Modeling and Simulation of Bentonite Extrusion in a Saturated Planar Fracture

Transcription

1 UCB-NE-5110 Modeling and Simulation of Bentonite Extrusion in a Saturated Planar Fracture R.A.Borrelli and J.Ahn Department of Nuclear Engineering University of California Berkeley Berkeley, CA April 2007

2 The authors invite comments and would appreciate being notified of any errors in the report. R.A.Borrelli J.Ahn Department of Nuclear Engineering University of California Berkeley Berkeley, CA ii-

3 CONTENTS 1. INTRODUCTION EXPERIMENTAL STUDIES FOR BENTONITE EXTRUSION IN A PLANAR FRACTURE EXPANSION OF BENTONITE IN A SATURATED PLANAR FRACTURE MODELING OF BENTONITE EXTRUSION OBJECTIE MODEL CONFIGURATION OID RATIO MODEL SPACE MATHEMATICAL MODEL FOR BENTONITE EXTRUSION BENTONITE EXTRUSION MODEL BASED ON CONTINUITY EQUATIONS Governing Equation for oid Ratio Distribution Governing Equation for Expansion of Bentonite Tip SIDE CONDITIONS PARAMETERS Permeability Coefficient of Compressibility Diffusion Coefficient Analogue OTHER BENTONITE MODELING EFFORTS Modeling by erbeke, et al. [1997], Ahn, et al. [1999] Modeling by Kim [2004] Modeling by Kanno et al. [2001] COMPARISON OF SABRE MODELS WITH KANNO ET AL. [2001] MODEL RESULTS OERIEW OF NUMERICAL SCHEME alidation of SABRE6 with Experiment by Kanno and Iwata [2003] Methodology SABRE6 Execution DISCUSSION PHYSICAL CONDITIONS CONTRIBUTING TO EXPANSION SABRE6 ALIDATION WITH EXPERIMENT BY KANNO AND IWATA [2003] Tip Expansion Solid olume Fraction Extended Modeling ASSUMPTIONS AND LIMITATIONS oid Ratio at the Tip of Expansion Frictional Effects CONCLUSION REFERENCES I MODEL BASED ON DEFORMING ELEMENT IN FIXED COORDINATES II MODEL BASED ON FIXED ELEMENTS IN FIXED COORDINATES III NUMERICAL SCHEME FOR SABRE iii-

4 FIGURES FIGURE 1: EXPERIMENTAL APPARATUS [KANNO AND WAKAMATSU, 1991], [KANNO AND MATSUMOTO, 1997]... 2 FIGURE 2: PHOTOGRAPHS OF THE BENTONITE EXTRUSION EXPERIMENT [KANNO AND WAKAMATSU, 1991], [KANNO AND MATSUMOTO, 1997]... 2 FIGURE 3: REPRODUCED EXPERIMENTAL SOLID OLUME FRACTION DATA [KANNO AND IWATA, 2003]... 3 FIGURE 4: UNIT CELLS THAT FORM THE MONTMORILLONITE MINERAL... 4 FIGURE 5: ATOMIC ARRANGEMENT OF MONTMORILLONITE PARTICLE... 5 FIGURE 6: DIAGRAM OF THE OSMOTIC PRESSURE CONCEPT... 5 FIGURE 7: SEMI PERMEABLE MEMBRANE UTILIZED FOR OSMOTIC PRESSURE CONCEPT... 6 FIGURE 8: ARRANGEMENT OF TWO MONTMORILLONITE PARTICLES IN A SATURATED SYSTEM... 7 FIGURE 9: REPRESENTATIE OLUME OF SATURATED BENTONITE FIGURE 10: BENTONITE EXTRUSION INTO A SATURATED FRACTURE FIGURE 11: RELATIONSHIP BETWEEN OID RATIO AND LOG K [LAMBÉ AND WHITMAN, 1969] FIGURE 12: RELATIONSHIP BETWEEN OID RATIO AND EFFECTIE STRESS [TERZAGHI, ET AL., 1996] FIGURE 13: EXPERIMENTAL SOLID OLUME FRACTION DATA [KANNO AND IWATA, 2003] FIGURE 14: EXPERIMENTAL SOLID DIFFUSIITY DATA [KANNO AND IWATA, 2003] FIGURE 15: OID DIFFUSIITY FITTING TO EXPERIMENTAL DATA FOR ADJUSTABLE PARAMETER DETERMINATION (D 0,G) FOR E T = FIGURE 16: SABRE6 SIMULATION OF THE TIP LOCATION FOR EXPERIMENTAL CONDITIONS PROIDED BY KANNO AND IWATA [2003] FIGURE 17: SOLID OLUME FRACTION FOR SABRE6 WITH EXPERIMENTAL DATA BY KANNO AND IWATA [2003] FIGURE I.01: DEFORMING ELEMENT IN FIXED COORDINATES FIGURE I.02: DOMAIN UTILIZED FOR TIP EXPANSION MODEL FIGURE II.01: FIXED ELEMENT IN FIXED COORDINATES iv-

5 TABLES TABLE I: OID DIFFUSIITY CURE FITTING PARAMETERS v-

6 1. Introduction Isolation of high level radioactive waste (HLW) is based on the utilization of a multibarrier concept, including both engineered and natural components; to achieve this objective assures that the failure of an individual component will not result in total failure of the entire system. Engineered barrier systems (EBS) include the waste form, usually consisting of a stable matrix of vitrified glass, cement, or bitumen, corrosion resistant packaging, and a buffer material surrounding the waste package, in order to retard groundwater flow and limit radionuclide transport. Compacted bentonite will be utilized as buffer material in a water saturated repository. In the saturated repository, the function of the buffer is to isolate the waste packages from near field processes, including corrosion and degradation as a result of groundwater infiltration. Compacted bentonite will be utilized for the buffer primarily due to three very favorable characteristics: low permeability, high sorption capacity, and high swelling capacity. A material that demonstrates a low permeability will result in a highly retarded groundwater flow. Bentonite particles possess an overall negative charge and a high surface area and thus exhibit a high sorption capacity. Perhaps the most important feature of this material is that bentonite swells greatly beyond its initial volume when in contact with water. This swelling effect could result in extrusion of the buffer into the fracture, effectively sealing the fracture and thus significantly reducing releases of radionuclides to the far field. Bentonite expansion into a fracture presents an important domain in the repository. Radionuclides released from the waste canister will migrate into the fracture and be released into the far field and subsequently into the biosphere. The bentonite buffer can both limit groundwater flow and retard radionuclide migration. Therefore, characterization of bentonite extrusion in this region is critical with respect to repository performance assessment. This report will present the derivation and simulation of a bentonite extrusion model for a saturated planar fracture Experimental Studies for Bentonite Extrusion in a Planar Fracture The underlying basis for the bentonite modeling conducted for this dissertation began with an experiment designed in order to determine the location of the extruding bentonite tip in a saturated fracture [Kanno and Wakamatsu, 1991], [Kanno and Matsumoto, 1997]. The experimental apparatus is shown in Figure

7 camera water supply pump spacer balance fracture balance load cell Figure 1: Experimental apparatus [Kanno and Wakamatsu, 1991], [Kanno and Matsumoto, 1997] In this experiment, a specimen of bentonite was confined in a non deformable volume with a diameter of 50 mm. When placed in contact with water, the specimen expanded into a horizontal planar gap of constant width filled with water. Photographs of this expansion were taken by a camera that was mounted above the experimental apparatus. The location of tip was the only data measured for this experiment. The pictures were analyzed to measure the radial bentonite expansion as a function of time. Some of the photographs of the expanding bentonite are provided in Figure 2. Figure 2: Photographs of the bentonite extrusion experiment [Kanno and Wakamatsu, 1991], [Kanno and Matsumoto, 1997] The bentonite specimen is shown by the leftmost photograph in Figure 2. Subsequent photographs show the specimen in various stages of expansion with increasing time from left to right. The experiment showed that the bentonite specimen expanded at a rate proportional to the square root of time. This experiment also demonstrated that the size of the aperture does not affect the speed of the expanding bentonite tip at sizes larger than 1.5 mm. The empirical relationship was obtained as [Kanno and Wakamatsu, 1991], [Kanno and Matsumoto, 1997] R t 1.3 t R 0 0 t 100h. (1) Further experimentation was designed to simulate bentonite extrusion [Kanno, et al., 2001]. In this experiment, a rock mass was simulated by a test cell consisting of two Plexiglas plates. An annular slot between the plates served as the horizontal fracture. Fracture aperture was determined by stainless steel plates sandwiched between the Plexiglas. A bentonite specimen of 50 mm in diameter and 50 mm in height was installed into a hole drilled into the test cell. Water was then supplied to the specimen. The bentonite then expanded due to contact with the water and extruded into the simulated fracture. Results for the migration distance of the -2-

8 extruding bentonite were obtained at various times for an aperture of.5 mm and 1.5 mm. For this experiment, migration distance also behaved linearly with the square root of time. Results also were obtained for the spatial distribution of solid material at selected times. Kanno et al. [2001] also proposed that friction will affect extrusion in fractures that exhibit an aperture below 1 mm. Another experiment was designed to measure spatial density distributions for extruding bentonite in a planar fracture, recorded by computer tomography, have recently been made available [Kanno and Iwata, 2003]. For this experiment, additional data for swelling pressure and for permeability were determined by the measurement of an experimental solid diffusion coefficient. The experimental results are reproduced and shown in Figure 3. Figure 3: Reproduced experimental solid volume fraction data [Kanno and Iwata, 2003] The solid volume fraction is defined as S. (2) The solid volume fraction and the void ratio are related, by definition, as 1. (3) 1 e Due to the measurement of this solid diffusion coefficient, further modeling efforts could be endeavored, which is the motivation for this report. The derivation and the validation of this model is contained in

9 1.2. Expansion of Bentonite in a Saturated Planar Fracture Bentonite exhibits a unique ability to swell greatly beyond its initial volume when in contact with water. This section will explain the manner in which bentonite expands when contacted with water. Colloids are classified as particles with a diameter in the range of 1 m to 1 nm [van Olphen, 1963]. Such particles exhibit a large surface area relative to volume; therefore, surface properties dominate colloid behavior. Montmorillonite is the primary mineral in bentonite, comprising an octahedral alumina layer sandwiched between two tetrahedral silica layers [Mitchell, 1993]. The alumina unit cell and the silica unit cell are shown in Figure 4. O Si O or OH Al Figure 4: Unit cells that form the montmorillonite mineral The montmorillonite mineral exhibits an overall negative charge due to isomorphous substitution of magnesium atoms with 2+ valence for every six aluminum atoms with 3+ valance. Without isomorphous substitution, the composition of the montmorillonite mineral is Si 8 Al 4 O 20 (OH) 4 nh 2 O, and with isomorphous substitution, the composition of the montmorillonite mineral is Si 8 (Al 3.34 Mg.66 )O 20 (OH) 4 nh 2 O [Mitchell, 1993]. and This negative charge balance on a montmorillonite particle can be determined as Si8 Al3.34 Mg.66 O20 OH 4, (4) Si 3.34 Al.66 Mg 20 O 4 OH, (5) (6) A schematic diagram of the atomic arrangement of the montmorillonite particle given by (4) is shown in Figure

10 10 O 4 Si 6 O + 4 OH 6 Al + 1 Mg 6 O + 4 OH 4 Si 10 O Figure 5: Atomic arrangement of montmorillonite particle Expansion of the bentonite structure can be described by the flow of water through the material due to the osmotic pressure concept. A diagram of the osmotic pressure concept is provided in Figure 6. h h Solution Solvent Solution Solvent Solution Solvent Osmosis Figure 6: Diagram of the osmotic pressure concept In Figure 6, a solution is shown in the left chamber and is separated from a pure solvent in the right chamber by a semi permeable membrane. This membrane allows passage of the smaller solvent molecules, but the membrane also prevents the passage of the larger solute molecules of higher molecular weight [Mahan, 1987], [Miller and Low, 1990]. This semi permeable membrane is enlarged and shown in Figure 7, for emphasis, with smaller solvent molecules and larger solute molecules. -5-

11 Figure 7: Semi permeable membrane utilized for osmotic pressure concept Based on the configuration shown in Figure 7, in the left chamber, due to the presence of the solute, the chemical potential of the solvent is less than the chemical potential of the solvent in the right chamber. Because the solute cannot pass through the membrane in order to equalize concentration in both chambers of this system, the solvent will flow from the right chamber to the left chamber. This flow is defined as osmosis. Therefore, the solute concentration in the left chamber will decrease, and the height of fluid in the left chamber will rise. A difference in hydrostatic pressure between the chambers will develop. The free energy of the solvent will vary directly with this pressure and inversely with the concentration, thus allowing a reduction with respect to the imbalance of the chemical potential of the solvent in both chambers. The flow due to osmosis will continue until the chemical potential of the solvent in each chamber is equal [Mitchell, 1993]. The osmotic flow can be prevented by applying pressure on the fluid in the left chamber. The pressure required to maintain an equal level of fluid in each chamber is defined as the osmotic pressure (). The osmotic pressure can be calculated in terms of the height between the fluid in each chamber (h), the density of the solution (), and gravity (g) as gh. (7) Based on experimental measurements of dilute solutions, van t Hoff developed a relationship between osmotic pressure, and concentration (c) that is analogous to the ideal gas law as [Mitchell, 1993] cr. (8) The universal gas constant is given by (R), and the absolute temperature is given by (). For multiple species, the van t Hoff equation is modified as [Mitchell, 1993] i i A i B R c c. (9) Solutions with different concentrations are given by (c A ) and (c B ), and (i) represents each species. -6-

12 In a water saturated environment containing bentonite, although actual semi permeable membranes do not exist, the concept can be applied on a microscopic level in order to describe the expansion of the bulk bentonite. Montmorillonite particles are assumed to be arranged in the saturated system as shown by Figure 8. c 0 2d c C, c A c 0 Figure 8: Arrangement of two montmorillonite particles in a saturated system In Figure 8, two montmorillonite particles are assumed to be arranged in a parallel formation in a saturated system of groundwater, separated by a distance (2d). Each particle exhibits a thin, planar shape, with negative charges. A single cation and anion species of the same valance exists in the system. At a sufficient distance from each particle, an equilibrium state is assumed, where the concentration of the cations and the anions are equal, indicated in Figure 8 by (c 0 ). The following assumptions are applied to the system [Mitchell, 1993] The ions in the system are assumed to be point charges; therefore, no interactions occur between the ions in the groundwater. The negative charge exhibited by each montmorillonite particle is distributed uniformly. The dielectric constant is independent of position. -7-

13 Each montmorillonite particles behaves independently. The attraction of electrical forces near the surface of the montmorillonite particle results in a high concentration of cations. Due to this high concentration, the thermal energy of the cations attempt to diffuse to regions of lower concentrations. Diffusion of the cations is somewhat prevented, however, because the electrical force is stronger. The balance of Coulomb electrical attraction and thermal diffusion leads to a diffuse layer of cations, with the concentration highest at the surface and gradually decreasing with distance from the surface. The negatively charged particles surface and distributed positive charge form the cations in the fluid phase are together defined as the diffuse double layer [van Olphen, 1963]. Figure 8 shows two montmorillonite particles in which the diffuse double layers overlap. The effect of a restrictive membrane can be considered due to the influence of the negatively charged montmorillonite particles on the cations in the groundwater. This restrictive membrane is shown in Figure 8 at the midplane between the two montmorillonite particles. Since the cations are not free to diffuse due to the electrical forces, concentration differences responsible for osmotic pressures are developed. The difference in osmotic pressure at the midplane of the montmorillonite particles and in the equilibrium solution surrounding the particles causes a repulsive pressure, or a swelling pressure. In order to equalize the chemical potential of the groundwater between the particles, more groundwater will flow into this region, indicated by the arrows in Figure 8, therefore causing the distance between the montmorillonite particles to increase. As a result, the entire bulk bentonite structure will swell. Based on the van t Hoff equation (9) the swelling pressure can be calculated as C A 0 0 R c c c c. (10) The concentrations of cations and anions at the midplane between the montmorillonite particles are given by (c C ) and (c A ), respectively. The concentrations of cations and anions in the equilibrium solution are given by (c 0 + ) and (c 0 ), respectively. At equilibrium, the concentrations of cations and of anions are the same, and, for dilute solutions the following relationship between the concentrations can be expressed as 2 c c c c c. (11) C A Therefore, the swelling pressure can be obtained in terms of the cation concentration at the midplane between the montmorillonite particles and the equilibrium concentration as c c C 0 R c0 2 c0 cc. (12) The principle of bulk bentonite extrusion by water movement is the basis for the modeling of bentonite extrusion in a water saturated planar fracture in this study. -8-

14 1.3. Modeling of Bentonite Extrusion Modeling efforts were conducted in order to describe the results from the experiment discussed in 1.1 [erbeke, et al., 1997], [Ahn, et al., 1999]. Because the experiment recorded the location of the extruding bentonite tip as a function of time, subsequent modeling contains a critical feature in that the spatial domain moves with time. The derivation of this model included the application of Terzaghi s one dimensional consolidation theory [Terzaghi, 1931]. Darcy s law was also applied [Terzaghi, et al., 1996]. The important feature of the consolidation theory involves the equivalence of the swelling pressure occurring as a result of bentonite in contact with water with the effective stress in the soil skeleton. For erbeke, et al. [1997] and Ahn, et al., [1999], modeling of bentonite extrusion was predicated on the principle that the driving force governing expansion of bentonite into a saturated planar fracture is based on the observation of bulk water flow in the system, due to the swelling pressure as discussed in 1.2. A model was first developed based on Narasimhan and Witherspoon [1977], in order to describe the mass flow of groundwater in a variably saturated deformable heterogeneous porous media. An important concept of modeling concerns the definition of the representative element volume utilized for the groundwater mass conservation equations. In Narasimhan and Witherspoon [1977], the representative elemental volume is chosen such that it contains a constant volume of solid material and a variable void volume. The model developed in erbeke, et al. [1997] and Ahn, et al. [1999] however, included more parameters than actually measured in the experiment, such as swelling pressure, permeability, and void ratio at the tip of bentonite extrusion. Therefore, standard relationships from soil mechanics were utilized [Bear, 1972], [Terzaghi, et al., 1996]. These relationships provide a model for void ratio expressed in terms of permeability and for void ratio expressed in terms of effective stress. These relationships also provide a way to express the coefficient of compressibility, as defined by Terzaghi, et al. [1996]. A parametric study showed that the constants appearing in the soil mechanics relationships could be consolidated into two, adjustable parameters [Shinoda, 2001]. Due the lack of measured data in the experiment, however, new adjustments to the model showed that differing combinations of adjustable parameters, along with varying the void ratio at the tip of bentonite extrusion could fit the experimental data. Therefore, Shinoda [2001] concluded that the study produced non unique results when compared to the experiment. Another model was developed for the extrusion of bentonite, based on the theory of clay particle diffusion, which proposes that the basic mechanism of extrusion is the free expansion of the bentonite particle structure due to its swelling ability [Kanno, et al., 2001]. Therefore, the driving force governing the movement of solid particles into the saturated planar fracture is defined as the gradient of the swelling pressure. The model derived by Kanno, et al. [2001] did not include a derivation in order to describe the location of the extruding tip of bentonite expansion. This solid particle diffusion model was compared with by experimental results; however, the validation procedure required untested extrapolations based on experimental data in limited ranges. -9-

15 In summary, the first models developed by erbeke, et al. [1997] and Ahn et al. [1999] include more parameters than actually measured in the experiment by Kanno and Wakamatsu [1991] and Kanno and Matsumoto [1997]. These models utilized empirical relationships established in the soil mechanics literature. Further experimental studies were performed to fill this gap by Kanno et al. [2001]. In this study, a model was also developed for bentonite extrusion by conceptualizing the movement of the solid particles as a diffusion process. This model, however, cannot trace the tip of extruding bentonite. Thus, a model for bentonite extrusion needs to be developed by modifying the models derived by erbeke et al. [1997] and Ahn et al. [1999] and by incorporating the parameters measured in the experiment by Kanno and Iwata [2003] Objective This report presents the derivation and analysis of a mathematical model for the expansion of bentonite in a saturated, planar fracture. The model has been developed based on Terzaghi s one dimensional consolidation theory and on Darcy s law [Terzaghi, 1931], [Terzaghi, et al., 1996]. The following items are included. A discussion of the physical processes affecting bentonite extrusion ( 2). The derivation of the mathematical model for bentonite extrusion ( 3). A comparison to an independently developed mathematical model [Kanno, et al., 2001], [Kanno and Iwata, 2003] ( 3). alidation of the model with the experimental data shown by Figure 3 ( 4). Discussion and analysis of the results ( 5). Conclusions drawn from this study ( 6). -10-

16 2. Model Configuration This section will address the physical processes upon which the modeling of bentonite extrusion in a saturated planar fracture is based. The model space will be defined. Bentonite experiments will also be summarized oid Ratio The parameter used to characterize bentonite behavior first must be defined. Included in Figure 9, is a general, schematic design of an arbitrary volume of bentonite upon which the mathematical model is conceptualized. Solid Phase ( S ) oid Phase ( ) S oid Ratio, e S Figure 9: Representative volume of saturated bentonite This volume of bentonite consists of two distinct phases: the solid phase, consisting of the bentonite particles, and the void phase, which can contain air, water, another gas entirely, or any combination of these substances. For the current analysis, the void volume will be filled completely with groundwater only. The total volume of this representative element is expressed as the sum of the solid phase volume and the void phase volume. The void ratio (e) is defined as the ratio of the void volume to the solid volume. Bentonite extrusion behavior will be described and will be modeled in terms of the void ratio as defined in Figure Model Space The domain in which the governing equations will be derived is shown in Figure 10. fracture bentonite L water flow 0 R 0 R(t) +r Figure 10: Bentonite extrusion into a saturated fracture -11-

17 Figure 10 shows a cylindrical cross section of bentonite that is intersected by a horizontal planar fracture. The critical feature of this particular analysis focuses on nature of the domain. The initial radius of the bentonite (t = 0) is shown at R 0. This location also indicates the intersection between the bulk bentonite and the fracture. Since bentonite expands when contacted with groundwater, so the domain of interest also moves. Therefore, at an arbitrary time (t), the domain under analysis is given as: R 0 r R(t). Groundwater will be taken up at the tip of the bentonite, R(t). The groundwater will then flow into the bulk bentonite region, resulting in swelling of the bentonite and subsequent expansion of the tip into the fracture. Since HLW waste packages are modeled as cylinders, a cylindrical geometry thus forms the basis of the domain of interest. The bentonite completely encloses the waste package; the domain is defined as the cylindrical cross section of the bentonite with the fracture. Due the manner in which the fracture intersects the EBS, analyses are conducted for bentonite behavior in the +r direction only. Due to waste package geometry, radial symmetry is assumed. The fracture is conceptualized for a planar geometry, and bentonite extrusion is assumed to be slow. Therefore, the azimuthal dependence is also neglected. The motivation for radionuclide transport modeling is provided by the saturated repository concept. Therefore, both the bulk bentonite and the fracture are assumed to be saturated with groundwater. For mathematical simplicity, the fracture aperture is assumed to be constant. Frictional effects in the extrusion with a fracture that exhibits an aperture above 1 mm diminish to a negligible level, and the extrusion is dominated by the free swelling of the bentonite [Kanno, et al., 2001]. Therefore, the effects of friction are neglected for this bentonite extrusion model. -12-

18 3. Mathematical Model for Bentonite Extrusion This section presents the full derivation for a model to describe bentonite extrusion in a saturated planar fracture. The model consists of two, coupled differential equations. This section will also address the conditions required for the model, based on the design of the system, as well as a discussion of the field parameters in the model. A mathematical analysis of this model with an independently developed model is also provided Bentonite Extrusion Model Based on Continuity Equations Bentonite extrusion into a saturated planar fracture is derived based on the characterization of groundwater mass flow. The volumetric domain for this derivation is defined as a fixed volume element in fixed coordinates, which was shown in Figure Governing Equation for oid Ratio Distribution This derivation utilizes the equations of continuity for a fluid and for a solid, for a one dimensional cylindrical coordinate system, respectively, as [Gambolati, 1973] 1 e e rv W, t 0, R0 r R t r r 1 e t 1 e (13) and rv S, t 0, R0 r R t r r 1 e t 1 e, (14) where W v : water velocity L T v S : solid velocity L T. This derivation is based on the assumption that the bentonite particles are incompressible [Gambolati, 1973]. Water is also assumed to be incompressible; therefore, the specific gravity of water () can be removed from the derivatives in (3.01) and canceled. Each of the four derivatives in the two continuity equations is computed to obtain e e 1 rv 1 e r 1 e 1 e r r 1 e t W W 2 v (15) and -13-

19 v S S rv 1 e. (16) r 1 e 1 e r r 1 e t Equations (15) and (16) are then added to obtain rv rv e 1 e 1 W 1 1 S vw vs 0. (17) r 1 e r 1 e 1 e r r 1 e r r Darcy s law is introduced as e 1e kg vw vs h, (18) where k : permeability 2 L L g : gravitational constant 2 T : viscosity coefficient of water M LT h : hydraulic head L. The hydraulic head is further defined as p h z. (19) M L 1 p : pore water pressure 2 2 T L z : elevation head L M : specific weight of water 2 2 L T For this analysis, only a horizontal fracture is considered. Therefore, z = 0 [erbeke, et al., 1997]. Furthermore, the assumption is made that water density us a function of pore water pressure only [Narasimhan and Witherspoon, 1977]. Therefore, Darcy s law can be expressed in terms of pore water pressure as -14-

20 e 1e k vw vs p, (20) and e k p 1 e r vw vs. (21) Darcy s law (21) is differentiated in the cylindrical coordinate system to obtain rv rv e e 1 W e 1 S 1 k p vw vs r r 1 e 1 e r r 1 e r r r r r. (22) The derivative of Darcy s law (22) is subtracted from (17) to obtain 1 rv S 1 k p r r r r r r. (23) An explicit form of the solid velocity can be obtained from this result as v S k p. (24) r This expression for the solid velocity is substituted into the expression for Darcy s law (20) in order to obtain an explicit representation for the water velocity, shown as v W 1 k p. (25) e r The expression for water velocity (25) is then substituted into the continuity equation for water (13) to obtain 1 k 1 p 1 e r, t 0, R 2 0 r R t r r 1 e r 1 e t. (26) In order to obtain a governing equation explicitly in terms of the void ratio, the effective stress in the system is considered. The effective stress is defined as the net stress acting on the soil skeleton [Narasimhan and Witherspoon, 1977]. The void ratio is not directly a function on the pore water pressure; instead, the void ratio is dependent on the effective stress in the system, and this effective stress is further dependent on the pore water pressure [Narasimhan and Witherspoon, 1977]. Effective stress is expressed as -15-

21 p. (27) M L 1 : effective stress 2 2 T L M L 1 : total stress 2 2 T L The total stress acts on the entire area, which includes the soil skeleton and the voids; the pore water pressure acts on total area that does not include the solid particle contact area [Terzaghi, 1931], [Terzaghi, et. al., 1996]. In addition, the assumption is made that the total stress in the system at any point does not change with time [Narasimhan and Witherspoon, 1977]. Therefore, this assumption indicates that the changes in pore water pressure are fully converted to an equivalent change in the effective stress, shown as p. (28) However, this statement might not necessarily hold for all cases: for extremely dry soils, no equivalence exists, but even for fully saturated soils, the equivalence might only be partial [Narasimhan and Witherspoon, 1977]. A modified form of (27) was developed in order to include the consideration that the change in pore water pressure might not be fully equivalent to a change in effective stress [Narasimhan and Witherspoon, 1977] p 0 1. (29) Bishop s parameter () is an empirically based coefficient that exhibits a nonlinear relationship to saturation. For a fully saturated medium, the parameter evaluates to unity. The effective stress can be differentiated with respect to pore water pressure as d d. (30) dp dp d p dp The derivative of the void ratio with respect to the effective stress is a common parameter occurring in the soil mechanics literature, and this term can be utilized to formulate the governing equation for bentonite explicitly in terms of the void ratio [Lambé and Whitman, 1969], [Bear, 1972], [Terzaghi, et al., 1996]. Empirical data obtained from uniaxial loading experiments of various types of clays reveal that the relationship between void ratio and the logarithm of effective stress is -16-

22 approximately linear [Narasimhan and Witherspoon, 1977]. This relationship is a fairly simplified determination of very complex processes, as void ratio is a nonlinear function of effective stress. Any empirical data exhibits the inherent assumption that strains are negligible in the intermediate and minor principal stress directions, and all possible strains occur in the major principal stress direction only [Narasimhan and Witherspoon, 1977]. The empirical relationship between void ratio and effective stress is defined by determining the slope of the best fit on a plot of void ratio versus logarithm of effective stress. This slope is referred to as the swelling index and is represented as [Narasimhan and Witherspoon, 1977] de de d d ln CS 2.303a. (31) d log d d ln d log The coefficient of compressibility is therefore defined as [Terzaghi, 1931], [Terzaghi, et. al., 1996] a de d. (32) This result is based on the additional assumption that the void ratio is a function of the effective stress only. The empirical data initially obtained that lead to the determination of the swelling index is based on the condition that the soil attained equilibrium with the loading before parameters were measured [Narasimhan and Witherspoon, 1977]. Therefore, further application of this concept to any subsequent analysis must be based on the assumption that the time to reach this equilibrium state is small and thus does not need to be considered. The definition of the coefficient of compressibility and the derivative of effective stress with respect to the pore water pressure can be utilized to transform the expression from the pore water pressure to the void ratio as p p e 1 e 1 e 1. (33) r e r a r a r Therefore the model is expressed explicitly in terms of the void ratio as 1 e 1 k 1 e r, t 0, R0 r R t 1 e t r r a 1 e r 2. (34) The behavior of bentonite in a fully saturated medium is thus expressed in terms of the void ratio (e), as shown by (34) where permeability (k) and the coefficient of compressibility (a ) are both functions of the void ratio. -17-

23 Governing Equation for Expansion of Bentonite Tip Since bentonite extrusion is fundamentally described by the mass conservation of groundwater flow, the domain will change with time. Therefore, another model must be developed in order to characterize the expansion of bentonite in the fracture. Bentonite will expand when in contact with groundwater. This tip expansion model is based upon the mass flow of groundwater from the fracture into the domain at the expanding tip, and upon the net mass flow of groundwater into the bulk bentonite at the intersection of the bulk region with the fracture. The domain utilized for this analysis was defined in Figure 10. The fracture aperture exhibits a width L. In this model, the groundwater is assumed to flow into the bulk bentonite, as indicated by Figure 10. Initially, the fracture consists of water only. The bulk region only contains saturated bentonite, but no groundwater flow is assumed; therefore, the region exhibits a constant void ratio. The domain is defined as R 0 r R(t), where R 0 indicates the intersection between the bulk bentonite and the fracture, and R(t) indicates the expanding tip of bentonite. At the intersection between the bulk bentonite and the fracture, the expansion occurs as groundwater contacts the tip and flows into the bulk bentonite region. The overall rate of change of the mass of water in this domain is expressed as d dt W t m R m 0 Rt. (35) W t : time dependent volume of water in the domain 3 L m R 0 : mass flow rate of water at the intersection between the bulk bentonite and the M fracture T m Rt : mass flow rate of water at the expanding tip of bentonite The consideration at the tip is based on the concept that the bentonite expands due to the uptake of water. The driving force that causes water to flow into the domain is the pressure in the bentonite structure, between individual solid particles. Therefore, the net mass flow of water through the boundary does not solely depend on the water entering the domain from the fracture, but, instead on the net mass flow of water through this boundary. The net mass of water through the tip boundary can then be defined as m m m. (36) R t R t R t M T -18-

24 The + superscript indicates the flow of water coming into the domain, and the superscript indicates the flow of water from the domain to the fracture. The mass of water flowing from the fracture into the domain is evaluated as m 2R t L qn t. (37) Rt Rt q : flow rate of water L T n : unit normal vector in direction of bentonite extrusion An equal quantity of bentonite expands into the fracture due to the uptake of water into the domain. The mass of water in this quantity of bentonite is expressed as e m 2R Rt t L qn t. (38) 1 e Rt Therefore the net mass of water flow at the boundary R(t) be obtained by adding (37) and (38) as e 1 mrt 2 Rt L q n Rt t 2 R t L q n t 2 R t L q n t 1e Rt 1e Rt. (39) Assuming an infinitesimal time interval, the mass flow of water through the boundary at R(t) is therefore obtained as 1 m 2R Rt t L qn. (40) 1 e Rt The net mass flow of water through the interface between the bulk bentonite and the fracture can then be defined as R0 R0 R0 m m m. (41) The + superscript indicates the flow of water from the fracture into the bulk bentonite region, and the superscript indicates the flow of water from the bulk bentonite region into the fracture. The mass of water flowing into the bulk bentonite region is calculated as m 2R L qn t. (42) R0 0 R0-19-

25 The mass of water contained in the volume of bentonite that flows into the fracture from the bulk region is calculated as e m R 2R 0 0L q n t. (43) 1 e R0 Therefore the net mass of water flow at the boundary R 0 can be obtained by adding (42) and (43) as 1 mr 2R 0 0L qn t. (44) 1 e R0 Assuming the infinitesimal time interval, the mass flow of water through the boundary at R 0 is expressed as 1 m R 2R 0 0L qn. (45) 1 e R0 Based on (35), the mass conservation equation for the domain can be expressed from (40) and (45) as d 1 1 t 2R L qn 2R t L qn R Rt W 0 dt 1e 0 1e. (46) Water is assumed to be incompressible; the density term can be removed from the time derivative on the left side of the equation, and subsequently canceled. The mass conservation equation is then expressed as d 1 1 t 2 R L q n 2 R t L q n R Rt W 0 dt 1e 0 1e. (47) By definition of the domain, the total volume in the domain is determined to be the sum of the water volume and the solid volume as and t t t. (48) W S The expression for the volume in (48) is differentiated with respect to time to obtain d d d dt dt dt t W t S t (49) d d d dt dt dt W t t S t. (50) -20-

26 The expression in (50) is then substituted into (47) to obtain d d 1 1 t t 2 R L q n 2 R t L q n S 0 R Rt dt dt 1e 0 1e. (51) Bentonite will flow from the bulk region into the domain at the intersection R 0, and bentonite will also flow from the domain out to the fracture at R(t). Therefore, the net solid volume flow can be obtained as d 1 1 t 2 R Lq n 2 R t L q n R Rt S 0 dt 1e 0 1e. (52) The expression for the solid volume flow rate (52) is substituted into (51) to obtain d 1 1 t 2 R0 L q n 2 Rt L q n dt 1 e R0 1 e Rt. (53) 1 1 2R L q n 2R t L q n 0 1e R0 1e The rate of change of the total volume in the domain is therefore expressed as d 2 t dt 1 e 2R t L q n Rt R t. (54) Based on the definition of the domain, the total volume can be calculated as t L R t R (55) The expression in (55) is differentiated with respect to time to obtain d d t 2 LRt Rt dt dt. (56) The derivative given by (56) is substituted into (54), and the flow rate is expressed in terms of the moving tip as d 2 2R t L Rt 2 Rt L q n dt 1 e Rt (57) and d 2 Rt q n dt 1 e Rt. (58) Darcy s law is applied for the cylindrical coordinate system as -21-

27 kg h q. (59) r The hydraulic head is defined as p h z. (60) For this analysis, the elevation head (z) is neglected because the domain is defined as a planar fracture. Darcy s law is therefore expressed as k p q. (61) r By applying the transformation given by (33), Darcy s law is expressed in terms of the void ratio as k e q. (62) a r This expression for Darcy s law is substituted into (58) in order to represent the movement of the expanding bentonite tip in terms of the void ratio as d 2 k e Rt, t 0 dt. (63) 1 e a r Rt Equation (34) coupled with equation (63) describe the extrusion of bentonite in a saturated planar fracture. Results from this system of equation can give the void distribution over the spatial domain at selected times, and the location of the moving tip in the domain at selected times. This model is referred to as SABRE6: saturated bentonite radial expansion Side Conditions Proper initial and boundary conditions for the system must be established. These conditions are based on the model space configuration defined in 2.2 and the domain defined by Figure 10. Initially (t = 0), bentonite has not yet expanded into the fracture. The properties of bentonite are assumed to be homogeneous [Gambolati, 1973]. The bentonite thus exhibits a uniform void ratio. Therefore, the initial condition is prescribed as e r,0 e 0 r R, R 0 R. (64)

28 Groundwater flows from the fracture into the bulk bentonite region, causing the tip to expand into the fracture, and therefore the domain size increases. The assumption is made that the water entering the bulk region distributes homogeneously; therefore, the average void ratio in this region can be approximated by the initial condition [Ahn, et al., 1999] as e R,t e t 0. (65) 0 0 This assumption is valid when the volume of water that has entered the bulk bentonite region (0 r R 0 ) is negligible to the total void volume in the domain. This assumption was verified by numerical analyses [Ahn, et al., 1999]. Bentonite at the tip expands due to the uptake of water. The swelling pressure, between individual solid particles, provides the driving force that causes water to flow into the domain. Therefore, since the coupled system derived in 3.1 is predicated on the behavior of the bentonite state at the tip, the assumption is made that the bentonite exhibits a void ratio no greater than the consistency limit at this location [Boisson, 1989], [Low, 1992]. Above this limit, bentonite will behave as a suspension of colloids in water, and the current models will not be applicable. The condition at the moving tip is prescribed as T e R t,t e t 0. (66) These conditions are based on the inherent assumptions that expansion occurs indefinitely and that there is a sufficient amount of bentonite available for this indefinite expansion [Ahn, et al., 1999] Parameters The coupled system derived in 3.1 exhibits nonlinearity due to the dependence of the permeability and of the coefficient of swelling on the void ratio. This section will contain a discussion regarding the methods of treatment with respect to these parameters Permeability Permeability is a measure of the ease with which a fluid moves through a porous medium. This parameter is a property of the porous medium only; therefore, permeability is dependent on the porosity, and, subsequently, the void ratio of the medium through which the fluid flows. Bentonite exhibits a low permeability, and, as a result, groundwater flow will be highly retarded in this medium. Permeability usually is expressed as a symmetrical second order tensor, but, for this analysis, the parameter is considered to be a scalar for an isotropic medium [Narasimhan and Witherspoon, 1977]. For saturated systems, permeability is also a function of the effective stress. Due to Terzaghi s one dimensional consolidation theory, the effective stress is a function of the pore water pressure [Terzaghi, 1931], [Terzaghi, et al., 1996]. Experimental studies have shown that for fine grained materials such as clay, the logarithm of permeability exhibits a linear relationship with void ratio, and, therefore, permeability can be expressed as [Lambé and Whitman, 1969] -23-

29 2.303 e e k k0 exp CK k 0. (67) The slope of the best fitting straight line for void ratio versus the logarithm of permeability is given by C K, and e k 0 is an arbitrarily selected data point with respect to the void ratio. Experimental data with regards to the relationship between void ratio and hydraulic conductivity (K) can be utilized in order to obtain information about permeability. Hydraulic conductivity is defined as kg K. (68) An example of the relationship between void ratio and hydraulic conductivity for various types of sand is shown in Figure 11. Figure 11: Relationship between void ratio and log K [Lambé and Whitman, 1969] The slope of each line could then be utilized to obtain C K Coefficient of Compressibility The coefficient of compressibility (a ) was introduced in 3.1. This parameter was derived as a result of the empirical relationship between void ratio and effective stress. The swelling index (C S ) was defined as the slope of the best fitting straight line for this empirical relationship (31), which can also be represented by an exponential function as [Terzaghi, et al., 1996] e e 0 exp CS 0. (69) -24-

30 Therefore, the coefficient of compressibility can be represented as a C e e S 0 exp. (70) CS Another example, with respect to the relationship between void ratio and effective stress is shown in Figure 12. Figure 12: Relationship between void ratio and effective stress [Terzaghi, et al., 1996] Figure 12 contains a plot of a uniaxial compression test. A piston applies stress and the specimen is compressed from a to b, resulting in the compression curve. The piston is removed, and the specimen rebounds to c. The stress is applied and compression occurs to d. The stress is released, and the specimen rebounds to e. Based on this compression test, swelling of the specimen is indicated by the curves bc, de, and fg. Furthermore, analysis of a large number of uniaxial test data indicates that the relationship between void ratio versus effective stress is approximately a straight line for most of the void ratio range [Narasimhan and Witherspoon, 1977]. Therefore, the swelling index (C S ) -25-

31 can be represented by the best fitting slope of these lines, and, in turn the coefficient of compressibility (a ) could be obtained Diffusion Coefficient Analogue Based on the derivation in 3.1, the parameters occurring in the spatial derivative of the void ratio governing equation can be considered as an analogue to a diffusion coefficient. By equation (34), this diffusion coefficient analogue can be expressed as D 1 k D0 exp Ge 1 e a 1 e. (71) The diffusion coefficient analogue (D ) is termed the void diffusivity. Based on the empirical relationships shown by equations (67), (68), and (69), the constants D 0 and G are expressed as k e e D0 exp k CS CS CK (72) and 1 1 G C K C S. (73) Based on the discussions of the empirical relationships for the permeability and for the coefficient of compressibility, the void diffusivity contains many constants. Additional study showed that the parameters appearing the void diffusivity could be consolidated into two, adjustable constants [Shinoda, 2001]. Therefore, in terms of a numerical scheme devised to generate the solution to the coupled systems, the adjustable parameters are only D 0 and G. Utilization of two, adjustable parameters, should lend to a more favorable comparison in terms of validation analyses with experimental data. Application of all the constants, by (67), (68), and (69), while based on established relationships in the soil mechanics literature, are not specifically formulated for any specific bentonite extrusion experiment, for which the motivation for the development of the model was initially based Other Bentonite Modeling Efforts This section briefly addresses other models developed in order to describe bentonite extrusion into a saturated planar fracture. Two of the models are essentially progenitors to the model derived in A third model was developed independently. All of the models were developed in order to describe the experiment by Kanno and Wakamatsu [1991] and Kanno and Matsumoto [1997]. -26-

32 Modeling by erbeke, et al. [1997], Ahn, et al. [1999] The driving force for bentonite extrusion conceptualized by this model involved the observation of groundwater flow [erbeke, et al., 1997], [Ahn, et al., 1999]. The model is given as 1 e 1 k e r 1 e t r r a r (74) and d k e Rt dt a r Rt. (75) The derivation for this model is presented in Appendix I. This model is referred to as SABRE4. alidation of this model with the experiment by Kanno and Wakamatsu [1991] and Kanno and Matsumoto [1997], however, could not be made due to a lack of experimental data, such as the void ratio at the tip, swelling pressure, and permeability [Ahn et al., 1999]. As an alternative, relevant parameters were estimated from parametric studies available in the literature [Lambé and Whitman, 1969], [Bear, 1972], [Terzaghi, et al., 1996]. The lack of the additional experimental data led to difficulty with fitting parameters in the model to the experimental data. This problematic fit resulted in subsequent comparisons of numerical results from the model to the experimental data that were not favorable for the cases that were analyzed [Ahn et al., 1999] Modeling by Kim [2004] The model developed by Kim [2004] is similar to the SABRE4 model, but the volumetric element was defined differently. The model is given as 1 e 1 k e r t r r a r 1 e 2 (76) and d k e Rt dt a r Rt. (77) The derivation for this model is presented in Appendix II. This model is referred to as SABRE

33 This model was not compared to the experimental data, but results of numerical experiments of both models by Kim [2004] showed that the change in the representative volume element did not provide a significant impact Modeling by Kanno et al. [2001] The mathematical model developed by Kanno et al. [2001] was based on the diffusion of solid particles in the system; i.e., the free expansion of bentonite particle structure due to its swelling ability [Kanno, et. al., 2001]. The model is a non linear, diffusion type equation for the solid volume fraction, with a diffusion coefficient analogue based on the permeability of water and the compressibility of the solid phase. This model did not include consideration of the moving bentonite tip, due to the conceptualization of the diffusion of solid particles. To determine the compressibility required by the model, Kanno et al. [2001] measured swelling pressure as a function of dry density in the range of 1.5 g/cm 3 to 2.0 g/cm 3. For the numerical simulation of the model, the swelling pressure for a dry density of 0.3 g/cm 3 was required. Thus, untested extrapolations of the experimental data into this lower region were utilized. iscosity values for bentonite gel also were not based on experimental measurements Comparison of SABRE models with Kanno et al. [2001] Model This section will demonstrate that the SABRE6 model, derived in 3.1 and the model developed by Kanno et al. [2001] are mathematically equivalent. This solid diffusion model is expressed as [Kanno et al., 2001] t D S. (78) The diffusion coefficient analogue is call the solid diffusivity and is given as D S k d d. (79) M L 1 : swelling pressure 2 2 T L This model does not incorporate an additional differential equation in order to consider tip location. The solid volume fraction is defined as S. (80) The solid volume fraction and the void ratio are related in the following manner, by definition, as -28-