 1996]. • The permeabilities of individual fractures cannot be distinguished from the bulk permeability of fractured rock in the immediate vicinity of a single-hole test interval [Guzman and Neuman, 1996; Rlman et al., 1998; Illman and Neuman, 2000b]. • Air permeabilities are poorly correlated with fracture densities (Figure 2.7), as is known to be the case for hydraulic conductivities at many water-saturated fractured rock sites worldwide [Neuman, 1987]. • Along the boreholes, there is a poor correlation between the location and density of fractures and local permeabilities from single-hole tests (Figure 2.11). • The geostatistical interpretation of air permeability and fracture density data [Chen et al., 1997] demonstrates that both parameters are characterized by very different spatial structure functions (variograms) and (Figure 2.9) and kriged estimates are quite distinct (Figure 2.17). 70 All these observations provide further support for the conclusion of Neuman [1987] that the permeability of fractured rocks cannot be reliably predicted from information about fracture geometry (density, trace lengths, orientations, apertures and their roughness), but must be determined directly by means of hydraulic and/or pneumatic tests. In addition, the fractures appear to form a dense interconnected threedimensional network (Table 2.2, Figure 2.5). As a result, it is reasonable to represent fractured rock at the site as a single equivalent porous continuum. Inertial effects were observed by Guzman and Neuman [1996] in only a few singlehole test intervals intersected by widely open fractures. Enhanced permeability due to slip flow (the Klinkenberg effect) appears to be of little relevance to the interpretation of pneumatic tests at the ALRS [Guzman and Neuman, 1996; Illman et al., 1998]. I further assume that airflow at the ALRS takes place under isothermal conditions, ignoring adiabatic effects and differences in temperatures between injected and ambient air in the rock. Variation of atmospheric pressure at the ground surface clearly affect pressure in the rock under both ambient and test conditions. They should be taken into account to fully describe pressure responses during pneumatic tests. 71 Chapter 3 NUMERICAL INVERSE METHODOLOGY The theoretical discussion on inverse methodology follows the principles and defini- tions established by Neuman [1973], Neuman and Yakowitz [1979], Neuman et al. [1980], Neuman [1980], Carrera [1984] and Carrera and Neuman [1986a, b, c]. 3.1 Background 3.1.1 Governing equations Gas flow in porous medium is governed by the following partial differential equation over a three-dimensional domain St with boundary F [cf. Bear, 1972, page 200], v . (kp vp + / J a (kgp 2q.= o op OZ\/J at (3.1) subject to initial and generalized boundary conditions P = po on Si at t = 0, (3.2) kPVp) • n = V (p f — p) qb along F (--- (3.3) where p is absolute air pressure [M LT 2 ], p is air density [M/L 3 ], ,a is air dynamic viscosity [M I LT], k is air permeability [L 2 ], 0 is air-filled porosity [—], qu, is a source term [M/L 3 T], n is unit vector normal to r, p f is prescribed boundary air pressure [M LT 2 ] on the boundary r, qb is prescribed air mass flux [M/L 2 T] normal to the boundary F, y [T I L] is a parameter controlling the type of boundary conditions (first or second type if y 0 or y oo, respectively; third type otherwise) and g is 72 acceleration due to gravity [L/T 2 ; 9.8 m/s 2 ]. The absolute air pressure p [N1m 2 ] and air density p [kg/m 3 ] are related through the equation of state PM P = ZRT (3.4) where Z is a dimensionless compressibility factor depending on air pressure and tem- ° perature, M is molecular mass [kg], T is absolute temperature [ K] and R is the universal gas constant [8.314 J/( ° K • mol)]. In the case of isothermal gas flow (T is taken to be constant), the problem is nonlinear due to dependence of p and on pressure p. For the ranges of pressure and temperature monitored during pneumatic test, the dimensionless compressibility factor, Z, is approximately constant and equal to 1. To characterize the borehole storage effect of the injection interval, the mass flow rate Q from the interval into the medium (proportional to qa, in Equation 3.1) satisfies the following mass-balance equation [e.g Neuman, 1988a] dM dl (3.5) where Q in3 is the rate of air mass injected in the packed-off interval and M is the air mass confined in the injection interval. The right side of the last equation can be rewritten as dM dM dp„ , dp dp w = vw (3.6) = dpw dt dpw dl dt where pw is air pressure at the injection interval and V,„ [L 3 ] is nominal volume of the injection interval. Since by definition, air compressibility C a [LT 2 /M] is (3.7) one obtains 73 Qi 713 — Q =17",„pC„ dp, (3.8) dt If the dimensionless compressibility factor, Z, in (3.4) is approximately constant, Ca is approximately equal to the inverse of absolute air pressure. Assuming that the pressures of injected air and air within the borehole interval are equal, the respective air densities are also equal (T taken to be constant). Therefore, one can write a volumetric version of (3.8) dp,, at — Q' = V„,ClaV„ 1 dp„, dt (3.9) where Q' = Q I p, [L 3 171 ]. The term on the right side in (3.9) represents volumetric rate of change in storage within the injection interval. The borehole storage depends on air compressibility, and decreases with the increase of absolute air pressure in the injection interval. During air injection, the storage effect also decreases at late times when air pressure in the injection interval stabilizes. During pressure recovery, qi n3 = 0 and theoretically there is no borehole storage effect. The borehole storage effect of the observation interval can be defined in a similar way Q , t vwcia dt dp„, v 1 di:6 p,, dt (3.10) where Q/2„,,, and Q' t are the rate volumetric airflow into and out of the observation interval. If there is no flow from the packed-off interval into the medium (Q' = 0 in 3.9 or Q'o„t = 0 in 3.10), the following dimensionless expression can be derived from (3.9) or (3.10) 42-5 exp — 1 (3.11) 74 where A 75 = P po P° is dimensionless pressure buildup, 7= t 9 '—n' is dimensionless time. - - When pneumatic test data are analyzed on a log-log plot of pressure versus time, the borehole storage effect is characterized by an exponential curve. Only when pressure buildups are small (Ap 1), T is also close to 1 and (3.11) can be approximated up -- to first-order as T (3.12) and the characteristic borehole storage curve on a log-log plot is a straight line with unit gradient, which is similar to the case of a constant fluid compressibility [e.g Neuman, 19884 Therefore, pressure records having significant pressure buildups at early times might demonstrate deviations from the linear behavior due to the impact of air compressibility on borehole storage. 3.1.2 Inverse problem As defined in previous section, the governing air pressure p (x, t) depends, among other factors, on the spatial distribution of air permeability, k (x), and air-filled porosity, (x), throughout Q. One can express this relationship through p (x,t) = where a = [a', al T (a) (3.13) is a vector of discrete k and 0 values, a', as well as other parameters describing their spatial distribution, a", and the forward operator ,F is a functional that maps a into p (x, t) through Equations 3.1 - 3.4. When the forward problem is solved numerically, ,F is a matrix and (3.13) is a set of algebraic equations. The inverse problem can be defined formally as solving (3.13) for a based on a knowledge of the state variable p (x,t) a = p (x, t)] (3.14) 75 where is an inverse operator. Neuman [1973] categorized methods of solving inverse problem into direct and indirect. Direct methods employ a direct inversion of the forward operator „F in line with Equation 3.14. This typically leads to a system of inverse equations in which the parameters a are the dependent variables. Quite often the system is overdetermined, in which case it is solved approximately by optimization. When the system of direct inverse equations is underdetermined, one typically augments it with constraints and/or includes a penalty criterion in the objective function to be minimized. Neuman [1973] proposed to base this penalty function on prior measurements of the parameters and/or prior ideas concerning their mode of spatial distribution (various degrees/modes of smoothness). Indirect methods adjust the parameters a in the forward problem (Equation 3.13) so as to minimize the difference between computed and measured p(x,t) values at discrete points in space-time. This is usually done iteratively by non-linear optimization. Adding a penalty criterion based on prior information [Neuman, 1980; Carrera and Neuman, 1986a, b, c] helps insure that the resulting parameters are well-defined. According to Hadamard [1932], a well-posed mathematical problem is one for which a solution exists, is unique and stable. Although the forward problem defined by Equations 3.1 — 3.4 is generally well-posed, the corresponding inverse problem tends to be ill-posed. This is due to lack of sufficient information about the state of the system (pressure, flux), measurement and interpolation errors, as well as computational errors associated with solving the forward problem. This can lead to non-unique and unstable inverse solutions [Carrera and Neuman, 1986b]. Stability of the inverse solution requires that small variations in the state variable p(x,t) produce only small variations in the estimated parameters a [Carrera and Neuman, 1986b]. Mathematically, this means that for every positive E, there exists a positive S such that 76 11/33. (x, t) —P2 (x, t) II <€ > Hal — a2II <S (3.15) or the more rigorous Lipschitz condition Hai — a2 11 < clIn. (x,t) —p2 (x, t) II (3.16) where p i (x, t) /3 2 (x, t), a la 2 , c is a constant and 11' 11 is a norm of appropriate space for p and a. Typically, a Hilbert space LP (R) is applied, L) .7=- {f (x) E R 1 Ilf (RA' = [f If (x)1 2 dx] 2 } (3.17) R where 11f (R)Il < oc and R is a real-valued domain. The notion of uniqueness implies that if there are two solutions to the inverse problem Equation 3.14, a l and a 2 , then Hai — adl 0 11Pi (x,t) —P2 (x, t) H 0 (3.18) If the solution of the inverse problem is unique, different parameter sets will produce different pressures. Identifiability differs from uniqueness, implying that different spatial distributions of the parameters within a given subdomain, a l a 2 , produce different spatial con- figuration of the state variable, p i (x, t) p 2 (x, t), or 11 .13 1 (x, t) — P2 (x, t)11 0 Hai — az (x) (.1 (3.19) Chicone and Gerlach [1987] proved theoretically that, for linear elliptic partial differ- ential equations, the unknown spatially non-uniform conductance parameter (in my case k) is identifiable within a given subdomain of the model region if every char- acteristic that crosses its boundary stays in the subdomain. Dietrich and Newsom 77 [1990] extended the idea to the time-averaged transient case. In the case of most hy- draulic and pneumatic tests, the parameters are usually not identifiable at locations far from a pumping or injection wells. Identifiability is a necessary but not a sufficient condition for well-posedness of the inverse problems. The existence of a unique and stable solution requires that when the problem is solved by optimization, the objec- tive function has a well-defined global minimum, which depends continuously on the parameters [Carrera and Neuman, 1986b]. I already mentioned that the inverse problem can be properly posed by augmenting its statement with supplementary information about the system. This may include prior information about the parameters or constrains on their sign and magnitudes. Neuman [1973] and Carrera and Neuman [1986a, b, c] demonstrate theoretically, and by examples, that ill-posed inverse problems can be made well-posed through such means. A functional representation of parameters across the model domain is termed parameterization. The number of discrete coefficients required to parameterize a given medium can vary from 1 when the medium is assumed to be uniform to as many as (or more than) the number of nodes or elements in the computational grid. For best results, the parameterization must be complex and flexible enough to represent adequately the spatial non-uniformity of the medium, and simple enough to yield a well-posed inverse problem. The number of unknown parameters must never exceed the number of observations (of both system state variables and parameters). The difference between the former and the latter is referred to as degrees of freedom. In general, as the number of degrees of freedom goes up, one can reproduce observed behavior more accurately, but at the expense of loosing confidence in the parameter estimates. Parameterization can be either deterministic or stochastic (geostatistical). The simplest deterministic approach is to assume the medium is uniform, which greatly enhances well-posedness but causes all details about medium heterogeneity to be lost. 78 Some such details can be recovered by adopting piecewise constant parameterization, known as zonation in which the model region is subdivided into a set of zones assigned uniform properties [Carrera and Neuman, 1986c]. Another compromise is to define model parameters as low-order (e.g. linear or quadratic) polynomial functions of space. Other popular options include interpolation based on inverse-distance or piecewise weighting, splines, etc. The number of coefficients (discrete parameters) utilized in the characterization of medium non-uniformity defines the degree of parameterization. Stochastic (or geostatistical) parameterization provides a parsimonious way to capture medium heterogeneity at a relatively high resolution with only a small number of unknown parameters. The underlying idea is to represent spatially varying medium properties as random fields, conditioned on discrete measurements of these properties. Both analytical [Rubin and Dagan, 1987] and numerical [de Marsily, 1978; Clifton and Neuman, 1982; Kit anidis and Vomvoris, 1983; Hoeksema and Kit anidis, 1984; Carrera and Glorioso, 1991; Sun and Yeh, 1992; Yeh et al., 1996; Yeh and Zhang, 1996; Zhang and Yeh, 1997; Gómez -Herncindez et al., 1997; Capilla et al., 1997] geostatistical methods have been described in the literature. Zimmerman et al. [1998] provide an excellent description of many of these methods, as well as a thorough comparison among them based on a series of specially designed synthetic test cases. I adopt one of these methods, based on the idea of pilot points, as proposed by de Marsily [1978]. 3.2 Stochastic numerical inverse model My stochastic inverse model is based on the geostatistical pilot-point method of parameterization [de Marsily, 1978], coupled with a maximum likelihood definition of the inverse problem [Carrera and Neuman, 1986a]. Whereas the latter authors include prior information about parameter values in their statement of the problem, I find it possible to pose the problem well, and solve it reliably, without the use of such 79 information (though its inclusion could, in principle, improve my estimates). 3.2.1 Pilot point formulation In the pilot-point method of de Marsily [1978], each medium property is viewed as a spatially correlated random field. A smooth estimate of this field is obtained by kriging of real and fictitious measurements of medium properties at discrete spatial points. De Marsily refers to the locations of fictitious measurements as "pilot points." As I use the method to analyze cross-hole pneumatic test data that do not represent point measurements, I do not include any real but only fictitious (pilot) points in my inverse model. Though I could, in principle, include the single-hole pneumatic test data as real measurements to better condition my inverse solution, I have purposefully not done so in order to explore the ability of my stochastic inverse solver to analyze cross-hole pneumatic test data without the help of single-hole data. A joint analysis of both sets of data would be an important future extension of my work. The location of pilot points is arbitrary and can be assessed either on the basis of site knowledge [de Marsily et al., 1984; Certes and de Marsily, 1991] or by optimization [La Venue and Pickens, 1992]. The unknown values of medium properties at the pilot points are estimated by the inverse algorithm. Let the unknown random property of the medium, z, be a non-homogeneous random field with homogeneous spatial increments (also called in the geostatistical literature "intrinsic" random field of "second order") such that E [z (xp + h) — z (xp)] = 0 var [z (xp + h) — z (xP)1 -= 2-yz (PO (3.20) (3.21) where z is defined at a discrete set of locations xp, called pilot points. The theoretical variogram -y z (111111) depends only on separation distance 11h11 between the pilot points, 80 xp . Existence of mean and variance of increments does not imply their in the case of z. Consistent with the geostatistical analysis of single-hole permeability data, I utilize an omni-directional power variogram -y.(11h11)= allh11 13 (3.22) where a (a > 0) and 0 (0 < < 2) are constants. In some of inverse analyses, I treat a and [3 as unknown parameters to be estimated jointly with log permeability and porosity at the pilot points by the inverse model. A linear kriging estimator at a point x o is defined as (xolz) = > (3.23) (xo, xP) zi; Zi == Z ( Xpi ) i=1 where P is the number of pilot points and A i (x o ) are so-called kriging weights. The latter are computed so as to insure that the estimator is exact (reproduces exactly values at the pilot pints), unbiased and "best" (minimizes the estimation variance). This leads to the following "ordinary kriging equations" for A z [Deutsch and Journel, 1992], 'yz (xpi—xp) i(xo)+/(xo) = - Yz (11xPi — P (3.24) j= 1 >2, A i (x o ) = 1 i=1 (3.25) where i (x 0 ) is a Lagrange multiplier (determined jointly with A i ). Kriging produces a maximum likelihood estimate of z provided that the field is multivariate normal and Y. (IN ) is fully defined. Since I treat the parameters of -y, as unknowns - to be determined jointly with values at the pilot points, my solution is not strictly 81 maximum likelihood even if the estimated variables are multivariate Gaussian. The kriging variance at point x o is given by 6-.2 (x0) = > (xPi xo) (x0) 1 (x0) — 1 (3.26) i=1 I apply these equations independently to log io k and log io ck, ignoring any crosscorrelation between these two variables. The vector of estimated unknown parameters a include discrete log io k and log io q5 values, a', as well as the parameters a and 3 of their power variogram models, a". The kriging estimates in Equation 3.23 are influenced by /3 but not by a. Therefore, a cannot be estimated directly by the inverse model, but it can be evaluated a posteriori from the sample variogram of kriging estimates within the computational domain. 3.2.2 Maximum likelihood approach Define a vector of residuals r (x, t; a) = p* (x, t) — (a) = pt (x, t) — p (x, t; a) (3.27) where p* and p are vectors of measured and simulated system responses (in my case pressures) at points (x, t) in space-time, respectively. In general, both p and r depend in a nonlinear fashion on a. The vectors pt, p, and r have dimensions N equal to the number of pressure observations included in the model (I shall refer to these as match points in Chapters 4 and 5). The vector a has dimension M equal to the number of un- known parameters. The residuals r are affected by observation and simulation errors. According to Carrera and Neuman [19864 observation errors represent differences between field measurements and their unknown "true" counterparts. Simulation errors are differences between simulated and unknown "true" system responses which are due to conceptual and computational errors. Conceptual errors arise from incor- rect description of physicochemical processes and their mathematical representation 82 (mass-balance equations, initial and boundary conditions, equations of state) as well as improper parameterization of medium properties. Computational errors arise from numerical approximations and numerical round-off errors. Both types of error have random and systematic components. A detailed analysis of various error types and their effect on inverse solutions is given by Carrera [1984]. The likelihood function L(alp*) of unknown parameters a given measurements p* and a forward model ‘F is proportional to the conditional probability P (p*Ia) of observing p* given a. The maximum likelihood estimate of a, a, is defined by the maximum of its likelihood function L(alp*), L(alp*). The likelihood concept implies that parameters a are unknown deterministic quantities, and their uncertainty is due to insufficient measurement data p* as well as observation and simulations errors [Carrera and Neuman, 1986a]. If r is characterized by only random, non-systematic components, and if a and r are mutually independent, then L(alp*) = P (P * 1a) = P (r) = P(1 3* — P) (3.28) If P (r) is Gaussian with mean E[r] = 0 and covariance matrix C r ,. = E[rr T ], one obtains 2 1 r T C;r1 r L(alp*) = (27r 1C„1) - 21 exp -- (3.29) Since the logarithm is a monotonically increasing function of its argument, it is convenient to search for the minimum of the function S defined as S = —2 log [L(alp*)] = = r T C;r1 r + log (C„) + log (27r) (3.30) Decomposing the covariance matrix C„ into unknown scaling factor, 3, and known "weight" matrix, V„, such that C r ,. = s r2 V,,, the last equation can be rewritten as 83 rTv-ir = 2" + log (1V„I) N log + log (27r) S The minimizing S is equivalent to minimizing the objective function 0 given by r = (3.31) Sr T v r-rl r (3.32) Since the errors r are assumed to be uncorrelated, the matrix V„ (respectively C„) has a diagonal form and the objective function simplifies to a weighted least-square problem rTwr (3.33) where W is a diagonal weight matrix. The objective function can also be rewritten as 0 = — wij (/):.; —P(x0i,ti; a)) 2(3.34) where Np is the number of observation points with coordinates x o and NT is the number of measurements in time t. The total number of observations N is the product of Np and NT. The diagonal terms of W represent the relative weight of each measurement in the optimization process. If measurement errors (diagonal terms of C„) were known, the diagonal terms of W could be defined as w = 1/4 . 3.2.3 Optimization Overview of optimization algorithms Various automated optimization algorithms can be employed for the minimization of the objective function 0 (a). All optimization algorithms require an initial estimate of the unknown parameter vector a o . They then perturb this estimate by some La so as to bring about a reduction in 0, and continue doing so iteratively until a given 84 convergence criterion is satisfied. Methods differ from each other by the way this process is accomplished. Taylor expansion of the objective function to second order about a current estimate a3 of the unknown parameter vector a at iteration j yields 1 0 (a3+1 ) -= 0 (a3 )+V a 0 (a )•(a3+1 — a 3 )+— (a +1 — a 3 )-V a2 0 ( 2 3 ) (a3+1 — a 3 ) (3.35) where a3+1 is the vector of parameters at iteration j 1. Here Aa = a 3+1 — a 3 , Va 0 (a ) is the gradient vector and V E2, 0 (a 3 ) the Hessian matrix of I. Depending on how many terms are retained in Equation 3.35, the search is either zero-, first-, or second-order. A zero-order search seeks the minimum of 0 without making any assumptions about its topography, which has advantages as well as disadvantages. The downhill simplex and simulated-annealing techniques are two examples [Press et al., 1992]. The former is very easy to program and implement, while the second is very efficient in avoiding local minima. First-order (gradient) search methods require computing of the gradient vector Va 0 (a3 ). The simplest but least efficient is the steepest descent method [e.g. Chavent, 1971]. An algorithm proposed by Powell [1964] searches along conjugate directions without calculating explicitly the derivatives of 0. There are several other conjugate-gradient algorithms, but the one proposed by Fletcher and Reeves [1964] was demonstrated by Carrera [1984] and Carrera and Neuman [1986b] to be the most efficient. Second-order (inverse-Hessian) search methods require evaluating the Hessian matrix 'ÇqL 0 (ai ). Direct computation of the Hessian matrix is not generally practical. Most second-order methods therefore approximate it on the basis of the Jacobian (sensitivity) matrix of derivatives, which consists of simulated pressure responses with respect of the unknown model parameters. Consequently, the computational effort required for each iteration is much larger than that for first-order search methods. It 85 is demonstrated in the literature [Cooley, 1985; Zou et al., 1993; Carrera et al., 1997; Finsterle, 1999] that the overall rate of convergence and computational efficiency of second-order methods are generally higher than those of first-order methods. More efficient and robust optimization algorithms can be developed through a combination of different-order search methods. For example, the Levenberg-Marquardt method, described below, combines first- and second-order search methods. Alcolea et al. [2000] propose an algorithm that jointly uses the Levenberg-Marquardt and simulated-annealing techniques. Levenberg-Marquardt algorithm The Levenberg-Marquardt algorithm is a robust method of nonlinear optimization that combines the inverse-Hessian and steepest-descent methods. Inverse-Hessian is a second-order optimization method which is very efficient close to the minimum. Steepest-descent is a first-order (gradient-based) method which is more robust, albeit inefficient. The algorithm proposed by Levenberg [1944] and later improved by Marquardt [1963] allows smooth transition between the two methods in the process of optimization. It has become almost a standard of nonlinear optimization and has been implemented in most groundwater parameter estimation programs such as MODFLOWP [Hill, 1992], PEST [Doherty et al., 1994], UCODE [Poeter and Hill, 1997] and iTOUGH2 [Finsterle, 1999]. Starting from an initial parameter estimate a o , the algorithm minimizes the objective function iteratively until convergence is achieved. In my case, the objective function 1) is (see Equation 3.33) Np NT Np NT = (Pi*j (1. = r T Wr = tv,jri2 J i=1 j=1 i=1 j=1 P(X0i7 tj; a)) 2 (3.36) Sufficiently close to the minimum, 43, and its gradient can be approximated roughly to second order by 86 , 1 = 0 (a i )d-V a 0 (ai )•(ai+i — ai )-k( — 2 Va 0 (a i — ai )•V a2 0 (ai ) (ai+i — ai ) (3.37) ) = V.0 (ai) • (a 3+1 — ai) V a2 0 (ai) (ai + 1 — ai) (3.38) Close to the minimum, this quadratic approximation of 0 should be satisfactory, and the inverse-Hessian method allows calculating the minimum within a single step by setting Aa =- [Va l. (a)] 1 Va 0 (a) (3.39) where Aa = a 3+1 — a i . However, if the approximation is not satisfactory, an alternative is to use steepest descent which is more robust far from the minimum, La = —cV a 0 (a) (3.40) where c is a vector of unknown positive constants. From Equation 3.36, the gradient of 0 is given by D (a) (a) Np NT ap(x0i, ti; a) = 2 Yjw iip(x0i, t i ; a)) Da jacid j=1 , / = 1, N (3.41) Taking second derivative, 32 0 (a) Np NT wii[ aP(X0i,t jj a) ap(x 0i , t j ; a) act /&tn., i=1 j=1 ,\ a 2 p((xoi, ti; a)1 03:j — p(xoi, tj; a)) p ai a am 1 N,m N (3.42) The second term in Equation 3.42, which consists of second derivatives of p with respect to a, is generally ignored. Close to the minimum, this is justified because 87 (p7 — p(t , a)) is close to zero and second derivatives of p are negligible compared to i first derivatives. Considering second-derivatives would have a destabilizing effect if there are significant measurement errors (outliers) or if the model fails to simulate the system response with a high enough accuracy [Press et al., 1992]. To calculate the second derivatives would also be computationally intensive. It is therefore appropriate to replace Equation 3.42 by a20 ( a ) [ap(x0i, actiaa ni [ act/ t 3 ; ap(xo„ t 3 ; Dam i, ... , ,i ,m = 1, N (3.43) Consider the Jacobian matrix as J = V a r (x, t; a) = —V a p (x, t;a) (i.e., Ji+i , / = ar(c oi ,ti; a) ap(x0i,tj;a) a., a., ) and rewrite Equations 3.41 and 3.42 in matrix notation as Va 0 = —2JWr (3.44) V a2 0 = 2J T WJ (3.45) where J T WJ is a so-called "normal" or curvature matrix. Substituting the last two expressions into Equations 3.39 and 3.40, yields Aa = [J T WJ] J T Wr La = 2cJ T Wr (3.46) (3.47) Marquardt proposed that the vector of unknown constants c in the steepest descent equation (3.40) be related to the diagonal elements of [J T WJ]1 , multiplied by some positive fudge factor a, called the Marquardt parameter. Defining a square diagonal matrix D as 3.48) D= [J T WJ 1J( ii 88 thus yields 1 c = aDTD 2 — (3.49) Marquardt also suggested combining Equations 3.46 and 3.47 into one expression Aa = [J T WJ + aD T D] 1 J T Wr (3.50) Depending on the Marquardt parameter a, the algorithm ranges from inverse-Hessian (a — 0) to steepest descent (a >> I). At each iteration, the algorithm searches for an a that minimizes the objective function. The largest element of aD T D is typically called the Marquardt parameter A. In many problems, the observations and optimized parameters differ significantly in magnitude. The same may happen to the elements of J. To minimize round- off errors, it is advisable to scale the Jacobian matrix J by the diagonal matrix D according to Aa. = [(JD) T W (JD) + aD T D1 1 (JD) T WDr (3.51) When a is large and the algorithm is close to steepest descent, convergence may be accelerated by defining an upgrade vector (Aa, where ( is an adjustment factor given by [Doherty et al., 1994] C= rTDTWJDAa AaTDTJTWJDAa (3.52) 3.2.4 Linearized analysis of estimation errors In the inverse methodology, the analysis of estimation errors is of critical importance. Here I follow the linearized analysis of estimation errors proposed by Carrera and 89 Neuman [1986a]. It assumes that the forward model .F (a) is linear close to the max- imum likelihood parameter estimate, 'a, and parameter estimation errors are multiGaussian. Therefore, the parameter estimation errors are fully characterized by their mean (equal to zero) and covariance matrix —aa E (E ,—aa — E [(a — a) (a — a.)1). The generalization of the Cramér-Rao inequality for the multivariate case demonstrates that E aa is such that E aa — F -1 is semi-positive definite, where F is the Fisher information matrix [SiIvey, 1975]. Therefore, F -1 defines 'a lower bound' for the covariance matrix of estimation errors. In terms of the maximum likelihood function, F is estimated as [SiIvey, 1975] F= 2 1 E [ 82 0a 2 (3.53) where S is defined at the optimum -6... The Fisher information matrix F represents a measure of information about the parameters that is contained in the inverted data, and E aa is a measure of estimation uncertainty. The more information is contained in the data, the less uncertain are the parameter estimates. Ignoring second order terms, the partial derivatives of S with respect to pneumatic parameters are 325 c) (Ory , (Or ) _ aa 2 — , , --e.k., 1 - 2j T c7.7,1 j (3.54) and as a result F = J T C,7,1 J (3.55) The Fisher information matrix F may become singular when some of the parameter estimates are highly correlated. The correlation between estimates will render the corresponding matrix components in J, and respectively in F, to be also correlated. If F is not singular, the covariance matrix of estimation errors E aa can be defined as E aa = F -1 = [J T C 77,1 J] 1 = s, [J T V 777,1 J] -1(3.56) 90 When the scaling factor of covariance matrix .9 72. is unknown, it can be assessed a posteriori, after optimizations have been completed as s2 = r T V„r rTWr (N — M) (N — M) (3.57) The 100(1 —10% confidence region of a is defined as S (a) — S (a) < where N — M) (3.58) (M, N — M) is a quantile of the F-distribution. The shape of the con- fidence region in general is arbitrary, defined by the contour of S on level S (a) + s r2 MF1 _,(M,N — M). Due to our assumption that the model ,F (a) is linear in the vicinity of a (.F (a) (a) (aaS:) (a — a)), the confidence region can be approxi- mated by an ellipsoid defined as (a — a) T E;c,1 (a —)< MFi _„ (M, N — M) (3.59) The linearization produces an overprediction of the size of the confidence region along the longer ellipsoidal axes [Carrera, 1984; Carrera and Neuman, 1986b]. The diago- nal terms of E aa define the variances of estimation errors for respective parameters. The parameter uncertainties can be represented by separate 100(1 — n)% confidence intervals defined as — (N — M) , = 1, M (3.60) where cr i is a square root of the i'th diagonal terms of E aa and t i „ /2 (N — M) is a quantile of Student's t-distribution. When the estimation errors are highly correlated, the variances of estimation errors significantly underestimate parameter uncertainties, and a computation of simultaneous confidence intervals is required [Carrera, 1984; Carrera and Neuman, 1986b; Vecchia and Cooley, 1987]. 91 3.2.5 Calculation of sensitivity matrix Three different methods can be employed to compute the sensitivity, or Jacobian, matrix J required for implementation of the Levenberg-Marquardt algorithm. Partial derivatives in the sensitivity matrix can be replaced by their finite dif- ference approximation. This is easy to implement, but is computationally intensive because it requires a series of forward model runs (M + 1 or 2M + 1 for forward and central difference schemes, respectively, M being the number of unknown parame- ters). However, the model runs are independent and can be conducted in parallel, which allows efficient simulation on multi-processor computers. Most importantly, this method does not require any change in the forward simulator. It is also possible to compute sensitivity coefficients directly by differentiating the original partial differential equation with respect to the unknown parameters. The resultant sensitivity equation is similar to the original equation and therefore a model similar to the forward model can be used to compute the sensitivity coefficients. Additional computational and programming efforts are required to calculate the source term of the sensitivity equation. The method requires M +1 model runs, which again can be easily parallelized. Compared to the finite-difference method, this approach is more computationally intensive, but more accurate. Carrera et al. [1990c] proposed an efficient approach which computes the sensitivity matrix simultaneously with the solution of a nonlinear forward problem. However, this approach requires further programming modifications in the code of the forward simulator. A third option is to use the adjoint-state method proposed by Chavent [1971] and extended by Neuman [1980] to finite-elements. This method is widely used in combination with first-order algorithms because it requires only 2 model runs to compute the gradient of the objective function with respect to the unknown parameters. To compute the sensitivity matrix, the method needs Np + 1 model runs (Np being the number of measurement points), which renders it computationally intensive 92 when Np > M, as is typically the case. The implementation of the adjoint-state method requires additional computational and programming efforts to calculate the parameters of the adjoint-state equation. In the case of our governing equation (3.1), the adjoint-state partial differential equation has the form of a convective-diffusion equation [Sun, 1994], which renders its numerical solution more difficult. 3.2.6 Model identification criteria Carrera and Neuman [1986b] used model identification criteria to differentiate be- tween models having different structures (parameterizations). These criteria support the principle of parsimony in that among alternative models producing similar re- sults, that with the smallest number of unknown parameters is generally ranked the highest. The four criteria, due to Akaike [1974], Akaike [1977], Hannan [1980] and Kashyap [1982], are (in the same order) AIC (â) = S (â) + 2M BIC * = S (a) (3.61) M ln N (3.62) Om (â) = S (a) + 2M ln (ln N) dm (a) = S (a) (3.63) M ln (27r ) + 1n11 1 1 (3.64) where â is a maximum likelihood estimate of the parameter vector a, and S is from Equation 3.31. 93 3.3 Numerical inverse model for ALRS A three-dimensional numerical inverse model was created to interpret single-hole and cross-hole pneumatic tests at the ALRS and to identify the spatial distribution of fractured rock properties. The inverse model is based on an indirect nonlinear approach and incorporates a forward flow simulator coupled to a parameter-estimating algorithm, which are used in conjunction with various parameterization schemes. 3.3.1 Computational domain The three-dimensional computational domain measures 63 m in the x direction, 54 m in the y direction and 45 m in the z direction, encompassing a rock volume of 153,090 m 3 (Figure 3.1). The medium is treated as a uniform or non-uniform continuum that is locally isotropic. Flow through the medium is taken to be single-phase isothermal. Medium properties to be evaluated include air permeability k [m 2 ] and air-filled porosity 0 [m 3 /m 3 ]. 3.3.2 Boundary and initial conditions As I consider only single-phase airflow, the saturation of air and associated pneumatic properties of the rock, remain constant during each simulation. The side and bottom boundaries of the flow model are impermeable to airflow. These boundaries are placed sufficiently far from the injection and observation borehole intervals to have virtually no effect on simulated pneumatic tests. The top boundary coincides with the ground surface and is maintained at a constant and uniform barometric pressure. During the cross-hole pneumatic tests, the absolute barometric pressure at the ALRS was not measured; records exist only for relative variations of barometric pressure. In my inverse simulations, I assumed that the barometric pressure is equal to 100 kPa, which is the average barometric pressure at mean sea level. In fact, the ALRS is at an elevation of 1,200 m above mean sea level, and the average barometric pressure 94 o -10 N -20 3 -30 -40 FIGURE 3.1. Computational domain and explicitly simulated boreholes. 95 should be approximately 87 kPa, but can vary with meteorological conditions. I performed inverse analyses on the influence of this barometric pressure discrepancy on the parameter estimates. Barometric pressure fluctuated during each pneumatic test; in some of these tests (e.g. cross-hole test PP4) the fluctuations were significant enough to influence substantially air pressure responses in some of the monitoring intervals. For the moment, barometric pressure fluctuations have not been simulated. It could be that had I incorporated these fluctuations in the inverse model, my estimates would have been somewhat different (especially in the region close to ground surface). Initial air pressure was set equal to the barometric pressure throughout the flow domain. 3.3.3 Borehole effects Due to the high compressibility of air, and the non-uniform nature of the rock, I expected both borehole storage and conductance to have an important impact on airflow and storage in the modeled region. The total volume of open boreholes is 4.1 m 3 , which is small in comparison to the rock volume between the boreholes, approximately 10 x 10 x 10 (1,000) m 3 , within which most of airflow takes place. Nevertheless, its effect is important. Borehole effects, especially conductance, are often ignored in the interpretation of field tests. There is little information in the literature about the effect that open borehole intervals may have on pressure propagation and response during such tests. Paillet [1993] noted that the drilling of an additional observation borehole had an effect on drawdowns during an aquifer test [see also Pickens et al., 1987; Carrera et al., 1996]. In Illman et al. [1998] and Illman and Vesselinov [1998], I demonstrated through numerical simulations that the presence of open borehole intervals has a considerable impact not only on pressure propagation through the site, but also on measured pressure responses within monitoring borehole intervals during cross-hole pneumatic tests. Similar results were later presented by Huang et al. 96 [1999]. My numerical model accounts directly for conductive and storage effects of all open borehole intervals by treating them as high-permeability and high-porosity cylinders of finite length and radius. How exactly the boreholes are incorporated in the model is described in the next section. 3.3.4 Computational grid The grid generator X3D [Cherry et al., 1996; Trease et al., 1996] was adopted to automatically subdivide the computational domain into tetrahedral elements. Grids were created in a manner that enhances the computational efficiency of the FEHM simulator, described below. One of the computational grids employed in my study is illustrated, by means of two-dimensional images, in Figures 3.2 and 3.3. Figure 3.2 shows three views of the grid perpendicular to the x-y, x-z and y-z planes. As the grid in the vicinity of boreholes is relatively fine, the corresponding areas appear dark in the figures. Figure 3.3 shows four cross-sectional views of the grid along vertical planes that contain selected boreholes. Since the grid is three-dimensional, its intersections with these planes do not necessarily occur along nodal points (i.e., what may appear as nodes in the figure need not be such). The grid shown in Figures 3.2 and 3.3 is employed for inverse analyses of both single-hole and cross-hole pneumatic tests performed by injecting air into borehole Y2. Other grids used in my study are very similar. All of them can be divided into three zones (Figures 3.2 and 3.3): (1) a regular grid at the center of the modeled area, which measures 30 x 20 x 25 (15,000) m 3 and has a node spacing of 1 x 1 x 1 m 3 ; (2) a surrounding regular grid haying a node spacing of 3 x 3 x 3 m 3 ; and (3) a much finer and more complex unstructured grid surrounding each borehole. In the last mentioned zone, different discretizations are applied around injection and monitoring boreholes. As a result, different grids are adopted for the simulation of air injection into different boreholes. I have interpreted pneumatic tests performed in 97 0A0A0.40n0n [email protected] orOMMOCIDOODOO co0AL.0„u0J040 141414 4141 h 1410414 0, 1 100 Aosge4e,TuTotwsroarAtio ••t•• • • -4A1 , o COCO [email protected]@ ' • i.:13 104 041 . L n11 . 1nnnnnnnAIDA6,1 . _4a0n __ A.. A. A. &VP iogooroa n 5 1 1414 OF „ •41035h2.1 141414 410 - r - 210L0200 4F 1004 4141 141410 141414 410 *00* 1111,1111,11. o 0 7 V? 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Joidaislolol ,.. .',:inrancqp2,Imagigi t.a.m.v.,...aa ...... EgIn's_N AlDmvmmomn .. .11 N''% "";r 3 val.:Mani wool leai •-attar .i?" r" f ',-.47*""' s•gp *** ' . 43' gomegmE .. . EMPARMI-MMIINEE ,rIMEMEMEMEMISE. ragigragiMallir 1 21 21 APPPISiMieSin 91 61 6 1611 00 409 1 1211102M 6 010 1 0§01 1 Waterline AMAMBO-OR - - o L.P 6 IMPE 6 ,10 ISE. mammal Cu o [w] z 99 4 different boreholes (Y2, X2, Z3 and W3) and, for each case, I have used a different computational grid. In each grid, the number of nodes and tetrahedral elements are respectively: Y2 — 39,264 and 228,035; X2 — 39,288 and 227,961; Z3 — 39,797 and 231,366; W3 — 43,812 and 254,310. The number of nodes and tetrahedral elements is largest in the case of injection borehole W3 because it is longer than the other injection holes. Node spacing at the center of the computational domain, where most of airflow takes place, is equivalent to the 1-m support scale of the majority of single-hole tests (Chapter 2). A 5-m long segment of the three-dimensional computational grid along an injection borehole is shown in Figure 3.4. This grid is wider and finer than those associated with monitoring boreholes so as to allow accurate resolution of the relatively highpressure gradients that develop around the former. However, the grids around both injection and monitoring boreholes have similar structures. Along the central axis of each borehole, nodes are spaced 0.5 m apart. The surrounding grid is designed so that the sum of computational volumes associated with these nodes is close to the actual volume of the borehole. Additional nodes are located along radii that are perpendicular to this axis within each borehole grid. The intervals between these nodes grow sequentially with distance from the borehole axis by a factor of 1.6, thus forming a geometric series. The number of rays and nodes associated with the injection borehole are larger than those associated with monitoring boreholes. Each borehole grid is additionally refined near the ground surface so as to obtain an accurate resolution of conditions near this atmospheric boundary. Where boreholes are located close to each other (as in the cases of V2 and the W-series of boreholes; W2A and W2; W1 and Yl; W3 and Y3), the grid between them is made finer in order to resolve correctly processes that take place within this grid volume. The most complicated of the grid structures is that representing the region between boreholes V2 and W2, which intersect each other (Figure 3.3). To simulate the effect of open borehole intervals on pressure propagation, these [w z ] I I f. .frlin.:ea- -n-4" .w L. . - -n.- me- - - VIn.-.nI L . . nn I i i P r- 1 n : -.n IN i N . - - 4PP:11 'Alln-•T ' 7!41n7•7 ' 7- 1W E.z-,. ;. - Ew-.,oviav.: n:170......7,E.,......„_;n,;17,..-1".....z.-.;-z-z.-z - .7.I.K.2..... -7- . n711`n 1n111!..,•& nnn:_l i -ii n:WI o W/ MI nWATo 1 PZ/, A -nMa 11 n 1 - III . .1 IP ur nn, M ItIEMPn --"IIMMIIn --"IIIMI -- [ w] A o AIMILy. nn-svallW rti ni-aqi !Am. 40 n 4 401, nnlf k3.0 n ''n40VOP''' 4 , 1.6 - [w] z 111k1 101 intervals are treated as porous media having much higher permeability and porosity than the surrounding rock. The permeability and porosity of nodes along the open borehole intervals are set to 3.23 x 10 m 2 and 1.0 m 3 /m 3 , respectively; these correspond to an ideal tube with radius equal to that of a typical borehole. The permeability and porosity of instrumented borehole intervals are set to 3.23 x 10 -5 m 2 and 0.5 m 3 /m 3 , respectively. At the intersection of boreholes V2 and W2, a lower permeability of 10 -1 0 m 2 is assigned to avoid numerical difficulties; this value is still orders of magnitude higher than that of the surrounding rock. Packers are assigned zero permeability and a porosity of 10 -5 m 3 /m 3 . Figures 3.5 and 3.6 show examples of permeability and air-filled porosity distributions within the computational domain. In the figures, open borehole intervals are shown to be of high permeability/porosity, and packers of low permeability/porosity. The spatial distributions of log lo k and log lo 0 in the figures are those shown earlier in Figures 2.18 (page 58) and 2.26 (page 66), respectively. In most inverse runs, the values of permeability and porosity along boreholes were kept constant as specified above. In a few instances, I allowed the inverse model to optimize the porosity of either the injection interval or both the injection and the observation intervals. I also ran test cases in which the borehole effects were excluded by specifying values of permeability and porosity along boreholes equal to those of the surrounding rock. The dimensionless borehole storage coefficient CD is directly related to the porosity 0„, along the borehole intervals. In the case of three-dimensional airflow from a point source, CD is defined [Inman et al., 1998] as CD = C, p ll (3.65) where C„, = Vs Ca [m 5 /N] is storage coefficient of the injection interval, Vs [m 3 ] is effective storage volume, Ca [m 2 /N] is air compressibility (as discussed in Section cv 102 In (-V [ w] z in co ri in •zi Lc, Lc) Lri (z) cc> N- 103 Ca) 0,1 mom N .._ E u x CD CO cr N (0 CO 104 3.1.1, equal approximately to 1/p where p is absolute air pressure), Vu, [m 3 ] is nominal volume of the injection interval and q,. is effective air-filled porosity of the surrounding rock medium (for uniform medium, o r is simply equal to the air-filled porosity Since I define the effective storage volume Vs as 0). Vu,0„„ the above equation takes the form CD Vs Ow =3 O r 30, (3.66) The effective porosity of the borehole interval, çb w , is allowed to take on values in excess of 1.0, as a way of accommodating effective interval volumes that are larger than those built into the model. In this way, Ow can account for air storage not only in the injection interval, but also in the surrounding rock. The net result is a complex three-dimensional grid which represents quite accurately the geometry, flow properties, and storage capabilities of vertical and inclined boreholes at the ALRS; is capable of resolving medium heterogeneity on a support scale of 1 m across the site; is able to represent, with a high degree of resolution, steep gradients around the injection test interval, as well as pressure interference between boreholes, no matter how closely spaced; and assures smooth transition between fine borehole grids having radial structures and surrounding coarser grids having regular structures. Such complex three-dimensional grids incorporating boreholes with various spatial orientations have not been previously applied in the hydrogeological research [see also Trease et al., 1996]. For example, Carrera et al. [1996] simulated boreholes on a much coarser computational grid utilizing one-dimensional elements interconnecting nodes located along the boreholes. A fine grid around boreholes is required to accurately represent flow from and into both injection and observation intervals. 105 3.3.5 Program FEHM I simulated the pneumatic test data by means of a three-dimensional finite-element / finite-volume heat- and mass-transfer code, FEHM version 96-05-07, developed at Los Alamos National Laboratory, New Mexico [Zyvoloski et al., 1988, 1996, 1997]. My decision to use FEHM was based in part on the ability of this code to simulate nonisothermal two-phase flow of air and water in dual porosity and/or dual permeability continua, and to account for discrete fractures, should the need to do so arise (it never did). The code is employed to simulate three-dimensional isothermal gas flow defined by Equations 3.1 - 3.4. The problem is nonlinear due to dependence of air compressibility, Ca , and air viscosity, ,a, on absolute air pressure, p. Variations in air viscosity are ignored in FEHM, which is justified for the ranges of pressure and temperature recorded during the pneumatic tests. Air dynamic viscosity ti, is set equal to 1.82 x 10 -8 Pa • s [Zyvoloski et al., 1997]. Spatial discretization of the governing equations is done on a three-dimensional computational grid of tetrahedral elements. Since the medium is assumed to be isotropic, the code is applied using its finite-volume (integrated finite-difference) option which is computationally more efficient than the finite-elements. The volume associated with each node is defined using three-dimensional Voronoï diagrams based on Delaunay tessellation [Watson, 1981]. Model parameters, here air permeability k and air-filled porosity 0, are defined at the nodes of the grid, not over the elements. Parameters are viewed as averages over control volumes associated with respective nodes. Modifying Equation 3.1 by replacing spatial and time derivatives with their finite-difference approximations and multiplying by the finite volume V3 , one obtains an equation representing local mass balance for node j at time step s, 106 wk i=1 ILL (Pis) 2 Szi qiVi = (PI p;s ) P:Ali ps3 Si; + kjg 1LL p s.-1 3 At, V (3.67) Here superscripts and subscripts define time step and node, respectively (e.g. ps3 -1 represents air pressure calculated at node j at time step s — 1), W is the number of nodes in the neighborhood of node j, ki 3 is the upgradient air permeability between nodes i and j, Al i 3 and Si j are respectively the distance and effective flow area between nodes i and j, Sz3 is the effective area for gravity flow associated with node j, and At s is the size of the s'th time step. Parameters W, V3 , A/ ii , Si3 and Sz , are obtained from the Voronoi: grid [Zyvoloski et al., 1997]. The time derivative is discretized using a fully implicit backward difference scheme, which is unconditionally stable and first-order accurate in time; for nonlinear problems, it is generally preferred to the higher-order central-difference scheme. Rearranging the mass balance equation (3.67) and taking into account initial conditions, boundary conditions and equation of state 3.4, a system of nonlinear algebraic equations is obtained for time step s [Zyvoloski et al., 1997] A(Ps)Ps = b(p5) (3.68) where A is a so-called "conductance" matrix and b is a vector representing boundary conditions and internal sources. Both A and b depend in a nonlinear fashion on air pressure p at the nodes. In the most general case, A is a square matrix of size U x U, and p and b are vectors of size U, where U is the number of nodes used to discretize the computational domain. The system of nonlinear algebraic equations is solved by means of the Newton-Raphson iterative method. Let d(Ps) = A(Ps)Ps — b(Ps (3.69) be a vector of residuals. The residuals are to be minimized iteratively. For the m'th iteration, the last equation can be rewritten as 107 dm(P s ) = Am(P s )Pis-ri — bni(P s ) (3.70) The residual for the next, m + l'st iteration is estimated to first order with respect to d m (ps) through clin+i( ps) = dm(Ps) + (vpd(P 5 )) A PL+ 1 (3.71) where Ap m s +1 = p m ' +1 — p m s and (V p d(ps)) m is a Jacobian square matrix of size Ux U, representing the partial derivatives of residuals d with respect to the current estimate of unknown pressures p, (Vpd(P s )) — adm(Ps) Op (3.72) Assuming that d m+i = 0, Equation 3.71 is solved for Ap m+1 by inverting (V p d(ps)), = [(Vpd(138 ))] 1 €1 .(P s ) (3.73) In general, the Jacobian matrix (V p d) m is neither symmetric nor sparse. In FEHM, successive iterations are performed until the L2 norm of residuals d(ps) becomes less than a prescribed tolerance. The code uses the algorithm GZSOLVE [Zyvoloski and Robinson, 1995] to provide a robust implementation of the Newton-Raphson sparse systems of nonlinear equations. Based on the convergence of the Newton-Raphson method, FEHM employs a semi-automatic time-step control. If convergence is not achieved within a given number of iterations, the time step is reduced by a certain factor and the procedure repeated. If convergence is attained with a specified number of iterations, the next time step is incremented by a certain factor. The user also specifies limits for the minimum required and maximum allowed number of iterations in each time step. 108 3.3.6 Parameterization The interpretation of pneumatic tests is performed assuming that the medium is either uniform or non-uniform. In the first case, the unknown parameters are uniform values of air permeability k and air-filled porosity 0 throughout the computational domain. It provides information about medium properties on the scale of the cross-hole test, and allows comparing the numerical model with analytical type-curve methods for the interpretation of both single-hole and cross-hole tests. For the simultaneous interpretation of cross-hole tests, I treat the medium as a non-uniform stochastic continuum. In some cases, I do so for air permeability while keeping air-filled porosity uniform, in other cases, for both air permeability and airfilled porosity. I have also explored some other approaches to characterize medium non-uniformity. The geostatistical analysis in Chapter 2 shows that fractured rock properties at the ALRS can be viewed as spatially correlated (regionalized) random variables, and the rock can be represented as a stochastic continuum. Based on this finding, I define spatial variations in the unknown medium properties using a stochastic inverse model based on the pilot-point method. Pilot points are located within the tested region where the monitoring intervals are concentrated. Most are located along boreholes; some are between borehole in- tervals. The number of pilot points varies between the inverse runs from 32 to 72. At each pilot point, there is an unknown log-transformed air permeability and air-filled porosity. Admissible ranges for the estimates are set between —20 and —10 for log lo k and between —5 and —0.5 for log io 0. The inverse model provides covariances and confidence intervals for the corresponding estimation errors. I treat the variogram parameters as unknowns, estimating the exponent 0 either separately or simultaneously with unknown medium properties at the pilot points. In the simultaneous case, I assign an upper limit of 1 on the values of 0, in order to 109 avoid convergence problems in the forward problem. The constant a of the power variogram model does not influence the kriging estimates, and therefore cannot be evaluated directly by the inverse model. However, a can be evaluated a posteriori from the sample variogram of the kriging estimates obtained by the inverse model. In my inverse model, kriging is performed by means of the geostatistical code GSTAT Pebesma and Wesseling [1998]. GSTAT is preferred to other kriging programs such as GSLIB [Deutsch and Journel, 1992] due to its simple input and output, easy batch-mode processing, option to simulate over unstructured grids, and (most importantly) ability to perform global kriging which in my case, is computationally much more efficient than employing a search neighborhood. For example, over a given structured grid with given input data, GSLIB requires 6 hours and GSTAT 2 minutes to compute the kriging field. 3.3.7 Program PEST A detailed description of the PEST code is given in its manual [Doherty et al., 1994]. Here I discuss only major features of the code, which are required to run the numerical inverse model. The code uses a variant of the Levenberg-Marquardt algorithm to estimate model parameters a by minimizing .1) (a). For each unknown parameter, one must specify an initial value and a weight. PEST allows the parameters to be internally (1) scaled and offset, (2) log transformed, (3) tied to another parameter, or (4) fixed. PEST also requires information about an initial value of the Marquardt parameter a and a series of parameters defining the optimization of (1. (a) for each iteration. There are two ways to limit the variation of optimized parameters in PEST. The first one is to define upper and lower bounds for each parameter. The second is to use prior information with weights equal to the inverse variances of prior estimation errors. In my case, prior kriging errors are mutually correlated, a situation that PEST 110 cannot handle. This is one reason why I do not rely on prior information. PEST allows limiting the amount by which a parameter is accepted to change during any iteration. This helps the optimization of highly nonlinear models. Two types of limits can be employed: factor and relative. Let a i be an unknown parameter and a io its current "best" estimate. In the factor-limited case, the plausible range of a i for the next iteration is given by ai E [ fi aiofil if ai o > 0 , azoi ., a, E [ azoji, — fi if aio < 0 (3.74) where fi is the maximum allowed factor change (fi > 1). Factor-limited parameters can not change sign. In the case of relative limits, the plausible range of a i is (3.75) where r i is the maximum allowed factor change (r i > 0). It is not recommended to use r i > 1, which allows parameters to change substantially, including a change in sign or becoming zero. The magnitude of parameter upgrade vector is defined such as neither of plausible parameter ranges is violated. PEST uses a technique proposed by Cooley [1983] to reduce parameter over- adjustment and to damper potential oscillations in parameter changes. Let Aa i be the maximum change for the i'th parameter within the current iteration. In the previous iteration, the actual parameter upgrade was associated with a maximum change Aa 03 for the j'th parameter. The new parameter upgrade vector is defined as •Aa, where 6 is such that 6 _ 4i : I ii ff s:— { 33, - -FF S where 11 (3.76) 111 s = z= j Aao, if i j 0 (3.77) Accurate computation of derivatives of observations with respect to parameters is critical for success of the Levenberg-Marquardt algorithm. PEST calculated derivatives directly using a finite-difference method, which requires a series of simulation runs. For each optimized parameter, it employs either a forward or a central difference scheme. For given group of parameters, PEST can switch from forward to central differences during the optimization when the relative reduction in the objective function between two successive iterations is less than a given value. In the case of the central scheme, PEST can calculate the derivatives using three different approaches. The most accurate approach is to fit a parabola to three parameter-observation pairs and then compute its derivative. Another approach is to fit a straight line instead of a parabola, which is recommended for cases when simulated observations can not be calculated with good precision. The third possibility is to use standard central differences, which ignores the middle parameter-observation pair. For each group of parameters, increments for derivative calculations can be defined (1) absolutely, (2) in relation to current parameter values, (3) in relation to the current maximum values for the parameters. To terminate the optimization process, PEST utilizes 7 criteria: 1. The objective function 0 becomes zero. 2. A given number of iterations has elapsed since the lowest objective functions 0 was achieved. 3. A given number of local minima for 0 are within a specified distance in param- eter space of each other relative to the lowest 0. 4. The gradient of the objective functions 0 with respect to all parameters is zero. 112 5. The upgrade vector components for each parameter is zero. 6. The maximum relative change of parameters within a given number of iterations is less than a specified tolerance. 7. The number of iterations exceeds a specified limit. After termination of the optimization process, if the normal matrix is non-singular, PEST computes the covariance matrix of optimized parameters. It defines the sepa- rate 95% confidence intervals of optimized parameters as discussed in Section 3.2.4. It also calculates the correlation matrix and performs eigenvector analysis of the covariance matrix. Since the covariance matrix is positive definite, eigenvalues are real and eigenvectors are mutually orthogonal, representing the axes of an M-dimensional probability ellipsoid (M being the number of optimized parameters). As explained earlier, the error analysis assumes that (a) the measurements are mutually uncorrelated, (b) the estimation errors are Gaussian, and (c) the model is linear. In my case, none of these assumptions are expected to be fulfilled and, therefore, I consider the corresponding statistics merely as crude approximations. I have modified PEST to calculate the four model identification criteria discussed in Section 3.2.6. 3.3.8 Parallel processing Enhanced computational efficiency can be achieved by parallelizing the evaluation of the Jacobian matrix J. Doherty [1997] created a parallel UNIX version of PEST. I have modified this parallel version so as to utilize more optimally the computational resources of a standard UNIX multi-processor environment. I have further altered PEST to allow efficient restarting of the optimization process, if and when it ter- minates prematurely, so as to virtually eliminate loss of computational time. The 113 parallelized version of the inverse model has been run on the University of Arizona SGI Origin 2000 32-processor cluster. 3.3.9 Code interaction My numerical inverse model incorporates FEHM, PEST and GSTAT. In addition, I use X3D to generate the computational grids, GMV [Ortega, 1995] to visually analyze them, and TECP LOT [Amtec, 1997] to present results graphically. I have supplemented these codes with a series of data-processing programs, which facilitate the generating, reformatting, handling, analysis and visualization of massive input and output data files. PEST is designed to interact with any kind of external simulator. Its features allow not only easy generation of one or a number of input files (which include the current values of optimized parameters) but also elaborate reading of one or a number of output files (which contain the simulation results for defined parameters). Since my inverse model is rather large and the corresponding FEHM input and output files are huge, these very nice features of PEST could not be fully implemented. Hence, additional pre- and post-processing codes for FEHM inputs and outputs were developed to allow direct and e ffi cient (in terms of disk-space and computer-time) interfacing of PEST, GSTAT and FEHM. 3.4 Linearization of governing equations The numerical inverse model solves the equations governing gas flow in their original nonlinear form (Equations 3.1 — 3.4, page 71). However, the analytical type-curve results of [Illman et al., 1998], Inman and Neuman [2000b] and Illman and Neuman [2000a] with which I compare some of my numerical inverse estimates are obtained using a linearized version of the governing equations. Ignoring the gravity and source 114 terms, the so called p and p 2 based linear representations of (3.1) for spatially uniform medium are defined as [Inman et al., 1998] • p based: V 2p 1.10Ca Op kp at (3.78) POCa 3p 2 kp at (3.79) • p 2 based: v2,n2 where air compressibility C a is equal to h 5 (see Section 3.1.1) and 19 is some constant - - absolute air pressure; Illman et al. [1998] took p to be the barometric pressure. During - single-hole and cross-hole pneumatic tests, pressure buildups not only were significant relative to the barometric pressure but also varied substantially in time and space, and therefore, so did air compressibility. The assumption that air compressibility is constant should impact the analytical identification of medium flow and storage properties as well as the storage properties of the packed-off intervals. 115 Chapter 4 INTERPRETATION OF SINGLE-HOLE TESTS As discussed in Chapter 2, Guzman et al. [1994, 1996] and Guzman and Neuman [1996] conducted a series of multiple-step single-hole pneumatic tests at the ALRS. Guzman et al. [1996] applied steady-state formulae for single-phase airflow in a uni- form, isotropic porous continuum to interpret late data from each step of an injection test. Illman et al. [1998] and Illman and Neuman [2000b] employed transient type- curves for single-phase airflow in a similar continuum to analyze all data from the first step of several injection tests as well as some later steps. Their analysis did not - allow reliable identification of air-filled porosity and borehole storage coefficient. Here I describe inverse analyses of four single-hole tests conducted by Guzman and Neuman [1996] (Tables 4.1 and 4.2). In each case, I present a separate analysis of pressure data from the first step of the test (labeled A) and of data from the entire test, including multiple injection steps (labeled A, B, C, D) and recovery (labeled R). My analyses of the first step consider three cases: (1) no open borehole intervals, (2) an open injection interval, and (3) open intervals in all boreholes. In the first case, I ignore open borehole effects; permeability and porosity are uniform over the entire computational grid (including nodes along boreholes). In the second case, I consider the effect of an open injection interval by assigning to it high permeability and high porosity values. In the last case, the effects of all open borehole intervals are considered by treating them as high-permeability and high-porosity porous cylinders. To analyze the single-hole tests numerically, a set of match points is defined for each injection step and recovery. The match points are distributed more or less evenly along a log-transformed time axis. On the average, there are 10 match points per injection step. These are assigned unit weights w i when analyzing pressure data from 116 4.1. Single-hole pneumatic tests analyzed by inverse modeling [after Guzman et al., 1996]. TABLE JG0921 JGC0609 JHB0612 JJA0616 TABLE [m] Distance from top of Y2 to center of injection interval [m] 2.0 1.0 1.0 1.0 16.10 13.85 15.81 17.77 Length of injection interval Test 4.2. Single-hole test data analyzed by inverse modeling [after Guzman et al., 1996]. Test JG0921 Injection step A B R JGC0609 A B C D R JHB0612 A B C R JJA0616 A B C R Duration [min] 66.0 63.05 11.12 144.9 161.1 220.9 195.1 286.0 109.65 100.05 95.25 62.05 91.9 132.1 100.0 726.0 Injection rate Injection rate [cm3Imin] 400.7 1983.6 0.0 499.2 999.8 1501.0 1801.2 0.0 502.3 1201.1 1951.7 0.0 300.3 800.8 1301.0 0.0 [kg I s] 8.014 x 10 -6 3.967 x 10 -5 0.0 9.984 x 10 -6 2.000 x 10 -5 3.002 x 10 -5 3.602 x 10 -5 0.0 1.005 x 10 -5 2.402 x 10 -5 3.903 x 10 -5 0.0 6.001 x 10 -6 1.602 x 10 -5 2.602 x 10 -5 0.0 117 the first step of a test. When analyzing pressure data from all steps simultaneously, the weights w i are made inversely proportional to the observed pressure value p:. Applying weights inversely proportional to the observed pressures is similar to defining the objective function as the sum of squared differences between the logarithms of observed measurements 137 and simulated pressures pi [Doherty et al., 1994]. This type of objective function allows better representation of weak pressure responses that characterize early-time pressure buildup and recovery. In my inverse analysis of single-hole tests, the medium is assumed to be uniform. The vector a consists of either two unknown parameters, air permeability k and air- filled porosity 0, or three parameters k, 0 and the effective borehole porosity Ow of the injection interval, as defined in Chapter 3, Equations 3.65 and 3.66, page 101. 118 4.1 Single-hole test JG0921 Parameters estimated for test JG0921 by both analytical and numerical approaches are summarized in Table 4.3. In the table, and through the rest of the text, the + range represents separate 95% confidence intervals identified by my numerical inverse model. Steady-state interpretation of single-hole test JG0921A by means of an analytical formula gives a pneumatic permeability of 2.8 x 10- 14 m2 [Guzman et al., 1996], and transient type-curve analysis based on the spherical flow model gives 2.6 x 10 -14 M 2 [Inman et al., 1998]; neither of these two analyses have yielded air-filled porosity estimates. The type-curve interpretation by Illman et al. [1998] did not produce satisfactory match of the measured pressure responses. The matches between pressure values computed by the numerical inverse model, and measured values from the first injection step of the test, are depicted in Figure 4.1. When open borehole intervals (including that used for injection) are not considered in the simulation, my numerical inverse model yields a match that is not entirely satisfactory (Figure 4.1), with air permeability k = 2.3 x 10 -14 + 2.6 x 10 -16 m 2 and air-filled porosity 0 = 4.5 x 10 -1 + 1.9 x 10 -3 . The porosity estimate appears to be too high for fractures. When only steady-state pressure data are included in the inverse analysis, the match at early time is poor, yielding k = 2.8 x 10- 14 m2 and 0 = 4.6 x 10 -3 , respectively. Here the estimate of porosity is based entirely on the time at which steady state commences, which I have specified to be 0.008 days, as indicated by open circles in Figure 4.1. The model is therefore quite insensitive to 0 and fails to yield a finite confidence interval for either parameter since the normal matrix becomes singular. When the effect of all open borehole intervals is included in the analysis, and the effective porosity of the injection interval is allowed to vary simultaneously with k and 0, the match improves significantly, yielding k = 2.2 x 10 -14 + 4.4 x 10 -16 m 2 , 0 = 6.7 x 10 -3 1 4.7 x 10 -3 and 0„ = 7.0 x 10 -1 + 6.7 x 10 -2 . The large confidence interval associated with 0 (of the same order as its estimate) shows low sensitivity to 119 TABLE Type of analysis Analytical steady-state (A) Analytical transient - Spherical flow (A) Inverse modeling - No open intervals (A) 4.3. Parameter estimates for test JG0921. k [m 2 ] 2.8 x 10 -14 95 [m3/m3] Ow [77/3 /m3 ] CD 4.5 x 10 -1 + 1.9 x 10 -3 4.6 x 10 -3 6.7 x 10 -3 ± 4.7 x 10 -3 1.4 x 10 -2 ± 1.7 x 10 -3 7.0 x 10 -1 ± 6.7 x 10 -3 8.0 x 10 -1 + 4.6 x 10 -2 3.5 x 10 1 [-] 2.6 x 10 -14 2.3x 10 -14 ± 2.6 x 10 -16 - No open intervals (A; steady-state) 2.8 x 10 -14 - All intervals open (A) 2.2 x 10 -14 ± 4.4 x 10 -16 - All intervals open (A,B,R)2.4x 10 -14 ± 7.1 x 10 -16 1.9 x 10 1 this parameter and indicates that its estimate is highly uncertain. It reflects the fact that transient data is influenced by borehole storage effects. Eigenvalues and eigenvectors of the covariance matrix of the parameter estimates are listed in Table 4.4. The first eigenvector is dominated entirely by permeability, indicating that this parameter is very well defined. The second and third eigenvectors show correlation between 0 and Ow , rendering both estimates less certain. As discussed in Section 3.4, the analytical type-curves of [Illman et al., 1998] considered air compressibility to be a constant. However, the increase of absolute air pressure at the injection interval decreases the interval storage as shown by Equation 3.9 (page 73). Figure 4.2 depicts relative changes in storage during injection steps and recovery due to changes in air compressibility, plotted versus time measured relative to the end of the preceding step. Since the relative pressure buildups during injection steps were not significant compared to initial air pressure, changes in borehole storage were not substantial. However, the effect of air compressibility on borehole storage appears to be important for obtaining good matches between observed and numerically simulated pressures during the first injection step. Figure 4.3 depicts pressure during each step and recovery, relative to that established during the preceding step, plotted versus time measured relative to the end of the preceding step. We see a very distinct unit slope during the first step (labeled A) 120 4.4. Eigenanalysis of covariance matrix of estimation errors obtained from interpretation of the first step of test JG0921. TABLE Parameters k [m 2 ] [m3 /m3 ] [m3 /m 3 ] Eigenvalues Eigenvectors 1.000 3.0424 x 10 -14 5.7346 x 10 -14 1.5993 x 10 -32 5.4264 x 10 -14 —0.8766 —0.4812 2.2001 x 10 -6 —3.5632 x 10 -14 —0.4812 0.8766 1.0564 x 10 -5 TABLE 4.5. Eigenanalysis of covariance matrix of estimation errors obtained from interpretation of all the injection steps and recovery of test J 00921. Parameters k [m 2 ] q5 [770 Im 3 ] [m 3 /m3 ] Eigenvalues Eigenvectors 1.000 —2.3431 x 10 -13 —2.7675 x 10 -15 8.2641 x 10 -32 —2.3433 x 10 -13 —1.000 —6.1345 x 10 -3 6.6215 x 10 -7 1.3300 x 10 -15 —6.1345 x 10 -3 1.000 5.1651 x 10 -4 of the test. To the extent that storage affects the other two pressure records in the figure, this effect is not clearly discernible. Air storage in the injection interval has theoretically no effect on recovery data (see Section 3.1.1), which should therefore be ideal for the estimation of air-filled porosity and borehole storage coeffi cient. I therefore expect a simultaneous analysis of pressure data from the entire test to yield a more reliable estimate of parameters than is possible based only on data from the first step. A reasonably good fit of my model (which now accounts for all open borehole intervals) to the entire two-step pressure buildup and recovery record is shown in Figure 4.4. The corresponding parameter estimates are k = 2.4 x 10 -14 + 7.1 x 10 -16 m2 , o „:„ 1.4 x 10 -2 + 1.7 x 10 -3 and q5„, = 8.0 x 10 -1 + 4.6 x 10 -2 . Table 4.5 shows that each eigenvector of the corresponding estimation covariance matrix is now dominating almost entirely by one parameter, showing lack of correlation between the estimates and suggesting that they are of high quality. 121 Ts co SCI• co 32 G) U) C— .0 4E1 4-1 CCI -° 0 fa C C .13 . CD a) • .• 2 o o .c 12 o o — 0 Z Z •0 1." o 111 1 1 CO (0• itt• Ii r - I Ii CO iji I Ii Ii CO T [ecP1] d I • 122 Q) c73 4 CID (3) 0) r-4 Ca) cn Cs.1 c7). ra) LLi 4-4 f24-i 0 • • 13 Cd 0 • • Cd 03.) > LLI :17's (1) 86 i$3 , 0 -4, (1.) • cn u 71.) c a.) L. f0 (L) cll blD cd 0 o • Cl) (1) O .4_> o - C.) C-1-1 0 LJJ • > o : o (3.) cc ca (1, g 4, 0 , a II aid • I a a o o co 1,- co in Tr 6 6 6 6 6 6 cn ai cn cri a6eicns alotiaioq u! °Bump amiele LU d id al f I co co cn O 7r. '7a ;-4 123 o o < aC 0<01 tr.! c!) CZ+ [edl] ainsseid angelau 124 125 4.2 Single-hole test JGC0609 Results of analyzing pressure data from test JGC0609 by various methods are listed in Tables 4.3. Steady-state analysis gives a pneumatic permeability of k = 2.0 x 10 -15 m 2 [Guzman et al., 1996], and transient type-curve analysis with a spherical flow model yields k = 2.9x 10 -15 m 2 [Illman et al., 1998]. As for the previous test, the type-curves could not match properly the entire pressure record [Illman et al., 1998]. Numerical inverse results are compared with measured values in Figure 4.5. In the absence of open borehole intervals, the inverse model yields a poor fit with k = 1.8 x 10 -15 +3.9 x 10 -15 m 2 and 0 = 5.0>< 10 -1 +4.2 x 10 -2 ; upon including the injection interval, the fit improves dramatically to yield k = 1.6 x 10 -15 + 2.6 x 10' 7 m 2 , q5 = 4.8 x 10 -3 +9.4 x 10 -3 and 0„, = 1.3+2.6 x 10 -2 ; incorporating the storage effects of all open boreholes results in an equally good fit with k = 1.6 x10 -15 +1.4 x 10 -17 m 2 , 0 = 5.3 x 10 -3 + 5.1 x 10 -3 and Ow = 1.3 + 1.9 x 10 -2 . As I have already discussed in Chapter 3 (Equations 3.65 and 3.66 on page 101), Ow is allowed to take on values in excess of 1.0, as a way of accounting for an effective borehole volume that is larger than the one originally built into the computational grid. Therefore, a borehole porosity in excess of 1 is plausible, implying that the effective storage volume Vs of the injection interval exceeds its nominal volume Vto . The excess is most probably due to openings in the surrounding rock. However, the available single-hole test data do not allow me to distinguish unambiguously between the roles that the borehole and the surrounding rock (fractures and matrix) may play in controlling the observed air storage effect. Despite the good fit, the air-filled porosity estimates do not seem reliable because of the relatively large confidence intervals. Indeed, for the last match, the eigenanalysis results displayed in Table 4.7 indicate that there is a correlation between 0 and 0.„„ most probably due to the influence of storage effects during the transient period of the test. Figure 4.6 presents changes in borehole storage due to air compressibility versus 126 TABLE 4.6. Parameter k [m2] Type of analysis 2.0 x 10_ 15 Analytical steady-state (A) Analytical transient - Spherical flow (A) 2.9 x 10 -15 Inverse modeling 1.8 x 10 -15 ± - No open intervals (A) 3.9 x 10 -15 - Open injection interval (A) 1.6 x 10 -15 ± 2.6 x 10 -17 - All intervals open (A) 1.6 x 10 -15 + 1.3 x 10 -17 - All intervals open (A,B,C,R) 1.7 x 10 -15 + 4.1 x 10 -17 estimates for test JGC0609. Ow [M3 I M3 ] 0 [m 3 / 77-43 ] 5.0 x 10 -1 ± 4.2 x 10 -2 4.8 x 10 -3 ± 9.4 x 10 -3 5.5 x 10 -3 ± 4.7 x 10 -3 3.6 x 10 -3 ± 2.8 x 10 -4 CD [-] 1.3 ± 2.6 x 10 -2 9.0 x 10 1 1.3 ± 1.7 x 10 -2 7.9 x 10 1 1.5 ± 6.3 x 10 -2 1.4 x 10 2 relative time. Pressure buildups were much higher than the initial air pressure, and as a result, changes in borehole storage were substantial even during the first injection steps. In a similar way, the variation of air compressibility modifies the medium storage in the vicinity of injection interval. This explains the poor matches obtained by type-curves, and therefore, the analytical estimates are biased. The effect of air compressibility on storage is important for obtaining good matches and reliable parameter estimates by means of the numerical inverse model. Figure 4.7 depicts relative pressure versus relative time for injection steps 1 - 4 (labeled A - D, respectively) and recovery (labeled R). Here again we see that only data corresponding to step A exhibit an unambiguous one-to-one slope at early time, while all other data appear to be less influenced by storage. I therefore expect a simultaneous analysis of pressure data from the entire test to yield a more reliable estimate of parameters, especially air-filled porosity, than is possible based only on data from the first step. A fit of my model to the entire four-step pressure record, including recovery data, yields a good match (Figure 4.8) with k = 1.7 x 10 - ' 5 + 4.1 x 10 - ' 7 m 2 , 0 = 3.6 x 10 -3 + 2.8 x 10 -4 and çbv, = 1.5 + 6.3 x 10 -2 . The model appears sensitive to all three parameters (Table 4.8) whose estimates seem reasonable. 127 o o o [e cPi] d 128 o LL1 CD 6 :4 4--1 Cf.) (1) C-40 0 • C••C LL1 CCS CD 0 • o C.) -4, 7:1 •474, C.? Ca Cf, • C.= LL1 4:7) ry• ctte 0 CO 4-4 CT) CO r's CD a6eiols aloqa.mq u! a6uetio angelm 0 129 0 0 0 ujr NNW 0 1- 1+ -zr uJ cp < co 1:1 i < + 0 C •I LU [edl] ainssaid 130 o 0 T [ecP1] d 131 TABLE 4.7. Eigenanalysis of covariance matrix of estimation errors obtained from interpretation of the first step of test JGC0609. Eigenvectors Parameters 4.9941 x 10 -16 2.9243 x 10 -15 1.000 k [m 2 ] —0.9758 —0.2188 2.9627 x 10 -15 95 [m3/m3] 1.5247 x 10 -16 —0.2188 0.9758 [m 3 /m 3 ] Ow 1.7338 x 10 -6 6.7666 x 10 -5 Eigenvalues 3.0691 x 10 -36 TABLE 4.8. Eigenanalysis of covariance matrix of estimation errors obtained from interpretation of all the injection steps and recovery of test JGC0609. Eigenvectors Parameters k [m 2] —1.1142 x 10 -13 1.3489 x 10 -16 1.000 —4.8913 x 10 -4 —1.000 —1.1142 x 10 -13 qS [m3irn3] 1.000 —4.8913 x 10 -4 -16 —1.8939 x 10 ] [rn3 /rn 3 9.7135 x 10-4 1.8990 x 10 -8 Eigenvalues 1.6206 x 10 -34 132 4.3 Single-hole test JHB0612 Table 4.9 lists parameters obtained by various methods of analysis from pressure data recorded during test JHB0612. Steady-state analysis gives k = 4.8 x 10-14 m2 [Guzman et al., 1996], and transient type-curve analysis using a spherical flow model yields k -= 6.5 x 10 -14 m 2 and using a radial flow model yields k = 1.3 x 10 -13 m 2 , 0 = 4.0 x 10 -5 and CD -= 1.0 x 10 4 [Illman et al., 1998]. Both type-curve models did not match the observed pressures record [Inman et al., 1998]: the spherical model failed at the late times, the radial model at the early times. Illman et al. [1998] also applied type-curves considering a single horizontal or vertical fracture intersects the injection interval; these models also failed to represent the pressure record. Figure 4.9 indicates that inverse analysis without open boreholes gives a poor fit with k = 5.2 x 10 -14 + 1.1 x 10 -16 m 2 and 0 = 5.0 x 10 -1 1 1.2 x 10 -3 ; incorporating the injection interval yields a much improved fit with k = 4.0 x 10 -14 +4.4 x 10' m 2 , 0 = 8.1 x 10 - 2 + 1.6 x 10 6 and 0„ = 1.2 + 3.4 x 10 6 (the huge confidence intervals reflecting a virtual lack of sensitivity to 0 and q5 , which are negatively correlated with each other); and including all open borehole intervals gives an equally good fit with k = 4.1 x 10- 14 m2, 0 8.8 x 10 -2 and 0„, = 1.2 (here the normal matrix as defined in Equation 3.56 on page 89 is singular and cannot be inverted to compute the covariance matrix, due to correlation between q5 and 0,,). The porosity estimates are clearly unreliable. Similarly to the case of test J 00921, pressure buildups were small compared to the initial air pressure, and therefore, changes in borehole storage due to air com- pressibility were minor (Figure 4.10). Plots of relative pressure versus relative time for injection steps 1 - 3 (labeled A - C, respectively) and recovery (labeled R) are shown in Figure 4.11. As before, only data corresponding to step A exhibit an unambiguous borehole storage effect at early time, while all other data appear to be less influenced by this effect. A simultaneous analysis of pressure data from all stages of 133 TABLE Type of analysis Analytical steady-state (A) Analytical transient - Spherical flow (A) - Radial flow (A) Inverse modeling - No open intervals (A) 4.9. Parameter estimates for test J11B0612. k [m 2 ] g5 [m 3 /m,3] 4.8 x 10 -14 6.5 x 10 -14 1.3 X 10 -13 5.2 x 10 -14 + 1.1 X 10 -16 - Open injection interval (A) 4.0 x 10 -14 ± 4.4 x 10 -16 4.1 x 10 -14 - All intervals open (A) - All intervals open (A,B,C,R) 3.9 x 10 -14 ± 1.6 x 10 -15 Ow [m 3 i m 3 ] 4.0 x 10 -5 5.0 x 10 -1 ± 1.2 x 10 -3 8.1 x 10 -2 ± 1.6 X 10 6 8.8 x 10 -2 9.6 x 10 -2 ± 7.0 x 10 -3 CD [-] 1.0 X 10 4 1.2 ± 1.6 x 10 6 4.9 1.2 1.3 ± 2.2 x 10 -2 4.5 4.5 TABLE 4.10. Eigenanalysis of covariance matrix of estimation errors obtained from interpretation of all the injection steps and recovery of test JHB0612. Eigenvectors Parameters 9.6439 x 10 -15 -7.5978 x 10 -16 k [m 2 ] 1.000 -2.2232 x 10 -2 -0.9998 0 [m 3 /m3 ] 9.6247 x 10 -15 -2.2232 x 10 0.9998 -16 -2 9.7400 x 10 Ow [rn3irn3] 6.1682 x 10 -6 1.1690 x 10 -2 Eigenvalues 6.3155 x 10 -31 the test should therefore yield a reliable estimate of all parameters, including air-filled porosity. A joint analysis of pressure data from all three steps of the test, including recovery, gives a good match (Figure 4.12) with k = 3.9 x 10 -14 + 1.6 x 10 -15 m 2 , 0 = 9.6 x 10 -2 + 7.0 x 10 -3 and 0„ = 1.3 + 2.2 x 10 -2 . As implied by the eigenvectors in Table 4.10, the model now is sensitive to all three parameters, whose estimates seem reasonable. 134 b , • CO CO T T 1 :1' ci T T 4-- CO CD ci ci [e clm] d ci • 135 cf) O Li, a) E ...= Q) > •447, as 71) Ce --c, cc, c=:, .472 = c.) n– n a.) n—n p—, (L) -4- Q) cn (1) tO ' cd 4- 0 C/) ,:t O LLI 5. cu 0.) › O o c...) -- c) c. o -cs ,sm <4. ca o co Q) cu • 7D Cd 0 ,_. c.) .— a) L) tv • ›— . —4 / . E 7!) '—' Li, P4 0 O o , - a) a) 6 co a) 6 I,- co In cr) a) a) 6 6 6 a6eicns alogaioq u! a6uego annelaN cr) 6 —c' t 136 [ed)i] einssaid anweim 137 0•• T- 1 I li 1 I I iiiiIiiilliii i I IN. CD ILO ii• CO i i i 1 I CV 1 1 1 I I i T le dm] d i 1 I 138 4.4 Single-hole test JJA0616 Results for test JJA0616 are listed in Tables 4.11. The steady-state analysis of Guzman et al. [1996] gives k = 5.6 x 10 -15 m 2 ; a transient type-curve analysis with a spherical flow model results in a relatively poor fit with k = 8.0 x 10 -15 m 2 [Illman et al., 1998]; and a similar analysis with a radial flow model yields an equally poor fit with k = 7.1 x 10 -15 m 2 , q5 = 3.1 x 10 -2 and CD = 1.0 X10 1 [Illman et al., 1998]. Numerical inverse results are compared with measured values in Figure 4.13. Upon ignoring open borehole intervals in my numerical inverse model, I obtain a poor fit with k = 5.4 x 10 -15 +7.9 x 10 -16 m 2 and q5 = 5.0 x 10 -1 +3.9 x 10 -2 . The fit improves greatly when I include the injection interval, leading to k = 4.0 x 10 -15 +7.8 x 10' m 2 , = 1.3 x 10 -1 + 5.8 x 10 -3 and 0 = 1.1 + 5.2 x 10 -2 ; the fit is equally good when I incorporate all open borehole intervals, yielding k = 4.1 x 10 -15 + 7.1 x 10 -17 m 2 , = 1.3x 10'+6.4x 10 and q5,,, = 1.1+3.9x 10 -2 . Here the numerical inverse model is capable of producing estimates with low errors for all three parameters based solely on data from the first injection step. This conclusion is confirmed by an eigenanalysis of the corresponding covariance matrix (Table 4.12). Figure 4.14 presents relative change of storage due to variation of air compressibility versus relative time. Similarly to the case of test JGC0609, obtained pressures buildups were much higher than the initial air pressure, and therefore, changes in borehole storage due to changes in air compressibility were substantial. This may explain the poor matches obtained by analytical type-curves. The effect of air compressibility on storage is important to obtain good matches by means of the numerical inverse model. Still, borehole storage dominates pressure transients and makes it difficult to obtain reliable estimates of air-filled porosity; identified value of q5 is much too high for fractures. Plots of relative pressure versus relative time for various injection steps in Figure 4.15 demonstrate that data corresponding to the first injection step (A) exhibit very strong borehole storage effect. Therefore, though the model calculates low estimation 139 errors, the porosity estimates are not reliable. Fitting the inverse model to data from all three steps of the test, including its recovery stage, leads to a good overall fit (Figure 4.16) with k = 4.3 x 10 - ' 5 + 1.9 x 10 -16 m 2 , 0 = 1.8 x 10 -2 + 3.9 x 10 -4 and 0,,, = 1.6 + 7.2 x 10 -2 . The eigenanalysis of covariance matrix demonstrates reliable estimation for all parameters (Table 4.13). The estimated 0 value is now much lower, and can be considered as representative of fracture properties. 140 TABLE 4.11. Parameter estimates for test JJA0616. k [m 2 ] Ow [m3/m3] 0 [m3/m3] 5.6 x 10 -15 Type of analysis Analytical steady-state (A) Analytical transient - Spherical flow (A) - Radial flow (A) Inverse modeling - No open intervals (A) 8.0 x 10 -15 7.1 x 10 -15 5.4 x 10 -15 ± 7.9 x 10 -16 4.0 x 10 -15 ± - Open injection interval (A) 7.8 x 10 -17 4.1 x 10 -15 + - All intervals open (A) 7.1 x 10 -17 - All intervals open (A,B,C,D,R) 4.3 x 10 -15 ± 1.9 x 10 -15 CD 3.1 x 10 -2 5.0 x 10 -1 ± 3.9 x 10 -2 1.3 x 10 -1 ± 5.8 x 10 -3 1.3 x 10 -1 + 6.4 x 10 -3 1.8 x 10 -2 ± 3.9 x 10 -4 [-] 1.0 x 10 1 1.1 ± 5.2 x 10 -2 2.8 1.1 + 3.9 x 10 -2 2.8 1.6+ 7.2 x 10 -2 3.0 x 10 1 TABLE 4.12. Eigenanalysis of covariance matrix of estimation errors obtained from interpretation of the first step of test JJA0616. Parameters k [m 2 ] 0 [m3 /m3 ] Ow [rn 3 /m3 1 Eigenvalues Eigenvectors 1.000 3.5461 x 10 -15 3.8852 x 10 -16 8.9518 x 10 -34 3.5628 x 10 -15 -0.9983 -5.8609 x 10 -2 7.0241 x 10 -8 -1.8002 x 10 -16 -5.8609 x 10 -2 0.9983 2.9162 x 10 -4 TABLE 4.13. Eigenanalysis of covariance matrix of estimation errors obtained from interpretation of all the injection steps and recovery of test JJA0616. Eigenvectors Parameters -4.4528 x 10 -16 3.5963 x 10 -13 1.000 k [m 2 ] 6.9652 x 10 -4 -1.000 -13 3.5963 x 10 ] 0 [m3 /rn 3 -2 1.000 6.9652 x 10 1.9479 x 10 -16 ] Ow [m 3 /rn3 1.2518 x 10-3 -8 3.6030 x 10 -33 3.5004 x 10 Eigenvalues 141 Cn1 EedIll d 142 0 e's'l I=MM I=MMM•111 MIMI= :u ==0111= MINIM == NM IMMIM MIM =MI MN= IMMMMM MMIn IMMM MM/ MI/1 /W. 1.1111....111.1111W MUM 71.1111 MIME 11=111111111MW L•N•• MO., BEM Ili 11111 IErA11111ow 7 31111I 111111"1111111 1111111111111111 II 11111111 ,.. 0 Q) s. 0 • T I • I.0 e— <-. 0 11 or) 1.0 co o aBeJols alogaJoq a6uego anReim o co o > 7.) 143 LU 0 0 W 1.0 0 9 LLI LLJ [e cpi] ainssaid anneieN 144 o 0 T [edm] d 145 4.5 Discussion \ Transient multi-step single-hole pneumatic tests in unsaturated fractured tuffs at the ALRS are amenable to analysis by means of a three-dimensional numerical inverse model which considers only isothermal single-phase airflow and treats the rock as a uniform, isotropic porous continuum representative of interconnected fractures. Bore- hole storage depends on compressibility of air in the injection interval, and decreases with the increase of absolute air pressure. Borehole storage effect influences pressure transients during the first step of each test and makes it difficult to obtain reliable estimates of air-filled porosity and borehole storage coefficient from such data. Pressure transients during subsequent injection steps are less affected by borehole storage than are those during the first step. Recovery data are theoretically free of borehole storage effects. A consideration of the effect of air compressibility on storage and a joint analysis of pressure data from all stages of a single-hole pneumatic test are therefore essential for the reliable estimation of air-filled porosity and borehole storage coefficient. The values of 0 estimated using complete pressure records for the four tests range from 9.6 x 10 -2 to 3.6 x 10 -3 ; the respective estimates of CD are between 4.5 and 140. The storage effect observed during single-hole tests seems related not only to the injection interval, but also to openings in the surrounding rock. Open borehole intervals surrounding the injection interval have little impact on pressures within the latter. Air permeabilities obtained by our inverse method are comparable to those obtained by means of steady-state formulae [Guzman et al., 1996] and transient type-curve analyses [Illman et al., 1998]. Due to linearization of airflow equations, analytical transient type-curves [Illman et al., 1998; Illman and Neuman, 2000b] did not match observed pressure records and did not allow reliable identification of air-filled porosity and borehole storage coefficient. 146 Chapter 5 INTERPRETATION OF CROSS-HOLE TESTS Illman et al. [1998] and Illman [1999] conducted 44 cross-hole pneumatic tests at the ALRS. To date, only one of the cross-hole tests, labeled PP4, has been fully analyzed by means of analytically derived type-curves [Illman et al., 1998; Illm,an, 1999; Inman and Neuman, 2000a]. Due to the pronounced non-uniformity of the fractured rock at the site, this approach allows only separate interpretation of pressure records from individual observation intervals, under the assumption that air-filled fractures form a uniform continuum. Here, I perform comprehensive numerical inverse analyses of five of the cross-hole tests labeled PP4, PP5, PP6, PP7 and PP8 (Table 5.1). Tests PP4, PP5, PP6 and PP7 were conducted with a packer configuration shown in Figure 5.1. In test PP8, the packers in X3 and Y3 were placed lower along the inclined boreholes by 5.5 and 5.0 m, respectively. In the 16 tested boreholes (Xl, X2, X3, Yl, Y2, Y3, Z1, Z2, Z3, Vi, V2, V3, Wl, W2, W2A and W3), there were 37 monitored intervals with lengths between 0.5 and 42.6 m. During tests PP4 and PP8, air was injected at a constant mass rate of 1 x 10 -3 kg/s (50 //min under standard conditions) into the middle interval of borehole Y2. During tests PP5, PP6 and PP7, air was injected into the middle intervals of boreholes X2, Z3 and W3, respectively. In the latter three tests, the injection consisted of two constant rate steps; however, I process only data from the first step, at a mass-rate of 1 x 10 -4 kg/s (5 l I min). The injection intervals were 1 m (PP7) and 2 m (PP4, PP5, PP6 and PP8) in length (Table 5.1). Distances between injection and observation intervals varied from 1 to 30 m (Figure 5.1). Parts of this analysis have been presented earlier by Illman et al. [1998], Illman and Vesselinov [1998], Vesselinov et al. [1999], Neuman et al. [2000], Vesselinov et al. 147 TABLE 5.1. Point-to-point cross-hole tests interpreted by numerical inverse model Cross-hole test Injection interval PP4 PP5 PP6 PP7 PP8 Y2-2 X2-2 Z3-2 W3-2 Y2-2 Length Duration Injection rate Injection rate [m] [days] 2.0 2.0 2.0 1.0 2.0 3.95 1.95 1.70 1.90 0.84 [1Imin] 50.0 5.0 5.0 5.0 50.0 1.0x10-3 0.1x10-3 0.1x10-3 0.1x10-3 1.0x10-3 [kgls] [2000a], Vesselinov et al. [2000b] and Chen et al. [2000]. The labeling of test intervals in my dissertation differs from that of Illman et al. [1998]. In both studies, when a single packer is used to isolate a borehole segment, the corresponding interval is named after the borehole (e.g. X1). In the case of multiple intervals, these are labeled by Illman et al. [1998] from top to bottom as UMB, UML and UMLB (e.g. X2U, X2M, X2B), respectively. Instead, I numbered test intervals sequentially from top to bottom (e.g. X2-1, X2-2, X2-3). 5.1 Qualitative analysis During the cross-hole tests, pressure and temperature data were collected by means of 38 measuring devices located below the respective confining packers [Illman et al., 1998]. As boreholes V2 and W2 are interconnected, the intervals V2-3 and W2 are linked, forming one relatively large and complex observation interval (Figure 5.1). Therefore, the actual number of test intervals is 37. Four of the intervals—Y1-1, V21, V2-3/W2 and W3-3—gave unreliable readings in all the cross-hole tests [Illman et al., 1998] (initially, data from interval Y1-1 have been interpreted analytically by Illman et al. [1998] and Illman [1999]; later, Illman and Neuman [2000a] excluded Y1-1 from their analysis). Most probably, due to a measurement problem, interval Y1-3 produced responses that are perfectly correlated with variations in atmospheric pressure. Pressure data from these five monitoring intervals are ignored in my study. This leaves 32 test intervals-1 injection and 31 monitoring. Discrete pressure data 148 0 -10 N -20 3 -30 -40 FIGURE 5.1. Location of monitoring intervals and packers along tested boreholes. 149 recorded in these intervals are shown by small solid dots on log-log plots in Figures 5.2 - 5.6. Pressure measurements are relative, representing changes in absolute air pressure with time. Pressure data were collected every 20 s during each test [Illman et al., 1998]. I filtered the recorded data so that the minimum time separation between two consequent measurements in these figures is (log t 2 - log t 1 ) > 5 x 10 -3 , where t i and t 2 are in days. In Figures 5.2 - 5.6 and all similar figures presented later, vertical axes show relative pressure buildup in kPa, and horizontal axes show elapsed time from the start of the test in days, respectively. In the case of test PP4, pressure responses due to air-injection into Y2-2 (Figure 5.2) are well-defined in all 32 intervals. Barometric pressure varied significantly and dropped by more than 1 kPa during this test (Figure 5.7). Late-time pressure data in intervals X3, Y3-1, Y3-2, Z1, Z2-1, Z2-2, Z2-3, Z2-4 and Z3-3 are visibly affected by this barometric pressure decline. Measured pressure responses in some of the intervals and the atmospheric pressure are plotted versus time in Figure 5.7. Injection into interval X2-2 during test PP5 produces responses in most intervals (Figure 5.3). However, 11 of these pressure records (those corresponding to intervals X3, Y1-2, Y3-2, Z1, Z2-1, Z2-2, Z2-3, Z2-4, Z3-3, W2A-4 and W3-1) are difficult to interpret. Pressure data from nine intervals during test PP6-X3, Y1-2, Y3-2, Z1, Z2-2, Z2- 3, Z2-4, Z3-3 and W3-2 (Figure 5.4)-are weak and difficult to interpret; most of these monitoring intervals are close to the injection interval, Z3-2. During test PP7 (injection into W3-2), pressure responses in all but three of the monitoring intervals Y1-2, W2A-1 and W2A-4-are amenable to interpretation (Figure 5.5). Test PP8 is similar to test PP4 and all corresponding pressure responses, except that in X3, are well defined (Figure 5.6). In contrast with PP4, there are no significant variations in barometric pressure during test PP8. As a result, there are no late time pressure drops in intervals Y3-1, Y3-2, Z1, Z2-1, Z2-2, Z2-3, Z2-4 and Z3-3 (Figure 5.2) which were affected by barometric pressure during test PP4 (Figure 5.2). In summary, the number of intervals that are amenable to analysis are as follows: PP4 - 32; PP5 - 150 X1 X2 - 3 10° 10° o 10' 10 0 10.0 10'10° Y2-2 X3 1 00 10. 10 " • •.: :47 . 10' - 0 110 ./fe-e.NNikt 1. • • • 100 1OE° 10' 10' 10° 10' 10-2 lo- 1U 10 zi 1U 0 10 Z2-2 8:0 - eAstr.e*4.9 10 • 10 Z2-1 0.4 0.2 10 10 10' 10 • 10' 10 4 10 -°- 10° 1U 100Z2-3 10'• 10 010-1 Z2-4 P 10° 10 0 10' o 10°10' 10" 10' 10° Z3-1 10 1 ,0 10 ° 10 • 10' 10' 10 0 10 2 10° Z3-3 10' o. 10 2 . 10° V1 10-210-' 10 210'10° 10° V3-2 V3-1 10° 40 • 10' 10' o 30 o . • • 10' 10 10' 10' FIGURE 5.2. Log-log plots of observed pressure buildups (dots; kPa) versus time (days) and match points (circles) during test PP4. 151 X1 100X2-1 1 0' 1 0' 1 0 -0 10" 10' 1.5 10' 10 -0 10 -2 0.5 10° 10' 0.08 0.06 0.06 0.04 1 0' 10 -2 1o 2 10 -0 10' 1 0' 1OE' 10 • 10 -2 10 -0 . . 16' - ' 16 0 (':1 W2A-1 02 0 00 0.15 10° OA 10' 10 - ' 0.05 • 10 -310 •• • •• 10' 1 0° 10-0 10' 10 0 W3-1 W2A-4 0.3 0.2 10' 0.1 10 0 10 -2 1 0 01 0 010' 10° 10 -0 1o 0 10' 1 0° 10 0 io' lo' lo° FIGURE 5.3. Log-log plots of observed pressure buildups (dots; kPa) versus time (days) and match points (circles) during test PP5. 1 52 X1 X2-2 X2-3 10' 1 0 -0 10 -2 1 00 10 10 -210" 10 0o' Y2-1 Y2-2 0.6 0.4 0.6 0.2 __sf• 0.5 10' 10" Y3-2 0.6 0.4 10 -210° 1 0 -2 10 Y3-3 16-0 15'100 10 -2 0.2 1 0 -4 10.0 1 0 -0 16-3 Z2 - 2 16-2 16' -Co. 0.2 io' Z3-2 0.2 52 0.15 51 0.1 50 100 49 48 c; 0.05 180 V3-1 0.3 0.2 0.3 0.2 0.1 0.1 18-0 . 18:r; V3-3 Iv 16. 0.2 0.1 10' 10' 10" 10 ° 10' 0.15 0.1 0,1 .• • • lo° 10' 0.6 0.4 101' 10 0 ._ 10" o 0.05 .• - 10" 10 -210° W2A-4 10; 10' 1 0 01 00 10" W2A- 1 0.3 0.2 W2A- 3 100 41°4°4 W1 8:1 10-2 • 11.. 10 -0 .:••• 10° 0.3 0.2 10-0 W3-1 10° W3-2 10 0.1 0.2 • ' • '1'1:: ' o' 10 -0 10" 10 10010 10° 10' 18-0 18' 16° 10 -2 o 10° Log-log plots of observed pressure buildups (dots; k P a) versus time (days) and match points (circles) during test PP6. FIGURE 5.4. 153 X2-2 X2-3 10' . • : 10' Y2-1 10. 10 - ' 10° L 10 - ' 10-0 10' ‘7r. 10 3 10' 10' . ' ,. 2.:n, , , 10' V3-1 10' 10' , 10' V3-2 10 0 10' • 10 . ° 102 :"1-A 10' 10° 10-' __ • •••••-.0.;..grf) 10 100 -..--: • . 10" f i: • .. " ...r:.:,'.• : • 10'10010'10 ° V3-3 eta* to 10' , - 10' 10° to-0 W2A-3 10° 10 - ' 10' 10 - ' 10 -2 10 o 10 1 0 10 10 16 10' 1 0 ' 10 ° 1 0 310' 10' 10° 5.5. Log-log plots of observed pressure buildups (dots; k P a) versus time (days) and match points (circles) during test PP7. FIGURE 154 X2-1 X1 10 ° 10° 10" 10' 10 10 010 -° o' X3 . . • •: • 10 1 100 10 4• • • , • 0-3 . ° • 10° 80 • • • • 10 -' 10 - s to' 10 - Y2-3 10' 10° 10' Z2-2 Z1 10 4 10 -0 10 .310" 10 - ' 10' Z2-4 10' • 10" 10 -0 • -1&" Z3-1 Z3-2 10° • 16' Z3-3 10' 10° 10 -° 10' 1 0 -0 10 -3 10 0 V2-2 10' 10° 10° 10' 10' V3-2 V3-1 10' 10' 10° .erw - 10° e° 10' W2A- 4 10 10° W3-1 10° ° 10' 10 -1 10 - ° 10' 10 10- 10' 16' 16' 16 ' - 10 - ° 10" 16 1 0 ' -0 - Log-log plots of observed pressure buildups (dots; kPa) versus time (days) and match points (circles) during test PP8. FIGURE 5.6. 155 1 ,646p,66,64NA66,66646A666,AA64,666.66446.6.61664, 114466,6, 6•A6:6A& A A A A A ▪ A O • O -1 Y1-3 Z2-1 Z2-2 Z2-3 Z2-4 barometric 0 1 2 3 4 Time [days] FIGURE 5.7. Measured variations in barometric pressure and pressure responses in some of intervals during test PP4. 21; PP6 - 23; PP7 - 29; and PP8 - 31. The total number of such intervals is 136. Illman et al. [1998] measured variations in air temperature during each pneumatic test at ground level and within most monitoring intervals. Air temperatures in the intervals varied insignificantly during the tests, except in the injection interval (Y2-2) during tests PP4 and PP8. Some of the temperature data from PP4 are presented in Figure 5.8; temperature data from PP8 look similar. Though the injected air was much warmer than ambient air in the rock, air injection at a high rate causes temperature in the injection interval to decrease. This could be either an adiabatic effect (rapid expansion of injected compressed air) or an evaporation effect (two-phase vapor flow) or both. The effect is localized and not observed in the surrounding 156 el - atmosphere I I injected 11 Y1-3 0 Y2-2 i \I 12i -1 i i i 0 I I j i i I \ I\ I \ \ I \ il1 k l n i Ii 2 1 I 3 1 , \ l, I 4 Time [days] FIGURE 5.8. Some air temperature measurements collected during test PP4. intervals (Figure 5.8). Overall, most monitoring intervals respond consistently to air injection during each pneumatic test. Since injection took place from several intervals during these tests, one can conclude that air-filled fractures at the ALRS are pneumatically well connected. Most air pressure responses show behavior typical of a three-dimensional continuum. Pressure in most injection intervals (Y2-2 in Figure 5.2, X2-2 in Figure 5.3, Z3-2 in Figure 5.4, Y2-2 in Figure 5.6) exhibits a decline at late time. This can be attributed to two-phase flow (as suggested by Guzman and Neuman [1996] and supported by the above temperature data), barometric pressure variations or instrumentation problems. The exception is injection interval W3-2 during test PP7 157 (Figure 5.5). Slopes of the early-time response curves are close to 1:1, demonstrating significant storage effect in the injection intervals and in some monitoring intervals located close to the injection interval (e.g. X2-2, X2-3, Y2-1, Y2-3, Y3-3 and V3-3 during PP4; Figure 5.2). The pressure record of the injection interval during PP6 (Z3-2 on Figure 5.4) shows an exponential pressure increase at early times which reflects the impact of the variable air compressibility (Ca ;---' 1h3, ) on borehole stor- 11 age as described in Section 3.1.1 (Equation 3.11, page 73). Some of the responses show early slopes close to 1:2—Y3-1, Z1, Z2-1, Z2-2, Z2-3, Z2-4 and Z3-3 during tests PP4, PP7 and PP8—which suggests that they are intersected by highly perme- able zones. In these intervals and in those intervals located close to them, measured records are clearly influenced by barometric pressure variations, suggesting that the highly permeable zones are connected to the ground surface. The data indicate highly permeable zones between some of the intervals producing very similar pressure measurements in all tests: X1 and W3-2; Y3-1, Z2-3, Z2-4 and Z3-3. The X3 records are very weak, implying locally reduced permeability. It thus appears that the medium is highly non-uniform. High-permeable zones around Y3-1 and the Z boreholes seem connected to the ground surface. The existence of these highly permeable zones is confirmed by geostatistical analysis of air permeability (Figure page 2.18, page 58) and fracture density (Figure 2.22, page 62) data in Chapter 2. 158 5.2 Inverse analysis of cross-hole tests To invert numerically cross-hole test data, match points are defined subjectively for portions of the measured pressure records that clearly demonstrate response due to air injection. Variations in pressure that result from either a change in barometric pressure or measurement problems are ignored. The match points are distributed more or less evenly along the log-transformed time axis so as to capture both early and late time responses correctly. The number of match points defined for each test are: PP4 - 252; PP5 - 132; PP6 - 105; PP7 - 155; PP8 - 210. They are indicated by open circles in Figures 5.2 - 5.6. Each point is assigned a weight of 1. The same set of match points and weights are applied in all inverse analyses. In the inverse model, all open borehole intervals at the site, regardless of their response, are simulated as high-permeability and high-porosity cylinders. I performed inverse analyses of the pneumatic tests by treating the medium as if it was either spatially uniform or non-uniform. Inverse runs were conducted in parallel on the University of Arizona 32-processor SCI Origin 2000 supercomputer. 5.2.1 Uniform medium I started by analyzing separately pressure data from each monitoring interval while treating the medium as if it was uniform across the site. Two uniform parameters were estimated on the basis of each individual pressure record in a test interval: log io transformed air permeability k and air-filled porosity q5. This interpretation is similar to the analytical type-curve analysis performed on some of the same data by Inman et al. [1998], Illman [1999] and Illman and Neuman [2000a]. On average, each numerical inversion required about 50 forward simulations, and the effective computational time was about 4 hours. The resulting matches between simulated and observed pressures for the five crosshole pneumatic tests are shown in Figures 5.9 - 5.13. Some of these matches are 159 very poor due to instrumentation problems and barometric pressure effects; some are of intermediate quality; and some are good to excellent. Tables 5.2 — 5.6 list log io k and log io 0 estimates and their respective separate 95% confidence limits. All permeability estimates appear to be reasonable. Unrealistically high porosity estimates (log io ck = —0.3, equal to the prescribed upper bound) were obtained for some intervals that appear to be pneumatically connected to the atmosphere. The corresponding estimation uncertainties are large, and fits between calculated and observed responses are poor. These data are ignored in the rest of my analysis. The remaining estimates are characterized by low uncertainties. Summary statistics for log lo k and log io q5 estimates are presented in Tables 5.7 and 5.8. 160 X1 1 o° X2-1 10° X2-2 X2-3 10° 10" 10° 10 0 10" 10° 10' 10' 10 -0 10° 1 0' 5.9. Separate matches between observed (small dots) and simulated (thick curves) responses for test PP4 assuming uniform medium. FIGURE 161 X2 - 2 X1 X2-3 10 ' 10° 2.5 2 1.5 10.0 1 0 •° lo Z2-2 lw 10 lo 0.08 0.04 1/1 10 1 0. 0 W3-1 W2A-4 0.3 0.2 1 0 .0 10 2 r 10' 0.1 1 0° 5.10. Separate matches between observed (small dots) and simulated (thick curves) responses for test PP5 assuming uniform medium. FIGURE 162 X2-2 X1 X2-3 1 0' 10' 0.8 1 0 -2 0.7 0.6 10 -3 0.2 0.5 1 0 -s 0.4 0.6 0.4 0.2 10 0 1'0° 16' Z2-2 0.3 0.25 0.1 0.2 10' 0.2 10° 10' 10° los 1o0 10' o° 10' 10 -210' 10° 5.11. Separate matches between observed (small dots) and simulated (thick curves) responses for test PP6 assuming uniform medium. FIGURE 163 X1 X2-1 X2 - 2 X2-3 10' 10 0 0.8 0.6 ••• • 10 0 0.4 100 0.2 10 0 10 4 1 0' 10 -0 10' 10 -0 10' 10 .0 10 -2 10' 10' 10' 5.12. Separate matches between observed (small dots) and simulated (thick curves) responses for test PP7 assuming uniform medium. FIGURE 164 X2-2 X2-3 1 00 10' 1 0 -2 10 0 10 -3 10' 10 -a o' 5.13. Separate matches between observed (small dots) and simulated (thick curves) responses for test PP8 assuming uniform medium. FIGURE 165 TABLE 5.2. Parameters identified for cross-hole test PP4 treating the medium as spatially uniform. log io k [m 2 ] Interval k [rn 2 ] loglo 0 [m 3 /m 3 ] 0 [m 3 /m 3 ] X1 -14.08+ 0.007 8.3 x 10 -15 -2.12 1 0.021 7.7 x 10 -3 X2-1 1.1 x 10 -14 -13.94 ± 0.017 2.6 x 10 -2 -1.59 ± 0.054 X2-2 -14.26+ 0.050 5.5 x 10 -15 1.2 x 10 -2 -1.93 ± 0.170 X2-3 -2.04 ± 0.131 8.7 x 10 -15 -14.06 ± 0.054 9.2 x 10 -3 -12.18 + 0.032 6.6 x 10 -13 X3 -0.30 ± 0.111 5.0 x 10 -1 Y1-2 9.4 x 10 -15 -1.21 + 0.047 -14.02 ± 0.017 6.2 x 10 -2 Y2-1 9.3 x 10 -15 -1.20+ 0.140 -14.03 ± 0.039 6.3 x 10 -2 Y2-2 2.1 x 10 -14 -1.22 ± 0.049 -13.68 ± 0.001 6.0 x 10 -2 Y2-3 5.0 x 10 -15 -1.70 + 0.266 2.0 x 10 -2 -14.30+ 0.073 -1.64 ± 0.018 Y3-1 -13.24 + 0.015 5.8 x 10 -14 2.3 x 10 -2 Y3-2 6.6 x 10 -14 -1.01 ± 0.047 -13.18 ± 0.012 9.8 x 10 -2 -1.19 + 0.127 2.1 x 10 -14 Y3-3 6.4 x 10 -2 -13.68 ± 0.032 Z1 -12.42 ± 0.029 -1.22 ± 0.028 3.8 x 10 -13 6.0 x 10 -2 -1.28 + 0.151 Z2-1 -13.19 + 0.047 6.5 x 10 -14 5.2 x 10 -2 2.4 x 10 -13 -1.47 ± 0.027 Z2-2 -12.62 ± 0.018 3.4 x 10 -2 Z2-3 -11.52 ± 0.090 3.0 x 10 -12 -0.30+ 0.080 5.0 x 10 -1 1.2 x 10 -12 Z2-4 -11.92 1 1.528 -0.30 + 0.670 5.0 x 10 -1 1.4 x 10 -14 -0.99 + 0.035 Z3-1 -13.87+ 0.017 1.0 x 10 -1 1.6 x 10 Z3-2 -13.80+ 0.015 -14 -1.39 ± 0.037 4.1 x 10 -2 -12.58+ 0.099 2.6 x 10 -13 -0.52 + 0.101 Z3-3 3.0 x 10 -1 -14.11 ± 0.027 -1.90 ± 0.150 V1 7.8 x 10 -15 1.3 x 10 -2 -1.94 ± 0.075 -13.95+ 0.021 1.1 x 10 -14 V2-2 1.1 x 10 -2 -1.45 + 0.058 -13.65+ 0.014 2.3 x 10 -14 V3-1 3.5 x 10 -2 -2.29 + 0.066 -14.21 + 0.007 V3-2 6.1 x 10 -15 5.1 x 10 -3 -14.42 ± 0.041 -1.62 ± 0.109 -15 2.4 x 10 -2 V3-3 3.8 x 10 -1.82 -13.94 ± 0.025 1.1 x 10 + 0.077 W1 -14 1.5 x 10 -2 W2A-1 -13.36 1 0.011 4.3 x 10 -14 -1.55 + 0.042 2.9 x 10 -2 -14.11 ± 0.041 -1.48 ± 0.119 W2A-2 7.7 x 10 -15 3.3 x 10 -2 -1.35 ± 0.043 W2A-3 -13.82 ± 0.018 1.5 x 10 -14 4.5 x 10 -2 -1.22 ± 0.050 1.2 x 10 -14 W2A-4 -13.93 ± 0.024 6.0 x 10 -2 -15 -2.08 ± 0.038 -14.21 ± 0.025 6.2 x 10 8.3 x 10 -3 W3-1 ± 0.114 10 -15 -2.08 -14.08 1 0.048 8.4 8.3 x 10-3 x W3-2 166 TABLE 5.3. Parameters identified for cross-hole test PP5 treating the medium as spatially uniform. Interval logio k [m2] X1 -13.72 ± 0.010 X2-1 -13.83 ± 0.033 X2-2 -14.90 ± 0.112 X2-3 -14.31 + 0.045 X3 Y1-2 -14.18 + 0.124 Y2-1 Y2-2 -14.16 ± 0.077 Y2-3 -14.81 1 0.034 Y3-1 -13.48 ± 0.136 Y3-2 Y3-3 -13.35 ± 0.101 Z1 Z2-1 Z2-2 Z2-3 Z2-4 Z3-1 -13.42 + 0.132 Z3-2 -13.43 ± 0.094 Z3-3 V1 -14.01 ± 0.039 V2-2 -13.73 ± 0.031 -13.90 + 0.097 V3-1 -13.99 ± 0.023 V3-2 -15.15 ± 0.117 V3-3 W1 -13.91 + 0.045 -12.22 ± 0.024 W2A-1 W2A-2 -14.55 ± 0.040 W2A-3 -13.80 ± 0.090 W2A-4 W3-1 -13.94 ± 0.038 W3-2 k [m2] 1.9 x 10 -14 1.5 x 10 -14 1.3 x 10 -15 4.9 x 10 -15 logio 0 [m 3 /m 3 ] -1.34 ± 0.030 -1.10 ± 0.129 -2.92 ± 0.741 -1.87 ± 0.097 0 [m 3 /m 3 ] 4.6 x 10 -2 7.9 x 10 -2 1.2 x 10 -3 1.3 x 10 -2 6.7 x 10 -15 6.9 x 10 -15 1.6 x 10 -15 3.3 x 10 -14 -2.07 ± 0.154 -1.98 ± 0.263 -2.78 ± 0.045 -1.33 ± 0.138 8.5 x 10 -3 1.1 x 10 -2 1.7 x 10 -3 4.7 x 10 -2 4.5 x 10 -14 -1.68 1 0.203 2.1 x 10 -2 3.8 x 10 -14 3.8 x 10 -14 -2.20 ± 0.127 -2.06 ± 0.108 6.4 x 10 -3 8.6 x 10 -3 9.7 x 10 -15 1.9 x 10 -14 1.3 x 10 -14 1.0 X 10 -14 7.1 x 10 -16 1.2 x 10 -14 6.1 x 10 -13 2.8 x 10 -15 1.6 x 10 -14 -1.75 ± 0.169 -1.42 ± 0.079 -0.98 ± 0.399 -1.46 ± 0.163 -2.79 ± 0.146 -1.67+ 0.093 -1.68 ± 0.170 -1.84 ± 0.068 -1.46 ± 0.215 1.8 x 10 -2 3.8 x 10 -2 1.0 x 10 -1 3.4 x 10 -2 1.6 x 10 -3 2.1 x 10 -2 2.1 x 10 -2 1.5 x 10 -2 3.5 x 10 -2 1.2 x 10 -14 -1.61 ± 0.114 2.5 x 10-2 167 TABLE 5.4. Parameters identified for cross-hole test PP6 treating the medium as spatially uniform. k [ m 2] loglo 0 [m 3 /m 3 ] logio k [m 2 ] Interval 0 [m/m] -14 -2.28 X1 -13.51 ± 0.038 3.1 x 10 + 0.040 5.2 x 10 -3 -14 -1.47 ± X2-1 3.2 x 10 0.016 3.4 x 10 -2 -13.50 ± 0.015 1.6 x 10 2.6 x 10 -2 X2-2 -13.80 ± 0.030 -14 -1.59 ± 0.009 -14 -1.47 X2-3 -13.31 + 0.076 5.0 x 10 ± 0.033 3.4 x 10 -2 X3 Y1-2 Y2-1 Y2-2 Y2-3 Y3-1 Y3-2 Y3-3 Z1 Z2-1 Z2-2 Z2-3 Z2-4 Z3-1 Z3-2 Z3-3 V1 V2-2 V3-1 V3-2 V3-3 WT1 W2A-1 W2A-2 W2A-3 W2A-4 W3-1 W3-2 -13.78 + 2.309 -13.58 + 0.755 -13.98 ± 1.499 -13.27± 0.033 1.6 x 10 -14 2.6 x 10 -14 1.0 X 10 -14 5.4 x 10 -14 -1.55 ± 1.454 -1.50 ± 0.729 -2.18 + 1.164 -1.20 + 0.024 2.8 x 10 -2 3.1 x 10 -2 6.5 x 10 -3 6.3 x 10 -2 -13.27 + 0.032 5.4 x 10 -14 -0.58 ± 0.013 2.6 x 10 -1 -12.00 + 0.043 1.0 X 10 -12 -1.15 ± 0.060 7.0 x 10 -2 -14.40 -14.22 + 0.006 4.0 x 10 -15 6.0 x 10 -15 -1.52 -5.00 ± 0.100 3.0 x 10 -2 1.0 X 10 -5 -13.43 ± 0.028 -13.71 ± 0.033 -13.97 ± 0.418 -13.66 ± 0.015 -14.13 ± 0.057 -13.89 1 0.015 -13.49 ± 1.966 -13.61 ± 0.062 -13.89 ± 0.097 -13.84 1 0.098 -13.89 ± 0.033 3.7 x 10 -14 2.0 x 10 -14 1.1 X 10 -14 2.2 x 10 -14 7.5 x 10 -15 1.3 x 10 -14 3.2 x 10 -14 2.4 x 10 -14 1.3 x 10 -14 1.4 x 10 -14 1.3 x 10 -14 -2.14± 0.063 -2.41 ± 0.058 -2.02 + 0.472 -2.31 + 0.049 7.2 x i0 3.8 x 10 -3 9.4 x 10 -3 4.9 x 10 -3 2.4 x 10 -2 2.0 x 10 -2 1.0 X 10 -2 5.9 x 10 -3 8.9 x 10 -3 1.2 x 10 -2 2.3 x 10-3 -1.63 1 0.014 -1.70 ± 0.026 -2.00 1 0.642 -2.23 ± 0.037 -2.05 ± 0.052 -1.92 ± 0.029 -2.64 ± 0.008 - 168 TABLE 5.5. Parameters identified for cross-hole test PP7 treating the medium as spatially uni form. k [m 2] logio 0 [m 3 /m 3 ] log io k [m 2 ] Interval 0 [m 3 im 3 ] 2.5 X 1 0 -16 -5.00 1.0 x 10 -5 X1 -15.61 -14.03 ± 0.016 -1.89 ± 0.021 1.3 x 10 -2 X2-1 9.3 X 1 0 - ' 5 -14.00 + 0.012 1.0 X i o -14 -1.62 ± 0.024 2.4 x 10 -2 X2-2 -1.91 ± 0.098 1.2 x 10 -2 X2-3 -14.04 + 0.055 9.2 X i o -15 X 0 -12 -12.00 ± 0.076 1.0 1 -0.63 ± 0.091 2.4 x 10 -1 X3 Y1-2 Y2-1 -14.63 + 0.001 2.3 X i o -15 -2.29 ± 0.000 5.1 x io -3 1.2 x i o -14 -2.21 + 0.100 6.1 x 1 0 -3 Y2-2 -13.92 + 0.075 -2.02 ± 0.036 Y2-3 -14.31 + 0.138 4.9 X 1 0 - ' 5 9.6 x i o -3 1.0 x 1 0 -2 1.5 X -1.99 ± 0.030 Y3-1 -13.82 + 0.063 -1.24 ± 0.018 1.1 X 1 0 -13 5.7 x 1 0 -2 Y3-2 -12.96 + 0.030 2.7 x 1 0 -2 5.3 X b o - ' 4 -1.58 1 0.060 Y3-3 -13.28 + 0.128 1.9 X i o - ' 4 -2.06 ± 0.349 8.7 x i o -3 Z1 -13.72 + 0.481 -1.95 + 0.146 1.1 x 10 -2 Z2-1 -13.36 ± 0.224 4.3 X i o - i 4 7.8 X i o -14 -1.19 ± 0.926 Z2-2 -13.11 ± 1.380 6.5 x 10 -2 7.1 X i o -1- 3 -0.30 + 0.334 5.0 x 10 -1 Z2-3 -12.15 + 0.516 2.3 X i o - ' 3 2.3 x 1 0 -1 -0.64 ± 0.495 Z2-4 -12.64 + 0.367 2.7 x 1 0 -2 Z3-1 -13.88 + 0.004 1.3 X i o - ' 4 -1.56 ± 0.003 -1.28 + 0.047 -12.74 + 0.612 1.8 X Z3-2 5.3 x 10 -2 2.2 X -0.49 ± 0.281 3.3 x 10 -1 Z3-3 -12.66 ± 0.374 1.1 X -1.83 ± 0.071 1.5 x 10 -2 V1 -13.96 ± 0.023 -2.13 ± 0.083 7.5 x 10 -3 V2-2 -14.08 ± 0.065 8.3 X 10 -15 1.6 x 10 -2 9.0 X io -- ' 5 -1.79 ± 0.090 -14.05 ± 0.074 V3-1 1.2 X 1 0 -14 -2.03 ± 0.130 9.3 x 10 -3 V3-2 -13.93 + 0.090 9.9 X i o -15 -1.71 ± 0.023 1.9 x 10 -2 -14.00 ± 0.044 V3-3 -2.26 ± 0.071 6.4 X i o -15 5.5 x 10 -3 -14.20 + 0.062 W1 W2A-1 2.6 x 10 -14 -1.97 + 0.071 1.1 x 10 -2 W2A-2 -13.59 ± 0.060 -1.29 ± 0.059 5.1 x 10 -2 -13.52 ± 0.112 3.0 x 10 -14 W2A-3 W2A-4 1.6 x 10 -2 2.5 x 10 -15 -1.80 ± 0.377 -14.61 + 0.094 W3-1 3.2 x 10 -14 -0.30 ± 0.846 5.0 x 10-1 W3-2 -13.50 ± 0.021 169 TABLE 5.6. Parameters identified for cross-hole test PP8 treating the medium as spatially uni form. k [m2] logio 0 [m3 /m3 ] log lo k [m 2 ] Interval 45 [m 3 /m3 ] -2.12 + 0.018 7.5 x 10 -3 8.1 x 10 -15 -14.09 ± 0.008 X1 2.6 x 10 -2 1.1 x 10 -14 -1.59 ± 0.032 -13.96 ± 0.013 X2-1 1.3 x 10 -2 -14.31 ± 0.047 4.8 x 10 -15 -1.90 ± 0.111 X2-2 -2.02 + 0.082 -14.11 + 0.045 7.7 x 10 -15 9.5 x 10 -3 X2-3 X3 -1.19 ± 0.020 -14.12 ± 0.013 6.4 x 10 -2 Y1-2 7.5 x 10 -15 -14.07± 0.058 -1.16 ± 0.142 6.9 x 10 -2 8.5 x 10 -15 Y2-1 -1.12 ± 0.100 2.1 x 10 - ' 4 7.6 x 10 -2 -13.68 + 0.005 Y2-2 -1.62 ± 0.264 2.4 x 10 -2 4.3 x 10 -15 Y2-3 -14.37 ± 0.089 8.3 x 10 -2 -1.08 ± 0.005 -13.20 ± 0.004 6.4 x 10 -14 Y3-1 1.7 x 10 -1 -0.76 ± 0.148 -13.41 + 0.027 3.9 x 10 -14 Y3-2 -1.21 + 0.075 1.5 x 10 -14 6.1 x 10 -2 -13.83 ± 0.026 Y3-3 -1.18 ± 0.047 2.8 x 10 -13 6.7 x 10 -2 Z1 -12.55 ± 0.052 -1.18 ± 0.069 6.7 x 10 -2 -13.29 ± 0.025 5.2 x 10 -14 Z2-1 -1.40 ± 0.019 4.0 x 10 -2 2.1 x 10 -13 -12.68 ± 0.014 Z2-2 3.1 x 10 -12 -0.30 ± 0.028 5.0 x 10 -1 -11.50 ± 0.039 Z2-3 4.7 x 10 -1 1.0 x 10 -12 -0.33 ± 0.450 Z2-4 -11.98 + 0.843 1.3 x 10 -14 8.5 x 10 -2 -1.07 ± 0.057 Z3-1 -13.89 + 0.042 4.1 x 10 -2 1.6 x 10 -14 -1.39 + 0.034 Z3-2 -13.81 ± 0.018 3.0 x 10 -1 2.7 x 10 - ' 3 -0.52 ± 0.054 Z3-3 -12.56 ± 0.058 1.4 x 10 -2 -1.87 + 0.059 6.5 x 10 -15 -14.18 + 0.015 V1 1.1 X 10 -2 1.1 x 10 -14 -1.96 + 0.070 V2-2 -13.95 ± 0.029 2.5 x 10 -2 1.1 X 10 -14 -1.60 ± 0.159 -13.98 + 0.040 V3-1 -2.31 ± 0.099 4.9 x 10 -3 6.1 x 10 -15 -14.21 + 0.019 V3-2 2.7 x 10 -15 3.3 x 10 -2 -1.48 ± 0.082 -14.57 ± 0.048 V3-3 1.2 x 10 -2 1.1 x 10 -14 -1.91 ± 0.041 -13.94 + 0.017 W1 -1.62 ± 0.056 2.4 x 10 -2 4.4 x 10 -14 W2A-1 -13.36 1 0.022 -1.46 ± 0.088 3.5 x 10 -2 7.0 x 10 -15 W2A-2 -14.15 ± 0.040 4.5 x 10 -2 1.5 x 10 -14 -1.35 ± 0.046 -13.83 ± 0.027 W2A-3 -1.24 ± 0.024 1.1 x 10 -14 5.8 x 10 -2 W2A-4 -13.94 ± 0.026 1.0 X 10 -2 -1.98 + 0.106 5.2 x 10 -15 -14.28 ± 0.109 W3-1 8.8 x 10-3 -2.06 + 0.038 7.0 x 10 -15 -14.15 + 0.023 W3-2 170 TABLE 5.7. Summary statistics for log lo k [m 2 ] identified for cross-hole tests treating the medium as spatially uniform. Minimum Test Sample size Maximum Mean Variance CV PP4 -14.42 -11.52 -13.57 0.568 32 -0.0555 -12.22 PP5 21 -15.15 -13.94 0.397 -0.0452 -14.4 -12.0 PP6 -13.66 0.220 -0.0343 23 PP7 -15.61 -12.0 29 -13.67 0.560 -0.0562 -14.57 -11.5 -13.68 PP8 31 0.530 -0.0533 -11.5 All -15.61 -13.69 0.479 136 -0.0505 TABLE 5.8. Summary statistics for log io 0 [m 3 /m 3 ] identified for cross-hole tests treating the medium as spatially uniform. Test Sample size Minimum Maximum Mean Variance CV PP4 -2.29 29 -0.52 -1.53 0.168 -0.267 PP5 21 -2.92 -0.98 -1.81 0.279 -0.292 PP6 22 -2.64 -0.58 -1.80 0.237 -0.271 PP7 26 -2.29 -0.49 -1.67 0.254 -0.302 -1.46 -2.31 PP8 30 -0.33 0.225 -0.326 128 -2.92 0.241 -0.301 All -0.33 -1.63 It is of interest to note that the injection intervals in cross-hole test PP4 (Y22) and single-hole test JG0921 (discussed in Chapter 4) virtually coincide. Though the injection rate during PP4 (1 x 10 -3 kg/a) had significantly exceeded that during JG0921 (8.014 x 10 -6 and 3.967 x 10 -5 kg/a), both tests yielded similar permeabilities and porosities for the injection interval (Tables 5.2 and 4.11). The estimates are k = 2.1 x 10' and 0 = 6.0 x 10 - 2 for the cross-hole test, and k = 2.4 x 10' and 0 = 1.4 x 10 -2 for the single-hole test. Some of the poor matches obtained for test PP4 (Figure 5.9) can be significantly improved by adding an adjustable borehole storage parameter to both the injection and observation intervals. In my analysis of PP4, borehole porosities along intervals are fixed as defined in Chapter 3 (Equations 3.65 and 3.66, page 101). In my analysis of single-hole tests (Chapter 4), there was only one adjustable borehole storage pa- rameter at the injection interval, Ow . Here I consider two such parameters, log io 0„/ and log lo 0„°. The new matches for eight of the intervals are depicted in Figure 5.14, and the corresponding parameters are listed in Table 5.9. Variations in the two stor- 171 X2-3 Y2-1 io ' 10° 10 -1 10 -2 10 -.3 10 -2 16-1 W2A-2 '-i"o°' 10 1 lo° 10° 10 1 - 1 0° 10 -1 10 -1 10 -2 10 -2 10 210 110° FIGURE 5.14. Separate matches between observed (small dots) and simulated (thick curves) responses for test PP4 assuming uniform medium and adjusting borehole storage parameters along injection and observation intervals. TABLE 5.9. Parameters identified for cross-hole test PP4 treating the medium as spatially uniform and adjusting borehole storage at injection and observation intervals. loglo 0 {m 3 /m3 Interval logio k [m 2 logio OL, [770/m 3 ] loglo O w° [m 3 /m 3 ] X2-2 -14.18 ± 0.010 -3.03 ± 1.484 1.83 ± 0.210 1.37 ± 0.544 X2-3 -2.69 ± 1.327 -13.95 ± 0.038 1.97 ± 0.531 1.42 ± 0.754 Y2-1 -14.00 -1.69 1.75 1.30 Y2-3 -14.21 ± 0.013 1.71 + 0.343 -2.57 ± 1.176 0.97 + 0.474 -2.99 ± 1.224 2.59 1 0.118 2.67 ± 0.073 Y3-3 -13.47 ± 0.013 -14.26 ± -3.83 ± 1.353 1.46 ± 0.065 V3-3 0.016 1.97 ± 0.028 1.82 W2A-2 -14.00 -2.38 1.86 -2.59 ± 0.018 W3-2 -13.96 ± 0.002 1.96 ± 0.009 1.85 ± 0.012 ] ] age parameters are an indication of variations in effective geometry of test intervals and medium heterogeneity. To check the influence of borehole conductance and storage effects, the analysis of test PP4 was repeated without including open borehole intervals in the model. Pressure in each test interval is calculated as the average of pressures at all nodes along the interval. Estimates of log io k and log io q obtained with and without borehole effects are compared in Figure 5.15. Neglecting the influence of boreholes causes a systematic increase in the estimates of both parameters by a factor of about 1.4. This means that open test intervals enhance the conductive and storage properties of the 172 i] t 1Mth borehole effects FIGURE 5.15. Estimates of pneumatic properties for test PP4 obtained with and without borehole effects. medium and must not be ignored. As I have already discussed in Chapter 3, a constant uniform pressure was specified at the top boundary of the computational region. Absolute barometric pressure was not measured at the ALRS during the cross-hole pneumatic tests. I therefore assumed that barometric pressure was equal to 100 kPa, which is the average barometric pressure at sea level. In fact, ALRS is at an elevation of 1,200 m above sea level, and the average barometric pressure should be lower, approximately 87 kPa, but can vary with meteorological conditions. To test the effect of this barometric pressure discrepancy on my estimates, I repeated the inverse analysis of test PP4 with a barometric pressure equal to 87 kPa. The two sets of estimates are compared in Figure 5.16. The smaller barometric pressure produced larger estimates of k by a factor of about 1.1, but did not affect the log io 0 estimates (except for a few intervals indicated in Figure 5.16 which are characterized by poor matches and uncertain estimates). The factor 1.1 is close to the ratio between the adopted barometric pressures (101/87 = 1.16). Thus, uncertainty in barometric pressure may cause a small systematic error in log io k estimates, which can be corrected a posteriori. Using analytical type-curves 173 based on linearized airflow equations, Illman and Neuman [2000a] however obtained opposite results. They concluded that a decrease in barometric pressure would not alter k estimates but would decrease 0 estimates by a factor of 1.1. To investigate this discrepancy, I examine the non-linearized governing airflow equations (3.1 — 3.4, page 71) neglecting the gravity term for consistency with analysis of Illman and Neuman [2000a]. An increase or a decrease in barometric pressure produces a similar change in air pressures within the rock during a pneumatic test. Absolute air pressure and air density are proportional to each other, as indicated by Equation 3.4. The pressure change would not modify significantly the gradient Vp and time derivatives 3p/at of air density p; it would influence predominantly the product of k and p in Equations 3.1 and 3.3. Therefore, a decrease in barometric pressure would produce an equivalent increase in permeability estimates. This is consistent with the results yielded by my numerical inverse model. On the other hand, the linearized p-based equation (3.78 on page 114) applied by Illman and Neuman [2000a] produces different estimates for different initial barometric pressures only due to change in the air compressibility C.. Illman and Neuman [2000a] took Ca to be a constant in space and time, and equal to the inverse of barometric pressure. Therefore, the analytical results are biased by linearization of the governing airflow equations. To date, analytical type-curves were used to analyze test PP4 [Illman et al., 1998; Illman, 1999; Illman and Neuman, 2000a]. The most up-to-date results are those given in [Illman and Neuman, 2000a]. The analytical and numerical estimates of log lo k and log io 0 are compared in Figure 5.17. Though type-curve results for k consistently exceed numerical results by a factor of about 1.7, the two data sets correlate quite well if one ignores intervals Z1, Z2-2, Z2-3, Z2-4 and Z3-3 (which, as discussed earlier, are located in a highly permeable zone and connected to the atmosphere). The difference can be due in part to round-off errors in both the numerical and analytical methods (the analytical solution relies on numerical inversion of the Laplace transform). When open monitoring boreholes are excluded from the numer- 174 1E+0 1E-11 1E-1 1E-2 1E-15 1E 15 1E-14 1E-13 1E-12 Barometric pressure 100 kPa 1E-11 1E-3 1E-3 1E-2 1E-1 1E+0 Barometric pressure 100 kPa FIGURE 5.16. Estimates of pneumatic properties for test PP4 obtained for different initial barometric pressure. ical model, the systematic difference between numerical and analytical k estimates is reduced to a factor of 1.4. This suggests that type-curve analysis assigns higher permeabilities to the rock so as to compensate for its inability to fully account for the influence of open borehole intervals on pressure. Further, the analytical solution solves a linearized version of the governing equation (3.78, page 114) as discussed in Section 3.4, in which air compressibility is taken to be a constant. During cross-hole tests, pressure buildups were significant in magnitude compared to the barometric pressure, and varied substantially in time and space. Air compressibility is approxi- mately equal to the inverse of absolute air pressure (see Section 3.1.1), and therefore, also varied considerably. This can be another reason for the discrepancy between numerical and analytical estimates. Other differences between the two methods of solution include the effect of constant-pressure boundary on the ground surface, and the need to prescribe arbitrary boundary conditions in the numerical model. Overall, there is a good agreement between log io q5 estimates obtained by numerical and analytical methods, except for intervals Y2-3, Y3-1, Z2-1, Z2-2, Z2-3, Z2-4, Z3-3 and V3-3. This is due to the ambiguous pressure records in these intervals: stor- 175 1E-11 - 1E+° 1E-1 Air-filled porosity [m3/ mil - 1E-12 - ir Z 1E-13 ZF5-4 - Z -n '3 1E-14 1E-4 Zp 1E-15 1E15 1E-14 1E-13 1E-12 1E-11 1E-5 1E-5 Numerical solution FIGURE 1E-4 1E-3 1E-2 1E-1 1E+0 Numerical solution 5.17. Analytically versus numerically derived pneumatic properties for test PP4. age effects heavily influences Y2-3 and V3-3; the Z intervals are located in a highly permeable zone connected to the atmosphere. I attempted to interpret pressure records from all monitoring intervals simultaneously while treating the rock as uniform [Illman et al., 1998]. This led to very poor matches, which is a sign of medium non-uniformity. 176 5.2.2 Non-uniform medium I used the stochastic inverse model described in Chapter 3 to interpret simultaneously pressure records from each of cross-hole tests PP4, PP5 and PP6. Here I describe in detail the inversion of test PP4. I also performed a simultaneous inversion of pressure records from all three tests. These analyses amount to high-resolution stochastic imaging, or pneumatic tomography, of the fractured rock mass, an idea proposed over a decade ago by Neuman [1987] in connection with hydraulic cross-hole tests in saturated fractured crystalline rocks at the Oracle site near Tucson [Hsieh et al., 1985]. The stochastic inverse model regards medium properties as spatially correlated random fields, characterized by variogram models with adjustable parameters, and conditioned on adjustable pneumatic parameters at a set of pilot points. In my study, I utilize between 32 and 72 pilot points located subjectively in the computational domain. Most points are placed along test intervals. To test the influence of pilotpoint locations, I inverted the same test data (PP4) using two different sets of pilot points with equal size. In the first case, all pilot points are located at middle of test intervals, while in the second case, the points are in middle between the injection and corresponding observation intervals (plus there is again a pilot point at center of injection interval). The first case produced much more satisfactory results than the second one in terms of objective function and estimation errors. This suggests that pressure responses in monitoring intervals are influenced substantially by the properties of the medium in the vicinity of corresponding intervals. Locating the pilot points predominantly along boreholes allows also better comparison between inverse estimates and the available ALRS data (single-hole air permeability, core matrix airfilled porosity, fracture density) which are also defined along the boreholes. The inverse model requires specifying initial values for the unknown parameters. In all numerical inversion presented below, initial values for log io k and log lo 0 were 177 set equal to the geometric average of each, as obtained from the earlier uniform inverse interpretations of PP4 pressure data. The numerical inversion started with spatially uniform values that were then successively perturbed and adjusted throughout the optimization process. The exponents of power variograms of the parameter fields were estimated both simultaneously with and independently of the other unknowns. When the exponents were assessed simultaneously with pneumatic parameters, their initial values for both log io k and log io ci) were set equal to 0.45, the value obtained through geostatistical analysis of single-hole air permeability data (Chapter 2). To check the influence of initial values on final parameter estimates, the initial values were either decreased by an order of magnitude, or set equal to random values uniformly distributed from —16 to —13 for log io k and from —3 to —1 for log io q5. This analysis proved that initial values have a minor effect on the final estimates. All numerical inversions converged in less than 50 iterations. Depending on the number of tests and unknown parameters considered simultaneously, the inversions required from 1,000 to 5,000 forward model runs which took from 10 to 50 hours of computational time when using a supercomputer with 32 processors. Cross-hole test PP4 Before starting the inverse analysis, I simulated test PP4 using the kriging estimates of air permeability (Figure 2.18, page 58) and matrix air-filled porosity (Figure 2.26, page 66) discussed in Chapter 2. The calculated pressure responses differed markedly from those observed in the field [Illman et al., 1998]. An attempt to improve the matches using trial and error proved to be difficult and unsuccessful. I also tried to parameterize the unknown spatial medium properties through firstand second-order polynomials. These attempts failed. The stochastic inverse model was applied to a set of 32 pilot points located within each of the 32 intervals that had responded during test PP4. Most of the pilot points were located at the centers of the corresponding intervals. In the guard intervals, 178 0 -10 N -20 3 -30 -40 FIGURE 5.18. Three-dimensional representation of 32 pilot points. Y2-1 and Y2-3, the points are offset from the center toward injection interval Y2-2, to define better medium properties between these intervals. In the long interval V1, the point is offset toward the ground surface where most airflow takes place during pneumatic tests. The location of pilot points is depicted in Figure 5.18. First, I treated log io k as non-uniform and log io 0 as uniform. The log io k random field was described through unknown values at pilot points and an unknown exponent of the corresponding power variogram. Also unknown was the uniform value of log lo 0, leading to 34 unknowns. Figure 5.19 shows simultaneous matches between computed and monitored pressures in all 32 intervals obtained with the corresponding 179 parameter estimates. The sum of squared pressure residuals 0 was 679 kPa 2 . Most of the computed responses capture in a satisfactory manner the late-time observed pressures. However, there are substantial deviations at early time, which suggests that the uniform log io 0 assumption is not realistic. The corresponding kriged estimate of log io k is shown by a three-dimensional fence diagram in Figure 5.20; the exponent of the power variogram was estimated to be equal to 1, its specified upper bound. The uniform log io 0 estimate was equal to —2.0. The region depicted in Figure 5.20 and in all similar figures presented later has dimensions 45 x 33 x 33 (49,005) m 3 , and includes all the test boreholes. To take into account spatial non-uniformity of log io 0, the number of unknowns was increased to 66 by including log-transformed air-filled porosities at the 32 pilot points, and the exponent of their power variogram. Figure 5.21 shows the resulting simultaneous matches between computed and recorded pressures. Compared to the previous case with uniform log io 0 (Figure 5.19), the fits are significantly improved, especially at early time. I compared the new matches with those obtained through separate analysis of individual responses (Figure 5.9). The sum of squared differences, 0, between computed and recorded pressures for all intervals was equal to 131.2 kPcg in the first case and 104.4 kPa 2 in the second case. Both the number of unknowns and the number of match points were the same in the two cases. Upon comparing the matches in Figures 5.9 and 5.21, one notes that for some of the intervals—X2-2, Y2-3, Y3-2, Z3-2, V3-3 and Wl—the simultaneous matches are better. The simultaneous matches for intervals Y3-1, Z1, Z2-1, Z2-2, Z2-3, Z2-4 and Z3-3 under- or over-estimate the observed pressure responses; however, the simulated curves represent more adequately the slope of observed responses compared to the separate analysis. A histogram of residuals between computed and recorded pressures obtained by stochastic inverse model is shown in Figure 5.22. The figure suggests that the population distribution of residuals is not normal, which is confirmed by a Kolmogorov- 180 X2-1 X1 X2 - 3 10° X3 10° 10 - ' 10 ° 110 1 0-2 10' 100 102 10 -2 10' 10 0 10 • Y2-3 O 1 0° Y3-1 io° Y3-2 10' 10° 10' 10 10° ' 10 -0 10. 0 -2 104 10 -2 10-' 10 0 10' Z1 10-' 10° 8:0 io° 1 0 ' 10 0 Z2-3 10 10 4 r 10 100 10' 10 -0 10 Z2-2 10' 0.2 10 4 10 210 -110 01 0 ' 10 0 Z2-1 10-' 102 Z3-1 10° o Z2-4 r 10 _1 10' 10 0 10° 1 0' 10' 10 10° 1 • ' 5.19. Matches between observed (small dots) and simulated (thick curves) responses obtained by simultaneous inversion of PP4 data with 32 pilot points; log io FIGURE is assumed to be uniform. 181 30 -10 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20 FIGURE 5.20. Three-dimensional representation of kriged log io k estimated by simultaneous inversion of PP4 data with 32 pilot points; log io 0 is assumed to be uniform. 182 X2-1 X1 10° X2-3 10° 10° 10 -2 10° 110 100 10" 10' 10° 10° 10 -0 10 -0 10" 10 Z2-4 10° o 10 -1 10 -2 10 -3 10 -2 10' 10° 10° 10° 10 -1 FIGURE 5.21. Matches between observed (small dots) and simulated (thick curves) responses obtained by simultaneous inversion of PP4 data with 32 pilot points. 183 Smirnov test. Though most of the residuals are close to zero, their sample distribution exhibits heavy tails due to the matches for intervals Xl, X2-3, V3-1, V3-3 and W2A-1. To improve these matches, the number of pilot points would have to be increased. The exponents 0 of power variograms are estimated both simultaneously and independently with the rest of the unknowns, and in both cases, the best current estimate was /3 = 1, the specified upper limit. Setting the exponent of the power variogram to values lower than 1 for both log io k and log io qS causes the matches to deteriorate. Unfortunately, I could not extend my analysis to values of 0 higher than 1. The pilot-point estimates of log io k and log io 0, as well as associated separate 95% confidence limits, are listed in Table 5.10. The points are identified by the intervals in which they are located. The confidence limits are computed from the diagonal terms of the covariance matrix of estimation errors E aa . E aa is computed using Equation 3.56 (page 89) without including prior information. Prior information would most probably have reduced the estimation errors. However, as we are going to see later, there are other factors that influence estimation uncertainty. Most log io k estimates seam reasonable and are associated with relatively small errors. Though four of them reach the specified upper (-10) and lower (-20) limits, their errors are nevertheless low. Compared to log lo k, the log io 0 estimates have larger errors; more of them are at the specified upper (-5) and lower (-0.3) limits. The covariance matrix of estimation errors at pilot points, E„,„ is depicted in Figure 5.23 as a two-dimensional map of square pixels. The parameters are numbered in the order they are listed in Table 5.10. The first set of 32 parameters describes air permeabilities, the second set air-filled porosities. Thin solid lines separate between matrix blocks associated with these two sets of parameters. Figure 5.24 displays the corresponding correlation matrix. One notes that although the cross-covariances of porosity estimation errors (upper right block in Figure 5.23) are higher than those of permeability (lower left block in Figure 5.23), the cross-correlations of porosity 184 80 60 40 20 0 0 0 *000 % *00 *t50 \ *00 '60 .6‘ 0 7 0*00 7 •tS 0 •0 FIGURE 5.22. Histogram of residuals between observed and simulated responses. 185 5.10. Parameters identified at the pilot points by numerical inversion of data from cross-hole test PP4 using 32 pilot points. TABLE Interval X1 X2-1 X2-2 X2-3 X3 Y1 Y2-1 Y2-2 Y2-3 Y3-1 Y3-2 Y3-3 Z1 Z2-1 Z2-2 Z2-3 Z2-4 Z3-1 Z3-2 Z3-3 V1 V2 V3-1 V3-2 V3-3 W1 W2-1 W2-2 W2-3 W2-4 W3-1 W3-2 logn k [m 2 ] -11.60 ± 0.23 -15.27 + 0.17 -15.29 ± 0.27 -16.01 ± 0.17 -20.00 + 0.30 -17.76 + 0.32 -16.61 ± 0.17 -13.05 ± 0.03 -14.56 ± 0.07 -12.95 ± 0.17 -20.00 ± 0.25 -20.00 + 0.64 -10.00 ± 0.37 -19.06 1 0.41 -14.37 + 0.33 -14.42 ± 0.38 -14.35 ± 0.44 -12.40 ± 0.41 -11.93 ± 0.20 -13.15 ± 0.36 -18.62 ± 0.23 -15.07 ± 0.37 -11.29 ± 0.18 -15.44 ± 0.07 -14.45 ± 0.66 -12.96 ± 0.74 -18.93 ± 0.46 -16.70 ± 0.15 -14.18 ± 0.18 -13.81 ± 0.14 -16.61 ± 0.03 -14.60 ± 0.19 logio q5 [m 3 /m3 ] -2.17 ± 0.87 -2.40 ± 0.22 -3.81 ± 0.48 -5.00 ± 0.45 -5.00 1 0.97 -2.72 ± 0.33 -2.53 ± 0.40 -2.38 ± 0.27 -0.30 ± 0.55 -5.00 ± 0.70 -5.00 ± 0.40 -5.00 ± 1.16 -1.25 ± 1.07 -5.00 ± 0.55 -1.63 ± 0.96 -2.76 + 0.79 -1.84 ± 0.93 -0.30 ± 0.46 -0.41 ± 0.25 -0.30 ± 1.15 -5.00 ± 0.78 -4.59 ± 0.94 -1.02 ± 0.39 -4.23 ± 0.16 -1.17 ± 1.15 -1.68 ± 1.43 -5.00 ± 0.76 -4.18 ± 0.51 -2.18 ± 0.37 -1.65 ± 0.44 -2.88 ± 0.58 -5.00 ± 0.51 186 estimation errors (upper right block in Figure 5.24) are lower than those of perme- ability (lower left block in Figure 5.24). The higher cross-covariances of porosity errors are consistent with the higher uncertainties of porosity estimates. The high cross-correlation between permeability errors suggests that the separate confidence intervals, discussed above, underestimate log io k uncertainties. The correct estimation of log lo k errors require simultaneous, in my case also nonlinear, confidence intervals which I have not estimated because this would be computationally very demanding. Figures 5.23 and 5.24 suggest that cross-covariances/cross-correlations between permeability and porosity errors (upper left or lower right blocks) are smaller than those between permeability errors (lower left block) and those between porosity errors (upper right block). However, there is cross-correlation (close to 1) between permeability and porosity errors at some of the pilot points (diagonal terms of upper left matrix block in Figure 5.24). This is as a result of the correlation between estimated parameters through the governing airflow equations. Absolute components of the covariance matrix of log io k and log io c/5 estimation errors are plotted versus spatial distance between the corresponding pilot points in Figure 5.25. The corresponding absolute correlation matrix components are plotted in Figure 5.26. The two figures suggest that correlation between estimation errors of log lo k and log lo 0 does not depend on the spatial configuration of the pilot points. Figure 5.27 depicts eigenvector components of the covariance matrix E cia . The eigenvectors are ordered according to the magnitude of their corresponding eigenvalues; that with smallest (2 x 10 -7 ) eigenvalue is numbered 1, and that with highest eigenvalue (1.3) is numbered 64. A thin solid line separate the first 32 components of each eigenvector, which correspond to log io k, from the next 32 components, which correspond to log io q5. Parameters associated with small eigenvalues are less uncer- tain than those associated with large eigenvalues. The components of each eigenvector represent the relative contribution to it by various parameter estimates. For example, the first elgenvector (with the smallest eigenvalue) is associated almost entirely 187 0.25 6 0.20 0.15 0.10 5 0.05 -0.00 -0.05 -0.10 -0.15 -0.20 -0.25 2 1 10 20 30 40 50 60 Parameters FIGURE 5.23. Two-dimensional representation errors at the pilot points. of covariance matrix of estimation 188 1.00 0.90 0.80 0.70 0.60 - 0.50 0.40 0.30 0.20 0.10 -0.00 -0.10 -0.20 -0.30 -0.40 -0.50 -0.60 -0.70 -0.80 -0.90 -1.00 10 20 30 40 50 60 Parameters FIGURE 5.24. Two-dimensional representation of correlation matrix of estimation errors at the pilot points. • 189 0.14 logio k [ 1112 ] 0.12 0.1 g u c (a i 0.08 o u g 0.06 Ti u) _o <0.04 * • • s• • • • • • •re s •.• • 8 • ..• • • • • • ••• .•:• •:• . :11,• • •4 eik :.,•* ..: . . : • 0.02 •• • f 10 O. 20 30 10 Distance between pilot points [m] 20 30 Distance between pilot points [m] Absolute components of the estimation covariance matrix versus dis- FIGURE 5.25. tance between the pilot points. 1.0 0.9 0 1?; 0.8 o 4=co 0.7 o 0.6 o .tO 0.5 a) 5 04 . u =-' 0.3 '4) 5 A C 0.2 •ct 0 .1 - 1.0 • I9g1q 0.9 4 is 0.8 •• • • • • k Ern 2 1 • •• • • s •••• •...• • • • • • • • • • •: . • • • •. log10 o [m 3 /m 3 ] - i3 tcu cu 0 .i657 oo • •• • • . *. • • •• • :••• • • • . .. . • . • .s. .• • • • .••••• • • •.1. . • • % . • • %.••• • • •.; . •••.u: . ••., • • :• • . • : : ..4* . 112, • .1„. _.. •%. . si. .. • 8... e.b -i .. ...„ .. • 1, . ..: • . ••• ,:f.• • .% .1.1.: ^ ,%.: .. • .....•• fti. ...s ..,..:.r. • .• •• ....._ . _„,„„%.:-,.....:: . • , I : • • 72 , , : Distance between pilot points o to • 5 0.4 0 23 0.3 '5 m u) 0.2 < 0.1 [m] FIGURE 5.26. Absolute components of the estimation correlation matrix versus distance between the pilot points. 190 with the eighth parameter, which is the permeability at the injection interval Y2-2. Therefore, the corresponding estimate is the least uncertain (as one can also see in Table 5.10). The last eigenvector (with the largest eigenvalue) is associated with a number of porosities which are poorly estimated and cross-correlated. Figure 5.27 demonstrates that eigenvectors associated with the smaller eigenvalues are dominated by permeability estimates. Therefore, the permeabilities are estimated with less uncertainty than porosities. The largest absolute eigenvector components associated with all parameter estimates are shown in Figure 5.28. The larger is this component for a given parameter, the smaller is its estimation error. The magnitude of largest absolute eigenvector components agrees with the size of confidence intervals in Table 5.10. Kriged images of estimated random log io k and log io 0 fields are depicted by threedimensional fence diagrams in Figures 5.29 and 5.30, respectively. These images were generated with power variograms having exponents equal to 1. The two pneumatic parameters exhibit comparable patterns of spatial variability, in that regions of high and low permeability correspond quite closely. Posterior sample variograms of the high-resolution estimates in Figures 5.29 and 5.30 are presented in Figure 5.31. Both increase more or less linearly with separation distance. The coefficients a of log io k and log io ç variograms are equal to 0.1 and 0.026, respectively. 191 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 -0.00 -0.10 -0.20 -0.30 -0.40 -0.50 -0.60 -0.70 -0.80 -0.90 -1.00 10 20 30 40 50 60 Eigenvectors FIGURE 5.27. Two-dimensional representation of eigenvectors of covariance matrix of estimation errors at the pilot points. 192 1.0 0.0 Parameters FIGURE 5.28. Maximum absolute eigenvector components associated with parameter estimates. 193 30 -10 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20 FIGURE 5.29. Three-dimensional representation of kriged log io k estimated by simul- taneous inversion of PP4 data with 32 pilot points. 194 - 10 1.5 -2 -2.5 -3 -3.5 -4 -4.5 -5 FIGURE 5.30. Three-dimensional representation of kriged log io estimated by simultaneous inversion of PP4 data with 32 pilot points. 195 2.5 2.0 - - -0- logio k [m 2 ] - _ -G logio 0 [m /m 3 ] 0... 3 / a X) f3" 1.5 0.0 0 5 10 Distance [m] 15 20 FIGURE 5.31. Sample variograms of kriged log io k and log lo 0 estimates obtained by simultaneous inversion of PP4 data with 32 pilot points. 196 Figures 5.32, 5.33 and 5.34 show what happens when PP5, PP6 and PP7 are sim- ulated using the above log io k and log io 0 estimates. There are substantial discrepancies between computed and measured pressures, especially in observation intervals close to injection intervals. However, the calculated curves do not differ substantially from the measured records for intervals close to the Y2 borehole. This is so because during test PP4, air was injected into borehole Y2 so that pressure responses close this borehole were distinct, allowing the local medium non-uniformity to be estimated with higher accuracy compared to the rest of computational domain (Table 5.10). To improve the matches between simulated and observed pressures, I analyzed test PP4 with the aid of 64 pilot points. The additional points were placed between and along test intervals. Figure 5.35 shows the resultant simultaneous matches between computed and recorded pressures. The sum of squared pressure residuals, 1), is now only 17 kPa 2 . Compared to the case of 32 points (Figure 5.21), the new matches are much better in most intervals, especially in X2-3, Y3-1, Y3-3, Z2-1, Z3-1, V1, V3-1, V3-3, W2A-1, W2A-2, W2A-3 and W3-2. Some of the latter intervals are within the previously defined highly permeable zone. I conclude that the numerical inverse model is capable of representing in a satisfactory manner the highly non-uniform fracture network at ALRS. Adding pilot points improves the matches, but does not change the overall spatial distribution of log io k and log io 0, which remain similar to those obtained using 32 pilot points (Figures 5.29 and 5.30). 197 X1 X2-3 10° 1 0' 10° 10 0 10" 10 to ° 10' 0 10° V3-2 10° 10 •; 10 1 o' 100 10' 10" 10° 10 0 10' 10° W2A-2 10" 10' 10 102 10 -a 104 10° 10' 10 -° W3-2 10° W3-1 10' 10 0 .f-44 100 10 10 16' 10 1 0' 10 0 PP5 applying parameter fields estimated by simultaneous inversion of PP4 data with 32 pilot points. FIGURE 5.32. Forward simulation of test 198 X2-1 X2-2 X2-3 10" 10 10' 10 10 10 • .3 Y3-2 10' 10 10 10' 10' 100 10 10' 30 20 10 -0 10 1 00 10" 10 -1 10 -0 10 10' 10 -1 10' 10' 10. 0 1 0 -3 10' 10' 10 -0 10" 5.33. Forward simulation of test PP6 applying parameter fields estimated by simultaneous inversion of PP4 data with 32 pilot points. FIGURE 199 X1 1 0' o • °- 10° 10010' 10" 10° Y2 - 1 10" 10 10 -0 10' 10' 10 -21 o° Y2-3 10°- 10 10' 10" 10" 10' 10 -0 10 4 10 -0 16' • W2A-1 10° 16-0 10-'10° 16" 4° 1 1 0° W2A-2 10° 10' 10" ro' 10° 10 - ' 5.34. Forward simulation of test PP7 applying parameter fields estimated by simultaneous inversion of PP4 data with 32 pilot points. FIGURE 200 X2-1 X1 X2-2 X2-3 1 00 10° 10' 10 -0 10 -0 110 100 10° 1 0' 10 -0 1 0' 0.4 0.2 1 0 -2 o' n 10' v1 FIGURE 5.35. Matches between observed (small dots) and simulated (thick curves) responses obtained by simultaneous inversion of PP4 data with 64 pilot points. 201 Cross-hole tests PP4, PP5 and PP6 The stochastic inverse model was used to interpret simultaneously pressure records in all test intervals from three tests labeled PP4, PP5 and PP6, during which air was injected at different rates into isolated intervals in boreholes Y2, X2 and Z3, respectively. I employed 72 pilot points located along some of the intervals as well as between them and the injection intervals. The location of the pilot points is depicted in Figure 5.36. There was a total of 146 unknown parameters including log lo k and log io 4) at 72 pilot points and two variogram exponents. Resulting estimates of log io k and log io 0 and associated separate 95% confidence limits are listed in Table 5.11. Pilot points located along the boreholes are numbered from top to bottom. In most of the longer observation intervals, there are more than one pilot points. Points marked by letters A, B, C, D and E are located as follows: A — between Y2-2 and X2-2; B — between Y2-2 and Z3-2; C — next to X2-2; D — next to Z2-2; and E between X1 and W3-2. Most log io k estimates appear to be reasonable and are associated with relatively small errors; this is especially true for estimates at pilot points along the boreholes. Only eight estimates reach the specified upper (-10) and lower (-20) limits, but their confidence limits are narrow. Compared to log io k, the log io ck estimates have much wider confidence intervals and a larger number of them (12) have reached the specified upper (-5) and lower (-0.3) limits. Figures 5.37, 5.38 and 5.39 show simultaneous matches between computed and recorded pressures during tests PP4, PP5 and PP6, respectively. Of the 96 sets of observed records in the three figures (32 corresponding to each of the three tests), only 76 include match points. This not withstanding, only a small number of the 96 matches are poor (due to barometric pressure effects and instrumentation problems), the rest being of intermediate, good to excellent quality. The sum of squared pressure residuals, II), for all three tests is only 133.2 kPa 2 . To this PP4 contributes 45.4, PP5 82.7, and PP6 5.1 kPa 2 . We recall that when we matched individual responses under 202 0 -10 N -20 3 -30 -40 FIGURE 5.36. Three-dimensional representation of 72 pilot points. 203 the assumption that the medium is uniform (Figures 5.9, 5.10, 5.11), the sum of all squared pressure differences at the same match points was equal to 173.9 kPa 2 of which 104.4 was contributed by PP4, 68.8 by PP5 and 0.7 by PP6. Figures 5.40 and 5.41 show three-dimensional perspectives of estimated log io k and log lo 0 fields, respectively. These estimates are similar to those obtained earlier through inversion of PP4 test data alone (Figures 5.29 and 5.30). 204 TABLE 5.11. Parameters identified at the pilot points by numerical inversion of data from cross-hole tests PP4, PP5 and PP6 using 72 pilot points. Interval X1 X2-0 X2-1 X2-2 X2-3 X2-9 X3-0 X3-1 Y1 Y2-0 Y2-1 Y2-2 Y2-3 Y2-4 Y3-0 Y3-1 Y3-2 Y3-3 Y3-4 Z1 Z2-0 Z2-1 Z2-2 Z2-3 Z2-4 Z3-0 Z3-1 Z3-2 Z3-3 Z3-4 V1-0 V1-1 V2 V3-1 V3-2 V3-3 WI W2-1 W2-2 W2-3 W2-4 W3-0 W3-1 W3-2 W3-3 AO Al A2 A3 A4 AS A6 A7 A8 BO Bl B2 B3 B4 B5 B6 B7 B8 Cl C2 C3 C4 D1 D2 D3 D4 El log, k [m a ] -12.99 + 0.91 -13.67 + 0.56 -18.08 + 0.29 -13.19 + 0.10 -16.92 + 0.25 -14.24 + 0.36 -12.50 + 0.28 -15.16 + 0.78 -15.42 + 0.37 -16.88 + 0.43 -16.82 + 0.36 -12.70 + 0.10 -14.94 + 0.14 -19.95 + 0.38 -15.01 + 0.48 -14.13 + 0.98 -14.87 + 0.95 -20.00 ± 0.63 -18.11 + 0.87 -15.02 + 0.35 -17.54 + 0.72 -14.01 + 0.62 -14.07 ± 0.66 -14.57 + 0.67 -13.95 + 0.56 -12.63 + 0.63 -14.87 1 0.38 -13.51 + 0.17 -15.79 + 0.53 -16.68 ± 0.46 -20.00 + 0.36 -10.00 + 0.62 -14.33 + 0.80 -10.00 + 0.32 -14.55 + 0.60 -20.00 ± 0.75 -14.34 ± 0.64 -20.00 + 0.96 -13.52 + 0.33 -12.26 + 0.39 -10.00 + 0.49 -19.79 + 0.02 -15.93 + 1.18 -16.01 + 2.95 -20.00 + 1.35 -17.28 + 0.28 -13.42 + 0.39 -15.26 + 0.55 -13.14 + 0.59 -19.02 + 0.63 -14.92 + 0.34 -13.14 + 0.69 -12.11 + 0.68 -14.47 + 0.82 -15.62 + 0.57 -14.24 + 0.55 -19.51 + 1.19 -15.60 + 1.03 -13.92 + 1.15 -15.95 + 0.63 -14.73 + 1.28 -13.29 + 1.79 -18.29 + 1.19 -12.87 + 1.11 -14.81 + 1.06 -14.23 + 1.18 -18.09 + 1.39 -14.60 + 0.98 -17.05 + 1.84 -13.20 + 1.43 -15.73 + 1.79 -11.24 1 0.98 4, [rn 3 /m 3 ] -2.33 + 1.67 -0.30 + 1.32 -5.00 + 1.50 -3.95 ± 0.41 -5.00 ± 2.11 -5.00 ± 0.87 -2.54 ± 1.78 -2.87 1 2.01 -0.30 ± 0.87 -5.00 + 0.93 -1.11 + 1.27 -5.00 + 0.40 -0.66 ± 1.07 -4.58 1 1.22 -3.04 1 4.40 -2.61 + 3.03 -2.81 + 1.48 -3.04 ± 2.05 -2.67 ± 1.46 -2.61 + 1.27 -2.73 + 2.14 -2.46 + 3.70 -2.47 ± 2.32 -2.50 + 1.31 -1.83 + 1.23 -0.30 + 1.72 -1.36 + 1.13 -5.00 ± 0.61 -4.32 ± 1.85 -1.54 ± 1.50 -3.97 + 2.21 -0.30 ± 2.12 -1.70 + 2.27 -0.39 ± 0.80 -1.67 ± 0.86 -4.90 ± 1.80 -2.67 + 3.18 -4.01 + 2.24 -2.43 ± 1.73 -2.46 ± 1.21 -4.98 ± 0.93 -3.75 + 1.24 -3.36 + 4.95 -3.57 + 4.49 -4.11 + 3.43 -2.52 + 1.59 -1.61 + 1.07 -1.67 ± 2.42 -1.36 + 2.30 -2.90 + 2.33 -0.44 + 2.51 -1.64 ± 2.96 -0.30 1 2.62 -1.96 + 2.97 -2.83 + 1.24 -1.81 ±3.17 -4.34 + 2.00 -1.11 + 2.79 -3.01 + 2.91 -3.52 + 2.76 -1.99 + 2.68 -0.89 1 3.31 -2.74 ± 3.20 -2.63 + 2.02 -2.90 ± 2.42 -3.53 + 2.99 -2.77 + 3.11 -3.33 + 3.80 -5.00 + 3.21 -2.73 + 2.84 -5.00 + 2.91 -3.19 ± 2.73 log" 205 X2 - 1 X1 X2-3 10 .0 1 0' 10' 10.0 1 0° 10 1 0° 10 .0 5.37. Matches between observed (small dots) and simulated (thick curves) responses for test PP4 obtained by simultaneous inversion of PP4, PP5 and PP6 data. FIGURE 206 X1 X2-2 0.4 0.3 0.2 0.1 100 10' 50 10' 1 0' 10 0 10' 1 0° 1 0' lo ' 10 -2 10' 10' FIGURE 5.38. Matches between observed (small dots) and simulated (thick curves) responses for test PP5 obtained by simultaneous inversion of PP4, PP5 and PP6 data. 207 X2 - 3 X1 X3 10' • .y. 1 10' 10° 0.2 16r° 10 16' 16° 16' 16 -2 16' 16° Z1 0.1'5 0.1 0.05 - Z2-4 0.25 0.2 10° 0.15 0.1 10' 5 10' 10' 10 10 V3-1 0.3 0.2 02 8:4 0.2 0.1 0.1 ,...01°* 1 0 ' 16 4 W2A-1 ' 1 0° 0.15 0.1 0.05 1 0.76 0.5 W2A-4 10' 0.25 10 -2 10' 10° 10' 5.39. Matches between observed (small dots) and simulated (thick curves) responses for test PP6 obtained by simultaneous inversion of PP4, PP5 and PP6 data. FIGURE 208 30 -10 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20 Three-dimensional representation of kriged log io k estimated by simultaneous inversion of PP4, PP5 and PP6 data. FIGURE 5.40. 209 -10 FIGURE 5.41. Three-dimensional representation of multaneous inversion of PP4, PP5 and PP6 data. kriged log lo estimated by si- 210 5.3 Discussion 5.3.1 Comparison between numerical and analytical analyses Illman and Neuman [2000a] developed analytical type-curves for the interpretation of cross-hole pneumatic tests. Their results are comparable to those of my inverse model. Nevertheless, there are major differences between the two approaches. Whereas analytical type-curves represent a linearized solution, the numerical model solves directly the nonlinear governing airflow equation. Whereas analytical models assume that air compressibility is constant, the numerical model accounts for its dependence on absolute air pressure. Whereas the particular type-curves used by the above authors treat the injection interval as a point, my numerical model represents its real shape and volume. Whereas the type-curves account indirectly for storage of air in pressure monitoring intervals, my numerical model does so directly and in all open intervals of each borehole at the site (not only those in which pressure is analyzed). Whereas the type curves assume that pressure equalizes instantaneously along each monitoring interval, my numerical model allows for rapid airflow and pressure equalization within each open borehole interval by modeling it more realistically as a high-permeability and high-porosity medium. My numerical model additionally accounts for the variation in borehole storage due to variation in air compressibility. My numerical model considers the effect of an atmospheric boundary at the ground surface, though at the expense of introducing artificial no-flow boundaries at the sides and the bottom of the modeled rock volume. The presence of open boreholes in the numerical model enhances pneumatic connection between the tested rock and the atmosphere. The most important advantage of the numerical model is that it allows resolving heterogeneities on a small scale, and to consider all test data simultaneously, thus yielding a high-resolution stochastic (tomographic) image of these heterogeneities. 211 5.3.2 Comparison between present stochastic inverse model and previous applications of pilot-point method The pilot-point methodology and its application were first described in the dissertation of Ghislain de Marsily [1978]. Later, the method was modified and used for the inverse analysis of several synthetic and real data sets [Certes and de Marsily, 1991; LaVenue and Pickens, 1992; RamaRao et a/., 1995; LaVenue et al., 1995]. There are, however, some important differences between my approach and those used previously. In the past, the pilot-point method was applied only to two-dimensional problems. I work in three dimensions. The method has not been used previously to identify the spatial distribution of storativities (or, equivalently, porosities). Kriging was commonly performed by utilizing not only pilot points but also actual measurements of model parameters. I include only pilot points in my inverse algorithm. Kriging was always performed using given variogram models with predetermined parameters based on actual data. I treat the variogram model parameters as unknowns to be determined by the inverse model. Parameters at pilot points were always treated as exact measurements. I view the pilot points as conditioning points of unknown stochastic fields, and corresponding values as uncertain estimates. La Venue and Pickens [1992] used sequential optimization to define the location of pilot points within the computational domain. However, their procedure does not allow identification of estimation uncertainty at the pilot points. I predefine the location of the pilot points and estimate the unknowns simultaneously. I also tried to optimize simultaneously both location and estimates of the pilot points. However, the optimization procedure became computationally very demanding. I believe, however, that this may be feasible and should be pursued in the future. 212 5.3.3 Comparison between different models In the previous section, I described various stochastic inverse models utilizing different sets of pilot points, and analyzing different test data. All produced comparable results in terms of matches and spatial distributions of parameter estimates. This demonstrates a certain robustness of my hydrogeologic conceptualization and analysis. Alternative stochastic inverse models can be compared using the four model identification criteria introduced by Carrera and Neuman [1986a] and discussed in Chapter 3. The values of the four criteria calculated for models of test PP4 using 32 and 64 points as well as the model of PP4, PP5 and PP6 using 72 points are listed in Table 5.12. The table also presents the optimum values of the sum of squared residuals €1. and natural log determinant of the covariance matrix of estimation errors E act . All four criteria identify the inverse model interpreting simultaneously tests PP4, PP5 and PP6 as the best by a wide margin. This reflects the fact that using a relatively small number of pilot points (unknowns), the model provides a satisfactory representation of observed pressures during all three tests. The other 2 models interpret only PP4 data. Compared to the model of PP4 with 32 points, the PP4 model with 64 points produces a much better representation of observed pressures, but is less parsimonious. These two models produce close criteria estimates. The first three criteria—AIC, BIC and Om—rank the 64 pilot-point model higher than the 32 pilotpoint model, and only the criterion dm suggests the opposite. Carrera and Neuman [1986c] and Samper and Neuman [1989c] suggested that the dm criterion is the most suitable for model discrimination because it is the most sensitive to model parsimony. 5.3.4 Scale effects In Tables 5.13 and 5.14, I list statistics of air permeability and air-filled porosity estimated by different methods. For log io k, single-hole data represent the combined set of 1-m and 3-m support data as discussed in Chapter 2. The separate analysis of 213 TABLE 5.12. Comparison of different numerical inverse models using model identification criteria (numbers in brackets show model ranking). Models Criteria PP4 PP4 PP4, PP5, PP6 Number of pilot points 32 64 72 Number of unknowns 128 64 144 Number of match points 252 252 489 131.2 cl) 17.03 133.2 -9 -348 -158 In lEaal AIC -1131 (3) -1396 (2) -2879 (1) BIC -905 (3) -944 (2) -2275 (1) -1040 (3) -1214 (2) -2642 (1) Sb m dm -1013 (2) -2382 (1) -832 (3) individual pressure records is based on data from cross-hole tests PP4, PP5, PP6, PP7 and PP8 (Tables 5.2 - 5.6); the statistics are derived from 136 and 128 estimates of log lo k and log io 0, respectively (Tables 5.7 and 5.8). The stochastic numerical inverse model was used to analyze all pressure records from PP4, PP5 and PP6 separately from each test and simultaneously from all three tests; the listed log lo k and log lo çb statistics are calculated from high-resolution stochastic estimates over a portion of the computational region presented in the three-dimensional fence diagrams above (for example Figures 5.29 and 5.30). This three-dimensional region (45 x 33 x 33 (49,005) m 3 ) encompasses test boreholes and predominant portion of the flow during cross-hole tests. It is assumed that this portion of tested rock was characterized by means of the stochastic inverse analysis of the test data. The same region was used also for the kriging of the ALRS data in Chapter 2 (Figures 2.18 - 2.26, pages 58 - 66). The region includes 53,176 nodes over a regular structured grid with resolution 1 x 1 x 1 (1) m 3 ; the number of nodes define the sample size. Geostatistical analysis of single-hole data and stochastic inverse analyses of tests PP4 and PP5, as well as of tests PP4, PP5 and PP6 simultaneously, produce comparable values of the mean and variance of log ic) k and log io 0. The inversion of test PP6 only could not define that well overall medium heterogeneity since not all test intervals produced reliable records. Compared to the rest of stochastic models, corre- 214 TABLE 5.13. Summary statistics for log io k [ml identified using different data and methods. Source Sample size Mean Variance CV 227 —15.22 Single-hole tests (uniform medium) 0.87 —0.061 Cross-hole tests Uniform medium - PP4,5,6,7,8 Non-uniform medium PP4 (32 pilot points) PP4 (64 pilot points) - - - - - PP5 PP6 PP4,5,6 TABLE 5.14. Summary statistics for and methods. Sample size Source Cross-hole tests 136 —13.69 0.48 —0.051 53176 53176 53176 53176 53176 —15.15 —15.07 —15.26 —14.02 —15.69 3.17 1.95 1.64 0.64 2.48 —0.117 —0.093 —0.084 —0.057 —0.100 log io 0 [m 3 /m 3 J identified using different data Mean Variance CV 128 —1.63 0.24 —0.301 53176 53176 53176 53176 53176 —3.00 —2.47 —3.07 —1.82 —2.98 1.45 1.24 2.12 0.66 1.00 —0.402 —0.450 —0.475 —0.469 —0.337 Uniform medium - PP4,5,6,7,8 Non-uniform medium PP4 (32 pilot points) PP4 (64 pilot points) - - - - - PP5 PP6 PP4,5,6 sponding mean log io k and log io 0 are higher since not only does injection take place in the highly permeable zone along Z borehole series, but also pressure responses are recorded predominantly in observation intervals located in this highly permeable zone. Compared to all other estimates, log io k and log io 0 obtained from numerical inverse analyses of individual pressure records, while treating the rock as being uniform, have a much higher mean value and much lower variance. Compared to the stochas- tic inverse model of tests PP4, PP5 and PP6 simultaneously, the uniform geometric mean of k is higher by exactly two orders of magnitude, and the log io k variance is 1.9 times lower; the uniform geometric mean of 0 is higher by more than one order of magnitude (a factor of 22), and the log io 0 variance is 4.2 times lower. Compared 215 to the single-hole data, the uniform mean and variance of log io k are 34 times higher and 1.8 times lower, respectively. This scale effect is evident also in the histograms presented in Figures 5.42 and 5.43. Permeability scale effects were reported by others [Neuman, 1990; Clauser, 1992; Rovey and Cherkauer, 1995; Sanchez-Vila et al., 1996; Meier et al., 1998; Samper and Garcia, 1998; Schulze-Makuch and Cherkauer, 1998; Martinez-Landa et al., 2000]. The permeability scale effect has been questioned by Zlotnik [1994] and Butler and Healey [1998] who suggested that it is due to an inadequate interpretation of small-scale field experiments such as slug and single-hole tests; they emphasized on the impact of skin effect on the small-scale estimates. However, in my analysis, the scale effect in the medium properties can be defined from cross-hole data alone, so these problems are not applicable. In addition, the consistency between single-hole estimates and stochastic inverse results suggest the former were adequately identified. The type-curve analysis of Illman et al. [1998] and Illman and Neuman [2000b] also indicates that the skin effect is negligible for the single-hole tests conducted at ALRS. Rovey [1998] and Meier et al. [1998] demonstrated through two-dimensional synthetic experiments that indeed scale effect in permeability may result from medium heterogeneity. To date, the analysis of single-hole tests has not provided air-filled porosity data for comparison with the high-resolution stochastic estimates. As described in Chapter 4, some porosity values were estimated on the basis of single-hole tests by means of numerical inversion, and they range from 9.6 x 10 -2 to 3.6 x 10 -3 . Therefore, it is not possible to confirm the scale effect in air-filled porosities with single-hole test estimates. Treating the rock as being uniform, when it is non-uniform, yields relatively high estimates of mean and low estimates of variance for both air permeability and airfilled porosity. Scale of inverse analyses is defined by the resolution of the medium non-uniformity over the computational domain. When the medium is viewed as being uniform, there is no resolution in medium heterogeneity and the scale of analysis is defined by the size of the computational domain. Clearly, this scale is much larger than 11111,111millIIII1 M111111111111111111M IlliIIIIIIIIIIIIIIIIIIII Al ILIMMIMMI m NI ME:1111111111111111111111 ih. r A.' IIIMIMINI IIn E MIE IIMMIM1111111111 A! nn .u. 1 Ilk re 'UW111111111111 A 1 111111=11111111111111 11 Dr AMP "41 1111111111111 MIIn IMM11111111 A 11111111111 V A Csi 6 6 Ci 6 Apuenbaij angelem o 217 u 111111 111111111 1111111111k\ ctZi 11111111111111 I CO C 4 O CO CO O Aouenbau angeiou 7 O 218 the support scale of singe-hole tests. As a result, there are substantial differences between the permeability and air-filled porosity statistics obtained from uniform analysis of single-hole and cross-hole test data. On the other hand, my stochastic numerical model resolves the non-uniformity of the medium at a scale (resolution) defined by the grid employed to compute kriging estimates. In our case, kriging is performed at the nodes of a computational grid. In the central portion of the computational domain, where most airflow takes place, the predominant distance between nodes is 1 m, which corresponds to the nominal support scale of single-hole tests. The grid along the boreholes has separation distances between nodes less than 1 m, but it accounts for a relatively small portion of the computational domain. Therefore, there is consistency between the scale of inverse analysis and the scale of single-hole data, and there is no scale effect in estimated medium properties. This is consistent with a earlier conclusion by Neuman [1990, 1994] that permeability (as well as dispersivity) scale effects diminish as medium non-uniformity is explicitly resolved by parameter estimation. 5.3.5 Comparison between spatial patterns of permeability obtained by various methods The overall pattern of spatial variability of log io k obtained from my stochastic inverse model (Figure 5.29, page 193 and Figure 5.40, page 208) is quite similar to that obtained from the kriging of single-hole data (Figure 2.18, page 58). There is good correlation between high-permeability and low-permeability zones. Though the range of values in Figure 2.18 is narrower than that in Figures 5.29 and 5.40, their mean values are very close (Table 5.13). The higher variances of high-resolution inverse estimates, compared to the single-hole data, can be due to several reasons. First, the domain size of high-resolution images is much larger than the rock region tested by the single-hole data. Since a power model with an exponent equal to 1 is applied, 219 the variance increases with domain size. Second, there are estimation errors associ- ated with the values at the pilot points. Therefore, the variance should represent not only spatial variability of the true unknown parameter field, but also the estimation errors. Third, the variance is influenced by the parameterization of the spatial distribution of medium properties. In the case of test PP4, the variance decreases as the number of pilot points increases (Table 5.13; see also Figure 5.45 discussed in the next section). The larger number of pilot points allows smoother representation of medium non-uniformity. However, the increase in the number of pilot points enlarges the uncertainty of pilot-point estimates. As already discussed in Chapter 2, the kriging estimates of fracture density are very different than those of air permeability obtained from single-hole data. The same is true for the high-resolution stochastic images of air permeability obtained from cross-hole data. In Figure 5.44, I compare air permeability profiles along four boreholes, obtained from single-hole tests and by the inverse model of cross-hole tests PP4, PP5 and PP6 simultaneously. Both sets of data represent values on a nominal scale of 1 x 1 x 1 m 3 . In borehole Y2, the two sets of log io k values correlate quite closely. This is so despite there being only 5 pilot points along the borehole. The same is true for borehole V2. In X2 and Y3, the inverse model captures only the general trend of log io k variations as defined by the single-hole data; both methods define a relatively high permeability zone close to the top and a relatively low permeability zone at the bottom. The numerical inverse model yielded power variogram exponents equal to 1, their specified upper limit. Geostatistical analysis of single-hole air permeability data in Chapter 2 yielded a lower exponent equal to 0.45. In Figure 5.45, I compare the sample variograms of 1-m and 3-m single-hole data against the sample variogram of kriging estimates from stochastic inverse models. The differences in the variograms are due to higher variability of inverse estimates. In the case of test PP4, the variogram estimates decrease as the number of pilot points increases. 220 ... X2 OS •n Y2 ... .. . .., 1.1. I .. -, V. 0.4 00 ... ... 0 • - - - - SH data -5 WE MN MN Inverse estimate I= .. . NM --- -1 .1•1 4 . N 1 - 1 . E .4 • N ,- - - - SH data n Inverse estimate -15 .41., • 444, -15 o 40 • - il 10, .. *I s. -20 i -20 -17 -18 -16 -15 log io k [m 2 ] -14 -1 3 - -18 -17 -16 -15 -14 -13 log lo k [m 2 ] - .... Y3 ... -10 n - n n SH data Inverse estimate • ..' ... - 40 •• 40 V2 - -5 - ) 10 t -10 - - E N-15 N - -20 - -20 - 1 / „„. - -25 / - - n ** .44 SH data 40. ,... ... ** " Inverse estimate . 1 1 * 1 -17 1 1 -16 1 1 1 1 -15 1 1 -14 1 -13 -12 log 10 k [m 2 ] FIGURE 5.44. Air permeability along boreholes estimated from single-hole tests and by simultaneous inversion of PP4, PP5 and PP6 data. 221 3.0 - - 2.5 1-m SH data 3-m SH data PP4 32 points PP4 64 points PP4, PP5, PP6 -- - 2.0 - 1.5 - E cr) o as > 1.0 0.5 - - 0 00 o log io k [m 2 ] , . FIGURE 5.45. Sample variograms of log lo k estimated from single-hole data and through stochastic inversion of cross-hole data. 222 In my study, high-resolution estimates of log io k have been obtained independently by means of two very different test and interpretation methods, using two independent sets of data with different support scales. Single-hole data yield local estimates of permeability in the close vicinity of the test intervals. Cross-hole test data represent three-dimensional flow spanning the rock on a relatively larger scale. Although very different, the methods of analysis are similar because the medium properties are viewed as spatially correlated random fields and their heterogeneity is resolved with equivalent resolution. 5.3.6 Comparison between permeability and porosity estimates I already mentioned that high-resolution images of log io k and log io 0 obtained by the numerical inverse model display similar three-dimensional patterns. This is seen on the three-dimensional fence diagrams in Figures 5.29 - 5.30 and 5.40 - 5.41, where regions of high and low permeability correspond quite closely to similar regions of porosity. The inverse estimates in the first two figures are plotted against each other (small dots) in Figure 5.46. Estimates of log io k and log io 0 at the pilot points are indicated by open circles. The plot suggests a correlation between log io k and log io 0, with a slope of approximately 1:2. Using very different methods of analysis, linear relationships between log io k and log io 0 with similar slopes have been estimated by laboratory analyses of sedimentary rock samples [cf. Nelson, 1994; Worthington, 1997]. More importantly, Guimera and Carrera [2000] reevaluated published data of a series of field tracer tests conducted in fractured rocks; their analysis suggested also a linear relationship between log io k and log io 0 with slope between 1:2 and 1:3. 5.3.7 Borehole effects I already noted that open borehole intervals impact the interpretation of pneumatic cross-hole tests due to preferential airflow through, and enhanced storage within, 223 -18 -16 -14 log io k [m 2 -12 -10 ] FIGURE 5.46. Air permeability versus air-filled porosity estimated by stochastic inversion of PP4 data at pilot points (open big circles) and within the computational domain. 224 these intervals. To examine this issue in more detail, I simulated test PP4 without boreholes using log io k and log io ck estimates from the inversion of PP4 with 64 pilot points. Calculated pressures with and without boreholes are compared in log-log and semi-log plots in Figures 5.47 and 5.48. The first figure emphasizes differences at early time, and the second at late time. It is clear that the presence of open borehole intervals has a considerable effect on pressure propagation through the site, and on pressure responses within intervals. These responses can be different in shape and either higher or lower than those calculated without open borehole intervals. The distribution of pressures across two-dimensional vertical sections through the numerical model at the end of test PP4 are shown in Figure 5.49 for the case where open borehole intervals are not accounted for, and in Figure 5.50 for the case where they are. The open borehole intervals, the packers, and their effects on pressure distribution are clearly evident in Figure 5.50. The figure also demonstrates that boreholes X2, X3 and Z3 are venting the system. Both figures show how the atmospheric boundary causes the pressure distributions to exhibit vertical asymmetry. The effects of lateral and bottom no-flow boundaries on pressure are seen to be slight. It is clear that borehole effects must be taken into consideration in the interpretation of pneumatic tests. 225 10' 10 1 01 10° , X3 10° 10' 1OE° 10 0 10 4 10° 10' 10 100 1 1 0° 5.47. Simulated responses with (solid curve) and without (dashed curve) borehole effects for cross-hole test PP4 iii log-log plots. FIGURE 226 X2 - 1 10 X2-2 X3 10 8 8 6 6 120 4 4 110 2 2 10 .2 10 90 10 2 Y3-3 10" 10° 12 10 8 6 4 2 10 10° Z2 - 3 0.3 0.2 0.1 0.2 0.1 0.2 and without (dashed curve) borehole effects for cross-hole test PP4 in semi-log plots. FIGURE 5.48. Simulated responses with (solid curve) 227 L( ) Q. 2 " a 6 6 6 6 6 Co g E o o cu E c0 a II N 2E E o 1 1 1 1 1 1 1 1 1 1 1 1 I 0 oj [ W Z 1 1 I 0 Co 0 7?" o 1I. I 1 o [W] Z 1 1 CO 1 1 1 71" 1 1- 228 73' ul ,- ,- o in O. 2 " ad d ci ci d 0.1 Q) o c'n el E ( 0 E o o -o 0 Q) 0 o 0 cJ I I I I I I I I I I I 1 I ttO 0 Q) Cnj c\J [w] z o °E X o I I 0 [ W] Z I Co I 1 1 229 Chapter 6 CONCLUSIONS The following major conclusions arise from my study: 1. It is possible to interpret single- and cross-hole pneumatic tests in unsaturated fractured tuffs at the ALRS by means of a three-dimensional numerical inverse model, which treats the rock as a locally isotropic, uniform or non-uniform continuum and air as a single mobile fluid phase. The rock continuum represents primarily interconnected air-filled fractures. 2. The inverse model is capable of accounting for storage and conductive effects of all existing open borehole intervals at the site, both monitored and not monitored. During cross-hole tests, these effects influence airflow significantly, and neglecting them produces a systematic bias in the estimation of pneumatic parameters. Single-hole tests are affected primarily by conductance and storage in the injection interval. 3. In the case of single-hole tests, transient records of multiple injection steps and recovery are amenable to simultaneous analysis by means of my inverse model. Storage of injection interval depends on air compressibility, and decreases with the increase of absolute air pressure. Borehole storage dominates pressure tran- sients during the first step of each test and makes it difficult to obtain reliable estimates of air-filled porosity and borehole storage coefficient from such data. Pressure transients during subsequent steps of the test are affected by borehole storage to a lesser extent than are those during the first step. More importantly, there is virtually no storage impact on recovery data. A consideration of variable air compressibility and a joint analysis of pressure data from all stages of 230 a single-hole pneumatic test are therefore essential for the reliable estimation of air-filled porosity and borehole storage coefficient. The storage effect ob- served during single-hole tests seems related not only to the injection interval, but also to openings in the surrounding rock. Air permeabilities obtained by our inverse method are comparable to those obtained by means of steady-state formulae [Guzman et al., 1996] and transient type-curve analyses [Illman et al., 1998]. Due to linearization of airflow equations, analytical type-curves [Illman and Neuman, 2000b] did not match observed pressure records and did not allow reliable identification of air-filled porosity and borehole storage effects. 4. Currently, the numerical inverse analysis has been performed for only four of the single-hole tests. The analysis of all the single-hole tests would provide air-filled porosity data for comparison with stochastic inverse results. The interpretation of single-hole data could be facilitated through numerical derivation of appro- priate type-curves, i.e. "numerical type-curves". 5. Qualitative analysis of data from cross-hole tests PP4, PP5, PP6, PP7 and PP8 demonstrates that the medium is well pneumatically interconnected and acts as a three-dimensional continuum. The medium contains highly permeable zones, some of which extend to the ground surface. 6. Assuming uniform medium, my three-dimensional inverse model is capable of analyzing independently individual pressure records from cross-hole tests PP4, PP5, PP6, PP7 and PP8. For test PP4, which is the only test analyzed to date by analytical type-curves, the latter and my numerical model produce similar results, though there is a systematic difference in permeability estimates. 7. Inman and Neuman [2000b, a] developed analytical type-curves for the inter- pretation of single- and cross-hole pneumatic test data which allow simple, fast and reliable assessment of pneumatic medium properties. Disadvantages of the 231 type-curve approach include reliance on linearized airflow equations; inability to consider the dependence of air compressibility on absolute air pressure; ability to account for borehole storage in only one monitoring interval at a time; inability to consider the effect of open boreholes on pressure distribution through the system; inability to consider barometric pressure effects; treatment of the rock as being uniform; and inability to analyze simultaneously pressure data from multiple monitoring intervals when the rock is non-uniform. 8. In contrast, my numerical inverse model solves the airflow equations in their original nonlinear form; considers the dependence of air compressibility on abso- lute air pressure; accounts directly for the ability of all boreholes, and packed-off borehole intervals, to store and conduct air through the system; can, in principle, account for atmospheric pressure fluctuations at the soil surface; provides kriged estimates of spatial variations in air permeability and air-filled porosity throughout the tested fractured rock volume; and can be applied simultaneously to pressure data from multiple borehole intervals as well as to multiple cross-hole tests. It has the disadvantage of being complex, difficult and time consuming. 9. Assuming non-uniform medium, my stochastic inverse model is able to produce three-dimensional high-resolution images of air permeability and air-filled porosity through simultaneous analysis of data from multiple borehole intervals and multiple cross-hole tests. This interpretation amounts to "pneumatic to- mography" or "high-resolution stochastic imaging" of the fractured rock mass, an idea proposed over a decade ago by Neuman [1987] in connection with hy- draulic cross-hole tests in saturated fractured crystalline rocks at the Oracle site near Tucson [Hsieh et al., 1985]. 10. The stochastic numerical inverse model is capable of representing in a satis- 232 factory manner observed pressure records during cross-hole tests. Stochastic estimates of air permeability and air-filled porosity obtained using data from different tests and pilot-point arrangements are similar, which demonstrates the robustness of our analysis and the proper conceptualization of hydrogeologic conditions at the site. 11. There is a good correlation between the spatial patterns of air permeability and air-filled porosity as obtained by the inverse procedure. All of these patterns differ markedly from the spatial pattern of fracture densities. A correlation analysis between log-transformed air permeability and log-transformed air-filled porosity yields a ratio close 1:2. 12. Air permeability data from single-hole pneumatic tests, as well as data con- cerning fracture density, overall water content, matrix porosity and matrix van Genuchten's a, are amenable to continuum geostatistical analysis. This means that each data set can be viewed as a sample from a random field, or stochastic continuum, as proposed for permeability over a decade ago by Neuman [1987, 1988b] and affirmed more recently by Tsang et al. [1996]. 13. The statistical and geostatistical analyses of single-hole air permeabilities at 1- m and 3-m support scale can be combined, which improves the characterization of medium non-uniformity. 14. Both geostatistical analysis of single-hole data and geostatistical inverse analysis of cross-hole data, yield a similar geometric mean and a similar spatial pattern of air permeability at the ALRS. This consistency demonstrates a certain ro- bustness of my hydrogeologic conceptualization and stochastic inverse analyses, and imbues me with some confidence that the obtained estimates constitute a valid representation of fracture pneumatic properties at the ALRS. However, the inverse estimates vary over a much broader range than do the single-hole 233 test results. This can explained as a result of (1) larger sample and domain size of inverse estimates, (2) errors in the inverse estimates and (3) parameterization of the spatial distribution of medium properties. The spatial correlation of air permeability is characterized by both methods with a power variogram, which is representative of a random fractal field with multiple scales of spatial correlation. To date, the analysis of single-hole tests has not provided enough air-filled porosity data for comparison with stochastic inverse results. 15. Analysis of individual pressure cross-hole records independently, assuming uniform medium, produces higher geometric means and lower variances for pneumatic parameters compared to those estimated from single-hole data and by stochastic analyses of cross-hole tests PP4, PP5 and PP6. This implies a pronounced scale effect for both air permeability and air-filled porosity. 16. Scale of inverse analyses is defined by the spatial resolution of medium nonuniformity over the computational domain. In the uniform case, there is no spatial resolution in medium heterogeneity and the scale of cross-hole test analysis is much larger than the support scale of singe-hole test data. This produces a scale effect. The stochastic numerical model resolves non-uniformity on a scale comparable to the support scale of single-hole test data. This eliminates any scale effect. These findings are consistent with earlier conclusion by Neuman [1990, 1994] that permeability scale effects diminish as the medium non-uniformity is explicitly resolved by parameter estimation. Permeability scale effects were also reported by others [Clauser, 1992; Rovey and Cherkauer, 1995; Sanchez-Vila et al., 1996; Meier et al., 1998; Samper and Garcia, 1998; Sch,ulze-Makuch and Cherkauer, 1998; Martinez-Landa et al., 2000]. 17. There are differences between my pilot-point approach and those used previously. The pilot-point method was applied only to two-dimensional problems. I 234 work in three-dimensions. The method has not been used previously to identify the spatial distribution of storativities (or, equivalently porosities). Kriging was commonly performed by utilizing not only pilot points but also actual measurements of model parameters. I include only pilot points in my inverse algorithm. Kriging was always performed using given variogram models with predetermined parameters based on actual data. I treat the variogram model parameters as unknowns to be determined by the inverse model. Parameters at pilot points were always treated as exact measurements. I view the pilot points as conditioning points of unknown stochastic fields, and the estimates as uncertain quantities. La Venue and Pickens [1992] used sequential optimization to define the location of pilot points within the computational domain. However, their procedure does not allow identification of estimation uncertainty at the pilot points. I predefine the location of the pilot points and estimate the unknowns simultaneously. I also tried to optimize simultaneously both location and estimates of the pilot points. However, the optimization became computationally very demanding. I believe, however, that this may be feasible and should be pursued in the future. 18. 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