ANALYSIS OF SPATIAL VARIABILITY OF 

ROCK-MASS QUALITY AND 

THERMAL-MECHANICAL EFFECTS IN GEOLOGIC 

DISPOSAL OF HIGH-LEVEL NUCLEAR WASTE 

Prepared for 

U.S. Nuclear Regulatory Commission 

Contract NRC–02–07–006 

Prepared by

Goodluck Ofoegbu 

Center for Nuclear Waste Regulatory Analyses 

San Antonio, Texas 

September 2011
ii 

ABSTRACT 

This report describes analyses performed from March 1998 through April 1999 to examine the 

effects of spatial variability of rock-mass quality on thermal-mechanical effects in geologic 

disposal of high-level nuclear waste. Spatial variability of rock-mass quality based on mapped 

fracture data is represented in complementary site- and drift-scale numerical finite element 

models of a conceptual waste disposal design. Analyses performed using the models indicate 

that a vertical temperature gradient centered at the disposal horizon combines with weaker 

lateral temperature gradients to create stress conditions that could favor rock damage close to 

disposal openings and in intervening pillars. Rock damage in the roof and floor of the openings 

could be reduced using appropriate ground support and is likely to occur more in rock of lower 

mechanical quality. However, ground support would not affect potential rock damage in the 

pillar, because such damage is driven by stress conditions outside the influence zone of typical 

ground support. Because of the stress control, rock damage in the pillar could occur more in 

rock of higher mechanical quality. The results show that ground support could be effective in 

stabilizing an underground opening used for disposal of high-level nuclear waste, if the ground 

support system is designed to withstand loading due to suppressed inelastic deformation and 

thermal expansion of the support elements and supported rock.
iii 

CONTENTS 

Section Page 

ABSTRACT ................................................................................................................................... ii 

FIGURES ..................................................................................................................................... iv 

TABLE .......................................................................................................................................... vi 

ACKNOWLEDGMENTS .............................................................................................................. vii 

1 INTRODUCTION ............................................................................................................ 1-1

2 ROCK-MASS QUALITY DATA ...................................................................................... 2-1 

3 MODEL DESCRIPTION ................................................................................................. 3-1 

 3.1 Model Geometry and Finite Element Discretization ........................................... 3-1 

 3.2 Material Models and Properties .......................................................................... 3-1 

 3.3 Initial and Boundary Conditions and Heat Source .............................................. 3-4 

 3.4 Analysis Procedure ............................................................................................ 3-4 

4 CALCULATED THERMAL-MECHANICAL RESPONSE ................................................ 4-1 

 4.1 Host Rock Temperature ..................................................................................... 4-1 

 4.2 Rock Damage Based on Inelastic Strain ............................................................ 4-1 

 4.3 Modeling with Single-Valued Rock-Mass Quality Index ................................... 4-10 

 4.4 Effects of Ground Support on Rock Damage ................................................... 4-11 

 4.5 Potential Loading of a Concrete Liner Ground Support.................................... 4-14 

5 SUMMARY OF INSIGHTS ............................................................................................. 5-1 

6 REFERENCES ............................................................................................................... 6-1 
iv 

FIGURES 

Figures Page 

1-1 Layout of the Conceptual Disposal Design (From DOE Figure 4-22, 1998) .................. 1-3 

2-1 Distribution of Rock-Mass Quality Q Along the Eastern Boundary of the 

 Conceptual Disposal Drift Array of DOE Figure 4-22 (1988) Based on the 

 “Full-Peripheral Data” Described in a DOE Contractor Report 

 CRWMS M&O Figure 40, 1997a) ................................................................................... 2-2 

3-1 Schematic Illustration of the Site-Scale Model Showing a 2,800-m Wide Disposal 

 Drift Array at a Depth of 300 m Below the Ground Surface ........................................... 3-2 

3-2 Geometry and Finite Element Discretization of the Drift-Scale Model ........................... 3-3 

3-3 History of Total Heat Source Sh Used To Represent the Thermal Load of Disposed 

 Nuclear Waste Based on CRWMS M&O Table V-1 (1997b) ......................................... 3-5 

4-1 Temperature Distributions Over the Conceptual Disposal Array at 25, 50, 100, 

 and 150 Years After Instantaneous Waste Loading of All Disposal Drifts ...................... 4-2 

4-2 Temperature Distributions Within Approximately 100 m Above and Below the 

 First 10 Disposal Drifts at the South End of the Site-Scale Model (Drifts #91–100) 

 at 50 and 150 Years After Instantaneous Waste Loading of All Disposal Drifts ............. 4-3 

4-3 Temperature Distributions Within 25 m Above and Below a Drift Based on the 

Drift-Scale Model at 150 Years After Instantaneous Waste Loading of All 

Disposal Drifts ................................................................................................................ 4-4 

4-4 Inelastic Strain Distribution at 150 Years for the Case of Drift Openings With Stiff 

 Ground Support. The Plot Arrangement is the Same as Described in Figure 2-1 ........ 4-5 

4-5 Inelastic Strain Distribution at 150 Years for the Case of Drift Openings With Degraded 

 Ground Support. The Plot Arrangement is the Same as Described in Figure 2-1 ........ 4-6 

4-6 Inelastic Strain Distribution at 150 Years Based on the Drift-Scale Model for Areas of 

 High Rock-Mass Quality ................................................................................................. 4-7 

4-7 Inelastic Strain Distribution at 150 Years Based on the Drift-Scale Model for Areas of 

 Low Rock-Mass Quality ................................................................................................. 4-8 

4-8 Schematic Illustration of Rock Deformation Mechanisms Due to Heat From a 

 Horizontal Array of Fully Loaded Waste Disposal Drifts ................................................ 4-9 

4-9 Inelastic Strain Distributions Between Drift Numbers 91–100 

(at the South End of the Disposal Array) Calculated Using Three Model Cases ......... 4-11 

4-10 Inelastic Strain Distributions Between Drift Numbers 41–50 

(in the Middle of the Disposal Array) Calculated Using Three Model Cases ................ 4-12 

4-11 Displacement History at the Roof, Floor, and North and South Sidewalls of a 

 Disposal Drift Without Ground Support Based on a Drift-Scale Model Analysis for 

 High-Q Areas ............................................................................................................... 4-13 

4-12 Inelastic Strain Distributions at 150 Years Based on Drift-Scale Modeling for 

High-Q Areas, Showing the Effects of Ground Support on Thermally Induced 

Rock Damage .............................................................................................................. 4-15

4-13 Distribution of Liner Pressure Around a Drift Circumference at 150 Years After 

 Instantaneous Waste Loading, Based on Highest-Q and Lowest-Q Drift-Scale 

 Models With Liner Fully Bonded to Rock ..................................................................... 4-16 

4-14 Variation of Liner Pressure With Angular Distance Around a Drift Circumference 

 at 150 Years After Instantaneous Waste Loading, Based on Highest-Q and Lowest-Q 

 Drift-Scale Models With Liner Fully Bonded to Rock ................................................... 4-17 
v 

FIGURES (continued) 

Figures Page 

4-15 Effects of the Liner-Rock Interface Model on the Calculated Liner Pressure 

 Showing Results for a Fully-Bonded Interface Model and a Frictional-Interface 

 Model With a Friction Coefficient of 0.6 ....................................................................... 4-18 
vi 

TABLES 

Tables Page 

3-1 Relationship Between Thermal Expansivity and Temperature....................................... 3-3 

3-2 Sequence of Thermal and Mechanical Analyses ........................................................... 3-5 

4-1 Vertical Thickness of Rock Subjected to Given Temperature Ranges at 150 Years 

 Based on the Site-Scale (Figure 4-2, lower plot) and Drift-Scale [Figure 4-3(a)] 

 Models ........................................................................................................................... 4-4 
vii 

ACKNOWLEDGMENTS 

This report describes work performed by the Center for Nuclear Waste Regulatory 

Analyses (CNWRA) and its contractors for the U.S. Nuclear Regulatory Commission (USNRC) 

under Contract No. NRC–02–07–006. The activities reported here were performed on behalf of 

the USNRC Office of Nuclear Material Safety and Safeguards, Division of High-Level Waste 

Repository Safety. This report is an independent product of CNWRA and does not necessarily 

reflect the view or regulatory position of USNRC. 

The author gratefully acknowledges B. Dasgupta for his technical review, E. Pearcy for his 

programmatic review and editorial review, and A. Ramos for her administrative support. 

Furthermore, the author gratefully acknowledges E. Pearcy and J. McMurry for encouraging him 

to search previous scientific notebooks for unpublished work. 

QUALITY OF DATA, ANALYSES, AND CODE DEVELOPMENT 

DATA: All CNWRA-generated original data contained in this report meet the quality assurance 

requirements described in the Geosciences and Engineering Division Quality Assurance 

Manual. Sources for other data should be consulted for determining the level of quality for 

those data. Computational calculations described in this report have been recorded in CNWRA 

Scientific Notebook Nos. 263 (Ofoegbu, 2008a) and 321 (Ofoegbu, 2000) and have been 

described in previous CNWRA publications (Ofoegbu, 1998b, 1999). 

References 

Ofoegbu, G.I. “Repository Scale Thermal-Merchanical 2-D Model.” Scientific Notebook 

No. 321. San Antonio, Texas: Center for Nuclear Waste Regulatory Analyses. 2000. 

Ofoegbu, G.I. “Variations of Drift Stability at the Proposed Yucca Mountain Repository.” 

Proceedings of the 37th U.S. Rock Mechanics Symposium, June 6–9, 1999. Vail, Colorado: 

Rock Mechanics for Industry. 1999. 

Ofoegbu, G.I. “Repository Scale Thermal-Merchanical Model.” Scientific Notebook No. 263. 

San Antonio, Texas: Center for Nuclear Waste Regulatory Analyses. 2008a. 

Ofoegbu, G.I. “Effects of Spatial and Temporal Variations of Rock-Mass Quality and Integrity of 

Support on Distributions of Potential Instability at the Proposed Yucca Mountain Repository for 

High-Level Nuclear Waste.” San Antonio, Texas: CNWRA. 1998b. 
1-1 

1 INTRODUCTION 

The potential for occurrence of a zone of damaged rock around underground openings used for 

geologic disposal of high-level radioactive waste is a key consideration for the disposal system 

design. Rock within the zone is referred to as damaged in the sense that the capability of the 

rock to contribute to waste isolation may have degraded somewhat by the formation of a 

fracture or crack population that did not exist prior to the excavation, waste emplacement, or 

other activities related to the repository development. The rock is considered damaged relative 

to the initial state because of the effects of the fractures or cracks on mechanical load-bearing 

capacity or on water flow. For example, Tsang, et al. (2005) defined the damaged zone as a 

zone in which the flow and transport properties of the rock have changed significantly relative to 

the initial rock-mass condition. However, defining rock damage with respect to the hydrological 

or mechanical effect could complicate damage characterization because a damage intensity 

great enough to cause a hydrological effect may be insignificant to mechanical stability. 

Therefore, in this work damage intensity is described in terms of the magnitude of inelastic 

strain, which could be used to derive indices for characterizing the hydrological or 

mechanical effects. 

Damage due to cracks or fractures is a characteristic of brittle rocks that have low matrix 

porosity or permeability but may support fluid flow through a network of cracks or fractures. As 

Tsang, et al. (2005) described, other forms of damage, such as pore dilation, may occur in other 

rock types. In brittle rocks, damage is triggered by stress changes large enough to overcome 

the rock strength. Such a stress change may result from excavation, thermal loading, or water 

pressure change. However, discussion and evaluations of rock damage around excavations 

typically have been dominated by consideration of damage due to excavation. There have been 

numerous references to the excavation damaged zone (EDZ) and several international 

meetings and activities aimed at understanding the characteristics of the EDZ, such as those 

Tsang, et al. (2005) described. In contrast, rock damage due to thermal load has not received 

much attention, though the extent and duration of significant temperature change due to 

disposed radioactive waste suggest that thermally induced damage could be significant. For 

example, Andersson (2007) described in-situ heater test results that show rock spalling due to a 

simulated thermal load in a shaft in granitic rock was approximately three times the spalling 

caused by excavation of the shaft. 

Rock deformation due to thermal load could consist of free thermal expansion or a response to 

the stress change due to suppressed thermal expansion. The latter occurs if the rock is heated 

nonuniformly and, consequently, tries to expand by different amounts at neighboring points. 

Nonuniform heating of the host rock of a geologic repository is inevitable because nuclear waste 

containers with varying heat output capacities are placed at discrete locations in the rock at 

discrete times, and heat produced by each container flows into the rock at a finite rate controlled 

by the host rock thermal conductivity. Consequently, the host rock is heated by an amount that 

varies spatially and temporally. Therefore, potential expansion of the rock is partially 

suppressed by different amounts in different directions due to neighborhood resistance, 

resulting in nonuniform increase in rock stress. The rock deforms as a result of adjusting to a 

new equilibrium state. Additionally, the thermal load causes pore fluids to try to expand by 

different amounts at neighboring points because of spatially varying temperature, which results 

in fluid pressure increase if the fluids cannot flow freely. The excess fluid pressure causes a 

change in the rock stress and loading conditions, and the rock deforms to adjust to a new 

equilibrium state. The excess pore pressure dissipates with time at a rate that depends on the 

fluid conductivities, compressibility of the solids skeleton, and distance to drainage boundaries. 
1-2 

Rock deformations can be elastic and therefore fully reversible, or they can be inelastic. 

Inelastic deformations may manifest through formation of new cracks, slip or opening or closing 

of existing cracks, or particle morphology changes not associated with cracking. In brittle rocks, 

such as various types of igneous, metamorphic, and indurated sedimentary rocks, inelastic 

deformation is dominated by cracking and is associated with damage because of the effect on 

mechanical strength and stiffness and on rock characteristics important to waste isolation, such 

as hydrologic and transport properties. 

The intensity and extent of thermally induced damage depend on the thermal load and rockmass thermal and mechanical properties. The thermal load and rock-mass thermal properties 

determine the magnitude and gradients of temperature, which combine with the rock-mass 

stiffness to determine the thermal stress, as Timoshenko and Goodier (1970, Chapter 13, 

Article 153) explained. The rock-mass strength determines damage occurrence for a given 

stress state. 

The work described in this report was performed from March 1998 through April 1999 to 

examine how the effects of spatial variability of rock-mass mechanical properties may affect 

thermally induced damage. The analyses were based on a conceptual design by the 

U.S. Department of Energy (DOE) for geologic disposal of radioactive waste at Yucca Mountain, 

Nevada (DOE, 1998, Figure 4-22). The conceptual design considered a horizontal array of 

waste disposal drifts approximately normal to an access main tunnel (Figure 1-1). The access 

tunnel in the conceptual design coincided with an existing tunnel that DOE had excavated to 

provide access for exploration of the rock-mass being considered as host rock of the disposal 

openings. Fracture data from the walls of the exploratory tunnel were used to estimate values 

of the rock-mass quality, Q, index as described in Civilian Radioactive Waste Management 

System Management and Operating Contractor (CRWMS M&O) (1997a, Figure 40). For the 

work described in this report, the Q data were used to define a potential variation of the 

rock-mass strength and stiffness in the north-south direction along the access tunnel based on a 

procedure Hoek and Brown (1997) described. The resulting mechanical property variation was 

projected uniformly in the east-west direction along the disposal drifts and, therefore, was 

assumed to apply to the entire host rock-mass of the conceptual design. The mechanical 

property variation was used to calculate variations of thermally induced rock damage in the 

north-south direction among the disposal drifts. The evaluation of rock damage focused on the 

mechanical effects of thermal expansion of solids but did not include any effects due to thermal 

expansion of water or air. 

Spatial relationships among the Q data, exploratory tunnel, and conceptual disposal drift array 

are described in Chapter 2 of this report. Chapter 3 describes the finite element models used 

for the analyses, and Chapter 4 describes the calculated results. Insights based on the analyses 

are summarized in Chapter 5. 
1-3 

Figure 1-1. Layout of the Conceptual Disposal Design (From DOE, 1998, Figure 4-22) 
2-1 

2 ROCK-MASS QUALITY Q DATA 

The exploratory tunnel, as described in DOE (1998, Figure 4-22), consists of a gently sloping 

north-south tunnel segment (east main) connected to the ground surface through the “north 

ramp” at the north and “south ramp” at the south. In the conceptual design (Figure 1-1), the east 

main connects to the eastern end of a series of disposal drifts oriented approximately east-west. 

The disposal drifts form a horizontal array. The eastern boundary of the array coincides with the 

east main but extends approximately 1 km [0.62 mi] to the north beyond the east main. The 

exploratory tunnel (east main and north and south ramps) covers a total distance of 

approximately 8 km [5 mi]. In the conceptual design, the disposal array consists of 105 drifts 

spaced at 28 m [91.9 ft] center to center. The drift array occupies a distance of approximately 

2 km [1.24 mi] along the east main and approximately 1 km [0.62 mi] beyond the north end of 

the tunnel (Figure 1-1). The Q data cover the entire exploratory tunnel as shown in 

CRWMS M&O (1997a, Figure 40). Therefore, because of the geometrical relationship between 

the exploratory tunnel and the conceptual disposal array, the Q data cover the east boundary of 

approximately the southern two-thirds of the disposal array. 

For modeling described in this report, the Q data were projected uniformly westward to the 

western end of the disposal array, the value at the north end of the east main (Figure 1-1) was 

projected uniformly northward to the end of the array, and all Q values were projected uniformly 

in the vertical direction. The implications of the spatial projections of the Q data will be 

discussed subsequently under “Model Description.” The resulting north-south (i.e., normal to 

the disposal drift array) variation of Q values is described in Figure 2-1. The model domain is a 

vertical north-south rectangle normal to the disposal drifts. Figure 2-1 illustrates the variation of 

Q values within a 35-m [115-ft]-high strip normal to the drifts and centered on the drift axis, each 

drift represented by a 5-m [16.4-ft]-high opening (white squares in Figure 2-1). To facilitate 

presentation, the strip is divided into 10 segments, each representing 10 drifts according to the 

numbers shown at the beginning and end of each segment in Figure 2-1. Drift number 100 is at 

the south end of the modeled array, and drift number 1 is at the north end. As described earlier, 

the Q value for drift number 32 (corresponding to the north end of the east main) was projected 

uniformly northward to obtain Q values for drift numbers 1–31 because of the lack of Q data for 

the northern one-third of the conceptual disposal array. The Q values in Figure 2-1 were based 

on the “full-peripheral data” of CRWMS M&O (1997a, Figure 40). 

As Figure 2-1 shows, the rock-mass quality index attains maximum values in the middle of the 

array between drift numbers 31–70. The value of Q attains a maximum of approximately 13.6 

between drift numbers 47 and 48 and a minimum of approximately 0.73 between drift numbers 

98 and 99. The spatial variation of Q values depicted in Figure 2-1 was incorporated explicitly 

into a numerical finite element model as described in Chapter 3. 
2-2 

Figure 2-1. Distribution of Rock-Mass Quality Q Along the Eastern Boundary of the 

Conceptual Disposal Drift Array of DOE (1988, Figure 4-22) Based on the “Full-Peripheral 

Data” Described in CRWMS M&O (1997a, Figure 40). The Figure Shows a 35-m 

[114.8-ft]-High Strip Centered on the Drift Axis. P The Strip is Divided Into 10 Segments, 

Each Representing 10 Drifts (White Squares) According to the Numbers Shown at the 

Beginning and End of Each Segment. Drift Number 100 Is at the South End of the Array, 

and Number 1 Is at the North End. 
3-1 

3 MODEL DESCRIPTION 

3.1 Model Geometry and Finite Element Discretization 

Analyses were performed at two spatial scales. A site-scale model that includes the entire 

disposal drift array was used to study potential variations of thermal-mechanical response within 

the array, and a drift-scale model was used to study the response at the spatial scale of an 

individual drift. The site-scale model (Figure 3-1) consists of a vertical north-south rectangle 

normal to the drift axis of the conceptual disposal array. The rectangle is 3,200 m [10,499 ft] 

wide in the north-south direction {i.e., 2,800 m [9,186 ft]} representing 100 drifts, plus a 200-m 

[656-ft] extension at the north and south ends, and is 1,000 m [3,281 ft] high. Each drift was 

represented in the site-scale model as a 5-m [16.4-ft] square opening with center 

(i.e., intersection with drift axis) at a depth of 302.5 m [992.4 ft] below the ground surface. 

The drift-scale model (Figure 3-2) consists of a 28-m [92-ft]-wide by 195-m [640-ft]-high 

rectangle with spatial coordinates consistent with the site-scale model, as described in 

Figure 3-2(a). The model extends 95 m [312 ft] vertically above the roof and below the floor of a 

5-m [16.4-ft]-diameter circular opening that represents the drift. The site-scale model was used 

to calculate temperature and displacement histories at the top, base, and left and right sides of 

the drift-scale model. Subsequently, the calculated histories from the site-scale model were 

used to define boundary conditions for the drift-scale model. 

The site-scale and drift-scale models were based on plane strain idealization, which assumes 

that mechanical properties do not vary in the drift direction and in-situ stress components are 

normal and parallel to the drift axes. These assumptions permit modeling to examine the 

effects of mechanical property variability (using the modeled north-south variation) on damage 

intensity and extent. The modeling is expected to lead to insights regarding how thermally 

induced rock damage may vary among relatively high and low quality areas within a rock-mass, 

even if the results do not directly quantify the damage intensity and extent for any individual site. 

The finite element models are based on eight-node quadrilateral elements with reduced 

integration and occasional six-node triangles for mesh refinement (e.g., Figure 3-2). For the 

site-scale model, beam elements with high stiffness were placed at the edges of the drift 

openings to represent a concrete liner ground support. In the drift-scale models, a 20-cm 

[7.9-in]-thick ring of material around the 5-m [16.4-ft]-diameter circle [shown in Figure 3-2(c) as 

two rings of solid elements] was assigned properties of concrete to represent a concrete liner 

ground support. The interface between the concrete liner elements and surrounding rock was 

modeled as fully bonded, except for the model variation described in Section 4.5 where the 

interface was modeled as frictional. In another variation of the drift-scale model described in 

Section 4.4, the 20-cm [7.9-in]-thick ring was fully bonded to and assigned the same properties 

as the surrounding rock to simulate a 5-m [16.4-ft]-diameter circular opening without 

ground support. 

3.2 Material Models and Properties 

Heat flow was simulated based on thermal conduction and using material properties from 

CRWMS M&O (1997b). The report gave a rock density of 2,274 kg/m3

 [142 lb/ft3]; thermal 

conductivity of 2.1 J/s.m.K; and a specific heat capacity of approximately 2.14, 10.5, or 

2.18 MJ/m3

.K at a temperature of 25–94, 94–114, or greater than 114 °C [77–201, 201–237, or 

greater than 237 °F], respectively. The increased specific heat value in the temperature range 
3-2 

Figure 3-1. Schematic Illustration of the Site-Scale Model Showing a 2,800-m 

[9,186 ft]-Wide Disposal Drift Array at a Depth of 300 m [984 ft] Below the Ground Surface 

of 94–114 °C [201–237 °F] simulates approximately the effects of latent heat of vaporization. 

Mechanical deformation was modeled as elastic-plastic with a rock strength defined in terms of 

the Drucker-Prager yield criterion. The mechanical parameters are thermal expansivity, which 

varies with temperature as described in Table 3-1 based on CRWMS M&O (1997c); Poisson’s 

ratio, set to a constant value of 0.21 based on CRWMS M&O (1997c); and Young’s modulus, 

and Mohr-Coulomb friction angle and cohesion, which were assigned spatially variable values 

using their relationships with the Q rock-mass quality index, based on a procedure Hoek and 

Brown (1997) described. 

Two drift-scale models centered on drift numbers 48 and 85 were set up to represent areas of 

maximum and minimum Q values. Each drift-scale model extended laterally from the middle of 

the contiguous south-side pillar to the middle of the contiguous north-side pillar, giving a model 

width of 28 m [92 ft], as described in Figure 3-2(b). Drift number 48 represents maximum Q, 

and number 85 represents minimum Q (Figure 2-1). The absolute minimum Q occurs between 

drift numbers 98 and 99. However, the minimum Q model was set up around drift number 85 to 

avoid lateral thermal gradients due to end effects, which the site-scale model showed to be 

strong in the vicinity of drift numbers 98 and 99. 
3-3 

Figure 3-2. Geometry and Finite Element Discretization of the Drift-Scale Model Showing 

(a) Entire Drift-Scale Model; (b) a Rectangular Zone of Interest Around a 5-m 

[16.4-ft]-Diameter Circular Opening, Representing the Drift; (c) Finite Element Model of a 

20-cm [7.9-in]-Thick Concrete Liner, Representing Ground Support; and (d) Finite 

Element Model of the Drift Section, Representing the Excavated Rock. The Vertical 

Coordinate y Is the Same as “Up” and y = 0 at 1,000 m [3,281 ft] Below the Ground 

Surface in the Site-Scale Model. 

Table 3-1. Relationship Between Thermal Expansivity and Temperature* 

Temperature °C Thermal expansivity (10-6/K) 

0.0 5.07 

29 5.07 

51 7.30 

98 7.30 

102 8.19 

148 8.19 

152 8.97 

200 8.97 

*CRWMS M&O. “Repository Ground Support Analysis for Viability Assessment.” 

BCAA00000–01717–0200–00004, Revision 00. LSN Accession Number MOL.19971210.0093. Las Vegas, Nevada: 

Civilian Radioactive Waste Management System Management and Operating Contractor. 1997. 
3-4 

3.3 Initial and Boundary Conditions and Heat Source 

The initial temperature was set to 18.7 °C [65.7 °F] at the ground surface and increased with 

depth at rates of 0.02, 0.018, 0.03, or 0.008 °C/m [0.0034, 0.003, 0.0051, or 0.0014 °F/ft] for 

depth ranges of 0–150, 150–400, 400–700, and 700–1,000 m [492–1,312; 1,312–2,297; and 

2,297–3,281 ft], respectively, based on CRWMS M&O (1997b). The initial vertical stress was 

determined using an average rock density of 2,274 kg/m3 [142 lb/ft3

], which gave a vertical 

stress of 0.022 MPa/m [0.000973 ksi/ft] of depth below the ground surface. The initial 

horizontal-to-vertical-stress ratio was set to the zero-lateral-strain value of ν/(1-ν), where 

ν = 0.21 is the Poisson’s ratio. 

Boundary conditions for the site-scale model (Figure 3-1) consist of fixed initial-value 

temperature at all exterior boundaries, no lateral displacement at the vertical boundaries, no 

vertical displacement at the model base, and free surface conditions at the ground surface. 

Boundary conditions for the drift-scale model [Figure 3-2(a)] were set using temperature and 

displacement histories calculated in the site-scale model at the drift-scale model exterior 

boundary nodes. 

Thermal load due to disposed nuclear waste was represented using a volumetric heat source 

applied to the rock volume within each drift section. It was assumed that all disposed waste in 

the conceptual design would be loaded into the drifts instantaneously at zero time. Therefore, 

the volumetric heat source was calculated as Sh/Vd, where Vd is the total drift volume based on 

an average drift length of 1,080 m [3,543 ft], as specified in CRWMS M&O (1997b), and Sh is 

the total heat output per unit time from all disposed waste. The history of Sh is plotted in 

Figure 3-3 using information from CRWMS M&O (1997b, Table V-1). 

3.4 Analysis Procedure 

Each analysis performed using the site-scale or drift-scale model consisted of a heat 

conduction analysis to calculate temperature distribution histories that were used as input to a 

mechanical analysis to calculate time-dependent stresses, displacements, and damage 

(Table 3-2). The heat conduction analysis consisted of two steps. First, a dummy step lasting 

2 × 10-6 seconds was performed to synchronize thermal and mechanical analysis times, but no 

thermal change occurred during the step. Second, a thermal analysis step lasting 150 years 

was performed to calculate the thermal effects of the volumetric heat source described in 

Section 3.3. 

The mechanical analysis consisted of four steps. The first two steps were isothermal at 

the initial temperature state and time-independent but assigned a nominal duration of 

1 × 10−6

 seconds each. The first step was used to establish the initial static equilibrium state 

under gravitational loading and boundary restraint. Thereafter, drift excavation was simulated 

during the second step by removing materials within the drift opening {elements within the 

5 by 5-m [16.4 by 16.4-ft] squares in Figure 3-1 for the site-scale model or elements shown in 

Figure 3-2(d) for the drift-scale model} and calculating a new static equilibrium state. The third 

mechanical step lasted for 150 years. Temperature distribution histories calculated from the 

thermal analysis were applied during this step and the mechanical response to the temperature 

distributions was calculated. The fourth step was time independent but was assigned a nominal 

duration of 1 year. The step was executed based on the thermal state at the end of the third 

step to explore mechanical effects of ground support degradation over 150 years. To examine 
3-5 

Figure 3-3. History of Total Heat Source Sh Used To Represent the Thermal Load of 

Disposed Nuclear Waste Based on CRWMS M&O (1997b, Table V–1) 

Table 3-2. Sequence of Thermal and Mechanical Analyses 

Time (year) Thermal Analysis Mechanical Analysis 

0–1 × 10−6 Dummy step—No thermal change Initial static equilibrium under 

gravitational loading and boundary 

restraint—No thermal change 

1 × 10−6–2 × 10−6 Dummy step—No thermal change Drift excavation simulated with 

pre-installed ground support. No 

thermal change 

2 × 10−6–150 Thermal analysis step using heat 

source history in Figure 3-3 

as input 

Mechanical response with 

excavated drift and ground support 

No time increment No thermal change from end of 

previous step 

Ground support elements removed 

during a nominal time period of one 

year (150–151 years) to simulate 

degradation of ground support 

the effects of ground support degradation on rock damage, elements representing ground 

support were removed during the step. 
4-1 

4 CALCULATED THERMAL-MECHANICAL RESPONSE 

The finite element models were used to calculate the effects of waste disposal thermal load on 

rock temperature, stress, and damage and the effects of rock-mass quality variability on 

thermally induced mechanical effects. Results of the analysis are described in this chapter. 

4.1 Host Rock Temperature 

Temperature distributions calculated using the site-scale model (e.g., Figures 4-1 and 4-2) 

indicate that a vertical temperature gradient would develop in the host rock of a nuclear waste 

repository, such as that modeled, and will likely be sustained for a long time during the heating 

period. Lateral temperature gradients also may occur, especially close to the ends of the 

disposal array (Figures 4-1 and 4-2) or within close proximity of the openings through a zone 

approximately two to three drift diameters above and below the openings [Figure 4-3(b)]. The 

lateral gradients are likely to be stronger at early times (e.g., Figures 4-1 and 4-2), are likely to 

weaken as temperature increases (e.g., Figure 4-2), and could strengthen again during the 

cooling period (not shown in this work). Results calculated using the drift-scale model 

[Figure 4-3(b)] indicate that the lateral temperature gradients could be sustained for a long time 

close to the opening and in the interdrift pillar areas near the elevation of the openings. 

Temperature gradients are important to rock stress as explained in Timoshenko and Goodier 

(1970, Chapter 13, Article 153) and to rock damage as illustrated by Ofoegbu and 

Curran (1987). 

Temperature distributions calculated using the site-scale and drift-scale models are similar as 

can be shown by comparing the lower plot of Figure 4-2 against Figure 4-3(a) and using 

information such as that described in Table 4-1. The comparison indicates that the drift-scale 

model can be relied upon to simulate the temperature distribution around individual drifts in the 

disposal array except for approximately five drifts at the north and south ends of the array. 

Simulating thermal-mechanical behavior near an end drift would entail using a drift-scale model 

with appropriate boundary conditions for the end drift. 

4.2 Rock Damage Based on Inelastic Strain 

As discussed in Chapter 1, inelastic straining of brittle rocks, such as several types of igneous, 

metamorphic, and indurated sedimentary rocks, is dominated by cracking. Therefore, inelastic 

straining is associated with damage because of the effects of cracks on mechanical strength 

and stiffness and on rock characteristics important to waste isolation, such as hydrologic and 

transport properties. Intuitively, the damage intensity associated with a given magnitude of 

inelastic strain should increase as the inelastic strain magnitude increases. Therefore, 

magnitudes of inelastic strain calculated using a numerical model could be interpreted to 

indicate relative intensities of damage, without quantifying the damage intensity in terms of 

crack or fracture density or connectivity. Therefore, for the analyses described in this report, 

relative damage was described in terms of the Euclidean norm of the inelastic strain tensor 

accumulated through the simulated history and hereafter referred to simply as “inelastic strain.” 

Plots of accumulated inelastic strain at 150 years shown in Figures 4-4 through 4-7 indicate that 

the heat load due to disposed waste could cause rock damage close to the drift openings, 

especially in the roof and floor areas, and in the interdrift pillars. Potential occurrence of rock 

damage in the roof and floor of the openings and in the pillars could be explained by considering 

the effects of the disposal geometry on mechanisms of thermally induced deformation, as 
4-2 

Figure 4-1. Temperature Distributions Over the Conceptual Disposal Array at 25, 50, 100, 

and 150 Years After Instantaneous Waste Loading of All Disposal Drifts 
4-3 

Figure 4-2. Temperature Distributions Within Approximately 100 m [328 ft] Above and 

Below the First 10 Disposal Drifts at the South End of the Site-Scale Model (Drifts 91–100) 

at 50 and 150 Years After Instantaneous Waste Loading of All Disposal Drifts 
4-4 

Figure 4-3. Temperature Distributions Within (a) 95 m [312 ft] and (b) 25 m [82 ft] Above 

and Below a Drift Based on the Drift-Scale Model at 150 Years After Instantaneous Waste 

Loading of All Disposal Drifts 

Table 4-1. Vertical Thickness of Rock Subjected to Given Temperature Ranges at 150 Years 

Based on the Site-Scale (Figure 4-2, lower plot) and Drift-Scale [Figure 4-3(a)] Models 

Temperature Range (°C) 

Rock Thickness Based on SiteScale Model in Figure 4-2, North 

Boundary of Lower Plot (m) 

Rock Thickness Based on 

Drift-Scale Model in 

Figure 4-3(a) (m) 

110–139 42.8 43.0 

100–110 19.2 19.8 

90–100 23.5 23.7 

80–90 25.7 26.4 

70–80 29.9 29.7 

60–70 32.1 33.6 
4-5 

Figure 4-4. Inelastic Strain Distribution at 150 Years for the Case of Drift Openings With 

Stiff Ground Support. The Plot Arrangement Is the Same as Described in Figure 2-1. 
4-6 

Figure 4-5. Inelastic Strain Distribution at 150 Years for the Case of Drift Openings With 

Degraded Ground Support. The Plot Arrangement Is the Same as Described in 

Figure 2-1. 
4-7 

Figure 4-6. Inelastic Strain Distribution at 150 Years Based on the Drift-Scale Model for 

Areas of High Rock-Mass Quality 
4-8 

Figure 4-7. Inelastic Strain Distribution at 150 Years Based on the Drift-Scale Model for 

Areas of Low Rock-Mass Quality 

Ofoegbu (2001) described. The shape of the disposal array and the relative closeness of the 

ground surface (Figure 3-1) allow a limited amount of upward expansion but negligible 

expansion downward or laterally. Therefore, rock directly above a typical disposal drift can 

expand upward more than less heated rock in the pillar and impose an upward drag (tension) on 

the pillar rock, as illustrated in Figure 4-8. Also, because the disposal array extends laterally 

much more than vertically and lateral expansion is suppressed more than vertical expansion, 

the lateral component of thermal stress is much greater than the vertical component. Close to 

the openings, the effects of the increased lateral stress are reinforced by stress concentrations 
4-9 

Figure 4-8. Schematic Illustration of Rock Deformation Mechanisms Due to Heat From a 

Horizontal Array of Fully Loaded Waste Disposal Drifts. The Disposal Thermal Load and 

Rock-Mass Thermal-Mechanical Properties Determine the Deformation Magnitudes and 

Type (i.e., Elastic or Inelastic) 

at the roof and floor but diminished by stress concentrations at the sidewalls. These stress 

conditions, which depend only on the disposal geometry, favor the development of a 

reverse-faulting style deformation mechanism in the roof and floor areas of the drifts and in the 

pillars, but a strike-slip faulting style deformation mechanism near the drift sidewall (Figure 4-8). 

Whereas the deformation style is determined by the disposal geometry, the magnitude and type 

(i.e., elastic or inelastic) of deformation are determined by the thermal load and rock-mass 

mechanical properties. As described in Chapter 1, the thermal load and rock mechanical 

properties determine the magnitude of thermal stress and whether the stress is sufficient to 

cause inelastic deformation. Thus, rock damage in the pillars is stress controlled and, therefore, 

more intense in higher rock-mass quality areas where the rock stiffness is large enough to 

generate stresses sufficient to cause rock failure. Conditions that increase rock stress will likely 

increase the potential for pillar damage, more so in higher quality (higher stiffness) than lower 
4-10 

quality (lower stiffness) rock. In contrast, rock damage in the roof and floor, though stress 

controlled, is influenced by support conditions at the boundaries of the opening. Therefore, the 

loss of ground support increased damage in the roof and floor, especially in areas of low 

rock-mass quality. For example, compare the increase in inelastic strain in the roof and floor 

areas of drift numbers 81–100 between the end of supported-opening conditions (Figure 4-4) 

and the degraded-support conditions (Figure 4-5). Also, compare the inelastic strain increase 

between the left and right plots of Figure 4-7 with the hardly noticeable change between the left 

and right plots of Figure 4-6. As the comparisons show, conditions that reduce confinement at 

the boundary of an opening will likely increase rock damage in the roof and floor, more in lower 

quality than in higher quality rock. 

Therefore, the disposal thermal load could cause rock damage in the roof and floor of 

underground openings and in the intervening pillars. Rock damage in the roof and floor could 

affect the mechanical stability of the openings, is influenced by ground support conditions, and 

will likely occur more in rock of lower mechanical quality. In contrast, rock damage in the pillar 

is stress controlled and could occur more in rock of higher mechanical quality. Furthermore, 

rock damage in the pillar will likely be insufficient to affect mechanical stability of the openings, 

but could affect water flow (e.g., Ofoegbu, 2001). 

4.3 Modeling with Single-Valued Rock-Mass Quality Index 

Additional calculations were performed to evaluate the use of single-valued representations of 

the rock-mass quality index. Analyses were performed using two modified site-scale models 

with Q assigned a constant value everywhere. Q was assigned the minimum value everywhere 

in one case and the maximum value in the other case. Results calculated using the two model 

cases were compared with the results described in Section 4.2. 

The comparison is illustrated in Figure 4-9 for drift numbers 91–100 and Figure 4-10 for drift 

numbers 41–50. The comparison could be used to evaluate different approaches for assessing 

the effects of spatial variability of a parameter, such as Q, on a response, such as rock damage. 

Typically, a statistical approach for the assessment may consist of performing analysis with a 

series of drift-scale models, each assigned a constant Q value based on a statistical distribution 

of Q and averaging the calculated response somehow to obtain a representation of the 

average response. 

For the results in Figures 4-9 and 4-10, the objective of such statistical analysis would be to 

obtain a representation of case 1(a) using a statistical combination of the results for cases 

2(a) and 3(a), or to obtain a representation of case 1(b) using a statistical combination of cases 

2(b) and 3(b). As Figures 4-9 and 4-10 indicate, deriving an adequate representation of case 

1(a) using 2(a) and 3(a) or case 1(b) using 2(b) and 3(b) will be difficult. However, the 

adequacy of a representation depends on the intended use of the representation. Figures 4-9 

and 4-10 suggest that information based on a statistical model of spatially variable 

thermal-mechanical response could be evaluated using calculations based on a 

thermal-mechanical model that includes aspects of the spatial variability of material properties. 

4.4 Effects of Ground Support on Rock Damage 

A variant of the drift-scale model that does not include ground support was used for additional 

analysis to explore the effects of ground support on rock damage due to thermal load. In the 
4-11 

Figure 4-9. Inelastic Strain Distributions Between Drift Numbers 91–100 (at the South 

End of the Disposal Array) Calculated Using Three Model Cases: (i) Q Applied as a 

Spatial Variable as Described in Chapter 2; (ii) Q Assigned the Minimum Value 

Everywhere, and (iii) Q Assigned the Maximum Value Everywhere. Plots Represent 

Conditions for (i) Drifts With Stiff Ground Support and (ii) Drifts With Degraded Support. 
4-12 

Figure 4-10. Inelastic Strain Distributions Between Drift Numbers 41–50 (in the Middle of 

the Disposal Array) Calculated Using Three Model Cases: (i) Q Applied as a Spatial 

Variable, as Described in Chapter 2, (ii) Q Assigned the Minimum Value Everywhere, and 

(iii) Q Assigned the Maximum Vale Everywhere. Plots Represent Conditions for (i) Drifts 

With Stiff Ground Support and (ii) Drifts With Degraded Support. 

 
4-13 

model variant, the two rings of solid elements originally set up to represent concrete liner 

[Figure 3-2(c)] were assigned rock properties and fully bonded to surrounding elements. The 

resulting model represents a 5-m [16.4-ft]-diameter circular opening (same as the other driftscale models) without ground support. High-Q and low-Q versions of this model variant were 

implemented, but an analysis using the low-Q variant did not execute to completion, because of 

numerical instability associated with excessive inelastic deformation at approximately 1 year 

after instantaneous waste loading. 

Typical displacement histories calculated using the high-Q, nonsupported drift model variant 

(Figure 4-11) indicate that thermal loading caused the simulated drift to displace southward and 

upward. The calculated displacements included a drift convergence (decrease in diameter) of 

approximately 20 mm [0.79 in]. The horizontal convergence occurred quickly during the first 

approximately 20 years and remained constant thereafter. The vertical convergence, in 

contrast, occurred more slowly but reached a constant value after approximately 50–60 years. 

The convergence due to excavation {approximately 0.2 mm [0.0079 in] horizontal and 1.8 mm 

[0.071 in] vertical} is negligible relative to the convergence due to thermal load, as 

Figure 4-11 indicates. 

Figure 4-11. Displacement History at the Roof, Floor, and North and South Sidewalls of a 

Disposal Drift Without Ground Support Based on a Drift-Scale Model Analysis for High-Q 

Areas. Vertical Displacement Is Positive Upward, and Horizontal Displacement Is 

Positive Northward. 
4-14 

The inelastic strain distribution calculated for the unsupported opening through 150 years is 

compared (Figure 4-12) with the inelastic strain distributions calculated for an opening with 

ground support, before and after removal of the support elements (as described in Section 3.4) 

to simulate support degradation. The results indicate that a concrete-liner ground support likely 

will not affect pillar damage but will have appreciable effects on rock damage close to the drift 

opening. This calculated result is consistent with the expected effects of a concrete-liner ground 

support on stress conditions. A concrete liner applies stress directly at the edge of the opening, 

thereby creating a confining condition that decreases rock failure near the opening. 

In contrast, the zone of influence of the concrete-liner stress does not extend far enough into the 

rock to have any effect on stress-controlled rock failure in the pillar. Therefore, a concrete-liner 

ground support could reduce rock damage near the opening but is not likely to affect rock 

damage in the pillar. In general, a ground support system could affect rock damage only within 

the mechanical influence zone of the ground support system. 

4.5 Potential Loading of a Concrete Liner Ground Support 

The drift-scale model setup included documentation of the concrete-liner pressure history 

through each analysis. Concrete liner pressure may result from suppressed inelastic 

deformation of the rock or suppressed thermal expansion of the liner or rock. For example, the 

calculated maximum inelastic strain magnitude was 3.78 microstrain for a high-Q model with 

concrete-liner ground support and 13.5 microstrain for the same model without ground support. 

This difference, which is also described in Figure 4-12, indicates that potential inelastic 

deformation of rock adjacent to a thermally loaded drift could be suppressed by ground support, 

such as a concrete liner. The ground support element would be subjected to loading due to the 

suppressed inelastic deformation. Additional loading of the ground support may result from 

suppressed thermal expansion of the support element or surrounding rock. 

As shown in Figures 4-13 through 4-15, the concrete-liner pressure may vary around the liner 

circumference and could be affected by the rock-mass mechanical properties. Generally, the 

lowest Q model gave a greater concrete-liner pressure than the highest Q model. Because the 

pressure due to suppressed thermal expansion of the liner or rock should be greater in the 

highest Q than lowest Q model,1

 the greater concrete-liner pressure in the lowest Q model 

indicates that suppressed inelastic deformation is the prevalent cause of concrete-liner pressure 

in the models.2

The liner pressure also is affected by frictional resistance at the liner–rock interface, as 

illustrated in Figure 4-15. Generally, liner pressure approaches a hydrostatic (i.e., equal all 

around) distribution as the liner–rock interface approaches frictionless. The liner pressure 

decreases in the roof and floor areas and increases in the sidewalls as the liner–rock friction 

coefficient decreases. For non-zero friction, the liner pressure in the roof and floor exceeds the 

liner pressure in the sidewalls by an amount that increases as the friction coefficient increases. 

A model that assumes the liner is bonded to the rock will give a maximum value for the liner 

pressure in the roof and floor and a minimum value in the sidewalls. 

 

1

Both models have the same temperature distribution and concrete-liner properties, but the rock-mass stiffness is 

greater in the highest Q model than in the lowest Q model. Therefore, pressure due to suppressed thermal 

expansion of the liner or rock should be greater in the highest Q model. 

2

As discussed in Section 4.2 and illustrated in Figures 4-5 and 4-7 (compared with 4-6), greater inelastic deformation 

will occur adjacent to a heated drift in lower Q than in higher Q areas. 
4-15 

Figure 4-12. Inelastic Strain Distributions at 150 Years Based on Drift-Scale Modeling for 

High-Q Areas, Showing the Effects of Ground Support on Thermally Induced Rock 

Damage. Results Are Shown for Cases of Supported Opening, Degraded Support 

(Simulated by Removing Elements After 150 Years), and No Ground Support for the 

Simulated 150-Year Period. 
4-16 

Figure 4-13. Distribution of Liner Pressure Around a Drift Circumference at 150 Years 

After Instantaneous Waste Loading, Based on Highest Q and Lowest Q Drift-Scale 

Models With Liner Fully Bonded to Rock. The Magnitude of Liner Pressure Is Plotted as 

a Proportional Radial Distance from the Liner–Rock Interface. 
4-17 

Figure 4-14. Variation of Liner Pressure With Angular Distance Around a Drift 

Circumference at 150 Years After Instantaneous Waste Loading, Based on Highest Q and 

Lowest Q Drift-Scale Models With Liner Fully Bonded to Rock. 
4-18 

Figure 4-15. Effects of the Liner–Rock Interface Model on the Calculated Liner Pressure 

Showing Results for a Fully Bonded Interface Model and a Frictional-Interface Model With 

a Friction Coefficient of 0.6. 
5-1 

5 SUMMARY OF INSIGHTS 

Thermal-mechanical analyses of a conceptual waste disposal design were performed using a 

series of site-scale and drift-scale models to examine characteristics of potential rock-mass 

mechanical response to the repository thermal load. The site-scale models included a 

representation of spatial variability of mechanical properties based on sitewide fracture mapping 

and were used to examine variability of mechanical response over the disposal area. The 

drift-scale models, in contrast, allowed a closer examination of the response at the scale of an 

individual disposal opening. Insights based on the analyses are summarized as follows. 

The results indicate the occurrence of a strong vertical temperature gradient in the host rock 

centered at the disposal horizon. The vertical temperature gradient will likely be sustained 

through the heating period. A weaker lateral temperature gradient also will occur, especially 

close to the disposal openings and near the lateral boundaries of the disposal zone. 

Temperature gradients control development of stress conditions that could cause rock damage. 

Rock damage due to thermal load could occur in the roof and floor areas of the disposal 

openings and in the intervening pillars. Rock damage in the roof and floor could be reduced 

with appropriate ground support and is likely to occur more in rock of lower mechanical quality. 

However, ground support is not likely to affect potential rock damage in the pillar, because such 

damage is driven by stress conditions outside the influence zone of typical ground support. 

Because of the stress control, rock damage in the pillar could occur more in rock of higher 

mechanical quality. 

A site-scale thermal-mechanical model could provide insights regarding the effects of spatial 

variability of mechanical properties and a means of evaluating models of spatial variability of 

mechanical response based on statistical analysis of responses calculated using 

constant-property mechanical models. 

An underground opening used for nuclear waste disposal could be stabilized using ground 

support designed to reduce inelastic deformation of the rock and, therefore, withstand loading 

due to suppressed inelastic deformation in addition to loading due to thermal expansion of the 

rock and support system. The magnitude of such loading can be calculated through numerical 

thermal-mechanical modeling.
6-1 

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