A MODELING ATTEMPT TO DETERMINE THE RELATIONSHIP BETWEEN MACROPOROSITY AND DEFORMATION MODULUS OF TUFF B. Burcin Avar, Scott Wilson, Ltd., Chesterfield, Derbyshire, U.K. Nick Hudyma, School of Engineering, University of North Florida, Jacksonville, FL, USA ABSTRACT A physical and numerical modeling investigation was undertaken in an attempt to determine the relationship between macroporosity and deformation modulus. The physical models consisted of cubic plaster of Paris specimens with two different macroporosity configurations; open ended cylindrical holes (2D specimens) and spherical cavities (3D specimens). The numerical modeling consisted of uniaxial compression testing under plain-strain conditions using FLAC-2D on models containing circular holes. Results show that cavity/hole size, shape, and location within the specimen have an effect on the deformation modulus. The data provided an upper and lower bound of deformation modulus as a function of macroporosity. The upper bound is a linear regression from the 3D physical models and the lower bound is a logarithmic regression from the 2D physical models and lithophysal tuff test results. The development of such bounding values will enable engineers to have confidence when assigning deformation modulus values in numerical models. RÉSUMÉ Une investigation physique de modelage et numérique a été entreprise dans une tentative pour déterminer la relation entre modulus de macroporosity et déformation. Les modèles physiques ont consisté en le plâtre cubique de spécimens de Paris avec deux configurations de macroporosity différentes ; ouvrir a terminé des trous cylindriques (2D spécimens) et les cavités sphériques (3D spécimens). Le modelage numérique a consisté en l'essai de compression uniaxe sous les conditions de simple-tension utilisant FLAC 2D sur les modèles contenant des trous circulaires. Les résultats montrent à cette taille de cavité/trou, à la forme, et à l'emplacement dans le spécimen a un effet sur le modulus de déformation. Les données ont fourni une limite supérieure et plus basse de modulus de déformation comme une fonction de macroporosity. La limite supérieure est une régression linéaire du 3D modèles physiques et la limite plus basse sont une régression logarithmique du 2D modèles physiques et les résultats de test de tuff de lithophysal. Le développement de tel limité des valeurs rendront capable des ingénieurs avoir confiance en assignant modulus de déformation estime dans les modèles numériques. 1. INTRODUCTION Porosity within a rock specimen or a rock mass has considerable effect on its strength and stiffness. Typically porosity is microscopic, consisting of spaces between individual mineral grains. However there are lithologies that have macroscopic pores, which can be seen with an unaided eye. Examples of such lithologies include vuggy or karstic limestone, vesicular basalt, and lithophysal tuff. The lithophysae or lithophysal cavities were formed by trapped pockets of air within the volcanic ash that forms the tuff. From a practical standpoint, it is important to determine the variation in deformation modulus of the rock as a function of macroscopic porosity so that appropriate values can be input into numerical modeling programs used for design. To quantify the change in deformation modulus with increasing macroporosity, both physical and numerical modeling studies were conducted. Unconfined compression tests were conducted on cube shaped specimen of plaster of Paris containing open cylindrical holes (2D specimens), spherical Styrofoam inclusions (3D specimens). The numerical modeling performed was plane strain finite difference modeling using FLAC 2D. In addition, specimens of lithophysal tuff were also tested.

The deformation modulus used in this study is akin to Young’s or elastic modulus. The term Young’s modulus is strictly appropriate only for intact rock specimens because Young’s modulus is a function of cracks, cavities and inclusions (Schultz, 1996). Therefore, the slope of stress-strain curve for specimens containing cavities is termed deformation modulus. The computed deformation modulus is a function of the macroporosity of the plaster of Paris specimens and the total porosity of the lithophysal tuff specimens. Macroporosity as applied to the physical models is the ratio of the volume (3D) or area (2D) of voids to the volume or cross sectional area of the specimen. The total porosity of the specimens includes both microscopic porosity and macroporosity. 2. PHYSICAL MODELING Two sets of physical models were used as part of this study: 2D plaster of Paris specimens and 3D plaster of Paris specimens. Difficulties obtaining cylindrical specimens of lithophysal tuff have been documented (Martin et al, 1995), so cube shaped specimens were

Sea to Sky Geotechnique 2006

510

produced to match the shape of the lithophysal tuff specimens used. The plaster specimens were prepared in an aluminium mold and had side lengths of approximately 152.4 mm. 2.1 2D Physical Models The 2D plaster specimens contained cylindrical holes extending through the specimen. The specimens were tested perpendicular to the specimen faces through which the holes passed. Twenty three specimens were produced with either an uniform hole distribution (6 specimens) or random hole distribution (17 specimens). The uniform hole distribution specimens had either 36 holes (4 specimens) or 9 holes (2 specimens). The random hole distribution specimens had either 20 holes (1 specimen) or 24 holes (16 specimens). Hole diameters used for the specimens included 6.35, 9.5, 15.9, 19.05, 25.4, and 31.75 mm. Macroporosity of each of the specimens was calculated using the area of the holes divided by the cross-sectional area of the specimen. Figure 1 shows examples of a 9-hole structured specimen, a 36-hole structured specimen and a 24-hole random specimen. Further details on the 2D physical models can be found in Avar (2002).

Figure 1. Examples of 9-hole, 36-hole, and 24 random hole 2D models. Arrows indicate loading orientation. 2.2 3D Physical Models The 3D plaster specimens contained spherical Styrofoam inclusions within the specimens. The specimens had either structured inclusions or random inclusions. The structured inclusions occurred in a central (solar system) configuration (2 specimens) or a stacked configuration where inclusions were stacked at the four corners of the cube (3 specimens). For the structured inclusions, the void diameters ranged from 101.6 mm to 25.4 mm. Three types of specimens containing randomly placed spherical inclusions were tested: specimens containing small voids (25.4 and 38.1 mm diameter), specimens containing large voids (50.8 and 63.5 mm diameter), and specimens containing both small and large voids (24.1, 38.1, 50.8, and 63.5 mm diameter). A total of 9 specimens were tested with randomly placed voids; 4 small void specimens, 3 large void specimens, and 2 specimens containing large and small voids. Further details on the 3D physical models can be found in Avar (2003). Figure 2 shows schematics of cuts through the center of the three types of 3D specimens. The dashed

lines indicate the location of voids not exposed in the central cut.

Figure 2. Examples of central, stacked, and random configurations for 3D models. Arrows indicate loading orientation. 3. NUMERICAL MODELING A 2D finite difference model (FLAC 2D) using a linear elastic material model was used investigate the influence of macroporosity on the deformation modulus. The cavities were circular in cross section and had diameters ranging in size from 6 mm to 25.4 mm. Two scenarios were modeled; in the first scenario cavities were not allowed to overlap and a minimum distance of 3 mm was maintained between cavities. In the second scenario, the cavities were allowed to overlap which represented coalesced lithophysal cavities that are seen in the tuff specimens. All models contained 24 holes. 4. LITHOPHYSAL TUFF SPECIMENS Ten lithophysal tuff specimens were tested as a comparison to the physical and numerical modeling. Each specimen was roughly cube shaped with side lengths between 100 and 150 mm. The size of the lithophysal cavities on the surface of the specimens ranged between approximately 1 and 50 mm. The cavities seen on the surface can be categorized into two types, single cavities, and multiple cavities that have coalesced together forming irregular shapes. The specimens were obtained from various outcrops on the Nevada Test Site. Examples of specimens and cavities types are shown in Figure 3.

Figure 3. Examples of tuff specimens showing a single cavity and coalesced multiple cavities.

Sea to Sky Geotechnique 2006

511

5. RELATIONSHIP BETWEEN MACROSCOPIC POROSITY AND DEFORMATION MODULUS Each of the physical and numerical models provides insight into how macroporosity affects the deformation modulus of specimens. Presentation of the results from individual models is followed by discussion and results from combining all data to show the relationship between macroscopic porosity and deformation modulus. 5.1 2D Physical Models Figure 4 presents the experimental results from the 2D physical models. Macroporosities for the models ranged from approximately 5 to 45 percent. The structured hole arrangement provides the stiffest response whereas random hole arrangement provides the least stiff response For specimens with the same number of random holes, the smaller the holes the stiffer the specimen. At low porosities, less than approximately 5%, there is no difference between structured and random holes. At higher porosities, there is a wider spread of modulus values. These trends indicate that both hole location and hole size are important factors when assessing the influence of macroporosity on deformation modulus. A regression analysis on the data yielded a best fit logarithmic regression with an R2 value of 0.82. 5.2 3D Physical Models Figure 5 presents the experimental results from the 3D models. Macroporosities for the models ranged from approximately 5 to 35 percent. In general, structured cavity arrangements provide the stiffest response. At low porosities, less than approximately 5%, there is no difference between structured and random cavities. There is a higher spread in modulus values at high porosities than at low porosities. These trends indicate that cavity location is an important factor that affects deformation modulus. A regression analysis on the data yielded a linear regression with an R2 value of 0.83.

Figure 4. Results from 2D physical model testing.

Figure 5. Results from 3D physical model testing 5.3 Numerical Modeling Figure 6 presents the numerical modeling results from the 2D FLAC runs. Macroporosities for the models ranged from approximately 5 to 45 percent. Since a linear elastic model was used, the results were plotted as normalized modulus versus porosity. Discrete or non contacting holes provide the stiffest response. Also, for specimens with the same number of random holes, the smaller the holes the stiffer the specimen. These trends indicate that both hole shape and size is an important factor when assessing deformation modulus. A regression analysis on the data yielded a linear regression with an R2 value of 0.96.

Figure 6. Results from numerical modeling.

Sea to Sky Geotechnique 2006

512

5.4 Lithophysal Tuff Specimens Figure 7 presents the results from the lithophysal tuff testing. Details of the testing can be found in Avar (2002). There is considerable scatter in the modulus values at the higher porosities, possibly indicating that cavity location is an important factor. A regression analysis on the data yielded a logarithmic regression with an R2 value of 0.73.

Figure 7. Results from lithophysal tuff testing. 5.5 Combined Normalized Data To compare results from physical modeling, numerical modeling, and lithophysal rock testing, the data required normalization so that all data fell on the same scale. For each dataset, normalization was made to either the zero porosity (linear regression) or 1% porosity (logarithmic regression) from each of the determined regression lines. Figure 8 presents the normalized datasets.

Figure 8. Normalized data sets with upper and lower bounds.

There is an obvious upper and lower bound to the four datasets. The upper bound is defined by the straight line regression from the 3D physical models and the lower bound is defined by the logarithmic regression from combining the normalized lithophysal tuff and 2D physical models. Since linear and logarithmic lines defined the data set, there is a large range of possible values for the deformation modulus at a given porosity. For example, the porosity with the highest range of possible modulus values is 13%, where the normalized deformation modulus varies from 0.76 at the upper bound to 0.37 at the lower bound. However at low porosity and high porosity, the difference between the upper and lower bounds are minimal. The range of porosities for the lithophysal tuff from this study is approximately 13% to 33%. The 13% porosity mark represents the highest possible range of modulus values but as the porosity increases to 33%, the range of modulus values is 0.38 to 0.14. 6. CONCLUSIONS The presence of macroscopic porosity in lithophysal tuff has a detrimental effect on its engineering properties, most notably deformation modulus. This study was undertaken to define the relationship between deformation modulus and macroporosity using physical modeling, numerical modeling, and lithophysal tuff testing. From the physical models, cavity/hole location and size have an influence the deformation modulus. The numerical modeling showed cavity/hole size and shape have an influence on the deformation modulus. Test results from the lithophysal tuff show that cavity/hole size and location have an influence on the deformation modulus. The data from the physical models, numerical models, and tuff testing was normalized using best fit regression lines from each data set. Plotting the normalized data revealed the data could be constrained with an upper and lower bound, which provides limits to the possible range of deformation modulus as a function of macroporosity. The upper bound was a linear regression based on the 3D physical models and the lower bound was a logarithmic regression based on the 2D and lithophysal tuff data. References Avar B. B., 2002. Numerical, experimental investigation of

deformation and strength properties of lithophysae-rich tuff and analog materials, Ph.D. dissertation, University of Nevada, Las Vegas, NV, USA, 134 pp

Sea to Sky Geotechnique 2006

513

Avar, B. B., 2003. The influence of lithophysal porosity on the in-situ stress-strain properties of Topopah Spring tuff, Final Technical Report, U.S. DOE/UCCSN Cooperative Agreement, DE-FC28-98NV12081, 42 pp. Available at http://hrc.nevada.edu/QA/Report/TR-02-008.pdf

Martin, R.J., Price, R.H., Boyd, P.J., Noel, J.S., 1995. Bulk and mechanical properties of the Paintbrush Tuff recovered from Borehole USW NRG-7/7A: data report. Sandia National Labs Report, SAND94-1996

Schultz, R. A, 1996. Relative scale and the strength and deformability of rock masses, Journal of Structural Geology, Volume 18, Issue 9, pp 1139-1149

Sea to Sky Geotechnique 2006

514